TOPICS IN NUCLEAR STRUCTUR

\J U I... '" "~* 1 -

VOLUME Proceedings of XVI Winter School B ie lsko - B ia ł a , POLAND 20 February - 4 March, 1978

edited by J. STYCZEŃ and R.KULESSA

I N S T I T U T E O F N U C L E A R P H Y S I C S J A G I E L L O N I A N U N I V E R S I T Y K R A K Ó W 1 9 7 8

*APO«T No

SELECTED TOPICS

IN NUCLEAR STRUCTURE

PROCEEDINGS OF XVI WINTER SCHOOL

V o l u m e 1

February 20 - March 4, 1978 Bielsko - Biała, POLAND

Edited by : J.Stycztri and R. Kutou

Cracow, June 1978

NAKŁADEM INSTYTUTU FIZYKI JĄDROWEJ W KRAKOWIE UL'. RADZIKOWSKIEGO 152

Kopię kserograficzną, druk i oprawę wykonano w IFJ Kraków

Wydanie I Zam. 147/78 Nakład 300 egz.

Preface

It was for the sixteenth time already that a group of physicists fron the Institute of Nuclear Physics in Cracow and the Institute or Physics of the JagelIonian University had organized a Winter School on Nuclear Physics fron February 20 to March 4, 1978, But, after several years of being used to Zft-kopane as a place of these meetings, for the first time this School was held in Bielsko-Biała, a town south-nest to Cracow, near to the Beskidy mountains.

The main aim of the School was that the lectures given there cover the most djtianiic trends in the low energy nuclear physica and provide the participants with a fresh insight into thci present statue of a number of basic problems and research work described often literally hot from the laboratory. The quality of the physics at the School speaks for itself and will be apparent for the reader of the proceedings,

Thr laterial presented by the speakers on the School was very large. Consequently, two volumes of the Proceedings had to be prepared. They were reproduced by photo-offset, and the submitted manuscripts were included without much of editing. Any technical shortcomings are hoped to be compensated by the intention to make t2ie valuable material available for the readers soon after the School.

V/e would like to thank dr Z.Stachura for hie help in col¬ lecting the manuscripts and express our gratitude for his re¬ marks. Special thanks go to Mme J.Kozarska for typing some and retyping some other manuscripts and Mr ff.Starzecki and Mr J.Wrze-siński for their help in making corrections.

J. Styczeń1 and R.Kulessa

Kraków, April 10, 1978

III

SCHOOL HOSTS Institute of Nuclear Physics, Cracow Institute of Physics, JagelIonian University, Cracow

SCHOOL ADDRESS Orfrodek Wdrażania Postępu Technicznego w Energetyce Blelsko-Blała, ul. Brygadzistów 170

ORGANIZING COMMITTEE Chairmen:

R. Kulessa A.Z.Hrynklewlcz

Members:

E, Bożek B.Styezeri M.Lach J.Styczeii TC.Potempa S.Szymczyk M.Ryblcka B.VJodniecka Z.Stachura , J.Wrzesiriski TC.Starzeeki K.Zuber

Secretary:

Z. Natkanlec A. Kęsek

CONTENTS

Volume I

1. Opening

REMARKS ON THE HOLE OP PHYSICS A.Z.Hrynklewlez . . . . . J

2. High Spin States and Yrast Trapa

HIGH SPIN ROTATIONS OF NUCLEI WITH THE HARMONIC OSCILLATOR POTENTIAL M.Cerkaski and Z.Szymariskl 13

EXPERIMENTAL V.'ORK ON HIGH SPIN ISO^RS AND POSSIBLE YRAST TRAPS G.Sletten . . . . . . . . . . . . . . 25

HIGH SPIN ISOMERIC STATES IN 152Dy F,A,Beck, C.Gehrlnger, J.C.Merdlnger, J.P.Vivien, E.Boftek J.Styczeń . *»3

NUCLEUS OF VERY HIGH SPIN STATES. MICROSCOPIC DESCRIPTION M.PłQB2a.1czak i»9

THE STUDY OF HIGH SPIN ISOMERIC STATES IN MULTIPLICITY EXPERIMENTS WITH 12C INDUCED REACTIONS D.Hageroan r t . . . . . . 114

HIGH SPIN STATES IN TH1') GROUND STATE - AND SIDE BANDS IN 156Dy, 162Er AND 168Hf INVESTIGATED THROUGH PROTON - AND 14N INDUCED REACTIONS

J.Vervier . 118

3. Nuclear Reaction

EVIDENCE FOR SYSTEMATICAL FEATURES IN PROTON ELASTIC SCATTERING

RELATED TO NUCLEAR STRUCTURE

E.Colombo, R.De Leo, J.L.Escudle, E.Fabric!,

S.Mlchelettl. M.Plgnanelll, F.Resminl,

A.Tarrats . . . . . . . . . . . . . . . . . . . . . . 123

6L1 INDUCED REACTIONS WELL ABOVE THE COULOMB BARRIER

JflJagtrzebaKl . . . . . 136

A MICROSCOPIC APPROACH TO THE DESCRIPTION OF THF GIGANT

MULTIPLE RESONANCES IN LIGHT DEFORMED NUCLEI

K.W.Schmld 167

ON THE INFLUENCE OF THE SHELL NUCLEAR STRUCTURE ON THE

DIFFUSION PROCESS

V.G.Kortavenlto. 221

HEAVY-ION EXPERIMENTS ON THE MP TANDEM AT ORSAY. PARTICLE

CORRELATION STUDIES AND MASS MEASUREMENTS ON EXOTIC

NUCLEI

P. Roussel . . • '. . . . 224

ON INELASTIC SCATTERING CALCULATIONS

W.J.G.Thi.issen Zk7

ELASTIC TRANSFER REACTIONS

H.G.Bohlen 250

THE BREAK-UP OF COMPLEX PARTICLES INTO CONTINUUM

A.Budzanovrskl 263

ANGULAR MOMENTUM IN HEAVY-ION REACTIONS

H.OeBchler , 276

BACK-ANGLE ANOMALIES AND MOLECULAR RESONANCE PHENOMENA IN

HEAVY-ION COLLISIONS

K. Eberhard . '. . 308

INFLUENCE OF CHANNEL COUPLING ON HIGH EXCITED STATES I. Rotter ; 363

VI

PRESENT STATUS OP THE 16 MeV TANDEM PROJECT AT LABORATORI

W4ZIONALI DI LEGNAnO /PAPOVA/

C.Slgnorinl 372

Volume II

4. Collective and Sinple Particle Properties of Nuclei

IN-BEAM GAMMA-RAY SPECTROSCOPY WITH 100-160 MeV <*»S IN WEDIltti-LIGHT NUCLEI C.Slgnorlnl . . . . . . . . . 381

NATURE OF THE 0 + LEVELS AND SHAPE TRANSITIONS IN THE Ge

AND Pt REGIONS

M. Vergnes

GAMMA-RAY SPECTROSCOPY IN MEDIUM-LIGHT NUCLEI

J. F.Sharpey-Schafer

HIGH-SPIN NEUTRON PARTICLE-HOLE STATES IN EVEN N=28 ISOTONES

J . Styczeń **6'

IN-BEAM INVESTIGATION OF TEL? N=82 NUCLEUS 143Pra

F. S t a r y *»77

SHAPE TRANSITION IN THE ODD Tb

G . w i n t e r , P . K e m n l t z , J . C o r i n g , L . F u n k e , E . r i l l

S . E l f s t r O r a , S . A . H J o r t h , A . J o h n s o n , T h . L i n d b l a d . . . . 1,82

RECENT EXPSRBiENTS ON THE SIWPE OF FISSION BAHlilER

V.Me tag k$°

ON-LINE ALPHA SPECTROSCOPY ON 1 GeV PROTON-BEAM

FROM SYNCHROCYCLOTRON

J. Korcilckl , 533

QUASIPARTICLE SPECTRA ABOVE THE YRAST LINE

R.Bcngtsson, S.Frouendorf . . . . . . . 551

DISCUSSION OF TH£ CPANIOTD EARTnEE-POCK-BOGOLYUBOV »!FTHOD

IN TEW!S CF SIMPLIFIED MODEL

S.Civiolc, J.Dudek. Z.Szymariski 588

THE QUASIM0LECU1AR MODEL IN TRANSITIONAL NUCLEI

G.Leander . , , . . . . . . . ..621

VII

PARTICLE-ROTOR UODEL DESCRIPTIOH OP ODD-MASS TRANSITIONAL NUCLEI J.Rełt8tad 658

SHAPE OP PLATINUM NUCLEI AROUND A-190 F.DOnau . . . . . . . . . . . . . . . 683

ODD-EVEN EFFFCT IN THE NUCLEAR SHELL-MODEL POR NUCLEI TTITS N«28 AND N-CO A. Bf. Land a » 68?

ANGULAR MOHENTUU PROJECTED TAVE-FUNCTIONS R.Bengtason. H.B.Hakansson . . . . . . 7Ok

WARD-LIKE IDENTITIES, CLUSTER-VIBRATIONAL MODEL AND QDASIROTATIORAL PATTERN V. Paar 715

5. Heavy-Ion Collisions-Positron Production: Ouaal Molecules

IN-BEAM EU!CTRON AND POSITRON SPECTROaCOPT AFTER HEAVY-ION COLLISIONS H.Backe 823

EXPERIMENTS ON K-HOLE AND POSITRON PRODUCTION IN COLLISIONS OP HEAVY IONS H. Bokemeyer 8 M

AN INVESTIGATION OP QUASI-M0LECULE3 IN HEAVY-ION COLLISIONS. QUASI-HOLECULAR ROENTGEN RADIATION K.H.Kaun 859

6. Cloalng Beaark» J. F. Shamey-Sohaf er , 8 8 1

LIST OP PARTICIPANTS

VIII

1. OPENING

RBMAEKS OX THE ROUE OF PHTSICS

A.I. Hrynklowles

Institute of Huolear Physios,Craoew

For all of us pbysles Is an exciting adventure. Bat doing

physios is not only great fun - it is also an Important sad

responsible task.

Tonight I would llko to Bako a fow remarks on the laportaa-

es of physios, on its rolo in tbo sodom world. Z aa afraid

that ay reaarks will sound qulto trivial to pbysiolsts, out sevw-

ral faots about physios aro not roallsod by sooloty or even by

solontlsts working in otbor fields. So thoy aro worth repeat¬

ing.

I aa going to toueh upon throo toplos: physios and othor

natural solonces, physios and technology and physios and sooloty.

I shall only bo ablo to mention sons of tho problems, It will bo

impossible to dlsouss thoa in dotall.

Physios and other aoieneos

There is no doubt that physios 1* the most fundamental sol-

ence and as such it forms a solid base for tho development of

other natural sciences such as ohealstry, biology, astrophyslos

and geophysics. Those and many othor scientific fields utilise

more and more broadly tho methods of physios and measuring de¬

vices constructed by physicists. Pbysioal apparatuses are con¬

tinuously improving, they are becoming more sensitive and preolso.

Their sensitivity is due e.g. to the fact that the detootlon of

nuclear radiation makes it possible to traoe individual atoms in

an investigated saaple. The precision of measuring some physi¬

cal quantities now reaches 10"*1?

Physical methods are also booomlng more sophisticated and

measuring devices more complex and that Is why thoy cannot bo

applied In other fields without tbs participation of a physloist.

Examples from tho field of medicine suoh as tbo application

of SQUID In magnotooardlography or magnetoenoephalography, aedl-

eal diagnosis by nuolear magnetic resonance methods or the ap¬

plication of protons or plons from high energy accelerators In

the irradiation of Internal tumors prove my point, that not me¬

dical doctors but only physicists are able to find tbe field of

application for snob sophisticated methods, to Introduce and to

apply then and to help other specialists in the interpretation

of results. The physicist is prepared to play the role of an

inter-disciplinary link between the different fields of scienti¬

fic activity.

In the history of physics, like In other sciences, different

periods caa be indicated, liany scientific Ideas were prosecuted

in the Middle Ages. In modern times physios has also served as

a target for the Interference of Ideology,

The idea that physics is a closed subject has been put for¬

ward at various times. I would like to recall tbe story of Max

Planck, When he was a young physicist, an older friend of bis

tried to discourage him by saying that in physics everything in¬

teresting had already been discovered, physics bad become a very

dull and uninteresting subject and that tbe only thing be would

have to do, was to dust the old apparatuses built by somebody

else.

At the beginning of our century Max Planck himself opened

a new chapter in the history of physics and tbe development ot

physics was nevar so bright and magnificent as it was in the years

following,Great new syntheses were made In tbe form of tbe theor¬

ies of relativity and ef quantum mechanics. Discoveries of tbe

atomic and nuclear structure, Investigations of tbe elementary

particles, fast development of solid state physics completely

changed our Image of the surrounding world. Today we know much

more but we are less self-complacent and self-confident than

physicists at tbe end of tbe nineteenth century.

It may sound paradoxloal but one can find some analogy be¬

tween the state of science today and In ancient times. At the

time of Aristotele^it was believed that tbe whole world was nade

of 4 element*: earth, water, air and fire* Their nature was un¬

known and tbe relationship between then was not understood. To¬

day we believe that nature can be described In teras of 4 funda-

mental interactions: strong, weak, electromagnetic and gravlta-

tlonal, but again ire do not know tbelr nature or the relation¬

ship between them. The new accelerators planned for the near

future, such as the Russian synchrotron with a 20 km - circular

path, are being designed with the aim of alscoverlng the Inter¬

mediate boson and to find the link between strong and weak Inter¬

actions. With these new accelerators the answer will be probab¬

ly found as to whether partons or quarks are the ultimate cons¬

tituents of matter and If they oan be obtained In the unbound

state. Most physlolsts now believe that there Is an end to the

investigation of the structure of matter.

Large scientific projects In physics, such as elementary

particles research, can be realised only within the framework of

international collaboration. This Is connected with the full

opennes of scientific research. All new results 'n fundamental

research are published in international scientific Journals.

More important results are communicated by mall or even by phone

almost Immediately after being obtained. Multinational research

groups are working together with huge and expensive experimental

facilities. International Institutes such as CERK or the Joint

Institute for Nuclear Research in Dubna have been organized.In¬

ternational collaboration in fundamental research plays an im¬

portant role in creating an atmosphere of mutual understanding

and respect. Scientists contribute to detente in International

relations and pave the way for other contacts.

Physics and technology

The first great inventions of man: the wheel, the lever,

windmill design, ship building and the construction of many me¬

chanical tools were not based on scientific research. This was

"pre-scientific" technology. Even beat engines were designed

before thermodynamics was formulated. The general laws of ther¬

modynamics and the notions of energy and entropy were introduced

in order to understand the principles of operation and the limi¬

tations of thermal engines .

Since physios started to explore the mloroworld and "oon-

•on sense* reasoning and macrosooplc models started to fall, the development of technology bad to be based on scientific re¬ search.

NUCLEAR ENERGY Is the result of tbe research of Skłodowska? Curie, Rutherford, Fermi, and many others, ELECTRONIC TECHNOLOGY come8 froa the discovery of the electron by Thomson, ELECTROMAG¬ NETIC WAVES were discovered by Hertz on the basis of tbe Maxwell equations, TRANSISTORS started from the quantum theory of solid state, LASERS are the result of tbe works of Barrow, Prokhorov and Townee on the stimulated emission of radiation, theoretical¬ ly described by Einstein*

One can give many more such examples whlob have tbe common feature that every great revolution in technology is based on the results of scientific research. They were not ordered by a minister, by some businessman or by an Industrial manager, who was looking for new sources of energy, for new industrial tech¬ nologies or was eager to Improve transport and communication.

Physicists are in most cases entirely unaware of the tech¬ nical consequences of their work. There Is a story about 1(1-cbael Faraday. Just after he bad discovered electromagnetic induction a member of the government visited him and asked what tbe point was of all the things he was doing. Faraday answered: "I do not know, but I am sure that your successors will some day levy a tax on it."

But apart from the direct applications of scientific dis¬ coveries, technology makes use of the methods developed by phy¬ sicists and of the measuring devices constructed by them. Phy¬ sicists in the process of research are often confronted with some unresolved technical problems and are forced to find their solution by themselves.

Physicists pursuing pure scientific results have solved "en passant" many problems in the field of material engineering, high-voltage technique, vacuum technique, low-temperature tech¬ nique, electronics, control and automation. We should keep In mind, for instance, that computers developed from the electronics of nuclear physios.

It is tbeae rery Important new techniques, which are the

"toy-products" of pure research that can be and ought to be quick¬

ly applied In practice.

The direct application of fundamental discoveries is possible

only In the most dereloped countries, where the cap between the

level of scientific research and the level of technology Is nar¬

row. In other oountrles where, because of old-fashioned techno¬

logies, this gap is nucb wider, scientific discoveries In moat

cases cannot find direct application and scientists oannot be

blamed for that. However, the "by-produots" of fundamental re¬

search, suob as new methods, new devioes and new technologies

should be quickly transferee to Industry and to other branobes

of the national eoonomy.

Let me now tell you a few words about the role of Physios

In a world facing an energy crisis. Conventional fossil fuels

such as petrol and ooal, which are being consumed right now at

a rate a million times higher than their rate of formation, will

be exhousted sooner than later. This is unavoidable and one can

only discuss whether it Is going to happen in 20 or 200 years,

I am pretty sure that the only way to prevent their oncoming ca¬

tastrophe lies in nuolear energy. We will have to endure the

period of nuolear energy until new sources of energy become

available.

Thus the research concerning new types of power reactors:

breeders and high-temperature reactors,should be intensified.

Such breeders will make possible the effective use of the large

resources of uranium and thorium by oonvertlon of U into fis¬

sioning 239Pu and 232Th into 233U, High-temperature reaotors

will be very Important souroes of high-temperature heat for many

technological processes.

In the long range searoh for energy for the future,work on

the controlled thermonuolear synthesis will be continued. Fur¬

ther researoh on the more effeotlve use of solar and geothermal

energy or of the energy of winds, sea tides and sea ourrents

should be pursued. In solving all of these problems physiolsts

will play the leading role.

In faoing the energy orlsls we were forced not only to look

for new sources of energy but also to develop ways of more eoo-

nomlcal consumption of existing resources* Less energy consum¬

ing technologies need to be developed, we have to inorease the

efficiency of energy conversion and transport, we have to limit

energy losses by Improving the insulting materials and architec¬

ture of building and by the utilization of energy lost In brak¬

ing. And here again Is a broad field of aotlvlty for physi¬

cists*

It seens to ne that some changes in the programme for

training physicists are needed. We have to draw more attention

to some Important problems in classical physios such ae thermo¬

dynamics, hydro -and aerodynamics, or the problems of friction,

which are now treated without much care and which can be very

Important for future physicists.

Physics and society

Every science plays a creative role in our culture. Funda¬

mental research in physics, stimulated by man's curiosity as to

how nature Is ordered and to what laws It is subjected, has a

particular significance for our Image of the world.Great dis¬

coveries, especially great syntheses in physics, have exerted

a powerful influence on the development of philosophy.

Fundamental research is one of the forms of the man's in¬

tellectual activity and as such does not need to be justified

by practical considerations. It may be said, however, that the

practical importance of physical research is an additional sti¬

mulus for scientists to undertake It and to carry it on.

The ever Increasing role of science In almost every field

of life has caused a reaction In society, characterized by cri-

ticlsB of scientists and their research. From time to time the

question Is raised If such rapid development of science Is re¬

ally necessary and if the expenses for science are justified.

Actually the rate of development of toienoe Is very high

at present. If the nuaber of scientific workers, published pa-

pers and scientific journals can be considered lndloatory of de¬

velopment, then we can say that scientific production doubles

every 10 or 11 years. Such a growth rate cannot last for long.

Comparison with the global Increase in population, which doubles

approximately every 30 /ears, leads to the absurd conclusion that

in the not so distant future every inhabitant of the Earth would

be a scientist. Thus the saturation point,the first symptom

of which can already be observed, haa to oome.

Quite frequently the opinion Is expressed that physics 1J

exceptionally expensive. Probably it costs nor* than other sci¬

ences, although the construction and equipment of a nodern ra-

dioaatronoralcal center, or a center for biological research

would cost as much as a large accelerator for physios research.

Fundamental researoh, apparently very expensive, does not

seem such when Its costs are compared with the budgets of the

great powers and especially with that part of the budgets spent

on armaments. It has been estimated that the cost of all funda¬

mental research In physics since Archimedes until the present

day has not exceded 40 billion dollars, which is approximately

the gross national product of the 0SA for ten days or the amount

of money spent there on armaments In a few months. However,

the cost of fundamental research in physics although not very

high when compared to funds used for other human activities, Is

not negligible and physicists cannot avoid dialogue with society.

The wise popularization of scientific results Is extremely Im¬

portant. It should be one of the prime duties of a scientist.

We have to make a constant effort to convince the public

that science serves humanity and that It is nonsens* to claln

that solenoe is responsible for all the evil on the Earth.

Population growth, man-made changes In the environment,

the approaching shortage of ran materials result in the fact

that there Is no turning back from the road taken by mankind

and that only science can help us to avoid or lessen the con¬

sequences of the trend we are following and to diminish the price

paid by us for progress.

It is worthwhile to quote the words of Welstkopf froa his

book "Physics In the Twentieth Century " : - "Science cannot

develop unless It Is pursued for the sake of pure knowledge and

insight. It will not survive unless It Is used Intensely aad

wisely for the betterment of humanity". The great successes of

physics in the twentieth century have developed In physicists

a kind of superiority conplex. Physicists have to rid them¬

selves of some of their arogance and submit to public control.

Freedom of thought and discussions, so necessary for the

development o'i pure science does not mean total freedom of ex¬

perimentation. Scientific discovery nay carry with It potential

hazards. Nuclear energy and ger.etlc engineering are well known

examples. But scientists today are well aware of these hazards,

the sources of danger are thoroughly studied and remedial means

are Introduced In advance.

Emotional objections against the development of nuclear

sources of energy arises mainly frori the Hiroshine complex. Of

course power reactors, and radioactive waste disposal do present

some hazard, but they are much less dangerous then other man-

made sources of disaster.

It Is true that science provided us with the knowledge

which made It possible to devise the new devilish arms of today;

chemical, nuclear and biological, but is it science that should

be blamed, for the sins of society ?

There Is certainly no doubt that the most Important task

of humanity is to avoid the disaster of a nuclear or bacterio¬

logical war, that Is, to maintain peace. We should not spoil

this unique and extraordinary experiment which nature has start¬

ed on Earth - we should not spoil Ufa.

2. HIGH-SPIN STATES AND YRAST TRAPS

• HIGH SPIN ROTATIONS OF NUCLEI WITH THE HARMONIC OSCILLATOR POTENTIAL

I * M.Cerkaskl and ZjSzyianakl, Institute for Nuclear Research,

Warsaw, Poland

-A .Introduction

Calculations ot the nuclear properties at high angular mo-Bentun \}-9] have been performed recently* They are based on the liquid drop model ot a nuoleus £l} and/or on the assumption of the single particle shell structure ot the nucleonlc motion* The calculations are usually complicated and Involve long computer codes*

In this article ve shall discuss general trends in fast rotating nuclei in the approximation of the harmonic oscillator potential. We shall set that using the Bohr Hottelson simplified version £i<33 of the rigorous solution of Valatin £ n 3 one can perform a rather simple analysis of the rotational bands, struc¬ ture of the yrast l ine, moments of Inertia etc. in the rotating nucleus C12»133 •""

— While the precision f i t to experimental data in actual nu¬ clei i s not the purpose of this paper, one can s t i l l hope to reach some general understanding within the model of the simple rela¬ tions resulting in nuclei at high spin* /. . . ,

2* Minimisation of energy

Following ref* £iil w ahall start with the familiar cranking model Hamlltonlan

H where h (k) , the singla particle contribution to H for the k-th particle Is related to the"true" single particle Hsmiltonian h £k) is the well-known way

Hera, ui •denotes the angular velocity of rotation and

tit* projection of the particle angular aoaentua on the rotation axis (1) .

Using tba standard notation of the h.o. faraonie osdlla-tor7 craation and annihilation operators b^-OO •»*

-1,2,3 | k-1,2,... A ) ve ay aay wita

Mow, folIONlnc Bohr and Mottalson [}Q] «a snail oait tba saoond tan la (frU) as it contains a saall factor ~ C^-^t and aots batvaan diffarant aajor h«o» shall* (fl,H+2 ) • Iba resul¬ ting Uaalltonian to ba dlagonalizad is ttaarafora

index k has been oatltted for slatplicity. fitploylnf the unitary traasforaatioc

4*= Aft*** + 4M +

we obtain attar soae standard aanipulationa b teras of the noraal aodes

(2. ?;

expressed in

= tąaX with

and

*"V <3e*«2 <o &4-*J

ej <J Total eigenvalue E of fl say ba written as

with

z-, - z: <*/ C4A. * / ; / o # i Y-occ. TT'

where sunaation runa over the occupied states V £ L . « . lowtst •nergy siganstates of h ^ ) and -y* • 1, o< OT /Z •

Calculating angular aoaentuB

ve say obtain an invars* relation

whara

has tha aaaning of thr aaxiauB angular aoaantua I that can ba raachad for a given configuration specified by quantities Z~, _, and 4£ys ^cf.#q. (j2»1i)) •

A straightforward calculation gives then

where

f

ainlnlsatlon of this expression with respect to nuc¬ lear shape, i . e . with respect to O3tJcJ£ andcJf fulf i l l ing the constraint of constant rolua*

o 3

gives

Employing eqs C2.17J to 02.20 ) on* obtains a final expression for the •inimised energy

which is valid tor fixed configuration specified by the three quantities <2J 2 ^ Z^ at any I,

Finally, 'the application of methods of group theory 0s the Elliot group of unitary transformation of the h.'o. potential £14"]) leads to definite expressions for the three sums £,> 2 and Z A ^characterising the nucleonic configuration } in terms of the total occupation

W'sO ' ĆH- last unfilled h.o. shell, n - number of particles in the M - th shell,) and the two parameters 2 %/*- characterising the irreducible representations of the SU (3) group D^»15"] • Final expressions for the extremal energy bands are

*"* o Ś x- & ;, ., and . « -ł^

OSX Tim tare* possibilities E^, E^, and Ey correspond to

the rotations about three principal axes of inertia.

3* Results and conclusions

Foraulw £2.23,) to (2*25) determine the energy of the rotational bands for each configuration specified by three quantities 2T, ?i andy*. 4 Quantity £ can be calculated from eq. (2*22 ) vhile the choi -st of the irreducible representations (źj/tjot the group SV &) baa to be based on aysaetry arguments.

Let us illustrate our procedure by few examples. Let us first take 4 Identical nudeons & the (tp)- shell (M-3) • It is assvsMd that the lower X • 0, 1, and 2 shells are comple-

tely f i l led . V* obtain i"rom eq. Ć2.22)

Ł £ (2.0 Z U sr O

£contrary to eq. <2»22.) there i s ao factor 2 1B front of the mm sinoe we ara dealing with one typ* of nueleons only} • Jfcnr, sj try arguments £see mfs . O M 5 1 ) t e l l us that the sort sj trie state for four particles can be characterised by the Toting diagram £22] i . e . the diagram Indeed, the completely symmetric diagram { 4 ] 1* net permitted for on* type of nucleons because i f the Paull principle. To each box of the above diagram there oorrasponds a phonos diagram £}] in the N • 3 shell* So, the particle symmetry {22} contains some of the diagrams contained in the representation produot

All the irreducible 30 (3 ) representatlons^jthat belong siaul-taneously to this product and to the particle diagram {22] are de¬ noted by the symbol ® and called pletysm. One can find out in tills case (see refs D6.173)

G1 <g) {22} = (i,2) + fr, I) + & O) + (S,2) f * (2,3)4 (3,2*4 0,«) + O>') )

By a direct check with formulae £2.23) to £2.25 ) one can see that the representation (%,/*) m I 8* 2 ) corresponds to lowest energy. Generally, the lowest energy representations are those that have largest possible 7i and/•«.. The three lowest energy bands corresponding to this representation are illustrated in Flg.1 in the plot of E 3 versus X2 , leading to a linear dependence. Similar arguments lead us to higher bands with (7\,/<) • £ H » 1 ) and £14,0) which are created if one or two particles are promoted to the next Cjimh ) shell. Band £7,1 ) it also shown in Flg.1.

rig.2 illustrates an analogous plot of tr versus X2 for the system of 12 neutrons and 12 protons £2Slg nucleus ) • In this ease the ground state bands are characterised by

which correspond to oomplete f i l l ing of the. H-0 mad M-1 h.*. shells plus 4 particles in the £sd) - shell <M«2) . The corres¬ ponding Zi t"\ and >t can be found by aimilar arguments aa descri-

Fig. 1. Dependence of B* on calculated for the system filled up to 4 identical paxticlss in th« Cfp) shell. Kuabers in paran-theais daaote the O- ") representation of Sir O ) . Traat states are marked Taj solid l ine. Representation fX[O «C9«0) belongs to particle symmetry { }

Fig, 2. Same as In Flg.1 for the systea of 12 neutrons and 12 protons £ 2T4g) • Banda "L" ara marked with solid Una, while bands "M" and "H" - with dashed Una,

btd above* Higher bands corresponding to 1, or U particles pro* aoted to the C*P) C M ) shell are characterised \>y(r,*,M ) -(65,11,3) and £68,20,0}; respectively. They «r* also Illustrated in Flg.2.

Fig* 3 represents the saae dependence la tbe oase of the S versus I representation* Ona can easily saa by examination of eq. C2*21") that tbs curve I (I) azblblts an inflection point at

X = X4. « {7rr~z£ (3. *i) For I « I « which is usually tht easa in physical situations the curve looks slallsr to a parabola* Hovrar, the dyosaioal •oaant of inertia

% C O ^ is yarying silently in the band. Fig.A Illustrates this variation as oostpared with tha static aoaent of inertia, /3~ttMX which is ooąputed under the assuaption of a rigid rotation of a nucleas of given shape. We have

Finally, Fig.5 shows the dependenoe of the yrast states on tha nuclear selfconslstent deformation for the case of the 2Stg systea. Tha Nilsson deformation paraaeters (£,f) an related in the usual way to the three h.o. frequencies a, } ^ and c->t which are determined froa the selfconsistent values given for any I by eqs. C2.19) and C2.20) <>ith the help of eq. C2.18J)) .

Suaalng up our results we can conclude that the aodel of a roteting harmonic oscillator leads to the yrast line composed of several bands* Within eaoh band the systea tends to acquire oblate shape which is sxlally syaaetrlc with respect to rotation axis £end point of the band ) • Futher Increase of angular ao-aentua beyond the end point of the band is only possible when the nuclaonie configur*tlon changes and new bands appear* Since the energy la a aonotonously Increasing function of angular aoaentua there is no chance for tha existence of the yrast traps in the case of a pure h*o. potential* The effective aoaent of Inertia ^dvn ^^* governs the variation of energy within each band is very dose to the rigid-body value aoaent of Inertia / 3 # t - t corresponding to the aotual nuclear shape* *

20

64.

63.

2 4 6 8 10 12 U 16

Fig, 3. S«M as in Fig. 2 but in tbt I versus I mprtMntatloiu Solid 11M oorrtaponds to yrast states*

21

- o - - o - o — J,Ł*,t. • o- o—o— f~

•o- . *»1

40

2 4 6 8 10 12 . 14 16 f

Pig* k. Mounts of Inertia for various bands labelled by tha ( ) mabtrs. Solid 11M denotes '"3-d— » while dashed lino

t o r * 2I>II * C— text)I t o r

22

5i Trast trajectory for th» 2Sff ayctaa In thł •atloa plaa*. Solid Unas danotc jrrast lina* Tha ^ oorraspond to various banda Csea tart } f

- dtfor-

References

1. S.Cohen, F.Plaail, and V.J;Swiąteeki, AnniPhys. H.T. §2/197*) 557.

2. A,Bohr and Bifl*Mottelson, Proc Hotel Syaposiw,Ronneby, 1974; H I aim Physic* Scripta 10Ą (1974) 13*

3. A»Bofcr and B.R*Mottelson, invited lecture at Internat.Conf.on Hucl.Structure,Tokyo,1977.

k, IUB«ngtaaonrl#S#LarBBonł6*Ł*and«ry P»N811«F>S,C«VllMK)( and Z^Szjaańakl, Phys.Latt. SSL 0975) 301*

5. lUNaargaard and V.V.Pa«hk«vich.Phv».L«tt. 593 ^1975) 218. 6* K«N«tr(aard,V»V(Pashk«vicb and S»Frauiidogf.lucl.Phy«.A262

^1976) 61. 7. C.G.And«r»«on,S.E.Lar«»on,0.L«andar,P#M811«r,3«0.llll*»on,

K.Poaorakl and Z.Szj«ańaki, Nud. Phys. A268 (1f76) 205. 8. A.FaaMltrtK.R»Sandbya 0>YlfP.GrtaMr,K,V*Scbaidt and R.R*

Hilton, Nud.Phy*. A256 ^976) 106. 9. K.Httrgaard,H*Tdlcl,łI.Pło*zajczak and A.Faa*darv Nuel.Pby*.

A287 ^1977) *8. 10. A.Bohr and B«R.Mottalaon, Nuclear Structurt, rol.2,B*nJaBln,

Ncv Jfork, 1975, p.85 and tt. 11. J.G.?alatlnrProc.Roy.Soc. ^2§ Ć1956) 132. 12. Z.S2y»ań»klf lacturaa dali-vwrcd at teal* c'tti da Physlqo*

Tbforlqu»,Laa Houch»»,1977,to b» publlabad. 13. N.OKrkaakl and Z.SzyaanBki, to ba publiabad. 14. J.P.Blllot,Proc.Roy.Soc. ^ 0958)128,562. 15* K.T.H»cht,ini Saltctad Topics In Nuclear Spaotroaoopy, ad. B.J»Varbaar(Norta»Holland |hibl.Co#tAastard«a»1964. 16. D.B.U.ttlawood,D>a Theory of Group Cbaractera and Matrix Re-

presentatlona ot Groups, Clarendon Prasa, Oxford,1950. 17* BtO.Wybouraa,S)—ttry Principles and Atomic Spaotrosoopy,

Wiley Intarseianoa.

Experimental work on high spin

isomers and possible yrast traps

Geirr Sletten

Let me start the lecture by quoting T. Ericson in his in¬

troductory talk, "Frontiers in Nuclear Physics" in Tokyo last

September: "Our understanding of nuclei at low excitation and in

the stable regions of the periodic system is much too good by now

to allow for drastic changes".

The areas of qualitative surprises according to him will

occur when nuclei are in some extreme condition or are being

probed in some extreme or particularly delicate way.

Among the areas specified by Ericson that are of particular

interest at this winter school are the very neutron rich, or

neutron deficient nuclei at the edge of stability, or nuclei

forced into extremely rapid rotations. We know that heavy ion

fusion reactions at high energies provide both rapid rotations

and neutron deficient nuclei at the same time, therefore a probe

is accessable: Will there by any surprises?

Maybe it is to drive expectations too far to as!-, such ;•..<.-

stions at all. After all, the subject of the Icctjr. is i.i:

spin isomers and possible yrast traps, and the existence of the

latter was already proposed by Bohr and Mottelson about 4 years

ago. It might, however, still have been a surprise to some where

the exciting isomers would be.

At very high angular momenta the classical centrifugal

effect might dominate the nuclear deformation and give rise to

oblate shapes where the large total angular momentum is aligned

a)ong the symmetry axis. In this mode the angular momentum is

composed by successive alignment of particle spins along the

symmetry axis rather than by collective rotation .

25

On the average the yrast states will have energies propor¬

tional to the reciprocal of the rigid moment of inertia for

rotation about the oblate symmetry axis multiplied by I . In

the case of such an oblate deformation the locus of yrast states

will be formed of levels of single particle character rather than

of collective nature/ and the transitions along the yrast line

will thus be of single particle character. In this situation the

decay along the yraat line might have to proceed by a high multi-

polarity electric or magnetic transition and thereby have a

considerable delay. For sufficiently large hindrance such yrast

traps might also decay by the emission of an a-particle.

During the last year a group from the Niels Bohr Institute

collaborating with physicists from GSI in Darmstadt has con¬

ducted a systematic search for delayed y-cascades with high mul¬

tiplicity . Ideally one could think that high spin isomers

would decay in rather long direct y-ray cascades to the ground

state and that the identification of such delayed cascades would

provide the signature of "super dizzy" decay hindered states.

Well aware that the unambiguous identification of a real yrast

trap would involve spectroscopic information on the entire y-ray

cascade, we considered it most important to first find nuclei

where detailed studies later could be performed. The experimen¬

tal philosophy was therefore to investigate as many nuclei as

possible with the highest possible sensitivity within the allocated

accelerator time.

The search was performed with a 16 Nal detector arrangement

shown in fig. 1. The 5 cm x 5 cm Nal scintillation counters were

operated in coincidence with 16 ns resolving time and well shielded

26

16 Na I

DETECTORS

TARGET

N •si

Fi<i. '. . Schematic drawinq of the lft Nal multiplicity filter

with tarqet and catcher positions indicated. The Nal

detectors arc in real well collimated. A target ladder

with 1r) positions and a scrips of 15 romotely dispo-

r..ililr> c.itchcrs arc omitted for simplicity.

from each other to prevent crosstalk. The detectors were all

looking at a catcher foil placed on the beam axis in order to

stop recoiling evaporation residues from a heavy ion reaction.

The targets were placed about 15 cm upstream in a cylindrical

lead shield thereby reducing the prompt Y-ray flux by a large

factor. In order to be detected the isomers must survive about

10 ns during their flight to the catcher. The catcher material

is a rolled foil of 12 mg/cm2 Pb metal with a 10 mm hole

concentric with the beam axi&. Because of multiple scattering

in the target, evaporation residues will have an angular sptead

with half-width of 61/0 "* 3-5° for the ' mg/cm targets commonly

used in the present experiments. At the beam energies used,

4.7 - 5.2 MeV/amu of 40Ar, 50Ti and 65Cu, the beam particles

themselves have an angular half width of 6 w 2 ^0.4° and are

therefore interacting very little with the catcher. In contrast

about 60% of the evaporation residues are stopped on the catcher.

The beam particles are at these energies below the Coulomb barrier

for the Pb catcher material and therefore only induce y-rays

from Coulomb excitation.

The pulsed mode of the ONILAC beam offers additional sensi¬

tivity for detection of delayed y-radiation. The macrostructure

of the beam is about 6 ms bursts at 20 ras intervals. The micro-

structure within the 6 ms is 1-2 ns bursts with 37 ns intervals.

A time to amplitude converter (TAC) was stopped by the microstruc-

cure beam bursts after having been started by delayed high-fold

coincidences during the 37 ns intervals. Another TAC, or sampling

unit was started at the end of the macrostructure burst, but stop¬

ped by delayed high-fold coincidences.

28

Fig. 2 shows the time distribution of four and higher fold

coincidences between the sixteen 5 cm x 5 cm Nal detectors in

the nanosecond region. The upper curve shows the result from

the interaction of 2 30 MeV 50Ti with a 1 0 0Mo target and the QC

lower the result when the target is Mo.

The striking difference between the two curves illustrates

a situation where no delayed cascades are detected as opposed to

a case, Mo + Ti, where an isomer decaying by a long y-ray

cascade is detected.

When the catcher foil is removed the detectors are only

sensitive to the radiation that comes from a section of about

4 cm along the recoil axis. Rough half life estimates can there¬

fore be obtained by comparison of the number of counts from iso-

mers that decay in flight to the number of counts with the cat¬

cher in position.

In the reaction Mo + Ti the time spectrum is composed

of at least two isomers. The bump is due to a short-lived isomer

(T. ._ "fc 1.5 ns) in Gd and the more flat part to several iso¬

mers in Gd. This interpretation is supported by excitation

functions and Y-ray spectra.

The sensitivity of the detection technique with respect t-c

delayed y-cascades with multiplicities of 10 or higher is of the

order 10 jjb for half lives in the interval 10 ns to 5 ms and 1 vb

in the interval from 5 ms to 10 min.

About 120 different combinations of target isotopes and pro¬

jectiles were investigated, giving the possibility of studying

about 200 residual nuclei in the region from Ba to Pb at high

spin (£ 1. 70 11 provided by the reaction) . Out of these, 20

29

O

O

W) Z O

o

10 TIME (ns)

Fig. 2. Time spectra of four-fold coincidences. Beam 50Ti at

230 MeV on 1 0 0Mo and95Mo targets.

30

different compound systems gave unambiguous evidence of high

spin isomerism with multiplicity between 8 and 18. These re¬

sults are summarized in table I and their occurrence on the

nuclear chart shown in Fig. 3.

It should be noted that because of neutron evaporation

the isomeric states themselves most probably belong to nuclei

3 to 5 neutrons away from the compound system. Both proton and

alpha paticle emission may complicate the assignment of the

iso/neric nucleus.

A striking feature of Fig. 3 is the clustering of isomers

in the region 64iZ-71 and 82-N-88 and is considered the main

result of these experiments. The island of high spin isomers

identified is covered by detailed calculations by D0ssing et

4) al. who in a theoretical search find yrast traps connected

with strongly oblate and weakly prolate shapes in this area.

The theoretical predictions of ref. 4 indicate isomers with spins

from 20 to about 60 in the empirical island of isomers. There¬

fore the multiplicities observed might seem unexpectedly low.

One obvious reason for this deficiency is the occurrence of

several isomers along the decay path.

If an intervening isomer has a half life comparable to or

longer than the time constant of the multiplicity filter, the

cascade will be chopped off, completely or partly, resulting in

an apparent low multiplicity.

Experiments to illuminate this have been carried out by

remeasurements of several isomers with an extended resolving

time for the multiplicity filter. Results for 16, 25 and 400 ns

resolving time in the reaction Mo + Ti are shown in fig. 4.

Because of random problems it will be at the cost of detection

Compound nuclei formed by

heavy ion bombardments

- stability

== 10 min

Rotational limit

Fig. 3. Region of the nuclear chart searched for delayed

high-multiplicity y-ray events. Compound systems

found through bombardement with 4Oftr, 50Ti and65Cu

are indicated with shaded squares. Cases where posi¬

tive results were obtained are shown with filled

squares. The final nuclei are llKely to have 3-5

neutrons less. The approximate borderline of defor¬

med nuclei has been defined by Z.*-/c, .&3.0. The 2

l'p/rn = 1 line is taken from Nucl.Phys. h 170, 321,

1970.

10s

104

ioJ

z

8 102

10'

I *400ns

Fig. 4. Fold-coincidence distributions for 3 values of the

multiplicity resolving-time. The corresponding calcu¬

lated multiplicities are 12, 15 and 19 for the re-

solvning times 16 ns, 25 ns and 400 ns respectively.

33

sensitivity to extend the resolving time beyond 25 ns. The

increase in the extracted multiplicities from 12 to 19 is indi¬

cative of a relatively long-lived isomer along the decay chain.

In this particular reaction we know from the results of

ref.5'6' and detailed spectroscopy in the reaction 3 Ba{ O.xn)

ref.7' that the high multiplicity isomer is in 147Gd. Counted

in sequence from the ground state there are isomers of 22, 5, 27

and 650 ns. The reaction 100Mo + 50Ti indicates at least one

i&omer with T 1 / 2 I 50 ns feeding these.

147 A rather complex sequence of isomers as in Gd might

therefore obscure the highest spin isomers and in particular

their spin values for the present experimental approach.

The obvious experiments to do in order to obtain the com¬

plete structure along the yrast line would of course be tradi¬

tional coincidence measurements supplied with angular distribu¬

tions and polarization measurements. Since spectroscopic infor¬

mation in the region we are dealing with is scarce, it will

amount to years of work to proceed by this way.

A more direct way to further characterize the isomers iden¬

tified, would be to measure their total Y-ray decay energy and

thereby obtain an approximate value of their spin . Present

results of experiments with a Ge(Li) counter working in coinci¬

dence with a 20 cm x 24.5 cm Nal crystal have given promising

results along these lines. Ref. 9.

Fig. 5 shows the experimental arrangement schematically.

A titanium-50 beam from the GSI UNILAC strikes a target which

is placed in a cylindrical lead shield upstream of the large

Nal crystal. Recoiling evaporation residues are stopped on a o n ft

Pb catcher 25 cm downstream where they can be viewed by a

Fig. 5. Schematic drawing of the Ge(Li)-Nal assembly used

for measurements of total Y-cascade energies. The

catcher foil Is a metallic 208Pb foil of 12 mg/cm

thickness with a 10 mm circular hole for beam passage.

The Go(1.1) has 22* with 1.8 keV FWHM at 1.3 MeV.

Ui

Ge(Li) detector intruding at 90°. The bean la stopped far

outside of the Nal crystal. Since the Hal crystal consists

of two separate parts and operates in coincidence with the

Ge(Li) an enhanced sensitivity to high multiplicity cascades

is obtained.

A Y-ray spectrum obtained in the reaction 104Pd + 50Tl

Is shown in fig. 6 and contains transitions from Er, Ho and Dy

residual nuclei. Assignments to elements are made by changing

targets to 103Rh and 102fl04Ru and by excitation functions.

The total y-ray energy sum corresponding to selected peaks in

the delayed Ge(Li)-spectrum is obtained as well as their time

distributions.

Also by this technique sequential lsomers present a diffi¬

culty which, however, can be overcome by careful analysis of the

multiparameter data.

Two time to amplitude converters started by Ge(Li) pulses

and the Nal sum pulse respectively, but both stopped by the beam

microstructure pulses, permit selection of delayed Y-rays. A

third TAC between the Ge(Li) and the sum crystal will respond

with a single peak in the time distribution if the selected

Y-transition and the corresponding sum is a prompt cascade.

Interruption of the cascade by a longerlived ioomer will In addi¬

tion to the peak give rise to a decaying tail on the time spectrum.

Further analysis of the 6-parameter event-by-event data

obtained in this experiment is in progress9' and will provide

cascade energy sums as well as the single transition energies

together with half lives of intervening lsomers in the decay chain.

Hopefully this experimental approach will supply sufficient infor-

36

02 03 0.4 05

TRANSITATION 0.6 O?

ENERGY (M*V)

08 09 10 1.1

l04 50T Fig. 6. Delayed radiation from the reaction l04Pd + 50Tl at

220 HeV observed aa 3 fold coincidence and correspon¬

ding to the total Y-enerqy sum.

mation for a description of the structure of the isomers iden¬

tified in the N-82 region.

Very rough excitation functions have been obtained for

10 of the lsomers. In spite of the limited quality of these

data it has been possible to make aost probable assignment of

the isomeric nuclei. These assignments indicate that in the

reactions where the highest multiplicities are observed (table I)

then the most probable isomerlc nucleus has 82 or 83 neutrons.

Similarly, the reactions showing multiplicities 8-12 are pro¬

bably due to lsomers in nuclei with N - 84.

That this trend Is the same as in the results of Dassing

might be fortuitous, but could on the other hand Indicate that

the isomers in this region have pronounced oblate shapes and ori¬

ginate from traps with I > 40. Measurement of the isomeric ex¬

citation energy E* should through the relation:

E* = Ch2/2J) • I2

give a rough measure of the spin I and test this possibility.

If the results will indicate a low excitation energy the

structure of the lsomers is obviously another.

Since the isomeric nuclei all lie near the N=82 shell and

do not deviate significantly from sphericity at low angular

momenta it might be possible to explain them In terms of spheri¬

cal particle-hole configurations. Such isomers are observed

near the Z»82, N«126 shells10'. It is hoped that the new expe¬

riments outlined in this report will bring the needed evidence

on the nature of the isomers and a critical test for the theories.

38

TABLE I, Isomeri observed tn nioosncoDd time rajv'c for compound nuclei with 56 iZ* 82 produced by JBAJ-, "Ti, and "Cu projectiles

Con-.pound nucleus

»Gd' KGd* *G<S«

•!^4 MDy ' Ho*1

' ! <Er' "'Sr "Er "Er "Er 'TVR "Tm

'"Tn. '"Tn, lslYb "°Yb "'Lu '"Lu "'Lu

Projcclila (Cu, MoV)«

"TH20*) "TH178) "ArdOM

*Ti (225) "Tl (225) "Ar(103) **n (212) "Ti(215> "71(215) «Ar(193) •Ar(173) "AN193) •sCu(28Bt «>Cu(2eO) >*Ti(215) "TK215) •»Cu(270) «>Cu(275) "Tl (230) "Cu(2751 "Cu(275) "Cu(275)

Multiplicity*

B . 2 1 1 . 2 13.2

15.2 13.2 12.2 15.2 18.2 15l2 17.2 8 . 2

11.3 15.2 15.2 15.2 12.3 13.2 H . 2 10.2 12.2 12.2 8 . 4

<n»'

1,6 100 100

700 400 30 40

250 B0

. . . 50

. . . 30 40 50

. . . 40 35

. . . 50 so

... 'Mean projectile energy In target. ""Derived from 4:5- tnd 5:6-foId ration aaeumir^; •

sharp multiplicity ( R e f . 1 2 ) . c Estimate based on yield ratios with u d without

ciuDhcr foil. Approximate uncertainty . 50%, ^GeffLD y-ray spectrum has been obtained. For the

"°G<1 compound nucioun the two lsomcrs T^ * 1.5 ns and 7" ,„ « 100 ns belong to "'Gd and '"Gd, respectively. In the other cascs no assignment has been possible be¬ cause linlo is known about the lower-spin sequence of yrast transitions.

'Estimated cross section for formation of the lonj-llved Uomer, which belongs to '"Gd, Is 6 . 3 mb u compared to 50 mb (or the total ' " ^ ^ cross section ( R e f . 1 3 ) .

As an appendix I would like to summarize experiments

aiming at detection of possible a-decay from yrast traps.

During the search for delayed y-ray cascades also two solid

state surface barrier detectors were observing the decaying

evaporation residues on the catcher foil. These counters were

only sensitive when a cascade with 3 or more y-rays were de¬

tected by the multiplicity filter. Because of the smaller

solid angle the sensitivity was of the order 100 yb for

10 ns < T^ ,, < 5 ms.

No delayed a-decays above these limits were observed in

the region searched. Fig. 3.

A supplementary search with higher sensitivity and the

possibility to observe half lives from 1 ns down to about 10

picoseconds has been performed by Kohlmuyer and co-workers

also at GSI. Their detection technique is displayed in fig. 7

and consists of a position sensitive counter telescope. The

counters are arranged in such a manner that they are shadsd off

from prompt reactions in the target, but as soon as a recoiling

nucleus leaves the target it can emit o-particles into the

counter telescope.

No delayed a-particles were detected in the area where

delayed high multiplicity cascades were found. For a solid

angle corresponding to a half life of 100 ps the cross section

limits were typically 1-6 ub, but with limits of 9.5 gb and 18.0

vh for the C.N. i54Dy and 153Ho respectively.

Reactions with Xe on a range of targets produced compound

nuclei roughly along the T. ,, % 10 m m boundary in fig.3 from

0 1 2 3

collimator degrader target Faraday cup

A E - Position sensitive detector

E - Position sensitive detector

Detector shielding

Fig. 7. Experimental situation for detection of u-decay in- vents particles scattered in the reaction chamber

flight. The E counter is lOOiim silicon position to enter the detectors. Extreme care was taken as

sensitive detector and the E counter 900um thick to the target flatness (even within 0.1 mm) and ad-

*• also position senaitive. The dotoctor shield pre- justmont of detector-target angle.

Z=60 to Z=83. None of these indicated a-decays in the half

life range down to 10 ps. Cross section limits for 100 ps

isomers were from 1-11 yb.

References

1) T.E.O. Ericson, Introductory talk at the International Conference on Nuclear Structure, Tokyo, September 5-10, 1977.

2) A. Bohr and B.R. Mottelson, Physica Scripta, Vol. 10A, 1974.

3) J. Pedersen et al., Phys. Rev. Lett. 3_9, 990-993, 1977.

4) T. Dossing, K. NeergSrd, K. Matsuyanagi and Hsi-Chen Chang, Phys.Rev.Lett. 39, 1395-1397, 1977.

5) Z. Haratym et al., Nucl.Phys. A276, 299, 1977.

6) P. Kleinheinz et al., Proc. International Symposium on High-Spin States and Nuclear Structure, Dresden, Sept. 1977.

7) 0. Bakander et al., Preliminary results at NBI, 1978.

8) P.O. Tj0m et al., Phys.Lett. 72B, 439, 1978.

9) J. Pedersen et al., Work in progress NBI-GSI, 1978.

10) D. Horn et al., Phys.Rev.Lett 3£, 389-391, 1977.

11) B. Kohlmeyer et al., GSI-Harburg unpublished results, 1978.

12) G.B. Hagemann, R. Broda, B. Herskind, M. Ishihara, S. Ogaza and H. Ryde, Nucl.Phys. A245, 166, 1975.

13) S. Delia Negra, H. Gauvin, H. Jungclas, Y. LeBeyec and M. Lefort, Z.Phys. A282, 65, 1977.

H K 2 SPUr ISOIIBRIC STUBS IH 152Dy * F.Beck, C.Gohringer, J.C.Merdinger, J.P.Vivien Centre de Recherches Hucleaires, Strasbourg

E .Botek. J.Styczeń Institute of Huclear Physics, Cracow

In the last years, a great deal of high spin experimental and theoretical work has been devoted to study of the yrast traps, and till now there is no clear experimental evidence of their existance.

There are two main interests in the research and studies of the yrast trapst

1/ One can provide information on their structure that is an important question which may help in understanding their nature in the terms of the single particle picture.

ii/ Their decay modes. Do they decay by gamma emission only, or by particle emission too ? On the other hand such isomers offer for experimentalists natu¬ ral means for detailed epectroscopic investigations due to their particular selective decay ways ±n comparison with a large num¬ ber of channels open in the decay of a compound system. The de¬ cay pictures may depend on shapes of nuclei in the ground and low spin states. For example, the decay of a deformed nucleus via electromagnetic radiations might exhibit delayed spectrum characteristic for a rotational-like pattern, while for a sphe¬ rical nucleus decay scheme may be fairly complicated. Op to now, scarce information on existence of high-spin isomeric states has been obtained by lifetime measurements. The Darmetadt-Kopenha-gen group [1 ] studied a number close to 100 particle-target com¬ binations leading to formation of about 200 residual nuclei at

Presented by E.Bożek

high spin excited states. In theix experimental review the

authors discussed an island of high spin isomers in the region

64^ Z „< 71 and S 4: 82. In about 20 different compound

systems, the isomers were observed with lifetimes between one

and a few hundreds of nanoseconds with multiplicities between 8

and 18. Some of these isomers can be simply explained as the

shell model isomeric states f2,5]. Therefore further detailed

spectrr>scopic studies are needed in order to perform a more

quantitative analysis and to determine the "true isomeric traps"

predicted by theory [4, 5, 6}. Nevertheless, a fair agreement

between the experimental data [1} and theory [4] concerning the

localization of this island can be considered [5). From the

theoretical calculations the following conclusion can be drawn:

1. In many nuclei are found more than one or two traps.

2. Generally the Woods-Saxon potential seems to produce more

traps than the modified harmonic oscillator potential.

5. In some cases both potentials give similar result.

These calculations provide three isomeric traps in Dy.

Recently one of these isomeric states with a spin between 14

and 18 4f at an energy of the order of 5 MeV was found in this

nucleus [7]. In order to make any conclusion on the structure

of this isomer whether it is an isomeric trap or not, one needs

more experimental information than only the energy and the life¬

time* For the first time this state was observed in the

^Gd/o<f, 6n/ reaction [73 an<i i t s lifetime was measured to be T1/2 ~ 6 0 n s* Unfortunetely, due to a strong hf interaction

in the gadolinum target the analysis of angular distributions

of deexciting Y-rays could not give spins of the involved sta¬

tes.

Here we report on preliminary data obtained in experiments

carried out with the 0 beam of the tandem accelerator of the

Centre de Recherches Nucleaires in Strasbourg. Natural CeO2

of 200 ng/cm was evaporated on a thick Pb backing. Using

the 14OC«/16O, 4n/152Dy reaction at incident energy of 86 MeV

0 ions, we performed extensive spectroscopic studies: excita¬

tion functions, two dimensional spectra of f - y (T) and

Tf- y /T»H/« The level scheme as established la thia work is shown in Pig. 1. The presently measured time delayed f -ray spectra confirme the existence of the lsomer at the energy /5O5O + x/ keV reported in Ref.£7} and furnish more precise value of its half-life ^/2 - 53*4 - 5«0 as. Moreover we have found a new isomeric state lying by 1 MeV higher /see Fig.1/ that the first one and its half-life was found to be equal to 8,5 - 0.7 ns. AD example of the decay curve for the second iso¬ meric state is shown in Fig. 2. Since the energy of the first isomeric state is not accurately known as the low energy direct transition deexcitating this level, the energy of the second isomeric state is not known too; the energy gap between thea

93<.5<

Pig. 1 Level scheme of 1^2Dy obtaintd In th« 1W)0«/160,*n/ reaction

Fig. 2

The experimental decay curve for the second isomeric state at 6077 • i keV

being equal to 1044 keV. Now it is to early to say whether these isomers could correspond to the two isomers predicted by the Warsaw-Lund group [4-J which are calculated to have the same excitation energy of 1.13 UeV. If the 53 ns isomer is identical with that observed in Ref.[1] seen in AT induced reaction, than the lifetime estimated in this reference is significantly shorter and the multiplicity is overestimated. The same effect was observed in the Gd isotopes [37.

In order to obtain more information on the structure of the first isomeric level, preliminary measurements of the g-fac¬ tor were performed using a differential PAD technique. In this experiment the target was placed in a 12 kOe external magnetic field perpendicular to the detection plane; the field direction being reversed every one minute. The two Ge/Li/ detectors were placed at the angles 45° and 90° with aspect to the beam direc¬ tion. The y - f coincidence time spectra were measured for the two field directions and combined in the known expression /ne¬ glecting the A^ term

*1 " *I * 3/4 ^ sin 2 / tJL * XI' — 3

where H^ are the numbers of counts for the two field directions /up and down/, Ag is the angular distribution coefficient, c j. = gyO^H^S is the Laxmor precession frequency and A 9 is the beam bending angle. This measurement was considered as a testing experiment to determine the experimental conditions in which the hf perturbation effects due to the radiation damage are decreased. From the fit of the above expression to the ex¬ perimental points /obtained at the room temperature we found (0^ £3 "1.28 .108sec. which gives g/3^2.1 where is the para¬ magnetic correction factor. Assuming that the dysprosium atom implanted into lead has electronic structure of Dy^+, the ef¬ fective magnetic field acting on the Dy nuclei is He£« * A H e I t , where R = 6.02 /if one neglects crystal field effects/. Taking this value of /3 one gets g ~ 0.35. To draw any con¬ clusions on the magnetic properties of this isomeric state, one needs more accurate value of the product fi g and and addition¬ al information on the influence of the environment on the para¬ magnetic properties of dysprosium atoms.

References

[1] J.Pedersen, B.B.Back, F.M.Bernthal, S.Sj^rnholm, J.Borggreen, O.Christensen, F.Folkmann, B.Herskind, T.l.Khoo, U.Neiman, F.POhlhofer, G.Sletten, Phye.Rev.Lett. ,JJ /1977/ 990.

[23 D.Horn, O.HSusser, T.Faestermann, t.B.McDonald, T.E.Alexander, J.P.Beene, C.J.Herrlander, Phys.Sev.Lett. *2 /1977/ J89.

[33 R.Broda, M.Ogawa, S.Lunaxdi, M.R.Maier, P.J.Daly, P.Ceinbeinz, Zeitschr.Phys. / in press/.

£4] It.Cerkaski, J.Dudek, Z.Szymaiiski, C.G.Andersson, G.Leander, S.Aberg, S.G.Rilsson, I.Ragnarsson, Phys.Lett. 2PJ& /1977/ 9.

[5] T.Djtesing, K.Heergard, Shi-Chen Chang, Int. S7ap0si.UK on High -Spin States and Huelear Structure, Dresden 1977, p.95«

[6] A.Faessler, H.Płoszajczak, I.R.S.DeTi, Phys.Rev.Lett. ^8 /1976/ 1028.

[7] J.F.Janson, Z.Sujkoirski, D.Chmielewaka, E.J.lleijer, Int. Conf. for Nuol. Stability, Carges 1976, P.J82.

NUCLEUS OF VERY HIGH SPIN STATES MICROSCOPIC DESCRIPTION **/

Marok PloszaJezsJc Institute of Nuclear Physios, Cracow, Poland.

Interest in the study of the nuclear rotation at very high angular momenta increased significantly after the suggestion of Bohr and Mottelson £ij , oonoernlng the possible existence of yrast isomers with the high angular momentum (often called yrast traps ) . These states would open new possibilities for the "classical" nuclear speotroscopy, allowing to resolve the single t -lines in the deexoitation spectrum of the nucleus at high excitation energy and spin. By observing the discrete transitions to the yrast trap one might hope therefore to learn about the nuclear coupling scheme at the excitation energies close to the desintegration point for nucleus (by fission or emission of fragments ) , Suggestion of Bohr and Mottelson was in fact initialized by calculations of the nuclear properties at high spins performed by Cohen, Plasil and śwlątecki £2] in the classical rotating liquid drop model (RLDM). These authors found that as a result of competition between the surface and Coulomb energies from one side and the rotational energy from other side, the nuoleus at(l = 0 spheric/ increases its deform¬ ation with increasing angular momenta rotating oblate around its symmetry axis. Such a shape is no more stable at spins sufficiently high to dominate the surface and Coulomb energies by the rotation term. In this case one expects rather sudden change of the deformation (from oblate to prolate) accompanied by a change of the rotation axis from z-axi« (symmetry axis) to x-axis. At still higherlnucleus fissions in this configu¬ ration.

Collective rotation about the symmetry axis,though classicaly

possible, is forbidden In the quanta! system, since in this

case potential is static and one camaot detect changes of the

electric field ol nucleus caused by the rotation. The total

excitation and angular nonentun might be therefore obtained by

• individual single particle excitations, sane as in the closed

shell nuclei, ver« one in fact detect many isomers.

In this lecture I viii discuss vainly two points. First

part oonoerns the studies of nuclear properties hidden in the

deformation energy surfaces (DES] . These energy landscapes

will be obtained either using the Strutinsky shell correction

approach adopted for the calculations at high spins, or using

the cranking Hartree-Fook-Bogolyubov theory (CHFB) , By studying

these landscapes, we will try to draw general conclusions con¬

cerning the shape (and its instabilities leading to the so

called giant back-bending) as well as the stability with respect

to the fission mode, by looking to the height of the fission

barriers at high angular momenta.

In the second part of this lecture X would like to discuss

more in details the structure of yrast isomers or traps at

high spins. Ve will see how the different microscopic mecha¬

nisms may lead to the lowering of the nuclear configuration

with spin J with respect to the neighbouring 1-1 and 1-2 con¬

figurations. Competition between these mechanisms as well as

the possible inaccuracy of the theoretical approaches using

the effective two-body forces adjusted to the nuclear properties

at low spins will be presented. Finally the simple, more

phenomenological method which would ba able to predict with the

reasonable acouracy the structure of a wave function, spins,

parities and excitation energies of the Isomers along the yrast

line will be given. This approach should be treated as a

usefull tool for both experimentalists and tbeoretltian. It

allows to learn about the fine details of the yrast line, in

absence of the microscopic theory which would use the trust¬

worthy Hamiltonian.

50

11^ Reformation enercy_surfaoes_at very high angular momenta

Nuclear properties at high spins embodied JJI the energy

landscapes have been studied first In the RID - model by

Berkeley group f2] . In the RLDM description, at I = 0 nucleus

is spherical increasing its deformation (oblate) with increasing

spin. In this stag* of rotation nucleus is spinning around its

oblate symmetry axis f(i>0> T = - 60°, see Fig. i) . At spins

higher than the oritloal angular momentum for the stability

of such oblate shape, nucleus changes Its form from oblate

to prolate (ft?0, t ~ 0") passing through the variety of

trlaxlal ( f<0") shapes. One knows however, that only the

nuclei with closed shells posses the spherloal shape In the

ground state (i = O) . Other nuclei exhibit more or less

oblate (y)

Y«n • * Z Y *0

prolate (Z)

prolate (y)

oblate (xl

Fig. 1. An illustration of the physical situation in the

plane. The rotating modes of the nucleus at the three axes are

shown. For each of the rotating states the symmetry axis is

indicated in the bracket.

51

pronouncsdquadrupole deformation, and this effect is uniquely

determined by the shell structure. The rotation, 'which In its

largest part is described by the classical, RLDM - expression,

tends to diminish the role of shell effects ( this statement might

be formulated even before starting the detail calculations) .

We are however concerned with a question: how stable is the shell

structure at high angular momenta, is it negligible at spins

corresponding to the fission instability in RLDM, how much

change predictions of RLDM concerning the trajectory of lowest

minimum as a function of the angular momentum eto. However,

to check these points on* needs the detailed calculations of

the shell structure at high angular momenta.

There was two method proposed till now to ••timate the shell

effects at high spins: Strutinsky shell correction approach

extended for If 0 £3-5] and the microscopic CHF(B) methodf using the pairing and quadrupole - quadrupole (P + QQ) force

as the residual interaction £6,7} . Generally, to describe

the rotation of ellipsoid around any rotation axis one has

to construct the total energy expression not only in the sector

- 60° -+ 0° (as necessary in the RLDM) but also for t -deform¬

ations between 0° and 120° (see Pig. 1 J , Other values of t*

are excluded due to the symmetry of the wave function with

respect to the rotation on angle T\ around x-axis [8] .

Discussion of the methods used to construot the DES we start

from the shell correction approach.

II. 1. Strutinokv approach at I > 0 and T = 0

Strutinsky renormalization at finite angular momenta will

be discussed here very briefly and only in oase of vanishing

temperature ( T = 0 limit } . Details can be found anyway in

Refa. 4,5 and references quoted therein. The single particle,

potential describing the nucleus in the rotating frame

oonsists out of the kinetic term t, trace of smooth two-body

interaction V(frt f, l>) and the cranking term • o j , . W ,

which in the classical theories has the meaning of the angular

velocity, is a Lagrange multiplier fixing the value of the

angular momentum.

It is convenient to define three kinds of energies in the

rotating frame for the namlltonlan (II.1.1.) . These are:

l} The independent particle energy In the rotating frame:

-; 11 1.2.

where £., denotes the eigenvalues of ° and the summation

goes over the occupied proton and neutron states,

The "smooth" Independent partiole energy:

where 4(£j is the Strutinsky smooth density of

levels [8],

iii) The "classical" energy in the rotating frame:

11,.».

where ltLp is the LOM - expression for the ground

state energy and J^, is the rigid body moment of

inertia. For each of these energies K 4 one dcfinei

the angular momentum:

and energy in the laboratory system:

The Strutinalcy renormalization of the total angular moaentuB

and energy at finite angular momentum might be therefore written

as:

II 1.7

Equation* (ll 1.?) simplify in case of the realistic potential*

like the Saxon-Voods average potantial.' Cue to the locality of

thi* potential (exospt of tho •pin-orbit ooupling term vhich is

anyway small) the smooth total angular momentum equal* to the

classical expression [9J :

II 1.8

and consequently at finite I, the difference between the

•classical* and the smooth independent particle energies

amounts to the difference at 1 = 0 :

In this oase the eq. (II.1.7) simplify:

One has to stress however, that expression (II.1.1O) are

striotly fullfilled is oase of the local potentials only. In

case of oommonly used S-V potential the differenoe between

*tt ( £, IT, ) and I£m.((^,Tt CO ) is of the order of few percents L8,5J.

II.2.^Calculations with the Saxon-Voods potential

First calculations in the Strutinsky shell correction approach at high spins, have been performed using the Nilsson s.p. potential C3,4] . Soon, it has been recognized that the velocity dependent! -Ł - term in the Nilsson model gives the unphysioal enlargement of the smooth moment of inertia J . Ihis enlargement cannot be compensated by the subtraction of the averaged value of •*• in each oscillator shell. Also the redefinition of the -C*-term : I'-^NCM**) taking JL = 1 instead of thejusual d = 1/2 does not remove this deficiency. In this case ~$ — ?r;> * as required, but the agreement for the ground state properties of nuclei, described, using the Nilsson model with the parameters of 1-s and newly defined i. terms refitted, is worse than in the case of standart o4 = 1/2 f"9J. Solution of this defect might be probably found in the new fit of energies of high - j, high - N orbitals in the Nilsson model which are too close to the Fermi surface in the comparison with tho S-W model. Till now, nobody tried to perform this tedious analysis and improve the Nilsson model in this way. Therefore, the trustworthy calculations of DBS have to use the S-V model at high spins as a source of s.p. energies in the Strutinsky procedure. These calculations have been performed first by Ne«rgard et al £^3* These authors used the average potential:

L * i + V + V +V which ia a SUB of kinetic tara i K,-„ I *>P> potential Vtt4 depending on the position of the nuclson in the potential well only, spin - orbit part of th« average potential /Jo in its ' radial part defined aa the derivation of S-V potential and the

55

Coulomb potential T approximated by the potential of a unifor¬ mly charged nucleus with Z - 1 protons. The basis states for eigenfunctions of K- was chosen as the eigenstates of the three dimensional harmonic oscillator:

IŁ; JX 2.2

where /£(£) denotes the spin wave function and Yn (>0 i a

the normalized harmonic oscillator wave function give in terms of the Hermite polynomial H_ as:

Il2.it

In the above equations, m is the nucleon mass, (*)„ repre¬ sents the oscillator constant and n denotes the number of oscillator quanta. Xn calculations of Hef. 5 only the basis states which satisfy the following "energy" conditions are taken into account:

with NQ = 10. The oscillator energy ktj is given here by:

for neutrons II 2.6

protons

The dependence of the smoothed sum of s.p. energies (il 1.3)

on the smoothing parameter f va" Investigated in details. We

find generally the good plateau for the function S f j ^ ( f ) at

f = 11.5 MeV, and therefore this value was used in all calcu¬

lations discussed in this lecture. Details concerning- the results

for the DES, reported in this chapter can be found: for the

transitional, and actinide regions in Kef. 10 and for the rare-

•arth region in Refs 5 , 1 1 .

W00S-SAXON

12** Fig. 2. Contour plots of the deformation enercies of ' Te_2 as

a function of the deformation parameters (b and T"(-CO*£ jfi 4Z0'/

for various angular momenta using the S-V potential. The shaded

area indicates the deepest minimum in the energy plane. The

energies MeV in the plane are measured relative to the deepest

minimum for each angular momentum. Beside each figure the value

of the energy of the minimum relative to the energy of tho spho-

rical liquid drop with angular momentum z«ro is indicated.

57

Deformation energy surfaces at I = 30, 50, 70, 90*. for \2.h Zg Te are shown in Fig. 2. Ibis nucleus exhibits at high spins typical tendencies for this region of nuclear chart. At 1 = O _2Te is almost spherical. Vith increasing I it atreches along the t- -60° line till (b ~ -0.65 and at I £ 90 k nucleus fissions passing the triaxial shapes. The shell effects are not strong enough to form a stable, prolate ( T~O') configuration

124 and the high spin behaviour of 5 2 T e *-• dominated by the "classical" term in the total energy. This statement holds in most of nuclei in the transitional region (A SS 1OO - 1*»0) . In this region of nuclear chart the quadrupole deformations corresponding to I & •'•cilIT a r e 8 e n e r a l i v l^ge ( (3 ~ -0.6) and often one might see the stable configurations at T » -60° with the ratio of axes R^tR^* 1:2. This region of superdefor-med nuclei at very high angular momenta was early suggested by Cohen, Plasil and święcicki C21 o n t h e basis of their calcu¬ lations in the RLDM. The often seen difference for the transi¬ tional nuclei between the RI-DM results of Ref. 2 and our results including the shell effects is the rapid and large changes of deformation with X. This effect is caused by the strong stabilizing role of shell correction at the configura¬ tions corresponding to the ratio of axes of nuclear potential being the simple numbers (e.g. R :RA =2:3 corresponds to f»,~ - O.iłO while R :R = 1:2 to p — -0.6k) . Therefore at any of these configurations nucleus will be stabilized in wider range of I. This effect might be clearly seen in Fig. 3, where the transition energies E2, M2 between tha yrast statea are

11S plotted versus the total angular momentum for i;2Te66" H*re

one sees three "discontinuities" resulting in the decrease of the transition energy &Ej= E.- Ey_Ł which is reciprocally proportional to the moment of inertia. First discontinuity at 3 ~ 3** Ł corresponds to the change of the shape from the nearly spherical configuration to the configuration with ratio of axes R :Rj_ = 2:3 (,f~ -0.it) . The second transition Is seen at I ~ kZ i. and is caused by the change of the configuration fron

2J3 to =1 :2 ( (b ~ -0.61) )

UJ

ul

20 40 60 TOTAL ANGULAR MOMENTUM I M

Fig. 3. Transition energies versus the total angular momentum 118 for rp^e66" T h e d e e P oiinima in the curve correspond to the

drastic change of the quadrupolo deformation lat T - -60 ), resulting in the enlargement of the niomont of Inertia J >f the nucleus. This rapid increase ,0!?'. J causes the decrease of the transition E - ' iv,:

The last transitiou/S^een at if. •" 6k Ł ie called usually the giant - bachbełitli.Jijj iilnd is due to the inotubility of the rota-

^ changp of the ting, oblate liquid drop resulting in ą ^ ft.r.ja.tj shape (from obiato to prolato)and tft ViiSZsi-'ti iSrs -from - 60° to O0} . These the I dependence of the be tested in tho exporimii along the yrast line

t h e

SUB ( « changes [jroditptions concerning

c dot'ortiatS'jinyj at high 1 could trsnaitioi) energy

Of the ł -cascade

59

would be measured for different total angular momenta, trans¬

feree! to the final nucleus. In case that the "trap" in the

curve &£ f ) would exist at high I then the increase of the

total angular momentum in the final nucleus obtained for exam¬

ple by increased bombarding energy of projectile would cause

the increase of the Y" -multiplicity and the constancy of the

largest j -transition energy emitted from yrast line. Such

experiment have been recently planed by the ftis group [12] .

For nuclei In the rare-earth region, examples of the neutron-

deficient 16°Cd<3<; (Fig.'* ) and the well deformed '^eLi^g (Fig.5)

from the middle of this region aro worth to study. |jjGd

prolate in the ground state ( f = 0 ) with rather moderate ft

deformation. Already at relatively low angular momenta the

WOODS-SAXON

Fig. k. Deformation energy surfaces of g?Gdgg f° r various

angular momenta using the Saxon-Woods potential. For details

see caption of Fig. 2.

60

lovest minimum has T =. - 60° and ft /%/0.1 . It is rather stable and persists up to I — 60 Ł . At I - 70fc the equilibrium configuration corresponds to the triaxial shape (V«» -30°]. At still higher angular momenta (I ** 90Ł] the lowes.t minimum has £>~ 0.7, X ** 0° and corresponds to the fission isomeric minimum. Other, neutron deficient light rare-earth nuclei show similar tendencies: the oblate line is reached at relatively low spins ( ~ 20b) , and the magnitude of the oblate deformation is small. This region of periodic table was early suggested by the authors of Uefs 3,4 as promising for the discovery of tho oblate yrast isomers. This prediction is also confirmed in the more elaborate, Strutinsky type of calculations, using tho Saxon-lioods s.p. energiesC53 . Recently is son* of light, neutron-deficient rare-earth nuclei the isomeric states with rather high angular momenta have been discovered fi3-i6j.

Nuclei from the middle of the rare-earth reeion are repre¬ sented here by 16sŁr<)8 (see Fi6»5 ) • The trajectory of the lovest minimum in the ft- "f plane looks very different than that of the transitional (see Fig. 2) and light, neutron deficient rare-earth nuclei (compare Fig.k j . At I s 0, these nuclei have a large ((^ ~ 0.3, T » 0°) and very stable quadrupole deformation. With increasing I, one might see the shrinking combined with the building of the positive f -deformation (see the DES at I = 30 fc in Fig. 5]. This shrinking is connected with the fact that higher angular momenta states in the nucleus are build by the subsequent alignment of the angular momenta of the individual nucleons with the rotation a;is. This is achieved by putting the particle orbits in the plane perpendi¬ cular to the x-y plane i.e. by occupying the s.p. states with small positive or negative quadrupole moments and leaving tho states with generally large positive quadrupole moment, in this way the positive quadrupole moment of the nucleus decreases with increasing I. At I = 50^ one observes that the trajec-- o tory of the lowest minimum bends once more back to the j = 0 line, however, at (1 -deformations lower than in the ground state. At I « 70fc the centrifugal forces are already strong

61

WOODS-SMOM

Fig. 5. Deformation energy surfaces of 6gEr98 f o r various angular momenta using the Saxon-Voods potential. For details see caption of Fig. 2.

enough to push the deepest mininum toward the negative X -de¬ formations. They are however still not sufficiently strong to form the m-łn-imiim at T= - 60° line at I smaller than the fission limit. Therefore this region jf nuclei sneins to be less favourable for the existence of yrast traps having the struoture as predicted originally by Bohr and Mottelaon fi].

Till now we have assumed that the yrast isomers might be formed in the oblate nucleus spinning around the symmetry axis only. Xn fact the same properties of the nuclear potential as seen along the T» - 60° line is aohieved also in the prolate

nucleus spinning around the symmetry axis ( (ł J 0 , J = 120° . The reason, that we discarded this kind of rotation as a favour¬ able way to produce yraot traps, was the classical rotational energy term. Rotation around the prolate symmetry axis corre¬ sponds to the spinning about the axis of the minimal inertia, and therefore the rotation energy for V = 120 is maximal. Thus, Judging from the "classical* LDM-resulta, the formation of yraot lsomers in the configurations at T = 120° is impos¬ sible. Shell effects however might modify the clasaical picture substantially. Especially favourable condifcvon-s for the rota¬ tion around the prolate symmetry axis are at the end of the rare-earth region (N = 110-120) , where the proton shells j'Z = 82) are almost closed. Neutrons fill gradually the high- BO mem¬ bers of the i 13/2 a n d ^-0/2 o r b i* a l J f o r t n o prolate deforma¬ tion (see Pig. 6j. The s.p. states which belong to the

tlMeVJ

4(- 99,,1/2-

UMeVJ

025 PROLATE OBLATE

Fig. 6. Nilsson single particle levels for neutrons with a hexadecapole deformation |a,=0,0. On the left hand side s.p. levels are plotted as a function of the prolate |3Ł -deformation, while on the right hand side as a function of obiato A Ł -defor¬ mation.

SHELL CORRECTION 1=0 SHELL CORRECTION 1=30

0.3?

Kk, 1U 120 t28 136

N(2=80l

« « t H2 120 1M 136 N(2=80)

SHELL CORRECTION

•at

Fig. 7. The shell oorreetlon a E(ft^t) for spin values I = O,3O,

60 and 90 ohovn as a function of the neutron number and the

deformation. The proton number is kept constant (Z = 801.

Results have been obtained with the Saxon-Voods potential. The

positive fb stands for the rotation about the b<elate syanetry

axis. The shaded area represents configurations with negative

shell corrections and the spacing between the contours is 1 HeV.

and K n/2 multiplets are bunched for small C> . Therefore, even for a very small cranking frequency splitting of these states will be large and consequently many levels will cross the Fermi level. In this way we can build up a very high angular momentum without loosing much energy. The opposite situation is seen for the oblate deformation. Here the last members of the i 1 o / 2

a n d "" a/o multiplets have low ulj . Therefore, the formation of an angular momentum as large: as on the prolate line, requires much higher s.p. excitations. The same feature is clearly seen also on Fig. 7. For the ground state, the shell correotion in Fig. 7 exhibits a very deep minimum at the spherical configuration for the neutron number Us 126. With increasing of the angular momentum the minimum at f = 120° is energetically more favourable. Kor example, at I = 30 fc. and N = 116 the minimum at T = 120° is situated 4 'ioV below the minimum at T = - 60°. At still higher angular momenta this clear preference of the rotation of the prolate nucleus around its symmetry axis is not seen any more and the nucleus prefers to rotate around the oblate symmetry axis. Thus, this asymetrie changes of the shell correction at T- 'Ł0 a"f( - 6o lines may lead to the formation of a classicaly forbidden con¬ figuration. First evidence for the existence of these kind of isomors have been given by Khoo et al.£i7-20j in -,- \\I .nu and ~z H-f- 105« These nuclei will be further discussed in chapter VI of this lecture.

To see the interplay of the conflicting t en.-1 en c ft; shown by the shell corrections and the classical RLXi - energy we present in Fig. 8 the DES in the sector -60°< T < 120° for g21Ibii6' The shaded area represents the lowest energy minimum. The energy of this minimum is normalized to zero. Only two minima exist at I = 30. The lowest one at 0.05 and S~ - 120° lies 7 MeV below the fission isomeric minimum at (S — 0.5 and f~ 0 . These two minima are separated by the mountain with the height of 12 MeV. With an increase of the angular momentum a new local minimum' on the f =-60° line appears. At J. = *ł0 this minimum is 2 MeV above the lowest minimum at (* ~ 0.05 and T —' 120° and 2 MeV below the fission isomeric minimum. At a till higher

Fig. 8. The contour plots of the deformation energies of

S2l>b1i6 a s a f u n c t i ° n of the deformation parameters |* and T (-60 < Tf" £ 120°) for various angular momenta using the

SV potential.

Fig.2,

Details can be found in the description of

angular momenta ( I <*» 50fcjone may see a drastic change of the

shape. The lowest minimum is now at ft — 0.28 and f s. -60°

while the minimum at the t = 120b l i n e ifi 1 M e V a b o v e« TOe

fission isomeric minimum at (i "- 0.5 and f" 0° is atill

2 MeV above the oblate minimum and the height of the barrier

which separates them is k MeV. At around I = 60 the minimum

at J " 120 is completely washed out and one sees only two

minima, namely at ffc - 0.25, "f =-60° and Q> ~- O.55, T *• 0°.

Tlie minimum at T = 120° line appear* in g2Pbii6 a t 81na11

fb -deformation ( ft ~ 0.05J . This effect is also obtained

in the Strutinsky calculation* for other nuclei at the end

66

of the rare-earth region and might be understand in a following-

way. For the sector of y -deformation from - 60° to 0°, the

nucleus is rotating around the axis of the of the large moment

of inertia. Therefore, the increase of the quadrupole deforma¬

tion is energetically profitable for high spins because in this

way the rotational energy term is minimized. Xn contrast an

increase of /! for $" - 120 means a decrease of the moment

of inertia. Therefore, the rotational energy leads to an in¬

crease of the total energy if one enlarges the deformation Tor

rotations around the prolate symmetry axis. Thus, ono may expect

that for high spin states ( I = 30 - 50k) the lowest minimum for

some isotopes may be found at T» 120° only for small (1 .

For the large quadrupolo deformations the difference between

the rotation energies at Tf« 120° and X — -60 is so Iai7;e

th«.t the nucleus lias to have a negative J . Nuclei showing 1 OR

same tendencies as g,Pb should be looked for in heavier,

neutron — deficient rare—earth nuclei and in the neutron defi¬

cient lead isotopes.

The properties of DES at high spins changes once more by

passing the shell closure at K = 126 and Z = 82. Similarly,

to the light rare—earth isotopes with neutron number slightly

larger than N = 82, Strutinsky calculations with a Saxon-l.oods

s.p. energies predict the existence of the island of oblate

isomers. This might be seen in the upper part of the Fig. 9.

Here, the trajectory of the lowest minimum in the ft -f plane

is plotted for various angular momenta in gg'lai20 a n d fcfc'!a1V'

Also in these nuclei the magnitude of the /J deformation along

the t =-60° line is small and consequently the giant-back-

bending takes place from the almost spherical ( T ' -60° ) to

the prolate (_ b~ 0° )configuration. Further increase of the

number of protons and neutrons in the open shells stabilizes

the prolate shape in the ground states. This is demonstrated

for the actinide in case of gn^lili2 (lower part in Fig. 9/

which shows a weaker tendency for the rotation around thr>

oblate symmetry axis than it is seen in the case of its sister-

isotope with neutron number N =

Ro

Increase of th» angular

Th

Fig. 9. The hodograps obtained for Ra, Ra (upper figure)

and 2 2 Th, 2-*2Th flower figure) by minimizing the total energy

( II 1.1;)with respect to (J , V deformation parameters.

232 momentum in „.Th is followed by the decrease of the quadrupole

deformation. This shrinking is not accompanied however by the

change of | deformation from zero to some value in the sector

0 -• 60 as it was the case in the well deformed - 6SEr9S

(Fig' 5)• The "f - ° no more corresponds to the deformation of

the lowest minimum at I ~- 80S. , However, the jump to the

fission isomeric minimum appears at I ~ 90< . Thorium isotope

with 13U neutrons shrinks and moves toward the t =-60° line

already at I > kOfc. At I = 6ot the V-deformation of the

lowest minimum is close to V =-60 line. Giant backbending

is seen In this nucleus above X = 9OŁ from the *>*njtmifn at C>~ 0.09, f =-60°.

Results of this chapter might be summarized as follows: i) The .promising candidates to look tor the high spin isomerlo

states corresponding to the axially - symmetric nucleus spinning around the symmetry axis are - neutron deficient and stable nuclei at the beginning and in the middle of the transitional region £1QJ neutron deficient, light rare-earth nuclei£3-53, light, neutron-defioient actinidas £3,10j (oblate imomera)

- neutron deficient nuclei at the end of the transitional region C21] and neutron defioient, heavy rare-earth nuclei fi1J /prolate i»o»eree/

ii) Temperature effects which smooth the shell irregularities had not been taken into account. This deficiency is prob-bable not important for angular moa«nta belov Ictttr ijl heavier nuclei, since the particle emission from the yrast line is less probable Z 223 and the nucleus looses its excitation energy by emitting the particles from the fis¬ sion isomeric minimum or in the pro-equilibrium processes. If anyway included, they would change the result* in the direction of the RLDM predictions of Cohen, Plasil and sviatecki.

III. Angular momentum dependence_of the_fission_barriers

Data discussed in chapter XX have been obtained for the DES, using the Strutinsky shell correction at finite I with a Saxon-Woods potential parametrized by the quadrupole defor¬ mation parameters fi and o . This parametrization might be sufficient for the discussion of the high spin statcG well below the I C R I T for the "giant back-bending". The latter is specified by the difference between the energies of oblate and fission isomeric minima and theproper description of the

fission od^mum cannot neglect the formation of a neck. Also

tlie experinRital data concerning the formation and stability of

compound nuclei reflect some uncertainties of the measured

critical angular momenta £i,23j. Therefore, the proper para-

metrization of the average nuclear potential by inclusion, of

the necking parametrization that would allow to describe with

a required accuracy shell effects in the fission minimum and in

consequence to estimate the nuclear stability with respect to

fission would be of the great value. These kind of calculations

are however, complementary to the estimate* of the stability

of rotating nuoleus with respect to the particle emission

(protonu or neutrons) and the ol -partiole emission. The

particle emission from rotating nucleus was studied till now

by Dossing et al,f2i*] in case-of the spherical nucleus. Esti¬

mates, made by the same authors, of the influence that has

the deformation of the potential well on the limiting angular

momentum for the emission of particles from a given s.p. confi¬

guration suggest that the assumption of sphericity is not suf¬

ficient. In the heavy nuclei (A 2. 100) the particle emission

from the cold nucleus might be important only from the fission

isoraeric minimum. The same conclusion might be drawn from the

Ref. 22. Even less informations is available for the e>t -part¬

icle emission probability from the high spin states, though in

heavy nuclei the oC -particle separation energy is much smal¬

ler than the particle separation energies C25J . Therefore, our

calculations of the stability that nuclei possess with respect

to the fission mode does not give the limit for the nuclear

stability. Theory as well as results discussed in this chapter

cover in part the results presented in Refs C26-2&].

Ill, 1. Description of spheroidal, dumb-bell and diamond shapes

Shell structure studies performed in the fission isomeric

minimum concerns the axlally symmetric shapes. The full difi¬

ssion of DES at high spins should definitely include the

• -degree of freedom. However, if one restricts himself to

70

the study of the fission configuration at high spins and the

giant - backbending, the comparison of the data obtained in the

whole (b - T plane with data ut 1 = 0 ° including the necking

parametrization is sufficient. Ve might have this hope since

the nucleus seems to fission always vith a axially symmetric

shape.

The Saxon-Woods potential used has a constant normal deri¬

vative along the nuclear surfacef29], This surface is described

in accordance with ref. 30 by:

ft + -o w / ni 1.1

for the necked shapes and by

for the diamond like shapes. In the above used cylindrical

coordinates H,) Vf W :

tt - e(c Coif

' 1 III 1.3

the diamond and necked shapes coincide with the ellipsoidal

shape for B s 0. oL factor in eq. (Ill 1«3) have been intro¬

duced to scale the x,y,z - coordinates with respect to the

u,vfw used. A simple and straightforward parainetrization

of all these shapes might be done by an extension of the urual

fi- I parametrization of ellipsoids. As a new parameter we use

T" , which is the ratio of the neck cross section and the

ellipsoid cross section. Some of shapes available in this

parametrization are shown in Fig. 10. At T = 1 we jet the

ellipsoidal shapes, for <r > 1 diamont like shape which rc-minde

the form of the ellipsoid with a positive hexadecapole defor¬

mation A M . For 0 C t- <1 we obtain the necked shapes.

< cc

125

1.00

0.75

0.50

0.25

0.00

/ f^ i /K i /Hf /K1,i \l/ \i/ \lr U/1

/TVi /N,1 /T\,1 _/Kl AT/ \ i / U/'1 \J/'

rft i ft\.i ft>j ft|,i ~Cl/ ixr Uj 1111

ety m.i Cni _dii . cJJ cp tp^ ^ i r

cfcp ctz! ct/,i ctf.i

4 1 •$• -^ 41 •

Ay A\I

A J A a Vpr" W r

m' (A1

0.0 0.2 0.4 0.6 0.8 BETA

10

Fig. 10. Axial and reflection symmetric shapes in ft- T para

metrization. The units for the axial and radial shape coor¬

dinates are marked beside the figure.

It is also usefull to define the "equivalent radius" of the

ellipsoid:

which by the definition fullfils:

III

III 1.5

for Rw=1. V in eq. (ill 1.5) is a volvune of the ellipsoid

The cross—section of the necking shape at V=0 might be expres¬

sed in terms of the equivalent radius in the following way:

S(w-O) = III 1.6

In case of the axially symmetric shapes one obtains:

n3.

*n A in 1.7

CaIculating the volume contained in necking shape ( ,T/ in terms

of coefficients A,B characterizing the nuclear surface in eq.

(Ill 1.1j :

III 1.8

and expressing them by radius r, R one obtains:

5"( 1 - r J R, - B III 1.9

It is clear from eq. (ill 1.9) that for r = 1, parameter I5 has to be equal zero. Correspondence betveen the commonly used Peuli parajnetrization C^"i I (see ^ o r example Ref. 30/ and the one used here f (i,1?) might be found by expressing A, B parameters in terms of both Cf k. and fl, T parameters. For the «»t C *** one obtains £30]:

III 1.10

111,2. Relations betveen the oscillator frequencies for the necking and diamond like potentials

The standart procedure to get the relations between the oscilator frequencies in any potential is to require that density distribution generated by the given nuclear potential is consistent with the spatial spreading of the generating field. In other words density distributions in x, y, z directions are connected with axis of the potential by following relations:

III 2.1 where >c Ł #C w, } y =• •£ V f Ł B * W Distributions Ś. U. ^ ; S V ? j V W ? might be obtained by the straightforward integration. In case of the necking shape ve get:

/ «,*"> s \ dvj \ d v \ M. dlu. - -5? ( A 1 1 - — J J J Iff ? -*;

<voŁ> - v-i (J A f I B )

III 2.2 where the integration limits are given byt

[ (Ą Jl)( M3w*j 1^* L - l ] III 2.3

Substitution of (ill 2.2) to eqs.(lll 2.i) gives for the ratio of the oscillator frequencies.

These relations might be rewritten in a form explitly including the volume conservation of the potential. Thus are expressed by parameters p, q and frequency CO, in a following way:

III 2.5 f

75

III. 3. Results of the calculations for the_fission barriers

To construct the DES at high spins ve used the Strutinsky

shell correction approach as described in chapter IX.1 vith

the s.p. energies obtained by diagonalization of the average

field ( Saxon-Woods potential/ parametrized by deformations

ft , r or fi>; TJ" . In the classical energy v« used the droplet

formula with the Lysakil parameter* f3i] restricting the droplet

model expression to the sun of the volume, surface and Coulosb

terms.

UQUN) ENUOV

100„ Fig. 11. Contour plots of toe RLEM-energy in £

momenta X = 50,60,70 and 80 & plotted versus A

i at angular

r - deforma¬

tions for J = 0 . The shaded area represents the minimum and

the equienergy lines are given in step of 2 MeV.

In Figa 11 and 12 the DES using the HUDM-expression Fig.11

and the total energy are shown for 44Ru<6• Nuclei in this

region are expected to have large stability against tbe fission

process and, therefore, might be good candidates to look for

the auperdeformed configurations £2j or yra»t isomers.

LIQUID DROP ENERGY <2r

Fig. 12. Contour plots of the total energy in

momenta I s 50,60,70 and 8 o £ plotted versus

at angular

r-defonnations

for Energy landscape has been obtained using the

eqs.(ll 1.7} with s.p. energies given by the SW potential, para¬

metrized by p> , r deformations. The shaded area represent

the minima of the total energy. Equienergy lines are assigned

in step of 2 NeV. The classical part of the total energy have

been normalized to zero at spherical configuration and 1 = 0 .

77

At I = 50 the lowest minimum oi" both classical and total

energies approximately coincide at {b " 0.4 and r *•" 0.95.

second minimum of the total energy is around 2 MeV above the

lowest one and its position differs from the LD-minimuni. For

I = 60, the LD-minimum is shifted to - 0.6 and r-0.95, while

the minima in the total energy stabilized by shell effects are

situated at roughly same deformations as it was at I = 50 fc .

However,the deepest minimum has now p -» O.65 and r ~ 0.8.

Fission barrier in RLDM - description disappear somewhere

between I = 70 and 80 fc. . Shell effects shows clearly the

stabilizing role at the fission configuration (at I s 70 the

lowest minimum of the total energy is surrounded by the " Z MeV

barrier ) . They also lead to the shift of the total energy

minima with respect to the LD - valley in the direction of

smaller r-parameters (larger neck ) , At I —80t the fission

minimum is situated at A * 0.95 a n d r "** O*** a n d i s surrounded

by the barrier of less than 2 MeV. We assume always 2 MeV

barrier for the limit of the stability against fission that

nucleus posses at high spins. Anyway, fission barriers

smaller than 2 MeV cannot be trusted, because the inaccuracies

inherent in the Strutinsky prescription (20.5 MeVJand because

of the absence of the thermal excitations of the system which

might destroy easily such barriers. Shell effects in nuclei

with A ff 100, though change the detailed position of the lowest

minimum,do not lead to the large stabilizations of the fission

isomeric minimum comparing with the RLDM - estimates [2j, Thus,

with a reasonable accuracy the fission stability limit in the

light, transitional nuclei might be given by studying the

"classical" part of the total energy alone. The same result

comes out for transitional nuclei from the studies of DES in

the C» - 0 plane fio] (See discussion in Chapt. Xl). The shell

correction in the rare-earth and actinide regions changed

however substantially the DES at high 'spins leading to the

spiral like shape of the trajectory of the loweat minimum for

different X or, to the formation of the nucleus spinning

78

around the prolate symmetry axis. The stabilizing role of shell effects in these regions of periodic table might also be seen In Table 1, where the critical values of angular momenta, obtained

Table 1. Nucleus

170y, 7010 I88p 78Pt

232Th

2^0

I C R I T for HŁDM

8łfc 78 fc

80 K

58 fc

ICRITln the present

Ukt

12". fc 122 <v

using the RLDM and th« shell correction approach for finite I, are compared. It is vorth to notice that the different sets of the LD-parameters as proposed in lief. 2, 31 and 32 do not change much the predictions for •rrrr'T •'•n *^e "classicaJ-" model. One should keep in mind that I--.— in the "classical" and in the

L«I\X i. Strutinsky calculations have different meanings. !„.,_„ in

wlvj- i. RLDM is assigned to the angular momentum corresponding to the dlsappearence of the fission barrier. While !„„.„ in the Strutinsky calculations is assigned to the 2 MeV barrier height. It is obvious from examples shown in the table 1, that the sta¬ bility of the nucleus with respect to the fission process incre¬ ases significantly when the shell effects are included. This 232 2*tO is especially appealing in both aotinide nuclei: Th and Pu, where I C R I T increases by around ŁłO L due to the shell effects.

In Fig. 13 the shell correction for ^ Q Y ^ Q O i s P l o t t e d f o r

angular momenta X = 0,1+0,80 Jc as a function of ft ( r deformations at i = 0 . In this calculation the pairing correction was not included. Yb which is situated in the middle of the rare earth region has a very stable quadrupole deformation and l= 0°.

79

SHELL CORRECTION 12

Fig. 13. Contour plots of the shell correction for 7 0Yb 1Oo

at angular momenta I = 0,40, 80fc. in the space of deformation

paraneters (i , r at f = 0*. The equienergy lines are giver.

in the step of 2 MeV. The shaded areas represent the local

minima.

80

At I = 40fc the deepest minimum has I" = 0° but this nonaxial deformation is not very different from fl* = 0°, and, moreover, the DES in the sector of negative T -deformations is flat. At I — 8ofe the fission isomeric configuration is once more domina¬ ting and therefore nucleus tends to have T ~ 0°. Thus, in this nucleus we may study the shell correction at T = 0° for low as well as high spins being sure that tha studied shapes correspond or they are close to the equilibrium shape. At I = 0, the shell correction exhibits three well pronounced minima at 0^^0.22, r ~ 0.88,rt>~ 0.7, r - 1.2 and ft~ 0.8, r - 0.6 with the similar depth of •»«- 4 MoV. Thus, all extremal configurations for the shell correction are not ellipsoidal. At I = 4ofc all these minima preserve. It in Interesting to notice, that the minimum at fb" 0.7, <r~ 1.2 /diamond like shape] , though does not lead to sue minimal moment of inertia, is not destroyed. Minimum, originally at fo~ 0.22 and r •» 0.88, is shifted towards smaller |J —values and smaller r. It is also first dissolved (at I i 8ot ). Also for the minimum at the diamond -like configuration one observes the tendency toward smaller fi . This tendency is however weaker than in the latter case.

III. 4. Results for the giant-backbending

Critical angular momenta I"!,™ *'or t n e rapid transition between the oblate ( T =-60° , g> 7 O) and prolate C V"=0°, (V > O / configurations have been estimated in several transitional(rare-earth and actinide nuclei f28]. To check the predictions of calculations concerning •'•CHTT' tliat u s e d either Nilssonpi^J o r

Saxon-Woods [5,10,11] s»p. potentials, parametrized by (1 - •" quadrupole deformations we have performed the detailed Łtudies of the DES using both ft-1" and fc-i* T" parametrizations of the nuclear surface described by the S-V potential. Details of the models used can be found in chapters 11.1, II.2 and III.1, III.2 of this lecture or in Refs 5,28 . Results for few nuclei

81

are shown in Table 2. The common effect for all studied cases

Table 2

Nucleus , Saxon-Voods . Saxon-Vooda

122S2J 86 fc 7k

68

150 G d 88 fc 66

68

210 p o 84 K. 6k fc.

232^ . 112*. 90

86

9k

88

84

112

fc fc fc

K

ia the significant decrease of the I C H J T for the oblate-prolate

transition, caused by the lowering of the fission isomerio mini¬

mum by r-minimization of the total enei-f.-y. The r-minimization

lowers energy minima often by around 10 HeV or more (see for

example values of the shell correction at I = 80 in Fig. 13 for

fc -- 0.75) and it seems in heavier nuclei ( A >120) at high spins,

that the exact value of r-deformation is more important than

the proper IT -value in the minimum. Due to the flattening

of the total energy in the sector of 1 between 0 and -60 ,

caused by the rotation term in the RLD-energy, the mistake due

to the missing T -minimization is less than 5 MeV (see Fig. 5

at I s 70, 90fc). I„_-T for all studied nuclei are by around

10-20 units of angular momentum smaller than obtained in Refs

£3-5] and flO, 1 ti. They are generally close to the critical values

for the triaxial instability, as predicted by Cohen, Plasil and

sviatecki [2]. Tha regions of nuclei and angular momenta, pre¬

dicted to be especially favourable for the existence of yrast

traps, are not changed by the inclusion of r-minimizatlon,

since in all studied nuclei the triaxial instability and the

subsequent giant-baclcbending is not seen below I •«• 60 It.

The main results of this chapter could be summarized, as follows:

82

1/ Inclusion of necking and diamond - like ( similar to

sfrapes with a positive ft,, deformation] shapes is extremly

important in the estimation of the fission instability at

high angular momenta.

11/ Whan both shapes Included, the triaxial Instability and

the giant - backbending seen at lower X, closer to the

values given in Ref. 2.

IV. CHF(B) approach for the calculations of DES at I>0, T = 0

The alternative approach to the Strutinsky shell correction

approach at finite I la the cranking Hartree-Fock(-BogolyubovJ

(CHFB) theory. Because of the Immense computer time, that this

purely microsoopio calculations require, the CJIF(B/ theory was

till now applied in a restricted s.p. basis and with the sche¬

matic two-body forces £0,', j. Here one calculates the DES as

the- expectation value of the many body Mamiltonian with a Slater

determinant of cranked o.p. wave functions obtained using a

potential with different deformations P and T • Ve will

present here only very shortly the main ingredients of the

method and the approximation used. Details can be found in

Ref s[6,7,33,3<tJ.

IV.1. Cranking-Hartree-Fock-Bogolyobov theory

The trial wave function is constructed with the help of the

cranked Nilsson Hamiltonian:

4£ „jW* j£ t,t ty

where;

S.p. basis s tates I t t ^ ' »? are the eigenstates of the operator and k^fai • They are given by [35»6]j

rL This symmetrization of the basis states is important when the pairing is included both in the model Hamiltonian H, and in the realistic Hamiltonian, Xn this case the dimension of the UFB matrix is reduoed twice. Phases in eqs.(lV 1.3) are de¬ fined as fc = (-)**"łftfc r</tf and )«•>, IS"? are spherical shell model states and its time - reversed counter part, Hamiltonian H^ is defined with the spherical s.p, energies of Kumar and Baranger £36] and <i-c parameters are introduced to ensure equal radii for protons and neutrons;

f/A) * iv i.u

The Lagreuige multiplier CJ_ in eq,(lV 1,i) is determined from the condition:

IV 1.5

The many body Hamiltonian used is taken to be the schematic pairing and quadrupole (P + QQ / force model C36J-'

r IV 1.6

Further, the pairing interaction is assumed to be zero. This

simplification is explained by two reasons:

i/ we are interested in the region of very high angular

momenta states where the pairing is believed to be

broken by Coriolis forces. CHFB calculations of Ref.37

shows that the pairing correction is decreased by factor

four between 1 = 0 and I — 25 fc, (estimations have been

done in case of 2/-EV/.

ii/ at high spins the pairing interaction is significantly

reduced, and therefore the HFB approximation is not

sufficient, giving to less correlations £38] .

Finally the DES is written as:

IV 1.7

where -c*wL '•Pi ' -s si v e n by the energy of the homoge¬

neously charged triaxial ellipsoid with a constant volume Q38J,

The wave function | P>f i, 3^ has the angular momentum fixed by

the condition (lV.1.5) and is constructed as a Slater deter¬

minant built out of the energetically lowest s.p. wave func¬

tions of the model Hamiltonian (IV 1.1). Calculations are

performed in two oscilator shells for protons (N = ^,5)

neutrons (N = 5,6) for the core JQ?'T- T h e core contribution

to the amount of inertia is belived to be absorbed in the

strength of the pairing and quadrupole-quadrupole forces as well

as in thra choice of the spherical s.p. energies [3^3• Parame¬

ters ox the forces have been also adjusted[7] to include consis¬

tently the Coulomb term in eq.(lV 1.7) without worse agreement

for the quadrapole deformation of the rare-earth nuclei in the

ground state.

IV. 2. Survey of results

Before we will discuss results obtained for DES at high /joins,

let us make the critical presentation of the. adventages and

shortcomings of the CIJFfB) approach ae proposed in Refs C^jTt

i/ Due to the ft T" -parametrization of the HF - field, the

fission configuration cannot be described. Inclusion of

the two - center oscillator basis vould however require

much larger s.p. space than used to describe properly

configurations with large deformations. In this case

also the more realistic forces should be taken into

account e.g. Skyrme forcesj^39j. This project is at

present impossible to perform due to computer limitations.

At I = 0 the HF calculations vith the Skyrme interaction

have proved, that the Strutinsky shell correction approach

is able to reproduce the HF - results[40].

ii/ In the microscopic model applied in Hefs 6,7 , the rela¬

tively small s.p. basis was used. For the rare - earth

nuclei we find, that this model gives the accurate des¬

cription of the states with ft j£ O.U - 0.5 and angular

momenta I 5 60 - 701, For deformations and spins larger

than given limits, the s.p. basis should be extended,

iii/ The effective force* and s.p. space have been adjusted

for the nuclei in the rare-earth region.

iv/ Sinoe we omit the pairing interaction, our results

should not include states vith I £. 20t.

v/ We have found in our calculations, performed within the

limits given in Cii) the correct asymptotic moment of

inertia in agreement with results obtained for the Saxon-

Voods potential. This is connected with the fact that

the basis states with high-J for proton N = 6 and neutron

N = 7 shells have been omitted. These states are too o

much depressed by the 1 - term.

NEUTRONS (N=100.A=170l

wo—

f —

^J^X^SJI^^K^^*^

NILSSON

WOOO-SAXON

— — KUMAR-BARANGEfl

05 10

ROTATIONAL FREQUENCY 15

Fig, 1**. The total angular momentum I g h obtained by summing up

the s.p. angular momenta of the occupied levels. Results for

I are plotted against the rotational frequency for the neu¬

trons (N = 100) in 1'QYJ» 1OO. The functions Iah((*> = 0,«J '

are given for-the Niloson (dashed line) and Saxon-Woods

(dashed-dotted line) single particle potentials, and for the

Kumar-Baranger Hamiltonian (solid line). The arrows mark the

value of W for vhich the first member of the indicated multi-

plet crosses the Fermi level.

8?

Comparison of properties of DES at high spins, obtained usin/j CIIFCB ) approach and the shell correction method with both Kilsson and Suxon-'noods s.p, energies, have been done in Ref. o. Vithin the limitations discussed in (i) -(ivj, both methods show similar results for the trajectory of the lovest minimum. Prediction of the CHF model with QQ - forces, for the existence of the axially symmetric prolate or oblate nuclei spinning about the symmetry axis, are shown in Fig. 15, Here, the

68

60 Q

64 •

i i

•

76

72

i 0

0

& .

0

0

0

0

0

0

0

98

0

0

0

80 o 0

0

0

0

102

o 0

0

0

o 0

0

0

0

0

0

0

106

0

0

0

0

0

0

0

0

0

0

110

20t><l

0

o

l i t

<

0

118

50*

76 82 86 90 N-

Fig. 15. Nuclei in the rare earth region are plotted with their charge Z against their neutron nuciber K. The shaded squares indicate nuclei which rotate around an oblate symmetry axis ( "J =_6O j for angular momenta SOtŁ i 1 $ 50 fc . Open points indicate nuclei which rotate around a prolate symmetry axis (IT = 120°). The full points give the approximate position of the oblate island of isomcrs found in Ilefs 13-16 .

88

cars-earth part of the nuclear chart is given with marked these even-even isotopes which for angular momenta 2Ot I ^ kok. rotate around the symmetry axis. Two islands of such

nuclei might be seen easily. The first one, at N — 84 and Z ~ 66 correappnde to the oblate nuclei, the second one at N**106 and Z — 76 depict the prolate systems »pinning around its symmetry axis. Both islands have been recently localized experimentally. The oblate one in the Ris^-Darmatadt [13,1*1}, Groningen-świerk ti5j, and Krakow-Strasbourg [163 collaborations and the prolate one by the Michigan group £.1V—2oJ. There is generally a good agreement between the theory and experiment for the centroid of these islands. Theory prediots however more chances for yrast isomers in the heavier, rare-earth nuclei than found till now experimentally. To see if this is a systematic discrepancy one needs more carefull experimental studies of yrast states in this region.

V. Unified description of yrast isomers at verj high angular momenta In this chapter we are going to discuss the formation of

yrast isomers at high spins. These states can arise from the statistical fluctuations in the distribution of the s.p.ener¬ gies and angular momentum projections onto the symmetry axis, of the states around the Fermi level £1,33* Probability for this is large when the slope of the energy vs. angular momen¬ tum curve is snail. Since the slope of the yrast line in RLDM is proportional to the angular momentum, therefore on* expects larger chanoe for traps at small X. Quantal fluctua¬ tions in the moment of inertia may however modify the smooth increase of the slope of yrast line significantly. They lead to the decrease of the slope in comparison with the RLDM -predictions when the shell correction to the energy is daoraa-sing with increasing X. Using the cranking model for 1 * 0 states at f =-60° and t* =120° lines one obtains the total

energy and angular momentum which are not continue*! functions

of the cranking frequency (0 . Therefore, applyi-og tbis model

we are not able to describe all states along tli? yrast line,

corresponding to the rotation of the axially symmetric shape

around the symmetry axis. A straightforward recipy, proposed

by the Lund-tfarsaw group [3]» to construct the missing states,

is to form the 1p - 1h, 2p - 2h excitations with respect to the

configurations given in the cranking model. Minimizing the

energy of such states with respect to the deformation parame¬

ters of the average field, one might hope to get the energies

not far off from the selfconsistent values. This requirement

is crucial for the pairing deformations Ap , A n at high

spins, when the effective strengtli of the pairing interaction

G'-' ( X = neutrons and protons ) is close to the critical

values G C^ T, 0^^^. corresponding to A n = 0 in the BCS theory.

In this region of angular momenta it is rather sensless to con¬

struct the non-selfconsistent particle - hole excitations with

respect to the BCS (or BCS + RPA) - solutions, build on the top

of the optimal states of the cranking model, since gaps depend

very strong on the small changes of the occupation probabilities

for the states near the Fermi level, A correct way of including

the BCS + RPA pairing in this model would be to sol-re the gap -

equation for optimal states as well as for the n-p-h excitations

with respect to the optimal configurations. In this prescription

one assumes, that we can trust the BCS model at these spins.

This assumption is however not too good in the critical.

region j

A special situation oonduoive to traps of the 2p - 2h type

arises when the densities of neutron and proton states differ

very much near the Fermi surface. Then, the probability for

such traps is high in between two neighbouring optimal states

that differ in the configuration of this kind of particles

which have the dense s.p. spectrum near the Fermi level. At

the point where the change of the configuration appears, there

is often a large transfer of angular Momentum that has to be

compensated by other particles in order to obtain all inter¬

mediate .spins.

90

The alternative meohanism for the production of traps is hidden in the properties of the residual two-body interactional^ . This interaction is smaller than the one-body effective interac¬ tion. Xn changes however more rapidly with angular momentum. The normalized, diagonal, effective matrix elements of the two-body interaction are strongly attractive for 0 aligned confi¬ guration and 180° paired configuration between the two s.p.

-I Vi w e-*M

Fig. 16. Effeotive diagonal matrix elements Fj <Ktjj*TTlVK*<i)*JT> divided by the average value ^ for the particles in the same j-ahell, plotted in T = O flower curve) and T = 1 (upper curve ) channels as a function of the aogle &12 between the'a.p. angular momenta Dł33 .

momenta. In these cases the a.p. vave functions have the largest overlap. In contrast for the angle 90° the overlap between the nucleonic wave functions is small and, consequently the attraction is weak (in the T = O channel) or one gets the repulsion (T = 1 channel). For particles in the different j-orbits, the appropriate symmetry of the effective matrix elements of V,allowing for their classifications, is the d~ jil-jv «• J parity. For even j , one finds the saae behav¬ iour as in case of particles in identical orbits. For odd j , matrix elements are strongly attractive for the paired configu¬ ration only. With decreasing angle between the angular momenta of the interacting particles (from 180 to O ) these matrix elements are less attractive.

Thtsats properties of the effective matrix elements of V cause the lowering of 7 = 0 and J = J states with respect to tha neighbouring J - configurations in the two-particle system. It also explains the angular momentum and coupling scheme for the particles in the ground state of the even-even nuclei and the appearence of the isomers in the aligned few-quasi-partiele configurations in the "spherical" nuclei around g2Pt> ( e*6« the 18* isomer in fl^°128 n a v i nK ^ne configuration f- j/j,) »(v^u/i) with a half life of k$ sec. ) . TŁese properties of of the two-body forces enables to understand the formation of the yrast isomers in the oblate nucleus spinning around the symmetry axis at very high angular momenta [jt2J, \k5J . Increasing the angular momemtum we have to excite the particles from the occupied s.p. levels with the small or negative angular momen¬ tum projections onto the symmetry axis, into the states with large, positive projections. In this way the overlap of the nucleonic wave functions increases, as well as the total quadru-pole moment decrease*. Consequently, the binding energy of such aligned, very high spin configurations decreases. Th±s might lead to the lowering of the total energy of particular configu¬ rations with respect to the configurations with smaller angular momentum and to the formation .of yrast traps Dt2], One has to

keep in mind, that the total s.p. energy Increases while moving

partioles from the oooupied s.p. states to unoccupied one with

larger projection onto the symmetry axis. Thus the MONA .

mechanism (Maximization of the overlap of nucleonio wave func¬

tions by alignment ) might not be responsible for the formation

of yrast isomers in every s.p. configuration with the large

overlap of s.p. wave functions. Xn order to see which of the

above discussed high spin isomers (statistical- or MONA - traps)

are more frequent we have worked out a unified, microscopic

description, based on the CHFB - theory whereby both kinds of

isomers appear as special oases [33]• This theory have been

applied in Refs 33, 3k to oblate, light neutron deficient,

and in fief. 11 to the prolate, heavy neutron deficient, rare-

earth nuclei. Details of thj method can be found in Ref. 33.

V, 1. Description of the theory

The total energy of the system for a given angular momentum

I has been obtained by minimizing the energy expression:

i.i

This enargy is a sum of the expectation value of. the many body

Hamiltonian \\ and the Coulomb energy term, oaloulated for the

homogeneously charged drop with a shape parametrized by the

quadrupole ji«.; V" and hexadecapole {1«|« deformations [¥£].

As a many body Hamiltonian we used the P + QQ Hamiltonian of

Kumar and Baranger £j6J, supplemented with a hexadecapole t»rm:

^C-v ,..

where

For the description of the ground state properties the hexade-capole deformation plays an important role. Henoe, there is no reason to believe that the Hi,0 term will be unimportant at high spins. Since the model is restricted to the studies of the yrast traps in the axially symmetric nuclei, we have used only Y o component of the hexadecapole forces.

Hamiltonian in eqs. (V 1.i) and (V 1.2 ) is defined in shells N = 4,5 for protons and N = 5,6 for neutrons with a newly adjusted strength parameters [333 of *ne pairing,quadrupole and hexadecapole forces, to account for the Coulomb term in eq.(V 1.1) The trial wave funotion i, (fbŁ, ft«|o_, if ) is a HF solution for the model Hamiltonian:

defined in shells: N = 3-8 for neutrons and N = 2-7 for protons. Average, s.p., potential h. in eq.(v 1.3 ) is given by the Nilsson expression:

{ k = * ^

where rv( if the spherical e,p. part and parameters *L^ for protons and neutrons are introduced to ensure equal radii for protons and neutrons. The cranking frequency CO in eq. (V t. is determined by the condition:

•ecuil.

along the T =-6o° and If = 120° lines only. Eq. (v 1.5) specifies rather the bands of the allowed °0 -parameters than the one frequency of rotation" ŁOj , for a given angular momeutum J . . Consequently, the solution of eq. (V 1.5) might not be found for all X values by occupying the lowest s.p. states only. This deficiency of the cranking model have been discussed in chapter XV.

In T_ , which is obtained in 6 shells fpr both protons and neutrons, we omit all the s.p. wave functions which in the spherical limit do not correspond to shells N = ^,5 (for protons) and M = 5,6 (for neutrons), Suoh a restricted wave packet is then used to find the expectation value of H in eq. (V 1.i).

Separable infinite range quadrupole and hexadecapole forces, though simple, cause problems due to the too strong dependence of its matrix elements on the diameter of the a,p. orbit. Since these foroes have been used in six shells for protons and neutrons we need some scaling prescription for the radial parts of the quadrupole and hexadecapole forces. The prescription used was:

V 1.6

where

T I k for protons ; - 1

5 for neutrons For the calculations at high spins X £. 20^- the pairing interaction have be«n neglected. It Is anyway not large at these spins. Detail studies of the pair breaking at such high angular momenta would involve high - multipole components of the pair field as well higher order corrections to the HFB

95

method (see discussion in chapter IV/. There are, however,

still many unsolved fundamental questions to be solved before

these corrections can be included in the satisfactory way.

Minimization ot the total energy (V 1.1) might be performed

only in the space of the so called "optimal" configurations

corresponding to the sloping Fermi surface obtained in (V 1.5J.

To construct all the angular momenta states along the yrast

line one might construct all the 1p - 1h, 2p - 2h excitations

with respect to the "optimal" s+ates. Such states are there¬

fore given by the Slater determinant of the corresponding

"optimal" state modified by the exchange of one /two/ nucloonic

wave function(s) below the Fermi surface by one /two/ s.p. wave

function(a) from above £33• The statistical fluctuations in

the distribution of these exchanged states and their spins are

thus filled by the above reclpy. Hence the total energy Cv 1. \)

have to be minimized for each X not only with rospeot to the

deformation parameters but also with respect to all np - nh

exoitations from the optimal states giving the total angular

momentum. It has been found numerically f33J that it is

enough to Inolude the 1p - 1h and 2p - 2h excitations only.

V. 2. Discussion of the results for yrast traps

For the calculations at the beginning of the rare-earth

region we have adopted )C;L0 = 65'A1' MeV and JL,,0 = 50*A

1#/tMeV

as the values of the strength of quadrupole and hexadecapole

forces respectively. As the typical example for the light

neutron - deficient rare-earth nucleus we consider 70Yb •

This nucleus is oblate (&">C. f=-60°j for the wide range

of the angular momenta [7j D3]. Thus, the model discussed

in sect. 1 of this chapter might be applied here succesfully.

Characteristic features in this nucleus are the large changes

in the hexadecapole deformation ( calculated from the hexadeca¬

pole moments assuming a homogeneous matter distribution) above

96

X u U ANGULAR MOMENTUM

Fig. 17. The excitation energies as well as the quadrupole Q 2 Q

(solid circles) and the hexadecapole Qj^ moments (open circles)

of the yrast states are shown as a function of the total angular

momentum for ~. Yb ac , The yrast trnr>s aro narked by encircled

" + " or"-" where + (-) denotes ti.v. i-uoiiive IXIC^L.L'VC ) parity

states. The points in squares correspond to the "optimal"

configuration.

J = Zk, The series of the yra

29~, 30~» 32", 33" is due to the variation in tiie

deformation from 0.0 to - O.CWt. The quadrupole deforniation is

here a rather smooth function of the total angular momentum.

Since, there is no clear correlation between variations in the

hexadecapole moment and the appearance of the yrast traps, one

can probably conclude that the MONA mechanism does not play an

important role here.

ANGULAR MOMENTUM [hi

Fig. 18. The excitation energies as veil as the quadrupole

moments Q o n (solid circles) of the yrast states are shown as

a function of the total angular momentum for %0\B gg» The

results have been obtained without the hexadecapole term in

the trial wave function and in the many-body Hamiltonian.

For details see the caption of Fig. 17.

The structure of the yrast line changes violently if the

hexadecapole force is switched off. Once more, the yrast

isomers at J = 22 +, 2k+, 25 +, 29" and 39" are definitely

formed by the statistical variation in the s.p. density matrix

for the s.p. orbits near the Fermi surface. For these angular

momenta the total overlap of the nucleonio wave fucntions

which is proportional to the mass quadrupole moment is roughly constant. Tne only significant variation of the overlap can be seen for the state 7 = "}k* which is the isomeric one, The excitation energy of this state is lower than the energy of the X = 33" state but is higher than the energy of 3 = J2~staze,

The transition 3k* -* 3Z~ is however retarted since it is of the 2p - 2h type.

Generally in this reg-ion ve find that the statistical traps are much more frequent than the MOKA—traps. This statement holds providing that there is no significant o -force contri¬ bution in the nucleon-nucleon residual interaction at such high angular motntnta. ( Our calculations includes only the long range part of the nuclear Hainiltonian.) Moreover, ve observe that traps above angular momentum ~ Uofe are rather infrequent. This effect is probably duo to the large slope of the yrast line above I s 4ot , whioh reduces the importance of th« shell fluctuations [3 1 £333 • ^ince the slope of the yrast line is roughly inversely proportional to the moment of inertia, which itself is proportional to /. therefore the limiting ang-ular momentum for the existence of yrast isoniers will be shifted toward smaller values in lighter nuclei. The simple estiEiate in the mass region of A •»100 would give I . of the order of

TTIfl UL IS - 25 fe • Thus. even if nuclei in this region would have the oblate (f =-60°) configuration at J > I ^ ^ tha yrast isomers will not be formed.

Nuclei at the end of the rare—earth region prefer to keep the prolate shape even at high spins tH,3i]. Thus, theoretical¬ ly, one expects formation of vlu "classically" unfavoured prolate systems spinning around the symmetry axis ( see the discussion in chapters II and IV , due to the strong shell effects at (i "7 O , V = 120° line. The typical example for these tendencies is the nucleus S2Pb11ć (FiS* '?) • Similarly to other nuclei in this region, most of yrast isomers can be seen at spin less than I = 30 fc, . There are, however, also very high spin isomers at J = 3S~ and at J = 58". Since the lowest minimum of the total energy for this nucleus oor»

99

we will not discuss respond* to T " 120° line above I ~

the "low" angular momenta Isomera ( ~3 = 20+, 30+ and J = Zk~,

27~ ) being the shape isomers.

20 -

;» -

i w. -

5 -

—

y ***

• • V

•

**

•

20 28 36 U 52 GO

ANGULAR MOMENTUM (hi Fig. 19. The excitation energies of the yrast states at

Y*s 120 are shown as a function of the total angular momentum

for 82Pbii6* T h e PO:Urts in squares denote "optimal" states.

The yrast traps are marked by encircled "+" or "-" where + ( -)

denotes the positive negative parity states.

The isomeric state at 3"* = 58" is of the particular interest.

Formation of this state is connected with the rapid decrease

of the Q/JQ (Fig. 20) and a very small increase of the Q 2 Q

moment. Thus, the hexadecapole deformation of the nucleus

100

is charged drastically while going from J = 58" isomers

(6,^0.0) to the lower ~f = tfV", 56+, 55* states. This variation

X a SB U 52 BD

ANGULAR MOMENTUM Ih) Fig. 20. The quadrupole Q 2 0 (solid oiroles} and the hexadeoapole Q^Q moments (open ciroleo] of the yrast states at t = 120° are shown as a function of the total angular momentum for 198 82Pb1i6.

of «. 6N -deforsnation causes such large ohange of the position

of the s.p. levels near the Fermi surface that the J = 58

Laomer cannot decay. Changes of the hexadecapole deformation

might be seen in this nucleus already at J =30*. At higher

spins the two minima at (V, > 0 and ^ ~ 0.0 compete and,

therefore, one may see a very large change of the bexaBeoapole

moment in the states along the yrast line.

101

In both nuclei presented in this section, the inclusion of the hexadecapole term is crucial for the understanding of the structure along the yrast line. Changes of the |)<j -deformation are even larger than the changes of the quadrupole deformation, though the magnitude of the hexadecapole term in the total energy-is rather small. It would be interested to compare these results with those obtained in the shell correction approach with the average potential parametrized by 0*1 h a n d P>v t* deformations.

VI, Simple model for the description of high spin isomers Every description of nuclear properties at high spins involves

some basic assumptions about the fprm of the effective nucleon-nuoleon forces. Constructing: the lkunil tonian f one uses certain fundamental symmetries.(resulting from our experience or intui¬ tion about the space, time, interaction etc.) which are believed to be fullfilled for the nucleus, and which restrict number of independent components. Coupling constants, measuring the rela¬ tive importance of the different components in the Hair.iltonian, are usually deduced by fitting seme nuclear obsorvablcs ( such as the multipole moments, energies of the lowest exoited states, transition probabilities, branching ratios etc. ) with this Hamiltonian and using some better or verse founded, approximate solution of thf nuclear many-body problem. If such a satisfy¬ ing approach to the many-body problem is known, then depending on the kind of experimental data to be explained, it is possible to leave the troublesome though more fundamental general II;. eiil-tonian and use some schematic, effective Hamiltonians.• These Hamiltoni ana are adjusted to the part of the observables only. If one desires to go beyond the space of the fitted data then the effective Hamiltonian should be modified by readjusting at least the force constants if not the form of the Mamiłtonian itself. Consequently, a priori there is no reason to believe in any universality of tho effective Hamilton!ans outside the bounds formed by the observablee defining this Hamiltonian.

iort

Consequences of this scepticism for the description of nucleus at high spins will be a subject- of this chapter.

VI. 1. Effective Hamiltonian for high spin states

Let us assume that the nuclear Hamiltonian consists out of the kinetic term i and the two-body force V only. Then, the total energy in the usually used IIF(B) approach might be written as:

vi 1.1

where 9 is the "generalized density matrix":

S'J VI 1.2

fullfilling the condition C = y . c> and 2Ł in eq.(VI -\.2) is the usual s.p. density and the pairing tensor correspond¬ ingly (Relation o 5c is valid only for the HF approach) , The"generalized density" J defines, for a given effective Hamiltonian, the HF ( B) field:

TV VI 1.3

(More phenomenological approaches start with fitting A- instead of the effective many-body Hamiltonian.)

Physical states correspond to the solution for J , mini¬ mizing the total energy (VI 1.i) along the lines formed by the constraining opeartors Ą. :

103

i -- i, ... tN vi uk

ot. S might be the angular momentum, proton and neutron numbers, multipole moments etc. Excited states in this picture are obtained for the same Hapiiltonian. by minimization of eq. (VX 1.1) along new lines tfoi: 3 . In this approach one assumes that the Hamlltonian is universal for the ground state as well as for all excited configurations. This risky assumption might not be true and what is even worth we are not able to get rid of them in the frame of HF(B) theory. Let us take an example of widely discussed traps in ~ Hf£lł-2,0 «JC,*<łJ . High -E isomers, found in this nucleus have been explained as two, four and six quasi-particle excitation with respect to the g.s. In these states q.p. are moving close to the equatorial plane of the prolate core, having their angular momenta aligned with the core-symmetry axis. One might conclude basing on the HFB calculations that, the total excitation energy is in this nuclei absorbed by few excited q.p. leaving the core unchanged ft6j . It is therefore reasonable to separate, in the total energy, the core states from the states of the excited fragment. Consequently, the eq.(VI 1.i) for the high spin, yrast isomers looks like:

vi t.5

First two terms correspond to the diagonalization of the Hamiltonian in the subspace Ac of core-states. Second

10*

line in expression (vi 1.5) gives the contribution to the energy

from the fragment-states subspace Xt • Finally the last term

represent the interaction between X and X_. Largest contribu¬

tion to the excitation energy of this configuration oomes from

the last term representing the interaction of A oore-partioles

with the A- fragment-particles. It has the significant and

large component from the aligned configurations of the fragment

and core particle*. This component of the two-body force is of

no importance in the ground state due to the symmetry imposed

on the occupation probabilities of different s.p. states by

pairing interaction. As we know from the Sohiffer's analyzis

E*3> **83 i the diagonal effective matrix element of the nuoleon-

nucleon interaction is some cases even larger for the aligned

than for the paired configurations. For 1 = 0 and low spin,

collective excitations the nuoleons saturates their forces in

pairs or quartets. Thus there is a need for the strong binding

in the configurations wjth the angle 180° between s.p. angular

momenta of the interacting particles (pairing configuration),

Different orientation of the nuoleonic orbits in the space

might be accounted for by the long range component in the force

having the well known bell-shape as a function of the classical

angle between spins of interacting particles. These requirements

are funfilled in the schematic P + QQ model, which has the

attractive, L = 0, short range component and the long range

quadrupole-quadrupole force. It is also the secret of the

succeses that the P + fiQ model had over the last 20 years in

explaining the speotroscopies data. However, this model ( or

similar approaches using the average 3.p. field and the pairing

interaction ) is probably insufficient for the description of

the excited configurations with many particles aligned with the

symmetry axis of the core. Xn these states the short range

component in the aligned configurations is needed. Strong

binding in these configurations cannot be accounted for by the

, quadrupole interaction. This is demonstrated in Tables 3 and

4 where the contribution from the S -force, and the absolut*

105

value of the ratio of the o ~ and quadrupole — force contribu¬ tions to the Schiffer'a matrix elements are compared. The values

t = j2) T „ 1

®12 1 7 °

<£7(QtJ - 1.89

|<<ry<G>/ 2 « 1 2

150

-0.6

1.03

120

-0.3

2 . 5

90

-0.17

0.37 .

6 0

- 0 .

0 .

1

83

30

- 0 .

0 .

ok

06

10

- 0 .

0 .

0 1

0 1

Table

912 1 7 °

COCG.J -U.3«»

l<f>/<a>l 29

150

-1 .6

16

120

-1 .08

5>*

90

-1 .12

14

60

-1 .51

75.5

30

-2.89

29

10

-8.58

57

of the delta - and quadrupole force components have been taken from Molinari et ał.pł8j. These authors fitted succeafully the dependence of the Schiffer's matrix elements as a function of the angle 6,, between the interacting particles using the S + QQ force model only. In the T = 1 channel the ©..„ -dependence of the Schiffers matrix elements is fitted with a reasonable accuracy by the QQ - forces. Xn this channel however, the binding at small angles is estremly weak. Much more interesting is the T = 0 channel. Here the & -force dominates 10-100 times the quadrupole components. Consequently, it is no hope to describe the »12 = 0° configuration with L = 0 pairing and Q-Q forces. Ev«n stronger 'domination of Vhm O -force over the Q-Q component of the residual two-body interaction is seen for j f j Ł .

106

From the above arguments it might be surprising the quite good

agreement obtained for the energies spins and parities of the

yrast isomers reported in Refs 3k, k6, k7, k9. Part of tne

discussed defficiencies of the effective forces used is probably-

absorbed in the average s.p. potentials which have the effective

s.p. energies. This would also explain the superiority of the

statistical traps over the MONA traps as found in Ref. 33.

VI. 2. Simple model for the description of yrast isomers

• Xn the absence of the satisfactory forces, for the desoription

of the interaction between particles in the high spin isomeric

configurations, it is valueablo to study the aimpio models.

From eq.(vi I.5J it is clear that if the deformation of the g.s.

and the isomerio configurations is same, then the Q for tho

excited states can be approximately found by diagonalizing the

Hamiltonian adjusted for tho g.s. - properties. Tho relatively

stronger binding of configurations with tho excited particle*

aligned with the core-symmetry axis can be studied then phenomo-

nologically. This may be achieved by looking to the value of the

angle S^

the core: angle S^ between the particle in the s.p. state li> and

where r

^ V I 2 - 1

j. and Oil i are here the angular momentum and their projec¬

tion onto the symmetry axis for thj particle It ? . For small

S o o r the last term in eq. (VI 1.5) lowers the total energy

significantly what may lead to the formation of traps. Let us

'show some details for the Lltt isomers. Detailed calcul

of this type in other nuclei will be published elsewhere

107

Using the HFB - theory and model described in Ref. k6 one finds lowest q.p. excitations in ~J3f. They are given in

£C) Table 5 together with angles 9 co_e calculated for a dominating j-component in the wave funotion /fiJ /T, £ > . By selecting from all possible 2 q.p. configurations those with combination of

IS5Ł2JL O*1" 7/2+ 9/2" 5/2~ 7/2" 9/2+ 7/2+ 1/2" UŁ/-, p p n n n n n

Vcore 28° 25° 51° *5° 50° 60° 64°

eploore' p-oore' o n e obtaijis as the moot probable candidates for isomero the 2 q. - proton state 8" (7/2*, 9/2~) and 2 q. -neutron state 6+ (5/2^, 7/2^ J . Experimentally they are seen at 1559 KeV and 1333 KeV with half - lifes T< ,^ = 9.BM,3 and Tjy2 = 9.5IŁ3 respectively. The second 8" state at 1860.3 KeV is also easy to predict as (7/2^1 9/2n) configuration. For' the U q.p. states, four states have the comparable sum of

7 / 2 P 9/2; 5/2; 7/2; Jff = 14-7/2^ 9/2; 9/2Q 5/2n J V = 15+

7/2; 9/zl 9/2* 7/2; J* =16"

Isomeri of \k~ state ^ A/i~ ^OI/-* comes from the faot that the h q.p. states with smaller angular momenta includes the 1/2^ state which lowers largely the binding of the *ł q.p. configuration. For the 6 q.p. - states the 22~ state has smallest sum of ®p2core* I ł i s therefore also isomeric with T1/2 = ^3 ui . Using' this method one can predict easily the angular momentum, parity of the isomers as well as the multiple appearence certain high J - states. This method is suocesfull

108

providing that in the g.s. the interaction used allows for

a proper description of the HFB - field.

++/ The theory as well as the calculations presented in this

lecture have been obtained as a result of the active collabo¬

ration with the group of Prof. A. Faessler from IKP - Jttlich.

109

References

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k. K. Neergard, V,V. Pashkevich and S. Frauendorf, Kucl. Phys. A262 (1976J 61.

5. K. Neergard, H. Toki, M. Pl.oszajczak and A, Facsslcr, Nuol. Phys. A287 /1977A8

6. A. Kaesalor, K.R. Sandhya Devi, K. Grtiiruaer, K.lt. Sclunid and R.R. Hilton, Nuol. Phys. A2£6 (1976) 10c.

7. M. Płoszajczak, K.R. Sondliya Devi, A, Faessler. Z. i'hysik A282 (1977) 267.

8. V.V. Pashkevich and S. Frauendorf, Yad. Fiz. 2£ (i97^) 1122.

9. G.'Leander, private comraiinication (i977K

10. M. Płoszajczak, H. Toki, A. Faeasler, accepted for publi¬ cation in J. Physics G ( 1978).

11. M. Ploszajozak, H. Toki, A. Faessler, Jfilich preprint (1977J . M. Płoszajczalc, H. Toki, A. Faessler, Contribution to the International Symposium 011 Hi/jh-Spin States and Nuclear Structure, Dresden, September 1977, p.93.

12. S.Ogaza private communication (1973).

110

13. J. Pedersen, B.B. Back, F.M. Berathal, S. Bjjfrnholm, J. Borggreen, 0. Christensen, F. Folkmann, B, Herskind, T.L. Khoo, M. Neiznan, F. Pfthlhofer, G. Sletter, Phys. Rev. Lett. 39 (1977) 990.

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15. J.F. Jansen, Z. SujkowskA, S. Chmielewska and R.J. Meijer, Proc. 3rd International Conference on Nuclei far from stability, Cargese 1976, p.

16. E. Bożek, results of Kraków-Strasbourg collaboration reported at XVI Winter School in Bielsko-Biaia ( 1978j.

17. T, L. Khoo, J.C. Waddington, R.A. O.'Neil, Z. Preibisz, D.G. Burke. H.W. Johns, Phys. Rev. Lett. 28 (.1972) 1717.

18. T.L. Khoo, F.M. Bernthal, R.A. Warner, G.F. bortach, G. Hamilton, Phys. Rev, Lett. 1 0975) 1256.

19. T.L. Khoo, F.M, Bernthal, R.G.H. Robertson, R.A. Warner, Phys. Rev. Lett. 21 (.1976) 823.

ZQ. T.L. Khoo, G. L^vhoiden, Phys. Lett. 67B (1977) 271.

21. T. Dossing, K. Neergard, K. Matsuyanagi, Hsi-Chen Chang, Phys. Rev. Lett. 22 (1977) 1395.

22. M. Cerkaski, J. Dudek, Z. Szymański, C.G. Andersson, S. Aberg, S.G. Nilsson, I. Ragnarsson, Phys. Lett. 72B (1977

23. H. Gauvin, Y. Le Beyec, Nuci. Phys. A22JJ 097^) 103.

2^. T. Dossing, S, Frauendorf and H. Sohultz, Nucl. Phys. A287 (1977) 137.

25. M. Faber, M. Ploszajozak, to be published.

111

26. M. Faber, A. Faessler, M. Płoszajczak, H. Toki, Phys. Lett. 70B (1977) 399.

27. M. Fafcer, K. Plcszajczak and A. Faessler, Contribution to the Conference on Nuclear Structure, Edinburgh ( 1978}.

28. M. Faber, II. Ploszajczak and A. Faessler, to be published.

29. J. Damgaard, H. C. Pauli, V.V. Pashkevich, V.M. Strutinsky, Nucl. Phys. A135 (.1969) **32.

30. H.C. Pauli, Phys. Rep. 2, n°. 2 (1973) 35.

31. V.D. Myoro, V.J. Swiatocki, Arkiv f. Fysik 2Ś. Ci967) 32»3.

32. II.C. Pauli, T. Lodergerber, Nuol. Phys. A175 (1971J 5^5.

33. M. Ploszajczak, A. Faeesler, G. Leander, S.G. Nilsson, to be published in Nuci. Phys. /t978/

"}k. A. Faessler, M. Pioszajczak, H. Tok^ , M. Wakai, Contribu¬ tion to the International Conference on the Nuclear Structure, Tokyo, September 1977.

35. A. Goodman, Nuci. Phys. A23O £.197^ *t66.

36. K. Kvunar and M. Baranger, Nuci. Phys. A110 ^196Sj 529.

37. A. Faessler and M. Ploszajczak, to be published.

38. B.C. Carlson, J. Math. Phys. Z_ (.1961 ) hh\.

39. T.H.R. Skyrme, Phil Mag. ± (1956) T.H.H. SkyrWG, Nucl. Phys. £ (1959) 615.

^0. M. Brack and P. Quentin, Kochester 1973, Vol. I., p. 231.

41. C.G. Andersson and J. Krumlinde, Nucl. Phys. A291 (1977J 21.

112

kZ. A. Faessler, M. Płoszajczak, K.R.S. Devi, Pliys. Rev. Lett. 26 (1976) 1028.

43. J.P. Sohiffer, Ann. Phyc. 66 (1971) 798.

kh. A. Eaessler, Proceedings of the XX Xnternational School on Nuclear Physios, Predeal 1976.

1*5. G. Leander, Nuol. Phys. A219 (197b) 2^5.

V6. A. Faesaler and M. Płoszajozak, Phya. Rev. £^6 (1977) 2032.

k7. A. Aberg. Lunfl preprint (i977).

^8. A. Molinari, M.B. Johnson, H.A. Bethe, V.M. Alborioo, Nuci. Phy«. A239 (1975) 5.

kS. G. Leander, In the Proceedings of the XVI Winter Sohool in Bielsko-Biala (1978).

50, M. Płosząjczak and A. Maj, to be published.

The study of high spin icomeric states in multiplicity

experiments with C induced reactions.

D.Hageman .Kernfysisch Versneller Icstituut.Groningen,

The Netherlands.

!After the existence of an island of isomers was proposed

in the region of neutron deficient nuclei above the closed

neutron shell N-82 , it was necessary to determine the

final nuclei before a proper comparison with theoretical

predictions was possible.

We used a sixteen Nal detector multiplicity filter in

coincidence with a Ge(Li) detector.The electronic systea

was designed to record per event up to sixteen coincidences

between the Ce(Li) and the Hal detectors,the gaana-rav

energies dissipated in the Nal and Ce(Li) detectors and

the Ge(Li) and Nal timing signals.A sinplified block scheme

is given in fig.1.The k-fold coincidences were registered

by the OCR which generated for each event a 16-bit word.

fig.l Siaplified block

scheme of the electronics

used for Multiplicity

experiments.

•(•Collaborators: K.J.A. de Voigt , J.r.W. Jansen , u.

E.H. du Marchie tran Voorthuysen, Z.Sujkowski.

1)J. Pedersen et at. .Phys.Rev.Utt. 39 (1977) 990.

nil

For each event the DCS bit pattern ,the Ge(Li) and Nal energy

and timing tignals were written event by event on magnetic

tape.The 0-6 fold Ge(Li) spectra were updated in core as

well as the Ge(Li) and Hal time spectra and two I and 2 fold

Ge(Li) spectra with software time gates on the Ge(Li) and

Nal time.By setting those gates on'Ge(Li)proapt-HaIdelayed',

respectively 'Ge(Li)delayed~NaIprompt',it is possible to

observe on line vith the Ge(Li) detector separately for 1

and 2 fold coincidence the feeding and the decay of an

"somer.An example is shown in fig.2.This figure illustrates

that with the present set up it is possible to obtain a

clear distinction between feeding and deexciting gamma-rays

during a two hours experiment with a SnA C beam.

C , 4 n ) D » E«60M«V 100cm* O«(U) - l6Nol multiplicity fillv . I- fold cdncldMKtt

Ey(l«VI

fig 2. Gamma-ray spectra obtained from the 100cm Ge(H)

detector in coincidence with one out of 16 Nal detectors.

Timing constraints were chosen as to observe the feeding

(top) and decay (bottom) of the 60 n» isomer in Dy.

115

Various target-beaa energy combinations were chosen to study isomeric states in the above aentioned region. The final nuclei are sumiarized in table 1 for three

12 different C beaa energies.

Outgoing Beam energy 70 Hev , Targets: particle. 139^ 14Jpr M ^ 1 4 4 M 146^ 150,., 154 M

3n 4n 5n

4n 5n 6n p4n

Sn 6n 7n p4n p5n

147_ 149_ Eu Tb H 6 E u Beaa energy l 3 9 U U I P r

'*6Eu U 8 T b U 5 E u

Bean energy I39La U 2 H d

149Dy '*5Eu '"Eu

U8Tb"

l50Dy ';

SO Hev , I44„. W Ha

l5IDy l!

101 Hev USd "

, Targctf: t6H. 1 **«-» RO am

152Er

l5łHo

, Targets: '6Md '**S

'5IDy I53Dy l5lEr I50Dy 15 i2Dy

t5lHo I50Ho

15«to '57Er

,46S> I5Sr 155Er

l 5 0 S . l57Er 156Er

162Yb

,49SB

'57Er

I54Gd

160Tb

I 5 O S . I58Er l57Er

12 Table I. The final nuclei studied with C induced reactions; Isoaeric states are found in the underlined nuclei.

The data ara not completely analyzed untill now.Isoaeric states are found in the underlined nuclei.As can be understood fron the nuaber of feeding and deexciting gaaaa-rays, these states are low lying states (<3 Mev) for the odd-even nuclei.

116

The isomers in Dy end Dy are also found by Stefanini

et al. .The case of By is extensively studied in the

K.V.I. . This established the existence of an 60 ns

isoner above 5 Mev and a new 13 ns isoaer about l'Jic*

higher. The analysis in tens of gaana-ray identification,

nuclear lifetimes and multiplicities is still in progress.

2)A.M.Stefanini et al..Nucl.Phys. A258 (1976) 34.

S.Lunardi et al..International synposiua on high spin

states and nuclear structure, Dresden 1977.

3)J.F.W. Jansen et al.,K.V.I. Annual Report 1975,46.

HIGH-SPIH STATES HI THE GROUND-STATE ASD SIDE-BANDS DT 156Dy,162Er

AND 168Hf INVESTIGATED THROUGH PROTON-AND 1^l - INDUCED REACTIONS

Jean Vervier

Institut de Physique Corpusculair, B 1J48 Louvain-La-Neuve, Belgium

Ground - state and side-bands in the nuclei Py, Er and

Hf have been investigated through the following reactions: 1 5 % b / p , W 156Dy, 165Ho/p,4n/ 1 6 2Er and 159Tb/i4lf, 5n/ 168Hf.

The proton and n beams were produced by the OTCLONB

isochronous cyclotron of Louvain-la-lfeuve, and standard techniques

in the field of on-line gamma-ray spectroscopy were used. The use

of proton beams allowed to populate not only the yrast states, but

also non-yrast levels with measurable intensities for their decay

\ gamma-rays.

The following main conclusions have been obtained. In Dy,

a "twin" backbending has been observed in the ground-state and

A -bands, and low-spin members of the "upper" band, which is

thought to be responsible for this "twin" backbending by band cros¬

sing, have been excited. The results have been described in the

framework of the rotation vibrations model with an "upper" band

with the variation of the moment of inertia and with reductions

of the ground - /h and ground - V- interactions. In Er, the

odd-spin members of the negations parity octupole band between

1~ and 13* have been populated, and the even-spin levels of

this band between 4~ and 12" have been discovered. Both se¬

quences of states are reasonably well reproduced by the current

models proposed for describing the octupole band. In Hf, the

yrast band has been excited up to 20*, and on yrare 14* level

has been discovered which is probably a member of the ground-state

band after its crossing with the upper band responsible for back-

bending. The "fine structures" of backbending in the even-even 5 = 96 isotoneB, i . e . the fact that backbending occurs at 16* in 164Er and 166Tb and at 14+ in 168Hf and 170W, i s qualitatively accounted for by a nodel where the influence of protons OŁ back-bending i s introduced through their effect in the nuclear deforma¬ tion.

The works contained in this contributions are described in the following papers:

- R.Il. Lieder et a l . , Phys. Lett. 49J, 161 /197V. - T.E1. Masri et a l . , Zeitshr. Phys. A274. 113 /1975/. - T.E1. Uasri et a l . , Rucl. Phys. A271. 1J3 /1976/. - F.W.N. de Boer et a l . , Bucl.Phys. A290. 173 /1977/. - T.E1. llasri et a l . , Wucl.Phys. A279. 223 /1977/. - R. Janssens et a l . , Nucl.Phys. A283. 493 /1977/. - C. Michel et a l . , Nucl.Phys. /to be published/.

3. NUCLEAR REACTIONS

EVIDENCE FOR SYSTEMATICAL FEATURES IH PROTOH ELASTIC SCATTERIHG RELATED

TO NUCLEAR STRUCTURE

E -Colombo. R.DeLeotj .L.Escudie 'E.Fabrici .S -Mi che l e t t i .M.P ignane l l i .

F.Resniini, and A.Tarrats*

. Isti tuto di Fisica dell'Universit3 di Mi lano and IstiCuto Nazionale di Fisica Nucleare, Sezione di Milano. Milano, Italy

In this communication some recent results ' on proton elas t ic Bcat-tering, obtained within an extensive experimental program which is being carried out at the Milan AVF cyclotron, are reported.

While e las t ic scattering data on medium and heavy nuclei have generally been well accounted for by the optical model, the situation for nuclei with A ^ 40 is far less satisfactory and the fi ts obtained using conventional local optical potentials are much poorer. Often rather questionable optical model parameters are required. Moreover, in the lack of sufficently systematic data, each nucleus was considered a case by i tself , characterized by i t s own individual structure, at least as far as disagreement with optical model predictions is concerned.

A typical example of the unsatisfactory fi ts often given by the optical model for light nuclei is shown, for N, in Fig.l . Both the differential cross section and the analyzing power are fairly well reproduced at 26 MeV by an optical model potential with standard geometries (dotted l ines) , however a potential with the same geometrical parameters cannot reproduce the differen¬ t i a l cross section at higher energies. An improvement can be obtained using a non-conventional geometry, as shown by the dashed curves which were obtai¬ ned using a spin-orbit radius parameter r • 1.62 fm which is unphysically large. The corresponding f i t to the polarzation data i s , however, very poor. Similar results have been obtained using non-convenctional geometries for the imaginary term in the potential . The same difficulties in f i t t ing large angles data using conventional local optical model potentials have been

, , 16„ d) J , 40„ (5) also found for 0 and for Ca

Permanent address: Is t i tu to di Fisica dell'UniversitS, Bari, Italy 'Permanent address: Departement de Physique Nucleaires, CEN, Saclay, France

123

Alei

-as

16

10 r

2sMeV

39.2

'* 1 1 1 1 i* 1 1 t 1 1 1 t 1 \ 1 1

60 120

Fig.l — Proton angular distributions (analysing pcwer and differential cross sections ratio to Rutherford) compared with optical model predictions, showing a typical, for light nuclei, disagreenent between experiment and calculation. Dotted and dashed lines are the result of calculations discussed in the text; full lines of an optical model f i t limited to the forward angles ( S< 90*)

180

Experimental results •

In an effort aimed at clarifying the existing experimental picture, we

have measured the angular distributions of 35.2 MeV protons scattered from

the following 45 target nuclei: 9Be, l°'lh

2 8Si. 31P,

1 2 ' 1 3

3 5 ' 3 7

1 9 F , 2 ° ' 2 2 N e , 23Na. 2 4 '

C1. 5 1

5 A ' 5 6 F e , 5 9 c o ,

V,

64.66,68,70

5 5 «n,

^

nuclei: 23Ha, 3 l P. 39K, 45Sc. 5O'53'5*Cr « d 62'64Hi the measurement ha.

been repeated at 29.8 MeV. These latter data, together other angular dis¬

tributions found in the literature, form a second set of 38 angular dis¬

tributions at incident energies close to 30 MeV for nuclei ia the same

mass region ( 9 < A < 70 ) . The energy dependence ha* been investigated

121*

Fig.2 - Differential cross sections relative to the Rutherford cross sections,for the incident energies reported on the right side. The full lines are the result of a vi¬ sual f i t to the experimental points. The energ} dependence of the backward maximum is clearly visible.

180

cm

between IS and 45 MeV, in few MeV steps, for N and 0 :the relative cross sections obtained are shown in Fig.2 and Fig.3. Data for differential cross

iveral er ,(3,6,5)

sections at several energies for 0, Si and Ca are already available in the literature

Fairly complete angular distributions have been run, data being taken typically at 5* intervals, between 15 - 20* and 170* laboratory angles. Statistical errors are neglegeable except, in some cases, at very backward angles, while absolute errors are well within 10 Z.

Phenomenological feature* From a mere inspection of this large body of data,a sistematic picture

emerges, which can be summarized as follows: 1 - Many nuclei exhibit a pronounced anomaly consisting in an unexpectedly

Fig.3 - Differential crosi sectiont relative to the Rutherford cross section*, for the incident energies reported on the right side. The full lines are the result, of a vi¬ sual fit to the experimental points. The energy dependence of the backward maximum is clearly visible.

cm large yield at backward angles. Some examples of the resulting angular dis¬

tribution are given in Fig. 4, for nuclei in the oxygen region.

2 - The effect is evident in two mass regions close to 0 and Ca, as

shown both in Fig. U and in the lower part of Fig. 5. Lack of data on more

nuclei in the A-40 region prevents a full appreciation of the extent of the

effect there. The angular position of the backward maximum is given, as a

function of A, in the upper part of the Fig. 5.

3 - The effect becooes evident (round 25 tfeV and is s t i l l present at 45 MeV,

at least in the nuclei ( N, 0, 0, *°Ca) so far investigated. The angu¬

lar position of the backward maximum stays essentially fixed. This feature

126

KT-

30 60 90 120 150

Fig.4 - Differential cross section! for proton elastic scattering from different target nuclei in.the 0 region. The l.nei drawn through the experxuental point* are only for eye guide. The itrong mass dependence of the backward maximum is shown.

m is clearly anomalous with respect to any standard optical model prediction.

(4.5) As in previous studies we have found that the backward maximum cannot be

fitted, above 28-30 MeV, with an optical model calculation with physically

acceptable values of the parameters.

4 - The effect looks more likely related to the structure rather than to th-

dimensions of the nucleus. Support for this statement comes from the large

127

170*

160°-

150'-

140*

1 1

—

—

— a

ee eee a

a

a •

I 1

.

a*

a

a

i

aaa

a

1

1

a a a a

ee

352

i

i i

i

a* aa

' a a * a

a a

MeV

i i

i a

—

r to

E E

1.0

0.1

n i i r

«• • •

-a a

a

-^30 MeV

A A

M *

35.2 MeV

1

Pig.5 - The upper part ahova the masi dependence of the angular poaitioo of the back¬ ward maxima. The lower part ibowi the *»** dependence of the peak; value of the croat. aectioo of the backward maz-ima for two incident energie*. The valuee of l / lj . where S-ia the quadrupole deformation paraaeter, are also given, as croaaea for even-even nuclei and trianplei for odd-A nuclei.

10 20 30 40 50 60 70 A

difference found for the couple of isobars °A- °Ca and 6Cr- Fe, where the

closed shell nuclei show the larger backward yield. Besides(the effect

ia coopletely absent in strongly deformed nuclei like ' Ne and Mg, as

i t can be readily appreciated for Me froa Fi». (,.

5 - It also turns out that the anomaly is not limited to the aforoentioned

backward region. In this respect we have ao far performed a careful examioa-

128

200

cm.

Fig.7 - Croif section difference betwe¬ en an optical model fit to the forward part of the angular distribution and the results of a folding model calculation, as a function of the angle.

Fig.6 - Comparison of the experimental integrated cross sections(full points) with those calculated by using a folding model potential (see text).

tion only of the data for nuclei up to A - 22. The striking result one ob¬ tains for the cross sections integrated over the forward angles is shown in Fig.6. The dashed line represents the predictions of a folding model calcula¬ tion , free from ad hoc adjusted parameters, and as such it gives a guideline for judging the extent of the effect. A closer analysis of the integrated cross sections reveals that the effect at forward angles is essentially centered around 6 • 30" being 30° - 40° wide as shown in Fig.7 in which is plotted the dif-cm ference between the cross section values given by an optical model fit to the forward part of the angular distribution only and the results of the folding model calculation. Comparison of Fig.s 4 and 6 shows inmediately that those nuclei for which the backward effect is most enhanced, e.g. C, * N,

' 0, are also characterized by an anomalously high forward cross section.

Spurred by the evidence outlined in point (4) above, we have made seve¬

ral attempts to find, if any, quantitative correlations with the collective

properties of the target nuclei. In this connection we repirt a striking re¬

sult obtained in the comparison between the cross section at the backward

maximum, O, . , and &,, the quadrupole deformation parameter. The latter.

I 2C

being deduced, for even-even nuclei, from B(E2, 2-*0 ) values, can be in¬

terpreted both as deformation or transition stre-gth. The values of 0.1 Z/ł¬

at 35.2 MeV (crosses in the lover part of Fig. 2) , give almost exactly the

observed O ^ ^ value, at least within the accepted errors of the B(E2)*s.

A similar agreement, but not of the same quality, has been found for odd-A

nuclei. In this case ji- is actually a deformation parameter, being deduced (9)

from ground state electric quadrupole moments. This impressive agreement,

which could hardly be fortuitous, indicates that whichever process is respon¬

sible for the effect* reported here, i t must be heavily dependent upon some

fundamental nuclear parameter like the degree of collectivity.

Data analysis

The analysis of the collected data is under way from different approaches

including phase shift* analysis, optical and folding model potentials and

two-step* processes.

Significative results have been so far obtained in the phase shifts

analysis which has -been performed with the aim to ascertain if some and

which partial wave could be responsible for the effect observed. Results have

been up to now obtained for nuclei with A<22 and only relatively to the

mass dependence of the effect; the energy dependence is s t i l l being investigated.'

The following procedure was adopted in the analysis: the phase shifts

derived from the folding model calculations,i.e. the same which produce the

integrated cross sections shown in Fig.6,have been used as a "reference" set.

An optical model fit to forward angles only,up to 0 =100" .gave for each

nucleus a starting set of phase shifts. This init ial set fits of course the

forward angle enhancement of cross sect ions.To remove uncertainties about

conclusions two rather different optical model potentials were used.The

analysis was then performed,using the program SNOOPY,and trying to fit the

full angular distributions.Ten partial waves were taken into account. To

130

Fig.8 - Decrease of the X value

when only one partial wave is va¬

ried at a tine.

minimize ambiguities, f irst the

partial waves, for each l-value

up to 9, were singularly searched

on, in order to find out for every

nucleus those waves which have the

largest effect in decrasing the v2 initial 7. . It turned out, rather

interesting!;-, as shown for ccrae nuclei in Fig.8, that initial decreases of

Z by a factor between 20 and 100 are obtained with just one L-value, the or¬

der of which increases smoothly as A increases. For example, they are L-2 or 3 9 16 19 22

from Be up to 0, then L-3 or U up to F, and finally L-4 or 5 up to Ne.

A similar search was also performed on every combination of two L-values.

Coupling with just the L«0 wave improves the f i t , while coupling with other

waves does not. The final search was then, made by letting every partial wave

up to L=9 to vary in succession and starting for each nucleus with the one

with the largest effect on X . In those cases where two waves play a

similar role for a given nucleus, like L-2,3 or L"3,4 e tc . ,

both possible successions were tried, although in the end no significant

differences existed between the two. The final results are presented in

Fig.9, as a function of A,in terms of the partial cross section pertaining

to each wave up to L«5. Continuous lines are the folding model "reference"

values, dashed lines the result of the search. The other optical model star¬

ting set,referred to above, produced also similar diagrams. ~

131

8 10 12 14 16 18 20 22

Fig.10 - Integrated cross sections, at 35.2 MeV proton energy, pertaining to

each partial wave, plotted against mass number. Continuous lines are the folding model "reference" values, while dashed lines give the phase shifts analysis results .

v2 The fi ts are of very good quality, final values of Z are beween 5 and

10 for a 10Z allowed error in the cross sections. We have not plotted, as a

rule, the starting values given by the forward angles optical model f i t , since

they are in most cases undistinguishable fron the final results . I t tumes

out, in fact, that the fi t to the full angular distributions is accomplished

by very small variations in the real and imaginary part of the phase shif ts ,

i . e . a few degrees in O and a lev percent in T), at most, thus leaving the

132

partial cross sections <J^ substantially unchanged. Only for the L-3 waves

Lvaries somewhat, and the i n i t i a l values are shown by crosses in the rele¬

vant diagram of Fig.9.

From this phase shifts analysis we can draw the following conclusions:

i) - to a very large degree of confidence the same part ia l waves responsible

for the forward enhancement are also involved in the backward effect.

i i ) - the fi ts to the full angular distributions, which are indeed excel¬

lent, are accomplished by small variations of the real and imaginary

parts of the phase shif ts . These variations show up, for every nucleus,

as a definite increase in some partial O^, at least with respect to the

folding model predictions.

i i i ) - judging from Fig. 4, the waves involved are'neither L"0, nor L»l,

but s tar t being L*2 and L-3 for nuclei below oxygen. Gradually they shift

to L"3 and L*4 with increasing A. The L-5 wave seems playing some role on¬

ly after A«18. In this connection we recall that at 35.2 MeV the angular

momentum of the grazing wave goes from L=2 or L-3 up to L-S in going from

9Be to 22Ne.

Similarly good f i ts have been obtained by A.M.Kobos using a proton

optical model potential including terms, both real and imaginary, depending

explicitly on angular momentum.

Coupled channels analysis have been attempted both coupling low-lying

inelastic channels and deuteron channels. The coupling of deuteron channels

is known Co produce sizeable effects on proton scattering on light nuclei .

A maximum at about the correct angular position is in fact obtained in the

coherent sum of one and two-step (p,d,p) processes.Difficulties for this mo¬

del, however, arise from the fact that a strong backward maximum is obtained

also for collective nuclei, which experimentally do not show the effect.

Similarly unsatisfactory results have been obtained in some test cases

133

of coupled channels calculations coupling Co low-lying inelastic states,

la these calculations the elastic scattering cross section is decreased in

a way roughly proportional to j3. This could be useful in describing collective

nuclei after having obtained a proper description of closed shell nuclei.

Conclusions

As a summary ve believe that at least four clear indications can be retai¬

ned from the .present experiment:

a) - The anomalies found may very well be the main reason of the knovn fai¬

lures of the optical model description ot proton elastic scattering on light

nuclei at incident energies above 30 MeV.

b) - The anomalies are systematically related to the shell structure of the

target nucleus and hence they should find an explanation within a framework

as general as the shell and optical models are.

c) ~ The close correlation with the collective parameters points out that a

macroscopic description of the nuclear states involved in the reaction pro¬

cess, or their couplings, should be an important part of a proper theory

of the effect.

d) - The effect does not involve, for any nucleus, one single, specific,

partial wave. The fact that the waves closest to the grazing

value are, for every nucleus, the most important may have some significance,

also in view of the correlation existing with nuclear deformation parameters.

While the data so far collected are certainly sufficient for starting a

major theoretical investigation, more experiments at different energies are

needed, together with more data around the second closed shell. These expe¬

riment will take place in the coming months.

REFERENCES

1 - E.Colombo, R. De Leo, J .L.Escudie , E . F a b r i c i , S . M i c h e l e t t i , M.Pignane l l i ,

F.Resmini, and A.Tarrats , Phys. Rev. Let t , to be published

2 - E.Colombo', R.De Leo, J .L .Escudie , E .Fabr ic i , S . M i c h e l e t t i , M.Pignanell i

F.Resmini, Proceedings Int .Conf.on Nuclear Structure , Tokyo, 1977

3 - N.M.Clarke, E.J .Burge, D.A.Smith and J.C.Dore, Nucl. Phys. A157 (1970)145

4 - H.V.T.van Oers and J.M.Cameron, Phys. Rev. 184(1969)1061

5 - E.E.Gross, R.H.Bassel , L.N.Blumberg, B.J.Morton, A.VanDerWoule and A.Zucker,

• S u c i . Phys. A102(1967)673

6 - R.De, Leo, G.D'Erasmo, A.Pantaleo, G.Pasquarie l lo , G . V i e i t i , M.Pignanel l i ,

and H.V.Geramb, t o be published

7 - J .L.EscudiS, and A.Tarrats , the code E l i s e , Compte Rendu d ' A c t i v i t e ,

Report CEA-N-1861, Saclay (1975)181

8 - S.Raman, H.T.Milner, and F.H.Stelson, to be published on Atomic and

Nucl. Tables

.9 - G.H.Fuller, and V.tf.Cohen, Nucl. Data Tables, A5,(1969)433

10 - A.M.Kobos, private comnunication and

A.M.Kobos, and R.S.Mackintosh, to be published

11 - R.S.Mackintosh, Nucl. Phys. A230(1974)175 and Phys.Lett. 62B(1976)127

6Li UTDUCBD REACTIONS WELL ABOVE THB COULOMB

BARRIER

Jerzy Jaatrzebski Institute for Huclear Research, Świerk near Warsaw *

and Indiana Onivarsity Cyclotron Facility, Bloomington, Indiana

Abstract

Raoent experimental data available for nuclear reactions induced by the Li projectile 10 UeV/nucleon are discussed. induced by the Li projectile at energies about and above

* Permanent and present address

-IHTR0D0CTI05

The experimental investigations of the mechaniea of

reactions induced by Li ions are much less nuaerous for

energies exceeding 10 Ue7/nucleon than for lighter or slightly

heavier projectiles. It is only quite recently that these

reactions were investigated in the 50 - 150 MeV energy range.

It is well known that for lower energies and especially

in the vicinity of the Coulomb barrier the loosely bound d-cC

structure of this projectile affects strongly the transfer

probability of Li "components". We shall try to answer the

question: does this structure influence also the reaction

pattern for bombarding energies exceeding many times the

Coulomb barrier.

In the reactions induced by light projectiles like protons

or alpha particles the preequilibrium nucleon emission plays an

important role. Recent experimental data obtained for Li indu¬

ced reactions indicate that at high bombarding energies the

preequilibrium effects may present a major factor limiting the

complete fusion cross section also for this projectile.

In reactions induced by ions slightly heavier than Li two

problems are of particular interest. One relating to the limi¬

tations of the complete fusion cross section due to the entrance

channel dynamic, structural limitation in the compound nucleus

itself or, for higher energies, the compound nucleus yrast line

limitation. The second problem, which is still waiting for a

full and general explanation, is the origin of fast, near beam

*37

velocity alpha particles observed with an appreciable cross

section in the collisions of complex nuclei well above the

Coulomb barrier ' ). Also for projectiles only slightly heavier

than Li (e.g. C, see ref. 3) products with energies and

angular distributions characteristic of deeply inelastic colli¬

sions were observed for bombarding energies about and above

10 HeV/nucleon.

Experimental studies of all these phenomena for a projectile

with an intermediate mass are interesting and have only been

partly explored. In the present talk I should like to survey

some new, mostly unpublished data concerning reactions induced

by Li ions at energies well above the Coulomb barrier. The

major part of the presented results has been obtained in the

course of the last two years by different groups working at the

Indiana University Cyclotron Facility.

ELASTIC SCATTERING

12

Elastic scattering data for targets ranging from C to

Pb at 50.6 MeV bombarding energy were reported recently ).

These data indicate that at this energy the °Li projectile

behaves like other strongly absorbed ions with no distinct

diffractive oscillations in the angular disxribution for targets

with A > 40. At about 10 MeV/nucleon the angular distribution

for Li is intermediate between that for ^He and heavier projec¬

tiles. However, the best fit optical model parameters indicate

that the diffuseness of the absorptive potential is by about

30% greater for the Li projectile than for heavier ions.

138

Different conclusions were reached in ref. 5 for the

bombarding energy of about 22 MeV/nucleon and low mass (A»28 )

target. In the quoted work the shapes of the angular distri¬

butions and the resultant optical model analyses were compared

12

with those for C induced reactions at 15 HeV/nucleon bom¬

barding energy. The Li data revealed the presence of a nuclear 12

rainbow, characteristic of light ions but absent when C or 0 projectiles are involved. The conclusion was that Li scat-

12 taring is quite similar to that of light ions while C behaves

like heavier ions. In the quoted reference, however, different 6 12

energies per nucleon were employed for the Li and C projec¬

tiles which made comparison somewhat ambiguous. The apparently

different result of refa. 4 and 5 indicates that more elastic

scattering data, in particular for heavier targets, are required

before a final conclusion mar be reached.

LIGHT REACTION PRODUCTS

Por lower bombarding energies, beside evaporation particle

the light particle spectra in Li induced reactions are domina-

ted by fast, foreward peaked alpha particles and deuterons ' )•

Both groups are characterized by bell shaped Bpectra centered

arround the beam velocity. The angle integrated cross section

for the beam-velocity alpha particles ia about double that for

fast deuterona, and for a given value of BL1/BQ (where BQ io

the Coulomb barrier ) it is independent of the target mass ).

For ET./E_=s 2 this cross section amounts to about 500 ab. hi 0

139

Recently ) the light particle spectra ranging from Z=1 to Z=8 were measured by a five counter telescope with a 12 AUD front detector for 10° £ 6Lab Ś 150° and 56Pe, 90Zr and 197Au targets. The bombarding energy was 95 MeV. The angular distri-butlons, illustrated for foreward angles in fig. 1 for ? Zr are characteristic for all the targets investigated. Figure 2 shows the Z=1 and Z=2 particle spectra and fig. 3 the angular distributions of oC particles and deuterona for different energy intervals.

As for lower bombarding energies, the light particle spectra are dominated by Z*1 and Z«2 particles with the charac¬ teristic beam velocity peaks superposed on a continuous, more or less exponential distribution, particularly important in tho proton spectrum but alao observed in the alpha spectrum. These continuous distributions are an indication, at least in the proton case, of preequilibrium processes occurring.

6 T 7 The observed spectra of Li, Li and Be show features typical of simple direct reactions with clustering of strength at low excitations. This is confirmed by their strongly fore-ward peaked angular distributions.

The heavier light products reveal a gradual flattening of the angular distributions when moving away from the projectile. A similar behaviour was observed for other high energy light -heavy ions. However, in the energy spectra of these particles there was no evidence for strength clustering near the Coulomb -repulsion energies as observed in deeply inelastic processes. Although this would suggest a change in the nature of collisions

0 ° 1 0 ° 2 0 ° 3 0 " UP" 5 0 ° 6 0 °

1 0 0 0

100 JQ

E a o

10

1.0

0.1 r

0.01.

Zr(\i,X) E,ob=95.5MeV

--- total a

(a-evap.pk.i d

i i i °U , 0 ° 1 0 ° 2 0 ° 3 0 ° 4 0 ° 5 0 ° 6 0 °

8 lob

? i g . 1 L a b o r a t o r y a n g u l a r d i s t r i b u t i o n s f o r t h e l i g h t p r o d u c t s

o f t h e L i + Z r r e a c t i o n a t 9 5 U e V b o m b a r d i n g e n e r g y

/ R e t . 8 / .

3G.0 I

24.0

fZ.o

Lu

Cf -if

4

0.0

(..0

0.0

b.o

0.0

fc.o

ŁO

4© fao

Co too

3u-

4o to fo 'oo

2O +0 fcO «O IOO

Pig. 2 Th« Z-1 and Z»2 p«rticl» ipaetra ob««rr»d in th« 6Li+9OZr reaction /R«f. 8/ .

10

100 f \

e

1000

1000

S 500

Of

100

100F

100F-

E l ab=95.5MeV

Ex(MeV?=

\

\

\

150

20-30

-40

40-50

- 70-80J

- 80-90 (evapora¬ tion peak!

i i i i

1.0

10

E,ab=95.5MeV

Ev(MeV)=

-F100

f500

100

20-30

0° 10° 20° 30° 40° 50° 10° 20° 30° 40° 50°

e lab

Pig. 3 Angular distributions of aC -particles and deuterons

tor different energy intervals. Li + " Zr reaction at

95 MeV /fief. 8/.

1*3

6 12

between the Li and, e.g., C projectiles, we should wait with

a final conclusion till the Pe data are completely evaluated

or even till the Li reaction on a lighter target is studied.

It was shown in ref. 3 that the intensity of "fission like" 12

products in C induced reactions ia strongly dependent on the target masa and increases with decreasing target mass.

The emission of the beam-velocity alpha particles is not

specific for the Li projectile, and is commonly observed for 1 2 \

heavier ions ' ). However, the yield of the beam-velocity

deuterons is characteristic for the Li induced reactions and

indicate! the Influence of the loosely bound internal structure

of this projectile on the composition of the light particle

spectra.

The Li break-up on c£ and cL in the Coulomb and nuclear

fields was previously studied for lower bombarding energies 7 f ^ ) .

If we extrapolate the cross sections determined in the kinemati-

cally complete coincidence experiments 9 ) , we can expect that

the Li break-up without excitation of the target nucleus con¬

tributes substantially to the observed d and o( yields.

The higher yield of the fast pC particles as compared with

deuterons deserves a comment. For lower bombarding energies this

excess was attributed ) to the one nucleon stripping reactions

and subsequent decay of the oC unstable -*He and ^Li nuclei. This

explanation alone seems, however, unlikely at least for the

bombarding energies well above the Coulomb barrier. Prom fig. 1

it may be seen that, e.g., for 9 - 15° one nucleoa pick-up

reactions have cross sections about 25 times smaller than in the

case of fast deuterons. Although one nucleon stripping reactions,

due to the difference in Q value, may have higher cross sec¬

tions, it cannot be expected that they may completely compensate

for the 5O% difference between fast alpha particle and deuteron

intensities at that angle. One can, however, find at least three

other reasons for this difference 1

(i) Li break-up with three particles in the final state

(Li -* p + n + oi).

(ii) Enhancement of the deuteron transfer reactions as

compared with oC particle transfer.

(ill) Processes analogous to those observed in hoarier ion

induced reactions, where the beam-velocity oC particles

are observed with a significant strength.

HEAVY REACTIOH PRODUCTS

In all experiments reported so far the heavy reaction

products were identified by gamma ray counting techniques. Both

in-beam and activation methods were used, the targets ranging

from 27A1 to 232Th 1 1 " 2 0/ ) . The recoil ranges of the radioactive

reaction products were also measured 13< *_). The results of

ref. 14 shall be summarized here as an example of the study of

heavy reaction products.

Examples of the excitation functions and integral racoil

ranges, projected on the beam direction observed in the

and 6Li + 5 Fe reactions are shown in fig. 4a - eL . The excita¬

tion functions and recoil ranges of the reaction products with

600

400

200

} 5?Ni from'Ve . 54_

from Fe

Pip;, 'ta Cross sections and Integral recoil ranges vs. bombarding energy for the radio¬

active products of the Li + Fe ami Li -t- Fe reactions /after ref. 1ft/»

The lines passing through the cross section points are only to guide the eye.

The lines indicated in the recoil ranges part of the figures are calculated

assuming a full momentum transfer reaction (formation of the oompound nucleus).

R (mg/cm2) p o o p — -») « » i ł O M

at

CROSS SECTION (mb) _ N *

ro •> a i 5 o o o o 9 o

a a a

O

O R (mg/cmz) CROSS SECTION (mb)

R (mg/cm4)

&>

O

CROSS SECTION (mb)

— M * n O • ?* . . .? ? . 9 • • . . . ?

O

i s et O

R (mg/cm2) CROSS SECTION (mb)

,? 9 •?,?„,? 4 , i ,

a m a o

i

R (mg/cm2) CROSS SECTION (mb)

p R (mg/cm4) O O O O -> » » « [ > y p

CROSS SECTION (mb) _ IM > 9) 5

ia * a p 9 po 5 8

CT " Zprod * 2 i n d i c a t e t h a t a 1 1 t h e s e

products are reached by micleon evaporation from the compound

nucleus up to bonbarding energies by about 10-20 HeV higher

than those, corresponding to the excitation function peak value.

At higher bombarding energies the preequilibriua nucleon emission

manifests itself by a levelling-off of the recoil ranges and

characteristic high energy tails on the excitation functions.

At 75 KeV bombarding energy the cross section for the formation

of these nuclei in processes in which the equilibration is

preceded by the emission of a fast nucleon was estimated as

about 200 nb.

The excitation functions and recoil ranges of the reaction

products with k^g - ADrod ^ * ftnd ZCM " Zprod 2 *ndl-cate

that for bombarding energies higher than those corresponding to

one or two oC particle evaporation peaks also some non-compound

processes contribute to the excitation functions of all the

products. These processes account for about 20-30% of the obser¬

ved reaction cross section in the energy range of 55 - 95 MeV

and in this energy range they are not strongly dependent on the

bombarding energy within the accuracy of ref. 14 estimate.

From the discussion of light particle spectra it follows

that only fast protons (or undetected neutrons), deuterons and

alpha particles have intensities high enough to be considered

as main light partners in the non-compound processes observed.

One may expect that two mechanisms can give a similar behaviour

of the projected recoil ranges vs. bombarding energy, namely

150

the transfer reactions and the fusion reactions followed

by emission of a fast, preequilibrium particle.

It is worth noting that a similar behaviour of recoil

ranges was also observed in reactions induced by heavier pro-

jectiles ~ ) and interpreted as due to the transfer reactions.

However, also there the preequilibrium emission, including may

be also fast cC particles, cannot be excluded on the basis of

the available data.

In ref. 14 the combination of the in-beam and activation

methods has allowed the authors to determine the cross section

distribution for heavy reaction products as shown In fig. 5

and 6. From such distributions the mean number of nucleone

emitted from the composite system (fig. 7 ) and the mass

distributions (fig. 8 ) were derived. In fig. 8 the mass 32 27 distribution of the evaporation products from the S + Al

reaction is also shown ^ ) . The differences observed between

Li and S induced reactions are probably due to the direct

processes in the former reaction and to the enhanced oC parti-"32 cle evaporation in the J S induced reaction.

In fig. 9 the total observed cross section is compared

with the reaction cross section, calculated using the parabolic

model ) (r = 1.22), and with the cross section calculated using

s If the transfer of one of the Li clusters would >-* mainly responsible, the most probable o< particle or deut6i m

21 \ energies would be determined by the optimum reaction Q value )

which gives, for the grazing angle of detection, energies

corresponding to the beam velocity.

151

6Li • S6Fe ELAB=55MeV

27

26

25

21

23

» MM

it CM

a v

SJFE

II MM

2 Sitni!

28

Si CO

W FE

6 UMK

21 sice.

2

29

5 * HI

55 CO

12

16 U MM

10

8

»cu

trm

4 M CO

77

200 U MM

47

20

stcu

60 «co 270

211

47

30

H O I

43 S< CO

375

100 MHO

6

fICtf

2

46

8

32

C.N.

facs

• FE

26 28 Pig. 5 Distribution of heavy produoti froa the Ł1 * 5'?« rtaotlon *t 55 Her

bonbarllog «n«rgy. Sis Indicated eroas aeotlont (in oV) ax> sbtalscd froa a eoooth Interpolation of the exoltatloa function for each reaction profluot* Then both aotlratlos as£ la-bean data vara rtTallable, the crcaa aectlon detenslned by the BCtlTatlon aeasureoeat !• Indicated /Bef. 1A/»

S, L l * F E L A 8 = 1 C

26

25

2A

23

ia w

act

5

a OMeV

27

S2 PC

9 fBtsnx

35

25

26

M.CO

26 u w

75 won

80

12

29

f t CO

13 j ^ M FE ^

56 U M

125

62

6

M C O

S7 HI

5

98

180 U M K

140

35

W CV

4 tr co

200 M FC •"'•

260

48

17 30

(SOl

4 W CO

47

78 f t MM

II ucR

•ICU

ty'M mk'•-.

t

15

5 C M

32

• t u

C.N. - y\\ m'Ą

St FE

26 28 6 Dlatrlbutlon ot baavj praAneta rro» the Ł1 • T» naotlos at 100 »eT

boabardlaf energy. See aleo oeptlon to f i t , 5.

152

60 70 80 90 100 110 E* (MeV)

Fig. 7 The average mass (in a.m.u.j emitted va. compound

nucleus excitation energy for the Li + ^ Fe reaction.

In the calculations of the excitation energy it was

assumed that all reactions lead to complete fusion

/Ret. 14/.

153

400

200

400

E 200

o

o u\ 400 V)

S200 O

200

100

L

5.4

P i g .

6.9

7.4

6, . , 56_ 62-

U + Fe —- Cu E* = 65.9 MeV

E"=93.0 MeV

£ =106.6 MeV

Al —*- Cu

E* = 87.0 MeV

) 10 12

-A product

•14 —r~ 16

—r 18

Li

M distribution of tut roic;ior. products for the

Pe and 3 + 'A.1 reactions . '."no arrowe

i n d l c o t a average "evapora t ed" masa i c r R giver, e x c i t s

t i o n firjergy of the ccasou' .c ayetem. Sec oJao c a p t i o n

to f i g . 7 /Hef. 1 4 / .

3 — 2.0 Z O ł— (J UJ co cn o

u

^ n

A-bR parabolic B model

r

_

<

" " " ^ • ^

t

i i 1

i i i i

-&BsnR2( i-r£-)

o = i c ( AI/3 + A V 3 I f m

—

, * * • • " * ^ - . . . ^

T "" •~~«^ '

1

— - A

^ B i

1 1 I 1

0.010 0.020

Cłg. 9 Total observed heavy products cross section in the 6Li + 56Pe reaction vs. 1/ECM /Ref. 14/.

155

y the aiaple expression & R • 7c H ( 1 - -S— ), where Ve la the

Coulomb barrier with radius parameter rQ » 1.4 fm. The complete fusion cross section, defined ) as the part of the reactions which lead directly to the formation of a compound, thermally equilibrated system, were estimated in rat. 14 as 1.1 + 0.2 b at energies of about 10 HeV/nuclson, ( i.e. about 65% of the observed reaction cross otctionj. A comparable fraction of the reactions lead to conplete fusion at these energies in C induced reactione on Cu and Vi targets ).

PARIICLE-QA10IA OOIHCIDENCES

Recently new experimental information was gathered * ) on the contribution to the formation of heavy reaction products of the processes in which the beam-velocity oC particles and deute-rona are participating.

Figure 10 shows the gamma ray spectra in coincidence with these particles observed in the ćLi + 197Au reaction at 15 UeV bombarding energy. Similar measurements were also performed for the 5 Fe target. The coincidence gamma ray apectra are consistent with the assumption that the remaining energy of the projectile

197 197 is transferred to the composite Au + d or Au +oC system which subsequently decays by particle evaporation. The prelimi¬ nary estimation ) of the coincidence cross section for the beam velocity cC particles indicates that only about 15/» of these particles are in coincidence with gamma rays belonging to mercury isotopes. Por 75 MeV bombarding energy the absorption of the beam velocity deuteron leads to the excitation energy of

156

5. * § •O H

5

I !

ct 1

o a H v; m o

0 C

a

S 3

the

f»

O o

' O O-

a

o

c r - » <* vO B

C < O

a o B o

NUMBER OF COUNTS NUMBER OF COUNTS

s s « ^ * «

695.6

I 416.4

rn

•I (0

l 1

< C

O B

the °°Hg nucleus of 35 MeV, when this nucleus should decay

mainly by neutron evaporation, so the determined cross section

for mercury isotopes accounts for the total coincidence cross

section.

It is therefore probable that the important part of the

beam velocity deuterona and oC particles are due to the pro¬

jectile break-up without excitation of the target nucleus.

A Icinematicelly complete d-oC coincidence experiment is now

in progress ^ ) and it has for aim to check this particular

point.

Assuming that a similar fraction of the bean velocity

o( particleB and deuterons lead to the formation of ' Fe + d

and Fe +oC composite systems, the non-compound processes

observed in ref. 14 for this target cannot be due principal/

to the oC or d transfer reactions. As it was discussed previo¬

usly also p or n transfers seem to have intensities too small

to explain the importance of these processes. Therefore pro¬

bably the main part of the non-compound processes observed in

Li reactions well above the Coulomb barrier should be related

to the preequilibrium phenomena.

PISSIOH

Fission products from the Li + °'Au reaction were iden¬

tified in counter experiments in refs. 8,18 . Figure 11 shows

the angular distribution of these products at 95 MeV bombarding

energy ) and fig. 12 summarizes the available cross section

158

80

70

.w 60

J50

b -o

30

20

6Li+197Au,95MeV Fission fragments angular distribut

crfiss={219i20)mb

30° 60c 90° 120° 150° 8 C M

1*i Angular distribution of fission products observed in

Li + - Au reaction at 95 KeV bombarding energy /Kef. 8/.

159

500-

1100

50-

- Fission

-

•

—

/

-

-

i

cross

*

/

i

J sections y/*

/ °

ALICE a f / a ns

B, <-

\ Indiana o Karlsruhe

i i i

i • -

-

--

60 80 100 120 140 Eu(MeV)

160

Fig. 12 Cross Station of fission products obssrrtd in Łi+ iu rsaotion T S . boabkrdiag tnsrgjr. Th« osloulstsd cross ssotio&B w«r» obtained vith tha halp of ALICE cod* 8*27) /Rafs. B.18/.

160

data. The erosa sections measured In the Karlsruhe ) and

Indiana ) experiments differ by a factor ot about two. The

former experiment was, however, done only at a 90° geometry.

Also the cross sections were extracted from the comparison

with target X-ray Intensities, while the probabilities of

forming K-shell vacancies in heavy ion reactions are still

rather poorly known.

The Li induced fission studies have only been started

at the Indiana University Cyclotron Facility. A large program

is now under way ) having for aim determination of the

fission cross sections for different targets and bombarding

energies. The Li particle may appear to be a very convenient

projectile for these studies, since still quite high angular

momenta can be brought to the composite system and the extra¬

ction of fisBion cross sections is less ambiguous than for

heavier projectiles due to the absence of deeply inelastic

processes.

SUMMARY AND CONCHTSIONS

The reactions induced by the Li projectile at boabarding

energies exceeding 10 MeV/nuclaon were extensively studied

during the last two years. The elastic scattering and light,

heavy and fission reaction products were investigated. Although

a large part of the collected information is atlll in a preli¬

minary form, some features of the reaction mechanism already

emerge from the available data.

At energies well above the Coulomb barrier the Li projec¬

tile seems to behave In many respects as other Btrongly absorbed

ions with singularities which are related to its smaller mass

rather than to its loosely bound structure.

The observed cross section for heavy reaction products is

well reproduced by a simple geometric model with radius para¬

meter similar to that employed for heavier projectiles. Also

a similar fraction of the reactions lead to the complete fusion

of thia projectile with the target nuclei. In the case of Li,

however, the most important contribution to the non-compound

prooesees seems to be related to the preequillbrium omission,

whereas it is generally believed that multinucleon transfer

reactions aoount mainly for the direct processes in the case

of heavier projectiles. The last statement may, hovjever, not

be fully true in vue of the preequilibrium phenomena recently

discovered •*•*) also in 0 induced reactions.

A similar behaviour of the recoil ranges for Li and

slightly heavier projectiles indicates probably the same mecha¬

nism, i.e. preequilibrium emission, as mainly responsible for

the non-compound processes observed.

One of the most striking recent observations was the low

yield of coincidence between the beam velocity od particles

and gamma rays. This observation indicates that although the

Li break-up contributes substantially to the reaction cross

section in the spirit of the optical model definition, the

loosely bound d-cC structure of this projectile does not

affect strongly the cross sections of the heavy reaction

products.

162

The suspicious difference in the fast aC particles and

deuterons yield may be indicating that the phenomenon of emission

of the "preequilibrium", bean-velocity alpha particles currently

observed in reactions induced by heavier ions also occurs for

the Li projectile. Unfortunately other reasons may be found

for this difference what precludes a definite statement about

this phenomenon.

The marked difference between Li and, e.g., C induced

reactions is the absence of deeply inelastic collision*. Howt-

ver, lighter mass targets Bhould be investigated before a

definite conclusion ia drawn.

This survey is largely baaed on unpublished data obtained

recently at the Indiana University Cyclotron Facility. I should

like to thanlc professors H. Smith and S. Vigdor for the permis¬

sion to quote the unpublished and often preliminary results

obtained by their teams. Uy sincere thanks are also due to all

my American colleagues for the stimulating scientific atuosphere

at that University.

References

1 H.R. Britt and A.B. Quinton, Phys. Rev. 24 /1961/ 877 2 D.H.E. Gross and J. Wilezyńflki, Phys. Lett. 67B /1977/ 1 3 J.B. Natowitz, H.M. Hanboodiri and E.T. Chulick,

Phye. Rev. £13 /1976/ 171 4 L.T. Chua, P.D. Becchetti, J. Janecke and F.L. Milder,

Hucl. Phya. A273. /1976/ 243 5 R.U. DeVriea , D.A. Goldberg, J.M. Wataon, U.S. Zistsan

snd J.G. Cramer, Phye. Rev. Lett. 3_9 /1977/ 450 6 R.W. Ollerhead, C. Cnasman and D.A. Bromley, Phya. Rev.

134 /1964/ B74 7 K.O. Pfaiffer, E. Speth and K. Bethge, Sucl. Phye.

A206 /1973/ 545 8 S.E. Vigdor, C. Castaneda, H. Karwowsici, P.P. Singh,

H.A. Smith and J.D. Wiggins, Bull. Am. Phys. Soc. 2£ /1977/ 1003 5 S. Vigdor, prir. com.

9 D. Scholz, H. Gemmeke, L. Laseen, R. Ost and K. Bethge, Huol. Phys. A268 /1977/ 351

10 R. Ost, K. Bethge, H. Gemmeke, Ł, Lessen and D. Scholz, Z. Phyeik 266 /1974/ 369

11 H. Kamowaki, J. Jastrzębski, M. Sadler, P.P. Singh and Z. Horoz, unpubl. data

12 U, Blann, P.H. Lanzafame and R.A. Piscitelli, Phya. Rev. 1^2 /1964/ B700

13 O.K. Cline, Nuel. Phyn. A174 /1971/ 73 14 J. Jaatrzębski, H. Karwowski, M. Sadler and P.P. Singh,

to be publ. 15 A.K. Gaigalaa, S.Jha, H.A. Smith and T.E. Ward,

I.U.C.F. Ann. Rep. /1977/

16 CM. Castaneda, H.A. Smith, Jr., T.E. Ward and T.R. Sees,

Phys. Rev. C16 /1977/ 1437

17 J.G. Fleissner, D.A. Rakel, P.P. Venezia, E.G. Punk,

J.W. Mihelich and H.A. anith, I.0.C.P. Ann. Rep. /1977/

and to be pub.

18 J. Kropp, H. Klewe-Nebenius, H. Faust, J. Buschmann,

H. Rebel, H.J. Gils and K. Wiashak, Z. Physilc A280

/1977/ 61

19 R.S. Tickle, ff.S. Gray, H.C. Griffin and H.A. Smith,

I.U.C.F. Ann. Rep. /1977/

20 A. Fleury, These presentee a la Faeulte des Sciences

de l'Universite' de Bordeaux /1969/

21 P.J. Siemens, J.P. Bondorf, D.H.E. Grosa and P. Dlckmann,

Phys. Lett. j}6B /1971/ 24

22 H. Delagrange, A. Fleury, F. Hubert and G.ff. Simonoff,

Phys. Lett. 37B /1971/ 355

23 Franooise Hubert, These presentee a l'Universite de

Bordeaux I /1973/

24 R. Bimbot, D. Gardes and M.P. Rivet, Hucl. Phys. A189

/1972/ 193

25 F. Pulhofer, W.F.W. Schneider, F. Busch, J. Barette,

P. Braun-Munzinger, O.K. Gelbke and H.E. Wagner,

Phys.Rev. £!£ /1977/ 1010

26 T.D. Thomas, Phys. Rev. Jj£ /1959/ 703,

Subroutine in ALICE evaporation code, see ref. 27

27 M. Blann, COO-3494-29

28 M. Blann, Proc. Int. Conf. on Nuclear Physics, Munich

1970, Vol. 2, p. 657

29 J.B. Natowitz, E.T. Chulick and ili.fi. Namboodiri,

PhyB. Rev. _C6_/1972/ 2133

30 C. CaBtaneda, H. Smith, P.P. Singh, J. Jastrzębski, H. Karwowsfci and A.K. GaigalaB, to be publ.

31 H. Sinith, priv. com. 32 S. Vigdor and H. Ksrwowski, priv. com. 33 D.G. Sarantites, J.H. Barker, Ł. Westerberg, R.A. Dayras,

M.L. Halbert and D.C. Eensley, to be publ., see also II. Blann and M. Becterman - Lecture presented at the 10th School on Nuclear Phyaica, Mikołajki 1977 / COO-3494-33/

A Microscopic Approach to the Description of the Giant Multipole

Resonances in Light Deformed Nuclei*

by

K.W. Schmid

Institut fiir Kernphysik der Kernforschungsanlage Julich,

D-5170 Julich, West Germany

Abstraet-r A purely microscopic model for the description of the giant

multipole resonances in light deformed nuclei and their ex¬

citation via proton radiative capture reactions is applied

to study the F( P , Y ) Ne reaction. Cross sections as well

as angular distributions are calculated taking into account

the electric dipole as well as the electric quadrupole part

of the eletromagnetic interaction. Though quantitative agree¬

ment cannot be reached, the essential qualitative features

of both the experimental cross sections and angular distri¬

butions are reasonably well reproduced. It is seen that the

19 20

F(P,Y) Me reaction runs predominantly over the giant di¬

pole states. The isoscalar giant quadrupole resonance is

only weakly excited although intermediate states with large

B(E2) values are available. This result may help to under¬

stand why inelastic u-scattering experiments on certain light

nuclei do detect a rather different E2-strength distribution

than the corresponding proton radiative capture experiments.

167

I. Introduction

During the last 15 years the proton radiative capture reaction has

been well established as a useful tool for the experimental study of the

giant raultipole resonances (GMR) in nuclei " '. Within the same period,

some microscopic theories of photonuclear as well as particle scattering

processes have been developed and used to study these resonances with

some success * ' . However, while both the microscopic structure as

well as the exci tat ion mechanism of the GMR for a couple of spherical nu¬

clei seem to have been reasonably well understood, there have been only

a few attempts22'28"30^ to tackle the GMR-problem in deformed nuclei.

For the latter, just the (P,Y) data * ' display some very interesting featu¬

res from which one may hope to extract some information about the GMR,

in addition to what one can learn from the study of closed shell nuclei.

For example, the (p,f) excitation functions for deforraed nuclei show

usually more structure than for spherical ones. Furthermore, in addition

to the (p,Y0) decay to the groundstate, the transitions ieading to the

low-lying 2 member of the groundstste rotational band have also been

measured. These (p»Y,) cross sections are found to be of comparable

magnitude as the corresponding (p,y ) ones, but their angular distribu¬

tions are usually quite different.

These features can be at least partly understood from the- phenomeno-

logical point of view. So, for example, the hydrodynamical model '

predictes a splitting of the GMR in deformed nuclei into various modes

corresponding to the different possible angular momentum projections on

the intrinsic symmetry axis. Such splittings have been observed experimen¬

tally in various nuclei of the rare earth region32'. Now, obviously light

nuclei do not behave as nicely collective as rare earth ones, however.

168

still some structure in the GMR due to the deformed intrinsic shape is

to be expected. On the other hand, the differences in the observed (P.Y 0)

and (p,f,) excitation functions may be interpreted with the :ie1p of the

rotational model '. In deformed nuclei we do expect rotational bands

based on the groundstate as well as on excited states. Because of the

similar intrinsic structure within a band and of the different angular

momentum selection rules for the transitions involved, this would explain

why the (p.T0) and (p,y,) cross sections are of comparable magnitude but

show rather different angular distributions.

The essential difficulty for a microscopic treatment of the GMR in

deformed nuclei is the creation and handling of suitable many-nucieon

wavefunctions. In spherical nuclei, all the relevant many nucleon con¬

figurations, namely the lplh configurations yielding the "gross struc¬

ture", or even the 2p2h and 3p3h states being responsible for the inter¬

mediate and fine structure of the GMR, can be easily coupled to definite

angular momentum. The continuum is then incorporated, either directly

as in the continuum shell model calculations ', or indirectly as in

Feshbach's formalism ' or R-matrix theory '. In deformed (i.e. "open

shell") nuclei now neither the Hartree-Fock (HF) determinant nor the

particle-hole configurations with respect to it have definite angular

momenta. Hence, apart from the cases where shell model wavefunctions

are available ', or where the deformation has simply been neglected ',

one either has to forget about the continuum and to be content with a

description of the nuclear structure (bound state) part of the GMR in

terms of intrinsic Tamm-Dancoff (TD) or Random-Phase (RPA) wavefunctions ',

or one is forced to perform numerically complicated angular momentum pro¬

jections.

169

The first attempt to use angular momentum projected deformed wave-

functions in ( P , Y ) and (p,n) reactions on light nuclei has been made by

Afnan '. Using Feshbach's formalism, he first defines the basis of

bound state wavefunctions by performing a diagonalization of the total

Hamiltonian in the space of very limited lplh-configurations with respect

to the deformed HF-vacuum. The resulting intrinsic wavefunctions are then

projected to good total spin by using the limit of strong deformations '.

There are a couple of reasons for the poor agreement of Afnan's results

with experiment, at least as far as the GMR are concerned. First, besides

the strong limitations of the modelspace, the "bad" effective Hamiltonian

and some severe approximations in the reaction formalism, the limit of

strong defromations is hardly justifiable in light sd-she!1 nuclei. Second,

the procedure of projection after the variation itself has some severe

shortcomings. Not only, that one overestimates the rotational structure

of the spectra, but one also runs into troubles with spurious admixtures

due to rotations of the system and, connected with this, with a possible

multipole appearance of the same states.

Only recently " ' we have proposed a microscopic model for the

description of the GMR in light deformed nuclei and their excitation via

proton radiative capture reactions which overcomes all these difficulties.

28) Similar to Afnan ' also in our model one starts with a version u' Fesf-

bach's projection operator formalism for nuclear reactions in oro<er tc

split the GMR-problem into three separate parts.

The first part, one is then left with, is a pure nuclear

problem and involves the calculation of wavefunctions for the bound states

of the target and the compound systems. For this purpose we use linear-

combinations of angular momentum projected deformed hole- anc particle-

170

hole states for the target and the compound nuclei, respectively, Both

sets of configurations are taken with respect to the deformed HF-vacuum

of the compound system, but opposite to Afnan's ' prescription, angular

momentum projection is performed before diagonalizing the Hamiltonian.

The second problem to be solved is a continuum problem and requires

a description of a continuum nucleon moving in the field of the target

nucleus. For this purpose the scattering waves of a real Saxon-Woods

potential are coupled to the angular momentum projected deformed hole

states mentioned above. Orthogonality between the bound and the continuum

wavefunctions is ensured using an orthogonal ization method proposed by

Wang and Shakin40'.

The third and final part of the GMR-problem consists then out of a

description of the coupling between the bound and continuum states. Here

standard methods known from the treatment of the GMR in closed shell

nuclei are adapted.

In ref. 39 this model has been applied to the electric dipole (GDR)

part of the giant resonances in Ne as seen via the F(p,y) Ne radia¬

tive capture reaction. It could be shown that the essential features of

both the nuclear structure as well as of the excitation mechanism of the

20

GDR in Ne are at least qualitatively well described by our model. How¬

ever, besides being restricted to the electric dipole part of the electro¬

magnetic interaction, the calculations made use of a couple of approxi¬

mations which were not completely satisfying and have hence been removed

in the meantime '.

So, in ref. 39, the *inal states of the reaction, i.e. the Ne

groundstate band, had been approximated by simple angular momentum pro-

171

jected HF-states, although, due to the projection, we do expect some mixing between the HF-determinant and the angular momentum srcjectea particle-hole (ph)-configurations with respect to it. The resulting gorundstate correlations could, even if they do not give drastic con¬ tributions to the energies of the Ne groundstate band, nevertheless

4ł'§ have some effect on the electromagnetic properties of the states ' and therefore on the radiative capture cross sections. In ref. 41 we have therefore included this effect.

In ref. 39 we had furthermore restricted the mixing within tne particle-hole states to configurations with either odd or even angular momentum projection K on the intrinsic symmetry axis, i.e. even :X's

had been required. Hence, although the angular momentum projection is performed before applying the many body Hamiltonian to the different configurations, the resulting states may still contain some rotational spurious admixture since the rotation operator connects states witn even

431 41' and odd K-quantum numbers '. Hence, in our latest wort ' we die not impose any restriction on aK.

In addition, the calculations in ref. 39 neglected the fact that the orthogonal ization procedure gives rise to an one body interaction terin via which the "quasibound" states, i.e. those single partic'te orbits which are used as basis for the calculation of the Dound states Bia wnich have hence to be taken out from the solutions of the Saxon-woods poten¬ tial in order to ensure orthogonality, may decay into tne contini/u.ti. This point, which had also been neglected in ref. 40, has lately been stressed by Micklinghoff '. In the improved version of our model 4"' these one cody decay terms have now been lak.en into account.

172

Finally, being interested only in the GDR, the calculations in

ref. 39 were restricted to the negative parity states with isospin T = 1.

In the meantime also the electric quadrupole part of the photon-nucleon

interaction has been included and therefore in addition also the posi¬

tive parity states with both T = 0 and T = 1 have been studied. This aim

required also an extension of the energy interval for the incident pro¬

tons towards higher energies.

Summarizing, one may say, that the improved calculations ', which

will be discussed here, had essentially two main goals. First, by removing

a couple of approximations we could test their influence on our former

results and thereby get some hints whether the quantitative discrepancies

between the theoretical and the experimental data are due to the limita¬

tions of our model or mainly caused by our lack of knowledge about the

effective interaction actually to be used for our problem. The second aim

of this work was to investigate additionally to the GDR also the giant

20 quadrupole part (GQR) of the GMR ir. Ne as seen via the proton raaiative

19

capture reaction on F. While there is no doubt, that tne proton radia¬

tive capture reactions are an excellent tool for the study of the GDR,

there has been lately a lot of discussion or it's suitability to investi-8 I1 4=i-4t>) gate the higher multipole resonances ' *' " . This discussion was

caused by some discrepancies of the experimental data obtained by inelastic

^-scattering and radiative capture reactions, respectively. We think, that

our calculations can ,7iake s useful contribution to tne solution of this

problem.

173

II. REACTION FORMALISM AND CONFIGURATION SPACES

The theoretical model which will be presented in these lectures has 37-39 41^ already been described in detail elesewhere '. However, in order to

make our calculations understandable, it will be necessary to repeat at

least the main lines of the theory. This will be done in the following

sections.

A. T-matrix for nucleon radiative capture reactions

The T-matrix for nucleon radiative capture reactions is given in first

order perturbation theory by

TT = <I|HY|f+>, (1)

where |I> is the final state of the capture process and H the electro¬

magnetic interaction. The initial state |y+> is an eigenstate of the

nuclear Hamiltonian H

(E ( + ) - H) \f*> = 0 (2)

and has to fulfil the proper boundary conditions for the entrance channel

under consideration.

In order to evaluate the T-matrix (1) we shall make use of Festibach's

projection operator formalism ' for nuclear reactions, which allows a

transparent classification of the states of the nuclear Hilbert space

according to their configurations. Following closely a version of this

formalism, which was specifically designed for the study of photonuclear

reactions by Wang and Shakin21^, we shall first divide the total Hilbert

space into three orthogonal subspaces P, d, and x. The continuum space P

will contain all those configurations formed by one nucleon in the con-

tinuum coupled to all the states of the mass A-l system which we are

going to treat explicitly in the calculations. The capture reaction

is then supposed to go through a set of selected states of the mass A

compound system. These states define the "doorway" space d. Finally the

rest of the Hilbert space is collected in the subspace x which is sup¬

posed to have only marginal influence on the reaction being considered.

Using standard techniques, the T-matrix for the capture process

leading from some entrance channel 'C > out of the P-space to some final

state jl> of the compound system is then given by

TY = <I|ft j C + > y (3)

where the "effective" nuclear and electromagnetic interactions are defined

as

ft = H + H T^C— H (4)

and

Hv = ;;_ + H p ^ — H( , (5)

respectively. The continuum space P is defined by some effective Hami)-

tonian tf

(E(+)-Hpp)!C+. = 0 , (6)

in terms of which the Greens function G ' is given by

Eq. (3) is a formal exact expression involving very complicated operators

ana can nence obviously not be used as such for practical calculations.

Tnerefore we have to introduce a couple of approximations.

175

Following Wang and Shakin ', we shall first assume, that the sub-

spaces P and d are designed in such a way, that the states of the residual

subspace x cannot decay directly to the final states via the electro-

magentic interaction and that they are furthermore not connected to the

P-space through the nuclear interaction. If this is true (or at least

approximately fulfilled), then the "effective" operators ft and ftp. can

be replaced by the usual H and H„j, respectively. He shall furthermore

assume, that we know a representation |d> of the doorway space

Left to be considered is then the doorway-doorway coupling via the x-space,

which should take care of the following two effects. First, it should

describe the so called "spreading-width" resulting from the coupling of

the doorway states |d> to more complicated bound A-nucleon configurations,

and second, it should take into account the absorption due to those

channels |C+> of eq. (6) which will not be treated explicitly in the cal¬

culations.

As discussed in ref. 39 the doorway states we are going to use here

are of very complicated structure and contain much more degrees of free¬

dom than for example the Tamm-Oancoff- or Random-Phase-wavefunctions

usually taken as doorways in the treatment of closed shell nuclei. Hence,

in our case, the spreading width will be much less important than in the

latter and the x-space will be predominantly used for the inclusion of

absorption effects due to the neglected channels. Since nothing more

special about the x-space is known, we shall assume in the following

that the doorway-doorway coupling via the x-space is small and can be

approximated by an energy and state independent constant & - i r /2,

176

an assumption which is supported by the small values L% = 0 and T% *.

100 keV actually been used later on in the calculations.

Using the above approximations the T-matrix of eq. (3) can now be

written as

where the "final state interaction" tera is given by

F Id " < ł l HT

G J" ) H Pdl d > •

and the "width and shift" matrix has the for*

H,d. - «(dd*)[E-Ed-4n+f rjQ - <d|HdpGj-)Hpd|d'> (1).}

In the following always the foni (9} of the T-matrix will be used.

B. Bound states

For the evaluation of the T-natrix (9) two classes of bound states

have to be calculated: the bound states of the M S S A compound system,

i .e . the final states |I> and the doorway states |d>, and furthermore

those states of the rass A-l target system and its mirror nucleus which

are later on to be coupled to a continuum nucieon in order to form the

P-space.

Considering these bound states we start as usual in nuclear structure

theory by defining a model space in terms of a f in i te orthonormil set of

spherical single particle states {Ca*, et, .,.) .He then assume that the

177

effective many nucieon Hamiltonian appropriate for this modelspace is

known and can be written as a sum of only one and two body terns

H ' | b t(ab) C* Cb • \ \^ v(acbd) C* C* Cd Cb (12)

Dealing with defomed nuclei, our next task is now to extract from the

Hamiitonian (12) the deformed average f ield each nucieon of the mass A

compound system feels due to its interaction with al l tire other nucieons.

Using standard Hartree-Fock theory (HF) ' , the selfconsistent deformed

orbits {a*, a t . • • •} of this average f ield can be expanded in terms of

our spherical basis

A • I *ai <£ . (13)

and the va.tatiDital parameters Afl1 be determined by minimizing the expectation

value of the total Harailtonian (12) with respect to the reference state

I > = n at |o> , (14)

isF '

in which the nucieons are occupying the A energetically lowest orbits up

to some level F. Any excitation of the system in the intrinsic frame can

then be viewed as a creation of particles and holes across the surface F.

The simplest of these excitations are the one particle-one hole (lplh)

configurations |Mo"1> * aM 'J* <15>

Now, for N * Z nuclei, i f we do not allow for —ty parity and isospin.

mixing in the expansion (13), the HF-vacuum (14) has positive parity and

total isospin T < Tz » 0. Assuming furthermore axial synKtry of the

orbits (13) the reference determinant has in addition a definite angular

momentum projection K * 0 and is even under time reversal, tn the l imit

of these approximation furthermore the lplh-states (15) can be easily

coupled to good total isospin and, by taking linear combinations, can

be nade to have definite properties under parity and time reversal

operations.

Of course, neither the reference state (14) nor the lplh-configu-

rations (15) with respect to i t have a definite angular momentum. In

order to restore the required rotational symmetry, we must therefore

project (14) and (>15) to good total spin. This can be done using

V1liars projection operators50'

P(JMK) « 14+1 / da DjJL(ft) R(Q) , (16)

where ft(n) is the usual rotation operator and DjL(n) the rotation Mtrix.

Actually, because of the assumed axial symmetry, for our problem the

threefold Integral of eq. (16) can always be reduced to a single inte¬

gral involving only one polar angle e.

Applying now the projection operator (16) on our intrinsic configu¬

rations (14) and (15) we obtain a set of new basis states, now having

the desired angular momentum and isospin quantum numbers. From (14) we

get

|J***MT«TZ«O> » P(JMK-0 |> (17)

and (15) yields

where ft and a indicate the time reversal partners of the states M and a,

respectively.

ITS

He shall then define the final states |I> and the doorway states

|d> of the compound system as linear combinations of the configurations

(17) and (18):

(-)J.+l) c£j |J~+HT-TZ«O>

M>F

where the t i lde indicates that no summation over the isospin quantum

numbers is being performed and the Kronecker symbol « ( ( - ) , + 1 ) in front

of the f i rs t tern is due to the tine reversal properties of ifw NT-

reference state.

Obviously, due to the angular momentum projection, the configurations

(17) and (18) do not fora an orthonoraal set. In order to obtain the con¬

figuration nixing degrees of freedom C in eq. (19), therefore, instead

of a simple diagonalization, a matrix equation for the total Hani 1 tonian

H has to be solved:

( H ^ - E 0 * 1 ) CJ*T ' 0 . (20)

Here H and N denote the energy and overlap matrix in the configura¬

tion space (17), (18).

It should be pointed out at this place, that performing the angular

momentum projection before the variation of the configuration mixing

degrees of freedon is essential out of two reasons. First, a projection

after a diagonalization of H in the intrinsic space (14), (15) would

neglect a possible angular momentiM dependence of the correlations and

hence always overestimate the rotational structure of the states under

consideration '. Second, and this is even more serious, the intrinsic

18O

configurations can be easily connected via the rotation operator. Hence, performing a diagonal1zat1on of H 1n the Intrinsic space would allow the systea to gain energy just by rotations. Such spurious admixtures due to rotations of the systea can only be amićed If the angular aoaentua pro¬ jection is performed before the variation as Indicated by eq. (20).

We would furthermore like to mention, that the spurious admixtures due to the center of mass motion, which always appear in nuclear struc¬ ture calculations If more than one major shell is involved ', can In our model at least approximately be eliminated as indicated in ref. 39.

Left to be considered in this section are the bound states of the mass A-l system. Here we shall use angular momentum projected hole states

h fP(W" a)« ol a<)^"J)* 5l>J (21) as configurations and obtain the configuration mixing again by solving a matrix equation similar to eq (20). The resulting wavefunctions for the mass A-l systems have then the force

|h> - I c£„ la"1 I"Q> . (22)

C. Continuum states

The channel states |c(hs)> are supposed to be given by coupling the partial scattering waves |s> of a continuum noc1eon to the bound states |h> of the target nucleus (or Its mirror partner). As in ref. 39 we shall treat these states closely following the prescription of Wang and Shikin ' based on the work of Auerbach et *1. 5 ZK

«aU

In this prescription one starts by defining a preliminary channel vector

|r.c(hs)) - I <IJ<Jr|MI«ieM > b V ) |(h)IHr>, (23)

where |(h)IM.> is one of the angular momentum projected deformed hole states of the previous section and b*(r) creates a nucleon with the quan¬ tum numbers s = ('tslsJs*s) łt ttłe distance r. Since this nucleon wight be In a bound orbital the channel vectors (23) are clearly not orthogonal to the bound states of the d-space and can therefore not be used as such for the definition of the P-space. We shall come back to this problem below.

The partial scattering waves |s> are now defined as solutions of some effective one body Hamiitonian n\ which is assumed to be local and furthermore diagonal in the partial wive indices s. As usual, this Hamii-tontan is assumed to be real and of Saxon-Woods form U£(r).

The radial part of the partial wave s is then given by

v f (r) « exp {*ił$(E)ł ^(r), (24) where %S(E) is the phaseshift of the indicated partial wave and v|(r) is the regular solution of the SchrSdinger equation

s(r) " ° (Z5)

with E • k /Zv (n being the reduced mass of the nucleon plus target system).

The corresponding Greens function can be written as

g*[ (r.r1) • -« v^(rt) {^(ry) t 1 v^frj} , (26)

where 8!(r) denotes the irregular solution of the differential eq. (25).

182

Using eqs. (23) and (24) we may now define a preliminary channel

state by -±

ur-2

O )*> - / dr r2 - ^ — |r,c(hs)). (27)

As already mentioned above, the space spanned by (27) is obviously

not orthogonal to the d-space. In order to ensure the required orthogo¬

nality between the P- and the d-space the channel states (27) have there¬

fore to be modified. This can be done by excluding all those single

particle states, which are used as a basis for the d-space from the

spectrum of the Saxon-Woods potential ft, a procedure, which has been

proposed by Wang and ShaMn ' and will be roughly summarized below.

Be (E+ - ?!) |t*> • 0 (28)

the integral form of the SchrSdinger eq. (25), g* the corresponding

Greens operator and |a> a bound state of some arbitrary local one body

potential, having the same angular momentum and isospin quantum numbers

as |t+>. Excluding |a> from the spectrum \t*> is then equivalent to

solving a modified Schrodinger equation

(E+-h) |E+> - 0 (29)

with

h » (l-|o><»l) ft(l-|o><a|) (30)

instead of the original one (28).

Inserting (30) in eq. (29) we obtain immediately

(E+-f() |E+> = -|a><a|f!|E+> , (31)

and, since <a|E+> s 0

1B3

|E*> - R+ \t> (32)

with

R+ , i . gV<°l (33)

<a[g \a>

similarly the Greens function g+ corresponding to the Modified Schrddinger

eq. (29) is related to the original function g* by

9+ - R+ t • (34)

As can be seen directly froM eq. (31), the exclusion of the state

|a> frM the spectruM of ft gives rise to an one body tera

(35).

describing the decay of the "bound" orbit |a> into the Modified continuua

|E+> via U. This tent, which has been neglected in the original work of

Wang and ShaMn 4 0' as well as in ref. 39, has lately been found44' to be

of considerable iMportance and will hence be taken into acount in the

following.

For |a> being a bound eigenstate of ft the operator R+ reduces to

l-|a><a| and leaves \t*> unchanged. In this case, furthermore, the one

body tera (35) is zero. The only effect of the orthogonal izati on consists

then out of an additional term -(|a><a|)/(E-Ea) substracting the bound

state contribution fro* the unmodified Greens function gf\

The extension of the above described Method to cases where More than

one state has to be reaoved is straightforward and will hence not be given

explicitly at this place.

16*

The essential approximation made in this section is the neglect of the channel to channel coupling. This approximation is supposed to be reasonable, 1f the channel states vary only smoothly with the energy, i.e., if they do not contain any single particle resonance. However, if a resonance appears, one can always extract from its wave function a quasibound state which can then be included in the d-space and projected out from the P-space using the above described method. The remaining modified continuuM will not contain this resonance any more.

In the following, we shall always use the modified wave and Green's functions defined by eqs. (32) and (34). A channel state 1s thus

|c(hs)*> - / dr rŁ -*— |r,c(hs)), (36) o

where v^(r) is related to v^(r) by eq. (32) and may have been modified by possibly appearing resonances.

D. Evaluation of the T-matrix

Having defined all the states involved we can now proceed to calculate the different terms of the T-matrix (9). For this purpose we shall first express the electromagnetic interaction H by its multipole components taken, as usual, in the long wave length limit. We shall then restrict ourselves to the consideration of only the electric multipoles Q of this expansion.

The reduced matrix element of the Q -part of H leading from an entrance channel |c+> of the form (36) to some final state |I> of the compound nucleus is then given by

<l!lQL||c(hs)*>» e i6stE) / dr r2 f j ; c ( r ) ^ (37)

where f\;cM

is the form factor

{ ; c (L) Rb(r) rL (38)

and ftj. (L) collects all the terms coning from angular momentum and isospin coupling and from the angular momentum projection. The sum runs over all single particle states of the d-space basis. Note, that the use of the modified scattering waves (32) in «$. (37) already ensures the orthogo¬ nality of the continuum with respect to the bound states and that hence in the definition (38) the original channel vectors of eq. (23) can be used. Thouąh lengthy due to the angular momentum projection operators, the cal¬ culation of the fiT. (L) is straightforward and can be handled using similar techniques as for the bound states in section I IB.

In a similar way as the direct term (37) also the transmission coefficients representing the coupling of the P to the d-spare via the nuclear interaction can be calculated. One obtains

<d|Hdp|c(hs)+> = e<6s<E> J dr rZ Frf;c(r) M 2 . . (39)

In case of a 6-interaction the form factor can be written as a sum of one and two body terms:

Fd.e(r) = <d|Hdp|r,c(hs)>

-- I xJ|C y r ) fts(r) (40) + . 1 *S5 V r ) V ' Re(r> •

Again, the sums run over all single particle states of the d-space

186

basis and all r-independent terms are collected in v| „ and A? „. Note, U,C U,C

that the one body tern of eq. (40) involving the Woods-Saxon Hamiltonian

frs(r) 1s due to the orthogonalization procedure and describes the possible

decay of the d-space states into the continue* via an one body potential.

These one body decay terms had been neglected in refs. 39 and 40.

Left to be considered is the nondiagonal part of the width and shift

matrix H of eq. (11) and the final state interaction term F of eq. (10).

Using the above results one obtains

(41)

and

I / dr r fi;c(r) / dr' r< 9[ ("V.r f) Fc.d(r')

where the sum runs over all channels |c(hs)> included in the calculation.

Note, that due to the orthogonalization procedure 9^ has not the usual

Saxon-Woods form (26) but is the modified Greens function of eq. (34).

Having calculated the T-matrix of eq (9), the cross sections and

angular distributions of the considered radiative capture reaction can

be easily obtained. An explicit formula for the differential cross section

has been given in ref. 39 and will hence not be repeated here.

187

H I . APPLICATION TO THE 19F(p,T)20He REACTION1

The above described model has now been used for the investigation of

the Multipole resonances of Ne as seen via the F(P,T) He proton radi¬

ative capture reaction. For this purpose according to section IIB first a

couple of calculations for the bound states of the target and compound

systems had to be performed. The model space for these calculations was

defined by a spherical single particle basis including all the Op-, lsOd-,

and Of-osciliator orbits (oscillator length b « 1.79 fR). As in ref. 39,

the lp-orbits have not been included in this space, since the p-waves of the

Woods-Saxon potential used later on to define the continuum states did not

show any resonance behaviour which would have permitted the extraction of

"quasibound" lp-orbits according to section I1C.

Additionally to the assumptions of section IIB we have assumed that

there is no orbital mixing between different major shells in the Hartree-

Fock expansion (13). This assumption allows to replace the one body

kinetic energy terms of the Hamiltonian (12) by spherical single particle

energies relative to 0. For the two body part of (12) then, as in ref.

39), the modified surface delta interaction53' (MS01) has been used.

As far as only the lsOd-orbits are involved, the parameters of the

MSDt (ATa[0 - .77 HeV, A T e l • .95 MeV. B T i 0 * -2.51 MeV and B U I • .37 MeV)

and the appropriate single particle energies U^, * -4.49 MeV, c. od5/2 i%m -3.16 MeV and e . • 1.05 MeV) have been taken without modification from

3/2 the shell model work of Halbert et ai.54'. It should be pointed dut that

th's is an effective Hamiltonian, specifically designed for calculations

inside an lsOd-shell basis. There is no reason why the same force should

be able to describe also the interactions between nucleons of different

188

major shells. Nevertheless, in ref. 39 the above set of MSDI-parameters

had been used for all the two body matrix elements needed for the bound

state calculations.

Therefore, in order to reproduce the l/2+-l/2~ energy spacing of the

A = 19 (T=l/2) system, we had to use in ref. 39 single particle energies

for the Op-orbits which were about 3 MeV higher than the experimental ones.

In the present work we have now kept the experimental values (e. Op3/2 =

21.74 MeV and c. = -15.60 MeV) and have performed the necessary renor-Opl/2

malization of the Hamiltonian by changing the monopole terms 8 T of the

Op-lsOd MSDI into BTxQ{p,sd) * -1.475 MeV and BTi!l(p,sd) « 1.075 MeV.

Obviously similar renormalizations are to be expected also for the

lsOd-Of and all the other major shell crossing matrix elements involved

in the calculations. However, while for the renormalization of the Op-lsOd

interaction the A*19 parity doublet can be used, there is no such experi¬

mental information available which would restrict the range of possible

renormalizations for the other shell crossing matrix elements. Hence, ex¬

cept for the above given modification of the Op-lsOd force, also in the

present work all needed two body matrix elements have been calculated using

the sd-shell parameters of the MSDI.

Left to be considered are the single particle energies of the Of-

orbits. Since the results of ref. 39 had supported the lower one of the

two sets considered there, at least if used together with the given MSOI,

but since they had also given some hints that probably slightly larger

values should have been taken, in the present work the values

8.5 MeV and cQf =12.5 MeV have been adapted. '7/2

'5/2

Using the above defined Hamiltonian H j now a Hartree-Fock (HF) cal-

189

culation for Z0Ne has been performed. The resulting selfconsistent de¬

formed single particle energies are given in Fig. l a ) . Because of the

above definition of H ^ , ths positive pjr i ty states are identical to

those obtained in ref. 39. Nearly identical results have also been ob¬

tained for the negative parity states of the Op-shell, since for the*

the use of different single particle energies has been compensated by

a renormalization of the two body matrix elements in the above described

way. On the other hand the negative parity states of the Of-shell have

here been calculated using the sane two body but a different one body

tern for the Hamiltonian and hence differ fro* the results of ref. 39.

I t should be pointed out, that because of our approximations in section

1 IB each of the seifconsistent orbits 1s fourfold degenerate. Note further¬

more, that although no direct nixing between the Op- and the Of-orbits in

the expansion (13) has been allowed, nevertheless both shells are pola¬

rized due to their interactions with the nucleons inside the lsOd-sheli.

The HF orbits (13) have now been taken to construct the HF-vacuum

(14) as well as the lplh-configurations (15) and the one hole-excitations

with respect to i t . Coupling these intrinsic configurations to good total

isospin T and applying Vi liars projection operator (16) then the angular

momentum projected configurations (17), (18) and (21) have been obtained

which build up the configuration spaces for the compound and the target

system respectively.

For the isovector (T«l) negative parity states of Tłe as 1n ref. 1

al l the lplh-excitations of the form (18) leading either from the Op-shell

into the empty lsOd-orbits or from the occupied part of the lsOd- into the

Of-shell have been included. However, opposite to ref. 39, here no restriction

on the K-quantum numbers of the intrinsic configurations has been made. As a

»9O

a) b) cl

'"017/2 " vV— **~ \ \ 6 57 7/2-

V33E »2-u Via£

tnlj) Sph Edef Ik")

0 l 5 / 2 J2I0

It 051 \ N e

(398) 125

52 3 516 498

- I I I

PHF PTD EXP J"

r

«

# ^

I t

>2_ it

57"

A*I9 T«1/2

1

ł ! 177

3/2'

s/r 3«"

1265.

EXP

1608 .1575 „ ; •

EXP THEOR J"

Figure 1: Groumhtate proptrt.es of ?0N» anil of the * - 19. T - 1/? tyt are displayed, fart a) of ihe figure sho« th* om» body tfrm

*?nh ° ' I f t* "^^^ b o d y i n tP r»ction Hdd and the selftanti itent of^rd single particle energy spectrum r fcsuiting frem * tr-pc-FnO cAtculrition for N»> u*inq thi% Hamt I tonidn. At I • pn.'rfjtt?s »vr- qivpn m HPV Part b) prcsent^ the *)rpun<lit»te »rJ of NP Th > throrftical pnrrijy values obtained by prc-ting drtfjaK»r (twimpntym front only the Kirt rpp«f oc !• rpfprpncP

• pr*'ni n.in ( I ('i*r .«ri/ ^ onprtrrft wi t h t ^ '•v p rp^g It )nn f ron thp

(PTO) and »ith the enpertmentai d i u " 1 (tXP). Again a l l the energies are given in HeV. Furthermore this part of the figure thews the dynamical quadrupole moments O j? of the ground-state Mnd as obtained fro* the calculated and measured R(E?)* inns i t ions via eq. ( * * ) . These moments are given in units of efnJ P«rt c) cf the figure shows f inal ly the low energy spectra of t-p A • 19, T • 1/7 system. The theoretical energies obtained frcn the annular momentum projected deformed hole states of eq. ill) are compared »ith eiperimcnt ' . Again a l l the energies are given in units of HeV.

consequence all the M-values fron 0 up to ±20 do now contribute to the

dipole doorway states {19).

Except for the Ne groundstate band, which had been approximated

by simple angular momentum projected HF-states (17), positive parity

states of the compound system had not been considered in ref. 39. In the

present work now not only the goundstate band but also the higher exci¬

ted positive parity states needed as doorways for the quadrupole part

of the T-matrix (9) have been calculated. For this purpose besides the

reference state (17) all the lplh-excitations (18) inside the lsOd-shell

as well as leading from the Op- into the Of-shell coupled to total iso-

spin T>0 and T«l have been taken into account. As for the negative pari¬

ty states also here no restriction on AK has been imposed.

Finally, for the mass A-l (T=l/2) system all possible hole excitations

of the type (21) have been included.

The dimensions of the such defined configuration spaces are listed

in Table 1. Note, that the projected HF-vacuum because of its special sym¬

metry does contribute only to the isoscalar (T=0) positive parity states with

even angular momenta.

The above set up for the Hamiltonian H^ causes a problem concerning 20 the calculation of the positive parity states of the Ne compound system.

Being an iffec*ive sd-shell force, the chosen MSOI is supposed to fit the

low lying positive parity states of 20Ne inside an lsOd-shell basis and

hence already includes the net effect of the 2 hu core excitations on the

low energy spectrum. On the other hand we have now taken into account such

core excitations explicitly. Allowing for some residual interaction between

them and the configurations Inside the lsOd-shell, one would therefore

198

J"

Dim

J"

01m

j "

Dim

J"

Dim

A - 19 T = 1/2 T z ' ±1/2

1/2"

2

3/2"

3

5/2*

3

l/2+

1

3/2+

1

5/2+

1

A = 20 T = 1 Tz = 0 ir = - 1

r

23

2"

35

3"

42

-

-

-

-

-

-

A - 20 T « 0 T 2 * 0 i t * + l

0+

9

1+

24

2+

39

3+

47

4+

52

-

-

A * 20 T * 1 1Z * 0 ir ' +1

0+

8

1+

24

2*

38

3+

47

4+

51

-

-

Table 1: The dimensions of the different subspaces of the d-space used in our calculations are presented. In case of the A = 19 T = 1/2 system ( F, He) this dimension is given by the number of "hole"-configurations (eq. (21)). In case of the A = 20 T = 0 system 20 { Ne) the dimension equals the number of "particle-hole"-confi-gurations of eq. (18). In case of positive parity states with isospin T * 0 and even angular momenta the Hartree-Fock reference state (eq. (17)) is included.

l»

doublecount their influence on the )ow excited states. In order to avoid

this doublecounting we have assumed in the present work that there

is no direct coupling between the two different types of excitations via

the energy matrix -,n eq. (20), i.e., we have considered also the Op-Of

interaction as an effective force already including the net effect of the

particles inside the sd-shell on the 2 hu core excitations. Note, that

nevertheless the 2 hu excitations are not independent from the nucleons

inside the lsOd-shell since the latter do polarize both the Op- as well

as the Of-orbits via the average HF-field.

The results of the bound state calculations are summarized in Fig.

lb) to 4. Part b) of Fig. 1 presents the groundstate band of Ne the

two lowest states of which have later on been used as final states for

the radiative capture process under consideration. The theoretical energy

values obtained by projecting angular momentum from only the HF-vacuum

according to eq. (19) (PHF) are compared with those resulting from the

multideterminantal wave functions (19) (PTO) and with the experimental

data (EXP) '. As can be seen, the groundstate correlations due to the

mixing of the PHF-states (17) with the lplh-excitations (18), which had

been neglected in ref. 39, cause an energy gain of about 350 keV, which

is a considerable but not too drastic improvement with respect to the un-

correlated PHF-spectrum. A slight improvement due to the groundstate

correlations is also obtained for the "dynamic" quadrupole moments Q???

which are related to the calculated and measured B(E«.,-va1ues by the

definition

B(E2;I-r)]1/Z <I2I1)000>"1 . (44)

Note, that because we did not allow for a direct mixing between the 2 hu>-

and the 0 .hm-excitations, for the calculation of the dynamic quadrupole

moments of the ground state band an effective extra charge 6 for both

protons and neutrons has been introduced. As usual in sd-sheil calcula¬

tions ' a value of 6 « .5e has been adapted.

Fig. lc) lists the low energy spectrum of the A*19 (T*l/2) system

as obtained from the angular momentum projected deformed hole wave

functions (22) and compares it with the experimental spectra of the two

mirror nuclei F and Me '. Because of the similar choice of H^ the

theoretical spectrum is almost identical with that of ref. 39. Both, the

positive and the negative parity band of the A*19 system are reasonably

well reproduced. The same holds for the Ne- Ne mass difference. All

the six states shown in the Fig. have later on been used for the defini¬

tion of the channel states needed in the continuum calculations. Actually 19 19 there are 12 macrochannels ]h> because both the Ne and the F spectra

have been used, the latter being obtained by substracting the experimental 19 19 F- Ne mass difference from the theoretical spectrum.

Fig. 2 shows the energies of the calculated isovector J" * 1* and the

isoscalar and isovector j" * 2 + states and the reduced B(Ex)-va1ues for

their transitions to the 0 + ground state of Ne. As usual for the calcu¬

lation of the B(El)-va1ues effective charges of 1/2 e for the protons and

-1/2 e for the neutrons have been used. On the other hand the 2fi« E2-transi-

tions listed here have been calculated without an effective charge. Ob¬

viously because of our neglection of the direct 2ftu-0n*> mixing one could

think of introducing an effective extra charge 8 not only for the sd- but

also for the Op-0f-transitions. We shall discuss the possible influence

of such an effective charge on the final capture cross sections below. Note

for the moment, that the choice of for example s • .5e would increase the

B(E2)-va1ues in Figs. 2 to 4 by a factor 4 but leave the isovector transi¬

tions unchanged.

195

- T I f I " I I I I I I I I | I I I I

Reduced BlED-value ją [mbl

T=1 1.0

0.5

|f r T i i i i i i i i i T

Reduced B(E21 -value IP (mbJŁ

Full lines T=0 — Dashed lines: T=1

V... i i :, TI i i r i i

15 20 25 X

AE [MeV] 35

I 40

Figure 2: Excitation energies of various Q-space states |d> and the reduced 8(E»)-va1ues for their v-tr»ns1tfons to the 0* groundstate of Z0He are presented. The upper part of the figure lists the B(E1)-values for the groundstate transitions of the negative parity states Mith isospin T • 1 and angular noMentua J • 1. tn the lower part of the figure the B(E2)-va1ues for the groundstate transitions of the positive parity states with angular amentia J « 2 are shown. Full lines represent transitions fro* states with isospin T » 0, dashed lines transitions froa states with T • 1. All B(El)-values are given in units of «b. theB(E2)'s in 10[«b]2.

196

As expected because of the similar choice of the Hamiltonian H.d the

results for the isovector J n= l" states show the same qualitative features

as those obtained in ref. 39. However, considerable quantitative changes

are detected. As far as the high energy part of the spectrum is consi¬

dered, these differences are mainly due to the different Of single part¬

icle energies used in the present work, while for the somewhat smaller

changes of the low energy spectrum the slightly different choice of tha

Op-lsOd Hamiltonian as well as the groundstate correlations and the ur.-

restricted aK-mixing are repsonsible. Nevertheless the wave functions

of the individual states are rather similar as in ref. 39 and hence as

we shall see below the resulting dipole part of the F(p,Y ) Ne cross

section looks not much different from that obtained in ref. 39.

As can be seen furthermore from Fig. 2 we find a couple of isoscalar

J" = 2 + states with strong B(E2)-transitions to the Z0He groundstate widely

spread over about the same energy region as the J" = 1 dipole states.

The isovector quadrupole states are found to appear at energies which are

about 10 to 12 MeV higher than those of the isoscalar quadrupole resonances,

and are spread over an even larger energy interval than the latter ones.

Figs. 3 and 4 present the El and E2 transitions to the first excited

2 + state of Ne. Here now dipole transitions from the j" = l" as well as

from 2" and 3" states and quadrupole transitions from the j"1 = 2* as well as

from 0+, 1+, 3+ and 4 + states are possible. As far as the positive parity

states are concerned all the spectra show similar features as that for

the 2 + ground state transitions discussed above. Always both the isoscalar

as well as the isovector resonances are spread over a wide energy range

being separated from each other by about 10-12 MeV. For the dipole transi¬

tions to the first excited state of Ne again the same qualitative feature

as in ref. 39 are obtained although as for the l" groundstate transitions

197

1 1 I 1 1 1 1 1

\

u Xs - I

i

• 1 i i i

1

T

f' T 1 i f

i i i i i i i

7

j t 1 i 1 1 I '

i i

i i i

i i i i i i T T i i i i i . . | _

Reduced 8(ED-value S

Dotted : J r = r Dashed J"=2" Full :Ja=3"

| 1 , , 1 . , , . . • i • i |

Reduced B(E2|-value l o ( m b J l _

< i" - n* 2

1

1

Full T = 0 Dashed:T=1

T i

t

r 1 1 i i i i i i i 1 1 1 1 I t

i

J ni 1 1 1 1 1

Reduced B|E2) -value

1 Full.T=0 Dashed :T=1

I , , 1 ! i i i 1 1 1 1 |

15

Fiqurt 3:

20 25 30

-AElMeVl-35 (JO

Encltttton tntrgiu [ivV] of Mrłout 0-ip»ce t t t t e i |4> and tnt reduced B(E>)-va1uet for their transitions to the f i rs t melted J ' . 2* (T . 0) state of ? V ire plotted. As (noun In the upper part of the figure, in this case negative parity states with isospln T • 1 and angular Kmenu J • 1 (dotted lines), J • 2 (dashed lines) and J • 3 ( ful l lines) contribute to the dipole part of the electromagnetic interaction. I (E2)-transitlons are aliooeif f ro* stttes with isotpłn T • 0 or T • 1 and angular "omfnn between 0 and 4. In this figure "» give the results of our calculations for the J* • 0* states (•iOdle of the figure) aid the J* • 1* states (lover part of the figure). As In f ig . 2 full Hues represent tht transitions fro* states with itospfn T • 0, dashed l i n n those fro* states •Uh T • 1.

198

' I ' 1 " I I I i I I I i r

J I

Reduced B(E2) -value lolmbl9-

/ = 2 +

Full:T=O

T Dashed:T=1

T • '

T

T T I 1 !

j_L i T r i i i i i r ( i i i i r i i \

Reduced B(E2)-value

T •

T « I

/ =3 +

Full T=0 Dashed:T=1

I 1 « I I t IJ ! J ~\ T I j T I I I I T T T f 1

Reduced B (E2)-value _ i i i i r

«V2 I Full:T=O Dashed:T=1

JhrłJ 1A I I I I

15 20 25 30

AEtMeVl— 35

Figure 4: Fig. 3 is continued with a listing of the B(E2)-transitions from states with JT « 2 + {upper part), JT • 3* (middle) and J" • 4* (lower part) to the 2* member of the 20Ne groundstate band. Again full lines represent transitions with AT • 0, dashed lines those with AT * 1.

considerable quantitative changes with respect to our former results are

detected.

It should be pointed out that all the states of Figs. Z to 4, except

perhaps the low lying negative parity ones which have been obtained with

a renormalized force, could obviously be shifted by may be a few HeV,

since our above discussed lack of knowledge of the effective Hamiitonian

actually to be used for our problem does not allow more than qualitative

statements on their positions and B(Ex)-transitions. However, just these

qualitative features of the spectra should not change very drastically if

the necessary but unfortunately unknown renormaiizations of the effective

d-space Han11ton1an would be performed.

Left to be considered are now the continuum wave functions and their

coupling to the bound states according to the sections IIC and 1ID, re¬

spectively. For the definition of the partial waves of the continuum nu¬

ci eons we have chosen a real Woods-Saxon potential the parameters of which

have been taken as In ref. 39 without modification from the work of Afnan '.

They are UQ « -50.5 MeV, U l s * -6.85 MeV, aQ * .7 fm and R • Rc<wl • Rts *

1.25 x (19) ' fm. The partial wave expansion has been restricted to the

s, p, d and f waves of this potential since only these will be relevant for

the capture process under consideration. The orthogonality of the continuum

states with respect to the bound states of the d-space has been ensured

by the orthogonalization procedure described in section IIC, and the

channel states (36) have then been constructed by coupling the modified

scattering waves (32) of the continuum proton or neutron to the low lying

states of "F or 9Ne respectively. Incident proton energies from 2 up to

26.6 MeV have been considered. The energy interval from 2 to 11 MeV has

been discretized in steps of 75 keV while for the higher energies up to

200

26.6 MeV a steprise of 150 keV has been used. For the two body part of

the continuum-bound state interaction Hp. as in ref- 39 a delta force

with the parameters given by Wang and Shakin ' has been taken. These are

VQ = 613 MeV • fm , a = .865 and b = .135. However, opposite to our pre¬

vious work, we have now included also the one body terms of H.^ resulting

from the orthogonalization procedure of section IIC. Note, that since the

bound states of the chosen Woods-Saxon potential are almost identical to

the corresponding oscillator states of the d-space basis, non negligible

one body terms are only to be expected for the d.^-proton and the f ,^-

and fg/g Proton and neutron orbits. Finally, as already mentioned in

section 1IA the doorway-doorway coupling via the complicated x-space has

been approximated irrespectively of the parity and the isospin of the door¬

way states by an energy independent constant shift s^ » 0 and width rx =

100 keV, the latter being about 30 % smaller than the value used in our

former calculations '.

With the above parameters now the total cross sections, the 90°-

yield curves and the angular distributions for both the 19F(p,Yo)20Ne(0+)

19 20 * + and the F(p,Yj) Ne (2 ) capture reactions have been calculated. In all

these calculations the final state interaction term (FSI) of eq. (10) has

been taken into account. Due to the inclusion of the one body terms of

Hpj this term has slightly increased with respect to our former result

but is still found to be negligible. Note, that the magnitude of the FSI

as well as the size of the direct part of the T-matrix (9) depend strongly

on the actual cho1' of the continuum and the doorway space. As less

complete with respect to the electromagnetic interaction H the chosen

doorway state is, as more important become both the FSI as well as the

direct term. As in ref. 39 also in the present work we have examined the

201

validity of the isolated resonance approximation (IRA), in which the

width and shift matrix M of eq. ( H ) is supposed to be diagonal in the

doorway space as, compared to the full calculation, in which H is in¬

verted for all the considered energies (MIA). The results of these

calculations are summarized in Figs. 5 - 1 4 and in Table Z.

Fig. 5 displays the 90°-yield curves for the 19F{p,To)20Ne(0'f)

radiative capture reaction with proton energies between Z MeV and 13

MeV. The theoretical results obtained using the isolated resonance approxi¬

mation (IRA) as well as by inverting the width and shift matrix are com¬

pared with each other and with the experimental data measured by the

Argonne group '. As can be seen both theoretical approaches yield almost

identical results for incident energies up to about 5.5 MeV. They repro¬

duce the strong experimental peak at 5.15 MeV rather well but fail to

give the right energy spacing between the three peaks below. This defi¬

ciency could most probably be overcome by an additional renormalization

of the effective Op-lsOd interaction. Above 5.5 MeV the agreement becomes

worse. Here the IRA-results show only one big bump spread over an energy

range of about 2 MeV. The inversion of the width and shift matrix splits

this bump into three separate peaks such improving the agreement with

experiment which nevertheless can only be called qualitative for this

part of the spectrum. Compared to our previous results there is now less

strength concentrated in the region above 5.5 MeV. This is caused by the

different choice of the Of-single particle energies as well as by the

one body terms of the continuum-bound state interaction H p. involving

the Of-orbits. It should be pointed out, that the quadrupole contributions

to the yield curves of Fig. 5 are always smaller than the thickness of

the lines and hence almost negligible. That the E2-part of the electro-

202

40 19 Hp.Yo) Ne(g.s)

: IRA*FSI

: MIA+FSI

(E1(T=1»-E2(T=0)-E2lT=1)|

EXP

31 41 51 61 71 81 91 101 111 121 13

Ep [MeVl

Figure 5: 90°-yield-curves for the I9F(p,Y )2 0 Ne (groundstate) radiative

capture reaction with proton energies between 2 MeV and 13 MeV

are displayed. The experimental data of the Argonne group '

(open dots) ire compared with the results of our calculations.

The full curve has been obtained using the isolated resonance

approximation (IRA) for the propagator of the r-natrf« (eq. (llj).

For the calculation of the dashed curve these propagator has

been inverted (MIA). In both the calculations the final state

interaction term (FSI) of eq. (10) has been included and dipole

as well *s quadrupole transitions have been considered.

2O3

magnetic interaction contributes at all can clearly be seen from the

non vanishing a,, a, and a^ coefficients of the angular distribution

(43) which are shown together with the &2 coefficient in Fig. 6. Since

the angular distributions are very sensitive to fine detail :c *.h<»

underlying nuclear structure, the agreement with the esperM-T./Ml '•'

is here obviously much worse than for the yield curves, rieverthci;~ ,

the general trends of the angular distributions are not too badly repro¬

duced at least if one takes into account that the present calculations

are purely microscopic ones. As already mentioned, a more quantitative

description of the fine structure can only be expected if some more

information about the effective Hamiltonian H.^ could be obtained.

For energies above 13 MeV only the isolated resonance approxima¬

tion has been used. Unfortunately in this energy region no experimental

data are available. Therefore the Figs. 7 and 8 show only theoretical

results. Fig. 7 compares the 90°-yield curves for the F(p,r0) reaction

with protons between 13 MeV and 26.6 MeV as obtained from a pure dipole

approximation with the results of the calculations including the quadru-

pole part of the electromagnetic interaction. As can be seen, the E2-

contributions are still small but tend to increase now with the incident

proton energy. This is also seen from the corresponding angular distri¬

butions given in Fig. 8.

Fig. 9 presents the total E2-cross section of the r.-channel for the

whole energy interval from 2 MeV to 26.6 MeV. Below 12 MeV this cross

section is always smaller than about one microbarn, although just in this

energy region a couple of isoscalar J* = 2* states with strong B{E2)-

transitions to the groundstate of Ne is available as seen in Fig. 2.

20*

_0 o

-1

o

-1

1 0

-1

• •

;: SK ; :

V

Figure 6:

3l Al 51 61 71 81 91 101 111 121 13

E p [MeV] The angular distribution of the 19F(p,t0)

ZONe(g.s.) radiative capture reaction with proton energies between 2 MeV and 13 HeV are presented. We g've the a, coefficients of eq. (43). As In fig. 5 the open dots represent the experimental data6' while the full and dashed curves show the results of the isolated resonance (IM) and the matrix Inversion (MIA) approximations respectively.

205

o O Ol

•urn

1V(p.Yo)20Ne(g.s.)

IRA+FSI E1(T=1)-E2(T=0)-E2(T = 1}

— :IRA*FSI PURE

13 ' 115 ' 117 123' 125

EplMeVl Figure 7: Fig. 5 is continued for proton energies between 13 MeV and

27 MeV. For these energies experimental data are not known. For the dashed curve only the dipole part of the electromagne¬ tic Interaction has been included while the full curve takes Into account also the quadrupole transitions. Both calcula¬ tions have been performed using the isolated resonance approxi¬ mation (IRA) ana include the final state interaction tera (FSC).

206

Above 12 MeV the total E2-cross section then continuously increases

showing some structure in the region of 17 to 19 MeV which is due to

the isovector resonances shown in Fig. 2 and reaches finally its maxi¬

mum of about 6 microbarns at the end of the considered energy interval.

Since the contribution of the quadrupole operator to the yield curves

may be roughly estimated dividing the total E2 cross section by a factor

4-n it becomes obvious why we did not see them in Fig. 5. Note, that even

the above mentioned possible introduction of an effective extra charge

B = .5e for both protons and neutrons would not change this feature.

Leaving the isovector part of the spectrum unchanged such a choice of

B would indeed increase the low ene-gy part of the E2-corss section by

about a factor 4, but even then the qi/adrupole contributions to the

yield curves of Fig. 5 would still be smaller than half a microbarn.

This result resoves at least qualitatively the puzzle why inelastic •.-

scattering experiments on certain light nuclei do see a very different

E2-strength distribution than the corresponding (p,v) radiative capture

reactions. Obviously the (a,a') experiments do excite all states with a

large transition density. Hence for example all the isoscalar j" = 2*

states with strong B(E2)-transitions will be excited whatever their indi¬

vidual structure may be. On the other hand the (p,>) reactions are muc*)

more selective on the individual structure of the states involved, for the

19 +

special case of F because of the target spin 1/2 only d-waves cf t*ie

incoming proton can couple with the target in order to yield J' = ?

states. Our numerical results show now, that such configurations are only

weakly coupled to the (p pf) structure of the 2 hw excitations at least

for the continuum-bound state interaction H p. used in the present work.

Hence at least that part of the isoscalar quadrupole resonances of Ne

207

-1

-1 13 H5 ' 117 ' 119 ' 121 ' I23' I25

206

EplMeV] figure 8: The angular distribution of the l9F(p,To)?0«e(g.s.) reaction with

protons between 13 MeV and 27 HeV energy are plotted. As in Fig. 7 the dashed curves result from a pure El-calculation while the full curves take into account also the contributions fro* the quadrupole transitions. Again the isolated resonance appronł-Mtion (IWtFSI) has been used.

' /

LU

19F(p.Yo)20Ne(g.s.) Total E2-cross-section

:IRA*FSI :MIA+FSI

125

19 20 Figure 9: The quadrupole part of the total F(p,t ) Ne(g.s.) cross section

for proton energies between 2 MeV and 27 MeV is presented. The

full curve reprints again the results of the iRA+FSI-approximation

while for the calculation of the dashed curve the full matrix in¬

version (M1A+FSI) has been performed.

having predominantly (p pf) structure is only weakly excited in the

capture reaction.

The results for the 19F(p,Y1)20Ne*(Z+) reaction leading to the 2 +

number of the Ne groundstate band are shown in Figs. 10 - 14. Since they

show very similar features as those for the y channel we will not discuss

them in detail but sketch only the essential points. As can be seen from

Figs. 11 - 13 the quadrupole contributions to the y, channel are even

smaller than those to the Y 0 channel even though for the transitions to

the 2 + state not only the 2 + but also all the 0+, I4 and 3+ states of

Figs. 3 and 1 have been included. As the almost vanishing a^, a3 and afl

coefficients of the angular distributions [Fig. 11) show, this result

agrees very well with the experimental data '. As expected because of the

higher density of the 2" and 3" states with respect to the 1" ones, here

the isolated resonance approximation is much less justified than for the

y -channel. As can be seen in Fig. 10 the inversion of the width and shift

matrix yields a drastic redistribution of the strength with respect to the

results obtained using the IRA. The quantitative agreement with the ex¬

perimental data ' is for both the yield curves as well as the angular

distributions of the (p,Yi) reaction worse than for the y -channel. This

is to be expected because of the more complicated underlying structures.

Nevertheless it should be pointed out, that the essential qualitative

features of the Yj-channel, as for example the with respect to the > -

channel completely different angular distribution are reasonably well

reproduced in our calculations. A more quantitative agreement would as

already mentioned above require a better knowledge of the effective

Hamiltonian actually to be used for our problem.

Finally, in Table 2, we give the integrated total (Y, P 0 ) cross sections

210

o O cn

c;

1 9 F ( P > Y 1 ) 2 0 N G * ( 2 + ) • IRA+FSI

: MIA+FSI (E1(T=1}-E2(T=O)-E2(T=1))

EXP

5l 6| 7l 8\ 9l 101 Ep[MeV]

Figure 10: Same as in Fig. 5 but for the 19F(p,y,)Z0Ne*(2+) radiative capture reaction with the 2 + member of the Ne groundstate band as final

state.

2 1 1

+1

0 0

-1 +1

o|o

-1 +1

0 0

-1 +1

0 0

-1

1

Am

V"

31 U\ 5l

t • V

T

• •

•

6l

°2

IńAf-S A^ \ ^Nj \ if'*'*"'

°3

7l 8l 9l 10l 111

•

• • • • •

:;

• •

ft/

12l 13

—

Ep I MeVl-Figure 11: Same as in Fig. 6 but for the ^(p.yj^Ne*^*) reaction.

212

JD

OK • o r

: IRA'FSI Et(T=»-e(T=O-E2fr=1l

- - IRA'FSI PURE El (T=T)

h5 ' H7 ' 119 ' fei r 123' t2S'

Ep IMeVl

o,

n re1 h7 ' h9' &T

EplMeVl-

Figure 12: Same as in Fig. 7 but for trie Figure 13: Same as in Fig. 8 but for the

19>r(P.Y1)20Ne*(2+) react ion. l 9F(p,Yj)2°Ne*(2+) reaction.

LU

I

19F{p.Y,(™Ne"(2*) Total E2-cross-section

IRA«FSI MIA'FSI

15 ' I? ' V ' I n ' k3 ' IB

Figure 14: Same as in Fig. 9 but for the 19F(p,r.)20Ne*(2+) reaction.

213

T a b l e 2 : T h e t h e o r e t i c a l a n d e x p e r i m e n t a l v a l u e s f o r t h e i n t e g r a t e d t o t a l

( f i P ) c r o s s s e c t i o n s a r e g i v e n . T h e l o w e r b o u n d o f t h e i n t e g r a l

w a s g i v e n b y t h e y - e n e r g y c o r r e s p o n d i n g t o a n i n c i d e n t p r o t o n

e n e r g y o f 2 M e V f o r a l l t h e t h e o r e t i c a l c a l c u l a t i o n s , w h i l e t h e

e x p e r i m e n t a l i n t e g r a l s a r e b a s e d o n a p r o t o n e n e r g y l a r g e r t h a n

2 . 8 8 M e V f o r t h e Y a n d l a r g e r t h a n 4 . 1 M e V f o r t h e Y j - c r o s s -

s e c t i o n . T h e u p p e r l i m i t s o f t h e t h e o r e t i c a l i n t e g r a l s w e r e g i v e n

b y proton e n e r g i e s o f 1 0 . 9 2 M e V ( a ) o r 2 6 . 6 0 M e V ( b ) . T h e e x p e r i ¬

m e n t a l i n t e g r a l g o e s u p t o a p r o t o n e n e r g y o f 1 2 . 8 8 M e V ( c ) . T h e

e x p e r i m e n t a l d a t a h a v e b e e n t a k e n f r o m t h e A r g o n n e g r o u p '.

Method

IRA

+FSI

MIA

+FSI

EXP

0+

(Y )

2+

(V

( Y . )

0+

<v 2+

4+

(v2)

0+

(v0)

2+

Jf

r

r. 2+

r, 2", 3"

1 " , 2 " , 3 " . 0 + , 1 + , 2 + , 3 +

3"

3 " , 2 + , 3 +

r r, 2+

r, 2", 3"

1 " , 2 " , 3 " , 0 + , 1 + , 2 + . 3 +

3"

3 " , 2 + , 3 +

all

all

E » - p o l e s

El

E l + E 2

El

E l + E 2

El

E l + E 2

El

E l + E 2

El

E l + E 2

El

E l + E 2

all

all

U p p e r L i m .

(a) (b)

(a)

(b)

(a)

(b)

(a)

(b) j

(a)

(b)

(a) (b)

(a)

(a)

(a)

(a)

(a)

(a)

(c)

1 8 . 5

2 4 . 6

1 8 . 6

2 7 . 5

7.2

1 0 . 5

7.2

1 0 . 6

.6

.9

.6

.9

1 8 . 5

1 8 . 7

4.7

4.7

.5

.5

2 4 . 8

9.6

21<t

as obtained from our calculations and compare thera with the experimental

data '. While for the Y0"Channel the agreement is not too bad, for the

Yj-channel only half of the measured strength is obtained. That means,

that both the positions as well as the B(Ex)-transitions are still not

sufficiently well described, and points again on the improper choice of

the effective Hamilton!an.

IV. CONCLUSION

About a year ago we had proposed ' a microscopic model for the de¬

scription of the giant roultipole resonances (GMR) in i^ght deformed nuclei

and their excitation via proton radiative capture reactions. This model

had then been applied to the giant dipole part (GOR) of the GMR in tie

as seen via the F(p,-y) Ne reaction. The results had been in surprisingly

good agreement with the experimental data. However, besides being restricted

to the GDR, these calculations had made use of a couple of approximations

which were not completely satisfying.

In the present work most of these approximations have been removed.

So, the groundstate correlations due to the mixing of the angular momentum

projected Hartree-Fock vacuum and the projected particle-hole configurations

have been included and no more restriction on the K-quantum numbers of the

intrinsic configurations has been made. Furthermore the one body decay

terms ' due to the orthogonalization of the continuum with respect to

the bound states have been taken into account and additionally to the GOR

also the giant quadrupole part (GQR) of the GMR has been studied.

215

All these improvements do have considerable effects on the quanti¬

tative results. However, the qualitative features of both the cross

sections as well as the angular distributions obtained in ref. 39 re¬

mained unchanged.

The quantitative discrepancies between our results and the experi¬

mental data ' can not be explained by these approximations. There are

strong hints that they are mainly due to an improper choice of the effec¬

tive nuclear interaction. Further improvements can hence only be expected

if some more reliable information about this interaction can be obtained.

Considering our results for the GQR in Ne, it is seen, that,

although a couple of isospin zero states with large B(E2) values is

available in the region between 17 and 25 MeV excitation energy, the iso-?fl i q on

scalar GQR in ^ N e is only weakly excited by the F(p,r) Ne reaction.

This result resolves at least qualitatively the puzzle why inelastic a-

scattering experiments on certain light nuclei obtain a totally different

E2-strength distribution than the corresponding radiative capture reactions.

It is obvious that the (a,a1) experiments do excite all those states with

a large transition density. Hence all the j" = 2*, T = 0 states which have

strong B(E2) values will be strongly excited whatever their individual

structure may be. On the other hand the (P,Y) reactions are much more

selective on the specific structure of the states involved. In the spe¬

cial case of F because of the target spin JT = l/2+ only d-waves can

couple with the target to yield ji" = 2+ states. Our numerical results show

that such configurations do only weakly couple to intermediate states of

predominantly (p~ pf) structure, at least via the conventional nuclear

interaction we have used. Hence at least the {p pf) part of the GQR

in Ne is only weakly excited via proton radiative capture.

216

However, a final answer to this puzzle cannot be given at the

present state. We shall still have to check whether the resonances de¬

tected in inelastic a-scattering really have the (p pf) structure

assumed here or whether they consist out of more complicated configu¬

rations like for example 2p2h-excitations from the p into the sd-shell.

This will be done in the future by calculating the transition densities

of our bound A-nucleon states and using them as spectroscopic information

for a DWBA analysis of the (a,*') reaction. Such an sn;»'ysis could per¬

haps also help to learn something about the single particle energies and

the effective interaction involving states of the pf-shell. It is clear

from the present work that such informations are necessary to obtain a

more quantitative description of the radiative capture reactions on light

sd-shell nuclei.

The work being presented in these lectures has been done in colla¬

boration with Or. G. Do Dang from Orsay, France.

References

1) H.E. Gove, A.E. Litherland, and R. Batchelor, Nuci. Phys. 26, 480 (1961). 2) N.W. Tanner, G.C. Thomas, and E.D. Earie, Nucl. Phys. 52, 29 (1964). 3) R.G. Alias, S.S. Hanna, L. Meyer-Schiitzmeister, R.E. Segel, P.P. Singh,

and 2. Vager, Phys. Rev. Lett. U, 628 (1964). 4) R.G. Alias, S.S. Hanna, L. Meyer-Schiitzmeister, and R.E. Segel, Nucl. Phys.

58, 122 (1964). 5) P.P. Singh, R.E. Segel, L. Meyer-Schiitzmeister, S.S. Hanna, and R.G. Alias,

Nucl. Phys. 65, 577 (1965). 6) R.E. Segel, Z. Vager, L. Meyer-Schutzmeister, P.P. Singh, and R.G. Alias,

Nucl. Phys. Ą93, 31 (1967). 7) "Proceedings of the International Conference on Photonudear Reactions and

Applications, Asilomar, 1973", edited by B.L. Berman (Lawrence Livermore Laboratory, Univ. of California, 1973).

8) S.S. Hanna, in "Proceedings of the International Conference en Nuclear Structure Spectroscopy", Amsterdam, 1973, ed. by H.P. Blok and A.E.L. Diepenbrink (Scholar's Press, Amsterdam, 1974), Vol. 2, p. 219.

9) S.S. Hanna, H.F. Glavish, J.R. Calarco, R. La Canna, I. Kuhlmann, and D.G. Mavis, in "Proceedings of the International Symposium on Highly Excited States of Nuclei", Jiilich, 1975 (unpublished), Vol. 1, p. S.

10) M.N. Harakoh, P. Paul, and K.A. Snover, Phys. Rev. £ U , 998 (1976); M.N. Harakeh, P. Paul, and P. Goroditzky, ibid, p. 1008.

11) P. Paul, Vol. 2, P. 72 of ref. 9). 12) O.P. Elliott and B.H. Flowers, Proc. R. Soc. A242, 57 (1957). 13) G.E. Brown and M. Bolsterli, Phys. Rev. Lett. 3, 472 (1959). 14) G.E. Brown, L. Castillejo, and J.A. Evans, Nucl. Phys. 22, 1 (1961). 15) V. Gil let and N. Vinh Mau, Nucl. Phys. 54_, 321 (1964). 16) R.H. Lemmer and C M . Shakin, Ann. Phys. (N.Y.) 2_7, 13 (1964). 17) B.Buck and A.D. Hill, Nucl. Phys. A95, 271 (1967). 18) V. Gil let, M.A. Melkanoff, and J. Raynal, Nucl. Phys. A97, 631 (1967).

218

19) J. Raynal, M.A. Heikanoff, and T. Sawada, Nuci. Phys. A101, 369 (1967). 20) 8.M. Spicer, Adv. Nuci. Phys. 2, 1 (1969). 21) W.L. Hang and C M . Shakin, Phys. Rev. £5, 1898 (1972). 22) J. Birkholz. Nucl. Phys. A189, 385 (1972). 23) S. KrewaJd, J. Birkholz, A. Faessler, and J. Speth, Phys. Rev. Lett.

33, 1386 (1974). 24) E. Grecksch, W. Knupfer and M.G. Huber, Nuovo Cim. Lett. J4, 505 (1975). 25) W. KnUpfer and M.G. Huber, Z. Phys. Ą276, 99 (1976). 26) S. Krewaid, V. Klemt, J. Speth, and A. Faessler, Nucl. Phys. A (in press). 27) O.S. Dehesa, S. Krewald, J. Speth, and A. Faessler, Phys. Rev. C (in press). 28) I.R. Afnan, Phys. Rev. 163, 1016 (1967); see also G. Pisent and F. Zardi,

N. Cim. 48, 174 (1967). 29) M. Maragoni and A.M. Saruis, Nucl. Phys. Ą166, 397 (1971). 30) D. Zawischa and J. Speth, Phys. Rev. Lett. 36, 843 (1976). 31) M. Danos, Nucl. Phys. 5_, 23 (1958); K. Okamoto, Phys. Rev. HO. 143 (1958);

M. Danos and E.G. Fuller, Ann. Rev. Nucl. Sci, 25, 29 (1965); M. Danos and W. Greiner, Phys. Lett. 8, 113 (1964).

32) See for example, E.G. Fuller and E. Hayward, Nucl. Phys. 30, 613 (1962). 33) A. Bohr and B.R. Mottelson, K. Dan. Vid. Selsk. Mat. Phys. Nedd. V_, No.

16 (1953). 34) Many Body Description of Nuclear Structure and Reactions, in Proc. of the

Int. School Enr. Fermi, XXXVI, ed. by C. Bloch (Academic Press, N.Y., 1966); C. Mahaux and HA. Weidenmiiller, "Shell Model Approach to Nuclear REactions" (North Holland, Amsterdam/N.Y., 1969).

35) H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958); 19_, 287 (1962). 36) E. P. Wigner, L. Eisenbud, Phys. Rev. 72, 49 (1947);

A.M. Lane, R.G. Thomas, Rev. Mod. Phys. 30, 257 (1958); A.A. Ayad, D.J. Rowe, Nuci. Phys. A218, 307 (1974).

37) K.W. Schmid, G. Do Dang, Z. Phys. Ą276, 233 (1976). 38) K.W. Schraid, G. Do Dang, Phys. Lett. 66B, 5 (1977).

219

39) K.W. Schmid, G. Do Dang, Phys. Rev. CIS, 1515 (1977).

40) W.L. Wang, CM. Shakin, Phys. Lett. 32B, 421 (1970).

41) K.W. Schmid, G. Do Dang, to be published.

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149 (1974).

43) K.W. Schmid, H. Miither, Phys. Rev. C16, 2050 (1977).

44) K.A. Snover, E.G. Adeiberger, and D.R. Brown, Phys. Rev. Lett. 32, 1061

(1974).

46) K.T. Knopfle, G.J. Wagner, H. Breuer, M. Rogge, and C Mayer-Bbricke,

Phyi. Rev. Lett. 35, 779 (1975).

47) E. Kuhimann, E. Ventura, J.R. Calarco, D.G. Mavis and S.S. Hanna, Phys.

Rev. Cll. 1525 (1975).

48) K.T. Knopfie, private communication.

49) See for example: G. Ripka, Adv. Nucl. Phys. 1, 183 (1968).

50) F. Villars, see ref. 34).

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S. GirauC, Nucl. Phys. U, 373 (1965).

52) N. Auerbach, J. Hiifner, A.K. Kerman, and C M . Shakin, Rev. Mod. Phys.

44, 48 (1972).

53) P.W.M. Glaudemans, P.J. Brussard, and B.H. Wldenthal, Nucl. Phys. A102,

593 (1967).

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55) F. Ajzenberg-Selove, Nucl. Phys. A19C, 1 (1972).

On the Influence of the Shell Huclear Structure on the

Diffusion Process

Kartavenko V.G.

Joint Institute for Nuclear Research

This short report is devoted to the investigation of the

shell structure influence of the intermediate nuclear system,

formed in the process of collision of heavy ions with atomic

nuclei, on the charge distribution.

1. The role of shell structure in heavy ion reactions is

quite important both from the point of view of clearing out

nuclear reaction mechanism and from the point of view of

new element synthesis.

2. The nuclear structure can be exhibited in different

ways:

i) in the processes of N,<i - evaporation

ii) during the fission of the formed compaund nucleus or

excited resultants

iii) at the preequilibrium stage, when the formed interme¬

diate nuclear system relaxes and decays.

3. Y/e observed the last, less studied process and some

aspects of its applications to the analysis of the interaction

of ratner heavy ions ( r\l and heavier ions; with imer.-.c -an

and heavy nuclei. That is because!

221

i) just such reactions have lately been intensively studied experimentally

ii) the collision of such complex nuclear systems cannot be described within the traditional direct or competely equilibrium ways of reactions.

iii} the problem of relaxation processes in such systems itself is interesting enough.

4* 33ie diffusion approach is- applied for the description of the evolution of the intermediate nuclear system. The approach seemed to be extremely useful tear toe analysis of nonequilibrium processes in heavy ion reactions. We have mainly used lioretto'e variant' ' of the diffusion approach, modelling system's relaxa¬ tion in the space of asymmetry coordinates (the atomic number 2 of one of the fragments) by means of stochastic process

obeying the Master equation:

for the population V H ^ M (the probability of finding the system at time t within configuration of the asymmetry £ ), The macroscopic transition probabilities ^ti> between i'

and 2 configurations are connected with the potential energy v4 of the system. The latter includes the binding energy of

the system, Coulomb interaction of subsystems and the rotational energy. The observed values, i.e. the charge and angular distri¬ butions, the kinetic energies of the resultants and Y -multi¬ plicities in heavy ion reactions can be calculated with the help of W Ł .

222

5. The valuation of the shell structure influence upon the diffusion process has been given in the symplest way. Calculating vŁ , we have substituted the liquid drop values of the binding energy for the experimentally known values (or their extrapolation' ' ) . Of course,the calculations of \/£ on the basis of the two-Centre Shell Model would have been more correct, but such calculations are extremely diffucult, so we have confined overselves to the simplest variant.

6. As a result of the analysis it is shown that the shell structure influences the diffusion process to a great extent, leads to even-odd structure of VX^ (4) enhances the yields of isotopes with even number of protons compared -with the yielrto of their odd neighbours and causes still greater enhancement near the magic numbers. The last point may be very important for new element synthesis.

Ill 7. It is seen by the example of the reactions Ar+Złw f

/U+A3 /Ve + Tk w , Xe+Sn,, Xe + Au m , that the diffusive mechanism may be rather significant for the explana¬ tion of the experimentally found irregularities of charge distributions.

References

1. L.G.Moretto and R.Schmltt. Journal de Physique, £5, 109, 1976.

2. Atomic Data and Nuclear Data Tables. 17, 1976. 3. R.Gupta. Particles and Nuclei. 8, 717, 1977. 4. R.Kalpakchieva a.o. Z.Phys. A, 282, 253t 1977. 5. V.V.Volkov. Particles and Nuclei. 6, 1040, 1975. 6. A.Gobbi a.o. Preprint GSI-P-5-77, Darmstadt, 1977.

223

Heavy Ion experiments on the 1*P tandem at Orsay :

Particle correlation studies and mass measurements on exotic nuclei

P. ROUSSEL

Instltut de Physique Nuclealre, BP n°1, 91406 Orsay, France

This talk will be about two heavy-ion experiments performeo at tne

tandem tV In Orsay and which both used a similar experimental setting

based on a magnetic spectrometer' Of coureo they are not the only HI exoe-

riments performed on the tandem, even with the spectrometer, even with the

physicists who were Involved into these experiments lat a time or as a

leading participant) and who are : M. BERNAS, C. OETRAZ, F. OIAf, R.FASBftC,

E. KASMY, M. LANGEVIN, F. NAULIN. A.O. PANAGIOTOU, E. PLAGMOL, F.POUGHEDN,

G. ROTBARD, P. ROUSSEL, M. ROY-STEPHAN, B. SAGHAI, J. VERNOTTE.

I. EXPERIMENTAL SETTING.

The spectrometer Is a 160° magnet, with a mean radius o-f 700 nun, a

gap at the mean radius of 70 mm and with a mechanical width of the pole

pieces of 300 mm. Its index is 1/2 and it may be recalled that tills inoex

brings the double focusing property of the magnet : In the symmetrical

position, the Image and object distances are r i/2/tg 111/2^) •». 490 mm in

both vertical and horizontal planes. The span of analysed momentum is tb\

the angular admittance is ±4.5°x ±1.5° [vertical). The energy resolution

is E/flE-v-2000 for an horizontal opening of ±2.5°. The kinematical correc¬

tion can be achieved by a mechanical increase of both object and Image

distances up to a correction of K «^- ^ • 0.5V° i.e. •v .30. DP dB

Equipment of the focal space.

It is based on the use of gaz counters (1)[2) the main features of

which are as follows (flg.1) ! they are single wlra proportional counters

using the charge division method. Tha central wire ha» a diameter of ISu

and a resistance of t< 40ft/cm. The advantage of such a low resistance is

that the carbon bllding on the wire (due to tht cracking of the gaz) may

221*

alter the energy response of the counter but not the Dosition one. The

counters are filled with pure propane at a pressure from 20 Torr to 70

Torr, and are used with Ó high voltage of ^ 850 V for the whole span of

pressures. On fig.2 is given an energy spectra for the use at 70 Torr.

I

Fig.1, Scheme of the counter

Fig.2. Energy loss spectrum obtained in ths study of tne J0( C

reaction at 66 MeV and at 8i2c*21°25. This spectrum was e

the signal from a PSD [10x50 tun2) behind the counter.

d fty

225

When events produced with a very low cross-section are to be iden¬

tified among millions of other ones, like in the search for exotic nuclei,

a redondant measurement of the parameters is needed. Two identical counters

(working at 70 Torr) are then put together and associated to an array of

position sensitive detectors in the focal space of the magnat (fig.3).

Reconstltutlon of heavy ion trajectories in a magnetic spectrometer

In many circonstances, it is necessary to have both a large solid

angle and a precise knowledge of the reaction angle. If two position mea¬

surements are made (one at least being in the image space of the magnet)

It is possible to determine the two parameters which caracterlze the tra¬

jectory coming from a point object : the angle 9 and the magnetic rigidity

Bp *. Ths two gaz counters are separated apart as Indicated in fie-4- 1&

I

Foil

Doublt rMif>iv*-wifi rountmr

Wlw

Fig.3. Scheme of the focal space of the magnet when used for the search

for exotic nuclei.

%(ith three position measurements (one of them being in the object space

of the magnet) It would be possible, in principle, to determine these

two parameters even with a finite bBam emittance by the use of appro¬

priate conditions for the focussing of the beam.

226

reduce the angular spreading in the first counter from multiple scattering. Its thickness has Oeen reduced by the use of thinner windows (400 ug/cm2. •+ 100 pg/cm2) and a lower pressure of the gaz (70 Torr • 20 Torr). The vertical arrangement is shown on fig.5. With this geometry, the vertical spread of the trajectories at the place of the two counters is well con-troled since they correspond to the image of the beam spot on the target and of the entrance slits. A further advantage is that It allows for a reasonably good time pick-off on the first counter (low pressure, concen¬ tration of the trajectories near the wire).

For a magnet with an Index, the trajectories corresponding to the same incidence 8 but with different Bp are focussed on a virtual point

Fig.4. Scheme of the geometry used for the constitution of the spectra In the displaced focal plane. From the measured positions on the two counters O.M.'X, and 0,M,=X-, the position in the displaced plane

" " 1 i. Ł i X2-O2A2 is given by : 0f1 = OA^ • A , ^ . „ ^

image I as shown on fig.6. Taking advantage of the constant angular magni¬ fication M»1, the two positions X1 and

follow the relation i measured on the two cojnters

d/0 • 9d (see fig.6)

from which 9 can be extracted

6- X_ -a X with a • 1»d/D

Fig.5. General arrangement in the vercKdl t/J-jne. Four t ra jector ies I I

4) grazing tho bean spot edp.e:, on tl.o t.jryet ..irO thr !.•'•;'! of tht

s l l t a nave been vl5uoli?et) to i l l u s t r a t e the adv.int-i,-'--. -.'^ ' .hi t

arrangement. A l l distances are f i ve " > i n .

Fig.S. General arrangement in the horizontal plar.e. The two centers G

and CUXp as well as the focal plane (along O.X,,], u>ave DL-e:n assi.

to be perpendicular to the medium trajectory. Tt>*_ ^xistencc of t

virtual point image 1^ ( S ? G text) and the use c; the V J J U C 1 fir

angular magnification learl", to the relation: X.,-*,, -- — .'•, • fr.d it

which e « X_ - a X, can be calculated.

228

I t can be shown that the abscissa of the intercept of a t ra jectory with a

focal plane (usually a displaced focal plane due to the kinematical fac-

torJ not para l le l to the counters (see f ig .4 ) is given by "Bp"» *c

where a, b, are the constants which define the posit ion of the focal

plane.

Some results obtained with th is device are shown on f ig .7 to 10. On

f i g . 10, i t is interest ing to note that the peaks (e.g. at channel "V130

or 160) which are narrow on part b) with a small angular aperture, and

broad on part a) with the f u l l anr,le range,correspond to reactions with a

d i f fe rent Kinenwtica] factor, dence to the ( C, ' C) reaction an d i f ferent

target nuclei ( ' C backing or r-i present in the target made with 113} •

To summarize the per f onitances. wtijcn łiavE. Liepr, łound wi 'h th t f-ro^tnt

device lut . • b'iy that two 11 mtvjoLri.fiL'fits are n o * Iht f.'r:>t on<- wjt1"' sr

occuracy of 10 to IT". t'io lecond one with 'J tr; t i . '• t lno E,ii::-ji i'^ d( 1 i -

v e r c d v ; l t r . , ' , t '• T O C Ł . T h e o f i r J f r t n . . i Ł - r ' . - " i i ' i t i " r : i . j f j t a ; j . i ' T ^ M . " ; ' o r •:

f u l l a p e r t u r e c ' ^.. 2 y ( t h i & o r u l d t . f : e a s i l y J ' i c r f . i s v ^ ' i ' i ^ ' i t ; i ^ ' . i i ' - ^ " - i n ^ i '

T o i n ^ i ^ o t f e t h e p c i - c i L ' I i - t ^ ' f - n ^ i - j 1 " ' r>^ t M . - ^-r-1 c ^ ' ^ . i i . ' . v J : . c - I

h e a v i e r i c n o , w e r i ^ v s s f . o * r . c r ^ : . i e : t h e fi> i ' ' . - I -T i ' . r ^ , ' ^ : . ' r •'l-0 ' ' : ' . ' •

' c r c h t l e n s e n t h e n r t c ' . r t ( t h j r . 1 a i m : t r ' ''r-l . . ; ' . ' ś f C ' ' - - : .. . " V , : - ' . :

t r . r s e t ; r - . c t r . i r r . c . r I t ' o i i o s m : ; ; a : p i >.: : ^: i ', • ' •••' ' f ł i j r i . r , ^ ' . r , . ' . _ j ';••••

i i C 560 S?C

Pig.7. Position spectra on the first and second counters for a rwrrow ope¬ ning [0.17°) of the entrance slits o the magnet.

229

TABLE I

Angular stragling in a gaz counter

of different heavy ions

ion

Coopteur d* 30

%

A/Ca

Kr

Compteur de 101

A/Ca

Kr

energie incidente E MeV

) ug

70

180

500

300

1000

> Kg

100

300

500

100

300

1000

perte d'energie AK MeV

1.5

0.75

3.0

12.5

9.0

2.5

1.5

1.0

5.5

4.5

3.0

AE/E Z

2.2

0.4

0.6

It.

0.9

2.5

0.5

0.2

5.5

1.5

0.3

dispersion aneulaire

69*

0.3

0.12

(0.012)

0.1

(0.03)

(0.034)

(0.01)

(0.007)

0.17

(0.06)

(0.017)

230

0 f . 40 SO MMI • F.40 20MHI 1 F.39SOMHr i 'M S9 2OMMI

F . 38 20 MMI

60

Fig.6. Spectrum of the calculated ^ „ t e

given p o s i t i o n of the peaK -rom e l a s t i c s c a t t e r i n g on f c l d . r i vc

spect ra are shown corresponainj j to f i v t p e t i t i o n s sprerjC eiori;- t '

used foca l p ione . An angular opening o ' j . 1 7 ° i s cStfirn.a w i t h t f *

entrance s l i t s of the magnet.

calibration

9. Spectrum of the calculated angle 8c ootained with the C ions tucteU In the study of the °Q{ O, ŁC) Ne reaction at 66 MeV. The slits of the magnet define an angular opening of 5.?' thr; center of whlcn corresponds to a reaction angle of B°.

Fig.10. C spectrum calculated in the displaced focal plane a) for the full angular range (5.2°) and b) for the events selected by a narrow win¬ dow on 8 . The excitation energies of the strongly populated states in Ne have been indicated.

232

II. PARTICLE CORRELATION STUDIES (s»e (3) and (4) and references therein)

The aim of the following experiments was to reach a deeper Knowledge

of the reaction mechanism in the case of a heavy ion Induced a transfer

reaction. Correlations experiments have been performed In order to measure

the polarisation of the residual nucleus in the reaction 16O(160,11C)20Ne"-»

a • 0 . Two types of polarisation could be axpactad :

(I) from a naive classical view of the reaction with all the fragments and

the transfered a in the reaction plans, one would expect a polarisation

perpendicular to the react-Ion plane as the result of the orbiting a. Tra¬

jectories with positive and negative deflection angles would than corres¬

pond to positive or negative polarization (m« *J or m» -J).

(II) It can be shown that a plane wave calculation leads to the population

of a magnetic substate m« 0 on an axis of quantization along the recoil

axis (i.e. _ln the reaction plane). DWBA calculations as well as experimen¬

tal results obtained with the ( Ll.t) reaction leads to similar results.

It must be noted that predictions (i) and (11) are not compatible

because (1) correspond to one vector among the many of those which are

necessary to represent the case (11)

r*c.«l at i i

In this particular case where all the fragments Involved have a

spin* zero, with the only exception of the "residual* Ne* in the first

step of this sequential reaction, the correlation function u(e,<f) measured

between the 1ZC and either the a particle or the 0 is given by

.c | Pj j^OJl* (1)

where all the information on the mechanism is included in the components

pm of the polarisation tensor which are normalized : C|pT|2 - 1 •

This formula is valid for any quantization axis but Is simplified if

this axis is taken perpendicular to the reaction plane since the selection

233

rul8 (Bohr theorem) J-m»evan (natural parity states) applies and reduces the number of P . ' s '

It Is worth noting that if there Is only one p"1 / O;|pj| -1^ u i s lnde-psndant of V: uiOJ.If lpT|2* IpT**1]z - 1 u can be factorized ul6,«P • «'(e)x«'(«f). In particular i f | p j | 2 * | p " J | 2 - 1 , #»«c f«'«W« i« H e f««'f>"< f ł « » « :

2Jr e.E) sin2J(6J • 2 (2)

It appaars from (2} that the correlation pattam is vary sensitive to a Is the dominant term. For example :

glvas already a modulation of 30t.

i t appears from u i tnat roe c small admixtures of p" If p |p-J|2 - 0.37 |p"^| « 0-03 givi

It is important to note from formula (11 or (2) that the exchange of all the p1? by pT* leaves the correlation unchanged and hence it is not possible to measure the sign of the polarization (as could be expactad since the a has no spin).

The results of a first experiment have already been published (4) and will be Just recalled i the levels of Ne Known to have an a struc¬ ture are preferentially excited (fig.11). OWBA calculations (fig.12) re¬ produce reasonably wall the angular distribution of the studied levels (those with a large cross section and which decay by an a emission) but cannot account for the observed strong polarization along an axis perpen¬ dicular to the reaction plane (fig.13, table II) .

.300 -

200 -

100 -

300 400 900 Chow*

Fig.11. 12C energy spectrum measured at 19* for the reaction 160(160.1ZC)20Ne at 68 MeV incident energy.

10'

- • • o '

I01

10'

icr

5 10 15 20 25 30 35

Fig.12. Comparison of experimental angular distributions and EFR OMBA cal¬

culations. Optical modal parameters usad are : VR«17 MaV, W^-7.2

MeV. A-0.49 fm, AS-O.1S fm, (^«1.35 fm. Rj-1.27 fm, for the Inci¬

dent channel and VR-17 rteV, Wj-5.6 MeV, A -0.49 fm, Aj-0.15 fm.

R_«1.35 fm, R--1.27 fm for the exit channel. The form factors have

been calculated as those of elightly bound stetas CB-0.4 ftaVJ for

the fu l l line curves. Predictions with more strongly bound states

(B-1.5 MeV) are also shown [dashed line) to indicate the Influence

of the form factor. A radius t"o-1.35 fm and a diffuseness a-0.65 fm

have been used for the Woods Saxon wells.

Tablell . Populations |pm |2 of the different magnetic

substates on a quantization axis perpendicular to the

reaction plane.

E • B.79 MeV J* - 6*

m •

•20.5

-6

86

90

-4

3

3

-2

1

1

0

1

1

2

2

1

4

0

0

5

S

1

E • 1.45 HeV

17.S*

20.5*

-5

SG

90

-3

11

2

- 1

11

0

1

3

0

3

1

2

S

1?

6

*M

0.80

0.60

Q40

0.20

0 30 60 90 CO l50if 0 30 60 90 120 l5Oe

In reoction ptont In parpendiculor plane

Fig.13. Comparison between a DWBA prediction (»olld line) and experimental

results for tha 12C-160 angular corralatlont measured in the sequen¬

tial reaction 160(160,12CJ20N«*-» o»160 at 68 fteV Incident energy for

the 6* (6.79 MeV) etate of the 2 0 N B . The dotted linet correspond to

the least squares search (based on equation (1) from which the values

of the populations \p?\2 given In table Ilare extracted).

A new experiment has been performed in order to see how would evo-

luate the observed polarization with the C angle. The procedure of re-

constitutlon of trajectories has been used and the angular distribution

of the polarization of Ne has been measured for &\2C between 5* and 21*

and around 45* (6_ „ t90*). The scheme of the experiment is given on fig.14.

Two bldlmenslonal spectra (E,P) from the position sensitive detector

(PSO) gated with signals selecting a given Ne level and a window on the

calculated 6 are shown on fig.15 and 16. On fig.17 appears the Kinematlcal

plot (corresponding to fig.16) which is necessary to build the correlation

function as shown on fig.18.

The analysis of the data is in progress and a preliminary result 1B

given on fig. 19 for tha level at 6.78 (lev 6*. It appears that the populo- V.

tion of the m-J magnetic substate stays very high on a wide span of C

angles. The horizontal line correspond to the plane wave prediction which,

on this axis of quantization (perpendicular to the reaction plane), should

not be very different from a OWBA prediction. All predictions must converge

at 6*0* on the indicated point (geometry II of Lltherland and Ferguson).

This preliminary result seems to confirm trie already published one

and different Interpretations will be attempted.

236

12,

RSD.

BEAM

OR

•E,. AND 9,e

*o l6o £. AND ft,

Fig.14. Scheme of the experiment.

III. EXOTIC NUCLEI.

This paragraph will deal with experiments done on the W tandem In

Orsay. In order to measure the mass excess of some light exotic nuclei and

possibly the excitation energy of their first excited state by relative Q-

value measurements. By exotic nuclei It Is meant nuclei far from the sta¬

bility line.

These measurements are to be compared with the predictions of mass

formulas which may differ, one from the other or as compared to the experi¬

mental value, when one goes further away from the stability line. Mass-

formulas can then be Improved with such measurement.

IB • The experiments have all used a beam of 0 6 at 91 fieV with an

intensity between 600 nA (Faraday cup) and 300 nA. The latter to increase

the lifetime of the targets. Targets of 2Al30 enriched at 90* of 180 and 14 2

of C enriched at 80% have been uBed with thicknesses from 70 to 120 ug/cm resulting in an energy resolution from 80 to 180 k»V.

The two experimental difficulties are 1) to identify a few nuclei

produced with very small cross-sections among millions of neighbouring ones

(elastic scattering for Instance) and this requires redondant measurements

of the different parameters which identify the nuclei i 11) to meaiur* the

0 value of the few well Identified events, whan the energy calibration is

obtained from different Known reactions, usually with different klnmatlcal

factors.

237

4

mi « iti n

I l l I > II 1

t urn II 11* unii t$mm u Ś iti tii •

, ' i f lT" ' t.111 'Ali. * 1 tW 1 i mi it

i i

i imnu i MVHMI i «•

11 i •i tu u

Vi '

I I I 1 1

fllł 1 1 i min

iniint *• i itm i

t mi i IHI 11

i i i

i \ %

••• i

i t

i

• t

i

i t

*

n i <

i

i

NOU.ICM

>•> •• l>

*>

I I a a

h g I Z 2 u 0 a a o

1 2

% i I I | 5

• • • • • • .11 HI Ml III M I . » K II II •

E • T .

i * •• • : • t ś

< • ' ' ; ,-• '* i •; ;• • ^ :* - t •?' v r? £ ? * b L r i ' •'"' • ' X r i • ' . ;

t :.3h

--eSsS.a: . i i "

: B-

s : - ^ 1

Fig. 16* An exainpls of the correlation axperlmantal data*.

. . . . . . . • . . . . « . . •.«. . . • • . * . , • . . t . . . . «fa — * . . . * . .

X

•c c o a

8 o u

o "a

. ••• U.

21*0

79MeV (6

e l 2 . = 13-75

0.401-

0.20!TJ In • u -

I 0.40 •-

0.20 L-

a , = I7°5O

i> 3 1

3 h il n

I90<:i0 1 1 J 1

/I n

C Z'J £0 SO 120 150y 0 30 60 90 120 150Lp

Pig.18 - Histogram of a correlation obtained after the steps of the analysis indicated on fig.16 and 17.

• 0.75u

J-0.50 8.79MeV (6+)

0.25- P'c--e waves

Preliminary results for the angular distributions of the polarisa¬ tion of Ne (the quantization axis i s perpendicular to the reac¬ tion plane).

tki

It has to be pointed out that there Is always some doubt that an observed peak Is not the ground state but may correspond to an excited state which could be favoured by Klnematlcal matching conditions. Only several convergent arguments may lead to eliminate this possibility.

The experimental method Is detailed in ref.(S) and is based on the use of the set of counters shown on fig 3 . Two AE measurements are used and two or three position measurements are necessary to eliminate events with "wrong" trajectories. "Wrong" meaning for instance they they nave suffered a scattering on the residual gaz In the magnet chamber or (more likely) during their path through the first parts of the set of counters. Flg.S.* shows a E-iE map, cleaned in this way, obtained In the study of the 1flO(18O.19N)17F reaction In order to measure the 19N mass. Fig. 2,r shows the "Bp" spectrum of th« few N evants together with that of two of the reac¬ tions [among v 10) which have been used to calibrate this spectrum.£r«f g~l

« I nf»fi u..•»....mi

|a*4M«m 111 I III I I * I 12 2 u n*

•••*» t 1 it I 11

I l*łł*

l)**lt I I — - ,-Iłatl l««*M>l***>2l I 4W UK

II •••Ml

l»t»MM*<iJ !»•/•••• FMI »MfMM«l M»TW 1 il 12 lt»«T*9 1)1 t U l l l I

21 I I 1

Fig.20 - The upper figure shows part of the^E-E display in the region where 19N ions are expected, for one of the four 5cm long position-sensitive Si detectors. The ions observed are identified in the sketch drawn below. For channel* where more than 10 counts are recorded, symbols are used. Their meaning is indicated at the top of the figure.

N counts Ą

nO*VM2K

91 M»v

1000

.77 .74 .75 Magnetic rigidity (r.m)

Fig.21 - Position spectra of the particles identified as 1 9N 17N, and 1 70 (see fig.20). The l 7N and *?0 are used to calibrate the first proportional counter. The peaks are labeled by the residual nucleus and its excitation energy. The counter was set in the kineraa-tically corrected focal plane of the lsO{1BO,^tS)17F reaction. Accordingly, the peaks corresponding to other reactions are broadened. One clearly sees the gaps between the Si detectors which are in coincidence with the proportional counters. The cluster of events in the upper spectrum is assigned to the 1 8O( 1 8O, 1 9N) 1 7F ground state transition. (Note that particle energy increases from right to left).

The measured mass : 15.81 ± O.O9 MoV is situated in the vicinity of three predicted values at 16.27 MeV, 15.32 MeV and 16.35 MeV and the experimental error is much smaller than the difference between these predictions.

The two other studied reactions are 0( 0, 0) 0 [see ref.(7)J and UC(18O.17C)15O [see ref. (8)J . The 150 Bg spectrum from the first one is shown on fig. (22) together with the O spectrum obtained with an ordinary 2*l 0 target with no Ł 80. It is seen that it is the presence of 160 in the target which do no permit a clear observation of excit¬ ed states of 2 10. The measured O.S. mass 6.153 - 0.070 MeV is in good agreement with the only one previously published at 8.122 - 0.075 MeV.

Pig.22. Th» Bo spectra for 1 50 from the 180(180,150)210 and 16O(1SO,15O)19O reaotiona.

2kk

(C) 21.76

4

(B) (A) 21.27 20.86 (M.V)

4 4

mm nn nn n 789 .794 .799

Magnetic rigidity

Counh

4 •

2

(T.m.)

Pig.23 - Position spectrum of the C ions observed in the Łi*c(18o,17C)15O reaction at 91 MeV.

For the last studied reaction, fig.(23) shows the B£ spectrum of the 14 events (50 nb/pr) identified as C. The arrows A and B correspond to two predictions [(A) Jelley-Cerny : modified shell model ; (B) Garvey-Kelson)J and the arrow (C) corresponds to the threshold for the decay C ;> C + n. Since this rather inconclusive spectrum was obtained,

the mass of C has been measured by Nolen (Heidelberg-Michigan 48

S.V.) using the same reaction on Ca which gives ten times larger a cross-section. The G.S. mass was given at 21.023 and an excited state was formed at 0.292 MeV. It appears then that the strongest peak at 21.300 - .070 in our spectrum could be the excited level observed by Nolen et al. and predicted at 21.315.

To conclude, it can be said that the use of the method of reconstitution of trajectoires could bring an improvement by the possible use of a larger solid angle and a better occuracy in the calibration for measuring the mass of exotic nuclei.

2*5

References

(1) B.Saghal et P.Rouasel, Nuci. Inst. rieth. _141_ (1977) 93.

(2) P.Rou9sel, M.Bernas, F.Olaf, F.Naulin, F.Poughaon, G.Rotbard, M.Roy-

Stephon, to be published In Nucl. Inst. and Meth.

(3) B.Fabbro. Thesis Orsay (1976) unpublished.

(4) F.pougheon at el., J . Phys. (Paris), Lett. 21,11977) 417.

(5) P.Haulin, Tbti i i (3o cycle) Paris-Orsay 1975 (unpublished)

(6) C.Detraz et a l . , Cargese (.Corsica) Mai 1976 p.248

and C.Detrar et a l . , Phy». Rev. C 15 (1977) 1738 (7) F.Naulin «t a l . To be published Phys.Rev.C (Jan. or £eb.) (8) F.Naulin et a l . Conference in Florence (1977) p.65

OK IKELASTIC SCATTESHTG CALCULATIONS

W.J.G. Thijssen

leohnloal University, Eindhoven, The Hetherlands

Doing macroscopio rotational ezoitatlon calculations of

permanently deformed axial symmetrio nuolei by inelastio scat¬

tering by the standard coupled channel (o.c.) method, ona often

enoounters the problem of a very fast incrtas of the computation

time, implying more ezoited states or more partial waves.

Another approaoh, which often reduoes the computation tlM ta

the adiabatic method, in whioh one assumes that the target

orientation is fixed during the interaction. Both methods antt

widely used and disoussed, see for instance Tamara ' and caff—

erences there in.

In the adiabatic method one introduces an additional eam-

stant of motion, the projection nu of the projectile angnlaw

momentum i on the nuclear symmetry axis. This is the I ' M —

that in the adiabatio calculation the o.c. basis |ljIJV > ta

replaced by the adiabatic basis |l3m3JM> , with 1 the orbital

angular momentum, I the target angular momentum and J the

total angular momentum with magnetic quantum number H. In this

adiabatio basis I is not a good quantum number, so one cannot

easily include limited angular momentum transfer AI in the

adiabatic calculation as in the o.o. calculations. Nevertheless,

2) it can be done as is shown by Schulte and Verhaar ' and Schul-

te /. In their few state adiabatic approach they decreased

the computation time by roughly a factor 10 with respect to

2%7

tho o.o. calculations. This a quite satisfactory result, BO

one would like to extend the limited area of validity of the

adiabatic method. A way to loolc at this problem is a classi¬

cal one. In the adiabatio method we take the target to be fixed

during the interaction. If that is not quite true, a possible

solution ia to take the target fixed during some projeotilepath

interval «id adjust the target direction, and so on.

If oua forgets the physios, one can do the same in sta¬

tionary quantummaohanloal calculations by dividing the radial

integration path into a number of intervals. Vow it is possi¬

ble to adjust the adiabatio radial wavefunotions every interval.

To realise this adjustment, we generalize in the c.c. basis

the in- and outgoing waves (adiabatio and o.o.) to within the

interaction region. To oaloulate the S-matrix elements, one

expands the radial wavefunctions as in- and outgoing waves.

How we do the same every interval with these generalized waves.

It turnes out that the coefficients of the fa- and outgoing

waves behave very similar in the c.o. and adiabatio cases, in

contrary to the wavefunotions themselves. We use tfils simi¬

larity to adjust the adiabatio wavefunctions.

We are working now on the realization of this method in

actual calculations, but I can show you a result of a very

simple calculation, which we made to test this method.

We made a two-ohannel calculation in which the adiabatic cal¬

culation was done in the c.c. basis, making the adjustment

very simple. The order of magnitude of the parameters is

realistic

248

The essential parameters are:

- Proton scattering: B = 40 MeV

- Woods-Saxon potential: Vo -= 50 He?

B « 5 fm.

a » 0.6 trma

-A2 « 0.2.

- Exoltationenergy * 5 MeV.

- Ho Coulomb excitation.

The resulting S-soatterlng matrioes are given below. We see

a good agreement of the adjusted adlabatio matrix with the

o.o. matrix with adjustment in 5 intervals. The adjusted

result converges essentially to the c.c. result with increasing

number of intervals.

S-matrix; (1=6,8.)

/ 0.75 + 1.-0.003

\-0.09 + 1.-0.65

/ 0.72 + 1.-0.02

1-0.11 + 1.-0.67

•0.003+ 1.-0.53

-0.70 + 1.-0.47

-0,09 + 1.-0.65]

0.72 + 1.-0.21/

-0.11 + i.-0.67\

0.69 + 1.-0.21/

-0.70 + i.-0.47\

0.47 + 1.-0.23/

c.c.

adjusted

adiabatic

adiabatic

References

1/ T. Taruura, Rev.Uod.PhyB. 37 (1965) 679

2/ B.J. Yerhaar and A.U. Schulte, Phye.Lett. 67B (1977) 381

3/ A.M. Schulte, Thesis, Eindhoven Technical University

Elastic Transfer Reactions

H. G. Bohlen

Hahn-Meltner-Institut fttr Kernforschung Berlin

1. Introduction

In the elastic scattering of heavy Ions with similar masses the

transfer of the mass difference leads to the same nuclear

system. This process Is called elastic transfer. The transfer

reaction amplitude interferes with the direct scattering

amplitude.

There is strong experimental evidence for the elastic transfer (1):

1. Deep (interference) structure In the angular distribution

at energies in the vicinity of the Coulomb barrier

2. Backward rise in the angular distribution at higher energies

3. Direct scattering and transfer move apart, if the incident

energy is raised

4. Constant phase at 90° (maximum or minimum)

5. Phase systematics at 90° in the angular distribution

a) Dependence on the fermion/boson core property

b) Dependence on the bound state 1 value of the exchanged

particle.

It is also possible to fit the angular distributions with an

optical potential (2), but the fit parameters are very unusual.

Especially the absorption would be very weak. The imaginary

part of the potential used in ref. (2)is only a narrow spike

at the nuclear surface. The structure of the angular distri¬

bution also behaves quite different for optical potential alone

than the interference structure in dependence on the energy.

The elastic transfer is now a well established reaction

mechanism, and this interpretation has been successfully

applied to many data. It belongs to a class of reactions,

where interfering amplitudes, resulting from exchange effects.

250

play an important role. A list of scattering systems typical

for a special mechanism is given below in A. In B those things

are summarized which can be learned from this type of reaction,

and in C several methods of calculation with a different degree

of sophistication are compiled.

Applications:

1) Elastic scattering: 16O(a,cc)16O , 28Si(16O,16O)28Si (ref.3,4)

2) Elastic transfer: 12C(13C,13C) 12C + 12C(13C,12C)13C (ref.1)

3) Inelastic transfer: 17O(16O,16O) " o j ^ + ^oC^O^O*.^) 160 (ref.5)

4) Symmetric transfer: 12C(14N,'3C) 13N + 12C(14N,13N) 13C (ref.6)

5) Asymmetric transfer: 14C(16O,'8O) 12C + 14C(16O,12C) 180 (ref.7)

6) Two-step-process: Inelastic scattering

two-nucleon-transfer

charge-exchange

Better information about:

1) Reaction mechanism with interfering amplitudes

2) Molecular states in nuclear scattering systems

3) Parity dependent potential

4) Phase between interfering amplitudes

5) Accurate spectroscopic amplitudes

6) Importance of higher order processes

Calculations:

1) Complete antisymmetrization of the system (time dependent

HF-calculations)(8)

2) Partial antisVnmetrization (cluster wave functions)(9)

3) Coupled channel calculation (10)

4) Molecular orbitals, two-state-approximation (11)

5) Interference with DWBA-amplitudes (12)

251

2. The general problem of (antl-)svmmetrizatlon In heavy

Ion scattering

The full antisymmetrization of a heavy ion scattering system

is very difficult and has been done so far only for few cases (8).

Usually one applies the Paul! principle only to the internal

wave functions and neglects the permutations between nucleons

of different nuclei. There is normally one term which becomes

important, namely the permutation of all nucleons of the light

nucleus with the same number of protons and neutrons in the heavy

one. This part of the wave function corresponds to the original

one with the cm-coordinate R inverted. The total wave function

is (anti-)symmetrized with respect to the core exchange (in the

a + 40Ca scattering the "core" would be the a-particle):

ft - 1/2{1I(R) + <-)AlC<-R>}. The second part of the wave function

gives rise to contributions to the cross section at backward

angles. From this point of view the elastic transfer follows

directly from the Paul! principle, and belongs to the same

class of reactions as Mott scattering.

In some cases also other reaction channels can not be neglected, 7 8

e.g. in the a + LI scattering. Here the Be + t channel is

strongly coupled to the elastic channel and must be taken

into account (13).

3. Interfering amplitudes in the PWBA-formallsm

The transfer amplitude can be calculated with the DWBA-formalism

and must be added coherently to the direct term. This is a

straigt forward method, which can be applied also to reactions

like asymmetric transfer, *.g. 1 4C( 1 6O, 1 2C) 1 80 + 1 4C( 1 6O, 1 8O) 1 2C.

It is a perturbative approach up to first order. The phase

between the Interfering amplitudes is obtained, if the transition

amplitude is calculated from the symmetrized wave function:

where V is the core-core potential and V. is the bound state

potential of core X. The wave function V is decomposed into

bound state and scattering wave functions:

252

The final result is (12) (for anqular momentum transfer

Z The sign (-)L arises from the relation \£rt fr^/f"*^^' "i«' '*'. S is the spectroscopic factor

The second direct term is small and can be incorporated in the

first one. If l._ >0 , the transfer amplitude must be added 17 16

incoherently, as shown in the 0 + 0 elastic s< attering (14) .

The treatment of the inelastic scattering in this formalism is

straight forward. The distorting potential V(R) must be re¬

placed by (2l+1)"10Ro |j[ Ylm< $,p) ( 1 ) .

If the direct amplitude describes already a transfer, e.g. 1 3C( 1 5N, 1 4N) 1 4C , the sign of the second transfer amplitud

depends also on the channel spin and isospin.

4. The LCNO-model

In the LCNO-model molecular wave functions are used instead of

nuclear bound state wave functions. This results in a non-

perturbative approach. The molecular wave functions (ft(r,R)

around the two cores are built up from a linear combination

of nuclear orbitals (LCNO) <P(r1 ,2) (in the following boson

cores and an 1 = 0 bound state are assumed):

where the scattering waves %(R) have been split up into the

gerade and ungerade part: x • Xo + Xu • ant* t n e relation

xg/u (" R ) " ± xg/u ( R ) h a s b e e n u s e d*

253

For 1 + O the molecular wave functions are yf'/t s 7S (^ ^

The eigen energy e of the molecular wave function is dependent

on R and on the g/u-syrranetry and acts as a potential in the

SchrSdinger equation of the scattering waves X_/u:

£-j£ 7/ The symmetry -dependent part of e(R) Is called exchange energy

or parity dependent potential. It has a different sign for even

and odd partial waves, which leads to a dominance of only one

parity of partial waveB in the angular distribution. The

SchrSdinger-equations above for g and u parity can be

written for all partial waves, if e »U(R) i s replaced by

2EB + K(R) + <-)LJ(R) , (EB binding energy)

A/JO , *i*V(*ó

exchange potential (DWBA form factor)

After the calculation of the scattering amplitudes ^ a/ u < t n e

total cross section is obtained by g^ = |f + f uj2. The cross

section for the direct scattering and the transfer process can

be calculated, if the sign of J(R) is fixed to + or - for

all partial waves (11):

11 - . - i2 d o . _ _ 1 ic c |2

The DWBA-transfer amplitude can be derived from the SchrSdinger

equation above by treating the exchange potential J(R) as

perturbation in first order.

5. LCNO-calculatlon for two-particle exchange

The molecular wave function of two particles Is constructed as

a product of two single particle molecular wave functions.

There are two different asymptotic situations for the bound

state:

1) both particles are bound to one core (ionic channel)

2) the two particles are bound to different cores (covalent

channel) .

254

For example in the system Si + Si the elastic scattering

represents the ionic channel and the n-transfer ^SK^Si.^SiJ^Si

is the covalent channel. The angular distributions have a

maximum at 90° (fig. 2) for the fermions Si , because the

channel spin is zero. Therefore, only the antisymmetric part

of the spin wave function is allowed (see also section 6) and

only even partial waves contribute (the total wave function

must be antisymmetric for this channel).

In the two-particle exchange four scattering amplitudes must

be calculated/ which correspond to the four linear combinations

of molecular wave functions ++, —,+-,-+. The exchange potential

is the sum of the single particle exchange potentials.

The cross sections are calculated from the scattering amplitudes

in the following way:

The LCNO-approach describes in the ionic channel simultaneous

transfer only. Recent coupled channel calculations (10) have

shown, that the sequential transfer is of equal importance.

6. Discussion of the phase between direct and transfer

amplitude

It is necessary for the understanding of the oscillatory

structure of the angular distributions to know the phase

between both interfering amplitudes. In the elastic transfer

this phase is real (+ or - sign) except for the case, that an

imaginary part of the exchange potential is introduced. A

negative exchange potential lowers the Coulomb barrier, and

the partial waves, which are calculated with -|j(R)|,are

stronger absorbedthan those calculated with +|J(R)|. The latter

ones then dominate in the angular distribution.

The sign in front of |J(R)| is determined by two symmetries

255

Fig. 1 Coordinate system

1 I* / 1

AAAa m ]

II \ DWB* cdculMwn :

W »o* wo" iso* wo* •e.

in ?ft ? Q 9Q Fig. 2 The covalent reaction channel Si( Si, *Si) SI

has a maximum at 90°, because only even partial

waves are contributing.

256

in the one-particle-transfer:

1. (-) : boson-fermion property of the cores

2. t-)1""1!: this sign emerges from the overlap Y ^ (rjJYjjj^ (r-|)

of the bound state wave functions, it determines

the g-u-property of the molecular wave functions

(besides the - combination).

Generally the lowest m^-value produces the largest overlap of

T-i*^ • A s i n atomic physics the molecular bound state can be

classified in o-,ir-,... bond for mi = 0,1,••• . An additional

minus sign arises from the negative sign of the bound state

potential, and the + or - sign of the linear combinations of

the molecular wave function is also in front of J(R) . The

final result for the sign is +(_)A+l-mi+1 #

It is also possible to describe a hole in a closed shell nucleus

as a valence nucleon instead of using a fermlon core. The

exchange energy has then the opposite sign than that of a

particle. Therefore, one obtains the same final sign as for a

fermion core.

The above considerations are also valid in the DWBA-formalism

besides for the sign (-)ml . It is replaced, bv (-)rat , where

mt is the projection quantum number of 1 . But only the term

with 1 = 0 remains coherent with the direct amplitude in the

cross section:

^ 2

If a two particle transfer is considered, which leads to two

members of an isospin multiplett oimaia C, e.g. 1 c (' 5N, I4N) UC,

the total channel spin S and isospin T enter into the phase (15):

Only the relative phase between f. and f, is important here,

f 1 and f2 have different spectroscopic amplitudes 0A/J< /

which might have different signs. This can introduce another

minus sign between f1 and t^ (15).

For the asymmetric transfer li!:e 14C(16O,'2C)180 the transfer

amplitudes are added up in the following j»ay (7):

257

In all cases the even (odd) partial waves dominate! if the phase

between the two interfering amplitudes is +(-) - Many data have

been interpreted consistently with these phase rules.

7. The strength of the exchange potential and higher order

contributions

The exchange integral J(R) can be evaluated with the Buttle-

Goldfarb approximation (16). It has the Yukawa-type shape for

J ( R )

with a • /2mEB/h decay constant

6 spectroscopic amplitude with e = S spectr.factor

N normalisation constant

A better calculation performs the integration of the exchange

potential numerically in the outside region. The inner region

is not important, because it is masked from the absorption.

Since the spectroscopic factor enters into the strength of J(R),

the depth of the interference structure is very sensitive on it.

Very precise measurements of (9N) can be made (17).

The importance of the elastic transfer reaction is directly

connected with the range of the exchange potential, given by o.

In general the strength at the nuclear surface decreases with

higher mass transfer and higher binding energy. Several exchange

potentials are plotted in fig. 3. Recent calculations of Baye (18)

demonstrate the splitting of the parity dependent potential for

very asymmetric systems like 1 60 + 28Si . The available data (4)

of the elastic scattering of this system show a small backward

rise at large angles, which was fitted with a Regge-pole analysis.

The relative large cross section at backward angles can be

physically interpreted as contribution of the parity dependent

potential (fig. 4).

If the exchange potential is strong, higher order effects

(multiple transfer) can be investigated with the LCNO-model.

858

i—r—i—i i—r—i—)—i—I—I—r"

2 -

,»Sj-30Sj

,BN.i60 Exchange Potential -I Bum* • GoMfaibl

Exchange Potential (numint»gra»iofi)

Fig. 3 Exchange potentials for different systems, which

are calculated by numerical integration (right)

or with the Buttle-Goldfarb approximation (left).

M

m

10

10

-3

10 -9

10

i i i i i i i i i i i i i i i — r r

• • •! » ' 30 60 90 120 ISO 180

Fig. 4 Elastic transfer calculation for 160 + 28S1. E L a b - 50 MeV.

The direct and transfer cross sections are indicated

by dashed lines, data points are taken fro* ref. (4).

260

The transfer amplitude must be expanded in a power series of the spectroscopic factor: f^r = Sfi(8) + S3f2(8) + ... Higher order effects can be seen in the cross section as deviations from the constant atr/s2 . These effects are expected for systems, where the exchange potential is not small compared to the optical potential. The systems N + O and 1 60 + 1 70 are good candidates for this effect (fig. 3). The analysis of the latter system has shown, that higher order contributions are present for a shallow potential, whereas they disappear with a deep potential (19). The Bame investigation has been made for the C + 0 system with coupled channel calculations (20), where the effąct of multi-step transfer has

ELab " time is much faster than the collision time. On the other hand CGcalculations for the system a + 7Li could not confirm the multiple transfer mechanism, which was derived from time arguments (21).

8; Conclusions

The elastic transfer reaction belongs to a more general class of reactions, which has to be calculated with an (anti -) symme¬ trized wave function with respect to the "core" exchange. A parity dependent potential emerges from this ansatz, which may be important in a more general way for the elastic scattering and transfer reactions. At large angles the direct scattering amplitude is strongly reduced by absorption, while the elastic transfer has its maximum strength at these angles. Therefore, even a large mass transfer may compete with elastic scattering in this region and give rise to a backward rise of the cross section.

The interference structure has been investigated carefully and phase rules have been compiled for the elastic and other symmetrized transfer reactions. The non-perturbative approach with the LCNO-model has allowed to investigate higher order effects. Possible multi-step contributions are present in the systems 12C + 1 60 and 1 60 + 1 70 , where the strength of the exchange potential is rather large and the core-core potential is well iciown from other data.

261

References

(1) W. von Oertzen and H.G. Bohlen, Phys. Rep. |9C (1975)

(2) G. Delic, preprint LBL-3492 (1975)

(3) W. Stinkel and K. Wilderrauth, Phys. Let. 4JB (1972) 439

(4) P. Braun-Munzinger et al., Phys. Rev. Let. 313 (1977) 944

(5) C K . Gelbke et al., Hucl. Phys. A219 (1974) 253

(6) J. Barrette et al., Nuci. Phys. A261 (1976) 491

(7) W.F.W. Schneider et al.. Nuci. Phys. A251 (1975) 331

(8) S.E. Koonin et al., Phys. Rev. CJ_5 (1977) 1359

(9) D. Clement et al., Phys. Let. 5_5B (1975) 19

(10) G. Baur and H,H. Wolter, Phys. Let. 6JSB (1976) 248

(11) W. von Oertzen and W. Norenberg, Nuci. Phys. A2O7 (1973) 113

(12) G. Baur and C.K. Gelbke, Nuci. Phys. A2O4 (1973) 138

(13) F.Becker and I. Strykowski, Nuci. Phys. A289 (1977) 446

(14) C.K. Gelbke et al., Phys. Let. 4_3B (1973) 284

(15) A. Gamp et al., Nuci. Phys. A25O (1975) 341

(16) P.J.A. Buttle and L.J.B. Goldfarb, Nucl. Phys. 7_8 (1966) 4O9

(17) H.P. Gubler et al., Nuci. Phys. A284 (1977) 114

(18) D. Baye et al., Nuci. Phys. A289 (1977) 511

(19) H.G. Bohlen and W. Norenberg, Phys. Let. 4_9B (1974) 227

(20) H. yoshida. Nuci. Phys. A257 (1976) 348

(21) H. Kelleter et al., Nuci. Phys. A210 (1973) 5O2

The Break-Op 01 Complex Particles Into Continuum

A. fludzanowekl

Inst i tute of Nuclear Physics, Cracow Inst i tute of Physics, Jagellonlan University

Break-up of complex nuclear project i les In the f ield of the target nucleus lias been studied for quite a long time. In fact, the f i r s t idea of the break-up of deutjron in the coulomb field can be traced back to the paper of Oppentielmer and Phil l ips / 1 9 3 5 / 1 ' . It i s quite natural that most of the experimental evidence obtained so far on break-up phenomena concerns systems with low binding energies namelyj deuteron /n+p/, Ll/d+*He/

9 8

and Be/n+ He/. A review of the existing experiments together

with excellent description of the break-up phenomenon in the frame of the CWiiA approximation with a good account of the

2/ older theories has been given by Laur and Trautmann . Recently, in connection with studies of preequillbrlum

phenomena in nuclear reactions, a vast ajnount of continuous spectra of particles emitted in reactions induced by fast project i les /50 - 140 L.eV/ has been measured in a broad enerj;) and angular range. At forward reaction angles these spectra show a dist inct peak of the width of several tens of MeV centered around the energy C ^ - ""%)a, £Q. where ""V-

. - -.«_ o-the mass and energy of the Incident project i le , „in accordance with a ;-iraple "spectator particle" model these broad peaks can be ascribed to the break-up process of the Incoming particle. Some examples of the break-up peaks are shown in f ig . 1 taken from ref. 3 / / /d = n+p/, f i g . 2 taken from ref. *^ /?iie = d+p/, f ig . 3 taken from ref. 5 ' /3Ue sr <Up/ and f ig . 4 taken from ref. ' / Li = d_+ (X/ where the underlined symbol Indicates the observed particle .

263

Fig. 1. Double differential cross-sections for the Ni/d,xp/ reaction at Ed = 80 MeV taken from ref. 3 / ' .

Fig. 2. Double differential cross-sections for the 90 3

Zr/ He.xp/ reaction at E_ = 70 UeV taken from ref. * / . 'ae

26U

Tig. 3. Double differential cross-sections for the Zr/ Ile.xd/ reaction at £» = 70 MeV taken from

ret. V. 3He

rttf//

Fig, 4. Spectrum on deuterons from reactions Induced by Li

on A12O3 target at Kg B 35 keV taken from ref. Li

6/

In order to specify the problem let us introduce the fol¬

lowing definitions:

a + A — > b 4 (c-f

("observed] (Unobserved\

a - Incoming projectile, A - target nucleus, b - the observed

reaction product, c - the remaining unobserved part of tbe

projectile a /a « b + c/. If tbe measured particle b Is

elastlcally scattered only and c + A undergoes any reaction

fulfilling concervation laws then we call such a process

inclusive break-up. If A is left in the ground state then

tbe process is called elastic break-up. If c + A system

forms any ohannel different from the elastic one then we have

the Inelastic break-up. In terms of the double-differential

cross section for the observed particle b we can write:

/i/

Let us first consider the elastic break-up. General

reaction theory provides us with expression:

dis-

where V^_ - velocity of the incoming particle, Pr - phase

space factor for the final 3 particle state and T;r - transi¬

tion matrix elements. Neglecting for simplicity spins of the

interacting particles and assuming a. y?a ire have;

266

where mc, m , pc and p^ indicate masses and linear momenta of particles c and b respectively. Tho standardyDWBA approximation In the post interaction form gives for l', ',£ :

where /V indicates the appropriate distorted wave function, Ycfj- the potential of the interaction between particles c and

b, jfrfc fa. and ćpĄ internal wave functions of particles b, c, a and target A In their ground states respectively, Expanding

/(_ ' In partial waves and integrating over the angles of emission of the unobserved particle c wo obtain /a-Cfc<iĄ-an incoherent sum of contributions to olastlc break-up from different partial waves of the unobserved particle namely ' :

where

/ O /

Eq. / 6 / ivas obtained in the so called zero range approximation which reduces the calculation of / matrix elements to 3

g / dimensional integral. OQ Indicates the zero range constant , _4 (T) f f t

(rj IS a special function which- allows for tbe finite range effects in the so called local energy approximation . denotes the radial part of tbe wave function of the particle c scattered on target A. If c is a neutral particle /neutron/ then outside of the nuclear radius for r > It we have:

267

where It, > ^ f and JWi-» Indicate tbe spherical Dessel

function.outgoing Hanko1 function and diagonal S matrix element

of tbe scattered neutron respectively.

Using the unitarity of the full S matrix for the n - A

system and neglecting tbe contribution to stripping Integral

from tbe lntorlor of tbe nucleus the following eq. bas been

derived for the inelastic break-up cross-section by Baur,

Trailtunn and Eoesel *i

where Vj£ and U^ indicate total reaction and total elastic

cross-section for tbe partial wave 1 In tbe n + A system,

Tim iB £ l v e o D> °4* /6/« ~^t*n is t n e s a m e a 9 S l v e n by 6<1* /6/ with -fl replaced by jl . This last term ia called sometimes the unscattered part of tbe elastic break-up.

A good Insight into tbe physics of tbe break-up processes

can be obtained with tbe so called "spectator particle" ap¬

proximation. The basic assumptions of this model are illustra¬

ted in fig. 5.

The essential assumption of the spectator model is that

only particle C Interacts with the target nucleus. Thus,

particles a and b are represented by plane waves. Starting

from tbe DV/BA expression /*/ C. Baur11' bas obtained the

following formula for the elastic break-up T matrix:

268

CL

Fig. 5. The spectator particle graph.

where t Is the off energy shell t matrix element for the

o - A interaction,' A = po.~p{r Is the momentum transfer,

& — momentum of the outgoing particle c, Q~ fór&j-h."Plr

and fo^fct) indicates the Internal momentum distribution In

the ground state of particle a. Neglecting the off shell

dependence of t and Integrating over the angles of emission

of the unobserved particle c we get a simplified expression

for the elastic break-up cross-section?

/10/

where * Is an adjustable constant.

For practical purposes ~f(Q.) can be easily evaluated

using the harmonic oscillator wave function Tor the relative

motion of particles b and c inside a.

269

ID case of the

where

relative motion we have:

fc&) / u /

The harmonic oscillator frequency

can be evaluated from tbe known values of the mean square 1°/ radius of the matter distribution of the particle a .

Recently tbe break-up process has been observed also for strongly bound projectile namely Of particle /neutron binding energy 20.G UeV/10//. In fig. 6 taken from 1 0 / ' the inclusive spectrum of hellons from Nl /Or 2~/ reaction

'Lair

measured at Jflllch Isochronous Cyclotron /li^. = 172.5 iieV/

Is shown.

[ i\_ 122 w i

Fig. 6. Inclusive He spectrum from reactions Induced by 172.5 UeV alpha particles on 60Ni target at Q. . = = 4.5°. Arrow Indicates the three body threshold. From ref. 1 0 / .

270

The three sharp lines on the right hand side of the spectrum correspond to bound state stripping to if,/,, and lgn/o

single neutron states in Ni which are strongly excited due to the angular momentum mismatch between Incoming alphas and outgoing helions. A broad peak centered around 122 ileV cor¬ responds to tne break-up process. In fig. 7 the double dif-ferential cross-section for the N1/0T,2V reaction at Ea- = 172.5 Ł2eV is shown. The measured spectra where summed in 6 MeV bins so that the bound state stripping peaks are averaged out. As can be seen the break-up peak is < Nearly seen at forward angles and practically vanishes around 20°. The oross-section for helion energies around the break-up peak decrease with increasing reaction angle by several orders of magnitude. For lower energy helions the drop off of the cross-section with angle Is much smaller so that for angles higher than 30° the spectrum takes a shape typical for the preequillbriutii process. Inclusive break-up cross-sections calculated using DiVEA theory with commonly accepted parameters /local energy approximation/ are shown by continuous lines. As can be seen at forward angles the inclusive break-up process can explain the position, width and the absolute value of the cross-section at the broad peak. a.t lower helion energies the discrepancy is too large to be removed by any parameter fitting so that other mechanisms are needed to explain the data e.g. the preequillbrlum model in the angular dependent form ' or multistep direct and compound model . The contribution of different partial waves of the transferred neutron to the break-up peak at forward angles are shown In fig. 8. The dashed line Indicates the elastic break-up. It is seen that higher £ values dominate the peak with increasing n lie energy. Thus the reaction selectively exclts favoured £, values. This opens a possibility to use / ?~/ reaction

for studies of the single neutron strength distribution. We note also that the elastic break-up accounts only for about 25% of the total inclusive cross-section.

871

Fig. 7. Double differential cross-sectlont for the 2N1/ lie/

reaction. Pull lines Indicate results of the D1VBA

calculations. The energies corresponding to the

ground state transition and three body threshold are

Indicated by arrows. From ref. i0'.

"Another interesting aspect of this reaction is the behaviour

of the cross-section at the three body threshold. If the

neutron energy approaches zero the elastic break-up crose-

-section tends to zero, so at EQ = O the total break-up

Is pure Inelastic. Calculation shows that the strongest con¬

tribution at the threshold comes from absorption of neutrons

with 1 » 4 and 5.

Some physical Insight into the break-up mechanism can be

obtained from the simple spectator model. In fig. 9 fit

Fig. 8. Calculated contributions of various neutron partial waves to the total Inclusive cross-section for the break-up process /full lines/. The "elastic break-up Is indicated by dashed line. From ref. 1 0 / / .

Fig. 9. Spectator model fits to the inclusive cross-section at 'Jla, = 5.5°. Full line indicate total Inclusive coulomb corrected cross-section. Dotted line pure inelastic. Dashed line pure elastic. Dot dashed total inclusive without coulomb diBtortlon.

273

obtained with formula /10/ at 5,5° deg lab is shown /dashed l ine/ . The internal momentum distribution has been calculated from the harmonic oscillator function assaaing ^/T^-~y ? 2,74 fm in accordance with electron scattering data. As can be seen the spectator peak is shifted by about 10 JleV towards lower energies relative to the experimental one / th i s curve lias been calculated, assuming a coulomb energy shift ~x. 13 MeV as compared to the y Ep rule / . The following correction are proposed:

1. Inclusion of the inelastic break-up cross-section by adding to the expression /10/ the following quantity;

where C is an adjustable constant, ]fQr takes Into account the phase spaoe density for '.lelion only /two body process/ and "jO* represents the Internal momentum distribution as usually. Adjusting two constants «; and C we get the total inclusive cross-section represented by the dot-dashed line. The partial inelastic contribution is represented by dotted line. The characteristic deficiency of the obtained f i t /dot-dashed l ine/ is that i t gives too much scattering on the left shoulder of the break-up peak. This can bo understood as an effect of the coulomb field on the outgoing hellons. The rutherford cross--section is proportional to ycj1" • T n i s means that hel ions

with loner energies are Bore strongly deflected from the an^le of observation by the coulomb field. Vie can easily allow for this effect by multiplying the cross-section calculated so far

T/F /"here ry- indicates the cncrcv of hclion tk t th h l i t th b k k and tZpeak represents the hel ion energy at the break-up peak.

The resulting fit is shown by the continuous line. As can be seen the fit is quite promising although wo shall not forget that i t contains two adjustable parameters.

27<»

In conclusion we can say that the break-up process i s a es tabl ished mechanism in nuclear react ions which occurs for complex incident p a r t i c l e s i r r espec t ive of t he i r binding e n e i j i e s . The i n e l a s t i c part of the break-up process may prove to be a useful tool for nuclear speciroscopy providing the off energy aliell sca t t e r ing amplitude wi l l oe well understood t h e o r e t i c a l l y .

Jtef erences

1. J.'d. Oppenheimer and M. P h i l l i p s , Phys. Rev. 4j5, "GO /1930/ . 2. G. ii aur and D. Trautmsnn, Physics Sieports 2oC, 293 /197o/ , .3. J.R. "in, C,C. Chang and II.n. Holmgren, Proc. In(,. Conf.

"uclear S t ruc ture , Contributed Papers, Tokyo trJ77, p. ~J!2. i . N. !.iatsiioka, .L. Stiiraizu, K. Uosono, ',". s a i t o , ii, iiontio,

N. Nakanislu, 1'. Toba, .-t. Go Lo ,ind i . Ohtani, Pror1. Irit. Conf. Nuclear S t ruc ture , 'JontrUnited iMpars, Tokyo 1977 p. 074.

3. >;. Matsuoltrt, A. Shimizu, ii. UOHODO , T. i a i t o , '.I. i indo , N. Kakanishi, i'. Toba, A.. Goto ami F. Olitani, Proc. I n t . Con/1. Nuclear S t ruc ture , Contributed Papers, Tokyo 1977 p. 374.

6. K.P. Artemov, 1V.Z. Goldberg, I .P . Petrov, I.K. ^erikov, •v'.P. Uudakov, '.i:.A. Timofiejev, ft. ;.olsk;i and J . ozmiaer, J. of Nucl. Phys. / i n russ ian / 22.. 2 4 i ; / l " - / .

7. G. Baur and D, rrautmann, Z. Pliysik 207, 103 /1974/ . 8. J.R. Shepard, Vi. 11. Zimmerman, J.J. i raushaar , Nucl. Phys.

fŁ275, 189 /1977/ . 9. J.1C. Dickens, R.K. Drisiso, f .C. Perey and G.U. ia tc i i l e r ,

Physics Let ters 15_, 337 /1965 / . 10. A. ŁiudzanowsŁci, G. Baur, C. Alder l ies ten , J. 3ojowald,

C. Mayer-BOriclce, .V. Oelert , ' . Turek, F. Koesel and D, Trautmann, Phys. liev. Letters /1978/ to be published.

11. C-. Baur, Z. Physik A277. 147 /1976/.

12. vi. Bohr, B.K. iiottelson, Nuclear Structure Vol. 1, p. 220, W.A. Senjawin, Inc. 1969.

13. G. Mantzouranis, H.A. "i7eidenjaailer and D. Agassi, 7.. Physik A276. 145 /1976/, U. Machner, Ins t i tu te fttr Iiernphysik der KFA Jaiick, Progress fieport /1978/.

14. E. Feshbach, Proc. I n t . Conf. on Nuclear Ueaction Mechanisms, Varenna June 1 3 - 1 7 1977, p. 1.

275

ANGULAR MOMENTUM IN HEAVY-ION REACTIONS

H.OESCHLER Centre de Recherches Nuclśaires, F - 67037 STRASBOURG Cedex

The title needs lome explanations as I don't intend to cover the whole subject of heavy-ion physics. On the contrary I have selected three different themes which are all connected by thi; standard figure used in heavy-ion physics. : the distribution of thr partial reaction cross sections <r. versus / . Formerly with light-ion reactions there was a simple division of all reactions into compound-nucleus formation and direct processes.

Now in heavy-ion physics more types of processes are observed :

r 276

Even this variety of fusion-evaporation, fuaion-fission, deep

inelastic collisions or quasi-fission and quasi-elastic is not complete. In

the scattering between heavy ions usually a strong Coulomb excitation of

the low-lying states occurs. Therefore one has to add the dashed line.

Now it should be precisely defined what is meant by the reaction cross

section. The solid line neglecting the Coulomb excitation or the dashed

one as the strict definition of a_ .

xl

The first part will deal with this problem : How to obtain the

reaction cross section from the elastic scattering in cases where a

Coulomb excitation is present. The Coulomb excited states can not be separated

from the elastic scattering in usual counter experiment. Can this "generalized"

elastic scattering be used in a consistent way ? Several methods to calculate

the reaction cross section from heavy-ion scattering are available. Not all

should be applied to cases where Coulomb excitation is present.

In the second part heavy-ion induced transfer reactions above

the barrier will be discussed. Most of the experimental results observed

in the reaction Si on Te agree with other transfer reactions proceeding

by one-step. Yet a detailed study reveals that this reaction exhibits

characteristics similar to deep inelastic collisions only explainable by

multiple interactions. The analysis of the data is carried out in the framework

of the diffusion model. The evolution from quasi-elastic to deep inelastic

collision will be discussed.

In the third part several experiments concerning the compound

formation with high angular momenta will be summarized. It will be

restricted to one nucleus Er which is well studied by various reactions

and different experimental methods.

I. Determination of the Reaction Cross Section from Elastic Scattering in

the Presence of Coulomb Excitation

II. Heavy Ion Induced Transfer Reactions a* a Diffusion Process

III. Decay of the Compound Nucleus Er

277

I. DETERMINATION OF THE REACTION CROSS SECT iU.V i'ROM

ELASTIC SCATTERING IN TT}E PRESENCE OF COULOMB '

EXCITATION *

Elastic scattering of heavy ions often shows a Fresnel-type

diffraction pattern. The analogy with optics has lead to a simple method

to deduce the reaction cross section :1he qjartfr-point recipe. Other methods

are known and the optical-model iruilyaig is probably '.h'j most widely usvd.

In the scattering with heavier projectiles th? Coulomb excitation

of low-lying st.ites can become very important. In snmc cis? th': flux

going into an ir.clastic channel may be as strong as all lh<? other ibsrjr J/UVP

channels together. In experiments the inelastic channel c^n normally -.ici

be separated from the elastic. One is nol measuring strictly elastic

scattering and the value a deduced from it is not the reaction cross R

section. Still this procedure may be consistent and a is then the absorption

of the other channels. It is the aim of this talk to verify this point and also

to find out which method to determine a should be used.

K

Let me illustrate the problem with two examples. In fig. 1 the

results of coupled-channels calculations are given. The dashed lines show

the "elastic" scattering as usually measured in counter experiments, i .e .

it contains the inelastic excitation to the 2 -states. In order to have well

defined terms we call it "generalized elastic scattering". The vaiue cr

deduced from it will be called reduced reaction cross section. The full

curves in fig.l show the strictly elastic scattering, which will give the

reaction cross section in its strict sense as the sum of all non-elastic channels.

* This work has been performed in collaboration with H. L. Hartley, Institut

de Physique Nuclźaire, Orsay. Several calculations were carried out by

D.L.Hillis, Niels Bohr Institut, Copenhagen and K. S. Sim, Centre de

Recherches Nucleaircs, Strasbourg.

278

The effect of the Coulomb excitation on

the elastic scattering. The full curves

represent the strictly elastic scattering

and the dashed ones the sum of elastic

ar.d inelastic scattering (generalized

elastic scattering)

The effect of the Coulomb excitation can be seen by a deviation from

Rutherford scattering already at forward angles.

We will base our conclusions on coupled-channels calculations ' '

•whvih we used as experimental data. Actually we started with a Q3D-expe-

riment at Heidelberg where we had excellent separation. But a6 you will

see in the following for several general conclusions a precision is needed

which can never be reached in an experiment. The parameters used in tho

coupled-channels calculations were based on the experiments. In the calculations

only the excitation to the first 2 state is included. The calculations

contain of course all interference terms present also "in nature".

The coupled-channels calculations give the reaction cross

section a_ in its strict sense and the flux going into the Coulomb excited

channel,Thus the reduced reaction cross section 'o' which corresponds to the

generalized elastic scattering can be defined by :

• ! • •

The values <r and ~ obtained from the coupled-channels calculations are

our "standards". We may now ask if the application of method X to.thc

strictly elastic scattering gives the value a and the analysis of the

generalized elastic scattering by method X reproduces the value aa.

We have chosen three methods to determine the reaction cross

section from elastic scattering : the quarter-point recipe, the optical-

model analysis and the sum-of-difference method. 279

Methode 1 : The Quarter-Point Recipe

In the framework of this method one determines the angle 0, i. at which the intensity hae dropped to one quarter of the Coulomb scattering

1 o (0 / ) = — c (0 , ). (2^

el 1/4 4 Ruth. 1/4 ' Within semiclaseical scattering theory and a sharp cut-off absorption modal g y p this angle is connected with the "grazing" angular momentum /

graz via

Here, »J is the usual Sommerfeld parameter. The reaction cvoht section Oj* ' i« then given by :

if (4)

Thi« method is baaed on the analogy to optics whore in Fresnel dif¬ fraction of light on a half-plane the intensity haa dropped to one quarter at the

geometrical shadow line. The application to nuclear scattering problems is due to Blair 2 ' and Frahn3' .

The values <rR' and o^ ' obtained from the analysis of the strictly elastic and generalized elastic scattering with this method are given in table 1 . The values of the strictly elastic scattering a ^ are much lower than the standard a . One should remember that the quarter-point recipe is based on a sharp absorption. This is evidently not fulfilled in the case of Coulomb excitation which occurs also at very fa'r distances.

Table 1 : Comp»rliem of Different Method* to Determine ReAClianCro** Section*

Btrlctly •:a«lLc

ftncrttlicd

Xcaction

r

" s .

" o .

Ms»

tMov)

i l . S

lot. 3

3 C T. I3«.S

I l .S

lOt.l

'"Te 1) , .5

(mb)

11*1

1,4.

, 3 »

(mb)

117

1U1

" »

, 0 / 4 )

(mb)

171.

740

-•(1/4)

(mb)

. 4 .

I U I

H I

CO JJ.I

n.«

44.5

-2. i

t .5

i:.3

-r l.mb)

not

l0>4

vou (mk)

(13

I t M

•77

nt I t . i

7.1

11.1

-LI

-0.4

! J _

SOD *R ftnbl

11*1

I94E.

m o

-SOD

(trbl

• Jt

1.71

•11

•

0.1

1.2

Ł1

O.i

U

280

&) A I* iht <Ult«r«Kc* In percent tMt*««i th*

In connection with the failure of the quarter-point recipe, I like to illustrate the effect of Coulomb excitation onto the normal Fresnel diffraction. This is nicely demonstrated by a work of Don Hillis studying the elastic scattering of C on several Nd isotopes . Nd has a closed neutron shell and is the most spherical of these nuclei. With increasing mass the nuclei have larger deformations and thus stronger Coulomb excitation. An effect of the increasing Coulomb excitation is seen in the washing out of the oscillations and in a. reduction of the "Coulomb bump" (Fig. 2). The disappearence of the Coulomb bump is a rather important fact, as it is typical in heavy-ion scattering (see e.g. the scattering of S on Te displayed in fig.l ), In fig. 3 there is an example of elastic scaitering of

AT on heavy nuclei . No "Coulomb bump" is seen and this can be

Fig.2 The strictly clastic scattering of C on various Nd isotopes ahow the reduction of the "Coulomb bump" and the disappearance of the oscillations at forward angles as influence of the Coulomb excitation which increases with the mass of the Nd isotopes (from ref.4).

reproduced by optical model analysis but the Fresnel model gives rather strong oscillations. The existence of a sharp absorption radius will necessarily cause the Fresnel oscillations. The inverted statement tells you thŁt missing oscillations in elastic scattering are caused by a soft abeorption or coupling effects and that consequently the quarter-point recipe cannot be used. This can be seen from table 1 where also the value cr for the generalized elastic scattering of S on Te deviates by 22 "» from the right value.

The quarter-point recipe should not be used in the pretence of Coulomb excitation neither for strictly nor for generalized elastic scattering.

281

OJ

O.OI

0.001

266 MeV

•W'lOJ .,•,,•0111 y , • > • > > '

-inwu

10 50 60 70 80

10

0.1

OOI

286 MeV

_L

• EXPftlMfNI — OfllCAl WOO! I

- V 68.0 -WI19 ..•0,'O.MD , , . . , . UW

30 to SO 60 70 80

dog)

Fig.3 : Generalized elastic scattering of T"Ar on heavy nuclei showing

no "Coulomb bump" and no diffraction pattern (from ref.5).

. 40

Methode 2 : Optical-model analysis

We have 'jCŁttŁd the elastic scattering of the coupled-channels

calculations using the optical-model code GENOA . The nuclear

potential was parametrized in the usual Woods-Saxon form.. The result

of fitting the strict'y elastic scattering of O on Nd'is shown in fig. 4 aj.

The fit is not very good. There are e.g. difficulties in reproducing the

oscillations at forward angles. In the upper part of fig.4 a)the difference

between the two results is shown. Besides a small region it docs not exceed

the 5 % level. The results fitting the generalized elastic scattering of O

on Nd as given in fig. 4 bj, demonstrates an excellent fit over the whole

angular region. The differences are of the order of 2-3 ",c . This is a case

showing typical Fresnel.diffraction. In fig. 5 a) and bjthe optical-model

analysis of the scattering of S on Te are given. Kven the strictly

elastic scattering (fig. 5 a), which fUready at very forward angles falls below-

the Rutherford scattering, can be fairly well reproduced. The pai-ametcrs

obtained are not too strange. The fit to the generalized elastic scattering

(fig. 5 b) is excellent. The differences are always below the 3 °i level

282

strict ty elastic scattering

• coupled - chonnrlt catcutouens ,\

— opticol - mottet U\

\

20*

• S7.

- 5 %

\

\

generalired elastc scattering

• coupled-chonneli colculct<o

— opticol-model fit

20'

Fig.4 a and b. : Optical model fit to the strictly clastic and gcneraiizpd

elastic scattering of O on Nd. The upper parts show

the deviations between the coupled-channels calculations

anu the optical-model fits.

aoi

- " - • - -

32S

strictly clastic scattering

coupled - chonnelt calculation*

optical - fnotlel fit

•5% a

- 5 %

2Oa SD° B0° 100°

penerolizrt elostc scattering

• couplrd - chonnelc cotcolotion

— optical-model tit

UP 60° 10* »0»

Fig. 5 a and b. : Strictly and generalized elastic scattering of S on Te.

See caption of fig. 4.

Now one might already guess what the comparison between the

values a and? obtained from the optical-model analysis and the correct R K

values of the coupled-channels calculations will be. The results are given

in table 1. The optical model applied to the strictly elastic scattering give

value? which are too low. The analysis of the generalized elastic scattering

•how excellent agreement with the coupled-channel results.

At first glance one might be astonished that the optical model

analysis of the strictly elastic scattering give*-wrong res-olts.This is not a fault

of the optical model but a fault of its usual application with a Woods-Saxon

form factor. This form factor can only describe short-range interactions,

whereas the Coulomb excitation is a long-range interaction ana we should

not wonder that the results don't agree. If one insists in fitting this type

of elastic scattering one has to use long-range absorptive potentials. 1 7 8) think two are on the market ' .

But as we have seen this effort is actually not needed as the

analysis of the generalized elastic scattering gives reliably the reduced

reaction cross section. And in most ol the cases one is interested in this

quantity, which is in some sense the absorption due to strong interaction.

Up to now we can only recommend one procedure the optical -

model analysis of the generalized elastic scattering. Before I continue to

other methods I would like to'commerxt on this- point^'The first remark concerns

the parameters. The values obtained by fitting the generalized elastic

scattering are nearly identical to the ones used as input to the coupled-

channels calculations.

This result tray be an advice for those using coupled-channels

calculations : As start parameters one should use the values obtained from

the fit to the generalized elastic scattering.

As the analysis of the generalized elastic scattering works so

nicely one might ask if also the partial reaction cross sections of each f-value are

reliable. The different curves of the a. are shown in fig. 6. The points and

crosses refer to the coupled-channels calculations. The full points represent

the total absorption o, which goes up to 200 n. For the high £-vaiues

the absorption is only due to Coulomb excitation (crosses). The fit with the

optical model to the strictly elastic scattering (full line) does not reproduce

the points. Subtracting the Coulomb excitation from the total absorption,

displayed as open points , represent* the reduced reaction cross section.

The optical-model fit to the generalized elastic scattering o. (dashed line)

reproduces this curve very good.

28I»

•a £

b"

20

15

10

4

5

0

A / A / A

' • / r / /

/

3 2 s + 130

Te

• coupled - channels

x Coulomb excitation

. • (5-0,(2*). 5, \ — optical-model fit

" \ — optical-model fit

E x 139.5 MeV Lab

calculation 01

0 , ( 2 * )

to strictly elastic c f

to generalized elastic Of"

-

-

' " " " •

• r p - - , . ,,, ,,, . , - ., - , ...

20 60 SO 100 120 WO 160 190

Fig. 6 : Partial reaction cross sections o. of the coupled-channels

calculations and the optical-model fits. Only the fit to the generalized

elastic scattering "a. agrees with the corresponding curve a,

of the coupled-channels calculations.

Thus the optical-model analysis of the generalized elastic

scattering can be well recommended. It does not only give the reduced

reaction cross section but also its partial values for the different angular

momenta.

Methode 3 : Sum-of-Differences method

This method dates back to I 9C5 to two articles from Holdeman 9) and Thaler and has recently been applied to heavy ions by Wojciechowski

et al. . In this method the reaction cross section is obtained by integrating

the differences between the Rutherford scattering and the elastic scattering

over all angles. In the paper of Wojciechowski et al. a classical derivation

of thi "> Tarmula is given. The geometrical cross section TT b. contains the

total absorption and the elastic scattering between 0 and 180° :

»?• (5)

The impact parameter b' has to be large enough that all absorption

take place inside n b. . The impact parameter b and the deflection angle

0 are connected by the classical relation Z ! Z 2 e Z 1

b i = - T i T - c o t z°i ' <6>

whert Z and Z_ are the atomic numbers of target and projectile, e is the

charge and E the kinetic energy in the center-of-mass system.

Substituting b in eq.(5) by eq.(6) gives :

2 Z Z e^ 2n n

o i

O O

ffR ' / / [ 'Ruth'0' " ° ^ d" W 1

as the left side of eq.(7) is the Rutherford cross section integrated

from 6, to 180 . The angle 0. has to be chosen that for angles smaller than

6 the elastic scattering cr is equal to the Rutherford scattering cr ] K.utn.

This formula implied the name, sum-of-differences method (SOD).

Written in a slightly different way as Zv IT

/

this formula is quantum-mechanically correct and model independent.

Carrying out the transition £ -+ 0 which has to be done with a model e.g.

the optical model, the authors have shown that the correction term is small

From the procedure it is evident that if this methods works it gives correct

values both for the strictly elastic and for the generalized elastic scattering.

The results are summarized in table 1 . For all the reactions both for the

strictly elastic and the generalized elastic scattering a very good agreement

with the coupled-channel calculations is obtained.

Thus from this result one might conclude that the SOD method

is the best and most preferable. Here one has to remember that the comparison

was made to the precision of the computer codes. Tbe practical use requires

286

to

i 0

-10

some remarks. It is not the problem that you don't have on your computer

file i program doing the integration for you, but the problem that you have

to choose an angle 6^ v/hat might depend very much on the errors of the

experimental data. In order to illustrate this problem we have plotted in

fig. 7 the integrand of eq.(8) for every angle. It demonstrates that the main

contributions in the scattering of S on Te aitse from the range around

the grazing angle. The error bars have beer, drawn under the assumption

that the scattering cross sections were known to ±\"!t . This is already a

very optimistic situation. One can see that for the generalized clastic

scattering the choice of Oj causes no problems and that the errors at the

"orward angles do not influence the reaction cross section very much. Y<?1 the

integration of the strictly elastic scattering is evidently very difficult. The

choice of 0 is rather critical or even impossible v/Hfo these errors bars.

A rather important contribution to the reaction cross section arises

from the forward angle region.

Consequently the SOD method is for practical use also restricted '

to generalized elastic scattering.

32_ 130. S + Te

• ttnetly clastic tcottennfl

. generalized elastic scattering

Error bars represent on "error" of

80° ~w~ ito8' no-

Fig. 7

The integrand of eq (8) vs

angle shows that the main

contributions to o" and -SOD . . *

<r_ arise from the range

around the grazing angJe.

Some error bars are shown

representing an assumed

uncertainty of i fe in o (0).

e,r

287

Summary

The quarter-point recipe should not be used in heavy-ions

scattering neither for strictly nor for generalized elastitt scattering.

The optical model with the usual Woods-Saxon form factor

can only be applied to generalized elastic scattering. Then it gives also

the correct reduced partial reaction cross sections "a^ .

The sum-of -difference* method is for practical cases

restricted to generalized elastic •cattering.

References

1) code CHUCK, written by P.D. Kunz, University of Colorado

Z) J.S.Blair, Phys. Rev. 91(1 954) 1 21 8

3) V/.E.Frahn, Nucl. Phys. 75(1966)577

4) D.L.Hillis, E.E.Gross, D.C.Hensley, C.R.Bingham, F . T . Baker and

A.Scott, Phys.Rev. CJ_6(1977) 1467

5) J.R.Birkelund, J.R.Huizenga, H.Freiesleben, K. L.Wolf, J. P. Unik

and V.E.Viola ; Phys.Rev.CJ2 (1976) 133

6) Code GENOA, written by F.G. Perey, version by B.S.Nilsson,

Niels Bohr Institute, Copenhagen

7) A.J.Baltz, S.K.Kauffmann, N. K. Glendenning and K. Pruesi, Phy».

Rev. Lett. 40 (1978) 20

8) W.G.Love, T Terasawa and G. R.Satchler, Phy«.Rev. Lett. 22.(1977) 6

9) J.T.Holdeman and R.M. Thaler, Phys.Rev. Lett. M_(1965)8I ;

J.T.Holdeman and R.M.Thaler, Phys.Rev. 1 9, (196S) B 1186

10) H. Wojciechowski, D. E. Gustafson, L. R. Medsker and R.H. Davis ;

Phys. Lett. 63B (1976) 413

H. Wojciechowski, L.R. Medsker and R.H. Davis, Phys.Rev. Cl 6 (1977) 1767

288

n. HEAVY ION INDUCED TRANSFER REACTIONS AS A DIFFUSION PROCESS *

In this part I would like to talk about the properties of transfer rea¬ ctions induced by heavy ions, i . e . quasi-elastic processes wqth transfer of a few

nucleons. One reaction which we studied at Strasbourg, will be discussed In detail as this reaction shows features similar to those seen in other transfer processes ; but some observations are clearly in contrast to the results of the other studies. This reaction is Si on Te which was measured at 140 MeV incident beam energy at the MP tandem.

The light reaction products (around mass 28) were identified by a time-of-flight setup. At SO cm from the target the particles traversed a 10 ug/cm thick carbon foil inclined at 45° and the emerging secondary electrons were accelerated onto a double channel plate to give a fast start signal. After a flight path of 108 cm the particles were stopped in a solid-state detector which provided a stop signal and energy inforrv?tion. A time resolution of 250 ps was obtained under beam conditions. No separation for the different Z-values was provided as the Q-values allow in nearly all cases a distinction between the isobars.

Recently P.Engelstein has improved the se^up considerably. We are now using a vertical carbon foil and the emerging electrons are bent by the field of a permanent magnet onto the chan.-el plate. In addition we have now currently installed an ionization chamber to get the Z-values, too.

32 In a recent bearntime using S we reach a time resolution of I 60-200 ps. For calibration purposes we used a Br beam and in fig.l the elastic scattering of the two Br isotopes 79 and 81 on a thin gold target can be seen. By changing the terminal voltage slightly one can switch from one icotope to the other having the same magnetic rigidity. The curves are projections onto the mass axis and both isotopes are well separated. With a flight path of 140 cm we obtained a A A/A of about 1/60.

* This work has been performed in collaboration with J. P.Coffin, P.Engelstein, A.Gallmann, K.S.Sim and P. Wagner, Centre de Recherches Nuclćaires and University Louis Pasteur, Strasbourg. Partly published in Phys.Lett. 71 B (1977) 63 .

289

Fig . I

Energy-mass diagram of elastic 79 81

scattering of Br and Br on a

gold target. The solid l ines are

£ the projections on the mass axis

demonstrating the separation of

mass 79 and 81 ,

Studying the grost properties of quasi-elastic reactions

one observes the cross sections concentrated around a certain energy

(optimum Q-value) ' c lose to the elastic peak and only few nudeons are

transferred (Fig .2) . In f i g . 3 some energy spectra of the one-nucleon

stripping reaction demonstrate the concentration of the cross section

within about 10 MeV excitation energy. The individual states are not

resolved. The differences observed at different angles will be discussed

later on.

120-

(MW) •»

(0

•to*'50*

..Ułł i*

'A;

M 25 2« » 21 2i JO

MASS

Fig.2.: Two-dimensional spectrum of the reaction 28Si on ' 3°Te

Fig. 3 : Energy spectra of mass 27 at several J--•0 290 angles.

"Si*'KU

\ elosttc tcatttring

100

(tnb/ir)

ID

. all Irnlir

pick-up

•tripping

Fig. 4

Angular distributions of elastic scattering

and quasi-elastic reactions. An optical-

model fit to the elastic scattering is

represented as a thick line in the upper

part of the figure. The curves through

the data, points of the transfer reactions

are to guide the eye.

20 60

In fig. 4 a survey of the results obtained with the reaction

Si on Te at 140 MeV is given. The upper point* represent the elastic

•cattering and the solid line a fit with the optical-model code GENOA '.

Below the sum of all transfer reactions exhibit a bell shape angular

distribution. This is typical for peripheral collisions. For small impact

parameters (large angles) compound formation takes place and at large

distances the strong Coulomb force prevents an interaction. The high

Somtnerfeld parameter *[ = 51 allows apparently the use of semiclassical

trajectories. The maximum of the angular distribution is about 10° forward

to the grazing angle and lies close to the "Coulomb bump". Similar shapes

are obtained for the individual transfer processes. Not more than four

nucleons are transferred in the stripping reactions and the cross sections

are decreasing with the number of transferred nucieons. As pick-up processes

only the one- and two-neutron transfer is observed.

The experimental <Q> -values are somewhat lower than the

predictions from the simple formula assuming clastical trajectories '.

But they reproduce the relative differences between the isotopes.

Thus all these results exhibit the same behaviour as seen in

other transfer reactions above the barrier induced mostly by O.

291

Fig.5 a)

Experimental <Q>-values vs cm angle for all

the transfer reactions. The dotted lines represent

calculations based on the matching of Coulomb

trajectories including "recoil effect" ' .

Fig. 5 b)

Contour plot of the one-nucleon transfer reaction

indicating the similarity with deep inelastic

processes.

Studying the optimum Q-value in more detail one has found

that the optimum Q-value is changing slightly with angle. This has been

discovered experimentally at Argonne ' and has been explained by the

shift in the center of gravity of the two nuclei at the instant of interaction.

The contribution caused by this effect, the authors called it "recoil effect",

is angular dependent. This "recoil-formula" has been verified in some

experiments just above the barrier .

The < Q^-values observed in the reaction Si on Te exhibit

quite a different behaviour (Fig. 5 a). Around the grazing angle the highest

<Q> -values are obtained and they are somewhat lower than the predictions 3) . of the "recoil formula" ' shown as dotted lines. At backward angles the

( Q > -values are slightly decreasing in agreement with the semi-classical

formula. But at forward angles a much stronger decrease is seen which Is

in contrast to the predictions and to other experimental studies.

292

The decreasing < Q>-values indicate a great difference between

the reaction Si on Te and other transfer reactions studied around the same

E/B ratio of 1.2

These strongly decreasing <Q> -values at forward angles are

interpreted as the onset of friction. This will become clearer if the results

are shown in another way as double-differential cross sections versus energy

and angle. (Fig. 5 b). Now the similarity with deep inelastic reactions is

evident. Most of the cross section is centered around the grazing angle and

decreasing to forward and backward angles. The projection onto the

abscissa i . e . the angular distribution, is a symmetrical bell shape (see also

fig. 4). The widths in the energy distribution of the outgoing particles at

backward and forward angles are quite different. It seems that a fraction of

the cross section follows a line from the maximum to forward angle* and

to lower kinetic energies. This part is Interpreted as due to multiple

interactions known from deep inelastic collisions . The two nuclei form

a rotating di-nuclear system and due to the longer contact time the outgoing

particles are bert to forward angles (without reaching negative angles in

this reaction). Simultaneously the multiple interactions increase the

excitation energy of the products.

In the following I would like to discuss several projections of fig. 5b.

The first projections are the angular distributions for different slices in

the excitation energy (Fig. 6). The lowest excitation energies demonstrate

10

1

0.1

Ont-nuclcon stripping slices in sxcitotion «ntrgy

A . h*- 5M*V

\ X'^-k-IO-IBMtY.

15-20M(V-

20-25 M«V-

.-30MtV

30-35MtV

100*

Fig. 6

Variation of the angular distributions

for different windows in the excita¬

tion energy showing the evolution

to the multistep processes.

293

5)

nice bell-shaped angular distributions. For higher excitation energies the

shape is broadened and the maximum is shifted to forward angles. Finally

for the highest excitation energies the angular distributions are rather flat

with some forward peaking. A spectroscopist would use this figure to

demonstrate the onset of multi-step processes. Studying the transition to

individual states the appearance of forward peaking has been observed

experimentally and successfully explained with tnultistep calculations

The observation of multiple interactions has guided us to analyse

the data in the same way as deep inelastic processes, i . e . by assuming a

diffusion mechanism. This is an alternative way which might be compared

with other possible descriptions, e.g. based on semi-classical trajectories

or on DWBA theory. In order to extract quantitative results we applied the

diffusion model in th« form proposed by Norenberg ' •

Without going into details I would like to remind you of some formulas

of the diffusion mechanism describing e.g. the heat transport or the Brownian

motion. The evolution In time of the quantity P where P is the distribution

of interest, is given by the differential equation

+ D„ J2P

The coefficient v represents the drift in the coordinate x and

D characterizes the diffusion. If the two parameters v and D do not

depend on time, a simple solution is given by a gaussian distribution :

t P(*,t) = exp

A nice illustration is the time dependence of a wave packet.

For time 0 it is a i -function and due to the drift is moves to x within

the time T and due to the diffusion D the distribution is spread 4)

UO

0 *„

29*

In nuclear physics one can study the diffusion both for the energy dissipation and for the mass transport. In a complete analysis four coefficients will be obtained. This was done for the reaction Ar on 232 6)

Th. . From the present knowledge a more complicated analysis should be carried out distinguishing the energy dissipation due to tangential and radial friction . This will not be done in the following and the aim is to compare the values obtained from a quasi-elastic reaction with those of a deep-inelastic collision.

The four coefficients to be determined are hidden in the following observable* :

V A . t = <AA

D, .t * 5- al (t) x = E,A . x z *

The most important point is the introduction of a time scale. We used the simple assumption of a rotating di-nuclear system where the difference between the grazing angle 0 and the angle of observation 0 .

B 6 6 gtux • obs. serves as a time scale

1

In order to calculate the rotational frequency w we took the moment of inertia of the nuclei in a rolling condition. There was certainly not enough time to form a sticking condition. The angular momentum of 57 ft represents not the average / -value of the quasi-elastic reactions but the lower limit / Ł obtained via

To extract the coefficients of the energy dissipation from the experimental data in fig. 7 the variance of the energy distributions is given versus the deflection angle. The angles are plotted inversely that the contact time increases from left to right. This figure characterize two domains :

(i) At the backward angles and around the grazing angle (x) a nearly constant width is seen corresponding to a direct transfer process.

(ii) At the forward angles (o) the width it increasing with the constant time. This linear dependence (solid line) is In agreement with the simple diffusion model.

»95

Fig. 7 a and b) . : Derivation of the diffusion and drift coefficients for the

energy dissipation from the one-nucleon stripping reaction.

The slope of this line allows to obtain the diffusion coefficient

Dg as a represents Ug.t and the abscissa the contact time t.

The relation of the variance versus the <Q}-values (Fig. 7b)

reflects the same behaviour. The crosses (x) at the backward angle*

characterize a constant width and a small energy loss . The forward angles (o)

exhibit a linear relation between increasing energy loss and width. This

figure supports the diffusion model as for energy loss zero the width has

to be zero. From the slope of the line the ratio D ^ V _ can be obtained as

a is proportional to D £ . t and is equal to v_ . t .

The coefficients obtained for the energy dissipation are :

_ = 5.8 x lO 2 3 MeV2 s"1

and v = 2.7 x MeV

I would like to mention a discrepancy with the simple diffusion model.

The ratio Dg/vg should be equal to the temperature of the system which is

1.1 MeV. This difference arises from the fact that fa the diffusion model

all quantum-mechanical effect are neglected. They cause an additional

spreading and the "pure11 diffusion coefficient is then smaller :

In order to extract the coefficients of the mass transport thing*

are more difficult. Here the angles is no longer a good time scale as the

different Coulomb fields in the exit channels cause a deflection to different

angles for the same impact parameters. Therefore the energy loss was

296

-4

-2 < <3

Moss transport coefficients u i eoft.Sł.*" '"' ecm'59-80

Cl ^ . 6 5 . 5 ° 11.1 ecm»71.«o

20

Fig. 8 Derivation of the mass transport coefficients.

*) Transferred n u n & A vs energy loss

b) Variance in mass vi energy loss

8)

ELot,(MtV) 40

chosen as time scale. I agree that it is rather optimistic to analyse the transfer of a few nudeons as a statistical process.

In fig. 8 a)the mean value of the transferred mass and in fig. 8 b)the variance of the mass distribution are plotted versus the energy loss . The results extracted at several angles agree very well. Here the line has to pass the origin as it represents elastic scattering : no mass transfer, no energy loss and no width. The slope in fig. 8 a)indicate! that the stripping processes are favoured. The study of the contact potential between two spheres show that it is energetically preferable to increase asymmetry, i . e . to favour the stripping processes. A similar problem can be studied with two soap bubbles connected by a small tube; the big one is getting bigger until the small one disappears.

Fig.9 The variance of the charge distribution vs the energy loss ihowi the relation between the general behaviour of deep

9) inelastic processes ' (dashed line) and the result of the transfer reaction Si on Te (dotted line) (cf. also fig. 8 b)

100 200 300 E ^ ( M s V i *rr

Usually the mass transfer is studied in deep inelastic reactions,

with a energy loss of a few hundred MeV. In fig. 9 the width of the charge

distribution between really heavy ions is shown. The dashed line marks 9)

the "universal curve" obtained in several reactions and the tiny small

dotted line is extracted in our analysis. It agrees very well with the

"universal curves".

Tram the slopes in fig. 8 a and b the mass transport coefficients

are derived :

D. = 2.1 x l O 2 2 s"1

2? -1 vA =-2.4 x i o " • .

In order to compare our results with those obtained for the

reaction Ar on Th we proceeded in the same way to extract a coupling

strength y . The diffusion coefficients can be related to a coupling strength

y, a mean energy loss per step A , an excitation energy ETx and factors o,p

taking into account the phase space :

If we make the highly questionable assumption of y_, = y , i . e .

the energy dissipation proceeds with the same coupling strength as the mass

transport one obtains

V = 2.5

and & = 3 . 6 MeV .

This is in good agreement with y = 2.07 obtained from the

deep inelastic reaction Ar + Th.

These values are calculated only to compare them with the Ar

on Th reaction. The relatively large value of £ ,the mean energy loss per step

indicate already that one has mainly radial friction and that the assumption

VE = y. is not fulfilled 7 '

Analysing the energy dissipation I only showed the one-nucleon

stripping reaction. In fig. 10 a and b the result for all the observed transfer

channel exhibit the same tendency a constant width around 0 and increasing

298

100

50

1 i

outgoing

- • 25 ' 27 o 29

•

a

i i

1 mass

/* j ^ /

1

1

t 1 ' 1

1

'groi

Z ~ 50

varianct around 0 g r „

sot 60 ec A Cm "groi

Fig. 10 a andb)

Dependance of the variance in the

energy distribution on the deflection

angle for all exit channels.

c)

Values of the variance obtained

around 0 vs the transferred graz.

mass A A.

- 4 - 2 0 2 transferred nucleons A A

width at the forward angles. I would like to stress that-this effect cannot

originate from particle evaporation after the transfer : It has been shown '

that equal temperature can be assumed for both partners and consequently

the excitation energy is divided according to the mass, i. e. nearly all

excitation energy is taken by the heavy partner. This is supported by the

fact that the pick-up processes exhibit the same tendency as the stripping

reactions which would not be the case if the observed variance* are

disturbed by sequential particle emission.

An interesting fact which I would like to point out, l.« the

constant width around the grazing angle which Is different for etch channel.

Plotting these widths versus the number of transferred nucleons a simple

relation is seen (Fig.) 0 c). It indicates that the more-nucleon transfer

proceeds with more interaction steps than the one-aucleon transfer.

*99

ID fig. 11 the cross sections of the different transfer channels

show a smooth envelope as known from deep inelastic collisions. This is

a rather interesting point and Iwould like to compare this rermlt of Si on Te

with the results obtained by the transfer processes induced with O and S

on Te (fig. 1Z). All three reactions are at comparable energies. The

cross sections were normalized to the one-nucleon stripping reaction.

The overall character is the same for all the three curves : steep fall-off

for the pick-up reactions, enhancement of the stripping processes. But

a remarkable difference is seen in the four-nucleon stripping reactions.

The O-induced processes are by a factor four stronger than the ones 28 32

induced by Si and S. The ground-state Q-value• do not restrict any

of t'.ietn. As the four nucleons are probably an a-particle in all the reactions

the only spectroscopy enters due to the different projectiles. Indeed the 1 A 7 ft %"}

ct-spectroscopic factor in O is 50 % higher than in Si and S. But

this does not explain a factor four. We think that different mechanisms may

resolve this discrepancy. From the analysis I presented it is likely that 28 32

the four-nucleon transfers induced by Si and S proceed via sequential

transfers washing out all the structure. Whereas the O-induced reactions

may proceed via an a-tranafer.

24 26 26 30

100 -

4

b

—

—

-

Fig.II

Cross section of the various mass transfer

channel*.

-2 0 +2

300

.•-100 3

<

10

MSi

0 i

•

1 1

* T e E lob

. , 39MeV

• HOMeV

• 90MeV

v\

-

t i

-4 -2 0

AA

Fig.12

Cross sections of various mass transfer

channels induced by G, Si and

S on Te. A remarkable difference

is seen in the four-nv.cIcon stripping

reaction.

Before I finish the discussion of the transfer reactions I would like

show two preliminary results of calculations to describe the extreme situations.

One extreme is the one-step process calculated in the DWBA framework.

We used the code DWUCK with a generalised form factor ' and as we

are not calculating transitions tDindividual states we have to introduce a

strength function. The results are given in fig. 13 a) as double-differential

cross sections showing Uie expected behaviour of a bell-shaped angular

distribution arou.id the grazing angle and < Q^ -values decreasing with

angle. Thus in complete agreement with the model based on semiclassical

trajectories.

The other extreme situation is a completely statistical process.

Grange and Richert from Strasbourg calculated this reactions with the diffusion

model used for deep inelastic reactions (The model of the common market

(Copenhagen, Munich, Oi say)with the linear response theory and classical

t-ajectories from the Wilczynski model }. Vet at these low energies this

model is questionable. A preliminary result is chown in fig. 13 b).

The deep inelastic part can be characterized by a line from the grazing angle

to forward angles and to smaller kinetic energies. The superposition of

these two calculations describe qualitatively our experimental observations

(fig. 5 b). A quantitative description canno' be expected from these simple

calculations. 301

1.1-

I — MO-Ui

M

fO-

U

' łO

W-

to*

DWBA

w

eem wC Fig. 13 a) Result of DWBA calculations for the one-nuci eon stripping process plotted as double differential cross sections vs energy and angle

b) Result of the diffusion-model calculation displayed in the same way.

28,

I30„

Summary We have analyzed quasi-elastic processe induced by ""Si on

"Te above the barrier with a diffusion model. The extracted coefficients agree astonishingly well with those obtained from the deep inelastic collision

AT on Th. I would like to stress that no obvious distinction between the domain of one-step processes and the region of multiple interactions can be made showing a continuous evolution. Several observations are indicating the onset of multiple interactions which is seen at a relatively low energy (E/V =1 ,2 ) .

The discussion in the framework of a diffusion model is certainly not the only possible one. For example the observed enhancement of the stripping reactions is a common features of the optimum Q-values based on semiclassical trajectories, of DWBA calculations and of the contact potential discussed briefly. Several observations e .g. the decreasing <Q>-va¬ lues at forward angles, can certainly not be explained by semi-classical trajectories nor by DWBA calculations.

References

t) P.R.Christensen, V.I.Manko, F.D.Becchetti and R.J.Nickles, Nucl. Phys. A207(1973)33.

2) Code GENOA by F. Percy, version by B.Nilsson, Niels Bohr Institute, Copenhagen,

3) J.P.Schiffer. H.J.Kórner, R.H.Siemssen, K.W.Jones and A.Schwarzschild, Phys.Lett. 44B (1973) 47 and N.Anantaraman, K.Katori and J.P.Schiffer, Symposium on Heavy-Ion Transfer Reactions, Argonne National Laboratory (1973) i Argonne, niinois.

4) J.Wilczynski, Phys.Lett. 47B (1973) 484 5) For example, D.K.Scott, Proc. of the Int.Conf. on Nuclear Physics,

Munich (1973), edited by J.de Boer and H. Ming. 6) W.NÓrenberg, Phys.Lett. 52B, 289, 1974 and J.Phys.3.7.0976) C5-I4I 7) S.Ayik, B.Schfirmann and W.NÓrenberg , Z.Physik, A277 (1976) 299

S.Ayik, B.Schfirmann and W.NÓrenberg, Z.Physik, A279 (1976) 145 G. WolBchin. and W.NÓrenberg, Z.Physik, A284 (1978) 209 S.Ayik, G. Wolschin and W.NÓrenberg, preprint Max Planck Institut, Heidelberg MPIH - 1977 - V39

8) L.G.Moretto and R.Schmitt, J.Phya. 3_7, (1976) C5-109 9) J.R.Huizenga, J.R. Birkelund, W. V.Schroden, K.L. WoU and V. E. Viola,

Phys.Rev.Lett. .37 (1976) 885 10) H.Kamitsubo, Proc. of the Symposium on Macroscopic Features of

Heavy-Ion Collisions, Argonne National Laboratory (1976), Argonne, Illinois

11) H.Oeschler, G. B.Hagetnąnn, M. L.Halbert and B.Herskind, Nucl. Phys. A266J1976)262.

12) H.Hofmann and C.Ngo, preprint Institut de Physique Nucleaire Orsay IPNO-RC-77-06 and references therein.

13) K.Siwek-Wilczynska and J.Wilczynski, Nucl. Phys. A264 (1976) 11 5

. DECAY OF THE COMPOUND NUCLEUS ' 6 2 Er *

In this talk I will summarize several experiments etudying the

complex Er :

Chalk River ^

! 4 6 E L = 65-110 MeV 1 6O + ! 4 6 Nd L

3 2 S + 1 3 0 T e E, = 120-1 63 MeV measured : fusion-evaporation cross sections via y-technique

V-multiplicities Copenhagen-GS1 Darmstadt

4 0 Ar + 1 2 2Sn E L = 3. 6 - 5.9 MeV/c 8 6Kr + 7 6Ge E. = 3. 6 - 3.9 MeV/c

measured : y-multiplicities

rel. fusion-evaporation crosi sections

Heidelberg »

3 ZS + 1 3 0 Te E. =163 MeV JU

measured : energies and masses of heavy fragments

What i» a well behaving compound nucleus ?

It decays independent of its formation

For the angular momentum range below 60 h this is nicely demontfntec

by the Chalk River data. The excitation functions of several decay channels

vs the excitation energy agree for the compound formation with O on

Nd and S on Te. Also the measured y-multiplicities for these

two reactions indicate that the maximal spia of the compound nucleus

* These experiment are carried out in collaboration with J. Barrett*.

A.Gamp and H. L.Harney, Max Planck Institut fu> Kernphysik, Heidelberg

30*

Fig. l I of a fu»ion-evaporation process 'max determined from y-multiplicity measure¬ ment* v» excitation energy of the compound nucleus formed by

86Kr on Ge (solid points) and Ar on Sn (open points)

(from ref. 2)

IO 4A to M too no uo EXCITATION EHCTOł IHtV)

increases with "]f E _ - V, (proportional to angular momentum brought

162,

cm Into the system).

At the GS1, Darmstadt the composite system i M E r was formed

with higher angular momenta and higher excitation energies. The results

of the v-multiplicity measurement* of the reactions Ar on Sn

and 86Kr on Ge are reported in a recent letter '. The main features

as shown in fig. 1 can be summarized as :

(i) At low energies £ increase* proportional to / rai(without f-window and without Jf>

cri(.). (ii) The values obtained for 8 Kr on Geand Ar on Sn agree with the

usual compound formation and show no entrance channel effects, (iii) A sharp upper limit of 65 h Is seen for the fusion-evaporation process

in the reaction Ar on Sn. Two explanations for (iii) are possible ; Either for /-values

higher than 65 h no compound system can be formed or the compound nuclei with spins greater than 65 h decay by fission.

Fig.2 shows the result* of a time-of-flight experiment carried out at the highest energy of the MP tandem at Heidelberg. Besides the elastic and quasi-elastic region a group of events are concentrated around mass 60. near half of the mass of the compound nucleus. The energies of .these products can be described by the Coulomb repulsion between two spheres (solid line in fig. 2). For symmetric fragmentation the total kinetic energies agree with the Viola systematic! 3 \ The projection on the mass axis exhibits a distribution with a mean value 79.5 u (A C N / Z * 81) and a width of 22 u-All these observation* allow the Interpretation of a fUaioning compound nucleus.

305

» 30 łO tO tO PO PO

-09

M«M fptctrum 4

M -

X •

30 W CO 10 WO IK •fan

Fig. 2 j Energy-mass diagram of the reaction i

S on Te. The solid line represents the Coulomb repulsion between two spheres with the given parameters. Below the mass spectrum of the fission events is shown.

The onset of fission around this energy can be expected from the rotating liquid-drop model . In fig. 3 the lines of stability for fission barrier of 0 and 8 MeV are shown. For the nucleus 162 Er the fission

4) barrier reaches 8 MeV at / = 69 n according to the CPS predictions For higher ^-values fission of the compound nucleus is dominating and the low spin compound nuclei decay preferentially by r article evaporation. This limit is slightly higher than the value of 65 n found in the y-multiplicity measurements.

Fig. 3 Lines of stability calculated by the rotating liquid drop model . The two solid lines indicate the angular momenta at which the fission barrier has reached 0 and 8 MeV, The nucleus Er is marked by an open circle

I 100

«0

CO

10

20

0

/

/

f

1

i

/

/ /

7 B,-8MeV-i \

if 11 100 200 )00

A

306

In fig. 4 the a a distributions of the reaction S on Te at

1 63.5 MeV are shown for various reaction mechanisms. The full curve

represents the total absorption obtained from the optical-mode'l analysis

of the elastic scattering. The dashed curve marks the fraction going into

compound formation. It results from an optical-model calculation with a

"fusion potential" (r = 1.07 fm). The calculated fusion-evaporation cross section

agrees well with the experimental one . The dash-dotted line indicates the fission

probability j ~ / \ calculated from the rotating liquid-drop model.

The overlap between the two curves yields the hatched area, the fission

cross section starting around f - 70 Ik. The calculated area yields about

half the value of the measured fission cross section indicating that the

calculated fission barrier is too high.

Fig. 4

Partial reaction cross section for

the total absorption and the

compound formation. The hatched

area represents the fusion-fission

process.

162, The compound nucleus Er is a well-behaving system. No

entrance-channel effects are observed in the excitation functions of several

evaporation processes for / -va lues below 60 ft. For angular momenta

higher than 65 ti the compound nucleus decays by fission. This is observed

directly in the reaction S on Te and agrees with the limit of the fusion-

evaporation process found in the y-multiplicity experiments with Ar on 1ZZSn. This limit of 65 fi is slightly lower than expected from the

rotating liquid-drop model.

References

1) B.Herskind, ANL-Report PHY-76-2, Vol.1 (1976) 385

B.Andrews, I.Beene, C.Broude, J.Ferguson, O.Hausser, B.Herskind,

M.Lone and D.Ward, to be published.

2) H.C.Britt et a l . , Phys.Rev.Lett.32. (1977) 145B

3) "Nuclear Fission" by R. Vandenbosch and J.R.Hulienga, Academic Press

4) S.Cohen, F.Plasil and W. J.Swlatecki, Ann. ofPhys. 82(1974) 557

Back-Angle Anomalies and Molecular Resonance Phenomena in

Heavy-Ion Collisions

K.A. Eberhard, Sektion Physik der Universitat MUnchen,

D-8O46 Garching

This lecture will cover two main topics, which have been

of particular interest to heavy-ion physicists during the

last few years. First I will try to review the experimental

situation on the so-called anomalous large-angle scattering

(ALAS). This will Include recent high-precision data for the

scattering of Lithium and Oxygen projectiles. Theoretically,

the recent success of a squared Hoods-Saxon formfactor to

reproduce ct+ ' Ca scattering at all energies, where data

are available, will be reported.

In the second part of this lecture we will discuss some

of the existing evidence for molecular resonance phenomena.

I. Elastic and Inelastic scattering at backward angles

1.1 Introduction

At the beginning of a lecture on back-angle phenomena it

maybe allowed to recall that the beginning of nuclear physics

is marked by the most surprising experimental observation

(in 1911) that a-particles could be scattered back from a

thin gold foil. Rutherford reports that to him it was as

unbelievable as firing a cannon-ball against a newspaper and

the cannon-ball came back. This was the discovery of the

308

atomie nucleus. More than half a century later, we are

again surprised by a-scattering at backward angles:

for some target nuclei the cross section at backward angles

for elastic and inelastic scattering is enhanced by up to

two orders of magnitude as compared to neighbouring target

nuclei. This time, it is the large intensity of these

backward scattered particles rather than the backscattering

process as such. Although we don't expect that the under¬

standing of the backward enhancement of the cross section

will lead to a similarly important fundamental discovery,

nevertheless it may hold - along with the observed inter¬

mediate structures in excitation functions - the key to

the understanding of high lying structure in nuclei (such

as nuclear molecular states) which are accessible these

days in heavy-ion collisions.

Before turning to the anomalous cases let us ask what

we do expect for the scattering of a projectile from a

target nucleus, such as sketched schematically in Fig.1.

Fig.1 Scattering of a projectile from a target nucleus -

schematically.

309

For large impact parameters b there is Coulomb scattering

only. For smaller impact parameters and for energies above

the Coulomb barrier the trajectory of the particle will be

determined by the Coulomb and the nuclear force. As indic¬

ated by the dashed line the trajectory can also lead to

negative scattering angles. (that this is really the case,

has been shown recently in deep-inelastic heavy-ion ex¬

periments by measuring the different circular polarization

for positive and negative deflection angles.1') Fig.2

shows the relation between the impact parameter

Deflection Function

Tl

Fig.2 Deflection function.

b and the scattering angles e . in general, small impact

parameters correspond to large scattering angles. This 'simple

classical point of view of the scattering process allows two

conclusions: first, it can be seen from the deflection function

that at the break from pure Coulomb scattering (the dashed

line) a large range of impact parameters will give rise to

scattering to the same angle 6 , thus leading to an enhancement

of the cross section at this angle. This is usually called

the Coulomb-nuclear rainbow. For one of th« curves a second

310

(nuclear rainbow) occurs at smaller impact parameters. Both

rainbows have been observed in numerous experimental angular

distributions, the nuclear rainbow only at relatively high

energies. It can also be seen from the deflection function

that scattering to large angles may occur for impact para¬

meters larger than zero, as indicated by the deflection funct¬

ion going to negative angles S. The second conclusion is

that with decreasing impact parameter the penetration of

projectile and target nucleus gets larger. This leads to

increasing absorption for small impact parameters b, i.e.

large scattering angles, thus leading to rapidly decreasing

cross sections with angle for elastic scattering.

The only major feature of the elastic angular distrib¬ utions, which cannot be described classically, Is the "diffract¬ ion pattern". Due to the fact that the angular momentum is guantized the scattering amplitude f(8) can be expressed in terms of a partial wave expansion:

£ (21+1) (S.-1) P. (cos6) (1) 4=o * Ł

with • e2iSl

Eq.(1) is the nuclear scattering amplitude for non-identical spin zero particles. Since the reflection coefficients n, and the phase shift 6^ are smooth functions with energy (in most cases), it can be seen immediately from Eq.(i) that the cross section a = |f(6)|2 will give rise to oscillations in the angular distributions. Since the odd-and-even partial waves are becoming more and more out of phase with Increasing angle, contributions from odd-and-even partial waves will cancel each other and, thus, will lead to an overall decrease with angle.

Typical examples are shown In Fig.3 for the Coulomb-

nuclear rainbow and the diffraction pattern for the scattering

of a-particles from Zr. An example Cor the second or nuclear

rainbow is seen most clearly In Figs.13c and 13d in the inter¬

mediate angular range for the scattering of a-particles from

4OCa.

1.2 Systematic! of back-angle anomalies

It has become uaual to call back-angle cross sections

anomalous if they are significantly larger than those for

"normal" cases, as e.g. those in Fig.3. In this section a

survey is given on the dependence of the back-angle anomaly

on 1) the masses of projectile and target, 2) the shell

structure of the nuclei involved, 3) inelastically excited

states, and 4) the range of bombarding energies for which

this anomaly is observed.

Projectiles and targets

Back-angle enhanced cross sections were firs.t reported

for the elastic scattering of a-particles from 16O and 12C

In 1959 (Ref.2). Many a-scattering data have been taken since;

the most dramatic enhancements were found in the vicinity

of 160 and 40Ca. For 3He projectiles the backward enhancement

Is about one order of magnitude less pronounced than for a-

particles. For target nuclei In the mass region A • 28 - 64,

Figs.4 and 5 sunmarize the observed experimental backward

enhancement for a- and He-scattering, respectively.3)

The elastic scattering cross section has been integrated

between 140° and 180°. A clear dependence on the shell

structure of th« target nuclei can be seen. This will be

discussed below.

312

10"

\<r

\QT

10

10"

,-3

o i+^Z r Elastic Scaiiorisig

Dała Optical Model

O 60 120 SCATTERING ANGLE

+90Z

180

Fig.3 Angular distribution for o+ Zr elastic scattering at bombarding energies of EQ(lab) * 21, 23.4 and 25 MeV. The solid curve shows an optical-model curve, from Ref.6.)

2.0

1.0

ja J0.5

0.2

0.1

0.05-

30

i'Ca He

I 180° dcr 140°

OPTICAL MODEL-TREND

•-c-

40 50 60 ATOMIC WEIGHT OF TARGET [amu]

Fig.4 The e las t ic scattering cross section for alpha

particles integrated over © > 140 . averaged over

aeveral energies around 2 5 + 2 MeV. The solid line

indicates the overall trend, the dashed line shows

the prediction of an optical model with a smooth A-

dependence in the parameters. Open circles are used

for targets with two or more neutrons in the f-p she l l ,

*"»ild points for others. (Taken from Ref.3.)

10

'He

: OPT MODEL'vAj I TREND ?- N.; UQ

'Ar

30 40 50 60 ATOMIC WEIGHT OF TARGET(amu)

Fig.5 Back-angle data for He-scattering. The conventions

used are the same as in Fig.4. (Taken from Ref.3.)

For heavier projectiles than a-particles only a very

few back-angle data are available. For Li+ °Ca and for

16O+

served

Si pronounced back-angle enhancements have been ob¬

4' 5' (Figs.6 and 7).

For targets heavier than Calciun no backward enhanced

cross sections were observed. Careful searches were under¬

taken in the Ni- and the Zr-mass region ' , where completely

"normal" angular distributions were observed for all isotopes

and neighbouring nuclei investigated.

Shell effects

A dramatic dependence on the shell structure of the

colliding nuclei is observed. As can be seen from Figs.4

and 5 the back-angle cross section is substantially smaller

315

10"

6Li ELAST. SCATT. E6y(lab) = 30MeV

ł DATA — OM-Calc. -

40 80 120 160 SGATT. ANGLE Qc.m.

Fig.6 Angular distributions for Li elastic scattering from Ca and Ca. The lines represent two different sets

of optical aodel potentials. (Taken from Hef.4.)

316

10 V br

i i i i i i i i i i i i i t i i i

28Sj (160 I60 ) 28S

E,Db«50MeV

E|Qb=55MeV

I I t! t I I I I I I L I -5 1 1 120 150 180 30 60

Fig.7 Angular distributions for the elastic scattering of

160 on 28S1 at Elafa - 50 and 55 MeV. Solid lines a

optical-model plus Regge-pole fits. (Rcf.Sb.)

317

if at least one pair of i-j,2 neutrons or protons is added 4O to the shell-closed nucleus Ca. This is most clearly seen

for 40Ca and 4 4Ca and for Ar and 4 0Ar. V M s systematic also holds in the vicinity of 1 6O. 8 ) In Fig.6 6Li+4OCa and Li+ Ca are compared. Again, the cross section for Ca is about one order of magnitude larger at backward angles. A comparison of the cross sections and the ratios between 4 OCa and 4 4Ca for 3He, 4He and 6Li projectiles at backward angles i» shown in Table I. The enhancement is strongest for °Ca and only a factor of 5 for 3He. The relatively low absolute cross section for Li Is probably due to the break¬ up of Li into a+d since scattering at backward angles corresponds to large momentum transfers. It is interesting

Table I

Comparison of angle-integrated cross sections for He, a,

and Li projectiles. The integrated cross sections f\ 80°

a, . = I d 0 were obtained at bombarding energies l n t ^140° ,

23-27 MeV for a particles, 28-29 MeV for He, and 3O MeV for 6Li.

°int(llbł Ratio

Projectile 40Ca 44Ca 4OCa/44Ca

3He

Li

40

2000

7

8 80

0 .5

S

25

14

318

to note that this palr-of-neutron systematics also holds

if the respective particle Is the projectile. In Fig.8

a comparison is made of the scattering of 16O and 18O

from 28Si at 180°. 5 b ) The difference in cross section at

this angle is 5«1O3 in favour of 1 60.

LU

1

Inelastic Scattering

Generally, the inelastic data are in close relation to

the anomalies observed for elastic scattering. Various

examples for backward enhancements of inelastic scattering

are shown in Figs. 9-11 and 14 - 16. The experimental data

for a+ ca inelastic scattering ' ' up to excitation energies

of about 7.5 MeV, indicate a decrease of the backward enhance¬

ment of the cross section with increasing spin of the final

l 60(28Si, l 60)28Si E|Qb=875MeV

MONITOR: 1000

l80(28Si, l80)28Si E|ab=875MeV;

M0NIT0R;4800

40 60 80 0 20 CHANNEL NUMBER

80

Fig.8 Spectra at e 0 (6_u - 180 ) for the reactions

3 CM 16O(28Si,16O)28Si and 18O(28Si,180)28Si. (Ref.Sb.)

Fig. 9 Experimental differential cross sections for the

inelastic scattering of 29.0 MeV o-particles on

Ca and the ratio to Rutherford scattering for the

elastic scattering. (Ref.9.)

320

73

KT*

6Li Eteb(6U)-30MeV

łDoło — CC-Calculations

K)H

icr*

r Jicr*

I-

I I I 1 I I I | I I I | I I I | ! I

40 80 120 160

Pig.10 Angular distributions of Li elastic and inelastic scattering from 40ca and AiC* shown together with t results of coupled-channels calculations. (Ref.4.)

3*1

O* 30* 60" 90" ł£-0* ISO*

Fig. 11 Angular distributions for inelastic scattering of 16O+28Si at E l a b = 50 and 55 MeV. (Ref.5b.)

322

state (see Fig.9). An exception is the scattering to the

+ 40

2 state at 3.90 MeV in Ca, which shows a strong decrease

toward backward angles, which is stronger than for any

other state investigated. No obvious explanation for this

behaviour has been found. Of particular interest are the

two 0 + states at 3.3S and 7.30 MeV in 40Ca, for which

a backward enhancement was found which is almost identical

in shape with the elastic scattering. 42 44 48 Inelastic a-scattering cross sections for the ' ' Ca

Isotopes ' are generally smaller at backward angles as

4O compared to those for Ca. The same behaviour is also

observed for the inelastic Li+ ' Ca scattering as shown in

Fig.10. Inelastic scattering5* of 160 to the first excited

(2 ) state of Si (Fig.11) shows a similar enhancement of

the cross section at backward angles as the elastic one

(shown in Fig.7).

In this connection it is interesting to note that no

spin dependence of the target nuclei has been found for the

back-angle cross sections for neither the anomalous nor the

normal cases. A detailed investigation, including a+ V

with a ground-state spin of 1=6 is given in ref. 3.

Energy range of anomalous back-angle scattering

The only systematic investigation of the energy range,

for which enhanced back-angle cross sections have been observed,

has been done for ot+ Ca scattering. From the combined data

sets of various groups ' the following picture is

obtained: at energies just above the Coulomb barrier

(Ect-6 MeV) the compound elastic cross section rises steeply

to its maximum value of about 1OO mb at 9 MeV and then

decreases rapidly to about 1 mb or less at 18 MeV as shown

383

by Bisson et al. Since the compound cross sections are symmetric about 90° back-angle enhanced cross sections obviously are not anomalous. However, the addition of a Hauser-Feshbach term in the optical model to account for the compound contribution did not yield a satisfactory descript¬ ion of the data at backward angles. So the effect of anomalous large back-angle cross sections starts somewhere in this energy region. Strongest backward enhancements are observed between about 20 and 30 MeV incident energy. A careful experimental determination of the compound contrib¬ utions in this energy range yielded a maximum compound

19) contribution of about 2 % . The back-angle cross

sections are therefore in the category of direct reactions.

At energies between 4O and 62 MeV a recent study of the Louvain-Cracow-Munich collaboration and the Miinster group

fint 300

200

100

, . . .

Ipb)

K • /

Elast " 1 . • .

i i i i

a**°Ca

łnelast.13")

N

. < . ,

1 1 1 ł

Scatt.

—

\ —

50 60 Ea(lab! [MeVl

Fig.12 Back-angle integrated cross section for elastic and inelastic <x+40Ca scattering. Data were taken from Ref. 18.

321*

showed that the back-angle cross sections drop rapidly

above about 54 MeV. Back-angle integrated cross sections

between 14O° and 130° are shown in Fig.12 for the elastic

and Inelastic scattering of a-particles from °Ca. A broad

maximum of the cross section both for elastic and inelastic

scattering to the 3~ 3.73 MeV state is observed between

about 45 and 54 MeV. A similar behaviour is observed for 44

a+ Ca; the absolute cross sections, however, are about one

order of magnitude smaller. This behaviour is discussed

in more detail in the next chapter.

1.3 Optical model with Moods-Saxon potential squared

Several attempts were made during the last ten years

to describe the back-angle anomalies theoretically. These

studies concentrated on potential scattering using optical

model and folding potentials, exchange effects, angular

momentum dependent absorption in the optical model potential,

and quasi-molecular resonances. A list of references to

these numerous studies can be found in the proceedings of

the "First Louvain-Cracow Seminar on a-nucleus interaction" ®'

(1973) and the recent articles by Grotowski ' '.

Here, only the recent results of the Louvain-Cracow-

Munich collaboration on a+ Ca and a+ Ca will be discussed.

For the first time, these authors were able to describe the

40 o+ Ca scattering over the entire energy range experimentally

investigated (up to 166 MeV) with the optical model using

fixed geometry and smoothly varying real and Imaginary potent¬

ial depths. In addition/ the sharp and irregular structures

at intermediate angles of the experimental angular distrib¬

utions, which have been a particular problem in earlier

325

studies, are astonishingly well reproduced. The apparent success of this study over earlier ones is based on the use of a squared Woods-Saxon potential for the real and imaginary part. The use of this form of the potential is suggestive from folding and double-folding models (for the real part). Other authors have used Woods-Saxon formfactors raised to some other power v (e.g. v=2.65 ' ) and have obtained

good description for similar a-scattering data; the energy 24) ranges , however, are somewhat smaller.

In the following, a brief summary of the comprehensive 1 fi \ analysis of the Louvain-Cracow-Munich collaboration

Is given. The 6-parameter optical potential of Eq.(2) was used.

V(r) = Vc(r) - Uof2(Rr,ar) - i W ^ 2 ! ^ ^ ) (2) with

-1 f (R,a) = (1 +exp ^ S ) (2a)

The radius parameter is R=rQ«AT'. fm, and the Coulomb radius is

Rc=1.3 • Ąj/ fm. The parameters used for o+ Ca calculations

shown in Figs. 13a through 13e by the dashed lines are:

0 o • 198.6(1-0.OO168Ea) MeV wo • Rr -

ar " Rl * al "

(2.99+0. 1.37 1.29 1.75

1.00

>AT fm

•*i fm

288 E ) MeV ' 3 fm

^ 3 fm

The linear dependence of WQ could only be established for energies smaller than 62 MeV, as only three angular distributions widely separated in energy are experlment-

326

KTL

.1OU

JOT1

ISO Fig. 13 a.

Figs. 13a - 13e Experimental data for ot+ Ca elastic scattering

and a comparison with theoretical cross sections

calculated with two different optical-model

potentials (Ref.18). Data are from Refs.14-18;

calculations are from Ref.18. 327

JO

"I—'—•—1 ""• ' 120 60 WO

r i * . ts b.

328

.10-

120 150 . 1B0 Fig. 13 e.

329

160 Fig. 13 d.

330

FIS- 1> «.

33»

ally available above 62 MeV; at these energies (100,

141.7 and 1G6 MeV) w took on the values 25.14, 23.25 and

22.88 MeV, respectively.

For comparison with a standard Woods-Saxon potential

we list Eqs. (3) and (4), which give the half-way radius

R. ,, and the 10 - 90 % distance a..,, describing the

diffuscnoss of the potential in its surface part for

the Woods-Saxon squared potential:

R. ,- = R + a £n (/2-1) fm (3)

a10-90 = a P n I ('/^O-1)/(>'1O/IJ-1) I fm (4)

As can be seen from Figs. 13a - 13e

it is possible to get a reasonable description of the

energy behaviour of the elastic scattering of o-particlcs

40 44 from Ca (and Ca , not shown here) using the optical

model with Woods-Saxon r.quared formf actors. In particular,

the energy dependence of the anomalous large angle

scattering is properly reproduced. In addition, DWBA calculat¬

ions for excitation of the 3.73 MeV (3~) state in 40Ca per¬

formed with the same potentials and with a value of the

collective deformation parameter fixed at 6,=0.22, give a

good description of the rapid energy variation of the

experimental data (shown in Figs. 14 to 16) . The

Value of 3 3 is taken from the literature and is compatible

with previous estimates.

We still follow ref.18 in studying the properties of

optical potentials in terms of the three-turning point

WKD approximation of Brink and Takigawa for complex potent-

25 ) ial scattering. In this approximation, the semi-

sc classical scattering amplitude f (•") splits into two parts

332

I a. •o

3~(3.73MEV) 40»0MEV

io-'J

.1OU

\

120 180

Fig. 1Ł.

Figs.14-16 Experimental data for inelastic a-t40Ca scan r:nc.

DtfBA calculations using two different opt ical-r.cćcl-potentials are shown by the dotted and full curve (Ref.18).

333

.10°

2 E

H i •o

10-11

10-L

3" (3.73 MEV)

JO"1

0 CMtdegl

120 1

150 160

g. 15.

33«»

10Ł

B

-10L

3" (3.73 MEV)

5 •o

toil 54. O MEV

10-LJ

10-Lj

62.OMEV

F i g . 16.

335

fT(6) and f„(£). The barrier term f„ is essentially the I B o

usual WKB scattering amplitude with reflection on the

external turning point, including a (generally small)

correction for barrier penetration. The internal contrib¬

ution f- describes reflection on the Internal classical

turning point which is reached by tunneling of the Incoming

particle through the barrier of the effective nuclear*

Coulomb+centrifugal potential. For discussion of the

isotopic effect of a-scattering at backward angles for

Ca and Ca, the relative cross sections a^ * IfjfB))

and a_ = |£_(B)| are shown in Fig.17. The actual cross section o

for comparison with the experimental data is given by

o S C = ifjO) + £B(6) |2.

As can be seen from Fig.17 the barrier term fD

c

dominates at small scattering angles and becomes more

important as energy increases. On the other hand, the

internal contribution fj dominates at large angles at

low energy; for scattering energies of 36.2 MeV and

49.5 MeV, shown in the figure, both amplitudes inter¬

fere constructively at forward and backward angles and

almost cancel each other at intermediate angles. This

demonstrates the importance of a careful choice of the

potential formfactor in order to obtain smoothly varying

potential parameters, as the scattering amplitude is built

up from two components which are sensitive to the values

of the potential in different parts of space and which

vary differently with energy. The squared Woods-Saxon form-

factor appears to be suitable for this purpose. From this

it is readily understood that the use of unappropriate

formfactors, such as the usual Woods-Saxon one seems to

be in this case, leads to discontinuities In the potential

parameters as a function of energy, when the relative

336

importance of both components varies markedly in the in¬

vestigated angular range.

We now turn to the question of the observed large

differences in the cross sections at backward angles for

4O 44 o+ Ca and a+ Ca. On the bottom of Fig.17 a calculation

44

of dj , aB and o s c is shown for the scattering of a+ Ca

at 36.2 MeV. As one can see the internal part of the

scattering amplitude is much smaller than for Ca (on top

of Fig.17) and now the barrier part dominates even at back-44

ward angles. This is due to a larger absorption for a+ Ca 40

as compared to a-*- Ca which was found to be necessary to

obtain a good description of the a+ Ca scattering results.

Fig.18 illustrates this difference. For two different

potentials A and B (A is the one of eq.(2) and B contains

in addition to volume absorption a surface absorption

18)

term ) the volume integrals are shown for the real (Jy)

and imaginary (Jw) parts of the potentials A and B. Whereas

the integrals of the real part of the potential are almost 4O 44

identical for Ca and Ca, the volume integrals for the imaginary potential differ considerably. As can be seen, the

40 absorption in a+ Ca scattering is considerably reduced for

lower indicent energies. For higher energies, a-par'_icius

are always less absorbed in Ca than in Ca, but the

energy dependence looks similar. This reflects the known

fact that although the backward enhancement of the cross

section in Ca does not exist above about 55 MeV, cross

sections for a+ Ca are always smaller than for a+ Ca in

the entire region of investigated scattering energies.

The critical dependence of the contribution from the

internal turning point o- is shown explicitly in Fig.19,

25) taken from Brink and Takigawa. The reflection coefficients

T\t (sec Eq.(D) are plotted as a function of t for the

337

30 . 60 90 120

Fig.17 Angular distribution o g c calculated from the semiclassical

scattering amplitude together with angular distributions

aI and o B calculated from internal and barrier amplitudes,

respectively (Ref.18).

400.

300

200.

•100.

(MeV. fm3 )

'Ca(A)

iE 4AT JVwD(PotB) 4A

ICO.

50.

50 i—

Fig.18 Energy dependence of volume Integrals of the real part

(upper picture) and Imaginary part (lower picture) of

the potentials A and B, respectively (Ref.18).

939

Fig.19 Comparison of the effect of different absorption

depths on the scattering amplitudes for the internal

and external turning-point contributions. (Taken from

Ref .25.)

barrier part and the internal part, using absorption strengths

of W Q = 26.6 MeV and 16.6 MeV. As can be seen, the different

absorption have practically no effect on the barrier part

but strongly change the reflection coefficients for the

internal part. It simply reflects the fact that the

partial waves reflected from the internal turning point

have to penetrate deeper inside the nucleus than for the

barrier part. A useful analytical expression for the strength

of the internal contribution as a function of absorption is

given by Brink and Takigawa 25) and is shown in Eq.(5).

(5)

For the conventions used we refer to ref. 25 . For

a+4OCa scattering at 29 MeV Eg.(5) reduces to

In I ~ exp(-0.16 Wo )

This simple and useful equation reproduces the difference

seen in Fig.19 to within a few percent.

In conclusion, the effect of back-angle enhanced

cross sections seems to be due to an enhanced internal

contribution f. of the semi-classical amplitude which in

turn is due to reduced absorption. It is felt that too

little is known theoretically about the absorption, to

answer the open questions such as the one why only pairs

of additional nucleons (s. section on shell effects) destroy

a backward enhancement or why no enhanced cross sections

are observed above the Ca-region. A comprehensive study

of all observed back-angle enhanced c.voss sections,

including projectiles of Li and 0, in terms of a squared

Woods-Saxon potential (or raised to some other power v) is

underway.

The energy dependence of the observed back-angle

anomalies are discussed in the following second part of

this lecture, where nuclear molecular phenomena are

considered.

II. Molecular Resonance Phenomena

During the last few years a great boom of experimental

papers have been published pointing to the existence of

nuclear resonance phenomena in heavy-ion collisions. For

a review of this presently very fast growing field we

refer to the Proceedings of the 1977-conference on

3V1

"Nuclear-Molecular Phenomena", which include excellent

review articles by Bromley, Siemssen, Greiner, and others.

No attempt is made here, to review the vast experimental

body of data; we will be interested only in the molecular

resonance phenomena in comparison with the back-angle

anomalies described in the previous chapter.

The hypothesis of a formation of nuclear molecules in

heavy-ion reactions was first introduced In the pioneering

27) work of Bromley and his collaborators about 20 years ago.

The idea was born from the observation of clearly nonstatist-

12 12 ical structure in C+ C-induced cross sections near the

Coulomb barrier. In this very first heavy-ion experiment

with a tandem accelerator the authors observed correlated

structure between various exit channels (n,p,a,y) following

the 1 C+ C bombardment. Similar investigations for

16 16 0+ O were negative in the sense that no such structure

was observed.

A rather simple picture was made for the nuclear

molecule, which still is the classical idea until present:

during the collision of the two C nuclei the condition

is met that the Coulomb and nuclear forces match the

centrifugal force for a certain impact parameter, say,

the one corresponding to grazing waves. This will lead

to some rotation of the two C nuclei around the coaaon

axis. After some time of rotation, the two nuclei will

either collapse lnta a compound nucleus or they will

keep their Identity, i.e. they will not fuse, but rather

decay back into the elastic or the Inelastic channel*.

In case of compound-nucleus formation one will observe

possible resonances through the evaporating particles;

otherwise one would expect resonances in the elastic and/or

inelastic channels. Since the two nuclei stay spend a

3*2

relatively long time in close contact, inelastic excitation

becomes likely.

To describe the molecular resonances theoretically

one needs a proper potential. A certain problem is, that

the resonances occur at high level densities, so they can

couple to many other nuclear states available. In other

words, this involves absorption and makes any predictions

concerning the properties of the molecular resonances

difficult because of the uncertainty with respect to the

absorption. Only the position of the resonances can

approximately be determined with some accuracy (from

the real potential). The proper treatment of the absorp¬

tion remains to be the main problem until present (see

below).

We conclude the historical part with a final remark

from our present point of view. For the classical nuclear

molecule, as described above, one would expect a rotational

band with a spin sequence of 0 + - 2 + - 4 ... for identical

particels such as C+ C. This, however, was not observed,

but instead some 2 and 4 states were found which did not

follow this rotational band. The explanation was event-

ually given by Imanishi and coworkers and Greiner

29) and coworkers who assumed that the observed resonances are

formed through excitation of one or both of the C

nuclei to the first 2 state during the collision.

The kinetic energy is then decreased and stationary states in the

potential are I populated as is illustrated schematically, e.g. in

Fig.26 of Ref.30. This alsc explains, that similar resonances

were not observed for 0 + 0 since the first excited states

are higher in energy than for C and are less collective.

The obvious extension of the measurements to higher

energies immediately confronted the search for possible

resonances with large statistical fluctuatxons in the excitat¬

ion functions. In the range of statistical fluctuations the

cross section changes drastically with energy,

with large peak-to-valley ratios. The strongly

fluctuating background makes it very difficult to separate

nonstatistical processes from the statistical fluctuating

ones. Even if careful measurements and analyses point to the

nonstatistical nature of some of the structures/ it is

usually impossible to extract any detailed information about

the resonances because of the coherent background. It thus

seems to be less promising to look for resonances in "indiv¬

idual" excitation functions which correspond to one angle

and one particular channel.

To look for nuclear molecules, three experimentally

observed effects seem to be most interesting:

1. oscillation in fusion cross sections

2. back-angle anomalies in elastic and inelastic scattering

3. structures in y-yield excitation functions (or equival-

ently angle-Integrated particle cress sections).

All these results, collected by many workers, are generally

not understood. Vie will look here at these results only from

the point of view of nuclear molecules.

Let us start with the idsion cross sections. Figs.2O

and 21 show the well known fusion measurements for

1 2C +1 2c, 1 2 O 1 6 0 , 1 2C +

1 8O. and 1 2C +'9F. oscillations are

observed for 1 2 O 1 2 C and for ^2C+^0, but not for 12C+ O

and C+ F. Also a 30 % larger cross section from C+ 0

compared to C+ 0 is observed. Thi» difference of 30 %

seems to be understandable in terms of different radii for O

and 0 as pointed out by Schiffer. ' But where come the

oscillations from? One may reason, that the surface partial

in

s« "bO.8

0.6

I:

1.0

0.8

0.6

0.4

m I

l8o+l2c

i ... i ... i ... i ... i . . . i ... i ... i

14 18 22 26 30 34 Ec.m.<MeV)

Fig.20 Results of the fusion measurements on three systems.

12 12 The open circles for C+ C represent measurements

at 6° only/ where the total cross section was obtained

by assuming a constant angular distribution. The solid

lines are fits using the Glas and Mosel model. The

dashed line for 1 2C+ 1 2C 1» elastic scattering (Refs.

31,32).

3*5

(0 c o

JQ

b

1 O

1.2

1.0

0.8

0.6

0.4

0.2

O

*

_

mmam

-/A

1

y

K

: t

• •••«•>••»*«

. 1

c\ \

;?+ c+

t

1 • J

«S1^

\

\v\ \ \

c

I . I .

• 1 • 1 • i

r+ C — l 8 0 + l2C

m

m

—

v "•••••: v -1 . 1 . ^ ^ -

0.04 0.06 0.08 aiO 0.12 0.14

Fig.21 Smooth curves drawn through the fusion data in four

similar systems (Refs.31,32).

waves are weakly absorbed only ana, thus, can show these oscillations (of course, they don't have to). This points to a connection with the back-angle anomaly phenomena, described in chapter Z of this talk, which also seems to depend on the transparency of the surface partial waves. It therefore seemed to be interesting to us to look for the fusion cross sections for o+ °Ca and ot+ Ca, since the elastic and Inelastic scattering for these two systc— show the largest anomaly observed (see Flgs.13a-.13e).

g+Ą0'Ą4C& Fusion Data The experiment was performed in the tandem laboratory

at the UniversitSt zu Xoln.33) Two Y-detectors were placed at 9O° to the beam as schematically shown in Fig. 22. A monitor detector at 5° enabled a careful control of the normalization as a function of energy. Fig.23 gives a survey over the available residual nuclei found In the bombardment of a+ Ca between incident energies of 8 through

Cotlimator 5mm0

1 fli Monitor

Fig.22 Set-up of the a+40'44ca fusion experiment at Koln -schematically.

3*7

~»Ca"

Ecm(MeV)

25:

•20-

•15;

-10-

5 :

O-

Fig.23 Open channels for a+ Ca induced reactions. The

Coulomb barriers for the various decay channels are

taken into account. The dashed lines indicate the

range of E_M# where experimental data were taken.

about 25 MeV(c.m.). Coulomb barriers for the various decay

channels are taken into account. The energy region studied

here is indicated by the dashed horizontal lines.

Typical Y-spectra obtained for a+ Ca and a+ Ca at

24.2 MeV (lab.) are shown in Fig.24. The ground state trans¬

itions, added up co form the fusion cross section, are given

in Table III. A preliminary analysis of the experiment

yielded the fusion cross sections shown in Fig.25. At some

energies , peaks of the order of 1O \ appear in the fusion

cross sections. If one arbitrarily assignes spins to

the structures indicated in Fig. 25 they follow

348

i -

i E *

| SIIKeV • 1 1 .Source ' > t

NJ 1 21

o* ł 0Co eT> 9o* E«* 24.2

7

M«V

15

•

16.

• SIIKtV t 6

5 7 1 J 6 | 9 | 10

«»**Ca e v . 90* Ei« 24.2McV

Fig.24 Y-spectra following a+ °Ca (top) and a+ Ca bomb¬ ardment at 24.Z MeV(lab). The assignments of the lines indicated by numbers in the figure are given in Table II .

a straight line in an energy-vs.-l(1+1) plot. Interesting enough, this rotational band is very close to that of Santo and coworkers who have extracted spin assignments from the shape of the back-angle elastic angular distributions. Our (open circles) and their results are.shown in Fig.26. Optical model calculations for the total reaction cross section obtained with the potentials of the Louvain-Cracow-Munich group18' (potential 1) and the Basel group24' give smooth curves;with the reaction cross section about 30 t larger than the measured fusion cross section.

3*9

Table II

Corresponding a-transitions In Fig.24

Peak Energy

(keV) number .Nucleus

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1

2

3

4

5

6

7

8

»

0

437

1157

1227.5

1524

1644.5

1728

1829

1921

1961

2301

2555

2714.5

2814

2954

322S

3736

228

382

437

887.5

1047

1119

1156

1429

1505

1841

42 Ca 43

42. Sc

42

43

Ca I

Ca

Ca

:Ca

Sc 42,

36

Ca

Cl

'Ca 42 Ca 40, Ca (2. escape)

42 Ca 40.

40, Ca (1. escape)

'ca

46

46, SC

Tl

42

46

Ca

Tl

46,

46, Ti

44 Tl

Ca

U 35,

35.

Ca

CX

350

Table III

List of 14 discrete Y'transitions that are sunned to fora the

total fusion cross section

Nucleus

43SC

4 2 S c

42Ca

4 0Ca

39K

3 6Ar

- •

(3*

<r-

Transitions (J*-

j )152j (|-

- * • • " • '

- O+)1525

- 0+)3737

\ )2814; ( |

- O+)197O

• | )472; ( f -

( | ' | )"- | )188

? - O+)2455

- | ) 3597

\ >84S;

4; <<|'

(keV)

( | - | )88O;

| ) - | > 2 4 5 9

He would like to emphasize that the structures in the fusion

cross section are small. On the other hand, a preliminary

analysis of some of the strongest a+*4Ca lines yielded

excitation functions which vary smoothly with energy and

do not show any 10 % structures. The analysis for a+ Ca

is not yet completed; the (preliminary) value of the fusion

cross section as compared to Ca is about IS % larger at

20 HeV and about 5 % larger at 27 HeV(lab.).

351

1 1 1 1 1 r-

<TF (mb)

1000

800

600

oc+40Ca Fusion

• <

20 25 E,ab{MeV)

10 . 15

Fig.25 Preliminary results for o+"uCa fusion cross sections.

One of the strongest transitions observed for a+ Ca

is the 72O keV transition from the 5/2*(0.88 MeV) to the

3/2+(0.15 MeV) state in 3Sc. The results are shown In Fig.27.

Structures are also seen here at about 13, 14.5, and 17 MeV.

Search for a Band- mixing of 16O+28Si and a+<0Ca In 4 4T1*

The strongest backward enhancements found to-date are

those for 16O+28Si and a+40Ca elastic and inelastic scattering.

Both systems would form the same compound nucleus Ti .

As shown before (Fig. 26>» the structures in the fusion cross

sections and the shape of the back-angle angular distributions

for a+ Ca are consistent with an interpretation as a rotational

352

E" IM

50

(0

30

20

10.

I"T eV)

-

•

•

-I f "• 1 1

/

/

I n

1 1

1 1

,,

,'

i 1 r

oca (•ElQSł-Scott-\oFusion

-

•

•

:

i i i i i i i j i ł i f i i I t i i i

0 100 200 300 Ł00 500 600

Fig.26 Energy-vs.4(£+t)-plot for the s t ruc tures in Fig.25 and from back-angle data of e l a s t i c scat ter ing (Ref.14).

i i i i i i r~ 40Ca(a,p)i3Sc"

3"oof 0 Y = 9O°

200

. •

100

0.B8

.t

0.152 g.s.-

98%

-5/2*

2%

'•I I I 1 I I I I I

3/2 ' 712'

15 20 25 EŁ<lłllMeVJ

Fig.27 Cross sections for the 40Ca(a,p>*3Sc* reaction from a ymeasurement of the 720 keV transition at 90°. Angular isotropy has been assumed.

353

band. For 0+ Si 'pronounced structures in the elastic and

Inelastic excitation function at 180°have been observed by

Braun-Munzinger et al.. * These results are shown in Fig.28.

Originally the bumps in the excitation functions wers inter¬

preted as corresponding to even spin values only. The

more recent assignment by the same authors call now for

a spin sequence of 9- ? - ? - 17 - 22 - 24 for the six

pronounced structures in the excitation function . The

grazing l-values for 1 6O+ 2 8Si are shown In Fig.29 along with the

latest spin values given above. Also included in this

Figure is the band for a+*°Ca. As can be seen both bands

cross at about 33 MeV excitation energy in Tl at an

l-value of about 12. This means that if we interpret

these trajectories as band structures there should be two

broad 12+ states at about the same energy. Naively, we

ao»

. aoz -

O.OV -

aoo

•

-

I

\

1 i 1

} 1 J

i

i \ \ \ ! I, I1

j

1

f \

h j . f V ^ V \j łl I

%

«&< V

I

' i !

> i i

ecm=»e

i !

' V, ł'

20

Fig.28 Excitation functions at 9CM = 180° for elastic scattering of 1 60 from 28Si {Ref.5b).

35U

I i i i i i i i i i

50

40

30

20

o Fusion

U '6o+28Si)72ro.2'^J

- , , , i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 100 200 300 400 500 600

Fig.29 Same as in Fig.26; in addition the results for 16O+28Si

from Ref.34 are included.

would expect these states to mix, so that each resulting

state would have a reasonable c lap with both the 16O+28Si

and the a+ Ca channels and should form doorways between

one channel and the other. One not completely agrcable feature

of this crossing, however, is that the equivalent (lab)-

bombarding energy in the O channel is only 34 MeV, which

is close to the Coulomb barrier, »o that small cross sect¬

ions would not be unexpected. If there were observable

effects of this band crossing, one would expect that if 2fi 16

one bombarded Si with 0 in this energy region one would

see enhanced cross section* in the Q+ c*q a channel and

that at back angi.ee the o-particle angular distribution

should look like the square of the t«1O Legcndre polynomial.

Recently we have carried out this experiment at the

355

Beschleunigerlaboratorium in Munich, together with John

Cramer who has previously reported on this experiment.

Fig.30 summarizes the experimental results. The cross

sections are generally small and Hauser-Feshbach calculations

(not shown in the figure) do reproduce both the magnitude

and the energy dependence of the cross section. A slight

enhancement is seen at E16 (cm) % 22 MeV (corresponding to MM 0

33 MeV in Ti*) in the experimental data. The Hauser-

Feshbach calculations also show a slight enhancement at

this energy, which arises fran the perfect matching of the

entrance and exit transmission coefficients at this energy.

In the angular distributions at 22 MeV no 1-12 behaviour

is observed; the angular dependence is rather smooth. It

is therefore concluded that within the experimental uncer¬

tainties there is no indication of the mixing of these two bands.

I I I T

20

l70° r=/ dolg.s.—5.28MeV)

130°

TT*. I 15 30 20 25

Ec.m.(MeV) Fig.30 Angle-integrated cross section for the reaction

356

Final Remarks

In this chapter on molecular resonance phenomena

only a few data and a few theoretical aspects have been

mentioned. For instance, the probably strongest evidence

for nuclear molecules, i.e. the recent 12C+12C and 12C+16O

inelastic data by the Stony Brook group are left out

completely. Also the very successful theoretical work of 291 Greiner, Scheid et al. ' and many other groups, efforts of

many years, are not at all discussed here.

We have tried to look at the molecular resonance pheno¬

mena from the point of view of the back-angle anomalies only..

The only clearcut case to-date where the angular distribut¬

ions show back-angle enhanced cross sections and the

excitation functions' molecular-type structures is the

elastic and inelastic scattering of 0+ Si. The spin

assignment to the observed structures, however, is not cleair art

present. High precision back-angle angular distributions

(which are difficult to take) aro n< iod to settle the

speculation on the spins of these structures.

4O For a+ Ca the fusion data and the back-angle angular

distributions (i.e. the P^-form at some energies) can be

arranged in one rotational band, if the spins for the bumps

in the fusion data again are assigned from energy-vs. -1 (I+T)l

systematics only. These structures are seen between incidcrrt

energies of 10 and about 18 MeV, where a spin assignment from

elastic scattering at backward angles is not possible because

of large Ericson type fluctuations in this energy range.

357

On the basis of the present data - back-angle anomalies

and molecular phenomena - the only rigorous statement we can

make. Is that the absorption for the resonating partial

waves has to be small. These waves are always within a

few "h of the grazing value. Therefore, surface transparent

potentials seen to be a necessity, which In turn points

to the Importance of shell effects in understanding the

observed phenomena.

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C. Lauterbach, H. Puchta, and U. Lynen, Phys. Rev. Letters

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A. Weidinger, Phys. Rev. C9, 1813 (1974)

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H. LUdecke, R. Santo, and R. Stock, Phys. Rev. Letters 26,

694 (1972)

9) H. Schmeing andR. Santo, Phys. Letters 3_3B, 219 (1970)

10) J.G. Cramer, K.A. Eberhard, J.S. Eck, and W. Trombik, Phys.

Rev. C8, 625 (1973)

359

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13) R. Stock, G. Gaul, R. Santo, H. Bernas, B. Harvey, 0. Hendrle, J. Mahoney, J. Sherman, J. Steyaert, and M. Zisman, Phys. Rev. C6, 1226 (1972)

14) H. Lohner, H. Eickhoff, D. Frekers, G. Gaul, K. Poppensieker, R. Santo, A.G. Drentje, and Ł.H. Put, Preprint MUnster IKP-9-77, unpublished

15) G. Gaul, H. LUdecke, R. Santo, H. Schmeing, and R. Stork, Nuci. Phys. A137, 177 (1969)

16) H. Eickhoff, D. Frekers, H. Lohner, K. Poppejnsieker, R. Santo, G. Gaul/ C. Mayer-Bć5ricke, and P. Turek, Nuci. Phys. A252, 333 (1975)

17) D.A. Goldberg, Phys. Lett. 55B, 59 (1975) I. Brissaud and M.K. Brussel, J. Phys. CJ5, 481 (1977)

18) Th. Oelbar, Ch. Gregolre, G. Paic, R. Ceulenecr, F. Michel, R. Vanderpoorten, A. Budzanowski, H. Dąbrowski, L. Freindl, K. Grotowski, S. Micek, R. Planeta, A. Strzalkowski, and X.A. Eberhard, preprint 1978

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20) R. Ceuleneer, R. Oeschler, H. Fuchs, K.A. Eberhard, and K. Grotowski, Proceedings of the First Louvain-Cracow Seminar on the Alpha-Nucleus Interaction, ed. by A. Budzanowski, INP Rep. No. 870/PL, Cracow 1973, unpublished

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360

22) K. Grotowski, Proceedings of the Second EPS Nuclear Physics

Divisional Conference - Radial Shape of Nuclei, ed. by

A. Budzanowski and A. Kapuscik, Cracow 1976, unpublished

23) F. Michel and R. Vanderpoorten, Phys. Rev. C_1£, 142 (1977)

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I. Sick, preprint 1978

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26) Nuclear Molecular Phenomena, Proceedings of the International

Conference on Resonance* in Heavy-Ion Reactions, Hvar,

Jugoslavia, May 30 - June 3, 1977, ed. Nikola Clndro

(North-Holland Publ. Cornp., Amsterdam - New York - Oxford, 1978)

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Letters 4, 365 (1960) 1; Phys. Rev. ^21' 8 7 B <1961)

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25, 176 (1970); for a recent review see: W. Greiner, Ref.26

30) D.A. Bromley, Ref. 26

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T.R. Ophel, and B. Zeldman, Phys. Rev. Letters 2£» 4 0 & (1976);

and P. Sperr, T.H. Braid, Y. Eisen, D.G. Kovar, F.W. Prosser,

J.P. Schiffer, S.L. Tabor, and S. Vigdor, Phys. Rev. Letters

37, 321 (1976)

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Collectivity of Medium and Heavy Nuclei, Institute of

Nuclear Study, Univ. of Tokyo, Japan, Sept. 20-25, 1976

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and V. zobel, to be published

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35) J.G. Cramer, Invited paper, Symposium on Heavy-Ion Elastic Scattering, Univ. of Rochester, OSA, Oct. 25-26, 1977

36) T.H. Cornier, J. Applegate, G.M. Berkowitz, P. Braun-Mun¬ zinger, P.M. Cormier, J.W. Harris, C M . Jachcinski, Ł.L. Lee, J. Barrett*, and H.E. Wegner, Phys. Rev. Letters 3JL< 9*° (1977); and C M . Jochcinski, T.M. Cormier, P. Braun-Munzinger, G.M. Berkowitz, P.M. Cormier, M. Gai, and J.W. Harris, preprint 1978 (subn. to Phys. Letters)

Influence of Channel Coupling on High E::cited States

I. Rotter Zentralinstitut fur Kernforsohung Rossendorf, DBH-SO51 Dresden

1, The problea

The investigation of the properties of high lying excited states in nuclei is one of the central problens of nuclear physics studies. The high excited states in nuclei lie mostly above thresholds for particle decay, i.e. they are embedded in the continuun .

The question arises how the wavefunction of such a discrete state embedded in the continuun looks like. The v/avefunction constructed froc bound single-particle wavefunctions contains not all properties of this resonance state (or decaying state) since it does not have the correct aeynptotic behaviour. This function usually used in describing the structure properties of nuclear states can not describe the coupling of the states to the continuun anS thdr decay.

Recently, a formulation of the continue shell codel {CZ\.) has been given which allows to describe the properties of discrete states by taking into account the coaplicated configurations liże in a usual nuclear structure calculation as well as the cov.;;lin; of the discrete states to the continuuc by a couulec char.nels

2) nethod . According to this CS1!, the wavefunction of an isolated resonance state or an isolated decaying state is given by

OC £+' = <l + Gp ' H p o Ó p . (1)

Here, H is the Hnrailtonian of the whole systen with bound as well as scattering single-particle wavefucctions. It is

H * QQ + ^PP + **PQ + ^QP ^^

363

with HQQ = QHQ and so on. The operator Q projects onto the

space of discrete states while the operator P projects onto the

space of scattering etates with one particle in the continuuc.

It is

and

P - 1 - Q. (4)

The functions cj> R are eigenfunctions of the operator HQQ,

lil:e in a usual nuclear structure calculation. The Green function

for the notion of a particle in the continuun is

G £ + ) • P (E - Hpp)"1 P. (6)

The eigenfunctions T of the operator Hp?,

HPP

can be calculated by a usual coupled channels nethod. The coupling

between the P- and Q-space is described-by

(2 - H??)«R<+> = K p Q 4 H . (8)

!:ere, « ^ + ) = G ,+ ) H p Q $ R is the second tern in eq. (1). The

function (1) is similar to the resonance wavefunction introduced

by Kohaiu: and Saruis .

The wavefunction J? R of an isoleted resonance state or sn

isolated decaying state contains, according to eq. (1), the v:sve-

function cf> p. This wavefunction is doninstin£ inside the nucleus.

This fact has been used in describing the structure properties of

nuclear states during all the years by neglecting the second tera

of eq. (1) altogether. The second tern describes the coupling of the

resonance state to the surrounding continuun, i. e. the asycptotic

behaviour. It contains the eigenfunctions ($> -, of the operator

HQQ as well as the eigenfunctions ? (2) of the operator Ep-,:

The wavefunctions ?C(S) contain the coupling of the different

channels c or the r.i.-cing of the different scattering wavefunctionc

in the sane nanner as the v/avefunctions C -j contain the internal or

bomd-state tr,i::ii:g. Consequently, the channel coupling determines,

like the internal Ki;:ing, the properties of states lying o rove decay

thresholds. The degree of influence will depend on the degree of frag¬

mentation of the basic state wavefunctions in the eigersfunctlona Cp . -

If the fragmentation is snail and the overlap between q>p and ?

is large then the influence of the channel coupling is e;:pecten tc

be snail in contrast to the opposite case. The influence v/ill be

relatively large if Couloab interaction plays an important role te-

cause of its long ran-e. Thus, channel coupling is e::pecteu to play

a role in the protles of iaocpin rr.ir.ir.g of nuclear states. This v.vll

be discussed in the following.

2. Isos-in ni::in.~ of reeonence states

i.c tv. e::anpie, the excitation of the iscopin foriidt'.er snolo-;

resonance 1 = 3/2~, T = 3/2 at 15.1 - e">' in the r.irror •./.zlci 1 "i 13 "

.'. and C will be considered here. Ca lev lat i one have oeen oeri'orr.eC

cy Ari::a and Yoshida 'on the basis of the ?eshbec:. theory. In

these calculations, isoapin ri::ir.g is treated fcy pertv.rcstior. theory

and the rauli principle for bound and scstterinr wavefurvctionc '.cs

been violated. The result of these calculations v;aa t;.3t ^-ternal

isospin sirring alone can not e::plain tr.e observed decay v.'idtr.c.

365

Only when an isoteneor conponent in the charge dependent residual interaction and nixine of the continuun states are taken into account, the decay widths calculated were comparable v/ith the e?:periraental observed values. Investigations of the nirror K1 transi-tiona in aass 13 nuclei have shown however that there is no evidence for an ieotensor cooponent in the electronagnetic interaction . Further, the results obtained in the CSU led to the conclusion that external nixinc of resonance states with different isospin nay be lorGe indeed "cut does not ciye tne oail) contribution to the widths if the resonances are isolated '. Therefore, an investiga¬ tion of tha widths of the 3/2", 3/2 resonance state? ot 15.1 i'«V e::citotion energy in the ^C and hi nuclei is e::pected to show the role the coupling effects via the continuun are playinj.

To Lie 1

'.Vidth V., of the 3/2", 3/2 resonance state at 15.1 I eV in the nirror 11 n

nuclei lJC and Ji'.. a, b - parameters in eq. (12) v/ith VQ = c-50 i.:ev re-3.

0

0 .

0 .

C.

8

5

2

0.2

0.5

0 .8

2 .

2.

3.

V,

1

7

0

13 : :

6.

3 .

5.

4

9

9

5

r i

.38+0

R

.81 5

13,,

* 0.86+0.12 5

1.55 7>

3. Calculations for the 3/2" 3/2 resonance state at 15.1 i»eV in the nirror nuclei C and I?

The widths I „(E) of isolated resonance states are defined 2)

In the CSU by the eigenvalues of the operator '

KQQ HQQ + HQP + HPQ (10)

366

which is effective in the Q space after the coupling to the con¬ tinuum has been taken into account. The eigenvalues are

H - I rR~\ - | VR .<&R| K°f | <|>R> . (11) E R - SJJ(ER) « Ug(Bg) • UR is the enercy snd I H x

VR i s tlie T'idt!l o f t h e resonance state. The calculations discussed here have been performed with the

nucleon-nucleon interaction

VCr, - r2) - -V0(s + b p'g) 5 {£, - r2) (12)

and parameters of the iVoods-3a:con potential oinilar to those used usually for the A * 16 systen: V(1*O) « 56.35 "eV, V(l=1) =. 57.^7 :.:eV, V(l»2) = 54.65 i:eV and V,_U-1) « 9.7C lleV, V,_(l«2) = 5.27 :.:eV for

is is ^2 13 both neutrons and protons. The configuration spsce of the C, C end H nuclei is the whole 1p-shell.

The widths f\ obtained in the nuaei".cal CDl'. calculations for the 3/2~ 3/2 levels in the A » 13 nuclei are in the sane order of aarpiitude as the experimental values (talle 1) slthour-i: neit:;er an isotensor cor.ponent nor even a Coulor.; tera in the rec.dual interaction has been introduced er.plicitely. The width is nonvor..'.c;.ir." because of the differences in the ceutror; end proton v;overur.ct.'.or.3. These differences are caused by tr.e sin;;le-particle Coulo:..: inter¬ action. In the follOT/ins the influence ?- the dr.fferunt r.i::!.-:; tr.c eouplinc effects on the width, will be ć;.scusced.

The CSi'. calculations jive a scall isos-,ir. z'.i.y.Lr.-- of tr.e •.vavii-functionB q) „ (table 2 ) . This result is in accordar.ee w.tr. the results of other calculations. The difference to other .'.ocelo con¬ sists however in the fact that the aave-unctior. of c recorterxo ctct? in the CSL: is J2 R which differs from <$ .;• the additional terr;. (SJ). The isospin purity of SI _ is determined not only by Internal r.i:;_.r-taken into account in the wavefunctions Cf> but also ty c:-.3rrjei coupling contained in the vvavefunctiors 7 (2) eppearir.- i:. t:.e tor:

(9). The CSLI calculations show that the cc::plin~ of ti.e open icoc."-:1

367

for the

Table 2

I8os?in inpurity in the wave function Q -, =

3/2", 3/2 level at 15.1 I'-eV in 13C and 1 % . *

T. is the isospin, 3., the enersy of the state before

. Paraneters in eq. (12): V Q = 650 VeV fnr, a = 0.5, b = 0.5.

i

1

2

3

4

5

G

7

3

V 5.

6.

10.

13.

14.

21.

21.

26.

Z 1

7

2

-

3

1

5

7

2T,

1

1

1

1

3

1

3

3

0.0014

-0.0006

-0.0003

-0.0015

1.0000

-0.0001

-0.0000

0.0000

=1As.v

4.9

6.8

10.6

13.3

14.2

21.1

21.5

2S.7

1

1

1

1

3

1

3

3

-0.0018

-0.0007

C.00C5

-0.0C17

1.0000

0.0001

-0.0000

0.0000

feu-bidden channel to closed lsoepin allowed channels enlarges the

width of the 3/2", 3/2 level by at least an order of oacnitude

(table 3). This neans that interncl nixing and channel coupling

are in lil:e manner important for tiie couplinc of the resonance

state to the continuum.

In further calculations, the -orrect consideration of the

Pauli principle has been shown to ce ioportant for the widths

of resonance states, especially if they are snail (table 4).

368

Table 3

Width V- of the 3/2~, 3/2 resonance state at 15.1 ZeV in ^ 3

in eq. (12): V = 650 LieV fa , a = 0.3, b = 0.2.

iluciber of channels

6

1

2

3

3

4

5

12 Corresponding states of C (0+,0)

;:

"

::

7.

X

X

, (0 ,0 ) , , ( 0 , 1 ) , (2+ ,0)1

;: :: y.

X

7. 3:

7. 7.

7. :: ;:

7 Zl

( 2 + ' 1 ) 1 ( 2 + ' 1 ) 2

;: ;:

VnA-eY

6.44

0.19

0.37

0 . Jlh

C.42

1.C1

4.63

This is connected with the appearance of additional resonances In

the cross section if the Peuli principle is violate:!.

Thus, the CCI.: calculations show that the charge dependent

aatri:: elements calculated with einple forces nay be lar~e clt-

houch the isospin ni.;:inc of the resonance states incide the

nucleus is saell. This result is in accordance ivith the er.peri-

cental data frorn detailed investigations on the z.irror nuclei

C and II . It is due to ' oth the differences in the wave-

l'unctions of neutrons and protons and the cou;:lin- of tno ico-

spin forbidden channels to isospin allowed channels. lecouoe of

the differences in the neutron and proton v/avefunctions the reso¬

nance state wavefunctions (1) of the r/.irror states are not relateJ

by the iaospin raising or lowering operator in contrast to the

wavefunctiono <$>-,* ?iie differences ore caused raainly ty ti:e Coulo:.

3in,-;le-;5article interaction since in our calculations for the

A = 12 and 13 nuclei the paraneters of the •.Voods-3a::o:i potential

for neutrons and protons were chosen to be the sane. The diffe¬

rences in the neutron and proton wavefunctiono together with

369

channel coupling characterize the asymptotic behaviour of the

resonarce wavefunction (1).

Table 4

'.Vidths of the 3/2~ resonance state in 13K at 3^c<c'^ = 12.0 LleV

without and with Pauli principle, T~ is the dorainatincjisospin.

I'arcneters in eo. (12): V Q = 650 I.leV fn , a = 0.6, b = 0.2.

neeononce state

l'o,

1 2 3 4 5 6 7 O

4.

. 4i:/i;ev 2 n

3.5 9.5 11.7

15.1 19.6

23.3

24.9

30.9

Conclusions

T, 3

1 1 1 3 1 1 3 3

without Pauli principle

7H/l:eV

7740

60 532 132 96

3G4 104 116

v/ith Paulx principle

TR/l:eV

2342

102 326 11 140

234 0.5 0.7

The CG.': calculations have shown that an isolated resonance

state is characterised not only by the confi^urational niriin-

(internal ni::iric) of the bound Gtates but in the satr.e nanner by

the nii;:inc of the different channels (channel coupling). Inter¬

nal nixinc to'.-.es place with bound states lyinc lower as v/ell ss

h.;;lier in energy as the considered resonance state, lv. the ca;:.e

"•.anner, couplir.j; to other channels plays a role irrespective of

whether they are open or closed.

If the coupling of a state to the continuu- is ta'.ien into

account, then an additional tern appears in the wavefunction as

well as a width f"\ and an energy chift 4 E_, relative to -she

370

value E^ calculated in a bound state nodel. Thus, channel coupling is contained in the wavefunction ŁJZJJ, the position 3^ and the width P p of a resonance state. If the width is soall, then channel coupl¬ ing nay change this small value considerably. This means that channel coupling is important if widths of resonance states with lar¬ ge fragmentation of the configurations are calculated or if selec¬ tion rules act. In these cases, closed channels with a large over¬ lap to the resonance state will play an important role. Although states below thresholds for particle decay have /"""., = O the energy shift £ Ep caused by the coupling to closed channels ie nonvanis-hinc. Therefore, channel coupling nay be important for the position of states also if they are bound.

Hefsrer.ceo 1) C. Llohau;: and H. A. Y/eidenrsUller, Shell-I.iodel Approach to

ITucleer Reactions, Amsterdam 1969

2) K.'.V, I a rc , I . Hotter , and J . Kohn, Hucl. ?hys. A275 (1977) 111

3) C. I.Iahau:: and A.i:. Saruis , :.Tucl. Iftys. A17? (1971) 103

4) A. Ariaa and Z. Yoshida, ITucl. Phys. A161 (1971) 402

5) P..2. Marrs, Z,G. Adelbercer, and i'.. A. Snover, Fnys. nev. C16 (1977) G1

6) I . Hotter , to be published

7) G.L'. Ter.r.;er in : lluclear Spectroscopy and reac t ions , ed. Cerny, 3 (1974) 61

\

PBBSENT STATUS OP THE 16 UV TAMDEM PROJECT AT LABORATORI

HAZIOHALI DI LEGNARO /PADOVA/

C. Slgnorlni

Istltuto dl Flsica dell'Universito e INFN /Padova/

Laboratorl Nazlonall dl Legnaro /Padova,Italy/

In tbe present talk I will briefly present tbe project of

the 16 UV Tandem of tbe Laboratorl Nazlonali dl Legnaro near

Padova and will speak about tbe actual status.

The accelerator, whose most lnportant parameters art. list¬

ed In table 1, was ordered at the end of 1975 at BVEC /High

Voltage Engineering Co., Burlington, Uass., OSA/. It has basi-

oally the same structure and lsngtb /24 m/ of the MP-Tandems,

presently operating up to about 13 UV, except for the tank dia¬

meter /7.6 m Instead of 5,5 m/.

With this accelerator a large research programme based on

heavy Ion beans will be developed. For this reason tbe machine

will be equlped with sources able to deliver reasonable currents

/ 50 nA/ of negative Ions of nearly any atomic species, name¬

ly: sputter source, duoplasmatron with lithium exchange channel,

off axis duoplasmatron. Tbe Injector /I50 kV pre-acceleratlon

voltage/ has an excellent mass resolution /better than 1/100/

as tested at GIC /General Ionex Co., Mass., USA/, the company

which built this apparatus as subcontractor of HVEC. Tbe high

voltage terminal will house both foil and gas stripper; a second

solid stripper will also be installed In the high energy side.

For more extensive technical details in tbe Installation

the reader is reffered to Ref. £ij.

Sketches of the future planed situation of tbe Legnaro La¬

boratories and of tbe building of the Tandem accelerator are,

shown in Figs, 1 and 2.

Concerning the Tandem building of rather classical concep¬

tion one can see bow two experimental areas totally Independent

will be build, one of then very heavily shielded /walls of 1.5m

thickness/ for "hot" beams. In the target rooms 6 bean lines

372

CLASS AND GUEST ROOMS m HEAT AND WATER CENTRAL

Fig. 1. Sketch of the future planned situation of the Legnaro Laboratories

GROUND FLOOR PLANT

SFS RECOVERY'" ^ SYSTEM I r ELECTRIC]

POWER

if 2n.. . ^..iASflW-'UX-.-.-^ A—

1 INJECTOR

tHIV '

EXPERIMENTAL —I AREA

P i f . 2 . Th» cround floor plant ot th» XTO 18 MV Tandva Aoo«l«r«tor

37«»

TABLE I

Parameters of th» ITC - Tandem

Terminal voltage range without tubes

Terminal voltage range with tubes

Analyzed protons

Analyzed Iodine

Analyzed chlorine

Terminal voltage stability

Totale charging current wltb

Laddertron

0-20 UV

0-16 UV

SuA at 3 UV

lOuA at 7.5 UV

5uA at 16 UV

0.1 puA at 16 UV

/most probable charge state/

0.25 puA at 16 UV

/most probable charge state/

50& transmission with 0.25 jiA Injected

i 1.5 kV

800 JLA

/3 each side/ will be initially built.

The effective start of the building construction was In

June 1977. The tank built by an Italian company: Bellell S.p.A.

Mantova has already been totally welded. The first part of the

building with the accelerator vault should be ready In June 1978.

/In this month, February 1978, the foundations are ready and

the big concrete walls start beelng erected/. So that starting

from this time the tank will be positioned and then / October

1978/ the accelerator mounting will begin. By the end of 1979

375

the Machine should be mounted and tested «o that aost probably the beginning of 1980 will be tbe date of the first experiments with tbe Tandem.

One of the peculiarity of the machine Is tbe charging sys¬ tem consisting of two laddertrons /see also Hot, ffi each de-llTerlng 400 pA current /down charge • up charge/. HVEC fens de¬ veloped an horizontal version following the work developed at Daresbury /2] for a vertical version of this device. A final suc¬ cessful acceptance test of tbe first laddertron was performed at HVEC plant In October 1977 with the data quoted In table II.

TABUS II

Laddertron aooeptance tests at HVEC In MPO

Total ooluan 1/2 oolumn

86 psl 86 pso sre pressure

13 MV 8 MV Terminal voltage

170 jiA 400 pA Laddertron total current

20 kV 36 kV inductor voltage laddsrtron speed

12 a/s 12 a/s

i 1.2 kV - 1.6 kV ripple /G.V.M. reading/

In connection with the realization of the 16 UV Tandea, a group of people fr<m Ullano University /leader P. Resnlnl / have studied the possibility of having a superconductive cyclo¬ tron to bs used as postacoelsrator for the Tanden. The parame¬ ter of the cyclotron are given In table III and tbe relative planed position with respeot to the Tandem la shown in rig.3. Presently, a model /l;l for tbe RP and 1:6 for the magnetic field/ has been successfully tested by February 1978 confirming the feasibility of the project.

SUPERCONDUCTING CYCLOTRON

Q.D. QUADRUPOLE DOUBLET

S.M. STEERING MAGNET

M 90° BENDING MAGNET

B.M. 225°( t3°) BENDING MAGNET

0 5m

Pig.3 . Position of Tandem and superconductive cyolotron

TABLE III

Parameters of tbe superconductive cyclotron

max. energy 55 MeV/Nucl. —» 10 UeV/Nucl. •In. energy 1/4 of sax. energy K - UE/Z2 • 500 0.1 % < E/E <0.2 % Ealttanoe /A/ t * an arad <A <6 an wad

lf ATande« < 1 5 •»'«"» Currenti 1011 - 1012 particle* tee"1

If tbe Tandesj delivers 1012 - 1013 partlolet łec"1

Construction data

POLE DIA: 1.8 • 22 kGausa < B < 41 kGauss /3 sectors - spiral shape/ Aooeleratlon Dees: 3 with 100 kV peak roltage Deflection: electrostatic 120 or 140 kV on" Bunching of the beam lnjeoted Into the cyclotron before Tandea at 400-500 kV level with two cavities

Frequency 21 or 63 MHz Ay il.5° or S3° Efflolenoy 35 % or 6 %

after Tandsn eleotrostatlo chopper operating on a frequency looked to oyclotron frequency.

References

1. C.Slgnorlnl, Revue de Physique Appllquee 12 /1977/ 1361.

2. T.W.Altken, T.R.Cbarlesirorth, Daresbury Lab. Report, DL/MSF/TU 1 3 .

378

SELECTED TOPICS

IN NUCLEAR STRUCTURE

PROCEEDINGS OF XVI WINTER SCHOOL

V o l u m e 2

February 20 - March 4, 1978 Bielsko - Biała, POLAND

Edited by : J.Styczeri and R. Kulessa

Cracow, June 1978

NAKŁADEM INSTYTUTU FIZYKI JĄDROWEJ W KRAKOWIE

UL. RADZIKOWSKIEGO 152

Kopię kserograficzną, druk i oprawę wykonano w IFJ Kraków

Wydanie I Zam. 147/78 Nakład 30"

Preface

It was for the sixteenth time already that a group of physicists from the Institute of Nuclear Physics in Cracow and the Institute of Physics of the JagelIonian University had organized a Winter School on Nuclear Physics from February 20 to March 4, 1978. But, after several years of being used to Za¬ kopane as a place of these meetings, for the first time this School was held In Bielflko-Biała, a town south-west to Cracow, near to the Beskidy mountains.

The main aim of the School was that the leetures given there cover the most dynamic trends In the low energy nuclear physios and provide the participants with a fresh insight into the present status of a number of basic problems end research work described often literally hot from the laboratory. The quality of the physics at the School speaks for itself and will be apparent for the reader of the proceedings.

The material presented by the speakers on the School was very large. Consequently, two volumes of the Proceedings had to be prepared. They were reproduced by photo-offset, and the submitted manuscripts were included without much of editing. Any technical shortcomings are hoped to be compensated by the Intention to make the valuable material available for the readers soon after the School,

We would like to thank dr Z.Stacbura for hie help In col¬ lecting the manuscripts and express our gratitude for his re¬ marks. Special thanks go to Mme J.Kozarska for typing some and retyping some other manuscripts and Mr W.Starzeckl and Mr J.Wrze-sidski for their help in making corrections.

J.Styczeń and R.Kulessa

Kraków, April 10, 19J8

III

SCHOOL HOSTS Institute of Nuclear Physics, Cracow Institute of Physios, JagelIonian University, Cracow

SCHOOL ADDRESS Ośrodek Wdrażania Postępu Technicznego w Energetyce Bielsko-Blała, ul. Brygadzistów 170

ORGANIZING COMMITTEE cnairaen:

R, Kulessa A,Z.Hrynklewlcz

Members]

E.Bożek B.Styczeń M.Lach J,Styczeń W.Potempa S.Szyaczyk M.Ryblcka B.TTodnleoka Z.Staohura J.Wrzesirfaki W.Starzecki K.Zuber

Secretary?

Z. Natkanleo A. K«sek

CONTENTS

Voluae I

1.

REMARKS ON THE ROLE OF PHYSICS A.ft.Hrynklewlot . . . • . . . . . . . . • > . . . . . 3

2. High Spin Stat»» and Yra»t Trapa

HIGH SPIN ROTATIONS OP NUCLEI WITH THE HARMONIC OSCILLATOR POTENTIAL M.Cerkaski and fl.Szyaati»kl 13

EXPERIMENTAL WORK ON HIGH SPIN ISOMERS AND POSSIBLE TRAST TRAPS G.Sletten * . . . 25

%IGH SPIN ISOMERIC STATES IN 152Dy F.A.Beck, C.Gehrlnger, J.C.Merdlnger, J.P.ViTien, E.Botek J.Styozeri *J

NUCLEUS OP VERY HICH SPIN STATES. MICROSCOPIC DESCRIPTION M.PłQ8zaf1ozak k9

THE STUDY OP HIGH SPIN ISOMERIC STATES IN MULTIPLICITY EXPERIMENTS WITH 12C INDUCED REACTIONS D.Hagenan . . . . . . . . . . . . . . . . . . . . . . 11%

HIGH SPIN STATES IN THK GROUND STATE - AND SIDE BANDS IN 166Dy, 162Er AND 168Hf INVESTIGATED THROUGH PROTON - AND 14N INDUCED REACTIONS

J.Vervler fi8

3. Nuclear Reaction

EVIDENCE POR SYSTEMATICAL FEATURES IN PROTON ELASTIC SCATTERING

RELATED TO NUCLEAR STRUCTURE

E.Colombo, R.De Leo, J.L.EBoudie, E.Fabricl,

S.Mlohelettl. M.Pignanelll, F.Resmini,

A.Tarrata . . . . . 123

6L1 INDUCED REACT! ONS JflELL ABOVE THE COULOMB BARRIER

J.Jagtrzeb>ltl . . . . . . . . . . . . . . . . . . . . 136

A MICROSCOPIC APPROACH TO THE DESCRIPTION OP THE GIGANT

MULTIPLE RESONANCES IN LIGHT DEFORMED NUCLEI

K.ff.Sohmld . . . . . . . . . . . . . . 167

ON THE INFLUENCE OF THE SHELL NUCLEAR STRUCTURE ON THE

DIFFUSION PROCESS

V.G.Kortavenko 221

HEAVY-ION EXPERIMENTS ON THE MP TANDCM AT ORSAY. PARTICLE

CORRELATION STUDIES AND MASS MEASUREMENTS ON EXOTIC

NUCLEI

P.Uoussel 224

ON INELASTIC SCATTERING CALCULATIONS

W.J.G.Thir1asen. Zk7

ELASTIC TRANSFER REACTIONS

H.G.Bohlen . 250

THE BREAK-UP OP COMPLEX PARTICLES INTO CONTINUUM

A.BudsanowBitl 263

ANGULAR MOMENTUM IN HEAVY-ION REACTIONS

H. Oesohler , 276

BACK-ANGLE ANOMALIES AND MOLECULAR RESONANCE PHENOMENA ±N

HEAVY-ION COLLISIONS

K.Eberhard . . . « . . , , . . . . . . 308

INFLUENCE OF CHANNEL COUPLING ON EIGH EXCITED STATES

I.Rotter , . . . , 363

VI

PRESENT STATUS OF THE 16 MeV TANDEM PROJECT AT LABORATORI

VAZIONALI DI LEGNARO /VAPOVA/

C.Signorinl 372

Volume II

4. Collective and Single Particle Properties of Nuclei

IN-BEAM GAMMA-RAY SPECTROSCOPY WITH JOO-16O MeV ot ' 8 IN

MEDIUM-LIGHT NUCLEI

C.Slgnorlnl 381

NATURE OF THE 0 + LEVELS AND SHAPE TRANSITIONS IN THE Ge

AND Pt REGIONS

M. Vergnes .<»t2

GAMMA-RAY SPECTROSCOPY IN MEDIUM-LIGHT NUCLEI

J. F. Sharpey-Schafer *»3i

HIGH-SPIN NEUTRON PARTICLE-FOLE STITES IN EVEN N=28 ISOTONES

J. Styczeń 461

IN-BEAM INVE TIGVTION OF THF N=82 NUCLrUS 1 4 3Pm

F . S t a r y 477

SHAPE TRANSITION IN THE ODD Tb NCCLFI

G.winter, P.Kemnltz, J.rorlng, L.Funke, K.rill

S.FlfstrOm, S.A.Hjorth, A.Johnson, Th.Mndblad . . . . i»g2

KECENT EXPERIMENTS ON THE SBAPF OF FISSION BARRIER V.Me tap, , J»90

ON-LINE ALPHA SPECTROSCOPY ON 1 GeV PROTON-BEAM

FROM SYNCHROCYCLOTRON

J. Kormlckl . 533

QUASIPARTICLE SPFCTRA ABOVE THE YRAST LINE

R.Bengtsson, S.Frauendorf . 551

DISCUSSION OF THF CRANKKP a^RTREE-FOCK-BOGOLYUBOV KfFTHOD

IN TERMS OF SIMPLIFIED MOPFL

S.Cwlok, J.Duriek. Z.Szymańskl 588

THE QUASIMOLECUL.VR MODEL IN TRANSITIONAL NUCLEI

G.Leander . ', , . . . . . . . . . 6 2 1

VII

PARTICLE-ROTOR MODEL DESCRIPTION OF ODD-MASS TRANSITIONAL NUCLEI .f.Rekstad 658

SHAPE OP PLATINUM NUCLEI AROUND A-190 F.Dflnau . . . . . . . . . . . . . 683

ODD-EVEN EFFECT IN THE NUCLEAR SHELL-MODEL FOR NUCLEI

WITH N-28 AND N-6O

A. Batanda 687

ANGULAR MOMENTUM PROJECTED WAVE-FUNCTIONS R.Bengtsson. H.B.Hakansson . . . . . . . . 70%

WARD-LIKE IDENTITIES, CLUSTER-VIBHAIIOHAL MODEL AND qUASIROTATIONAL PATTEHH V. Paar 715

5. Heavy-Ion Colllalons-Posltron Production; Quasi Molecules

IN-B5AM ELECTRON AND POSITRON SPECTROflCOPT AFTER HEAVT-ION COLLISIONS H.Backe 823

EXPERIMENTS ON K-HOLE AND POSITRON PRODUCTION IN COLLISIONS OF HEAVY IONS H.Bokemeyer . . . . . . . . . . . . . . . . . . . . . Ski

AN INVESTIGATION OF QUASI-MOLECULES IN HEAVY-ION COLLISIONS. QUASI-MOLECULAR ROENTGEN RADIATION K.H.Kauh 859

8. Closing Eetrki

J.F.Sharpev-Sohafer * . • • • . • . , . . . , . . , , 881

LIST OF PARTICIPANTS , 885

VIII

4. COLLECTIVE AND SINGLE PARTICLE PROPERTIES OF NUCLEI

IN BEAM GAMMA RAY SPECTROSCOPY WITH 100-160 MEV

a'S IN MEDIUM LIGHT NUCLEI

C.Signorini

Istituto di Fisica dell'Oniversita, and

IMFM, Padova (Italy)

1 - INTRODUCTION

The topic of theae seminars concerns the interaction of rather energetics

a-partldes with nuclei of the 1 f 7/2 shell. The msin attention will be focus¬

ed on the measurements of cross sections with in beam y-ruy spectroscopy techni¬

ques following the bombardment with a-particles In the energy range 90 to 160

MeV. These experiments are presently under Investigation from people of the

Physics Institute of the Padova University (Italy) and the Institute of Nuclear

Physics, KPA Jfillch (Germany) at the JOllch lsocronus cyclotron.

The results, new from some aspects, give to me the opportunity first to review

the field of the interaction of rather high energetic projectiles (of any type:

«, K, p, o» heavy ions) with rather light nuclei and second to try to under¬

stand this rather new field of nuclear physics.

In fact only very recently nuclear physics studies with high energetic projec¬

tile have started partly because new facilities have started to be in operation

(LAMPF, SIN) and partly because some typical high energy facilities are becoming

obsolete for the study of the elementary particle Interactions (Saturn*, the

Sincrocyclotron at CERN, SREL, ANL).

Just anticipating some of the results extensively discussed later one can say

that one start*to get a real feeling of what la going on at this intermediate

energy. One aees a link between the very low energy Interaction where the in¬

coming particle fuses lntc the nucleus (compound nucleus formation) and the very

high energy interaction where only the interaction with single nucleon* or single

elementary particles. Is present.

Of course we are only at the beginning of this very large and open research

fleldi but maybe we »x» on the correct way of understanding the phenomena new

from some respects.

381

2 - DESCRIPTION OF THE EXPERIMENTAL DATA

The experimental data are the absolute cross-sections for the production

of all possible end nuclei following the bombardment of 5 kFe and S 0Cr with

E =90 to 160 MeV in 10 KeV steps. The cross-sections are deduced by the detection a

of "in beast y-rays'. Before describing in details the experiments and the re¬

sults It is worthwhile to speak about the background of the experiments.

2.1 - The Background

The present experiment was suggested partly from the experimental data known

at the time of the beginning of the experiment with elementary particles projec¬

tiles (like: Ti, K, p ) and partly with heavy nuclei (A>4) in the 1 f 7/2 shell

and in lighter nuclei (s-d shell).

The first experiments with stopped kuons/BAS 72/ fast negative pions/JAC 73/,

/LIN 74/SEG 76/ slow positive and negative pions/ASH 74/ stopped negative pions

/ULL 74/ and 100 MeV protons/CHA 74/ were reporting to observe the end nuclei

differing from the target by one or more a-particles with a higher cross-section.

These new phenomena wore of course triggering quite a lot of speculation about.

(Actually this fact has been somehow redimensioned by later experiments as dis¬

cussed in the next chapters).

Extensive experiments performed with heavy ion projectiles (6<A«32) at bomb¬

arding energies ranging from ~20 to about 100 MeV (in part done from the Padova-

-Mtlnich collaboration, see for example: /SIG 78/ and references quoted therein)

were discovering systematically very high spin states in the 1 f 7/2 shell.

The ot-particles were a good link between the two set»of experimental data be¬

cause the large binding energy of this projectile let It resemble more to an

elementary particle (nucleon, meson) than any other one and it can be well suited

(maybe by a-a semlfree scattering) to excite ot-like structure. On the other hand

this projectile starts to have a considerable mass (compared always to elementary

particles) so all the excitations in the final nucleus involving angular momen¬

tum effects (high spin states) start to play a considerable role.

There was finally a last typical experimental fact behind this investigation,

namely the concrete possibility of observing in beam y-rays following the bomb¬

ardment of light nuclei with a-particles of considerable energy (above 100 MeV).

In these nuclei, in fact, at difference from the heavy ones, the lower coulomb

382

barrier allows that also the charged particlea are emitted and consequently

many more exit channels are open.

2.2 The Experiment

The experiment was performed at the Isochronus Cyclotron of the Institute

of Nuclear Physics (XFA - Jflllch, Germany).

The "y-ray spectra were recorded with large volume Ge(Ll) detectors at 90°

and 125° to the beam direction at E -90 to 260 MeV (practically the whole

energy range of the cyclotron) in 10 MeV steps.

Some initial care had to be taken In order to record good y-ray spectra:

1: No collimators were used at any position except for focusing of the bean

on the target;

2: The beam, with a diameter of O m m at the target site was stopped on a beam

dump inside a 2 o long concrete wall;

3: The target chamber was rather large K 3 cm at 'east from the beam center

line to the next wall) to avoid the background produced by the beam diverg¬

ing from the target towards the wall;

4: The detectors were heavily shielded with Pb from the y-rays not originating

from the target;

5: No y-ray detection was possible at angles smaller than 90° since the detectors

were practically paralysed (most probably neutron background);

6: The target adopted for the final measurements were rather thick (1,10 mg ca> )

in order to avoid that too many nuclei produced were decaying in flight.

With targets of M mg cm the Y-ray spectra were much poorer because of the

strong doppler broadening.

The experimental data consist of:

1 - Cross-section measurements deduced by in beam Y-rays with the targets 5*Fe

and 5*Ni. The absolute precision is around 20*. The main error sources ori¬

ginate from the current measurement with the target (MOt) and the absolute

detector efficiency calibration (MOt).

2 - Activity measurements, in the minute region, and off beam measurements, in

the nanosecond region. But nothing special was observed from both these data.

3 - Gamma-gamma coincidence data at E *150 MeV with the **Fe target.

A typical Y-ray spectra from the reactions sl|Fe+a and 5*Ki+a are shown in the

fig la and lb. For the 5*Fe target all the final nuclei and all the observable

gammas were analyzed, while for the Ni the first analysis reported in this talk

383

IT (lain

Fig. la) la bca» Y-r*y« '«>• t)w reaction

PI9. lb) In b*aa Y-r«yi fro* th* rascŁlon

was limited only to the even-even end nuclei.

Before entering into the details of the excitation functions analyzed soae

general aspects of the reactions under investigation can be underlined.

First of all: with the care mentioned above the In beaa Tray spectroscopy

with rather light targets and high a-beaa energy Is quite feasible without big

experimental troubles.

Rather high ..pin states are populated in both reactions. In the **Fe+a reaction

one observes in i0Cr(2a emission) up to J71-!!* in the direct spectrum (610 JceV

Y-ray 1I+-1O+ transition). From the y-T coincidences also the S0Cr 12 states,

the highest spin reported until now in this region only from(several) HI work,

is observed. The same situation happens in the saNi+a reaction leading to *cCr

(now 3a emission)where again the 11* state is observed. This means that quite

a lot of angular momentum enters into the (maybe) compound nucleus.

The picture is quite similar to what observed with HI Induced reactions.

One can compare the 5*F«+a system with the l>JC»+"o and 3tMg+i2S ones where

the same compound nucleus Is formed (fig.2a and 2b). One see* essentially the

same final nuclei populated laying around the N«Z+2 line as the target nucleus.

The a spectrum has more Y-lines and this is due most probably to the less selec¬

tivity of this projectile. In the three reactions compared one has similar or¬

bital angular momentum involved (but different excitation energies) as shown

in fig.3

Mainly the yrast states of the final nucleus are populated; but in some cases

also non yrast states are populated with large cross-section; see for example

in ihFe+a the level at 378 JceV In 5 3Mu (op emission) with spin 5/2" which decays

to the g.s. 7/2 . More over the cross-section for the population of this last

level behaves differently with respect to the other levels; it looks more flat,

(maybe more direct effects).

The cross-sections for the population of several levels in final nuclei:

even-even, odd-odd and odd ones are shown in the fig.4a,b,c,d for the system 5l*Fe+a. The fig.5 shows only the even-even nuclei populated in the 5*Ni+a system.

The smallest cross-sections observed by the present experimental technique are

around 5mb.

E, (M.V )

Fig. 2a) In beam Y-r«y« from the reaction *JCa+"o

Pig. 2b) In baaa Y-ray» £ro« the reaction **Ng+'*S

386

100

50

130 MeV

42Ca+ łC0

/ 125 Mev

Fig. (-MX

3) Coapound nucleus excitation energy

50 i. angular •oaentua) of the

OD O*

E.<M«V>

Fig. 4«. Excitation fur.rtlons for single y-rays In the reaction

I . mm £•<•*«>

Tigs. 4% and c.

Pig. 4d.

389

150

50

\

"ft (2a)

N - • f I I

-. T ] r 1 , i r

Hi

•

E« (HcV)

E« (M*V>

too

E.(M«VI >50

O a)

»•—o* i n I 4<—2*lll«7>

. >—:.<«l-W«—<*imt) g" " f*" I -tł-iK-ia* CSIOI.

Fltj. 5) Excitation functions for single y-rays in the reaction 58Ni+a

In many cases one sees all the yrast states known; in some cases the very

high spins were not particularly searched since the main interest was fo¬

cused in the total cross-section, and in its determination, the knowledge

of these states is not relevant.

In both reactions the nuclei which would be produced by la, 2a, 3a and 4a

emission from the compound nucleus are clearly observed, of course since

with this type of spectroscopy only the end nucleus is investigated, one

cannot decide whether a particular nucleus is populated by a specific com¬

bination of the outgoing nuclcons, as for example' 3o or another one like

I2C or 2a2p2n (on this specific point we will cone back later on).

The total cross-sections for the production of the different end nuclei

versus bombarding energy are shown in the fig. 6a,b for the two systems

studied.

ZOO

E.CMeVi

Pig. 6) Total cross sections observed in the reactionts*Fe-Ki, s*Ni*a

391

The general picture is for several respects similar to what observed In similar cases. See for exaaple the reactions <a,xn+yP) on "*Au shown in fig.7 (studied at jOllch) /DJA 74/. The decreasing cross-sections can be In¬ terpreted as the'talls" foreseen by the precoapound missions. But the in¬ creasing cross-sections cannot be interpreted as thresholds: i.e. the begin¬ ning of a new compound ealtelon since they open accordingly to the data of fig.8 too late of at least 20 to 30 MeV. This fact, which will be anyhow ex¬ tensively discussed at the end of the sealner, suggest* that aaybe soae new phenoaana are observed and they need soae new interpretation.

Pig. 7) Total cross sections observed in the reaction lł7Au«a

392

3 - PRELIMINARY DISCUSSION OF THE DATA

Some of the experimented results obtained seen new. We will try to inter-prete the present data from o particles bombardment in the frame of the data existing from other high energetic particles (E>100 MeV) which are anyhow not too many. Since we are in a relatively new field we will try somehow to revlw the present status of understanding similar experiments in order to try to come to some conclusion on the experiments with the a beam.

The data concerning the present experiment are optically arranged In the fig. 9 in order to see' at once the relative position of the nuclei populated and the relative cross-sections from the darkness of the squares identifying the different end nuclei. The data are presented only at an intccMdiate ener¬ gy of 120 MeV since the cross-sections do not change drastically with the ener¬ gy. This way of representing the data Indicates the main flow direction of the outgoing particles and allow a quick comparison between reactions Induced by different particles. The sane way of presenting the results will be adopted also for the data from other experiments. "

4 - THE QUESTION OF HI EMISSION AND THE ALAS SCATTERING

Before discussing other phenomena it is better to discuss whether the ano¬ malies observed could be interpreted as due to HI emission (3<x-'zC, 4a«"o, 2<łpn»10B ) of course of non statistical type since this last one should hap¬ pens much earlier (see fig.8). Such a mechanism has been in fact invoqued to explain the so called Anomalous Large Angle Scattering, observed,i.e.,in '"ca /BUD 64/.indeed heavy ion emission (*Li, *Be, łiC) with cross-sections of yb/str have been observed.

In any case we have searched expressly for 12C, " o and l0B looking at pos¬ sible Y-lines coming from the first excited stated of these nuclei. For I2C and "o there might be troubles due to possible tPppler broadening of 4.4 and 6.6 MeV lines; the 10B case is easier since the first excited state has 717 keV. The cross section for the production in bean cf these y-lines is smaller than 5 mb so we can exclude with a certain confidence the emission of heavy ions.

393

I i I i \ i

i »

i »

c o • •4 U U o 01 0) c ii

li I i"

• » • » * r « 3

I I I 5 . 3 -u •o

lii i i

•i O

•3 a 5S ł

& t

I I

o S

s I

5 - THE INTERACTION OF", K, p ETC. HI WITH THE NUCLEI

It seems worthwhile to try to understand the present experimental data in

the frame of the existing results with other high energetic particles (elenen-

tary and more complex). It is not said that this is the best and most correct

way to interprete the data but, since not so much information is known in this

field, this is probable the unique way we have: essentially let us see the

possible common features for this type of interactions.

There are data existing with pions, kuons , protons, alpha and a large

variety of heavy ions. The masses of the different projectiles are the fol¬

lowing:

* K p a HI

.15 .50 1.0 4.0 M l (masses in Amu)

We know from the high energy physics that the v and K Interact strongly (like

the nucleons) and, apart from the problems connected with the production of a

beam of similar particles, they differ only in the mass (the interaction is

basically the same). Moreover the lifetime of both particles W O seconds)

is so long that for these type of experiments these particles can be conside¬

red as stable.

If one then try to compare all these different particles one has essentially

projectiles of very different masses with quite different angular aosenta in¬

volved.

With HI one has very large angular momenta involved while on the opposite

is well known that the pions are absorbed and Interact in an s-state (I.e. with

zero orbital angular momentum involved). One might expect maybe, from what is

known from high energy physics, that the IT and K interact Bore singularly with

nucleons rather than the other particles. But also this difference can be for¬

gotten for example from the data obtained with pions at rest /(ILL 74/, at 60

KeV /ASH 74/ and higher energies: 220 HeV /JAC 75/, 3B0 HeV /JAC 73/, where one

observes clearly nultinucleon removal from the target and not only scattering

and single nucleon removal (typical of direct nucleon interaction!.

From these considerations one can assume to deal essentially with particles

which have basically the sane nature.

Let us now see the peculiarities of what observed.

395

6 - THE FIRST DATA (The"so-called" enhanced a-emission?)

We start with the first data with n and K. (BAR72, JAC 73, LIN 74, ASH 74,

ULL 74, SEG 76) and 100 MeV proton /CHA74/ since they first raised the question

of a possible enhancement in the a-particles emission. Two examples are shown

in fig.)0a (60Ni+n~, 380 MeV) and 10b (56Fe,se Ni+p, 100 MeV) where practically

only the nuclei differing from the target by 1 alpha particle were observed or

analyzed. In the proton cast following some previous work /WAL 66, KKO 70/ it

is proposed that an incident nucleon initially interacts with one nucleon in

thr nucleus and then Ifollowing this nucleon-nucleon encounter) one nucleon

usually leaves the nucleus carrying away significant fraction of the energy.

This single large energy collision occurs with a probability of 0.8. Therefore

i hi' data are not explained by the evaporation model. This process leaves ap¬

proximately half energy (50 MeV) to the nucleus. Therefore it is already pro¬

posed the necessity of a mechanism for decreasing the average excitation of

the target nucleus below that which woul^, be achieved by absorbing totally the

incident particle. It is important to start with these first experiments be¬

cause this story of a possible a emission enhancement mad' some rumor a couple

of years ago. Now, as shown later, the fact is totally redimensioned.

7 - THE "SECOND GENERATION" EXPERIMENTS WITH 71 AND p .

A successive series of experiments have been done at LAMP/JAC 75/.with 2 'O MeV

'.' «nd 200 MeV p on 5 8 r t°Ni, and /SAD 77/ with E =304 164 MeV on Ni isotopes. P

The data are shown in the usual way in fig. 11 for TT beam a.id in fig. 12a and 12b

for the p beam.

In the data of /JAC 75/ one essentially sees how not only a removal is active

but practically many other channels of roultinuclcon removal are open.

There are 5.2 to 5.4 average nucleons removed with the TT beam and 4.1 to 4.5>

with F at 200 MeV (in the data /CHA 74/ with E =100 MeV one had less nucleons P

removed ("v.3.211. The flow of the outgoing particles -eems to go along the bottom

of the valley of stability and therefore the "a nuclei" (the even even ones)

might seem to show up better (since moreover all the y-ray strength is concen¬

trated only in one y-line: the 2 -0 transition).

The protons data were taken at 80, 100, 136, 164 MeV; so for the 5BKi target

they extend and overlap the previous measurement /CKA 74/.

396

a)

b)

c)

r> 10mb

4mb<

2mb<f<4mb ' • % : • .

i

60 Ni+n"(519MeV)

Fig.10a) Nuclei populated in the reactions '°Ni+n

» ) ^ ^ «>Z0mb

•T.

(i i»tomb

bl 2mb««<10mb

C>

Fig.10b) Nuclei populated in the reactions stFe, 5<Nl+p

397

O t O

50 «««i0 CJ:«««-.7O

Cl: 21 •••••40 , t ,

"Sc

ł ) MNi*ir ł(220MeV)

1S«C«30

O 1-t«IO . * "

- i r

l» 1 b) "N,*n-(220Mev1

B Cl *Nl»lł ł

, T i

•SC

tOOMevi

• t i

"Mi,

" I .

*Mr

" t .

"kł.

•cr

"N,

"c»

"tr

r-

*•?«

, N .

75

" F .

N "Ft

>

••r

%t "Fe

>

*NI+p<200MeV)

U) Nuclei populated in the reactions "NI + T T , s»"°Nl+p

PROTON NUMBER-Z ii St

r • I

i

AVERAGE NUMBER OF NUCLEONS REMOVED

a 0 U M 4 a K-

3 n 01

•D M O 01 It 3 rt M i H-3 tu

3 c o It

a Mt Eeri

3

0 a

« o 0 Mi

r ł 3" a o o u 01

01 It n rt H-0 3 01

0 tr tn It

3 to o. 3

5 O

8 f t

rr

w

Many exit channels are observed besides the a ones and some have a total cross section larger (fig.12a). From the sum of all the observed channels 60% to 90* of the total reaction cross section is observed. The nuclei populated lay around the line of stability. It is significant to compare the 56Ni with the 6<1Ni nu¬ cleus (fig.12b). This explains the strong a-removal cross section observed with 5eNi in the previous experiment.

The average nucleons removed increase with bombarding energy (fig.12b) with a slope of 0.015 nucleons/HeV and not with -vO.l nucl/MeV as expected by the eva¬ poration model. It is suggested first a pre-equilibrium phase, i.e. a fast p or nin the exit channel and then a successive evaporation. This is suggested by the "*Ni data where the (p,pxn) is mainly observed, and not the (p,xn) typical compound,with more than 50% of the total cross section. Calculations made for this reaction /GAD 77/ with the pre-equilibrlum excitation model foresee 'vi.5 fast nucleons emitted in the»e reactions, 3.4 to 4.3 average nucleons emitted and, at the energy reported not a's emission but rather 2p2n.

8 - THE PRE-EQUILIBRIUM EMISSION

The last piece of information comes from the work /GAD 77b/ on the nearby region with A=90. The cross-sections from the reactions: "sr+p, E «30?8S MeV, "y+p, E -3Of85 MeV, '°Zr+p, E »9ł86 MeV have been analyzed with calculations based on pre-equilibriua + evaporation processes. An example of the data is reported in fig. 13. The relevant points of this work are the following:

a) in order to reproduce the experimental data one needs to assume a preformation probability for ot-particles Inside the nucleus of about 10%. This is shown by the difference between the continuua and broken lines in the fig. 13. b) The experimental cross sections present two minima with a diffference of 130 MeV. The second minimum can be Interpreted (as predicted by the calculations) as the opening of the next threshold, where, instead of 2 protons and 2 neutrons bounded in one a particle, one has the four nucleons coming out separately. Por this process one need* roughly 30 MeV energy more due to the fact that the bin¬ ding energy of one ct particle must be additionally given to the nucleus in or¬ der to see 2p2n (inctead of one a) cosing out.

400

90 Zr t P

M

«•-

10°-:

-o-

/ %

/

-o-

23 T

-o-

\

\ \

\

\ \

• •

-o-

v> 61

r

A / / / / / /

("*)

/

1 44

\

\

72

Y

/ /

w to ;o to to

Fig. 13) Data (experimental and theoretical) from the reaction 'cZr+p. The

number above the abscissa line are the effective thresholds compu¬

ted as explained in the text.

9 - TENTATIVE INTERPRETATION OF THE REACTIONS 5kFe+a AND s*Ni-ta (E =90łl60 MeV) a

After the analysis of the "analogous" reactions induced by elementary par¬ ticles like n and K, protons and heavy ions, one can try to interprete the phenomena presently studied.

The strongest cross sections observed as for example: 2a and a2p, (SBNi+a system) , ap, 2a , a2p, a2pn (5<#Fe+a system) and in this last case some weak ones like 2pn, <jp2n, apn, 2p decrease slowly at higher bombarding energy or are rather flat. This is rather typical of the pre compound emission which shows up with a flat tail at higher energy once the pure evaporation cross section ilocrcases. In these cases the evaporation part (with the typical bell shaped cross section) cannot be observed because shows up at an energy <<90 MeV; in Huvio processes not so many particles are Involved and the thresholds open at lower energies.

Mwayts (for small and large cross sections) many a-partlcles are i-mttti'd. In fact nearly each cross section is labelled with at least la. and in many cases up to 4a are observed. This could be a temptation to Interprete these plionomena with an enhancement of a-emission (similar to the first TI and p induced reactions). Most probably these processes with a particles in the exit channels ire the most favoured one for the deexitation of the two nuclei 5(1Fe and 5*Ni. The main flow of the outcoming particles is running essentially around the N=Z+2 line (5"Fe, 50Cr, *'Ti, Ca) and touches marginally the N-Z+4 line (52Cr, uliTi); and just around these nuclei runs the valley of stability in the 1 f 7/2 shell. That means that the outcoming particles essentially are streaming out in the direction expected from simple considerations. So this is not a new pheno¬ menon.

Something new seems to appear in the cross sections which increase more or less rapidly like 2a2p, 3a. 4a (both systems) and 2apn, 2ap in the sl<Fe+a system [but one should remember that the cross sections leading to non even-even nuclei in the stNi+a system have not yet analyzed] . These processes are not connected to the emission of heavy ions like ' 2C (3a). " o (4a). I OB (2apn). As already discussed if these nuclei are emitted the cross section it much smal¬ ler than 5mb on all of the three cases. And this of course excludes any possibi¬ lity of breaking up the nucleus (fission) as on the other hand well expected. In all these cases it appears that a "threshold- opens. Within the statistical

402

model (evaporation processes) these thresholds can be simply predicted, within maybe a couple of MeV error, by the formula

Ethr " W n ) "

with E k i n<n) = 1 MeV, E x(core)s 2 MeV, and ^coal the coulomb barrier for the p and a particles. This formula la a rather good starting point as observed several tiaes. In par¬ ticular it check*quite well also for the system *°Zr+p previously discussed. The table compares the thresholds experimentally observed with the calculated onejunder the assumption that the maximum number of a particles are emitted (Op-On) and that successively la particle is broken, '2 etc... of course the different columns differ each other from i<28 MeV which is the binding energy of one a-particle.

All these processes cannot be described as the emissione of a particles In the maximum number allowed since the thresholds observed are at much higher energy than expected (we are between 10 to 30 MeV! off). The next step is to see whether 2p2n are emitted instead of a (at "v-28 MeV higher).

EFFECTIVE THRESHOLDS

!*Fe*X

2apn 3a

2a2p 4a

SIHiłO

2a2p 3a 4a

EXP (MeV)

80 85

85-90 M05

A.70 80 90

+Op-On

59 48 58 63

55 47 63

CALCULATED [MeV) +2p2n +4p4n

87 76 86 91

83 75 91

115 104 114 119

111 103 119

+6p6n

132

147

»_•»

131 147

1.03

This cannot be excluded also from the recent data in the '°Zr+p reactions.

This alternative interpretation in some cases is not too bad, like 3a and

4a emissione in 58Ni+a, but in other cases fails, like 2apn, 3a, 4a in

5*Fe+a. Moreover the next peak (where the next a particle is broken) is prac¬

tically not observed. One exception might be tlje 3a in "Ni+a, where the dip

at 130 MeV might be interpreted as the next valley, (which anyhow does not

appear 30 MeV higher, 80+30-110 MeV, but 50 MeV!).

One can conclude that even this interpretation of the data explains only

partly the phenomena observed. An alternative mechanism (not in opposition

and from many aspects complementary) could be a quasi free a-a scattering

(= knock out) mechanism. The incoming a-partlcles scatter on a performed a

particles of the surfice of the nucleus. Then one of the two a with = one-

half of the energy penetrate into the nucleus starting the intranuclear cas¬

cade (precompound + compound). This means that all the cross sections are

shifted to higher energies (since not all the energy of the a particle is

left to the nucleus). In particular the processes with 2a in the exit chan¬

nel should appear 20-40 MeV higher as one really observes.

The other processes with only one a particle in the exit channel, since

this fact appears definitely at lower energy, cannot show up in the present

experiment. Moreover at these lower energies, most probably this semifree

scattering does not appear yet and one has more compound nucleus processes.

All the phenomena related to the clustering of a-nuclei, discussed in

details in the next point, support this picture.

10 - THE a CLUSTERING PHENOMENA RELATED TO THE PRESENT EXPERIMENT

Since there are several facts related to this experiment, it is maybe worth¬

while to give a larger introduction to this point following also a recent work

/HOD 77/.

One of the simplest reasons of the possible existence of the a-substructure in

the nuclei is due to the fact that the binding energies of different particles

are:

d t- T a

2.2 8.5 7.7 26.2 MeV

In addition the average radius of an a is 1.6 fn, of a d 2.2 fn (nucleon radius

0.4 fm) and the average nuclear distance is 1.5 ftn. From these data an a parti-

UOk

cle may exist (i.e. = strong overlap of the 4 nucleons shell model wave func¬

tions) inside the nucleus.

Beside this, there are several other facts to be mentioned:

a) Some nuclear matter calculations recently performed /BRI 73/ in order to

see the possibility of nucleons condensation into a clusters do not foresee

any particular clustering at the normal density of nuclear matter. Only if

this density is reduced of a factor 3 there comes a point where alpha-parti¬

cle condensation can appear; this can practically happen only at the surface

of the nucleus where the nuclear matter density drops rapidly to zero.

This speaks for possible existence of a particles at th« surface of the nuclei.

b) A preformation probability of around 10% has to be invoqued not only in the

experiments like the '°Zr+p previously mentioned /GAD 77b/ but also in other

type of experimental data. The analysis of several (p,a) and (n,a) knock-out

type reactions /MIL 74/ need to lnvoque a preformation probability of 0.2 to

0.4.

In agreement with these data if one include also the analysis of the a decay

(heavy nuclei) one again /BON 74/ obtains a preformation probability for an

a particle >10».

c) A third experimental consideration cooes from recent data from the reaction

'°Zr+a at 140 MeV / WUJ 77/ where the outcoming a-particles were directly de¬

tected. From the fig.14 one sees that at forward angles many high energetics

""Zr + w .Ea* 140 MeV

Fig. 14) Alpha particles observed

'in the reaction '*Zt*a

a are emitted. And also for this reactions two types of mechanisms are propos¬

ed: one which occurs on a fast time scale retaining the initial dynamical

information and a second which results in an equilibration process.

In conclusione one can say that there is a large evidence of the existence of

a-particles in the surface of the nuclei so that this semifree scattering is

quite likely to occur.

11 - REACTION AT VERY HIGH ENERGY (SATURNE)

Some experimental evidences of the reaction mechanism proposed for the

a-reactions studied in the present work come also from some experimental data

from high energy p and a interactions at Saclay.

The data are from the reactions ""Ca+p at 210 and 600 MeV /ALA 73/ /ART 75/

and from the reactions *'co and stFe*xj at 600 MeV /ALA 74/. See figs. 15,16,17.

The important fact, if one looks at the figs.15 and 17, is that the panorama

of the nuclei populated with the '"ca target and protons of 230 and 600 Mev

practically does not change. This means that the projectile deposits very

small energy and practically always the same reguardless its own velocity ior

energy) .

Xa+p

= 230 MeV

"Co

1 l i Er(M»V) • * * trl««rl

Fig. 15) Gamma ray spectra from the react ion llOCa+p

It 06

The situation in the system 5tFe+a at OIMJ AeV is also very similar to s*Fe+tj

at 150 MeV (Fig.9) and this of course supports the previous assumptions.

In conclusion there are really several evidences which support a reaction

mechanism where at higher bombarding energies the interaction of the projecti¬

les (in this respect p or a are identical,'and most probably all Tt, K) is al¬

ways more localized at the surface of the nucleus and practically leaves al¬

ways the same energy at disposition of the compound nucleus.

200 <00 600 » 0 BOO C00 VtD Enarai*

2 0 0 .

59,

§« = 600 MeV

200'

FP-KX

= 600 MeV

•**1 400 600 BCD WD B00 CANAUX

Fig. 16) Gamma ray spectra from the reactions 5?CO*TJ and vsFe*a

kot

Fig. 17) Nuclei populated in the reactions '''ca+p, 5łCo,

12 - REACTIONS WITH BI

There is a final aspect in thi* comparison with other "analogous" reactions that I would like to bring. This is with the two BI induced reactions *'ca+"o and 2kMg+32S where the saae compound nucleus as in 51"Fe+a is fonted. The BI data are coming from Masureaents lnpublished, done by our group at Munich Tan¬ dem. As already shown in fig.3 in these three reactions at the bombarding ener¬ gies studied one has similar angular aostenta but different excitation energies, so it is not so clear versus which quantity the measured cross section should be plotted (actually the best way would be a bidlmensional plot but somehow difficult to be visualized). There are some problems with the absolute values of the 3 cross sections, so, maybe, one need some (very) empirical and crude nor¬ malization. In the plot of fig.tS, versus excitation energies one sees that for example the 3a from the BI reactions comes much early than the 3a from sllFe+a reaction. This is a very good sign to support the theory that in the second case the process is semidirect, and should appear much later than in the BI reactions where the process of compound nucleus type appear where expected.

U08

50-

compound **Ni

500

"Fe + a

50 100 150 tjjjIM.ir,

« »0 [„„(WO

Fig. 18) Cross sections versus excitation energy (the compound nucleus is

the sane in the 3 cases)

But if one looks at the plot versus the transferred angular momentum (approximat¬

ely the same for the 3 reactions (fig. 19) all the 3 curves of a reaction like the

3a emission (anomalous in 5lTe+a systeml are roughly in the sane position. This

indicates that the parameter angular momentum practically never taken into ac¬

count in all the previous considerations may play an important role.

13 - CONCLUSIONS

As conclusive remarks on these a data on sllFe and SBNi I would summarize

the, following points;

a) There are some cross sections observed whose behaviour with energy cannot

be simply explained only in the frame of the existing models.

b) This "anomalous" behaviour can be partly interpreted with a semi-free a-a

scattering on the surface of the nucleus.

50-

* — **«•<»

20

'* 40

Z'"

W

n

•o e

/7

40

•0

\

' u,.

Fig. 19) Cro** sactions versus angular •onentum transfer (the compound nu¬

cleus is the sane in the 3 cases)

This is in agreement with the indication that at higher boabarding energy the

interaction of any bonbarding particle is always sore localized on few nucleons

of the nucleus and deposit into the nucleus 'always a smaller fraction of its

energy.

c) There are still sone aspects of the interaction related with the angular

•BomentuB transfer not yet totally understood.

* * *

I would like to aknowledge the collaboration of all collegues of the

Institute of Nuclear Physics (J&lich) and, in particular, of Prof. O.W.B.

Schult , who made this experiment possible as well as of Dr. Spolaore, who

analyzed very carefully the data.

(The most of the pictures of this seminar were extracted from her doctor

work at Padova University).

k\0

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/GAD 77/ Gadloli E., E.Gadioli Erba, G.Tagllaferri; International Confe¬ rence in Nuclear Physics, Tokyo 1977, P.725.

/GAD 77b/ Gadloli E., E.Gadioli Erba, J.J.Began; Phys.Rev. 16C_ (1977) 1404.

/BOD 77/ Hodgson P.E.; Alpha-clustering in Nuclei in:The uncertainty Prin¬ ciple and Foundations of Quantum Mechanics (J.Wiley and So., London 1977) ed. W.C.Price and S.S.Chlssick , p.405.

/JAC 73/ Jackson H.E. et al.; Phys.Rev.Lett. 31 (1973) 1353.

/JAC 75/ Jackson B.E. et al.; Phys.Rev.Lett. 35_ (197S) 641.

/KRO 70/ Kroll P.R. and N.S.Hall; Phys.Rev. Cl (1970) 138.

/LIN 74/ Lind V.G. et al.; Phys.Rev.Lett. 32. (1974) 479.

/MIL 74/ Mllazzo-Colli L., G.M.Braga Marcazzan, M.Milazzo, C.Slgnorlni; Nucl.Phys. A216 (1974) 274.

/SAD 77/ Sadler M. et al.; Phys.Rev.Lett. 38 (1977) 950.

/SEC 76/ Segel R.E. et al.; Phys.Rev. 3C (1976) 1566.

/SIG 78/ Signorini C . M.Morando, G.Fortuna, A.M.Stefanlni; IV E.P.S. Nuclear Physics Divisional Conference "Physics of Medina-Light Nuclei", Flo¬ rence 1977 (CoBposltorl, Bologna 1978) ed. P.Blasl, R.A.Rlccl.

/ULL 74/ Ullrich B. et al.; Phys.Rev.Lett. 3_3 (1974) 433.

/WAL 66/ Wall N.S. and P.G.Ross; Phys.Rev. 15£ (1966)- 811.

/WUJ 77/ Wu J.R., C.C. Chang, N.D.Bolagren; International Conference in Nuclear Physics, Tokyo 1977, p. 726.

Nature of the 0* levels and shape transitions in the Ge and Pt regions

M. Vergnes Institut de Physique Nucleaire, BP n°1, 91406 Orsay, France

My talk shall be mainly devoted to shape traasitions and to the correlated sub¬ ject of the nature of the 0+ levels in the Ge and Pt regions.

A few years ago we decided in Orsay to try to gather as much experimental re¬ sults as possible on nuclei in these two transition regions using transfer reactions. The difficulty of many of the experiments performed were generally high coulomb barrier and/or large negative Q value. On the other hand, because of the complicated structure of the nuclei studied the level density was generally rather high. These experiments could be performed using our super-MP tandem which has the interesting -and rare- fea¬ ture of working routinely') at 13 MV, giving p and d beams of 26 MeV and lie and a beams of 39 MeV. The emitted particles were analyzed by a split pole spectrometer equipped in its focal plane with 8 home made solid state position sensitive detector*. The energy resolution was always between 20 and 9 keV and that permitted to observe, even at low energy, many new levels unresolved before. As an example the first figure shows the re* gion of the second 2* level in 68Ge. Our resolution has permitted to resolve it as a doublet and we have shown') that the second member is the first 0* excited level of 6 Ge, previously unknown.

My talk shall be divided into two parts : the first one devoted to our experiments concerning the Ge region, the second one devoted to possible interpretations of our re¬ sults and also in some cases of results obtained in other laboratories, in the Pt region.

I. Nature of the low-lying 0ł level in the Ge isotopes.

It is well known that there exists in the even Ge and Se nuclei and also in other even nuclei in the same region, a low lying 0* level. In the Ge, the energy of this 0* level appears (fig.2) to vary regularly with N the number of neutrons and it becomes the first excited level in Ge (N«40) . The nature of this level and more generally the exact structure of the nuclei in this region is still a puzzle. It should be recalled ro better understand the interest of a nucleus like 2Ge, that only very few nuclei ( 'O, "0Ca,72Ge,90Er,96Zr,98Mo) are known with a 0+ first excited level.

In fig.3 are shown the ratios of the energies of the levels divided by the energy of the first 2* level. It is clear that the variation with N for the 0* level is very different from that observed for all the other levels and that the ratio becomes very small (1.17 and 0.83) for °Ge and 7 2Ge, showing clearly that this level if not a puoncn state. There is, as can be seen in fig.3, a 0+ level close to the position expected for the 0+, 2-phonons level, but in 68-7<>-72Ge it is not the first excited C* level, but the second. Incidentally this second 0* level in 7 aGe was first observed by us 3). This is true also of the first 0 level of 6 Ge, as already said.

In the past, up to 1976, several people have tried to reproduce theoretically the position and sometimes the electromagnetic properties of the low-lying 0; level. Briefly, to &ive a few examples, Gneuss and Greiner4), Larsson and Ragnarsson ) and others using different recipes, calculated potential energy surfaces and minima in these surfaces, sometimes obtaining the corresponding energy spectra... De Vries from Utrecht ) performed quite complicated q-p plus phonon coupling calculations... To make a long story short, the low lying 0 is never correctly+or convincingly reproduced, but the other levels and even the second experimental 0 excited level are more or less repro¬ duced.

In order to try to shed a light on the nature of this rather mysterious O2 level,

1*12

1.753 „• AE(FWH-M) » * 10 ktV

177S

Gt(i,t) Gt

f ig 1

3.

2-

GE nucUi

n n "rrrrr* "r

i* n u n n x

fig 2 fig 3

II

CM • 4 II

O

II

00 o II

z

(O m u

kik

we have cried to observe it in as many experiments as possible and to measure the ratio 8"O(0:)/o(0 . ) . The experiments done in Orsajr>3»7) are summarized in figure 4. The dotted lines show (t,p) experiments performed recently at Los Alamos®)-

Our first and certainly more illuminating experiment has been the (3He,d) reac¬ tion on the Ga isotopes^)• Before this experiment there had been a proposed shell model explanation for the O2 level in 7 2Ge. The explanation was ) that it was exactly of the same nature as in 90Zr (the 40pof Zr being replaced by 40n in Ge), that is to say a neutron configuration mixing of (pl/2)2 and (g9/2)2, due to pairing and orthogonal to the 0+g.s. On the other hand an experiment done at Mac Masters : 71Ga(d,t), shows10) that the occupation probabilities of the pl/2 and g9/2 orbitals are about the same as in Ge 72 (and also Zr 90). This leads to the scheme given in fig.S. It is clear that, if we perform the (3He,d) reaction on 7IGa, the operator can only change the proton configuration, but not the neutron one. Therefore ve should populate only the g.s. and not at all the excited 0 . We performed the reaction mainly to verify if that was true. It is not and the R value is very large : 0.7. Therefore we conclude that the above des¬ cription of the two 0 levels in 72Ge is wrong. It is also clear that the excited 0 level is definitely not a collective state but is mainly a rather simple shell model state. I am going now to outline rapidly a model similar to the one of Lobner,Hade roan, Monahan, Fournier et al. ) but based on a simple configuration mixing of protons ins¬ tead of neutrons. This model is very schematically summarized in fig.6.

The order of the shell model orbitals is given in fig.6 and it should be recalled that the p3/2 orbital should be filled at Z -32. In fact it is well known that all but 3 of the many odd Z nuclei between Z -29 and 37 have a 3/2" g.s. indicating that the p3/2 orbitals is in fact not filled and that the additional protons go on another orbital.Wo have made the simplifying hypothesis that they go only on the f5/2 orbital.

We admit in this very crude model that the neutrons configuration is the same for the level| of the nuclei with the same value of N, at least the ground states and the excited 0 levels, and that there is a configuration mixing for the protons on onl, two orbitals : p3/2 and fS/2. With these very drastic hypothesis the wave functions for the nuclei 71Ga and the two 0 levels of 72Ge may be written :

<l>Ga -a(p3/2) 3 + 6(p3/2) . (f5/2)*

2 (f5/2)2

0 0 72

„ „ -B' (p3/2) 1 < -a ' ( P 3/2) 2 (f5/2)2 e0

0

and similarly for 71|Ge

>--. 6"(p3/2)2 (£5/2)2

-„ . 0 0 0 74 8 > s

„ ^ -6"(p3/2) ' t -o"(p3/2) i ! ( f5 /2) 2

eo+ 0 0 0

We have only in fact 3 free parameters to be determined : <x ,a' and a".This has been done by using : 1) the occupation numbers measured for the p3/2 and fS/2 orbitals in our 7'Ga(d,JHe)7°Zn reaction ; 2) the ratio R in the reaction 7lGa(JHe,d)72Ce i 3) the ratio R in the reaction 71>Ge(p,t)72Ge. The values obtained are : a 2-0.87, a' 2-0.37, a"2 «0.03. We then have a model, very crude of course as I said, and we can try a test by using it for predicting othir experimental quantities. The results of chis test are shown in Table I.

g«/2

Ptpi/D*

36 n

32 P

gi/2

pt/2 a(p 1/2)2

36 n

32 P

T] Seo» M

gf/2

pi/2 a(p i/2)2

36 n

32 P

• • •

pi/2

FK5.5

f 5/2

PI/2

28 P

to n

TJ

26 P

to n

'Go.

Our model

Z=32-

g ta p i/i r sn

P ł/2

FK5 6

Table I

Comparison between quantities determined experimentally in several reactions and the corresponding values predicted by our simple model. R is the ratio a(C>2)/o(0t s ) .The underlined values have been measured by us in Orsay or Los Alamos.

Reaction

7ICa(3He,d)72Ge

id

id

7*Ce(t,p)71|Ce

73G=(P,d)72Ge

7sAs(d,3He)7*Ge

7'JAs(p,a)72Ce

Quantity measured

CSS-O*g.s.

G. Gj

K

K

R

R

Values

Exp. MODEL

3. 1 3.2

5.9 5.8

25? 2SZ

% o o i\l V.

607. 592

It can be seen that the data are reproduced quantitatively and it is therefore reasonable to conclude that there is at least some truth in our description. In fact,?, recenl calculation ) of Faessler and co-workers shows that the low-lying O2 level of Ge can be reproduced at the right position, but only if they include in a generalized

generator coordinate method the 2 quasi-particle states for protons, those tor neutrons having pratically no influence. This is a theoretical confirmation of our empirical des¬ cription.

In summary it seects well founded to say that in 7 ?Ge, vjhich otherwise bi-haves as 3 quite soft vibrator more or less spherical, the low lying 0 2 level is mainly a very simple shell model or 2 q-p proton state.

II. Shape transition in the Ge and Ga isotopes-

How that we have an idea about the structure of 2Ge and its 0 Jevtls, we may stop ignoring the neutrons and try to see what is their influence on the structure of neighbouring nuclei. It is clear that there is in these nuclei an important coupling between the protons and the neutrons. 1 shall illustrate this point by two examples :

1) As a function of N the 9/2+ level of the odd-N nuclei go down in energy and becomes the g.s around N=40, in agreement with shell model. What is anomalous is that the same thing happens'^) for the 9/2+ level of the odd-Z nuclei, which seems to imply a strong interaction between protons and neutrons, at least for this orbital-

2) Several nuclear properties, some directly related to protons, show very in-portant variations as a function of N, with extremum at N=A0, as can be seen in fig-?.

The preceding results may be explained either by a sub-shell closure at N«40, which is the more natural hypothesis, or by a shape transition which leads also, as is well knowi. •*), to extremum in many nuclear properties.

As far as shape transitions or deformation in these nuclei are concerned, what is the theoretical situation ?

N

FKS. •. P f l U l »«rp W T Feck mlrwliHaiM for «•» w >

a) The possibility that an important structural change occurs between 7zGe and 7kGe would explain the fact that in the calculation of Faessler and co-workers") they are unable to reproduce even the right ordering of the yrast levels in 7<lGe, with the parameters used successfully for 72Ge and 70Zn.

b) We have ourselves preformed H.F calculations ) using the Skyrae III interac¬ tion and we get the curves shown in fig.8, **Ge being oblate, 7*Ge prolate and 72Ce being quite soft and probably more or less spherical.

14 c) Recent calculations of Kumar ) , using a new deformed q-p basis with only

one ajustable parameter, would indicate Ge as spherical, 72Ge being soft oblate and 7"Ge deformed oblate.

Other results that I have no time to mention in details, in some odd nuclei,like decoupled bands, low energy positive parity levels... are explained by admitting a pro¬ late even core.

It if clear that the models foresee shape transitions and deformations... but the shapes are not the same for the different models. It is.clear, however, that the hypo¬ thesis of a shape transition in the Ge nuclei around N-40 does not appear as unreasona¬ ble.

Experimentally, how would it be possible to make a clear choice between a sub-shell closure at N-40 and a shape transition? This is possible by a comparison of the (p,t) and (t,p) reactions. It is well known, as you can see in fig.9, that when you cross a magic neutron number N like in the Pb isotopes, the population of 0 excited le¬ vels is large in both (p,t) and (t,p) only for the magic final nucleus and R is large above N for 'p,t) and under for (t,p). The reason is, in (p,t), that when the target neutron number n is n>N, the g.s is obtained by picking pairs on the orbitals above N and the excited 0 by picking them on internal orbitals ; but when n£N, the pairs can only be picked on internal orbitals and only the g.s is appreciably populated. The same is true when adding pairs, in (t,p).

If the large value of R in (p,t) for 72Ge was due to the subshell closure at N« 40, we would expect R to be of the same order for 70Ge(t,p)72Ge.

On the other hand, when you go through a shape transition as in the So isotopes, the maximum in R does not occur'-1) at the same N value for (p,t) and (t,p). This is shown in fig.9 and the reason is as follows : the g.s of iiSm is spherical, the g.s of «gSa is deformed, therefore the g.s *•* g.s transitions are hindered both in (p,t) and (t,p). The excited 0 are "shape isomers" and are spherical in »oSn> and deformed in ««Sm. Strong transitions (large R values) are therefore observed in (p.t) for ,oSa + («Sn and in (t,p) for (gSm* joSm.

The (p,t) experiments on the Ge isotopes have been perforaed ) in Orsay at 26 MeV and the (t,p) experiments very recently®) in Los Alamos at 17 MeV. The results are shown in fig.10. It is clear that they look very much the sane as those observed for the Sm isotopes, and not at all like those observed for the Pb isotopes.

The effect observed for the even Sm isotopes when going, between N«88 and N-90, from a spherical to a deformed shape, has been known for quite a long tiae. A similar effect has been observed more recently'6) for the odd Eu isotopes, at the saae neutron numbers, the explanation being the same. With that in mind, we have also perforaed the (t,p) reaction at Los Alamos on the two Ca isotopes. The results are coapared in fig.II to those corresponding to the Eu isotopes. The similarity is as striking here as the one observed above between the even Sm and Ge isotopes : in the reaction *^Ca(t.p)jj^Ga, the L-0 g.s+g.s transition dominates the whole spectrum, but in the reaction JJCaCt.p^JCa, the L-0 strength is divided between 3 levels, only 302 going to the g.s. We therefore conclude from the above evidence that it seeas that a shape transition takes place be¬ tween N-40 and N-42 in the Ge and Ga isotopes. The values of R observed for the even Ge are smaller than for the even Sa isotopes and the g.s*g.s transition im not completely hindered in the Jl0Ga(t,p)JjGa reaction, as it is in the

l|,EuCt,p)1 "Eu reaction. It is therefore clear that the overlap between the g.s wave functions is larger in the Ge.Ca

122 124 126 128 N (final nucleus)

88 90 92 9t N( final nucleus)

U20

t7)

00

I . . . 1111 i t i i i 111 i

100 6S 31 Go «So 73

12 Go

100

us «6 fc

FK3.11

422

transition at K-40-42 than it is in the Sm, Eu transition at N-88-90.

III. What shapes 7

Let us come back now to our very crude and naive model of paragraph I. We have seen that it accounts already for many experimental results. We have very recently per¬ formed') the (d,sHe) reaction at 26 MeV on the Ge isotopes, in order to measure the po¬ pulations of the proton orbitals in these nuclei. These populations are shown in Table II for the p(3/2+l/2) and f5/2 orbitals.

Table II

Occupation numbers for the fS/2 and p( 1/2+3/2) proton orbitals in the Ge isotopes. The values within parentheses are obtained from our model wave functions.

occupatioti"*1^^ nurafrfr* ** "-

<f5/2>

<p(l/2+3/2)>

38

1.24

2.95

40

1.34 (1.26)

2.78 (2.74)

42

2.20 (1.94)

1.87 (2.06)

44

2.40

1.65

Here also, we observe the evidence of a structural change between N-40 and N«42, one proton which was in the p(3/2) orbital for the lighter Ge jumping then on the f5/2 orbital. The results of our model are also shown in Table II and we see that it gives a fair account of the change observed between 72Ge and 7l4Ge. We must also keep in mind that the strong population of the excited 0 level in the 72Ge(t,p)"lGe reaction is re¬ produced by the model. It is therefore tempting to look more carefully at the Ge wave functions and to see what is the change between N-40 and 42.

We see easily that going from N"40 to N-42 the components with more than 2 parti¬ cles on any orbital disappear. It is possible to show?) that the same is true for reaso¬ nable Ga wave functions. I might of course veil be wrong but that could correspond to a change from a standard shell model coupling scheme for Ge to a deformed coupling scheme for 7"Ge (like the Nilsson model), where no more than 2 particles are allowed on a given orbital. The transition would then be a "spherical"* "deformed" one, like for the Sm isotopes, but more gradual. This would be in agreement with the conclusion of ref.12, that the deformation in this region is maximum for N«42.

IV. Nature of the 0* levels in the Pt isotopes.

Kumar and Baranger ) have calculated long ago that an oblate to prolate shape transition takes place, when going from tl.e heavy to the light isotopes of Pt. Experi¬ mental evidence seems to support this prediction and the transition is believed to take place between A>188 and A-186. However some properties of the heavy Pt nuclei have been described recently using a rigid triaxial rotor model'**).

The ratios of the energies of the levels of the even Pt nuclei, divided by the energy of the first 2 level, are shown in fig.12. From the behaviour of this ratio as a function of the mass number A, it is clear that the levels can be arranged in • quasi-ground state band (spins : 0,2,4,6), a quasi Y band (spins : 2,3,4) and a quasi K-0 band, (spins : 0,2). In the Sm isotopes, as was shown before, low lying 0* levels have been beautifully observed as strongly populated in the (p,t) and (t,p) reactions. This well known phenomenon is due to the rapid change in 6 when going fron N-88 to N-90 and the

H23

C(h»V)

use

too*

sto

tt«««M

na tu III IM ni »2 m ni Fi(.12 Energy levels of the Pt isotopes. «:

W IK W) Wt

6* r*''

4 5% °*~*Ś

'!

..5-<25*/a

0+www

190

<10"J

-"rvn J*1.7%

5TT

m m

n

rif13 | qu*it trxtd «t«c« *«< T •*•*•, ••< Ik* 0* Icvtli okl«r»«4 kr ut i* tht (f it) n i t t l u . Tk* aapiilltieio *r* (Ivan ! • ><rct>t «( tkt ! • • l»t««iitj ••< t iv t l i »r<-vioutly «nkK0w» art i*4icata4 ky a «t«r<

EXP K-B

rig.14 U.y.li of "'ft. On tk* I aft '• ««»«tt»tmcal, on tka tight : coapuc<4 %y Cuaar «*4 Satasfar.

levels have therefore been called "fi shape isomers". The 6 parameter changes only slow¬ ly in the Pt isotopes, but the Y parameter could change quite rapidly ; it vould there¬ fore be possible in principle, if there were deep valleys in the Y direction, to observe 0 levels due to "Y shape-isomerism" relatively strongly fed in (p,t). This reaction has been studied in Orsay ') at 26 MeV on the 3 lightest targets normally presently avai¬ lable : I9SPt, l9*Pt and even '*2Pt (0.78Z in natural Pt). The resolution was 13 keV (F.W.H.M.). Several 0 levels have been observed in each isotope. Mo one has a strength larger than 7Z of the ground state one and the g. s-*-g.s intensities are about the same in the 3 cases (within I0Z). It can be concluded therefore that there is no evidence for y isomerism .

In order to try to observe this effect, and to go as far a* possible towards the neutron deficient isotopes, we have managed^) to prepare a quantity of the order of 40 Mg of l'°Pt (0.01Z in the natural Ptl) using an isotope separator in Orsay. A target has been prepared by deposition and drying of a dropplet of • chlorhydric solution of Pt onto a carbon backing. We have used this target last'week, and even with the relati¬ vely poor enrichment and resolution we did observe the g.» of J"Pt. Preliminary analy¬ sis permits to estimate at S 67. the ratio of the O2 cross section to that observed for the g.s. There seems therefore, even in this case, to be no evidence of Y shape isotnerism or shape transition.

The (p,t) reaction has permitted to measure the population of the known 0 levels, and to observe new ones, due to the very characteristic L-0 angular distribu¬ tions- Our results are shown in fig.13, with the 3 quasi-b tids discussed previously. Populations are indicated for the 0 levels, and levels previously unknown are indica¬ ted by a star. We see clearly 3 different sets or sequences of 0+ excited levels in th< 3 Pt isotopes. We can first try to understand the nature of these levels in the light of our (p,t) results and of their Y decay properties only.

The first O2 excited level was known previously in the 3 isotopes and generally interpreted as+a 6 vibration. This level decays mostly to the 2~ level and the ratio P»B(E2, 0+->- 2y)/B(E2, 0 •2t.8.) is of the order of 10. Such a ratio is predicted foi the Oy(K»0,2 Y-phonons) by uavydov20) and Beliak and Zaikin2') in an approximate treat merit of the Bohr-Mottelson Hamiltonian+for small vibrations of an axial rotor. The cor¬ responding predicted ratio P for the Og level is 0.1. This is in favour of a Y interpre¬ tation of the O2 level. This opinion is confirmed by the following argument : the nucleus 192Pt is rather well described by the Kumar and Baranger calculations") as shown in fig. 14. The predicted ratio P is 6.5 and the wave function of the level has a 651 over¬ lap with 3 spherical phonons, corresponding to a 2Y~phonons vibration. The first excited O2 level in the Pt isotopes studied could therefore be interpreted as corresponding mainly to the y vibration.

The second, 0*, excited level was known in l'"Pt at 1.48 MeV. It is not measura¬ bly populated in our (p,t) reaction.! Cizewski, from B.N.L, has shown ) that this level in ' 9<1Pt decays mainly to the 2RtS>, ratio P % 0.1, to compare to the theoretical "esti¬ mates of Davydov20) and Eeliak and Zaikin21). The 03 second excited level could there¬ fore be interpreted as corresponding mainly to the 6 vibration.

The ratio P for the third, ot, excited level, measured by Cizewski is of the or¬ der of 1 in lsl<Pt : this level does not seem to be observed in the decay > of the gold nuclei, when the other 0* levels arę observed, and it has apparently been much more strongly populated than the other 0 levels, including the g.», in a low resolution stu¬ dy of the J''Au(p.a)1'"Pt reaction25). According to these results, it would appear to be quite different from the other 0 levels.

. The stability of the energy and population of this level in the 3 Pt isotopes is very striking.

In order to try to precise the nature of the different 0 levels, we have decided to do all the possible simple reactions going to one selected nucleus, '''Pt : (p,o) , (p,d) and (3He,d) ; (p,t) being already studied.

One of the most disturbing result was the apparently very strong excitation of several 0 levels in the ' "Au(p,O)"ltPt reaction2'). This experiment has been done very recently in Oriay at 25 MeV, with 25 keV resolution ' ) . It appears that the peaks previously observed with low resolution do not correspond to the 0 levels and that the 0 levels are only moderately populated (see table III). The i i5Pt(p,d)'f*Pt reaction has also been performed very recently^7) in Orsay at 25 MeV. The preliminary results concerning the 0 levels are given in table III.

Table XII

E ex

0

1.27

1.48

1.54

1.89

(P,t)

integ

100

2

<0.1

6

2.4

(p,<*>

V 100

25

obscured

22

obscured

(P,d)

100

3

<0.4

13

(»He,d)

C*S

100

28

(91)

3

(62)

The (3He,d) and (a,t) experiments on '"ir were programmed in Orsay but could not yet be performed. I have however data taken at Me Masters at 25 and 27 MeV respective¬ ly26) but only one angle for each reaction that D.G.Burke was nice enough Co send to us. He gave me the permission to show it in this talk, although he of course considers it as very preliminary with measurement at only one angle.

The DWBA analysis, performed by us for the (3He,d) reaction at 50*, gives spec-troscopic factors with large error bars of course, but at least two, possibly 3, 0* levels -beside the g.s- would appear as strongly populated, which is a relatively rare occurence in single particle transfer reactioaB. The results are shown in Table III to¬ gether with the results of the (p,t) reaction") for completeness. It Must be kept in mind that the 1.48 MeV 0+ level is very close to (and impossible to separate from) a 7~ level at 1.485 MeV which i»_ clearly strongly populated in the (p,a) reaction. This could also be true (hi 1/2 transfer) in the (JHe,d) reaction -in the absence of angular distribution- and it explains the parenthesis in Table III. (A 11/2" level J£ strongly populated28) in (3He,d) on '*z0s and 1 > 00s). The same could possibly ba true for the 1.89 MeV 0+ level which appears to be strongly populated in the (5He,d) reaction at 50°.

The sun of that cross sections measured26) for all the final 0+ levels shown in Table III in the ' "Ir(a,t)''''Pt reaction is of the same order as the cross section mea¬ sured B ) , at practically the same energy and angle, for the l *20s(a,t)'*3Irg., reaction. This leads to the tentative scheme shown in fig.15. **!0s is known to be a prolate nu¬ cleus; this seems also to be true for l'3Ir, fron a comparison of the Measured magnetic moment of the g.s with theoretical values computed using the Nilsson model29). As said previously the ground state of ' '''Pt is believed to be oblate. That would be in agree¬ ment with the results sumurized in fi^.15, which look similar to those shown previously for the (t.p) reaction on the Ga or Eu isotopes and are in favour of a shape transition between " 3 I r and lfl>Pt. The strong population of the 1.48 and 1.89 MeV 0* levels in the ( He,d) reaction has to be confirmed by future experiments with angular distribu¬ tions; these levels, if really strongly populated, should have a shape sore similar to

426

22%

100 3/2*

193i

FIG. 15

the ' 9 3Ir g_ 8 than to the 1''*Ptg_s, that is to say prolate or possibly spherical.

To conclude my talk on these Pt nuclei, 1 would like to compare our (p,t) expe¬ rimental results to a very recent partly preliminary, and as yet unpublished, calcula¬ tion of F.Iachello and O.Scholten-"), using the interacting boson model of collective states, but taking explicitly into account the proton and neutron degrees of freedom. They were nice enough to permit me to show it in this talk. The model used in this par~ ticular case is quite similar in some aspects to the rigid triaxial rotor with Y"30*, but he produces 0 + excited levels which do not exist in the rigid triaxial rotor model32).

In fig.16 you can see a comparison between their calculated 0 energies and the experimental energies. (The 0 first observed by us are shown in «+circle). The agree¬ ment is quite good. It should be remarked however that only four 0 levels (including the g.s) are predicted, when 5 are known in ''"'Pt. It sees* clear (tee Table IV) that the missing level in l "Pt is the 1.48 MeV level. It is not observed in our (p,t) expe¬ riment, bu{ there are very clear evidences in the litterature 3 ) , showing that it is -really a 0 level. This level is not reproduced by the node1,which cou^d iaply that it is not a collective state. From the wave function* the first excited 0 theoretical le¬ vel has a 3 phonon character in agreement with our eapirical analysis leading to a 2 Y phonon attribution and the second one has a 2 phonon character, that is to say it looks like a (5 vibration.

"127

Table IV

Final

nucleus

•»»Pt

l* 2Pt

"°Pt

l MPt

State (theoretical

classification]

ot ot

ot ot or ot

ot ot ot 0*2

ot ot ot 0\

Energy

Tb Exp

0

1131

1495

1853

0

1123

1573

1966

0

932

1662

1944

0 _,

1265

1480

1546

1890

0

1196

(1542)

1617

0

922

1670

Tb

100

0/-03

4

1.6

100

0.04

it

0.8

100

0.05

4.1

0.3

100

0.9

strength

Exp

100

1.7

<0.1

6

2.4

100

<2.5

0.6

5

100

5

7

100

(6)

The comparison of the experimental and calculated (p,t) intensities in Table IV shows a quite good agreement for the theoretical 0s level, but the negligible theoreti¬ cal population of the O2 first excited level is not in agreement with experiment.

The inclusion of higher order terms in the expression of the 2 neutron transfer operator would however be expected to improve the agreement in this case.

The computed ratio : V « a(22)/a(2|) is compared in Table V to the experimental ratio. The important and systematic variation is rather well reproduced.

Table V

A

Pexp

rth

190

25Z

33Z

192

15Z

20Z

194

IIZ

I0Z

196

4Z+

IZ

ref.34.

U28

7 8 Pt Energies

• • exp

theo

- - 0 3

/o;

E + 2

+ 1.5

+ 1

188 192 196

f i g 16

References

1) M.Vergnes, Nucl. I n s t . 146 . (1977) ,81 .

2) F.Guilbaut, D.Ardouin, R.Tamisier, P.Avignor., M.Vergnss, G. Rot bard, G.Berrier, R .Se l tz , Phys. Rev. £15., (1977) ,894. F.Guilbaut, D.Ardouin, J.Uzureau, P.Avignon, R.Tamisier, G.Rotbard, M.Vergnes, Y.Deschamps, G.Berrier, R .Se l tz , Phys. Rev. C i6 . ( l977) .1840 .

3) D.Ardouin, R.Tamisier, G.Berrier, J .Ka l i fa , G.Sotbard and M.Vergnes, Phys. Rev. C l I . (1975),1649.

D.Ardouin, R.Tamisier, M.Vergnes, G.Rotbard, J . K a l i f a , G.Berrier, B.Gramsaticos, Phys. Rev. C12,(1975),1745•

4) G.Gneuss, L.V.Bernus, U.Schneider and W.Greiner, Colloque sur l e s Noyaux de Transi¬ t i o n , Orsay 1971, report IN2P3, unpublished, p .53 .

5) S.E.Larsson, G.Leander, I.Ragnarsson and N.G.Alenius, Nuci. Phys- A26I . (1976) ,77.

6) DeVries, Thes i s , Utrecht 1976 and pr ivate conmunication.

7) G.Rotbard, M.Vergnes, G.La Rana, J .Vernotte , J . K a l i f a , G.Berrier, F.Guilbaut, R.Tamisier and J.F.A.Van Hienen, Phys. Rev. Cl^,(1977),1825 and to be published.

1*29

8) M.N.Vergnes, G.Rotbard, F .Gui lbau t , D.Ardouin, C.Lebrun, E.R.Flynn, D.Hansen and S.D.Orbesen. Phys .Le t t . 72fl,(1978),447 and to be pub l i shed .

9) U.Eberth, J . E b e r t h , E.Eube and V.Zobel, Nucl. Phys. A257,(1976) ,285, J.Haderman and A.C.Res te r , Nucl. Phys. A231,(1974) ,120. W.G.Monahan, R.G.Arns, Phys. Rev. JJS4, (1969) , 1 135, K.E.G.LÓbner, G.Danhauser, D.J.Donahue, O.Hauser, R.L.Hershberger , R . L u t t e r , W.Klinger, W.Witthun, Z. Physik A274,(1975),251. G.C.Bal l , R .Fournier , J .Kroen, T.H.Hsu and B.Hird, Nucl. Phys. A23I , (1974) ,334.

10) D.A.Dohan and R.G.Summers G i l l , Nucl. Phys. A24I , (1975) ,61 .

11) M.Didong, H.M'uther, K.Goeke and A . F a e s s l e r , Phys. Rev. C14, (1976) , l 189.

12) M.Behar, A .F i l ev i ch , G.Garcia Bennudez and M.A. J . M a r i s c o t i , Proceedings of the In¬ t e r n a t i o n a l Conference on Nuclear S t r u c t u r e , Tokyo, September 5-10, 1977, p .292 .

13) K.Kumar, Colloque sur l e s Noyaux de T r a n s i t i o n , Orsay, 1971, repor t IN2P3, un¬ publ i shed , p .35 .

14) K.Kumar, p r i v a t e communication, to be publ ished in Journa l of Physics C.

15) S.Hinds, J . H . B j e r r e g a a r d , O.Hansen and O.Nathan, Phys. L e t t . J_4, ( I96f>\45. J .H .B je r r egaa rd , O.Hansen and O.Nathan, Nucl. Phys. 86, (1966) ,145. J.R.Maxw.-U, C.M.Reynolds and N.M.Hintz, Phys- Rev. 151 , ( 1966 ) , 1000. H.Debenham and N.M.Hintz, Phys. Rev. L e t t . 25., (1970) ,44 .

Ifa) H.Taketani , H.L.Sharma and N.M.Hintz, Proc. I n t . Conf. Nuclear Phys i c s , Munich, J . de Boer, H.J.Mang Eds. North Holland Publ . Co. Amsterdam, V o l . 1 , ( 1 9 7 3 ; , 2 2 9 . D.C.Burke, E.R.Flynn, J.D.Sherman and J .W.Sunier , Nucl. Phys. A258, ( l97b) ,118 .

17) K.Kumar and M.Barangei, Nucl. Phys. AI22, (1968) ,273-

18) T.L.Khoo, F.M.Bernthal , C.L.Dors, M.P i ipa r inen , S.Saha, P .J .Daly and J . M e y e r - t e r -Vehn, Phys. L e t t . 60B,(1976),341 and M.S.U.C.L.-195 (1975).

19) M.Vergnes, G.Rotbard, J . K a l i f a , J . V e r n o t t e , G-Ber r i e r , R . S e l t z , H.L.Sharraa and S.M.Hintz, B.A.P.S. I I , 2 1 , 8 , (1976) ,959 and to be pub l i shed .

20) A.S.Davydov, Nucl. Phys. 24_, (I 96 I) ,682 .

21) V . l .Be l i ak and D.A.Zaikin, Nucl. Phys. ^ 0 , ( 1 9 6 2 ) , 4 4 2 .

22) M.Ca i l l i au , T h e s i s , Orsay, 1974.

23) R.Sel tz and N.M.Hintz, Rapport LYCEN 7302, La Toussui re (1973).

24) J .A.Cizewski , R.F.Casten, G.J.Smith and W.R.Kane, p r i v a t e communication and B.A.P.S. I I , 2 1 , 4 , ( l 9 7 6 ) , p . 5 5 8 , DG8.

25) W.Henning, R.Muller , K.E.Relim, M.Richter , P.Rohter and H . S c h a l l e r , Munich Annual

Repor t , ( 1972) ,p .95 .

26) D.G.Burke, p r i v a t e communication of unpublished r e s u l t s i s g r a t e f u l l y acknowledged.

27) M.Vergnes, G.Rotbard, G .Be r r i e r , J . K a l i f a , J . V e m o t t e , R . S e l t z , Y.Deschaaps, to be publ ished.

28) R.H.Pr ice , D.C.Burke and M.W.Johns, Nucl. Phys. Al?6, ( 1 9 7 ' ) , 3 3 8 .

29) A.B'acklin, Colloque sur les Noyaux de T r a n s i t i o n , Orsay (1971), report 1S2P3, un¬ publ i shed , p . 163.

30) We g r a t e f u l l y acknowledge the C.S.N.S.M. SIDONIE s e p a r a t o r group, and p a r t i c u l a r l y M.MEUNIER M.LIGONNIERE, G.MOROY and Y.LEGOUX from the I . P . N . , for the p r e p a r a t i o n of the ' 9 ° P t t a r g e t .

31) F . I a c h e l l o , i n : Proc. 1974 Amsterdam Conf. on Nuclear s t r u c t u r e and spectrc-scopy (Scho la r ' ? P r e s s , Amsterdam, 1974) p .163 .

32) A.Arima and F . I a c h e l l o , Phys. Rev. L e t t . 40 , (1978) ,385 .

33) R.L.Auble, Nuclear Data B_7, (1972) , 95 .

34) P.Deason and F.M.Bernthal , M.S.U, p r i v a t e communication.

Gamma-ray spectroscopy in medium-light nuclei

J.F. Sharpey-Schafer'

Oliver Lodge Laboratory, The University of Liverpool, U.K.

§) Introduction

Before I start on the amin part of these lectures, which will be about techniques

in Y-ray spectroscopy followed by results we have obtained recently on odd nuclei near

''"Ca, I would like to try to answer the question: "is present day y-Tay spectroscopy

nuclear botany gone mad?" That many beautiful and subtle experimental techniques

have evolved in "r-ray studies no-one would deny, and this aspect is certainly a major

attraction for me personally. However we should « lso examine how the great increase

in experimental data has increased our physical understanding of nuclei.

Developments have been especially marked in the s-d shell where ten years or so

ago, apart from a few well studied nuclei like 20Ne and 2uMg, only the spins of the

first few l2vels in any nucleus were known and the structure of the nucleus was

described by a relatively crude phenoraological model: e.g. either the rotational

model for 19-A- 25 or a weak coupling model for 30 i A £ 38. The nuclei near 26Si

were difficult to describe although *s»?9Si showed signs of rotational structure

associated with oblate deformation. Sophisticated calculations were confined mostly

to N • 2 even-even nuclei where model spaces could be reduced to a manageable size

by the use of symmetry arguments. There were, for example, many erudite discussions

of the failure of theory to predict the lowest Kn = 2* ban-1 an 2 Mg at the correct

excitation energy while the problems of obtaining even a remotely reasonable des¬

cription of odd and odd-odd nuclei received scant attention.

The use of GefLi) detectors has changed the amount of available nuclear data

drastically. The very good energy resolution of these detectors meant firstly that

very complex y-ray decay schemes couid be studied and secondlv that Doppler shift

techniques for measuring lifetimes of nuclear states really became widely applicable

for Lhe first time. The low absolute efficiency of <ie(Li) detectors caused a decline

in the use of coincidence correlation techniques fcr measuring spins and the devel¬

opment •jf methods'' of utilizing >-ray ang'.lar distributions taken in singles exper¬

iments using reactions such as (o,n) and (p,n) to KIV< a high degree of alignment to

the nucleus. The realization ' that the ncasui fitcnt of linear polarization could be

used to resolve many of the standard ambiguities that bedeviled v-ray angular

+ Colleagues associated with the previously unpublished work in these lectures are

A.M. Al-Naser, A.H. Behbehani, L.L. Green, A.N. James, C.J. Lister, P.J. Nolan,

N.R.F. Rammo, H.M. Sheppard, L. Zybert and R. Zybert.

correlation measurements and the construction of polarimeters with high sensit- ;

ivi ties and good energy resolution, has lead to rigorous spin assignments being made

for as many as 25 levels in some nuclei. 1

The first really successful attempt to get a good shell model fit to the levels

in a non even-even s-d shell nucleus was for ^Ma i Preedom and Wildenthal ' used

the semi-realistic Kuo interaction to make a calculation for six particles in the full

s-d space outside a closed '^0 core. The Kuo matrix elements were then varied to give

the best obtainable fit to known positive parity levels in nuclei lighter than 22Sa

and an extremely good fit (fig. 1) to the levels in 22Na was obtained using these

adjusted matrix element*. The next advance was made by Rex Wbi tonead at Glasgow who

solved ' the problem of making very big shell model calculations by utilizing the tri-

diagonalization procedure of Lanczos. This calculation technique allowed shell model

calculations in the full s-d space to be extended to nuclei in the middle of the

shell near mass 28.

Using modified Kuo interactions extraordinarily good agreement has been found

between the shell model levels and experimentally known levels. Not only are the

energies of the levels very well reproduced but good agreement it obtained to most of

the measured E2 and Ml transition strengths. It must be said however that an

occasional member of the old school of Racha coefficient manipulators has aired the

possibility that the big 3bell model calculations loose the physical insight given

by simpler representations! It must be admitted that some paper* on shell model

calculations are a bit indigestible at first reading, one major problem being the

sheer volume of information calculated and tested against a similar wealth of exper¬

imental data. I would claim that the shell model calculations enhance our physical

understanding in that they allow a deeper microscopic insight into why and how the

simple models work and where their limitations lie.

As a first example of this insight consider the positive parity levels of 22Na

shown in fig. 1. In a simple phenomenological model the levels may be interpreted in

terms of rotational states by coupling parallel and anti-parallel the odd neutron and

odd proton in the different Nilsson orbits. Outside the closed 1 60 core (fig. 2) the

fl - 1/2 Nilsson orbit 6 is full and a proton and a neutron are in orbit 7 with fi • 3/2

which may couple to form either a K* • 3* band (fig. 2(a)) or i K' • 0* band (with

T • 0 and I components, fig. 2(b)). The experimental identification of these bands

is shown in fig. 1. Another group of states with spins I*, 2*, 3*, 4* are linked

and spectulatively labelled K* » I*. The collective model predicts )* bands either

by promoting a particle from orbit 7 (0 - 3/2*) to orbit 9 (fi - 1/2*) and coupling the

spins anti-parallel to Kn • I* and T - 0 (fig. 2 ( O ) , or by promoting a particle to

orbit 5 (Cl « 5/2*) again with anti-parallel coupling (fig. 2(d)). With the large

deformation (B « 0.50) deduced for 22Na it is expected that the configuration in

fig. 2(c) with the particle in the sf , Q - 1/2* orbit 9 should lie lower in energy

5100 5063

tno

4524

4360

4071

3942

3707

- < 7 + ) -

•4+,T=!--v]

6 * -

3060 2969 3+ —-_--J

1934 1952 1937

1528

B91

657

563

• O+T--I -

• » • - - .

Experiment

3* d

"~ - - --4932

^.. 4563

4152

! 3S23 L~--—1-"- - - 3794

' \ "3229

2EC2

--2602

2056

1853

1471

- 6 6 3

- - 6 6 3

2 2 K

7*

5 +

4 +

/+.T-1

6+ 1 +

2 +

1 +

3 +

5 +

Shell Mods!

FIG. 1

; .D .

8 I]

s\.

V7

9 Sj i )'2

5 .,

6

"Ocore WTKJTfJ P

(a) K*= 3+

r77//t77A1 K7777i/Z

T=0 and 1

«

7///

t ^* * T

I////

—%— t &

K*-1+ , T = 0

(c)

•+

* 1 + K*=1

(d)

n

—^—0—

V//I PIG. 2

than the configurations where all the particles are still in d, ,, orbitals as shown in

fig. 2(d). Repeating"' the shell model calculations of Preedom and Wildenthal ' using

the Glasgow-Manchester code ' the contributions of different shell configurations to

the predicted levels in 2zNa may be examined. These are presented as bar graphs in

fig. 3 with the number of particles in the d..., s ,_ and d,,_ orbits being given as

n, r.- ' and n, respectively. Thus 312 in fig. 3 means that 3 particles are in d5/2' sl/2 d3/2 the d ,„ orbit, one is in the 8,/, orbit and 2 are in the d_,, orbit. In fig. 3 only

configurations which occur consistently with percentages greater than 1% are represented

and although other configurations do arise they are always less than 5X.

The grouping together of levels in fig. 3 has been made by putting the lowest energy

state of given spin in the first group, the second loveat state in the second group, etc.

In like fashion experimental states may be matched to theoretical states as shown by the

dotted lines in fig. 1. This procedure accounts for all the positive parity states below

5.1 MeV assuming the spin of the 4524 keV level is 7*, the A.770 keV level is 3* and the

4296 and 4583 keV levels have negative parity. It is immediately noticeable that the

states labelled K « 3 in fig. I have very similar shell model structures in fig. 3 and

that these are again very similar to the structures with K « 0. Thus most of the K » 3

and K - 0 levels displayed on the left hand side of fig. 3 have strong 600 components as

expected from the rotational model pictures given in fig. 2(a) and (b). On the upper

right hand side of fig. 3 are the configurations of the levels that have spectulatively

been labelled K * I in fig. 1. It can be seen that the 600 configuration is almost

negligible for levels in this "band" supporting the collective model hypothesis that

the configuration is given by fig. 2(c) and not by fig. I(d) where 600 would be expected

to be very significant. In contrast a group of levels beginning with a I state cal¬

culated at 3794 keV (to be identified with the experimental level at 3942 keV?) has a

large component of 600 (fig. 3, bottom right hand corner) and may be the K* « 1 band

head with the configuration of fig. l(d).

A similar discussion of the "rotational" structure in 23Na and 2l(Mg has been made

by Watt et al ' . They present (fig. 4) their calculated wavefunctions in 2<tMg by

plotting the number of particles in the d.,_ "-hell N3/1 »g*inst the number in the d,,.

shell N,,,. It can be seen in fig. 4 that the K* • 0 + ground state band hat very siailar

occupancy numbers for J • 0, 2, 4, 6 and B. Similarly states which are identified as

being members of the lowest K - 2 band also have configurations similar to each other.

One of the most interesting states in fig. 4 is th^ lowest 8 level which is predicted

to have most of the particles condensed into the a, , shell. This state aay be identi¬

fied with the 8 + level at 11.86 MeVl2>. On the Nilsson model the lowest intrinsic 8*

state would have two particles from the n • 3/2 orbit number 7 promoted into the

ft « 5/2 orbit number 5 so that all the spins of the particles in orbits 7 and 5 were

coupled parallel. Or in simple terms, the 3 ground state of zzNa coupled to the 5

ground state configuration of 26A1 to give J" - 8 . Thus the collective aodel predicts

«»35

a:\ r.': v;. sncn vi 1.20221 3:2 % E : O : C ' :'O i',1 212 i I'D 22:

J 3 _ j_

ż »? -g r-i 11 n PI n — n n - 1

5 n ;

20

o 1-20

fin

n n 11 - 2 0

O

I Q° 7

4563 * 522 „EL I I i-. n •—. n >-

3/2 [211 J

1 2?5 503 .JJIJ

0 663 657

•20 0

~Ł O

n lsoB n

2 2OS6

4

<07"i, n 5

^710^CL

nfln-nn .J,

20 ' -aaJXl 0

20 0

20

0

1 IC5?,

2f33

n 3 / 2

3 4160

I- 20

C

20

n l.

fl n fl n _ - 20

20

- 0

20

0 3/2 [211 ] + 1/2 [251 ] K= 2

n 1-20

0 20 1-2

3/2 [ 2 U j t 5/2 [202] K-1

FIG. 3

an 8 intrinsic state with most of the nucleons condensed in the d_ ,_ orbit. A similar level with a condensed d, /. configuration is the lowest 13/2* level in ^Mg

13) + at 5.461 MeV ' which has two particles in orbit 5, coupled to give the 5 ground state configuration of 26A1, and a 3/2 proton hole in orbit 7 all coupled to give J11 » 13/2 . The shell model shows us that although the intrinsic levels of the collective model exist there will be no simple extension of the "band" as levels with higher spin can only be made by changing the intrinsic shell model configuration. These condensed states can only arise because the Preedom and Wildenthal inter¬ action gives much lower effective d, ., single-particle energies than the Kuo inter¬ action.

14) It is also possible to understand the early observation that the Kuo inter¬

action gives level ordering such that members of each rotational band have the correct relative energy separation but that the band heads are at the wrong energies with respect to each other. The resultant muddling of the levels makes the level sequence predicted by the Kuo interaction quite different from that observed experimentally. The large shell model calculations show that the wave functions for the levels of a "band" have very similar occupancy numbers (fig. 4) or configurations (fig. 3) hence, given a slightly incorrect set of matrix elements, the energies of members of a band will all be shifted in the same way but their relative energies will be preserved. Different "bands" have very different occupancy numbers and will be shifted in totally different ways by an inadequate interaction.

Ordinarily shell model calculations use a charge independent interaction so that mirror nuclei are predicted to have identical level schemes. Experimentally the level* in mirror nuclei in the s-d shell differ by 100 to 200 keV which is the sort of accuracy being achieved with the big calculations and modified effective interactions. To improve this accuracy Kelvin et al have recently included Coulomb matrix elements in calculations on K « 24 nuclei. The improvement in the agreement with experiment achieved is impressive, as seen for calculations in 2<4Na and z<łAl shown in fig. 5.

In summary, the shell model calculations in the s-d shell have become so good that, for this region, it is probably advisable to calculate what levels might be seen in any proposed experiment, in order to determine just what the physics interest is, before starting the experiment. For me the beautiful agreement between experiment and theory justifies all the hard work and ingenuity put into the spectroscopic measure¬ ments giving the experimental data. Experience shows that phenomological aodela are far from useless - they give the initial understanding and a skeleton on which to hang the microscopic large shell model calculations. They have also historically given vital clues on how to make large calculations work when they intially have appeared to give predictions at total variance with reality. We should therefore continue to develop our experimental techniques and make interpretations using intuitive isodels in regions which do not have the harmony of experiaent and theory seen in the s-d shell.

437

FIG. 4

Mg

t dufum (or ttjtcl in I'Mtcalcuhu-o will, the Vrccoum W<l> 1nl inir;j.in>n SMICS »c labelled by,/, ndiciici the i Ih tute of ar.f<i)3t momenlum y. The two di^oml tolid linci JiTii': :iic teflon of the dupain auihWc <o c ipl ipj sutct. The broken !ln« definr the iUowcd region! for / » 10 and y • 12 datci Only IUICI whi.-li cjn be unjriWguuini)' ai.^nrd to loun rial fc-.i.^t have been thown a> belonglne lo the banili.

No l£»p!) NolCWCI NolPWIAl Al (CWC 1

FIG. 5 Spectra of 2UN« and 2UM calculated with the Preedom-Wildenthal (PW)

and Chung-Wildenthal plus empirical Coulomb (CWC) interactions.

438

12 y-ray techniques

2.1 Measurement of spins

The measurement of spins by means of observing y-ray anisotropies depends on being

able to find a method of aligning the state so that the population of the magnetic sub-

states is not isotropic. The main methods used are:

a) using y-y correlations as in the Method I of Litherland and Ferguson

b) using geometries symmetrical about the beam axis so that outgoing particles are

detected near 0 or 180 to the beam direction and y-ray angular distributions are

measured in coincidence with the particles (Method II )

c) using aligning reactions, where the outing particle* from a reaction carry much

less angular momentum than the incoming projectiles, and measuring y-ray angular

distributions in singles. Examples of such reactions are (a,n) and (p,n) reactions

which have negative Q-values.

Thus states may be populated just above threshold so that the outing neutron has a

very low energy. The population of the magnetic substates may then be calculated using

the statistical model with a code such as MANDY. It is also found that (a,p) reactions

with sufficiently negative Q-values and heavy-ion induced fusion-evaporation reactions

produce alignment in the states they populate.

Given an aligned state, the angular distributions of its deexciting y-rays to states

of known spin J, may be measured. These are then fitted with possible spin J. and

electromagnetic mixing ratio 6 hypotheses. For many years it appeared to be a matter of

luck whether or not it was possible then to deduce unambiguous values of J. and 6. It

was noticed that for many transitions it was not possible to distinguish between two

values of J. for the initial state. For example, suppose the real physical situation is

that J. • J, • 2 so that the decay is pure quadrupole, then it is found that Chic has

che same angular distribution as a mixed transition with J. - J, and arctan 6 - -65 .

This assumes maximum alignment and uses the sign convention of Rose and Brink .

Similarly a pure dipole transition with J. • J, + 1 has the same angular distribution

as J. » J, - 1 with arctan 6 « -6°. These ambiguities were first systematically tab-1 17)

ulated by Peter Twin who pointed out that they could often be resolved by an accurate

measurement of the linear polarization P of the y-rays at 90 to the beam direction.

Thus for the pure quadrupole transition J\ « J • 2 full alignment gives P = +0.8 where¬

as the J. = J, arctan 6 = -45° solution has P a -0.4 assuming the initial and final

states have che same parity. A change of parity would change the signs of both polar¬

izations. Thus a relatively inaccurate measurement of the polarization is sufficient

lo determine a unique spin value for J. and also the parity IT. An example of such a

pure quadrupole angular distribution with its relevant x2 Cits is shown in fig. 6(a)

for the 4015 keV ground state transition in M1Ca.

15

8 y

- / <—

-I 1 </ /

/

I \ —i

1 1

- i

i u

Q>

H 1 :

I r I :

a o o

o

O UL

u

kko

Us-

\ \

7"/ r ,••'•

! ! • j_^.

If a transition is pure dipole in character then for J. = J, + I, maximum align¬

ment and ir. = ir, we find P * -0.4 while for J. - J - I and arc tan 6 = -6°, P = -0.2

and a very accurate measurement is required if the polarization is to resolve the spin

ambiguity. Examples of data for El and MI transitions are shown in figs. 6(c) and

6(d) respectively.

The tables of ambiguities show that for mixed dipole quadrupole transitions

where 6 is large no ambiguity exists. This is illustrated in fig. 6(b) where the

angular distribution for the ground stnte decay of the 4343 keV level in '•'Ca is shown.

The angular distribution has strong a2 and ak terms but the polarization measurement

is extremely inaccurate as the y-ray h&j such a high energy. Only J. » 9/2 is found

to tit the data and tan" S - -85°. The polarization gives no information though it

turns out that positive parity can be ruled out on transition strength grounds.

The experimental techniques to make accurate angular distribution and polarization

measurements should be well known by now. It is generally advisable to employ an 18 19 20) escape suppression shield ' ' in conjunction with the large Ge(Li) detector

measuring the angular distributions in order to simplify spectra with Ey > 2 MeV and

to reduce the Compton background so that the angular distributions of weak peaks in the

spectra may be measured with good statistical accuracy. Often these weak peaks contain

much of the interesting physics.

4 21) Polarimeters consisting of 3-Ge(Li) detectors are usually ' employed and must

be very accurately calibrated. Other polarimeters have been discussed by Twin but

in practice have often not given sufficiently accurate numbers to give many unambiguous

spin assignments. Before making polarization measurements it is usually advisable to

measure rough angular distributions so that an estimate can be made of the spin

possibilities and the accuracy needed on the polarization measurements in order to

obtain an unambiguous result. A positive advantage of the long times required to make

accurate polarization measurements is that it ensures that the angular distributions

may be measured very accurately for even the weakest transitions. Often 'the spin of a

level may be determined from a strong transition and then this known spin used in

conjunction with the angular distribution for a weak y-ray to give a measurement of 6

which has physical importance.

The discussion so far might give the impression that measuring unique J was

becoming relatively straight forward. Things have improved but serious problems remain

and these centre on the question of the degree of alignment achieved in experiments.

Even in the most ideal experiment there is some population of higher substates so that

the measured coefficients of p2<Cos 6) and Pt,(Cos 6) in an angular distribution are

reduced by factors a and a from those expected with full alignment.

231 Geometrical considerations require 1 >a 2 >o1) >0. The problem is to find what

a and a are in a given reaction. This .nay be done by calculating the population

kkt

2) parameters of the substates with the statistical model and then using these para-

24) meters as input data to be fitted together with the experimental data . This works

reasonably well for (a,n) and (o,p) reactions to nuclei with A £ 4 0 as for both

reactions I - 0 and I » 1 outgoing waves dominate, the transmission coefficients for

1=2 waves being 20 to 40 times smaller. If the levels are being fed by higher

levels then the change in alignment may be calculated using the statistical model

directly or by using the U coefficient of Rose and Brink . For levels being fed by

dipole or quadrupole cascades the correction is very small for those cases we have

examined. For A£50 the I • 2 partial waves make significant contributions and it is

not so clear that the statistical model will be sufficiently accurate. A comparison

of a2 and a^ predicted by the statistical model programme MAHDY and experimental values

obtained from known E2 transitions in the nuclei 61Ni, 6 3» 6S 6 7Zn, 67Ca has been made 25) at Liverpool by Abdulrasoul Al-Naser . A direct comparison of a2 (MANDY - experiment)

is Bhown in fig. 7(a) where no error has been put on the calculated a2. It can be seen

that there is a very big scatter of results which is not strongly dependent on the

energy above threshold for the state. If a I5Z error is lut on the a2 predicted by

MANDY (fig. 7(b)) then the distribution becomes more reasonable although there are still

cases where the experimental alignment is significantly higher than expected. These

cases could be caused by feeding from higher spin states. Certainly the MANDY pred¬

ictions of a should be regarded as having an error of about I5Z.

An alternative approach is to assume that the population parameters have a

Gaussian dependence on |m|. The width of the Gaussian may then be treated as a para¬

meter to be searched for. In fig. 7(c) the same experimental data are displayed

on a plot of au against a . The relationship between <»2 and a^ predicted by Gaussians

of different wiuths for J - 2 and J » 6 are shown a* dotted lines. The data show that

au is usually bigger than that predicted by a simple Gaussian. To obtain a larger au there has to be a higher population in the higher substates than given by a single

Gaussian distribution. A consiftant method of treating the alignnent has yet to be

agreed.

2.2 Decay schemes and lifetimes

I would now like to consider some improvement* in Y~r coincidence techniques. In

making such measurements to establish decay scheme! it is usually thought that it is

essential to get the counters as near the target as possible to improve coincidence

rates and statistics. In general it is a good principle to improve the signal to noise

ratio in any experiment. In a y-y experiment if the peaks of interest are weak then,

if the gates are put on this peak, the Compton background and its subtraction dominate

the statistics. If the gates are put on a large peak to look for the small one, it

cannot be seen because of background and lack of statistics. Both situation* can be

improved by sacrificing coincidence rate and having at least one Ge(Li) detector

mounted in an escape suppression shield. Fig. 8(a) shows a spectrum in an unsuppressed

kk2

I \ I

W O u

i i

r i

n

E = Energy

0 I<E<2 a E > 2

i n i

above

1

threshold

nil*

(MeV)

J - I

(a)

d l IT9 MKTA 1 Ml

I

-

-12 -10 -8 -6 -A -2 0 „MANDY ^Exp

2 A 6 8

(Standard deviation)

E=Energy above threshold (MeV)

-3 -2 MANOY

- 1 0 1 2 3

?* (Standard deviation)

( c )

FIG. 7

4*3

o

o <r UJ 03

2100

1800

1500

1200

Ca total spectrum in Ge(Li) with NO escape supression

4000

3000

2000

8S0 900 9S0

CHflNNEL NUMBER ( a )

1000

FIG. 8

800 SSO 900 950 CHflNN'EL NUMBER

( b )

Ge(Li) for all Y-rays in coincidence with Y-rays detected in a suppressed Ge(Li).

Fig. 8(b) shows the suppressed spectrum in coincidence with all the y-rays in the

unsuppressed Ge(Li). It can be seen that the signal to noise is far superior in the

suppressed spectrum so that it is possible to observe the Y~ray peaks from high spin

negative parity states in lt3Ca which are produced by the "*°Ar(a,n) reaction. Gates

can be put on these peaks and their position in the decay scheme established.

Heavy ion reactions suffer from the problem that they invariably populate

cascades of states and hence there are problems in measuring lifetimes in the DSA.M

regions. It the feeding is accurately known corrections can be made for this in

favourable cases. There is however the problem of the feeding time due to the

reaction process and unobserved Y~rays. This problem can also arise in reactions such

as l|()Ar(a,n)''3Ca where the population of higher spin states are determined by angular

momentum considerations rather than by thresholds due to the Q-values. A method of

elminating the feeding problem is to use a y-y coincidence requirement to determine

the recoil velocity when the state is populated. This is done by putting the Y

detectors at 0 and 135 to the beam direction so that the Doppler shift in one

detector1 for Yj populating level 2 (fig. 9) will, by the setting of suitable gates,

define the initial recoil velocity for level 2 so that the Doppler shift of Y 2 C Ł n

be measured in coincidence with Yj. The unshifted position of y2 m ay De determined by gating on the stopped part of Y[ always assuming the line shape is suitable. Clearly the analysis is simplified if the recoils are restricted to a narrow cone by the reaction and if v/c of the recoils is large enough for electronic stopping to dominate so that scattering due to nuclear stopping can be neglected.

We discovered that the '"'Ar (9Be, 'in)'*^Ti reaction was ideally suitable to invest¬ igate this development of the WAM. The line shapes at 0° of the 1344 keV <!0* to 8+) and 1597 keV (8 to 6 ) transitions in u£lTi are shown in fig. JO. In fig. 11 the line-shape at 1350 of the 1597 keV y-ray in ehown in coincidence with (a) the stopped part and (b) a section of the moving part of the 1344 keV line at 0°. The accuracy of the lifetimes is limited by statistics in the present data which took 24 hours to collect. Longer runs are in hand.

This configuration for y-y coincidence experiments has advantages for establishing the order of the y-rays in the decay scheme. If a gate is put on a stopped component of a y-ray then all decays below this will also have to be stopped while y-rays above this may have shifted components. Thus a gate on the stopped part of the 1597 keV y-ray shows that some of the coincidence 1344 keV line is shifted (fig. 12) so that the 1344 keV y-ray is above the 1597 keV decay in contrast to the ordering suggested when these y-rays were first observed.

Clearly a similar y-y gating technique can be used to eliminate unknown feeding in RDM plunger experiments. In this case the gate must be set on the moving peak in the first y-ray in the cascade as the stopped peak contains no information as all

7-ray line shapes at 0°

Side feeding

/a/ Schematic deoay seheme of levels produced in a nucleus by a heavy Ion fusion-evapo¬ ration reaction show-ing side feeding components, /b/ The Doppler broadened line shapes of y-rays at 0<s 0° if side feeding times are lees than the slowing down time /sohematic/. In a y-- y/colneidence experiment the side feeding components in •fni /A andVii •** b* eliminated by gating on \.. If an approp¬ riate part of the line shape of •{. is chosen, this will define an initial^average recoil velocity v .

(a) (b) FIG. 9

S1M03 ŚO d38WflN

UU6

"70

BO

V)

u

fc

HO

10

S

*5Ti Er=135°

( a ) LINE SHAPES IN COINCIDENCE WITH THE STOPPED PART OF THE 1344keV PEAK

1250 1300 1350 1«K> l«50 1500 ISSO 1600 16S0 1700 17S0 1800

CHANNEL NUHBEB

O CM

( b ) LINE SHAPES IN COINCIDENCE WITH THE FASTEST MOVING PART OF THE 1344keV PEAK

1250 1300 13S0 IW0 IttO 1500 ISSO WO 1K0 1700 17S0 IMO

FIG. 11

en

70

60

SO

00 CO 00

Ti 9 y = 1 3 5 SPECTRUM IN COINCIDENCE WITH THE STOPPED PEAK OF THE 1597 keV tf-RAY

o CM

30

CO

oo 00 CM

SSO 1000 iOSO 1100 11S0 1200 1250 1300 13S0 l<i00 IUSO 1500 1S50

CHRNNEL NUMBER

FIG. 12

subsequent peaks in coincidence with it will also be stopped. Clearly one problem in

such y-y coincidence techniques is lack of statistics and in an attempt to solve some

of these problems we are hoping to use electronics developed by Kandiah at Harwell

which employ time variant filters with DC coupled amplifiers and will usefully count

at rates up to 100 kHz.

2.3 Measurement of reaction thresholds

One other advantage of using Ce(Li) detectors in an escape suppression shield is

the very low backgrounds that can be easily achieved. For instance above 1.46 MeV,

where a background y-ray due to ''"K decaying to U®AT is usually seen from concrete in

the surrounding building, the background in an escape suppressed Ge(Li) is essentially

zero. These instruments are therefore ideal for detecting the onact of reaction

thresholds. We have recently used the technique to measure the (? endpoint energy of

the superallowed Fermi decay of 38nlK. In fig. 13 (a) we show the appearance of the

2168 keV y-rays in 3BAr from the decay of the ground state of 7.64 min. 3BK produced

(fig. 13(b)) in the 3BAr(p,n)38K reaction. A solid 38Ar target29' is used and the

threshold can be detected to better than 200 eV. The problem however is of calibrating

the bean energy so that this energy resolution can be utilized experimentally. Fig.

I3(c) shows the thresholds measured for !1<N(p,n) 1(<0 which was used as a calibration,

and the 38Ar(p,n)38K reaction. The other calibration reaction used was I0B(p,n)30C.

The excitation energy of the 0 isomer is obtained from y-ray decays in 38K itself.

The final value obtained for the endpeint energy of the 3BlBK decay is E «

5020.71 ±0.85 keV where the biggest contribution to the error is the uncertainties in

the calibration energies.

v §3 y-ray spectroscopy of 39»'<1K and u l» u 3Ca

For the past three years we have been carrying out a programme on the -y-ray spec¬

troscopy of nuclei populated by beams of a-particles on solid isotopically pure

targets of 3^>38>u^Ar. The advantages of solid as opposed to gas targets are that the

DSAM can be used to measure many interesting lifetimes and that very ciean targets can TO \

be made. The targets are formed by cooling a 250 um thick gold backing down to I2K

by using a closed circuit helium refrigerator operating on a Sterling cycle.

3.I The nucleus 3SK

The 36Ar(a,p)39K reaction is a very clean way of populating levels in 3?K as

competing channels are extremely weak. We are therefore able to study the v-decays of

43 excited states in this nucleus. We have measured the lifetimes of 24 of these

levels and assigned unique spins and parities to 16 levels. The lifetimes and decays

of many levels have been previously studied . There is also some very nice

particle work on this nucleus which together with our spin assignments show

that many negative parity states are well described by Bernstein's model .

In this model a d- . hole is coupled to the 3 and 5~ levels of the l(0Ca

(a)

" o-

2IM

imai i n n trafi na aliSn "i 350 LOU IAZ SO0 VX

I iimimAn i ii i MI I I IVt 1.00 IAO iOO

li i Mani I I nn J50 tOO (SO WO 550

Chonnel number

1 0 ' | >• 17990 MOOO WOK) W f i

T i l

7 6i mm

FIG. 13

1.50

core giving 8 levels with J S 13/2 . We observe almost twice this number of levels

which requires ' consideration of 3p - 4h states to account for the surplus.

At the moment there is some diasgreement about the spin assignments for some of

the higher spin states seen in heavy ion reactions. Originally the level at 5718 keV

was assumed to be 13/2 . Data we took gave an unambiguous spin assignment of

9/2 , the 13/2 solution being rejected. Recent measurements using Y~ray angular

distribution and polarization techniques very similar to ours, but populating the

levels with the 28Si(16O,ap)39K reaction, give a spin !3/2~ for the 5718 keV level.

There is some debate about how the alignment in the Ca,p) and the heavy-ion reactions

should be described. Measurements are at present in hand in an attempt to get an

agreed set of spins for the higher levels in 39K.

3.2 The nucleus M 1K

In the compilation of Endt and van der Leun only the spins of the ground and

first two excited states are listed as being known. Recent heavy-ion work

established the existence of states of high spin which were observed via Y-ray

cascades involving both positive and negative parity states. We have measured '

the lifetimes of 14 levels in ***K and assigned spins and parities to 10 levels

between 1.5 and 3.0 MeV (fig. 14).

The most obvious feature of the positive parity states is the close grouping of

levels around 1.6 MeV and 2.5 MeV. This structure would be expected in a model where

the d.,» proton hole is coupled weakly to the pair of iy>2 neutrons giving rise to the (f7/o>? -,+ •>+ i* A+ levels in ll2Ca which are at 0, 1.52, 2.75 and 3.19 MeV respect-

ively. A calculation of positive parity states in 'K has been made by Pellegrini

using matrix elements derived by fitting the (f_. ) ^ levels in 1<2Ca and the (f.,.) g>

(d,..)"' levels in ''"K. The results of this calculation are shown in fig. 15 and are

compared with the experimentally observed levels assuming that the 1594 keV level it

1/2* and the 2756 keV levels is 5/2*. It can be seen that Pellegrini's calculation

gives too much splitting of the levels indicating that the particle-hole interaction

is too strong. We have repeated the calculation but using particle-hole matrix

elements that fit an average of levels in U CK and lt2K and effectively weaken the

interaction. The result is shown on the right hand side of fig. 15 and it can be seen

that the experimentally observed bunching of the levels is reproduced. The E2 tran¬

sition strengths observed are comparable with the core transitions in Ll2C» with the

exception of a strongly enhanced 24 ±5 Wu transition from the 7/2 level at 2509 keV

to the 3/2+ level at 1560 keV.

The properties of the negative parity levels in ^1K are very sinilar to the

( f ? / 2 )3 levels in "3Sc.

3.3 The nucleus ^

We populated this nucleus with the 3eAr(atn)<tlCa reaction and have aeasured Che

V

f V

i

e

.10 a

1123

<D

—»

s

I" E ,40 .

s

a

-

s

s

•

e m

p

s>

• f c -

s

S9II

m

.to.

8

j

a

<0

;

I2

s

500

s S

Hi" R J-i

Positive Parity States In K

I 1 UJ Z UJ

2 2 o

u X Ul

3 -

3/2

5/2

7/2

t/2

Pellegrini ca(c

8

EXPT This Work cale

J 15/2

11/2 9/2 13/2 7/2 11/2

3/2 5/2 7/2

1/2

3/2

Ol

lifetimes29* of 22 level* and assigned spina and parities to 17 levels. The wealth of information now available on this nucleus should hopefully allow a reasonably good description of the physical properties of the nucleus to be made. I attempt to do this in fig. 16. Firstly the main single particle states are separated oat consisting of the 2p3.2, 2pJ/2, Jg^j particles and I d ^ and 2s^ 2 holes. The 5/2~, 7/2 , 9/2* and 11/2" levels at 2576, 2959, 3677 and 4015 keV respectively might be members of a K* - 3/2" 5p - 4ta band based on the veil know deformed 3/2" state at 2463 keV. The

- — +30 9/2 to 7/2 transition with and C2 strength of 16 _ g Wu is the only in-band tran¬ sition observed as out of band frays dominate the decay modes.

The positive parity levels are best interpreted * ' as being an tyt2 neutron coupled to the 3~ octupolc vibrations 1 state and the 5~ p - b state in w0Ca. The states with tha 3" core are the 3400, 3050, 2606, 2684, 3201, 3369 and 3914 keV levels with spinsl/2*. 3/2<+\ 5/2*, 7/2*, 9/2*, 11/2* and 13/2* respectively. The 11/2* ievel*5) decays to the 7/2~ ground state of hlC* with an 13 strength of 24 __ j Wu which compares with 2912 Wu for the core iransition. The members of the multiplet lie on a beauti¬ fully smooth curve in a plot of excitation energy vs. spin which is characteristic of weak coupling. There are three within multiplet transitions: the 13/2 •* 11/2 , 11/2* •+ 911* and 3/2<+) •* 5/2*. Using the formula given in ref.46) it is possible from the Ml strengths to work out (gc - g_) 2 where gc is the core and g_ the particle g-factor. The three transitions give (gc - gp) - ±(0.346 10.043), ±(0.277 ±0.009) and ±(0.22810.030) respectively. Taking g_ - -0.46 for the f?., neutron we obtain g^ • -0.1110.04, -0.18310.009 and -0.23 ±0.03 or gc - -0.81 t 0.04, -0.554 ±0.009 end -0.6910.03. The experimental value47' for the g-factor of the 3" state at 3.74 MeV in '•"Ca is |g(3~)| - 0.1510.07. The 5* • iJ/2 multiplet is identified with the levels 3740, 3495, 3614, 3976, 4519, 3830 and 5219 which have spins (3/2, 5/2)*, 5/2*, 7/2*. 11/2*. 13/2*, 15/2* and (17/2*). The 9/2* member of the multiplet has not been identified and there is a spare 7/2* level at 3974 keV! The within multiplet tran¬ sitions are 17/2* + 15/2*. 13/2* + 11/2* and 13/2* * 15/2*. Using the same method as before gives gc » X).02, -0.2710.06 and -0.18 + 0.09 or gc • <-0.94, -0.65 ±0.06 and -0.7410.09. These may be compared . u measurements*?) of |g(5~)| • 0.61 ±0.1 or 0.31 ±0.05.

There is some evidence of mixing between tht states of the two anil tip let* as many of the states are joined by fairly strong Ml transitions. The strongest of these is the 13/2* •* 13/2* with an Ml strength of 350 ± 220 mUu.

We are left with a fairly simple sero order picture of *łCa which accounts for the majority of states below 4 MeV. What is clear is that a very large shell model space will be required to describe in a microscopic manner the properties of l>1Ca that we see as spherical, deformed and vibrational states.

3.4 The nucleus

let only ia tha «Ar(«,s()*aCe zmsxtis* very prolific im prałaci** Trays it U

KU Ul

ŁO

Ex (MeV)

4.0

3.0

2.0

5279-

4492

4975-5219-

4519-

-17/2*

-13/2*

4015-11/2*

11/2-4343- -9/2"

3974 7 / 2

(3/2.S/2)-

K*=3/2" I deformed "band"

O 0* L7/2"

Ca

FIG. 16

also a very effective way of neutron damaging Ge(Li) detectors! In l*3Ca we have studied the decay schemes of 46 levels, measured 23 lifetimes and established unique values for 20 spins and parities. As in '''Ca we would hope, with this amount of data, to be able to build a reasonably complete zero order description of the lower levels. However as there are now three neutrons outside the '*0Ca core the structure becomes even more complex.

The negative parity states formed by the (f-, ) ^ configuration are at 0, 373, 593, 1678, 2094 and 2754 keV with spins 7/2~, 5/2", 3/2", 11/2", 9/2" and 15/2". The main 2p_ ,, single particle strength is in the level at 2046 keV. We establish other negative parity levels with spinji 5/2 , 7/2 , 9/2 , 11/2 , 13/2 , 15/2 , 17/2 or 13/2" and 19/2" or 15/2" at 1931, 2067, 2250, 2754, 3050, 3662, 4394, 5155 and 5931 keV respectively. The partial decay scheme far these levels is shown in fig. 17 and a spectrum showing some of the y-rays was given previously (fig. 8 ) . If these level? are considered to belong to a negative parity rotational "band" then it is found that the out of band E2 transition Btrengths for the 5/2 -* 5/2 and 7/2 -* 7/2 r-rays are enhanced at about 10 Wu. Preliminary measurements of the in-band strengths for the higher spin members indicate that these are strong E2 transitions. The Nilsson model would indicate that a deformed band based on a 5p - 2h configuration should have K * 3/2 . A j " « 3/2 state which is seen only weakly in the (d,p) stripping reaction and might be similar in stricture to the 3/2 deformed state at 2463 keV in ^ C a is at an excitation energy of 2103 keV in <<3Ca. A deformed band based on the level at 2!O3 keV might therefore account for the non (f.,,) levels shown in fig. 17.

We have established the spins of a series of positive parity levels whose decays are shown in fig. 18. The spins of some of these levels had been correctly conjectured in previous work ' and were assumed to belong to a K * 3/2 deformed band in analogy to other bands based on d.j,2 holes. Recently Nann et a I have shown that the lowest states with 11/2* £ J* S 17/2* are strongly populated in the *llK(a,d)ll3Ca reaction with L • 6. This suggests that these Btates are made up largely of config¬ urations of the type (f

7/2)j „ 7 • ^ 3 / 2 ^ ' O u r lifetime and mixing ratio measure¬ ments establish that while the levels with j " S 9/2* are joined by strongly enhanced E2 transitions (fig. 18) the lowest levels with J* ł 11/2* decay with weak E2 strengths. The Ml decays from these higher spin levels are strong and are consistent with the ( f 7 / 2 ) 4 x (d3/2^"' Picture. We observe the decays of second 13/2* and IS/2* states which were also observed in the (a,d) reaction. These two levels are associated with reasonably strong E2 decays and lie in a better straight line on 1 E y vi. J(J + I) plot with the levels with J11 5 9/2* than the lowest 13/2* and 15/2* levels. But the Ml strengths indicate that these levels are all strongly mixed and do not have a doainating K - 3/2 structure. We have observed further positive parity states decaying to the 15/2 and 17/2 levels which we are presently investigating.

776 5931

5155

4394

3662

3050

2754

2250 2067 1931 1678

593 373 •

0

27

1984

1373

1076

1877

\

1695

156Q 1337v

i

7

141

146.1

2

801

908

\

1493 761 y

732

612 ^

! \

1

15/2,19 '2

43 Ca

T 17/2,13/2

15/2"

13/2"

11/2"

15/2~

9/2 7/21 5/2_ 11/2"

3/2"

5/2"

7/2"

FIG. 17

00 Ex{keV) 4591 —

4186-

3944-

3505-3371-

2951-2754-

2410-

2094-1902—1

1678-

1394-

990-

593-

373-

(a) 7- ray energies in keV

1827

1015

.021

617'

507

1693 .962

1067857

1076

508

542

836

815

1190 573

751 S54 617

381

X7 405

(b) B(E2) in W-u- (c) B(M1) in m Wu-.17/2*

00 V

15/2"

L-9/2"

11/2" oo

+1

ss si?

00

in V

CM CM

V

15/2*

15/2+

13/2* 13/2*

11/2*

19/r

7/2+

5/2*

3/2*

220

43,

3/2"

5/2"

7/2"

FIG. 18

Ca

54 Conclusions

My own conclusion is that careful and detailed work in Y~ray spectroscopy can

not only provide compilers of nuclear data something to compile, it can give us a

rather complete view of a nucleus from which elegant physical descriptions can be

derived. These phenomological descriptions can then form a basis for a proper

microscopic understanding of the nuclear level structure.

Acknowledgements

I would like to thank all my colleagues at Liverpool for their help in producing

the material for these lectures.

References

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4) P.A. Butler et al, N.I.M. JOB (1973) 497

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f>) R . R . W h i t e h e a d , N u c l . P h y s . A I 8 2 ( 1 9 7 2 ) 2 9 0

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8) J.D. MacArthur ct al, Can. J. Phys. .54 (1976) 1134

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10) A. Watt, D. Kelvin and R.R. Whitehead, Phys. Lett. 6_3B (1976) 383

11) A. Watt, D. Kelvin and R.R. W.iitehead, Phvs. Lett. 6_3B (1976) 366

I?) D. Branford, M.J. Spooner and I.F. Wright, Part. & Nucfei 4 (1972) 231

13) P.A. Butler ec al, J. Phys. £1_ (1975) 663

14) B.J. Cole, A. Watt and R.R. Whitehead, Phys. Lett. _45B (1973) -29

1 5 ) D. K e l u t n , A. W a t t a n d R . R . W h i t e h e a d , J . P i i y s . £ 3 ( 1 9 7 7 1 ] 3 3 9

1 6 ) H . J . R o s e a n d D .M. B r i n k , R e v . M o d . P h y s . 39_ ( 1 9 6 7 ) 306

17) P . J . T w i n , N . I . M . _[0ft ( 1 9 7 3 ) 4 8 1 a n d L i v e r p o o l P h y s . D e p t . I n t e r n a l R e p o r t S o . 17I - .

I P ) r . K . A l e x a n d e r e t a l , N . I . M . 6_5 ( 1 9 6 8 ) 1 6 8

1 9 ) J . F . S h a r p e y - S c h a f e r e t s ! , N u c i . P h y s . A i e 7 , , 9 7 1 ) 6 0 2

2 0 ) J . K o n i j n e t a l , N . I . M . [ 0 9 ( 1 9 7 3 ) 8 3

2 1 1 H . H . E g g e r . h u i s c n e t a l , K u c i . K h y s . A 2 8 5 ( 1 9 7 7 1 167

2 2 ) P . J . T w i n , P r o c . I n t . C o n f . o n N u c l . S t r u c t u r e a n d S p e c t r o s c c p y . A t n s t e r d a r a (\Q7&)

Vn 1 . : , 3 Hi

23) G.A.P. Engelbertink et al, N.I.M. J_V3 (1977) 161

24) A.M. James, P.J. Twin and P.A. Butler, N.I.M. _l_lj> (1974) 105

25) A.M. Al-Naser and L.P. Ekstrom (private communication)

26) G. Fortuna et al, Nuovo Cim. 34A (1976) 321

27) K. Kandiah, Radiation Measuremetits in Nucl. Power (Inst. of Phys., London, 1966)

28) J.F. Sharpey-Schafer et al. N.I.M. J^5 (1976) 583

29) C-J. Lister et al, J. Phys. G2 (1976) 577 30) A.K. James et al, J. Phy». G4 (1978) in press 30 J.L. Durell et al, Hucl. Phys. A219 (1974) 1 32) R.H. Boyd, A. Mijnerey and G.D. Gunn, Nuci. Pbys. A281 (1977) 405 and

references cherein 33) A.M. Bern»tein, Ann. of Phys. 69 (1972) 19 34) J.J. Kołata et al, Phys. Rev. C9 (1974) 953 35) P.J. Nolan et al, J. Phys. C2 (1975) L33 36) H.H. Eggenhuisen et al, Nucl. Phys. (1978) in press 37) P.M. Endt and C. van der Leun, Nuci. Phys. A214 (1973) 1 38) P. Gorodetzky et al, Phys. Rev. Lett. 21 (1973) 1067 39) K.P. Lieb et al, Nuci. Phys. A223 (1974) 445 40) J.U. Olncss et al, Phys. Rev. £!_[ (1975) 110 41) C.J. Lister, A.M. Al-Naser, M.J. Maynard and P.J. Nolan, J. Phys. C3

(1977) L267 42) C.J. Litter ct al, J. Phys. G4 (1978) in press 43) F. Pellegrini, Nuovo Cim. £8B (1967) 155 44) M.J.A. de Voigt, D. Cline and R.H. Horoshko, Phys. Rev. C1£ (1974) 1798 45) C.J. Lister et al, J. Phys. G3 (1977) L75 46) A. de Shalit, Phys. Rev. J22 (1961) 1530 47) N. Benezer-Koller et al, Bull. Am. Pbys. Soc. J5 (1970) 1666 and

Ibid 21 (1972) 931 48) N.G. Arlenius et a l , Nuovo Cits. 8A (1972) 147 49) A.R. Polletti et al , Phys. Rev. £T3 (1976) 1 ISO 50) J. Styczeń et al , Nuci. Phys. A262 (1976) 317 51) H. Nairn, V.S. Chien and A. Sana, Nuci. Phys. A292 (1977) 205

HIGH-SPIH HEUTROH PABTICLE-EOIE STAIES IH EVEN H=28 ISOTOHES

Jan Styczeń *

Institute of Unclear Physics, Cracow

1. Introduction

During this lecture I will be mainly concerned with the nuclei of ^°Ti, ^2Cr and ^*Fe. These three even H=28 itotones belong to the fp shell nuclei, their low-lying excitations have been eipeoted to involve pure or nearly pure /^n/2^1 *7Pe con¬ figurations /n=2, 4, -2, respectively/ since Ca is known as a rell closed magic nucleus. Hence, the spectra of the low-ly¬ ing states in ^ Ti and ^"Fe should be equivalent as the two pro¬ tons in the former and two proton-holes in the latter nucleus are in the first order responsible for these excitations. This is nearly the case as can be seen in 7ig.1. However, the high¬ er energy excitations, beyond the 6* maximum spin value of the /1f„/?/ configuration are quite different in both nuclei which

is shown later on. For many years a great deal of effort, both experimental

and theoretical has gone into attempts to obtain a clear under¬ standing of the structure of the fp shell nuclei. From a theoretical point of view, progress has been rather slow due mainly to the difficulties associated with applying the shell model to this mass region. During the last years the nuclei of the sd shell have been studied extensively and euocess-

x Collegues associated with the previously unpublished work in this lecture are: F.A.Beck, E.Botek, C.Gehringer, B.Haas, J.C.Merdinger, A.Muller-Arnke, T.Pawlat, lf« Schuls, Z.Stachu-T*, P.Taras, M.Touleaonde, J.P.Vivien.

A

3

2

1 -

-6*

•U* .6* •2* -6*

52Cr Fig. 1

The experimental states of the /1 ^n/2^ configuration in the even H=28 isotones of ^°Ti, "cr and ^*Fe

fully by means of complete diagonal!zatione of shell model Ha-miltonians [1, 2], The success of the model can be attributed to significant advances in computation methods, and to the fact that it has been possible to determine appropriate effective in¬ teractions by least-squares fits to the spectra of a range of nuclei, usually with the Euo [3] reaction matrix elements as starting point.

For heavier nuclei, i.e. those of the fp shell, however, the number of configurations within the complete model space in¬ creases so rapidly owing to the larger number of active parti¬ cles and valence shells that severe truncations of the space

tin #ip S6_

are neoessary. Usually a core of Ca, Ca or -^Sl 1 B em¬ ployed and diagonalizations are carried out within the lowest one or two /P*/2» Pi/2» *5/2^°* ^*7/2^"** configurations £*J

From the experimental point of view, nuclei of this region

i»62

hare been investigated since a long time: In the 1960** the data were essentially originating from light ion beau /A */ and charged particle or V"-ray /with VaJ counters/ works. Starting from the beginning of the 1970's beams of heavy ions /A ^ 4/ have been currently available from several electrosta¬ tic Tandem accelerators. This and a tremendous improvement of ln-beam }f-ray spectroscopy due to large volume high resolu¬ tion Ge/Li/ detectors, fast electronics pulsed beams etc., has allowed to observe many new levels, mainly of high spin va¬ lues. Therefore a new interest in the spectroscopy of the If?/? nuclei was strongly pushed forward.

The H«28 isotones ^°Ti, ^2Cr and ^P» have been the object of numerous experimental studies and theoretical calculations £5 - 7] • Assuming that only pure /If7/2^ configurations are responsible for the low-energy states, one expects to observe ex¬ cited states of positive parity with spin I up to 6 /n* -2/ and 8 /nx4/. All these states have indeed been observed in "li [5] and ^ T e [7] whereas in ^2Cr a state with J^ « 8 + was recently suggested by Foletti et al. [3] and Berinde et al. £6], and con¬ firmed unambiguously in the present work.

In this lecture, experimental evidence is presented for the existence of positive parity levels in these nuclei, with spins exceeding the maximum angular momenta allowed by the /tn/2^ representation. The presence of such high-spin states can be . understood by allowing a single neutron to be excited from th« closed ^fn/o shell into the 2Px/2 shell. For example, in ^Tl, by coupling the maximum angular momentum In=5 of such a neu¬ tron particle-hole exoitation to the aligned proton ^*o/2' configuration /I57 «6/, states with spins up to 11* are possible. Whereas if only protons were to be excited even over the whole fp-ahell, the highest spin reached would be 6*.

It is shown here that shell model calculations allowing ex¬ citations of one particle to the higher orbits of the fp shell are able to account for the observed level schemes and decay pro¬ perties. The calculations lead to configuration'admixtures in the fn/2 levels of the order of 20%. These admixtures are vi-

tal for explaining the strengths and types of a nusbar of transi¬ tions in ^ T i and 52Cr.

2. Experiment The experiments were performed at the Strasbourg IIP Tandem

accelerator and the Cracow U-120 cyclotron by means of / ot ,2nj" / reactions using targets of enriched Ca, ^ T i and ' Cr, respecti¬ vely. This kind of reaction, leading to high spin states, has the advantage compared to heavy ion reactions that very few outgoing channels are open, which greatly facilitates spectroscopic work. In the case of the Ca+oć reaction all the strong V- -ray lines correspond to transitions in ^ Ti /see Fig.2/, whereas the other

, 1 KT-r hf.u

J I i ; • ' • < • • '

£*£?&> (I

Pig.2 Examples of singles and coincidence j* -ray spectra taken with 120 am? and 60 o r Ge/Li/ detectors with an energy resolu¬ tion of 2 keV for 1333 k«7 of 60Co

are mostly due to reactions on Impurities in the target.

The following measurements were performed:

1/ excitation functions of V* -rays /see Pig.3/ »t alpha

beam energies E^ • 24 •£> 33 BeV;

ii/ y* -ray angular distributions with a Ge/Li/ detector;

iii/ ?•- t coincidences with two Ge/Li/ detectors positioned

at different sets of angles In order to determine D.C.O.

ratios [93;

iv/ linear polarization of V--rays using a Compton polari-

meter [,10];

v/ time distributions of delayed y--rays using standard

pulsed beam techniques.

2.0I-

s UJ

5 a.

161-

Z

w

ST. 12

10

I I I

"Ti

/

/ / /

1 L_ 1 1

i i

—

1121 kW

B03MT

231N»V 406 k*V

1939 N V '

52SIMV w ~. »

1122 MV

24 25 26 27 ALPHA BOMBARDING ENERGY

Pig. 3

Gamma-ray, yields in the

*8Ca /0/,2n/50 Ti reaction

measured for two incident

beam energies. All the

X- -ray Intensities were

normalised to the intensi¬

ty of the 2+-«»0+ transi¬

tion at lower energy of

the alpha bean.

The conpton polarineter was constructed of three Ge/Li/

detectors. A 69 cur Ge/Li/ detector was positioned in the hori¬

zontal plane at 90° to the beam direction. Compton-scattered

f~ -rays were detected by two Ge/Li/ detectors of 50 car act¬

ing as polarization analysers, one lying in the reaction plane

and the other perpendicularly to it. The size of the scattering

detector was appropriate for the detection of f-rays exceed¬

ing 0.9 MeVj for experiments involving lower-energy p -rays

the scatterer was replaced by a 20 car Ge/Li/ detector. The

mechanical frame allowed the polarimeter to rotate about the

target chamber with the relative crystal positions held fixed.

The j«- -ray spectra from the scatterer and from the analyzer

detectors were digitally summed after conversion by the analog-to

-digital converters /ADC/; this system allowed for easy gain ad-

justement between the three detectors. An anticoincidence sys¬

tem allowed for rejection of the 511 keV ^ -ray in the analy¬

zers which could result from pair production in the ecatterer.

The polarimeter was calibrated with well-known E2 transi¬

tions emitted by nuclear levels strongly aligned in /pfp'/ re¬

actions on ^Fe, 24Mg, 28Si and 1 2C targets.

In lifetime measurements of a few up to a few tenths of

nanoseconds,the naturally pulsed alpha beam of the U-120 Cra¬

cow's cyclotron was used /pulse width e 3 ns, repetition time

* 90 ns, which depends on the energy of the beam. When longer

lifetimes were to be measured, then a f- - f- technique with

4 NaJ/Tl/ counters and a Ge/Li/ detector were applied. Pulses

of the NaJ counters were mixed up and put in a fast coincidence

with the pulsed beam. These coincidences were gating the next

time-to-pulse-hight converter /TAC/ provided with a stop pul¬

se of the Ge/Li/ detector and the start pulse of the sum of the

NaJ/Tl/ counters /?ig.4/. Time-delayed gamma-ray spectra

wire stored in a PDP-11 mini-computer. The decay curves

/Fig. 5/ were obtained from the delayed f- -ray Intensities

of the time-calibrated p -ray spectra. The accurate time ca¬

libration was obtained from the enhanced chance t" ~ f- coin¬

cidence peaks at the beam-bursts /see insert to Fig. 4/.

-QGe(Li)

PDP-11

Pig. 4 A. sehenatic representation of the arrangement for lifetiae measurements. The Insert shows a part of the TAC2 time spec-trum. W1, W2 etc.. denote the windows put on this spsetrua to obtain„.callibrated time delayed r -ray spectra.

10-

1«6fc*V GAMMA-RAV

X= 517145 ns

t-H TIME (micreweond*)

The delayed curre for the 146 keT V-- transition in

Simultaneous fits to the angular distribution coefficients,

the linear polarization data and the D.C.O. ratios allowed unique

spin and parity assignments for the majority of the new levels

/Fig.6/ and yielded mixing ratios for the £ -ray transitions

as well as spin-alignment attenuation factors. However, for some

transitions in ' Ti and or the angular distributions and li¬

near polarizations could not be measured accurately because of

overlapping / -rays. In such cases it was not possible to re¬

solve the /I+V - I and /I-1/ - I ambiguities [i"Q. Therefore

the suggested spin and parity assignments /given in parentheses/

are supported by excitation function arguments. In the case of

^Fe, the initial alignment of the 6528 keV level whose mean

life was determined to be 517 - 45 ns /see Fig.5/ which can be

compared wita the lifetime value of 525 - 10 ns[i5, 18), was not

preserved due to hyperfine interactions. Consequently, the an¬

gular distributions of the r -rays issuing from the 6J82 iceV

10'r

10'

°Ti

0 l °o

Fig. 6 An example of a simulta¬ neous ^ 2 analysis of the distribution, D.C.O. -ratio and linear polari¬ zation results for the 2937 IceV transition in 5% -80° -40° 0°

arc tan 6 40° 80°

U 68

and 6528 keY levels were nearly isotropic and It wae not pos¬

sible to give spin-parity assignments for the two states. The

presence of three higher excited states, which axe weakly popu¬

lated, is clearly established by the ^f-jf coincidence data.

The level scheme as observed in the present work for ^°Ti is

shown in Fig. 7 and in Fig. 8 together with those for "cr and

5"

The level scheme of 48,

Fig. 7

^ Ti nucleus aa obtained with the help of

the "^Ca / <X ,2n/ reaction. The numbers beside the transitions

indicate their relative intensities. To the right are plotted

levels which were observed as weekly populated in the reaction.

Their gamma decays were too weak and could not serve in spin

and parity determinations.

>*69

« Disoussion

Energy levels and electromagnetic properties of the even N = 28 isotones were calculated J19]in a model space that allows at most one nucleon in either the j=2p»*2» ^Pi/2 or ^^5/2 s^e^* Three sets of two-body matrix elements enter- into such a calcu¬ lation:

i/ the forty matrix elements <tn/2 d-jl i *7/2 ^ 2 ^ » as

deduced by Johnstone and Benson [12], where d-j and j 2

refer to one of the j-subshells specified above, ii/ the twelve matrix elements ^'7/2 '7/2!v I J7/2 ^ as

renormalized by Euo and Brown [13]• &nd which couple the states to the single-particle excitations, and

iii/ the /*n/2' interaction. For this last interaction as well as the single particle energies the values of Ref. [12] with certain small modifications [14] were adopted.

MrV<

12

11

10

9

8

7

6

-tr -ir

Pig. 8

Comparison of experimental level schemes with theoretical pre¬ dictions based on the shell model calculations in which one f neutron is promoted to tha whole /fp/ shell.

"•70

In Fig. 6 the experimental and theoretical results on energy levels in ^!Ti, ' Or and <*?e axe compared. The left-hand part in each spectrum exhibits levels which are predominant¬ ly ft n/o/1 t whereas the right-hand parts of the calculated spec¬ tra show the lowest-lying intruder levels of each spin between Mr* and the maximum spin that is possible in the present model space.

Since the correspondence "between experimental and calculat-CQ CO

ed spectra for Ti and ' Cr seems to be well established, a brief discussion of some characteristic properties of their wa¬ ve-functions will be given below.

The coupling of /f?/?/21 configurations and single-particle excitations has a noticeable effect on the structure of the sta¬ tes belonging to the ^n/2^ b a n d« On the average these sta¬ tes are only 80% /tj,*/1' On the other band, this coupling leads to only extremely small /tn^f^ admixtures in the 4 + and 6+ intruder states of 50Ti and in the 4 +, 5+, 6* and 8* intru¬ der states of "Cr. All intruder states consist of more than 99.5£ of neutron excitations. The maximum-spin state in each nucleus is uniquely formed by the aligned coupling of a ^3/2 neutron to the maximum spin fo/2 configuration. All other in¬ truder levels contain more than 9OJS 21*3/2 excitation, whereas the 1fcj2 component amounts to 1% with the exception of the 4 +

and 5+ states where it lies in the 5% range. The present notion of the wave-functions is further sup¬

ported by comparison of the measured electromagnetic decay pro¬ perties with the calculated ones. In table 1 are listed these properties for ^ Ti. It is known that in the pure fn/2 model a large isoscalar polarization charge of 0.9e is needed to explain the yrast E2 transitions. With an admixture of 2056 of other configurations provided by the present calculations, a polarization charge of only 0.5e to 0.6e is necessary to give results in agreement with experiment. All intruder states de¬ cay almost purely by the M1 mode. Theory agrees with experiment in showing very small multipole mixing ratios in the main branch of the deo&y. This branch is always the transition to the near-

Table 1 : Electromagnetic transitions in Ti. B(E2) values were calculated using nucleon charges

e = 1 . 6 and e , = 0. 6 TT *

Transition Branching r»tio | % \ Multipole mixing ratio B(E2) | e^fm* |

I I |keV| exp calc. expC calc. exp. calc.

Ej Oj 1554 0 "J 66 i 8 82.75

4j 2 ( 2675 1554 l a 6 0 ! 10 8 1 - 8 3

6 4 3198 2675 J 34.2 +_ 1.2 39.06

7 6 6135 5510* not observed 0 .03

5 5070 not observed 0 . 7 2

6 3198 100 9 9 . 2 5 0 .141 +_0.025 0 . 0 3 7

8 7 6539 6135 94+^3 9 3 . 7 3 0 . 0 1 7 + 0 . 0 0 9 0 . 0 1 3 • l i """

6 5510 not observed 2.51

6 3198 6+_3 3.76

9. 8 6769 6539 100 98.80 0 . 0 3 5 ^ 0 . 0 1 5 0.010

7 ( 6135 < 19 1.20

10j 9 ( 7570 6769 100 99.91 0 . 0 4 4 ^ 0 . 0 1 8 0.028

8j 6539 < 15 0.09 (llj) 10j 8790 7570 100 99.27 0.17 ^ 0. 10 0.047

9 j 6769 < 12 0.73

* Mff.lel * Caloolatvd l«y«l positions 0 Rot* and Brink pbaaa oonvantlon

est intruder level of lower spin except for the decay of the 7* state to the predicted 6+ intruder state which has not been ob¬ served. This is precisely what is expected from the calculation. The corresponding 3(1 matrix element is exceptionally small be¬ cause of an accidental cancellation between the contribution due to the magnetic moments of fy/2 proton and neutron and the contri¬ bution of opposite sign due to the P3/2 neutron. The E2 branch to the predicted 5+ level is also calculated to be very weak. The fact that this transition has not been observed can be considered as a further indication that configuration admixtures are present in the 6^ state. If one assumed this state to be pure fn/j, then the 7+ state could decay to it via Ml only owing to the very small admixture of the fc/2 neutron excitation. Consequently, this transition rate would be smaller by a factor of 500 and the 7 +—*5 + E2 branch would b« observed to be stronger than the 7+-*-6* branch.

The ' Cr nucleus is particularly interesting since it is the only even H=28 system in which configurational purity can be test¬ ed by applying two selection rules of the fo/2 model:

1/ no U1 transitions are possible between j n states formed of one kind particle configurations

ii/ within the half filled shell, E2 transitions can only ta¬ ke place between states differing in seniority by two units.

Experiment shows, contrary to rule /i/, a weak 2| -+2\ M1-transition /see table 2/. It also shows, in sharp contrast to rule /ii/, strong E2 transitions connecting both lowest 4* levels with the first 2* and the first 6+ level thus indicating strong seniority mixing in these 4* states. The results of the present calculations agree qualitatively with these observations. As in ^°Ti the "allowed" E2 transitions are well reproduced with a polarization charge of 0.6e. As far as the decay of the intruder states 1 B concerned, experiment and theory agree that, in general, they decay in the H1 mode with small E2 admixtures. As in 50Ti, there exists an HI transition in 52Cr which direct¬ ly indicates configuration adnixturas in th« highest state of

,ble 2 : Electromagnetic transition! in Cr diacunect in the text. Effective charge* are taken a* In table I.

h

4 2

2 2

6,

9 ,

Transition

lt

2 , «,

21 4 2

Zl

4 2

4,

8 2 7 ,

|keV |

2370

2767

2965

3115

6454

1434

23 7Q

1434

2767

2370

1434

0

2767

2370

5825

5398

4750

Branching

«cp.»

100

1.6 + 0.2

98 .4+0.2

99.5 +0.2

0.5 + 0.2

1 . 0 8 + 0 . 0 3

98.92 + 0.03

2 6 + 1

< 15

74 + 1

ratio |«4 | B(M1) | 10"3

calc. exp.

100

6.0

94.0

< 0.01

0.1

9 9 . 7 o . 6 1 ; : ; 0.2

0.3

99.7

7 .6 140 » o

3.2

89.2 2 0 * ' 2

'.uil calc.

3.4

54.8

33.3

B(EZ)|

exp.

83 + 17

73 + 28 7 3 -15

138 t 46 O O 2 6 t o : o i 5

2 8 . 8 + 1 . 7

5 8 . 7 + 2 . 0

, 0 0 + " 0

< 343

2, 4i e fm |

calc.

113.0

10.5

131.8

0.012

12. Q

89.7

54.7

64.7

Ref. for

mean life

| 7 |

|8|.|ift,|

lifrl

|T||e|

1*1

* RefJlT I except for the 6454 keV level.

the /fn/2/4 configuration. This ia the 1704 keV Ml decay of the 9+ to the &t level. The transition would be strictly for¬ bidden if the 8^ state were pure ^7/2' and *ne 9+ single par¬ ticle excitation did not contain any ^5/2 components. With the fc/2 admixtures in the 9+ state, resulting from the present cal¬ culations, the strength of this transition still would be U- or¬ ders of magnitude smaller than observed if 8^ were pure /If^ro/ •

The third K=28 system that was investigated is ^4Fe. How¬ ever no definite spin assignments have been made for high spin states in this nucleus. Besides the levels at 6332. 6528 and

40 16 6725 keV, which were also seen in the Ca / 0, 2pt- / reaction [15], the levels at 7506 and 8022 iceV were observed aa decaying to the levels at 6282 and 6528 keV, respectively. Shell model calculations for high spin states give results that are very different from those obtained for the proton particle-hole con¬ jugate system ^ Ti and which do not follow the simple systema-tics found for ^ Ti and ^ Cr. In the excitation energy region between 6 and 8 UeV in which there are states with spins great¬ er than 8, the calculations reveal numerous states with spins between 8+ and 11*. Because of the scarcity of experimental data, the correspondence cannot be as easily established as for -*°Ti and ^ Cr. In these states not only proton excitations oc¬ cur with a finite probability, but also neutron excitations tc the 1fe/2 sad 2p^ /p orbits become important. In Hef.[15] the g-factor of the long-lived 6528 keV /10+/ level is reported tj be +0.78 - 0.02. This value can be compared with the value +0.66 which was computed Pi9] for the lowest-lying 10* state.

References

[1} E.C.Halbert, J.B.McGrory, B.H.Wildental, and S.P.Pandy a, in Advances in Nuclear Physics, edited by H.Baranger and E.Vogt, Plenum Press, New York, 1971, Vol.f, p. 324-.

[2] B.J.Cole, A.Watt, and R.R.Whitehead, J.Phys. ±2. / 197V 137*.

[3j T.T.S.Kuo, Nucl.Phys. A103 /1967/ 7.

H.G.Benson, J.P.Johnstone, Can.J.Phys. ££ /1976/ 1683*

475

[51 R.L.Auble, Nuci.Data Sheets 12 /1976/ 291. [6] A.Berinde, R.O.Dumitru, M.Grecescu, I.Keamu, C.Protop,

N.Scintei, C.M.Simionescu, B.Heits, H.W.Schuh, P.Von Brentano, K.O.Zell, Nucl.Phys. A284- /1977/ 65, and references therein.

[?l B.A.Brown, D.B.Fossan, J.tl.Mc Donald, and K.A.Snover, Phys.Rev. C_2 /197V 1055 and references therein.

is] A.R.Poletti, B.A.Brown, D.B.Fossan and L.K.ffiarturton, Pnys.Rev. CIO / 197H/ 2J29.

[9] J.A.Grau, Z.W.Gratowski, F.A.Rickey.P.C.Simms and R.M.Steffen, Ph7e.Uev.Lett. ^2 /197V 677.

10j F.A.Beck, T.Byrski, A.Hnipper and J.P.Vivien, Phys.Rev.CTJ /1976/ 1792.

11^ L.P.Ekstrom, H.H.Eggenhuisec, G.A.r.LngelbertinJi, J.A.J.Her¬ mans, and H.J.M.Aarts, Nucl.Fhys. A28^ /1977/ 157-

12] I.P.Johnstone and H.G.Benson, J.Phye.G: Hucl.Phys. ^ /1977/ L69.

J13j T.T.S.Kuo and G.E.Brown, Nucl.Phys. A11» /1963/ 241. [141 A.MUller-Arnke and R.D.Lawson, to be published. l15l J.W.Noe, D.F.Geesaman, P.Gural and G.D.Sprcuse, Bull.Am.Phys.

Soc. 22 /1977/ 528; and Proceedings Topical Conf. on Physics of Medium-Light Nuclei, Florence /1977/ p.23.

[i6j S.W.Sprague, R.G.Arns, B.J.Brunner, S.E.Caldwell and. C.K.Hozsa, Phys.Rev. W /1971/ 207^.

[17] R.P.Yaffe and R.A.Meyer, Phj-s, Rev. £16 /1977/ 1581. !18] E.Bożek, T.Pawłat, Z.Stachura, J.Styczeń, F.A.Beck, C.Grhria-

ger, B.Haas, J.C.Merdinger, N.Schulz, M.Toulenionde , J.P.Vivien, Proceedings Topical Conf. on Physics of Medium-Licit "uclei , Florence /1977/ p.27.

';19J B.Haas, F.A.Beck, C.Gehringer, J.C.Merdinger, M.S.ochul;, P.Toulemonde, J.F.Vivien, J.Stycaeń, E.Bożek, Z.Stachura, T.Pawłat, I.A.MUller-Aracke, to be published .

In - ceam Investigation of the N = 82 Nucleus 14^Pm

H. Prade, L. Kaubler, U. Hagemann, H.U. Jager,

M. Kirchbach, L. Schneider and ?. Stary

Zentralinstitut fiir Kernforschung Hossendorf, Bereich 2

The in-beam investigations • of high-spin states in ^?m per¬

formed at the Rossendorf cyclotron have been completed by measuring

the y-ray linear polarization. In these experiments, two different

polarimeterB have been used. The first one consisted of two cylin¬

drical Ge(Li) detectors having volumes of about 20 cm , respecti¬

vely. The other polarimeter was a parallel plate 3e(Li) detector

with an active volume of 27 x 27 x 5 mm"'.

Our new data allow to conclude definite spin and parity values for

the levels at 2060.2, 2881.9, 3075.6 and 3376.7 keV. ?or a series

of further levels nore accurate 3pin-parity assignments can be pro¬

posed. All these results are summarized in the level scheme shown

in fig.1.

There has been published a shell-model investigation by Wildeathalr'

which successfully describes the spins of the ground and first exci¬

ted states of K=82 isotonefl as a function of mass. In these calcu¬

lations, the 14 configurations of the types (Igy/p' ^5/2^ ar-&

(1g7/2» 2 dS/2' ^2d3/2' 3S1/'O have been taken into account, and

the two-body part of the Hamiltonian has been parametrized in terms

of the modified surface delta interaction. We adopted the shell-

model approach of Wildenthal and calculated the whole spectrum of

even-parity states in *3Pm. One obtains (fig.2) a level sequence

which coincides with the experimental data. The excitation energies

are predicted a bit too small.

Pig. 1. Level eoheme of ą-(d, 2n) reactions.

as obtained from (cC, 2n) and

1*78

rrr 7X13

SHELL-MÓDL EXPERIMENT

7/2' 7/2* 272.0

?ig. 2. Comparison of calculated and experimental energies fcr

positive parity states. In the calculated level scheme

only the lowest three states are shown for each spin value.

CD O

T«bl* 1. Contributions {%) of configurations In tha wave funottona

"7/2

5

6

6

7

5

7

4

6

7

4

S

5

a e

configuration

"5/2

5

4

4

3

5

3

6

. 5

4

6

2

«

3

M3/2

1

1

1

1

1

3"i/2

1

1

t

•1

1

51

0.3

1.2

1.4

0.5

0.1

0.2

0.2

30.7

0.1

0.2

0.3

-

0.4

56.7

71

0.4

2.5

0.6

0.6

0.4

0.6

0.4

0.3

70.6

1.0

0.6

22.8

0.2

-

h

1.3

2.6

3.6

3.9

0.1

0.3

0.1

8.9

1.7

0.1

0.2

0.1

6.9

70.4

11

0.2

1.5

53.9

0.3

-

-

0.2

0.3

. 0.3

11.8

0.3

-

31.2

-

9l

1.2

3.7

2.9

1.9

0.6

1.5

-

7.3

17.5

0.1

4.1

-

0.3

59.1

>2

0.4

47.1

0.9

0.6

0.9

1.5

9.4

1.4

5.0

0.2

32.4

0.1

0.1

0.1

21/b

h

2.1

3.9

1.0

3.0

1.1

5.3

0.?

13.0

61.9

0.1

0.4

6.7

0.6

-

V

4.1

3.8

4.2

3.2

0.8

1.7

0.1

5«.O

11.9

0.4

0.5

0.1

-

15.1

'2

1.3

4.6

0.8

1.9

1.3

5.8

0.4

5.3

72.8

-

-

0.7

-

5.J

" i

1.1

3.9

0.5

0.7

2-3

4.1

0.2

0.2

BO. 9

-

3.3

2.9

-

*

}3

2.0

10.0

1.1

4.1

1.9

2.6

3.6

14.4

55.9

-

0.8

0.5

0.8

2.3

«,

1.4

3.0

1.0

1.3

1.5

2.0

.

1.1

88.2

-

0.4

-

-

1.6

1.3

.4.4

1.3

0.2

-

6.2

81.8

0.2

-

_

-

3.8

1.0

3.1

3.4

0.2

0.1

_

07.8

0.6

-

-

In our earlier discussions 1'2' the lowest states with f = 7/2+,

11/2 , 15/2+ and 17/2 were explained oy coupling a g-7/2 proton

to the 0 +, 2 +, 4 + ar.d 6+ levels of the core, respectively. There¬

fore, the configuration Os-7/o) ^ ^ 5 / ? ' should be dominate in

the 7/2+, 11/2 + and 15/2+ states, whereas a (1g7//2)°(2d5/2)5

structure is expected for the 17/2+ level. For the 15/2+ state,

ir. particular, this assumption is supported also by the g-factor

of the state . The shell-tnodel calculation confirms these con¬

clusions (table 1) and ahov/e that also the lowest 13/2+ state be¬

longs to the (1g7/2) (2d^/2) r.'ultiplet. The positive parity states

above 4 VeV rr.ay arise from the coupling of a dr / or £7/2 pr-^or,

state to aligned states of the ( h ^ ^ ) two-particle excitation.

The oód-?arity states (fig.1) are ,iost probably to be explainer

by coupling of one h..^ proton to the C , 2 , 4 and 6 core stri¬

des. T'ow, Tore extended shell-Tiodel calculations inclu-iinc the

1 h ^ / , shell are in progress.

References

. '•;. I'rii-?, 'J. iisge^ann, L. K'lurler, L. Schr.eider ar.-: :•'. .rtary, rric. of the Internat. ~yn:posLu:r. on I:i .fi-~pir.- .'tTCes, l:res".e:.

:r:>33' C!??7N> 23

i. '.V, A."-~'re.j tsci:eff, I1. );2 5e".?.;".r.f 1. "'.abler, li:,-, L. Jc::::eider, 7. Zt~.vy, rr"-c. oT the "~. "asuri^.r. Tchool or. .': ;cl. " I'ukloo.'.i::- -'ir. rr=ss).

•>.. :. -. .Vi" -er.thr.1, T'.ys. '.ev, Lett. 22 '"i?'r0 111 ?

X.I. .'_•• i 1-

Shape transition In the odd Tb nuclei

G. V/lnter, ?. Kenmitz, J. Dorlng, L. Funker Zentrallnstitut fUr Kernforschung

Rossendorf, 8051 Dresden

3. Slfstrom, S.A. Hjorth, A.Johnson and Th. Lindblad Research Institute of Physios Stockholm

Experiments

The odd mass transitional nuclei 15; !:b [1] and 151Tb [2] have been studied In the (<*.,2n), (*,4n) snd (T,3n) reactions. Zxoitaticn functions, angular distributions ar;o coincidences of the v-rsys were measures using Ge(Ll) spectrometers. Also conversion electrons and delayed v-r2ys were recorded.

The nucleus

1 53 In the K=8B nucleus Tb levels of positive parity are identified up to spin 27/2. The level schene is shown in fig. 1 . ;..ost of the observed states 3re interpreted as rseinbers of rotatior.sl 'zzni' characterized by the Kllsson configurations 3/2+[41i], 5/2*[-C4i and 7/2 [404^. Furthermore, s sequence of r.egstive-pr-rity ;t?-tas ic remarkably well described in a CoriOiis ccuplinn- orlculation inrlui ing al l configurations of the h11 ,~ -'iisson nult iplet . In this cal¬ culation the moment of inertia parameter varies with collective anpular raomentum in the seme way as ir. the ground st?te bsnd :f 152

C-d. Using the same prescription for calculating rotational ener¬ gies, the Irregular level spaclngs within the 5/2+[<s02] and 7/2+r^04 bands are well explained. Results of this calculations for the ne¬ gative- and positive-parity s tates , respectively, sre shown in fi.rł. 2 and 3 .

1 * 5 1 The nucleus Tb

The level scheme of the K=86 nucleus 151Tb Is shown in fig. 4 . A 25s isomerlc 11/2" level was found to decay via levels of 5/2+

and 3/2+ to the 1/2+ ground s ta te . The isoner and 15/2+ state are

482

(3/2'ltnil

1 51 -ig. 1 Level scheme of Tb. The width of the arrows is E measure

of the transition intensity as observed in the (<*,£r.) re¬

action.

3 r i

5 ~ 2-1 1 ui

I ,.M-

- ^ "?" nff-•w-

0-

?y?-

EXPERIhCNT 50F' ROTOR RtOO ROTOR

CALCULATION

Zxperlmental ans oslculateć energies for negstive-7s.rit;

B 2 U3 2V2 ZS/2 SPIN I ( I 2 SCALE)

W2

3 Analysis of the rotational er.erries for the bands built on 7/2+[4O4J and 5/£T[402]. ...e curves conrect fr.e results of tho calculation. Experiments! data are shown by dots.

S162.7

(3S/2-)

Fig. 4 Level scheme of ?b. The width of the arrows is a raessure

of the transition Intensity as observed in the («,4n) re¬

action at E^= 51 I.:eV.

fed through two similar level sequences v/ich have been established

u? to (35/2~) and (39/2)+, respectively, another level sequence

ranges from 33/2^+^ to (45/2+). The levels up to 11/2" are inter¬

preted as shell model excitations of the odd proton. In fig. 5 is

shown how the levels above the isocier can be related with excita¬

tions of the doubly-even nuclei 14S '150Gd. The states 11/2" to

23/2~, are interpreted as a coupling of the h ^ / - proton to the 150Gd ground state multlplet. The states between 15/2+ and 27/2+

arise from a coupling of the h,.,/, Droton to the sequence of nega-' — 1 50

tive parity states build on the 3 level in x Gd. The incomplete alignment ( 3~(x)1Th11 /?)i5/2 m a j r b e r e l a t e d t 0 a strong h11 / , d5/2 component in the 3" state. The high-spin states are mainly ascribed to excitations of the 5 valence nucleons outside a magic ft^g

core [4] . Three-particle oluster3, C^-H/2*'f7/2l"19/2^27/2~ a n d

^'irll11/2vh9/2vl'13/2^33/2+ ' r e s P e c t i v e l y » a r e assumed to persist as

stable components within the high-spin multiplets. These clusters

are formed by maximisation of the overlap of nucleonic wave func¬

tion by alignment of the angular momenta (fcONA). The structure

based on the 33/2^+^ level shows a rather good agreement with a

( f 7 / 2 )2 multiplet (fig. 5). The spacings within the 25/2" to 35/2"

sequence are characteristic for a (£7/0) multiplet. The same pat¬

tern is observed within the even parity multiplet (29/2+ - 39/2+)

where the 3~ core excitation is coupled to the states above 25/2".

The comparison of both multiplets with a calculated (*7/p) multi¬

plet is shown in fig. 6 .

Conclusions

1 53 The N=88 nucleus, ' Tb, can bo understood In the framework of the

rotor model If a soft core Is Included. In contrast to these results

no hints for rotational bands were obtained for 1^1Tb. Considering

the levels of the yrast region up to 5 MeV, the transition from

spherical to deformed nuclei is locates between the isotopes with

86 and 88 neutrons.

U 86

MeV

O1-

(17-)

(13-)

3-

I

66

(39Q*)

35Q* 22_V<f7,2>2 „

7/2 )*•...

15/2- 19/2*

_ 4 _ 11/2- _ 15/2*

irh l 1 /2 3"®irh1V2

151

Fig. 5 Comnarison of the excitation energies within the strcnrl.y populated level sequences in Tb nd in the doubly-even nuclei 150Gd and 148Gd. The shell model assigrjnent st.-r.ć for the proposed nain components.

3D

2

MeV

4

1

0

THEORY

15/2-

\

9/2" 11/2- ^

211-

5/2" 7/2, ~

U9/-

\

^ ( 3 / 2 " ) 5/2" 7/2-

' " 8 5

EXPERIMENT

^ , 2

00+)

>. 8 +

\V 7 + \ \ 6*

1 9/2 '7/2 "9/2

1?hn/2 I

(35/2-)

(29/2-) 31/2-

27/2-

25/2-^^-^23/2-

. 151T C*86 65'

(39/2^

35/2+

31/2+

29/2+

^ 8 6 Fig. 6 Comparison between experimental onJ calculated energies

within the ( fy / 2 ) 3 multiplets of 14l)G(Pnnd within high-spin multlpleta in '50Gd find 151Tb.

1. G. '.Vinter et a l . Nuci. Ptyś. in press.

2. ?. Kemnitz et a i . Kuci. Fhys. submitted

3. D.R. Haenni ano ? .? . Sugihara, Phys. Rev. C16 (1977) 120

4. ?. Kleinheinz et a l . Proc. Int . Conf. on nuclear structure, Tokyo 1977 ?.S64. Proc. Int . Symposium on high-spin states and nuclear structure, Dresden, ZfK-336 (1977) p.25

5. G.:. Hellas: :;ucl. I'ata Sheets 19 (1976) 337

lecent experiments on the snape of t&e fission barrier

V. Metag

Kax-Planck-Institut fur Kernpnysik, Heidelberg, West-Germany

Introuuction

In tbese lectures I would like to summarize the most recent

experimental results on the shape of tne fission barrier wite

particular emphasis on the work aone at EeidelDerg.

Since the first observation of spontaneously fissioning isomers

oy Folikenov ana coworkers in Dubna in 1*62 a large amount of ex¬

perimental eviaence nas Deen accuiuuLHtea in favour cf the concept

of a uouble huii.,,ed fission barrier. Zne tneorelical foundation for

this picture was provioeo by V. Strutinstry wuo calculated the

potential energy of actiniae nuclei as a function of

arid showeu that the existence of metaatable aeformea states Lay be

due to the interplay of single particle effects ana the energy

associated with the defornation of bulk nuclear matter. Strutinsky's

v.ork has demonstrated that a quantitative understanding of the snape

uf the fission barrier is not only of direct relevance for z:.e

fission process itself, but also for ti.e interaction of scell

structure ana collective motion in general wnicn is a central issue

of nuclear structure. It is no exaggeration to s&y tnat tais con¬

current experimental and theoretical uevelopment has brought fissicn

bacK to the mainstream of nuclear pnysics. Tnis subject has recently

been reviewed in various articles f i y t C P /.

In the following I will give a brief ana extremely simplified intro¬

duction into the field not adressed to experts on the :ifsion

process.

The double numped structure is only a smai.1 althou^n essential

part of the total fission barrier as iLLustrated in fig. 1. Here,

the binding energy of the system is plotted as a function of the

deformation of the fissioning nucleus and the distance between tne

two separated fragments, respectively. ?ission occurs either spon¬

taneously fron, the ground state oy penetrating the potential carrier

or n,ay De inuuced through particle in.;jact providing enough excitation

energy to overcome the barrier.The ai:':'erence of about 200 MeV in

the oinuing energies of the two separótea fragments on toe one hand

and tue fissioning nucleus on tiie otner nand appears taainLy as the

^ 90

kinetic energy of the two fission fragments which are - starting

from tne scission point - driven apart by their mutual Coulomb

repulsion. This energy balance is fairly well described by the

liquid drop moael (LSM). In the early stage of tne fission process

(lower part óf fig. 1) the UM fission barrier is determined by the

interplay of Coulomb ana surface energy leading to a flat maximum

in the potential energy- at the saddle point.

The liquid drop model implies a uniform distribution of

nucleons in pnase space. Any deviation from this distribution

(shell effects) leads to the well known discrepancies between

experimentally observed binaing energies and tnose calculated with

tne liquid drop model. Nuclei with certain magic nucleon nuaoers

like T>b show an increased stability by up to 10 MeV. These

magic nucleon numbers are found not only for spherical shapes but

also in distorted nuclei at certain deformations as shown in fig. 2.

Here, the eigenvalues of a harmonic oscillator potential are plotted

as a function of the major to minor axis ratio c/a of a rotational

symmetric ellipsoid. Whenever c/a is the ratio of i two maia.ll

integers, considerable degeneracy appears in the level scheme, in

particular for c/a=2. In a real nucleus|- the harmonic oscillator

is certainly an oversimplified moael -j this degeneracy disappears,

but tiiere is still a tendency for level densities to be high

at some deformations and low at others. The magic nucleon numbers

for spherical shapes are different from those delated to the de¬

formation c/a=2:1. For an actiniae nucleus of tnis deformation fig.2

gives a magic neutron number of 140 whicc will later be compared

with experimental eviuence. Fig. 2 illustrates that periodic changes

in the density of single particle states occur not only as a function

of nucleon nuu.ber at a given deformation but also as a function of

deformation for a fixed nucleon nuriber. These changes in the level

density are related to fluctuations in the binaing energy. A decrease

in the level density implies an increase in the binding energy since

the nucleons occupy more tightly bound orbits. Vice versa a bunching

of levels leads to a decrease of the binaing energy. Strutinsky has

given a prescription of how to aeauce shell corrections to the

nuclear binding energy from these single particle schemes which

then have to be superimposed on top of the smooth liquid drop defor¬

mation -energy. For a low level density the nucleus is tightly bound

-1800

> -1900

i ui-2000

PO

TEN

TIA

L

5

0

_

*

/

-

\ . . SCISSION POINT

\ EHEROł REtEASE N. s: ?00 MEV

) O CD C=>

\

\ / i II0U1D WOP Y'x V / ENERGY \

1

-S B

2

Roiio ot o«es, c/a

• ' "* " • — — - 1 . 4

1 • ł i 1 i

3

i E l :a—«=; ——-_ ——fc^.

4

. 1

1 QO

DEFORMATION 06

Ot'wmalion < Q8 1.0

Fig.t Total energy of a fiaaioning «yat«io «• a fur.otion of the distance between the ••parated fragmenta /upper part/. Enlarged view of the deformation energy in the. early •tage of the fisalon process /lower part/

Fig.2 Energy levels of a harmonio oeoil-lator potential for prolate aphoroldal deformations. The partiole number* of the olosed shell* are indioated for a sphere and for a spheroid whose major axi* 1* twice its ainor axis. /Taken from ref.33/

and the snell correction to the energy is negative wnereas for

a high level density the shell correction is positive. As

illustrated in a scnematic way in fig. 3 SUCH a superposition

of defora.ation dependent shell corrections onto the liquid drop

codel fission barrier yields (in the case of actinide nuclei) a

barrier with two humps ana a second metastable minimum in bet¬

ween at a larger deformation than the grounastate in the first

minin-um and with an excitation energy of 2-3 MeV. In large scale

numerical calculations the potential energy of the nuclei is cal¬

culated as a function of all relevant shape parameters anu the

simplified plot of fig. 3 should rather be considered a cut

through sucu potential energy surfaces along the fission path.

The picture of a aouble humped fiseion baxrier does, indeed,

explain trie major features of fiseion isoii.ers:

i) taeir excitation energy of about 2 MftV

ii) tneir relative stability against f-decay guaranteed by

the inner carrier

iii) the appreciably shorter belflives for spontaneous, fission

from tne isomeric state as compared to fission from the

groundstate because of the smaller barrier that has to

be penetrated in the fissioa process.

Intermediate structure in the prompt and delayed fission pro¬

babilities, v.uicn I will un.it in the present a\scussion, are

also readily interpreted in terms of a aouole numped fission

Darrier union again, is a strong indication for tue validity

of tuis concept.

Pig. 4 contains tne most recent compilation of fission isoaeric

halflives and snows that these isonierB occur only j.n a ratner

limited region (island) of tne nuclear chart. Tnis supports

the picture of a douDle humped fission barrier in a qualitative

way since only for proton nuii.bere y2 * Z 6 37 tne liquid drop

model barrier exnibits a flat maximum so that superimposed shell

corrections lead to two humps of comparable neigut. The cluster¬

ing of fission'ison.erB at neutron numbers 141 A K fc151 is

qualitatively consistent with the variation of the calculated

neutron snell corrections at trie second n.inin.uii with neutron

nuiLDer. As schematically illustrated in fig. 5 the strongest

negative shell correction expected for the magic neutron number

- 5 o

5

1

-

/ / / /

N \

\ \

\

a. z c r-

i

to <B

O deformation

Is 5

V7 Superposition of sue 1.1 correctioi:s o;, tc^ oi' xi.e liquia

uoei aefortation energy (c&cneu curve) yieluing

Guule uuii.yeci fis: i^n carrier for actiniae nuclei.

•

94 Pu

93Np

92 U

91 Pa

96 Cm

35Am

30 ns

5 ns

34 ns 37ps

97Bk

35ps

l ips 110 ns

98Cf

55ns tOps

160ns

6 ns 500 ps

116ns

9.5 ns 600ns

15ns

900ps

2.6 ns

40 ns

180ns SOps

1.5 ps

3.4 ns

195ns

820ns

42 ns

14ms

30ns 21ps

2 ns

>nons *5ps

5-Sjis

50ns 3.6 ns

13ns

1.0ms

60ns

U9

640 ns

380ps

150

N >

73 ps

SOns

151

Fig. 4 Fbrt oi' tue nuclear ciifirt giving tne lifaiflives of a l l fii-.Riu;. iBOL.ers knov/n at present. Tv.o values for t;.e san.e nucleus incicate spia isoaeric stat ts in tue se-cona jLiniL.um.

energy for magic neutron jE

number I

max

deformation \ \

?ig. b Double Lui-peu fiesior. barriers for a Łacie ar.c non-n.a ;ic actiniae nucleus, respectively; aeŁcnstratinę ti.e effect on tne fisrion ison.eric he If life.

yields tne relatively deepest second LininuE, the largest outer barrier ana consequently tne longest htilflife for spontaneous fission frcn tne isou.eric state. 7.itn increasing cistance froit tne tagic neutron nur-ber tne second minimus Deccii.es less pronounced tne heiflives get si^orter ana arop below lue detection lin.it sc xauX ooservatiie fission isoiaers are concentrated arouna tne j>.a*ic neutron number r;ni'jh has Dten determined to No = 14o fret, a f i t to fission isoLerxc nalf-lives plotted as a function of neutron nuŁber in i'i/c. 6. Tfcis result is in qualitative ajireeaent with He =140 given by tne

oscillator pot-.ntisl (fi. .2) ano corresponas v/itn gaps in single pprticie spectra at Xue se-

cuna Łiiiin.uci (fip. 7) oDtaii.c-j iu refinea calculations by vuriuas authors usin£ more realist ic nuclear potentials.

11 Tun uei'orii.ation of fission lson.eric states

All tnę t^ycriiLentL-1 reuuits diGCifted so fur pri/Viae only iruiract t LXi. suuln concLusive eviaence for tne CAistence of a uouoie nuji.peo iission D&rrier. ;. airect ana aefimte proof re-airts Xue n.easurea.ent ol the aeioraation of a iistion iaoi. eric state v.mcii suoulu oe a snt pe isoaer as a i'onseqaeiace of tiit iouDle naapeu structure of tne barrier. The pioneering step in tiiis direction cas oeen tsuen Cy Suetnt et al . Tney founo fiK.WtnRt a rotational Dana is basea not only on ttie grouno-state biX also on tne fission isoi:eric state of ru by Łieasuring aelayea ctiinciaences oetweec transitions v.itiiit» tńis D: no ana isuL.eric fission. A siDiilar feipericent nas u.eanwiiile oeen per-:ori..ea on ^^U by XLe cocDined Seattle-Copenhagen group. Tne ii.ot,eiit of inertia of tnis b&nd is tore than twice as &ig as ti»8t of tne groundstate rotational or nd and even larger ti.an tne n.on.ent of inertif* of a rigid rutor at grouncstate defor-n,ation. Tnis war. ounsicerea evidence for a strong cefortastion cf txje fission isoi..eric state altnougb no quantitative vt-lue of ti*is uei'ort.atiori could be ueuuceo fron. tnie Łeasurement since tne iuvjti.ent of inertia is onl;; a todel aeptnaent function o? oefon^ation. On ti.e otner u&.nu, & n.t siij-eEient of tne B(E2) values of tnese transitions Vroulc. oirectly give tne ueforŁ&tion since tney are related to tne quadrupolen.oment of tne nucleus in tne fission ii-oii.eric state via tne v^ell eEtablisnea r t t a -tional Łouel. I t was clear tnŁt tne n-easureiL^nt of tuese B{E2)

- 2

_ - 6 U)

O)

o

- 8

- 1 0

-12

a.

odd-odd Am

odd-even " s Am

even-odd Pu \

uo

even-even

f J U5

neutron-number

150

; - ^ L M i v e s o f 1'ic-^i-;.:. i r . . : : ••]-.-: i : . >••->, A:... :

1*97

NEUTRONS

4-1

o

-2

NIX NILSSON MOSEL MdLLER NILSSON NIX (1973)

Pig. 7 Neutron single particle levels at tae defor&ation

corresponding to the second minimiv, calculated by

various autnors. (from ref. 34)

Pig. 6 Hotatioaal bands built upon the groundatate and fission tsomeric stats of 2l*°Pu. Excitation energies, spine, and halflives of the rotatioaal levels are indicated.

1*98

values v.as aot fetaible since only tae ioentification of these rotationi-1 states toclc already about 14 days of beec tin.e.A com¬ pletely new technique had to be developed for iteaeuring tóe lifetimes or excited nuclear states vmicn are expected to be of the oraer of eoce 10-30 pe using theoretical estiiL&tes for the quedrupolemoiLent. This ner; technique which will be described in the following exploits the fact tnat the rotational transi¬ tions in heavy nuclei are highly converted and that fast /.uger cascades following 3uch transitions lead to high ionic charge states. lie) The cnarge plungar

The principle of the nev. n,ettoa is explained in fig. y. A recoil ion excited in a nuclear reaction leaves the target witii the lov. equilibrium charge of 1+. After some picoseconds a converted transition occurs and a conversion electron (long arrow) is emitted from the recoil ion followed by a sequence of Au«er electrona froa the Auger cascade in the atomic snells initiated by tne internal conversion process (small arrows). ThUB, the lo-ft charge recoil ion is convertea into a highly charged recoil, ion wituin a time snort compared to the nuclear lifetimes of interest (10 s). Tne main point of our new technique is that at some distance froa tne target we nave placed a tnin carbon foil. By passing t m s foil the nignly charged recoil ions re-cbpture the electrons Io3t in the Au^er cascaae and eaerge ?ritii tne equilibriiuij charge. There ere now two possibilities: Eitt-er the nuclear de-excitation occurs prior to reaching the chyrge-resetting foil - s*ior.n in the ucper part of fig.y-tnen e low charge ion is observed or tt,e converted transition occurs after passing the carbon foil - ehov.n in tt.e lor.-er part of fig. * - then a higiily chargeo. ion is aetecti-d. Thwef tne cnarge aietrioution of the recoil ions consists of two con.-pletely separate components with relative intensities depending on the tii.e of tise nucle&r aecey. 3y aeaeuiing the caarb-e distribution behina the caroon foil as a function of its distance to the target one can therefore aetentine tne de-excitation tike of tne nuuleŁT level decaying tarougi. a con¬ verted transition wituout ever oc«erving tiiie transition directljr. e measure v.hat happens in tue nucŁeus by ooeerving the icpact on the atomic shells v.hici. ie c u e tore easily

1(99

yield o.

n I

target \

carbon foil yield

H

0* 10* 20* 30* charge state

H

fig-

0* 10* 20* 30* charge stole

The principle of the "uherge plunger" Łethod. (L-low charge COB.ponent, H-uici. charge cocponent)

accessible in the experiment. Ooviously, a certain analogy exists to the plunger technique for/"-transitions. In tti&t case, a ttiick plunger is used to stop trie recoil ions ana tne intensities of Doppler shifted m u iinsnii'ted •jf'-r&ys are conpared, v.hereas in our case a thin jlun :er foil is used ana tne intensities of the ni^L cr.arge ana lov; cnarge recoil ions pat-sing th plunger ajrangen.ent are conpare a.For this reason we have called the new method the "charge plunger technique" ( CPT ). After outlining the principle of the method and Djfore ais-cussing first results I woulu like to describe in more detail xue above mentioned Auger" cascades in the atomic shells. If a converted transition occurs in the excited nucleus wnile recoiling into vacuuis an inner shell vacancy e.g. in the L-sne'Ll is proouced. (The transitions TiitL.m tne rott.tionsl bt.na in the second HiiniŁuni are too low in energy to undergo k-conversion). As illustrated in fig. 10 tnis vacancy Lay eitner be filled by x-ray emission or - with a relative probability oi :-.Dout 50 $ in the actinide region - by fo Anger process.

•too

roentgen fluores -ctnst

Auger z

cascade f s

tStcp

Augtr process

hole from internal conversion

Pig. 10 Schematic illustration of ti.e first two steps Auger caecaae.

an

In the lat ter case an electron from trie fii-sceli juu.pe into tee L-snell v.nile simultaneously an auuition&l electron is euiitt :ri fropi tne K - or ci "tier sheLls. Tue cne Lioie xu tne L-siit.i.J. u&s tuns ot en re^LŁi-et. by tv.v uoi.es ir, ^i^iier she Via. Tiiis procei-s - lurtner GoupLicntea by tee occurrence of fast Coater-Xronig transitions -procjeos li-.e an a v&i; nciie to :as cuter :.ue LIB ana converts a 1 actiniae ;c;i into a 1.1 charged racoil ion miti. an av€ri.;;e charge of1 14*. This

10} been ooservea Dy ue Ttieclsv.ilc v>tio mvestigatea tae

2 7 2 7 Distribution of -"Np ions recoilinc into vacuun. t'ron. tue J. -aeoay of Am. The charge uist: ioutions snov.n ii rit> 11 extiiiiit even m^.ner CŁ; rges aue to furtner nuclear tr:-ri:-itions tuat tame place in cascBue v.ita the fin-t one. Avenge cder^es of 21+ ana 2o+ ana v.ictxis of 6 - o cnarge units nave been

2 "5 V determined for Np ions uncergoing 2 rnd 3 coneecutivc-converted trsnsitions, respectively. i',e nave continued SUCH investi,n'tioii8 by procacing iii^ciy cŁargeo. rtcoii. xens in bean, .-ma by t;easurin/; tr«eir cu:rre aistrioution tiiroai/n deflection in a E;a n<_tic field (fi?v 12) After pasijin^ a col Łiautor xue ions ere - aepenuing on tl.eir cnergo ntPte - stopveti t t ui'ferfcnt positions of a recoil ion coi. Lector vjnere taey &re eitner iaentil'itQ airectiy or their specific aecay n.oae. ( * , "f -aecay.fipeion) • 501

100

10

2AiAm

10)-

- w

0

Ti

r i

i i

' / / ;

/ i \ t i t

t

u

1 /

/ ; r

"—^

y -J^3~ / \

i

/

i 1

(3 t

t ł

i f

21 26

•I I

\ \ »\ <\

hĄ -\ \\ : '' 'A ' " i • • • i 1 ' i l

1 6 10 15 20 25 30 W 1S 20 M JO JS

Charge Charge

11 Cnarge uistriouticns of -"lip ions undergoing up to J> converted transitions after «l -aecav of An,

B = 15.8 KG

detector foil

tube

detector holder

f i r . 1t Ł-.;,eri.i."CtPi arrriit-erxr.x :'sr ti.e ::"n c'irer. - i c t r i o u t i . as cy u of LC cticn ot 1-x.e r^..-...L ;:.r.;:..'-tia i'it-ld (t.uZzi.e^ K.-ea) v^r. ic; .! t^

i r.? i:i ^

502

In the de-excitation of a rotations1 bana in an even-even nucleus wnich is appreciably fed up to spin 10+ in an (ei, jin) reaction, even 5 consecutive converted transitions Lay oc^ur, leaaing tc still nigner cnarge states than ob¬ served in tne source experiment af snov.n iri fig. 13 for

CK ions from tne J Pu (ti,in) reaction. As inaicated by dashed lines the hign charge part of tne cistrioJtion has been unfolded into tne individual contrioutions fron. 1,2,3 ana n.ore consecutive converted transitions using tne known centre positions anu v.iatns of tnese co&ponents. Taking trie Known conversion coefficients into account tne aeasurea relative intensities of tne components uirtctly refLect tne siue feeding intensities into the various rotational states.

Tne essential test for tne feaaioiLity of the tecunicue was to demonstrate tnat trie nitjnly ci.hn;ea recoil io'is QO inuceu regain nil the electrons lost in tne Auger process whtn passing the thin caroon l'o.l.Ttie proof is ci in fip. 14 v.nicii snows the charge distribution of Cm ions

proaucea in tne Fu (d,3:i) reaction at 27 MeV boaDaraing energy (ti^us avoiding the population of any Icnsr livea h excited isoaeric state) after passing a 3 pLg/cjiiC csroon foil at a distance of 1.5 naii (fLignt tin.e 2.4 ns). (Tnis aistance has Deen cnosen singe tae directly populated groundstste oand nas to De ae-ey.citea to <C 10 J after 2.4 ns for any reasonable as3un.ption aoout tne quaQruoo LemcŁent of uCi:.).

Tne iracti^n of jiigŁ cu&rge states (>10+) representing jb 1 of a l l recoil ions prior to reaching tne csroon fuil i s r t -auoee to < b ' 10~ • Trie observea effectiveness of tne charge resetting csn oe understood on the basis of calculated electron c; pture cress sections of i.i*;*.i.y strippeu luns. ?ror: ti.ese c&lculateo cross sectioiłS i t can also De concluded tnt't consecutive single-electron capture is tne cor..iaant t:,uce in tr.e cr^r.je resetting prccess.

l ib Test exr.erii',.er.ts v-iti. tne cttarge-:/Łju:t?er tecr.r.icue

Sefore a[..pL,yintf tne cuiiTge pi.unj.er teciitiir.ue to tut- i!.ea.Eurentnt of me auaaru^olen.on.ent or1 a fissicn isor.eric stfite v.e nave

503

la.3n)

20* 30* Ui* chorgt state

Pig. 13 Charge cistribution of 240Cn, iono fron, the 23'Pu(«(, Jn)

reaction at 33 ; eV, aecoaposea int^ the contributions

fron. several consecutive converted transitions

1

10*

0 20 40 H M

distanc* along dtttctar faM (mm)

14 Cnarge ciBtrioution of 2*°Cc. recoil ions for a caroon lo i i - t a r t • , . ttvnce oi1 1.5 nx.^ueR.onstrPtin*; tne efficiency of tiie cnsrpe resettinc:

13) 2 4 O tested it with Cm-nuclei whose groundstate qua&rupole-Łoaent can rather reliably be extrapolated fron. existing systeciities. ?ig. 15 snows charge aistrioutions of Cm ions measured mitn tne arrangement of fig. 12 and a cnarge resetting foil pLaced at distances of 86, 1faO, and 1500 UL froc. the target. With increasing distance from the target (longer flight tin,e) the nun.ber of highly charged recoil ions i.e. the nun-ber of excited rotational states cecreases. Only rotational states still excitea after passing the carbon foil lead to high charge states. The large yield of nich cnarge rscoil ions at small distances is gradually shifted to the equilioriua chsrge peak for large distances. For further analysis we consider the percentpge of hi^n charge recoil ions plotted in fii-. 16 as a f mction of tne tercet-carbon foil distance. Using tne Vnovin recoil velocity the distance scale is converted into a aecay tin.e. Tne short lived aecay is attributed to tiie de-exuitation of the grounc-state rotational Dana in Cm wnile tiie long livea component v.ith a relative intensity of 10 io is sue to tiie delayed feeding of the rotation:; 1 bano. tnrough tne decay of a longer liveo. spin isomer (10 ns^T^,„6.100 ns).Tiiis contnoution to txie charge aistribution cannot be a background effect since tne charge resetting' is effective to better th^n * as discussec above. *• The curve v.uich describes the calculated ae-excitation of the rotational bana tnrcugh consecutive E2 in-band transitioas is tne result of a least squares fit to the experimental points, v.'itn unov.n conversion coefficient oł unć transition energiesii-the rotational haiflives

fcinu tiius tne t o t a l ae -exc i t s t ion t i ae are (witi.in tne rotati„riL-. n.uael) only a fiJiiction of one parameter - ti.e qu&arapu Leii.cii.ent-wiiicn iLa - ti.us ce aeaucea i'roi.. suci. a l i t . Tne resu i l of Q =(1£.C _+ 0.5) i s in u,ooa agreement witi. tne syster.fit- JS

1 of ground.-tate qaacrupolfcii.ou.entp in tije actiniae region in fic. 17.

505

1J, CaargL uistrzoation of 24OCJŁ ions from tae 2'3Pu (rf.in) reaction Hieasured for various distances of the carbon foil to twe terget.

f

u J S Ł .

u

•1

\

1

. - - .

i - . . . . . . . .

•fig

506

240 1t> Fercentf>ge of i^igtily ctiargea Cn. recoil ions as a function of ti»e distance between tBrget and carboti foil . Tee curve is a least-BCUpree f i t of a caecoue calculeticn to trie u&ta points jrieluing a nuadrupole EiOtent of (1Ł.0 i 0.5)b anc allov.in,~ in aaoition for a contribution fror:. a 1-ng livea isou.eric state.

20

12

10

8

6

-

Th

T I Ro

Ro

Th

I Ro

I

U

i Th

I

I I Th

U

i u i

Pu

f U

1

Cm

ł

ii Pu

I

Cm i

Pu

1

CmCm

i 1

i

i

T i Cf

-

•

-

1

230 240 A

250

- p

260

?ig. 17 QuaarupuLfcir.'jJi.eiitG of actiniae nuclei in tneir groundstate aa a function or' r.ass nur.ber. Apart fron. ti.e v ine for

CŁ tne quF.arupoltmoir,eats are aeaueci fron. i(E^) values of the 2+ -» 0+ transitions ana t&tcen froii. tne compiiption of re i". 14. )

tłlPuM.JnI J»Cin \

Q.«T2b I •omgnc contrtMtkm flubctradcd

.Kt '• •

\ I :

\ j

(Wane* tagit-carbon Mlljml

1b Decay curves for ti.e contriDUtions of inaivit;Uiil 24C rotHtional lev&ls to tt.e cnr-rge uistrioutioa of Cn.

recoil ions.

507

By unfolding trie charge ciistriDution measured for several distances of tne caroon foil to tiie target into the com¬ ponents for 1,2,3. and n.ore consecutive converted transitions tne naiflives of tne inaiviuual rotational levels may be ae-tenr.xned fron. the decay curves of tne respective contributions to the cuarge distributions sn v.n in fig.- 18. The curves are again the result of a cascade calculation with! halflives of 154 ps, 71 ps ana 41 ps for the 2 +, 4 + and 6+ states wnich are consistent with tne quadrupolecoiLent of 12 b. Tne half-lives of tnese states are obtained taking the usual corrections for siue-feeuing ana in-band feeding into account. SunjLarizing, with this experiment on Cm we have aemonstrateu tnat reliaDle results on the quaarupoleiLOiiient ana the half-iiveti of inaiviuual rotational levels may be obtained with the charge plunger technique.

lie The ouaarupolen.on.ent of tne to pa fission isuaer in J Pu After this auccesful test the technique has been applied to a study of the rotational band feeding tne fission ison.er of -^Pu proauceci by a {«t,>n) reaction. Tuis isoi.er in an odu-even nucleus was Belectea as a first case oecause of its convenient naif life (T-j/p = 8 is) and its large proouction crors section ( (5a •» 20 UD) despite of tne oovious aisaavan-tages in the aata analysis which will be aiseusted later. The experimental arrangement - similar to tnat of fig. 12 - le suawn in fig. 191. After passing the charge-resetting foil in distances of 15 pm up to 5 BJU anu deflection iu a E.agrietic field tne isoiiieric recoil, ions are stopped on tne ooiique sections of a detector arrangement. To clarify tne geoceiry fig. 20 shoTi-s cuts tnrough the set up in forwara ana sice-wara direction. Since the flight tir.e of approxiniHteiy ICC r.s is short con.ps.reo to the isoaeric haLflife of a ps n.o::t of tne fission oeca. s occur on tne recoil ion coaectjr nr.u are registered v?itrj fission track: detector foils wnici; ccn.-pletely cover Zue uetector arrangement. Tue Horizontal uipk is siiieldeu froŁ pron.pt fission events in trie target an: fracn.ents froc. fission in rlignt so tnat even at tcrwarc angles only delayea fission frugniente are cetectec. Tim fission tracks are n.aae visiole to tne unŁiuet eye ueing xue

508

2*1

detector foil

H- 5 cm

Pig. 1^ Experimental set up for the measurement of onarge distributions of fission isoaeric recoil ions as a function of the distance between target ami carbon foil.

fission tsorrnr

fission fragments

fission tsonwr

1 cm

fission fragments •

dtltctorfoil ?

cut in forward direction

cut in

stuvwmu one iion

Pig. 20 Cuts through the detector arrangement of fig. "ly in forward ana sideward direction.

509

Xi.-:-, spsrk scanning technique. At forv.aru angles xhe horizontal •jis,; is spl i t into two parts to provide an outlet for the

oL-oeam. Pig. 21 j is a photograph of the experimental set up ana snows tne pLunger arrangeD-ent in front wnich contains the stretcnea target ana carbon foi l . Tne flight paths of is„r.eric recoil ions are indicated.

Tne charge oistrioution of fission isomeric recoil ions n.easureu for several distances of tne caroon foil to the -target are aispiayea in fig. 22 . The observed fraction of nigh cnarge ions i s plotted in fig. 23 as a function of the flight tin.e. Tne snort lived decay is attributed to tne de-e>.citation of the rotational bsnu based on the fission i s ta te . 'Iue long livea cor;.ponent is assignee to tz<e aelayea f'.;euing uf tnis rotationrl bana fron. the decay of wn hithertc uf.'./jo'.vn ; /is spin isoc.er locnti.c li. trie second itiinin.ui of c-jJI-\i. In the Lower part of fig. 23 the short lived decay is j/Lotteo verRiif -m extended tiu.e scale after substruction o:' lue lorjg livro cotuponent. For comparison the rt-pulte of caecaue caL ;ul;tiuns for tne oe-eAuitetion of tue rot&tion;-l Dni.u arf ^ivtn ;«,c;ain a.-'nui;.ing pjre E^-tr^nsitions wit;,in the Darju ana a ąuharupoLenioii.ent of 'b c. For hn oau-ev&n nucious as 'Jl-u tr.e rotcational lif 'tii..es are, novever, not only £en::.itive to tne -.u;iuru;.,o Len.on.ent but also tc the K-value or' tne banu (equfl to tne spin of tne Dana ne&a) au:j even i:.i.-re to ti.e tii:.ount of M1/E2 admixture. Since experimental im\,:T..c:tiou on Uiese quantities it; not'yet avril^uie lor the rotationć-1 Dyna in tne seconc n.init.iui, of Pu the 1.1 strerjftt

lias oeen consiaerea v-it^m tne rotational Kout L for tv.u l i c i t m c situations ihyrf'OtGrized by paralLel anu antipar^l i t l coup Ling of t:ie orbital ana spin angular ir.on.enta. By f i t t i r ^ tne corres-pcnainR c." L julstt-.ci Qe-ex j i t s t ion curves to the t;xptrxi;.f utel decay <. iff o re tit vuLues for t:.e quearupolen.on.ent i-.n- cecucea ;inu Lis-teó if. ttsbio I for K ^ 5/2. ri.-. Lw; rives ti.-- result of c.n unfolcin»{ anf-lyt=ii ix. v.i.ioh t:"Cj ucjay curvir." ^•:' t:.c various contrioutions to ti.e „t.;-2\~-? uistributi^n ax-c- ••ri: iyccj5 sepcirately yieiuiii^ n;.i:'liv%E of (*'ć ± ;)pa r . (c _+ 's)ys, ana (a _+ 2)ps for tne three lov.ect rotatiun:>l Ptf-.tes. Tnis resalt excluaep K = 1/2 &na 'i/?-banes

510

i -ii-..t..-r :i.u of tt.e pli.uii.;i.-r ana u e t s c t J r a r r a n g e m e n t . The t r : . j>:^t3i -ii-n of tut: f i s s i o n i owners* and the be? • l i t ' ,:LL'. .n n r L tu: i • ::i t o i l .

•» I . .UM

no

u

tot

•0

DO

no

100

m

1 1 1

\ MX

, Y . . - ^ ? w » . . . . turn

•10 (im

1

W^^^—. o" tf »" w *o* o* nr nr 90' 40'

ftw|> ttaia

Re dint ri onti^ns of fission isoiiieriu recoil ions ii.eur-urea at ui; feretit aistEinces of tłie carbon foil to ine tai-fjot and Decomposed into thu contributions from RPV r:>l couvcrteu transitions.

WO 300 500 2000 4000 distance target-carbonfotUMml

so no dwtance target-carbon foi t pml

23 (a) Fraction of uigaly ciiargea recoil ions as a function

of the carbon foil-target uistance. (b) Fraction of

hit ily onarged recoil ions corrected for the contribution

of the 3 ne isonier. The theoretical curves represent

cascade calculations for different K values of tue ro¬

tational bana anu a qm-drupolecodent of 3t>.0 b

512

" *U (O .3n} 2 " m P u

100 dtcoy time [p«J

Pig'. 24 Decay curves for tne contributions of inuiviauf.1 rotations I levels t^ trie crir-rge cintrioutior. cl' fission isoŁeric recoil l^ns. TLe curves are tj.e result oi cancnot calculation.

since for tiiese K values the lowest transitions are sc LOW in energy tnat L-conversion oecun.es energetically inpossicle, tńus prolonging tne decay tin.e b;,r at least a factor tr r -e o ir.pared to tne preceeain^ transitions in contrast to C D -servatior..

The resulting cuearupoieuion;ents of tsole I between 24 - ;o D exceed by ±'Ł r tx e value of {11.0 ± 0 . 5 ) D nea?urec fcr xi.e groandstates of ^ ^ ru ana ^^'"Pu, ana are in gocj a.nreec.€:.1 y.ith tneoretical estimates 'based on trie Strutins<y crcceaure. If tne Sflap« of t.ue nucleus in Xhe is^n-^ric sts.tt is uescrioea oy a prolate spneroid the deaucea v&ii.ef of tr-e quaurupcleii.o-nient correspond to axis ratios 1.7 *o/a 6. 2.0 -f.uicn are -irrespective of the systematic uncertainties aiscussec pDcve-larger tuan tne crounastate uef'-rn.atior. of c/a = (1.3 +. 0.0^>). Ti^us, tiiis experiment proviaes tae quantitative proof tnat fission isomers are euape isomerE as suggested in tne picture of a double nuaped fission barrier Dy V. Strutinsky.

513

In oraer to avoid tne complications in the aata analysis

encountered for ode-even nuclei a modified enarge plunger

arrangement using electrostatic fields is preeentl/ tested at

Heiuelberg for measuring tine charge distritmtious of even-

even isjji.ere whict. are unfortunately all too short lived for

ueflection in a magnetic field.

table 1 experiment

2 3 8U(«,3n) 2 3 9 mPu, T1/2-8us

QotbJ

k-T fl-A-1/2 fl-A+1/2 without Ml

5/2

111

9/2

11/2

24b<C ground

238 p u

240Pu

theory

36.21 2.8

33.01 2.5

33.8 t 2.5

34.0 i 2.6

lo< 36b ^-state

32.9 13.0

24.7 t 3.2

23.913.4

24.1 i 3.6

-> 1.7<c/a<2.0

Qo-(1!0iQ5)bl , Qo . (1Ut05)bjC / a ' ( U t 0 (

fission isomet

23Bpu

240 Pu

ref.17

37.6 b 38.2 b

ref.18

343 b 35.0 b

36.41

32.91

33.3 1

34.2 i

)5)

2B

2.5

2.6

2.6

I I ó. The quadrupolenioirient of the 40 ps fission isomer in

Por completeness let me mention a totalLy different approach to Łeasure the quadrupolemoment of a fission isomeric state pursued by G. Sletten and myself at the Niels-Bohr-Institute. Hotational lifetimes are inferred froa. the branching ratio for spontaneous fission and electromagnetic decay of rotational levels in the second minimum. The method requires an isoner with a very snort halflife so xnat rotational de-excitation ana fission lifetimes are comparable. Since the Ei.-lifetiu.es are again expected to lie in tne picosecond range (fig. 25 ) the oest candidate is tue ison.er in • Pu with a fission half-life of (37 ± 4)ps. Depending on the size of the quaarupole-znoijent fiscion can more or less favourably compete with the electromagnetic decay of ine rotation;;! states.

\

\

Q0«35b.

V

3 in

\

\

W

12

33

6*

*•

» 30 50 Ouodrupoł* moment Qolb)

Fig. 2b (a) Tne rotetionel brna on tue fission isct..eric sv i t of 2^bPu. Partial halflives 1'cr (s'n-.czroa.i-.p,nr. t ic uecay are calculeted v,itu tue rotsti^nHl koaei esi.-ic.it.. a quedrupolemon.ent of 35 o, (b) Ti.? yieia rr-tic of fission fron. rothtional levels to ti.at of trie C+ ctutc calculated for cifferer.t quaurupolec.orente.

5«5

On tne right siae of fig. 25 the fission yield from all e>cited states relative to that from the O ground state is plotted as a function of the quadrupolemon.ent. The larger the quadrupolemoment the faster will tie the in-band E2 transitions ana the smaller is the contriDution of excited rotatioru-1 levels to tae observed fission decays. The two components in the fission yield may be separated by measuring tne angular cistrioution of the delayed fission products. Frag¬ ments Iron, the spontaneous fission of rotational states v ith B ir.iv.jn spii, I are emitted witn a characteristic auculnr t;i;:tribution rel;'tiv» to the bunm aais, provideo trie filigna.c-nt oj' tjiesc .'-t'.tec oDtninea in tiie nucleiir reaction is pre--

i ouriu,; theLr lil'etiiiie v.hicb has been confinted in a U' oxperin-ent . Tiie loi.ut/t 0+ state oecaye isotro-

picL. iiy, u.<t tnę uiąner e;An rtatct c^ntritiute increasingly lu mi &ni putropic pattern to the extent that tijfy unucrgo l'i: ilon. Tije renuiting net anieolropy v.nicn c<>n De e>puriii,-int''lly is tj.u^ niamLy & function of tne ii ,; erjt txircu^i; tne branci^iiig ratio betw«f n ES- trć:nailions Hfiu spontaneous fission. Pig. ?b enows the t.easureo an./alar ai :p lnujt iuc oi oeiayea fission fragments in coapen^on witn CLilcuLateu aistrioutions assui.ing ailferent values of lue juaorupcieniu;..fynt. 3est pgreeaent nitu ooservatior. is found

+14 for Q = J7_' y b, v.nicL is in fair agre-.ment v.it^ t<.e result lor •'JI"Pu ootained witc the charge plunger technique.

516

2.5

2.0

15

1.0

1 ' • 1 • ' 1 ' ' I

6

0 Ib] &

. 3 7 '

\ 20 *0

T •

—

•8 b

j .

to

/ 1 / /

• 1 ' ' 1

/

/

i i

) -

/

V

I • . I . • I • . I • • I i • , 1

0* 30* 60* 90* 120* 150* Xtff

Fig. 26 ExperiiLeiital points of tee normalized angular

distribution in comparison witn calcuLBtions for

three values of tne quadrupolemoment. The inset snov.e 2

the X. -distribution of least squpres fits to the ex¬

perimental data.

517

III Speetroscopy ot low lying states in tnę second minimum The analysis of tbe charge plunger data on v Pu has shown how bauly some knowledge of tne single particle states at the defonuation of the second minimum is needed. Such information by itself is valuable since it sioula allbw a crucial test of the single particle models underlying the Strutinsky type calculations. These models have never been checked so far for the large deformations of a fission isomeric state.- There are mainly two possibilities for identifying 9ingle particle states in tue second mininaun: either by measuring the g-factor of tne fission isuHKric state or by directly observing the rotational transitions populating the fispion iaomeric state in an ouu-even nucleus. Both approaches have been pursued at Heidelberg.

Ilia The ^-factor of firgion isotteric states The pioneering work on g-factors of fission isoihers has b*en done at Copuouagen. Kalish et. al" investigated tue i fission isorters xn 2^'Pu with halflives of 110 ns and 1.1 fie. The

finje of their d: ta is, however, not completely clear because of sotie aifficulties in reproaucing tne reeuits. Fig. 21 sriov.s hov,, at least in principle, tne spin of a fission isomeric state may be inferred from the measured g-factor.pro¬ vided it is nearly a pure tingle particle state. The sign of the R-f:iCtor characterizes the Nilsson orbit an to wnetner the orbital and spin angular momenta couple parallel or anti-parallel. The experimental technique also usea tt Heidelberg and illu¬ strated in fi*;. gb is to measure the time dependent angular distribution of delayed fission fragnents when an external magnetic field B is applied perpendicular to the plane of the detectors and the target. The exponential decay observed in the intervpls between the bursts of a pulsed beaa is moaulated by twice |the Lannonprecess^on frequency w^ = gB u M /% which determines the g-factor. The main difficulty in these experiments is to preserve tne alignment of the nucleus ootained in the nuclear reaction during the lifetime of the ison.eric state. If the alignment is destroyed througn hyperfine interactions in the target the angular distribution will become isotropie and the modulation of the decay curve disappears. Two techniques

518

0.8

2 0.6 u £ i 0.4

0.2

-0.2

-0.4

-0.6

j _ I 1*1 1*1

-

n

-

-

-• i i i i

-

= A-1/2

-

"

-

A • 1/2

-

1 i

V2 3/2 5/2 7/2 9/2 (1/2 13/2

I

Fig. 27 g-factors of single particle states inceforaeu cad-neutron nuclei calculated r'or parallel ana ant i -parallel coupling of the orbital anc spin angular momenta, respectively (from ref. c) .

fission counter §1 N(t)

B o

a. Z

Fig. 2b Experimental arrangement (scnecatic) for neasurinp

g-factors vnitn the spin rotation n.ethod. On tx.e ri .-.t,

the modulation of tne decay curves is iitustratet..

to preserve the alignment have been tried. As indicated in

fig. 28,' fission isonieric recoil ions were, in the one case,

implanted into a cubic Pb lattice in the hope that at a lattice

site they will not experience strong electric field gradients^

thereby avoiding attenuation of the alignment through inter¬

actions with their large quadrupolemoment. Recoiling into

crystaline Pb does, however, not exoludeVinstantaneous dis¬

locations due to the stopping of the recoil ions and cumulative

radiation damage from fission fragments. The other approach

has been to use metallic U-targets heated up to 1000° C close

to the melting point. At these temperatures the 0-lattice is

transformed to a cubic configuration and, furthermore, the

disturbing quadrupole-interactionsshoula be diminished througn

a diffusion of the radiation damage sites. Applying both techniques

the spin rotation patterns shown in fig. 2a nave been observed

for different magnetic field settings. The difference in counts

between the two detectors divided by the sum of counts should

in first approximation give a simple sinusoidal time dependence.

The only statistical relevant oscillations observed at 1.1 KG

for a U-target of dOO C could unfortunately not be reproduced

for other magnetic field settings.

At present, attempts to decouple the disturbing quadrupole-

interactions by applying a longitudinal magnetic field are

pursued.

Summarizing, one has to adniit that up to now there are no

reliable and reproducable experimental data on g-factors of

fission isomeric states.

Ill b Conversion electrons from transitions in the second

Spin isomers located in the second minimum which feed the

rotational band on the fission isomeric state with some time

delay open up the possibility to study these rotational tran¬

sitions using the shadow method which has been described in

detail at this school oy Dr. Backe.

The charge distributions of 2'yPu fission isomeric recoil ions

measured with the charge plunger technique at large distances

of the carbon foil to the target have revealed the existence

of a 3 ns spin isomeric state,(see lie). Its aecay curve is

plotted separately in fig. 50. The decay of this isotier into

520

238,

- U/Pb sandwich target 20°C BU560G

0.2

0.0

-0.2

• 1^0.0

-0.2

0.2

0.0

-0.2

, T% = (8.QgQ.S)jis

U-metallic target 20 C BUlikG -

U-metallic target 1900°C B t=UkG

U -metallic target 900°C BJ=600G'

1 I

IT 5 K)

time Cps] 15

Pig. 2y Tin.e-aifJ'erntiftI spin rotation patterns octainea for

the b jis fission ison.eric stnte in Pu &t several

magnetic lield settings using two aiflerent tec.

for pi'eserving tiie nuclear alignment.

Ztrfallsztit

•Pit;. '$Q l)c-cay curve of n spin ieut.er feeding tixe rotational DHUQ on ti.e t'iseion isoaeric state of CJ3* witii tiiK cuarge plunger teciinioue.

JPu, n.easurea

delayed fission detector

beam Pb shielding coils of solenoid

Si (Li) detector

target orbit of an electron

catcher for recoil ions nucleus in flight

31 Experimental arrangement for studying electrons from

tbe decay of short lived (1-30 ns) states in the

second minimurc..

522

2^u the rotational band on the Pu 0 us fistioning isoner has been stuuied as a first case in collaborstion v.ith Drs. Backe ana Richter using tneir solenoia electron spectrometer with small modifications as shown in fig., 31 . • Pu nuclei excited in the 3 us state through the 2 ^ T J

(o(.,3ii)reaction recoil from the target and tne conversion electrons emitted in tneir aecay in flight are transported through the magnetic field to a Si (Li) detector v.iiicii is placed in about 40 cm distance from the target in order to reuuce the Y~ a nó x-ray background. The recoil ions - then in the 8 fis state - are stopped in a catcner foil down-stream ana their subsequent fission aecay is registsred vritn an annular detector. Geometrical shielding has boen exploited twice in this sot up. The electron detrctor is shielued fron. ^-electrons by placing tne target 1. 5 rsx up-stream una the fission uutector is shiclceci from prompt fission events in the target by mounting it ineiae tne t,ole of the annular detector. A spectrum of electrons n.easurea in aei:yod coin-ciaence witi. fission is snovm i-. fig. Yd • In two inueycnuent runs lines in particular at 11,13t ana 1J ^eV nave o-en re¬ produced decaying ''-ith a half life of T-w^ = (7-b +. 1.5)/us ir. agresment witn t^e b ps haiflife of trie fig: i^n isa.er. Ihe quantitative analysis cf tx;is rpi.-ctr'UL is rtiii in progress witn tte aic *" i) to establisn a rotational, at^ay scuzme ii) to determine tne jnultipolerity ana thus trie M/Ł";- i;.i/.i:ig

ratio of the transitions t'run tue relative intensities of the conversion line8

If at least two rotational transitions are identified trie moc.eiit of inertia of the Dana &nu - even tio:e iii.portsnt - tr.e spin of the fission isutier can be uniquely detera.11.ed. Pig. 33 snows tne energy spectrum of ulectrons r.ttocttu in coinciaence vitn oelajea fission in tne * °U (ot, ii) retiction. On top of consiaerabLe oa^Kgrouna cair.li uue to coi.vercicn electrons from fisf-ion i'r£iSL.er;ts electron lines arc oDservea at the energies earlier reported by S^ecnt et pi. Tut. •'- -»2 ana possibly b iso trie ó + -s<;+ transition or txie rjte.tior.al bsna on tiie fission lsoceric state in Pu are ooviousiy fee in the decay of a hituerto unknown spir; is^r.&r. Tne lniertftiu.;

\i [o.3nl "'"Pu

80

-20

*i—i—i—i—[—i—i—c—r—i—i—i—i i i—i—i—i—i i i i i—i—r

1

« I w 20

\ '

Hi'""" : \ N

N ... r -U 2S

ilxil

-f-1—i i l | i l

10 20 30 40 electron energy IkeVl

Pig. j2 Energy spectrum of delayed electrons measured in

coincidence with delayed fission in the irradiation

of 25bU with 33 L'.eV ^-particles.

Fig. 33 Spectrum of delayed elctrons from the

reaction observed in coincidence with delayed fission.

52*1

feature of th is spectrum i s tee line at 1 ij keV which Lay be interpreted as the Jl component of the 2+ -» 0+ t ransi t ion not ooserved m tne ."'unica experiment because of severe back¬ ground from £-electrons. The iaentif icat ion of tne 2+ ->0+

t ransi t ion v.ouia completely eliminate tne alternative inter¬ pretation of tne lines observed by Specht et al as rotational

t rans i t ions v/itnin a bane, en a K-iso&er in the f i r s t mini&.uii 21) as suggested by Grecnukhin '

Tnese two exar..,les show tnat witn tr.is type of experiments in teres t ing and important Lpectroncopic informstion on low lying s tates in tne second nAuinam will ejijei-ge ^n tne future.

IV Fissi .n carr ier nei :.ts fr. i. fl-ct layec fiFEiou

After discussing problems reu t< o to ti-e uefcn..ation of tr.p si.aue lswiLeric states ti.e reat of the iectjre will w? WJ-voted to ti.e question of ti;e ourrier i^-oi^nts. "'r.«se (.ave ^eneraiiv been aeteni.i.iea oy n.ensur:.tu-; t'lfsiur. pri. b i ic i i in^s as a functiot. of e..jitaticr. energ;, . i f resonance s t r i c t ires occur due to ni^x.ly excitea vion-t i .i.ai stat- a m ti.-.- secr..o Biiniii.ui. ti.e analysis -f sue.-. ar-< tfi^v.ni_x. 1 an, not t-oii.i; '.J review teri-. yi-,lu3 i;ji'cn:.utio. --i jotn ocirri'.rs. i t ;,as ct -r. noteu tut-t tiie i.ei^nts of li.e n.ner oa r ru r s ii, GeteriMiie ; ir. tiiis wi-.y sr.ov, a sybte:..-.:ti'j aeviLtic:. fro-. c-Lcun-t a « [ : e s , psrtioalrjr-L;; evii.t:.t tor Ta is - t -pes ( "Ti.-arion :\ LV " ) . T:.e li:.ite^. ii.ic.cer o: avaiLf: ui.e f :.-• t s I.ŁŁ, Ł.-V.\ /c-r, so ::.r res t r ic ted t a i s ^oi.L-ariaon tv IU; wi f IU;.K ir.e t:t •u i i i iv l i . .e . In "ci e fciLOWiiig 1 v.i L 1 uiscu.iS a uev. net,;.'-'- to uerive :i.~fi^E barriers of n-iclci far off tue i.tóciLity Line by sir.i Le jros? section n.easureu.ents exploiting fissior. fuiLcwi.g eie.tr^:i capxure uecay.

Tne principle uf tne ro-uv/ n.etnoa is expis inea in -if,' ;"4- A neutron deficient &.cti.-:ide nucleus witi. ,..ruti.n :iui. oer ^ anu ru-utron cur.Der '."decays vic. electron capture oeoay into u,e aaugŁter nucleus (Z - 1, ii + 1). ~t?.t'-s u> to a;i e.'.oit^tivii energy ,- iven by tt.e Q.-vr: LU6 ii.i.;as L;.e K-uJ...-i.ri.: er.er y ure pop^i.ateG ana n.aj either aecay oy Y"-e::iSBicn evt,.tuauy t~ tue gr^un-st&te or v.itij a certain prooabij.it;- ai?^. ;;y :'issi^.... In part icular , for aoti^iae HUCLCI &o--ut 1C - tui r . t i s av.- ;, :'r_:.

525

the stability line t'ne Q£-value becomes cou,pars>bie ?:itn t ie

fission barrier heignts so tr,at the fission decay gets nore

ana more important. The time scale for such a fist-ion process

is governed DV thi electron capture decay with halflives

of soi:ie sec to ain Wulcu i s long comparea to the suDsequent

;,roL.pt or even aelayea fission fron. the isoaeric state.

Tiie proDPDiiity for fission foliowana eleertron capture aecay

can be determined ex^erit-eutaJ-ly by' measuring trie cross section

for tne ueL&yeu fission activity ana for the electron capture

uecay of the precursor nucleus. On the otner hana, this pro-

DaDility may ot cHicuiiJtea anu is I'ounc to oe a sensitive

I'uuction of tXie fisaion carrier ,aran.ettrs. T'je jjioauct of

the integrated Ferai function f ar.u the ^-strengtn function Ś& accountB for ttje pupu lotion of \tu> e/.citea states ana XL" ffctor

factor |—t- aescriDeB \i,o pj-r-.Dubility for t:.eir fisrion decay.

Tuene tertiiB h!<ve tu \J>I integrated ever tue full accessible ex-

energy ruurv unu <jivi<ien Dy a nom.ali^ation integral-

The n.ain ewrry i;f j/'.'iaence of tnę interrana shov.ii on the

ricj.t raae if '. ac. 34 atoms fron. th^ exponential variation

of ti.e ,.iutu • .. for s^otorrier since the

VlMtV] Spottwng noch Elektroneneintang

E

IZ-1.N-U dtforantion

j /lla,E)S,IE)^M= dE p . OECF . o , 'fit Ó£C 7><Q(-E)S,(E) dE /(Q,-BK-Er - j ^ dE

PEC.F- zp^ = f (a, .E j.hui )

Fig. 34 Illustration of fission aecay following electron capture.

526

f Ff exhibits only a siiifil variation witii excitt.tion energy. Furthermore, to simplify the analysis tne A-strength function may De considerea constant for neutron ceficient heavy nuclei above a Kn v.n cut off energy (J (c = «:t>/V A*KeV) whicn seems justifieu by the results of tne ISOi-Lh «;roup ' . Since an extrapolation of measured barri-r iic i,:nts to neutron deficient Fu isotopes - to oe discussed in the foŁi^.-.ing -clearly indicates that tne inn&r barrier is ui;;uer ti.ar. tue outer one the folaeo. fission proof-unity p£ctt essentially a function o;uj f ti.e Q -v:-tue, of tne inner carrier E, ,

- v a l u e s a r e r e t i a o i y an^ i t s curvj ' tar<? r e a i c t i ; u oy

r:- r.;;y ^ w « -,&:r for iLutae v . i t . i.>;

, Dy about _+ 200 keV and curvature en&rnes! li W s t p n a a r d t e c l j ; , i q u e s f o r n u c i l i : 1 .1.1; U.a e t>Ł/ i l i t , v 11.-if? ao n o t vpry c o n e i o e r t i b l ; / by i:.ort tnarj 4; 100 >.&V. I I J I E LH'-'IIP ttic- h e i ł c u t E, o f ihr> i n n ^ r D y r r i c r SIB tj,>- or.lv fr'."-? ut-ru-n . e t e r t o oe ae te r fMi ico froi *Lx;o n,';ai-:u r ea : i : i on : r IJ'-DJ a f o l l o w i n g eLpc t ro r i cai t u r e u e c n y . Ixje s e n s i t i v i t y ol1 t ne //.r-tjjoo 3., 11 . u . - t r f t a 1;, : 1,;. Zb ••:. Biiowo c o n t o u r l i n .e l o r ti.n i i s s i o n p r ^ o . ' - u a l i t y a s ^ : j ; , u t o f E. anu Q^ f o r a iixea c u r v a t u r e - a e r , - ; o;' 900 K « 7 . Ł V n

E.IMeVl

OrlMf»l

Fig. 35 The probability of fission following electron capture decay calculated as a function of tne height E. of the inner barrier and tbe Q-value for electron capture decay assuming a fixed curvature energy of yOO keV. 527

a uaIflife 1\/z

tne experimental fission probability is only determined

within a factor 3 a barrier aeigot may be determined to

_+ 400 Jce7, allowing for tne above mentioned uncertainties

in % U , and Q g. •

Fig. 36 snotvH tne decay curve of a fission activity with

= (5> + 7) s and a fiseion probability

= (1.3 -oi^'^O~'i' * •aas been próauceo. by irradiating

*^'Hp witn 104 L'.eV J-particles of the Karlsruhe isochronous

cyclotron ana is assigned to the electron capture decay of J Aa folLov/&a by fiaaion of Pu. Proii. the measured Pfr.r ~

value a i.eigi.t of (5-3 i 0.4)KeV is aeaucea for tne inner

barrier of Pu. Usin;- puolisnea cross section obtained by

WiO lJuuna yr^up a si>;,.U-r ar.elysis gives a fission barrier

&f =(b'7 i O.V) Ui'iV for ^Hfn. '£,0X1. barritra exeeeu tneore-

by about 2-'i KeV.

10

i UJ

UJ O to

237 Np +104 MeV ot

TV2 = (55±7)sec

(3-=( 5±1)nb

so 150

TIME fsed 250

fig.

528

36 Decay curve of t&e activity assigned to the electron capture decay of ' Am followed by fisBion of 232Pu.

Pig. 37 gives a sumne.ry of all experimental inner and outer fission barriers for Th, U, Pu, and Cm isotopes as a function of neutron nun-ber including the two newly determined barrier heights. Por comparison static fission barriers calculated

27,2«) with different single particle potentials are also given. The heights of the inner barriers have been lowered by corrections due to ")f"-aeforination vcii.au are, nowever, insignificant for neutron deficient nuclei. All calculations reproduce the outer barriers fairly well, but systematicslly underestimate the inner barriers by up to 3 !• eV. This has previously be<:n noted in par¬ ticular for Th nuclei ("Tn-nTjoi;.aiy") but is cle&rly visiDle for U nuclei as well. Tr.e nev/ly determined barrier neiguts estaoLisn the at-.riis ait ration for neutron celicient Pu isotopes. Attntunts nave been Bif-ue to s.lve tr.e •'"Tn-a.-oriiLly" by intro-ducing a triple-hun.pea fissiori carrier c-'»-/ ;•.

o

c

i o o o o o o

""•—• " — - o «if»et rtocw A tWffltr hOll It!

T h O Hamtr »«c

Cm

o o c o

136 140 U4 U8 152 Neutron Number

Ł 8 w" 6 ,_ i ? 2

S 6

ś * 2

o o o o o i

Th «M>S<»o

Hoatt Martn Ołf Foldtd Sw>'-w«

Pu

Cm

136 K0 1U U« 152 Neutron Number

Pig. 37 JSeasured heights of the inner and outer fission barrier as a function of neutron number in comparison with theoretical predictions from shell correction calculations based on different single particle potentials ( from ref. 23 ).

529

Por such

a barrier fission isomers snould not be observable ainoe

states in the well between tae first and second barrier will

decay back to the groundstate by y—emission, while states in

tne ratner shallow well between the second and tnird saddle

are too short lived to detect tneir fission dacay. Fission

isorters have so far not been observed in Th nuclei - altncugh

photo fission data imply their existence - but a 37 ps fission

isoriier exists in ' Pu ae discussed above. Tnus, the concept

of a triple iiumped barrier seems to fail at least for the Pu

isotopes and, consequently, uoes not explain the general deviations

between calculated and experimental barrier heights found

throughout the Pu to Th region. Instead, tne discrepancy

appears to point to defects in the calculation of

fission barriers and, in auaition, rises serious aoubte

on txie predictive power of these calculations in mass regioris

further av.ay from the well stuuieci actinide nuclei as e.g.

cujjerneavy nuclei. Possibly, either »•• modifications of

the Stratinsicy procedure or new concepts like dynamic •in)

fission bnrriers 'may help to solve the discrepancy.

V Conclusion

Al'tur qualitative experimental evidence in support of the

aoublc nuciped fission barrier concept has been accumulated

for several years a quantitative proof cas oeen provided by

trie measurement of the quadrupolemon.ent of fission isomeric

sta.tes. While attepmts to measure magnetic moments of fission

isomers have failea so far information on single particle

states is expe^tec from electron epectroscopy of low lying

levels in the second mininnnL. Discrepancies between experimental

and theoretical barrier heights, now also established for

neutron deficient Pu isotopes, indicate severe defects in the

calculation of fission barriers.

Acknowledgements

All eperiments described in this article have been performed

together with Dietrich Habs who has stm^iy influenced the

research program of our group witi ingenious iieas. The

friendly collaboration with H.Baclce, P.Paul, //.Pedersen,

L.Richter, O.Schatz, P.Singer, G.Sletten, H.J.Specht, and

G.Ulfert, who have all contributed to the results presented

here, is gratefully acknowledged.

530

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?r,) P. Hornsh/ij, B.R. Erdsl, P.G. Hansen, E. Jonson, K. Alek-

lett, G. Nyman; Nucl. Phys. A 2J9 (1975) 15.

i-X) V.I. Kuznetsov, N.K. Skobelev, G.N. Flerov; Sov. J. Kucl.

Phys. b O967) 191.

27) P. Holler, J.R. Nix; Nucl. Phys. A 229 (197*0 269.

Physics and Chemistry of Fission 197J. Proceedings of a

Symposium Rochester, New York 13-17 August 1973. Interna¬

tional Atomic Energy Agency, Vienna, 197**. Vol. If p. 10'

2 u ) H.C. Pauli; Phys. Rep. 7 (1973) 35.

2y } S.E. Larsson, G. Leander,

Physics and Chemistry of Fission 1973. Proceedings of a

Symposium Rochester, New York 13-1.7 August 1973- Interna¬

tional Atomic Energy Agency, Vienna, 1971*. Vol. 1, u.17('

•JC) A- Gavron, H.C. Britt, J.B. Wilhelmy; Phys. Rev. C13

(1976) 2577.

31) V.E. Zhuchko, A.V. Ignatyuk, Y.B. Ostapenko, G.N. Smiren-

kin, A.S. Soldatov, Y.M. Tsipenyuk; JETP Lett, Vol. 22,

(1975) 118.

32) H.C. Pauli, T. Ledergerber

Physics and Chemistry of Fission 1973. Proceedings of a

Symposium Rochester, New York 13*17 August 1975. Interna¬

tional Atomic Energy Agency, Vienna, 197^. Vol. If p. 4c'.

33) J. H. :;ix Ana. Hev. Nucl. Soi 22 (I-J72) 65

34) R.Vaadenbosch,P.A.Russo, G.Sletten, K.Kehta,

rnys.Rev. C a (1973) 10B0

532

OH LIMB ALPHA SPBCTROSCOPY ON 1 GeV PHOTON BEAM FROM SYNCHROCYCLOTRON*

J. Kormlckl

Joint Institute for Nuclear Research, Dubna, 0SSB and

Institute of Nuclear Physios, Cracow

In the collaboration of the Joint Institute for Nuclear Research in Dubaa and Leningrad Institute of Nuclear Physics In Gatchina, the arrangement for the investigation of nuclei laying far fron the beta-stability line is put in operation.

The arrangenent is based on the synchrocyclotron of LINP and the mass-separator IRIS working "on-line" with l GeV pro¬ ton beaa from synohrocyolotron.

The purpose of the projeot Is to study the alpha decay, delayed proton emission, decay schemes and quantum character¬ i s t i c s of excited states of short living nuclei, laying far from the beta-stability l ine.

In the present talk I would like to report the experimental results of our f irst investigations concerning the alpha-decay of nuclei laying above the closed neutron shell with N » 82. Preliminary results of these investigations were published in the Proceedings of the XVIII Conference on Nuclear Spectrosco-

i a/ py and Nuclear Structure in Alma-Ata, USSR ' ' , Experimental data concerning the alpha decay of heavy

rare-earth elements were reported in many papers, mainly by group of R.O.Uaofarlone and K.S.Toth. All existing data were

lecture la based on references1' and 1' and

533

obtained in reactions with heavy ions. The assignment was

based on the excitation functions and in some cases addition¬

al y on the parent-daughter relationships*

In our experiment alpha emiters are produced in spallation-

reactions on Tungsten or Tantalum, and we have a possibility

to select Isobars with a definite mass number A. Having this

oportunity we decided to remeasuro alpha-decays of heavy rare-

earth elements at the some time serening for new alpha ealters.

Recently the similar attempt has been realized for Yb lzotopcs

on ISOLDE facility at CBHN10//.

EXPERIMENTAL ARRANGEMENT

Proton beam with the energy of 1 GeV from the synchro¬

cyclotron Is transported to the target room to Irradiate

Tantalum target combined with the ion source of mass separator

(Pig. 1). The products of spallation reactions Ta (p,Xn,Yp)

are ionized in the ion source and separated accordingly to

a mass number A by means of "on-line" mass separator tilth the

deflection for 55°. The separated ion beam passing through

a snitohyard is dlrrected to a collector in the experimental

room.

The ion source based on the surface lonlaation, developed

in Dubna by V.I.Ralko, G.Bayer, and A.Piotronskl allotted to

kepp the high temperature of the target (3000°K) *'. The high

temperature, due to the rise of a diffusion velocity, high

ionlsation coefficient and short hold up time give the high

yield and the snail delay In the ion source. The yield

curves of our ion source Hill be discussed further.

Ions having a definite mass number A are collected on

a aliminium backing of a rotating collector (Pig.2). Using this

collector and an alpha dedectlon system, both controled by

a small computer, one is able to determine the half-lives of

alpha emiters ranging from O.i sec to about 200 sec. The lower

IRIS EXPERIMENTAL

ROOM ROTATIWG

COLLECTOR1

Fig. 1. IBIS experimental arrangement.

RTG OR •£ DETECTOR

ION BEAM

Fig. 2. Rotatiag collector.

535

limit of this range la defined by a time needed to transport

the backing from Ion beam to tbe detector, and tbe higher one

Is connected with a repetition time of tbe rotating collector.

Tbe experiment Is fully controled by computer program from

tbe small computer M-400 (see Fig.3). Signal from tbe computer

starts tbe motor transporting tbe sample from i*n beam to tbe

detector. Then 4 aeries of measurements each consisting of

elgbt 512 channels alpha-speotra recorded as one 4096 channels

speotrua In U-400 computer arc started. After each of tbls

4 series, 4096 channels speotrua Is transferred to another

computer (EC-1020) and reoorded on magnetic tape. After tbe

storing of all four 4096 channels spectra on tbe magnetic tape,

tbe next sample is transported to the detector and tbe cycle Is

repeated. Collected spectra can be displayed during tbe

experiment on tbe sorean.

Tine of tbe collection of tbe 512 channels spectra In each

group, and tbe number of groups can be chosen and introduced

to the prograa from key-board during tbe experlaent. Time of

the collection of ions on tbe collector aay be chosen as smal¬

ler or equal to tbe full tiae of tbe storage and recording of

four 4096 channels spectra. For the transfer of the 4096

spectrum 1 sec is reserved. For one mass-number A, usually

several hundred of such series of 32 of 512 channels spectra

are collected.

Data handling is arranged with tbe help of HP 2116 C computer

with Tectronlz point display. 512 channels spectra reoorded

on the tape was summed in tbe order, first to first, second to

second, and so on, by the computer prograa using the code re¬

corded in first 10 channels to sign the speotrum. Resulting

32 spectra are used aa 32 points for half-lives calculations.

For energy determination the suma of 32 such spectra Is used.

In Fig. 9 where tbe sunm spectrum for A • 153 is presented, tbe

tipical energy resolution being 23 keV, and obtained using

10 mm diameter silicon surface barier detector is shown.

OD

DISPLAY CONTROL

EC 1020

"OMPUTER

Fig. 3. Scheme of the control of the experiment.

-10B

-107

-106

-105

-10* /

• 1 0 3

• in* 1U

i

YIELD

/ i

' ©

/ w

155 1 1 •

for Yb ^ ^ \

/ ai im /

/

» 3 t

— THEORY(RUOSTAM) K3« RELATIVE

—POWDER TARGET

I SURFACE IONIZER

160 165 A • i i i i i i i i i i i i

155

Fig. 4. Yields of Yb Isotopes.

53"

EXPERIMENTAL EESULTS

First I would comment about the yields .for tbe Ion source

with surface ionlsation, used in oar experiment. Tields were

calculated from the absolute Intensity of alpha lines for a

number of rare-earth elements.

Pig. 4 sbows tbe advantage of this type of ion source for

tbe study of short living isotopes. Tbe slope of tbe curve

representing the yields for the Yb isotopes is smaller than for

tbe ion source with powder target. That is most probably

conneoted with all processes responsible for a delay In target

and ion source. Due to that fact tbe absolute number of atoms

collected on tbe oolleotor per second in saturation for tbe

lightest Yb Isotopes are not smaller than on ISOLDE, though

in our facilities tbe masi of tbe target and tbe proton bean

arc much smaller than at ISOLDE . Obtained yields are in

agreement with the relative trend for spallatlon cross-sections,

calculated according to Budstan formula. Absolute yields for

Yb, To, and Er isotopes are shown In Pig. 5.

Results of the present investigations of alpha decays for

isobares with A-151 to A»157 will be discussed in order of mass

numbers.

A - 151

The measured energies fdr alpha decays of 151gHo, 151mHo, 151Dy, and half-lives for 151gHo and 15lBIHo ar» in agreement

with previously measured results (Fig. 6).

A * 152

Energies of alpha decays of 152Br, 1 5 2 " H O , 152gHo and 152Dy

agree with the values published previously. Half-lives obtained

in the present experiment are following: T4/j_ • 48.1 - 0.3 s

for 152Ho and T ^ , 9.7 i 0.1 a for 152Br (Fig. 7 and 8).

A - 153

The measured energies of alpha decays for 153Tm, 153Er, 1 eHo, 1S3"Hot and

i53Dy are In a good agreement with previous¬

ly reported values. The measured half-lives are following:

538

A/s

•KJ»

• 1 0 $

•m5

« *

•n3'

-10*

•X)

YIELDS

° y Q -X

/ / / *

/*

155

Er

"-" X. / / * /

Tm

% •

a -

Yb < ^

-Yb Tm

-Er

- RELATIVE SPALLATION CROSS-SECTIONS ACCROING TO Rt/DSfAM

tw 1 15 A

Fig. 5. Yields of Yb, Tm and Er liotopee.

CHANNEL 11Ó0 • " ~ " '2Ó0' ' ' ' 3Ó

P i g . 6 . Alpha spectrum of the i sobare with A-151.

33"

300

•150

•30

c

2

JJ

A=152

100 200 300 CMA'WU 400

« 7. Alpha spectrum of the i sol) are with A=152.

Fig. 8. Half-lives for isobare with A=152.

Ttyz - 35.6 - 0.2 • for 1 5 3Br and T</Ł - 1.65 i O.i • for 1 5 3Ta (F ig . 9 and 10). We do not observe any alpha branching con-

^ CO

neeted to the decay of Yb having 83 neutrons, In agreement with the same behaviour of other nuclldes with 83 neutrons. But on the other hand on the decay curve of alpha line with energy 4.105 UeV In 153Tm we see the long living ooaponent. Results of the repeated experiment with rlsed ratio of Yb to Ta iso¬ topes in the Ion source are shown in Pig. 11. The existence 153 of the component with T</Ł • 4.5 s in the deoay of Ta having T-y, - 1.65 s Is seen. On the basis of the parent-

153 1S3 daughter relationship of Yb and Ta this decay tlae we assigned to a /3+ decay of new Isotope Yb with T Vi - 4.5 - 0,7 s. The same conclusion Is found In the recent paper10/ A - 154

Energies of alpha decays for 154Yb, 154gTm, 1 5 4 ł Ta, 154Er, gHo, and Ho are In agreement with previous papers. We

measured following half- l ives: T*/i • 0»4 * 0.04 s for * 4Yb, Tt/i - 3.14 i 0.06 s for 15*BTm and T-yr • T.9 - 0.2 s for 154gTm. The last value differs significantly from the value 5 - 1 reported In reference ' (Fig. 12 and 13). A * 155

For Ł Yb, Tm, Er measured energies of alpha lines are in agreement with previous data. Half-life for Yb is Ty, - 1.80 i 0.7 s and for 155Tm Is H/, « 27.1 ± 0.08 s what

7/ 7/ i s in disagreement with earlier measured value 30 s ' ; See

Fig. 14 and 15. A - 156

The new value ot tbo energy of alpha deoay for 156Yb i s E - 4.687 UeV and half-live Is TA/2 - 25.1 ± 0.2 s . This energy differs significantly from the value E • 4.800 UeV

at

obtained by K.S.Toth and coworkora ' . This energy Is in agre¬ ement with the value known for alpha decay of Ert which we

100 • 200 300 CHANNELS

Fig. 9. Alpha speotrun of the isob&re with A=153.

10000 5000

1000

153 Er

Ti/2=35.6!0.1s

YIME(s)

50 100

Fig.lOa. Decay curve for the 4.671 MeV alpha line.

100-

10-

1-

\

\ X \ \

153Tm Ęt =5.105 Ms Ti/2=1.65t0.1

TIME(S)

S

0 10 20 30 Flg.lOb. becay curve tor the 5.105 .'.'eV alpha line.

1000 5001

100 50

A =153

= 5.105 MeV

2 ,= 1.6510.1 s 20 łimefs)

Pig.11. 153Yb f mo decay eeen f or the A O O T« l i ne .

100 200 300 400

Fig. 12. Alpha speotruia of the isobare vltfa A*154,

100 _ 15iTm s

= 5.032 keV 3.U10.06s

2 4 6 8 1.0

* \ .

50

Fig .13. Decay curves of 5.032 UeV and 4.955 UeV alpba l ines froa 154nTm and 15*

•120 S

(Ul|

»2<5

_:

> ~

c

60

i

-,T.,.,^^.rV.-V

r 55

3.97

3.9!

.o.

5 ,

1

1 • w l l

• I

1 1 1

•'"I

c r*

r

5 d w O

£

**

/li

2 A=155

l l : "J 1 1

*

n\ -_ » s s _ ^ ^ d ^ * O i/>

:' s i 5

.2 - s L " /I 1 : . J i n

0 100 200 300 CHANNEL 400

Fig.14. Alpha spectrum of the 1sobare with A*155.

A00 300 200

100

50

10

s

\ IV

\ A

K

i 55Yb E« = 5.205MeV Ti/2= 1.80 i 0.07s

k 1

\ \ TIMElsl

100

so

10

A=155 1S5Tm E^ = 4.458 MeV Ti/2= 27.1 »0.8s

50 TlHETsl i m

10 20

Fig.15. Decay curves of 5.205 MeV and 4.458 MeV alpha l ines from 155Yb and 155Tm.

observed for this mass number with the characteristic growth period preceding the decay with the half-l ife of parent Yb nuclei (Pig. 16 and 17), Por this mass number «e observed also alpha decay of Tm in agreement with earlier data. Alpha line from 1S6mTB having Tvj. • 19 a 7^ la probably weak In the relation to gTa and we have not aeen i t In our apectra. In the ganaa spectrum for this mass number we found the line with energy E • 116.1 keV and TVa » 25 i t belonging either to 156T» or to 1MBr or " V A - 1S7

The energy of alpha decay for Yb agree with previously measured value, balf-l lfe T%. • 35.9 - 1,0 a la ellghtly greater than In referenoe ',

A new alpha line with the energy B » 4.996 MeV and Tji • 3.S a found for tbla mass number we assigned to the alpha decay of Lu, and alpha line with the energy E a 5.105 UeV to the daughter 1S3Tn which have T'/i - 1.6S s (Pig. 19). The shape of the decay curve for Tn do not contradict to this interpretation. The presently measured alpha energy for

Lu disagree with the value reported in . Concluding we obtained the energies and half-lives for al l

short-living alpha emitters of rare-earth eleaenta and In the forthcoming communication we will complete theae data with experimental values ot alpha to total decay ratloa.

Table I shows a summary of our measurements.

•2700

•1500

•300 •HK

8

"3 2 o ,,-

* i

vrr.»rw»"t ~y

A=156

2Ó0 ISo

Ui

Fig.16. Alpha spectrum of the lsobare with A-156.

30

V \

\

_ 156 y t

Ti/2= I

\

A=156 C.667 MeV

25.3iO.1s

Pig.17. Decay curves of the 4.678 ifeV and 4.800 U»V alpha lines f r o 186Tb and 152Br.

200

WO

A=157

100 3 0 0 CNANtitt • • 4 0 0 '

Fig. 18. Alpha spoctrun of tho I sob aro with AJ»157.

r l \J T

Ml

"TT

TT

! 1 1 1 1 1 t 1 1

4]

ID i i i

E< T1

\

A=157 = 4.996 MeV

2t57LuS

\ 20

L i i i J Is)

Fig. 19. Decay curves of 4.996 lleV and 5.105 MeV alpba l ines from 157Lu and 153Tm.

Table I

Nucleus!""

151Dy

151Ho 151Ho 152Dy

1S2Ho 152Ho 152

153T 153

153'

153

153

! 153

154

154

154

154

Er

154

154

155

155

155

156

156

157

157

Ho

Ho

Er

tra

Yb

Ho

Ho

Er

Tm

Tm

Yb

Er

Tm

Yb

Tm

Yb

Yb

Lu

.067

4.517

4.607

3.628

4.386

4.454

4,800

3.467

3.908

4.008

4.671

5.105

no ot

3.724

3.938

4.170

4.955

5.032

5.333

4.012

4.458

5.205

4.233

4.687

4.504

4.996

t 0.005

t 0.005

Summary of alpha decay data

^Presentresults

i 0.005

t 0.005

i 0.005

- 0.005

i 0.005

- 0.005

t 0.005

36.0 - 1.5

47.0 i 2.0

48.1 i 0,3

9.7 t o.l

I 0.005

i 0.005 j 35.6 i 0.1

35.6 - 0.4 a

47.0 i 2.0 e

2.3 - 0.1 b

2.36± 0.16 o

52.3 - 0.5 a

10.3 * 0.5 •

6.4 - 0.2 b

2.0 - 0.1 a

9.3 - 0,5 a

36.0 - 1.0 s

i 0.005 j 1.65* 0.1 i 1.58± 0.15 a ]

1 1 T

t 0.005

i o.oos ± 0.005

i 0.005 ! 7.9 i 0.1 j

i 0.005 I 3.14^ 0.06 }

t 0.005 1 0.4 i 0.04 I

- 0.005 I - }

- 0.005 ! 27.1 t 0.8 i 4.. ' . 4 . . _ !

4.0 - 0.5 s

3.25* 1.0 n

11.8 t o.l m

i 0.005

i 0.005

i 0.005

i 0.005

i 0.005 JL

3- 0.07

75 ± 6 8

25.1 - 0.2

35.9 i 1.0

=:3,5 8 1

3.75- 0.05 O 5 i i 3 1

. 1

2.98- 0.20 s j

0.39± 0.04 8 !

5.3 i 0.3 m {

39 - 3 s

1.65* 0.15 s

80 i 5 s

25.8 i 1.0 sa] 34 i 3 B {

- *} i *''Recently reported In reference '

HEFERENCES

1 / V.P.Afanasjev, L.Kh.Batist , E.E.Berlovich, Yu.S.BllnnikOT, V.A.Bystrov, K.Ya.Groaov, Yu.V.Yelkln, V.G.Kalinnikov, T.Kozłowskl, J.Kormickl, K.A.Uezelev, F.V.Moroz, Yu.N.No-v ikov , S.Yu.Orlov, V.N.PanteleJev, A.G.Poljakov, V.I .Halko, E.Rurarz, V.K.Tarasov, N.0. Sbchlgalev, Yu.V.Yusbkevich, U .Jan ick i , M.Jahln, Proceedings of the XXVIII Conference on Nuclear Physics and Nuclear Structure , Alna-Ata 1978, M-L, "Nauka", 1976.

2 / V.P.AfanasJev, L.Kh.Bat is t , E.E.Berlovich, K.Ya.Gromov, V.G.Kall innikov, T.Kozłowskl, J.Kormickl, K.A.itezelev, P.V.Moroz, Yu.K.Kovlkov, V.V.PanteleJev, A.G.Poljakov, V.I .Raiko , E.P'irarz, V.K. Taras ov, Yu.V.Yushkevlch, Proce¬ edings of the XXVIII Conference on Nuclear Physlce and Nuclear Structure , Alma-Ata 1978, U-L, "Nauka", 1978.

3 / E .E.Berlovich, L.Kh.Batist at a l l . IzT.Akad.Nauk SSSR, S e r . F i z . , 40 (1976) 2036.

4 / G.Baler, A.PlotronBki at a l l . , P r e p r i n t JINR PG-5761, Oubna 1971.

5 / H.L.Ravn, L.C.Carraz, J.Denlmal, K.Kugler, X S k a r e s t o d , S .Sundel l , and L.Westgaard, N u c l . I n s t r . and Ueth . , 139 (1976) 267.

6 / R.D.Macfarlane, Phye.Rev., 136 (1964) 941.

7 / K.S.Toth, a.L.Hann, U.A.IJaz, Phys.Rev.,C4 '1971) 2223.

8 / K.S.Toth, H.L.Hahn, U.A.I jaz , and ff.U.Sampl«, Phys.Rev.C2 (1970) 1480.

9 / R.Gouvln, A.Le Beyec, and N . T . P o r l l t , Abstract submitted t o the European Conf. on Nuc l .Phys . , Alz-en>Provence,1972.

1 0 / E.Hagberg, P.G.Hansen, I.C.Hardy, P.Hornsh^J, B.Jonsson, S .Uattson, and P.Tidenand-Peterson, Nucl.Phys. A293 (1977)1 .

550

QUASIFAHTICLE SPECTRA ABOVE THE TRAST LIKE *

JJ. Bengtsson, / . ~ ' f C •' '

NORDITA, Copenhagen, Denmark •'

and S. Frauendorf " r u~

Central Institute for Nuclear Research, Rossendorf, GDR

Most of the experimental information about high spin states

concerns the yrast lino and its vicinity. In most experiments

predominantly the yrast states are populated. In recent time

it has also become possible to observe the non yrast states.

The first observations of these states were made in odd mass nuclei

/see e.g. the review [2"]/ but very recently also the identification

of non yrast states with high angular momentum in even-even nuclei

has been reported [5]. I shall present in this lecture a theoreti¬

cal approach to the spectra in tbe yrast region, more precisely,

to noncollective excitations above the yrast line of deformed ro¬

tating nuclei.

In the yrast region almost all excitation energy is needed to

build up the high angular momentum. The level density for such

a cold rotating nucleus is relatively low. This permits to employ

the concepts that have been developed for the analysis of the nu¬

clear spectrum near the ground state. The angular momentum may be

incorporated if one baseu the analysis on the Routhian

H'= H -i3 J CO

The material of the lecture is published in Ref.fi]

551

instead of the Hamiltonian H of the system. The Lagrange multi¬ plier ensures a finite value of the angular momentum J. The Routhian is readily recognized as the Hamiltonian in a frame of reference rotating with an angular frequency t3 . The ground state of H'(u>) corresponds to an yrast state.

The natural starting point of the theoretical analysis is the Hartree-Fock-Bogoljubov approach to H', which is usually called the (selfconsistent^cranking model. There are a number of investi¬ gations of the yrast line using this approximation or closely re¬ lated ones ( see e.g. the review [2j), about which it has been re¬ ported several times in the history of these Winter schools. Also this time we shall hear the reports by Dr. Płoszajczak and Professor Szymański about this kind of approach.

We choose a simpler and forget about all problems with a self-consistent determination of the deformation and the pairing. :n

this approximation H' becomes the Houthian of particles in the deformed potential and pair field

h'= h s p - A S + A(P+ + P) - u*,x , f 2) where h represents the deformed shell-model Hamiltonian, in sp our case the modified oscillator potential, and P+ is the opera¬ tor of the monopole pairfield, the strength of which is fixed by the gap parameter A . The expectation value of the particle number N is fixed by the chemical potential A. . We restrict our¬ selves to axial shapes and rotation about the :-axis perpendicular to the symmetry axis. Generalizations avoiding these restrictions are straightforward. The operator

hsp • hsp - " j X (?) will be rc-fer<.-d to as the single particle Routhian.

The field Routhian h' permits the calculation of one - , two - etc. quasiparticle excitations, whose energies raay be ex-

552

pressed by the corresponding quasiparticle energies. Like the analogue approach to the low lying excitations (10 = ty, familiar to all of us, it does neither permit to calculate the energy of the zero quasiparticle configuration, nor to describe collective excitations. On the other hand, these restrictions lead to a considerable reduction of the computational efforts and permit to represent the results in a very condensed way, namely the diagrams of quasi-paxticle energies,which contain the information about the whole excitation spectrum.

In experiment one measures the energy 2(1) as a function cf the angular momentum I. These data cannot directly be compared with the spectrum of h', which represents the energies in the ro¬ tating frame as a function of the angular frequency C~» . Usually in the HFB-ap .roaches the connection is established by adjusting k> so that the constraint < J..> = I is fullfilled and transform¬ ing the energies :rom the rotating frame into th^ laboratory system by means of tłu £q.(1). We have used the alternative way, namely to represent the °:c,"erinestal energies in th' form of the 3outhian E'(LO) defined by

E' (UJ) = Z (co)- co I x '•^} , '• ~)

which is the energy in the rotating system. The use of <^> icsttad of I in displaying the experimental ent-rgies has become quite fa¬ miliar since the discovery of backbending.

The quasiparticle formalism

Before going on, let me shortly summarize sone relevant rudi¬ ments of HFE - theory /more details can be found in the theoretical papers quoted in the review [2]. The quasiparticle excitations are defined by means of the quasiparticle states that arc the eigen¬ vectors of the quasiparticle 5cuthian "V.' . tte write "K as a

553

matrix in the basis of single particle states f ©c j_ > t which diagonalizes the single particle Routhian h' .

The single-particle Routhian h' is invariant with respect to a rotation D x around the x-axls by an angle IT .

D h' D" 1 = h* j D = e~Li x*. (5) x sp x sp ' x Ky i

Therefore the eigenvalues £^Ł and the eigenvectors I0'-,,) of h" may be classified according to their symmetry with respect sp to Dx.

Following Bohr and Mottelson [4] we call « the signature of the state, although they introduced this name not for oc but for the whole exponential factor. We prefere oc because this is an additive quantity. In the case of single particle states oi takes the values - 1/2.

The signature is also a good quantum number for the quasi-arti¬ cle states and, consequently, the total set of quasiparticle equa¬ tions splits into two uncoupled subsets.

where

(8 )

Hti - a t ' a r e t n e natrix«lements of the pairfield in the

chosen basis /see Ref [ i ] / .

The quasipartide Routhian obeys the symmetry

(C 1 0/ '

from which foliowe that there are pairs of states with

55<»

and one needs to solve the eigenvalue problem (7) only for one

signature, say cc - 1/2.

The only further general symmetry of the problem is the parity

07 . Therefore, we can classify the quasiparticle states with res¬

pect to ot and Tf , and in oui case of the modified oscillator

potential also with respect to N, the total number of oscillator

quanta. Fig.1 shows a diagram of the quasiparticle levels as func¬

tions of the angular frequency OJ . In order to have a full clas¬

sification of the levels, v.e label each with the asymptotic quantum

numbers of the modified oscillator potential. These labels are

of course only relevant for <*> = 0, but also for u, 4 0 we are

using these labels for the continuations of the levels.

The usual way to construct quesiparticle configurations from

the solutions of Eq. (7) is to consider only one half of the solu¬

tions /usually E^LI ^'0/ as the "physical" ones, which define the

quasiparticle operators fi^ and the vacuum. 'The one-, two-...

quasipsrticle configurations are obtained by acting with one, two,..

fij" on the vacuum and the excitation energies are given by the sums

of the corresponding quasiparticle energies E' . This way of

looking at the problem is the most simple one as long as there are

no crossings with negative energy levels. The situation becomes

more complicated when such crossings appear like in ?ig.1 at coy'coo =

0.55. but also in the case of the quasicrossing at u>/uo0 =0.035. '.Ve

feel that it is much easier to cope with th<;se problems if one ccn-

siders positive and negative energy solutions simultaneously.

The proper theoretical tool for handling the full set of quasi¬

particle levels is the generalised density matrix, which has been

introduced by Bloch and Messiah [5] .

55?

6.485hwo;A -.12hcoo:£-0.26 [521 1/23 [633 7/23

0.14 [523 5/2]

0.08

0.04

3

UJ

0

0.04-

0.08

0.12 [6425/23 [5213/23

0.16 [50511/23

0.02 0.04 0.06 0.C3 0.10

Fig.1. The quasi-Qeutron energies Z^ as lunctir'r..; :. rtt frequen¬ cy CO . The energy unit is i\ CJO = 4-1 A~''-/ :ieV. The levels wita c< = 1/2 and «. =-1/2 are drawn as full and dastec lines, respecti¬ vely. The levels are labeled by the oscillator ąus-itum numbers F.N i H .Alki valid for CO =0. If a single particle state at co =0 lies below the Fermi surface the negative energy solution is labeled with the quantum numbers. The parameters of the calculation are quoted at the top of the figure. The position of ?v corresponds to about 96 neutrons.

556

The operator C*L creates one particle in the state |ot j^acd |«t A > represents the state vector of a quasiparticle configuration, v/hica is classified by its total signature oct and the necessary ad¬ ditional quantum numbers A. The dfnsity matrix Tt a is iiagonal in the basis (7) and its eigenvalues are the quasiparticle occupa¬ tion numbers nau. » wni°k take the values 0 and 1. Tie refere to the corresponding levels as "free" and "occupied", respectively. Any quasiparticle configuration [ot . A > is ćf.tercined Ly its occupa¬ tion numbers n.rt(U. . However, there is a peculiarity th«* distin¬ guishes this scarj;-c froa the familiar occupation nu;r.ter r-j.-r-.senta-tion of feruions! If a level Qtu. is occupied,its partner - oCv. sust be free. Consequently always half of the levels are occupied and half of them are free. Thi3 follows directly from the cosauta-tion relations

Coti' ]= du/ , [C*t , CV'] =0 (11;

v;hicb imply

and consequently

n.otv = 1 - nKV (13) •

The restriction (13) i s due to the double dimension of the quasi¬

particle representation ( 7).

A further peculiarity also due to the double dimension concerns

expectation values. Ltt us e.g. consider (Jx^>

<A « . t | JX I A « t > - Z^ J * w < OC t A ) G i . CaL' I V >

By doing a commutation and a relabelling of the sum indices we obtain

55?

t|Jx| Ace,) = |

This may be written as

<Aoct j Jx ( A cLty = \ £ tr ( Joc *oc) (16 ) The matrix J, is the angular momentum in the double dimension re¬ presentation

which is block diagonal with rtspect to cc . If we change to the quasiparticle representation that diagonallzes 9tx an3 n we obtain

A) = J Z where the angular momentum expectation value of a ;uaripart state is defintć as

Thus one might say, that each occupied level contz.-.-/: - one half of itB angular comtntuni to the total value.

A similar argument con be oriven for any one-pi. .. ;• o...erb.tDr, for two-particle transfer operators, the total si^iatur-. <X . and the total parity as well.

The configuration with the lowest energy corr spends to the one with all negative energies filled. We call it the vacuun con¬ figuration | vac)> . By freeing a negative level and occupying its positive partner we obtain a one quasiparticle configuration.

558

Repeating this creation of "quasiparticle-quasihole" excitations

generates two-, three, etc. quasiparticle configurations, i'hs ex¬

citation energy of a configuration /«..£•£/> may be expressed by the

difference of the corresponding density matrices

A\<OtA> -$«OtA) -$« (vac) (20)

In the approximation (1) of noninteracting quasiparticles one ob¬

tains

e ' ( « t A ) = E ' [ « t A ] - f ' ( )

The factor w appears for the same reason as in Eq.(19). Thus if

we consider only the relative energies of quaslparticle configura¬

tions we can formally say that the total energy is equal to one

half of the sum of the quasiparticle energies of all occupied le¬

vels. For absolute energies this is of course wrong.

There is a useful property of the quasiparticle energies E~«.

which follows from the stationarity of the quasiparticle solutions

- ~ - ' < / J°< > . (22)

The negative slope is equal to the expectation value of the angular

momentum component along the x-axis, i.e. it measures the contribu¬

tion from each quasiparticle state to the total angular momentum 1^

along the x-axis. Therefore, knowing the excitation energies of the

quasiparticle configurations as functions of 6J we are able to cal¬

culate the additional amount of aligned angular momentum too.

The total signature «< . is an important quantity, because it

implies restrictions of the particle number and the spin of a con¬

figuration. Since

Dx2(irj = Dx(2ar) = e,~lJ>N (23)

it follows immediately that fw^A/ must be a superposition of

states v-ith particle numbers

N = 2<x t mod 2 . ( 24 )

Therefore, configurations with integer signature oc must be re¬

lated to even particle number and the ones with halfinteger a t

to odd particle number.

The selection rule concerning the angular momentum reads

I = oit mod 2 (25)

It is not generally valid but only applies to casis where the ro¬

tational symmetry around the •/.-&••'- <s sufficiently strongly viola¬

ted, so that collective rotation aoout this axis is possible. ~q.l'2'

then reflects the w.ll-known syznetrization of the wav~functions in

the rotor model, that derives from the fact that, for the class of

nuclear she es that we are considering, the intrinsic frane of re¬

ference is only determined up to a rotation of 5 about the intrin¬

sic axis. A detailed discussion of this point is given in Befs

p]> M • ^n thiB sense we associate a certain configuration I c*t A}

as function of to with a rotational band with the intrinsic quan¬

tum numbers A and the angular momenta I = <x . + 2n, n integer.

User's instructions fo:: the level diagrams

The quasiparticle energies as functions of the angular frequency

oo represent a very condensed piect of information. These diagrams

permit to calculate the relative energies and angular momenta of

the quasiparticle configurations just by combining the corresponding

luasiparticle energies. Since this is a rather simple operation,

we base the whole analysis on level diagrams like the ones shoira

in Figs 1 and 3. Ltt me briefly repeat the most important points

of the quasiparticle theory as a kind of user's instructions for the

560

diagrams' for a l l , who are not sc engaged in theory.

1/ A quasiparticle configuration i s defined by indicating the oc¬

cupied levels.

2/ If a level S' i s occupied i t s partner Z ' a i s f r e r .

5/ An additive quantity /i^errj, angular monentum, signature, par¬

t ic le number/ i ; equal to one half of the contributions of all

occupied levels.

4/ For a quasiparti :le ltvcl -^„ . ( . ^ ~tt n gativ- si ore is er:si r

to i t s angular noaentu;.-. cozpon'.rt along the z-axis.

5/ The total signature oi a. sorifiguration •ieterL-in&s i t s an,."'-ilir

momentum

I = oc,_ mod 2

and whether the systen has odd or even particle i:uaibsr

5 = 2 «. fc mod 2

6/ There is a vacuum config<iraticn, v.-hirc has finical -mr^f ar.d

int£eer ott . It corresponds to filling all neraxive levels,

provided that this leads to integer oc^.

7/ Since 7?e only calculate relative ^uanrities we define a r-;fc-ren-

-,e configuraticn, -which for scali os jorresponds to the vacuum

configuration.

6/ The ;arity of the refer^nc; state is + . The parity of an ex¬

cited confiirura-icn i; the product of the parities of all pairs

of levels that changed the occupation, where the parity of the

: air it defined as the parity of one of its levels.

In order to analyse the experimental spectra we mist deter¬

mine froa the data those quantities we can read out of the level

diagrams, namelly the angular frequency u> , the relative Pouthians

e' /excitation energies/ and the relative angular momenta i.

561

The experimental quantities

The angular frequency can be defined by the canonical relation

-Kco = f f x (26)

approximating the differential quotient by a quotient of finite

differences. This idea is not new, it has became rather familiar

since the discovery of back-bending /see e.g. review L2]/. lie use

an expression that also applies to cases, where it is necessary to

take into account the angular momentum K directed along the sym¬

metry axis Z.

*^U-' 1 (1 + 1J - 1 (1 - -11 (27)

Due to the symmetry properties, .. '-scribed by the signatures, v:e

always restrict ourselves tu sequencies with A I = 2, when defin¬

ing a rotational band. In Eq.(27^ we use

although K is well defined only in the limit CJ = 0. On the

other hand I is not very much dependent on the exact value of K

when I » K. The use of I + j corresponds to a quasiclassicai

treatment of the three dimensional rotation [4-] . As well knov.n

/see e.g. review 2 , there exist other possibilities to define

than £0.(27), which for small frequencies are leading to slightly

different results. Eqs (27} and (28) define the experimental

function I fw).

The transition I + 1 — 1 - 1 determines the value co(l)

and Ix(l) is calculated from expression (28) taken at mean value

I. Fig. 6 shows examples of such plots. We also introduce the

experimental value of the relative aligned angular momentum

i M = IX(W - I g M , 129) where I (to) is the angular momentum of the reference configuration,

O

which we are not going to calculate.

In a similar way the experimental Routhlan is dtfi-ieJ as

562

a>(I) IX(I)- E g (»{1)) (30)

where

E(I) = \ (E (I - 1)+ E(I + 1 » (31)

Compared to Eq.(4) we have here subtracted the quantity E*(OJ( I)), which is the Routhian of the reference" system. To avoid confusion we are using capital letters ( E', I etc.) to describe the total va¬ lue of a quantity, while small letters (e'( i etc.) are used for the value of a quantity relative to the reference system.

For not too high angular momenta it is natural to choose the vacuum as the reference configuration. Its signature is oc. = 0 and the parity is 7T = +, This corresponds to the ground state ro¬ tational band ( g - band ) of the even-even reference nucleus. From the g - band we calculate by means of Sqs (27),(28) and the re¬ lation

the reference quantities I (w)and E' (U J ) . For odd-mass nuclei 6 6 k

we prefere as a reference the mean values of I_ and E obtained from the two adjacent even nuclides. In the odd nuclei it is also necessary to correct for the odd-even mass difference, so we have e'(l,A) = E(l,A) -CJ(I.A) I X{I,A) - jj Eg(co( I,*) , A +

+ E ^ ( I , A ) , A-I)J + Aoe(A) (33) where A is the mass number and Ą Q e is the odd-even mass dif¬ ference.

If there occur drastic changes of the vacuum like e.g. the quasicrossing of the levels a, -b and b, -a at co = 0.036 u 0

in Pig. 1 , the vacuum configuration becomes a rather inconvenient reference. In such a case it is more useful to refere after the crossing, to the configuration which keeps the character of the

563

wavefunction before the crossing, i.e. to the two quasiparticle

configuration where a,b are occupied (also with oCj. = 0 and 3T = +J .

We call this configuration the ground state configuration or short¬

ly the g-configuration. We define the corresponding reference

quantities from the experiaental data by extrapolating the smooth

part of the g-band at low angular momenta through the crossing re¬

gion. If unperturbed states of the g-band arc known after the

crossing /which is usually not the case/ one might instead make-

an interpolation.

The smooth continuation of the g-band can be constructed in

a coi»(/inient way by means of the parametrizatlon

where T. and "l^ are constants. This is the Harris formula of

rotational spectra [&], which is known to be rather accurate as

long as there are no irregularities e.g. back bending. The cons¬

tants *t and *L are taken from the linear part of *J ^ = I/w

as function of w , which is the usual backbending plot.

Integrating Eq. (34} we obtain the reference Routfaian as

- 4 ji

The- last term enters due to the three dimensional character

of "che rotation. So we obtain from Eq.(28) for the g-band (X = o)

and this leads to I (o) = - * as seen in ?ig. 6. In order to

obtain E (I = 0) = 0 the lower boundery of the integral must

be put at w Q = \ | 0 .

In all experiments we have analysed 30 far, we have used the

Karris parametrization for the reference quantities I (OJ1) and

E' (OO ) . In order to interprete these expression in terms of

configurations let me use ?ig.1 as a representative example. As

long as co < 0.026 u» g-configuration ia obtained by filling

all negative Ievel3. Then it corresponds to a smooth continuation

of this configuration through the interaction region. After the

quasi-crossing ,the two levels a,b are occupied and -a, -to are

free. At still higher frequencies there appear further quasi-

crossings (ui = 0.045 co 0» 0.052 co ). In this region the

g-configuxation gradually looses its identity and can no longer

serve as a reference.

The experimental Routhians e'(uj) must be compared with the

theoretica"1 quaslparticlc excitation energies ebteistd when chang¬

ing g-configuration, described above, Into the configuration under

consideration. The aligned angular momentum i(u>) may be obtained

as the negative derivation of e*(w).

The parameters of the Harris formula %. and ^ are fitted

to the low spin part (i < 6*) of the g-band, i.e. they are ad¬

justed according to the behaviour of the levels before the first

quasi-crossing. In the crossing region and beyond it the low order

polynomials (J4^ and (35) continue the smooth trends and this

is just how the g-configuration is defined.

Choice of -parameters

Before comparing the experimental data with the calculations,

I must say something about the parameters of the field Routhian^].

There are two possible attitudes: either one calculates them from

a selfconsistent theory or one takes them from suitable experimental

data. As already said, we decided for the latter one.

In order to keep things simple, we choose fixed values of the

deformation (£= 0.26 axial shape, no hexadecupolejthe gaps [An =

^ = 0.12łi uj ss 0.9 MeV) for the interval of the angular fre¬ quency of $>•"> 6 0.05-K co ^1 30-ft), which we are going to analyse. The choice of the deformation may be justified by ex¬ perimental data on E2 - matrix elements for the region A ta 165 to which we concentrate and also by calculations of energy surfaces of rotating nuclei [7]. The choice of a constant gap may seem surprising, however the success in interpreting the data indicates that this is not a bad guess. Moreover, investigations of the dependence of A on co ,which take into account corrections for the particle number fluctuations lead to a rather slow decrease of A [1]. The chemical potential X is determined from the ex¬ pectation value of the particle number in the ground st& -e. The particle number does not change very much with u and the choice of X is not very critical for the structure of the level dia¬ grams.

Let me now illustrate our way of analysing the yrast spectra by means of a few representative examples.

One quasiparticle configurations in a nucleus with odd neutron number; 'Yb.

One-quasiparticle excitations are generated from the g-confi-guration by freeing an occupied level, say E' , and occupying its partners E ^ . According to our rules this corresponds to the excitation energy E^ and total signature c* the reference has oc t = 0 !).

The Routhians for the six observed bands are shown in Fig. 2. They are directly comparable with the levels in the upper half of the quasiparticle energy diagram /Pig. 1/. Each level in Fig.2 is calculated from a sequence of states with A I * 2 that is for states of the same signature. The two lowest configurations

366

ON

E' CMeVJ

1.0

0.5

T r T r

C52t i/2] [50511/2]

[5213/2] Yb 167

oc —1/2 _

C523 5/2]

Fig.2. The experimental Routhians B* in '167Yb. The full lines correspond to « * 1/2,

the dashed ones to a a - 1/2. If necessary experimental points with «, = - 1/2

are specially marked by uu encircled cross. Note the different scale in comparison

*ith Fig.1, -fvtJ =7.5 MLv I The data are taken from Ref. (11) .

are the bands[642 5/2; ot = 5] ( i.e. I = 5/2, 9/2, 13/2,...) and

[642 5/25 = - 1/2] (i.e. I = 7/2, 11/2, 15/2,...), which show

a clear signature splitting. Both levels are descenting rapidly

with an almost constant slope, corresponding to an aligned angular

r 1 1

momentum i = -de /dw « 4.6 and 3.3 for the ot = j and oc = -^

signature, respectively /determined for the interval O.K'KO>J-<'0.2/.

The agreement with the theoretical quasiparticle levels is good al¬

though the theoretical signature splitting is a little smaller

and starts at a somewhat higher value of <*i(») . The theoretical

slopes, determined for the same interval of *CJ aa above, give

i ca 4.1 and 3.7 respectively.

At *vw * 0.27 MeV the theoretical iuasiparticle levels [642 5/2]

interact with their negative partners. This interaction is closely

related to the backbending phenomenon /see the discussion for Er/.

In the case of the odd configuration [642 5/2; cc = j] the levels

-b and a are occupied, while b and -a are free* The contribu¬

tion of these levels to the total energy /or angular momentum/ will

therefore be | (EIb + Ea ) (or " 2 ~TZ3 ( E-b + Ea)) • w h i c h S o e 8

smoothly through the region of interaction. This expresses of

course that we deal with identical fermions, i.e. a canonical trans¬

formation icluding only the levels a and -b does not change the

state. The fact is commonly stated as "the blocking of the state

by the odd particle prevents backbending". The situation for [642

5/2; oc = - iy] is analogue.

As already mentioned we use as reference Routhian the g-configu-

ration, corresponding to an interpolation in the theoretical dia¬

gram between the levels a and -b and between b and -a through

the interaction region. Therefore the experimental analysis shows

a picture without the interaction, i.e. the experimental Routhian

of the configuration [642 5/2; u s 5] crosses the Fermi surface

without any perturbation.

568

In order to make this point completely clear let us write the theoretical value for the experimental Routhian. We need only to consider the contributions from the levels a,b, -a, -b, because the contributions from the remaining levels to the Routhian and the reference Routhian are the same and cancel each other. For GJ below the quasi-crossing we have for [642 5/2; «*• = Xl

\ \ ( E ^ • E;) and

and thus

After the crossing the reference Routhian is E' = [ E + E. ) and thus •'« E'- E; * J ( ; J) ; Now, as we use an interpolation between E^ and E ^ t w« get a smooth continuation of E' into E^fe through the interaction region. The experimental crossing point with the line e'= 0 lies at *<*; s 0.28 MeV and corresponds quite well with the value read from the theoretical diagram ł w = 0.25 MeV.

The [642 5/2; ox = - J -band is expected to intersect the line e'= 0 at -t«w*0.30 MeV. The experimental band is not observ¬ ed up to this frequency, but the continuation of the Routhian seems to indicate a somowhat higher frequency ( .u?*-0.J6 MeV).

The [525 5/2J -band starts with a very small slope, which in¬ creases with the frequency in good agreement with the theory. We cannot follow the experimental band all the way back to zero, but its continuation seems to cross the [642 5/2 ; <*• * - I-bands at roughly the same frequency as in the theory. The experiments do not give much information about the signature splitting since only one point is known for oi = - \ . Also for the higher lying le¬ vels there is a good agreement between experiment and theory.

569

For the [505 11/2; oC = - ^J - bands the energy relative to

the ground state is not known exactly, and we have placed it at

energy where it can be expected to lie. Notice especially the

[521 1/2; oi = - -x\- bands which show a large signature splitting

also at small frequencies. These bands form a K = •* band, and

for such a band the Routbians of the two have a finite slope

- a/2 in the limit w = 0, where a is the so called decoupling

factor.

One quasiparticle configurations in a nucleus with

odd proton number; ^Ho.

In the figures 3 and 4 we compare the experimental Routhians

with the quasipaxticle energy diagram for " H O . This nucleus

has an odd proton number, and therefore we see the proton levels.

With a few exceptions the theory reproduces the experimental re¬

sults (relative position of the levels, slopes and signature split¬

ting). The theoretical [411 1/2; 06 = - \ j levels lie about 200 >eV

too low for small frequencies and the theoretical [541 1/2; <*. = -±]

level lies much too high. One might speculate why the theoretical

\yv\ 1/2; « = j] level lies so high. This is an h 9/2 level, and

these levels might be badly fitted by the modified oscillator para¬

meters ( tt , fx), which essentially have been adjusted to reprodu¬

ce the levels originating from the h 11/2, g 7/2 and d 5/2, J/2

shells. It is also possible that an excitation to the [5^1 1/2;

oc a 2] level, which is a strongly deformation driving level, might

introduce an increase of the deformation. It is in fact possible

to get down the [5*H 1/2; o<= 5] level to the right position by

increasing £ from 0.26 to O.JO.

570

Fig.3. The quaslproton energies E'Wu. as functions of u) . Apart from the, different energy scale the same parameters and conventions as in Fig.1 are used. The position of the chemical potential cor¬ responds to about 67 protons. The points with the arrows are experi¬ mental Routhiana taken from Fig.4.

571

[MeVJ 1.5

1.0

[5237/2 J

.05 .10 .15 (MeV)

.20 .25 .30

Fig.4. The experimental Routhians in Ho *. Conventions like in Fig.3. The data are taken from Ref.(12).

572

Two quasiparticle excitations in an even-even nucleus:

Two-quasiparticle excitations are obtained from the g-confiru-

ration by freeing two occupied levels, say 2 _ a and E..^

and occupying their partners E' and B* . According to f

the rules, the excitation energy is equal to E* + E^, a

the total signature is ct t = oc + ot2 , which corresponds to in¬

teger spins and even particle number.

When analysing the experimental Routhian of an even-even nucleus

it is useful to plot e'(w)/2 instead of e'(o) , because this is to

be compared w1 h 1/2 (E^. + E ) what is Just the middle

between the two levels.

We see in experiment two crossing banda with positive parity

and oc £ = 0, the g - band and one to which we shall refer as the

s-band Stockholm, super. There is only a very weak interaction be¬

tween these bands at the crossing point at I a 16 and this allows

us to connect, in a unique way, the two branches belonging together,

as shown in Figs 5,6 showing e'(**0 and I(t»>).

•At low angular frequencies the g-band coincides with the re¬

ference Routhian defined by its Harris parametrization and lies

therefore at e'= 0 in the Fig. 5« Also in the region of the cros¬

sing with the s-band and beyond, the g-band is almost identical

with the reference Routhian, showing only small deviations. It

is also seen that the s-band behaves is a relatively regular way

and its Routhian is crossing the Routhian of the g-band.

As already discussed, our choice of the reference Routhian cor¬

responds to an interpolation between the levels a, -b and b, -a

in Fig.1. The s-band therefore lies at e'e E a • E^ before and

a* E_ a + E-b **ter tn® crossing and passes continuously through

the interaction region.

Actually the interaction seen in the theoretical diagrams must

be eliminated because it acts at a given value of <»> and aixee

573

10 E72

CMeV]

0.5

-0.5

1523 7/2Jo CA0A7/2L Er 164

O «-0 ®

C5235/2]n

g-band

n 16425/2 ] n

01 Fig.J*. The expcrimeutel Routhian3 in Er . The full lines correspond to et = 0 the dashed ones to <* = 1. If necessary experimental points with a. = 0 are specially marked by circles.

Fig.6. The angular momentum I as a function of to • The upper part shows only an yrast sequence. In the lower part both the yrast and the yrare se¬ quences are shown. The dashed lines smoothly connect the branches the con¬ stitute the s- and s-bands. The corresponding experimental points are al¬ so included. The data are taken from the references quoted, in Ref.[2] and from Ref.[3].

575

states with very different angular momentum. This leads at the crossing point to a strong dispersion of angular momentum, which is' a signal that the description based on h' becomes questionable. Moreover, it leads to spurious branches in the functions l(cS) /shown e.g. in Fig. 6/ connecting Ig(<j) and IS(<J) in such a way? that within the narrow frequency interval , where the levels inter¬ act, the two functions interchange character. Similar spurious branches appear in E'(w) and E(l). Indeed it has been shown [8] that the elimination of these pathalogies just leads to a similar interpolation between the levels as we use ad hoc.

If we follow the yrast line of ^*Er we see that it switches at I = 16 from the g-band to the s-band. This corresponds in Fig.6 showing I(o) to a jump from the point at łńw = 0.51 MeV (14* - 12*) in the g-band to the point at ftw = 0.25 BeV in the s-band, (i3+ - 16*). The transition 16*, - 14* is indicated by a circle (at+iw = 0.28 MeV and e'= 0.06 MeV). We see that the sharp backbending in ^ ) r arises simply from connecting two dif¬ ferent bands to the yrast sequence.

The experimental interaction between the states of the g-band and the s-band is small but finite. It is reflected by the small irregularities seen in Fig.6 for the transitions 1 8 * — • 16*—•I** — > 1 2 * in both the g-band and the s-band. A more carefull ana¬ lysis of this interaction and its relation to the theoretical in¬ teraction, which we have eliminated, is the key for understanding the systematics of the appearance of back-bending. I shall return to this point at the end of my lecture.

The aligned angular momentum of the s-band is about 8. This can be compared with the theoretical value 7.8, obtained by adding the contribution from the levels [642 5/2} M = - * ] and the value 7.9 obtained from the experimental [642 5/2; 0£ = *• jy] le¬ vels of

576

The frequency at which the s-band crosses the g-band is in

experiment 0.295 MeV and in the theoretical calculations 0.28 MeV.

The lowest odd parity configurations that can be constructed

are those involving the neutron levels [642 5/2] and [523 5/2].

Considering the signatures we can distinguish between four differ¬

ent configurations, namely

1/ [642 WZi <* = £ ] [525 5/2; et = \ ] with <*t = 1 and odd I.

2/ [642 5/2; ot» \ ] [523 5/2; oc= - J] with at = 0 and even I.

3/ [642 5/2? ot= - J] [523 5/2; *= \ ] with ott = 0 and even I.

4/ [642 5/2; a = - 3] [523 5/2; oc= - \\ with a t = -1 and odd I.

They are here ordered so that the one with the lowest Routhian

comes first. The difference in energy comes from the signature

splitting /mainly from [642 5/2 ] /.

When comparing with the experimental ftouthians we find three

closely lying bands /one with odd I and two with even 1/ whose

Eouthians agree well with those which can be constructed from the

quasiparticle energy diagrams by adding the quasiparticle energies

for the appropriate levels. Two of the experimental Houthians

very close together, namely the dashed /odd 1/ and the full

/even 1/ levels marked [642 5/2]n [523 5/2]n in Fig. 5, while

the third one is crossing the other two /stars in the figure/.

This behaviour is not exactly the one that can be exptcted from

the quasiparticle diagram. On the other hand we must take into

account that the residual interaction between the two quasipartlcles

might disturbe the pure quasiparticle picture.

From two of the levels, namely the dashed one, which we identi¬

fy as configuration 1/ and the full one, which we identify as con¬

figuration 2/ we can construct a rotational band /with A I = 1 /

•starting from a K = 5 bandhead. The Routhians of these configu¬

rations /if extrapolated to somewhat higher frequencies/ cross

577

the g-band Routhian at-hu = 0.36 MeV. This crossing can be identified with the crossing in the quasiparticle diagram of the [642 5/2; 06 = - 5J level vrith the [523 5/2; <* = - \\ levels at •h ui <« 0.046 ł\io0 / ~0.35 MeV/ -ud the energy 0.055 &u>Q. The continuation of the Routhian of the third observed band, identified as configuration 3/ has to cross the ground band Routhian at high¬ er frequency in agreement with the theoretical predictions. These circumstances as well as the energy of the Bouthians at the highest observed frequencies is in agreement with the identification of the three bands suggested above. If this asignement is correct it is obvious that the third band /stars in the Pig./ changes its character at low angular momentum /the lowest observed state is 1 = 8 / . .

The configurations 1/ - 4/ do not show backbending. The reason is the same as for "ib, since one of the excited quasiparticles occupies a [6*2 5/2] level.

One more odd parity band /with both signatures/ has been ob¬ served. Its properties are radically different frora those of the [642 5/2]Q [523 5/2]Q bands.. Its Routhiar lies higher, its moment of inertia is smaller, its aligned angular momentum is scalier and it evidently has a band head at I = 7. A comparison with the quasi-particle energy diagram suggests the assigaement ^404 7/2J J523 7/2] , i.e. the lowest two quasiproton configuration, not showing any signature splitting. This asignement is also confirmed by the (S -decay of the 1 6 ^ m /1s = 6"/ isomer [404 7/2] [523 5/2]Q

which proceeds to the 1785 keV state of 164Er. Also for the [404 7/2]p [523 7/2] states, we can distinguish between four different configurations, depending on how we combine the signatures. It is not yet clear whioh of these combinations correspond to the observ¬ ed states.

578

The Interaction between the g-and the a-band

The crossing between the s-and g-bands differs from other

crossings because the two bands hare the sane signature and parity.

This permits a finite Interaction between the bands. I »h»n now

come to a systematic investigation of this interaction, which

allows to understand the back—bending phenomenon more deeply.

We carried out calculations of the neutron level diagrams for

an inter?all of \ that covers the neutron numbers 90 N ^ 108.

Like in Fig. 1, there is always a pair <x = ~ 1/2 of levels from

the *-4ifx shell crossing its negative partners at a frequency of

0.03 u , < u <C 0.04 u3o . Therefore the described crossing

between the g-and the s-bands is general feature of the rare earth

region. Similar calculations for the proton system show that the

crossing from the proton s-band /built from n ^^ *PP«***

at higher frequencies. Therefore the first irregularity seen

in the even spin yrast sequence of even-even nuclei - be it

backbending or not - should be due to the crossing of the g-band

with' the neutron s-band.'This Interpretation is* of course equiva¬

lent with the alignement picture for back-bending first proposal

by Simon and Stephers \jf\.

If one follows, what happens with the s-configuration when A

goes to zero, one realizes that this excitation becomes the g-con-

figuration of one of the even neighbouring nuclei. The very

existence of the s-configuration as an excited configuration In

the same nucleus is a direct evidence for strong pair correlations. f

Therefore, from the experimental evidence for the existence of

tne s-band, like back-bending and similar irregularities of th«

yrast line, we can conclude that the pair correlations for th*

corresponding frequencies must still b« strong *n* our assumption

of a constant A are quits reasonable. This interpretation

implies that there must always exist th* coaplaasntary parts of

tbe g - and s-bands as real yrare states. Op to now such states have been identified in Iy156, Er16* and Os 1 8 6. Usually, only the yrast states have been found so fax.

The rather sharp crossing of tbe g - and s-bands seen in Er is an exception. There may be a stronger interaction between the bands changing the "Z" of the function I (to J of the Er yrast

162 levels into an "S"as demonstrated by the example Er in Pig.6 or it may even prevent back-bending.

The interpretation of the even spin yrast and yrare states in terms of the crossing g-and s-bands suggests the following analysisi of the experimental data: We assume smooth functions Eg{I) and Es (i) for tbe noninteracting g- and s-baads, respectively, and a constant interaction V between them. This leads to a 2x2 matrix with the eigenvalues Eyrast/yrare * By neanfi °* *ne ca~ conical relation co, = dE(l)/dI we find

yrast/yrare (I) + w (1)1 -/+ • + v Ł

(37) The two noninteraction bands cross at the angular momentum I„.

G

In the vicinity of this crossing point one may use the linear ap¬ proximation

h ' | c w + *o «* *. • where yc, j and IQ are parameters not depending on (O . It is seen in Fig. 6 , that the assumptions of a constant equal slopes for both curves and a constant aligned angular momentum J for the s-band are reasonably well fullfilled. In this linear approxima¬ tion Eq.( 37) becomes

where u> c is tbt frequency, at which the iaptjturbed Routhians

5S0

Eg'and E B ' are equal /see Fig. 5 /.

The experimental function i (u) can also be expressed 137

Eq.(39) and the linear approximation (38) for the reference Zito

Therefore it contains the saae parameters as appearing in these

expressions. Fig. 6a shows i (u>) for Br 1 6 2. She experimental

parameters of the two-level expression(39) are determined by

the following geometrical properties of the functions I (to) and t

- The point (lc, u>c) is the inflection point of the functions

VaBt/yxaxe^) « * l yrast/yrare(w) ' ^ the case of sharp

back-bending of the functions Iyj,airt (<->) and i y^g^C^) it

may also be determined from

where co^ is the frequency, where the curves start to bend back,

and co. , where they bend forward again.

- The parameter *tc is defined as

- The relative aligned angular momentom J of vthe s-bsnd is the

shift of the s-band with respect to the g-band in the plot !(<•>)

/see Fig. 6b/. It is also given as the asymptotic value of

t -rras.fc(OiJ) after the interaction region /set Fig. 6a/. A fur¬

ther possibility is ^ y r a s t ^ O * «'/'2* 7aA systematios of the

experimental values of i in the rare earth region is shown in

Fig. 7.

- The interaction V. If both the yra3t and yrare levels are identi¬

fied /Er 1 6\ '^^t Os186/, then |V| can be calculated from

the experimental level distances and the unperturbed distances

/the first term in the square root in Eg.. (39)/ In the case of

a sharp back-bending the shift CJ „ ^ ^ ( i ) - ^qC 1) at tne l M t

experiments! point before back-bending is used to calculate |T|

by means of Bq. (?9). If the function ^ „ t (w) i» relatively

58 *

smooth, it is better to determine |V| from the slope at

This relation implies the condition for the* accuranoe of back-bend¬

ing in the yrast line / dljj.^^ /<L*> < 0 /

1*1 < ft * £ (*3) The quantity on the right hand side is also included in Tig. 8,

which shows the systematics of |V| in the rare earth region.

All nuolides for which )V| lies below the line J2/ 4 i show

back-bending.

Let ne cone back to the theoretical description of the cross¬

ing between the g-and s-configuratlon bj aeans of the level diagrams.

As seen in Fig. 1 in the theory there is also an interaction, which

acta, hoiever, between states of equal u> instead of equal I. The

corresponding airing of the g-and the s-configurations, whose angu¬

lar momenta differ by $, expresses in a drastic way the violation of

angular momentum and leads to as incorrect description of the band

crossing. A.B already discussed, we eliminate the interaction by

using a smooth interpolation between the quasicrossing levels. In

this way we avoid the appearance of the unphysical branches. This

approach results of course only in a description of the non-inter¬

acting g- and s-bands.

Now we go a step further and analyse the theoretical level cros¬

sing /in Fig. 1 e.g. a and -b/ in terms of a constant interaction

7/2 and the two unperturbed crossing levels. The interaction be¬

tween two levels of equal signature is just one half of the distance

of closest approach. Snowing this value one can reconstruct from

the known interacting levels the non-interacting ones /E^° ,X°^/

588

From the two E^° that must be occupied in order to obtain the 8-configuration we determine j as sta of the negative slopes at u> . These are the theoretical values of j shown in Fig.7.

The experimental values of i scatter around the fitted smooth lines. The theoretical values reproduce the experimental trends reasonably well for 68 * Z « 74 and 90 ś n 4. 102, although they seem to be about 1-fc larger than the experimental ones. Large deviations are seen for Z * 66 and for 5 > 104, which are presumably due to the hezadecupole deformation not taken into account in the calculations.

The theoretical interaction between the s- and the g-configu-ration amounts to twice the one between the crossing levels i.e. it is equal to the distance of closest approach to the levels. The bad thing with this theoretical interaction is that it acts between states of different angular momentum. However, this does not mean that its value is meaningless. Actually one expects that the interaction acting between states of equal angular mo¬ mentum I , i.e. between different frequencies v **e - r/2+c and tJc + i fi <i , should be approximately equal to the inter¬ action at 'c , because the levels involved are not strongly u> _ dependent /constant slope/. For this reason we compare in Fig.8 the theoretical interaction matrix-element with the experimental value of the interaction between the s- and the g-bands obtained by the above described analysis.

The theoretical interaction is an oscillating function of A . The mechanism of these oscillations has bean discussed by Beng-tsson, Bamamoto and Mottelson [io]: The two crossing levels of given signature /in Fig. 1 a and - b,e.g./ may be expanded into the states at w x 0 ; the projection t being a good quantum number. The expansion coefficients c"K of the negati¬ ve level are connected with the c* of. the positive level

10

£

5

. -On

* • " •

Or *

MSDCM

oEr.

s A

aW>. « Ht, »W. o 0 *

w.th j r

Er łb.Ht.W (r)

N • 100 102 104 O6 tOt : «

Pig.

U 64 65 66 6.7 68 69

,7. The aligned angular momentum j of the s-band. The full refer to even-even nuclei. The open symbols show the sum

of the aligned angular momenta I of the lowest QC - V 2 and ot* s -1/2 bands arising from the i-fj/i states in some odd neutron nuclei. The thick straight lines are fitted to the experimental data. The thin line represents the theoretical values calculated with the parameters quoted in ?ig.1. The ordinate shows /\ and the corresponding particle number. The data are taken from the re¬ ferences quoted in Fig.6.

hu»-

6.4 6.5 6fi 6.7 68 6.9

Fig.8. The interaction between the g-and the s-bands in the rare earth region. In addition to the theoretical values of 7 /oscillat¬ ing curre/ the quantity / / $ Jc /smooth curves/ has been includ¬ ed. Its experimental values /thick lines/ are obtained from the fitted straight lines of j in Fig. 7 and ^ c « *50 *&>;*, which is a reasonable value for the region. The experimental values of V are calculated either from the slope at the inflection point /open symbole/ or directely from the shifts in the transition energies /full symbols/, the same data as in Fig. 7 are used.

584

by the approximate relation C~ »»(-) &J • ** ^ lies ia

the middle between the single particle levels £„ . and £„

the dominant contribution is caused by the tera proportional to

i * . If ^ increases the contributions of term with j •, M

J K H K J

grows. Because of the phasefactor (-) it has the opposite

sign and tends to cancel the dominant contribution. At X « £•*

complete cancellation is reached and 7 changes sign. Thus the

appearance of zeros of V is a quanta! interference effect.

As you can see in Fig.6 the experimental values of the Inter¬

action | V| follow rather closely the theoretical values. The

bumps at K * 92 and IT • 96 are dearly seen in experiment.

Around the zeros at N * 89, 95, 108 the experimental interaction

is strongly reduced. The theoretical bump at 5 as 104 is about

twice as high as seen in experiment. This night also be due to

the neglection of the hexadecupole deformation. The systematics

of the appearance of back-bending is explained by the oscillating

bahaviour of the theoretical interaction, i.e. by an interference

effect. The correlation of the experimental interaction with

the oscillations predicted by the theory is a rather compelling

evidence that the suggested interpretation of back-bending must

be correct.

Concluding this lecture I should like to remark that I pre¬

sented only few representative examples in order to illustrate

the main conceptions of our approach. We also investigated the

systematics of the BO called mixed positive parity bands in the

odd mass Er isotopes,the back-bending in some odd proton and

odd neutron nuclei, the proton-back-bending in Er ^ and the

E2 - matrix-elements in the crossing region of the g- and

s-bands [1], The agreement between the experimental data and

the theory is of the same order as for the discussed examples.

5«5

We have seen that the yrast spectra of both even-even and

odd Bass nuclei can be Interpreted in terms of the experimental

Routhians combined with the level diagram at least qualitative¬

ly, in important step is the classification of the high-spin

states with respect to parity and signature. Even without using

the theoretical diagrams one can establish useful relations be¬

tween the data, because the measurements of the odd mass spectra

provide a kind of"experimental quasiparticle energy diagrams".

The experimental level diagrams at **> 4 0 are a valuable

extension of the information about the levels in nonrotatiag

nuclei, because the rotating field probes the single particle

wave function, at least its characteristics responsible for build¬

ing up the angular momentum. Thus, the slope of the experimental

Routnian /i.e. the al: gned angular momentum/ provides additional

information for the identification of quasiparticle configurations,

especially when high angular momenta are involved.

I hope that the discussed examples also demonstrated the

need for experimental Information about the non-yrast states.

Especially in even-even nuclei the data are still very scarce.

Measurements of the yrare branches of the s- and g-bands would

be very interesting, because the information about these bands

extracted only from *;he yrast line is sometimes very inaccurate*

The situation is especially bad for nuclides that do not show

back-bending.

It is a challaage to try our method in the case of odd-odd

nuclei. For this purpose it is necessary to measure the bands

up to an angular momentum of about 20* /A 160/. Until now

suih data are not available.

586

References

[1] R.Bengtsson, S.Franendorf, Proceeding* of the International Symposium on High-Spin States and Huclear Structure 1977, p.7*-78| and Hucl.Phys. / to be published/* B.BengtSBon, 8. Frauendorf, Phys.Lett. / to be published/i I.Bohr, B.B.Mot-telson, Perspectires la the Study of Nuclei with High Angular Momentum, Proceedings of the International Conference on Huclear Structure, Tokyo 1977, Vol.11.

[2] R.M.Lleder, H.Ryde, Phenomena in Fast Rotating Huclei Publi¬ cation KFA Julich and Adrances in Huclear Physics /to be pu¬ blished/.

[3] O.C.listener, A.f.Sunyar, I . i.v Mateoslan, Preprint Brook-haven ff.L. 1977t G.O. Draculis, P.M.Walker, A.Johnston, Proceedings of the International Conference on Huclear Struc¬ ture, Tokyo 1977, Vol. I, p.415.

[ i ] A.Bohr, B.Mottelson, Buclear Structure Vol.11. [5l C.Bloch, A.Messiah, Hud.Phys. 22, 95 /1962/. [6] ' S.M.Harris, Phys. ReT. 12§, B5O9 /1965/. l

[7] G.Andersson, R.Bengtsson, S.S.Larsson, &.Łtander, F.MOller, S.G.Kilsson, I.Ragnarsson, S.Aberg, J.Dudek, B.Herlo-Pomoraka, K.Pomorski, Z.Smrmsnski, Iluol.Phys. A269. 205 /1976/i K.Nurgard, V.V. PaehkeTich, S.Frauendorf, Hucl.Phys. A262. 61 /1976/.

[8] I.Hamamoto, Hucl.Phys. A271. 15 /1976/. [9] F.S.Stephens, R.S.Simon, Hucl.Phys. A163. 257 /1973/.

[10] R.Bengtsson, I.Hamamoto, B.Mottelson, HORDITA-Preprint, Copenhagen 1977.

[11] T.Lindblad, Hucl.Phys. A238. 28? /1975/. [12] L.Funke, K.H.Kaun, P.Eemnitx, H.Sodan, G.Winter,

Hucl. Phys. Ą122, 576 /1972/.

58?

Discussion of the Cranked Hartree-Fock-Bogolyubov Method in Terms of Simplified Model

S.Cwiok Institute of Physiea, PL-OO-662 Warsaw, Koszykowa 75, c'

J.Ttadek institute of Theoretical Physics, PL-0O-Ó81 7/arsaw, ?!oża 69,

Z.3zrmański Institute for Nuclear Research, "L-OO-^S"! "Varsa.v, Hoża 69.

: The simplified model Ramiltonian describinc a ro¬ tating nucleus is solved within HFB formalism. Sinmlificatior.c ir tr.e Haniltonian, which are of purely algebraic nature, leac in consequence to relatively small matrices and thus solution.; of ?!?5 equations can be examined in details without much CODI-ruter effort, ./e believe that, desrite simplifications, all general featurus characteristic for the model itself are still contained in our approach and that the main conclusions we draw are valid for more realistic Hamiltonians as well.

588

I. Physical Effects and Choice of the Suitable Model

New experimental techniques developed recently in

connection with heavy ion reactions gave al6o possibility

to examine properties of high spin nuclear states excited

in heavy ion collisions. In particular, individual states

with angular momenta higher than I«20h were observed and

the corresponding deexcitation processes were studied in

detail in many nuclei.

The first step in understanding the nuclear proper¬

ties appearing at high angular velocities is to determine

experimentally the so called yrast band i.e. the sequence

of states possesing lowest energy at a given spin value.

The next step would then be to reproduce experimental data

basing on a suitable theoretical model. The better is *he

model the larger is the number of effects that can be a-

nalysed within the corresponding formalism. Consequently,

the list of effects we would like to account for should be

prerequisite for deciding what theoretical model work with.

Let us briefly remind the phenomena expected to play

most significant role in high spin nuclear rotation. Assume

the energy of a rotating nucleus can be expressed in the form

^ ion) o-o

where the first term in (1.1) is expected to depend mainly

upon the intrinsic degrees of freedom while the second one

accounts for the effect of rotation. Separation of this type

turns out to be good approximation when describing low 3pin

589

excitations. Its validity suggests generalization of the

classical concepts such as moment of inertia, <*(def._), or

the rotational frequency,to , in the quantal description

of motion. Assuming the energy of the systen, E, and the

expectation value of the angular raomentum squared

are most fundamental and most relevant quantities in the

quantum mechanical approach, both &(def.) and OJ can be

defined from classical analogies

If we present the known experimental data in terns of the

latter two parameters insteaa of 1 and 2 then the first

observation is that if varies significantly with I. Seve¬

ral possible mechanisms were invented in order to explair.

nature of this dependence, for instance

i. centrifugal stretching /effect through the nuclear

deformation/,

ii. rotation-vibration coupling /dynamical effect/,

iii. Coriolis antipairing effect /and possible phase

transitions/,

iv. rotational alignement

v. rapid shape transitions connected with aprearanee

of local minima in E VB. deformation curve,

v i . the so called gapless superconductivity effect /see

below/,

and possibly more other effects contributing to the back-

-bending behaviour of £*#(«*) curve; note that the mecha¬

nisms listed do not exclude each other and some of them

are closely related among themselves.

The experimental data, when represented in £ vs. I

picture show rather smooth behaviour. The same dependence

can be made much better visible if represented in & vs. u> .

plot where i t takes the form of a characteristic multivalued

s-shape /the "back-bending"/ proved to occur in many nuclei.

Each effect, from the mentioned above, can be thought to con¬

tribute to the "back-bending" of <?(«*> 1, although some of them

were argued to play less important ro..e in real nuclei. Thus

we see that the theoretics: model should incorporate prefe-

rably all of the mentioned possibilities so as to account for

the interplay among them. The so called "Cranked Hartre'-Fock

Bogolyubov" method seems to be one of the best practically

tracteble models with this respect. It is based on the nu¬

clear Hamiltonian which for practical purposes is written

usually in the form

where £ represents average nuclear field, ^M^J" the

suplementary two-body interactions and c and c are

particle creation and anihilation operators, respectively.

591

Expectation value ex the Kamiltonian within a set of trial

wave functions should then be minimized providing us with

the arproximate fonnula for the energy of the system. Ho¬

wever the wave function describing the system should in

principle be characterised by several quantum numbers such

as energy, particle number, angular momentum and/or its pro¬

jection /if allowed by the symmetry of the problem/ etc. It

turns out, on the other hand, that solution of the problem

simplifies considerably if we forget about additional sym¬

metries for the trial wave functions; in such a case extra

account must be taken for the conserved quantities. This can

be done by specifying constraints to the Hamiltonian and in¬

troducing Lagrange multiplier /another approximation!/ tech¬

nique. According to this technique, the requirement that

narticle number, N, and e.g. x-conponent of angular momen¬

tum are specified is accounted for by solving the related

problem for the auxiliary Hamiltonian

(1.5)

instead of that in eq. (1.4). Here h and u play a role of

the Lagrange multipliers.

Let us now be a bit more specific about the mathema¬

tical aspects of the model; we are going to present only

the main assumptions and formulation of the method /Sect.11/,

and then discuss the results /3ect.lV/ on the base of the

simplified Hamiltonian /Sect.III/. References to the rele¬

vant literature are ?iven in "Bibliographical Note" /Sect.V/.

592

II. Outline of the Method

Let us begin .ith a constrained Hamiltonian

H*

vhere the two terms ••.•ith the minus sirn •jonie frotr tre con¬

s t ra in t equations

< N > - N ';•"•)

We introduce 'he so called quasi nart; -:le rerrefentatior. t'

applyinr to Hamiltor.ian (2.1a) '.he Borol.rubov trarjcf rrsc* - tr.

Hers a. and o- denote the ouasi-ar t^cle ~reat:or, -ini a ' . : r . -

lation operators, re»re? • i velv, anc! the transformation —•::

:"ioients A— • and B— ^ have to be determined fro:r the ?cr:^.-

tion that the f i c t i t i ous objects, auasj .rart icles, are asjur

do not in terac t with each other, iurpose, we are wor-:*r.~ *.

the f in i t e number, n , of s ta tes Id )> = c//c> , oi.-'.Z, .

*) ?he approach involving quasiparticle reprejentation liz

to the equations of motion which are exactly the sane -is *.v

obtained after making use of the minimi2ation •nroieaure.

593

*y 'O C*

Then equalities (2.3a) and (2.3b) introduce 2nrf + 2n— »(2nJ real unknowns, since Arft- and B-4- are in general comple:: a/n^ matrices. Although the coefficients kd( and B-( are not all independent since 2n^ "orthogonality" relations among A-s and B-s follow immediately from the anticommutations of the quasiparticle operators / {t-'-fyi * «£// a n d (t»'ii\ *° a n d

also /Cj,C-J »r and {^uie/%) "°/ nevertheless Źn^ un¬ knowns remain still independent. This is equivalent to say that dimension of the apace we deal with was artificially doubled.

Note that Lagrange multipliers A and a enter solutions of the algebraic problem for A-s and 1-s and can be deter¬ mined from the relations (2.2a) and (2.2b) provided the co¬ efficients A4ł- and B^^ have been found first. In addition, it can easily be shown that « is identical with the angular velocity of rotation, u> . The latter observation makes it evident that for M = U> JO there is no degeneracy ..ith resnect to time reversal because u> enters our Hamil-tonian linearly. Consequently, there exist matrix elements connecting I<O and their time reversed, l3>, and so the di¬ mension of the matrices in question is n * n , in general.

Por purposes of this paper it will be enough to specify the interactions entering our constrained Hamiltonian, for instance, in the following way:

i. £ stands for the average field which we can put in the form of the deformed Woods-Saxon potential /denoting by

the corresponding sincle particle hamiltonian we have i m <•< | h.rfS |/S> / .

i i . <tf ( stands for the short range two-body force po-

59k

tential e.g. monopoly state independent pairing force

/ ^fi/S =-iGSU/i £rr *'i"^)»'/"fr) where G -pairing force strength constant and signf*)=1 for the

state M> and signf.0 =-1 for the corresponding time-

reversed image, /«T> .

Such a simplified Kamiltonian can still succesfully imitate

nucleon-nucleon interactions contributing to nuclear pheno¬

mena at high angular momenta.

3uppo3e, the average field possesses certain additional

symmetries, for example, it is symmetric about, say, 0, pjcis

and invariant with respect to 180° rotation about 0v axis

/the axi3 of rotation/. These additional symmetries allow to

perform the so called Goodman transformation vhich reveals

in this case imrortant symmetry of the i:73 equations. Let /* >

be the quasiparticle state with projection of the anrular

momentum /on 0_ axis/ eaual to SI. and let tk > be its time

reversed. Define /Goodman transformation/ new set of states by

(K)=

and the corresponding conjugate states by

?he indices k and k run here over the n, quasiparticle "Vi

states, the index K, takes n^/2 values since

}k> and JJT> enter (2.4a)pairwi3e only. Then index K runs over

the remaining n^/2 values. It can be shown that in the new

595

representation

f ~ .- T O (2,5)

and

form

to r.his symmetry HFB equations separate into the

(2.6a)

r

K'

for K = 1,2, ... nw/2 and L=1,2, ... n^/2. The Quantities 1^^

and 4„„ are oomnosed of A-s and B-s /see Appendix/. Ac a re-

suit we have to determine two matrices of the dimension

(n^/2 * n^/2) only, viz. A.. and Bj?^ '"roR 're firs*. .:et

of equations /eqs.(2. >a) and (2.5^ / i.e. jn^ real number^,

and the corresponding two, AT£* and B.,r froir, the 3econd set

of eauations /eqs.(2.7a^ and (2.7b^ / i.e. the remaining £r£

real numbers. It is important to observe that equations

(2.óa) and (2.6b) transform exactly into the equations (2."'a)

and (2.7y if we replace AK'L, 3pv and E L from the set I by

B-A~, kTj? and Er , respectively, from the set II, or that e-

ouations (2.7a) and (2.7b) transform into (2.óa) and (2.6b)

when replacing AC>£, B K£ and ££ from the set II by 5?L, A^ L

596

and (-Ej) » respectively, from the set I. Thus we can see

that it is sufficient to solve only one from the two 3ets

of equations and, as a consequence, we can limit our con¬

siderations to, say, set I only. Although we reduced the

effort to finding only jn^ unknowns from the original 4n^

unknowns, which was possible because of

i. choice of the representation in which all the matri¬

ces / A , B, \> ,4 / are real, and

ii. symmetry of the Hamiltonian which -illoweo to nerfora

Goodman transformation,

nevertheless it is worth emphasizing that HPB equations,

even in their final shape of (2.5a,bJ and (2.7a,b) form a

sets of nonlinear algebraic equations. Moreover, the E.

quantities have to be determined together with the corres¬

ponding A-s and B-s in a selfconsistent way. .'Jote here, that

if V, A , A and o» were known, then eqs.(2.6a,b) were for-

ma'lly equivalent to the usual algebraic eigenvalue problem;

this observation suggests the iterative procedure of finding

the solutions:

1. We start with reasonable trial values of v - & matrices

and diagonalize the matrix

finding in this way certain A-a, B-3 and EL-s.

2. V.'e recalulate v and & according to their definitions

which involve A-s and B-s and repeat the

sequence (1.)-*(2.) until the selfconoistent solution is

597

found. Taking into account the symmetry between set I and Bet II

/described earlier/ we can label eigenvalues in set I by

(\/2> BVj-f "* ' S4* ~Si' '•' > " E C ? ) and in 8et n by

(S n~, E^-, ... , E~, -E4, .... - E ^ . W e assume that the eigenvalues were already ordered so that they form a decreasing s quence.

Now there remains at least one more trouble. Both seta contain twice as many solutions as the number of degrees of freedom posoenscd by the system. This fact reflects doubling of the dimension of the space which we deal with and we see that interpretation of our solutions needs more carefull treat¬ ment. Let us examine, in this reepect, the simple case of pure monopole two-body pairing force in the limit u>-»0. It is easy to show that for the st?te independent monopole pairing

where Ł can be identified with usual BCS energy gep parameter /the last relation being valid even more generally for any par¬ ticular value of to £0/. In the limit c*>-»0, however, the approach preeented here is equivalent to the usual BCS formalism what can easily be demonstrated by inserting eq.(2.9) together with VK)li * £KSKK' into eq. (2.8) ; the resulting matrix can easily be disgonalised and we get

and

for the sets I and II , respectively. Now, any excitation

598

which for even particle number in the system has to be com¬

posed of two, four, ... - quasiparticlc excitations must, by

the definition, have total excitation energy higher than the

energy of the vacuum state. This requirement is fulfilled if

and only if the solutions (-Sr , - 3 ~ , ... , -H~.) and (-3.,

-E 2 , ... , -En ) are rejected as "unphysical". We will follow

this £5 post reinterpretation of the solutions also for u> tO

despite the fact that E, does not equal -E- any more. The

latter fact will slightly complicate the situation for larger

(O values aa we will see below.

III. The Simplified Model Hamiltonian

In this section we develop additional simplifications

of the interactions contained in Hamiltonian (2.1a). The pur¬

pose is

i. to simplify the algebraic form of HPB equations /see

eqs.(2.6a,b) and (2.7a,b) / so as to avoid big compu¬

tational effort

ii. to retain all important ingredients of the approach,

nom<_ly, presence of the aver&ge field with the aid of

non completely trivial two-body interaction, selfcon¬

sistency of the solutions and the effect of time-re¬

versal symmetry breaking due to rotation.

The presence of the single particle potential is simu¬

lated by introducing eigenvalues of a certain single particle

Hamiltonian ( £ —*• u JO, £ j, ) * ^et u s

599

that we are not interested here in qualitative description

of the effects appearing in any real nucleus; we concentrate

rather on the 3tudy of sone mathematical properties of HFB

equations and on the qualitative interpretation of their so¬

lutions. We try, on the other hand, be as close to the reality

as possible within our approximations. Keeping that in mind

let us observe the very characteristic configurations appea¬

ring for neutrons in the flare Earth nuclei, composed of some

positive parity states belonging to i /2 multiplet lying uong

the negative parity levels of "N-C" shell. This fact sug¬

gests to choose the 3ingle particle spectrum in the form of

the two sets of states: the first one, composed of 7 3tates

simulating i /2 multiplet and the second one, defined as

2*52. degenerate two-level model whose role would be to si¬

mulate the neutron "11=5" orbitals. The sin/rle partible con¬

figuration chosen by us is illustrated in fig.1, where also

the calculated Fermi level A /for u> =0/ is shown.

In addition to the average potential field we intro¬

duce two-body interaction in the form of the monopole pai¬

ring force with the strength constant G arbitrarily put e-

qual to 0.30 MeV. The results presented in the following

correspond to the total particle number N=12. The formalism

applied is essentially that described in preceding sections;

in the numerical example discussed below n^ «14 so that di¬

mension of matrices A and B is equal to 7 in this case.

IV, Results and Interpretation

Let us first examine the 3olutions E^ and Er /the latter

represented by daahed line* In tig.2/ ae function angular

relacity to.

600

—4 0—' •ft. U

-4.2-'

-7.2 —

- A f) O.«J ———

ultlplet

-6.3 —

-8.3 c:

3/2" multlplet

Degeneracy s2xS2

.. . X=-?.13

Degeneracy s 2x2

Plg.1 The Bingle partlole level aohome imitating H3/2 aultiplet and the negative

parity shell /(3/2)~ degenerate multiplet/. Hvuabers give the energy in MeV,

levels are oooupied by N«=1 ' particles, ft «4.

0.1 02 0.3 04 05 06 07 08

0.1 02 03 04 05 06 07 08

ftu

01 02 03 04 05 06 07 08 09

Fig. 2 Th» quaalpartiola alf*nvaluaa /in MaV/ oorraapondin* to 1 13/2 aniltlplat /upp*r part/, tb« tvo-l*v*l Modal /a*dlv« part/ and th» «n«rffy (ap^/bottoa part/ aa funotiona of an<ular valooity /Hvrln ItoV/. Nota th* oharaotariatlo ain«ularltlaa for łtir O.ltO MaV /gaplaaa auparoonduottrlty racioa/ and for fcu-O.6 MaV /pbaaa transition region/.

602

Although "physical" solutions are all positive for u»-»0,

we plot Es- as if they were negative in order to makre the

picture better readable. !The bottom part of the picture

illustrates the corresponding self-consistent & values.

Figure shows the two particularly interesting regions on

the uf axis where the singular behaviour of the functions

in question takes place;

i. discontinuouity in & / a.nA., obviously, ;n 2^ •• -nd

in Ev-s/ w hi =h in our numerical examrle corresponds

roughly to ftw =0.39 WeV,

ii. multivalued /but continuous/ behaviour of A as a func¬

tion ofLJ in the region where pairing correlations

vanish. This, in our example, takes place at -fiw- 0.61 .

Let us note that the latter effect leads also to the multi¬

valued behaviour of E^ and E=- as functions of u sine- to any

particular &. value corresponds there a definite se* o:"

and E*. Since however, this cannot be clearlv illustrated in Li

the scale of this figurs, we only mark the singularity by a

break in curves.

Ze are goinr to discuss the two effects separately ao

they correspond to different physical phenomena. In the fol¬

lowing they are referred to as the gapless superconductivity

and the phase transition effects, respectively.

Gajless superconductivity.

In order to have a better insight into the nature of

solutions corresponding to the first singularity, let us ex¬

amine in nore detail the results in fig.3. For this nurrose

it is convenient to transform the KFB equations into the e-

auivalent form valid after our simtlifieations

603

2 -?

-6

- 9

0.25 0 50 0 75 100 125 1 bO 1 75

3 L~cl.it ions to the e"us*ions U,*..-:) ano (ft. 'b

-j:-osFir,ps of continuous ^nc iacr.-cotted C\;:-

the try.i. lium'bers in parenthese- ~ive the c

•::' -1. u; / in l.'eV/. 3ehaviou.r o? '•« r Xvc^- ' •.

Le:-,I'.• : by n r t ra ipht l ine , };owevyr "Au - > (.•

s'r-'-ifichntly with increase of co . This i< •

hivicur of solutions arousó ^ u = 0.365 as 3

usir.f enlarged scale in f lp . ft.

-r- repr--rcrv: PC

-pp. r." ir,r \ - J u

: ' .r , i]y 1-t r^y-<

o-i'-vc- '"pf • m p .

t: :;:). ,"):,;• %

6Ol»

( 4 . 1 b )

Both c o n a t i o n . ; ^ 4 , 1 a , ' ) -,'in be rc-r^r-J'.1^ ' i : ~ • -n : * . : ' ; . : ''or

v a r i a b l e s A ^n3 Ó , a t .1 riven u> '.vin's:: h-T'"1 ";.~ ••• ••;:•;.•

t c r . I]xrrG3Gi r. - /i t>v ^ frorr, 4.h? f, r.^", "in"i *.r '• ^ *"^r." r t; - i

- 0 " - : » f i r . - i A = k < & ) t t ' k . ' k . W , ? ' : ^ ' : - - . T :•. --.'. -:.:-~

of ^ i.-.d 4 : o r r e : - i - , j ; r , - ".: •.r.<: ' : / " i ; : : t ; : r . . : -v- ri--•

tei Lv. t h i s fir-Jr-': by r ro s : ; : . - - ; 5f A ..M; •„•.-•. A ,6a; r^ r V.

s a s e w v a l i j e , l l o t e * . r . i t i " '-!-.'• r c : " j " " - . . ; . tr-.rc : ?)..(&) • -

^ a v e r ; v e r y ^ . u ^ h l i k 1 - a -'.r~--"••*.* 1 ; r . c - • ( . " . • ? > " . : /\ _ ( 'd j . -:

c o n r l i v i a t e a , r ^ u l t : v i l u e : r j . - . ? * . . - • ..••?••.••..••.•- . r . . : : . • . ' • . .

* ~ t r o r . ~ d ? T " ^ n : i 9 r . ? t -; 'OTi t o . * : . ^ r ' . . : •- ^!" ~: c: *••: " i t . ' . . " i -

f r e r j e r . 1 : " w r a u j - j .; J D J t a r . * . •. ': •"•. r".'-.:. ~r. -."" t r : c ? • . " • • • /

ir. ( \ , A ) r l a . - . t . T V . . ; t<=- . : • • . - • . - • c r . L - - t : r - r . : r , : ' - : • • ^ " t

t h e v i : i n i f - •? : ' t h o f l r r - ' . r ' . r . . . t : r. - ; ; r . t f . « j « f ' . ' - r " . 7 "'

. " c V / o o ^ r a r T - : ' : • ' . . " c.ni :'L~.''.' \r.: : r. . '. 1-ji: t r - . * e : : r - " . n * . £

l y i n . " : , - . . ; , ; - h o w i n - , th--. t ; J - -.- .1 v < ; r - h : - r : . ' . : : / ; r . r - . ' . a -

l u t i c r . s , s a y f A . , 4 , ) i n - : ( A ? , 4 ? ) f r . e r . we - - - . f . r -v.c-

605

j U_ __

Pig. * Illustration of difficulties faced when solving self-consistently HFB equations in the gapless super-conduotivlty region. The singular behaviour of > a ć * ) Is illustrated by comparing behaviour of the two full lines marked with the corresponding -tf CJ values 0.380 and 0.385 /MeV/. The upper part of this figur* shows that only two self-consistent solutions were obtained. This may ba viewed as follows first we solve eq.U.ib) finding Xu - ~Ku C*) with high nunerioal accuracy. Then we oaloulate Fc» G -(right-hand side of eij. A.la) taken at (4,^-w t&)). Zeros of ?. indicate the solutions /(A > ? O values solving the self-oonolotent problem/. There are only two zeros corresponding to A, ~ 1.3 MeV and ń Ł ~ 1.59 UeV respectively.

as i t i l lustrates fig.5 /upper part, continuous l ine / .

Consequently, if we now try to calculate the yrast line

i . e . E vs. I dependence then we face a real difficulty:

a big portion of the function £(I) remains undefined by

the "cranked HFB" procedure v/hen basing on the ground state

HPB configuration only. This means, in other words, an al¬

ternative: either H?E approach is irrelevant in this case

or the ground state configuration is not aieouate for rc-

neratinp; yrast s tates with spins frora 1^ to I . /i to 11 in

our example/.

Let us assume that the !l?E theory is s t i l l valid, and

thus le t us take into account, in addition to the selfcon-

s is tent solution for the ground state configuration also a

selfconsistent solution ^orro^'-oniiinr to the f i r s t cjxc:tr-'i

state which, in the case of co ='"., is ':za?tly eouivalent to

two-Tuacirarticlc "xc i ta t ; or.. T we ^Tlcuia^c the ••ner-'*v for

',r.is new cor.Ti r'.vritio:: /'::c . 1 c ->. o t. c i t E-, -' is a fui?t:o.-. of

w '-hen we ?ct the relation i llustrat-3'i by thr iasf.o.i i.r,^

in f i f . 5 . I t can be i mmo-J iatoly seen fro- the f;f*;-e *!nt

57 is lov.'er, at least ^or sone w values sat is ly:nr

* s 0.595 KcV, than the energy of the vacuum sia'<.'!

We thus observe that the vacuum state conf itru rat ion may vary

with u> or, in other words, "2qp" confipuraticn may have

zero excitation energy for c e r t a i n s despite t1 e fact that

4 ^0. This apparently unusual situation is now rcferrei to

as a napless superconductivity effect.

607

It was shown that validity of the cranking nodsl

procedure depends on whether the angular momentum spread

around the mean value <fl ^ =1 is snal; if there sre strong

admixtures of various engular momentum components in the

wave function then errors of the cranking model predictions

nay become very large. For most of the states in our example

the purity condition /small spread of 1/ is fulfilled rather

well. However states corresponding to repid "interchange of

structure" between the two bands / s ^ dashed nnd full lines

in fig.5 for *w ~0.4 MeV/ contain lerce contributions from

various angular momenta giving raise to the increase in spread

in I. Looking at fig.5 we can aee that the branch marked ",2,3

extends into 5 ,S' ,l' , ... ,11' while the sequence of states

12, 13, 14 ••• forms -E extension of the 2qp band 1° , J* ,

3° , ... ,11" . According to the interpretation vie would like

to follow here, the states 5'" ,6 , ... 11* which corres¬

pond.to rapid interchange of properties between the ground

state and 2qp branches are treated as spurious and are not

taken into account when constructing the yrast line. The in¬

terchange just mentioned manifests itself by rapid increase

in slope of the corresponding E vs.w curve. Actually, the

branch 5 ,6 , ..., 11 is alaost equally steep as that com¬

posed of 5 ,6'" , ..., 11*' . This results from another cros¬

sing lying nearby. !ffe shall not consider its effect here so

as not to complicate the discussion intentionally limited to

an isolated crossing. This particular effect is discussed by

us on the example of even simpler model Hsrniltonian based on

the two s.-ts of two-level models /forthcoming article/.

608

01 02 0.3 Q4 0.5 06 0? 08 09

20

10 r3"CZlX

u 8- 10" 11"/^

_ _ ^ I Ł . JgJŁJŁ~21f ?2. 2.3~. —g<* 01 0.2 03 0 4 05 06 07 06 09

I-iR.5 The energifss alonf, the Ri-cuiid s ta te band / l uU liiw-atid alone Ule 2cjp bond /d.fiohe4 l i n e / vs. -ftw /upper part / ' Kumbere denote oorrosponding opin VQIUO,. Bottom part i U u the values of a for given spins. Note the r«pld incraase in lOp in the branch 5 ' , 6 ' , . . . H, • n e a r fto-0.5 M.V due to - interact ion between some stataa near the actual Fermi l . v . l / the .e atat«t represented by th« two higheet-lylng daahad CUTV.B, f i g 2 /

609

The moment of inertia calculated on the base of the

results plotted in fig.5 ie illustrated in fig.6 showing

characteristic back-bending behaviour.

The same physical facts can be expressed eouivalently

by ploting E vs. I as it is done in fir. 7 for -the eround

3tate and for the excited 2qp configurations /the states

denoted 5'' , 6nl , ... were omitted in this figure'. Hcte

the characteristic band crossing picture which provides an

alternative illustration of the back-bending due to g-anless

superconductivity pnenoraenon. Obviously, there are other

types of band crossings possible as well /e.~. the eround

state and (i-vibration bands/ which however are not discussed

here.

Particle alignement. it is interesting to analyse pro¬

portions in which particles /or quasiparticles/ contribute

to the total angular momentum since from such an analysis

conclusions may be drawn about the microstructure of the a-

lignement process. Although the rearangement in the micro-

structure caused by the nuclear rotation depends upon the

individual characteristics of the system such as details of

the single particle spectrum /form of the average field/ or

on the properties of the residual interactions nevertheless

some typical properties of the process can be extracted.

610

DC

•o

CO

m r Ml

i

o 4»

i 1 P> •

I O

i

\D

611

I • -92

-93

-94

-96

-97

-99

-99

-100

-101

Splfll

ut

18

15 20 Spin I

Pig. 7 The same information as that plotted in fig. 5 but now plotted

in E vs. I pioture. Note the characteristic crossing of bands.

Observe also the smooth behaviour of A vs. I curve /upper part/.

The values of A correspond there to the yrast line defined

aooording to the prescription given in the text.

Let us remind that the axial symmetry of the system

about 0 axis was assumed and rotation around 0 axis was z x

cranked. Por to 0, even very small, theI5.xi/2 rarticles

start aligning almost immediately /the curves marked with

1 and T in fig.8/. Note however that we can hardly 3peak

about full alignement for t*» < uT since the quasiparticles

1 and 1 contribute to the total spin in rather different

proportions. At the same time 2 and 2 try to contribute co¬

herently but their contributions to the total spin decrease

markedly with increase in u). Note that the alignement, un¬

derstood as a coherent contribution of at least two quasi¬

particles takes place, in fact, after the first transition

point /co/ OA-zr, see fig.8/. We can say that the rotational

alignement and the gapless superconductivity effect are

strongly correlated and are merely two different descriptions

of the same situation, seen from different points of view.

Phase transition. Let us finally discuss that region

in u> axis where the pairing correlations disappear comple¬

tely due to fast rotation of the nucleus. The multivalued be¬

haviour of 6 treated as a function of w /see fig. 5/ is clo¬

sely related to the strength of the pairing force and the

form of the single particle spectrum. The effect of decrease

in G is vanishing of the multivalued behaviour whereas incre¬

ase of the pairing force strength from G*C30 MeV to G*0.45 MeV

causes strong enhancement as it can be inferred from compa¬

rison of the results in fig. 9 with those in fig.10. Note

however., that 4 vs. I dependence calculated along the yrast

line is smooth except for the immediate vicinity of the first

613

0.1 02 0.3 0.4 0.5 OB 07 06 09

Fig. 8 Illustration of the quasipartiole contributions to the total spin.

The only most contributing quasipartloles were taken Into account

in order to make this figure better readable. Note the most rapid

configuration change corresponding to 1 and T states.

0575 0.600 0.625 0.650

Fig. 9 "Phase transition region" - A vs. to behaviour for G « 0.30 MeV

/compare with fig. 10/. Niunbere give the corresponding spin values.

615

MS 0.70 075 OJO OJS 0*0 0S5 1.00 1.06 1.10 1.20 1.2S

10 Conparlaon of tlga. 9 and 10 illustrates the effect of pairing foroe strength on the behaviour of A vs. I in the phase transition region. Seorease In G may oause complete dlsappearanoe of the characteristic multivalued behaviour.

A =0 point, marked with "II" in fig. 11 /see fig.12/. Finally,

comparison of the functions A (I), E(I), E(u>) and I (us) is

given in fig.12. Behaviour of spin I vs. w shows clearly

that another back-bending in the •? vs. u> picture will be

observed in thi3 case in phase transition region. This can

easily be deduced if we remind the relation

} = fi /TTzTTj /to

equivalent to the de f in i t i on of f, eo. ( i .7b^ .

Concluding the d i scuas ion presented in thi<? ->3jT"r

we can 9 3y the fo l lowing:

i . There are po3s ib ie several mechanisms tha t may caus1"

e back-bending e f f e c t . Here we concentre tat or. th c 3C

ca l led gapies? superconduct iv i ty effect and en BC.T.-- :•:-

t a i l c s cf the phase t r a n p i : i o n / d i s ^ p p e - r s n e t D:" _ -a i r in r / .

i i . I t ic a Tueation o:' ir;::v;:u r-.; r m r r r ; : e : of '-':.<.• :v.;: fT

whether the (rapletc "upr-renn : \ j ; t ; v ; ty ':':'«.^t wil l xrn^'-i'j

disappearancc of t a i n n r or not . "T not ".h.n '.y.t- ba " i - t o " i . : ."

aj

a u e t o p : C e t r a n s i t i o n n a y a p r e a r r r o v ; 1 ' ? : t h r p . i i r : r . ~ . ' o : ' ^ 1 "

strength i s b:r enourr.. ?or too weaJ: r-i;.-: n r- no b^c>:-ron:iv."

"iue to rhase tranoit^on is ^xtc^tei i ^co r - i i r '.o "1?*.° Tr r-I

liscunced here.

i i i . 7he r-iple.3C cjper :on.iuc; t .vi ty nc ronrwic . vory cfter.

the s i tuat ion that "he F^riri e;:ergy A :c CIOJO ".o or.t. of '.ho

Single pa r t i c le lovely.

iv. Neither projection of par t ic le number nor rroject.on of

angular «o«n tua was accounted for In th i* papsr.

617

>

to § •3 0

i

(M

618

g i o

i 5 3

o

1

9

o I

(A3W) V (A»W)

619

V. Bibliographical note

The so called cranked HPB method wee proposed for

description of the back-bending effect in nuclei in pa¬

pers O~tQ • •^3e interpretation ot the results of the

method by considering, in addition to.-the ground state

band alBO a 2qp band can be found in papers £S,10j . Per

the details of the specific model Haciltonian and the re¬

sulting features of the selfconsistent HH3 solutions see

refs. [12, 13J .

References 1, H . R . D c l a f i , B . B a n a r j e e , H.J .Mang, P . R i n r ,

Phi 's . L e t t . ££3 (1973) 2 7 .

? . B . B s n s r j e e , H . J .Ming , P.Ririg, Huc l .PhyF. . . : '= . ( " " )~

3 . F .C .Bhargeva , N u c l . P h y s . A207 C>V7?) 2f- .

4 e Pc C.Bbargava , D. J . T h o u l e s s , I l u c l . P h y s . •'• " : {',"'i ) ; " .

5 , 3 ,Bose , J .Kruml inde , E.R.r.!ar3halek , P h y r . : e t : . ;' 3C'>"

6, C.Y.Chu, 2 . R . l ! a r s h s l e k , P .R in f , J.Kruir.lir,-; • ,

J . O.Raamunsen, Phi ' s .Rev. C±2. ( 1 9 7 5 ) 1 O 1 " .

7 , A.Goodman, N u c i . P h y s . A?30 (1974) i&£.

A.Goodman, N u c l . P h y s . \?-&5 (1976) 113-

'"° A, Paesp l i - r , K.R. Sandhi'-T Devi , ?. Gruma^r, >".. ••'. J ^hr: . ,

A . R . H i l t o n , N u c l . P h y s . A?-?6 (" 1976 ) 10c.

: " . . .Bohr , "4 50 Confe rence" , Koper.harer., ' '.".

iC, :;Hć.-n:anoto, TJucl.Phys. A2_2J_ ( 1976 ) I s . 1 •> . R, E?nf t s c o ' i , S .Prauendor f , Conference i:- :>\: - " •. -.r ?h% • •

lircsc'en, 1977.

IT. £«Szvmariaki, Lec tu re3 d e l i v e r e d --t XXX r: ;:-—; - j o : - - .

; u a e r Schoo l . Les Houchee, 1977.

1.:, S.Ćwiek, J .Dudek, Z.3zynn: ' .ski, t o be pul-: a-h-:-ż .

6 2 0

The quasimoiecuiar model in transitional nuclei

G. Leander

Department of Mathematical Physics, Lund Institute of Technology, Lund, Sweden

A review is made of some recent developments in the application

of the Bohr Hamiltonian tc transitional nuclei. The p.-imary concern is to understand the advantayes and limitations inherent to the basic model and its many variants.

621

1. Introduction

In 1952 Bohr ' tentatively suggested that the theory of molecules could be adapted to describe low-energy excitation moties in nuclei, and as it has turned out there is a great deal to learn from this by now very fa¬ miliar approach. The aim here is not to review or supplement the many successes of the model, but rather to examine the question of how far it may be possible to push on, to point out some loose ends of the theo¬ ry and some standing unanswered questions. On the technical side a few recent advances in implementing the model will be brought to attention and further developments suggested.

!n the nuclear analogue of a molecule, the ensemble of nucicons o<> a whole takes over the role Of the atomic centers in defining intrinsic axes, I.e. spatial orientation coordinates, and in'.rinsic collective coordinates. A few valence nucieons may play the role of the electrons. The dual role of the orbitals near the Fermi surface is then an approxi¬ mation referred to as violation of the Pauli principle. The Hamiitomar, becomes

H " Hcore <Sf > + Hpart ł Hint<S>

where oc are the collective coordinates.

It is assumed that they can be approximately separated out in a many-body nuclear Hamiltonian, but attempts to actually do so have met w.t little success '.

2. Core with no valence particles

The core collective coordinates are usually thought of as describing the shape of a sharp-surface liquid drop. With this picture, which of course excludes other low-lying collective modes e.g. of pairing type ', it is possible to make a multipole expansion whose lowest and most im¬ portant term is the quadrupole. There are five quadrupole coordinates. Three of them, the Euler angles 8{, define the orientation of the in-

622

trinsic axes and enter only into the rotational part of H . This

part is well established and will not be discussed further. It should

only be mentioned that the experimentally observed moments of inertia

? have not yet been convincingly accounted for. Earlier successes on

the basis of the modified oscillator model were recently discredited

when it was noticed that the £ term gives a spurious 30% effect •

The two intrinsic quadrupole coordinates are 6 (in the expansion of the

radius or alternatively r in the potential) for elongation and if for

axial asymmetry. In addition to the rotational part, H is assumed

to contain a potential V(jJ,|) and a vibrational kinetic energy

(2; T^.jf, « B>0(p,s

The general mathematical forms of V and T are discussed by Rohczinski"''.

The nwjor weakness of this assumption is that the three mass functions ;

the moments of inertia and the potential may also depend on the nuclear

state, reflecting the role of e.g. pairing or Cor-.ol is forces. Using

a variant cf the Strutinsky method it is possible to calculate the po¬

tential energy as a function of spin ', and results for the ratner e<-

treme case Ne are shown in f i g . ] . At spin 0 the potenf-al has a we'1. -

deformed prolate itiinimuTi, but at spin 8 it has changed character comple¬

tely under the influence from the Cor i ol i s force and there is a w e a H y

deforned oblate minimum instead . It should be noted that the t r a d i ¬

tion to oblate shape at higher spin cannot be contained in the m e ^ f a l

mass functions because the rotation at 1=8 takes place around the obla¬

te symmetry axis and this is not allowed for a collective core.

A dynamical treatment of the intrinsic deformation coordinates implies

that there should be excited vibrational states, but it is an open

question whether non-rotational low-lying excited states of transitio¬

nal nuclei can be described in this way.

There is a difficulty connected with the concept of shape itself. Con¬

sider a number of particles bound in a deformed shell model or self-

consistently generated potential. The fixed ...apt? can be defined e.g.

6 2 3

0.6

Fig. 1 . Potential energy of deformation for Ne, calculated with

single-particle potentials that are cranked around the x-axis at

frequencies corresponding to nuclear spin 0 and 8 respectively. The

x-axis is the syimetry axis at&*60° . The contour line separation is

1 HeV.

62U

as the expectation values of the single-particle quadrupole operators

&„ • Z (2z;2- X;'-^*)

However i t is obvious that neither these operators nor any other func¬ tions of the single-particle coordinates can be good quanties because they do not comnute with the Hami!tonian.for example

( 4 )

The non-zero standard deviations from the average values

(5) <r(0) = [<&*> - <<2>2]' /fc

can then be regarded as shape v i b r a t i o n a l a m p l i t u d e s wh ich a r e the

p o i n t a m p l i t u d e s no t of any l o w - e n e r q y Qjaaruc>ole mode but o f t h e 9

q u a d r u p o l e . A s i r cp lp c e l c j l a t ;r ,r f ' j ' t>,e m o d i f i e d o s c i l l a t e C ' v e ; A

0 ? n , expressed in terms of the usual deformation coordinate p

100 ( 6 )

and a c o r r e s p o n d i n g s m o u i t i n t h e Jf -d " . r e c t i o n . I n t r a n s i t i o n " I A ' P :

t h i s i s a s i z e a t i l e t r a c t i o n o f t h e t o t a l d e ' o r m a t i o n n r t h e d i s f . n r f o n

b e t w e e n p r o l a t e and o b l a U j , w h i L h l e a d s t o some p r o b l e m s . c i r s t ! > t h e r e

i s t h e t e c h n i c a l p r o b l e m of how t c e v a l u a t e t h e p o t e n t i a l and ruass f u n c ¬

t i o n s m i c r o s c o p K a l l y a t a g i v e n c e f o r m a t i o n . S e c o n d l y . \' t h e i o n - e p e 1 ' -

gy q ' j a d r u o o l e n o d e i5 t o be e n v i s a g e d as a s l i_ : shape v i D r a t i o n s j c e r -

i i iDosed on t h e f a s t g i a n t q u a d r u p o l e o n e , t h e n H seems l i k e l y t h a t i t s

a m p l i t u d e must be c o n s i d e r a b l y l a r g p r . ) r so, t h p o n l y o c t i o r , ' c a

t r a n s i t i o n a l n u c l e u s a p p e a r s t o be v i b r a t i o n ' , a c r o s s t i e e n t ' " e J f - p l i . -

ne snrt an e f f e c t i v e X c l o s e t o 3 0 ° . l >om t h ' s y o i n t o f v i e w i t •.; u^:z-

1 i n q wncn e x p e r i m e n t a l e v i d e n c e on odd -A n u c l e i ' n f . - . c a t e ^ V ' v a l u c s o f

say 1 6 ° , as w i l l be d i s c u s s e d f u r t h e r l ie l o w .

625

experiment theory

188 Hg

0.2

Fig. 2. The experimental energy levels and E2 decay scheme of the

early backbender ^4g are displayed together with calculations)

results obtained from a phenomenological collective Hamiltoni an with

mass functions of hydrodynamical type. The /3 deformation of the oblate

minimum is f i t ted to the experimental B(E2; 2*-*0*), i ts depth to

the ratio E(4+)/E{2+) and the free mass parameter B-240 K2 MeV1 to

E(2+). The ft deformation of the prolate minimum is f i t ted to the moment

of inertia in the excited band, giving too small in-band E2 rates.

Its depth and barrier against the oblate minimum are f i t ted to

and E(2£).

626

In comparing theory with experiment, the simplest approach is to assume a hydrodynamical functional form for the mass functions and to fit the potential to experimental data. A typical result is shown in fig.2. The fit is quite good, but the potential exhibits structure that is an or¬ der of magnitude too large compared with the structure that can be ob¬ tained from microscopic calculations. Alternatively one may use micro¬ scopically calculated potentials. This has been done in a systematic survey ' of the A«40-90 region with a simple prescription for the single mass parameter, namely B»A J! MeV" . The calculated splitting of the Jf-unstable phonon multiplets Is invariably much too small, although the basic trends are reproduced in some nuclei with characteristic structu¬ ral features, such as 76Ge (fig.3).

The empirical situation is then the following. From extensive earlier work we know that the low-energy properties of transitional nuclei are closely correlated to the size of calculated prolate-oblate potential-energy differences (see e.g. ref.7)). However, if the y-vibrational mode exists, then it cannot be the potential itself that is responsible.

In the /9-direction the calculated potentials often show larger varia¬ tions, and a shape dynamical theory can give better agreement. An ex¬ ample from the A=40-90 region is the position of the CU state in the titanium isotopes (fig.4). Furthermore the quantity E(4+)/E(2+) is plot¬ ted in fig.5. Its value in the symmetrically deformed rotational limit is 3.33 and in the spherical vibrational model 2, thus it is sensitive to the rotation-vibration interaction and the quality of the agreement in fig.5 is non-trivial. Particularly the variations for Ca and Ti are quantitatively reproduced by variations in the theoretical potential-energy surfaces. However, there are irregularities around the magic nu-cleon number 28,and in fact the 28 gap poses a problem with regard to the & degree of freedom. Strong shell structure persists in the M.O mo¬ del out to very large deformations, and the potential surface is corre¬ spondingly flat. Therefore one would expect large-amplitude vibrations, but experimentally the nucleus 2sN128 has a' the attribut;es of extreme sphericity.

In attempting to describe doubly even nuclei with H C O P e(gj) alone, the

627

•e -r

0 Ot U OJ OL

Fig. 3. The experimental spectrum of Ge is shown alongside a f i t ted

rotor spectrum and furthermore the solution of the Bohr Hamilton!in

(B«A H MeV" , 1<4) in the microscopically determined potential dis¬

played to the le f t . The lower part of the figure gives the V dependence

of the hydrodynamical rotational itoments of inertia (solid l ines) , and

another extemporaneously drawn dependence (dashed lines) that would

lead to a considerable depression of the T band.

628

Ex (MeV) 'Ti

3*

2*

- 0* 0* 0* ar Thea

c (and «J

are cMpamed with 1

Fig. 4. The experiment*! spectra of Ti and

the solutions of the Bohr Haailtonian {B«A *2 NeV"1, I i4 ) in Micro¬

scopically determined potentials that are also plotted. The calculated

second 0+ state of **Ti is mainly localized to the strongly defonwd

second winiBUB.

«Z9

9> O i i i 1 I T

2.5

2.0

1.5

•—• Thtory . . • • I I I ! i i i i i i

50 60 70 80 90

Fig. 5. The experimental ra t ios E(4*)/E(?*) in the fp shell nuclei are p lo t ted together w i th values emerging from

solut ions of the Bohr Mamiltonian (B«A H^ MeV ) in microscopical ly determined potentials.

next step is to allow for a non-hydrodynamical behaviour of the mass functions. For exaaple the spectrum of Ge would obviously be better reproduced by shape dynamical theory i f the Y dependence of the rotatio¬ nal inertial functions were to be adjusted in a way that is indicated qualitatively in the lower part of f ig .3 . Systematic calculations using microscopically determined mass functions have been carried out recent¬ ly by Rohozinski et al ' . They selected the xenon and barium isotopes as suitable objects for study, because the potential energy is essenti¬ ally V-unstable in this region. I t turns out that the shell structure effects on the mass functions can only lead to the required amount of splitt ing in the deformed phonon multiplets after some phenomenological adjustments of the scale. Concerning the qualitative nature of the spl i t t ing, the results are inconclusive because none of the Xe or Baiso¬ topes have pronounced spectral characteristics to be tested. The major success of the calculation lies in the reproduction of energy rat ios along the yrast l ine. An interesting observation is that the shell s t r u c t u r e i n the mass f u n c t i o n d i d not s t a b i l i z e the J j " -deformat ion *f>

any of the nuc le i s t u d i e d .

There fo re the c a l c u l a t e d e f f e c t i v e J r -va lues of a l l these n j c l e - l -e

c lose to 30° . For 1 2 6Xe and 1 3 4Ba t h i s p roper ty of trie c a l c u l a t e d wave

f u n c t i o n s has been tes ted aga ins t experiment t y coup l ing on an odd

h ^ , , p r o t o n , whereby the negat ive-Do1" ' ty s o e t t r a ? ' Cs anc J JŁ n

could be adequately reproduced .

Of prime i n t e r e s t , however, are t r a n s i t i o n a l nuc l r wnere the e\ de"ce

from odd-A spectra i n d i c a t e s V - v a l u e s c l o s e r to the a x i a l 1 /

l im i t s 0° or 60°. An appropriate example is Os, whe>-e Meyer-ie--'>eT. deduces the ef fect ive deformation fb*0.20, JT -16°. As mentioned anovp. such a small I r-value at th is A-deformation may not Be compatible » • ' / the existence of a low-energy IT-vibrat ions 1 mode. In the shape dynamical approach, where hydrodynamical type mass function are assumed and the potential is f i t ted to experimental 0s data, a completely unrealistic potential emerges from the f i t . This is seen in f ig . 6,

187 which also shows that the proton particle spectrum of l r is well reproduced from the f i t ted 0s potential. Kumar and Baranger '

186 have made an a priori more realistic calculation for Os with microscopically determined mass functions and potential, but

531

2000-

1000-

IktV) i

3000-

2000-

1000-

ExpWMMflt TM«ry

Fig. 6. A potential-energy surface fitted to reproduce experimental lac

properties of TC0*IIQ- The contour line separation 1s 1 MeV. A test of the calculated wave functions has been made by coupling on a particle in an hg . j shell with an f^.. shell located 3.S MeV above, and comparing the results with the h g . . spectral of ^ l r 1 1 0 .

632

then the deformation does not seen to be sufficiently stabilized. Al¬

though no odd-particle coupling calculations have been nade in connec¬

tion with this early work, many aspects of their results clearly show

that the calculated V-vibrational amplitudes are too large. The cited

nns V-value of the ground state is 23.3°. Furthermore the ratio E(4+)/

E(2+) is 3.17 experimentally but 2.76 in ref ' and the spacing in the

iT-band is E(3+) - E(2,+) » 144 keV experimentally, but 281 keV in

ref. ' . In the f i t ted potential of f ig .6 i t is precisely these quanti¬

ties which make the large potential gradients necessary.

The latter.or even better the V-band staggering parameter (E(4,») -

E{3+))/(E(3+) - E(2g+)), is a sensitive instrument for measuring the

quality of dynamical calculations with regard to the t degree xjf freedom.

Toe calculations with microscopically determined potential surfaces are

not yet technically adotuate, because as pointed out in ref^' they are

sensitive to the prescription for the pairing. As an example of the pos¬

sible role of pairing, Kumar ' has recently shown that the anomalously

low-lying 02 state in bt can be reproduced if the pairing collapses

in certain regions of deformation space.

To summari7e the present discussion of attempts to describe transitio¬

nal nuclei as vibrating and rotating collective cores, there s t i l l ex¬

ists very l i t t l e empirical evidence that the non-yrast vibrational mo¬

des are being treated in terms of the appropriate degrees of freedoms.

I t is an important task for the future to investigate whether more re¬

fined calculations can supply this evidence,and if not to pin-point the

flaws in the philosophy behind the calculations.

3. Core and one valence particle

In formulating the quasi-molecular model, Bohr pointed out that a few

valence nucleons may have to be treated separately. We will consider

the general case later and start by examining odd-mass nuclei, where i t

is obviously necessary to take into account explicitly the coupling of

the odd nucleon to the core ' ' ' z ' . During the last few years improve-

633

ments of sxperimental techniques have provided a great deal of new in¬ formation on odd-A spectra, and models based on the Bohr Hamiltonian have proved to be very useful for interpreting the data. In 1975 Meyer-ter-Vehn ' demonstrated the importance of the intrinsic If degree of freedom, and this has spurred on several developments of model techni¬ ques. In order to fully appreciate the near equivalence of all models starting from the assumptions of the Bohr Hamiltonian and in order to get a feel for the strengths and limitations of various formulations, let us now dig into technicalities starting from the Hamiltonian (1) which can be written

Here H ~ . is the single-particle Haroiltonian in an adiabatic core part field corresponding to some fixed set § of the collective coordinates. It is assumed that the total core field acting on the odd particle is determined by the core collective variables j< through some prescription satisfying the requirement of rotational invariance, for example

(8) V(c<,r)= U(r*{i + Z [jjj- \(

This total field can be expanded around a static field defined tyS^ the zero-order term goes into H t while the remaining terms consti tute H-jnt(8$)- For example, choosing 5 = 0 gives to first order in jg

where the potential in H*°° is U(r2).

This is the intermediate coupling Hamiltonian. It can be diagonalized in a basis where eigenstates |x R t ^ of H c o r e are coupled with eigen-states |nlji2,> of H t to states of good łotal angular momentum |T nil; R j I M ^ . In practice the basis space must be truncated with

regard to both the core and the particle spectrum. The eigenstates of the spherical shell model Hamiltonian H*"5 appear in (2j+l)-fold de-oar t

generate j-shells, and since all members of a j-shell are needed for the angular momentum coupling it is in practice possible to include on¬ ly a small number of j-shells. The major advantage of the intermediate coupling scheme is that all contributions to the matrix elements have a clear physical significance. H and H simply give the eigen-

CO' c Pa r Ł energies of the core and shell model Hamiltonians respectively as dia¬ gonal contributions. The radial part of H . can be determined from the shell model or to a fair approximation be replaced by the constant t= 40 MeV. The remaining terms in matrix elements of ¥ . are trivial geo¬

metrical factors and the reduced core matrix elements K^'^'fZtf t #^>

of the collective variables. Thus it is possible to perff"1 an inter¬

mediate coupling calculation to any theoretical or experimental core

whose energies, multipole moments and transition rates can be calcula¬

ted or measured. The most common method is to consider only the sjarjrj-

pole mode, which of course is a good approximation only when static oc-

tupoles and hexadecapoles are absent and the dynanrca! ones 1 if iic» •<•

energy , and further to approximate the core by a spnencal vipratc-r.

t> few other even simpler cores have recently been stud'ed by ^duat.a a^d

Sheline '. It is always possible tc s o l a c e d fe w o' tne m a t n < ele¬ ments by quantities observed experimentally m tie core, ^ H " severei-

ly improves the agreement between the coupling caic'a'.ior 3rc tne e«-

perinental proyerties of tne Cdd-S system.

The spherical phonon core takes :nto account tie * - H ^oiir^v: •(- ae-

grees of freedom embodied in the Bohr Harm 1 tor.iar. -Uhouci1- t^f f-ztt-r

states probably cannot be found if i^ysical cores, th-s need --et be

r" serious objection if it is simply a matter o* f raqmentat" ;r, ^.e ir

additional, e.g. single-particle, degrees o' 'reedom. *he r a s'.ronc

quadrupole matrix element from a higher model state car se ta>er. to si¬

mulate several weaker couplings. However, a spherical core :s net aDDrc-

priate for many transitional nuclei, which are believed to *ave rather

stably A -deformed although v-scrt collective potentials. A scheme ae-

vised especially for this situation is develooped in ref '. "he core

states, instead of being solutions to the Bohr hamiltonian in a ooten-

tial V ( ^ ) = 1/2 C p , are solutions in an arbitrary potential V,|(S )

and may therefore be referred to as deformed phonons. From a group theo¬

retical point of view they are completely analogous to the spherical

635

phonons and are consequently easy to handle. The method can be further generalized by including a B-dependent tern V ' ( p , j r ) in the collec¬ tive potential. I t is diagonalized simultaneously with H-n^. Then both V'(j3,Jr) and H i n t give off-diagonal contributions to the Haniltom'an matrix. In the l imit of inf ini te basis spaces this is completely equi¬ valent to f i r s t solving the Bohr Hamiltonian and then performing an in¬ termediate coupling calculation. In practice the device of a deformed phonon basis gives a more economical truncation up to the point where V'( j f l , i r ) begins to dominate over H. .. Let us examine the relative im¬ portance of these two terms since i t is interesting also from the phy¬ sical point of view. To this end a calculation is made where a j=11/2 particle is coupled to a jB-deformed V-unstable core with parameters characteristic of a transitional nucleus. In fig.7 the quantity <cos 3 is marked for each yrast state of the odd-A system. I t can be taken to define an effective * as

t = I arccos <"cos 3*^

Dut then u must be realized that the limits * -0° and i=60° cannot oe asproacned even i f there is a very deep ar.ially symmetric minimum in the collective potential, f i g . 7 shows that the odd particle ha?, a strong polarizing effect on the core through H. .. Jts nature on the dominant coupling scheme in the different odd-A states, can be strong coupled, favoured decoupled or unfavoured decoupled. V the core polarization in the 1=1/2 state were to be achieved instead for the 0 state of the core by a potential term proportional to cos It , the prolate-oblate difference would have to be about i Mev.

At f i r s t sight the large fluctuations in deformation for different odd-A states would seem to signify that dynamical cores are not very simi¬ lar to rigid ones. However, this is not so, as wil l be seen later on. Here we wil l merely inspect the transition from oblate to prolate in a dynamical core. Fig.8 shows results obtained with a j=9/? particle coupled to a core which again has a small but relatively stable £-de-•formation. The prolate-oblate potential energy difference is varied by means of a potential term proportional to cos 3f , so that the middle cf the abscissa corresponds to the ^-unstable case, with predominant-

636

<cos3j> -

1 3 5 7 9 11 13 15 17 19 21 I (h/2)

Fig. 7. A particle in an h 1 1 / 2 shell with an f^,2 shell located

5 MeV above has been coupled to air-unstable core with a small but

rather stabler-deformation. The quantity <cos 35> measuring the

effective * -deformation of the core is plotted for the lowest state

of each spin up to 21/2. Solid lines connect the low-spin, the un¬

favoured decoupled and the favoured decoupled states respectively.

637

o -

-8 1

-4 i

= O°)-V

-±. 0

j J_

u 60*) (MeV)

8

Fig. 8. The energy spectrum of a system consisting of an h g . , particle coupled to a core with a small but rather stable fi-deformation. The prolate-oblate potential-energy difference on the abscissa implies a stably prolate minimum on the far left , ł r- instabi l i t y at the mid¬ point and an oblate minimum on the right hand side.

638

1y prolate shapes to the lef t and oblate ones to the right. There is a transition from a decoupled band on the prolate side through the t r i -axial regime to a strong coupled band on the oblate side. The results are similar to those obtained for a rigid rotor when tt is varied from 0° to 60°. An important thing to notice is that large prolate-oblate potential-energy differences are required for the transition. Since the

•different types of bands are observed experimentally, i t appears that some transitional nuclei are shape stabilized in a way that cannot be accounted for by realistic potential terms alone.

In p r i n c i p l e the only l i m i t a t i o n of the in te rmed ia te coup l i ng HamiHo-

n ian (9) r e l a t i v e to the f u l l Hami l ton ian (1) l i e s in tne neg lec t of

second and h igher order terras in j j . In p r a c t i c e there may appea r son?

d i f f i c u l t i e s . U n t i l r e c e n t l y one of these was the treatment of p a i r - n g .

and the approach was e s s e n t i a l l y l imi . ted to j - s h e l l s we l l above or De-

low the Fermi su r face . However, a major breakthrough on t h i s po in t t"-as

been made by Donau and Frauendor-f J / , and t h i s w i l l be s t j d i e d -n de-

t a i i M a t e r . Another is t h a t hexadecapole deformat ion is d i f f i c j l ; tc

take i n t o account even to z e r c ' t h .-.rder and such s term may p lay a s ' - -

fiificant r o l e in some odd-A s p e c t r a . Most i e^ ious is the f a c t tha t • "

t r a n s i t i o n a l nuc le i where the ' " i r c t o ^ d f suadrupole c o r e - p a r t i c l e " i -

t e r a c t i o n term is more tnan a wes> >.-rt j r b a t i o r , nume r i ca l convergence

requ i res an increase o f the s i n q l e - D a r t u l e M; .T - . to dimens'Grs t ha t

soon become unmanageable. The reasor, for tfc.-s is obvious i f ortr cori-'-

ders the many d i f f e r e n t coup l ings t ha t a r i s e when a deformed s * e l ' "•'.-

del p o t e n t i a l is d iagona l i zed in a spher i ca l bas i s . F ig .9 crudely • ! ' . , -

s t r a t e s the e f f e c t . In the two spectra to the r i g h t a p a r t i c l e hat tee-

coupled to a schematic core c o n s i s t i n q of qround and gamma bands ^p to

sp in 10, w i t h energ ies and ma t r i x elements taken from a j f - uns tab le vib¬

r a t o r w i t h a £ - d e f o r m a t i o n of about 0 .16. For the middle spectrum the

t runca ted s i n g l e - p a r t i c l e space conta ins only the h„ , , s h e l l , wh i l e m

the r i g h t - h a n d one the f^ ,„ and f^,., s h e l l s have been inc luded at 1 MeV

and 3.5 MeV resp. above the h „ , p s h e l l . This i s r e a l i s t i c f o r protons

in the A=l90 mass r e g i o n . The f^,-, she l l a f f e c t s p r i m a r i l y the 1 = 7/?

l e v e l , wh i l e the more d i s t a n t f ^ , 2 she l l changes the whole charac te r

o f the spectrum. In a sense t h i s r e f l e c t s a p o l a r i z a t i o n of the core

toward the p r o l a t e s i d e , but i t should be r e a l i z e d t h a t a s i m i l a r e f -

639

MeVi

1.0 -

0.5

11

3 7 9 17 1

13

11

11 13

EXTENDED REFERENCE EXTENDED CORE MODEL PARTICLE SPACE SPACE SPACE

Fig. 9. Three j«9/Z spectra resulting from coupling calculations made on exactly the sarn premises except as regards the truncation of the basis space. Further details are given in the text and the caption to fig. 11.

640

feet persists with a deformed rigid core '. For comparison the spec-trim to the left in fig.9, shows the effect of including the ten addi¬ tional core states of lowest energy in the coupling calculation. A way of resolving? these difficulties is to make the Taylor expansion of the core field not around CT= 0 but around -a value of that in some sense corresponds to the static average of the core field and minimizes H i n t(<J). Since there is isotropy with regard to the spatial orienta¬ tion of the nucleus, this can be achieved only for the intrinsic coor¬ dinates of g . In the Intrinsic frame of reference the potential '3; is

(10) HS'l)- U(rł{f-aewyX'W •

and the zero-ordsr term from an expansion around 5= e,"&,.... T-vey' a deformed'shell model potential whose e'igenstates can be usec 'c<- the s ing le-par t ic le basis. The three d i f f i c u l t i e s mentioned for tne inter¬ mediate coupling scheme are now el iminate'd. F i r s t l y the-defomed soten-t j a l in H «. sp l i t s the degeneracies of th_e j - s h e l l and a BCS pair ing part t . ' • c a l c u l a t i o n on the deformed s i n g l e - p a r t i c l e spectrum can ta*e ' r t : account a ' s i t u a t i o n wnere l eve l s from the same j - s "he l l apoear o c t * i-bove and below the Fermi su r face . Secondly i t is poss ib le to • n c l j d e e . g . hexadecapole terms i n the s h e l l model p o t e n t i a l . T h i r d l y mos* o' the coup l ings between d i f f e r e n t s i n g i e - p a r t i d ? s ta tes ai-e absc-ceo - n -to the de^srmed s i n g l e - p a r t i c l e energ ies anc wave ' ^ n c t i c n s , and on ly a very small number of l eve l s near the Fermi sur face need to be i n t ! * -ded i n the bas i s . However, a new d i f f i c u l t y has appealed. The Harm I t a -nian i s now

<"\ H = Hcore<Sf) + C " " )

Here H i n t contains the par t ic le v ibrat ion coupling, while the par t ic le rotat ion coupling is marked as being present in the deformed sincPe-par t ic le Hamiltonian which refers not to the laboratory 'rame but to the i n t r i ns i c frame defined by the core. The d i f f i c u l t y is that no app¬ ropriate basis for diagonalizing (11) is known to ex is t . A simpli fying approximation, that we w i l l study in some d e t a i l , is to neglect the

vibrations around * , » , ,in other words the rigid core approxima¬ tion. Then Hłf)t disappears and Hcore becomes * rigid rotor Haniltonian

K »

where J£ is the angular momentu* of the core, * refers to the intrinsic axes and \ ire th^-constant moments of inertia. When diagonalizing in basis states that are not themselves eigenstates of the first term in (12), it is possible to choose the* in'such a way that the contributions from the particle-rotation interaction in the second term a»>e transform¬ ed away. This choice is the strong-coupling basis '. Then £ is most con¬ veniently expressed as the difference between the total angular momen¬ tum and the single-particle contribution j and the Hamiltonian (12) becomes i'3) H . I ( I > , ) - Z IK(e:) j , • j * ) /2

Here the particle-rotation interaction has been transformed into Hc o r e

instead and emerges in the form of a Conolis term - 2 I « J « / 3 « and a recoil term £ itf/2.3v . The recoil term has the appea ranee af" a single-particle term and could in principle be included in H* !f

T 8 v so i t would have to appear e x p l i c i t l y in the deformed shell model and i t is not correct to leave i t out completely. Recent developments

ID j

in the solution of (13) can be found in ref. ' .

T*ie detailed derivation of (13) has been carried through to emphasize three points:

i) The strong-coupling Hamiltonian (13) is identical with the Hamiltonian for intermediate coupling to a rigid core. The on¬ ly difference lies in the bases commonly used to diagonalne them.

i i ) The use of the strong-coupling basis does not mean that an adiabatic approximation is made, unless the single-particle basis is truncated to a single orbital. Generally there are non-adiabatic Coriolis and recoil coupling terms between the

61*2

single-part icle orb i ta ls .

i i i ) The pairing calculation usually of monopole type,is made on

the adiabatic single-part icle orbi ta ls. This may or may not

have connection with a hitherto unresolved d i f f i cu l t y of con¬

siderable theoretical import, namely that the Coriolis coup¬

l ing matrix elements seem to be anomalously weak in nany ex¬

perimental cases ' .

The models of Meyer-ter-Vehn ' and Faessler and Toki ' have the same

Hamiltonian as in (12), but technically they are hybrids between the

intermediate coupling and strong coupling approaches. Both involve the

strong-cojri ing representation for the sake of the pairing calculation,

which makt; .t necessary to have a r ig id core. In addition they are

both burdened with one of the disadvantages of the intermediate coup¬

l ing modeli namely a single-part icle basii space consisting of spheri¬

cal j - she l l s . The reason in the case of the Faessler-Tofci model is that

angular momentum recouplings can be made to allow for a variable moment

of inert ia in the core.

151 Recently Obnau and Frauendorf ' have devised a technique for perfo1--

ming a 'deformed' adiabatic Pairing calculat ion completely withir, the

framework of the intermediate coupling scheme, thereby retaminc i t s

advantage of allowing a completely arb i t rary set of core levels aid

quadrupole matrix elements. In addition they take into account the nor¬

mally neglected couplings between par t ic le and hole states, which gives

at any rate some contr ibut ion to the Coriol is attenuation ef fect . ; t

should be strongly emphasized that i t is t r i v i a l , f r om a computer pro¬

grammers point of view, to include their treatment of pairing in a con¬

ventional intermediate coupling code. The new parameters are the pair ing

gap A and the Fermi level A . F i r s t the usual intermediate coupling Ha¬

miltonian is diagonalized, with the only modification that the energies

of a l l core states are taken to be zero. Apart from truncation effects

th is is equivalent to a calculat ion of adiabatic s ingle-par t ic le levels

in a var iety of deformed s ta t ic shell model potentials corresponding to

the d i f fe rent orientations and shapes that are taken on by the rotat ing

and v ibrat ing core. From the eigenenergies 6 . ^ the quasipar t ide ener¬

gies and occupation amplitudes are given by standard formulas of BCS

pairing theory

Next the core-particle coupling Hamiltonian is formed in the basis de¬

fined by the eigenvectors "\and diagonaiized. Its matrix elements are

(16) H ij • £ i </ij * ("i j * v. V j) H ijr*

Here the core Hamiltonian, which was diagonal in the intermediate coup¬

ling basis, has been transformed to the new basis via the orthogonal

transformation defined by the vectors Yj • If matrix elements of the HI

and E2 operators were calculated in the usual way between the statesV-

after first diagonali2ation, they can now easily be calculated between

the non-adiabatic states. A pairing factor U-UJ + v.V; must be appended

to the contributions to the Ml matrix elements.

Let us next examine the snag in the strong-coupling approach, viz. the

rigid core approximation. It was stated above that the concept of a ri¬

gid core shape cannot be taken literally. The essence of the rigid core

approximation can be understood in the intermediate coupling approach.

There the core properties influence the odd-A spectrum through

i) the energies of the excite-* <">-e states,

it) the reduced multipole m a t n * dements between these states.

These two sets of quantities can be referred to as the total multipole

field of the core, and the rigid rotor can be viewed as a few-parameter

family of multipole fields which is useful to the extent that t can

be adjusted to reproduce the dominant features of the nuclear multipole

field. The parameters are the intrinsic multipole moments Q\ and the

three rotational moments of intertia 3-K , «. = 1,2,3. The multipole mo¬

ments usually taken into account are the quadrupoles. The octupoles and

higher odd moments lead to broken reflection symmetry and no core-par-

6kk

tide model has yet been formulated that allows for them. One possible

application of such a model would be to analyze bands built On the high¬

ly deformed third isomeric minimum in the lighter actinides. The hexa-

decapoles are easier to include because they do not break any symmetries

and in fact when there is a sizeable hexadecapole moment it may have im¬

portant effects on the odd-A spectrum. This is illustrated in fig.10,

where an h^,g par.ticleis coupled to a core with the deformation £.-

0.16. Hexadecapole deformation changes the energies of the adiabatu

single-particle levels and there are significant effects on the aid-A

spectrum which are analyzed in more detail in re< '. The three moments

of inertia are in principle free parameters.but they are often for con¬

venience connected to the quadrupole moments through a formula of hydrc-

dynamical type.

In order to get some insight into the differences between the multipole

fields of rigid and dynamical cores, let us study the quadrucole moae.

In the case of a nucleus with stable axial symmetry it is well-kno^n

that the effect of vibrations ot\ the ground and gamma bands a>"e simula¬

ted by a rigid rotor with a small effective S • The guadrupole matrix

elements can be closely reproduced . The vibrational bands, complete¬

ly absent ir, the rigid rotor, couple only weakly to these two lowest

bands.

Larger differences could be expected between a rigid rotcr and soljtions

of the Bohr Hamiltonian in the 5f -soft collective potential of" a transi¬

tional nucleus. A priori, one might suspect that isotropy with regard to

"8 could be important for the odd-A spectrum in analogy with the isotro¬

py with regard to the spatial orientation angles aivinq rise to the V.o-

riolis force.

Fig.11 shows the results of coupling a j=9/2 particle to schematic co¬

res whose excitation spectra include only the ground and gamma bjnds up

to spin 10. Their mass functions have the hydrodynamical functional form

and their parameters are typical for a transitional nucleus in the A»

190 nviss region. The label ff-unstable refers to the solution of the

Bohr Hamiltonian in a ^-deformed but JT-unstable potential. In the po¬

tential of the dynamical rotor the minimum in the A-direction is 4 MeV

6*5

1500 - .

1000

500

\

9 —

13 -li —

— 13

- 9

•— 13

li — — — n

3

55 (-

-05 0 05

0"

- 05 0 05

30°

-05 0 05

60' Hf. 10 The lowsr | iarl of I ho f l p u r e »how» a d l a b a t l c »]rif;li>-p u r t l c l e apocli-Q of o r b l t n l i wltti p r o d o i i n a l l y h , c o n t e n t ,

l c u l t c S At ( O 16 ind tllf Toront v a l u e * of ' ahŚ calculatocS At (. = O. 16 ind tllf Toront v a l u e * of 0 6

T l i H.O. a l i e l l model pararaotor* are K= 0 . 0 6 2 0 , *. z 0.61**, a p p r o f r i a te l o r protons 1b the A=130 t'eglori. 7ite upper part g i v e s the e x c i t u t l o n s p a c t r a wlion h p a r t i c l e r r a i t o occupy thpbe o r b i t a l * 19 couplod to > r l c i a t t - i a l l a l core wi th E ( 2 * ) : }UO keV.

MeVA

1.0 -

0.5 -

CORE ENERGIES

5 13

11

7-UNST RIGID ROTOR

RIGID ROTOR

RIGID ROTOR

QME'S

0 L-

7-UNST /-UNST DYNAM ROTOR

RIGID ROTOR

F ig . 11. Some j =9/2 spectra r e s u l t i n g from intermediate cou^ l inc

ca l cu la t i ons w i t h d i f f e r e n t sets of core energies and quad rup l e

mat r i x elements.The fo l l ow ing parameters are the same fo r a l l four

cases: E(2+) = 358 keV, B(E2; x R ->TR) = 0 , B(E2; 2 + - > 0 + ) = 0.419

k = 40 MeV, A = 193, Z = 79.

deeper at tf=30° than at S=0° and 60°. The * =30° ' r i g i d ro to r ' is cho¬

sen to have the same exc i ta t ion energy and quadrupole matrix element from

the ground state to the lowest 2+ state as was calculated in the "s-unstable'

case. The f i r s t two spectra of f i g 11 show the ef fec t of changing the co¬

re energies. The increased s t i f fness of the core rotat ional bands when

the ro ta t ion-v ibra t ion interact ion is removed is straightforwardly ref¬

lected in the odd-A spectrum. This is a major drawback of the r i g i d core

approximation. I t can be circumvented by means of a variable moment of

iner t ia , which however is tedious to implement when the strong-coup¬

l ing basis is used. The three right-hand spectra exh ib i t the ef fects of

changes in the quadrupole matrix elements. Only small sh i f ts of the odd-

A levels occur, and they can be interpreted to re f lec t the p o l a r i z a b i l i -

ty of the jr-unstable core. I f the is deformation of the equivalent rotor

core had been chosen s l i gh t l y in excess of 30°, the sh i f t s would have

been even smaller.

A few of the core quadrupole matrix elements are given in table 1. One

systematic difference arises from centr i fugal stretching of the dynami¬

cal cores. Another is that the gamma to ground band matrix elements are

largest from odd-spin states in the r i g id ro tor , while in the ir-unstab-

le core they are largest from the even-spin gamma-band members. The lat¬

ter difference has recently been studied experimentally by means of an 194 22\

analysis of Coulomb exci tat ion y ie lds in Pt ( ref '). Due to the

experimental level spacing i t is the transi t ions to the even-spin 5 -

band members that are important fo r the y i e l ds , and they were found to

be weak as the r i g i d rotor model predicts. However i t is of course not

excluded that the hindrance of interband t ransi t ions could be of non-

collective origin.

The fl-vibrational core states couple weakly to the ground- and <-bands

However there are higher Jf-vibrational states that couple strongly to

the i'-band .many of which do not have analogs in the rigid rotor. Their

effect on the odd-A levels is indicated in the lefthand spectrum of

f i g . S. The middle spectrum is identical with the one to the lef t in

f i g . 11.The yrast states are not shifted at a l l . However, already in the

9/22 i > r a r e s t a t e there is a significant change in the energy, because

its wave function has a large component from the 22+ state of the core

648

Tablet Some quadrupoie matrix elements for the three cores referred to in f igs. 9 and 11. They are scaled so that <0*//Q//2*> = -100, and the phases are consistent but not unique. All diagonal matrix elements are zero.

transition ^-unstable

ground * ground band

2 * 0

4*2

6*4

6*6

game * gamma band

3*2

4*2

5*3

5*4

6*4

ganma * ground band

2*2

3*4

4*4

5*6

6*6

-100

-165

-222

-275

138

-134

-160

108

199

123

87

127

106

-136

higher gartma * garma band

0*2

2*3

2*4

4*4

4*3

- 62

101

- 62

-136

-146

higher garrma * ground band

2*2

4*2

beta * ground band

0*2

2*0

2*4

beta * ganma band

2*2

0

0

27

- 12

- 46

34

dynam, rotor

-100

-164

-222

-274

157

-117

-156

139

176

122

105

102

122

-100

- 34

59

- 42

-167

-128

12

12

24

- 12

- 41

31

rigid rotor

-100

-156

-212

-255

158

-104

-145

145

138

120

118

70

126

- 61

-

-

-

-176

-1C0

-

27

-

-

-

-

6*9

which couples to the Jr-vibrational 0 + state (table 1).

In conclusion, a rigid core with the same effective 2T as the dynamical

core is quite adequate for describing e.g. the odd-A states populated

by the yrast cascade, or in general bands built primarily on the ground

band of the core, and it is fairly good for describing states built pri¬

marily on the y-band. The higher vibrational states become important

in more ambitious calculations. However, a low-lying vibrational octu-

pole may in some cases be very important because it couples directly to

the ground band of the core. For example, the low excitation energy of

the 13/2+ band heads above the 7/2" ground states in the N=83 isotones

cannot be reproduced by a rotor plus particle calculation with standard

shell model parameters. This may be because there are low-lying octupole

states in the N=82 cores. A perturbation calculation shows that the

[fjtp 3") 13/2+ configuration then comes down close to the (i 73/2 ® '

and that the direct coupling between the 0 and 3 states is of the

right magnitude to account for the downward shift '.

4. Core and many valence particles

As mentioned above, the 'stiffness' of the rigid rotor rotational bands

may have to be corrected for, whereas the full dynamical theory often

does better in this respect. However, the latter is also incapable of

reproducing 'upbending' or 'backbending' effects in the yrast line,

caused by the rotational alignment of core particles with high spins.

In order to take this mechanism into account it is necessary to ab¬

stract all the aligning particles from the core and to include them in

HpartOf eq- ™-

54 As a simple but instructive example we consider the nucleus og^pg- Fig. 12 shows the 1=0 potential surface and a spectrum obtained by solving a zero-valence-particle collective Hamiltonian in this potential,scaled so as to f i t the experimental position of the midpoint between the 2+

and 4+ levels. The small experimental E(4+)/E(2+) ratio is not reproduced. There is also another theoretical spectrum shown in f i g . 12 that results from a two-valence-hole calculation carried out by Paar '. In this

650

(MeW

(T-2- PART. (Paar)

EXR O-PART.

Fe 0.3

0.6 Fig. 12. The experimental excitation spectrum of Fe is shown together with a theoretical spectrum obtained by Paar from the Alaga model, and another one obtained from a zero-valence-particle collective Hamiltonian with a microscopically determined potential shown in the lower part of the figure. The energy scale in the spectrum from the latter calculation has been .adjusted for the sake of the present compari son.

651

calculation two proton holes in the f ^ shel l , located imnediately

below the Z=28 gap, are coupled to a spherical quadrupole vibrator,

and the description of particularly the yrast .states improves drasti¬

cally. In a basis /OR;iX where the two f , , j holes are coupled to spin

«X6 and then coupled with a core phonon of spin R to total spin I , the 54 composition of the lowest states in Fe is obtained as

| 0+> = 0.81 |0 0 ; 0> + 0.49 |2 2 ; 0> + . . .

| 2+> = 0.64 JO 2 ; 2>+ 0.50 | 2 0 ; 2> + . . .

( e*y = 0.45 | 0 4 ; 4> + 0.55 |2 2 ; 4> +

0.53 / 4 0 ; 4_>+ . . .

\t>+) = 0.79 ( 6 0 ; 6 ^ + 0.45 f 6 2 ; 6 ) + . . .

The second component in the 0 state reflects the shape polarization from the two holes. The angular momentum of the 2 state is about equally shared between the core and the valence particles, which is here characteristic for a collective rotation. A radical structural change has occured in the 6+ state, and to a lesser extent in the 4 . The two holes have aligned their spins and are almost completely responsible for generating the angular momentum. In this situation we know that the core is polarized to an oblate deformation ar.d that the angular momentum arises from a non-collective single-particle rotation around the symmetry axis. The mechanism underlying the change of the

Me potential in f i g . 1 above is completely analogous. There are twc protons and two neutrons in the d^.^ shell aligning to give a total angular momentum 5/2+ 3/2+ 5/2+ 3/2 = 8. The more collective rotational states are s t i l l Dredicted to exist but at higher energy. The two nuclei Fe and Ne have only a few particles or holes outside spherical cores with magic particle nunbers 28 and 8 respectively, and the lowest yrast states have a correspondingly simple structure. Let us consider some other situations that are beginning to be under¬ stood and whose study may lead to new approaches for more general problems.

652

In a we11-deformed prolate nucleus there may also exist favoured con¬

figurations where two quasiparticles align to give a spin of about J,

but the prolate deformation persists more or less unchanged. If the

energies of the ground band levels are4I(l*l) and the moment of

inertia is not significantly affected by the alignment, the energies

in the rotational band built on the aligned configuration are roughly

speaking^(I-J)(I-J+1) + Ej where Ej is the quasiparticle excitation

energy. Then the aligned band crosses the ground band at spin

(17) L L f T - I ^ S j )

and becomes yrast. In this situation i t may be that when the rotational frequency is large,it is no longer meaningful to perform the pairing calculation on the adiabatic single-particle orbitals, and i t is necessary instead to diagonalize the pairing and Coriolis terras

251 simultaneously ' . The present level of understanding does not l ie far beyond the unrealistically simple case outlined above, and the f ie ld lies open for new developments. One interesting apprcach is the study of so-called Routhians as proposed by Bengtsson and Frauendorf ' .

In very weakly deformed nuclei the collective degrees of freedom may play a secondary role compared to the valence particles. An example

212 is the nucleus Rn, whose yrast line has been charted experimentally up to spin 30 (ref. ' ) . The spectrum above spin 12 is shown in f i g . 13 together with a calculation that succeeds remarkably well in reproducing not only the energies but also the electromagnetic decay selection rules. Only the 154 ns 30+ isomer is not accounted for, since the calculation allows M2 decay to an experimentally unobserved 28" level. In the calculation the dynamics of the collective variables are completely ignored, both as regards the intrinsic coordinates and the Euler angles. The theoretical states are obtained by placing particles in deformed, axially symmetric shell model orbitals. The spin is taken to be the sum of al l single-particle projections on the symmetry axis and the energy is essentially the sum of the single-particle energies. All the

653

(MeV)

10

30*

- 29*

5 -

25+

- 22*

9+, 20"

17', 18"

15", 17'

12*. K +

Theory

Spherical

212 Rn

Experiment

[Horn et al )

Theory

Deformed

Fig. 13. The spectrum and yrast cascade for Rn down to the 12+

level . In the theoret ical spectra the- energy of this s ta te , which corresponds to f u l l spin alignment <:•* four protons occupying the hq , , she l l , is normalized to the experimental value. The calculat ion shown to the l e f t is made with a spherical core po ten t ia l . In the calculat ion to the r ight the deformation is adjusted for each con¬ f igurat ion so as to minimize the tota l energy. For th is case the low-multipole electromagnetic t ransi t ions allowed by elementary single-par t ic le selection rules are indicated and a l l levels relevant to the yrast cascade are included in tnę c lo t .

states are non-collective as regards the generation of angular momentum, but it is important that the static collective deformation be varied from state to state in a consistent way, i.e. to minimize the energy of each configuration. The theoretical spectrum to the left in fig. 13 shows that if the spherical shape of the lowest states is retained for all states, then not only the moment of inertia but also the micro-structure of the yrast line is badly reproduced by theory. The deformation of the highest states lies around £=-0.10, £.=0.02.

Each valence particle or hole in a given orbital has an energywise pre¬ ferred deformation, and the ensemble of valence particles and holes can be said to ineract with each other through the core polarization.

In the language of a more complete core-particle coupl ing calculation, there is an indirect interaction between the particles mediated by the coupling of each particle to the core through H. Ag). A pure shell model calculation with an inert core must take this into account through phenomenological effective interactions. Although the experinental properties of individual nuclei can often be accurately renroduced by such a device - see for example Blomavist's calculation ' for Ł3p and that of Horie and Ogawa ' fcr Fe - it would of course be much more instructive to be able to worv with more fundamentally derived inter¬ actions. For this reason the core-multiparticle coupling approach holds great promise for the future. It can be viewed as a unified auasi-molecular and shell model. The role of the core is tat.en into account neither through pure phenomenology nor through a configuration space of 'astronomical' dimensions. However, at the present stage there are initial difficulties to be overcome. On the technical side, the methods that exist to date are very primitive and more general techniques may have to wait a few years for the next generation of electronic computers. On the more fundamental side there is the violation of the Paul i prin¬ ciple, i.e. the dual role of the valence particles. These particles must also be considered as constituents of the core, because they influence it not only through H. t(jf). For example, in Paar's description of ^gFe,^ it was not possible to use a 'bare' -,Jti79 core.

655

In conclusion, although the simplest applications of Bohr's quasi-molecular model are beginning to grow whiskers from the point of view of progress in nuclear research, there are s t i l l some important points to be cleared up. More advanced applications may lead to exciting developments in the near future.

The material presented above leans on the work of many people, published or communicated privately for which I express the sincerest gratitude. Special acknowledgement is due to Prof. S.G. Nilsson and the other members of the Lund group, particula-" • Dr. I. Ragnarsson who has parti¬ cipated actively in preparing the ta'

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( 2 2 ) I . Y . L e e , 0 . C l i n e , P . A . B u t l e r , R .M. D i a m o n d , J . 0 . N e k t o n , R . : .

S i m o n and F . S . S t e p h e n s , P h y s . R e v . L e t t . 39 ( 1 9 7 : ) 684

( 2 3 ) J . K r u m l i n d e , p r i v a t e c o m m u n i c a t i o n

(24) V. Paar, I I Nuovo Cimento 32A (1976) 97

(25) R. Bengtsson and S. Frauendorf, prepr int

(26) D. Horn, 0. Hausser, T. Faestermann, A.S. McDonald, T. ' ' . Alexander

and J.R. Beene, Phys. Rev. L e t t . 39 (1977) 389

(27) H. Horie and K. Ogawa, Nuci. Phys. A216 (1973) 407

6 5 ?

PARTICLE-ROTOR MODEL DESCRIPTION OF ODD-MASS TRANSITIONAL NUCLEI

J. Rekstad, Institute of Physics, University of Oslo

The term "transitional" is often used on nuclei with properties which

fit neither into the schemes of well deformed nuclei nor the spherical

nuclei with only few particles outside closed shells. It would certainly

be better to classify these nuclei by the properties they have in common

than by the lack of properties. Even if they in some respect are very

individual, these nuclei in fact have what I will call characteristic

signatures which easily can be recognized and makes it reasonable to talk

about a specific physical system.

The most striking difference between the structures of spherical and

deformed nuclei is the occurence of rotational bands in the latter ones.

These bands have energies - proportional to the square of the angular

momentum 1.

In transitional nuclei we also find bands, but these bands are better

described by

"con " (1 - J-)2 (1)

where J represents the sum of angular momenta for all the particles

which cannot be projected out in a collective rotation.

J • U all particles

When plotting the energy versus angular momentum, these bands are

represented by parabolas with minimum point ac I ~ J. The decoupled bands

associated with the so-called intruder states gg^' h n / 2 a n d *

are well known examples, in fig. 1 are shown the decoupled i.,..-bands in

the N « 89 isotones. However, this type of bands seems not to be limited

to the intruder states.

When studying single nucleon transfer reactions one also sees a typical

difference between transitional and the well-deformed nuclei. In the latter

658

RELATIVE EXCITATION ENERGIES (keV)

to

8.

ones the spectroscopic strengths are spread widely on many states, while

in transitional nuclei one see essentially one strongly populated state of

each spin. These states are identical with the I = J states constituting

the minimum-points in the parabolas. In figs. 2 and 3 are shown examples

from the A ~ 100 and A ~ 150 regions.

The .Hamiltonian (1) may be written

H = A(l - J ) 2 » AI 2 + AJ2 - 2A-IJ /

Here, the second tern on the r.h.s. is the socalled recoil term, and the

third - the Coriolis term - describes the couplings betujsen single particle

excitations and rotation.

Trie- parabola structures te.l us '.':, :i both the Coriolis and the recoil

uffect are important when we art dej';:;.: with transitior.al nuclei. First

1 a;n going to discuss the recoil terr. before I go into some detail in the

calculations on two nuciei in the ma .->.:• 150 region. The participants in

'.his prcsject are

iron Bergen: Gunnar I.ip

Jan Lien

Oddmund Straume

from Oslo: Torgeir Lngeland

Magne Guttormsen

Finn Ingebretsen

Eivind Osnes

John Rekstad

The recoil term may be written

T,2 7 h 2

"recoil ° U J " 2J , ,, k k,ali part

K

6 •' C

0 200 £00 600 800 1000

Excitation energy (keV)

Fig 2. Spectroscopic factors from single-neutron transfer into the

Ru and Ru nuclei

Fig 3. Pick-up spectroscopic factors for the Sm and 3 Sm nuclei

661

Usually, the recoil term is considered as a part of the single particle

potential, e.g. since "•

j 2 * U + s ) 2 ~ I2 + 2£s

it could be included in the Nilsson potensiul by adjusting the coefficients

of the spin-orbit and the centrifugal terms.

This is certainly not possible if the second sum on the right hand side

contributes significantly, since this is a sum of two body operators.

Let us for simplicity assume two particles in time-reversed orbits with

j. + j? = 0. We obtain

ii'ii = • ji

This assumption is only fullfilled in the spherical limit, but we find that

also in deformed nuclei the contribution from the two-body part of the recoil

term is essential.

In fig. h is plotted the recoil energy of the 1/2 [660] and 3/2 [651]

Nilsson orbitals for different number of particles in the iji/o shell.

Pair correlations are not included. Maximum recoil energy for the state

with the odd particle occupying the orbital Q is achieved when either the

orbitals with ft-1 and Tj+1 are empty or both the fi + 1 and a * 2 orbitals

are occupied by pairs.

This strong dependence on the number of particles is removed when we

take pairing into account, as shown in fig. 5. The pairing gap A is

550 keV, approximately the half of actual value for the nuclei where the

""13/2 s t a t e s a r e observed. The pairing correlation provides that the

occupation probability is distributed on several substates, approaching

the situation in spherical nuclei where occupation of all substates is

equally probable. The result is that the odd particle is responsible for

allmost the whole recoil effect, thus this is nearly independent of the

number of pairs, e.g. the position of the Fermi level. The recoil energy

is positive both for particle states and for hole states. This is in

662

I

UJ z LU

O U UJ CC

800

400

0

S 800 1

z UJ

o UJ

er

* 13/2

No pairing

3 5 7 NUMBER OF PARTICLES

^

1 3 5 7 NUMBER OF PARTICLES

F i g 4 . R e c o i l e n e r g y f o r t h e l / 2 + ( 6 6 0 ) a n d 3 / 2 + f h 5 1 ) N i l s s o n s t a t e s

i n t h e n o - p a i r i n g c a s e ( £ 0 = 0 . 2 1 , t i / 2 3 =21 k e V )

663

iboo

O

a/- cfy iamet hractt

A =

Fig 5. The recoil energy of !.•,/, states with pairing included

contradiction to what we abtain when only the one-body part of the recoil

term is considered, as shown in fig. 6. Here, a is the ground state,

and 2. and fi, is a hole state and a particle state, respectively,

originating from the same spherical large j-state. As pointed out, the

one-body part of the recoil term may be taken care of by adjustments of

the single-particle potential, resulting in a change in the relative position

of the spherical states. Thus, if the recoil energy is assumed to be

positive for the particle state £•_ it.has to be negative for the hole

state ". and vice versa. This is not correct since the recoil energy is

positive for both classes of states.

The most direct way to verify the effect of the recoil terra is to

study the systematics in the chains of odd-mass isotopes. One consequence

is that states with small 'A values with dominating large j components

in the intrinsic wave functions are excited compared to states with

dominating small j conponents.

rigs. 7 and 8 show systematics of band head energies for the Zn, Ge

and Gd isotopes respectively. The .". = ; orbita Is from the 3rwi> ' i - * ••

and also froro h q. o do never come iow in excitation, even if these orbit.als

are empty in the nuclei on the left nand sidu ->i the- figures, ar.d filli-d

with particles in the nuclei to the riv;ht. The ground states an.- s-ail

f. states corresponding to low j values. This is considered is j result

of the recoil effect even if several other effect also raay contribute.

Finally, the recoil effect may also explain why the high spin states

(represented by the minimum points in the parabolas) only rarely occur as

ground states.

In transitional nuclei the Coriolis effect gives rise to strong coup 1;n^s

between single particle excitations and rotations. Except for the •.in-

explainable attenuation of the Coriolis matrix elements, this effect is

considered as well understood.

66 5

b)

ig 6.

•ft2

DEFORMATION

nuclear nuclear Level scheme Level scheme without recoil with recoil

included

The recoil effect when the two-body part is included. The recoil

energy is positive both for particle- and hole states.

Ge-isotop«

34 36 38 40 42'36 40 42 V. 46

NEUTRON NUMBER

Fig 7. Systemstics of band-head energies for the Zn and Ge isotopes

666

•r :R

GY

[k

z tu

TATI

ON

EX

CI

1000

800

600

400

200

• Ml'

' '3/2*

: - •

• 3/2*

. 5/2"

X \ \ \ i

° \

\

\ i

J

Gd- isotopes

łt/2- ^ 1/2*

'J

S^^a ° ' 3 / r

o- o 9/2*

n 3/2" 87 89 91 93 95

NEUTRON NUMBER

86 88 90 92 9Ł 96 NEUTRON NUMBER

Fig 8. Systematics of band-head energies for the Cd isotopes

667

To describe quantitatively the strong coupling phenomena that characterize

the structure of transitional nuclei requires a detailed knowledge of the

single-particle potential. When t'ae deformation decreases the quadrupole

term in the single-particle Hamiltonian plays a less dominating role and the

single-particle orbitals will be more sensitive to changes in the potential.

Thus, the usual technique of adjusting the bandhead energies without changing

the single-particle wave functions should not be adopted as a recipe for the

description of transitional nuclei.

We have been studying transitional nuclei in the mass regions A ss 75,

100 and 150. Here I will report on calculations on the Sm-isotopes Sm

and 3m, the first is typical transitional nucleus while the latter is

considered as a good rotator.

We assume an axial symmetric system and consider the Hatniltonian

- H. + H int rotor

where H. is taken as the Nilsson single particle Hamiltonian. Pair

correlations are taken into account by the BCS approximation .

In the present calculation the single-particle potential is determined

by interpolation between the experimental level schemes of spherical muclei

near the N « 82 and 126 magic numbers. Since the harmonic oscillator

potential applied by Nilsson is too simple to describe the interpolated

level scheme, we introduced strength parameters K for the spin-orbit

coupling which depend on the orbital angular momentum I. A good fit to

the interpolated level scheme shown in fig. 9, was achieved and should be

appropriate for the Sm and Sm nuclei. For comparison, also the

spherical limit of the most commonly used Nilsson scheme for neutrons in

this mass region is shown in fig. 9.

A quadrupole deformation e » 0.21, deduced from the B(E2) values

measured by Coulomb excitation of Sm, is used in the present calculation.

A hexadecapole deformation is determined to be e. - -0.055 from the energy

668

4.0 -

160 ISO

MASS NUMBEP 200

Fig 9. Procedure for for determination of single particle potential

parameters from the level schemes of spherical nuclei.

669

difference between the 11.'2~, 9/2~[514] and 11/2", ll/2~[505] states

observed in the Sm spectrum.

A change in the rotational parameter of more than a factor of two from

Sm to Sm makes it difficult to estimate a proper value of this para¬

meter for the Sm nucleus from the even-even neighbours. However, both

the observed 11/2 [505] rotational band and the i13/2 decoupled band

indicate rotational parameters of the order of Che Sm value. The

dependence on the single-particle structure, corresponding to different

"deformation-driving" effect of the different Nilsson orbitals, does not seem

to be of the same significance as reported for soim- N = 87 isocones.

In the present calculation the same r :ationa! parameter value was used

for all bands except for the 11/2 [305] ! ij. A good fit re the experimental

data was achieved with the parameters

11/2"[505] band: yr = 17 keV,

All other bands: — = 24 keV.

These values are fairly consistent with the chosen quadrup. K UŁ t orrui on o!

0.21.

The pairing gap parameter I = 1.1 MeV represents an average vjlue

deduced from odd-even mass differences in the actual region. The Fi-rmi

energy >. was allowed to vary within a range of values consistent with the

number of particles in the nucleus.

The calculations were performed without any attenuation of the Coriolis

force.

In fig. 10 are compared the results of the calculation with data from

single neutron transfer ractions. For the f7/2> ho/2 a n d N i / 1 s c a t e s

the overall agreement between experimental and theoretical pick-up spectro-

scopic factors is gratifying. The states have been sorted in groups according

to the dominating spherical component.

67O

10

0 5

10

0 5

10

05

-

i -

5"

1 -

-

1" 1" 2 2

9"

1

7-2

9" 2 •

11" 2

-

1

151 Sm

i

i

n-2

1

, THEORY

I

i

h 9

i

' 7 2

11" 2

1

1

i

1.0

0 5

1 0

05

10

05

151 Sm, EXPERIMENT

h i !

s-i" r, ll 7" 2

11"

I 500 1000 1500 0 500

EXCITATION ENERGY (keV)

1000

( ' • • • ; > . • ! i ••; •

' 1 ' !

:y.,v \- .i i n .i 1 . m i l . .i l i - u l . U i u l p i r l - - ; | i s p c i - t r o s c o p i c

11*

1

1500

The two ll/2~ states shown in the upper section of fig. 3 are the

ll/~[505] band head (261 keV) and the H / 2 ~ (1378 keV) member of the 9/2~[514]

rotational band, respectively. Both these states are originating from the

h u ,, spherical state situated below the N - 82 gap. These states carry

about 95Z of the 11/2~ strength observed in the pick-up reaction and interact

only weakly with the other negative parity states.

The states shown under the f-,2 label represent the ground state band

in Sm. Both the spectroscopic factors and the y-data support this inter¬

pretation. The main component in the intrinsic wave function of this band

is the 3/2~[532]Silsson orbital, though the experimental level scheme reflects

that the band is considerably perturbed. The 5/2 state is supressed in

energy below the 3/2 state representing the band head. In Gd, which

has a spectrum very similar to the Sm spectrum, the same band behaves

like a normal rotational band with the 3/2 state as the ground state.

It should be noticed that the strongly populated 9/2 state at 175 keV

does not belong to the ground state band but to the band labelled h,.,..

The 9/2 member of the ground state band is found at 295 keV. This inter¬

pretation is in agreement with the Coulomb excitation data, and is reproduced

in the present calculation.

The ho/2 band is partly decoupled, which is surprising since the h„,,

spherical state is situated in the middle of the N » 5 shell surrounded

by other negative parity states. In order to reproduce this decoupled

structure in the calculation, it is necessary to keep some distance between

the hg.2 and f? .„ states in the spherical limit, in accordance with the

observed level schemes of the N - 83 nuclei. If the Nilsson scheme of

Lamm is applied, the structure of this h... band cannot be reproduced.

The pick-up spectroscopic factors demonstrate the h9.^ character of the

intrinsic wave functions of this band, though the experimental data indicate

a somewhat stronger coupling to the f?.2 states than given by the model.

672

The T-dccay observed after the Hd(a,3n) Sm reaction follows mainly

the three rotational bands shown in fig. 11 where this are compared with

the results of the calculation.

While the.11/2 [503] band corresponds to a particle strongly coupled to

the rotating core, both the f^,, ground state band and in particular the

hg., band, show tendencies to be decoupled in the calculations.

Even if the ll/2~[505] band follows closely the 1(1+1) rule for the

lowest states, the calculation shows that the oand couples to some extent to

the 9/2 [514] band. To obtain the best energy fit, the rotational parameter

has to be somewhat increased (17 keV) compared to the value deduced under

the assumption of an unperturbed rotational band (14 keV).

The first levels in the ground state band (f . ) are very perturbed,

while the band members higher up in the

spectrum behave more like a normal rotational band. The reason for this

particular structure can be seen from the Coulomb excitation data listed

in table 1. In the Coulomb excitation process three states are strongly

populated: the 7/2~ state at 66 keV, the 9/2" state at 295 keV and the 5/2"

state at 168 keV, respectively. The first two states are members of the

ground state band, while the latter is the 5/2 [523] Nilssoti state. The

5/2 state occurs much lower in the excitation spectrum than the corresponding

state predicted by the model at 474 keV. Thus the Coriolis coupling between

this state and the low lying member of the ground state band is obviously

larger than obtained in the present calculation and is the eason why experi¬

mentally the 5/2 state is found to be the ground state.

In table 1 the theoretical B(E2) values are compared to the values

deduced from the Coulomb excitation data. There is good agreement between

the experimental anc" theoretical values. In particular, all the strong

transitions in the Coulomb excitation process are associated with considerable

theoretical B(E2) rates. The discrepancies in table 1 are not larger than

expected from the fact that in the ground state wave function a significant part

Table 1 Experimental and theoretical values for Couloab excitation

EjIkeV)

66

70

105

168

175

209

285

295

7/2"

5/2"

3/2"

5/2"

9/2"

7/2"

1/2"

9/2"

Experiment*'

B(E2) le

0.82 i 0

-

0.013

0.14 1 0

-

0.010

-

0.45 t 0

2barn2]

.08

.03

.04

Theory b )

B(E2) [e2barn2]

0.59

0.00

0.01

0.06

0.02

0.00

0.01

0.83

«) The values are found by (d.d1) studies except for the

105 k«V- and 209 k«V-tran»ition» where the (C1.C11) reaction ia n»cd

b) The nucleus radius is K_ • 1.25 A 1/3

two

1000

u z o s

500

W '

•*»-._ n~

I

Calk Cale. ITS ' * j "

f[MS].t.«d ' « — : : : : = £ E«> C«lc

h i - band

V

4 — ? ' , y

' :>C ?" a tli

, • •

r

Eip. Calc f 1 • ban*

nt\ fly-* I Ml »7

Ei» Cale

Ofhłf

Fig 11. Level scheme of Sm , deduced from the 15°Nd («,3n) Sn

reaction, compares with the results of the present calculations.

675

of the coupling to the 5/2~[523j is missing. One should notice that in

agreement with the model prediction none of the h.,. states are strongly

populated in the Coulomb excitation process. The members of the hq/n

band are displayed in fig. 12. The band exhibits a parabolalike structure

giving the appearance of a pure -j decoupled band with the Fermi level

positioned close to the U = 3/2 orbital. However, the minimum in the

parabola occurs at I ~ 5/2 whereas a normal decoupled band with pure

j = 9/2 would have minimum at I = j = 9/2. This reveals that the hq/?

band is actually strongly mixed with other orbitals. This is also reflected

in the low value (0.89) of the decoupling parameter for the 1/2 [530] orbital

calculated in the present model. It is very gratifying that the present

model produces this feature without any attenuation of the Coriolis term.

The y~decay studies are consistent with our interpretation of the hq/9

decoupled band. Strong transitions are seen between the 21/2 and 17/2

levels and between the 17/2 and the 13/2 levels. Further, the 13/2 state

decays into the 9/2 state at 175 keV and indicates that this state is member

it the n band.

To test this description the model was applied to the well-deformed

nucleus 3 Sm. However, the information about high spin states in this

nucleus was scares, thus we carried out an investigation of the band structure

of this nucleus by means of the Nd(a,n) Sra reaction. The experiments

were performed at the Tandem laboratory of the Niels Bohr institute in

Denmark.

The results of this study is shown in fig. '3. The fspectrum was dominated

by cascades corresponding to the 11/2 1505] band and a decoupled ijo/n b a n d-

The results of the calculations on this nucleus are compared as well to these

•y-data as transfer data reported in the literature.

The deformation parameters of Sm where found to be eo = 0,23 and c =-o 055

in contradiction to the investigation of Sm, we found it necessary to reduce

676

1600 -

KOO -

1200 -

O 1000

x

o 800 -

200 -

1 3 5 1 9 11 13 15 17 19 21 2 2 2 2 l 7 l 2 2 2 l

SPIN

Fig 12. The hg/2 band in 151Sm

677

EXCITATION ENERGY CkeV] _ en o o o

o o o

Ul

c* id co cr>

313.7 I -

Wl f n

en CO CD

O

CJ1

- 0 CA>

O

the Coriolis matrix elements, especially for the i ,, band. The

calculations were performed with a reduction factor

2 2 where (V - V ,) represents the difference in the number of p,-iirs in the

cores of the n and ft1 states. With this attenuation we achieved a very

good fit to the measured i1, ,9 band as shown in fig. I4.

In fig. 15 are compared the calculated spectroscopic factors of the

*"^Sm nucleus with the results of stripping and pick-up experiments.

The agreement is to our opinion gratifying.

As a conclusion one may say that the present version of the particle-

rotor model is useful to describe the structure of typical transitional

nuclei. The main reasons for this success are the explicit inclusion of

the recoil tern and modifications nade for the sinrle particle potential

to achieve agreement with "i'.t level structure of spherical nuclei.

I-

«p. CAU,

Fig 14. Comparison between the observed and calculated i , band in Sm

680

o v

I \-i

Hi*

1 -" V/> o ^

O V

t

O

On*

•J-U ". O ^T^rT-

68 i

= > • *

U o

V

a.

E

1

AH

T 1°

o

Sj Ó V^ "tn o tn 'vj ft \n "Wj O WJ "l« o ^?

» l i

I

I Vi

I 0>

O

10

682

SHAPE OF FLATHTOU NUCLEI AROUND A=19O

F. Dttnau ,> ,v_

ZfK Rossendorf, Berclch 2

The Interest in the shape of Pt nuclei results from the

suggestion that these nuclei should be a clear case for stable

triaxial deformation realized already in the lowest members of

the ground and excited bands. Microscopic calculations of the

potential energy surfaces in this mass region do not support

this suggestion because only very shallow minima are obtained

which cannot produce a stable triaxial shape. One could think

that the kinetic energy part of the collective Hamiltonian leads

to a concentration of the wave function in the triaxial region

0 < v < 60°. It is known, that the calculation of the inertia!

functions needed for the kinetic energy operator is much more

problematic as the procedure to get the potential energy surfao-.

Thus, the question whether stable triaxial nuclei really exist

or not is likewise the question how good are the currtnt micro¬

scopic calculations of the collective Hamiltonian for transitior.-

al nuclei. The consequence of the existence of stable triaxial

nuclei in the heavy mass region would mean that the theory is

basically wrong. Therefore the arguments concerning triaxial

shape have to be carefully proved. What are arguments for and

against a stable triaxiality in Pt nuclei ?

i/ The location of the second 2 + state which lies below the

first t-+ state. However, already the position of the

second 4 + is wrongly predicted in the triaxial rotor de¬

scription.

ii/ The successful application of the triaxial rotor plus

particle model [ij to the adjacent odd-A nuclei has been

68 T

considered as a strong argument for triaxial shape. It has

been recently shown [2] that the quite alternative assump¬

tion about the y -degree of freedom, namely the coupling

to a v-unstable core explains the observed band structures

in the odd-A Pt nuclei as well as the triaxial rotor core.

This statement includes also electromagnetic transitions,

iii/ The weak population of a second 0 + state of 194»196pt ^

a Coulomb excitation experiment f3]was interpreted as a

hint, that the 2 y -vibrational state should be high-

-lying, it means the nuclei are stiff against ^-vibra¬

tions. However, in the /p,t/ reaction^"} excited 0 +

states were found which car be nicely interpreted as 2 v -

vibrational states. During these lectures we learned by

Prof. Vergnes that in the systematic study of these 0 +

states in this mass region no evidence for a shape transi¬

tion has been found and the previous interpretation as

2i' - vibrational states could be confirmed.

Recently the question of triaxiality / y-softness or not/

was investigated by analysing the results of a Coulomb excitation

experiment. The y-yields and the B(E2)'s /extracted by coupled

channel analysis of this process/ have been compared to different

models, the /-vibrational degree of freedom of which ranges from

Y -soft to /-rigid. Looking at the values found for the

/ -band it is seen that the more stiff the *• -degree of freedom,

the better the agreement with experiment. A carefull obser¬

vation of this comparison show that the real fact involved is

the weakness of the interband transitions / V -band ->g.s. band/.

Just this is a very special feature of the rotational bands of

a jr = 30° triaxial rotor compared to other collective models as

for instance the Y-unstable core. To contribute to these Y

684

questions we analysed the results of / o£ , 2n/ reactions on

Os isotopes [5 ] from which a critical branching ratio

r = B(E2. »*- »)

B(E2, 4-'- 20

could be deduced. If one believes in a stable triaxial shape

of the Platinum nuclei then a value

r v=30° = ° ' 4 5

<J is expected. The Y-unstable core gives ry_uns-t; = 0.90.

The experimental values are

rexp = 1« 2»'". 1' 6

which cannot be understood in the collective models. The experi¬

ment was recently repeated /L.Funke and P.Kemnitz/ to find out

the M1-contribution of the t-'-* 4- transition. The experimental

value given above could be confirmed because of the small M1 -

admixture. This unexpected branching ratio says us that not only

the interband transitions 4'-» 4 are weaker as predicted in com¬

mon collective models /aside from the special feature of the

V = 30° triaxial rotor/. The intraband transitions within the

y -band are also weak.

The reason for these discrepancies seem to be not understand¬

able in terms of the collective degree of freedom alone. Now also

other experimental values for the BCE^s in Platinum are available

[6]. They slightly deviate from those extracted by the group/"3]•

However, the conclusions drawn remain valid especially the weak¬

ness of the interband transitions. The branching ratio r men¬

tioned above comes out within the errors.

Yet these experimental data pointed to a fact which maybe is

the key for the understanding of the Pt nuclei. The collective

models cannot reproduce the loss of collectivity seen in the g.s.

68s

band comparing the predictions of any collective model /including

triaxial rotor/ with the values experimentally found. Obviously,

2-quasiparticle states are mixed into the collective states. The

band crossing at I = 10 is known since a long time f5J« Our very

preliminary calculations are based on the assumption that 2qp

states mainly composed of an f^^/2/ <^o-a£i.&a&\ii.on. is admixed

to the collective states. For the interaction of the 2qp modes

and the collective states the common quadrupole coupling has been

applied. The interaction between the 2qp is a short range cor¬

relation like ^-function. The result was the following. The

admixture of 2qp components of course tends to describe the de¬

crease of collective transition strength. However, at the same

time the y= 30° triaxial core and the y-unstable core give

equivalent transition probabilities for the interband and intra-

band transitions. From these very preliminary calculation one

could say: the question whether Platinum is y-stiff or y-soft

cannot be decided in that way. Thus our conclusion is that the

nuclear shape in Pt is still unclear. There is no right to

prefer a certain model for the interpretation of available data.

REFERENCES

[ij J.Meyer-ter-Vehn, Nuci. Phys. A24-9. 111 /1975/.

[2] F.Dttnau, S.Frauendorf, Phys. Lett. .7JB, 263 /1977/.

[3] I.Y.Lee et al., Phys. Rev. Lett. 3_2, 684 /1977/.

I>] E.Seltz, Nucleonika 22, 33 /1977/i Proceedings of the

Masurian School 1976.

[5J S.A.Hjort et al., Kuci. Phys. A262. 328 /1976/.

[6] K.Stelzer, F.Rauch, Th.W.Elze, Ch.E.Gould, J.Idzko, G.E.Mit¬

chell, H.P.Nottrodt, R.Zoller, H.J.Wollersheim, H.Emling,

Phys. Lett, /to be published/.

686

ODD - EVEN EFFECT IH THE NUCLEAR SHELL - MODEL

FOR NUCLEI WITH N = 2 8 AND N = 5 0

' A . B a ł a n d a . ' ° •• ''A- - '

Institute of Physics, Jagellonian University, Cracow

The effective interaction energies were deduced from the experimental data for single neighbouring nuclei with N = 28 and N = 50. Odd-even effect, which was found, manifests itself by the dependence of the effective two-particle interac¬ tion energy on the number of particles which are filling the same orbit. The results were compared with the predictions obtained for £ -function interaction with a varied value of spin exchange term. Comparison with the experiment is given.

' • . . ' . J '.•• • - •

1. Introduction I t i s well known L1|2] that nuclei with N = 2o and i> = ?^

are well described by the nuclear shell-model in which tłr- ''J0?. and Sr can be treated as cores. Using the phenonenologic?! approach, in which the number of catrix-elenonts con be reduces by considering only the lowest dominant configurations, one c-.>r: investigate the two-body interactions defined by Vj = j ^ | 3-jJ |V|j-i Ćo ^/^ • These interactions aro treated thor: as parameters to f i t the experimental data. Any m.-iny-body matrix-element can be expressed as the linear cor.Mna-oion of Vj[3] . For any state described by a pure configuration snd energy Ej one gets an equation of the type

I 3 j Vj = f fEj, DHEE, EG)^ ' (1)

where a, are coefficients, DNEE denotes difference of the C nuclear binding energy [4] and E denotes the Coulomb energy

of the valence particles. If energy states are described by

687

the mixea conl'igurations, the equation (1) should be modified

a little. For a large number of states belonging to several

nuclei, one can obtain effective interactions which should re¬

produce positions of all the model-levels within the region in

question. Results obtained from the single neighbouring nuclei

[43 show the odd-even effect in the f„ <2 region. A similar

effect was observed in the previous work [53, where VQ and v

2

were different for ^°Zr and ^1Nb. The aim of this work is to

show the odd-even effect in the P-w2 S9/2 region (° ZrC6j, ' Nb

[7], No[8j). Results of the calculations presented here are

more exact and more complete than in [5J . The odd-even effect

is discussed. The S -function Interaction with e small spin-

exchange term can be taken as a aero-order approximation of the

two-particle interaction. Values predicted by this interaction

are compared with our results in sect. 3. Comparison with the

experiment is given in sect. 4.

2. Derivation of the effective two-body interactions

Notation, sign convention and method of obtaining the

affective two-body interactions from experimental data for the

simple case of fn/p orbit only, were presented in work [4-J .

In tnat case the set of equations of type (1) allows to calcu¬

late directly the two-body interactions. When two or more or¬

bits are included into considerations, the method is following.

Experimental energies corrected for DKBE's and single-particle

energies (or E also) should be equal to the eigen-values of

many-body interaction matrix. The method of calculation re¬

quires that the energy of any model-state should be expressed

as a combination of the parameters Vj (v^ denotes the necessary

two-body matrix-element; the number of the parameters V-, de¬

pends on the configuration space ). If the configuration space

is restricted to the 2p-wp and ^So/p orbits, one needs 9 two-

body parameters and 2 single particle energies. These parame¬

ters for proton-proton interaction ere

688

I g9/2 J > = VJ < P 1 / 2

2 J=0 |V! p 1 / 22 J=0> =

2 s 9 / 22 J=O> = v

J =

J=O <P 1 / 2 S9/2 J =* lV ' Pi/2 S9/2 2 J =*> = V 4M < P V 2 £ 9 / 2 J=? I V | P V 2 S 9 / 2 J=5

The single proton energies £ and £. , órea-ced aa cocctanrs, P e _ . were oaiculoted i'roE che energies of 1/2 and 9/2 states or

~v'i L5j vieldine the values

"r:e :::f;~h.oć -its :itre n

*::•.-£& t-vo values tc be constant because ?:;iv ror ćŁi"i'er^nt I' = 50 isotones. i'- st r-v.:.v-r'/;, .ve exp:-^; sr.7 .-nr-.iy-tody ixed ccni"i.-tursticEE 22 a sun of two "ntr

•.vnere

anć D,. • depends on s ingle p a r t i c l e ene rg ies . S t a r t i n g from the equation for the e isen-vs lue problem we can wri te down

H X = E X

K C

(6)

where the matrix A is "build up from the eicen-vectors ss

689

columns and N denotes the order of the interaction matrix for

specific spin value. After substituting the eq. 4 to the eq.7

one gets

Ek = h 1=1

where N

Ckl = Z d=1

and K

ck = Jj In our case the matrix 1).= cor.-.; ins onl;v zei-or: with the excep¬

tion of the element D ™ = - 2 <. =-2(E... " ^ n ) which cor¬

responds to the configuration V P-|/2 ^/^^' ~yr' ec*' b 5k

should te corrected for DKBE and single particle energies

{ or Z^ also ).

As an example for 9/2+ ground state in " 'I7b we have

where Gz = DEBE(Nb,Sr) + 5£ =(2128 + 2 8 ) keV. Ihe energy

of any odd-parity state in ^Sb orising from the p 1 / 2 g-^11"

configuration can be expressed as [1OJ

(12) Finally, we have a set of equations of the type

Ez = f ( V l t DMBB, £ p > £ g A ) (13)

which can be solved by the method proposed by Glaudemans et

690

al,£i1J. This method requires the knowledge of the initial

values of the parameters V-., which can be taken e.g. from ref.

[i2j. The initial values of V^ serve to construct the inter¬

action matrix* After diagonalizing procedure one gets sets of

eigen vectors(necessary to construct an equation of the type

fl3))and eigen values, which can be compared with the experimen¬

tal energies. If the required agreement is not achieved, these

values are treated as starting parameters for the next step of

iteration. Two computer programs NIOB and MOLB were written

to solve this problem for ° Kb and Mo especially.

The fitting procedure described above gives 3 set of pa¬

rameters the final values of which depend on the set of levels

taken into account. It should be mentioned at this point that

if we take into consideration different levels of some nucleus,

the final parameters do not change their values significant^

[*1* The fluctuations are much smaller than those observed

in the case when the parameters were obtained from levels be¬

longing to different nuclei. For more complicated level cche&o,

it was not obvious which level should be included into analysis.

In the firsc step all model-levels were taken into account.

For such a set we obtained a RMS deviation per level, which

was often large. Sffi we define as

BUS

where n denotes the number of levels and k the number of para¬

meters. In the next step we skipped some levels. The smaller

SMS value obtained in this step attests that the missed level

has more complicated structure or its interpretation was wrong.

691

3. Besults of calculation end discussion

The final values of the obtained'parameters are presented

in table 1 and 2. The mean interaction energies defined by

7(2)=X (2J + 1) V,/ 2 f2J + 1) (15) J even J even

are also given in table 1 and 2. A graphical representation

of the parameters is shown in Pig. 1.

Results show the odd-even effect which is prominent,

especially for VQ and V,. When the Coulomb energy is not

taken into account the parameters are shifted towards positive

values (less attractive interactions). To obtain nuclear

effective parameters which can be compared with the $ -force

interactions we should subtract a constant C„ from the values b

presented in table 2. The constant C is define, as follows o

Cg =<g2(TT)|V| g2(TT^>J -<g(fr)g(>0|V|

with J = 0,2,4,6 and b, where IT and U denote proton snd

neircron, respectively. It has been shown by Gross and Frenkel

that C equals to 327 keV [12] . Figure 1 also contains

values of neutron-proton interactions. These values were ob¬

tained for the f«p/~ orbit from Sc using the Pandya transfor¬

mation and those for gg/2 orbit were taken from the ref.[13J .

For Vc and V,. e.i. for the case when the interaction is o o repulsive, it seems that the effect changes the sign. One can

say -chat the odd-even effect manifests itself in such a way

•t'aaz the interaction energy of "Che two particles depends on

z:ic: number of particles which additionally are filling up the

zsm'j orbii. It does not mean that real interactions should

rave óhe same behaviour. The effect accounts for the assump¬

tions made in the calculations. We do not consider the origin

oi' this effect here, but only conaent on it in relation to

t <". 2

TABLE I Effective two-proton interaction enorgios for £n/p orbit (in keV). Only statistical errors ere indicated* V(2") is defined as a mean interaction energy for states with T = 1. E(£,o<-)8re predictions for <T-forces with spin-exchange term. For comparison between the E(<f,oŁ.)and others aeo taxt.

I !

V2 \ V

| vo

I " I 1 _

50 T 1

-2<j}4 + 26

-1^79 t 27

)

1

-2t93 ± *y -1Obf; + ^0

-,

^ r

-;.'biu + >+

-i . 'oy + ^1

- ^ i i >..•

'«-••• + v * J

S . _ -

j

-«.'«if^ + 7 2

- l ) o b + 2 9

-t>'- + 2 o

•'. ;•. + . T

- 7 -1

5^Fo

-2<tf;4 + 65

•J'J + 49

w? + ^h

j - 4 6

E(S.oc)

-2^55

-556

-273

-1?6

: -5^6

TABLE II

Effective two-proton interaction energies for P^/2 69/? configuration space(in keV). See comment to the Table I.

-2100 +150 -1772 + 30

-661 + 13

53 + 23

300 + 15

.479 + 9

-1865 ± 51

-688 + 24

48+23

400 + 26

-561 + 150

868 + 50

528 + 23

-495 + 46

827 + 69

525 ± 10

173 + 10

- -^

wFe E(6.«) MZr 2Mo

Fig. 1. Effective two-body interaction energies for

60/2 orbits. Only statistical errors are denoted*

The shift of parameters for 6g/2 orbit is explained

in text.

695

the works published earlier* It was noticed (14] that values of 1f„ -2 matrix-elements depend on dimensions of configuration space. This effect i s presented in Table III*

Table III . Matrix elements from ref.

lV ' f7/2 \ " i t h 1f?/2 8 n a ^3/2 o r b i t s

J I Pure *n/2 shell j f „ / 2 and some ' ! p I M nucleoris

0 2

L143

-3110

-360 80

-2800 -1290

0 340

f?/2 -configuration

-2110 -1110 -100

230

"The bigger the shell model space the smaller the ostrix ele¬ ments in absolute value attributed to 1f, 7/2 states* This comment together with the odd-even effect allows one to conclude that an enlargement of configuration space i s more important

' Ti - ^ r and for even nuclei like ' Ti, - ^ r and . Fe than for odd nuclei like y V and Mn. The differences between our parameters and the parameters obtained for the whole ^n/o"^.'? configuration space are smaller for odd isotones than for even ones.

A second comment concerns the work of Eisenstein and Kirson [17] • It has been shown that energy levels in fn / 2

region can be well reproduced by using a three-body interaction in addition to the two-body one. These calculations as well as ours were performed within the pure fn/ 2 configuration space. Maybe, the odd-even effect reflects some contributions of the three-body interaction, i f i t was necessary to take them into account*

It i s well known that § -force interaction can be taken as a zero order approximation of the two-body interaction.

696

Such an interaction with a small spin-exchange term has the form

V12 = zSfa - ?2)[(1 -oO + ot P( ? i • £ 2 >3

where coefficient cC is equal to about 0.2. The matrix

element which corresponds to such interaction can be calculated

from the relations (18)

X-'--^

where S = (-if sod ^ a r e coanected with tae radial sav-; function. The mean interaction energy £(2) defined by

i ( 2 > $ ( 2 J + 1) Vj/ f (2J + 1 )

satisfies the relation

¥(2)= Q (1 - &*/2) . ••

Jiormalization of d -forces was chosen in such a way, zr.sz

lation (21 was fulfiled for two-body interactions otceir. d -

°Sc emploing the Pandya transformation. Comparison cet.vee

the obtained parameters and predictions lor the S -force

interactions are shown in Fig. 2.

69-

y i ? ? i

Fig. 2. Comparison of the observed and calculated two-body interaction energies. The $ -forces with a small spin-exchange term was chosen as a zero-order approximation of the two-body interac¬ t ions .

The two-body interaction energies are plotted as e function of the angle 6-,,., where

r12 = (22)

is the angle between the two orbits of identical nucleons [19].

The smooth curves correspond to the different contributions

of the spin-exchange term. A proper separation of the VQ and

V,. values can be obtained with 06 ** 0.2. This comparison shows

that it is necessary to include some additional interaction to

the S -force one. This interaction should be repulsive at the

angles less than 120° and attractive for the angles close to

150°.

698

One can observe an interesting behaviour of the differ¬ ences between Vj for different nuclei, e.g. VoCFO-V lTO^X^ (FAVV^O* A similar behaviour is observed with the values predicted for the £ -forces with different contributions of the spin-exchange term.

ą.. Comparison with the experiment The number of levels predicted for fn/g r e S i o n within the

pure fo/p configuration is small. Therefore it is difficult to compare the experimental positions of levels with those calculated with the help of parameters derived from the same nuclei. A general conclusion is that the positions of 3/2~ states do not agree with the experiment. Admixture of 2p,/2 configuration is important for these states. On the other hand, it is interesting to calculate the positions of levels for these nuclei which have a few active nucleons outcide the closed core. As an illustration of this effect, Fig. 3 shows the experimental energies of 'V [20] compared with the stror.r; coupling model calculations "A" [21] and with the energies obtained using parameters derived from the present method "B". Quality of both calculated sets is similar.

In the Zr region of nuclei, energy levels of ' K b and Qp 'Mo were also calculated. Results are presented in Figs 4 and 5> The sets of energies denoted by A were calculated with the help of effective two-body parameters obtained by a f i t t i ng procedure for the whole 69/2 region [13]. For these nuclei, the model predicts a lot of levels . The agreement between the experimental [7,8] and calculated level positions obtained in these examples enable us to conclude that parameters derived from well known levels of some nucleus can be able to reproduce energies of other model levels not yet observed in this nucleus.

600

u

Z5-

15

10

0.5

-5/2

-7/2

-1/2

-•an -s/2

- an

• 9/2 •11/2

•5/2

• 7/2

' 3/2

15/2 — ' 9/2,11/2,0/2

7/2.9/2.11/2 9/2 "

/2

3/2

5/2

9/2

11/2

•J/2 • 5/2 • 7/2

EXP

H/2 ' 13/2

3«

. 5/2

. 7/2

•9/2 7/2

' 11/2

.15/2 • 3/2

9/2

1/2

9/2

• 11/2

-3 /2

5/2

• 7/2

B Fig* 3« Experimental and theoretical negative parity level

scheme for the *V« A - strong coupling model f21"), B - shell model and phenomenological parameters*

700

Ul

- 2 V 2 '

.Kir

• 9/2*

n/r-

ł-K/r-

inr-

i7/r-nrr~

9/r-

3/r-

5/r-

3/2*

11/2*

on*

5/2*

9/2*

7/2"

7/2 S/2 n/2

3/2 3/2 5/2

5/2

5/2

^ 1/2"

9/2*

-21/2* 3A67

3W0

.15/2" 2660 15/2" 9/2- 2631

• 21/r

5/r

EXP

• 9/2" 1791 9/2"-

9/2' 7/2*

3/2"

5/2"

1/2-

9/2-

1637 1581

1313

1187

1045

0

3/2" —

5/2

1/2-

9/2 7/2

- 9 / :

B

FiK« A-. Kxperimental and calculated positions of levels for 7 Nb. The version A was obtained with the parameters SIG from Table II fij] and B with our parameters.

701

UJ

5.0

4.0

3.5

3.0

2.5

-r-2"-

- 3r=

- 5*

0* 3*

-00) 5311

5150

• i r 4485

4327 •9" 4?50

- 2' 3930 -(4) 3871

3757 3752

• r 3623

33666

2' 30626 30055

• «* 2758.8

- 6* 26115 5" 2526

2520

• 6*

22824

• 2* 15O&7

EXP B

Fig. 5« Experimental and calculated positions of levels for ° Mo. See comment to Fig. ^ .

702

References 1} J.D. tic Cullen, B.P. Bayman and Larry Zamick, Phys. Rev*

134 (1964) 515 2) D.K. Gloeckner and F.J.D. Serduke, Nuci. Phys. A22O (1974)

477 3) A. de Shalit and I. Talmi, Nuclear Shell Theory Academic

Press, New York (1963) 4) A. Bałanda, Saport IPJ No 986/PL (1977) 5) A. Bałanda, Acta Physics Polonica B8 (1977) 501 6) J.B. Ball, M.W. Johns, K. Way, NDT, A8_ (1970) 407 7) A. Bałanda, R. Kulessa, W. Walus and J. Sieniawski, Acts

Physica Polonica, B2 (1976) 355 8) A. Bałanda, fi. -Kulessa, W. Waluś and Z. Stachura, IFJ

Annual Baport 1977 p.63; in print 9) M.W. Johns, J.Y. Park, S.M. Shafrth, D.M. Van Patter and

K. Way, NDT, A8 (1970) 373 10) H. Auerbach and I. Talmi, Nucl. Phys. 64 (1965) 458 11) P.W.M. Glaudemans, G. Wiechars and P.J. Brussaard, Nucl.

Phys. 6 (1964) 529 12) R. Gross and A. Frenkel, Nucl. Phys. A267 (1976) 85 13) F.J.D* Serduka, R.D. Lawson and D.H. Gloeckner, Nucl.

Phys. A256 (1976) 45 14) I. Talmi in Effective Interactions and Operators in Nuclei

ed. by B.B. Barrett, Springer-Verlag, Berlin (1975) p«64 15) P. Federman and S. Pittel, Nucl. Phys, A155 (1970) 161 16) J.B. Me Grory, B.H. Widenthal and E.C. Halbert, Phys. Bev.

<32 (1970) 186 17) I« Eisenstein 8nd U.W. Kirson, Phys. Letters 4?^ (1973) 315 18) M. Moinester, J.P. Schiffer and W.P. Alford, Phys. Bev.

179 (1969) 984 19") J.P. Schiffer, Annals of Physics 66 (1971") 798 20) J. Styczeń, Report IFJ No 917/PL (1976) 21) B. Haas, P. Taras and J. Styczeń, Nucl. Phys. A246 (1975)

141.

703

Angular momentum projected wave-functions

R. Bengtsson, Nordita, CK-2100 Copenhagen 0, Denmark

and

H.-B. HSkansson, Department of Mathematical Physics,

:, -. ' ( • j <• . Lund Institute of Technology,

P.O.Box 725, S-220 07 Lund 7, Sweden

Abstract. Angular momentum projection has become a vital link between

intrinsic model-wavefunctions and the physical states one intends to

describe. — --

We discuss in general terms some aspects of angular momentum projection

and present results from projection on e.g. cranking wavefunctions•

Mass densities and spectroscopic factors are also presented for some

casBs. f i 1

1. Introduction

Projections of various kinds have become an important connection

between the nuclear model wave-Functions and the physical states

one tries to describe. Well known examples are projection of particle

number in the BCS or HFB models and projection of angular momentum

for HF wave-functions. In some special cases also projection of isospin

can be necessary . We shall here only consider projection of angular

momentum from intrisically deformed nuclear states with or without

axial symmetry.

As we have employed an algebraic projection technique , rather than

the usually used numerical procedures, we have an efficient tool for

direct analysis of the projected wave functions. After presenting some

general ideas about angular momentum projection we give a few applications.

Some properties of the cranking model wavefunction are illuminated by

examples, mass densities for rotating nuclear states are displayed for

a few cases and finally applications to single particle transfer reactions

are discuBsed.

2. Angular momentum projection in general

Slater determinants are the simplest antisymmetric many particle wave-

functions that can be formed from single particle orbitals generated

by some nuclear model. In most cases one has to pay the price of some

broken symmetries in order to gain energy or simplicity in a single

particle model. The loss of well defined angular momentum, as is the

case for most nuclear momdels, has been considered as a serious

mutilation of the states, in particular when rotational properties are

705

studied. Projection of angular momentum has therefore bBen much used ime ar 2.4)

together with the Hartree Fock madel for a long time and recently been discussed in connection with the cranking model

Due to Peirels and Yoccoz we have the following projector for angular momentum

Here R(fi} is the operator which rotates any wavefunction through the Euler angles (1 • (a,8,r) and D^Cfi) is its matrix element in a \3fi> basis. Rigorously, PjL is a projector only when M«K, because then PKK * PKK * PKK* However, no anbiguity can occur when the state to be projected has good K, i.e. it has axial syrrmetry. Then one always can use P. , and get any value of the magnetic quantum number 1*1 in the laboratory frame by using the raisering or lowering operators J* -• ((J*K)(J±K+D) J on |JK> until the desired state |JM> is reached. As usual 5± « j t i5. However, when the intrinsic state is non-axial, as for instance a cranking wavefunction, problems may occur. Such a state, #_, can always be expanded in eigenstates of Jz and 3 '.

•n * I C«IM \*M>- (2) *J Vu Ail I

aJn Here a stand for all extra quantum numbers needed to distinguish between different states of the same J and M. The only proper projector which can project out a specified 3 from + D is

P3 - I (Ł (3)

706

Applying this on y^ we get •_ • P t_. Obviously, $- has not a well-

defined M.

It has been proposed that one might use the operator Pp, • 1 P ^ to

J get a state ^ out of </„. This operator is not a projector, as pointed

4] out by Sorensen . Beside that, it does not even give the correct

fractions of spins contained in *_. From eq. (2) we see that the total

strength of spin 3 is

while PM gives the result £ (.1 c ,„)2 which is incorrect because

" a H ^ 1 (£ c , r ) 2 • 1 if there is more than cne M value. aJ M a j n

If the intrinsic state conteins more than cne K.-value, it is evident

that the projection technique itself is unable to txtract a state |J,f>

with well defined values of both 0 anc! M, containing all the strength c-f a

given J. Tc derive such a state from e.g. a cranking v;evefunction, sore

further information is needed. One possibility of utilizing the projected

wavefunctions is the following

A state with given 3 can be written t • F ^,t\^>- By operating with

J on |JM> fcr each value of M until |0J> is reached we get a set of

wavefunctions |BJVJJJ> , where a^ indicates the original M-value. These

states are in general not orthogonal. However, it is always possible

to make an orthogonalization. This yields a limited number of orthogonal

states |a,J,M«J> which are all contained in the original wavefunction,

although with different M-valoes. If we consider these states as a

small subspace, selected by the intrinsic wavefunction, we can apply

the ideas used in projected HF or generetor coordinate methods. Thus

the selected states should be diagonalized with respect to a suitable

707

Haniltonian (in practice only matrix elements might be known), and the

lowest state would be the best one that is possible to extract from

the original intrinsic wavefunction. If several states of a given J is

desired, the subspace must be enlarged with additional states obtained

from projection on excited intrinsic configurations.

3. Angular momentum projection and the m-scheme

The importance of choosing a convenient representation is obvious

"for any one engaged in many particle calculations. The recent

development in the shell nodel calculations by Whitehead et al.

shows this most clearly. They work in the m-representation, i.e. with

uncoupled A-particle slater determinants;

aJ 1m 1T 19J 1m 1x 1 \^f}'*

where a. is the fermion creation operator and I - > is the particle jrm

vacuum. The isospinprojection T distinguish between protons and neutrons

and j m have their conventional meaning.

When projecting angular momentum we also have chosen this represntation '

Thus any deformed A-particle state 9- can be expressed in terms of

slater determir.ants, eq. (5) built up by elementary j-basis states

|jmr>. The projection of angular momentum can then be performed algebrai¬

cally by expoiting the rotational properties of each individual wave

function | jmr>. We will not describe this procedure in detail here, but

refer to ref. 2. The main advantage of this projection technique is

that one gets the projected wave functions explicitly in the very

convenient m-scheme representation, which facilitates the calculation

of all kinds of matrix elements, overlap integrals etc.

708

4.1 Angular momentum projection on cranking wave-functions

We have applied the proj'ection technique on cranking wave-functions in a

simple model space, consisting of the Mp • 1p. .„)-shell. With

4 particlesin the six s.p.-levels of this space it is possible to couple

to angular momentum J«0,1 and 2. Any 4-particle wavefunction constructed

in this space must therefore be a mixture of these angular momenta.

By assuring a prolate deformation E=0.4 a projection on the yrast cranking

configuration gives the result shown in fig. 1. We see that the 3=C

component is decreasing while the J-2 component is increasing with the

rotational frequency oi. The figure elso shews the distribution on

different proj'ection quanturr. numbers '•'. . (We hsve used the x-axis,

i.e. the rotational axis, as the cuar.tizaticn sxis cf the angular rorer.turr,.

Notice that for high rotatiral frequencies the dominating J=2 component

has M -0*2, meaning that w& h=ve c s^rcngly aligned state arc that

<0 > , will give a goco estirr-ate cf the ciarrarating angular rriomenturr.

It night be surprising that the yrest configuration also contains odd

values of 3 (in the case J = 1). hcv.ever, the only symmetry of the cranking

wavefunction in addition to the parity is the signature ', which

implies that the wavefunctions are eigenfunctiens to R (it) with eigen-

values +1 och -1. In our representation R tir)|Jf. > « (-1) 'x|Jrlx> and

thus the signature only imposes the restriction on M to be even or

odd (cf. fig. 1, where only even values of M appears). As soon as the

time-reversal syrmietry is broken, which is the case for ułO, we cannot

give any restrictions to J. The direct reason why we get any admixture

of odd 0 into the yrast configuration is that the prolate deformation

of the single particle potential mixes states from P*/2 with states

from Pj/2' F o r 'Further details we refer to ref. .

% 50 -

710

4.2 Mass densities

The nuclear mass density at a point r of the state t can be written as

p(r) • «(r-r') p(f.r') • <f|a£ a ^ , (6)

where a are the fermion operator which creates e nucleon at r.

Expansion in a specific basis |v> yields

4 • i (7)

r * I *v

so we £et the density as

pCf) - [ <(?)• (r)p IB) liV v v

where

p - <f|a+a |?> {£)

which is the so called density matrix. The non-diagonal elements of

p play an important role in creating deformations although they are

relatively few and mostly very smell. The m-scherre representation of

V, i.e. as an expansion in Slater determinants (eq. (5)), simplifies

the calculation of matrix elerrents considerably.

Figure 2 shows the mass distribution of ^C for different values of

the total angular momentum J and its projection on the z-axis (M). AIsc

the strongly deformed (oblate) intrinsic state is displayed. Note the

strictly spherical case 3-0 M«C, which should describe the ground state

of 1 2C.

The projection gives the angular momentum distribution in the intrinsic

state as: 34.1 \ J«0. 55.5 % 3=2 and 10.2 % J«4.

711

t Intrinsic K. = 0 state

Projected J = 4, l"l = 0 state

* ProjectBd J » M - 0 state

ł Projected J « 4, M = 3 state

+ Projected J * 4, P! * 4 state Fig. 2. The density distribution in various " c states obtained by angular orojection from the oblate deformed intrinsic K«Q state. In the upper left diagram the body-fixed symmetry axis, in the other diagrams the laboratory z axis, is horizontal. The J»4 distri¬ butions can be thought of as generated from the intrinsic distribution by rotating the latter around an axis perpendicular to the intrinsic symnetry axis and then letting this rotation axis precess around the z axis at an angle arccos{M//J(3+1)} (indicated by dash-dotted lines). The shaded areas mark the regions of maximum density for each state.

712

4.3 Overlap integrals

The m-scheme representation of angular momentum projec.ed states makes

the calculation of overlap integrals very sitnple. We have used projected

wavefunctions to calculate the overlap integrals needed to describe the

reaction C(p,2p) B. We shall here not deal with the reaction theory or

how the wavefunctions of the states involved in the reaction are calcula¬

ted, but sinply refers to ref. .

1C)

The relevant overlap integrals for the Ip,2p)-reaction can be symboli¬

cally written as <r,s,t„ (A-1)|*„ (A)> . They can be interpreted as the "f "i

amplitude of finding the knocked-out particle with spinprojection s at a position r and the residual nucleus in the state ł.. (A-1) while the target

3 3 f J nucleus is in the state *^ (A). Dnce the states ł^ (A-1) and 4>j (A) have

'i f i *\

beon detemined the evaluation of the overlap integral is trivial .

Between the ground states of C and B we get

<r.s,ł>3/2(11B)|łJ*C(12C)> = Z entj|ntj> « -O.S77C 1p3/2 - G.C342 2p where the coefficients a . are deterrining the sc celled spectroscopic

-? 2 12 factors C^Sn . • (2J*1) a^.. Between the C groundstate and the lowest 1/2" state of 1B we get the overlap C.52ee 1p,y- • 0.0441 2p1/2 and between

12 - 11 the C groundstate and the first excited 3/2 state of B 0.1644 1P3/2 •

0.0050 2p3-_. In the last two cases the coefficients a^^. are much smaller

than those obtained between the groundstates.

Since the magnitude of the reaction cross-section is strongly dependent on the overlap integrals it is interesting to compare the relative cross-sec¬ tions, viz. o : c,,-,- : ao/-,- . From ref. we have the thoeretical

g.s. 1/2 3/2 values 1.00 : 0.19 : 0.12 while the experiments by Bhowmik give 1.0C :

0.14 : 0.07.

Acknowlodgements

We are very greatful to Tore Berggren for close cooperation, especially what

concerns the treatment of the (p,2p)-reaction.

References.

1. H.-B. HSkansson, T. Berggren and R. Bengtsson, in preparation

fo r publ icat ion.

2. R. Bengtsson and H.-B. HSkansson, preprint Lund. 1978, to be published.

3. G. Ripka, Advances in Nuclear Physics, v o l . 111966), Plenum Press,

New York.

4. P.A. Sorenssn, Nucl. Phys. A?31C1977)475.

5. R.E. Peierls and 3. Yoccoz. Proc. Phys. Soc , A70C1957)363.

6. H.A. Lame and E. Boeker, Nucl. Phys. A111(1966)492.

7. A. Kanlah, Z. Physik, 216C1968)52.

6. P.P. Whitehead. A. Watt, e .J . Cole and I . Morrison, Advances in

Nuclear Physics, vo l . 9(1977), Pienin- Press, New York.

(J. P. Bengtsson and S. Frauendorf, to be published.

1C. T. Berggren, Nucl. Phys. 72(1965)337.

11. R. Bhowmik, C. C. Chang, J. - P. Didclez and H. D. Holmgreu.

Phys. Rev. C 13 ( 1976)2105.

WARD-LIKE IDENTITIES, CLUSTER-VIBRATIONAL MODEL

AND QDASIROTATIONAL PATTERN \

V. Paar ( '•_,^J_, G L | 'J / Prirodoslovno-matematicki fakultet, University of Zagreb

"Rudder Boskovic" Institute, Zagreb, Yugoslavia

1.1. Elementary Excitations in a Nuclear System

It is the old idea by Landau to describe a quantum-mechan¬

ical many-body system in terms of two basic types of elementary

excitations, "quasiparticle" and collective excitations, and

by their mutual interaction. This interaction plays the basic

role in creating physical properties, and hides many unexpected

features. Physicists usually like to consider more transparent

limiting cases, weak-coupling and strong-coupling limits, i.e.

the situation when the interaction between the elementary ex¬

citations is weak and strong, respectively. Landau has started

with these ideas in the theory of condensed matter, but soon

they have penetrated in other fields of quantum physics.

Bohr, Mottelson and Migdal have pushed it into nuclear physics.

What are elementary modes of excitation in a nucleus?

The first type, " quasiparticle1' type, is associated with inde¬

pendent motions in the average field such as spherical shell-

model states, Nllsson states, BCS quasipartioles etc.

The second type, collective type, is associated with different

kinds of collective degrees of freedom, such as vibrations.

715

HOW TO CALCULATE ZERO IN THOUSAND WAYS?

5-5

UJ

a.

x LU

a. o LU I

EXAMPLE :

/ M E A S U R E D N

\QUANTITY )

MODEL A-CALCULATION FOR

/ MEASURED \ e V QUANTITY ) " 5 ~

MODEL B: CALCULATION FOR

(QUANTITY0) - 6 * 4 - 4 - 6 * 0

MODEL O CALCULATION FOR

MODEL A IS OK

MODEL B IS OK.

- 6*5-5*1-1*2-2- 6 * 0 * 0 * 0 - 6 — ^ MODEL CIS OK.

Pie.

•7'

pairing vibrations, rotations, giant resonances etc.

The easiest wav_to reoognize and divide elementary nuclear

excitations into these two types is to do it phenomenologically.

That is the attitude we adopt in these lectures.

The other, more fundamental but much more difficult way is

to approach the problem microscopically in a self-consistent

way. In this case one tries to construct collective excitations

from the underlying shell-model structure, and then to use them

as the building blocks for the nuclear states. In this lecture

we shall not speak about such attempts.

1.2. "Extreme" Representations for the Muclear System

Speaking in terms of nuclear elementary excitations we

could consider familiar shell-model and collective model as

extreme representations.

i/ Shell-model as an "extreme representation": only shell-

model /i.e. "quasiparticle" in Landau's sense/degrees of

freedom are included.

In this approach shell-model configurations with

non-negligible influence on nuclear properties have to be

included in the configuration space. In this way the

Pauli principle is correctly taken into account. Exam¬

ples of this type of approach are large shell-model cal¬

culations. The difficulty, however, is that the number

of non-negligible configurations for most of cases is too

large to be practically handled. This requires rather

drastic truncation of the configuration space, whioh

717

affects thoee physical properties that are sensitive to

the configurations we have neglected; first of all,

these are some collective aspects of the nuclear behaviour.

In this way part of the physical features is lost.

11/ Collective model as an "extreme representation": only

collective degrees of freedom are included. It means

complete averaging over the under-lying shell-structure,

and thus the Pauli principle is completely neglected.

In the cases when the role of some shell-model configura¬

tions is more pronounced, it is obvious that such complete

averaging may not be a fair representation of the actual

situation, and again a part of physical features is lost.

1.J. Simple Old Particle-Vibration Coupling

A simple quantum-mechanical representation in the sense of

uandau with both "quasiparticle" and collective elementary

excitations included is a well-known particle-quadrupole vi¬

bration coupling £from somewhat different viewpoint this sub¬

ject is considered in the lecture by Georg Leander). The ele:-

mentary excitation of the first type is one single particle,

and of the second type quadrupole vibration. This representa¬

tion seems reasonable for odd-A nuclei such that the neigh-

boroing even-A nuclei show a pronounced low-lying quadrupole

vibration. Then the elementary modes of excitation in odd-A

nucleus are one single particle in any one of the available

single-particle configurations in the valence shell and quanta

718

of vibration, so called phonons. The basis states of this coupled representation are

where j denotes the angular momentum of the single-particle configuration in which there is a single particle, H is the number of phonons and E is their total angular momentum. Angular momentum j of tht single particle, and R of phonons are coupled to the total angular momentum of the coupled basis state. Practically one usually restricts the basis states to those with single—particle configurations from the valence shell and with up to a few phonons.

The interaction between the single-particle and the qua-drupole phonon is

Hm j^ f

Here p£ a n d &» a r e 'tłie creation and annihilation operators of phonons, B(£2)(Z^-^O*)y,g is the B(EI) value for transition from one- to zero- phonon state, and Ł ^ is the nuclear shell-model potential.

The basic processes in the particle-vibration coupled system is the emission and absorption of a phonon by a single particle. If a single particle Ijh} emits a phonon and simul¬ taneously Jumps into single-particle state |4ż the corre¬ sponding matrix element of the interaction is

719

with the conveniently defined ooupling strength a =

In the present discussion we assume that for every speoific

nucleus and valenoe shell the coupling strength a is oonstait

and we treat it as a parameter. Typical value for the periodic

table is a ft» 15.

The same matrix element corresponds to a process when

a single-particle | i ~^7 absorbs a phonon and becomes single

particle )£> .

" In diagrammatic representation these two processes are

presented in fig. 1.3.1

i 1

Ul

J

Fig.

A single particle is presented by a straight line ori¬

ented in the positive time direction chosen upwards , and

a phonon by a wavy line. Similarly, one presents single hole

by a straight line, oriented in the negative time direction.

Of course, in a language of wave functions the emission of

a phonon in diagram on l.h.s. of fig. 1.3.1 means the first-

order admixture of the component IjJ 12; j ^ to the zeroth-

order oomponent | J> . The amplitude of this admixed compo¬

nent is given by the produot of matrix element ^ ^

and of energy denominator s—u •. .

Here £, and €w are the single-particle energies of the

single-particle states Jj > and \ i'^> respectively, and

720

is the pbonon energy. This is nothing else but the first-order perturbation theory. Generally, diagrams are only a simple way to systemize and visualize contributions in the usual Rayleigh—SchrSdinger perturbation theory. All possible diagrams with n particle vibration coupling vertices corre¬ spond to the n-th order perturbation theory. Topology of dia¬ grams and simple rules how to calculate the corresponding con¬ tributions makes diagrams universal tool in treating quantum-mechanical problems by perturbation theory. The important aspect is also that specific classes of diagrams can be con¬ nected with a specific physical role.

In fig. 1.3.1 we have presented first order diagram which admizej Ił/ł&jł^' *° tfr^> • There are many more higher-order contributions which also contribute. Por example, in .the third order we have three diagrams, presented In fig.1.3*2

) )

Fig. 1.3.2

In the intermediate states the summation over all single-particle states available in the valence shell has to be per¬ formed.

Obviously, the amplitudes obtained by the diagonalisation

of the Hamiltonian matrix correspond to the sum of all corre¬

sponding diagrams) up to infinite order, but with no more pho-

nons in any of the intermediate states, than are included in

the basis for dlagonalization.

2.1 1-Forbidden 111 Transition in Zeroth-Order and Particle Vibration Coupling

We discuss here qualitative features of M1 transition,

-which in zeroth-order approximation takes place between single-

particle or hole states, with orbital angular momentum differ¬

ing for two units (A 1*2). Por example iy2 —*. s ^ 2 , jV, _• j,' et0#

Standard M1 operator consists of two parts: one part is

associated with the single-particles

and the other is associated with the vibrational core

The notation here has the usual meaning: J and t are the to¬

tal angular momentum and the spin operator for single-particle,

respectively, and R is the angular momentum operator of the

vibrating core. The quantities g« , g and gR are the stand¬

ard gyromagnetic ratios.

Thus, for standard.M1 operator, the M1 transitions between

single-particle states Mf} and |/f> with

722

are exactly forbidden, since the matrix element

For example:

This is due to

O

none of the operators in the standard M1 operator can change

the orbital angular momentum of the single particle. Therefore,

such transitions are called 1-forbidden transitions.

'.'.'hat happens with such M1 transition, which is 1-forbid¬

den in zeroth order, in the presence of the particle-vibration

interaction ? Por the case of intermediate coupling strength,

wave functions are rather mixed. Then there appear nonvanish-

ing contributions from the sizuable admixed components in the

wave functions of the initial and final state. Let us illus¬

trate this in the case of d^w — > a^, 1-forbidden transition.

Due to particle-vibration coupling we have now correlated wave

functions of the initial and final state ("dressed" single-

particle states):

The 1-forbiddeness for '^A^-^ISy/,^ M1 transition obviously

does not hold any more, since,for example, the matrix element

j3/2> (2.1.5)

is large; as are also the matrix elements of M1 operator be¬ tween several other components in the wave functions of the same initial and final state. Obviously, since the particle-vibration coupling strength is not weak, i.e. since the mix¬ tures in wave functions are large, the 1-forbićdeness does not hold any more. So we could expect this to be rather strong M1 transition. However, the result is surprising:

i/ We do a complete calculation: form a Hamiltonian matrix for the particle-vibration coupled system, diagonalizing it we obtain complex wave functions, and using these wave functions we calculate B(MI) value for transition of this type (l-for-bidden in zeroth-order approximation, i.e. for the largest component in the wave function of the initial and final state). The resulting B(MI) value is small, at least several orders of magnitude smaller than the TCeisskopf single-particle unit ; in apite of the fact that there are rather large individual partial contributions.

ii/ Experiments also show that the 111 transitions, inter¬ preted to be of this type, are sizeably hindered.

What leads to partial restoration of 1-forbiddeness in spite of complex oharacter of the wave functions of initial and final state, with sizeable components which break 1-forbiddeneos ?

2.2 1-Forbiddenesa for the Degenerate Oscillator Shell

The answer to the former question is: systematio quantum—

mechanieal destructive interference for the coupled particle-

vibration system. (V. Paar and S.Braat, Nuol.Phys. A (1978)>

v;e will present this destructive interference for the case

of simple model of degenerate oscillator shell, where it is

complete and physically particularly transparent.

In degenerate oscillator shell the single—particle states

Mtt1},{tj)tl&tyH)jl€t2/j+£>. are present, and all have

the same energy.

In zeroth-order approximation we have

M - o ;(2.2.1)

In the second-order of the perturbation theory for the particle

-vibration coupling the result is (after performing angular

momentum algebra and using analytical expressions for Gj-sym-

For the contribution from the single-particle^MI operator:

(2.2.7)

725

JOT the contribution from vibrational part cf M1 operator:

where t Av'*i'* - a, Ą

»iti 2,+a-ł f2.2.11>

The total M1 transition moment in the second order

thus contains 16 terms, 8 of which are positive and 8 negative. Since they appear in pairs of the same magnitude but of opposite sign, they cancel exactly; in the second-order of the perturba¬ tion theory destructive interference is complete:

^O. (2.2.13V

The same result applies to any order of the perturbation expansion; thus, due to complete destructive interference •within each order of the perturbation theory, the 1-forbiddeness is exactly restored for degenerate oscillator shell, although individual components give sizeable partial contributions to the corresponding B(E!1) value. If the interference were com-

726

pletely constructive, the resulting B(M1)value would be for the intermediate particle-vibration coupling strength very large, of the order of magnitude of Tfeisekopf unit.

2.3 1-rorblddeness for Unclear Single-Particle Shells

In the case of degenerate oscillator single-particle shell the cancellation between partial contribution to the W1 transi¬ tion nonent is complete, 1-forbiżćecess is exactly restored for the mixed ware functions generated by the partiole-vibration coupling.

In the case of nuclear single-particle shells i/ degeneracy is removed

ii/ some single-particle states of the group II,;}— 1 > , fl,j> , |l+2,j+1> , il+2,;j+2> are missing from the valence shell.

As a consequence, the destructive interference, though still present, is relaxed. Here we present a short survey of relevant shell-model situations:

The shells S-20 and 50-82: there appears d, ,0—* s^,0 1-forbidden M1 transition. Now some of the interferent terms discussed in the foraer section are missing, because we have sinsle-parxicle states: |lj> =/%>//^?//^)-/fl4/Ł>//^(/^-/4/Ł>. Hovr does the absence of |l, ,i-1> single-particle state affect the destructive interference ? Now the first AJ (eq. 2.2.4) is zero and the second is absent. The first As (eq. 2.2.6) is zero and the remaining three are absent. Finally, the first

727

Ae ^eq.2.2.101s zero, and the second is missing. Por the re¬ maining terms/ , which involve single-par¬ ticle states |l,j> , |l+2,j+1>, ll+2,J+2> , the total cancel¬ lation is not affected by the absence of 13-1 > . Thus, the deviation (i) does not effect destructive interference in the • second order. However, Uie .energy denominators are no longer equal, since they involve dependence on the energies of the

4 A intermediate s ta tes ; instead of /X7\». now we have.r~T—

WJ *} ~V* flow different terms enter in summation with different weights and therefore destructive inteference is still present, but not complete any more.

The shell 23-50. There appears JLr* ft/A 1-forbidden transition. In this case 4.».| , with the |1,J-1> , ll,;j>, 11+2, 3+1 > single-particle states present while |l+2,;}+2> is missing. How we have three unbalanced terms AJ (;j+1,j+i), A*s (j+'Sd+'i) a A (j+i)» but they involve mutal Incoherence: the first is negative and the other two positive. The remaining 8 term3 (t 2.V, Z.i.i, i.Zk^sanoel exactly for degenerate single-par¬ ticle energies, and approximately after removal of the degen¬ eracy.

The 50-82 shell. There appears * -» c{f/Ł 1-f orbidden transition.

The 82-126 shell. There appears A,.-» C' 1-f orbidden transition.

In the last two cases situation with destructive inter¬ ference la analoguous to that for f -^ p . transition.

726

As we see, the 1-forbiddeness, which is exactly restored for the degenerate oscillator single-partiole shell, is slightly relaxed for the realistic nuclear shells; thus the correspond¬ ing B(M1) values is still at least few orders of magnitude smaller than the W»isskopf single-particle unit. That is the reason why we classify such transitions as 1-forbidden also IŁ the presence of particle-quadrupole vibration coupling of in¬ termediate strength, where weak coupling is no longer estab¬ lished and the wave funotions exhibit pronounced complexity. In spite of large admixtures in the corresponding wave functions the contributions from the M1 operator are small.

2,k Asymptotic Cancellation for Corrections to the Zeroth-Order Tensor Ml Moment

In previous sections we have revealed a peculiar effect of destructive interference to the 1-forbidden M1 transition moment for the standard M1 operator. That is the reason why in such cases the contribution of the tensor component of M1 operator prevails, although rather small.

Namely, in addition to the standard M1 operator there Is a tensor part:

which partly incorporates the effects of the 1* oore polariza¬ tion and of the mesonic exchange current. This term has been

prtQ extensively discussed In the Pb region. Here we discuss it

-29

in the framework of the particle-vibration coupling for the

dominant role of the low-frequency quadrupole vibration.

In the zeroth-order approximation for 1-forbidden transi¬

tion we have , after working out the angular

momentum algebra

>

In the second order we have two types oi" contributions

'V.?3a.- and 3.3rant, Phys.Lett- 7^7 '', '"•-;.' 297) i

i/ Terms with summation over t~o intermediate single-

particle states. For this contribution we use label V.C.

(the reasons will become obvious later):

with the energy denominators:

li/ Terms with summation over one intermediate single-

particle state. Por this contribution we use label S.S.

If we restrict summation over

tions lAt£> and ll,{^ we have:

and only to (2.4.5.)

configura-

730

tit ~ / 2 > ) * ^ l^.«t-.D,

-.2.4.6/

** 2.4.15

rhus, we have four positive terms contributing to V.C. ten;

and four negative contributing to S.S. term. In the csse of

degenerate single-particle states ifi

and for the asymptotic limit i"^"° the destructive inter¬

ference between the 7.C. i 3.3. terns is complete. For finite

•y this destructive interference is partly relaxed but is still

very effective. To illustrate this, -ve present the partial

contributions to <l'j'nMT(Mi) ll£f)>

\

731

in units ti s.

and for i -*

i*

1(3',3')

T(j.j')

T(3'.3)

s*(j')

£

0

0

1 5

0

1 " 5

1

1 " TIT

1 " 75

19

in table 2 . 4 . 1 :

TABLE 2 .4 .

i 4

9x55 64

25x49" 4

15

25X7

1 " TO

3

1

4 " 75

17

235

£ 2.

16

100 19x5

10

1 3x49

4

2 " "3x55"

1

41 " 7075

1039

,1

i

4 9x49

16 9x121

|

1 7x99

5 ~ IT

5 " bxV7

2 " 3x77-

4 " IT

- JLŁ

3007 557?

i -1

(23) 1

I

1 - -5

0

1 7

As we see, for each M1 transition of the type il'= 1+2, i'= j+1> —•• |l,j> four positive and four negative terms are obtained, and destructive interference between them is very effective. For example, owing to this incoherence

732

the total second-order contribution to the 3/2+ —+ 1/2+

(i.e. d,/£ —• S V 2 ) 1-forbidden transition moment is 1/19

of what would be obtained in the case of constructive interfer¬

ence, ©lis reduction faotor is -ml for ~ —• % (i.e.f-;

TOST for ?/2+ -• ^ 2 + (i- 6-^^/! ) •»* W for

9/2* —* 7/2~ (i.e. njjj,"* {-j )• In the asymptotic limit of large

angular momentum, i £> 1, destructive interference is complete

and the corresponding M1 transition moment for the transition

|l'= 1+2, i'= 3+1> -» | l,j> is 0. On the other hand,the

sum of absolute values of all partial contributions to the

transition moment* i.e. the result we would get in the case

of complete constructive interference is 0.76, 0.53, 0.51, 0.51

and 0.50 for the transitions

and \!!=lWii'=j+i> -» ll^y } respectively.

Thus, in zeroth and second order we have

As an illustration, for d,/2 —* S^ / we have

For spherical and transitional nuclei

ft

so the second order contribution is s=i 20 times smaller than the

zeroth-order contribution, in spite of the intermediate

coupling strength. In the case of constructive interference

+ |S | ) = 4r ? > so the ratio of the second- and

733

zeroth-order terms would be

C2.4.19;

i.e. the second order contribution nould be of the same order

of magnitude as the zeroth—order term. This clearly shows the

pronounced role of destructive interference.

2.5 Diagrammatic Representation: Self gnerrles Versus Vertex

Corrections (v?ard Identity) for Tensor K1 Operator

The contributions labelled V.;. and S.2. in Sec.2.ft are

the terms contributing in the second-order of the perturbation

theory. Their labels denote vertex corrections and self ener¬

gies, respectively. These names we have given in accordance

with the analogous terms in quantum electrodynamics. The dia-

grams giving contributions <(fiV /I M ( w ) \\ 2i A.„ and

<£i'l' ti(*T(Mi)Ui>Sc a « presented in figs.2.5.1a

and b, respectively.

L MT

u —x '

Fig.2.

(a) (b)

The new basic element, in addition to those of Sec. 3.1

is the interaction with the external field l"j, contributing

to M1 operator. It is represented by a dotted horizontal

line, with x at the free end and label Mr refering to its

name.

Por example, diagram (a) of fig. 2.5.1 has the meaning:

single particle lj> emits a phonon and becomes | }")> » then

I j"> interacts with the external field- M T /meaning emission

of gamma partlole due to the part MT of the M1 operator/

and becomes lj*"> , then lj'"> absorbs the same phonon which

was at the. beglning emitted by |J>, and becomes lj>. In the

language of the wave functions the same diagram has the meaning

of the contribution to MV transition moment due to components

\f, 12; j>and I i"', 12; $"> admixed in the first order in

the wave function of the initial (l7>) and final state (Jj'>)

respectively.

In quantum electrodynamics in second order there appear

diagrams of the same type as those shown in fig. ?.5,1 but

straight line represents electron, and wavy line photon.

Also, the rules to perform calculations are different than here

/so called ?eynman rules/. Diagrams (a) and (b) are for obvious

reason called vertex corrections and self-energies, respectively.

In quantum electrodynamics the summation over momenta in the

intermediate states are performed, and this involves infinities

in partial contributions. The basic role is played by so oalled

VTard identity: self-energies and vertex corrections mutually

cancel in the limit of low momentum transfer. In other words

735

there is a very pronounced systematic destructive interference

in all orders of the perturbatipn theory, and this kills all

infinities which mutually cancel. This.is the key of the Quan¬

titative success of the quantum electrodynamics.

Do we have here^ in a simple particle-phonon coupling,

analogous cancellations ? Yes. This is what we see in table

.2.4. V for M1 transition moment <" l\M J _> . Hf.-re, vertex

correction and self-ene,rgy'contributions exactly cancel, now

in the -limit, of low. relative angular momentum;*transfer . ,:

'.- ~i"*\~~•• O since nucleus is a system with rotational sym-

'.metry. This is a Ward-like identity for a particle-phonon

. system.' Prom table 2-4.1 we. see also that cancellation be¬

tween vertex correction aed self-energy, though not exact for

physical low | valiies, still approximatly holds. This type

of cancellation is valid in any order of the perturbation.

This is seen from diagonalization results, which, as we men¬

tioned, present, a summation of all diagrams up* to infinite

order, with inclusion of the intermediate states contained in

the basis space.

The meaning of the Ward identity, i.e. cancellation be¬

tween vertex corrections and self-energies for 1-forbidden 5M

transitions in the usual language of wave functions is the

following: part of the M1 transition moment which is lost

owing to the deorease of the main component in the wave func¬

tion is recovered by the contributions from admixed components

in the wave functions. In the other words such M1 transition

moment does not depend on the degree of mixing in the wave

functions; different wave functions, starting from pure single-

736

-parricle states to more and more mixed wave function, give

the same result.

2.6 Kard-LiKe Identity for the Standard Ml Operator

, Cancellations discussed in sections 2.2 and 2.3 can also

be expressed as a 7?ard-like identity of a special type. In the

second order self-energies are zerc, beoause standard M1

operator cannot chance the orbital angular momentum: Namely

for self-energy in fig. 2.6.1 ' •

j'. M S p*M v | B

Fig. 2.6.1

because of anrular momentum conservation i'"~ 3, and

<t''t+2,i'*J+illMsft+MVlSHtj> - O. So the only nonvanishin* contribution is due to vertex correction in f ig . 2.6.2

J-.

Pig.2.6.2

J

This contribution, as we have snown for the degenerate oscilla¬

tor shell in the second order, is zero. So we have here a spe¬

cial type of Ward identity: self-energies are zero, sum of ver-

73?

tex oorrection Is zero, so the sum of self-energies and vertex

corrections is zero.

Concluding discussion of "1-forbldden" SJ* transitions

we may say that there is a strong destructive interference

between the second-order terms, and between the higher-order

terms in each order of the perturbation expansion, so prac¬

tically the zeroth-order result is preserved, although the

wave functions are rather mixed.

3.1 Ward-Like Identity for 52 ?ransitionB and Y.oments

Let us consider now the 22 aonent between the single-

particle states, and the effects of particle-vibration coupling.

In the partiole—vibration coupling the 32 operator consists

of two parts, the single-particle part

and the collective part

Here es#p* is the single-particle charee and

value corresponds to the transition from a one-phonon zo a zero-

phonon state of the vibrator.

Let us consider diagrams, i.e. perturbation terms, con¬

tributing to the E2 moment for the transition between tvc

single-particle states l^y-^j^ . There is a set of

diagrams which involve the interaction of a single particle

738

with the electromagnetic field Ms _(S2) at a certain point

P.Such diagrams will be referred to as being of the particle

type. To each such particle diagram corresponds a class of

induced collective diagrams with the only difference that the

absorption or the emission of a virtual phonon takes place at

the point P instead of the E2 interaction via Ms _(E2).

This phonon is created earlier, or annihilated later, respec¬

tively, by the electromagnetic field M T I B(E2), with all pos¬

sible time orderings. In figs. 3.1.1 and 3.1.2 we present all

particle diagrams and induced collective up to the third-order

in the particle-vibration coupling CV.Paar, Phys.Lett. 6OB,(1976) 232;,

The lowest order process is the zeroth-order particle-dia¬

gram B of fig. 3.1.1

Mv,B(E2) J

J

—x

B J J

MSP(E2)

P2 B*P ' * Fig. 3.1.1

The corresponding olass of first-order induced collective con¬

tributions consists of two diagrams, labelled P1 end P2.

These two diagrams are referred to as polarization effect.

For example, in diagram P^ the interaction with the electro¬

magnetic field lly-gCEa) creates a phonon, whioh is afterwards

absorbed by the single-particle fr whioh p * quently jumps into

the configuration 4- .

For diagrams B, P,, and P„ the sum of corresponding contri¬ butions may be presented in the form:

739

BP * B + P1 +-P2 = B • 2 e8'P (3-1-3) where *- Ą+ M 1 M

This is so called factorization theorem. We can express this by saying that diagrams P and P^ can be simply included in the diagram 3 by charging the particle charge es*p* into effective charge Jes'v'', diagrams V, and P„ can be completely included In the renormalization of the single-particle charge, i.e. in the charge polarization. In this way we have no more need to consider the induced col¬ lective diagrams. This we have presented on the r.h.s. of fig. 3.1.1. The ohange es'p*—• es*p* is symbolized by x—»(x)

In the seoond order we have three diagrams of the particle type, labelled V.,, S,, and S^ in fig. 3.1.2. They are of the same type as for example diagrams considered for M1 moment in Sec.2, but now the dotted line represents the interaction with the E2 operator K3 (E2) . V^ is the vertex correc¬ tion and Sj, s!J are the self-energies.

However, in the third-order perturbation theory now we have the corresponding classes of induced collective diagrams^ To particle diagram V,. corresponds a class of four induced colleotive diagrams V2, Vj, V^, V5< Take for example diagram V2. It looks like V,,, but at the point labelled F instead of dotted line representing interaction with the electromarr.etic operator M (E2) we have the absorption of virtual phonon by single particle. And this virtual phonon was created before, between the moment when the particle * emitted a virtual phonon

and becoming 4 and the moment whioh corresponds to the point P. Similarly, to the self-energy diagrams S1 and S, there

'correspond classes of induced collective diagrams Sg, S_, sj.»S5 and SÓ, SÓ, S^, SĆ» respectively.

As in the earlier case for diagrams B, P ,,, we have also here the siailar factorization; however, in the second and third order it is exact only in the asymptotic limit of deftener-rate oscillator shell, i.e. for degenerate single particle states.

Then follows a very interesting factorization theorems:

X

J Jl

. )

S,

i"

X

S',

V 2

s s

S1,

+ s ; + S3 + S^ + S^

• \_MviB f

MvlB

-r*

V5

(3.1.5)

(3.1.6)

S'

Fig. 3,1.2

Again, it is enough to consider only the partiole diagrams

(v.., S^, s!j) while the classes of induced collective diagrams

lead to exactly the same renormal1zation of effective charges,

as it is the case in the zeroth-order (fig. 3.1.1) . These

factorization we have presented on the r.h.s. of fig. 3.1.2.

In this way, up to the third order we have to consider only

four diagrams (BP, V, S, S*) . The contribution from BP, given

by the expression (3.1.3) presents a sizeable enhancement of

the zeroth—order transition moaent, by a factor *.e *-*.

In practical cases -n % 3 •? 5. L f

The contribution to E2 transition moment i~*i from

the seoond- and third- order processes, given by V, S, s' in

eqs.(3.1.4), ^3.1.5) and (,3.1.6) is equal to

(3.1.7)

Here the ^ »nd^2 summations are performed over all available

aingle-partiole states in the valenoe shell. The symbol

We oontider now (3.1.7) in the limit of large angular

momentum

fioation sinoe the spin-flip

The nonspin-flip aatrix elements in this sase are

^-".oo . Por large 4- we have essential slmpli-

inoe the spin-flip matrix element^

In this asymptotio limit some reoouplings appearing in.(3.1.7)

are equal to unity

(3.1.S)

while the others are < d. and therefore asymptotically zero.

Partial contributions to (3.1.7) corresponding to all possible

intermediate states are presented in table 3.1.1 for 22 tran-

sition ^ / = ^ - 5 ;

TABL7I 3.1.1

h

j

j

j

j

j

i

T

- 2

- 2

_ 9

+ 2

0 T

j

j

i

5

A

Jo

-

-

-

-

j

L

2

2

2

4

V

0

VI

0

0

If!

/ /

/ -§/l

3 Hi " 1 1 2

/

/

1 [Z - lii

. a / I 4 i 2

"Til

/

7*3

Thus V + S + S* = O,

i.e. vertex corrections and self-energies cancel exactly. Again, as in Sec. 2 . this is just a content of the Y;ard iden¬ tity for the low relative angular momentum transfer 4j= ~Ę -> O. This, together with the asymptotic condition of degenerate single-particle energies A£-*O can be expressed as

A ( I , £ ) - » O which is analogous to the limit of low four momentum transfer A k = & Cjb t)—> O Ł n quantum electrodynamics.

Again, normalization conditions represent a renormalizatim of the zeroth-order matrix element due to the reduction of the nain component in the wave function: the probability that a single particle will interact with the external electromagne¬ tic field is no longer unity, since it is reduced by the pro¬ bability of having a virtual phonon around the single particle. This reduction of probability of finding a bare single particle is equivalent to a change in normalization of the dressed sir.gle-particle state. This effect, reflected in the decrease of the experimental spectroscopic factor froa the unity reduces •the leading order bare plus polarization (3+?)Z2 moment. On the other hand vertex corrections act coherently with the zeroth and first order contributions '.3+P) and tend to restore what is lost due to self-energies, i.e. reduction of the ampli¬ tude for the zeroth-order component.

These two effects in the second and third order cancel exactly for the asymptotic limit A 1 -* -9 > A Ł -*>• O.

Therefore, a departure from the weak ooupling limit does not

effect the B+P contribution.

Analog discussion holds for the statio quadrupole moment

of a single particle in the presence of particle-vibration

coupling.

As we move away from the asymptotic limit, the selection

rules for the reduced matrix elements of Y2 an(3 f o r recoup-

lings relax, characteristics of specific shells appear, and the

situation partly loses its simplicity and transparency.

However, systematic destructive interference between vertex

corrections and self-energies remains, resulting in a tendency

towards approximate cancellation. Thus, although coupling

strength is not weak, the importance of higher-order terms is

largely diminished.

4.1 A Rule for the Sign of E2/M1 Mixing Ratio in Vibrations!

Yrast Band

The E2/M1 mixing ratio is defined as the ratio of E2 to

M1 transition amplitude

Here we derive the rule for o for transitions between the

yrast states of a vibrational band based on a unique-parity

state. On the yrast line, the angular momentum of the TT-phonon

state is maximal, i.e. R=2U. Particle-vibration coupling creates

a "quasi-rotational band" on the yrast line (see Sec. 5);

together with the main components in the wave functions it is

presented in fig. 4.1.1:

Ij.N 2N;I-j*2N> lj,N 2N.I-J • 2N-1>

lj.N-1 2N-2;I-j*2N-2>

lj,N-1 2N-2;l-j*2N-3>

,VN

I i

P2

v2

>

v,

Ij,

Pig. 4.1.1

We have two types of transitions:

i/ Transitions of the type AN = C, i.e. for which the

phonon number is the same in the largest component of the

initial and final state. In fig. 4.1.1 such transitions are

labelled P^, P^, ..., P^, ... .

ii/ Transitions of the type £ N = 1, i.e. for which the

phonon number differs by one in the largest componet of the

Initial and final state. In fig. 4.1.1 such transitions are

labelled V,, V„. .... Vw ... .

7U6

Now we have two types of mixing ratios:

and

In aocordanoe with the previously established systematic des¬

tructive interference in higher-order terms, in deriving the

rule for d we take only leading order diagrams (including

induced collective diagrams for 22 moments) , as presented

in fig. 4.1.2

||Mv,B(E2)

MSP(M1) iy + y^r

MSP(M1) MVIB(M1) ?| + X 1

if M M S

H i -Pig. 4.1.2

Por processes labelled ?„ general expressions? for the E2 and

Ml transition moments between the states of N-phonon multiplet,

of angular momenta fr+ 2N-1 and i + 2N after working out the

angular momentum algebra are:

Here, Q(<) is the quadrupole moment of the particle state fr,

which acts as a band head, nere we include only a standard M1

operator, since it is allowed.

In leading order, the effect of phonon anharmonicities, if

sizeable, can be included in formula (4.1.2) by changing

where Q(2N) is the speotro8copic quadrupole moment of the

N-phonon state of angular momentum 2N.

Por the transitions of the type V„ (transitions between

the multiplets for which the main components belong to the

N-phonon and (if-t} -phonoa multipletiwe have:

(4.1.5) -M 4 Jj

Using expressions 4.1.3 and 4.1.4 we get for transitions

of the type AN = 0:

Using expressions (4.1.5) and (4.1.6) we get for transi

tions of the type AN = 1:

Inwbotl/eqs. (4.1.6) and (4.1.7) the sign of the mixing Q(d\

ratio is given by the sign of • ^- lf- so generally our

result is

This is a rule obtained in the spherical representation.

let us compare this rule with the well-taiown rale for the

mixing ratio in the rotational model

7-& recall that in the rotational model the spectroscopic

quadrupole moment of the band head is related to the ground-

state deformation

<**•>> •$$»<*' *•'•">

In this way, the rule for the sign of the mixing ratio, obtained in the particle-vibration coupling for spherical nuclei, is compatible with that obtained for deformed nuclei in the rotational model. Modification (4.1.*) does not affect this conclusion since Q(K}V) <v QC12)~ - Ą .

Here we have considered the yrast band for unique parity band (for example 3 = 1f?/i, i ?/i , In the case of normal-parity states there are additional con¬ tributions arising from the mizinp of nultiplets based on dif¬ ferent single-particle states and eventually changing a band head.

Finally, we present results of recent experiment on 11^In which has determined several mixing ratio*on the unique parity yrast band based on the 9/2+ ground state. In this case i = S9/2 . Since Q(g9/£1J> 0 and g, - gp > 0, we have due to the spherical X -rule {h.1.8) prediction

This agrees with the experimental data (W.H.A.Hesselink, J.Bron, P.M.A. van der Kam, V.Paar, A. van Poelgeest and A.G.Zephot, Nucl.Phys. 1973)) presented in table 4.1.1 :

TABLE 4.1.1

W2} 15/2;

15/2;

17/2}

9/2; 11/2*

13/2} 15/2;

0

0

0

.42 i 0,

.52 i

. 1 5 '•

EXP

• 0.11 .20

• 0.09 - 0.16

TH/PEi?P/

0.45 0.16

0.33 0.12

TH/EXACT/

0.44

0.20

0.54

0.19

750

In the column d IHE0R1(PEHT.) we present the results obtained by using the simple formula (4.1.7) for 13/2^ -^ll/st (in this case N=l, i.e., the transition occurs between the states for which the largest components belong to the oue-phonon multiplet) and for 17/2^ •> 15/2* (H=2), and by using formula (4.1.8) for 11/2* * 9/2* (N=l) and for 15/2* ->-13/2^ (N=2).

In calculating electromagnetic moments the following char¬ ges and gyromagnetic ratios have been adopted: e '"*= 1.5, e T I B = 2 7 g = 1 g = 0 7 g s , gR 2.7, 1, gs 0.7 g s

f r e e, gR = 0.

5.1 Decoupled and Strongly Coupled Bands in the Particle-Vi¬ bration Coupling

Considerable interest has been recently shorn for "decoup¬ led" (with spin sequence j, ć~2, j+2, ...) and "strongly coup¬ led" (i, 5+1, i+2, ...) band structures for unique-parity states

2,- "5g/£> 1hti/j' n i 3 ^ ' *•* l R t r a n s i t i o n a l nuclei. ( 7/ 2 g/£ ti/j' Shese tands have been interpreted in terms of symmetric and asymnetric rotor-psrticle coupled models (see lecture by Georg Leander). ™e would like to show that these patterns emerge already in leading order in the particle-vibration coupling. Let us consider the energy splitting of the one-phonon multi¬ plet. The contribution from two second order diagrams is pre¬ sented in fig. 5.1.1

\

Fig.5.1.1

751

The contribution to the energy for the one-phonon multiplet

state of angular momentum I is

Here the important role plays the anharmonicity of the phonon.

As a measure of anharmonicity of the phonon we take its static

moment QXTTB^I) ' n *^e c a s e °- harmonic phonon it is zero.

Then we have a contribution to the energy shift of the

multiplet states already in first order. Namely, for vibrator

we have now the diagonal matrix element O 2 u ( b £ +

so we have an additional, first-order term

Taking for Q(j) the contributions in zeroth and first order

Sec. 3) we can write (5.1.2) as

i 12 a ijk?

The corresponding first order diagram is drara in fig. 5.1.2

j

Fig. 5.1.2

The total energy shift in the first and second order is

752

Now depending on the sign of product Q(j) * Q(2) V_ B . we have the following situation, taking into account signs of the corresponding &j coefficients:

i/ For Q(a). 0(2)^3 < 0 :

AE(I = i-2) = NEGATIVE TERM + POSITIVE TERM &E(I = j-1) = Positive + NEGATIVE AS(I = i) = POSITIVE + NEGATIVE + POSITIVE £2(l = 3+1) = NEGATIVE + NEGATIVE A2(l = i+2) = NEGATIVE + POSITIVE

The state I = j-t-1 is the only one which has coherent negative contributions from AS,.(l) and AE,,(l), and is mostly shifted down. Therefore, the state I = j+1 lies below I = S+2: a sequence of a strongly coupled" band j, j+1, j+2, ... is es¬ tablished.

ii/ For Q(j) Qfe1)y-[.g> 0 the last tern in expressions given in /i/ changes sign. In this case the states I = j+2 and I = j-2 both have coherent negative contributions froiu ^,3^(l) and AEp(l), and are mostly shifted dorm; therefore I = j+2 state lies below the state I = i+1. Also, since si s- łl \[t £ tl '2 i-Z t l Si2j+2 15 ) the state I = i-2 lies

below I = 3+2.

In this way a decoupled band sequence j, j-2, i+2f j+1, ... is established.

The same type of consideration of leading multiplets can be extended to the consideration of two-phonon, three-phonon,... multiplets. The general conclusion may be erpressed as the °QQ selection rull'CG.Alaga and V.Paar, Phyo.Lett. 61B (1976) 129)S

753

Strongly coupled and decoupled bands are developed by the

particles-vibration coupling for the product of quadrupole

moments

QU)- Q(2)VIB < °

and

respectively.

In fig. 5.1.3 we present the resulting yrast states of

calculation for the coupling of a particle to anhansonic phonon.

Anharmonicity ofiiie phonon is defined by the cubic and quartic (?Qr i+i+l T ; -t tp1) ""5"" [ flfl?*) /•/1N 7

anharmonicities >•» D J o PO) A -^ £. \P " /^ \OP/^AQ. A^C,2,k

Quadrupole moment of the phonon is approximately related to

the cubic anharmonicity: <3^2)wo^ lCbCB3.)C2t -*"Ot'ivm,

In the calculation we use Av 4 0.6, AK '= 0.3.

The single-particle state is j = h ^ ,„t and phonon energy

nu) = 1. Yrast states are presented as function of the par¬

ticle-vibration coupling strength a. V,'e see well developed

"strongly coupled" rotational-like band 11/2, 13/2, 15/2, 17/2,

19/2, 21/2, 23/2, 25/2, 27/2, 29/2, 31/2, ... for a> 0.

Then Q(^>0 (a>0 corresponds to a hole), Q ( 2 ) V I B ~ ~ A <"°>

so Q(3)'<K2)VIB<0.

On the other hand, for a<0 v/e have a decoupled band

with high-spin states sequence 11/2, 15/2, 13/2, 19/2, 17/2,

23/2, 21/2, 27/2, 25/2, 31/2, 29/2 This is again due to

tlQ rule sinoe: Q(j)<0 (a < 0 corresponds to a particle),

Q'V2-)VIB~-A21 < 0, so Q(j) • Q(2) V I B> 0.

47/2

-45 /2

43/2

41/2

39/2

37/2

12 a

755

Por quasiparticle states, the BCS correlations can be in¬

cluded in the screening of a, namely a ^ u j "" j ' a*

What is the leading order prediction of the particle-vi¬

bration coupling for the E2 transitions between the yrast

states In the case of decoupled and strongly coupled band ?

By taking into account the zeroth-order and induced first-

order contributions we get for transition moment between the

memben of the same multiplet, i.e. the states for which the

largest component has the same number of phonons

(5.1.4)

Pnr transitions between the states for which the largest com¬

ponent differ by one phonon we have

<t, AH ^-iji

Thus for Q(j)-Q(.2) V I B < 0, for which a "strongly coupled" band

is developed, we have constructive interference for AI = 1

transition I = 3+2K-»I = J+2N-1, and destructive interference

for AI = 2 transition I = 3+2N-* I = j+2N- . 'Therefore, as

in a normal rotational band, we get large B(E2) values for

transitions ... J+5-> j+4-*3+3 -»j+2 -,3+1 -,j.

On the other hand, for Q(3)'Q (2)yrB>0> for which a "decoupled"

band is developed, we get constructive interference for A I = ?

transition I = j+2N-*I = 3+2N-2 and destructive interference

756

for AI « 1 transition I • j+2H-» I • J+2N-1. Thus we have

large B (E2) values for stretched E2 transitions

.... 5+6-*i+>i-*j+2—* j, aa is characteristic of "decoupled"

bands.

In this way, the same QQ selection rule is valid both for

energies and E2 transition for unique-parity yrast states.

In a transparent way we see now a mechanism, which generates

"strongly coupled" and "decoupled" bands. The result appears

even more general than it is in the framework of particle-vi¬

bration coupling: any model which in some way, implicitly or

explicitly includes quadrupole moment of the core and of odd

particle will lead to the pattern of "strongly coupled" and

"decoupled" bands. Here lies the origin of the fact that these

features arise in different models, both in spherical and

deformed representation.

6.1 Some Examples of Leading-Order Bffects for a Few Particles

Coupled to Quadrupole Vibration

In more complicated cases than the previously considered

of one particle-vibration coupling.systematic destructive in¬

terference in higher orders also plays an important role.

Again this enables us to extract only a few leading-order dia¬

grams, and deduce from them in a simple way certain qualitative

features, leading to simple rules. Here we present two such

effects, one for the case of two and one of three particles

coupled to quadrupole vibration (such model will be refered

to as cluster-vibration model,and it will be a subject of Sec.B).

757

6.2 • Quadrupole Moment of the State in Even-Even Nuclei

as a Leading-Order Effect in the System of Tiro Particles

Coupled to Quadrupole Vibration

In the case of n particles (cluster of particles) coupled to

the quadrupole vibration the E2 operator again has two parts:

the single-particle part (3.1.1) and the collective part (3.1.2).

However, the single particle part involves now sum over all

particles included in the cluster:

(6.2.1)

For two particles coupled to vibrator, in 6.2.1 we have n = 2

How in a specif!-; problem which we consider we have two

possible situations (G.Alaga, V.Paar and V.Iopae, Phys.Lett.i3P

(1973) 459; V.Paar, Heavy-Ion, High Spin States and Nuclear Structure

Vol.2, IAEA., Vienna, 1975, p. 179; and to be published).

i/ 21 state has as the largest component the two-particle

zero-phonon state: |(j2)2^> or \(3^i2y^ •

ii/ 2^ state has as the largest component the two paired

particles coupled to one-phonon state: \{i2)o, 1 2; £\

Case /!/

Now we have a similar situation as for the quadrupole moment

of one single particle in the particle vibration coupling,

only instead of one, now we have two particles. The leading

order contributions are due to the zeroth-order particle- and

first order induced collective diagrams presented in fig. 6.2.1

X

I I

r -X rig. 6.2.1

758

Again, as in Sec. 3, the sum leads to the factorization B' + P, + Pg - B' (6.2.2)

The quadrupole moment is given by the shell-model quadrupole moment of two bare particles, but it is enhanced by a factor *vj (for definition of see Sec.3); in fact the shell model|quadru¬ pole moment for pure two particle configuration is enhanced by renormalizing the charge: es#p^ł?es'p", where practically

Ca3e /ll/ In this case we have no zero- and fir3t-order contributions

to the quadrupole moment, since 22 operator of particle type 1£S**>*(E2) obviously has zero matrix element for paired particle state <02)o||Ms*p^2)H (j2)oy = 0, and collective part of E2 operator 55 (22) has no diagonal matrix element for phonon state

<12|| M7IB(E2)!| 12> > 0. (6.2.3; The lowest-order contributions arise in the second order, and the corresponding diagrams with nonvanishing contributions are drawn in fig. 6.2.2

Fit. 6.2.2

759

Here by two close paralel lines we denote two paired par¬

ticles. This pair can be broken due to the partiole-vibration

interaction with simultaneous emission or absorption

of. a phonon, or due to the interaction with the electromagnetic fie

M S* P*("E2), for example first two diagrams on l.h.s.jor due to the

interaction with the electromagnetic field MS#P*(S2), for example th

lower vertex in the two diagrams on the r.h.s. of fig. 6.2.2.

To each diagram of the particle-type in fig.6.2.2 there

corresponds a class of 4 induced collective diagrams, involving

an absorbtion or emission of an intermediate phonon by the

collective E2 operator II (s?). Again, due to factorization

analogue to the one in case /i/ all 24 induced third^order

diagrams can be included in the particle diagrams of fig.6.2.2

with the charge renormalization &S'^'->/rl es"p*. In this way,

we have to calculate only 6 diagrams from fig. 6.2.2.

6.3 Semiquantitative Estimate for cfc? ) (case ii)

If we write down the contribution from six diagrams in

fig. 6.2.2, including 24 third-order collective diagrams in

charge renormalization ee'p^-* eS*p*, in the adiabatic limit

0jf we get a simple and useful approximate formula:

C6.3.O

760

Here~Y"T(;J) is the quadrupole moment of the neighbouring nucleus with one particle coupled to vibrations. A is the pairing gap

%- 2 HeV around "2 = 50) and oL is the pairing amplitude ZZ 1 around H = 50J. The coefficients N, corresponding

to a given pair of valence-shell particles are listed in Table 6.3.1 TABLE 6.3.1

Por each available t ,, o , „ \ o n ; o Nc

1 1

j"/ j / / /

Ifov; we present two examples of application of 6.3.1 for even Cd (Z = 48) and even 1e nuclei (z = 52).

Per Có vie have: j = go/2 > sno two available non-spin flip pairs in the valence shell: (fi/lfa'/l')2 an& (? yl Ą 5/1)2

To each of these pairs due to table 6.3.1 there corresponds :i? = h, ir£ = 4 so we have total ll£ = 8, F^ = 8. These are the only nonvanishing IT-coefficients. ] By taking Q(d)s 0 { % / 2 ) ^ °'8 e b froin or3d I n isotopes and£=2 IieV,c(= 4 v;e get, depending on the single-particle states included, results presented in Table 6.3.2

:i

TABLE 6.3.2

-05x ł

761

Por Te we have: J = g?/2> ^ t h e available nonspin-flip

pairs are (£y2*y2)2> (3i/2(d5/2^2 a n d (s1/2,d3/2)2: The corresponding N-coefficients follow from table 6.3.1 :

N, 1 N. 2» *l - 2; N* ł » c • '» « a = '» b ' c '

N^ = 4; and NT = 4, K • 4; respectively.

By taking j Q ( 3 ) S <^(S?/2)'5' °'5 e b f r o m o d d S Ł i s o t o P e s w e

depending on single-particle states included, results presented in Table 6.3.3

TABLE 6.3.3

Si*\]k-peJ[tck jUi* t 4M) o.i

VJe see that the basic role in generating quadrupole moment for / —2 N Cd is played by the (69/2 ) 2 configuration and for Te by

('Sn/2 d3/2^2* TnUB» ^ e result is for Te critically sensitive to the d,/2 single-particle state. If this state is not in¬ cluded, or placed too high, the wrong sign of theoretical qua-drupole moment of the quadrupole moment is obtained /experimen-

762

tally ^(2-))me < c / Analogous game should appear also in other

types of calculations, for example in shell-model calculations.

6.4 Qualitative Rule for

Combining approximate formulae for cases /i/, /ii/, and

the simple shell-model properties Q[(j2)2]> 0, Q ^ M ^ J ^ °»

where j> and i2 present a nonspin-flip pair i.e. 3 • 3„ ~ 2P*"

with opposite sj ;S for two holes, the following simple rule

for the sign and magnitude of the quadrupole moment is obtained:

Q. t 4*

<o, fiH^JtrJ

4V* Here J is the lowest single-particle state, and j' are avail¬

able single-particle states of the angular momenta j - 2J and/or

2 - j if j is of normal parity in this shell, and j* denotes

all normal—parity single-particle states in that shell if j

is of unique parity.

763

Application of this rule to different nuclei is presented

in table 6.4.1.

I

U*

U i i

U

1

1 J

1 1

1 •* 1 1

TABLE

0

6 . 4 . 1

SfjONDiAO

0

>O •

Taking into account that Q(2^) has the opposite sign than

the ground state deformation Q of the rotational model, in

this way we could get insight into creation of the effective

deformation.

Finally, we should like to stress, that these results de¬

rived from simple formulae and rules are in agreement with the

extensive computations based on diagonalization of the Eaailto—

nian matrix for a system of two particles coupled to quadrupole

vibration.

Here we have assumed that the main part of the anhanaoni-

city is generated by the (two-particle)- phonon coupling i.e.

764

that the quadrupole moment of phonon Q v n, is negligible

If not, one has to ddd x ^viB *° e above estimates. Factor

* is estimated from the above renormalization.

7.1 The I « j - 1 Anomaly as Leading-Order Sffect for Three

Particles Coupled to Vibration

The so called . I .*.. jai. an omal y. JjS- _.efiamoa_feajaire...ln_.odd

nuclei with three or more particles /holes/ in the valence

shell with the low-lying high-spin single-particle state

/j^> /for example f-y2> Sq/2» ^\\l^' I n s u c h c a s e s experi¬

mentally one observes low-lying state I = j-1, which may even

become a ground state /I * j-1 anomaly/.

This feature, which can be reproduced in the shell-model

calculations and in the particle-rotor calculations, we present

here as a leading-order effect in the model of coupling three

particles (cluster of three particles) tc the quadrupole vibrating

Let us consider the unique-parity states. In the zeroth-

order approximation the lowest lying state is | ó ) I * j ;

at the energy »«> above it there is a one phonon multiplet

I(i3)h 1 2 ; I ^ with I = j-2, j-1, j, 3+1, j+2, based on the

three-particle state /(j ) j ^ » at the pairing energy A above

)0 )d^ there are states of seniority three /(3^jl^» , In tha

presence of particle-vibration interaction the leading contri¬

bution to the splitting of one-phonon multiplet states is cue

to the second-order diagram presented in fig. 7.1.1.

765

Pig. 7.1.1

In the intermediate state, because of angular-momentum conser¬

vation, the contribution is only due to the configuration

(3 )l. The corresponding formula reads

So the energy shift is basically given by the square of three-

particle shell-model matrix element. After inserting the ex¬

pressions for coefficients of fractional parentage, these can

be brought into the form:

C '

The energy denominator iii 7.1.2 is negative; consequently,

H(l) is proportional to "^<ifyi\tfjAf>^ %, the I-dependent shift of multiplet states I ji i, given by the

second term in the upper line of eq. 7.1.2 , is proportional

to

766

f f By using analytical expressions for 6j-symbols, we obtain in

the asymptotio limit (j » i);

i.e. only the I * 3-1 state is shifted doim because of the

I~dependent term*- while the order members of the multiplet are

shifted up.

The 63-coefficients i\ £ ^j have for any

the same sign as in the asymptotic limit (e^.7.1.ft).

The first term in upper line of eg. 7.1.2 presents the

I-independent shift downward of states I = 3+1 > 3+2.

In the asymptotic limit j » 1 » the ordering of one-phonon

aultiplet levels is I = 2-1 , I • i-2 , I = j+1 , since

owing to»j7.1.2 the shift downward is proportional to 1 + 2

,. 120 a n d , - 60 r e s p e c t i v e l y . • (2i)

(2) 3 C2i 'Phis feature appears qualitatively also for any $> £ . ?or

n

example, in the case 3 = 7 the matrix elements 7.1.2 are

^ for I = I , i for I = 2J , TJ f or I = | , y£ t or I = | and 1 7 "* ">

i~- for I • j . Thus the ordering of the ty2 nultiplet states

is the following: |, ^, |, |, |. The situation is analogous

also for the other single particle states i £ | .

Thus, the lowering of the "anomalous" I » j-1 state appears

in the second-order of the three-particle-vibration coupling

as a consequence of the angular momentum algebra in the shell-

767

model three-particle matrix element*

There is also an additional second-order contribution to

energy splitting of one phonon multiplet, due to diagram in

fig. 7.1.2 which involves two-phonons in the intermediate state.

(J3)j 2

A

(a) (J3?J. (b) MSP(E2)

*"w—X

3 x

2 (J3)j i» • » *

I Pig. 7.1.2

J\ 1 I 1

. Pig. 7.1.3

How there is no restriction on the angular momentum of the in¬

termediate three-particle state. However, the term with the

smallest energy denominator (-"few/ is the one vrith the inter¬

mediate three-particle state (j )i\ its contribution is agair.

proportional to (,-)•!"' Jl * I i.e. it enhances the effect I* I p

of 7.1.1.

As we see, in the 3partiele-vibration model anomalous

I = j-1 state is of collective character |(j )jf 12; I^in ze-

roth order, but with a strong first-order admixture of the

| (j )l * 3-1/ three particle state. The character of this

state is reflected in the large B(B2) value for the 3-1 -*• j

transition. In the zeroth-order approximation this is a collec¬

tive transition from the one- to the zero-phononj state (diagram

in fig. 7.1.3a).

768

The corresponding contribution ia

m 1(7.1.5;

In first order, because of the diagram 7.1.3b, the contribu¬

tion is of the form

In ths case considered here, (^<~ Sj*^*** ^ ° (since

A >"fcuJ ) and a < 0 , ao (7.1.6) is positive, in

constructive interference with the zeroth-order term (7.1.3).

As a result, tho |j-£>-*| j)> E2 transition ia strongly

enhanced.

Thus, in the taodel of coupling three particles to the

quadrupple vibration we obtain aa loading-order effects the

"anomalous" low-lying I=j-1 state of a collective character.

At the same time the /l=j-l)>-> / I=j^ Ml transition is hindered

(this follows from a discussion analogous to that in Sec. 2).

7.2 The I = j-2 Anomaly as a Leading-Order Effect for Three

Particles Coupled to the Quadrupole Vibration

In Sec. 7.1 we discuss the lowering of the Iaj-1 state

for the (j ) configurations. In some nuclei with the low-lying

I=j-1 state there appears also the anomalous low-lying I=J-2

state; this is sometimes even a ground state and we call it

the 1=j-2 anomaly.

In the model of three particles coupled to the quadrupole

vibration the I=j-2 anomaly is due to the influence of the single-

769

particle configuration / j ^ » I J-2> auoh that

£..y i . + ku, if it exists. (If it lisa lower, then the

character of the lowest I*j-2 state is changed, it is no

longer | (j3) j,12; I»J-2> tout / [(j2)O,3'J j'«j-2> .)

If a single-particle state jj'^>/ |j^> lies above

£. + tu),, an additional second-order contribution comes from

the diagram of the type in fig. 7.1.1. but now with the

/ [(32)O,j'J I=j'^> as an intermediate state contributing to

the energy shift of a one-phoaoa multiplet state of angular

momentum I=j'. This second-order contribution is proportional

to the square of the matrix element

In the asymptotic limit (j^>l), we have

JL s (722)

This spin-flip selection rule, already used in Sec. 3, is s t i l l approximately effective for the real ist ic j ; for exampla, for

we hare

ThuB the only possible aizeable influence on the splitting of the one-phonon multiplet J( y)Jf12}l\is due to the possible presence of the single-particle configurations / j ' = j+2^> and/or / j ' - i -2> .

Let us consider the effect of the presence of t h e j j ' * j - 2 \ state (this i s usually the case, since the spin of j i s already rather high ) .

770

In second order, the contribution from the \{j )O, j ' j I=j ' \

cluster as an intermediate state, is

The matrix element (7.2.1), which causes the shift ot the 1=3-2

state, is even lareer than the matrix elercent (7.1.2), widen

fives rise to th« shift n* ~ol*.-i I=j-2 fswto. In the asymptotic

limit (j » 1 ) we have:

• J - ,'(7.2.4)

Consequently, with increasing particle-vibration coupling

strength a, the I=j-2 state may ba aaifted down even more than

the I=;j-1 state, if the single-particle conf igurat ionVia

available; the I=j-2 state then crosses even the I=j-1 and

1=3 states and becones the lowest state of the corresponding

parity. The collective nature of this transition is again

reflected in the strong |l=^-2%>-^ J I = ^ E 2 transition.

Thus, the conclusions arising from the leading processes

in the three-particla— vibration model are the following:

In the ca«o of the low-lying single-particle atata

| i ^- 5/2^ such that no other single-particle state j i'^

with gj + Leo - C , < 0 of the same parity exists

771

there arises:

i) the low-lying collective doublet I=j, j - 1 , if the single

particle \i'} = \i~2^ state of the same parity is not available

in the same valence shell. Such a situation may appear, for

example, for f"y2 (Z=25 and K=25) nuclei in the 20-28 shell

(no pT/p hole state is available in the oscillator shell

f7/2' P3/2 ł P l / 2 ' f5/2^» g3/2 ( Z = 4 7 a n d N = 4 7 ^ n u o l e i ^ 28-50 shell, h^yg (Z=79 and N=79) nuclei in the 50-82 shell.

i i ) The low-lying collective triplet I=j,j-1,j-2 appears if a single-particle state of jj '} =|j-2^ is available in the same valence shell. Such a situation nay appear, for oxaziplg, for t*,2 nuclei (Z=23, N=23, N=85), Ą ^ nuclei (Z=43 and N=43), and h*y_ nuclei (2=85). In these cases the available single-particle configurations | j ' ^ = [j-2^ of tho same parity are

P3/2' d5/2' &ni f5/2 ł rQ8P°c'tively«

Taking into account the usual single-particle spectrun,

the partner / j ' ^ =/j-2^ single-particle state appears in tho

specific case of nuclei with threa particles in the valence

shell or in a unique-parity subahell; thonrfhe triplet condi¬

tion is satisfied.

8.1 Conclusion: Coupling Strength Is ofc V.'eak, V.'ave Functions

are j oj simple, but still there is Something Very Simple...

In quantum mechanics the attempts made in Chapters 2-7 meet

immediate criticism:"coupling strength is not weak, wave functions

are not simple, so it does not make sense at all to use some per¬

turbation arguments". So, let's go to diagonal!zation of Hamil-

tonian matrix. We get numbers out of the computer, but whether

there are some simple processes which give dominant contributions,

which basically determine the properties of the coupled system,

large computation leaves hidden. On the other hand, diagrammatic

method opens the possibility of "easy" grouping and classification

of the parts of the perturbation expansion. Of course, a sum over

ril possible diagrams is equivalent to diagonalizing the Kamilto-

nian, but v/e can try to select leading classes of diagrams. These

diagrams then can provide a simple understanding of some features

of nuclear structure, and show that these features are basically

simple; thus, v/e could trace them in different modelf. If these

basic elements are the same in different models, which otherwise

may substantially differ, es;ecially at the first sight, we may

get very similar results for particular properties.

In the presence of particles /one or more/- vibration coupl¬

ing such an understanding of simple origin of soce properties

leads to certain predictions; rules and even semiquantitative esti¬

mates. In chapters 2 - 7 some of what we know are discussed, while

probably there are many more as yet unknown.

773

In the former Chapters we have clearly pointed to the sec¬

ret of origin of these simplicities: many higher order Serm

/self-energies and vertex corrections/tend to cancel due to de¬

structive interference and/or lead to renormalizations^a sort

of smoothing procedure by nature. The analog features are well

known in quantum electrodynamics; there the Ward identity expres¬

ses the destructive interference between self-energies and ver¬

tex corrections, which becomes exact in the limit of small four-

-momnetum transfer. It is interesting to consider similarities

and differences between two boson-fermion systems: electron-

photon and single particle-phonon. We. leave this to the reader

/with a comment that Y2 in the field-theoretical formalism reads

where oT"* is a creation operator for single particle \i"r^> /• a

Finally, our conclusion is that some nuclear properties are

basically given by specific diagrams /perturbation terms/ in the

presence of particle-vibration coupling. When interaction becomes

stronger, wave functions become more and more mixed, main compo¬

nents smaller and smaller, but a sum of^results remains rather in¬

sensitive to this, or changes in a very systematic way: what is

lost due to decrease of main components is recovered by many new

components. This is visualized in Fig. 8.1.1.

This is a content of Ward-like identity. It is probably worth¬

while to undertake systematic investigation of its role not only

in the particle-vibration models, which seem to be especially

transparent in this respect, but also in other models. This could

otter the possibility to find out what are essential ingredients

of a certain model with respect to certain physical properties.

SIMPLE W. F.

ONE OR A FEW DOMINANT CONTRIBUTION IN LEADING ORDERS-CONTRIBUTION FROM ONE OR FEW COMPONENTS IN SIMPLE WAVE FUNCTION

VERY MIXED WF APPROXIMATELY \

CONTRIBUTION FROM MAIN COMPONENTS

CONTRIBUTION FROM MANY ADMIXED COMPONENTS

Fig. 8.1.1

In other words different models may then arise as equivalent in

respect to processes which dominate desired property} while the

other aspects, in which they differ, may mostly be unimportant

due to systematic destructive interference, leading mostly to

calculate additional aeros in different ways /see Fig.1.1.1/.

Comment: Of course, eventual closeness of a singularity points

in energy denominators can at the first sight mask these simple

features, but experience for particle-vibration models shows

that also in such cases there is a smoothing procedure based on

alternation of sign in successive orders of the perturbation ex¬

pansion; so one can consider situation away from singularities •

and then extrapolate through singularity points. Comparison

with exact result /of diagonalization/ indicate that such a

procedure is.reasonable. The subject «f this comment is not dis¬

cussed in the present lecture.

775

B. OLDSTER-VIBRATION MODEL: INGREDIENTS,CALCULATIONS, BANDS

9.1. Cluster-Vibration Model: What it is ?

Cluster-vibration model is a simple realization of the

"Landau representation" /see Sec.1/ for the nuclear system.

With respect to a simple particle-vibration coupling it is

more complicated: instead of one particle, now there are more

particles /holes/ or BCS quasiparticles coupled to vibration.

These few particles, which we treat explicitly as fermion de¬

grees of freedom, we call cluster. It is nothing like a com¬

pact cluster /for example oC -cluster/; it is a dynamical clus¬

ter, for example of a few particles moving in the valence shell.

There is also one more additional element with respect to usual

particle-vibration coupling: the residual force between the par¬

ticles of the cluster. We take again the simplest but most im¬

portant component of the residual force: monopole pairing. Thus

we may say: In a veiy small part of a Hilbert space for a nu¬

cleus, for a cluster, Pauli principle is taken into account.

In the rest a complete averaging over the shell model degrees

of freedom is performed and description in terms of only col¬

lective variables assumed: here the role of Pauli principle be¬

tween the nucleons is completely ignored. Pauli principle in

the presence of particle-vibration coupling, in this simple mo¬

del indeed shows a richness of phenomena. Probably this is a

line of future progress in nuclear structure. Here we choose

everything as simple as possibles the simplest clusters /two

three or four particles, two or three BCS o.uasiparticles/, the

simplest dominant residual interaction /monopole pairing/, the

776

simplest dominant vibration /quadrupole/; still, the introduction

of the Pauli principle seems to introduce basic improvements with

respect to the usual particle /or BCS quasi-particle/ - vibration

models, as well as a key to better understandings how the physical

properties of a nuclear system are generated.

The background of the cluster-vibration model can be also

viewed as follows. What we would like to do is complete /read:

impossible/ shell-model calculation for a simple residual force:

short range pairing plus long range quadrupole-quadrupole force.

Ze mean complete shell-model calculation with all protons and all

neutrons with many proton and with many neutron shells. In front

of such an impossible task we do an intuitive /phenomenological/

step. 7,e take a few particles in a valence shell /or subshell/

and treat them explicitely, as shell-model particles confined to

move in the valence shell or subshell. This is our dynamical

cluster. All the other nucleons, jumping over all the many shells,

arc tr&ated by a complete averaging procedure; this complicated

cotion is apororcLmated by the effective vibrator. Its quanta,

phonons, here approximately describe the infinitely complex shell-

-model solutions of complete shell-model calculation in this spa¬

ce. Thus, what we get by the phenoiaenological simplifying step

from terribly big shell-model problem to the cluster-vibration

model is partly /for states which are predominantly zero-

-phonon/ like a very restricted shell-model calculation for a

few nucleons /cluster/ in the valence shell, but with.for shell-

model habits.unusual effective interaction: pairing plus noninstan-

taneous phonon exchange and phonon self energies. Partly /for

states which are predominantly phonon multiplets/ it corresponds

777

to the introduction in the basis states of large building

blocks incorporating combination of many shell-model confi¬

gurations. How good approximation of a real situation for our

redefined "core" is an effective vibrator ? What about the Pa-

uli principle violation between the particles of the cluster,

and the particles of the nucleon excitations in the internal

phonon structure.which may also be excited to the same shell

which we have reserved for the particles of the cluster ?

This part of Paul! principle is neglected in the model. How¬

ever, since many shells are available to nucleons from the vi-

brational core, and since we do expect the main contribution to

these vibrational excitations from the excitations within the

valence shell for larger number of other type of valence-shell

nucleons than those included in cluster,it seems that components

in the wave function of an effective vibrator which involve exci¬

tations to the valence shell of the cluster are not sizeable. '!e

do not calculate this because we do not assume any specific in¬

ternal structure of phonon. Therefore, this shortcoming of

the model does not seem more serious than the other assumptions

made, for example the one regarding the residual force or har¬

monic character of an effective vibrator.

What to include into the cluster ? In the present lecture

we consider the nuclei where the choice of cluster is straight¬

forward and simple. For example, if we have a few protons in a

proton valence shell and 10 neutrons in the neutron valence shell

the choice is obvious: these few protons in the valence shell will

be taken as a cluster. Whenever we have a few particles or holes

in valence shell or subshell, while the other valence shell

778

contains even number of nucleons closer to the half of the

shell, the choice of cluster seems unambiguous. Such cases will

be discussed in the present lecture with two- and three-particle

clusters.

For odd-even nuclei, we take a cluster in the valence shell

with odd number of particles.

If the cluster with odd number of particles contains five or

more particles /holes/, or if in even nuclei both valence shells,

for protons and neutrons, have .jur or more particles, we heve to

simplify a cluster. One rather rough way is to take a smaller

cluster /two-particle for even, three-particle for odd nuclei/

with renormalized single particles and phonon.

The second, better way is to approximate 5,7...-particle

cluster by a 1 plus 5 -quasiparticle cluster, and 4,6...-parti¬

cle cluster by a 2-quasiparticle cluster. Then we have a syste-

natics of calculated isotopes or isotones in dependence on the

number of particles in valence shell. Such a program, with a

particle-number projection, has been recently developed ,

but in these lecture we will not discuss these very interest¬

ing results.

The third way is to treat more complicated clusters /4,5

particles, 1 particle-3hole, etc./, involving even limited

mixed proton-neutron clusters. This, in fact, would mean to

include partly a phonon dressing in some smaller shell-model

calculations.

7 9

9.2. Hamiltonisn Matrix and Piagonalization

For the n-particle cluster the Hamiltonian reads:

| ( . ) f H P 9.2#1 Here HJ(L. describes the motion of n valence - shell parti¬

cles or holes and H represents the free quadrupole vibra¬

tions /phonons/. The third term is the familiar particle-vi¬

bration interaction (1.3.2) but now it is summed over all parti¬

cles of the cluster. Hp is the pairing residual force between

the n particles of the cluster.

In setting up the Hamiltonian matrix there appear the same

number of parameters as for example in the classical Kisslin-

ger-Sorensen model /1 quasiparticle-vibration coupling/:

- single-particle energies in the valence shell for cluster;

in principle can be taken as extracted from transfer data

- phonon energy tti » ia principle can be taken as the energy

of the 2^ state of the core though somewhat lower value usu¬

ally gives somewhat better result. This reflects additional

polarization of core.

- pairing strength G ; the usual estimate is G ~-^ •

- particle-vibration coupling strength; in principle can be

determined from B(E2)(2.|-». O^) and < *• 57 ^> ^ 50

by using estimate q. zfH £lls£ ,/0(EIX2*-»0.%« -In some regions optimal values"forYa show attenuation with res¬

pect to thiB simple estimate. This seems analogous feature to

the attenuation of the Coriolis - coupling in the rotational

model.

780

The Hamiltonian 9.2.1 is diagonalized in the basis

Here 0 is tne angular momentum of the n-partiole cluster, and

f ar* additional quantum numbers specifying the state of the

cluster. NR denote N phonons coupled to angular momentum R.

Cluster angular momentum J and phonon angular momentum R are

coupled to the total angular momentum I.

As a result of diagonalization we get energy spectra and

wave functions in the basis (9.2.2).

9.3. Electromagnetic Properties

Using wave functions obtained by the diagonalization of

cluster-vibration Hamiltonian, we can calculate different nuclear

properties, electromagnetic properties, spectroscopic factors for

transfer reactions etc.

Electromagnetic properties are obtained by calculating ma¬

trix elements of electromagnetic operators, using the wave func¬

tions of the cluster-vibration model.

The E2 and M1 operators have their single-pax tide and col¬

lective parts. They are the same as in the case of one particle

-vibration coupling, (3.1.1),(}. 1.2) and(<^.1.i), (2.1.2),(2.4.i) ,

respectively. The only difference again is that in the single-

particle part of the E2 and M1 operators we perform summation

over all particles of the cluster.

We comment on the effective charges and gyromagnetic rati a;

The single-particle charge includes the bare charge and the pola¬

rization charge in which way we simulate the possible excitations

781

of the particles of cluster to higher shelK, which is neglected in the model. Usually we take es*p'= 1.5 for protons and o.5 for neutrons, i.e. the polarization charge of 0.5 both for pro¬ tons and neutrons.

Instead of VB(E2)(2^*0$^ in (3.1.2) we usually employ the vibrational charge e

Wfl= i OK,,

Throughout the periodic table /except well deformed nuclei/ it is mostly between 2 to 3.

For the gyromagnetic ratio <Ł- we take the hydrodynamic 2.

estimate*' = » ; though this is the upper limit. For gyro-magnetic ratio cf we take the free values, 1 for protons, 0 for neutrons. For a- we use quenched values, as usual. We treat this as parameter. Usually it- is in the region 0.6 -- 0.8&gee, where ggree = 5.59 for protons - 3.82 for neutrons.

The tensor term with 9U is not included in most of calcu¬ lations. Where included, we use in accordance with the estima¬ te by Hamamoto for Pb region <f =0.24 ggIee.

10. Examples of Cluster-Vibration Coupling Calculations for Three Particle Clusters

Calculations have been performed for many nuclei in the re¬ gions A = 40 - 150 and A = 190 - 220. Here we present a few examples. '

10.1. Are "Decoupled" Bands in °*' °^Au Evidence for the Ro¬ tation Aliened Coupling to a Tri3xial Shape ?

As a first example we present results of calculation for '' 7-"Au. This is specifically interesting case since the ne-

78a

gative parity states show a so called "decoupled" band, while

the positive-parity states resemble partly to the weak-coupling

scheme. The negative - parity decoupled band sequence 11/2",

7/2^, 15/2^, ... has been recently interpxeted by Meyer ter Vehn

and Stephens as the evidence for the triaxial shape of Au nuclei.

Therefore it is interesting to compare the results of cluster -

vibration model for these nuclei /V.Paar, Ch. Vieu and J.S. Dio-

nisio, Nucl.Phys. A284- /1977/ 199/ with the results of other mo¬

dels. This comparison is presented in Figs. 10.1.1 and 10.1.2

for negative and positive parity levels, respectively. The choice

of a cluster for Au nuclei is unambiguous. In AiAu^^^ we have

5 proton holes in the 50 - 82 shell and 32 neutrons in the 82 -

126 shells. Obviously three proton holes in the 50 - 82 shell

present a cluster, while all the other excitations are included

in the effective vibrator. In the internal structure of the pho-

non the dominant role is probably played by the neutron excita¬

tions in the partly filled 82 - 126 valence shell.

As seen, the cluster - vibration model reproduces "decoupled"

negative - parity band based on 11/2" state. Also the shell-model

calculation by Hecht /Phys.Lett. ,5_8B /1975/ 255/ for three pro¬

ton holes in the 50 - 82 shell is successful in this respect. We

should also remind of the particle-anharmonic phonon model dis¬

cussed in Sec. 5t which also gives such a decoupled band. For

unique parity states particle-anharmonic vibrator model is a reaso¬

nable approximation to the cluster-vibration model for 11/2" band,

since h^u approximately factorizes out of three-hole clusters.

Thus, the interpretation that unique parity band in Au iso¬

topes is conclusive evidence of the triaxial shapes does not seem

783

MIAŁ ROTOR

TRMXIAL ROTOR

TEPHENS MEYER rtot* TERVEHN"

NEGATIVE RARITY LEVELS

11.11 tllllł

"^~"~ (1 ____

SHELL MODEL HECHT*1

«>Au CLUSTER-VIBRATOR

OUASRWTICLE-VIBRATOR

Pig. 10.1.1. Comparison of experimental and theoretical negative -parity states in 1 ^ » 1 ^ 5 A U ł Experimental data: radioactive decay /no label/, reaction studies /label**/, both radio¬ active decay and reaction studies Aabel* /

78U

POSITIVE PARITY LEVELS

- 1 1.5

- t »

- 11.4

IKAu " 'Au CLUSTER-VIBRATOR

OUASIFARTCLE -VIBRATOR

Fig.10.1.2. Comparison of experimental and theoretical positive-

parity states in

785

to be justified.

By using the cluster - vibration model with the same para-

metrization as for negative - parity states we have calculated

also the positive parity states. As seen from Fig. 10.1.2 agree¬

ment with the experiment is rather good. For these nuclei no

other calculations for positive-parity states are available. It

would be interesting to calculate the positive-parity states by

using the same triaxLal shapes as for negative-parity states.

Looking at Figs 10.1.1 and 10.1.2 , one can tentatively as¬

sign 16 negative - and 16 - positive parity calculated levels to

the experimental ones.

The comment on the theoretical situation around odd Au iso¬

topes is in a picturesque way presented in Fig. 10.1.3

PARTICLE (QUASI PARTICLE)

Fig. 10.1.3

786

The moral of the story about the normal and decoupled unique-parity bands may be: any model which implicitly /like cluster-vibration model, shell-model/ or explicitly /like particle-tri-»T"t"i rotor model, particle—enharmonic vibrator model/ includes correct signs of quadrupole moment of the band head and of the neighbouring even core may work for this band pattern.

(Therefore, it may be misleading to interpret these bands as the evidence for a certain model.

10.2. The I = j - 1 and I = .1 - 2 Anomaly in t„/2 ~ S h e 1 1 Nuclei

In fn/p nuclei the lowering of the 5/2" /I = 3-1/ and 5/2" /I = j-2/ states is a pronounced feature. This will be re¬ ferred to as the I = j-1 and I = j-2 anomaly-, respectively. In the odd frw 2- n u c l e i w4*0 one particle or hole in the 20 - 28 shell this anomalous lowering does not appear; the 7/2" state, arising from fn/2 i s sizeaW-y lower than the other negative-parity states. However, for odd nuclei with 3 particles in 20 - 28 shell there is a low-lying triplet 3/2", 5/i" > 7/2T , and in nuclei with 3 holes a low-lying doublet 5>/£~t 7/2"• La Fig. 10.2.1 we present a sketch of experimental situation for low-lying states.

In Sec. 7 we have seen that I = j-1 and I = 3-2 anomaly arise naturally in the cluster-vibration model as the leading or¬ der effect. In the Z N = 23 nuclei due to the fact that fy^ configuration in the shell above is available, and belongs to the same oscillator shell as jL.„ , we have the I = j-2 anomaly in addition to the I = 3-1, On the o ćher hand, for HQS Z = 25 nuclei, the pCj hole is not available and we do not have the

787

•7/2" •5/2"

2 5 M n

-5/r •7/2"

•7/2"

5jMn

5/2"

7/2"

DOUBLETS

7/2"

5/2"

•25

7/2" 5/2"

J72" •3/2" .5/2" "7/2"

-3/2"

•5/2"

-7/2"

TRIPLETS

•5/2"

-7/2"

23 A9w 23 v

23

.5/2"

23

-7/2 1

Fig. 10.2.1

788

i? 1r

7/2 3/2

1

Fig.10,2.2. Low-lying states in Z or N=23 nuclei as a function of particle-vibration coupling strength a

7ŻT 7/2~ 7/2"

T/r^s/r ~5/2~

a -0 .5 1 12 1.A 1.6 1.8

Fig.1O.2.3. Low-lying states in Z or N=23 nuclei as a function of particle-vibration coupling constants a

789

I = j-2 anomaly. Also, the lowering of anomalous states reflects

the collectivity of the nucleus, because it increases with in¬

creasing coupling constant a. This is reflected in experiment

in the fact that for singly closed-shell nuclei the anomalous

states are less shifted down.

In Fig.10.2.2 we present the result of calculation for low-

lying states in Z N = 23 nuclei as function of particle-vibra¬

tion coupling strength a. . Here £ ( p^yO - £ (-fv/Z ) = 5 MeV»

•k(0 =1.5 MeV.

In Fig.10.2.3 we present the result of calculation for low-

lying states in Z or H = 25 nuclei. Now f>, configuration is

absent, and we get no lowering of the I = j-2 = 3/Z state.

In Fig.10.2.4 we compare the spectrum calculated for ^Mn

/V.Paar, Nuovo Oimento 32A /1976/ 97/ with experiment and with

the other available calculations.

In Table 10.2.1 we present the comparison of calculated

and experimental electromagnetic properties of Mn

5/2-f -• 7/2-

3/2,,--• 7/2>J"

11/2,,- -> 7/2f

9/2^ -» 7/2.,"

3/2^ -» 5/2f

9/2^ -» 5/2f

15/2^ - • I I ^ "

TABLE 10.2.1

B(B2)(eb)2

THE

0.014

0.008

0.010

0.006

0.004

0.003

0.010

EXP

0.014(3)

0.015(5)

0.014(3)

0.009(2)

0.005

O.OO7

B ( M 1 >C%) 2

THE

0.001

0.008

0.062

EXP

0.004*1

790

E(MeV)

-3 /r

-B/1" -3/2"

-9/2"

-n/r

-15/2"

-9/2" -n/r

5/2" Ś.2"

-3/r

-5/2~

-3/2" -s/r -3/r

-5/2"

-7/r present

calculation OSNES LIPS AUERBACH McCULLEN OeSHALIT (SM.) McELLISTOEM (SM) BAYMAN (S.M.)

(SM) ZAMICK (SM)

-3/2

•7/2" LAWSON URETSKI

(SM)

-3m-

-vr 7/2" - 7 / 2 - 7/J-

RAJ KISSLINGEB CHUNG RUSTGI SORENSEN SINGH (SM)

(SM)

10.3. Odd-Ag Kucie!: Coexiatenoe. of "Weak Coupling" and I s.1-1 Anomaly

The variety of experimental data on 1°5t107*109^ iaotopes represents a challenge to the basic concepts of the shell-model and collective picture. Low-lying negative-parity states exhi¬ bit a multiplet pattern. The ground state 1/2" /it is tempting to interpret it as (^ / is followed by a doublet 3/2", 5/2" /it is tempting to interpret them as a weak-coupling multiplet iPiltL » 1 2J /'2' 5/^ and in the f o l l o w i n 6 group of states there appear 3/jT, 5/Z~, 7/2.", 1/2~, 9/Z~ /it is tempting to interpret them as [fyz , 20] Z,[fi/x ,22j 3/Z ,5/1 ,[fyz , 2 ^ 7/2 f 9/-i /. On the other hand positive-parity states reveal the I s j-1 anomaly: 7/2^ state lies below 9/2^ , although the only available positive-parity valence-shell configuration is qZj. . Furthermore the 7/2^ -^ 9/2^ E2 transition is strongly collective.

Among the properties which present challenge to the weak -coupling or intermediate coupling of one hole /or quasiparti-cle/ to vibration there are:

- Q(3/2;)< 0, ClĆ5/2p < 0 }

- U(H1) (3/2} -> 1/2}) large ;

and among those challenging the shell-model approach: - multiplet|pattern for negative parity states^

3/2} -+ 1/2}),

(7/2} -* 9/2}) only moderately retarded.

79a

It seens obvious that both the collective aspect /quadru-

pole phonons/ and the shell-model aspect /Paul! principle In

the valence shell have to be taken Into account. Such a cluster-

vibration model calculation performed for Ag /V.Paar, Nucl.

Pays. A211 /197J/ 29/ resolved these problems. In Fig. 10.3.1

9/2* • 7/2* •

0 1 1/2". THEORY EXP THEORY EXP

F i g . 1 0 . 3 . 1

793

we sketch a comparison between the calculated and experimental

negative - and positive - parity states.

In the comparison three additional experimental states are

introduced /11/2}, 13/2j, 5/2^/ identified after this calcu¬

lation was performed.

In Table 1O.3>1 we compare the experimental and calculated

electromagnetic data of °Ag:

3/27 -*-

5/27 ~* 3/2- -

5/2- -»

3/2^ - ^

5/27 -»

S/2' -

3

5/2- _*

3/2- -»

9/2} -> = = = = = =ss = = =

1/27

1/27

1/27

1/27

1/27 3/27

3/27

3/27

5/27

3/27

T/2+

1/27 3/27

5/27

T/2J

TABLE

a- === ========

" B (E2 ) ( e 2

EXP

0.111

0.107

0.00004

0.0058

0.0061

0.008

0.018

0.0011

0.0022

£0.219

10.3.1

THE

0.106

0.090

0.00003

0.005

0.004

0.O0T

0.004

0.004

0.038

0.132 S=S=£S£S!S= S=== == = == = ==

EXP

< 0

THE

-0.12

-0.39

===================== B(MI)

EXP

0.20

0.18

/vO.003

0.041

0.049

0.10

0.12

0.20

£0.030

„„.«„,,

EXP

-0.13

4.27

THE

0.35

0.16

0.01

0.07

0.11

0.39

0.35

0.23

0.022

THE

-0.08

4.50

79I*

10.4,The 7/2^ and 13/2^ Bands in i47Sm and the De¬

coupled Band - Normal Band Interplay

2>ne ^^Smft nucleus has three neutrons in the N=82-126

shell, which play the role of the cluster in the cluster-

vibration model. The calculation reproduces two pronounced

bands, the 7/2T ground-state band, sad the 1 3 / ^ band, in

agreement with experiment, as presented in fig. 11.6.1

(J. Kownacki, Z. Sujkowski, E. Hammaren, E. liukkonen,

M. Piiparinen, H. Hyde, T. Lindblad and V. Paar, to be

published).

Up to 15/2", the 7/2T ground-state band resembles the

well-kao?oi pattern of the ao-callsd decoupled band. There is

a stretched 1=2 B2 cascade j+4 = 15/2^ - j+2 = 11/2^ - j =

7/2", with the inversion of tho level ordering with respect

to the normal band: J+l = 9/2^ lies above 5+2 = 11/2^ and

j+3 = 13/2" lies above j+4 = 15/2~. furthermore, in the

energy region between the j = 7/2" and j+2 = ll/2~ state

there are two low-spin intruder states: j-2 = 3/2 and

i-1 = 5/2". The origin of such 1=3-1t j-2 anomalies has

already been discussed.

A decoupled-band pattern is partly revealed in the

calculated B(E2) values: B(E2)(15/2^ - 11/2^) = 0.99,

B(E2)(ll/2^ - 7/2") = 0.099; B(E2)(l3/2" - 15/2][) -

0.041, B(E2)(9/2^ - 11/2^) = 0.059 (eb) .

The 1=1 Ml transitions are only moderately hindered:

B(Ml)(l3/2^ - 15/C) = 0.110, B(KL)(9/2^ - 11/2J) = 0.039 „. It was shown in sec. 5.1 that in the case of a particle

coupled to the anharmonic vibrator the decoupled-band structure

appears when the product of the apeetroscopic quadrupole moment

of the band head and of the effective core is positive; if it

is negative, a normal-band structure appears. For the cluster-

vibration coupling, the situation becomes more complicated. In

the present case, sgn [Q(7/2i)] ^ J 1

795

= sgn CQ(f7/2)]< 0, and sga [QU^J = sga qfa7/2 '3 /2 J 1 j °» so Q(7/2~)*Q(2?")>0, and a decoupled band ia predicted.

Indeed, both the experiment and the cluster-vibration calcula¬

tion follow this pattern up to 15/2T.

The main components (>4#) in the wave functions of the

favoured states are the following:

Up to 3+4. = 15/2~, both theory and experiment give a decoupled band; above i t there appears, both theoretically and experimentally, a change of pattern: for signer spins, the band i s of the normal type. The ordering is j+5 = 17/2~, j+6 = 19/2", 3+7 = 21/2", j+8 = 23/2". Therefore, the normal

1=1 cascade appears; however, because of the closeness of the 19/2~ and 21/2** states (the distance between tneia i s bó.o iceV), the ground-state band cascade crosses the 19/2~ state, so we have 23/2" * 21/2" *17/2~ >15'/2~ » . . . .

Thus, in the ground-Btate band we have a decoupled-noraal irregularity for the spins j+8 = 23/2~^ I )> j+4 = 15/2".

The structure of the wave functions reveals a change in the pattern of the 15/2^ and 17/2^ states; in the 7/2^, 11/2T, and 15/2^ states, the fły2 clusters dominate, while in the 17/2~, 21/2~, and 23/2T states, the most important clusters

are f^/2 f5/2 a n d £l/2 ^9/2* However» t n e sensitivity to both the parametrizaticn and the truncation should be lcept in mind.

796

^^^^

=o.

The- c a l c u l a t e d s t a t i c :xc-lz'~~ • !•= i . i . . . - : . t - :" -'..'. • ; . - : c -

of the 7/2" baad are Q(7/2C) = -0.24, Q(ll/2~) = -0.37, Q(9/2~) = -0.29, Q(15/2j) = -0.62, Q(13/2^) = -0 .51 , Q(17/27) = -0.75 (eb), Q(l9/2~) = -0.44. This reflects an af¬ fective prolate deformation for the ground-state band, generated by the mechanism of the cluster-vibration coupling.

For the posit ive-pari ty band based on the 13/2 s ta te , only the s ta tes of the stretched B2 cascade have been observed: 25/2+ •* 21/2+ >17 /2 + •»13/2+. The present calculation predicts the decoupled band based on j=13/2 , in agreement with the ex¬ perimental data. The j+1 = 15/2, s ta te l i e s above j+2 = 17/2-,, 3+3 = 19/2^ l i e s above 3+4 = 21/2^ and j+5 = 23/2* above 3+6 = 25/2, . At the same time, the 1=2 E2 transi t ions within the band are stronger than the 1=1 t ransi t ions . The B(E2)

797

values for the favoured transitions 2!?/^ •*•

21/2* >17/2^, 17/2^ >13/2^ are 0.073, O.O69, and 0.073 (eb)2,

respectively, while for the unfavoured transitions 1 9 ^ -p 21/2^,

15/2* ^17/2^, the B(E2) values are 0.005 and 0.001 (eb)2,

respectively.

The largest components in the wave functions of the states

of the favoured branch of the decoupled band are the following:

The 15/2^, 19/2^, 23/2^ states, which belong to the un¬

favoured band, have not been observed as yet. In addition,

the 3-2=9/^ state, predicted as a regular decoupled-band

element between the 5=13/2^ and j+2=17/2^ states, has not been

detected in the present experiment.

By the simple criterion exemplified for the 7/2~ band,

the 13/2 band should be decoupled in the cluster-vibration

model. In this case sgnLQ(13/2^)] = sgnfcjd-j, ,2)J < 0,

sgn [Q(2*)] = sgn «[(f7/2 P^)*] < °> ao Q(13/2j)*Q(2*) > 0, a

condition for the generation of the decoupled band. In contrast

to the 7/2~ band, in the 13/2 band the j-2 low-spin intruder

is present, and the j-1 intruder is absent. The lowering of

the o-2 state in the cluster-vibration model is due to tho 2

f7/2il3/2 a n d f7/2P3/2i13/2 c o mP° n e n t s« On.the other hand,

no blocking of th 13/2 band appears, because 13/2 plays a

minor role and therefore the j-1 state is not lowered.

Anilogously to the situation for the 7/2" band, the states

of the 13/2 band correspond to the effective prolate deforma¬

tion; the effective static quadrupole moments of all members

of the favoured band are Q(13/2^) = -1.06, Q(l7/2*) = -O.99,

Q(21/2^) = -I.04, and for the members of the unfavoured band:

0.(15/2^ = -0.75, Q(19/2j) = -0.97, Q(23/2*) = -I.36 eb. The

corresponding magnetic dipole moments are -0.92, -1.18,

-1.94 and -1.14, -1.54, -2.25^- respectively.

798

10.5. A Few More Examples

Here we present results of a few more calculations in the

cluster-vibration model:

CO

jj^Fe,^ /three-proton cluster in 28-50 Bhellj V.Paar, Phys.Lett.

42B /1972/ 8/ - Fig.10.5.1 and Table 10.5.1.

E(M.V)

O/J".

0/i :)

(112; vz")

laaxan. C.ml.tt Hainamata McSrary 3 aaiticla ck»ta> EipKim.nl Maclarlana Wa<ialaw»>i Arima _ , -»iktalar

Malik ' No )raa | Schali , Shall - ma^al

Datarmad parlicla - ratar

Fig. 10.5.1

799

cc

§

" • • = = =

3/2;

5/2;

3/2-

5/2;

i/27 3/2;

5/2;

3/2-

5/2"

- 1/2;

-» 1/2;

-> 1/2;

-» 3/2;

==========

Particle -Rotor

0.00005

0.011

0.007

0.007

P - it

0.16

-0.20

-0.13

-0.06

TM31.K 1 0 .

B(F2)(e-b)2

Cluster Vibration

0.00053

0.010

0.013

0.007

Q (e • b)

C - V

0.12

-0.28

-0.10

-0.05

Kxper

0.00048±0

0.013 ±0

0.020 ±0

Exper

0.21 i

5 . 1

•

.00008

.002

.004

•

0.03

=__ =

P

0 .

0 .

p

0 .

- 0 .

1 .

- 0 .

0 .

= == — =

- n

0 3 1

0008

- It

34

01

14

50

35

B(Ml)(n.m.)2

C - V

0.023

0.002

0.0003

1 (n.m.)

0 - V

0.09

-0.22

0.51

0.19

0.04

= = ==:s====s=

Exper.

0.015

0.0020

Exper.

0.09

-0.16

0.85

0 . 6

fZzn,7 /thxee-neutron-hole cluster in 28-40 subshell-,(V.Paar

E.Coffou, U.Eberth and J.Eberth, J.Pfcys. G 2 /1S76/

Fig.10.5.2 and Table 10.5.2.

'£'•'

Fig. 10.5.

TABLE 10.F.2

Experimental and theoretical evidence for the identification

of the 3"/2i and 3"/22 states as predominantly collective

and cluster state, respectively

Spectroscopic factor

B E2 3g OS

Magnetic moment

Electric moment

3V2.J I

S"xpt

small

large

positive

o

= j - i

Cluster-vibration model

small

large

positive

negative

3"/22

Expt

large

small

0

?

cluster

Cluster vibration model

large

small

negative

positive

801

95u0 /three-neutron cluster in 50-82 shell;(R.A.Meyer, K.V

Marsh, D.S.Brenner and V.Paar, Phys.Rev. C16 /1977/

Fig. 10.5.3.

,(3/2. •',(3/2.

.pit. ' • . ( 3 / 2 .

' •^(1/2)

- ( 3 / 2 . - ( 3 / 2 ,

.(1/2)

'{?'!.

5/2

-Mi

-in

5/2) .5/2) 9/2)

5/2)

5/2) 5/2)

9/2 i

802

174X53 / t h r 8 e - P r o t < m cluster in 50-82 shell; V.Paar, Hucl.Phya. Ą2TL /1975/ 11/ Pig. 10.5.4 and 10.5.J.

TABLE 10.5.3

1

1/2J

3/2J 5/2;

1/2J

S/2J V2J 3/2J

*S*T4 !

P

1 •

- «*\ \

* in\ t -. w\ |

+ 1 -» 3/2j 5 - • 3/Z* J

!

•

j :

B

0.0«4

tp.050 0.070 0.007 0.112

0.040

CM)

p.r.

1 0 1 0 i 0

t 0 i 0

« (•

EH

-0.79 -0.T1

• r .

C b

.010 013

.008

001

002

Th»ory

0.069

0.034

0.07S 0.007 O.10S 0.037

0.027 0.026

Theory {

-0.87 1 -0.36 j 0.27 j

! 0

! 0

1 0

0

0

0

«(>«)(».«•)

Farp.r.

.0236 - 0.0012

.0072 t 0.0004

.14

.030

.086

.036

f (.n...) Exp.r.

2.91 2.02 t 0.1S 1.15 t O.Ot

2

fiwory

0.01

0.04

0.24

0.01

0.0s 0.06

Thtory

2.92 2.3« 1.20

E (MeV)

0 L

Pig. 10.5.4

211. 85J /three protons in Z >82 shell; V.Paar, Phys.Rev.C11

/1975/ 14J2/ Pigs 10.5.5a /calculations/, 10.5.6b /experiment/.

P03

EINW) !

to*

ro/2-

,23/2" 7/Ą ji ~3f2- MI-\\, J/air

[5g-v—isi'sir 9/2-

J/&-10.5.6 a

-ISO']

Fig. 10.5.6b

-nri

-nrri

"'At "At t7At

804

a /three-neutron-hole cluster in 50-82 shell; V.Paar and

B.K.S. Eoene, Z.Phys. A279 /1976/ 203/ ?ig.1O.5.5

E(M*VI

• 3"

- 5'

Fig. 10.5.5

11.1. Two-Particle Cluater-Vibration Coupling

Row we consider a few specific calculations for even-even

nuclei described in the cluster-vibration model by coupling two

particles in the valence shell or subshell to the quadrupole

vibration. Here a special emphasis we shall put on the creation

of rotational-like bands by the mechanism of cluster-vibration

•odel in spherical representation. In this way we get some in¬

sight into how the bands are generated in spherical and transi-

805

tional nuclei, both vita and without the band crossing.

: Ground State Band. 2% - Band and Additional statBn <- —'

11.2.

As a first example we consider g o ^ Q V Here we treat two

neutrons in the 82-126 shell as a cluster. The result of calcu¬

lation for positive-parity states is compared to the experimental

results in Pig. 11.2.1 /Xa.Ya.Berzin, M.R.Beitin, A.E. Krumnya

P.T.Prokofev, H.Rotter, H.Heiser, F.Stary and V.Paar, to be pub¬

lished/. In Table 11.2.1 we present a comparison of the availa¬

ble experimental and calculated electromagnetic properties of

TABLE

B(S2) (22-» Oj)

l (22-> 24)

l (8 2 ->6^

*(!82 ** 62?

EXP

0.088*0.010, 0.102*0.003

-0.07*0.15, -0.39*0.21

+0.26*0.04

0.04

HI + (28* 10) % E2

0.19*0.14, 1.1, 0.1

111

1 6-5 t i5

1

THE

0.092

-0.16

+0.22

0.06

Ml+28£ E2

0.32

Ml+0.4?* E2

24

4

806

2\-2*-

0*-

T H E O R Y o*

E X P E R I M E N T

Fig. 11.2.1

807

The quasivibrational situation with strong "stop-over" /0| ->2^, 22~*4» A"* A' A->QV and weak "o«8»-ov«r" E2 transitions arejgenerally reestablished in the cluster-vibra¬ tion model. In the simple vibrational model the 2^ state and the otj, 2|t Ą states would be interpreted as a one-phonon sta¬ te and a two-phonon triplet, respectively. The results of clus-ter-vibratio.i aodel partly resemble the features of such a situ¬ ation, although the wave functions are rather different. For i44Hd we get: B<E2) (2+•*<>}) = 0,092, B£B<0(0+->2}) « 0.070, B(B2)(2+ ">2^)= 0.070, B(E2)(*i ">2}) = 0.068, BCE2X2 2-* 0^>= 0.003. The 0^-^2^, 2|->2^ and 4^->2} B2 transitions are strong, conparable with the 2^->Q^ E2 transitions, though the corresponding B(E2) values are a few tinea smaller than in the case of the vibrational ratio/B V I B(E2)(O|, 2|, *% -» 2^= 2B(E2)(2^-*oy /• The "cross-over" transition 2^-^0^, which is strictly forbidden in the pure vibrational model, is only ap¬ proximately hindered in the cluster-vibration model; the calculat¬ ed B(B2) (2% ~* O}) amounts to 1% of the B(fe;)(^->O*). Large systematic differences between the cluster-vibration model and the pure vibrational model appear in the static B2 moments: these are zero in the vibrational model, while their magnitude, depend¬ ing on the shell-model situation, may be rather large in the cluster-vibration model.

In the cluster-vibration model calculation for ^ T d - the O^i 2*2 and 4* states are not based on two phonon excitations, as it might be implied starting from the vibrational picture, but are of rather mixed character; however, a sizeable part of each

808

wave function is given only by a few zero-and one - phonon

components. The pronounced deviation of the ot,, 2?,, 4^ model

states from the vibrational triplet states is reflected in their

energies: these model states lie rather far apart owing to their

different character. Particularly, the 0* state is pushed up

and the other states appear in this energy region /6+., 2t, J>*,

*2» 1 1 * 2» **V 5 /» This fact should be particularly emphasiz¬

ed, because the absence of Ot state close to 2+, and 4^ is some¬

times used as an argument against the presence of modes of the

vibrational type, A S we see, cluster - vibration model natural¬

ly generates this feature.

The cluster-vibration model, which employs spherical re¬

presentation, generally introduces the elements of the rotation

-like structure /bands/ around the yrast line, both in even -

and odd-A nuclei. The calculation for "Tld clearly reproduces stwo bands: the ground-state band ... .10^-^8+->6+->4+-^2+->oi,

with strong E2 transitions inside the band. The calculated

B(E2) values show a small gradual increase going up the band

except for the slight irregularity for 4+-*2+. This is due to

the pronounced admixture / W 3 5 %/ of the /£«, A cluster

to the 4 + state. This irregularity is also accompanied by the

irregularity in the energy spacing E(4+) - E (2+.) < E ( 2 + ) - E ( 0 + ) .

Above 4+ state the collectivity gets again stabilized.

The electric quadrupole moments of the ground-state band

in 44Nd have the same sign: they are all negative: Q(2^f) =

-0.16, (1(4+) = -0.44, q(e1{) =-0.72 , Q(8+) = -0.7C, Q(iO+)=

-0.60. This result of the cluster-vibration model in the termi¬

nology used in the deformed representation means that correla-

809

tions of the cluster-vibration model have generated an effective prolate deformation for the ground state band.

Let us ceaaent on the quadrupole moment of the 2^ state. This is, as explained in Sec.6, a leading order effect in the two-particle /cluster/ - vibration coupling. Using approximate formula /6.J.1/ we get

Here j is the lowest single-particle state for 1/ł4Hd), i is its nonspin-flip partner (;}'= ^ Furthermore %(i#z) - <i(7/2^.) 0 4 5 N d ) « -0.5 eb. By insert¬ ing the single-particle energy £#> = 1.3, we obtain <j(2^)= - \ 0.5 4i - 1.6|= - 0.18. It may be surprising how good the simple approximate formula reproduces the result computed by using complex wave function. The background is discussed in Sec. 7.

The cluster-vibration model also partly generates elements of the second band in the yrast region, the rotational-like elements of which are less pronounced. The calculation for ^?d generates a second band, reminiscent of the rotational Y" - band based on 2g. However, Q^2f }^0 (as a vibrational-like feature generated by the cluster-vibration model^, while the other members of the second band have Q 0.

As we see from Fig.11.2.1 in the region above 6} state there are many states In addition to two bands, both experimental¬ ly and theoretically.

810

11.3 B a n d a

In Zn also two positive-parity bands have been observed

/Fig. KJ.3.V.

S i

6*-

2*-

0*-

i • i ^

2*—1

0*—

-8*

EXP Pig. 11.3.1

THE.

Calculation in the cluster-vibration model /V.Lopac and

V.Paar, Nucl.Phys.^S22 /1978/ f71/produces the ground-state bend

...10* -^8^ —^-6^-»^-^2^-^0^ witn strong E2 transi¬

tions /2O - 30 w.u./. The second calculated band is based on

the 2^ state. It involves rather st^ong...8p -^6* - » * | -»22 E2

transitions / Rf10 w.u./. The E2 transitions between the

members of the two bands are mostly small / ?z 1 w.u. or smal¬

ler / . The spectroscopic quadrupole momenta of the members of

the g.s. band are large and negative. /The sign is a typical

shell-effect in the cluster-vibration model, see Sec.7/. The

second band does not exhibit regularity in quadrupole moments,

although i t shows tendency toward changing i t s sign with res¬

pect to the ground-state band.

I t is interesting to look at the structure of wave func-

811

t ions of the ground state band. The largest components / > 4%/ are:

|0}> = 0.701 / P | / 2 / 0 , 0 0 > + 0.35 I / F 2/ 2 / 0 , 0 0 >

+0.36 I /f5/2/0,00> - 0.31 I /Ą/z>'oi°°> -0.20 J /p|/2/2,12<> - 0.22 | /P ? / 2 P1/2/2,12>

= 0.63 | /Pj / 2 / 0,12> + 0.34 | Ą +0.32 j/f | / 2/0,12> - 0.271 -0.22 | / l | / 2 / 2,00> - 0.22 | /P ? / 2 P1 / 2 / 2,00>

0.59 | /Pj/g/ 0,24> + 0.33 +0.29 J / f ^ / O ^ ^ - 0.24 j /Ą/£/ O, -0.28J /P3 / 2 / 2,12)>- 0.30 | /P5y2, \$ty = 0.50 I /Ą/2/0.36^ + O.27J ZPf/2/ 0, 36> + 24 I / f t / 2 7 °'56> " °*20 I Z4/27 ° ' ? 6 ) -0.35 | /I3/2/ 2»24J> " °'*0| ^3/2 P 1 / 2 7 2 ' 2 ^ + ° ' 2 0

-0.38|/P 3 / 2f 5 / 2A,12> The four largest components in the wave functions of the Ożj, 2^, 4^, 6^ states exhibit a very simple basic structure: coherent proton pairs coupled with zero, one, two and three phonons, res¬ pectively, with the maximum alignment of phonos angular momenta. Thus, these states could be approximately presented as a proton pairinfa phonon 0^ coupled to aligned phonon quanta /0,1,2,3/. The proton pairing structure appears predominantly the same and the difference is only in the phonon part. We expect that main contribution in creating phonon have two quasi-neutron excitations: the 0^, 2^, 4^, 6^ states involve rather similar type of proton motion, the difference being largerly due to neutrons.

812

If we normalize the wave functions to these largest four components, we obtain the approximate factorization:

=[o.763 | /Ą/2/ 0> + 0.385/ /P?/2/0> + 0.390 [ f [ -0.3*2 / / g | / 2 / o j |00>

12\ y = jo.?6OJ /Pj/2/0J> + 0.385) -0.325J/e|/2/ 0}J | >

4 / ] I 24> = [O.768|(P|/2)O> +0.416 / (E 2 ^ ) 0>

It is obvious that these states contain approximately the same coherent two-proton-pair state, in which we recognize the well known pairing phonon. Thus, the elementary mode of the pro¬ ton pairing excitation may be defined approximately by

0>

Then, the leading order terms in the wave functions may be pre¬ sented as

*>»f0>p |00> , |2*>2|0>p

and they amount to more than 50^ ot the total wave function

for these states. The rest of each wave function partly ex¬

hibits a similar substructures the largest admixtures in the

Oif, 2^, tĄt 6 states can be approximately factorized as

I 2p > h 2 > , I 2 p >|00>, |2^ J 12>, { 2 ^ |24> , respecti¬

vely. Here 12} represents a superpo^+:ion of the^P^

813

2, ( P V 2 f 5 / 2 ^ and (Ą/z)Z broken Pairs of

coupled to angular momentum 2, coherent with xespect to the effective Q-Q force induced by phonon-exchange between the par¬ ticles. Thus, the building block I 2//^ corresponds to the well-known proton quadrupole-pairing phonon.

Thus, the ground state band is mainly built of vibrational band based on Jo S/C\ with the admixed band based on )2^k/p\ This provides some insight into the internal band structure.

11.4 Tw0 0+ Bands

We treat ^gCd as two proton-hole cluster in the 28 - 50 shell coupled to the quadruoole vibration. In this way we ob¬ tain by calculation two bands, the ground-state band and the second band based on 0^ state in agreement with recent experi¬ ment /V.Paar and R.A.Meyer, to be published/. In Fig. 11.4.1 we compare the experimental and theoretical situation

6* -r—6*

Fig. 11.4.1

In table 11,4.1 we compare the experimental and theoreti¬

cal branching ratios for iaterband and intraband transitions.

TABLE 11.4.1

d2 "*• H 22 _» O,

42 -» 22

4 2 -» 2n

61 -* ^

62 -, 6 62 —» 42

62 ^ 41

Branching ratio

EXP

73.6

26.4

86

7 6

99.97

0.03

56.4

9.8

33.9

THE

69 31 74 3 23 99.997 0.003

55 6

39

The structure of the wave functions reveals that the g.s. _2

band involves in main components the So/2 cluster while the

0 2 - band mostly involves combination of clustery P^/2 » —2 / —1 —1 \ p3/2 ' V?1/2 p3/2) * AccordinS1yi w« say that all components

in Ojj -band with ggyp cluster create g-band, and all components

in ot -band with p cluster create a p-band. Transitions be¬

tween the 0^ -band and OX -band eri.se due to admixing of p-band

into g-band for 0^ -band, and admixing of g-band into p-band

for 0 2 -band. This is evident from the complete wave functions

of the band-heads 0^ and ot, presented in table 11.4.2.

We see that the wave functions of the band-heads can be

approximately presented as

815

TABLE 11.4.2

(g9 £•

Pi /2- 2

P3/2-2

Pi/2"2

P3/2-2

Pi/2"2

P3/2-2

Pi/2"1

PS/2"2

Pi/2' 1

P3/2-2

Pi/2' 1

PV2-2

2 0,00> 0,20> 0,30> 2,i2> 2,22>

2,32> 4,24> 4,36> 6,36>

0,00

0,00

0,20

0,20

0,30

0,30

P3/2-1 2 » 1 2

2,12

P3/2-1 2 ' 3 2

2,32

P3/2-1 2 » 1 2

2,2 2

"g-band"

0.532 0.297

- 0.065 0.556

- 0.222 0.193 0.278

- 0.086

0.079

- e t P> admixed "p-bard"

- 0.158

- 0.179

- 0.090

- 0.107

- 0.016

- 0.020

+ 0.181

- 0.110

+ 0.067

- 0.042

+ 0.082

- 0.058

\o*> admixed "g-band"

0.128 0.096

- 0.064 0.249

- 0.136

0.091 0.153

- 0.058 0.049

1 P> "p-band"

+ 0.334

+ 0.354

+ 0.216

+ 0.195

+ 0.076

+ 0.096

- 0.497

+ 0.294

- 0.191

+ 0.118

- 0.271

- 0,271

TOTAL TOTAL

+ £!«>

816

The simple correlations of cluster-vibration model have generated a type of band mixing, with a mixing coefficient £• being some¬ what configuration-dependent, but mostly in the range between 1 1 Ę and ij. Thus the cluster-vibration model partly generates averag¬ ing over clusters and creates bands. 11.5. ^'IgFe: "Band-Crossing" at Low Spin

*Fe represent nuclei with clusters consisting of three and two proton holes, respectively, so they can be described on the same footing. Fig.11.5.1 shows the result of a calculation for two proton holes in the Z=28 shell, with the parametrization /a=1.2/ as used for the ^ M n calcula¬ tion /V.Paar, Nuovo Cim. "$2k /1976/ 97/. The resulting energy spectrum corresponds qualitatively to the experimental states. The calculated 0* ground state is followed by the 2^ first excited state. These are followed by a group of four states, 4^, 2g, 6^, and Og. The three calculated states which follow are 2t, "r%, and 3^. This pattern corresponds roughly to the experimental pattern 0*f Z"^, 4^, Og, 6^, 2t, 4g> 3-p

The main components in the wave functions of the low-ly¬ ing positive-parity states are

817

OB

ÓÓ j-7/2" BAND

DECOUPLED NORMAL"

E(MeV)

1 - 1

1 1

r* i

i i ••

i-i—

j *8

J-6

i-2

-

Jc2

J

EXP (a)

i*7

1*5

j * 3

j*A i*i i*2

-L

1

'i-i

j

THE EXP

j - i3/2 ł BAND 1 DECOUPLED'

i*

J*

J*

6 ' ^

A

2

(b) THE

ri«.11.5.1

\0t> =

The calculated E2 tranaitions between the low-lying yrast states are

On the other hand:

In this way, the calculation predicts the irregularity on the yrast line: the member of the ground-state band is lb*> ^2 not the 6* state. The d, state is the intruder state, based on the (f" ,„)6 cluster. The ground-state band is predicted to be

10* ^ 8t £ i * * 4 ^ 2 ! * °i* Iłle irregularity on the yrast between 4, and 6* resembles the well-known back-banding phe¬ nomenon, usually considered for higher-apin states; it appears already for low-spin value (4). for Fe, this feature was predicted by the cluster-vibration calculation (V. Paar, Mucl. Phys. A185 (1972) 544), and the latter was determined expe¬ rimentally

819

11.7. Conclusion; Cluster-Vibration Model vs. Quasi-

Rotational Features

In the second part (B) of this lecture we discussed

the cluster-vibration model, with special emphasis on some

of its consequences: We have demonstrated how rich the

cluster-vibration model is in different phenomena on and

around the yrast line. The appearance of regularities,

as well as of irregularities in odd- and even-A nuclei ("band-

crossing", "band mixing", "back-banding",...), and the mechanism

of their generation in the cluster-vibration model have been

discussed. It is hoped that such an approach in the spherical

representation of the cluster-vibration model, which includes

the dominant shell degree of freedom (cluster in the valence

shell, the Pauli principle in. this limited part of the Hilbert

space is completely included) and the dominant collective degree

of freedom (low-frequency quadrupole vibration), may lead to

a better understanding of these phenomena and their relationship

in different models.

The author expresses his gratitude to the Organizing

Committee of the XV Winter School on Nuclear Physics for

the inspiring atmosphere in Bielsko-Biala and for

the technical assistance in preparing this manusoript.

5. HEAVY-ION COLLISIONS - POSITRON PRODUCTION: QUASI MOLECULES

In beam electron and positron spectroseopy after heavy-ion collisions

H. Backe J ^ v" ~ Institut fiir Kernphysik der Technischen Hochschule Darmstadt, Darmstadt, Germany.

-Abstract -Some new aspects of in beam spectroscopy of electrons and positrons after heavy ion collisions are discussed. The experimental set-up used is essen¬ tially an electron transport system with normal conducting solenoidal coils and a Si(Li)-detector as energy dispersive element.-

Spectroscopy of low energy conversion electrons emitted from recoil nuc¬ lei in flight after {HI,xn)-reactions is possible with a longitudinal semicylindrical baffle between target and Si(Li)-detector. Special fea¬ tures of this method are high transmission and strong suppression of the prompt ć-electron background. This method was applied for the investi-gation of a subnanosecond 12 isomer in the Pt-nucleus by electron-gamma, electron-electron coincidence measurements as well as lifetime-measurements based on the detection of conversion electrons.

The first observation of positron creation in 1.1 GeV ° Pb + ° Pb collisions is reported. It is established that a dominant fraction of the positron yield is from none nuclear origins. __, • ... ,

1 . Introduction The interaction of heavy ions from currently used heavy ion accelerators with solid state targets gives rise to a lot of interesting physical processes including both nuclear and atomic aspects. Nuclear levels may be excited by Coulomb excitation or nuclear reactions. The observation of the deexcitation radiation gives us the opportunity to study the structure of the atomic nucleus. On the other hand, the interaction of the ions with the atomic electrons of the target nuclei leads to typical atomic processes as e.g. the ionization of inner shells with the possibi¬ lity to study quasimolecular systems existing there for a very short time period. Most of our knowledge about these processes so far has been ob¬ tained by "in beam" methods observing Y~ or x-ray radiation with high resolving Germanium or Silicon spectrometers. However, the addition of electron and positron spectroscopy has certainly also some very inter¬ esting aspects and may lead to important complementary information about those physical processes. For instance, the addition of conversion elec¬ tron spectroscopy to the Y-ray spectroscopy methods gives us the unique

823

possibility to observe 0 + •+ 0 + transitions or to determine the jnultipo-

iarity of certain transitions by the measurement of the conversion co¬

efficient. For highly converted transitions the conversion electron spec-

troscopy in principle is much more sensitive as -y-ray spectroscopy. How¬

ever, these low tnergy tra. sitions are often observed by the very intense

6-electron background resulting from the interaction of the charged beam

particles with the electrons of the target atoms. In Fig. 1 a calculated

6-electron spectrum is shown which is expected from the binary encounter

theory when 2o8Pb is bombarded with 9o MeV 160 ions [1]. In the low

energy part of the spectrum we have cross-sections as large as 1o barn/

keV which have to be compared with typical compound nuclear cross sections'

of 1 barn. For every conversion electron spectroscopy just in the low

energy part of the spectrum it is very important to get rid of this back¬

ground.

-10-

. K -shell

i . . i i i . . i

1 2 5 10 20 50 100200 500

Fig. 1 The 6-electron spectrum as calculated from the binary encounter

theory for 9o MeV 160 on 2o8Pb [1].

On the other hand this 6-electron "background" may be subject of inter¬

esting physical investigations. The high energy part of this spectrum in

principle contains information on the momentum distribution of electrons

in the combined quasimolecular system [2]. The observation of this effect

is of interest for the investigation of inner shell quasimolecul^r wave

functions in superheavy ion-atom collision systems. Very little

tal information concerning the high energetic 6-electron spectrum is

available presently.

Another very new aspect of in beam spectroscopy is the spectroscopy of

positrons created during the collision of very heavy ions. As expected

from theoretical calculations [see 3 and references cited therein] the

binding energy of the 1 so-state in an Uranium-Uranium collision may ex-

ceed twice the rest mass moc of the electron. If that state has a vacan¬

cy, a spontaneous positron emission should be observed, an effect closely

related to the very old Klein-paradox [4].Beside this qualitatively new

effect electron-positron pair creation should be also possible in the

very strong time dependent electric fields in close collisions of very

heavy ions.

Unfortunately there is another source of positrons which has to be ex¬

pected as background process. Every nuclear level excited e.g. by the

Coulomb excitation with transition energies greater than 2 moc =

1o2 2 MeV may deexcite by internal pair creation. This effect must be

carefully investigated and taken into account.

In this contribution two aspects of electron and positron spectroscopy

after heavy ion collisions are discussed in more detail. Both experiments

make use of essentially the same setup, a solenoidal electron (positron)

transport system with LN, cooled Si(Li)-detectors as the energy disper¬

sive element. The general features of this instrument are a high collec¬

tion efficiency and broad range energy acceptance which are necessary

tools for the detection of small cross sections (1o ... 1o barn) as

expected for certain conversion electron transitions, the high energetic

part of the S-electron spectrum or for the emission of positrons. In

chapter 2 a new method for the detection of ns-delayed conversion elec¬

trons after (HI,xn)-reactions with strong suppression of the prompt 5-

electron background is described utilizing the fact that delayed con¬

version electrons are emitted spatially separated from the target due to

the recoil velocity of the emitting product nuclei. This was done by

using a simple semicylindrical baffle in axis of the solenoidal electron

transport system.This method was applied for the spectroscopic investi¬

gation of the Pt-nucleus, as described in chapter 3. Electron-gamma

and electron-electron coincidence measurements as well as lifetime mea¬

surements based on the detection of conversion electrons have been per-. 1 ft ft

formed to investigate a 12 isomer in Pt in more detail experimentally.

In chapter 4 first experiments concerning positron spectroscopy after

collisions of very heavy ions near the Coulomb barrier are described. 825

For that reason the solenoidal transport system must be equiped with a

Nal detector assembly to detect positrons "stopped in the Si(Li) detector

by her 511 keV annihilation radiation and distinguish them this way

from electrons.

2. Conversion electron spectroscopy [5]

2.1 The experimental apparatus

The experimental apparatus used is essentially an electron transport

system with an LN2 cooled Si(Li)-diode as the energy dispersive element [5],

Similar devices exist in different laboratories e.g. also in Stockholm [€',

Amsterdam [7] or Rossendorf. Our instrument is installed at the MP Tandem

accelerator at the Max-Planck-Institut fur Kernphysik in Heidelberg and

a similar one at the GSI in Darmstadt for positron spectroscopy. The ap¬

paratus is shown in Fig. 2. The two solenoidal coils are powered by a

5o kw supply with a maximum current of 1ooo A. The magnetic field strength

on axis of the solenoid is about o.6 Tesla. The beam enters the vacuum

chamber perpendicular to this cut between the two coils. Electrons created

in the target spiral around the magnetic field lines to the Si(Li)-detec¬

tor located at the end of the long coil. A high collection efficiency

Ge(U) t DETECTOR

LEAD J TARGET ^COILS SHIELDING

TO VACUUM PUMP

10cm

Fi<3- 2 T h e experimental set-up. For any change which might affect the

vacuum, the Si(Li)-detector will be moved behind the bellows valve. The

target can be changed without breaking the vacuum in the solenoid. If

necessary, condensed vapors on the detector surface can be evaporated with

the aid of a heating coil.

of about 25% especially for low energy electrons is achieved with this

transport system. The Si(Li) diode must be protected very effectively

against the low energy part of the 5-electron spectrum without strong

suppression of the low energy conversion electrons to be investigated.

This can be performed with an experimental technique described in the

next subsection.

2.2 The recoil shadow technique

To select conversion electrons emitted from evaporation residues in

flight (produced after (HI,xn)-reactions) from prompt 6-electrons a spe¬

cial insert as shown in Fig. 3 is used. This makes use of the fact that

conversion electrons are usually emitted delayed (between ns and ps) and

TARGET BEAM ALUMINUM DIAPHRAGM

DETECTOR

PROMPT j z ELECTRON ORBIT

ORBIT OF AN ELECTRON EMITTED FROM A RECOIL NUCLEUS IN FLIGHT

Fig. 3 The recoil shadow technique. It is shown a cut through the electrc

transport system containing the beam and solenoid symmetry axis.

therefore spatially separated from the target. The essential part of the

insert behaves like a sernicyUnder. Its main plane contains the symmetry

axis of the solenoid and lies parallel to the target plane. The upstream

part of the solenoid is closed by this semicylindrical insert, the down¬

stream part is open. The target is moved upstream by a distance d from

the axis. Prompt electrons starting from the target have the tendency to

spiral back to the field lines they started from. While doing this they

collide in any case with the baffle and are finally absorbed. Electrons

which start in the downstream part of the spectrometer from recoil nuclei

have a certain probability to spiral to the electron detector without any

collisions with the diaphragm.

827

In Fig. 4 a spectrum is presented taken after the 176Yb{16o,xn)192"xPt reaction, at 85 Mev with this semicylindrical insert. The target was posi¬ tioned about 4.3 mm behind the longitudinal baffle in the shadow region, corresponding to a flight time of 1.3 ns. We clearly see electron lines up to energies as low as 15 keV with a very good peak to background ratio. The suppression factor of 6-electrons can be estimated to be better than 5 orders of magnitude at that energy.

18Oonl78Yb at 85MeV ELECTRON SINGLES SPECTRUM

Btif « » « V ? I t ( t t t eV8 if ^

se x x

t t Ł «T

400 500

Fig. 4 Electron spectrum taken with the recoil shadow technique.

The main features of this recoil shadow method are a sharp defined shadow region and a high detection efficiency of delayed low energy electrons. It is possible to adjust the target to a distance of o.2 mm relative to the longitudinal baffle without any dramatic change of the background. This distance defines the shortest lifetime which can be detected (16O-reactions : T ^ 2 = 7o ps, a-reactions : T 1 / 2 = 3oo ps) . The detection efficiency for electrons depends of course on several parameters: (i) the target position behind the longitudinal baffle, (ii) the recoil velocity of the emitting product nuclei, (iii) the diameter of the elec¬ tron orbits and (iv) the size of the electron detector. A peak detection efficiency up to 8% can be obtained under optimum condition. The main disadvantage of this method should also be mentioned namely the moderate line width at high electron energies due to the Doppler broadening be¬ cause of in flight emission of the electrons.

The recoil shadow technique can be applied for different physical ques¬ tions (i) Investigation of that part of a level scheme where highly conver-

. ted low energy transitions are present by the electron-gemma coin-828

cidence technique [8],

(ii) localization of subnanosecond high spin isomers by the electron-

electron coincidence technique ,

(iii) Lifetime measurements in the subnanosecond time region,

(iv) Spectroscopy in the second potential wall of fission isomers with

electron-fission fragment coincidences.

In the following chapter some of these applications are described in more

detail.

3. 188, Investigation of ' Pt by the electron-electron and electron-gamma

coincidence technique [9]

3.1 Localization of the isomer by electron-electron coincidence

In Fig. 4 the delayed conversion electron spectrum taken after the 176Yb(16O,4n)188Pt reaction at 85 MeV was already presented. In this spec¬

trum electron lines with energies as high as 55o keV can be identified

and the question arises: what is the reason of all these lines? For the

more neutron rich nuclei 19o'192pt it was well known [1o,11] that there

are irregularities in the positive parity band which were explained by -2 —2

the coupling of (u i., ., )- and (IT h. . ,, )-quasiparticles to the oblate

deformed core [12]. This effect results in a very small energy difference

between the 12+ and the 1o+ levels and hence in a relatively long half

life of about 1 ns for the 12+ state. On the other side for 188Pt it was

expected [1o] that the lifetime of the 12+ state is very short because

of the large energy difference of 329 keV between the 12+ and the 1o

state. To localize the isomers responsible for all that delayed trans-1 ft ft

itions in Pt electron-electron coincidences were taken applying for

both detectors the recoil shadow technique. The arrangement is shown in

Fig. 5. The Ge(Li)-detector is replaced by an Si(Li)-diode. Both Si(Li)-SKLO- TARGET LONGITUDINAL SI(LI)-DETECTOR / BAFFLE- SYSTEM DETECTOR

ELECTRON ORBIT BEAM

ELECTRON ORBIT

ALUMINUM

LEAD

HEAVY METAL 10 cm '

Fig. 5 Experimental arrangement for electron-electron coincidence measure¬

ments. For both Si(Li)-detectors the recoil shadow technique is applied.

detectors are able only to detect delayed conversion electrons. A coin¬ cidence is detected only if a level with a lifetime in the order of about a ns decays at least by two subsequent electron transitions. This method has the big advantage that only the decay chains of an isomeric level is observed free of all prompt cascades feeding that isomeric level. This method should therefore be well suited for the localization of isomers in the nanosecond time region. The coincidence spectra were taken in the usual way known from y-y experiments and gates were set at every conver¬ sion electron line. A typical electron-electron coincidence spectrum is shown in Fig. 6. This way we clearly identified beside a 7 -level at 1768 keV a 12+-level at 28o9.9 keV as isomeric. The decay chains of the 12 isomer are consistent with the new level scheme of Daly [13] (Fig. 7) investigated by the y-y coincidence technique which was kindly submitted to us prior to publication.

V c t-

m % O

1000

500

0

200

100

0

§ 1

•

f

k—

b

I

j

1

o J

1 -;

V ' J ,

,! 1'

100

g

i. . i

J i_i

I

1 1.

J •' f, r 1 i

till

O .1

8

•

6 1

s

"i

ŻÓ6

" » Y b ( « O 4n) B8Pt at S8MeV Electron-Electron Coincidence Meosurements Gate ot K 266

t

51

Goteot K417( io;-e;i»4;-

•L2O3

'I » V 8s

i i_i

. K 414 «L 340)

h 1 V

S

4Ó0

V !

o

5

fig- 6 Typical electron-electron coincidence spectra.

It decays by three chains: first through a 1o+ state at 2663.3 keV which feeds into the 8+ state of the y-band, secondly through the 1o+ state at 2437.1 keV which probably belongs to the ground band and thirdly through a 1o state at 27o2.1 keV which also decays into the ground band. The corresponding 1o7.8 keV line was not observed in the y-y experiment. But it can clearly be seen from Fig. 6 if a gate is put at the K 417 line only the 146.6 keV line appears while both lines are seen when the gate

H10

is on the 2+ •* 0 + transition. This level scheme is also supported by Electron-Ganuna-Coincidence measurements. In Fig. 8 the Gamma spectra taken with a 9o cm3 Ge(Li)-diode are shown with gates on the (L+M) 14 6.6 keV and L 1o7.8 keV conversion electron transitions.

Prom the K/E ratios (see Fig. 6) it can be concluded that the 146.6 keV and 1o7.8 keV lines are of E2 type.

" j W l '»' »**» Ml HMO

Mva eto* • *

Fig. 7 Level scheme of Pt . 188r

200

100

o o

:8

A !r V «T V ( Q4ń)^>tat8S**V ti ŁTT 1 1 Bectron-Oammo Colncktonc* , J i if «f

li 5 " L«M Un*s

• » it i / Gate at 107.8 keVL Lin*

Fig. 8 Typical electron-gamma coincidence spectra.

831

3.2 Lifetime-measurements

Half lifes can also be measured by the recoil shadow technique. For that

reason the target can be adjusted in beam direction relative to the longi¬

tudinal baffle by a micrometer screw as shown in Fig. 9. The lifetime

-:;COTLS

W&TER U COOLB^G C H M # £ L

Fig. 9 A cut perpendicular to the axis of the solenoid along the beam

axis. The target position is adjustable along the beam direction by a

micrometer screw.

O

U O in

u

176Yb(16O.4n)188Pt E =85MeV

^ > V 12*—10* No L 147

\ \

12- i o X N 7^5--: K2O3 :

t x 4 ;

4

d/mm

F-i(3- 1o Typical decay curves as taken with the recoil shadow technique

based on the detection of conversion electrons.

832

measurement technique was tested by the l 5 2Sm( 1 6O,xn) 1 6 2' 1 6 3' l 6 4yb reac¬

tion. In the even nuclei the lifetimes of the 2 + levels were measured [14

by the recoil distance method. There is agreement within the statistical

errors of both methods [5]. Typical decay curves for the 12 + •*• 1o+ L 147,

12 + 1o + L 1a8 and 7' + 5* K 2o3 transitions in 1 8 8Pt are shown in Fig. 1c

Half lifes of (62o - So) ps and (2oo - 2o) ps have been deduced for the

12 +- and 7~-levels respectively. With the known branching ratios [13]

of the decay of these levels and the half lifes the B(E2) values can

be determined. They are collected together with results of the neigh¬

bouring Pt nuclei in Table 1.

Table 1: Halflifes and B(E2)-values of 7~ and 12 states i n 186,188,19opt

I so top Level

(energy/keV)

L1/2 /ps B(E2)/e2b2

186 Pt

188 Pt

19o. Pt

7 (1953)

12+(281o)

7~<1768)

12+(2727)

7~(1631)

85 - 1o

62o - 5o

2oo - 2o

152o - 9o

77o - 14o

O.48 - 0.06

£ O.59 - 0.06

0.24 - o.o5

£ o.15 - o.oi

o.24 - o.o5

It is quite interesting to note that the sum of the B(E2)-values

(summed over all observed decay branches of the 12 + state) in 1 8 8Pt

is nearly a factor of 4 greater than in 1 9 oPt. On the other hand 1 8 6Pt

was also investigated by the same methods, but no isoir.eric 12 + state

could be identified; The reason for this behaviour may be the oblate-

prolate shape transition from Pt to Pt. The B(E2) values of the

7 -* 5 transitions agree qualitatively with theoretical calculations of

H. Toki, K. Neergard, P. Vogel and A. Faessler done with the semi-de¬

coupled model extended to the y-deformed core [15].

833

4. Positron spectroscopy [16]

There has been a long-standing interest in atomic physics and quantum

electrodynamics to the question of what happens physically to a bound

electron when the strength of the Coulomb potential exceeds Z > 1/a.

As noted in theoretical studies (for references see e.g. [3]), a quali¬

tatively new phenomenon is expected to occur when the binding energy of

the electron exceeds twice the electron mass. The filling of a vacancy

in such a situation may lead to spontaneous emission of positrons, a pro¬

cess related to the Klein paradoxon [4]. Although the experimental situa¬

tion to study this process cannot be realized in stable atoms presently,

the formation of quasimolecular states in U on 0 collisions near the Cou¬

lomb barrier, may provide a possibility to study this process. However,

in such systems dynamical induced positron creation with an electron either

in a vacant bound state or in the continuum may occur.

To investigate that proposed positron.emission two experiments have been

initiated at GSI in Darmstadt. One of these employs again a solenoidal

magnetic field to transport the positrons from the target 55 cm away to a

catcher (Fig. 11). The arrival of a positron is detected via the annihila-

C/cm

-NQI

11 Magnetic field distribution for the solenoid-positron transport

system. The insert shows the positron detection efficiency as a function

of energy.

tion radiation in colinear coincidences by two pairs of 3" x 3" Nal detec¬

tors. The principle features of this instrument are a large positron col¬

lection efficiency because of the magnetic mirror effect and a broad ener¬

gy band acceptance. Including the selection of only 511 keV full energy

peaks in the coincident Nal counters, the total efficiency attains an ap¬

proximately constant value of 2.6% over a range of positron energies of

about 1 MeV. Because the positron spectrum is expected to have only little

intensity above 1 MeV, the total cross section is measured by this method.

This way various target-projectile combinations as a function of beam

energy have been investigated. A typical Nal-coincidence spectrum taken

after the bombardment of a 2 mg/cm2 thick U target with a U-beam

of 4.7 MeV/u energy is shown in Fig. 12. In the Nal single spectrum no

I 1 o in

t 8 O

8J

6-

4

2

0-

NaJ1 coincidence spectrum (511keV window on NaJ2 and time window between Na J1

andNaJZset)

511 keV /peak

x (134 events)

J i 0 200 400 600 800 1000

Channel

12 A typical Nal coincidence spectrum.

511 keV full energy peak could be identified while in the NaI1 - Nal2

coincidence spectrum with the NaI2-gate on the 511 keV peak a nice 511 keV

line appears. The events in the full energy peak are defined as positrons.

Total cross sections for positron creation were measured using an Si sur¬

face barrier detector of 45° to the beam direction for normalization to

Rutherford scattering (see Fig. 13). The total cross section is plotted

as a function of the distance of closest approach 2a in a head on colli¬

sion in Fig. 14. -Z1 Z2 e 2 E77M7~

It is 2a = M1 + M 2 K1M2

e = 1.44 MeV fm

with Z1,M.,Z2,M2 charge and mass numbers of'the projectile and target,

respectively and E. the projectile energy. This parameter was chosen be¬

cause for 2a = const, the positron cross section for different target-pro¬

jectile combinations can*be compared for equal Rutherford cross section.

835

Magnetic field shielding

3eam

Fiq. 33 Arrangement of the Nal-, Ge(Li)- and surface barrier detectors around the target position, shown in a cut perpendicular to the solenoid axis.

2000

1000

500

200

•o 100

ii 5° o

20

10

5

-

-

-

•

- o

— i • « - 1 • • 1 --—i—•—i—i—

Total Cross Section

>

• \ = \ \

\ \ ł

208Pt,-2MU t \

"8u-"8u ": 238U-?08Pb "

-

"safe energy'Pb-Pb Pb—Oik :

L. . . , l . i , . . : 15 20 25

2a/fm Fig. 14 Total cross section for positron production for different projec¬ tile-target combinations as a function of the closest distance of approach in a head on collision 2a.

The following systems have been investigated Pb + Pb, Pb + U, U + Pb and U + U. The main feature of this measurement is a steep increase of positroi production with decreasing 2a with no characteristic slope change in the region where nuclear reactions are expected to set in. Another important feature is that the cross section for 2o8Pb + 238U changes by a factor

836

of about 4 in comparison to ° Pb + ° Pb but changes only a factor of 2 738 238

in comparison to U + 0. Because the background subtraction of posi¬

trons from nuclear processes is expected to be more difficult for the

U + 2 U system than for the 2o8Pb + 2o8Pb system, the latter one was

investigated first in more detail.

To keep the background as low as possible, positrons are measured in coin¬

cidence to scattered particles. This offers at the same time the possibili¬

ty to measure the differential cross section for positron production with

respect to the scattered ions. Scattered projectiles or recoils were de¬

tected in a plastic scintillator counter (Fig. 15) at two angles in the

A

i Plexiglas Light pipe

Beam

15 The plastic particle counter arrangement.

laboratory of (45 - 1o)° and (25.5 - 4.5)°. The y-ray spectrum was also

recorded in coincidence to the plastic particle counter by an 3" x 3" Nal

detector located near the target (see Fig. 13). From this spectrum the

evaluation of the nuclear background positrons becomes particularly streight

forward, because the principal source of nuclear positrons in this case is

the pair conversion decay of the Coulomb excited 3 (2.614 MeV) state in 2o8Pb. This state is also populated in the decay of 1 Pb. In Fig. 16 the

in beam and source y-ray spectra are compared, showing the similarity of

spectra above 1 MeV. Obtaining the ratio of positrons to 2.614 MeV gamma-

rays emitted by the 212Pb source placed in the target position, an-i multi¬

plying this ratio by the intensity of 2.614 MeV gamma rays observed in

837

Fig. 16 Gamma ray spectra from Pb +

with scattered projectiles or recoils at 212

Pb Lab Ion

collisions in coincidence

(45 - 1o)° and from the

Pb source.

beam, directly yields the positron intensity from the Coulomb excitation

of the 3~state. The determination of the background positron intensity from

such a direct comparison does not depend upon knowledge of the essential

experimental parameters such as the gamma-ray and positron detection effi¬

ciencies and the internal pair conversion coefficient, and includes con¬

tributions from secondary processes such as external pair production.

The measured differential cross sections are plotted in Fig. 17 for a

o a.

31

100

50

20-

10-

5

2

1

05

02

JOB 208 Pb* Pb

.2o=1S2 fm

i--f

\ X 15" 45"-

e,Lab : Blon

oCEMeas. •; — C E Cale. :

• CE-Backgr :

subtr. • --Theory

20 25 30 2a/fm

Fig- 17 Differential positron production cross sections for 2 o 8Pb + 2 o 8Pb

as a function of distance of closest approach in a head on collision 2a.

The insert shows the differential cross section as a function of the

scattering angle in the laboratory system with corresponding angle averages.

838

range of bombarding energies from 3.6 to 5.6 MeV/u. The positron yields

from the decay of the 3~-state and a 4+-state at 4.o9 MeV are also shown

in Fig. 17 together with Coulomb excitation calculations. For all these

measurements the contribution from nuclear Coulomb excitation alone is only

a fraction of the total positron yield and this fraction is particularly

small at the lowest bombarding energies. It is also shown in the insert of

Fig. 17 that the differential cross section, with respect to Sj^r for the

excess positron intensity over the Coulomb excitation contribution pos¬

sesses a more forward peaked angular distribution than that for positrons

from the deexcitation of the 3~-state.

Calculations by Reinhardt et al. [17] for positron production are shown

also in Fig. 17. The general agreement of the calculated cross sections

and projectile energy dependences with the measurements suggests that the

observed positrons are associated with those processes involving induced

positron emission by the very strong time varying electric fields present

in the quasimolecular collision system.

References

1. Folkmann, F., Borggreen, J., Kjelgaard, A.: Nucl. Instr. and Methods

119, 117 (1974)

2. Kozhuharov, C., Kienle, P., Jakubassa, D.H., and Kleber, M.:

Phys. Rev. Letters 3J3» 54o (1977)

3. Reinhardt, J. , and Greiner, W.: Rep. Prog. Phys. i£, 219 (1977)

4. Klein, 0.: Z. Physik 53_, 157 (1929)

5. Backe, H., Richter, L., Willwater, R., Kankeleit, E., Kuphal, E.,

Nakayama, Y., Martin, B.: Z. Physik in press

6. Lindblad, Th., Linden, C.G.: Nucl. Instr. and Methods J_2£» 3 9 7 O975)

7. Konijn, J., Posthumus, W.L., Gondsmit, P.F.A., Schiebaan, C.,

Geerke, H.P., Maarleveld, J.L., Andringa, J.H.S., Evers, G.J.:

Nucl. Instr. and Methods 1_2_9' 1 6 7 (1975)

8. Richter, L., Backe, H., Kankeleit, E., Weik, F., and Willwater, R.:

Phys. Letters 71B, 74 (1977)

9. Richter, L., Backe, H., Zeidler, S., Weik, F., and Willwater, R.:

to be published

1o. Piiparinen, M., Cunnane, J.C., Daly, P.J., Dors, C.L., Bernthal, F.M.,

and Khoo, T.L.: Phys. Rev. Letters 3±, 11lo (1975)

839

11. Hj'orth, S.A., Johnson, A., Lindblad, Th., Funke, L., Kemnitz, P.,

Winter, G.: Nuci. Phys. A262, 328 (1976)

12. Cunnane, J.C., Piiparinen, M., Daly, P.J., Dors, C.L., Khoo, T.L.,

and Bernthal, F.M.: Phys. Rev. CT3, 2197 (1976)

13. Daly, P.J., et al., to be published

14. Bochev, B., Karamian, S.A,, Kutsarova, T., Nadjakov, E.,

Oganessian, Yu.Ts.: Nuci. Phys. A267, 344 (1976)

15. Toki, H., Neergard, K., Vogel, P., and Faessler, A.:

Nuci. Phys. A279, 1 (1977)

16. Backe, H., Handschug, L., Hessberger, F., Kankeleit, E., fichter, L.,

Weik, F., Willwater, R. , Bokemeyer, H., Vincent, P., Nakayama, Y.,

Greenberg, J.S.: submitted to Phys. Rev. Letters

17. Reinhardt, J., Oberacker, V., Soff, G., Muller, B., and Greiner, W.;

to be published

Experiments on K-hole and positron production in collisions

of very heavy ions

H. Bokemeyer • , ,-••, , ., >

G S I, D-6100 Darmstadt ! • — .'

•I) Introduction

The collision of very heavy nuclei at energies around the Coulomb-

barrier opens not only a chance of enlarging our knowledge on nuclear

physics but also gives rise to some quantum-electrodynaaiic processes

which occur because of the deep and strongly time-varying Coulomb-

field of the two colliding nuclei with Zj + Z 2 » 137. - ^ )

I want to focus your attention in the lecture onto the behaviour of the

bound and unbound states of the electron during the collision. The electron

as well as his antiparticle the positron is governed by the

Dirac-equation. The equation predicts eigenstates during the collision

as schematically shown in the energy diagram of Fig. 1. The abscissa is

a time-scale. The nuclei reach their distance of closest approach at

t = 0. Also shown is the positive and negative energy continuum from

which the lower one is completely occupied in our world. The striking

effect is the diving of a bound-state energy level at Z 169 which

allows for spontaneous production of positron (process c) - which means:

an electron of the completely filled negative sea jumps into the possibly

ionized direct K-shell without energy consumption, the remaining hole

appears as a positron. This exciting effect has been predicted theoretically 1) 2)

independently by different groups: by Greiner et al. ', by Rein ' and

by Zel'dovich and Popov3' in 1969 and 1972 in close relation to Kleins

paradoxon '.

This process c is accompanied and hidden by other shown processes (a,b,d)

which have been investigated intensively in theoretical studies mainly

by the Frankfurt-School. There is first the ionization of the K-shell

and the inpart coherent process of induced transitions from the negative

sea into the empty shell or the positive continuum which is supposed

to represent the main contribution to positrons of such molecular type.

841

Positive Energy Continuum

Fig. 1 Schematic representation of pair production processes in

heavy ion collisions.

These transitions b and d are predicted to be drastically enlarged in magnitude compared to analogous atomic processes. This is because of the relativistic shrinking of the electron wave-functions in a potential of a Z-value about 170 - which is much larger than Z = 137, the limit for 'normal' electrodynamics - and the strong variation in time of the Coulomb-potential during the collision.

In the f i rs t part of my talk I would like to report on results for the K-hole production mainly the probability P(b) for producing a K-hole at a given impact parameter b and the projectile energy dependence. We used a Doppler-shift attenuation method which is familiar to nuclear physicists but quite new for this type of experiments. The data allow comparison with theory to settle the long and lively discussion con¬ cerning the K-hole probability which is by the way the crucial parameter for the observation of the spontaneous positron-production.

In the second part I wi l l deny on the observation of positrons of molecular type, an experiment done with an Orange-electron-sepctrometer,which data gave some insight into the projectile angular dependence and Z-dependence of the positron-production. The experiments have been carried out by a collaboration between the TU Munchen, GSI Darmstadt, the TH Darmstadt and the Yale University in case of the positron experiment and GSI Darmstadt, the Yale University and MPI Heidelberg in case of the K-hole-production experiment. About another positron experiment has been reported here already by Dr. H. Backe. All experiments have been carried out at the high-energy end of the UNILAC-HI-facility at GSI Darmstadt.

842

Before I now come to the experiments I would like to give you some feeling about the adequate parameters for a collision of HI (Fig. 2) for example U on Pb with an energy of 5.9 MeV/u which means 1.2 GeV total kinetic energy and a Z-value of the united atom of Z = 174 in whose Coulomb f ield the K-sheil radius of the united atoms shrinks to 150 fm. This has to be compared with the K-shell radius of Pb of about 1500 fm and the nuclear radius of 7 fm. The distance of closest approach, 2a, is about 16 fm, which means that the nuclei represent a combined charge-center even for the K-shell.

With an ion velocity of about 10 % of l ight velocity, the nuclei need for the passage of the K-shell region about 10" s which is short compared to the life-time of a K-hole. This is an important point for the K-hole-production experiment to which I come now.

S A.X.

T. • -f.l * -

Fig. 2 Relevant parameter for K-hole and positron production.

II) K-hoie-production6^: H. Bokemeyer, F. Bosch, H. Eroling, O.S. Greenberg, E. Grosse, D. Schwalm A) Principle of the method The physical quantity we were looking for is the probability P(b) for producing a K-hole in a collision with a given impact parameter b. The main idea of the experiment is that the Doppler-shift -c~— of the observed x-ray is uniquely related to the scattering angle of the emitting atom (Fig. 3). Because of the large life-time ^.hole °f a n°l e compared to the collision time the number of characteristic x-rays N(AE. ) at a given Doppler-shift is practically identical with the number of K-holes produced in this collision with" a corresponding scattering angle or impact parameter. Let me show this correspondence in the specific case of target-excitation using the experi-mental set up shown here. A 4.7 MeV/u Xe-beam hits the 1 mg/cm Pb-target and will be stopped in the Ni-backing. The characteristic Pb-X-Ray is observed in the intrinsic Ge-Detector at 0 . Because of the shown time-scale - the time needed for target passage is much larger than the X-ray-lifetime but much smaller than the stopping time - the emission of X-rays occurs practically before the ion looses energy^t any case before the ion comes to rest in the Ni-backing, which therefore only serves as a beam stopper. In case of a thick Pb-target, which is one further step in the experiment, the X-ray energy reflects furthermore the momentary velocity of the ion and therefore the projectile-energy dependence of the K-hole production.

Let suppose, the elastic scattering is a good approximation, which is actually the case, the target velocity in the CM-system is given by the velocity of the CM.

The projection of the target velocity onto the ^-direction is for (^-obser¬ vation v C H (1 - cos 0 ) with 0 the scattering angle of the projectile. This means, we have a Doppler-shift for 0°-observation proportional to (1 - cos e ). Notice the direct correspondence from AE to e , and with this to the impact parameter b because of the missing ^-dependence due to rotational symmetry. This is a special advantage of the 0°-observation. The X-ray-spectrum of the target has to have a sharp cut-off at maximum Doppler-shift of twice the CM-velocity. For a substantial amount of K-hole-production for large scattering angle e , that is large AE tłie characteristic X-ray line of the target-atom is appreciably broadened to higher energy and may look as sketched.

S)

4»t Tb M-

*™*V//>

Fig. 3 Principles of the experiment for K-hole production.

Actually the X-ray-spectrum at 0° is given by this formula (Fig. 4).

The X-ray intensity dNK of the observed Ka-line is porportional to the

fluorescence yield u„ times the cross-scetion da-u in the CM-system

and times the Jacobians for coordiante transformation. The cross-section

is given by the Rutherford-cross-section b-db times the probability for

K-hole-production P(b). This clearly shows, there is a 1 -correspondence

between the X-ray intensity at a given Doppler-shift AE and P(b) at the

corresponding impact parameter b. Note the extreme sensitivity of the

method for small values of b. If I adopt a b-scale onto the energy-

spectrum then b = 0 fm is at the maximum-shift and b = 40 fm is nearly

already in the unshifted line. This means the region of P(b) for b

smaller than 40 fm gives the main part of the line shape and the rest

only contributes to the unshifted line.

81*5

in

1 «L U k -S

Fig. 4 X-ray line-shape formula (all angles in CM-system).

The spectrum has of course to be folded with the apparatus function of the Ge-diode. But as the resolution is about .5 keV and the maximum shift about 6 keV this wi l l not change the spectrum drastically.

For 90°-observation (Fig. 3) the Dopplershift is given by AE /E = sin e cos * . Because of the left-right-symmetry the corresponding line shape has to be symmetric.

B) Experimental Results:

I pick out the data for the observation of the Pb Ka-lines in three typical systems (Fig. 5). In an assymetric system the Is atomic state of the heavier partner becomes the lso state of the united system and the Is level of the lighter partner the 2pl/2o-state. Therefore we have in case of observing the Pb-Ka-lines with the system Xe ->• Pb an example for lsu-ionization and with the system U-Pb for a 2pl/2c-ionization.

81*6

0.1

4. ł

-r/C - O.-l

[•v ••rfw*-

Fig. 5 Presented systems for K-hole-production and relevant parameters.

Let me show you now as an example the Pb-Ko-part of the Xe •+ Pb X-ray spectrum which is already background corrected. (F ig. 6). We observe a strong asymmetry of the l ine which reaches up to the maximum sh i f t of about 6 keV. The observed l ines are shifted and broadened compared to the Pb-Ka-lines of a radioactive Bi-source by an amount which is explainable by sa te l l i t e - l i nes caused by mult iple vacancies. This asymmetry which disappears completely for ^ " -ob¬ servation is the expected l ine shape explained before. Already a t th is point one can state without any data-reduction that there is an appreciable amount of K-hole-production for large scattering angle or small values of b. This is much more pointed out in the result ing d ist r ibut ions for P(b) (Fig. 7) which have been deduced by a l ine shape analysis on the mathematical basis I indicated in Fig. 4. Both systems, the 2so-example Xe •+ Pb and the 2pl/2o-example U •+ Pb show a remarkably strong enhancement of the K-hole production for small values of b which reaches even more than 80 % probabi l i ty in the U •+ Pb case. The absolute ordfnate scale is not a result of these measurements but is a function of the absolute cross-section which is taken from the data of Behncke et a l . 'a t GSI. This strong r ise of P(b) for small values of b is an outcome of the r e l a t i v i s t i c shrinking of the electrpri wave function and the extreme time-dependence

847

to

LU z z < o - J

cc LU 0 .

l/> —

1 O

r\li PIT

106

105

10'

103

io';

103

i n 2

- ' ' - 90°-BG

I.100). / v

/ K a 2

- 0°-BG •' \ - \ ; :--

$ • •

t'-1 •

— - ^ ' ' .

1

A / \

.A

-.

' • > •.

i i '<

-"" ^fli-SOURCE i Ni-BACKING

1 ! tU

69 71 73

1 ine 5

Ik If— /*lO)u ł

,v ,

i •

i PIP) 006

0.0Ł

102

\ 0

A \ .o \

\ , %

\ \ \

' • * • • • * ' - " • ' • " ' ' • * *

(_ I

75 77 79 i i

*' 20

hane of Pb for

1 1

10 5

136Xe

i i

ok=5b

v 5 0 pllro)

3 6 X e * 2 0 8P 4 7 MeV/A

• - ; .

••> • ^ • > . . -

81 83 Ex 1

0

+ 2 0 8 pb

100

b -

106

5 10s |

in

10'

103

1 0 2 •

:"

i n 2

[keV]

observed at 0° 1 2 '

(before and after background-subtraction) and 90" (after 2

background-subtraction). The target thickness was -1 mg/cm . Also shown is the measured background of the Ni-stopper (normalized) and the Ka17 lines of Pb of a radioactive Bi-source.

of the potential. It was one main characteristicum of the calculations of 8)

the Frankfurt group which in contradiction to the much smaller extra¬ polations from lighter systems predicted K-hole-probabilities of some %. The fa l l off for much smaller values of b has been explained with a rotational coupling mechanism. In the symmetric system Pb •+ Pb the extraction of a P(b)-curve from the spectrum is much more d i f f icu l t because of the overlap of the shifted and unshifted Ka-lines (Fig. 8). But we used the data to extract the projectile energy dependence of P(b) by using a thick Pb-target. What one observes is acomplete assyraetry of the shifted and unshifted lines which results from a loss of Doppler-shift because of the loss of projectile energy in the target. There is again a 1:1 correspondence between the Doppler-shift and the projectile energy. The strong curve results from a f i t with a P(b) approximated by an exponential model varying only in inten¬ sity but not in shape with the projectile energy, which had to f i t simultaneously the thin and thick target curves.

848

P(p)

0.06

0.04

0.02

it

E

0

I I T I i i i r

0

P(p)

0.5

0

\ 47 MeV/A

i i i i

50 100 p[fm]

1.0r-r

+ 2 0 8 p b _|

A.7 MeV/A

\ \ \

• I , ^ v

0 50 Plfm] 100

F i g . 7 Impact-parameter-dependence P(p) o f the K-hole p roduc t ion .

The abso lu te scale i s taken from 7 ) .

Fig. 8 Ka12-line shape for Pb for 2 0 8Pb + 2 0 8Pb observed at 0°

after background subtraction. The target-thickness was 2 p

57 mg/cm (upper part) and 0.5 mg/cm (lower part).

This first analysis gives a preliminary energy dependence of the K-hole. cross section of approximately E (Fig. 9), which can be compared with data from Behncke et al.9'in asymmetric systems taken in a conventional experiment by measuring the absolute cross section at different projectile energies. We are now on the way to prove, if this very simple and much less beam time consuming Doppler-shift method can be used for determination of projectile energy dependence of asymmetric systems as well.

850

5*

Fig. 9 Projectile energy dependence of the K-hole production (preliminary data). The data for Pb + Au and Pb + Mo are taken from 9).

851

I I I . Positron production:

H. Backe, E. Berdermann, H. Bokemeyer, M. Clemente.J.S.Greenberg, L. Handschug, F. HeBberger, E. Kankeleit, P. Kienle, Ch. Kozhuharov, V. Nakayama, L. Richter, P. Vincent, F. Weik, R. Willwater

We now come to the second experiment I wanted to present you, namely the ob¬ servation of positrons during the HI-collision. We are interested in this experiment especially for the Z-dependence of the positron-production which has been predicted very strong and the dependence of the positron-production from the scattering angle. Before I describe the experimental set up I should mention that there are s t i l l further known positron-sources. The dominant one is e , e"-pair-creation after Coulomb excitation of the nuclei or nuclear reactions. These positrons are not distinguishable from the dynamically induced positrons we are looking for and represent the main background. I t was one of the forthcomings to separate these by systematic observation of the y-ray spectra in coincidence to the scatterd particles. I shall come to this in detail later.

A) Experimental set up

The cross section for the positron-production is about 100 pb and the e+-energy-spectrum is expected as continuous. This means, the transport of the positive charged positrons out of the highly radioactive target-region onto an effectively shielded detector system is unavoidable. We used in this experiment a high transmission magnetic spectrometer of the iron-free Orange-type ' (Fig. 10). This gives the following experimental facts:

- The detector-system is shielded against direct radiation. The beam wil l be stopped in a cup inside the lead-shielding.

- The toroidal magnetic f ield produced by 60 coils separates for positive and negative charge, so that the much larger electron-component, mainly <5-electrons, is completely suppressed. This is the main advantage com¬ pared to the solenoid-spectrometer we used in the positron experiment reported this morning by Dr. Backe.

- The magnetic f ield focuses only a given momentum-band which has been enlarged to about 15 % for ^ by detector-dimension. This gives a direct energy information but on the other hand forces a time consuming point-by-point measurement to get a complete spectrum.

852

ring counters

Ge(Li)D

beam

target NaJ 3x3" PM

Fig. 10 Schematic drawing of the experimental set up of the positron-experiment in the orange-B-spectrometer.

- The spectrometer gives room for an annular-parallel-plate avalanche counter to detect the scattered particles for 13.5° < 0 Lab < 32.3°, which is subdivided into four rings. This counter works in coincidence to the e+-detector and enables discrimination against fission products and compound-reactions mainly originating from Oxygen-impurities in the target.

- A 3"x3" Nal-counter detects y-rays in coincidence to particle events in the ring-counter.

- The e+-detector is composed out of a 2"x4"-NaI embedded in a cylindrical scint i l lator, so that a positron-event is characterized by y coincidence between scinti l lator and one 511 keV annihilation y-quant in the Nal.

What is then told a positron event ? First there has to be a coincidence between Nal and the scinti l lator of the positron-counter which then reveals a quite clear Nal-energy-spectrum with a clearly visible 511-line. The overall e -detection efficiency comes out to be 0.026. Then one inquires an additional coincidence to one of the four ring-counter elements which f inal ly ends up in a rather clear 511-spectrum of the Nal-crystal.

853

B) Positrons from ruclear and non-nuclear oriqin:

As already mentioned, a main problem was the separation of positrons from nuclear processes like pair-creation after Coulomb excitation. A measure for the deqree of excitation of a nucleus in a qiven collision is the y-spectrum taken in coincidence to the scattered particle. In principle i t is possible to calculate the positron rate for internal pair-production from the measured y-spectra but this needs the knowledge of the multi-polarity of the transition and fai ls completely in the case of EO-transitions. We therefore used an experimental observation for the calibration of the nuclear positron part: the y-spectra (Fig.11 1 coticident to particle-counter show in their gross structure a quite similar shape of exponential character up to 7 MeV which is connected to the statistical character of the

Fig. 11 Smoothed y-ray spectra for the systems U+U, U+Pb, U+Au, U+Ta, U+La at 5.9 MeV/u.projectile energy coincident to a particle-counter events in Ring-counter 2 (RC2) (16.1°-20.2°) and ring-counter 4 (RC4) (26.9°-32.3°).

85V

transitions of the highly excited nuclei. And furthermore, the number of positrons per y-ray with an energy higher than 1.44 MeV, the minimum y-energy for an observable positron (Fig.12 ), remains approximately constant and independent of scattering angle for systems like U on La and U on Ta which are expected to contribute neglectable to molecular positrons because of the low Z-values whereas i t raises remarkably for the higher Z-systems. We took a value N +/N = (8.1 +0.9-10 as weighted average for the nuclear positrons which was further on used to subtract the nuclear part from the higher Z-systems by multiplying this value with the target-specific y-intensity. The resulting non-nuclear positron-production came out to be at least 50 % higher than the nuclear part in U-Pb and at least a factor of 2 in U-U.

The remaining non-nuclear part is shown in the next transparency (Fig. 13). The projectile was in any case U at 5.9 MeV/u and the energy window for the positrons (478 + ^3) keV. The data are drawn as a probability jr for production of positrons as a function of the scattering angle e

0CM in the CM-system. As we did not distinguish in the experiment between projectile and target nucleus, which is anyhow impossible in symmetric •collisions as U on U, we supposed the projectile to be detected in trans¬ forming from the laboratory-system to the CM-system which gives us an error

u-41

1

Fig. 12 Number of positrons (Eg+ = (478 +|? keV) per observed y-ray (E > 1.447 MeV) coincident to particle-ring-counter 1...4 for different systems U+La, U+Ta, U+Au, U+Pb, U+U.

855

10 ,-4

10"

10 ,-6 676 466 392 337 300 274

20 30 £0 50 60 70 %

54 Fig. 13 Positron production probability dP/dEe+ at (478 ± 53 keV) positron energy as a function of the CM-scattering angle of the projectile after subtraction of the nuclear part of positron production.

of about 20 % in g£ in this plot. There are also shown the results for Pb-Pb at again 5.9 MeV/u. As already explained this morning by Hartmut Backe , this is the only case where the nuclear positrons can be measured inde¬ pendently via the nearly radioactive decay of 2 1 2Pb, whose decay populates practically the same level as the Coulomb excitation does, so that the nuclear positron production could be subtracted much more accurately in the Pb-Pb-case.

I added into this picture the distance of closest approach R . in terms of the parameter a = half the distance of closest approach in a head on collision. ^ „ / a is a constant for all systems, if the projectile is scattered into the same CM-angle. Moreover, it indicates that the relative velocity+ is the same for all systems along vertical lines in this plot, if the radius vector is measured in terms of a and the pro¬ jectile energy per nucleort is the same - in other words vertical lines in this plot connect systems at same fcfnematic conditions.

•9 i

856

What positrons have we now observed ? The data wi l l f i t into the picture

of the so-called dynamically induced positrons - the processes d,e,f in 12)

my f i r s t transparency as i t is now proposed by the Frankfurt group '

The theoretical curves in Fig. 13 are taken from these calculations.

The theory also predicts an extremely strong Z-dependence of L , n about

18 for this dynamically induced positron production. I f one compares this

with the data one should eliminate the influence of different relative

velocities on the time variation of the potential in the different systems

(same kinematic conditions). This is done in this plot i f we compare the

systems along vertical lines which are as we have seen lines of some

relative velocity. I f we do th is, the distance of the hand-drawn curves

gives an n-value of about 17. in relatively good agreement with theory.

On the other hand, i f the picture of a cr i t ica l Z-value and cr i t ica l

distance for the diving process (c) would be right ' ' 'an additional

increase of the probability for the larger scattering angles in the

U-U-system might be expected but is not yet visible. A system with a

larger value for Z, + Z~ would be worthwhile but is at the moment not

possible.

I I I . Summary

I reported on two experiments, K-hole production and positron-production,

which were done with the goal to get some experimental insight on the

behaviour of electronic bound and continuum states in a strong and

strongly time-varying Coulomb f ie ld (Za >> 1). The data observed are

consistent with a theory which predicts strong transition enhancement

because of re lat iv is t ic shrinking of the electronic wave functions and

extremely time-varying Coulomb f ie ld .

The also predicted spontaneous positron-production in a diving process

has not yet been observed, although the existence of the necessary

large K-hole probability has been settled.

857

References

1) W. Pieper and W. Greiner, Z. Physik 218 (1969) 327

B. Muller, J. Rafelski, W. Greiner, Z. Physik 257 (1972) 62 and 183

2) D. Rein, Z. Physik 221. (1969) 423

3) Ya. B. Zel'dovich and V.S. Popov, Sov. Physics Uspekhi JL4 (1972) 673

4) 0. Klein, Z. Physik 5_3 (1929) 53

5) F. Beck, H. Steinwedel and G. Sussmann, Z. Physik 171 (1963) 189

6) J.S. Greenberg, H. Bokemeyer, H. Emling, E. Grosse, D. Schwalm and

F. Bosch, Phys. Rev. Lett. 39 (1977) 1404

7) H.H. Behncke, P. Armbruster, F. Folkmann, S. Hagmann, and

P.H. Mokler, Proceedings of the 10th Int. Conf, on the Physics

of Electronic and Atomic Collisions, Paris (1977), 156

8) G. Soff, W. Betz, G. Heiligenthal, J, Kirsch, B. Miiller, J. Reinhardt,

W. Greiner, Fizika 9 (77) 721

9) H.H. Behncke, P. Armbruster, F. Folkmann, S. Hagmann, J.L. Macdonald,

P.H. Mokler, priv. communication

10) E. Moll and E. Kankeleit, Nukleonik T_ (1965) 180

11) H. Backe, L. Handschug, F. Hessberger, E. Kankeleit, L. Richter,

F. Weik, R. Willwater, H. Bokemeyer, P. Vincent, Y. Nakayama, and

J.S. Greenberg, to be published in Phys. Rev. Lett., GSI-Bericht P-2-78

12) B. Muller, V. Oberacker, J. Reinhardt, G. Soff, W. Greiner, and

J. Rafelski, Fizika 9 (77) 765

13) J. Rafelski, and B. Muller, Phys. Lett. 65B (1975) 205

14) V.I. Lisin, M.S. Marinov and V.S. Popov, Phys. Lett. 69B'(1977) 141

15) D.H. Jakubassa, and M. Kleber, Z. Physik A277 (1976) 41

AS IffVESglGATIOK OP QUASI - E0LECPLS3 HT EEAVY IOIT COLLISIONS

QUASI - MOLECULAR ROSrTGEIT RADIATIOH

(Lecture JJotes)

K.H. Kaun '^„' - ; \.~ j

Joint Institute for nuclear Research, Dubna, D33R

In the recent years the problem of the observation of quasi-nc?. ecu? ss

and of the quasi-molecular Roentgen emission in heavy ion-attv: z~. '. ' i-

sion3 was very important in view of the electronic structure nf -,e:vy

qua3i-atoms with effective atomic numbers much higher than 1C'_ an:1 for

the observation of new processes of quantum electrodynamics in very

strong electromagnetic fields C^""0. ^ie observation of such • rooec.-e:

as the decay of the neutral vacuum in the overcriticcl field cf tvo

uranium nuclei is important net only for quantum eleetrc<!yr..i-.rics but

also for quantum field theories, if tlie ccurlir.- ccnsti.nts ere ::- t

small. Por e::ample, in this way the theory of r.jir. condenset i ~n Lr.

nuclear matter was stimulated. In the recent years the observntion of

quasi-molecular X-rays has shown that the qua^i-molecules can, in prin¬

ciple, be observed in heavy ion collisions. It is clear thct it the

present time the most exciting problem in this field of rhysics ir the

observation of positrons in the U + 0 collisions. At this '/Vinter 3Chool

we heard the lectures of Dr.Backe' and Dr.Bokemeyer from Darmstadt on

this important problem. In my talk I will give a review on the present

status of experimental studies of the transient formation cf quasi-mo-

lecules in heavy ion collisions and the quasi-molecular X-ray emission.

In recent years those investigations were a starting point for the pre •

sent positron experiments.

2. The general Quasi-Molecular (Q',1) Picture and the Kechanism of

T Ionisation

In the QK picture \_<f\ the collision is aanumed to be slow enough for

the electronic wavefunction in the atoms to adjust themselves, at each

859

internuclear separation H, to the molecular configuration appropriate for a diatomic molecule with nuclear charges separation R.

and Z„ and internuclear

COLLISION TIME . 10'"9 £. Pig. 1. The „scheme"of the transient formation of quasi-molecules in

heavy ion collisions.

This situation occurs if the adiabaticity parameter (v./u ) <ST1, where v^ IB the projectile velocity and ri s(2En/me)'1' is the orbital velocity of the n-shell electrons, and the collision system is nearly symmetrical (Z^«sZ_). An additional condition that makes the observa¬ tion of QM X-rays possible is that the distance of closest approach between the colliding nuclei is equal to or smaller than the classical Bohr radius of the n-shell in the united system. Prom these conditions the lower and upper limits for the projectile velocity (or energy) can be derived as follows [fQ:

1.2 x 10-3 10"4 Z/n,

where Z=SŁ Z.gi and n is the main quantum number of the given atomic shell. Pig.2. shows this "QM region" of projectile energies for symmetric collision systems (Z.,=Z2) and K-shell electrons (n=1) as a function of atomic number Z^. The collision systems investigated in our experi

\ 860

meats in Dabna [7] are placed bet¬

ween the two limits of this region

so that we can hope to observe

quasi-molecular phenomena and the

QM picture should be correct.

0 20 40 60 60- 100 ATOMIC NUMBER Z, • Z2

Pig. 2

As long as the two QM conditions are not violated in a heavy ion colli¬

sion it is believed that the inner shell electrons will follow nuclear

motion nearly adibatically into molecular levels, as sketched in Fig.3:

It is clear from our principal QM picture,- that the quantitative des¬

cription of adiabatic atomic collisions must be based on the QM two-

cen-fcre wave functions. Por the relativistic case of heavy QM systems

this problem was first investigated by B.MUller et al. £8] in the one-

electron approximation by solving the two-centre Dirac equation:

861

The true states of the dynamical collision system Y(t) are obtained as a solution of the relativistic form of the time-dependent Schrodir.ger equation:

In the time-dependent perturbation theory the solution of the tiir.e-dependent Schrodinger equation for a single electron state can be ex¬ panded in terms of the wavefunctions obtained from the stationary two-centre Dirac equation: (

-iJEn.au Upon substitution of this ansatz in the Schrodinger equation one is left with an infinite system of coupled charjiel equations:

an(i) -- jL. am(t)&

'.vhc ne 5c. (R) is rhe angular velocity and R is the relative radial velo¬ city of the two colliding nuclei. The summation over m includes also the continuum states. The first part in the interaction matrix elements is commonly called radial, the second one rotational or Coriolis coupling. The v.vo opera¬ tors exhibit different angular momentum Election rules: the rarfial coupling acts between states with the same A quantum number, whereas the rotational term couples states with anfTiiar momentum projection A differing by + 1. The rotational coupling between 2v3T - 2p<5~ states at closest approach R m i n and the radial coupling between 2p6" - 1s6" states are very important for vacancy production in inner shells, especially in the 1s6" -state. This is shown schematically in Pig. 4:

_ c

-+-

,'"*; elecłr

U.A. 862

Before we go into a detailed discussion of QM X-rays, we will discuss

the mechanism of the inner-shell ionisation. Pig. 5 shows the approxi¬

mate regions of validity for various models connected with inner-shell

vacancy production in ion-atom collisions [9]. ar small values of

Z-j/Z the inner-shell ionisation is dominated by direct Coulomb ioni¬

sation and various theoretical approximations (for example, CCA or

F.7BA) describe the experimental results

adequately.

In the other extreme case, when 2-j/Zp

is near unity and v./uv is small, the

molecular model applies. In this case

relatively intense continuum X-rey

emission can be identified as Q'.'. X-ray.1:.

The region in-between of Fir. 5 is stil7

no-man's land. .Ye will discu33 here

the K-shell vacancy production in t're

QU~recion [iO, 1i] .

Pig. 6 shows the typical dependence of the cross section for the ior.i

sation of E-shells on the atomic number Z? (target) in 150 ','eV Xe ion

collisions [12], ',ve see that the linii ter

collision partner become?; prefernb.7 v ioni¬

zed. Very similar trendo are ob.ser"e<i ;>; W'p , , ,

[ ! a l l of experiments at incidence ener.-ie:.

E^LV.eV/E and for nearly sy-nnetric col-

ision systems &.* ^^o» i «•

i

?ig. 7 shows schematically the correla¬

tion diagram of QM orbitals fcr a colli¬

sion Z1 + Z2 fZ 2^Z 1). Only levels rele¬

vant to K-vacancy formation are included.

According to the electron promotion ro-

h cesa in the QM model, vacancies from the

L-shell of projectiles Z.. (Xe ions) rnav

get across the 2pT -+• 2p6" rotational coupling to the 1s state of the

lower-Z2 collision partner. The ionisation of the H-shell of higier -

S1 Se projectiles is possible only by the radial coupling of 2p©- •*•

states. Por the-probability of vacancy transfer from the 2p<ror-

863

tital tó the 1sr state Meyerhof [13] gives the relation

0

::U) = 1/(1 + ^ t C i ) , '--here xoc (I., - I2^V1 a n d I1'I2 a r e t h e K~ s h e 1 1

ior.isatjon energies of the colliding atoms. It is clear that by elec¬

tron pror-otion via 2pT - 2p6" coupling the 1 s6" vacancies cf the sy-

;te;: (Z, + S_) can be formed and the QM KX-rays can be obtained only

cr?t--r the fomntion of 1s vacancies in the K-shell of the projectiles

ia pri.-icry collisions. The direct ionisation of inner-shells by Cou-

lo-ib excitation is important at higher incidence energies. In Refs.

[i4,15^ "the scaling properties of the 2p<y - 2p3i"rotational coupling

are discussed for symmetric collision systems. Meyerhof et al. fi5]

,~ive a scaling law for the ionisation cross section of inner shells:

= f

Pig. 8 shows the experimental K-vacancy production cross sections

scaled according to this scaling law for symmetric collision systems.

This picture is correct only for the electron promotion region

(E^j$1 KeV/Hj Z.jjc Z^) and gives very "pessimistic values" for the po¬

sitron production cross secticn in the U + U collisions.

10 10'3 I0"2 10"1

At the present time we know from new theoretical and also experimen¬

tal results that relativistic effects enhance the 1sO" ionisaticn by

direct Coulomb excitation in the O + U collisions f' J*

3. The Properties of the Quasi-Molecular Rpentrer. Radiation

The CLASSIFICATION OF THE Q.M X-RAY SPECTRA is schematically shovm in

Pig. 9. This scheme is correct only for the non-relativir-tic raas, if

the spin-orbital interaction is small, and for small intemuclear dis¬

tances. In analogy to the atomic Roentgen spectroscopy we will classify

the components of the Qi* X-rays as the LET-, LX- and KZ-radiation.

The QK LX-ray spectrum (and thus the appearance of intermediate qus.3i-

malecules, recognized by the QK X-rays!) was first found in 1972 by

Saris et al. [}7~\ in the Ar + Ar collisions in the process

-f Si

The experimental spectrum of this radiation is shown in Pig. 10.

Armbruster et al. [i8~} have found the QM MX-radiation in the I + Au

collisions (bhe quasi-molecular 4f —*-3d transitions at the X-ray ener¬

gy E j * 8KeV of the transiently formed superheavy quasl-malecule with

ZT + Z2 = 132.').

865

The experimental spectrum is shown in ?ig» 11.

".a. 9

Ar(2<0K«V)» Si

t 1.0 K«V 17<K«VSilK) •

X-RAY ENERGY

1.10

V IA ! V

\J

The KX component of Q" X-raya was obtained in 1973 in the 3r + 3r

collision in Stanford [19] and in the Ge + Ge collisions in Dubna p2Cj .

Our group at Dubna have been earring out, in the first place, experi¬

ments to study some aspects of atomic characteristics and QM KK-rsys

of very heavy and symmetric collision systems p1 - 23j such as

Hi + Hi C39 MeV and 57 HeV), Ge + Ge (54 MeV and 81 UeV), Kr + ICr

(42 MeV), Kb + fib (67 MeV and 96 MeV"), La + La (115 MeV) and Bi + Bi

(144 KeV and 172 KeV), Por these heavy colliding particles, the adia-

baticity condition is fulfilled better than in the case of collision

systems with lower Z, where the orbital velocities of electrons are

rather small and the observation of QM radiation is more difficult

due to the existence of some competing effects, which also show con¬

tinuous X-ray spectra* Pig. 12 shows a typical KX-ray spectrum obser¬

ved in our experiments [23] . U- was measured in the collisions of

67 MeV Fb iona with the atoms

of a "target made of metallic

pure niobium 1 mg cm thick.

Besides the intensive KX - li¬

nes of the Hb atoms- and the ab¬

sorber material (0,2 mm Cu), the

spectrum contains a continuous

intensity distribution, which

866

ranges approximately up to the united-atom KX-energy and is mainly for¬ med by QK transitions. We first 3howed in. our experiments with Ge, Hb and la ions that the QK KZ-ray oontinua consist of low—energy and high-energy components, denoted by us as C1 and C2, respectively. It is of interest to review the experimental evidence for this X-ray con¬ tinuum situated above all the atomic characteristic KX-lines and inter¬ preted as a QK KX-rsy spectrum. Since QLI X-raya form contir.ua or.e must mainly consider other continua with which QM X-rays night "03 con¬ fused or which could form a background under the QM spectra. In ;:eariy symmetric collisions of very heavy ions the 'CQTSTHUJCVS 3ACr.GflCU!:ii KA-DIATIOE Can be assigned to the following processes (in addition to the normal room background): Bremsstrahlung of secondary electrons (3EB), Hucleus-nucleus bremsstrahlung (HUB), Radiative electron capture (EEC), Compton scattering of Coulomb-excited nuclear T-rays. The latter type of background radiation was very small in all of our investigations (by using projectiles and targets with nuclei having1 high-lying excited states, Ei, Kr, Kb, Bi). For the brsmsatrahlunp: processes (SEB and HUB) we have made calculations [24, 25]. The dif¬ ferential cross section for production of bremsstrahlung from secon¬ dary electrons by impact of heavy ions (S1, Z^ ,A1) with target atonis (Zj.Ag) is given by the formula:

v;here dS^(E/,E1 )/clE<r - is the cross section for production of secon¬ dary electrons in the energy interval [E^,Ef + dEJ by an ion with the incidence energy E 1, and dY(Srf»fEr)/dEr - is the yield of bremsstrah-lung in the X-ray energy interval [sp, Er+ dEp] , induced by an (T-elec-tron of an energy E^.>Er. The cross section dĄ/E^-.E.. )/dE/- can be cal¬ culated by meana of the binary encounter approximation of Garcia et al. J26]. The nucleus-nucleus bremsstrahlung can be computed with good accuracy by using the classical theory of Alder et al. [27] . The elec¬ tric dipole E1 component of OTB has the highest intensity and contri¬ butions from higher electrical multipolarities and also interference effects can be neglected in our cases of investigations. The differen¬ tial cross section of the E1 component of HUB is:

867

of the velocity of the radiative system using the X-ray energy Doppler shift in the collisions Fb + 67 KeV Hb. The essence of this experiment is as follows (Pig. 17): As a function of the velocity of the radiative system, the energies of the X-rays emitted obtain a Doppler shift which can be determined by detecting X-rays at different angles with respect to the ion beam di¬ rection (30,31], The Doppler velocity characterizes the radiative sy-

Ce OETECTOR . 45*/135*

' t Not Doppto Ihiftrd bl ' . 1

—'.'•••i-cct r

c ) Dopolcr shifttd

i .r

d) Dopcltr shifted with » .

• '-r-t...,-!-

20 30 Ł0 SO 60 70 M Ex(90*)(K«Vl

stem irrespective of the details of the assumed production mechanism for the X-ray continuum. In particular, if radiation in a certain X-ray energy region,is believed to originate from Qiu processes, the Doppler velocity should be equal to the centre-of-mass velocity of the inter¬ mediate molecules. Fig. 18 shows the results of these measurements. Here we present the Hatios R of the normalized spectra at +90° and -90° and at 45° and 135°, as a function of the X-ray energy in the labora¬ tory system. Figure 18c showa that in the energy region of both conti-nua, C1 and C2, X_raye are emitted from systems having the velocity of the quasi-molecule Nb + Wa, From this fact we conclude that both components of the X-ray continuum, C1 and C2, originate from quasi- mo¬ lecular transitions. It is important to determine the Doppler shift of the contijiuum X-rays from heavy ion collisions also for another reason. In the interpreta¬ tion of the laboratory anisotropy of the continuum X-ray spectrum, knowledge of the Doppler velocity is needed in oxder to compare the

870

measured aniscrtropy with the theoretically predicted centre-of-mass

anisatropy of the quasi-molecular X-rays. We will return to this pro¬

blem later,

flow, let me come to the IETERPRETATIO1T OP THE TffO-COMPOHEKT STRUCTURE

of the quasi-molecular KX-ray spectra. Heinig et al. £32] tried to give

an explanation to the origin of the components, which may take place

during the transient formation of quasi-^olecules. The authors pointed

out that in all molecular correlation diagrams for medium atonic num¬

bers Z1 and Z„ reported till now, the 2p5*term shows a relative aini-

mum. Fig. 19 shows this effect in the correlation diagram for the qua¬

si-molecular states of the system lib + M'o as an example. The energies-

of the QM-states are calculated here by Truskova [33] by solving a

non-relativisitc problem with a two-centre potential and a fixed inter-

nuclear distance.

20 30 40 SO 60 1/ «,Nb-.,Nb

Pig, 20 shows that transitions from higher terms to the "2pff-minimum"

have in all cases (Ge, Kr, Kb, La) a higher energy than the transitions

owing to the characteristic lines. In addition, one can see that the

maximum binding energies smax(2P&>2Z') o f QM 2pff-levels agree with

the "endpoint energies" of the C1 components measured in our experiments

with Ge, Kr, lib und La ions. According to these suggestions the higher

intensity of the continuum 01 can be explained assuming that the va¬

cancies in the 2p5'minimum were filled mostly in a first collision,

whereas a second collision is assumed to produce the continuum C2. Under

871

Nb«67MeV Nb

0 20 <O 60 «0 «0 ENERGY (KeV)

o2

o'

X-RAY ENERGY /'

Ag-Ao. -/Nb-w :

/Br-NB / Bf-Br

/Ni-N> / Pt-Ni/Ni-Ft

r Ft-Fe i Cl-I*

i Co-Co / 1

(VI-41 -

SO 100 200 ATOMIC - NUMBER

The rapid variation of the anisotropy near the maximum energies of the 1s6" and 2pC states can be an important effect for the spec-troscopy of super heavy quasi-molecules. Por example, Wolfli et al. [44] found that the "turning point" of the T K E ^ ) function directly °:ives the corresponding K^ -transition energy of the total united atom (aee Fig. 26). In Ref. [45] the anisotropy effect ił(E ) was calculated in the dynamical theory for the system Wi + Hi at vari¬ ous bombarding energies. The results of these calculations are shown

in Pig. 27. The theoreti¬ cal curves reproduce the experimental data only by assuming the special alig¬ nment of 2p3T and 2p6~ sta¬ tes and by including the "slippage" effect. It ta¬ kes account of the fact, that the electronic mole¬ cular orbitals cannot fol¬ low the fast rotation of "the colliding system at

small intennieleai. distances. Further experimental and theoretical investigations are needed to clarify the anisotropy effect of the Q- X-ray spectra.

874

4. Conclusions

The experiments with medium heavy ions give a consistent picture of

the quasi-molecular X-ray 3pectra. The QM X-ray investigations give,

in principle, the possibility for the two-centre spectroscopy of mo¬

lecular orbitals of snperheavy quasimolecules in the atonic number

region Z>100, In the future such a sper-.troseopy will require coin¬

cidence experiments to observe the QM X-ray spectra under definite

impact parameters*

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1. Rafeisfc£ J.e.a., Phys. Rev.lett., 1971, v. 27, p. 958. 2 . flonoB B.C. , 0|t3T<j> /(3W ,T.feO, c.AZZi.

3. 3e^ib3o6uv J).E, nono6 B.C., W<+H -m-1> T.iOS, C<K>3. 4. Reinhard J., Greiner W.f Quanrun: Electrodynamics of Strong Fields.

Rep. Progr. of Physics, 1977, v. 40, p. 219. 5. Fano U., Lichten W., Phya. Rev. Lett., 1965, v. U , p. 627;

Barat '.!., Lichten W., Phys. Rev. A, 1972, v. 6, p. 211; Lichten '$,, The Quasi-Molekular tiodel of Atomic Collisions. In: Atomic Physics 4, Ed.G. zu Putlitz. K.Y., 1975.

6. Armbruster P. e.a., Physica Scripta A, 1974, v. 10, p. 175. 7. KayH k.r., Maacppacc U.s$panK B., 3MASł -19łł) T.S, c|2*ifc-8. duller B. e.a., Phys. Lett. B, 197", 47, p. 5;

Muller B. and Greiner '.V., Z. i:aJ--vforschung 1976, 31a, p. 1. 9. Madison D.H., Kerzbacher E., .'tjnic Inner-shell Processes, Ed. B.

Crasemann, Academic Pre3S, 1:'."5, Hew York, p. 2. 10. Briggs J.S., Macek J.H., J. i--:rS. 3, 1972, v. 5, p. 579;

-Fastrup B. e.a., J. Phys. 3, 1974, v. 7, p. L 206. 11. Keyerhof W.E., Taulbjer-j K., Ann. P.cv. Kucl. Sci. 1977, 27, p. 279. 12. Gippner P. e.a., Hucl. Phys. A, 1975, v. 245, p. 336. 13. Meyerhof W.E., Phys. Rev. Lett., 1973, v. 31, p. 1341. 14. Brigss J.3., Macek J.H., J. Phys. B, 1973, v. 6, p. 982. 15. Keyerhof W.E., ConLTi.Atom.Mol.Phys., 1975, v. 5, I'o. 2, p. 33;

Łleyerhof W. E.,Fhys. Rev. A, 1976, v. 14, p. 16". 16. Betz IV. e.a., ?roc. of the Second Intern. Conf. •.:. Znr.er Shell

Ionization phenomena. V.2.Invited Papers. Freiburg, 1-76, p. 79; see the- Lectures of H, Backe and H, Bokemeyer at thic '.Vinter School.

17. Saris P.'.V. e.a., Phys. Rev. Lett», 1972, v. 28, p. 717. 18. f.Iokler P.H. e.a., Phys. Lett.., 1972, v. 29, p. 827;

Mokler F.H. e.a., Proc. IX ICPEAC, Seattle, 1975. Univ. Was.i.ng-ton Press, Seattle 1976, p. 501; Kraft G. e.a., Phys. Rev. Lett., 1974, v. 33, p. 476; Polkmann P. e.a., Z.Phys. A, 1976, Bd. 276, S. 15.

19. Ueyerhof W.E. e.a., Phys. Rev. Lett., 1973, v. 30, p, 1279; Phys. Rev. Lett., 1974, v. 32, p. 502.

20. Gippner P. e.a., Preprint JINR E7-7636, 1973; Nucl.Phys. A, 1974, v, 230, p. 509.

?1. Clippner P. e.a., Proc.Inetern.Conf.Reactions between Complex Euclei.Nashville, June 1974. Preprint JH!R E7-8006, 1974; Phys. Lett. B, 1974, v. 52, p. 183.

22. Frank ff. e.a., Preprint JEJR E7-9029, 1975; P^ys. Lett. B, 1975,

v. 59, p. 4-1.

23. Kaun K.H. e.a., Preprint JESR E7-9629, 1976; Proc. II Intern. Conf.

Inner Shell Ionization Phenomena. 7, 2. Invited Papers. Freiburg,

1976, p. 68. ...

24. Folkmann P* e.a., Hucl. Instrum. and Meifiiods, 1974, v. 116, p. 487;

Folkmann F.J., Phys. E: Sci. Instram, 1975, v. 8, p. 429.

25. Gippner P., Communication JIHH E7-8843, 1975.

26. Garcia J.D., Phys. Hev., 1969, v. 117, p. 223; PhyB. Rev. A, 1970,

v. 1, p. 280,

27. Alder K. e.a., Hev .Mod ..Phys., 1956, v. 28, p. 432.

28. Kienle P. e.a., Phys. Hev. Lett., 1973, v. 31, p. 1099.

29. Betz R.D. e.a», Phys. Rev. Lett., 1975, v. 34, p. 1256.

30. Meyerhof W.K. e.a., Phys» Rev. A, 1957, v. 12, p. 2641.

31. Frank W. e.a., Preprint JDTR E7-9861, 1976; Z.Phys. A, 1976, Bd.

279, S. 213.

32. Heinig K.H. e.a., Phys. Lett. B, 1976, v. 60, p. 249.

33. Tpy£KO&x H.*., Coofuw,cHUt QUQU., ?AA- 4010+, AV*t>.

34. Heinig K.H. e.a., Preprint JEffi E7-9862, 1976; J. Phys. B, 1977,

. v. TO,p. 1321.

35. Macek J»H,, Brigga J.S., J.Phys. B, 1974, v. 7, p. 1312.

36. Smith R.K. e.a., J.Phys. B, 1975, v. 8, p. 75.

37. Heinig K.E.. e.a., Preprint ZfK Rossendorf, 1976; J. Pliys. 3, 1977.

38. Saris P.W., Hoogkamer 03i.P., Proa. FICAP. Berkeley, July 1976.

Ed.R.Marros e.a., Plenum Press, 1976;

Meyerhof V7.E. e.a., Proc. FICAP. Book of Abstracts, 1976.

Ed.R.Marrus e.a., p. 56.

39. Frank Vf. e.a., Communication JONR E7-9065, 1975; Preprint JI'JR

E7-9427, 1975; Z.Phys. A, 1976, Bd. 277, S. 333.

40. Miiller B. e.a., Phys. Lett. B, 1974, v. 49, p. 219; Phys. Rev.

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41. Greenberg J.S. e.a.,.Phys.Rev.Lett., 1974, v. 33, p. 473.

42. Kraft G. e.a., Phys. Rev.Lett., 1974, v. 33, p. 476;

Polkmann E. e.a., Z.Phys. A, 1976, Bd. 276, S. 15.

43. Thoe R.S. e.a., Phys. Rev. Lett., 1975, v. 34, p. 64.

44. W61fl.il W, e.a., Phys. Rev. Lett., 1976, v. 36, p. 309;

Wblfli ."*;, II Intern. School on Buclear Physics. Lecture Rote.

Predeal 1976.

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Betz E.D. e.a.f Abstracts U ICPEAC.Seattle, (1975. Vol. 2.

877

6. CLOSING REMARKS

CLOSING REMARKS

John Sharpey-Schafer

Oliver Lodge Laboratory, the University of Liverpool, U.K.

We are finishing a long and very interesting session so I am

going to keep my remarks very brief and not keep you from your

lunch. The quality of the physics at this Winter School has

spoken strongly for itself and this will be immediately apparent

to anyone reading the proceedings. The lectures have been so ar¬

ranged that speakers were given enough time to develop their topics

in a clear logical manner but, due to the skill of the organizers,

the time was only just sufficient so that speakers stuck to the

physics and had to avoid unnecessary discussions into tortuous

detail. This skill shown by our hosts has ensured a relaxed style

for the lectures and on atmosphere of businesslike informality.

The standard of the physics has been very high and the research

work described has often been literally hot from the laboratory.

Many very original new techniques have been described by experiment¬

alists. These techniques, which spanned most of the nuclear physics

touched on in this School, had a simplicity and elegance which has

been the hallmark of really special experiments in the past. Many

of these techniques had been developed in the larger centres but

it was encouraging that smaller laboratories had also made major

contributions. The experimental work described | in this school is an

indication of the richness and breadth of our subject and shows that

there is as much inginuity and talent about as there has ever been.

Nuclear Physics is not only flowering experimentally but it is

making rapid advances in theory as well. The wealth of the experi¬

mental data available has enabled theorists to home in on the es¬

sential degrees of freedom available to nuclei in different mass

881

regions, excitation energies and angular momenta. I was amaged that the clarity and beauty characteristic of the experimental contributions seemed to be an infection caught by the theorists as well 1 I found, to my shame, that several topics which I had with unjustifiable prejudice previously thought were dull and bor¬ ing actually contained very significant and exciting physics.

In short I would like to thank our Polish hosts for their skill in choosing such excellent speakres and the speakers them¬ selves for bringing such an atmosphere of interest and excitement to the subject of nuclear physics.

interest and excitement has not been confined to physics however - and I am not referring to the "tea house of the August moon" located in room 306 ! The excitement started before the school began with the application from on high of a translation operator to the School causing the switch of venue from Zakopane to 3ielsko-Biała. The organizing committee chaired by Reinhaxd Kulessa and also the staff of the Centre here must be congratula¬ ted on performing wonders at such short notice. Unfortunately even such geniuses of organization could not have guessed that the "red hot" physics discussed at our school would raise the local tempera¬ ture so much that it caused an unseasonal melting of the snow after the first weak !

On behalf of the foreign visitors I must say how grateful we are to our Polish hosts for their generous hospitality and for tak¬ ing such care and working so hard to make our time here enjoyable both from the physics and social points of view. Thanks from all participants must go to those in the Institute of Physics at Krakow and the Institute of Physics at the Jagellonian University who have made it all possible. In particular to Dr. Kulessa and his commit¬ tee for their organization, to Drs Styczeń and Stachura for the

. scientific programme and to the many members of the social comnit-

882

tee - particularly to all the beautiful ladies who attended to our every need if not our every want. Our thanks and appreciation must go to the director of the Ośrodek Wdrażania Postępu Technicz¬ nego w Energetyce and all of his charming staff who were always so cheerfully helpful.

This School has not only been a contribution to physics but it has been a further step towards the international friendship and understanding that this world needs so much.

LIST OP PARTICIPANTS

1. D. Auger 2. E. Backe

3. A. Bałanda 4. P, Baumann 5. R. Bengtsson 6. Z. Bochnackl T. H.C. Bohlen

8. H.Bokemeyer 9. E. Bożek

10. A. Budzanowski 11. D.C. Constantlnescu

12. S. Ćwiok 13. M. Dąbrowska 14. P. Decowskl 15. P. DBnaii

16. Ch, Droste 17. J. Dudek 18. R. Dymarz 19. K. Eberhard 20. C.A, Endulesou

21. P. Engelstein 22. S. Frauendorf

23. U. Garuska 24. A. Gófdf 25. A. Gyurkovlcb 26. D. Eageman

I.P.N. Orsay Institute of Nuolear Physics, T.B., Darmstadt Jaglellonian Uni7erslty, Cracow C.R.N., Strasbourg NORDITA, Copenhagen Institute of Nuclear Physics, Cracow Hahn-Meltner Institute for Nuclear Research, Berlin West G.S.I., Darmstadt Institute of Nuclear Physics, Cracow Institute of Nuclear Physics, Cracow Central Institute for Physics, Bucharest Technological University, Warsaw University of Warsaw University of Warsaw Central Institute for Nuclear Physics, Rosendorf, Dresden University of Warsaw University of Warsaw Institute of Nuclear Physics, Cracow University of Munich Central Institute for Physics, Bucharest C.R.N., Strasbourg Central Institute for Nuclear Physics, Rossendorf, Dresden University of Łódź M.Curłe-Skłodowska University, Lublin W.A.T., Warsaw K ,V , I., Gronlngen

885

27. E. Hammaren 28. A.Z. Hrynlciewiez 29. H. Hrynkiewioz 30. R. Janssens 31. L. Jarozyk 32. J. Kajfosz 33. W. Kamlńskl 34. K.Eaun

35. J. Konnieki 36. V.E. Kortavenko

37. J. Kownacki 38. K. Królas 39. R. Kulessa 40. J. Kuimlnski 41. M. Łach 42. G. Leander 43. J. Ludziejewski

44. A. Łukasiak 45. V. Metag 46. S. Micheletti 47. T. Morek 48. A. Moroni 49. W. Nazarewlcz 50. H. Oesohler 51. J. Okołowloz 52. E. Ondrusz 53. V. Paar 54. M. Ploszajczak 55. A. Potempa 56. Z. Prejblsz

57. B.J. Pustylnlk

University of Jyvaskyia Institute of Nuclear Physics, Cracow Jagiellonlan University, Cracow University of Louvaln-la-Neuve JaglelIonian University, Cracow Institute of Nuclear Physics, Cracow M.Curie-Skłodowska University, Lublin Joint Institute for Nuclear Research, Dubna Institute of Nuclear Physics, Cracow Joint Institute I'or Nuclear Research, Dubna Institute for Nuclear Research, Warsaw Jagiellonlan University, Cracow Jagiellonlan University, Cracow Silesian University, Katowice Institute of Nuclear Physios, Cracow University of Lund Institute for Nuclear Research, Świerk, Warsaw Institute for Nuclear Research, Warsaw Max-Planck Institute, Heidelberg Aldo Pontremoll Institute, Mllano University of Warsaw Aldo Pontremoll Institute, Mllano Technological University, Warsaw C.R.N., Strasbourg Jagiellonlan University, Cracow Institute of Nuclear Physics, Cracow Ruder Boskovic Institute, Zagreb Institute of Nuclear Physics, Cracow Institute of Nuclear Physios, Cracow Institute for Nuclear Research, Świerk, Warsaw Joint Institute for Nuclear Research, Dubna

886

58. J. Rekstad 59. I. Hotter

60. 61. 62.

63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73.

74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89.

P. B. K.

M. M. B. E. J. J. C. G. J. Z. P.

w. A. B. J. M. Z. Z. S. W.

Roussel Rozenfeld Rusek

Ryblcka Rzeszutko Sawicka

W.Schmld F.Sbarpey-Schafer Sienlawskl Slgnorlni Sletten Sromicki Stachura Stary

Starzeckl Strzałkowski Styczeń Styczeń Subotowicz Sujkowskl Szymański Szymczyk Śmlałowskl

W.J.G. Thijssen J. M, J. W. s. E.

Turkieiricz Vergnes Vervler Waluś Wiktor Will

University of Oslo Central Institute for Nuclear Physics, Rossendorf, Dresden I.P.N., Orsay University of Wrocław Institute for Nuclear Research Świerk, Warsaw Institute of Nuclear Physics, Cracow Institute of Nuclear Physics, Cracow Institute of Nuolear Physics, Cracow Institute for Nuclear Physics, Jallch University of Liverpool Institute of Nuolear Physics, Cracow University of Padova Niels-Bohr Institute, Copenhagen Jaglellonlan University, Craoow Institute of Nuclear Physics, Cracow Central Institute for Nuclear Physics, Rossendorf, Dresden Institute of Nuclear Physics, Cracow Jagiellonian University, Cracow Jagiellonlan University, Cracow Institute of Nuclear Physics, Cracow M.Curle-Skłodowska University, Lublin Institute for Nuclear Research, Warsaw Institute for Nuclear Research, Warsaw Institute of Nuclear Physics, Cracow University of Wrocław Technological University, Eindhoven Institute for Nuclear Research, Warsaw I.P.N., Orsay University of Louvaln-la-Neuve Jaglellonlan University, Cracow Institute of Nuclear Physios, Craoow Central Institute for Nuolear Physics, Rossendorf, Dresden

887

90. B. Wodniecka 91. P. Wodniecki 92. J. Wrzeslński 93. K. Zipper 94. K. Zuber 95. Z. Żelazny 96. IV. Żuk

Institute of Nuclear Physios, Cracow Institute of Nuclear Physics, Cracow Institute of Nuclear Physics, Cracow Silesian University, Katowice Institute of Nuclear Physics, Cracow university of Warsaw M.Curie-Skłodowska University, Lublin