Theoretical spectroscopy and the fp shell

PHYSICSREPORTS(Review Section of PhysicsLetters)70. No. 4 (1981) 235—314. NORTH-HOLLAND PUBLISHING COMPANY

THEORETICAL SPECTROSCOPY AND THE fp SHELL

A. POVESt and A. ZUKERLaboratoirede PhysiqueNucliaire Théorique,C.R.N. Strasbourg.BP2O. 67037StrasbourgCedexFrance

ReceivedSeptember 1980

Contents:

Introduction 237 5. The (fp)2 interaction andthespectrumof 425c 254

0. Notations 238 6. Comparisonwith exactdiagonalizations 2561. Consequencesof previous(fpr calculationson the vali- 7. Energy levels 260

dity of realistic forces and the use of effective model 8. Binding energies 298spaces 239 9. Electromagnetic transitions and some remarks on the1.1. Validity of the realistic interactions.The monopole wave-functions 300

effects 239 9.1. Electromagnetictransitions 3001.2. The choiceof model spaces 241 9.2. The wave-functions 302

2. Effective interactiontheory 242 10. Thebook of arguments 3042.1. Schematic review of quasidegenerateperturbation 10.1. Threetrends 304

theory 242 10.2. The variousintruders 3062.2. Linked clusterproperties 244 10.3. The threebody effects 3072.3. Non perturbativearguments 245 10.4. Couplingschemes 3082.4. Quasiconfigurations, perturbative dressing, core AppendixA 308

polarization,minimal choiceof model spaces 248 References 3133. f~

12quasiconfigurationsin thefp shell 2484. Energy denominators and higher order perturbation

theory 251

Abstract:The recently developedquasiconflgurationmethod is appliedto fp shell nuclei. Secondorder degenerateperturbationtheory is shown to be

sufficient to produce,for low lying states,thesameresultsaslargediagonalizationsin the full (f7/2p3/2p112f5/2)”space,due to theoperationof linkedcluster mechanisms.Realistic interactionswith minimal monopolechangesare shown to be successfulin reproducingspectra,binding energies,quadrupolemoments and transition rates. Largeshell model spacesare seen to exhibit typical many body behaviour.Quasiconflgurationsallowinsightinto theunderlyingcoupling schemes.

±Addressfrom October1980: Dpto de Fisica TeoricaUniversidadAutonoma,CantoBlanco, Madrid-34.Spain.

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A. PovesandA. Zuker,Theoreticalspectroscopyandthefr shell 237

Introduction

In a recentpaper [1] (hereafterreferredas I) we have presenteda version of quasidegeneratepertubationtheory designedto handlethe effective interactionproblem.The calculationsthatwe aregoing to discusshereneedonly the mostelementaryresultsof the theory andwerereadywell before I.If we havedelayedtheir publication,it is mainly to be ableto drawsomeconclusionsandmakesomeclaimsthat maybe takenseriouslywhenbackedby acompletetheory but soundextravagantotherwise.

Moreover, althoughthe presentstudy is indeed a piece of theoreticalspectroscopy,meant to becomparedwith data, it can also be interpretedas a large scalemodel, illustrating the reductionof amanybodyproblemin the full Hilbert spaceto a smallquasidegeneratediagonalization.

The small space(s)is (are)the f,2 configurations,adequate,as we shall see,to describethe nucleibetween

40Caand ~Ni. Somespecialcaseshavealreadybeenpublishedin ref. [2], which alsocontainsthe first basicideaon which the resultsin I areconstructed.

To summarizesomeof the aspectsof our approachto the problemit seemsadequateto collect in thisintroductionsomegeneralremarksaboutshell model work. As nothingwill be said that will not bebackedor illustratedin the bulk of the text, the tonemaysounda bit dogmatic,all the moreso in viewof somestatementsthat maynot follow currentthinking on the subject.

In dealing with nuclear spectra,two extremesshould be avoided: that of insisting on a rigorousformulationwithout free parameters,andthat of allowing for too manyparameters.In thefirst casetherisk is to losecontactwith the data,in the secondthe risk is to losequantitativecontactwith a rigorousformulation. To strike a useful interactivebalancebetweendata and theory,Occam’s razorcould beinvokedthrough two principles:

(1) Minimal tamperingwith the realisticinteractions.(2) Minimal choice of modelspaces.Concerning the choice of effective interactions, it is clear that the idea to proceedfrom the

nucleon—nucleonphaseshifts to the spectrais the only safeandrigorousstartingpoint. It maywell benecessaryto go beyonda careful Bruecknercalculation:non nucleonicdegreesof freedom,andmanybody forcesmaybe present.However,weareentitled to expectthat theseeffectsbe clearlydemandedby the data and not forced on them.Accordingto our experience,the bestway to detectthe needofsomethingnew and interesting,is to acceptonly thosechangesin the realistic interactionsthat areirrefutably necessary.

Concerningthe choiceof model spaceit shouldbe rememberedthat in all casesthe correctstartingpoint is thefull Hubertspace,andin no casea calculationcould be takenseriouslyif its purposeis notto comecloseto an exactsolution in that space.Datahavethe nice but treacheroushabit to makeusbelievethat a smalldiagonalization,or evenno diagonalizationat all, is sufficient to describemanyoftheinterestingstates.This habit is calledShell Model. The wholeproblemis to understandwhat are thefew states that get diagonalized. In any case they are dressed,sometimesvery heavily dressed,configurations.We call them quasiconfigurationsandit is the businessof aneffectiveinteractiontheoryas the onepresentedin I, to tailor the dresses.

The dressingis a prediagonalization,whoseobject is to decouplethe quasiconfigurationsfrom therest of the space.The initial coupling that is thus eliminated is expectedto be weak enough to betreatedin perturbationtheory,or at mostby truncatingdrasticallya hierarchyof linked clusterintegralequations,whichin lowestapproximationyield Brueckner—Hartree—Focktheory.

If the quasiconfigurationspaceobtainedin first approximationis still very large,afurther reductionmay be possible,as will be shown in this paper by reducingthe (fp)fl spaceto f,

2. Linked cluster

238 A. PovesandA. Zuker, Theoreticalspectroscopyandthefp shell

mechanismswill be operatingagain, that conventionaltruncationsdo not respectin general,therebyinviting severeinconsistencies.The only consistentway of truncatingdemandsthat a given subshellbeentirelyincludedor excluded.Wheneverpossiblethe latter choiceis obviously the best.

In our view, the only reasonto do largeshell modeldiagonalizationsis to learnhowto do smallones.Section0 introducessomenotationsoften used.Section 1 defines the spectroscopicproblem and reviews Pasquini’s unpublishedwork, putting

emphasison its influencein defining the two principlesabove.Section 2 is a commentedreview of someaspectsof I (ref. [1]). We introducethe very elementary

formulaewe shall needand the (not soelementary)interpretativeingredientsthat the resultswill need.Theaim is to makeit possibleto view shell model (perturbative)diagonalizationsas part of the generaldressingprocess,and the fp caseas an exampleof general(nuclear)manybody behaviour.

Section3 gives the explicit form of the effectiveinteractionfor ffl quasiconfigurations.Section4 exploressomeconvergenceproblemsandclarifies the fundamentalrole of a good choiceof

energydenominators.Section 5 is a brief commenton the interaction.Section 6 containsa comparisonof a few exactdiagonalizationwith the quasiconfigurationresults.Section 7 is a detailedpresentationof the energyschemesfor all ffl nuclei. Someargumentsemerge.Section8 dealswith the binding energies.Section9 dealswith transitionsprobabilitiesin someselectednuclei.Section 10 providesan overall view of the results,which turn out to havesomeinterestthat goes

beyondthe good or bad agreementwith the data.Appendix A gives a derivationof the generalizedparticlehole transformin the caseof threebody

dressing.

0. Notations

fp meansf712p312p112f512,f alonemeansf712, r (restof the shell) is a genericindex for p312p112f512.Configurationor rn-configuration: [ni, n2, . . .]. Set of stateshavingn1 particlesin orbit 1, n2 particles

in orbit 2 etc. n1 + n2 + — n. mT-configuration:[n1T1. n2T2, .. .]. Subsetof a configurationin whichthe wave-functionsof individual orbits arecoupledto good isospin T1. The whole is recoupledto goodisospin T,

[J] = 2J + 1, F= fT. [1’] = (2.1+ 1)(2T+ 1) (_)F = (_)J+T

W~= W~= normalizedantisymmetrizedtwo body matrix elementfor the arbitraryorbits rstu. Alsocalled (rsfVjtufT).

f = 0(x) meansthat f is at mostof orderx.

a~,a5 are the usual creation and destruction operatorsfor orbit s. When working in coupled

representationwe use French’snotation= a ~ B55. = (~)s+sza

A. PovesandA. Zuker,Theoreticalspectroscopyand thefp shell 239

1. Consequencesof previous (fp)N calculations on the validity of realistic forces and the useof effectivemodel spaces

Traditionally, nucleiwith 20 � Z < 28 andN � 28 havebeendescribedby assumingthe f712 orbit tobe partially filled with particlesinteractingthrougha forcedefinedby theexperimentallevelsin mass42dominatedby the f~,2configuration[3,4, 5].

Undertheseconditionsnucleiwith given numberof particlesaddedto40Ca havethe samespectrum

as nuclei with the same number of particles subtractedfrom 56Ni (cross conjugate symmetry).Furthermore,addingprotonsto ~Ca shouldproducespectraidenticalto that of the Ca isotopes.

Thefollowing list is representativeof the successesandfailuresof the simple ffl model.(i) Somecrossconjugatenuclei,e.g. ~‘Ti and50Cr havestrikingly similar spectra.Othersdo not, e.g.

47Ti and49Cr.(ii) The spectraof 50Ti and 42Caare expectedto be the sameandthey are,but this is not a success

but a puzzle: the0~and2~f2 levelsin 42Ca areknownto bestrongly mixed andthusshifted,through4p2h intruderswhich areabsentin 50Ti.

(iii) The BE2ratesfor the Yrast levelsarenicely given by the f~,2modelprovidedsome(ratherlarge

but not absurd)effectivechargesare used.For otherstatesthe situationis not sobright: the branchingrate

R — B(E2,4e~Ti,2~T~*2~)

— B(E2,50Cr,2~-+2t)

shouldof order 1 in anyf~,2calculation,independentlyof the force used,providedthe effectivecharge

for neutronsandprotonsareof the sameorder.ExperimentallyR � 25.(iv) The quadrupolemomentsin the f” model are alwayspositive. Experimentallythey are always

negative.(v) By and large, spectraobtained in the f~,2model are very good, but sometimesthey show

substantialdiscrepanciesfor low lying levelsthat are difficult to reconcilewith the good positioningoftheir neighbours.This is the case,for example,in the2~stateof ‘°~Sc(the Pandyatransformof the

42Scspectrum).

Theseproblemswere solved,at least partially, by configurationmixing calculationsof PasquiniandZuker [6,7] which includedall the orbits of the (fp) shell. Reference[61is unpublishedbut ref. [71containsa comprehensivesummary.The specific aspectsof f” spectroscopywill reappearthroughoutthe paper.The purposeof this section is to examine two questionsraised by the calculationsthatsuggestedfurtherstudy: the validity of the realisticinteractionsand the choiceof model spaces.

1.1. Validity of the realistic interactions. Themonopoleeffects

An interactionis called realisticif it is the resultof a BruecknerG matrix calculationwith a potentialV that reproducesthe N—N phaseshifts.

W,~ (rslGjtu) = (rs( V]tu) + ~ ~rslVIaI3Xa$IGItu) . (1.1)

,~,3

The orbits rstu arein the modelspace.The intermediatestatesa/3 involve unoccupiedorbits. The single

240 A. PovesandA. Zuker,Theoreticalspecroscopyandthefpshell

particleenergiese arethoseof some“suitably defined” singleparticlefield. The startingenergyÔE is aconstantthat may either summarize “off shell” effectsor be taken as an auxiliary quantity in thedefinition of the singleparticle field. It is the latter interpretationwe shall need.(Explanationscan befoundin I or in refs.[8—111.)

In eq. (1.1) the matrix elementW,~,to be usedin a shell model calculation is identifiedwith thecorrespondingG matrix element.In practice,somefurtherdressingof this barevalueis necessary,thecorrespondingprocessesand theunderlyingassumptionthat G behavesasa smoothpotentialwill beexaminedin section2.

The calculationsof refs.[6,7] weredonewith threedifferent realisticsetsof matrix elements:Kuo—Brown [8]using the HamadaJohnstonpotential (KB).Kahana—Lee—Scott[9] using theLee potential (KLS).OberlechnerandRichert [10]using the Oberlechnerpotential (OR).OberlechnerandLee usesumsof two termseparablepotentialsthat arepractically identical.Theyareobjectionablein that theydo not havethecorrectlong rangebehaviour(OPEPtail) but they allow veryprecise G matrix calculations. The Hamada—Johnstonpotential has a good tail but forces ap-proximationsin the G matrix. In spiteof thesedifferencesKB andKLS producematrix elementsthatare spectroscopicallyindistinguishable(see fig. 7 of ref. [7] for an example).To exhibit the mainlimitation of the realisticforcesit is convenientto separatethemonopolepart of theHamiltonian:

HmT (~~)a~+ (nt) ~ T~)2—T~—T~]

V,~= (1 + 6~~)-’(~[J]) ~ W~[J](1 — 55t(—y~

T); (~)= ~ n(n — 1) (1.2)

a=~(3V1+V°), b= V1— V°,sameforss,standtt,

which gives the averageenergyof the mT configurationsfor two shells s and t, and generalizestoseveralshells.Notice that n

5 and i’~areoperators.If only m configurationaveragesare necessary,theHamiltonianis

Hm = (~~5)Vss+ (~)V~:+ flsfl:Vis

-t (1.3)V,.,= (1 +o~~)~

1(~[F]) ~ [fi w~(1—

Hm andHmi~can be thoughtof asgeneralizationsof the French—Bansalformulae [12].The fact theygive configurationsaveragesis not entirely trivial [6,13]. The calculationsin refs.[6—7]show that thefollowing modificationsof the interactionsarenecessary

W~(KB)- W~(KB)- (_)T 300keV(1.4)

WI~(KB) -+ similar changes.Seesection5.

Thesechangesare strictly monopole,they do not dependon ther orbit (samefor pi,2p312f512)and they

are strictly identical to thoseneededin KLS. TheOR interactionwascalculatedwith smallerstartingenergyÔE in eq. (1.1), it leadsto greateroverall attractionneedingslightly larger/smallerchangesin the

A. PovesandA. Zuker, Theoreticalspectroscopyandthe foshell 241

T= lIT = 0 centroIds. Qualitatively the situation is identical. That the monopoleeffect is of over-whelming importancecan be gatheredby comparingtheconfigurationsf~%andf~r4 in ~Ni: after theVtr centroldin eq. (1.3) is changedaccordingto (1.4) the 4p 4h stateslose 10 MeV with respectto theclosedshell andbecomeexcited,startingat 5MeV insteadof dominatingthegroundstateandhavingan excitedclosedshell at 5 MeV.

Thatotherchangesarelesssignificantcanbeseenfrom the fairly good aspectof thespectraoncethemonopole behaviouris corrected[7]. A more careful search should unearthsome small beneficialmultipole (nonmonopole)effectsthough.

Mostof thebinding energyof a nucleuscomesfrom monopolecontributions.Theyareglobal effects,that cannotbe expectedto be right if thepotentialsdo not bind nuclearmatterproperly.The presentstateof theart [14]indicates that the Reid potential is better than Hamada—Johnstonin this respect.However, lowest order Bruecknertheory is not sufficient and it seemsincreasinglyclear that morerefinedapproximationswill improve thesituation but probably fall shortof reproducingthe observedsaturationpoint if true three body forces are absent.It is reassuringconceptuallythat the globalpropertiesof nuclei show in the spectraquite massivelydue to the amplifying effect of the numberoperators.For other multipoles, the amplification is much smaller and it is difficult to distinguishbetweendefectsin the potentialsand defectsin the diagonalizationsdue to poor definition of theeffective interaction (dressing). Some indications come from nuclei, such as ~Sc, which are verysensitive to certain off diagonal matrix elementsand are reproducedto within much better than100keV/level by KB and LKS without any modificationsbeyond (1.4). From such examples,it istempting to concludethat thereis little point in tamperingwith the non monopole behaviourof therealisticinteractions.

The useof betterpotentialsandmoresophisticatedtreatmentsof theenergydenominatorsin the Gmatrix is certainly warranted[11], but our remarksindicate that significant progressdemandsacomprehensiveview of the dressingprocessto include on the samefooting whateveris necessarytohavegood saturationproperties(i.e. good monopoles)andgood multipole details.

Minimal tamperingdoesnot meanthe realistic interactionsare well as they are. It meansthatphenomenologicaladjustmentsmust be sufficiently indispensableto teach us somethingbeyond thecosmeticimprovementthey may bring to thespectra.

It is worth noticing that for the p andsdshells,theKB andLKS interactionsseemto needrelativelyminor modificationsbeyondmonopoleeffects(that includeup to at least4 body forces)[15].

1.2. Thechokeofmodelspaces

The calculationsin ref. [6] reveal that most low lying statesare dominatedby the ground stateconfigurations:f”, f”r” (e.g.50Sc)or r” (Ni isotopes).Beyondthe Ni isotopesthe g

912 orbit comesdownatsuchaspeedthat it becomesdifficult to speakof fp spectroscopyalone.

In all cases,the addition of the next configuration(f”1r, r’r”~1, r”~1r1)brings greatbenefits.

There are many excited levels dominatedby thesestatesand in somecasestheir inclusion in thediagonalizationmakesit possible to explain or predict some interestingdata such as the massiveappearanceof 1~levelsin 50Sc and theexistenceof a 0~state(f~p

312)practicallydegeneratewith the“pairing vibration” (to all intents and purposes,an f~p~,2state) in ~‘~Ca. Far more interesting,however,is the dressingeffect of the newconfigurationson thegroundstateones(f”), which makesitpossibleto solve,atleastto a largeextent,the typeofproblemsraisedin points(i) to (v) above.To explainthis mechanismwe considerthe Ni isotopes.

242 A. PovesandA. Zuker,Theoreticalspectroscopyand thefp shell

A core, evenif it is a good “closed shell”, is neverinert. Addedparticleswill coupleto its particlehole degreesof freedom.In 160 and40Ca thesestatesare at someexcitationenergyof order 2 hw atleast (the giant resonances).In 56Ni, which is as good a closed shell as 160 and40Ca, thesestatesarepresentandare expectedto producesimilar effects (e.g.effectiveB(E2) charges).However, therearealso0 ho) particle holeexcitations(the 2~of ~Ni is at 2.7MeV). Couplingto them will be muchstronger(10—20% mixings) than to the giant resonances(1% mixings), but still weak enough to suggestperturbative treatment. These couplings involve admixturesof r3f’ components in dominant r2wave-functions. As we shall see, the suggestedperturbativetreatment is possible but it involvesintroducingthreebody forces and two body transitionoperators.Increasingthe size of the spacestoinclude moreconfigurationsturnedout to be a frustratingoperation.Most of the benefitswerewashedout or disappearedunlesstruly enormousincreaseswereallowed.The reasonscan beunderstoodeasily.Assumewe do some ffl + f~r+f”2r2 diagonalization.Strong pairing matrix elements((f~

12(VIf~,2(01))mainly) will be allowed to depresssubstantiallyf” but r’r will be left untouched.If we now movetothe f” + f?~~r+ f”

2r2 + f~3r3space, the diagonalizationwill producevery much the same that wasalreadyobtainedin the enormouslysmallerf” + f”~rspace(agood exampleof thisis 56Ni, seeref. [7]).

The only usefulindicationsprovidedby the largerspacesconcernedthe slow but efficient build up ofB(E2)strengthfor the lowesttransition.With KB, a calculationin 56Ni givesB(E2, ~ -~ O~)= 8 fm4 e2(effective chargeof 0.5 for the neutrons)for the r2 space.Whenr3f’ is addedthe ratejumpsto 55 fm4 e2andit becomes132fm4 e2 in an exact(f

712p312)’8 diagonalization,which maybe of somerelevancefor the

two statesinvolved in the transition (the experimentalvalue is 140±10 fm4 e2) but of none for theothers.The only interestof this exampleis to showthat the basicmechanismbeyondthe improvements,is the coupling to ph degreesof freedomandthat a very largediagonalizationmayprovidesomeaddedcoherenceto it.

The otherthing a largediagonalizationdoes,is to inducesomuchmixing thatoneis left wonderingabout the possiblemeaningof the ideaof dominanceof the groundstateconfiguration,which remainsby far the largestcomponentbut becomesa small fraction of the total wave-function.

The indicationsthat emergedfrom the work of PasquiniandZuker were quiteperplexing to a shellmodelphysicist: configurationmixing was strong andled to oversizematricesbefore the benefitsof anobviously perturbativemechanism,apparentin minimal diagonalizations,could be recovered.Clearly,therewas somethingwrong in the way the problemwas beinghandled.

2. Effective interactiontheory

2.1. Schematicreviewofquasidegenerateperturbationtheory

We shall make precise the notion of dressing. It is convenient to start by reproducing someelementaryresultsfrom I.

The full matrix representingtheeigenvalueproblemwill be written in block form:

Bi + (Il i/JI) (‘I %‘lJ)

(II VII’> _____________Ej +(JIVIJ)

(JIW> (JIW’>

A. Povesand A. Zuker, Theoreticalspectroscopyand thefp shell 243

The Hamiltonianis

H=H0+V. (2.1)

The diagonalmatrix H0 containsthe e terms.The natureof the splitting betweene~and(IJVJI) (samefor .1) will bediscussedlater: H0 is not necessarilyonebody. The statesI/f belongto themodel/exter-nal space.Capitals will alwaysbe repeatedand summedover. Lower caseletters i or / will refer toindividual membersof the i-space(model)or i-space(external). — —

The aim of the theoryis to decouplei and/ spaces.Let’s considernewstatesi andj definedas

(1 = (.1÷(iIV1JXJI

i 1) = — IIX1I %‘li> (2.2)SI’

S,j = S — Ej.

Then

(tl1)=0 (2.3)

(11HIJ) = (ii V1J>[1 +5ç5:] + O(W21s)= O(W2/e) (2.4)

wheref = 0(x) meansthatf is at mostof orderx, i.e. If Ix~<constant,and Wis sometypicalvaluefor(I~V]J), while s is sometypical valuefor s~. — —

Equations (2.3) and (2.4) indicate that the new i and /-spacescontain statesthat are strictlyorthogonaland that their matrix elementshavebeenreducedfrom 0(W) to O(W2/s)with respecttotheoriginal blocks. Inside thespacesorthonormalityis manifestlyviolated.To restoreit we redefinethestates:

(11 = ~+ (ii vlJ)(JI — ~(iJ %IJXJI V1’X’I2 SjS~j (2.5)

_) — IIXII VJ/) — ~IJ)(JIvIIXII ~‘1j~ 2 8Ij8Lr

(lit’) = ô~.+0(W4/r4), (/11’) = ö11+ 0(W

41e4) ~2.6)

(l1i)=0, (l]HIJ)=O(W2/s) (2.7)

(TIHI T’) = (iIHIi’> + .1(11V1JXJIVIi~[1.+8i1 Si’J

~1(iI VJIXII vlJXJI vji’) — 1 (iivJJ)(JI VJIXII Vu’> 2 82 s~

1s1., 2 eLreIJ ( . )

+ (ii V1JXJIV1J’XJ’I %‘li’> + 0(W4/s3).

244 A. PovesandA. Zuker, Theoreticalspectroscopyand thefp shell

Transformation(2.5) conservesthe importantpropertiesof orthogonalityanddecouplingof eqs.(2.3)and(2.4) as shown by eq. (2.7), but it doesnot restorefull orthonormality inside the blocks (eq. 2.6),although thesituationis “improved” with respectto what would obtain with states(2.2).

To chooseatransformationsuchas(2.5) ratherthan enforcestrict orthogonalityis a matterof greatsubtletyextensivelydiscussedin I. Someheuristicargumentsaregiven below.

2.2. Linkedclusterproperties

Considera non degeneratesystemfor which theexact groundstatehasawave-functionof the form

IO) I0)+AajIai)+Aasj,Iaf3i/>+(2.9)

= 0>+ A(l)Iph> + A(2)12p2h)+...

The state 0> is a closed shell, a,f3,... etc./ij,... etc. orbits are above/belowthe Fermi level. TheamplitudesAa,~.. . ,~... (generically:A(n)) areantisymmetricin the particleandhole indexes.

The amplitudesA(~)havethe following property

A(2)= A(lA(l)+A~) (2.10)

A(s)= A(l~.AwA(J)+ A(l~A(2)+ A~)

etc.

whereA~,)is linked, i.e., irreducibly non factorable.The factorizationsindicatedin (2.10) haveto be done in all possibleways, rememberingthat A(~)isantisymmetric,avoiding doublecounting andrespectingthe conservationlaws relatingthe particle andhole indexesof A(~).(SeeI.)

Equations(2.10) areaprofoundtheoremfirst discoveredby HugenholtzandHubbardin the contextof linked clusterperturbationtheory.It is however,independentof perturbativearguments(for a sketchof a simpleproof seeI, [17] containselaboratedetails).

Theideaof linked cluster(not necessarilyperturbation)theory is that thelinked contributionsto A(~)becomesoonmuchsmallerthan thefactorable(unlinked)ones.As aconsequence,a knowledgeof veryfew linked amplitudesA~,)becomessufficient to approximatethe full wave-function, i.e. includingmp—mhcomponentswith arbitrary m. (Technicalremark: in the presenceof hard core potentialsallamplitudesmust contain a fully linked componentto insure properscreening of the core in thewave-function. This difficulty is more conceptualthan practicaland in our caseit can be ignoredaltogetheras we shall be concernedwith the consequencesof linked cluster ideasin large but finitespacesin which the interactionhasalreadybeenscreenedandis definedby finite matrix elements.)

Orthonormalization(2.5) hasto be interpretedsymbolicallyas insuringthat (11 in eq. (2.2) could betreatedas “that part of the wave-functioncontaininginformationabout the amplitudesA(1) andA(2)”and not as a ip lh + 2p 2h approximationto the wave-function.

If a ip lh + 2p 2h approximationis to makestrict senseit is necessarythat

(i1i>= 1+I(iIVIJ>J2/e~,~1

(2.11)~ (il VIJ>I2/e~~ 1.

A.PovesandA. Zuke,~Theoreticalspectroxopyandthefo shell 245

Transformation(2.2)on the otherhandmakessenseprovided

(ii Vi J)/e~~41 Vj. (2.12)

The~ sign in (2.11)is unnecessaryin view of thesummationconventionbut is introducedto emphasizethe differencebetweenthe weak condition (2.12) and the strong condition (2.11), which are all thedifferencebetweenRayleigh—Schrödingerandlinked clusterperturbationtheories.It is ironical that thisdifferenceboils down to a matterof interpretation:if condition(2.11)is strictly enforced,normalization(2.5) is strictly correct. If (2.11)is relaxedin favourof (2.12),eq. (2.5) is symbolically correct. In bothcases(2.8) is the approximateresultto third order in the energy.

Thereis nothing symbolic in eq. (2.2) which is indeeda good decouplingoperationprovided it isunderstoodthatsomerecouplingalsotakesplace,ascanbe seenby examiningcarefully eq. (2.7).In theoriginal matrix, (il mixed with matrix elements0(W) to ip lh and 2p 2h states.After (2.2) (11 mixesO(W2/s)with the sameip lh and 2p2h states.In additionit mixesto thesameorderwith 3p 3h and4p 4h states,and thereare many more of these(at least in the many body case).The clue to thesymbolicmeaningof eq. (2.5) is that this extramixing carriessomenew information (thatwill go intoA~)andAu)) but also much that is known, e.g., going from a ip lh stateto a 3p3h stateis to a largeextentthe “same” asgoing from a OpOh to a 2p2h state.The things that arethe “same” contributetheA(

2,A(l)factorsto the A(3) amplitude.The orthonorinalizationin eq. (2.5) is chosen to preservethe structure(2.9) and (2.10) of the

wave-function. Strict orthonormalizationwould condemnthe wave-functionto a strict ip lh+ 2p2hstatusandforfeit insight into the higher amplitudes.

So far only thenon degeneratecasehasbeenconsideredasa particularapplicationof theequationsin section2.1 which describea generalquasidegeneratesystem.In view of the obviousformal analogybetweena wave-function and a dressedstate, it is quite plausible to conjecturethat much of theconfigurationmixing observedin large matrix diagonalizationsis the consequenceof hidden linkedclustermechanisms.Somepossibleexampleswill be shown in section4.

Somewordswill be saidnow aboutamoregeneralview of the decouplingof spaces.

2.3. Non perturbativearguments

The underlyingassumptionthat Wis is a smallnumberwill breakdown if the potentialis singular.Since our object is to formulate a unified prescriptionfor fully decouplingthe i-space(dressingthei-spacewith its full wardrobe),we haveto understandhow to dealwith this problem.It can be doneconvenientlyby introducingthegeneraltransformation

(11 = (il + (iIiIJXJI (2.13)

Ii> = Li> — II>(II VI,>. (2.14)

Here ~ is a (manybody)operatordefinedbetweenany two statesi and/. It is clearthat (1] and~J)are

manifestlyorthogonal.To imposeperfectdecouplingwe request

‘~liHti>=o (2.15)

which leadsafterelementaryalgebrato

246 A. PovesandA. Zuker, Theoreticalspectroscopyand thefp shell

e1(iI~’IJ>= (il VII> + (iI’ciJXJI VII) — (il VII)(IIi’I/) — (iji’IJ)(JI VJI>(IIi’[/) (2.16)

which is an integralequationfor the operatori~.It can be solvedby Neumannexpansionor by nonperturbativemethods.

Assuminga solutionof eq. (2.16)hasbeenfound, the i-stateshavebeencompletelydressedandtheyinteractthrough an effectiveHamiltonianwhich can be easilydiscoveredby relying on the existenceofa naturalbiorthogonalbasis

(2.17)

which makesit possibleto write

(l]IIIi’> = (iII~Ii’>+(ili’IJ>(JIVli’>. (2.18)

Since the i-stateshavenull matrix elementswith the rest of the full space(2.15), solving the secularproblemdefined by eq. (2.18) producesthe eigenbrasof the system.The non Hermiticity of matrix(2.18) meansthat the eigenketsare not the conjugateof the eigenbrasbut solutionsfor the conjugatematrix

(iIIIIi’> = (iIIIIP)+ (iJ VIfXJIiTIi’). (2.19)

An explicit reductionto Hermitianform is given in I.A transformationsuch as (2.13) and (2.14) is always possibleand always leadsto the exact equations(2.16) to (2.19). To give it the senseof a “dressing”dependson physicalconsiderations:state(ii in eq.(2.13)mustbe “primus inter pares”.Its amplitude(1 by definition) mustbesubstantiallylargerthanthatof any otherindividual component:

(iJiTIj) ~ 1 Vj. (2.20)

Otherwise,it is difficult to speak of a given state/ as dressing i, rather than the other way round.Condition (2.20) is weakerthan condition (2.12) anddefinesimplicitly the parametersthat haveto besmall to guaranteethe existenceof a dressingprocess.

Eq. (2.16) is generalandexact but it dependson condition(2.20) to be useful.The flexibility in thechoiceof model spaceallows to extendits rangeof applicability. We considersomeparticularsituationsthat amount to a coarseclassification.

If the model spacecan be reducedto one stateandeq. (2.20) holdswe havea normalFermi system.The single stateneednot be a closedshell or aFermi sea.It maywell consistof alinear combinationofSlaterdeterminantscoupledto good angularmomentumandisospin: i.e. a memberof a jfl configura-tion, the mostnaturalextensionof the ideaof normalsystemto openshell nuclei (seesections4 and6).

The breakdownof condition(2.20) for a single modelstatemayoccurin threedifferentways:(a) accidentallythroughlevel crossing.Two weakly interactinglevelsthat would normally contribute

to the dressingof each other may happento be sufficiently close in energy to producea strongadmixture.

(b) Statistically through line spreading.Some interesting(simple)statethat at low excitationwouldhavebecomea good eigenstateafterdressing,finds itself at higher energy,surroundedby a crowd of itsdressingpartners.It mixes strongly andacquiresa width (F’ in jargon).

A. PovesandA. Zuker,Theoreticalspectroscopyandthe/pshell 247

(c) Coherentlythroughcollectivity. This is a generalizedbreakdownof condition (2.20).Many statesin the representationthat is being used have equalclaims to priority and they haveto be treatedsymmetrically.The situationforces either introductionof a model spaceor changeof representation.This is thecasefor superconductivityor rotationalbehaviour.

The notion of model spaceand its implementationin eq. (2.16) are naturally designedto handlesituations(a) and (c). Line spreadingneedsan extensionof the theory.Heavyconfigurationmixing maybe presentin all casesanddoesnot imply theexistenceof collectivemotion.

Normality is associatedto an underlyingrepresentationof non interactingFermions.In the caseofsuperconductivityand rotational motion this picture can be recovered,at the price of violatingconservationlaws, by introducing Bogoliubovquasi-particlesor Nilsson orbits. The operationamountsto definea modelspacesincethe basicSlaterdeterminantis nowan intrinsic statestandingfor a bandof levels: degeneracyis presentin aseeminglynon degenerateproblem.Exactly the oppositehappensin aj” configuration:non degeneracyis presentin a seeminglydegeneratespace.

When(i~is restrictedto a singleSlaterdeterminantof deformedor numberviolating orbits,eq. (2.16)will (obviously!) takecareof restoringthe conservationprinciples.

A completetheorywould start by recastingeq. (2.16) in a manybody languageand thensolve tosomeapproximationthe resulting setof (manifestly linked!) non linear integralequations.Zabolitzkyhassucceededin giving numericalsolutionsfor non degeneratenuclearsystems,in the frameworkofthe expStheory [16].A relatively simple diagrammaticversion of the equationsfor this casecan befound in I.

The extensionof the equationsto the generaldegeneratecasehasnot beengiven so far, but somepreliminary remarksarepossible.

To fix ideas,we can think of the sdshell as the modelspacein which we arepreparedto diagonalize(sd)~matrices.The equationssuggestthat the two cores,160 and‘10Ca be treatedas particularmodelstateswith n = 0 and n = 24. Let us assumea good solution for 160 hasbeenobtainedthrough aselfconsistentBrueckner—Hartree—Fockcalculation.The correspondingG-matrix is then the effectiveinteraction for n = 0. Among other things, the dressingprocessin the presenceof sd particles,mustproducethe necessaryadditivesto thisoriginal G-matrix, so thatit becomestheselfconsistentG-matrixfor 40Caby the time the n 24 state is calculated.Among the additivestheremust be operatorstochangethe Pauli projectorsin the intermediatestates,andto ensurethe size readjustmentin the singleparticleorbits (in 160, hw 14—15 while in 40Ca,hw 10—12). This size effect is the most obviouswayto understandthat the two body matrix elementsneedan n-dependence,i.e. that effectivethreebodyforcesmustbe present.But thereis moreto it: Zabolitzky’s resultsalso showthat a goodG-matrix isnot enough,Bethe—Faddeevcorrelationsare also necessary.In our languagethis meansthat theeffective force has threebody ingredientsfrom the start (see I). And thereis worse:oncethe fullcalculationsin 160 and 40Ca are finished, little doubtscan remain that a true threebody force is alsonecessary[16].

Evenif we are not preparedto pay theprice of a full calculation up to Zabolitzky’s standardstoobtain solutionsof eq. (2.16) in the generaldegeneratecase(assumingwe were capable!),someveryuseful phenomenologyemerges.It has been shown by Cortes and Zuker [15] that a successfulcalculationin the sd shell demandsat leastfour bodyinteractions(effectiveor otherwise).Ourremarksamountto saythat this phenomenologyhassomefundamentalroots.

So far wehaveonly mentionedmechanismsthat will affect themonopolebehaviourof theeffectiveinteraction:in a closedshell the only operatorsgiving a non null expectationvaluearethosepresentinHm (eq. (2.3) extendedto manyshellsandmanybody forces).Whenparticlesareaddedto thecore,the

248 A. PovesandA. Zuker,Theoreticalspectroscopyandthe/pshell

global behaviourof the systeminvolves non-monopolepolarization effects. As they haveproduced

morethan their shareof headaches,a word will be said aboutthem in thenext subsection.2.4. Quaskonfigurations,perturbativedressing,corepolarization, minimalchoiceof modelspaces

Themodelstatesi will be takento be configurationsandthe correspondingdressedstatesi will becalledquasiconfigurations.

Integralequationsaresolvedin configurationspacewhile shell modelmatricesaresolvedin aspaceof (quasi)configurations.

The problem takenas a whole is always one of diagonalizinga large matrix, only the methodschange.There is oneinterphasemethod: perturbationtheory. It is expectedto makesensewhen thesingularitiesof the forcehavebeenbluntedsufficiently by the integralequationsor when thematrix hasnot yet reachedthe minimal dimensionsthat force a numericaldiagonalization.

Thereis a differencebetweena numericaldiagonalizationthat is allowed by computationalmethodsandonethat is forced by physicalcouplingschemes.

Corepolarizationis the mechanismthat couplesinternalstateswith coreexcitations.In spiteof longefforts, it is not clearyet, at leastin our opinion, whethercore polarizationbelongsto the realm ofintegralequationsor perturbationtheory. It seemsplausible,if not evident, that monopolepolarizabil-ity is a global effect relatedto selfconsistency(seethe last paragraphsof section2.3 and ref. [17]).Forthe othermultipoleswehavenothingto say.Nor do we haveanythingintelligent to proposeconcerningthe choiceof modelstates.Exceptthat it is uselessto do numericallywhat can be doneperturbatively.

3. f~quasiconligurationsin the fp shell

Given an interactionadequatefor full (fp)” calculations,we shall dressthe f~,2configurationsto workin amodelspace.

The Hamiltonianis

H = ~ r~n,— ~

(<U

r

A — + — / \r+r~ .‘ \ — IC C— a ,,.~, ,,~ — ~) ~ ~rstl4)— ~17f215/2P3/2P1/2

~rs= (1 + 8rs)_1~l2, r —JT, I fT (...)T = ()J+T, [I] = (2J + 1)(2T+ 1).

The f712 shell will be called I and r will standgenericallyfor p112p312f512.Thefirst of eqs.(2.2) reads:

(11 = (il + (il V1IQ>(QI + (il V2IPXPISiP (3.2)

i ~f”, q ~f”~r, p —f”2r2.

The notationi, p, q (or I, P, 0) includesall quantumnumbersnot explicitly shown.The denominatorsin (3.2) shouldnot be confusedwith the singleparticle energiesin (3.1). Only matrix elementsof typeWffrr (Wfffr) contributeto V

2(V1).

A. PovesandA. Zuker,Theoreticalspectroxopyandthe/pshell 249

Equation(2.8) is now

(JIDIHIJeQ)= (fWIHlf2fl> + ~ (.J_.+_i_)(f7t~IViIf”~r(fl))(f”~r(f1)IV1lf~(fl))

+ ~ (-~— + —~—) (f~~IV21f” r(12)X1r2(fl)IV

2[f”(fl)) + O(V3/s2) (3.3)

where

V1 = — .—~=~

v2 (3.4)

= (.f(F)l VIfr(F))

and

V2 = ~ [flV2Wr[(AA)r(BrBry]O

(3.5)W~= (f~(F)~VIr2(F)).

Thefirst stepis to eliminateall referenceto the r-spacein expression(3.3) andit can be achievedbyusing closurein the intermediatestatesat fixed IT I throughthe following generalexpressiondue toFrench[14]

~ (Kfl~~S~~f~ = [~]1/2 ~ ~ WvXKI1II(S°’X TA)~Ii~[1~)(3.6)

whereK, K’ andy arenontensorialquantumnumbersand

U(flufI’A; wv) = (_1) fl’+A[]h12{1l U

In our caseU it = p = 0 andexpression(3.6) reducesto the form

~ (dil IS~IIyIi)(yJZ IT°I1K ‘/1> = [f~]112(KflII[S° x T°]°~fK ‘(2). (3.7)

To derive(3.7) from (3.6) is to kill mosquitoeswith sledge-hammers.Obviously, this sum canbe madeonly if our energydenominatorsdo not dependon the index y.The r-operatorscanbe isolatedby recouplingandcontracted.The resultis:

(!71H117)= (f7IH+~HRl+~H~If~) (3.8)

with

HR1 = — ~ (_ 1)’ ~(I~j~l)aWfWf’[((AA~B)’(A(BB~’y]° (3.9)

250 A. PovesandA. Zuker, Theoreticalspectroscopyand the/pshell

HR2 = — ~ ~ [fl”

2(Wc)2[(AAy(BB)T]° (3.10)

I.i(I+L’~ L~(I+i~ (3.11)e~ 2 \E,~ Ej’~/ S2 2 \Egp E~pJ

if now we write HR1 in normalorderwe get

H — (2) i (3)R1 RI~ RI

where

H~1= —~ [I]”2(Wf)2[(AA)T(BBY1° (3.12)

H~J= ~ (_1y_r_r’[II~]hi’2w~wf’~ (~1)A[A]L~2{i~i, ~ [((AA)l~A)A (B(BBY)A]o. (3.13)

Hence,the matrix elementsbetweenquasiconfigurationscan be calculatedthrough the introductionof the effective Hamiltonian

.Heff = 11R1 + HR2~1

in the configurationspace.As it was our goal, all referenceto the r-spacehasdisappeared.The lp—lhdressinggives rise to two+ threebody renormalizationswhile the 2p—2h contributesonly to the twobody part.

Thereis anotherway to do things that allowsa finer sumover intermediatestates.We sketchit onlyfor 0 statesfor which we write the internalcouplingsexplicitly

f~r

First we usestandardangularmomentumalgebra to separatethe r-spacein each of the matrixelementsin eq. (3.3). Thenwe contractout the correspondingoperators(use (3.7)). We areleft with anexpressionof type (3.6) involving only f operators.The intermediatesumscan now be carriedout atfixed ~u,which allows to keepenergydenominatorsthat dependon the internal recouplingsof theintermediatestates.The operatorin eq. (3.9) becomes

{[(AA)~By[A(BByy}°—~ ~ [Qr] I/2(...)fl+r_~o [~1112U(flrfir, o)p){[AAIrBIr[A(BB)r1rY (3.14)

The price to pay is obvious:our Hamiltonianis no longer a manifestscalar,andit can only be usedinmatrix elementsbetweenstatesof good/1. Notice that if we sumover w in eq. (3.14)the arrow becomesan equalsign. We shall seethe conceptualimportanceof this resultwhen discussingthe denominatorproblem.

A.PovesandA. Zuke~~Theoreticalspectroxopyand the/pshell 251

Technicalnoteon numericalaspectsAngular momentumandWick algebraaregoodfor humans.Computerspreferuncoupledrepresen-

tationsandcarrycontractionsin theirown way.Eq. (3.13) is useful in analyticwork (seeAppendix A) but numerically it is better to programme

directly eq. (3.3)(or similar onesfor more shellsor higherorders)in the rn-scheme.The intermediatesumsdisappeartrivially and thecontractionsare simply doneby takingmatrix elementsfor n = 0, 1, 2etc. to obtainthe 0, 1, 2 bodyoperators.If fZ3 is takento beacoupledstate,thematrix elementsin eqs.(3.12) and (3.13) are generatedautomatically, with the addedadvantagethat they correspondtonormalized,orthogonalizedstates,anoperationthat remainsto be donein (3.13).

Coupledr statesare simply obtainedby diagonalizingsome operator,the most useful one, wefound, is thebaretwo bodyHamiltonian (H in (3.8)).

4. Energydenominatorsandhigherorderperturbationtheory

Let usconsiderthe third order termsin eq. (2.8), which we rewrite for convenience

~1(iI vliXiI V1J)(JIVu’) ~1(iI vlJXJIVIIXII Vu’)+ (iJ VTJ)(Jlv1i’XJ’l VJi’) = 0(W3/ 2) (412 e~’jei., 2 ~ 6LJ51’j’ S

Let’s assumenow that statesare diagonalin theoperatorscontainingmatrix elementsW~and W~.(both orbits equal),i.e., the matrix hasdiagonalblocks for eachconfigurationffl, r~r andf’~2r2.Thenthe first two terms disappearcompletelyand in the lastwe canhaveonly contributionsof thefollowingtypes

4(r( V’Jf 1r1)(f~rjIV1r r2)(f”~r2Ivir)= ED

(4.2)

-~(r I VIr~rXf”’rl1’lf”2r2)(r2r2I VIr) = P.M.

TheED (energydenominatortype)terminvolvesrescatteringbetweenconfigurationsat thesamelevel(r~r goes to f”’r, samefor f”2? which we havenot written). It may be important if particularcoherencecanbuild (think of theanalogybetweenf”~randa ph state).The ED termcanbe thoughtasrepairingthe baddenominatorchosenthat ignoresthis type of coherence.If we prediagonalizein thefull r’r space(all r), optimummendingobtains: the ED term goes,as we can incorporateit in thedenominatorW~for arbitrary r. This is why we call it ED.The RM (real mixing) term is what its name indicates: to make it disappearinto the energydenominatorswould needblocks largerthan fnrm.

Fortunately,coherencein this caseis unlikely asit is inefficient to try to reachthe samestatef”2?by acting oncewith V

2 and twice with V1 (eqs.(3.4) and (3.5)).From this discussionit follows that the first two terms in eq. (4.1) go if we usefor the energy

denominatorstheexactdiagonalvaluesof ouroriginal representation.The termED maygo by treatingr” configurationsasa singleentity. The termP.M cannotgo: it betterbesmall if we want to avoid a fulldiagonalization.


Termsthat can go will be calleddispersive,the others“real trouble”.Now weexaminethe choiceofdenominatorsthat may insuregood convergencein low order.Let us rewrite an expandedversionofthe decouplingcondition (2.4)

~T1HIj>= (ii VII)[1 + + (J~VII)— (ii Vu)] + (ul VNIJXJIVNJ1)— (il VNIIXIJ VNI1)+ O(W3/e2)

(4.3)= (Wzi/e)~+ (W2/~)q+ O(W3/e2).

Writing .VN stressesthat only strictly non diagonalcontributionsremain.The diagonaltermsA areallregrouped.They produceadispersiveeffect that can be simply cancelledby changingthedefinition of e

e -~ ~r+ A.

Whetherthe dispersiveterm shouldgo or remaindependson the relativemagnitudeof W andA:thereis no point in choosingvery detaileddenominatorsif A W or smaller.

If we areforced to changerepresentationit is becausethereis hiddendenominatortrouble,as couldhappenin term ED in eq. (4.2). Since the new representationinvolves diagonalizing,which impliesdilating the energiesinside the blocks,the dispersiveterm in eq. (4.3) will growpotentiallybigger whiletermsO(W2/e) should be smaller as the trouble hasbeenmadedispersive.It becomesimportant tochooseA well to insuregood decoupling.

Figs. la andlb illustratethe situation.Choosingindividual denominatorsin fig. la is useless,while itmay becomeanimportant factorin lb. The interestingquestionis: can we choosee differently in fig. la

__ I”:

Fig. 1. The full/broken lines standfor model/externallevels.

A.PovesandA. Zuker, Theoreticalspecizoscopyand the/p shell 253

to anticipatefig. lb andavoid an,alwayspainful, changein representation?*Without going into details,theansweris no: evenif we changerepresentation,a singlee is not easyto find in fig. lb asit dependstoo much on the dynamicsof the problem.Either we keepthe one we had before (the unchangeddistancebetweencentroIds),in which caseconvergenceis exactlywhat it was,or we changerepresen-tation andeliminatedispersion.

Onething is sure: not to taketheenergydenominatorsclose to thecentroIdsbetweenconfigurationsneedscompellingreasons.The only betterchoiceis to takesimply exactdenominators.

Fig. la is also a possibleillustration of the rn-schemesituation,in which everyblock is a singlestate.We expect something like fig. ic when moving to the coupled representation.Now we could usecentroIdsat fixed IT to havebetterenergydenominators.

In fig. ic we assumewe havecoupledto good IT by simply diagonalizingJ2 + T2. If we choose(theW~part of) H instead,thesituationis likely to be closerto fig. id. This may seemdisastrous,but it is ablessingin disguise.Having externallines within the model spacemeansthat the notion of quasi-degenerateperturbationtheoryis in trouble:it shouldbereplacedby non degenerateperturbationtheoryasthe figure showsclearly (rememberthat matrix elementsbetweenfull lines arenull now).

If we want to usethe simple formulationof theprecedingsectionwemustuseenergydenominatorsthat be constantover the intermediatestatesfor which it is possible to invoke closure. The mostsophisticatedchoice will be to keepcentroIdsat fixed IT and intermediatecouplingsas eq. (3.14)allows. If dispersiveeffectsareimportant a secondordercalculationwith thesedenominatorsmay bebetterthangoing to third orderwith thesimpler oneswe haveused.We describethemnow:

ApproximationI is to work atfixed IT centrolds.Sinceno closedexpressionsareknown for fixed J,it is necessaryto averagenumerically

s(ITnO)— s(ITnm)= (1xI~~Ir~h7x)— ~ (fn_mrm.JTyIHlp_mrmjyy). (4.4)

Theseexpressionswereusedonly for the four nuclei we can calculateanalytically ~Ca, 50Ti, ~Sc and54Co (seetheAppendix). It is goodto rememberthat thecalculationsweredone in the rn-schemeand

before most of the theory. Eq. (4.4) is sufficiently boring to incorporatein the programmesthat wethoughtashort bruteforce checkof its usewaswarranted.We shall discussits relevancelater.

ApproximationII is an averageoverJ to eq. (4.4). It can be done analytically through eqs. (2.2),from which it is easyto write

e(TnO)— s(Tnl)= — s~+ (n — 1)(aff—atr) + ~ — btr)[aT + /3] = zlr°+ q(Tn) (4.5)

where a and /3 are constantsdependingon the numberof statesfn_l with T= T~±~.A similarexpressionholdsfor m = 2, which wecall 2de°+ p(Tn).The n dependencein eq. (4.5) is theabsolutelyminimal requirementto get decentresults:table 1 offersconvincingproof.

ApproximationIII stemsfrom someunwantedfeaturesobservedat thebeginningof theshell whichamountedclearly to an overestimateof thedenominators.Theywerecorrectedby comparingwith theexactvaluesandmaybe interpretedasaslight systematicdispersiveeffect.Theadoptedexpressionsare

s(TnO)—s(Tnl)=4~°+q(Tn)+ ôr+(n — 1)(n — 2)L13 (46

s(TnO)— e(Tn2)= 2As°+p(Tn)+ 2(n— l)(n — 2)L13.

* Brueckner theory is secondorder perturbation theory in a prediagonalized2p—2h space.

254 A. Povesand A. Zuker,Theoreticalspectroscopyand the/p shell

Table 1Valuesof thequantitiesinvolved in the energydenominators

formulae(in MeV)

r = 2p3/2 lf~,2

2P112

—2.0 —6.5 —4.0aff— aft —0.39 —0.13 —0.34bff— b,, —0.20 —0.75 —0.25(Of, — a~)

~bfr brr) +0.46 —0.40 +0.06or +1.00 +2.00 +2.00~3r 0.0128 0.0224 0.0192

The denominators(4.6), calledDEIII from now on, areused throughoutthe calculation.The termsSrandA

3 are chosensoas the cancelin MCo. The valuesof all the quantitiesto be usedto computetheenergydenominatorsareshownin table1.

It shouldbe obviousthat if a~i— af,. equals400 keV while e~— ~ris 2 MeV (r = P3/2), by the time wehave10 particleseq. (4.5) tells us that the energydenominatoris not 2 but 6 MeV. It shouldbeequallyobviousthat a puresingle particlechoicefor the denominatorswill insuredivergenceof the series:A ineq. (4.3) will be 3 times largerthan s. This exampleshould be sufficient in making clear what is themajor sourceof convergencetrouble: the ED and RM third ordercorrectionsare totally negligiblywhen comparedto the disastersbaddenominatorscan cause.

5. The (fp)2 interactionandthespectrumof 42Sc

We usethe KB interactionwith the monopolechangesmentionedin section 1

V~(KB1)= V?f(KB) — 350keV

V~(KB1)=V~(KB)—110keV (5.1)

V~(KBl)= V~(KB) — (_)T 300keV Yr.

The centroIdsaredefinedin eq. (1.2).Corepolarizationas calculatedin ref. [8] was kept.We refer to the endof section2 for the (non?)

reasons.Inclusion of the g912 orbit is trivial but it was omitted: for most of the ffl region its statusas

well definedquasiparticleis non existent.In fig. 2 theexperimentalspectrumof

42Sc is comparedwith the (fr)2 diagonalizations.Beforepassingjudgementon the matrix elements,it is useful to rememberthat intrudersplay a strongrole, the T= 1

wave-functionsareof the form a(fr)2 + 13(fr)4(sd)2and /3 can be almost as big as a. As the intrudersdo not lead only to accidentalband crossing in few statesbut keepintruding, mostly in even—evennuclei, until midshell, their effect cannotbe accountedsimply by changing~ (I = 01 and 21 inparticular).The dressingshouldbe done properly andaccountfor statedependence.As we havenotdone it, we prefer to keep the interaction untouched:no dressingis better than bad dressing.However.. . we aregoingto contradictourselvesin threelines.

A. PovesandA. Zuker,Theoreticalspeciroscopyandthe/pshell 255

(fr)2- CALC.

KB KBI KB2 KB3 EXPE (MeV)

________ +6 _______ 6+0.0 6~ 6~ 6~

4~4- 4+ 4+ 4+4+

-1.0 ______ 342+ 2+ 2+5+ 3+ 2+

• 5+ 545+ 34 3+ _______

1~ 3+

-2.0_______ +

7+ I

1+-I- 7+ 7+

0+ 71 ______ 7+0+ 0+ 0~ 1+

-3.0

0~

Fig. 2. Experimental~Scspectrumversus(ft)2 results.

The interactionswehaveusedareKB : only third line in (5.1) (V~centroids).Only usedin tables5, 6 and 7KB1: definedin(5.l)KB2: W~(KBl)— 300keVfor F = 10, 30, KB1 centroIdskeptKB3: W~(KB2)— 200keY, KB1 centroIdskept.

The KB2 modification obeysto the feeling that the two levelsare definitely too high andthat it isdifficult to visualize intruding or dressingprocessesthat can move them sufficiently close to theirpositions.Notice that we stopquite short of making them agreetoo well with experiment.KB3 is avaguefeeling. Themodification doesnot rely on argumentsaboutaccountingfor intrudersabit andafterall. Simply, therearemanylittle things that improveandfew that get worse.Table2 shows thedifferent numbers.

In assessingthecalculationsit is important to bearin mindthe following remarks.Thechangesin centroldsareabsolutelyessentialto get resultsin secondorderperturbationtheory

anywherenearthe truth. By far themost importantcontributionscomefrom Hm in eq. (2.3). Monopoleisovectorpropertiesareof somewhatlesserimportancebut theyplaya role in binding energies.


Table 2(/(JT)IH[f

2(JT)> in MeV. Different choicesquotedin thetext

(iT) = 01 10 21 30 41 50 61 71)

42~ —3.20 —2.58 —1.61 —1.70 —0.40 —1.68 0.00 —2.57

KB —1.81 —0.52 —0.78 —0.21 —0.09 —0.50 0.22 —2.20KBI —1.92 —0.87 —0.89 —0.56 —0.20 —0.85 0.11 —2.55KB2 —1.92 —1.17 —0.89 —0.86 —0.20 —0.71 0.11 —2.45KB3 —1.92 —1.77 —1.09 —0.86 —0.19 —0.71 0.18 —2.45MG’ —2.22 —1.45 —1.15 —1.07 —0.36 —1.10 0.29 —2.42

‘Ref. [38].

The choiceDEIII for the energydenominatorsin eq. (4.6) is of monopolecharacter,and could beinterpretedas a slight modification to Hm. As mentioned,it is mainly dictatedby hints emergingfromthe exactresultsandit affectsvery slightly andglobally the bindingenergies.

Thechangesin KB2 aremostlycosmetic.They do seemdictatedby thedatain mass42 [34]andwereintroducedfor this reason,withoutconsiderationof their effect in the restof the spectra.

In section 7, only spectra calculatedwith KB2 and KB3 will be shown. With respect to the“fundamental”KB1 changesthe extragainscan be characterizedas follows:

KB2: very slight, but very systematicimprovements.They do not affect howeveranybasictrend.KB3: further improvementswith respect to KB2 (they will be seen in the figures) but not as

systematic.SometimesKB2 will give betterspectra.The differencebetweenKB1 and KB2 can be describedas quantitativelysimilar to that betweenKB2andKB3 exceptthat the spectraalwaysfavourKB2.

The multipole changesin Wffff we havementionedrepresentin our opinion adequateerror barsinthe matrix elements.Nothing can be gainedat the presentstagein introducinggreatermodifications.They couldimprove spectacularlyandspuriouslysomelevelsandtherebymaskthe real effectswe shalltry to unearth.

6. Comparisonwith exactdiagonalizations

All the calculationsaremadewith denominatorsDEIII of eq. (4.6). For reasonsof economywehaveonly diagonalizedexactly43Sc,43Ca, ~Ti (T = 0 states)and~Ca.

In fig. 3 we haveplotted the quasiconfiguration,exactandexperimentalspectraof 43Sc.The low lying triplet 3/2 5/2 7/2 is obviously of intrudernature.In what follows we shall “clean”

the figuresof suchstates.Thereis furthermorea triplet of “fr intruders”: the first calculated3/2, 1/2 and5/2 statesin the

full spacedo not show up in f3 (they do but higher up). Their structureis the following

3/2: 27% f3, 31% f2p312,...

1/2: 25% f3, (15% + 10%) f2p

312, 20% f2p

112,...

5/2: 46% f3, 28% f2f5/2,

Exceptfor the f2f512 configurationin 5/2, all theamplitudesarethat of a leadingcomponentandnot the

sumsfor the wholeconfiguration.

A. PovesandA. Zuker,Theoreticalspectroscopyand the/pshell 257

a I I IU) N.

HIC%JN)2

.9

U) I I I I~ to N.

I I I t~

I I I to..-I ~ I IIflL.. tI I I I~c’,iI I I I C_)I I I

CIII I I LiZ

r() N 0

lii

II liii I I I I IN. ~ 0)10~- ~ IO0)~,r N. K) If) K) N.

aINK)

I I Hil I I I L~l~.9

0

N. K) 0)10 IIII_ I IK) N.

~- ~- ~- If) ~- 0)I II I N- ~~ I~ ~III ‘2....I III I

I III I I .~ ‘CI III I CaI III I I CD .9

10

Ca Cf

IIN. K) 0),g) II I

N.I II III II II

II iiI II III II II II II II

I I I— ~. N .‘- 0

LucZl

I.LIm

258 A. PovesandA. Zuker, Theoreticalspectroscopyandthe/p shell

Thereis nothingunexpectedin thewave-functionsfor 3/2 and1/2-: in the first it is fairer to think of f3dressingfp

312, ratherthan theotherway round.In both, thedominanceof a single quasiconfigurationisdoubtful. In the 5/2 case_thereis nothingdoubtful: it is beautifullydominatedby f

3 terms.How comeitdoesnot appearin the f3 spectrum?Actually it is there, but some700keV higher up. In fig. id (insection4) we havepreparedfor this situation:an externallevel comesclose to a model doublet,anddressesit

I1~) In) + a11’r), 112 = in> + /3If’C’r). (6.1)

Notice that a and /3 may be smallish, and the dressingstill perturbative.As the two levels arequasidegenerate,theywill mix evenly.The final wave-functionsare

(lf~>+ ajf’~’r)) ±(If2~>+ 131f’C’r>)

L~E(MeV)

8

12~ _______ 12+

10+ 10+

68+ 8~

4

6±6+

4+

44-

2

2+24-

0 0+ 0-I--13.88 -13.81

KB3(fr)4 KB3-(f)4

Fig. 5. t’Ti spectra;seecaptionto fig. 3.

A. PovesandA. Zuker,Theoreticalspectroscopyandthe/p shell 259

It is only a matterof phasesto havea relatively purer statecoming down.If the intruder is so closethat a and$ arelargewe mayusesimilar argumentsto cometo thesameconclusion.

Oncetheseintrudershavebeen“dismissed”we can comparethe restof the levelscomingfrom bothdiagonalizations.It is fair to say that the agreementis quitegood. For 43Ca in fig. 4 theagreementisalso good.

In ~Ti in fig. 5, the situationis similar, as it is in fig. 6. Notice in ~Ca that it is the groundstatethatcomestoo high by some100keV ratherthan the restof thespectrumtoo low. This error in the groundstateis moresubstantialthanin theothernuclei wehaveshown. It is precisely this type of effect thatDEIII tries to suppress,or ratherto redistributeevenly: asa monopoleoperationit canhardly improveselectivelythepositions.

i~E(MeV) ~Ca

8+8~

4

3

6~6+

4+4+

4+2 4+

2~I

0 0+ 0+

-4.99 -4.87

KB3-(fr)4 KB3-(f)4

Fig.6. ~‘Caspectra; seecaption to fig. 3.


We concludethat, asfar asenergiesgo, quasiconfigurationswith the simplestdenominatorsin secondorderproducedeviationsthat arein no caseworsethan 100 keV.

Next we move to the wave-functions of ~Ti as obtained in the exact diagonalization.

0~, 49%f4+•.(56%) 8~, 40%f4+~~~(61%)

2~,44%f4+...(47%) 10k, 73%f4+...(75%)4+ 30%f4+...(41%) 12k, 95%f4+~~~(95%)

6~, 33%f4+~~~(50%)

Figures give the amplitudes of the main f4 component and in parenthesis the total f4 amplitude in anarbitrary seniority basis. The message is rather clear: linked cluster mechanisms must be at play forperturbative calculations to_produce the right energies when the true wave-functions look so badlymixed. In the model space t, dimensions are small and the diagonalizations are hardly necessary if thebasis chosen diagonalizes the W

0~~part of H in which case the mixing becomes perturbative except foraccidents (see fig. id). Notice that the quality of agreement in the level scheme is nearly independentofthe amount of mixing. Notice also, that calculatingin ~Cr will probably lead to a 10 or 20%dominanceof f

8 in the exact matrix diagonalization (if we are lucky, dimensions are 0(106)).From this, we may conclude a bit presumptuously, albeit safely, that if spectroscopy in the f” shell

cannot be explainedsatisfactorily,the blameprobably doesnot restwith our incapacity to diagonalizelarge matrices.

7. Energy levels

While readingthis sectionit is advisableto rememberthe remarksmadeat the end of section5.We presentthe resultsof f’C (42Sc),f’C (KB2) andf’C (KB3) calculationsandtheexperimentalspectrafor

all nJT in the region.This massof lines is perhapsdifficult to digestbut our aim is to provideevidenceabout effective interaction theory and realistic forces and the readermust be entitled to pick thesuccessesandfailuresby himself (but weshall offer somehelp).

There are three types of problems raised by the spectra:(1) Denominatortrouble: overall, the simplestacceptablechoiceDEIII makes a good job. Still,

therearesomesystematicerrorsdue to neglectof J dependencein the denominators.The mostvisibleis perhapsthe insufficient lowering of 3/2 statesin nuclei with low Tat the beginning of the shell. In~Sc similar troubleoccurredfor the1~statethatwas curedby the useof fixed J denominators.

(2) Intruders trouble. It is very manifest in even nuclei at the beginning of the shell: the low lyingstates are not sufficiently bound,whichmakesthe high spin ones appear too depressed. Intruder troubleis absent in the odd nuclei and beyond midshell.

(3) Real trouble, about which the calculationprovidesa statementbut not an answer.We would liketo suggest that not all the trouble rests necessarily with the theoreticians. Wewould urge checks onsome level assignmentsandmore carefulsearchfor someabsentees.Missinga level in a calculationisseldom a real problem. Producing a low lying one where none, is observed is far worse. Experimentallevels are taken from ref s. [18,19].Pairsof crossconjugate nuclei are grouped together.All f’C calculations use DEIII denominators except the quartet ~Ca, 50Ti, “~Scand54Cowhich usefixedJ averages.

We present only f” (42Sc) spectra,although f’C (54Co) would have been interesting to include for


profoundreasonswe discoveredonly after the figureswere drawn (they can be foundin refs. [5,6]).Other phenomenologicalchoicescan only producecosmeticchangeson thesetwo sets.Thef”(KBX)spectracontain variable two body forces (nT and even nJT dependence)and three body forces.Needlessto sayf” (KBX) resultslook ratherhorrible.Now theguidedtour

Mass43—5343Sc/53Fe(figs. 7 and8)

43Sc~E(MëV)

17— 17~

17 17

4

19151 — 13~ 1313 13

193 ie - 15

1515

11 —9

2 11 9~11 _______ 9~

9_ 11

I

0 7 7— 7 7

3 413 4/342SC-f KB2-f KB3-f EXP

Fig.7. ~Scspectra:fromleft toright; f~,2resultwiththematrixelementstakenfrom the42~experimentalspectrum;quasiconfigurationsresultwith KB2

interaction;id. with KB3 (definedin the text); experimentalspectrum.


53Fe~E(MeV)

19 15

13 1313 19

_______ 1313 19

II— —

11

11— 11—

2

_______ 9_

9—

9—

I

0 .T_ 7_ 7~

13 “13 “1342SC-f KB2-f KB3-f EXP

Fig. 8. 53Fe spectra;seecaptionto fig. 7.

Good overall agreement.High spin levels in the right order in both nuclei, including inversionandspacingof 9/2-—i 1/2.

43Ca/53Mn (figs. 9 and 10, see also fig. 4)Fine details missing. See 3/2 discussion in 47V.

A.PovesandA.Zuker,Theoretlcalspectroscopyandthe/psheil 263

~E(MeV) 43Ca

3 15

15

1515

9—

2

11 9~ 9~

3_

_______ 3—

3—5—

5_ 5_5—

0 7~ 7~ 7~

3 ~v3

42SC-f KB2-f KB3-f EXPFig. 9. ‘3Ca spectra;seecaptionto fig. 7.

264 A. PovesandA. Zuker, Theoretical.cpectroscopyandthe/p shell

53Mn

~E ( MeV)

I5 —

3 1515

15

2 9_

11_ 14— 11—

3_ 9_3_ _______ 11~

3—3

I5—

5—5_ 5

0 7~ 7_ 7_ 7_

4~7 4/42 ‘V

4.2

42SC-f ‘~ KB2-f~’ KB3-f ‘~ EXP

Fig. 10.53Mn spectra;seecaptionto fig. 7.


Mass44—52

~Ti/52Fe(figs. 11 and 12)

~E(MeV) ~Ti

8 12+

12+ _______ 12+ 10+104- _______ 12~

104-104-

8±

6 8~

8+ 8+

4 6~ 6+

6+ 6~

_______ A+ _______

4+

4+

2

2~’2+

2+ _______ 2~

0 0+ 0+ 0+ 0+

42SC-f4 KB2-f4 KB3-f4 EXP

Fig. 11. MTi spectra;seecaptionto fig. 7.

In ~Ti the high spinsarecorrectlypositioned:the low spin onesunbound.Typical intrudertrouble.In52Fe the effect disappears:notice the right relative position of 10~and 12~in agreementwith recentexperiments[20].

266 A.PovesandA. Zuker, Theoreticalspectroscopyandthe/p shell

~E(MeV) 52Fe

8

12~10-I- ______ 10+

______ 10+ . 12+12+ 12+

______ 8+

6 8+

6+ 6+

4 6+ 6+

4+~ 4+ 4+

4+

2

2+ 2+ ______ 2±

2+

0 0+ 0+ 0+ 0+

“‘12 “12

42SC-f’~ KB2-f KB3-f EXP

Fig. 12. 52Fe spectra;seecaptionto fig. 7.

A.PosesandA. Zuker, Theoreticalspectmscopyandthe/p shell 267

~Sc/52Mn(figs. 13 and14)The spectrumof ~Sc was oneof our early favourites(in the endwe got to like them all). Notice thecorrectpositioningof thehigh spins,missedby the 42Sc matrix elements.Explanation:in ‘~Ti,the 42Scmatrix elementsdo well with thehigh spinsbecausethey “account”for intrudermixing. In ~Scthereisno intrudermixing.52Mn is also quitedecent:noticethecorrectinversionto producethe6~GS.

LXE(MeV) 4~’scio4-11+

10~l~4

10+11+

11 +

9+ ______ 11+

3

9+9+

9+

2

______ 5+ ______ +7+ ______ -~-

7 5I ______ 7~: + 7+

— 3+ ~1— ______ f 3+4+ 3+ 1+

— 1+ 4+

6+ 4+ 4+

6+ 6+ 6~0 2± 2+ 2+ 2~

4 414 414

42SC-f KB2-f KB3-f EXPFig. 13. “Sc spectra;seecaptionto fig. 7.

268 A. PovesandA. Zuker, Theoreticalspectroscopyand the /p shell

52Mn

i~E(MeV)

u-I- ______ 10+

10+~11+

4iif

11+

9+

8± 9+3 9+ _____ f

_______ 8+8 ±

8+

2

______ 5+ 5±7-I- _____

5+ _5-l-

I______ 7+

7~ 3+3+ ______ 7± ______

4+ 3~ 3± 4±_______ + 4+ 1±

_____ 4+ _____ 2~

1-i-2+ 2~

0 2+ 6+ 64- 6+

42SC-f42 K82-f12 KB3-f12 EXP

Fig. 14. 52Mn spectra;seecaptionto fig. 7.

A. PosesandA. Zukei Theoreticalspectroscopyandthe/p shell 269

~Ca/52Cr(figs. 15 and 16)

Phenomenalintrudersin ~Ca.Perfectlyacceptable52Cr.1~E(MeV) 8~ 44Ca

5 8+

8+

8+4

64-

3

4+4+ ______ 6+ 6+

4+ 4+ 4+

4+

2 ______ 4+

24-

2+

2+ ______ 2+

I

0 0+ 0+ 0+ 0+NA 4l~1

42SC-f’ KB2-f’ KB3-f’ EXPFig. 15. “Ca spectra;seecaptionto fig. 7.

270 A. Povesand A. Zuker, Theoreticalspectroscopyandthe/p shell

52Cr

~E (MeV)

o ______8~

8+ 8* 8*

8+

5+ 5+ 5+4 5+

5+6+ 6+ 6+

4+ 4+ 6+

4+ 4+ 4+ 4*4+

4+

2 2+ _____2~

2+ _______

2+

0 0+ o-l- 0+


Fig. 36. 52Cr spectra;seecaptionto fig. 7.

A. PosesandA. Zuker, Theoreticalspectroscopyandthe/p shell 271

Mass45—5145Ti/5tMn (figs. 17 and 18)

See discussionof 3/2- in 47V. Chaotic high spin level orderingin 45Ti. Perhapsa little help fromexperimentalistsneeded.In 5tMn situationcompatiblewith data.

~E(MeV)25

8 _______________ 2527

2727

27 21________ 2123

21 ______

6 ______ 23

19 _______ 19 1923 ________

17 17 21

19 ________ ia ________ 194 _______ 19_ _______ 43_

17 _______ j7 _______ 17_______ 13 13 1713

15~ _______

_______ 15 151513

2 9

_________ 11~ _________ ___________________ 11— 11~ 11~

9— _______

3_3_

5—_______ 7_ _______ _______ 5_ _______ 5_

0 5~ 7~ 7~ 3:7


Fig. 17. “Ti spectra;seecaptionto fig. 7.

A. Povesand A. Zuker, Theoreticalspectroscopyand the/pshell

51Mn

L\E(MeV)

827

— 27

27

2723 23~23

21 _______ 21~6 23

21 21~

19 19 194

17 19 17 1715 171 _______ 13

_______ 13 _______ 13 _______ 15 159_ 5__ 5__ ______

_______ 15 3~ 133_ ______ 5_7________ 7—

7—75

2 _______ _______ 3:_______ 3_ _______ I11 3 3~

______ 9_ 11 ______3_ _________ 11 _________ 11_______ 9—

9_ _______ 9_

_______ 7_ 7________7_ _______ 5_

0 5~ 7 5~ 5

~lu42SC-f~~ KB2-f1~~ KB3-f EXP

Fig. 18. 51Mn spectra:see captionto fig. 7.

A. PosesandA. Zuker,Theoreticalspectroscopyandthe/pshell 273

45Sc/51Cr(figs. 19 and 20)45Sc is quite healthy.Again high spin statesin f” (42Sc) too high.5tCr: excellent candidateto denominatortrouble. Excited statesare shifted up by some 500 keVbecauseof overbindingin groundstate.

45L~E(MeV) Sc ______ 21 2123

_______ 23 236.20 21 —

5 21___23

4 19

1917 17

19

________ 1319

_______ 13 17 3153 15

17

15

1~ _______

2 5_

_______ 9 _______ 153_ 9__ _______ 5_ 9_

______ 15 9 1~11 3~ I 7_

f1 _______ 3 11

1 11—

0 7_ 7_ 7_ 7_4, 4,5

42 SC-f KB2-f KB3-f EXPFig. 19. “Scspectra;seecaptionto fig. 7.

274 A. PosesandA. Zuker, Theoreticalspectroscopyandthe/pshell

~E (MeV)_______ 15

15 13 1317 ______ 15 ______ 913 17 ______ 1311 17

_______ _______ 9~I19_ 11 —11

_______ 1~ _______ 15_,17_3 9— 9—

______ 7_ ______

_______ 11~ _______ 13 _______ 13 _______

_______ 13 3__ 79_ 3_

______ 1s ______ 5— 15______ _______ 3 ______ 13

15 _______ 5~ ______ 7

1~2

______ 9_ 3_______ 9_ 11 ______ 11______ 3=______ 5 7_

11— 1I_5—

9—I

0 7_ 7_ 7_ 7_

11 “11 ‘5,

42SC-f KB2-f KB3-f EXP

Fig. 20.51Cr spectra;seecaptionto fig. 7.

A. PosesandA. Zuker,Theoreticalspectroxopyandthe/pshell 275

45Ca/51V(figs. 21 and 22)Sameremarksfor this pair.

45COL~E(MeV)

153

15

1515

2_______ — 9_

11—

9_ 119—

3_ 11 11— 3_

3—3—

I

5—

5—

5—

0 7_ 7_ 7_ 7_

4l~

42SC-f~ KB2-f’~’ KB3-f’~ EXP

Fig. 21. 45Caspectra;seecaptionto fig. 7.

276 A. PosesandA. Zuker,Theoreticalspectroscopyand the/p shell

51v

L~E(MeV)

45 15

3 15

IS

3—

-9_ 9_

2 11~ 11_

11 9~

3_ 113—

3--

I — —

5 3

5

5—

0 7— 7_ 7—

41 “11 ‘VII42SC-f KB2-f KB3-f EXP

Fig. 22. ~‘V spectra;seecaptionto fig. 7.

A. PosesandA. Zuker,Theoreticalspecnoscopyand the/pshell 277

~V/5°Mn(figs. 23 and24)The T = 0 statesin ~V are quite good but shifted bodily with respectto the GS. Again probabledenominatortrouble.5°Mnis satisfactory.

L~E(MeV)

11+4 41+

8+ 8+

74 7+3

9+9+

9+

2

_______ 4+ 7+______ 1~ _______ 3+ 64-

3 4.3- ____ ____ ++

5+ 6 4+ ____________ 5+ ______ 63- -‘

______ 14- ______ 7+I ______ ______ 74- ______ +

6+ &- ______ ______

5+ 3+3+14- 3+

34-

0 0~T=1 0~T:1 04-T~1 04-T~1

42SC-f6 KB2-f6 KB3-f6 . EXP

Fig. 23. “V spectra;seecaptionto fig. 7.


50MnL~E(MeV)

3 84-9+

3+ _______

______ 2~ ______ 8~ 9~24- 9+

44-T~14+T~1 44-T-I

2 _____ + ~+ ______ 5+6±

4+ _1+3 _________

______ + 5+ 4+5

______ 1+3±

5+ ______

______ 2~T1 — 2~T~I ______ 4- 3*

7+ ______ 34. 24-T1 7*____ ___ 7 ____

64- 6+ 3+1+ 64- f~2+T:1

3+ _______

1+ 1± 2~1+5+ 5+

______ 5+5+~

0 0+ 0+ 0+ 0+


Fig. 24. ~ spectra;seecaptionto fig. 7.

A. PosesandA. Zuker, Theoreticalspectroscopyandthe/p shell 279

~Ti/5°Cr (figs. 25, 26, 27)The showpieceof crossconjugatesymmetry.Strongattenuationof intrudertroublein ~Ti. We haveanextrafigure for 50Cr to showthat thecalculationdoesa good job.The resultsfor this pairwill be calledArgumentno. 1.

4611~E(MeV)

12+

8 _______ 12~3- 124- 11+

123-11+ ______ ÷

11

104- 104-6 _______ 10+ %1

84- 8+ _______ 84- 84-

4

______ + 6+6 _______ 6+ 6~

4+ 4+ ______ +2 4+

______ + ~. ______22~

0 0+ 0+ 0+ 0+



280 A. Poses and A. Zuker, Theoretical spectroscopyand the/p shell

50Cr

~E(MeV)

8[ ______ 124-12±

124-11+ 124-

11 _________ 114-

144-

10+ 10~ 10~6 10+

8+ 84- 8+

4

______ ______ 6+ ______

6+ 6±

4± 43- 4+2 4+

2+ 2+ 2+2+

0 04- 04- o+

42SC-f1° KB2-f10 KB3-f1° EXP

Fig. 26. ~Cr spectra;seecaptionto fig. 7.

A. PosesandA. Zuker, Theoreticalspectroxopyand the/p shell 281

50Cr (OTHER STATES)

L~E(MeV) 6+ _____ 6~

4.5

______ 14- 2*3+

04-4.0

______ 2+ 1+ 4+5+ ______ ______

_______ 5+ _______ 1,22+ +0 5,6

5+ ______ 6+6+ 6+ 5+

_______ 2~6~3- 4+ ______

3+ 4+4+ 4

3.52+4+

4±4+ 4±

2+

3.0 2+4+

2+ _2+

2.5 2~4,Af~

42SC-f” KB2-f “ KB3-f q~j EXP

Fig. 27. 50Crspectra;seecaptionto fig. 7, statesnot included in fig. 26.

282 A. PosesandA. Zuker,Theoreticalspectroscopyand the/p shell

~Sc/50V(figs. 28 and29)Definitely good.Herewe shall speak of Argumentno. 2.To avoid unnecessary suspense we mention that the Argumentswill be discussedin section10.

i~E(MeV) 8~

84-

7-f-4+

______ 4- 7+

1.0 4± ______ 1~4+ 7+

5+5+ 4+

5+ 5+

4+ ______

2

0.5 ______ +5+ 2

3+5+ 5+ 5+

3-f-

3+ ____________ 4+ ______ 64- 6+

0.0 2~ 6+ 4+


Fig. 28. “Scspectra:seecaptionto fig. 7.

A. PosesandA. Zuker,Theoreticalspectroscopyand the/pshell 283

Soy

~E(MeV) 11+

11+11+

11+10+ 404-

10+

10+

3

9+ 9+ 9+

9+

2 ______ 8+8± 8+

8+

______ 5+ ______ 5+7+ 7+ 7+

5+4+

___ + +5+ 2+ ______ 3+3+ 5+ — 5+ 4+

_____ 4~ _____ 5+0 2+ 64- 6+ 6+

10 ~10 “10

42SC-f K82-f KB3-f EXP

Fig. 29. ~V spectra;seecaptionto fig. 7.

284 A. PosesandA. Zuker, Theoreticalspectroscopyandthe/p shell

‘~Ca,48Ti, 50Ti and54Co (figs. 30—33)

46Ca

i~E(MeV)

6+

3 64-4+

4+

6~ 6~

4+ 4+

2

2+

24- 2+

2+I

0 o4- 0+ 0+ 0+

6 V42SC-f KB2-f KB3-f EXP

Fig. 30. “Ca spectra;seecaptionto fig. 7.

Here we interrupt the guided tour for a first interlude.These nuclei are generalizedPandyaor particle-holetransformsof 42Sc and of each other. They areparticularin that

(i) Analyticexpressionsfor the energiesareavailable(ref. [2] andAppendix).(ii) The energydenominatorshavebeenevaluatedatfixed JT.

(iii) They representaquadrupletest of f~,2configuration(not quasi...!) calculations[3,4]: their

spectramust be identical to that of42Sc (~Ca,50Ti and ~Co) or obtainedby Pandyatransformation

(~Ti).

A. PosesandA. Zuker Theoreticalspectroxopyandthe/pshell 285

48sci~E(MeV)

3

1+1+ ______ 1+ 1-I.

2

7+ ______ 2+ ______ 7+ 24-

1 7+ 2+ 7+

3+ ______ 3+ 3+

_______

3-f2+ _______

______ 4- 5+ 4~ _______4~54- 4+ 5+ 5

0 6+ 6+ 64- 6+

8 ‘V842SC-f KB2-f KB3-f EXP

Fig. 31. “Scspectra;seecaptionto fig. 7.

The figures tell us that ~Caand50Ti passthe testwith flying colours.For~Sc,theagreementis still

very goodandfor MC0 it is at leastfair. If the MC0 two body levelsarePandyatransformedto obtain

~Sc, the2~movesto its right position but the 1~movesup. Furthermore,the0~(analogof ~Ca OS)comesdown strongly. (Weapologizefor the lack of figure, in fig. 31 the0~is not shownbut comesatthe right placein the quasiconfigurationresultsandabit too high in fe.) It is also easyto seethat theT= 1 statesof ~Co will not compareas well with ~Ca and

50Ti.

Thequalityof agreementof the f,2 model, is, overall,quitegood.By the standardsof 10 yearsagoit

shouldbe consideredamazing,especiallyin view of the mess~Sc is experimentally(the 1~is drownedin a seaof levels).Rememberthat the calculationsweredone 15 yearsago.

Nextwemention aratherremarkablefact: the KB (neitherKB2 norKB3: KB) W~matrix elements(given in table 2) look like an unmitigateddisasterwhencompareddirectly with

425c, ~Ca and 50Ti


50T IlxE ( MeV)

6+ 6+

3 6+ 6+

4+

4+ 4+ 4+

2 _____ 2+

2+24- _______ 24-

I

0 0± 0+ 0±

10 4, N

42SC-f KB2-f KB3-f EXP

Fig. 32. 5°’flspectra;seecaptionto fig. 7.

(slightly better in 54Co). However,their Pandyatransformto ~Scis quite good: the J = 2, 3, 4, 5 levelsaresimply fantastic,J = 7 is too low by some700 keV, J = 1 is too high by some1 MeV andJ = 0 is toolow by some 2 MeV. The situation is exactlyas in MCo: the KB matrix elementsgive betteragreementfor thegood spins(2, 3, 4, 5) and worse for the bad ones (0, 1).

It follows that a plausible spectrum of ~Sccan be obtainedfrom very differentmatrix elementsandthereis a hint that thedefectsto be found areeither42Sc-like (very bad2’) or MCo~like(very bad0~and1~).

Now we studyfig. 31 andnotice:Argumentno. 4: the very bad positioning of the 1~state for the bare KB, ~ matrix elementshasbeen corrected by the three body force.

A. PosesandA. Zuker, Theoreticalspectroscopyandthe/pshell 287

54Co

L~E(MeV)

6+

34+ 6+ 6~ 6+

4+ — 4+ 4±

5+2 ______ 4- 5+ ______

24- 3+2+ ______ ______ ÷

3+ 2~

I _____1-I-

______ +1-i. 1 1+

7+

7+

0 0+ 04- o± o4-

411 ~4A ‘Vi

42SC-f~ KB2-f ‘‘ KB3-f~’ EXP

Fig. 33. ~Cospectra;seecaptionto fig. 7.

In table 3 we give the V1 contributions(eq. (3.4)) of the different orbits to the levels. Notice the

overwhelmingf512 dominance,which shows,it needsmaybe, that modelspacesdependon dynamics:what could be the meaning of an (f712p312)” model space?

Table 3

Contributionsfrom HRI (eq. (3.4)) to theenergylevelsof ~Sc(mt. KB3)

\\JF 0~ 1~ 2~ 3~ 4~ 5~ 6~ 7~

2p~ 0.0 —0.21 —0.26 —0.22 —0.18 —0.24 —0.20 —0.13if,,2 0.0 —1.37 —0.39 —0.45 —0.38 —0.36 —0.44 —0.212p,,2 0.0 —0.09 —0.06 —0.08 —0.06 —0.03 —0.04 -0.04

288 A. Poses and A. Zuker,Theoreticalspectroscopyandthe/pshell

The high unperturbedpositionof the 1~is at the origin of denominatortrouble,curedby the use of

approximation I (eq. (4.4)), in approximation III, the state stops midway (too high by some500keV).

Furthermore:Argumentno. 4’: the very bad positioningof the 0~level hasbeencuredby a monopolechange(seesection5).

Now we moveto fig. 32.Argumentno. 1’: 50Ti hasa spectrumthat is very closeto that of 42Ca where intruders play a dominantrole. Quasiconfigurationsreproduceit well.

Now wemove backto fig. 30.Argumentno. 1”. How come~Cawith intruder influencehalf way between42Sc and 50Ti, hasstill thesamespectrum.Quasiconfigurationsarenot so goodhereBUT not asbadas they were in ~Ca.

Finally ~‘Co,in fig. 33, is quite nice and the calculationsare much closer to their experimental

47v~tE (MeV)

6 21 —

23 21 2323

23

4 19_19

1919~

_______ — _______

_______ 7

______15_ 0_ — ______ 1_______ _______ is— 13 15_______ _______ 15

2 9~ ______ 9~ 3~

9_ 9_ 7_

11 11— 11— _________ 11~3 9_

3_— 5_ — _______ 7_

0 ~- 7— ~—

42SC-f7 KB2-f7 KB3-i’~ EXP

Fig. 34. 47V spectra;seecaption to fig. 7.

A. PosesandA. Zuker, Theoreticalspectroscopyandthe/pshell 289

counterpartsthan in 42Sc. This is certainly due in part to the absenceof intruders.The remainingdiscrepanciescan beeitherdueto denominatortroublewhichapproximationI is likely to havereducedor to real trouble.Herewe haveArgumentno. 4”.Endof the first interlude.Secondinterludebegins.

Mass47—4947V/49Cr (figs. 34 and35)

49Cr

~E( MeV)15

15

4 15 15

15

15 15

______ 15

3 _______ i59—

_______ 13 — 13_______ 1~ 13 —_______ 15 7 7

_______ 3_7—

3_2 . 3_

________ 3—9_ —

_______ 9_ 9_~~11— 11~

3—9—

I

_______ 7_ _______ 7_ _______ — _______ —0 5_ 5— 5 5

42SC-f9 KB2-f9 KB3-4~ EXPFig. 35. “Cr spectra;seecaptionto fig. 7.


In spite of some missed details the picture is quite good. Of particular interest is the status of the 3/2

states.

Argumentno. 3In 43Ca,the calculated3/2 is too high (fig. 9). From the exactcalculation in (fr)3 we mayexpecta

very minor improvementin this level whichshowssignsof denominatortrouble.In no casewill it comelow enough.

In 45Ti the 3/2 is coming down experimentally.There is certainly denominatortrouble in thecalculationsandwe mayestimatea gain of a few hundredkeV that will not be sufficient.In 47V the 3/2 has become the GS. The calculations bring the level down but not enough. Exactcalculationsin the f

12+ f~’ spacedepressit a further300keV (refs. [6] and [7]).A transparentcaseofdenominatortrouble.The general trend of the 3/2 states can be summarized as follows:

Therearethreestatesplaying arole: fn, f”’r andintruder.Sometimesthe intruderis spectacular(caseof

43Sc in fig. 3), sometimesthef”1r state kicks around severely the ffl state(again 43Sc).However, thedressedf”, in spiteof the pushesandpulls of both intruders,survivesas a distinct entity in mostnuclei.In which caseit shows the following trend.

exp exp f”. trend.quality

43Sc 53Fe out43Cadown 53Mn up right+45Ti down 51Mn up right +45Scdown 51Crup right — (0)

45Caup ‘1V down wrong(right)— (0)47V down 49Cr up right 047Ti up 49V down right 047Scdown 49Ti up right 0

By down and up we simply mean the relative positions of the states with respect to the averagein thetwo nuclei.

By right andwrongwe denotethe trendof the calculations; by +/— wemeanthat the heavier/lighternucleusis muchbetterreproducedby the calculationsthan the lighter/heavierone.

By 0 we meanthat both agreequite well with experiment.The commentsto figs. 19 and22 showthat for thepairs 45Sc—51Crand 45Ca—51Vwe are justified in

selectingthe indicationsin parenthesis.For the timebeingwe only noticeaglobal trendreproducedbythe calculations.

This is the end of the second interlude. Guided tour resumes.

47Ti/49V (figs. 36 and37)For the 3/2 seeabove.47Ti is not bad.49V is distinctlygood.

A. PosesandA. Zuker,Theoreticalspectroxopyandthe/p shell 291

i~E(MeV) 2321

21 j9

19

19

19~19~

19~

17

47_ 17 15 _______

1513~

j3 13

3 ______ 11~ 7_ ______

________ 151 _________ 3~ ________ 11~

_______ 3 _______ 3 _______ 139 _____ 5_ ______ 15f5~ ______ 11_ ______ 1~~

_______11~ 15

9_ _______ I

9—3

3_ 1111 11— ______ 9

I3—

5_ 5_ 7_ 7_0 7 7 5_ 5

., ‘V.,

42SC-f’ KB2-f’ KB3-f’ EXP

Fig. 36. ~TI spectra;seecaptionto fig. 7.

292 A. PosesandA. Zuker, Theoreticalspectroscopyand the/p shell

~E(MeV)•19~

419

19

17 19_______ 17~17~

13_ _______ 11 _______ 13_ ___________________ ____________ 11~17_ 1I3 _______ 13 _______ 15

11 _______ 9 9 j315 _______ 7 _______ 119

5 ______

_______ 3 ______ 1I 11 9~119_ 7_15~ 9_ _______ 7_ _______

______ 11 ______ 9 ______

15 ______ 1515 7~

2 _______

3~ ______ I3_ 3_5—

9_ _______ _______ 3___________ 1~

1~ 5 5_5_

_______ 9—II —________ 9—11~ ___________ ___________

11~ II—3—

3—

3—

5— _______ 3_5_ 5_

0 7~ 7 7 7


Fig. 37. “V spectra;seecaptionto fig. 7.

A. PavesandA. Zuker, Theoreticalspectroxopyandthe/p shell 293

47Sc/’~Ti(figs. 38 and39)~Scis fair. In 49Ti thereis againaglobal shift of theexcitedstates(sameas in 51Cr and51V).

47sc~E(MeV)

I 9

4

I9

17 19_

____ 17~3 _______ 17~ 15

13 45

_______ 11~ 11— 1315 ______

7_ _______

______ I— 13 — —I3 ______ 5,7

152 ______ 9~ ______ ______

15159_ 9_

_______ 5_ _______ —_________ II—

5— 5_

11~ 11~_________ 1I

3—3_ 3_

0 7_ 7_ 7_ 7_

429C-f7 KB2-f7 KB3-’~’~ EXP

Fig. 38. 47SCspectra;seecaptionto fig. 7.

294 A. PosesandA. Zuker,Theoreticalspectroscopyandthe/p shell

1~E(MeV)19

19~49—

4

1717 17

11— 11~ 17~19

3 ______ 13 13

1315

______ ______ 515_ 15_

7-7_ ______ 5_ 7

9_ ______ 9=5

2 ______ 9 311 — 5

0 7 7 7_



A.PosesandA. Zuker,Theoreticalspecb’oxopyand the/pshell 295

Mass48~Cr (fig. 40)

~Cr

t~E(MeV) 8~(5.91)

8~ _____8~

1~

4

6+ 6~ _______ 6+ 64-

3

4+ 4+

2 4~t~4+

2+ 2+2+

I

2+

0 0+ 04- o~ 04-4,

42SC-f° K62-f° KB3-f~ EXP

Fig. 40. “Cr spectra;seecaptionto fig. 7.

296 A. Poses andA. Zuker,Theoreticalspectroscopyandthe/pshell

The experimentallevel schemein ref. [18]placesthe8~at 4.06MeV. In a first draftwe simply noticedthis exotic behaviour. Recently (remember these results were ready two years ago) the experimentalistshavecorrectedthis anomaly[21].They alsofind an uncertain 9~10÷at 7.06MeV. The calculationsgive10~at 6.61MeV, 9~at 7.17.

~V and~Ti (figs. 41 and42)We are happy to close this touron two definitelygood nuclei.The last sectionwill containan analysisofwhat we haveseen.

48v£~E(MeV)

11+______ 4- 114-

4

114-

3

9-I-

9+ 9+

8+94 84- _______ 8+

2 84-

7+ _______ 4- _______ 4- 7+

______ 5+ 74-1 3+ _______ 34-3+ .7-l• 3+

5+

6+ _______ 5+ _______ 5+ 6+

_____ 1-I. _____ 11: _____

4+ s-I. 4+ 2+44- 2+ 2+

0 24- 4+ 4+ 4+

‘V

42SC-f KB2-f KB3-f EXPFig. 41. “V spectra;seecaptionto fig. 7.

A.PosesandA. Zuker, Theoreticalspectroxopyandthe/pshell 297

481j

~E( MeV)

12+8

124-

11+11+

11+ II4-,124-10+

10+ ______

______ 104-6 104-

8+ 84- 8+ 84-

42+ 2+

2+ ______

_____ 3+ _____ 6+ _____

______ 64- 64- 3+______ 6+ 3+

3+ ______ 64- 6+

4+ _____ 2+ ______ 24- ______ 2+4+ 4+ 4+

2 2+

2+ 2+2+ ______ 2+

0 0+ 0+ 0+ 0+

42SC-f8 KB2-T8 KB3-f8 EXP

Fig. 42. “Ii spectra;seecaptionto fig. 7.

298 A. Pavesand A. Zuker, Theoreticalspectroscopyandthe/p shell

8. BInding energies

In table4 we presentthe nuclearbinding energiesobtainedwith f (42Sc)andf’~(KB3) calculations.The single particle field (BE(41Ca—40Ca)= e~)and Coulomb effects have been extracted from theexperimentalvalues[6].

Table 4Two body binding energieson a“Ca core(in MeV)

N ii 2T Exp j”(KB3) f”(’~Sc) N 2J 2T Exp j”(KB3) f”(42Sc)

2 0 2 3.11 2.74 3.11 9 7 7 15.05 15.63 23.023 7 3 2.68 2.57 3.08 7 5 23.47 24.19 31.70

7 1 6.82 6.73 7.17 7 3 29.70 30.24 38.184 0 4 5.45 4.87 6.19 5 1 34.49 34.71 42.13

4 2 8.10 8.24 9.15 10 0 6 25.92 26.45 36.840 0 14.57 13.81 14.34 12 4 30.47 31.37 41.68

5 7 5 4.96 4.52 6.17 0 2 38.71 39.49 49.847 3 10.99 10.86 12.66 11 7 5 32.97 34.24 47.565 1 15.60 15.52 16.78 7 3 39.35 40.69 54.06

6 0 6 6.54 6.40 9.26 5 1 43.71 44.84 58.178 4 11.33 11.50 14.09 12 0 4 42.77 44.58 61.350 2 20.30 19.94 22.24 4 2 45.56 47.18 64.34

7 7 7 5.45 5.85 9.22 0 0 51.11 52.70 69.527 5 13.55 13.69 17.91 13 7 3 48.92 51.32 72.077 3 20.68 20.67 24.39 7 1 53.05 55.43 76.155 1 24.72 24.60 28.34 14 0 2 57.67 60.87 85.88

8 0 8 7.04 7.45 12.28 15 7 1 63.38 67.07 96.6112 6 13.35 13.67 19.20 16 0 0 70.60 76.10 110.370 4 23.82 24.00 29.668 2 26.70 26.88 32.550 0 32.38 32.27 37.54

In the tableand in fig. 43 it is clearthat somewobbleoccursin thevalueBE(exp—th)at the beginningof the shell. Its correlation with intruder mixing is manifest when we notice, for example in mass 44,discrepancies of —760keV (T = 0) and —580 keV (T = 2) for even nuclei, compared with +140keV forthe odd T= 1.

The wobble is clearly superimposed on a broadseculareffect. If we try to correctit by expressionsofthe type

or

badfits areobtained.The choice

~1=cr2(2)+as(3) (8.1)

A. PavesandA. Zuker,Theoreticalspectroscopyandtheft’shell 299

BE(Exp-th) MeV INT.KB36

0

oTMIN

‘- +A ‘MIN

x TM,N4~2

a TMIN+34. • IMIN+4 0

A3.

A0

2.xA0

Ax x

1. 0

x AA o

o~DA X X AX°0. A

~Do°A

A x X0

—1

I I I I I I I I I I I I

2 4 6 8 10 l2 14 16

A -40Fig. 43. Binding energydifferencesbetweenexperimentandquasiconfigurationscalculation.

gives goodresultswith

cr2— —19±3 keV, a3—+13±1keV. (8.2)

Assigning an arbitrary error bar of 200 keV per point we obtain ~2/p= 1.05. No gainsresult from the

introductionof T dependence.Thecorrectiona

2to the two body centroIdsobtainedfrom KB3 in table2 is totally insignificant.Theinterestof the a3 value,on theotherhand, derivesfrom its existenceand its small size.

300 A. Poses and A. Zuker, Theoreticalspectroscopyand the/pshell

In refs. [6,7] it is shownthat the binding energiesgiven in table4 for r(42Sc)can be correctedquitewell by a cubic term. As the discrepancy in 56Ni is of 40 MeV comparedto 5.5MeV in f’~(KB3),it isclearthat the ai parameterwill beconsiderablylarger(andsowouldbe ~2/p).The effectiveinteractionsHRJ and HR

2 in eqs. (3.9) and (3.10) when added to the original bare Hamiltonian (H) produceimportantchangeson the two andthreebody centroIds.In particularHRI contributestermsin n

2 + n3,but they cancel exactly in 56Ni (easily checkedsince V, in eq. (3.4) cannot changethe energyof a(quasi!) closed shell). HR2 can only producen2 centroIds. However, through the n dependenceof theenergydenominators,higher powersof n arepotentiallypresent.

The fact that these internally generatedmonopole contributionsneedonly a very small a3 extra

correctionin (7.2) is thereforesignificant.In secondorder perturbationtheory, the only remainingprocessthat hasnot been included,that

could produce an n3 term would be mediatedby matrix elementsof thetype WSfffwhere s should be at

least in the hfp shell (derivation identical to that of eq. (3.9)). Thisprocesshasbeenmentionedtowardthe endof section2.3 andwould correspondto aselfconsistentsizereadjustmenteffect.

If the f712 quasiparticles do not wish to change size appreciably while the shell is being filled, the only

remedyleft would be to interpret a3 as a genuine three body force. The values in (7.2) could probablybe considered as a fairly reliableorderof magnitudeestimateof the threebody force (effectiveor true)generatedoutsidethe model space.

It should be noticed that the agreementwith experimentfor the binding energiesis comparabletothat of the spectra.In view of thevery largenuclearcorrelationenergies(70MeV in

56Ni, seetable4) itmayseemodd that the absoluteerrorson such largenumbers,be comparableto the relativeerrors inthe spectra.Binding energieshoweverhaveto be judgedon anenergyper particlebasis.It is directly inline with the spirit of linked clustertheory andwith ourinsistenceon the fact that thecoreis a memberof the modelspaceas the GS is a memberof the spectrum.

9. Electromagnetictransitionsandsomeremarkson thewave-functions

9.1. Electromagnetictransitions

We can write the operatorfor an electromagnetictransitionof multipolarity A in our notation andfor the (fp)-shell as

= ~ ~ (9.1)

where pp’ = if712,

2P3/2, if512,

2P1,2.

If now wewant to calculatea transitionbetweenquasiconfigurationswe haveto evaluate(f7J1lAIf~).Using the expression(3.2) for thef’ stateswe can write

~ffI[2AIf~)= (iJI’2A Ii’) +~ ~ifVIJXJII1AIi’) + —~—-(iIUA IJXJ’I Vu’) + termsin ~7A . O(W2/s2). (9.2)gil

We can sumover intermediatestates,andafter somealgebraobtain

!71 ia~I I!~= (j(F)IIflA IIi(r’)) + (_~i._+ __1_) (i(f)I (~A v)A I Ii’(r’)) + (9.3)

A. PosesandA. Zuker,Theoreticalspectroscopyandthe/p shell 301

Using expressions (9.1) and (3.4) for I? and V (only V1 in eq. (3.4) contributesto that order), eq.

(9.3), aftersomerecouplinghasthe following form:

/7IInAIIf~= (i(r)(IDAIIiF(rF))1 1 1 _______

+ ~ (__+ -_-_)(fIIDkIIr)~[A]”2 ~ (—1f’’~” [r]hI~2 W’’(i(F)II[A(A(BB)’~’)TIIi’(F’)). (9.4)

So the relevanttransition operatoramongquasiconfigurationsconsistsin the conventionalone forthe configurationsplus a two body effectivetransitionoperatoracting on configurations.Note that attheorderwestop,no one bodyeffectivetransitionoperator(effective charge)appears.So, the effect oftruncationis irreducibly of two body character.Neverthelessip—lh excitationswhen neitherp nor hbelongto the if

7,2 orbit give rise to theseeffectivecharges.~ matrix elementsinsteadof ~We present in tables5 and 6 the results for E2 transitions in some selectednuclei. We use

consistentlyan effective chargeof 0.5 e for protonsand neutrons.We compare our resultswithexperiment,includingalso the resultsof f~,2andpartial (fp) diagonalizationstakenfrom ref. [7].

Resultsfor theconjugatepair ~‘Ti/50Crshowthe following features

(i) The renormalizationis very importantandgenerallygoesin the right direction (seepoint (iii) insection1).

(ii) The quality of agreementwith experimentis similar for thequasiconfigurationsresultand thoseof partial diagonalizations.This can indicatethe needfor a higherorderdressingof theconfigurationsto calculatetransitions.

(iii) A very interestingresult is providedby thequadrupolemoment.In an ffl model, they havethewrong sign. Evidently no effective chargecan accountfor this. The changeinducedby the two bodyoperatordoesit andthe resultis quite closeto theexperimentalone.

The sametype of resultsareobtainedfor thepair47V/49Cr shown in table6.We havealsocalculatedthe transitionsin 50Ti (seealso table6), thevery good agreementwe find

corroboratesthe analysisof this spectrummadein section6.In table 7, we present the results for some Ml transitions.Although some improvementsare

obtained,mainly in 47V/49Cr the situation for ~1Ti/50Crremainsratherpoor. The sensitivity of Mitransitionsto very fine detailsof thewave-functionsexplainsthis partial defeat.

Table 5

E2transitionprobabilitiesin e2fm4

Nucleus .J-÷J, Exp~ /“(42Sc) f~+/~~Ir(KB~)b j”(KB3)

“‘fl 2t Ot 214±20~ 61.15 81.6 100.4

2~Ot 2.1±0.5 3.13 1.9 3.62~Ot 20±4 1.14 6.5 16.32~2t 182±41 20.60 96.7 105.4Q2(2t)~ —19±10~ 9.97 —6.9 —6.1

50Cr 2t Ot l82±50’~ 70.84 125.3 120.42~ Ot 8.5±3.0 8.56 9.4 15.42t 0~ 15±4 0.04 0.3 0.122t 2t 1.17~i~ 19.55 6.1 3.1Q2(2t)~ _

30~4~pd 7.32 —20.2 —19.7

a ref. [24]. b KB’ definedin section5.cref [25]. “ref. [26]. 9n efm

2.

302 A. PosesandA. Zuker, Theoreticalspectroscopyand the/pshell

Table 6E2 transitionprobabilities in e

2fm4

Nucleus J -~ J, Expa /“(KB) f” +r’r(KB’) f”(KB3)

5/2- 3/2 897~4~1 26.70 130.1 131.37/2 3/2 315±118” 27.23 63.1 54.711/2- 7/2 203~~ 70.40 124.9 106.7

“9Cr 7/2 5/2- 314±80’~ 102.70 207.4 186.6553±160~

9/2 7/2 307±13W 47.29 154.6 160.7373±130~1

11/2 7/2 135±45C 84.14 — 106.811/2- 9/2 511±250C

90~0d 43.26 — 95.3

Nucleus J—aJ1 Exp /‘°+fr+/”r2(KB’) id(KB~)C f”(KB3)

2~ O~66~8~ 80.98 72.6 67.6

4~ 2~ 601~ 81.00 73.0 57.9

6~ 4’ 33÷2’, — — 25.0

a ref. [30]. ‘ref. [29]. “R from ref. [311,r from ref. [321and8 from ref. [31].did but 8 from ref. [33]. eKB~definedin ref. [7]. ‘ref. [28].

5ref. [25]. “ref. [27].

Table 7Ml transitionprobabilities in p,,.

Nucleus I -“ J, Exp f”(”2Sc) /“ +/‘~‘r(KB’) j”(KB3)

“6’Fi 2t 2t 0.033±0.OP 1.77 0.26 1.092t 2t 0.19±0.03a 0.43 1.60 1.35

50Cr 2t 2t 0.42±0.05” 1.77 1.10 1.362t 2t

023~004b 0.43 0.30 0.255/2 3/2 0.07±0.01C 0.01 0.13 0.157/2 5/2 0.45±0.01C 0.001 0.14 0.209/2 7/2 >3 x 10” 0.39 — 0.0749Cr 7/2 5/2 0.14±0.04e 0.001 0.16 0.079/2 7/2 0.39±0.14~0.39 0.48 0.5311/2- 9/2 0.57±0.19a 0.41 0.50 0.38

a ref. [25]. b ref. [24]. C ref. [29]. dref. [30].

R and8 from ref. [31],s~from ref. [32].

9.2. Thewave-functions

The readerwho hasfollowed us this far knowsby nowthat we preferdiscrepancyto agreement.Wewould like to explorenow what can besaidaboutthe transitions.

First of all we would like to dismissany criticism aboutourchoiceof effectivecharges:an isoscalarvalueof 0.5 (Z/A) is for usauniversalconstantmadeplausibleby Mottelson’s arguments[35] that willbe changedthe day our knowledge of the dynamicsallows to do better*. As linked cluster theory

* Do we really want to spoil theresultsfor ~ in table6 by choosingsomeothervaluefor theeffectivecharges?We would like to add this to

our list of section8 asArgumentno. 4”.

A. PavesandA. Zuker,Theoreticalspectroscopyandthe/pshell 303

teaches,for thewhole of thefn model thereis oneone body effectivecharge(at most two: isovectorandisoscalar).Once it is chosenorcalculated,thereis no way of changingit. In that senseour choiceisasgood asany, astables5 and6 show.

Next we would like to makesomegeneralcommentsthat link wave-functionpropertiesto whatweseein the transitions.

(1) Energy denominatorproblems. If we refer to fig. lb in section4 we can understandthat underfavourablecircumstancesthe forcemay be suchthat the energyobtainedwith detaileddenominatorsmaybe very muchthesameastheonecomingfrom thecrudercentroIdapproximation.All our resultson energiesindicatethat this approximationis (surprisingly)good.The favourablecircumstanceswouldthenexist in thesensethat

= <i1 ~1.’~’11,11) t~1~ ‘~“I’) (9.5)

whereE is theconstantcentrold,and$., the amplitudeof the J-statein the I dressing.Whencalculatingtransitionthecorrespondingequalitieswould be

~ — (ij VIJXJI[IIi’) (ij V1J)(JI[lIi’) 9 616Al~— EL, —. S

If (9.5) is somehowright and (9.6) somehowwrong it would meana fairly poorwave-functionhasproduceda good energybut whenit is usedfor transitionsit will showits weaknesses.

As exact diagonalizationsin f” + f~t1~spaceswill producewave-functionsthat for all intents andpurposesamountto perturbativecalculationswith perfectdenominatorsfor the V1 partof the force only,we cangatherfrom tables5, 6 and7 that thereis morethan a lucky circumstancein thevalidity of eq.(9.5): thedifferencesbetweenquasiconfigurationsandexact diagonalizationsare negligible: eq. (9.6) isapparentlyvalid.

(2) Third order perturbation theory and transitions. The statementabove concerning the wave-functionsobtainedin a diagonalizationf” + f”

1r mustbeunderstoodproperly:it is the dressingduetothe V

1 forcethat is right. The binding energywill be invariablewrong.Furthermore:for even—evennuclei, pairingeffectscomingfrom V2 are important,andtheir neglectwillyield wrong spectra.In the even—oddcases,thespectradependlesson V2, but it would be a mistaketoconcludethat transitionswill be betterreproduced.

A secondordercalculationfor the transitionsis only sensitiveto V1 and the comparisonmadeinparagraph(1) betweenexactdiagonalizationandquasiconfigurationresultsmakessense.This doesnotmeanthesecondordercalculationis enough:pairingwill beallowedinto playonly in third order.

A recurrenteffect, takenfrom real life shell model calculationswill immediatelyexplainwhy pairingis essential.As an examplewe considerthewave-functionsof

40Ca in an (f712d3,2)

8model [36].The0~GS and3 excitedstatesare

I0~l) ~

13—i)= a’d~,2f7+2+ ~‘d~,2f~,2 (9.7)

BE3 (aa’ + $$‘ + a’$Xds,2lQ3If

7,2)a~a’ ~6—~6.


As f3’ maybe a ratherlargenumber,the $f3’ term is important.It is evident that in a perturbativeapproximationit appearsonly in third order. In somejargonstheeffect would be called an RPA correlation.

This is only anotherexampleof the danger of thinking only in terms of lowest order in perturbationtheory: if pairing is important for the secondorder energies,we cannotforget it for the transitionssimply because it comes in third order.

Unfortunately we have not followed our own adviceandonly secondorder calculationsaregiven inthis paper.

(3) Quadrupolemomentsandf” wave-functions.The two preceding remarks apply to first and second order dressing in the wave-function(secondand

third order energies and transitions). Nowwe turn to zerothorder in the wave-functions.Assumingwe havediagonalizedonly the termsin Wffff of the bareHamiltonian,we havebrokenthe

initial degeneracyand we havethe zerothorder wave-function.Our discussionin section4 indicatesthat this shouldallow us to view the problemas a non degenerateone. It is indeedthe case,and ourwave-functionsthat correspondto well identified experimentallevelsareindeedsolidly dominatedbyonecomponent(0(90%)) in this representation.

Although the improvementoverf”(42Sc) calculationsin tables 5 to 7 is quite clear in general,thereisone casein which it is impressive:for the quadrupolemoments.If onepausesfor a minuteto think onthe sort of mixing necessaryfor theffl (42Sc)statesto reversethe wrong signs, the task is seen to be verydifficult. If now werealize that the zerothorderwave-functionis not given by f”(42Sc) but by f”(KB3)(f” not fn), thingsbecomeseasier: the f”(KB3) wave-functionsdo not give largepositivequadrupolemomentsbut small ones.Half of the work is done: f~(KB3)(or any KBX) are very different fromf’ (~2Sc).

Perhaps,thisis the mostsignificant single exampleof the wisdomof minimumtampering(noticethatmonopoleeffectschangenothing in the wave-functions).

10. The book of arguments

10.1. Threetrends

Now that all the information has been presented, it is possible to extract somegeneral trends.Althoughwe havesingledout someparticularnuclei,all contributeto the list of argumentsin section7.

Argumentof type 1 dealwith even-evennuclei.At the beginning of the shell the spectra arecompressed.In 42Caand“~Cathe trouble is maximal. In~Ca it persists but there is clear improvement. In 50Ti the calculatedspectrumis practically perfect.Strangely enough 50Ti and42Ca have almost indistinguishable level schemes.

In ~Ti again the calculationdoesnot split sufficiently the high spin statesfrom the low ones.In ““‘Tithe improvementis clearand in 50Cr the situationis nearly perfect,as it is in ~Cr.

For these nuclei, cross conjugation and the simple assumptionthat 42Ca,~Ca, 50Ti and 54Co(T = 1)

have identicalspectraseemsto work quite well. This illusion is confirmed by the very nice schemesobtainedwith the 42Sc set of matrix elements.

However,we knowquitewell that intrudersfrom the sd shell arepresentandplaya determinantrolein positioningseverallevelsin the neighbourhoodof 40Ca(e.g.the2~statesin 42~Caareexperimentallyan even mixture of f” and intruder).We could expect thereforethat the choice of the experimental

A. Pavesand A. Zuker, Theoreticalspectroscopyand the j~shell 305

schemeof 42Sc amountsto a drastic renormalizationof the two body interaction to accountfor theintruders.As thequasiconfigurationresultsignoretheseeffects,it may comeasno surprisethat f”(425c)gives betterresultsthanf”(KBX) at the beginningof the shell.

Pushingthe argumentfurther,we noticethat very soonquasiconfigurationsbecomecompetitiveandoutperformthe f” (42Sc)calculation.But then, why is it that whensd intrudershavemovedout of thepicture, f” (4~Sc)keepsproducingsomespectrathat arealmost identicalto thoseof ffl (KBX) and theexperimentalones?The situationis particularlystriking in 50Ti, 52Cr, 52Fe.

Now we forget for a minute the particularf” (42Sc)calculationand keep in mind only the generalpredictionsof anf” modelaboutcrossconjugation,andcomparequasiconfigurationsandexperimenttofind thefollowing generaltrend:Trend 1: In even-evennuclei quasiconfigurationshavesomeproblemsat the beginningof the shell,mostprobablybecauseof neglectedintrudereffects.Starting atmass46 thecalculationsimproverapidlyandtowardsthemiddle of theshellthey do very well andevenmanagesomevery precisepositioningofisomersat theendof theshell (12” in 52Fe,not an accident,asweshall see).Whenmovinginto the shellthe internalfp dressingproducesalmostthe sameeffectson the spectraas thosedue to sd intrudersnear

Argumentsof type 2 dealwith odd—odd nuclei.Here thereare good theoreticalandexperimentalreasonsnot to expectintrudertrouble andindeed,starting in ~Sc, quasiconfigurations yield very good spectra.Not only the ground statespins areinvariably well reproduced,but the whole level sequencesappearin the right order with very fewinversions.Thereaderis askedto bewareof fictitious shifts, e.g.: the T= 0 spectrumof “~Vis nearlyperfect. It is the T= 1 0”~GS that is off position by some300keV as can be checkedthrough thebinding energytableand figure (section8).

Crossconjugationis oftenseverelybroken.The sensitivity to thesingle small changebetweenKB2andKB3 is ratherstrong andshowshow severelytheodd-oddnuclei test the interactions.Henceit isdifficult to assessthe relativeimportanceof denominatortroubleanderrorsin theforce.The situationis summarizedinTrend 2: In odd-oddnuclei quasiconfigurationsdo well from the beginningof theshell most probablydue to the nearabsenceof intrudereffects. Internal fp dressingproducesmany rearrangementsthatexplainquite systematicallythebreakdownof crossconjugatesymmetry.Thesenuclei aresevereteststhat demonstratethe soundnessof the realisticinteractions.

Argumentsof type3 dealwith odd-evennuclei.In even-evennuclei normal parity intrudersare presentand active, in odd-oddnuclei non-normalparity intrudersarepresent,but, perforce,nonactive.In even-oddnuclei, both normal andnon normalparity intrudersare present and the former may be quite active or not at all. Furthermore,theexperimentalinformationis often unreliable.Finally, evenwhenintrudershavegoneand experimentsareclear, thesenuclei haveidiosyncraticbehaviour.On thecomputationalside, they showmore oftenthantheirevenandodd counterpartsclearsignsof denominatortrouble.To this we may add that crossconjugationis broken,but it is difficult to discerna clear pattern,mainly due to the accumulationofuncertaintieswehavementioned.

To understandour complaints,let us start by examiningthe quartet43Ca, 53Mn, 45Ca, 51V, whichshouldhaveidenticalspectrain thef” model. 43Cais not very good inf3 andslightly worsewith f3(425c).If thereis anything like a patternto be found we would try the following for the f3 calculations:disregardfor thetime beingthe3/2, andnoticethat theotherspins(5/2-, 11/2, 9/2 and15/2) wouldwelcomean upwardpushof some200—300keY.This is very muchaneffect to be expectedfrom thesd

306 A. PosesandA. Zuker, Theoreticalspectroscopyand the/pshell

intruderswhich depressviolently the0~T= 1 GS of 42Caand areexpectedto actselectivelyon statesof low seniority (7/2 in 43Ca is the only statewith seniority 1).53Mn is peculiar: the f’3(42Sc) calculation is far more adequatethan it wasin 43Ca,andseemsto do aswell or betterthanf13. However:for quasiconfigurationsa trendis clear: the groundstateis underboundby some300keV. This time it is not the fault of the intrudersbut cleardenominatortrouble.45Ca: very much the samestory as in 43Ca: we would like to decompressthe spectrum(this time weforget about5/2 ratherthan3/2).51V: exactlythe samesituationfound in 53Mn.

Perhapsit is useful to call attentionon the spectraof 43Sc and53Fe: quasiconfigurationsdo a nearlyperfect job in the first and a good one in the second(notice againthe isomer19/2-) only marred byoverbindingof some300keY in the 7/2 GS. The ffl (42s) spectraare definitely poorerin both nuclei.More interestingfor our caseis thecleardisappearanceof a low lying 3/2 from both calculations,whileit exists in both nuclei. Which bringsus to the only Argument3 explicitly mentionedin section7.

The 3/2 statesare alwaysprominent low lying membersof the spectra.Experimentallythey keepgoing down until midshell (GS in 47V) and then go up again.Within denominatortrouble corrections,the trend (that breakscross conjugation)is well reproducedby the calculations.At a finer level itappearsthatwhensdintrudersarepresent,thecalculationsdo notmanageto depressthestatessufficiently,while theydo an increasinglybetterjob whenmovingaway from 40Ca.By thenfailurescan beattributedquitesafely to denominatortrouble.

It is difficult, for reasonsthat will be clearvery soon, to extracta trendthat parallelstrends1 and2.Forodd nuclei the resultcan only havethe following patchworkcharacter:sd intruders may or may notbe determinant at the beginning of the shell. Cross conjugate symmetry is broken. Quasiconfigurationsreproduce the trendseverywhere,sometimesfine details (isomers)areexplained,but thereis a strangetendencytowardglobal level shifts that can be attributedto sd intruderswhenthey arepresentandtodenominatortroublewhenthey areabsent.

At this point, Dr. Watsonfeelsquite gloomy while SherlockHolmesshowssignsof contentmentanddoesnot evenbotherto askfor the wave-functionsto explainwhy thingsareelementary.

10.2. Thevarious intruders

Already in trend1 it is quite obviousthat therearetwo sortsof intruders:thosecoming from sdshellexcitations(the intruderfrom below)andthosecomingfrom fp shell dressing(the intruderfrom above).Theseintruders,madeof manystates,manageto obscurethe picture, by detachingoncein a while asmall individual intruder that will wreck havocin somelevelsthrough local bandmixing. In fig. 3 for43Sc we haveexamplesof small intrudersfrom aboveand below having the samequantumnumbers(3/2). We havenot shown the wave-functionsof 42Sc, but 5~T = 0 is pushedaroundby an intruderfrom abovein very muchthe sameway 2~T = 1 is known to be the victim of an intruder from below.

Although they indulge occasionally in the disturbing practical joke of planting decoys, the bigintrudershavesolid secularhabitswhich includeextremerespectfor oneanother:whenoneis at work,the otheronewaits for its turn.

It so happens,that in even-evennuclei both intrudersdo the samejob, which, by and large, ismocked by a single two body f2 force. It explains why f”(42Sc) works so well for so long, whilequasiconfigurations(which containonly the intruderfrom above)wait until the intruderfrom below hascompletedits taskto takeover.

In odd-oddnuclei, the intruder from above acts alone, and the break-downof cross conjugate

A. PavesandA. Zuke~Theoreticalspectmscopyandthe/pshell 307

symmetryshowsthat its effect is not dueto a two body force. In even-oddnuclei, it would appearthatit is theconstanttwo body forcedue to both intrudersandthethreebody forcefrom above, that areatthe origin of the complexsituation.To someextentthis is true,but a closerinspectionwill bring somesurprises.

Theintruderfrom belowcan only be somethingof the form f”÷2d2.We arebeingratherschematic,

in that f” hasto be fully dressed, i.e. statessuchasf~1f512d~sj~,andmanyothers,play a role. Still, the

consequenceof thistypeof dressing,as can beeasilychecked,is an effective1+ 2 body force. This 1+2bodyforce is not constantandgiven onceandfor all in the

42Scspectrum.The energydenominatorswillhavea ferociousstatedependence(niT at least)if wewant to do well in low orderperturbationtheory.A constanttwo body forcemay be a good,butdeceptivesubstitutein theeven-evennuclei: in 43Ca itshowsalreadyits flaws.

The intruderfrom aboveis eitherf’2r2 or fn_1,.~As we shall showit is f’~’rby itself that carriesmostof the responsibilityfor thechangesasit producesthe 3 body force that is essentialfor breakingcrossconjugatesymmetry.In the4n and4n +2 nuclei,thef”2r2 dressingis important,but its influencediminisheswith increasingisospin. It is quite easyto realizethat in thepair~‘Fi—50Cr,thedifferencesarenot due to the small n dependencein thedenominatorof the two bodyforce,but to thestrong3 bodyeffect. Its efficacy can be judgedfrom the fact that theenergydenominatorsincreasethroughouttheshell, while the shifts insteadof diminishing becomestronger: the betterperturbativeconditionsarefulfilled, thestrongerthe effect. We haveherea cleansignatureof linked cluster theory.(Explanation:therearemanymorestatescontributingto the intruder.)

It remainsto be understood,or at leastshown, how a threebody forcecan do so remarkablythesamethingasa two body onein evennuclei.

10.3. Thethreebodyeffects

Argumentsof type4 showwhat the threebody forcedoes.Herewe can go ratherfast. In table3 it is shown that the spectrumof ~Sc is due to mixing throughHRI, ~ is unimportant.Had the same table been written for 50Ti the results would have beenidentical: it is mixing with fr thatproducestheenormousshift in the0~state.It maybe strangethat a 3body forceproducesan effect almostidentical to a two body pairingforce (asthe oneresponsibleforthe spectrumin 42Ca), but this is what happens.That the two body contractionin HR1 is not veryimportantcan be checkedfrom theanalyticexpressionsin theAppendix.

Furthermore:oneof theeffectsthat is clearly seenin table3 is theaverageshift of the T= 3 stateswith respectto the T = 4 state.

This isovectorchangeis quite similar to thestrongshift downwardof the T 1 stateswith respecttothe T=0 onesin 42Sc.

Evenmore strangely:thespectrumof MCo is quitecloseto the (fr)2 (or /~,not shown) spectruminfig. 1: the intruderfrom abovedresses42Sc andMCo in very muchthesameway. Rememberalso that,hadwe chosenthe ~Colevelsto calculate~Scwith aPandyatransform, the T= 4 statewould havebeentoo low by almost2 MeY, while we get ~Scand~Cosimultaneouslywell. The threebody partofHR1 seemsto producewhat’s neededdependingon thenucleus:for ~Scan isovectorshift suchas theonedueto the intruderfrom below in 42Sc, but without thedefectsthat f8(42Sc)has.For 50Ti we getnearlythesamespectrumasin 42Sc, but in additionwe haveperfecttransitionrates.For MCo, wehaveagainagood spectrumin which the isovectorshift and thestrong pairinghavedisappeared.

308 A. PosesandA. Zuker, Theoreticalspectroscopyand the/p shell

The threebody effectiveforce is really a jack of all trades, that producesdifferent effects in eachnucleus.For theenergies,in someeven-evencases,it can bluff usby taking the guiseof a constanttwobody force. But we can call the bluff in the odd nuclei and in the transitionrates(and quadrupolemoments).

10.4. Couplingschemes

In the competitionbetweenH and HRI in eq. (3.8), H is the winner in the fp shell: we call thissituation j—j coupling. Sometimes,however,f” containsf” componentsthat are dominatedby aneigenstateof H but mixed appreciablywith others.Here we move towardsan intermediatecouplingschemethat will be reachedwhenH andHR1 become equally important. It maywell happen,that insome cases, not in the fp shell, but elsewhere,HRI will overwhelm H. Then, and we are onlyspeculating, we shall speak, perhaps, of the rotational coupling scheme.This is the wild shot in theBook of Arguments.

Appendix A

Some familiarity with ref. [14] and its appendix [37] is usefulhere

A.1: Generalizedparticle—holetransform

As it is well known, for two body forces,the spectraof two particlesin a shell andthat of two holesin the completeshell areidentical.Whenquasiconfigurationsareused,we haveto deal with two + threebody effective operators,and then things becamemore complicated,but still it is possible to writeanalyticformulaefor the transformation.We shall study the renormalizationcoming from V1 excita-tionsthat is

1 rrr’ll/2HR

1 = — (—1)”’~” W’~W~’[((AA~B)’(A(BB)T’)’]°. (A.1)

The particle—holetransformconsistsin makinga changeof vacuum(in ourcasefrom 40Cato ~Ni),that implies the following change in the operators

A—~B B-~(-)2~’A (A.2)

the barred operators act on the new vacuum.Toget theright operatoron thenew vacuum,wefirst write (A. 1) in antinormalorder,andthen apply

the p—h transformabovewhich leavesus with the operatorson the new vacuumin normalorder. Tomakethe algebraclearerweshall refer to thegraphsof fig. 44.

We takethe coupledoperatorin (A.1) which can be associatedto graph(I) and recouple,sowe get

(I) = U(fFfF’; rF”)(—1)T~T’ (II). (A.3)

A. PavesandA. Zuker, Theoreticalspectroscopyandthe/p shell 309

B//\~1

Fig. 44. Schemesof theoperatorcouplings usedin thegeneralizationof particle—holetransform.

Usingnow thecoupledcommutator

[(BB)T,(AA)’1’~= —2[fl”~öp’.o — 4[FFh]1~12{~ I I

we obtain

(II) = _[fl]1~~2. 2~8~-~-8,~(III)

~4. (— i)~”’’”’~ U(f”JT”f ; if’) (IV) (A.4)

+(—1~~T’~(V).

310 A. Foveaand A. Zuker, Theoreticalspectroscopyand the/pshell

Combinationof (A.1), (A.2), (A.3) provide the generalexpressionin the new vacuum. As we areinterested in the two hole case, the three body part will not actso wejust write the two +onebodypart.

First, we can rewrite

(IV) = ~ U(ffff; AF~)[(BB)A(AA)A]o (A.5)

andafter somealgebraobtain

HRI = 2 [F] ,r21 ~A[(,~4)A(E,~)A]O[A]l/2 (A.6)

where

fA f fl= —4~[FFl]WI’1vr’(_1y+~ ~ f F f ~. (A.7)

r F’J

So we have found the level displacement in MC0due to the V, renormalization, given simply by ~A

A.2: GeneralizedPandyatransform

Now the changeof vacuumis doneonly for neutrons.This introducesgreatercomplexitybecauseoneis forced to work in n—p formalism.Anyway the taskis feasible.

The nondiagonalHamiltoniancan be written in isospin formalismas

V1 = — ç~=~ [F]1”2WF(RAAr)“ (RB)” ]0 + [(AA)T(BB’~’]°).

When going to n—pformalism and taking into accountthat now, protonandneutronfields commutewe can write

V1 = — ~ EJ]”

2(W~r{(AnAnY(BnB~nY+ ~ + samefor p} — ~

+ ~ ~ + (A1,A~)~(B~B~)’}). (A.8)

The change in operators associated to the vacuum change is

A~—t’A~ B~—*B~

A~—*B~ B~-~(-1)2tA~.

Wecan then write the Hamiltonian in terms of the barred operators (asno confusionis possiblewedrop the bar), introducingthe phases.Next stepis to performthe sumover intermediatestatesin eq.(3.15). Onemustbevery careful with phasefactorsdueto the differencein commutationrulesbetween

A. PosesandA. Zuke, Theoreticalspectrascopyandthe/pshell 311

B; N BNr’ AN

A~ A Bp

AP,~~,:N Ap~~,B p

Fig. 45. Schemesof the operatorcouplingsusedin thegeneralizationof Pandya’stransform.

n—n andn—p. Finally one can write (A.1) in the following form

1 Ff1111/2— — I j_1\f~~_J’L~~’J I_1\I~~’kARi — 2 ‘~ ‘~‘ [rJla k ~) (~k k k~

Where thereare eight possible crossings:(~1)~” is the phasecoming for the r-spaceoperatorsextraction;~k the phaseassociatedto the changeof operators;°k is thecoupledoperatorand flk thematrix elementassociatedto it usingexpression(A.8). Theoperators°k arepictured in fig. 45 andthe

312 A. PosesandA. Zuker,Theoreticalspectroscopyand the/pshell

correspondingphasesandfactorsare:

k 11/k c6k 11k

1 1 ()2f w’W~’,J, J’ even

2 0 1 (—1~’W~W~,Jeven

3 0 1 W~W~’4 1 (_)21 —W~W~,Jeven (A.10)5 1 1 (—1)’W’W~’,J’even6 0 ()2f (—)‘~~‘W’W~7 0 ()2f — W~W”,J even8 1 1 W~W~’,J, J’ even.

We shallnot studythe completeHamiltonianreferredto the newvacuumbut only its two body partleadingto the~Ca/~Scand50Ti spectra.So we haveto expressoperatorsk = 1 to k = 8 in normalorderandkeepthe contractionsup to two body terms. Clearly operatorsk = 4 andk= 7 do not give such acontraction.

After some work of commuting, recoupling and using sum rules for Racah coefficients weobtain

(k = 1)=2(_ly+~1’d~~~(A~B~)° (A.11)[f]l/

2

If J’ f”I—4 f r J ~.[JJ’Ar]”2[(AA)’~(BB)’~]°

~A f fj

f f J’l If f J} ft f A ‘~ ((A~A~)A(B~B~)A)o(k=2)=2(_1Y+’+f+~[J.PrA][u]{f r rflf r r f f o~J

(k = 5) = 2(~1~’- . (sameask= 2)

f f RIf f J’l(k = 3) = ()J’+3f_r[JJlrA]l/2 {~ r AJ it r A ,ç ((A~A~Y’(B~B~Y)°

(k = 6) = (...)J_f_r+l[AT]l/2 ~ ~} 5jj,[(A~A~)A(B~B~y]o

— ()J’±A±f_r[JJlAr]I/2[ff] {f f “i ff ~ il. Jf f Aj ((Ar,Ap)A(BpBp)A)Of r a-if r a-JEff a-i

+ (—)~‘~ (A1,B~)°6~~.

(k =8) = (1)J-f~r~

‘A [A]~2~

Now, rememberthat, only (A~A~)(B1,B0)operatorswill contribute to

50Ti, only (A~A~)(B~B~)to

A. Pavesand A. Zuker, Theoreticalspectroscopyandthe/p shell 313

~Ca, and only (A~A~)(B~B~)to ~Sc.Observingthis andputtingresults(A.11) and(A.10) in eq. (A.9)weget for the energycontributionof V1 excitationsto thesenuclei the following final expressions:~Ca.

F-Il ~f -I’ flAE(A1)= ~ W~’J

2. 2 ~f+ ,~ 4~(—it’ .~f r J }. [JJ’]W1W” (A.12)Jeven IJJ 1, even I

1,/Il

50Ti.

~E(A1)= ~ w.nI2f~+ ~WAhI

2_ ~ (—1~~~’[JJ’a-]- 1 ~T~}{f f J} g ~ Ai]. WJWJ’

~Sc.

~

— J 6 W~2ft f J l~ ~ If f J 1 If f J’] W~W”H j” 2 if f A15 ~ ~

11..f f A15 if r AjI 2

+2 ~ (~1)A~+~’~~[JJ’o’].ff f ‘ ~.~‘ff “ ~ff t A1~ ~JwJ~~.reven,J’ tf r a-j tj r a-) U f a-J

To get the spectraone must add to expressions(A.12) the parts coming from the original f~,2interaction and from the pairing renormalization, that, being both two body forces obey the usualparticle—holeor Pandyatransformsfollowing thecase.

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Theoretical spectroscopy and the fp shell

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Transcript of Theoretical spectroscopy and the fp shell