Nuclear Physics AI07 (1968) 1- -13 ; (~) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

TWO-NUCLEON TRANSFER AND PAIRING PHASE TRANSITION

R. A. B R O G L I A and C. R I E D E L t

The Niels Bohr Institute, University of Copenhayen, Denmark

and

B. S O R E N S E N

NORDITA, Copenhayen

Received 2 October 1967

Abstract : A sys tem o f n particles mov ing in two non-degenerate j-shells and interact ing via the pair ing force is amenable to an exact t reatment . This mode l exhibits a phase t ransi t ion, namely the one between no rma l and superconduct ive phases. The behaviour o f the cross section for the two-particle t ransfer react ion in the seniority zero channel is studied. Special a t tent ion is paid to the t rans i t ion region.

1. Introduction

The two-level model was used among others by H6gaasen-Feldman 1) to check the energy spectrum predicted by different nuclear models against an exact calcula- tion for the case of a pairing residual force.

Recently, attention has been paid to the existence of vibrational modes based on fields which create or annihilate two particles 2, 3). One such field is produced by the pairing interaction. The corresponding mode is the pairing vibration. This state can be reached in a two-particle transfer reaction. Experimentally, the pairing vibration has been observed as an uncoupled mode in the 2°6pb(t, p)2°8pb reaction 4) and coupled to other degrees of freedom (especially the quadrupole one) in different regions of the mass table 5). Theoretically, the main features of this degree of freedom were studied in the frame of the linearization method both for the superconductive 3) and non-superconductive cases 3,6). In the framework of the two-level model, we make a detailed analysis of the way in which the two-phonon pairing vibration dilutes into the one-phonon pairing mode as one goes out from a closed shell. We study also the behaviour of the matrix elements of the pair field in going from the non-superconductive to the superconductive phase. This knowledge can improve our understanding of the changes that take place in the pairing coupling scheme along the transition region.

2. Exact solution

In this section we follow closely ref. J). We consider a system consisting of pairs

t Pe rmanen t address: Central Inst i tute for Nuc lear Research , Ros sendo r f near Dresden, G e r m a n y ( D D R ) .

1 January 1968

2 R . A . BROGLIA et aL

of fermions, each of which occupies time-reversed orbits. Two j-shells are available which can have the same or different degeneracies. Let D be the distance between the shells. As the residual interaction, we use a pairing force with constant matrix elements. The Hamiltonian of the system is

H = ½D(N2- N,)-¼G(31 A + +32 A+)(), A, +J2 A2), (2.1)

where

_ 1 + + o A+ = x/'oil .>oE (--)J-"afma~..-m 4 2 [aj aj ]0,

N i = ~. aj,,ajm+ ( j = 1, 2), m

J = (2j + 1) ~r. (2.2)

Assuming n pairs to be present in the system, we can introduce an orthonormal basis

[m, n - m ) - 1 + m + n - m = Mm (A , ) ( A 2 ) [ 0 ) . ( 2 . 3 )

The matrix element of the Hamiltonian (2.1) in this basis is

<m', n - m ' l H I m , n - m >

= 6m,,,,[(n- 2m)D - G(m(O 1 + 1 - m) + ( n - rn)(O 2 + 1 - n + m))]

- - 6 m ' ( m - 1 ) G [ m ( 0 1 + 1 -- m)(n -- m + 1)(O 2 -- n + m)] ~

- 6,,,tm + ,)G[(m + 1)(O, - m)(n - m)(O 2 + 1 - n + m)] ~, (2.4)

where O ---- ½(2j+ 1) is the degeneracy of the j-shell. The overlap integrals that enter in the evaluation of the cross section of a two-

body transfer reaction are the matrix elements of the pair field operator T~ = A +. Each term is weighted with the amplitude of the corresponding components of the initial and final state wave functions and with geometrical factors 7). In the frame- work of the two-level model, we define the cross section with which a state is ex- cited in a two-body transfer process as the square of the matrix element of the operator

T = A + +A~'. (2.5)

This definition of the "two-body spectroscopic factor" is a reasonable one if the degeneracy of the two levels is equal, and if both have similar radial dependence.

In the basis (2.3), the operator defined by eq. (2.5) has matrix elements given by

<m', n+ l - m ' l T I m , n - m >

= a . , . E ( n - m + )(a2 - n + , , , ) l a d ½

+ 6 m ' ( m + 1)[( m + 1)(O1 -- m ) /O l ] ½.

From the commutation relation [Ni, A s] = 6fs2A +, one can

(2.6)

easily calculate the

T W O - N U C L E O N T R A N S F E R 3

occupation number of the two orbits

(a lNl la) = 2 ~ talC., m[ 2, m

(a[N2[~) ---- 2 ~ (n-m)[Cn, m[ 2, m

where the eigenfunctions of the total Hamiltonian are

I~) = ~ C,,mlm, n - m ) . m

(2.7)

(2.8)

3. Approximate treatment of the pairing Hamiltonian

In the following, we consider the two levels to have the same degeneracy, the lowest one to be completely filled in the limit of G = 0.

3.1. NON-SUPERCONDUCTIVE CASE

For small values of G (x = 2Gf2/D < 1, see below), the collective degree of freedom associated with the pairing Hamiltonian can be expressed as a combination of 2p-2h intrinsic excitations (with energy E = 2D). In the RPA, the energy of the pairing vibration is

co = 2W = 2 D( 1 - x ) ~, (3.1)

where x = 2Gf2/D gives a measure of the pairing strength as compared with the single-particle splitting. The quasi-boson creation operators are

( l-x) D-W D+W/'

F ~ - - GO ( A~- + A, ) (3.2) ( l - x ) * D - W ~ - W "

The conservation of the number of particles implies that in this phase A = ( T ) = 0. The fluctuations around this value correspond to the two-phonon state 3)

Ipv) = r:r lb>, (3.3)

with energy Eo + co, 10) being the correlated ground state. The intensity with which the ground state and the pairing vibration are excited

in a two-body stripping reaction is proportional to the following matrix element:

[(g.s.(n)l Zlg.s.(n - 1))l~pA = I(pv(n)[ Zlg.s.(n - 1))[~pA

= ( 1 -- X) ½, (3.4)

where ]g.s.(n--1)) = F~+[0) stands for the ground state of the system with n = 2(f2-1) particles. This matrix element approaches linearly to 1 as x ~ 0.

Assuming the backward going amplitudes to be zero, the state given by eq. (3.3)

4 R. A. BROGLIA et al.

has an energy equal to 09 = D ( 1 - x ) , the ground state now being the "closed-shell" system. The corresponding matrix elements of the pairing field do not depend on x and are both equal to 1.

As x increases, a~ decreases due to the fact that particles gain energy through the pairing force. The transition between the non-superconductive to the super- conductive phase takes place t for x = 1. This behaviour can be compared to the phase transition between spherical and deformed nuclei.

3.2. SUPERCONDUCTIVE CASE

To take into account the main effect of the pairing correlation for values of x > l, we make use of the quasi-particle transformation

+ Vj ( - )J -"a j_ , , , (3.5) O~jm = U j a i m -

Due to the symmetry of the model, the Fermi energy is equal to zero. The parameters U and V are

U 2 = V 2 = ½ ( I - l / x ) ,

U2 z = V 2 = ½(1+ 1/x). (3.6)

The quasi-particle energy is ~ = GI2. The system possesses now a permanent pairing distortion

x /~(0 lTI0) = A/O, (3.7)

where A = Gf2(1 - l/x2) ~ and I0) is the BCS ground state. The fluctuations around this equilibrium distortion are induced by the residual interaction among the quasi- particles

where

tt HR = H'p + Hp,

I-i; = Z +dr) ) ' , J

H; ' = ¼G( X x / ~ J ( d f - d J)) 2, J

d ; -

(3.8)

This residual interaction gives rise to two collective modes which are a combination of two-quasiparticle intrinsic excitations. The first corresponds to the one-phonon pairing vibration and has energy o~p = 2A. The second is the spurious state associated with the two-quasiparticle (v = 2) channel and has energy ogs = 0. Its wave function is proportional to N acting on the ground state of the Hamiltonian (3.8) [refs. 1,s)].

* For Jl ~J~, the phase transition takes place when the condition G(~F2~+X/F22)~/2D = 1 is satisfied.

TWO-NUCLEON TRANSFER 5

The creation operator of the pairing vibration is given by

1 Gg2 1 . d +

q~"+ = x~( l . - l l x~) ÷ { 2 G ~ - % x/2 ( 2 - d : )

G[2 1 } 2Gt2+Ogp x/2 ( ~ ¢ 2 - d l ) • (3.9)

The square of the two-body matrix elements is equal to

I<g.s.(n)lZlg.s.(n- 1)>12pA 1 ~ D 4 2

-- 0 - - 8( oo+o,)o, oo) '

1 1 (3.10b) I<pv(n)l TIg.s . (n- 1))i2pA = 2 X(X 2 - 1) * '

where n is the average number of pairs in the final nucleus. The same magnitudes in the BCS approximation are

I < g . s . ( n ) l T l g . s . ( n - 1 ) > 1 2 c s = t - ~ f2, (3.11a)

1 (3.11b) I<(v = 2)(n)l TIg.s . (n- 1)>l~cs = 2x-- i ,

where 1 . d +

corresponds to the boson creation operator %+ in the independent quasi-particle approximation.

TABLE 1

Limiting values of al and a2 (superconductive phase) in the RPA (BCS) approximation for x ~ 1 and x ~ o0

x ~ l

Kg.s.(n)lTIg.s.(n-1)>l ~ I<pv(n)lrlg.s.(n--1)>l ~

g.s.c. 1/16~Q m

BCS o ½

x ---->- ct3 g.s.c. -Q 0

BCS /2 0

In table 1, we give the limiting values of the squared amplitudes (3.10) and (3.11) for x ~ I and x -~ m. The corresponding values for the energy of the pairing mode are O)p = 0 and O)p = Gf2 = d'. For x ~ m, the symmetric two-shell system is equivalent to the degenerate model in which the only possible excitations are quasi- particle excitations, and no collective root exists.

6 R . A . BROGLIA et al.

4. Discussion of the results 4.1 . E X A C T S O L U T I O N

I f not explicitly mentioned, all our calculations were per formed for the case of a two-level symmetr ic system with degeneracies f21 = f22 = 20.

I 1.0[

0.8

0.6

0.4`

0.2

0.0

ICn, m l 2

l(g.s) I n = 2 0 I ~ x = 0 . 5

i l i l l

" I . +

I I . I b / l !1 I

]~i ~'\ ~\ ( n - m )

1.o

o.8

o.6~

0.4`

0.2

oo

I C n , m l 2

n = 2 0 x= 1.0

i /(g.s)

\ \ I \ ', ~ l " \ \ \ ( n - m )

2 "~ "~" ~" 6

1.o

o.B

0.6

ICn ,m l 2

n = 2 0 x : 2.0

0.4` c,/o;) cob ~.~1

/ x" ", / \ # / % / ~ , ~ . , , 'N . . . ~ / . / X ;'¢'T " V " . ) < ' . ~ \ ' ~ . " ' - . ( n - m )

o o i _ . / ~ \ / i \ - - , / " . . l . ,,'' ~ - . , ~ - .~ - . . - . . . . ,.x___ 0 2 4, 6 8 10 12

F i g . 1. S t r u c t u r e o f t h e e x a c t w a v e f u n c t i o n o f t h e g r o u n d s t a t e a n d f i rs t t w o exc i t ed 0 + s t a t e s f o r d i f f e ren t v a l u e s o f t h e p a r a m e t e r x .

In fig. 1, the magni tudes [Cm, nl 2 are displayed [see eq. (2.8)] corresponding to the ground state and first two excited states as a function of the number of excited

T W O - N U C L E O N T R A N S F E R "]

pairs (n -m) for the case of n = 20 (closed shell) and for different values of x. As x increases, the sharp Fermi surface gets more and more diffuse. The corresponding occupation numbers are given in table 2. In fig. 2, we plot the squared amplitudes of the ground state closed-shell system for different values of the parameter x.

1.0t ICn'rn 12 j ( x =0.5)

0"8 f~(X = 1.0 )

0.6 [-~

0.~ = .

(x =1.6)

0.2 = " ~

0.0 0 2 4 6 8 (n-m)

Fig. 2. Probability of having different numbers of excited pairs in the exact ground state. The st1 ucture of these wave functions shows how the Fermi surface gets more diffuse in going from the non-

superconductive to the superconductive phase (see also table 2).

The behaviour of a 1 oc I(g.s.(n)lTIg.s.(n-1)>l 2 and a2 oc I(pv(n)lTIg.s.(n-1)>12 near closed shells is displayed t in fig. 3. The interplay between the pairing type of cor- relation and the shell structure is well illustrated by the trend of tr 2 for different values of x. For x > 1.4, the crossing of the closed shell is not felt by the "cross section" where, as for x < 1.4, there is a sudden drop in tr 2. This is connected with the fact that in the non-superconductive phase, the pairing vibration is a two-phonon state [eq. (3.3)] whereas, in the superconductive case, it is a one-phonon type of excitation [eq. (3.9)], the closed shell being defined as a state with no phonons. In both cases, the two-body transfer operator apart from anharmonic terms is a

t We neglect any dependence of the cross section on reaction kinematics.

15

10

I I

I I

18

19

20

21

(n-l

)

× =

2.0

)

X=

1.5

)

x=1.4

) x=

1.3)

X=

1.2

)

X=

l.0

X=

08

X=

0.5

/ \

I<pv

{n)lT

Jgs(

n-1)

>12o

co'2

e

xact

/

\ /

/ .

.-"~

\

(× =

1.0

1 /"

.-

"

( x

: o.

8)

"" ..o

-'°"*'

\ ~\~\~t

X=x= o

.5~.2) I o

/- \\

\\\ \\ ',\\

\\x

t

\. (x

= l

a) .

_~.

\,&\~

\,

(X

1.5)

x ---

..x ....

.~< .

._.. x

_....

×

)...

(X =

2.0

) ..

~(~

..~

;.~

..~

f ....

........

........

........

........

....... l~

t --~-"

-4~

18

19

20

21

(n-l

)

Fig

. 3.

Th

e cr

oss

sect

ions

(rl

an

d

~r2

calc

ulat

ed w

ith

the

exac

t w

ave

fun

ctio

n a

re p

lott

ed f

or t

he c

ase

Ot

= -(

22 =

20

an

d

for

"tar

get

nucl

ei"

wit

h a

nu

mb

er o

f pa

rtic

les

n <>

20

(clo

sed-

shel

l sy

stem

). T

he

chan

ge i

n th

e co

upli

ng s

chem

e as

fun

ctio

n o

f x

is c

lear

ly s

ho

wn

by

(ra.

A

valu

e o

f x

> 1.

4 is

ne

eded

bef

ore

the

shel

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fect

s ar

e w

ash

ed o

ut.

Th

is f

eatu

re s

eem

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be

fou

nd

in

actu

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ucle

i xo

).

I=

0 0 r.

TWO-NUCLEON TRANSFER 9

one-phonon operator. I t is interesting to point out that the non-superconduct ive

type of behaviour of 0-2 is observed through the t ransi t ion region, and that a rather

large pair ing coupling constant is required (x ~ !.4) to wash out this shell depend-

ence.

I< pv in) I T 19s ( n-l} >l~exaet I<gs(n)l Tlgs (n-1)>12exact

1.0

0.5

X I I I

0 1 2 3

Fig. 4. Ratio tr~/a 1 of the exact cross sections as a function of x for the case of n = 20.

TABLE 2 Occupation numbers of the two levels corresponding to the exact ground state and first two excited

0 + states for different values ofx

g . s . Ox + 02 +

N~ iV, N~ N~ N~ N~

0.6 39.94 0.06 37.87 2.13 35.84 4.16 0.8 39.72 0.28 37.52 2.48 35.50 4.50 1.0 39.22 0.78 37.15 2.85 35.23 4.77 1.2 37.50 2.50 37.10 2.90 35.12 4.88 1.6 32.94 7.06 34.10 5.90 34.76 5.24 2.0 30.30 9.70 30.68 9.32 32.00 8.00

Experimental evidence for a non-superconduct ive behaviour of 0-2 as one crosses a closed shell is found in the 2°6 '2°apb(t , p) reaction 4'9) and in the 46'4aCa(t, p)

reactions 10),. For the superconductive phase, the corresponding experimental

evidence is found in different places of the mass table 11).

In fig. 4, the ratio a2/al is shown as a funct ion of x. This magni tude goes from

0 " 2 / 0 " 1 = 1 for X = 0 to 0"2/0-1 = 0 for large values of x and behaves smoothly

th roughout the t ransi t ion region.

t In the framework of the two j-shell model, one can use the (p½, g~) levels in the Pb case and the (f~, p~.) levels in the Ca case. The value of the x-parameter needed to obtain the pairing vibration at 4.9 MeV (~°aPb) and 5.5 MeV (4~Ca) is equal to 0.5 and 1.2, respectively.

10 R . A . BROGLIA et al.

In fig. 5 are p lo t ted the absolute values of a 1 and tr 2 as a funct ion of x. The in-

crease of a I does no t occur at the expense of the pa i r ing v ibra t ion "cross sec t ion"

only. As x increases, all the two-body J = 0 channel intensi ty is collected into the

g round state. This result is in agreement wi th the avai lable exper imenta l mater ia l 11)

and was first po in ted out by Yosh ida 12).

4.2. APPROXIMATE SOLUTIONS

In figs. 5 and 6 we p lo t the values of cr 1 and a2 for the different app rox ima t ions as

a funct ion of x. F o r the non-superconduct ive case, the R P A turns out to be a ra ther

p o o r app rox ima t ion for values of x ~> 0.7.

20

15

10

X 0 I

0 3 4

-7S

/ 1 2

Fig. 5. Absolute value of tr 1 and tr 2 for the different approximations used in diagonalizing the pairing Hamiltonian. For comparison, also the exact cross sections are plotted. The explanation of the drawing is given on fig. 6. The curve (-- • - • - ) corresponds to or1 = I< g.s. (n--1)1TIg.s. (n)>l'ecs

[<g.s.(n--1)l Zlg.s.(n) >12RPA •

F o r the superconduct ive phase, we see f rom fig. 6 tha t even if the R P A represents

an improvemen t over the BCS t rea tment , a z comes close to the empir ica l value only

when x ~ 2. F o r smaller values, the large differences po in t to s t rong anha rmon ic

effects. F o r x --, oo, the two-level system f21 = f22 = 20 goes in to the degenerate mode l

TWO+NUCLEON TRANSFER l I

with degeneracy f2 = 40. Consequent ly , the dis t inct ion between the two channels

1 and 2 is lost. F o r this system, the expecta t ion value [(g.s.(n)lTIg.s.(n-1)1> 2 = 21.

The R P A and BCS results are equal to 20 (see fig. 5). This is consis tent with the fact

tha t errors of o rder 1/D are expected in connect ion with the quas i -boson app rox ima-

t ion.

2.5

2.0

/i[' -x-x- I<gs(n)lTigs(n-1)>l 2 Exact i - - I<pv(n)lTIgs(n-1)>12

. . . . . I<pv(n)lTIgs(n-1)>12~'c l Super- I I<(v: 2)(n)l T [gs (n-1)>l 2csJ c°nductive . . . . . phase

, I

ii . . . . . i<pv(n)lTrgs(n_l)>pX, c } Nonsuper- t J =l<gs(n)lmlgs(n-1)>l~,c conductiVephase

1.5

1.0

o.+j 0

I

I I I t x \

• \ k \

I I

2 3

Fig. 6. The label gsc = RPA stands for ground state correlations. See also caption to fig. 5.

In o rder to check the influence o f g round state cor re la t ions on a l and a2, we show in fig. 7 the ra t ios

12 R. A. BROGLIA et al.

= I<g.s.(n)l Tlg .s . (n- 1))[2pA I<g.s.(n)l Zlg . s . (n- 1)>l~cs

( D' )~ = 1 - 8(2G~+~p)copGf2 2 '

I (pv(n) l r lg . s . (n- 1))12pA X = - ( 4 . 2 )

Y I((v = 2)(n)lTIg.s .(n- 1))]gcs x/x 2 - 1

(4.1)

The different behaviour of 6 and V is easily understood physically. The fact that the system has a permanent pairing distortion A (static field) defines the ground-state coupling scheme and so the value of tr 1 oc (A/G) 2. The inclusion of correlations among the quasi-particles gives rise to fluctuations around this static distortion and

\ \ . \ x / - - T

× 1 ~ I I I I L I

1.1 1 . 5 2 , 0

Fig. 7. Correct ion factors to tr t and tr I coming from ground state correlat ions.

thus to a collective degree of freedom, namely the pairing vibration. On the other hand, Ha has small self-consistent components. All the above physical results are valid for the case of lower degeneracies and also for f21 ~ f22.

The authors wish to thank Professor Ben R. Mottelson for many enlightening suggestions during the preparation of this paper and for carefully reading the manu- script. Discussions with Dr. T. Udagawa are also acknowledged.

TWO-NUCLEON TRANSFER 13

References

1) J. HOgaasen-Feldman, Nuclear Physics 28 (1961) 258 2) A. Bohr, Compt. Rend. du Congr. Int. de physique nucl6aire, Vol. I (Centre National de

la Recherche Scientifique, Paris, 1964) p. 487 3) D. R. B6s and R. A. Broglia, Nuclear Physics 80 (1966) 289 4) J. H. Bjerregaard, O. Hansen, O. Nathan and S. Hinds, Nuclear Physics 89 (1966) 337 5) J. H. Bjerregaard, O. Hansen and O. Nathan, Nuclear Physics 86 (1966) 145;

O. Hansen, private communication on 1~8, ~lsSn(t ' p) reactions 6) R. A. Broglia and C. Riedel, Nuclear Physics A92 (1967) 145 7) N. K. Glendenning, Phys. Rev. 137 (1965) B102 8) D. R. B6s, Nuclear Physics 49 (1963) 544 9) O. Hansen, private communication on the ~°sPb(t, p) reaction

10) J. H. Bjerregaard et al., Nuclear Physics A103 (1967) 33 11) S. Hinds e ta l . , Phys. Lett. 14 (1965) 48;

R. Middleton and D. J. Pullen, Nuclear Physics 51 (1964) 77 12) S. Yoshida, Nuclear Physics 33 (1962) 685