A microscopic description of fermion boson interaction

A Microscopic Description of Fermion Boson Interaction A. VAN EGMOND and K. A L L A A R T Natuurkundig Laboratorium, Vrije Universiteit, 1007 MC Amsterdam, The Netherlands

INTRODUCTION The interacting boson model (IBA) and its e x t e n s i o n to odd nuclei, the i n t e r a c t i n g b o s o n - f e r m i o n model (IBFA), have been quite succesful in d e s c r i b i n g nuclear p r o p e r ties in various regions of the p e r i o d i c table (Iachello, 1981). Since the p r e s e n t a tion of the model, much work has been done to u n d e r s t a n d the model as an approximation of the shell model (Iachello, 1981; Otsuka, 1978a, 1978b, 1981; Scholten, 1980; Arima, 1977; Ginocchio, 1980; Duval, 1981). The main aim of this work is to reduce the large number of p a r a m e t e r s w h i c h occur e s p e c i a l l y in the IBFA Hamiltonian, by using relations p r e d i c t e d by the shell model. Moreover, the origin of certain parts of the IBFA H a m i l t o n i a n is not always u n d e r s t o o d in detail (Talmi, 1981). In applications of the IBA or IBFA model one u s u a l l y retains a number of p a r a m e t e r s to be fitted to the e x p e r i m e n t a l data, such as boson energies, b o s o n - b o s o n interaction parameters, and a few b o s o n - f e r m i o n i n t e r a c t i o n p a r a m e t e r s (Iachello, 1981). In this paper we shall start from the usual assumption that the boson states represent fermion states built of coherent pairs with angular m o m e n t u m J = 0 (S-pair) and J = 2 (D-pair). Furthermore we shall assume that the structure of these coherent pairs is the same in the odd as in the even nuclei. For the computation of the m i c r o s c o p i c structure of these pairs we e m p l o y the broken pair model (Gambhir, 1969 ; Allaart, 1981). We shall consider p r o t o n D-pairs as well as neutron D-pairs, so that a connection may be made w i t h the IBA-2 model, w h i c h d i s t i n g u i s h e s b e t w e e n p r o t o n and neutron bosons. By e q u a t i n g m a t r i x elements of the shell model H a m i l t o n i a n w i t h i n the one b r o k e n pair space to those of the IBA-2 H a m i l t o n i a n w i t h i n the c o r r e s p o n d i n g boson model space: S u b s e q u e n t l y by a similar p r o c e d u r e for the odd nuclei the IBFA p a r a m e t e r s Fab . A 3 and A are calculated. The results p r e s e n t e d here are for Xe isotopes, for aD w h i c h a IBA-2 fits (Puddu, Scholen, Otsuka, 1980) as w e l l as IBFA c a l c u l a t i o n s (Cunningham, 1981) have been performed. We shall check h o w well certain relations b e t w e e n the coupling parameters, w h i c h are assumed in IBFA model fits (Iachello, 1981; Scholen, 1980; Cunningham, 1981) to reduce the number of free parameters, are fulfilled. C a l c u l a t e d m a g n i t u d e s shall be compared w i t h the fitted values. The f o r m a l i s m is p r e s e n t e d in sect. 2. N u m e r i c a l details concerning the model space and interaction are d i s c u s s e d in sect. 3. C a l c u l a t i o n s are carried out in a model space i n c l u d i n g up to four major shells in order to study the role of other than

405

406

A. van Egmond

and K. Allaart

valence shells in the d e t e r m i n a t i o n of the boson model parameters. Results p r e s e n t e d in sect. 4. A summary of the conclusions is given in sect. 5.

2.

are

FORMALISM

2. I Broken

pair model

and boson model.

The b r o k e n pair model (Gambhir, 1969, 1979), or e q u i v a l e n t l y the generalized seniority model (Talmi, 1971) is suitable to stress the important role of superfluid nucleon pairs, with angular m o m e n t u m J=0 (S-pairs) creates by the operator.

S "l"

~ ~- ~ a [ a a

The n o t a t i o n

is as usual

a =

ja);

(ha, Za,

a =

a ]

(2.1)

(Allaart, 1981; Akkermans, 1982; Gambhir, 1979); l (2Ja+l)2 and the brackets indicate angular m o m e n t u m

coupling.

A m i n i m i z a t i o n p r o c e d u r e is used to determine the values for the coefficients ~ in (2.1) to obtain a 0 + ground state that is a pure S-pair state. This zero-broken pair state can be w r i t t e n as (unnormalized)

(s ln 10>

<2.21

where p is the number of proton pairs and n the n u m b e r of neutron pairs. Next the shell model H a m i l t o n i a n is d i a g o n a l i z e d w i t h i n the space of the one broken pair states to obtain the lowest 2 + one-broken pair state (normalized)

+ BPM> 121,

=

rap% a J" ](2

~

piP2 ~PIP2 L

(S:) p-I (s:)nl 0 > +

1 P2 ]

[ + % ]c2 (s+)n_1(s~ )pl°>

(2.3)

E ~nln 2 an i an2 n,n 2 which

shall be written

as

(CTrD:S: + C~ D'~Sj') ~) "Fr (s*)P-ITr (S:)n-1 As a short n o t a t i o n

c

we shall write

I"~> + %

0 >

for this state

>

(2.4)

w i t h the operator D ~ n o r m a l i z e d such that C 2 is the weight of the one-brokenpart; p r o t o n - p a i r p a r t of (2.4) and C 2 the w e i g h ~ of the o n e - b r o k e n - n e u t r o n - p a i r

v

C 2 + C 2 = i.

The state (2.2) (after normalization) n o r m a l i z e d boson states

(pq-.)½ (s)

(~)~ (sd

and

(2.4)

Io >

are n o w m a p p e d

on the c o r r e s p o n d i n g

A Microscopic

Description

of Fermion

Boson

Interaction

407

and i

C d

(s

(s%)n [ 0 > + (p'

((p-l) :n') ~

1

Cd %(s)J" n-1 ( s%)p I 0 >

(n-l)')

~

A s s u m i n g that the m a t r i x elements of the shell model H a m i l t o n i a n between (2.2) and (2.4) are the same as the boson m a t r i x elements, using "i"

"

HIBA_2

= gdd d

+ EVdtd d ~ ~ + < Q9(2) "Q~

Q~2,~)

= d(p)j- ~s + s ~ ( p ) P P P P

(2)

(2.5)

the states

(2.6)

with

one obtains

the IBA-2 p a r a m e t e r s gPd : <"D~ I HSMI"D">p

where

E0 is the energy K :

(nTn~)-2

xp

"J~9_(2,~) (dpdp)

(2.7)

as - E0

of the S-pair i

+

state

(2.2),

and

<"D~]HSMI"D"v>

where n ( n ) is the smallest of the number of p r o t o n (neutron) p a r t i c l e s The IBA-I p a r a m e t e r gd is just the energy of the 2 +1 state (2.4). We shall

or holes.

also assume

X~ = (nTrn~)) ½ <"~gll ~'~h II

"~>< %)P %)nil

r2y 2 I1"~>-~

A similar p r o c e d u r e shall be followed for the odd nuclei. By coupling mion to the states (2.2) and (2.4) one obtains after n o r m a l i z a t i o n

la>= {N(a)7-½ a'ais )P (S;i)n ] 0 >

(2.s) an odd fer

(2.9)

and

lad,j> =

{

N(ad,j)

r_,

L n[

u a z]

S'+C" ralD~] (2) ~j-'~,//~tkp-1/~' , ~kn-l,,~k)] bTr}lka,TT, I0> ~ ~ k a ~j

(2. I0)

Obviously, we would like to map these states on the c o r r e s p o n d i n g IBFA states, but because of the fermion commutation rules they are not orthogonal. There is no way to o r t h o g o n a l i z e the states (2.9) w i t h o u t losing the simple m a p p i n g on the fermionboson space, but the overlaps may be n e g l e c t e d if they are small. The states lad;j> can be p u t orthogonal to the states la> and then the part of lad;j> that is o r t n o g e n a l to la> is identified with the corresponding IBFA state. The usual ferm i o n - b o s o n interaction

408

A. van Egmond and K. Allaart

VBF = ]~ Aa[ died) (0)(a~aaa) (0) ] (0) + ~ r abl\r<~'~ ÷ a ab +

Z abj

i aJb :

<~>>
I(aa~d)(J)
can be written in the form

(o)( ~ )(o)(o)

=

a

+

~ I~ ab ab

S~+

d

b

]

(J)/~. h(J) (0)

^

+z abj

a

(2.12)

using the mapping procedure one obtains

tab : - < ad;JblHsM I b > b

where

N

is the total number of boson

(2.13)

N -~

n

+ n

and R ab j = < ad;jlHsMibd; j >

where

- @ ab (Ed + E a )

E is the energy of the state la> a X from the even nucleus one may compute

ab

ab - X Fab/5

(-)

, cf.

(2.14)

(2.9). Taking the magnitude of

b2

(2.15)

and subsequently obtain

J

Aab+A

. a ~~.,/S

6ab =

j' -{a 2 ~} • M b(2j'+l)j b 2

(2. 16)

j,

which follows from the equality (2.11) and (2.12). Ja-Jb The r.h.s, of (2.13) does not necessarily obey thesymmetryrela- F = Fba (-) ab tion which must hold for V to be Hermitian. We shall check this BF

A Microscopic Description symmetry n u m e r i c a l l y quently

2.2

of Fermion Boson Interaction

in sect. 4. The q u a n t i t y

J Rab

is obviously symmetric;

409 conse-

Aj = ~ a " ab

Computational

technique.

The calculation of p r o p e r t i e s of states with many S-pairs may be carried out Straight forwardly by w r i t i n g out the S n states in terms of seniority-zero shell model states. Similarly states with one D-pair and a number of S-pairs may be written as a linear combination of seniority-two shell model states. Actually, if a proper minimization of the e n e r g y is carried out, states composed of S-pairs may provide very good approximations to low-seniority shell model calculations (Allaart and Boeker 1972; Van Gunsteren 1974). This computational method has been extensively described in the paper of Gambhir and colleagues (Gambhir 1969) in which this form a l i s m was called the Broken-Pair model. The basis of this technique, the SU(2) quasispin formalism, had been described earlier (Kerman 1961, McFarlane 1966). An alternative technique, w h i c h has the advantage that the resulting expressions are a bit more transparent, is related to the BCS model. Let us for the moment consider one kind of nucleons, say protons. The state built of n S-pairs is obviously n! times the coefficient of z n in the non-number conserving state vector IV(z)> = exp(zs "~1 I0> = ~ z n n

(n~) -I (stlnlo>

.

(2.17)

This state vector has the convenient p r o p e r t y that it is a v a c u u m state for suitab l y chosen objects. Namely, from the relation

ajm exp (zS t) I0> = z~j (-) j - m a ~ '-m exp(zS t) I0>

one finds that the operator

• ~ m )f j b j m = (ajm-Z~j(-)3-m aj,_

(2.18)

annihilates the state vector ( 2 . 1 7 ) ; the factor f~ is chosen well that equation ( 2 . 1 8 ) , together with the hermitian conjugate expression, forms a canonical transformation;

so

f. = { l + z 2 ~ } -½ . For z=l eq. ( 2 . 1 8 ) is 3 3 Bogoliubov-ValatiD t r a n s f o r m a t i o n for BCS q u a s i p a r t i c l e approximation one does not care about the correct number certain average over all values of n. We do keep track p a i r s by considering the coefficient of the p r o p e r power

just the well known operators. So in the BCS of pairs n, but takes a of the correct number of of z only.

The state with one proton D-pair in addition to n-i S-pairs is obviously the coef(n-l) ficient of z in the expression

~

D+I~ (z) > = plP2 = E zn n

f PlP2

(n.) -I Dt(s ')

f [dip d ~](2) l~(z)> Pl P2 1 P2

I 0> .

(2.19)

410

A. van Egmond

and K. Allaart

Again, for z=l, the state ( 2 . 1 9 ) is a BCS t w o - q u a s i p a r t i c l e state; a s u p e r p o s i t i o n of states w i t h d i f f e r e n t n u m b e r s of S-pairs. Matrix e l e m e n t s of one- or t w o - b o d y fermion operators b e t w e e n states (2.17 ~ or ( 2 . 1 9 ) may most c o n v e n i e n t l y be worked out by e x p r e s s i n g t h e m in terms of the o p e r a t o r s (2.]8 ), using the inverse transformation

J" = aJ m

(bq" + z(-) j-m~j bj,_m) f ; a = 3m 3 jm

The r e s u l t i n g e x p r e s s i o n s

seem,

at first sight,

( a"~ )%. 3m

(2. 20)

to be t r a n s c e d e n t a l

functions

(of

i

a c o m p l e x z) due to the a p p e a r a n c e of the factors f. = (I+z2~.2) -~. A glance at ' . 3 eqs. ( 2 . ~ 7 ) and ( 2 . 1 9 ) may convznce the reader however, that [hey must be finite p o l y n o m i a l s in z, the h i g h e s t power is the m a x i m u m number of pairs that may be stored w i t h i n the (finite) a d o p t e d shell model space. Because the complete e x p r e s sions of the m a t r i x elements are finite p o l y n o m i a l s in z, one does not need resid u u m i n t e g r a l s (around the origin in the c o m p l e x z plane) to pick out the coefficient of the p r o p e r p o w e r of z (that is the m a t r i x elements of a fixed number of pairs). Instead of the r e s i d u u m i n t e g r a l s (Ottaviani and Savoia 1970) more simple summations may be used, due to the f o l l o w i n g theorem, w h i c h one may easily verify. If F(z)

=

N n E C z n n=0

then KMAX C

n

=

E k=l

-n z k F(z k)

(2.21)

2zk - ] and KMAX is the largest of the two n u m b e r s n+l and KMAX " k N-n+l. T h e r e f o r e the r e s i d i u m i n t e g r a l s I (pqr..t) which occur in t h e a b o v e c i t e d n work of O t t a v i a n i and S a v o i a may always be replaced by finite sums L (pqr ..t) (Allaart and Van G u n s t e r e n 1974). D e t a i l e d formula's w h i c h we used to compute m a t r i x e l e m e n t s b e t w e e n states w i t h m a n y pairs (and w i t h an odd fermion c o u p l e d to them) are given in the appendix. with

z k = exp i

The reason w h y we have called this c o m p u t a t i o n a l t e c h n i q u e a bit more t r a n s p a r e n t then a s t r a i g h t f o r w a r d e x p a n s i o n in l o w - s e n i o r i t y shell model space, is that one may always obtain i m m e d i a t e l y the BCS model a p p r o x i m a t i o n s (by p u t t i n g z=l w h i c h results in all sums L in the exact e x p r e s s i o n s b e i n g equal to one). These BCS approximations, w h i c h are rather accurate for q u a n t i t i e s w h i c h vary only smoothly w i t h the n u m b e r of pairs, yield a good i m p r e s s i o n of how p h y s i c a l q u a n t i t i e s d e p e n d upon the filling of the shells, that is upon the c o m p o s i t i o n and the number of S-pairs. The relation b e t w e e n the BCS p a r a m e t e r s Ua, v a and the S-pair coefficients ~a should be fixed by the relations:

Ua + v a = 1 ;

va

u

- p ~a .

(2. 22)

a

where the p r o p o r t i o n a l i t y factor value of the n u m b e r of p a r t i c l e s 2 2 (2j +i) P ~a a a 2 2

1+0 ~a

Q is fixed by the c o n d i t i o n that the e x p e c t a t i o n in the BCS state is twice the n u m b e r of pairs: = 2n.

(2.23)

A Microscopic

Description

of Fermion

Boson Interaction

411

We shall always use parameters ~ , or corresponding u and v , w h i c h minimize exactly the energy of the state a with n S-pairs. So a the a composition of the S-pairs changes a bit from one isotope to another, as shown in table I . Only for very special interactions (Allaart 1981) the S-pair structure which one obtains by this method is riqorously constant. The structure of the D-pair is obtained subseq u e n t l y by d i a g o n a l i z a t i o n of the Shell Model H a m i l t o n i a n within the space of b r o k e n - p a i r basis states. The 2 + changes from one nucleus to another. We shall see however that in our calculations it is still remarkably constant for a whole series of isotopes, cf. fig. 2. The structure should not be mixed up with the structure of the collective 2 + states in terms of quasipartioles, which is then of course ~zot constant with varvina particle number (Allaart 1981).

2.3

Determination

of the S n state

The values for ~, or equivalently for u and v are fixed by the requirement that the S-pair state, which is now the product of the proton- and the neutron Spair state, is the ground state for the Hamiltonian. This means that the matrix elements between the S-pair state and the parts of the one broken pair states orthogonal to it vanish. M a t h e m a t i c a l l y this implies that a set of coupled "projected gap equationS"must be solved by means of an iterative procedure. The equations can be written in the form % 2UsVsE~-(~2-v2)~ = 0 s s s % where s may be a proton or a neutron orbit. E and ~ are functions of all (proton s and neutron) parameters u and v. They are given by s 2n-2 % E

= C Pl

Pl

+ ~ v 2 F(plPlnn0) n n

L(n) L2--~----+

2p-4 L(pplp ~ PP ~~ v

F(ppplP10 )

2p-2 L(PlP I)

r

^

I 2p-2 2L (plPl)

2p-4

2p-2

2p

q

) L
p q [ 2 v 2 vp q F(ppqq0)

2p-2 L(PlP 1 )

/ 2p-2

2p-2

Z q 2 V 2l t q ~LL ( P)l q ) - L ( Pql q ) - 'i, L(Pl)-L(P

2p-6 2p-4 ~L(pclpl)-L(pqpl)-

Pq

r 2p4 • G
2p2

L(n)2n2p-2 L

[ 2p-4 2p-2 IL(PPl)-L(PPl

/ 2p

Pl

2p-4

2p-2 2p-2\ L(p)

L (plPl)

2p-2 A

2p

/ 2p-2

2n-2 + ½ ~ ~VpV2F(ppnn0) pn

2p-2

(L (Pl) - (L (Pl)~2DP~J I+UpVpUqUq L ~ -

O) Pl Pl ½ Z ~ u v G(ppp p P IPl (u 2 -v 2 ) 2p-2 P :--Pl Pl Pl L(PlPl)

2p-4

o

412

A. van Egmond

and K. Allaart

In these expressions p , q and Pl denote p r o t o n orbits and n is a neutron orbit. The upper indices 2p and 2n in the p r o j e c t i o n sums L are the number of protons and the number of neutrons resp. The same expressions are valid for the neutrons, with p r o t o n and neutron indices in i n t e r c h a n g e d (p1+nl; p+-+n; cf+m; 2p+-+2n). When the values for u and v are fixed this way, the S-~ai9 state is an eigenstate of the Hamiltonian, because HSM does not mix it with the (seniority two) one broken pair configurations. TABLE

1

Amplitudes

~

in the Sv-pair

a

12 4Xe

12 6Xe

Ii 8Xe

]30Xe

132Me

13~Xe

g7/2 d5/2 h11/2 81/2 d3/2

0.61 0.73 0.16 0.21 0.17

0.60 0.72 0.17 0.22 0.18

0.60 0.72 0.18 0.23 0.19

0.60 0.72 0.19 0.24 0.20

0.59 0.71 0.19 0.25 0.21

0.59 0.71 0.19 0.26 0.22

3.

ADOPTED

SHFLL MODEL

INTERACTION

AND SINGLE

PARTICLE

ENERGIES

In Xe isotopes the most relevant shells for both p r o t o n and neutrons are those b e t w e e n magic numbers 50 and 82; these are the ig7/2, 2d5/2, lhll/2 , 3Sl/2 and 2dz/2 orbits. Because the role of the collective 2 + state is crucial in the p r e s e n t study, we included also other major shells which may add configurations to this state. All included shells and single-particle energies are listed in table 2. These energies of the valence shells have been derived from experimental data on the heaviest odd Sb isotopes (Lederer 1978) and the o n e - n e u t r o n - h o l e nucleus 131Sn (De Geer 1980). For the other shells the precise values of £ are not important; only the inverse of the e n e r g y - d i s t a n c e s to the valence shell~ area measure for the magnitude of their contribution to the collective state. In all calculations we have used harmonic oscillator wave functions with oscillator parameter d=b-l=0.4432 -i fm

TABLE orbit 2d3/2 1915/2 Iili/2 2g9/2 3pi/2 2f5/2 2p3/2 ii13/2 2f7/2 ih9/2

2

Adopted neutrons 11.6 11.5 ii.0 I0.0 6.7 6.1 5.7 4.95 4.35 3.2

single ~article

energies

protons

2d3/2 3si/2 9.0 8.5 8.2 7.4 6.7 5.6

for Xe isotopes

neutrons

Ihii/2 2d5/2 Ig7/2 ig9/2 2pi/2 if5/2 2p3/2 If7/2 id3/2 2si/2 id5/2

0.0

-0.3 -0. I -2.8 -2.4 -7.4 -7.7 -8.4 -8.5

(MeV)

protons

2.4 I 1.5 1.5 small valence 0.7 | 0.0 - 5.0 5.3 - 6.0 5.5 - 9.0 -13.0 -15.0 -18.0

J

space

A Microscopic As an e f f e c t i v e

Description

interaction

V(rl,r 2) =

we took

-~ T=0,1

of F e r m i o n

Boson

Interaction

simply a t w o - p a r a m e t e r

delta

413

force

4ZATP T 6(rl-r2),

(3.1)

where PT is the p r o j e c t i o n o p e r a t o r on isospin T[ This form is not u n r e a s o n a b l e b e c a u s e one finds e m p i r i c a l l y (Akkermans 1982) that the force m u s t have a short range. The T=I s t r e n g t h d e t e r m i n e s the amount of p a i r i n g c o r r e l a t i o n s (energy gap in the even spectra), the T=0 strength may then be fixed to p r o t o n - n e u t r o n muJtiplets in o d d - o d d nuclei. The spectra shown in fig. 1 for 13°Sn , 132Te and 50 80 52 82 132Sb indicate that our a d o p t e d v a l u e s A 1 = 27.5 MeV fm 3 and A 0 = 45 MeV fm 3 are 51 81 reasonable. 130

134

5o S F I 8 o 2.5

~6+)

132 _.

52 T e s2

}Z+

6+

51 b b 81

6+ o+

>

20 7-

.~-~

,1+

0 /

w m z

1.5

7 0

i+ 2+

2+

2+

I+

-~Iz+ ~÷ i

2÷(3+) ~

2+

'~

10

X ~J

)+ Q5

~\ ,(g?,2)v(,ir2, -I 3+

o+ ex p

Fig.

4. 4.1

[

0+ cole

o+ exp

o+ cole

}+

4+

--_~+

exp

~\

~\

calc

S p e c t r a w h i c h have been used to fix the two p a r a m e t e r s of the e f f e c t i v e interaction; the delta force (3.1). The single p a r t i c l e e n e r g i e s used here were e x t r a c t e d from i n f o r m a t i o n on 131Sn and odd sb isotopes, cf. table 2.

RESULTS The even Xe isotopes

We c a l c u l a t e d the S- and D-pair states for the i s o t o p e s 124-13~Xe w i t h the H a m i l t o n i a n given in sect. 3. The a m p l i t u d e s of the n e u t r o n c o n f i g u r a t i o n s in the D-pair o p e r a t o r show a r e m a r k a b l y small v a r i a t i o n as the n u m b e r of n e u t r o n s changes. This is p l o t t e d in fig. 2. This feature supports the idea that the d - b o s o n is the same o b j e c t in a sequence of nuclei. In a n a l o g y to the b o s o n model we call the e n e r g y d i f f e r e n c e b e t w e e n the states (2.2 and (2.3) gd" There values are listed in table 3 where also the n u m b e r s £z and e~ '

d

are given w h i c h one o b t a i n s w h e n only the p r o t o n p a r t or only the n e u t r o n p a r t of the D-pair is c o n s i d e r e d (after this state is c a l c u l a t e d as a m i x t u r e of p r o t o n and

414

A. van Egmond

and K. Allaart

0 8 - -

04

g 712 d 312

d512 s112 $1/2 d3/2

O0

L

i

i

,

J

i

~

,

-

,

d 312 d 312 g 712

d 512

~

~

d 5/2 d 312

-04

h11/2

hltl2

d5/2 d 5 / 2

-

'~'~

---4.-_

g ?/2 g 712

-08

I

I

/

I

I

I

I

14

12

10

8

6

4

2

NEUTRON H O L E 5

Fig.

2. A m p l i t u d e s of n e u t r o n c o m p o n e n t s in the D-pair o p e r a t o r in Xe isotopes for d i f f e r e n t n u m b e r of n e u t r o n holes.

neutron excitations, so that the full e f f e c t of the p r o t o n - n e u t r o n i n t e r a c t i o n is i n c l u d e d in the wave functions). The m a t r i x e l e m e n t that mixes the p r o t o n D-pair and the n e u t r o n D-pair is e q u a t e d to the c o r r e s p o n d i n g IBA-2 q u a n t i t y C C <(n n )~ to obtain values for <. The m a g n i t u d e s of C 2 and C 2 are in all cases ~ b e~ t w e eTT n ~ 0.4 6 and 7[_ g ~ V and gdV as well as those of K are 0.54. One may notice that the va 1 ues oz u n n a t u r a l l y large c o m p a r e d to e m p i r i c a l values from IBA-2 model fits (Puddu, 1980). TABLE

3

Computed

IBA P a r a m e t e r s

nucleus

124Xe

126Xe

128Xe

l$0xe

132Xe

134Xe

gd

(MeV)

-0.21

-0.26

-0.24

-0.15

-0.02

0.32

Cd

(MeV)

3.18

3.22

3.21

3.]5

3.02

2.81

gd

(MeV)

2.81

2.72

2.68

2.69

2.77

2.98

<

(MeV)

-0.92

-1.02

-1.12

-1.25

-1.44

-1.82

Due to the strong p r o t o n - n e u t r o n i n t e r a c t i o n the wave functions (2.3) contain about ten p e r c e n t e x c i t a t i o n s from one orbit to another one two major shells higher W h e t h e r this p e r c e n t a g e is c o r r e c t or that it should be a bit larger or smaller is i r r e l e v a n t to this discussion. All these small a d m i x t u r e s of the wave function add c o h e r e n t l y to the m a t r i x e l e m e n t w h i c h mixes the p r o t o n and n e u t r o n D-pair, i.e. to the large value of K. If only the p r o t o n p a r t or n e u t r o n p a r t of (2.3) is considered the large e x c i t a t i o n e n e r g y r e q u i r e d for the small c o m p o n e n t s is not compensated by a large p r o t o n - n e u t r o n i n t e r a c t i o n energy. This e x p l a i n s the large values for c d and c d. The v a l u e s of X, as d e f i n e d in eq. (2.8) are d i s p l a y e d in fig. 3. B e s i d e s the values c o m p u t e d in the space of several m a j o r shells, we have also p l o t t e d the v a l u e s of X w h i c h one o b t a i n s w h e n only c o n f i g u r a t i o n s w i t h i n one major shell are retained. These are often larger than those found in the large space.

A Microscopic

Description

of Fermion

Boson

Interaction

415

This should be expected because the large number of small components contribute coh e r e n t l y to the q u a d r u p o l e transition m a t r i x element (the denominator of (2.8) but there is no such coherence in the q u a d r u p o l e moment (the numerator). With only the valence shells p a r t of the wave function the p a r a m e t e r X ~ changes sign at 1~°Xe; the trend is then the same as when only the h 11/2 orbit were active. The absolute

x°:i 0

valSpace full 5pace

i

I

I

I

val 5pace Xv 04 f

full space

0.0 I

-OA i 0.4

~

0.0

- 04

112

/2'

val. space full

I

I

10

8

I

6

I

4

5pace

I

2

NEUTRON HOLES Fig.

3

C a l c u l a t e d values of k for Xe isotopes. The results "full space" one o b t a i n e d by including all orbits listed in table 2.

m a g n i t u d e s of X ~ would be about five times larger when only the h 11/2 shell were considered however. The values of X p which we compute are in qualitative agreement with the fit of Puddu, Scholten and Otsuka (Puddu, 1980) ; i.e. they are rather small.

4.2

The interaction

parameters

Fab

These p a r a m e t e r s were o b t a i n e d for 125Xe and 131Xe, using eq. (2.13). Their values are listed in table 3. As the odd particle is a neutron, one expects that the main contribution to F comes from the interaction with the proton part of the D-pair. Therefore we have ~ m p u t e d separately the q u a n t i t i e s F ~ and F ~ , whlch " correspond to the interaction with the p r o t o n - and with the n e u t r ~ part aDof the D-pair (2.3). These numbers, also listed in table ~, are computed with /N in eq. (2.13) replaced by / n and / n respectively. So the Fab are smaller than the averages of and F V ab" Note s ymme t r y F ab ~

that for neither [ba =

P~NP9-

N"

of the computed

(-)Ja-Jb

Fab

parameters

Fab , F ~ab and F V ab the (4.1)

416

A. van Egmond and K° Allaart TABLE ~

CALCULATED IBA PARAMETERS

F

IN Xe ISOTOPES

125Xe a

b

Fab

(-)3a-3bF ba

g7/2 g7/2 d5/2 d5/2 sl/2

d5/2 d3/2 sl/2 d3/2 d3/2

0.18 -0.20 -0.20 0.13 -0.01

0.ii -.016 -0.17 0.05 -0.01

F~ ab

(-)3a-3bF~ ba

F~ ab

(-)3a-3bF ~ ba

0.13 -.010 -0.ii 0.08 -0.02

0.04 -0.03 -0.04 0.04 -0.02

0.32 -0.52 -0.56 0.31 -0.006

0.33 -0.52 -0.56 0.31 -0.007

Ja-JbFv ba

Fz ab

0.06 -0.09 -0.06 0.005 -0.03

0.35 -0.87 -0.78 0.48 -0.44

131Xe Ja-Jb a

g7/2 g7/2 d5/2 d5/2 sl/2

b

d5/2 d3/2 si/2 d3/2 d3/2

Fab 0.22 -0.34 -0.30 0.21 -0.17

(-)

Fba 0.16 -0.35 -0.29 0.16 -0.16

F~ ab

(-)

0.15 -0.13 -0.12 0.09 -0.05

(_)Ja-JbF~ ba 0.35 -0.87 -0.78 0.48 -0.44

will be satisfied exactly, but for the strongest, the proton-neutron interaction part it is satisfied to a very good approximation. So it is meaningful to take the symmetrized average of the calculated numbers Ja-Jb ~(rab(Calc.) + (-)

Fba(Calc.)

and compare that with the IBFA model assumption

FaD =

F(UaUb-VaV b)

r (alj~ Y2[lb)

(4.2)

This comparison is shown in fig. 4. It appears that there is excellent agreement when the overall strength F = -0.13 MeV is employed, both for 125Xe and for 131Xe. This means that the factor /N in (2.13), which originates from the S-bosons, is very well confirmed. In vibrational models such a term is simulated in a somewhat more obscure way by taking the particle-vibration coupling strength proportional to the square root of the B(E2, 2~÷0+) ; (Bohr and Mottelson 1953, Kisslinger and S6rensen 1963, Lopac 1969}. Our conclusion is that the part of Fermion Boson or fermion-phonon coupling models is considered here is well established. We should mention a problem in the definition of F , although it appears not to ab be very important due to the small value of this quantity. The problem is, that when coupling the odd neutron to the D-pair state, one obtains states (2.10) which are not completely orthogonal to one another and they also have some overlap with the fermion + S-pairs state (2.9). These overlaps are in general less than ten percent. We have always orthogonalized the particle D-pair state to the state without a D-pair. We have not further orthogonalized the particle D-pair states among each

A Microscopic

Description

of Fermion Boson Interaction

other, because then the unambiguous correspondence fermion-boson coupled states gets lost.

417

between fermion states and

o.~

r°b l t O0 i

v

k

-05J

, +

T

~C-- I +

i +

V +

i I

I

I

off,2~gg L

I

I

I

I

+

+~ I

I

I

L

t 0 t ¶\\~\ O0 1

~

~

h

-05 Fig. 4.

Comparison of calculated values of Fab (full line) and the form adopted in IBFA model fits (eq. 4.2) with F = -0.13 MeV (dashed line).

L~

4.3

The exchange p a r a m e t e r s

A~ ab

I

The p a r a m e t e r s A j were computed following the procedure sketched in sect. 2, using ab eqs. (2.10 - 2.16). The n o n d i a g o n a l elements, a%b can be obtained directly from eq. (2.16), but for the diagonal elements a monopole contribution has to be taken into account. For this reason we shall consider first A j The calculated values are a#b " p l o t t e d in fig. 5, where they are compared w i t h an expression that has been derived m i c r o s c o p i c a l l y (Talmi, 1981), which is given by _ _ v<~j Va~(V j ~I Aj = A0 20 QjaQj b + -- -- + ab /2 j + 1 Ua/kU j

with

(4.3)

r2 Qja = (j jj 7 Y2 jj a).

In fig. 5, the computed values are plotted,

together with the values given by eq.

418

A. v a n

Egmond

and

K.

Allaart

MeV

A J 125Xg

1.0

3

a4:b

Q5 5

5

0

9

z'~\

e.

/

d'~

,,.

_

3 5

5

-0.5 3 v__l

\

- 1.0

\

7,3

7,5

7,1

53

3,1

5,1

(2Ja,2jb)

A j 131 acb X@

MeV

1.0 5 7

O.5

\

5

\\ /if\\

0

-0.5

- 1.0

7,5

7,3

7,1

5,3

5,1

3,1

(2ja,2Jb) Fig.

5. Comparison of calculated values Aa3b for a ~ b (full lines) and the values obtained from eq.(4.3) with A 0 = -2.17 M e V (dashed lines). The numbers in the figures indicate 2j.

A Microscopic

Description

of Fermion

Boson

Interaction

419

(4.3). The overall strength A 0 is chosen in such a way that the best agreement is o b t a i n e d w i t h the c o m p u t e d values. For 125Xe and for 131Xe the same value A 0 = 0 . 0 7 9 keV is obtained. The form (4.3) is d i f f e r e n t from the one that is often u s e d in IBFA model fits • (Cunningham, 1981); it d l. f f e r s by factor (u u u.2 ~ - 1 . w l, t h that a l t e r n a t l. v e formula a lJ ~ . . . the best a g r e e m e n t w i t h the c o m p u t e d values yle~ds a !~ whlch ±s three times larger 125 131 ., 0 . for Xe than for Xe. So one should p r e f e r Talml s e x p r e s s l o n (4.3). One may notlce from fig. 5, that the relation (4.3) i n d e e d d e s c r i b e s the rough c h a r a c t e r i s tics of the c o u p l i n g constant A] . We should add one remark however. The e x p r e s • ab slon (4.3) is only d e f i n e d if the angular m o m e n t u m 3 is that of one of the valence orbits. So there is no c o u n t e r p a r t for our c o m p u t e d value of A] with a=d3/2, b=g7/2 and j=9/2. The Ig9/2 orbit yields only a very small n u m b e r abwhen eq. (4.3) is a p p l i e d in this case. The 2g9/2 orbit may not simply be i n s e r t e d in eq. (4.3) for this case b e c a u s e such a remote orbit implies a la
Aj aa

+ A

J a a/5

we assume that the r e l a t i o n (4.3) also holds for the d i a g o n a l elements, with the m a g n i t u d e of i 0 o b t a i n e d for a%b. We then obtain p a r a m e t e r s A from the difference b e t w e e n the c o m p u t e d n u m b e r s and the n u m b e r s given by eq~ (4.3). These differences, full lines).

which should c o r r e s p o n d to In IBFA fits, one assumes

A

a

1 = A(2Ja+l) 2

A

-a a/5

,

are p l o t t e d

in fig.

5,

(the

(4.4)

One may notice from the c o m p a r i s o n shown in fig. 5 that our c a l c u l a t e d n u m b e r s agree quite well w i t h this relation for the value A = - 2 . 1 7 MeV. So it seems that the m o n o p o l e t e r m in the f e r m i o n - D - p a i r i n t e r a c t i o n is rather well p a r a m e t r i z e d by r e l a t i o n (4.4). Our c o n c l u s i o n is that, a l t h o u g h there are the d i s c r e p a n c i e s shown in figs. 5 and 6, the IBFA p h e n o m e n o g i e s a c c o u n t in general rather well for the i n t e r a c t i o n between an odd fermion and a D-pair. A nice result is that the v a l u e s for A and A o b t a i n e d in this analysis are the same for 125Xe and 131Xe, p r o v i d e d that 0 T a l m i ' s formula (4.3) is adopted. 5.

CONCLUSIONS

A s s u m i n g a c o r r e s p o n d e n c e of b o s o n model states and f e r m i o n - p a i r state we have computed quantities Fab , A 3, and A a . We c o m p a r e d our results w i t h those o b t a i n e d aD from formulas for the f e r m i o n - b o s o n i n t e r a c t i o n p a r a m e t e r s , w h i c h one e m p l o y s in IBFA model fits to reduce the n u m b e r of free p a r a m e t e r s . In order to obtain r e a l i s tic n u m b e r s for 125Xe and 131Xe we have u s e d a large m o d e l space to c o n s t r u c t the c o l l e c t i v e D - p a i r and a d o p t e d a simple b u t r e a s o n a b l e e f f e c t i v e shell m o d e l interaction. For the even i s o t o p e s we c a l c u l a t e d q u a n t i t i e s Ed' £d' X, w h i c h may be

420

A. van E g m o n d

I1 M eV

-4

1>

and K. Allaart

A

J 125Xg Aa ~ ~/'~ II

5 7 9 ~

7 h 3 5t~

r

?

j

-2

g 712

h 11/2

d 5/2

d 3/2

s 1/2

O

6 Aa

MeV

-4

15

J

131Xg

9

? • 5

7 9

g•'V 3

-2

h 11/2

Fig.

6.

g 7/2

:,

~ j"

,r

3 s ~

5~ d3/2

s1/2

C o m p a r i s o n of the c a l c u l a t e d monopole term and its form a d o p t e d in IBFA model-fits. The full lines are c o m p u t e d from eq. (2.16). The dashed lines c o r r e s p o n d to the IBFA form A =A/2j +i w i t h A=-2.17 MeV. The n u m b e r s in the a a figures indicate 2j.

r e l a t e d to IBA-2 parameters. It appears that £ Q and < are larger than the values o b t a i n e d in e m p i r i c a l fits. The latter values d are p o s s i b l y "renormalized" to a c c o u n t for a d m i x t u r e s of other than S- and D-pair s t r u c t u r e s (Otsuka, 1981a, 1981b). We find that the p a r a m e t e r s r b are quite well r e p r e s e n t e d by the formula (4.2 ), w h i c h one adopts in IBFA calculations; the over~ll strength F is the same for 125Xe and 131Xe. For the e x c h a n g e p a r a m e t e r s A 3, the formula ( 4 . 3 ) , which has been d e r i v e d by Talmi (Talmi, 1981), d e s c r i b e s am the overall b e h a v i o u r rather well, as shown in fig. 5. The strength A is also the same for b o t h odd isotopes. A n o t h e r formula that one u s u a l l y ad~pts in IBFA model fits, which is d i f f e r e n t 2 -I can only be compared with our from the one Talmi d e r i v e d by a factor (u UbU.) c a l c u l a t e d n u m b e r s if we assume a strengt~ ] A_ that is three times larger for 125Xe than for 131Xe. There is a class of p a r a m e t e r s A~b , n a m e l y for those values of j, w h i c h do not c o r r e s p o n d to a shell model orbit in the valence space, and with a~b, w h i c h is n e g l e c t e d in IBFA model fits. We find that this type of interaction p a r a m e t e r s is not e s s e n t i a l l y smaller than the others however; so this IBFA a p p r o c i m a t i o n seems not justified. The p a r a m e t e r s A and the o v e r a l l strength p a r a m e t e r A may describe rather well the m o n o p o l e a p a r t of the f e r m i o n - b o s o n i n t e r a c t i o n as can be seen in fig.

6.

A Microscopic Description

of Fermion Boson Interaction

421

We find the value A = -2.17 MeV. One may conclude that IBFA model fits employ a form of the Hamiltonian which has indeed many features of the interaction between an odd fermion and a D-pair in the broken pair model. The overall magnitudes appear independent of the number of particles. It remains to be seen how the "renormalization" by other broken pair structures may change the numbers. It seems also worth wile to investigate if another phenomenological term in the IBFA Hamiltonian can explain the discrepancies for

J Aab"

Acknowledgements: We thank professor E. Boeker for his stimulating interest in this work. This investigation was part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).

APPENDIX The matrix elements of the shell model Hamiltonian and the overlaps in the zero and one broken pair states are calculated using the generating function technique. Since we deal with both protons and neutrons we now introduce two complex variables x (for protons) and z (for neutrons). The generating functions for the zero-~and one broken pair states can be expressed in the x and z dependent operators b T, for which J~(x,z)> is the vacuum state. The S-pair state is now denoted as J~2n,2p> •

where 2n and 2p are the number of neutrons and protons resp.

l~(x,z)> = ~ z n x p l~2n,2p > pn

J~(plP2 JM) ; xz> = Z zn~+ll~2n,2p(plP2) pn

; JM>

= (PPl pp2 )-l[x BjM(plP2)-6J06plP2PluplVplX(X2-1) where

B@jM(ab) = [b#abb%](J) -

btP a a=

and

] IV(xz)>

(for protons)

1 % 3 a-ma (Uaa~ - x(-) Wee-Q)

Pa = (u2 + x2v2) ~ a

a

For odd nuclei we have

l~Cv,xz)>

=

X

zn+l~J~2n+l,2p(m)> =

Pn

zb J~Cxz)>

pn J~(abJc;jm) xz> = ~ z n+~ xPJ~2n+l,2p(abJc;jm) > = (0aQbPc)-l[z 3 v (JMJcm c ljm) . pn Mm

422

A. van Egmond and K. Allaart

b@B]M(ab ) + z2(z2-1) 6J0~ab~JjeaUaVabT + + jj-lucvcP(abJ ) 6jjbb~]x J" p l~(xz)> 7

IV(PlP2 Jn;jm) xz> : Z z n+Ix p+IIT$,2n+l,2p (plP2Jn;jm)> : (pplpp2Pn)-l[x2 z Mmr pn n (JMJnmn ijm) bJ" t (plP2) V BjM The projection sums

(x2-1)xz@J06plP2@jjn k L (pq...t)

k L (pq... t)

iuPl vPl b+]l*(xz)> ~

are defined as

2Ja+1 1 NMAX -k ~a (p) a NMAX Z (zn) n:l (pppq...pt)2

with zn = exp(i~ N~--X)

The maxtrix elements of the shell model Hamiltonian in the zero and one broken pair basis are given by 2n,2p

<~2n,2pIHl~2n,2p > = m00 <~2n,2p[Hl~2n,2p(nln2);JM>

<,

~2n,2p

(nln2);aM[Hi

= ~

[4

n u v

/M 2n'2p

J0[ nln 2 1 n I nl\ 00

~2n,2p (n3n4);

J~>

2n-2,2p -M02 (n3n4;nl)} + { n l ~ n 3 ; n 2 ~ n 4 }

2n,2m

= M22

)

2n-2'2P(n)h+M2n'2P(nln2)]

(nl -Moo

~S _6

l/

02

G.u v ~ , 2 p

(nln2n3n4) +L JO nln 2 I n I nl<102(n3n4;n

+ ~j0~nln2~n3n4

^ nlunlVnlUn3Vn3

^ / 2n,2p 2n-2,2p 2n-4,2p n u v u v |M (nln)-2M00(nln3)+M00(nln3)h i n I n I n 3 n3\ 00 3

2n,2p j _ { 2n,2p <~2n,2p(PlP2 ) ;JM]HI~J2n,2 p(nln 2) ;JM> = M22pn(nln2plp2J)+l~J06nln2nlunlvnl
A Microscopic Description of Fermion Boson Interaction 2n,2p 2n,2p ' + MII (n,n') <~2n+l,2p(n) IH192n+l,2p (n)> = @nn,M00~n)

<~

JnJn ,

2n,2p _ _ (n n~Jn~.jm)IH 1 (n)> = (n n n~Jn ) zn+l,ip i z J' '~2n+l,2p MI3 I z 5

/ 2n, 2p 2n-2,2ph ~ u v |M (nn) + @J06nln2 nn 3 1 n I nl\ 00 i -M00(nnl)]

+

_ / 2n,2p 2n-2,2p n.u v @.. {M (nn ;n )-M 1 (nn3;n I) 6J0~n]n 2 1 n I n I 3 3 n # 11 3 1 1 /

v n3~nn2 < 2n,2p 2n-2,2ph Un 3 n 3 P ( n l n 2 J ) 6nl M00(nn3)-M00(nn3) )

-Jj

---i - aj

u

v

P(n n J) ~

n3 n3

i 2

@

/ 2n,2p |M (nn

nln3 JJn~ ii

2;n3 )

2n-2,2p -MII (nn2;n3)}

<~2n+l,2p(nln2Jnn3;Jm) IHl~2n+l,2ptmlm2Jmm3;Jm)>=

2n,2p 2n-2,2p Ml3(n3'mlm2Jmm3;n')-Ml 13(n3'mlm2 Jmm3'n ))i,,

[ Jn 0 n[n 2 i n I n I@jj V

t

2n,2p M33(nln2Jnn3;mlm2Jmm3;J)

n3

v / 2n,2p 2n-2,2p )] - JnJ-lUn3 n3 ~(nln2J n) 6nln36jjn2
+ [ (nln2Jnn3)~(mlm2Jmm3) ]

2p 2n-2,2p + ~Jn06nln2~jjn3 n Iu n Iv n I~Jm U^~ mlm 2 ~.. m u m Iv ml[I~ n3m3[+M 00 (n3n 1m)-2M00(n3nlml) 3]m31 [

[ 2n,

2n-4,2p } [ 2n,2p 2n-2,2p 2n-4,2p n m )l] + M00(n3nlm I) + ~.3n jmq {M ( n m ;n m ) 2M11(n3m3;nlml) + M1 [ ii 3 3 1 1 1 (n3m3; i i ~j +

423

424 I_

A. van Egmond and K. Allaart

[" 2n,2p 6 6 6.. n u v " '-lu v P(mlm2Jm) 6 ~ IM00(m2nlml). Jn 0 nln 2 33n3 1 n I nlJm 3 m 3 m 3 mlm 3 jjm2 2n-2,2p 2n-4,2p } [ 2n,2p 2n-2,2p 2M00(m2nlml)+M00(m2nlm I) + 6. . IMll(m2n3;nlml)-2Mll(m2n3"nlml) 3m23n 3

-

+ Mll(m2m3;nlml)~]~+

+ JnJmj-2u

(nln2Jnn 3) *~* (mlm2Jmm3)

[ 6n2m> I MOO (n2n im I) v u v P(n.n.J )P(mlm2J m)@ (~ 6• d n3 n3 m3 m3 i z n nln 3 mlm 3 33n2 Jim2"

2n-2,2p 2n-4,2p ] [ 2n,2p 2n-2,2p 2n-4,2p -[] -2M00(n2nlml)+M00(n2nlml)>+6. . {-M (n~m~.n.m.)-2M11 (n2m2;nlml)+Mll (n2m2;nlml)j.] ] 3n23m2L ii z z' I 1 2n,2p _6plp2 p 1u91 vPl[[6nn , . = 6j06nn,a02 (p.p_;n)+(~ 1 2 Jr) 2n,2p 2n, 29-2] f 2n,2p 2n, 29-2 ]] M00(Pln)-M00(Pln)~ + 6.. {M.. (nn' ;pl)-Mll (nn';Pl)~J 3n3n,L Ii ^--i2p-2 / 2n 2n-2\ -JJ P(PlP2J) F(PlP2nn'J)L(PlP2)kUnUn'VplUp2L(nn')-VnVn ,Uplv92L(nn'} 2n-2 29-2 =@j J 6n n L(nln2n3)L(plp2 )P(plp2Jp). np 3 P(nln2Jn)F(plP2nln2J)u91v92Unlvn2 4: JnJpL(nln2n3)L(plp2)P(nln2Jn)6nnl n3 j j P(plP2Jp)P(n3n2Jp)F(plP2n3n2Jp)U

v u v .~6 06 n u v 6 PI P2 n3 n2 Jn nln 2 I n I n I jjn3

/ 2n,2p 2n-2,2p I+~95_ i~ (plp2%) F (piP2nn3Jp ) [6jp 06nn3
".-I29-2 [ / 2n-2 2n +Jp3 P (plp2Jp) F (plP2nn2Jp) L (plP2) lUnUn2Vp I%2~L (nln2n) -L (nln2n) ,I -

.

A Microscopic Description of Fermion Boson Interaction

425

/ 2n-4 2n-2 hi] - v v u v (L(n.n^n)-L(n.n_n)){l+~ _ ~ pl u v ~ • n n 2 Pl P2 \ i Z i Z ,]J JpO plP2 Pl Pl 3n3 ] {<~2n+l,2p(nln2Jnn3;Jm) IHI~(n) > - <~J (nln2Jnn3;Jm) IHI~(n) >~ 2n+l,2p 2n+],2p-2 2n+l,2p-2 (PI) f]

In the last term between ~ ~ 2p projection sums L(p,...)

there must be an additional index

Pl

in all proton

2n,2p-2 <~2n+l,2p(plP2Jn;jm) IHl~2n+l,2p(p3P4J'n ;jm)> = ~nn,~jj,P(p3P4J ) PlP3 P2P400(plP2 '

"

' 6

6

M

2n,2p-2 2n,2p-2 + P(p4p2J) 6 ± PlP36P2P46--'M'dd II(nn';PlP2)+@nn'P(PlP2J)P(P3P4J')6JJ'6P2P4MII(PlP3;P2n)

- JJ'

P

(~

E

J

J"]

P2P3 J"[Pl P4 P2 ~] In

j -j +J" F (p4Plnn, j,,) (-) P4 Pl /u i

' J

(-) Pl

P4

n

2n 2p-2 p4UplUnUn'L(nn')L(PlP2P4)+v

2n-2 2p-4 VnVn,L(nn' L (PIP2P 4) > P4 Pl v

2n 2p-4 2n-22p-2 \] - F(PlP4nn' J") (vP4 vP l un un' L(nn')L(plP2P4)+u P4 uPl VnVn,L(nn' )L(PlP2P4)

)]

+ 6nn'6JJ'P(PlP2J)P(P3P4 J)[u[plVp2Up3Vp4F(Pl p2 p3 p4~ + 41 G(plp2P3P4J){ 2p-2 2p-6 }] 2n uplUp2Up3Up4L(plP2P3P4)+VplVp2Vp3Vp4L(plP2P3P4) L(n) where

P (abJ)

is the antisymmetrisation operator

P(abJ) = [I + (-)Ja-Jb+J(a ~=*b)]

The expressions

M

are given by

k k,~ (ppnn0) • MOO (nln2"~PlP2- .)=Rk0 (nln2 . .)L £ (plp2 ..)+R0~ (plp2 . .)L (nln2 . .)+ ~ v2v2pnF pn p n k-2 i-2 L (nnln2) L (pplP2..)

n)

426

A. van Egmond and K. Allaart

k,i M02 (nm;nln 2-.plP2- • )=-2@nmnUnVnSnL]~

k-2 (plp 2..)L(nm)-~

9. 6jnJm(~ZnZmL(PlP2

. .)

~

~,

n'

{n

2v ,F(nmn'n'0)

(

(u n v m +u m v n )L(nmn' nln 2..)-u n ,Vn,G(nmn,n,

0) VnVmL(nmn,nln2

• .)_UnUm

k-2 k-2 ]~-2 L(nmn'n n ..)~;~-~ {5 (UnVm+UmVn)L(nmnln2..) ~ l~VpF(nmpp0) L(pplp 2 ) I 2 /I JnJm InZm "" P _ [ 2n-2,2p 2n-2,2p k,]~(nm,m,S)=L]~Rk 2(nmn'm,J)+P(nmJ)@nn,16mm,M00(nm) + 6jmJm 'M1 i (nm;n) / M22 k,]~ k-2 }.-2 M22pn(nln2PlP2J)=L(nln2)L(plP2J)P(nln2J) P(plP2J)u

u v v F(p p n n J) Pl nl P2 n2 i 2 i 2

k,i < k k-2 MII (nn' ;nln 2 • .plP2 • .)=6nn,En Un2L(nn'nln 2. • )-V2nL(nn'nln2..)hL

(p Ip 2 .

.)+n-I JnJn ,

m v2F(mmnn'0)\UnUn,L(mnn,nln2. .)_vnvn ,L(mnn,nln2. ) -½UmVmG(mmnn,0)L(mnn,nln 2 .) In

"

.

]]% [p ~,-2 k k-2 (UnVn,+Un,V n) jL(PlP 2--)+ pvp2F(ppnn'0)L(pplP2.. ) ][UnUn,L(nn'nln2..)-VnVn,L(nn'nln2..)] } k,i k,i --I k,~ Ml3(n,nln2Jn 3)=6J06nn3M02(nln 2;n 3)-Jn-IP(nln2J )6nn2n 3 M02(nln 3;n 2) ~Li rq_l{G

-

k-2

k-4

(nln2n3nJ)~Unlun2 v n 3u n L (n.n~n~n)-v i z~ n Iv n 2u n 3VnL (nln2n3n) ]

U U U

L n I n2 n3 n

(n,n~n~n) -u v v v L ± z J n I n 2 n 3 n (nln2n3n)

]}

k,~ M33 (nln2Jnn3,mlm2Jmm3;J)= ~jnJm (5n3m 3p (nln?n) <~nlm I(~n2m 2+JnJm P (nln2Jn) p (mlm2Jm) k-2 ]1 n]]M00 (nln2n 3) +6jj,P" (nln2Jn) [ <~n iml (~n2m2Mlk-2, (~n2m 2 6nlm 3 (~n3mlln3 [nl n2 J ' 1 (n3m3;nln 2) j JmJJ ~

]

{nl n2 Jn}

+ ~n3m3P(mlm2Jm)~n2m2Mll(nlml;n2n3) +3n~ImP(nln2Jn)P(mlm2J m)

n3 j J

6n3m1(Sn2m2

A Microscopic Description of Fermion Boson Interaction

427

k-2,1 k-21 k-2,Z Mll(nlm3;n2n3)+@nlm36n3mlMll (n2m2'nln~)+6jn2m26nlm3Mll (n3ml;nln2)]j

k,£ - - ml Jn j ] k,i +~JnJm~n3m3R22 (nln2mlm2Jn;n3)+[JnJmP(mlm2Jm)6n3ml{m3 Jm m2~R22(m3m2nln2Jn;n3 ) ]

+ [ni ~=~ mi ]+~(nln2Jn)~(mlm2Jm)~n~ m j~Inj2 nnl3 JJn][m2 ^ k,i(m3m2n3n2J;nl) fl j ml m 3 Jm j }J2@nln2R22

where k-2 L (abcde)

k, Z(abcdJ;e) =P (abJ)P (cdJ) [UaVbUcVd F (abcdJ) Lk-4 R22 (abcde) +IG (abcdJ) (UaUbUcUd) k-6 +VaVbVcVd L (abcde)

] L

k-4 k k-2 1 nn'[ 2 2 (nnn'n'0)L (nn 'n in2 ••) ^ 2Vn,VnF and R0(nln2..) = ~ nZvn2~nL(nnln2..)+ ~ n nn ' k-2 + UnVnUn,Vn,G(nnn'n'0)L(nn'nln 2--) ] The overlaps are <~2n,2pl~2n,2p > = L2PL 2n 2p/ 2n 2n-2 <~2n,2pl~2n,2p(nln2) ;JM> = @J0@nln 2nlunlvnlL : P(nln~)@nln3@n2n4 IL L(nln2) 2p/ 2n 2n-2 2n-4 + ~ _6 ~ nl u v n u v L
<~2n,2p(nln2 ) ;JMl~2n,2p(plP2) ;JM>=6jO_6plP2~nln2=inUplVplnlUnlVnl L2n

2n-2\/ 2p 2p-2 h (nl)-L(n I) j~L(Pl)-L(Pl) )

2n 2p <~2n+1,2p (n) l~2n+l ,2p (n')> = ~nn ,L(n) L

428

A. van Egmond and K. Allaart

[ / 2n 2n-2 ) <~2n+l 2p (n) [~2n+I 2p(nln2Jn3 ) ;Jm>=~nlu v 6 ~6 n 6 ~L(n_n )-L(n~n ) , , L n I n I a u n I 2 nn3\ 1 j i J . ( 2n 2n-2 h ! 2p - Jn--i Un3Vn3 L (nln2) -L (nln2)}P(nln2J) 6nln3 69 Jn L

<~2n+l,2p (nln2Jnn 3 ;jn9I~2n+l,2p (mlm2Jmm3; j m > = 2n-2 2n-4 n~u v 6_ ~6~ 6 ~m.u v 6 ^6 6 [L2(nlmln3)-2L (nlmln3) +L (nlmln3) ] I n.l n.l Jnu ~*.n~l z jjn3[ I m.l m~l Jmu m 1m~z ]]m 3 2p -

Jm 3 "$-lum3Vm3P(mlm2Jm) 6mlm36jj

[ n ~ m 2] L m2

-J J u ^n --I n 3Vn3~(nln2Jn) 6nln36jjn2 { n 3 ~=~ n 2 } L 2p 2n-2 2p[~ + L(nln2n3)L (nln2Jn) n I n2

6nlm16n2m26n3m36Jn Jm

+P(nln2Jn) P(m m~J )J J 6 6 I z m n m nlm 3 n3m I n2m 2

J

/ 2p 2p-2h 2n <~2n+1,2p(n') I~2n+l,2p(plp2Jn;Jn,m)>= 6nn ,6plP2 6dO~p.u v (n(p~)-L(p~))L(n') 1 Pl Pl \ I I 2n 2n-2 . _q • = ~n u v 6 ^6 (L(n n )-L(n v 3 <~2n+l,2p(nln2Jnn3;Jm)i~(PlP2JP n;3m)> L ' n I n I Jn U nln2\ i J In 3 )-Jn j u n3n 2n 2n-2. h^ L(nln2)-L(nln2/P(nln2Jn)6

~ . / 2p 2p-2\ 6 . r6 ~_ ~P.u v |L(p~)-L(pA)~ nln 3 33n{ plP2 dpU-I Pl Pl \ i I/

2n 2p-2 <~2n+l 2p(PlP2 Jn;jm) l~2n+l,2p(P3P4J'n';jm)>=6JJ'6nn'L(n){P(PlP2 J)6 6 L(plp 2) , PlP3 P2P4 +

[ 2p-4 2p-2 2p p.u v p
In all expressions with

a

and

b

{a ~=~ b} means: the last expression between the brackets {}

interchanged.

A Microscopic Description of Fermon Boson Interaction

429

REFERENCES Akkermans, J.N.L., Allaart, K. (1982). The empirical form of the effective nucleonnucleon interaction in a model space with correlated J = 0 pairs. Z. Physik A304, 245-255. Allaart, K., Boeker, E., (1972). FBCS for odd nuclei and the invers~gap equations. Application to N = 50 isotones. Nucl. Phys. A198, 33-66 Allaart, K., van Gunsteren, W.F. (1974). Projected Quasiparticle calculations in large model spaces. Nucl. Phys. A234, 53-60. Allaart, K. (1981). Are bQsons nucleon pairs? In F. Iachello (Ed.). Interactin~ bose-fermi systems in nuclei. Plenum, New York. 201-208. Arima, A., Otsuka, T., Iachello, F., Talmi, I. (1977). Collective nuclear states as symmetric couplings of proton and neutron excitations. Phys. Lett. 66B, 205-208. Bohr, A., Mottelson, B.R. (1953). Collective and individual particle aspects of nuclear structure. Mat. Fys. Medd. Dan. Vid. Selsk. 27 no. 16. Cunningham, M.A. (1981), Multilevel calculations in odd-A nuclei. Phys. Lett. 106B, 11-14. Duval, P.D., Barrett, B.R. (1981). Shell model determination of the interacting boson model parameters for two nondegenerate j-shells. Phys. Rev. C24 , 12721282. Gambhir, Y.K., Rimini, A., Weber, T. (1969). Number conserving approximation to the shell model. Phys. Rev. 188, 1573-1582. Gambhir, Y. K., Haq, S., Suri,.J.K. (1979). Generalized approximation to seniority shell model. Phys. Rev. C20 ,381-383. Geer, L.-E,de., Holm,G.B. (1980). Energy levels of 127'129'131Sn populated in the ~- decay of 127'129'131In. Phys. Rev. C22, 2163-2177. Ginocchio, J.N., Talmi, I. (1980). On the correspondence between fermion and boson states and operators. Nucl. Phys. A337, 431-444. Gunsteren, W van., Boeker, E., Allaart, K. (1974). The FBCS model and the inverse gap-equations applied to the Tin isotopes. Z. Phys. 267, 87-96. Kerman, A.K. (1961). Pairing forces and nuclear collective motion. Ann. Phys.12 300-329. Iachello, F. (1981). Present status of the Interacting Boson Model. In F. Iachello (ed.), Interacting bose-fermi s~§tems in nuclei.,Plenum,New York. p 273-283. Kisslinger, L.S., Sorensen, R.A. (1963). Spherical nuclei with simple residual forces. Rev. Mod. Phys. 35. 853-915. Lopac, V. (1969). Properties of ll9sb in the semi-microscopic model. Nucl. Phys. A138 , 19-52. Lederer, C.M., Shirley,V.S. (1978). Table of isotopes., J. wiley, New York. MacFarlane, M.H. (1966). Shell model Theory of identical nucleons. In P.D. Kunz, D.A. Lind and W.E. Brittin (eds.), Lectures in theoretical physics. University of Colorado press,Boulder,Colorado, p. 583-677. Otsuka,T., Arima, A., Iachello, F. (1978). Nuclear shell model and interacting bosons. Nucl. Phys. A309, 1-33. Otsuka, T., Arima, A., Iachello, F., Talmi, I. (1978). Shell model description of interacting bosons. Phys.Lett. 76B, 139-143 Otsuka, T. (1981). Rotational states and interacting bosons. N U C l o ~ h y s , A368, 244284. Otsuka, T. (1981). Microscopic basis of the p~oton neutron interacting boson model. Phys. Rev. Lett. 46, 710-713 Ottaviani, p.L., Savoia, M. (1970). One and three quasiparticle projected states for odd mass nuclei with a major closed shell. Nuovo Cim. 67A, 630-640. Puddu, G., Scholten, O., Otsuka,T. (1980). Collective quadrupole states of Xe, Ba and Ce in the interacting boson model. Nucl. Phys. A348, 109-124. Scholten, O. (1980). The interacting boson model and applications. P h . D . thesis, Groningen.

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and K. Allaart

Talmi, I. (1971). Generalized seniority and structure of semi-magic nuclei. Nucl. Phys. A172, 1-24. Talmi, I. (1981). Shell model origin of the fermion-boson exchange interaction. In F. Iachello (ed.) Interacting bose fermi systems in nuclei. Plenum, New York 229-243.