Microscopic structure of monopole and quadrupole bosons

Nuclear Physics A397 (1983) 102 114 t~ North-Holland Publishing Company

MICROSCOPIC

STRUCTURE

OF

MONOPOLE

AND

QUADRUPOLE

BOSONS E. MAGLIONE lstituto di Fisica dell ' Universitgt di Padova, and 1NFN, Lab. Naz. Legnaro, Italy

F. CATARA Istituto di Fisica dell 'Universith di Catania, and INFN, Sez. Catania, Italy

A. INSOLIA Istituto di Fisica dell "Universita di Lecce, and INFN, Sez. Catania, Italy

and A. VITTURI Istituto di Fisica dell "Universit~ di Padova, and INFN, Sez. Padova, Italy

Received 26 April 1982 (Revised 4 October 1982) Abstract: Within the pairing-plus-quadrupole model the wave function of the intrinsic state will be described as a product of basic fermion pairs, which are a linear combination of correlated monopole and quadrupole pairs only. The microscopic structure of these S- and D-pairs, as well as the mixture coefficient between them, are determined by minimizing the total energy within the field approximation. While the mixture between monopole and quadrupole pairs changes drastically to account for the transition from sphericity to static deformation, their internal structure remains fairly constant when varying the number of particles.

1. Introduction T h e successful a p p l i c a t i o n of the p h e n o m e n o l o g i c a l i n t e r a c t i n g b o s o n m o d e l 1) to the d e s c r i p t i o n of even a n d o d d low-lying nuclear spectra in different regions of the p e r i o d i c table still leaves o p e n the challenging p r o b l e m of the m i c r o s c o p i c f o u n d a t i o n of the model. M o s t of the suggested m i c r o s c o p i c a p p r o a c h e s have d e s c r i b e d the s- a n d d - b o s o n s in terms of c o r r e l a t e d pairs of particles c o u p l e d to a n g u l a r m o m e n t u m zero a n d two (S- a n d D-pairs), b u t have been m a i n l y restricted to the case of a single-j shell (or the s o m e h o w equivalent case of m a n y d e g e n e r a t e shells), where the choice of the s t r u c t u r e of the b o s o n s is u n i q u e ; different techniques have thus been suggested in o r d e r to m a p in this restricted s u b s p a c e the fermion h a m i l t o n i a n a n d o p e r a t o r s o n t o c o r r e s p o n d i n g b o s o n ones, b o t h for even 2,3) a n d o d d 4) nuclei. 102

E. Maglione et al. / Microscopic structure

103

In the more realistic case of non-degenerate shells the definition of the collective SD subspace is still a crucial point 5). A satisfactory solution to the problem has been obtained for spherical and transitional nuclei in the framework of the brokenpair approximation 6.7), where the low-lying states are described as a condensate of collective S-pairs, with the inclusion of a restricted (1 or 2) number of higher multipole or non-collective pairs. The method cannot, however, be applied to deformed nuclei. The aim of this paper is to shed some light on the latter subject. We extend to the case of deformed nuclei the formalism of the number-conserving BCS approach, by extending the nature of the basic boson with the inclusion of a quadr u p o l e component, but still neglecting higher multipole components. We therefore assume that the (unnormalized) wave function of the ground state of our system can be described, assuming axial symmetry, in the form s)

I~'.> = (r+)~lO>,

(1.1)

where n is the number of active pairs outside the closed shell and the basic pair operator F ÷ is assumed to be a linear combination of monopole and quadrupole collective pair operators F + = ctS + +flD +.

(1.2)

The collective pair creation operators S ÷ and D ÷ are microscopically defined as S+

=

v/ ~ ,,~ . a -~' ~ O 0+Ca,a~,

(1.3)

a

D+

=

~ Pab, n ~+~a.b~ L20

(1.4)

a>__b

where A +oo(a, a) = ~

A+(a'b}

"120

=

E ~jamja -- m l O 0 > c ~ , c+ - m, m

1 ./=--7-. m l g n \ ~ \ s- ' ' ' J , - - / X/1 +6a,b X?//a"/b "

ca,+mCb,+- m

(1.5)

(1.6)

in terms of the fermion creation operators c+. With the inclusion of the non-spherical D-pair, the pair condensate (1.1) is to be viewed as an intrinsic state, and the symmetry of this state will imply a spectrum with rotational band structure, up to I = 2n which is the m a x i m u m value of angular m o m e n t u m contained in the condensate. Compared to the broken-pair model, the present approach has not any limit to the number of allowed non-

104

E. Maglione et al. / Microscopic structure

seniority zero pairs, but these are restricted to the quadrupole collective ones, with the exclusion of higher muitipolarity ones and non-collective ones. The mixture coefficients ~ and /~, as well as the amplitudes ~a and //ab defining the structure of the collective pairs S and D, are then determined by minimizing, for the different numbers of particles, the hamiltonian in the subspace spanned by (1.1), i.e. by the condition

6~(TJ"IHITJ")~() = O,

(1.7)

0(7'.) being the norm of the state I~P.).

2. The pairing case

We first consider the case of pairing interaction of the form

Hp = - G A o+A 0,

(2.1)

where A ~ = 2 Lla-i ~._!~½A +(aa,)~ot0

.

(2.2)

a

Of course in this case our unnormalized trial wave function (1.1) for the ground state will only include S-pairs, i.e. it reduces to the form I~.) = (S +)"10).

(2.3)

In this case, therefore, our approach is equivalent to the number-conserving BCS or zeroth-order broken-pair model and to the formulation by De Takacsy 9). It represents an improvement to the one given by Otsuka and Arima lo) since the matrix elements of the hamiltonian between the trial wave function are exactly obtained without making use of the number operator approximation. The explicit formulae are given in the appendix. The results of the calculation for the ground-state energies are shown in fig. 1, as a function of the number of particles. The single-particle energies and the coupling constant are the same as in Kerman, Lawson and MacFarlane ~1), and are given in the figure caption. The results are compared to the energies obtained in an exact shell-model calculation and in the BCS approximation. For a better test of the model we give in tables la, b the occupation probabilities of the different singleparticle shells and the associated two-particle transfer spectroscopic amplitudes for cases with different number of particles. The comparison confirms the excellent

E. Maolione et al. / Microscopic structure i

,

i

i

i

PAIRINCASE G i

/i

105

i

I

.-,0

>. llJ

-3

~

EXACT

NpAtRS

Fig. 1. Ground-state energies obtained in the pair approximation (eq. (2.3))(dash-dotted line) for the pairing hamiltonian, for different number of particles. The single-particle states and corresponding energies are e(ds/2) = 0, /:(g7/2) = 0.22 MeV, e(sl/2) = 1.9 MeV, e(d3/2) - 2.2 MeV, e(h11/2) = 2.8 MeV. The coupling constant was taken as G = 0.187 MeV [-ref.tl)]. The results are compared with those obtained in an exact shell-model calculation (solid line) and in the BCS approximation (dotted line).

description of the ground-state wave function given by the pair a p p r o x i m a t i o n model in the case of pairing interaction. The variation of the structure of the S-pair is illustrated in fig. 2, where the different amplitudes are given for the different isotopes. The structure of the collective pair is fairly constant, and the small variation of the amplitudes are systematically related to the single-particle level distribution and to the m o v e m e n t of the Fermi surface. More precisely the amplitude associated with a definite state increases when the Fermi surface is getting close to that level and then will decrease as the level will be mainly occupied t. However, we have to mention that even small differences in the amplitudes lead to observable effects when the n u m b e r of pairs is large. Indeed, by fixing the amplitudes as those obtained in the one-pair case, in the middle of the shell one gets energy differences of the order of 1 MeV even if the overlap between the intrinsic wave functions of the pairs is large (say, 0.99). The effect of the shell structure is further clarified if we notice that in the case of degenerated shells the structure of the collective pair is constant and uniquely

t We observe that the same feature is present in the results of ref. 7) (see fig. 11 therein) and that the increase of collectivity of the pair with the filling of the shell seems to be related to the particular choice of the single-particle energies.

E. Maolione et al. / Microscopic structure

106

TABLE la Occupation probability coefficients Npair = 2

Npair = 5

Npair = 8

j

exact

pair approx,

BCS

exact

pair approx,

BCS

exact

-5 2 72 -1 2 _3 2 11 2

0.325 0.214 0.029 0.023 0.016

0.325 0.214 0.029 0.023 0.016

0.316 0.222 0.030 0.023 0.016

0.715 0.607 0.078 0.060 0.038

0.713 0.609 0.078 0.060 0.038

0.707 0.612 0.085 0.063 0.038

0.936 ¢).922 0.404 0.234 0.105

pair approx.

BCS

0.936 0.922 0.388 0.234 0.108

0.931 0.914 0.370 0.249 0.115

The occupation eoeflicients Vjz = (Tt.l~,.aS,.aj,.l~,)/O(~P.)(2j + 1) obtained in the pair approximation are compared with those obtained in the exact shell-model calculation and in the BCS approximation, for different cases. TABLE lb Two-particle transfer spectroscopic amplitudes Npair = 2

Npair = 5

Npair = 8

j

exact

pair approx,

BCS

exact

pair approx,

BCS

exact

pair approx.

BCS

52 22 !2 -3 2

0.900 0.876 0.168 0.213 0.305

0.900 0.877 0.168 0.213 0.305

0.891 0.890 0.171 0.215 0.306

0.926 1.135 0.271 0.338 0.469

0.928 1.135 0.271 0.337 0.468

0.937 1.134 0.282 0.347 0.472

0.445 0.571 0.592 0.641 0.743

0.438 0.561 0.575 0.639 0.762

0.523 0.687 0.547 0.660 0.806

A similar comparison to that detailed in the caption of table la is made for the two-particle transfer spectroscopic amplitudes Tj = (~,r[a+af]o/x/21~P,_ 1)/(O(~u,)O(~U, t)) 1/2.

determined and the pair approximation

yields the e x a c t result. It is f u r t h e r m o r e

e a s y to m a p t h e f e r m i o n p a i r o n t o a n s - b o s o n a n d t h e t w o - b o d y

boson hamil-

tonian of the IBM form (2.4)

Hbo s = WsS+S+UoS+S+SS, where ws = - G O

and

u 0 = G, gives t h e e x a c t

answer

2).

In the case of non-

d e g e n e r a t e shells t h e e q u i v a l e n t b o s o n h a m i l t o n i a n c o n t a i n s - h i g h e r o r d e r t e r m s in t h e i n t e r a c t i o n . T o test t h e i m p o r t a n c e o f t h e s e t e r m s we h a v e fitted the s e q u e n c e o f e x a c t g r o u n d - s t a t e e n e r g i e s w i t h a q u a d r a t i c h a m i l t o n i a n o f t h e f o r m (2.4), l e a v i n g ws a n d u o as a d j u s t a b l e p a r a m e t e r s . T h e r e s u l t i n g m e a n s q u a r e d e v i a t i o n is o f t h e o r d e r o f 250 keV.

107

E. Maglione et al. / Microscopic structure 1.C

.

.

.

0.8

. . ; l d 5/~)'

.

S - PAIR' P A I R I N G CASE

/~ (og7/2)

O.E 0.,~ 0..~ I

(2S1~21)2J

I

2

I

3

I

I

4

-

5

~ I

6

I

7

l

8

NpAIR Fig. 2. Microscopic amplitudes of the S-pair in the case of the pairing hamiltonian for the chain of different nuclei, corresponding to the results given in fig. 1.

W e finally m e n t i o n t h a t the w h o l e p r o c e d u r e c a n be easily g e n e r a l i z e d to the d e s c r i p t i o n of o d d nuclei, w i t h respect to t h e s e n i o r i t y - o n e states. I n this case, for e a c h t o t a l a n g u l a r m o m e n t u m j,, the trial w a v e f u n c t i o n will be o f the f o r m

le~f~)

(2.6)

= Ca+ (S + r l 0 ) ,

a n d a g a i n t h e s t r u c t u r e of t h e S - p a i r is d e t e r m i n e d by t h e e n e r g y m i n i m i z a t i o n p r o c e d u r e . T h e results o b t a i n e d are c o m p a r e d in t a b l e 2 w i t h t h o s e r e s u l t i n g for t h e e x a c t s h e l l - m o d e l c a l c u l a t i o n . As in t h e e v e n case t h e a p p r o x i m a t i o n results to be fairly g o o d especially for t h e r e l a t i v e energies, w h i l e t h e d i s c r e p a n c i e s in t h e a b s o l u t e v a l u e s j u s t reflect t h e c o r r e s p o n d i n g o n e s in the n e i g h b o u r i n g e v e n nuclei. I n all t h e cases the s t r u c t u r e of the S - p a i r is q u i t e close to t h a t o b t a i n e d in t h e

TABLE 2 Energies of seniority-one states (in MeV) Npart = j

exact

s_ 2 _2_7 ½ 3_ 2 12-t

- 2.412 - 2.401 -- 1.157 - 0.879 - 0.311

Npart = 11

7

pair appr. -

2.410 2.398 1.155 0.877 0.310

Np,,, = 15

exact

pair appr.

exact

pair appr.

- 1.566 - 1.690 -0.888 - 0.626 - 0.079

- 1.549 - 1.672 -0.880 - 0.617 - 0.070

2.619 2.426 1.515 1.766 2.301

2.653 2.460 1.558 1.812 2.352

Comparison of the energies of the lowest seniority-one states obtained in an exact shell-model calculation with those obtained in the pair approximation (see eq. (2.6)). The comparsion is made for cases corresponding to different numbers of particles.

108

E. Maglione et al. / Microscopic structure

neighbouring even nucleus, even if the Pauli principle due to the odd partible acts in an asymmetric way on the different components.

3. The pairing plus quadrupole case Let us move now to the case in which the effect of the long-range part of the residual interaction is strong enough to give rise not only to collective low-lying excited states of the system (i.e. surface vibrations around the spherical shape), but can lead also to a permanent shape distortion. In the case of axially symmetric quadrupole deformed nuclei this can be achieved by adding to the pairing term in the residual interaction a separable quadrupole-quadrupole term in the form 12) Hq --= --K E

Q2uQz.,

(3.1)

where the quadrupole operator is given by

Q2, = ,,~5- ~ (allrZY211b) B~;,b)

(3.2)

a,b

and, in terms of fermion operators,

B~ b~= ~
(3.3)

m, m '

Standard values of the coupling strength for the different isospin channels are given in the literature 12), but the value of course has to be renormalized if a restricted fermion space is assumed. We have chosen values of the coupling strengths (given in the caption to fig. 3) leading to a ratio ~c/G larger than the standard one. This has been necessary in order to obtain, in our qualitative example with only one kind of particle, a well deformed system at the middle of the shell with /3 = 0.24 (with A = 125) and an effective pairing gap A ~ 1.5 MeV. The calculation has been carried out within the field approximation, that is by replacing the operators A~ and Q in the pairing plus quadrupole hamiltonian ((2.1) and (3.1)) by their expectation values in the ground state. The explicit formulae for these matrix elements are given in the appendix. The results are shown in fig. 3. The system remain spherical at the beginning of the shell, then starts to be deformed from 3 pairs on. As a consequence the ground state will be build up of monopole pairs in the former cases, while it will become a mixture of monopole and quadrupole pairs in the latter ones. The structure of the S- and D-pairs, characterized by the amplitudes % and/3,b respectively, only show smooth variations with the number of particles, of the same magnitude as in the

E. Maglione et al. / Microscopic structure

109

1.0

{

0.5

/ I

I

I

I

0.6

(d s/2~. (d~V2).-~ ----....~

0.3

(d5/2.d3/2)~ (d5/2,g7/2)-,~.

(,7/2.d3/2)j.

I

f

I

I

-------

(h|V2)2 ~. I

I

~I

I

I

BeAt e f

I

I

(d5/2)~ 0.(

~/2)'

O.~

S PA,. NpAIR

Fig. 3. Microscopic structure of the basic pair (1.2) for the different number of particles in the case of the pairing plus quadrupole hamiltonian. The single-particle levels are the same as in the pure pairing case (see caption to fig. 1), with the coupling constants G = 0.249 MeV and x = 0.208 (Mw/h) 2 MeV, respectively. We show in (a) the amplitudes ct and fl of the S- and D-pairs, and in (b) and (c) the microscopic structure of the quadrupole and monopole pairs, respectively.

pure pairing case, and similarly the mixture coefficient remains fairly constant in the deformed region. Variation of this order results in energy differences of the order of 200 keV in the deformed region (see fig. 4). The smooth variation of the structure of the basis pair F in the deformed region suggests the possibility of mapping the pair F onto a boson 7 and of describing the ground-state energies in terms of an equivalent boson hamiltonian. Assuming a constant structure for the pair F the sequence of ground-state energies can be exactly reproduced by the equivalent boson hamiltonian Or

Hbos = ~

n=l

h.(~+)"7 ",

(3.4)

where the coefficients h. are given, in terms of the matrix elements of the fermion

110

E. Maglione et al. / Microscopic structure ,

i

i

i

,

,

i

\ \\\

-2

\

. \\

/ \

\

3

rd

order

boson

/, /~

"

.:

~-4 W -6 2

order~ o s o ~ \

harndtoman

~...~__~--Pa~r Approx. '

\ \

-8 NpAIRS Fig. 4. Ground-state energies obtained by the minimization procedure in the pairing plus quadrupole case (solid line). Details on the single-particle levels and coupling constants are given in captions to figs. 1 and 3. The dashed curve is the result obtained by keeping frozen the structure of the pair F, as resulting from the calculation with 6 pairs. The dot-dashed and the dotted curves are the predictions of the boson hamiltonian (3.4) truncated to two-body and three-body interactions, respectively.

hamiltonian, by the recursion formula

(r"lHIr"} h, -

n!

.~i m ! ,,= 1 ~(h,..

(3.5)

The m e t h o d is similar to that used by Lie and Holzwarth 13) and is actually a b o s o n expansion. In our case, the 7-boson being a particle-particle boson acting on a finite space determined by the total shell degeneracy QT, the n u m b e r of terms in (3.4) is finite. It is relevant to the discussions raised by the I B M to see whether the truncated second-order b o s o n hamiltonian (3.4) gives acceptable results. As apparent from fig. 4, the predictions of the second-order hamiltonian are very poor. The higher order terms seem to play an essential role, and at least the third-order term has to be included to get a reasonable agreement with the fermion calculation. O u r prescription for obtaining the I B M hamiltonian is not unique, and other m a p p i n g procedures m a y be more effective. Furthermore, there is still the possibility that a better agreement might be obtained by a "phenomenological" parametrized twob o d y boson hamiltonian, but. this would mean that the reciprocal transfer of information from the fermion to the b o s o n worlds is not straightforward.

E. Maglione et al. / Microscopic structure

111

4. Conclusions Within the pairing plus quadrupole model we have presented a method which can be used to determine dynamically the most favoured microscopic structure of monopole and quadrupole collective pairs. Even in the presence of a phase transition this structure results to be fairly constant and the transition is reflected only in the drastic change in the mixture between monopole and quadrupole pairs. This seems to substantiate, on a microscopic basis, one of the assumptions of the IBM. However, two other kinds of problems, also related to the basic assumptions of the model, are still open. The first one is that the constancy of S- and D-pairs can only be achieved if the Pauli principle is correctly taken into account. It is not clear whether a two-body boson hamiltonian in the corresponding sd space is sufficient to do the job. The results shown in the previous sections seem to indicate that the higher order terms play an essential role. The second one is whether this collective subspace is able to account for all the properties of the low-energy nuclear spectrum, or, on the other hand, whether the neglected components, even if small, are essential. The problem has been raised 8) especially for strongly deformed nuclei, and calculations done in the Nilsson + BCS approximation seem to show that for some physical quantities the effect of the higher multipole pairs cannot be neglected 14). This is obtained in spite of the fact that the S- and D-pairs are dominant components in the F-pair 15,~4), thus showing the sensitivity of some observables to even small components in the wave function. Within our number-conserving method this problem will be discussed in a forthcoming paper.

Appendix In this appendix we give the explicit formulae necessary to evaluate the matrix elements of the pairing plus quadrupole hamiltonian in the field approximation. We start from the expressions for the following commutators

[A d b), A ff" a,] =

( IL

_(_)j~+j~ / ( 2 I + l ) ( 2 L + l )

"ff ( l + ~ S )

fI

L

A}

E E a. . . . . Yo J, Jf Jg

x [6g,,6it,(6~6ea + 6,,a6e,) + 6S,,6gb(6ba6e~ + 6,b6~a)]B~b ~),

(A. 1 )

112

E. Maglione et al. / Microscopic structure

[B~6b,, A[o~c'a,] - Z qg(Aab, Icd; Lae)A~o a'e) L,e

= (_)~.+~. /(2A+1)(2I+1) X/ (1 + 3.,)

Z (IOAOILO>Z~/I+6.~

Leven

e

I A L}

+(a,e)

(A.2)

× J, Jc Jb (3cbfde+3baoce)AL° '

+ (c,d)]J, Ai~6f)] - ~ z(Jab, Lcd, Ief; Kgh)A~o °'n) lIAr6 b), A LO ghk

Jb+J~ / ( 2 J + l ) ( 2 L + l ) ( 2 I + l ) ghmn

j"jhI J, Jo L JKR

x~

(6e,.6fh+6mffeh )

X [(~an~)bm((~ac(~dg ~- (~ad(~cg) ~- (~am(~bn((~bd(~cg "~- (~bc(~dg)] ~*KO zl + (g, h)

a+(c,a)q A[o(~,f)] = O,

[[[A~db),A[,(o~'d'], A+(e'f)qLO J, "'MoA +(g'h)-I3=

(A.3)

(A.4) O,

(A.5)

and from the general operational expression

CD" = D"C+nD"-*[C, D]+½n(n-1)D"-z[[C, D], D ] + ....

(A.6)

By defining the basic pair operator F in the form (cf. expression (1.2)) F+ =

E

~A~I(ab)A+(a' *AO b)"

'

(A.7)

A=0,2

ab

and by using the previous expressions for the commutators, we can derive the following coupled recursion relations for the norm: = n < o l r " - ' r +"- ~1o> +½n(n-.l)

x ~ a(j"b)a(~)aTz)z(Jab, Lcd, lef; Kgh) JLIK abcd efoh

(A.8)

E. Maglione et al. / Microscopic structure

113

and for the matrix elements of A&b) and B!,“bb’operators

(olrn-

lA$b)f+nlo)

=

n

C

a~d)d,L

X

(- )ja+%uJdSbc)

(‘m’bd-

Jtl

LCd

+hb)(l

+6,,)

1) C af”)a(lef)

(OJT"-Lf+"-1(0)+3n(n-

LIK de /gh

x X(Jab, Led, Ief; Kgh)(O~~"-2A',4~'T+"--10)

64.9)

and (Olr”@!$)r+“)O)

= n C a~4p(Aab,

Icd; Lae)(OJT”-‘A~~“)Ti”(0).

(A. 10)

IL cde

In the case of pairing hamiltonian the basic pair I- contains only the Scomponent, and we can easily release the field approximation, by evaluating the exact matrix element of the hamiltonian using the recursion relation

W’

(O(r” - 2A($b)T

+ R - 1(oj

_

+ (2]:+1) + 4n2(n-

1)2

(2&+ 1)2

ayj4

6,,

(OlP-2r+n-210)

2(n c2j,+

-

’ )ab”“)abbb) 1jt2jb+ 1l

-

(Ol~“-2A,+d”,“‘A~~‘T+“-2)0)

~(Olrn-2sb”o”)r+n-2,0~

. (A.ll)

JO

References 1) Interacting Bose-Fermi systems in nuclei, ed. F. Iachello, (Plenum, NY, 1981) 2) T. Otsuka, A. Arima and F. lachello, Nucl. Phys. A309 (1978) 1 3) T. Otsuka, A. Arima, F. Iachello and I. Talmi, Phys. Lett. 66B (1977) 205; 76B (1978) 139; T. Suzuki, M. Fuyuki and K. Matsuyanagi, Prog. Theor. Phys. 61 (1979) 1682; R. A. Broglia, K. Matsuyanagi, H. Sotia and A. Vitturi, Nucl. Phys. A348 (1980) 237 4) 0. Scholten, The interacting boson approximation model and application, Ph. D. thesis, University of Groningen, 1980, unpublished; R. A. Broglia, E. Maglione and A. Vitturi, Nucl. Phys. A376 (1982) 45 5) A. Klein and M. Vallieres, Phys. Lett. 98B (1981) 5; M. R. Zirnbauer and D. Brink, Nucl. Phys. A384 (1982) 1 6) K. Allaart, in Interacting Bose-Fermi systems in nuclei, ed. F. Iachello (Plenum, NY, 1981) p. 201; J. N. L. Akkermans, E. Loriaux, K. Allaart and G. Bonsignori, preprint IFUB-81/12 (1981)

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