Symmetry breaking of the galilean invariance in superfluid nuclei and its connection with quadrupole pairing interactions

Nuclear Physics A542 (1992) 221-236 North-Holland

NUCLEAR PHYSICS A

Symmetry breaking of the galilean invariance in superfluid nuclei and its connection with quadrupole pairing interactions 0. Civitarese'~*, Amand Faessler and M .C. Licciardo'~*

In.çtiiui .für Theoretische Physik, Universitât Tübingen, D-7400, Tübingen, German-v Received 20 February 1991 i Revised 23 December 1991)

Abstrad : Effects due to quadrupole-pairing interactions are analyzed in the context of a self-consistent symmetry-restoring model. it is shown t-, iat the breaking ofa multipole galilean invariance leads to results which are similar to the ones obtained hy including quadrupole-pairing forces in the interaction among quasiparticle pairs. The model is applied to the description of low-lying quadrupole excitations in even-mass Sn-isotopes .

1. Introduction

The analysis of nuclear collective excitations in terms of separable multipole forces and the foundation of these theories in terms of elementary excitations has been very successful ") . The treatment of separable multipole forces in the random phase approximation (RPA) is nowadays a matter of textbooks 6-8 ) but it is still a very powerful method and the only one suitable for the description of collective excitations in heavy nuclei. It has been used to describe correlations in open shell nuclei 2,4,5,7,8), where the nuclear BCS model 1.6) is used to describe monopole-pairing effects . In addition to separable multipole-multipole forces, which originate in particle-hole-like long-range correlations, multipole-pairing forces have been introduced to describe excitations in nuclei nearby closed shells ") and in connection with the pairing isomerism '0). Although the inclusion of quadrupole-pai ring effects has been shown to be very useful for the description of excitations in nuclei with few valence particles 9) its treatment in spherical superconductive nuclei has not been completed yet. Recently '"'), the inclusion of quad rupol e-pairi ng correlations in superfluid spherical nuclei has been discussed in the framework of a perturbative Correspondence to : Dr . 0. Civitarese, Department of Physics, University of La Plata, C.C. 67 1900 La Plata, Argentina . Permanent address : Department of Physics, University of La Plata, 1900 La Plata, Argentina . Fellow of the CONICET, Argentina . 0375-9474/92/$05 .00 @ 1992 - Elsevier Science Publishers B.V. All rights reserved

222

0. Civiiarese ei aL / S~yiwnetiy breaking

12expansion ") and in terms of self-consistent theories ") . The results of ref. "), concerning the reduction of the quadrupole energy-weighted sum rule and the enhancement of the matrix elements for two-nucleon transfer processes, and the connection between dynamical boson effects and quadrupole-pairing forces discussed in refs. 12-' -") have renewed our interest in the subject. It is therefore of some importance to elucidate the role which is played by quadrupole-pairing effects when they are treated on the same level of approximation as the more conventional multipole- multi pole forces, namely : in the framework of the RPA and with the inclusion of monopole-pairing effects as they are described by the BCS method. We have taken the results of refs. 11 -6 ) as a motivation for the present work. We have focused our discussion of quadrupole-pairing correlations by extending symmetry P restoring techniques proposed by Belyaev ) and by Pyatov et A " -9) and lately applied to monopole-pairing vibrations ") . We have studied a connection between the above-mentioned quadru pole-pai ring effects and the velocity dependence of the CS quasiparticle mean field '). In this work we discuss this connection and we show to what extent quadrupole-pai ring forces do produce the same effects as the residual interactions which originate in the velocity dependence of the BCS mean field. The theoretical framework ! 7-19) which we have adopted to describe these 21). effects has been compared to other methods, in particular by Meycr ter Vehn In ref. 21 ) the shortcoming of standard RPA calculations, concerning the violation of translational and galilean invariances, is pointed out and a solution to it is proposed which is based on an e7.--.ct step-by-step elimination of RPA spurious states. Although the method of ref. 21) is suitable for the elimination of spurious states from the RPA basis, it is not concerned with the structure of the interactions . Since we are dealing with this problem, as we have said before, we have chosen the method of refs . 17-19) instead . The present approach allows us to construct residual interactions under conditions which are fixed by symmetries . The same method has recently been adopted by other authors 22 ). The details of the fermalism are presented in sect. 2. Results, for quadrupole excitations in even-mass Sn-isotopes, are presented and discussed in sect . 3. Conclusions are drawn in sect. 4.

2. Formalism In this section we shall briefly describe the conventional treatment of quadrupole (Q:! ) and quadrupole-pairing (P~-,) separable interactions in superfluid nuclei " " I Then we shall introduce a self-consisteat separable interaction, as it is given by the explicit breaking of the galilean invariEtnce, generalized to the description of quadrupole excitations . For the microscopic treatment of the residual multi pole-multipole interaction between quasiparticle pairs we have adopted the well-known quasiparticle random phase approximation (QRPA) method 2 ). Further considerations about the treatment of collective excitations will be based on Bohr and Mottelson's model of nuclear vibrations ').

0. Civitarese et aL / Symmetry breaking

223

2.1 . QRPA TREATMENT OF Q2 AND 1~, INTERACTIONS

Let us start with the definition of a separable muItipole-multipole interaction 2), A

H = Ho - g2 Y_

t

P2 j£ P2 ju _X2 1 Q2 ju Q2 ju 1£ M

which consists of a single-quasiparticle term, H0 , a separable P2 term and a separable A A Q2 term. The operators Q2,, and P2, are given in terms of coupled quasiparticle-pair configurations, namely A

A

Ho =I ENj , .

Ptli£ A

Q2 ju

=

E

ilJ2

= E

JIJ2

(2a)

P2(jlj2)[Uj,uj2 A''(jlj2 :2,g) - vj, vj2 A(j j j2 :211)]

(2b)

t->(jlj2)[A" (jlj2 :2g) + A(j j j2 :2,g)] ,

(2c)

where Nj = E aj,,, aj,,, ,

(3a)

ni

A"(j l j2 :2-u) =

1 [aj,a',] 2 -,/-I+ 5iI ii-

(3b)

A(ilh :2M) = (- 1) _' ([ ail_' a'j2 The quantities

j

1,(j j,)

(3c)

and P2(j]j2) ,, which appear in eq. (2) . are defined by

t2 (il j2) = 21 _1515(il Il r

2

Y-1 Il j2)«A + Si, j2(UjlPi2 +

P2(jlj2) = -,I;'(j, Il F(r)Y2 Il j2)"'/l + 451, j2

F(r) = r 2/ Ro2 .

5,

Uj2 Vil) 9

(4a) (4b) (40

We have used standard notation to denote quasiparticle creation (annihilation) operators aj (aj) t~j and vj BCS factors, quasiparticle energies Ej and the full set of quantum numbers (ji = (n, 1j)j ) which are needed to define a single-particle state, respectively . The quadrupole-pairing operator P2,, is a generalization of the monopole-pairing one 9.10,20). Both in Q and P we have neglected scattering terms which are proportional to a'a since we are interested in the QRPA treatment of the hamiltonian (1) ") . The coupling constantsg 2 and%2 are hereafter to be associated with the P, and Q2 channels of the hamiltonian H, respectively . The present form of the hamiltonian (1) coincides with the one adopted in refs. ' o, "). A more detailed discussion of its structure, in terms of an effective nuclear field theory 3 ), has been presented in ref. ") . For further details, concerning a possible double-counting of configurations, we shall refer the reader to ref. ") where this particular aspect of the problem has been discussed carefully . 2) We can write the QRPA equation of motion in its commutator form A

[H, F2,u (n)] = w,,F2 ,.L (n)

0. Civilarese el A / S~ynztnetry breaking

224

where w, represents the energy of the nth excited J' = 2' state generated by the action of the phonon creation operator

I

I ,";-

2 ,(n)=

J1

j2

[Xn'A"(j1j2 :21A) j i2

:2 .u)j Yj(n',A(j1j2 h

on the QRPA correlated ground state. The quantities Xj( n'j . and Yj( .",', are forward and backward-going amplitudes which together with the eigenvalues Wn are deter1,2 mined by casting H in its vibrational representation ) . This is done by expressing the commutator (5) in terms ,~f the following quantities :

Mp', r,(n)] = P2, ,

r

2 ,A

(n) ]

5,A

= 5,, -,A , ( - I (7)

which are obtained in the quasi-boson approximation and where 1 "" = 2 - 1

110J2)

il i2 (i+Ibil .i2)

2

0""

[X ( n ) + y(.#1 .) ] -i i2 11 .12 1)

E

P2(jlj2)_ X (. n .~ 1 ( 1 + é5 . . ) [U--Il U-J2 Y'"' jlj2 - Vil Vj2 JI .12 ilJ2 -Il J2

=2 1

P2(jlj2)_

il

+ &il j2)

[U-.11 UJ2. X'", M2

Vil Vj2

9

Y(. n .) ] .1112

After straightforward algebra we have obtained the following expressions for the forward- and backward-going amplitudes :

X(. n ') = -/I-/:!

I

( Ej , j2 - wn X[2X2 A

(n

0 t"(jlj2) + 92

(p (n

-(n) P201 j2) Vjl Vj2 1 O P2 (i 1 j2) 1~jl Uj 2 - 92 1~

Y'n' = ~ ,il j2

'E~j l

j2 -t- (On /

X[2X2 A

(i 1 j2) - 92

=(n ~P20J2)Uj^21 IP2 (i 1 j2) t~~ 17j2 + 92-

is

the energy of the unperturbed quasiparticle-pair configuration Q, 5j2)The eigenvalues (On and the quantities defined in eq. (8) are obtained by solving the following set of equations : where

Ej .

j2

=

~jl + Ei2

I (r) ) D(wn)

(p (n) " =(n)

0,

0. Civitarese et A / S~ytnmetry breaking

225

where the matrix elements of the QRPA matrix D(w,,) are given by: ~,(v)2E,, Di , = 1 - 4X2 Y, 2 ., E j, D12=292 D13= 2g2

V ),02 ( IV ) I

V),02( V)

1,

D, 1 = 4X2 D12 =

1

Î2( OP2(

- 292 E 1,

0

p2( 2

rf~4) `

+ (E,-w

V(

Ei , + Wn) U(v)

( Ei , - to

Ei, +£Ün )

(& - w,, + E,, E,, - WP?

+

+

E, + toi,

U(P) V(t, )2E,, E 2à, _ 0)2n

D'13 = 2 92

Y,1,

D31= 41y,

+ Y_T_2 ( OP 2( I") 1, ( E,, + w,, E,, -

D32 = 2 92 D33 =

Y,

9

9

2( ~V ) U( ) V V(P)2E, P2 2 _ (02 E L, n

1 - 292 E Ê 2( 2

,)

1,

( E,,+ O)n+ E, - o)#,)

In eq . (11) we have introduced the following notation : IV -=

01

9

j2)

T2( I") =

V ( I' ) = VjA2 1

U(P ) = Uj. ~h 1

t2(jlj2) , '~'il+ Oil j2

fi2( IV) =

Then, from the secular equation :

P201h) /IT 8j. i2 '

(12)

(13) det Il D(£On) Il = 0 , can determine the value of the ratios ~(n '/ A (n ' and O(n '/A( n ' as a function of eigenvalues tOn and of the amplitudes A (n '. The corresponding normalization

we the condition, for the amplitudes (9), can be written as: 02

2 _ [( X(.n . ) ) 2 + t5j, j2 ) J02

y( In .) )2] = 1 J2

(14)

and it fixes the value of A (n) . In the above derivation of the QRPA quantities we have assumed, for simplicity, that only one kind ofparticles (neutrons) are effectively coupled by the quadrupole correlations . The same sort of equations can be obtained if protons and neutrons are considered simultaneously. For the sake of the calculations which we have performed, for even-mass isotopes of Sn, the inclusion of protons is not relevant since they are in a normal (non-superconductive) phase .

0. Civitarese et al / Symmetry breaking

226

1.2 . MULTIPOLE GALILEAN INVARIANCE AND SELF-CONSISTENT QUADRUPOLE FORCES

In this subsection we shall discuss a possible connection between the effective hamiltonian (1) and a symmetry-breaking mechanism based on the generalization of the galilean invariance to A-pole shape vibrations . Our argument is based on well-established symmetry considerations') and it can be formulated as follows: the velocity dependence of the quasiparticle mean field') induces the appearance of multipole forces. These multipole forces, when treated in the QRPA, produce the same effects as those due to multipole pairing forces. Therefore, as we would like to show presently, the breaking of this generalized galilean invariance has a definite effect upon the structure of low-lying multipole excitations which could not be attributed to the usual residual interactionS 2 ) . The velocity dependence of the BCS quasiparticle mean field can be determined by evaluating the commutator : Q2,A oA pair H( 21A

where Q_jA '*-, = r J By keeping terms which are proportional to quasiparticl e-pair operators, namely : 2 y., _A

1110 , A(jl j2 :21L)l = Ej,j2 A"UJ2 H,) , AUJ2 :21L)]

we can write for the operator 0 p 0 pair 21.L

I

iI]:!

r

:2g)

= -Ej.j2A(j1j2 :2ji)

of eq. (15) the following expression:

t--(jlj2)EA. 12-

[A" (jlj2 :2g) - A(j l j 2 :2tLfl

.

(17)

Furthermore, the commutator:

MI

opaiT 21A

A -i

pA

JA - JA

f2 = 4

f2

t~(ilh)Eili2

~ 1 +5il j2

Jlj2

(18)

represents the symmetry breaking induced by the velocity dependence of the BCS mean field . In fact, one can introduce an effective interaction of the form: H'= H( , - yo Y_ O~,, 1£

0.,,£ .)

and subsequently one can evaluate the expectation value of the double commutator : A ([[H', Q2

A

(20) A 19 Q~2,1) = -f2 (1-2 YOP If the symmetry operation represented by the A -pole coordinate, r" YA,(A =2), is fully preserved by the effective hamiltonian (19), then the value of the coupling constant Y, can he determined. If (20) vatlishes, then Yo ,:--

2f2

(21)

0. Civilarese el aL / S~ytntnetry breaking

227

and a zero-energy mode will be obtained due to the action of the multipole operator A Q2,. on the correlatrd ground state. This condition is, of course, no longer compatible with the assumption that vibrations can be produced by density fluctuations of the mean field since it would imply a permanently deformed ground state. However, we can think of a situation where the symmetry is not fully enforced by the constraint represented by the effective interaction introduced in eq. (19) and that the extent of the symmetry breaking is represented by an effective value of the coupling constant, y, in the range 0 <_-- y -, y(, . In tLis fashion we can write the hamiltonian H = Ho - -y 1 O m 0-,,, - k,(y) 1 Q2,u Q2ji 1A

(22)

-)

AL

and treat it in the QRPA approach . In order to set up the corresponding equation of motion, let us introduce the commutator rolm , , ""I

where

R "" =

F. 2,L(n)] = 5,,L ,R

E

2M"(jlj2)

J 1J2

Ïl j2)

(n)

(23)

(X( n ) _ y,(n) jlj2

li(24)

M 2 (j1j2) ~ t-)(jij2)Ei l i2

At this point we can write again the QRPA equation of motion, as we have done before, and the corresponding system of equations can be written as : lh 2 .j2 R (n) 1 -4-y E 2(jlj2)2Ej .2 2

1

Eiij, - OJ ',

ilh

+A

i"(jlj2)IÎ1 2(jlj2) 2 tù,, 1 Eil2 J2 - (On JIJ2

1

4k.,(y) E -

-(jl_j2)_Î"(jlj2)2£o,, tÎ~ " 2 Eji2 j, Li2 , t 2 (jlj2)2Ej, j2 1 +A'" ) -4k,(y) E ') 2 0) _n JIJ2 E il i2

=0 ,

R'" )

4 -y 1,

1

where

1k)(j1j2) =

--

1 = 0,

(25)

_M"(jlj2)

_~i T 67.j2 From the secular equation corresponding to this set of coupled equations we can y(n X(n ' and ) . We determine the value of the eigenvalues tOn and of the amplitudes have obtained for these quantities the following results : i'l)

(11)

(n ', ,rym,(j~j2)+a )k,('!')t2(jlj2)11 j Ej, i2 - ÙJn _

I . ; 02 - = /_1M

1

(n (n)r-YM-'(j]j2)+a . )k,(Y)t2(jlj2) 1 _L 'Lun L-;il i2

1 (26)

228

0. Cïviîarese et aL / S~vnuneijy breaking

where

Again, in this case the normalization condition for the QRPA amplitudes, eq. (14), is used to determine the value of the quantity R" ). It might be useful to comment on some of the features which can be extracted from the QRPA equation of motion under the partial fulfillment of the symmetry requirement, eq. (20) . Firstly one can observe that eq. (20), in the limit Y,) =O, is exactly proportional to the energy-weighted sum rule (e .w.s.r.) of the quadrupole operator'); it means that the inclusion in the hamiltonian (19) of a term which partially restores the A -pole galilean invariance will reduce the value of this e .w.s.r. As we shall show in the next section, this reduction is also present when quadrupolepairing forces are introduced as residual interactions . Therefore, at least qualitatively, one could expect to find some analogies in the effects of both the quadrupole pairing force and the symmetry-restoring interaction introduced in eq. (22) . The second observation which we think is worth mentioning, is concerned with the fact that the symmetry breaking will be always present if the quasiparticle mean field, in the BCS approach, is chosen 'to represent the unperturbed quasiparticle motion. As we have said before this is the result of the velocity dependence of the BCS singlequasiparticle hamiltonian . Therefore, the inclusion of the symmetry restoring term would be superfluous before performing the BCS transformation " 6) . In this respect, the situation is the same which one can find in dealing with number-violating effects . The violation of the symmetry is enforced by the way in which the mean field is treated and not by the two-body interaction by itself. Moreover, it depends upon the way in which the collective excitations are defined. For the case of the particlenumber symmetry, quasi particle-pai r excitations would occur at zero energy since the particle-number operator itself contains such terms and because the BCS hamiltonian 14() does not commute with quasiparticle-pair components of the particlenumber operator 20) . Therefore we can argue that quasi particle-pai r configurations, which are included in A-pole vibrations, should be distinguished from those contributing to the permanent deformation of the ground state. In the limiting case of a permanently deformed system this statement will be completely correct since the first quadrupole mode can, of course, have zero energy . For spherical systems this situation will not show up hence -/ will not take the value yo . If a small admixture of the -y-interaction is allowed then the structure of low-lying quadrupole oscillations will be different from the one obtained -vvith pure quadrupole forces . This is the case of the solutions of eq. (25) . In fact, it can be easily shown that this system of equations, eq. (25), for a fixed value of y, determines Uh:? value k-,(y) =

yo (l. - yF(3» (yo - -y) F" ) '

(28)

0. Civilarese el aL / S~ymmetjy breaking

229

where =4

F"'

E -11 i2

(il j2) 2 E.l j2 Ej'il j2 - oi 2#l

')

for the effective quadrupole coupling constant k-,(y) . Therefore, for a complete symmetry breaking (y = 0) one can recover the well-known value of the quadrupole coupling constant 1 .2) . On the other hand, this quadrupole coupling constant will become infinitely repulsive if the symmetry is fully recovered with a critical behavior given by lim k-,(y)~--O(

Y - yo

\

Yo yo-Y

(29)

) -

We shall discuss some consequences of this result in the next section . To complete the introduction of the basic equations of the formalism let us define the expressions which we have used to compute reduced matrix elements of the quadrupole and quadrupole-pairing operators, as they are given by the QRPA approach . They can be written as : (n

-

Il Q2 Il O~ = Vf5E Il i2

(n 11 P, 110) = vr5-

21-)(jjj-» (1+13 i. i-, ) 2p.,(jlj-,) (I+5

(fi) 'l i2

+ Y (1 811 » , i2

1") . -

Uil Uj2'X /1

12

V11. V12 Y111 ") j2

(30)

respectively . The arguments which we have presented in this section, concerning a partial restoration of the A -pole galilean invariance for a BCS+ multipole-multipole hamiltonian at the QRPA level of approximation, lead to results which are rather similar to the ones of ref. '), although we have not considered explicitly the requirement of translational invariance . This assumption is justified, in the present approach, since we have taken the BCS quasiparticle mean field as the starting effective single-body mean field and we have only considered the consequences of its implicit velocity dependence upon the structure ofresidual interactions between quasiparticle pairs. 3. Results and discussion

We have selected, as an example for the application of the formalism of sect. 2, the case of quadrupole excitations in even-mass isotopes of Sn. Data about the excitation energy of the first excited J' = 2' states are available in the mass region 102 Sn --> 13 ' )Sn [refs. 23-26 )] which show a relatively constant value for these energies of the order of I MeV . Also, for these even-mass isotopes of Sn, a calculation of quadrupole transition matrix elements and two-particle quadrupole-pai ring matrix elements has been performed "), in the context of the QRPA and for separable

230

0. Civiiarese et aL

/ 5~, nititetri, breakipig

TABLE I Set of neutron single-particle states included in the calculations. The single-particle energy values, -,, have been taken from ref. ') . 1 0 1 0 2 0 1 0 1

i

4 2 4 0

-4.106 -0.200 0.861 1.322 2.506 2.878 7 .369 8.326

7 2

i

2

2 5 3

[MeV]

E,

7 2

quadrupole and quadrupole-pairing interactions . We have taken the results of ref. ") as a guide. For the single-particle basis we have adopted the set of neutron singleparticle states which are listed in table 1 . The single-particle energies have been taken from ref. 5). For the strength of the separable monopole-pairing channel, which is treated in the BCS approach, we have chosen the value Go = 19/A MeV. With this value the observed odd-even mass differenceS 27 ) are fairly well reproduced and the obtained BCS gap values (,J) are of the order of ') J = I I/A 1/2 MeV. Ail radial matrix elements have been computed by using harmonic-oscillator radial wave functions. We have included only neutron orbitals in our configuration space, as in ref. ") . Once the BCS parameters have been determined all the information which is needed to compute symmetry-breaking effects is given and we can compute the value of the symmetry-restoring parameter -yo . The values which we have obtained for this quantity are shown in fig. 1 . The mass dependence of -yo seemingly suggests 2 .5-,

1 .0 1 102

,

,

1 0-6

I

110

I

~

114

I

I

118

VAS S

I

I

122

I

I

126

~

~

130

Fig. 1 . Mass dependence of the syrnmetry-restoring parameter y,, eq . (21) .

0. Civitarese et al. / S,-mmeij~j, breaking

231

has a minimum nearby the middle of the shell. In order to determine the dependence of the strength of the separable qvadrupol e-quadrupole residual interaction upon the symmetry-restoring mechanism we have calculated the value k2(Y), of eq. (28), by fixing for different values of y the energy of the first excited 2' state. The results, for the case o f 114 Sn, are shown in fig. 2 As expected, from eq. (29), the full restoration of the symmetry (y = yo ) requires the use of a repulsive quadrupolequadrupole interaction of infinite strength in order to bring the first excited 2 -,- state at the desired energy. However if the value of y is chosen to be of the order of 60% of yo , the effective quadrupole- quadrupole coupling constant takes the value which results from the inclusion of quad rupole-pairing forces, as explained in sect. 2. The mass dependence of the effective parameter y* is shown in fig. 3. These results have been obtained by solving eqs. (25) and (28) for &j,, = E (2 ',) and by searching for the value of the quadrupole-quadrupole interaction which agrees with the value obtained from the solution of eq. (10). It is worth mentioning that the value of the ratio y*/ yo , which as we have indicated before suffices for the renormalization A the coupling constant for Q2 channels of eq. (22) in the same amount found for the case of the hamiltonian (1), is not far from the results which can be obtained in a fully degenerate single-shell situation . For this case one would obtain y*/ yo = 0.75, as is shown in fig. 3 . The strength of the quadrupole-pairing coupling constant92 of eq. (1) has been fixed, as in ref. "), at the value92 = 207TG,, . That the effective quadrupole- quadrupole coupling constantX 2 has similar values in both treatments, namely: (i) by including a quadrupole-pairing channel with strength92 or (ii) by 196 .

30

E

201

't

0

1

0

1

X2(~% + 0 2)

0-

\YO

Y 30 0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

yx 10'[ MeV-'

1 .2

fM-4]

1 .4

1 .6

Fig . 2 . Dependence on y of the effective coupling constant for the quad rupol e-quadru pole residual interaction, k,(-I) where y is the coupling constant corresponding to symmetry-restoring interactions . 4 The solid line corresponds to the solution of eq . (28) for the case of quadrupole excitations in " Sn . Horizontal dashed lines repre-ent the value of the coupling constants for pure quadrupole (X2(Q2)) and quadrupole plus quadrupole-pairing interactions (X2(P2+ Q2)) forces . Vertical arrows represent the values of the effective symmetry- restoring coupling constant (Y*) and the full symmetryrestoring parameter (yo ) .

0. Civitare.çe et aL / S~yPnmetî~y breaking

232 Il

~ ()

_~

Fig. 3. Mass dependence of the effective symmetry-restoring coupling constant (y*) in units of y(, . The horizontal dashed line represents the value of the ratio y*/y,, corresponding to the single-shell limit.

partially restoring the A-pole galilean invariance, with the inclusion of a residual interaction of strength y* - 0.6y, ) resulting from the velocity dependence of the BCS quasiparticle mean field, suggests a co-existence mechanism . It can be further explored by comparing vibrational properties associated with the first excited quadrupole excitations . After solving the corresponding set of QRPA equations in (i) and (ii), we have obtained a remarkable agreement between the effects associated with hamiltonians (1) and (22) . Both residual interactions reduce the B(E2 ; 0' --> 2 '1 ) by the same amount. This is shown in fig. 4. With the calculated B(E2) values, in both approximations, we can furthermore calculate quadrupole deformation parameters and the restoring-force parameter for the vibrational field '). The results are shown in figs. 5 and 6, respectively . We have compared the present results with the . 0r

Z U

I

0 .5-1

U. U

-

1

02

I

I

106

~

I

1 1, 0

-

1

7

1, 1 1

4

1

1

1 F)

VAS S

ik--A

S FR

'111- Al

P 2+ Q 2

.

1 22 1

I

~

126

I

i

1 30 1

Fig. 4. Mass dependence of the reduction factor for B(E2 ; 0+ - 2 +,) transitions calculated from the QRPA treatment of the hamiltonians (1) (P-"+ Q2) and (22) (SR), respectively . These results have been obtained by dividing the calculated values by the quantities obtained with pure quadrupole forces .

0. Civitarese et aL / Sytnrneto~y breaking

233

O. It

QZ C\j

0.10-

0 .05

'A

i 1

1

X~) O-OOt --T 102 106 110

`4

'L

"-~

)I

D

P2 + (~'

0 0

("'

118

12~

12F,

1,,4

~)

VA

Fig. 5. Mass dependence of the calculated deformation parameter, P2, for quadrupole transitions in even-mass Sn isotopes . Vertical lines represent the range of variation of the data taken from refs. 23,25,26). With Q2, P2 + Q2 and SIR we have indicated the results which we have obtained from the QRPA treatmeiit of pure quadrupole forces, with the inclusion of quadrupole-pai ring forces and with effective syrr.r,-.ct-.yrestoring interactions, respectively.

liquid-drop model results ') and with those obtained with pure quadrupole forces. Again, from the results which are shown in these figs . 5 and 6, we can observe that both hamiltonians (1) and (22) produce the same effects . It is also nice to see the overall agreement with the available data, particularly, for the case of the restoringforce parameter, since for it the large difference observed between data and the vibrational model predictions ') has been a puzzle for quite a time. The comparison between e.w.s.r. values, for quadrupole (Q2) and quadrupole-pairing (P-',) operators, obtained from the QRPA treatment of the hamiltonians (1) and (22), is shown in tables 2 and 3 . Both the reduction of e.w.s .r. values for Q2 and the enhancement of the matrix el--ments of X for transitions to the first excited 2' state, shown in these A~

A

A,

1000-1

100U

"'2

10 1 102

106

1 10

1 14

1 ~8

122

* '12

', 2F, ' ~~)

V AS 1-7 1

Fig. 6. Mass dependence of the restoring force parameter, G, for the models which we have described in the text . The labels are as in fig. 5. LDM represents liquid-drop model results.

0. Civitarese et aL / SYnitneirt, breaking

234

TABLE 2

Energy-weighted sum-rule values (e .w .s .r.), for quadrupole transitions in even-mass Sn isotopes . Results obtained by using pure quadrupole-quadrupole forces (Q2), with the inclusion of quadrupole-pairing forces (P'+Q2) and with symmetry restoring interactions (SR) are shown in columns 2-4, respecf 4] tively. All values are given in units [MeV- m e.w .s.r. A

Q2

P" + Q2

SR

104 106 108 110 112 114 116 118 120 122 124 126 128 130

5347 6320 7097 7714 8153 8450 8728 9072 9404 9564 9454 9013 8178 6953

2882 3073 3436 3771 4005 4248 4512 5052 5294 5673 5939 6059 6014 5646

2944 3472 3898 4238 4480 4644 4797 4987 5166 5253 5194 4646 4498 3818

tables 2 and 3, support the claim about a similarity between the correlations induced by the hamiltonians (1) and (22). In view of these results we can interpret the reduction in the e.w.s.r. value s for quadrupole transitions which is obtained by the inclusion of quadrupole-pairing forces ") as the obvious consequences of the partial restoration of the A -pole galilean invariance . In other words : the velocity dependence of the BCS quasiparticle hamiltonian is partially compensated by the additional velocity dependence introduced by the quadrupole-pairing residual interaction . In the present model this conclusion is self-explanatory. Similar results have been reported in ref. 22) . 4. Conclusions In this work we have studied a connection between quadrupole-pairing effects in spherical superconductive nuclei and the velocity dependence of the BCS quasiparticle mean field. In the framework of the QRPA approach we have computed: (i) the correlations which are induced by the explicit inclusion of quadrupole-pairing separable forces in a model Kimiltonian, and (ii) the correlations which originate in the partial restoration of the A-pole galilean invariance . We have shown that for the case of low-lying quadrupole excitations in even-mass Sn-isotopes both -interactions produce the same effects, namely : (a) the overall

0. Civitarese et A / S~ymmelry breaking

235

TABLE 3

Matrix elements for 0' - 2 1 transitions, in even-mass Sn isotopes, induced by the pair creation operator A'2 . QRPA results for pure quadrupole-quadrupole interactions (Q2), with the inclusion of quadrupole-pairing residual interactions (P2 + Q2) and with symmetry restoring interactions (SR) are shown in columns 2-4, respectively f3t 110+)12 1(2'11 1 2

A

Q2

P, + Q,

SR

104 106 108 110 112 114 116 118 120 122 124 126 128 130

0.136 0.141 0.134 0.116 0.101 0.095 0.091 0.075 0.051 0.028 0.011 0.002 0.001 0.002

0.313 0.304 0.280 0.244 0.217 0.204 0.192 0.162 0.120 0.079 0.046 0.023 0.008 0.001

0.290 0.280 0.260 0.228 0.204 0.193 0.183 0.159 0.126 0.095 0.073 0.061 0.054 0.045

reduction of the quadrupole EWSR, (b) the enhancement of the matrix elements for quadrupole-pair operators, (c) the reduction of the deformation parameter 82 as compared with the values obtained with pure quadrupole forces Q2, and (d) the increase in the values of the restoring force parameter C2 . The overall trend of the results which have been reported in ref. ") has been reproduced with an effective symmetry breaking represented by the value y =:: 0-6y", for the coupling constant ofthe symmetry-restoring interaction . As we have discussed before, the generation of quadrupole-pairing-like forces starting from the restoration of the generalized A-pile galilean invariance requires the adoption of the BCS mean field for the description of the quasiparticle mean field. The structure of the residual interactions which are obtained in this manner differs from the one which is obtained by expanding the quadrupole-pairing operator in the quasiparticle basis. However, by keeping terms which are proportional to quasiparticle pairs "), the QRPA treatment of both interactions leads to the same results. In the present approach the strength of the residual quadrupole interaction, k,(-y), shows the effect of the symmetry breaking more clearly than, for instance, in the approach of refs. 12-16) . Although the present approach is simpler than the approach of refs . 12-16), it has still a certain weakness since the choice of the BCS pair field, as a representation of the single-body mean field, should be accompanied by a convenient phenomenoT!-.*L_ logical choice of the coupling constants for the residu .,-!. 4- a

Fz.'-~s

i

236

0. Civitarese et A / S~yrnrnelrj, breaking

also been discussed in ref. ') in connection with the simultaneous treatment of translational and galilean invariances . However, since the main mechanism which we are discussing here is the reduction of the sum-rule values for A-pole shape vibrations induced by quadrupole-pairing-like channels the empirical determination of the corresponding reduction factor yields to less freedom in the choice of the above-mentioned coupling constants. This work has been supported by the Bundesministerium fOr Forschung und Technologie under contract No. 06TC390/91 . Two of us (O.C. and M .C.L.) would like to express their gratitude for the kind hospitality extended to them at the Institut far Theoretische Physik, Universit5t Tiibingen . We would like to thank Professor Ben Mottelson for useful comments and discussions . References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27)

A. Bohr and B.R . Mottelson, Nuclear structure (Benjamin, Reading, MA, 1975) Vol . 11, ch . 6 D.R . Bes, R.A . Broglia and B.S . Nilsson, Phys . Reports 16 (1975) 1 P.F . Borlignon, R.A. Broolia, D.R . Bes and R. Liotta, Phys . Reports 30 (1977) 305 L.S . Kisslinger and R.A . Sorensen, Rev. Mod. Phys . 35 (1963) 853 R.A . Uher and R.A . Sorensen, Nucl . Phys . 86 (1966) 1 P. Ring and P. Schuck, The nuclear many-body problem (Springer, N .Y ., 1980) ch . 6 and 8 D.R . Bes and R.A . Sorensen, Adv. Nucl . Phys ., vol. 2, ed . M . Baranger and E. Vogt (Plenum, New York, 1969) 129 R.A . Broglia, 0. Hansen and C. Riedel, Adv. Nucl . Phys . vol. 6, ed . M . Baranger and E. Vogt (Plenum, New York, 1973) 287 R.A . Broglia, D.R . Bes and B.S . Nilsson, Phys . Lett . B50 (1974) 213 1 . Ragnarsson and R.A . Broglia, Nucl . Phys . A263 (1976) 315 J. Dukelsky, G.G . Dussel and H.M . Sofia, Phys . Rev. C27 (1983) 2954 H. Sakamoto and T. Kishimoto, Nucl . Phys . A501 (1989) 205; A501 (1989) 242 and references therein T. Kishimoto and T. Tamura, Nucl . Phys . A192 (1972) 246 T. Kishimoto and T. Kammuri, Prog . Theor. Phys . 84 (1990) 470 S.I . Kinouchi, H . Sakamoto, T. Kubo and T. Kishimoto, Inst. for Nucl . Study, Univ . Tokyo, reprint I NS-815 (1990) H. Sakamoto and T. Kishimoto, preprint 1990, to be published S.T. Belyaev, Sov. J . Nucl . Phys . 4 (1967) 671 N.I . Pvatov and M .I . Chernei, Sov. J. Nucl . Phys . 1 6 (1973) 514 (Yad . Fiz. 16 (1972) 931) M .I . Baznat and N.I . Pyatov, Sov. J. Nuci . Phys . 21 (1975) 365 (Yad . Fiz. 21 (1975) 708) 0. Civitarese and M .C . Licciardo, Phys . Rev. C38 (1988) 967; C41 (1990) 1778 J. Meyer ter Vehn, Z. Phys . A289 (1979) 319 H . Sakamoto and T. Kishimoto, Nucl . Phys . A528 (1991) 73 and references therein N.G . Jonsson el aL, Nucl . Phys . A371 (1981) 333 H . Clement et aL, Proc . of the Int. Conf. on nuclear physics, Florence (1983) vol . 1, pag. 126 P.M . Endt, At . Data and Nucl . Data Tables 26 (1981) 47 Nucl . Data Sheets 32 (1981) 287; 35 (1982) 375; 36 (1982) 227; 37 (1982) 289; 38 (1983) 191 ; 38 (1983) 545 ; 41 (1984) 325; 41 (1984) 413; 49 (!986) 315 ; 51 (1987) 329; 52 (1987) 641 ; 53 (1988) 73 ; 57 (1989) 443 ; 58 (1989) 765 A.W . Wapstra, Nucl . Phys . A432 (1985) 1