Effects of pairing correlations on superdeformed bands in the A ≈ 150 region

Nuclear Physics North-Holland

EFFECTS

A509 (1990) 80- 116

OF PAIRING

CORRELATIONS

ON SUPERDEFORMED

BANDS

IN THE A=z 150 REGION Y.R. SHIMIZU’ The Niels Bohr Institute,

University

of Copenhagen,

E. VIGEZZI

and R.A.

DK-2100

Copenhagen

0, Denmark

BROGLIA

Dipartimento

di Fisica, Universitci di Milano, and INFN Sez. Milano, Via Celoria 16, 20133 Milano, Italy, and The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen 0, Denmark Received

12 April

1989

Abstract: A systematic investigation of the 5(‘) and 5”’ moments the mass A = 150 region has been carried out. Satisfactory data is obtained by taking pair correlations into account Nilsson-Strutinsky model with calculations performed within

of inertia for superdeformed bands in overall agreement with experimental supplementing the standard cranked the cranked BCS plus RPA approach.

1. Introduction The recent discovery of superdeformed bands opens a new, fascinating chapter in the study of nuclei under extreme conditions. In particular, concerning the interplay taking place in these configurations between shape deformations, pairing correlations and single-particle motion, as they express themselves in the kinematic and dynamic moments of inertia. The essential identity between J(l) and JC2) at the highest spins associated with the first observed superdeformed band ( 15’Dy), led to the suggestion that the nucleus was behaving in these states like a rigid rotor. However, the finite value and marked w-dependence of the quantity J(l)-J(*), associated with the newly observed superdeformed bands of 149Gd, “‘Gd and lSIDy, indicates

that even at the highest

rotational

frequencies

(hw L- 0.7-0.8 MeV), struc-

ture and dynamical effects still play a central role in determining the properties of the superdeformed bands. In the present paper, a systematic study of both the single-particle and collective aspects of all known superdeformed bands is presented for the first time, treating on equal footing both the kinematic and dynamic moment of inertia. Special emphasis is placed on the role played by the deformation of the mean field in which the nucleons move, as well as on the effects arising from the static and dynamic pairing correlations. These calculations are extended to other nuclei in the A = 150 mass region, which are expected to be particularly promising in developing superdeformed yrast configurations. Seminal work on the role played by selected orbits ’ On leave from Department 03759474/90/$03.50 (North-Holland)

@ Elsevier

of Physics, Science

Kyushu Publishers

University B.V.

33, Fukuoka

812, Japan.

Y.R. Shimizu

in these

configurations

has been

starting point of our work. In sect. 2 the two moments the calculations compared

et al. / Effects

is concisely

reviewed.

with the experimental

correlations

81

out in refs. 371071’),which

carried

of inertia

of pairing

are defined. The results

constitute

In sect. 3 the model are discussed

data. The conclusions

the

used in

in sect. 4, and

are collected

in sect. 5.

2. Two moments of inertia: J(‘) and J’*’ In order to investigate

the detailed

to study the two moments

properties

of inertia

of the rotational

motion,

it is useful

‘)

(2.1) J(*) = dI(o)/dw

(2.4

= -

defined for each rotational band in terms of the spin I, and of the rotational frequency w. The energies in the intrinsic and laboratory frames are denoted by E’ and E, respectively. According to eqs. (2.1) and (2.2) (see also fig. l), J(l) corresponds to the slope of a line going through the origin while J(*) corresponds to the local slope in the I-w plot. J(l) is sensitive to the single-particle alignments and J(*) to the properties of the collective rotational motion. Actually, the difference J(1) _ JW = _w -dJ”’ dw can be related to the “apparent” the o = 0 axis of the extrapolated J(*), through ia These relations The quantities E2-gamma

are shown

(2.3)

’

alignment, io [ref. *)I, defined as the intercept on line associated with the dynamic moment of inertia

= o(J(‘)(w)

- J(*)(w)) .

(2.4)

in fig. 1.

J(l) and

J(‘) can be related to the observed rays and to the nuclear spins according to

energy

E,

of the

J”‘-21/E,,

(2Sa)

J’*‘=4/AE,,

(2Sb)

and

where AE, denotes the difference of two consecutive y-rays. Thus J(l) depends on the spin assignments, while J (*) does not. Since spin assignments constitute one of the most uncertain outputs of the analysis of the present experimental data, the

82

Y.R. Shin&u

et al. / E”cfs

of pairing

, / I / /

/ i’ / i

I

-0, 0

correlations

.2 iiw

I .4

(MN)

Fig. 1. J(l), J(‘), and apparent alignment iO. The definitions of the two moments of inertia and of the apparent alignment (cf. eqs. (2.1)-(2.4)) are illustrated in the I-o plane. The figure, taken from ref. *), applies to the case of ‘59Er.

extracted value of Jc2) is considered to carry less uncertainty than that of J(l). On the other hand, J(l) contains more information as Jc2) can be obtained from J(l) through eq. (2.3). Furthermore, and from a theoretical point of view, J(” is rather insensitive to the detailed properties of e.g. single-particle orbits or change of for a theoretical deformations as compared to J ‘*) . It will then seem desirable investigation of the properties of the newly observed superdeformed bands to try to estimate both quantities on equal footing. In what follows, an attempt is made to carry out such a project. 3. Theoretical framework and details of the calculation The theoretical

calculations

of J(l)

and

1”’

are done

in two steps:

(1) the

deformation parameters are determined by the cranked Nilsson-Strutinsky method, and (2) the pairing correlations are evaluated making use of the cranked-KS plus RPA approach using the monopole pairing residual interaction. Although the details involved in both steps are well known, in what follows we summarize the main points

of the theory.

3.1. NILSSON-STRUTINSKY

CALCULATION

A recent discussion of the Nilsson-Strutinsky calculation applied to high-spin rotational bands has been carried out in ref. ‘)_ In the present paper we follow

Y.R. Shimizu et al. / Effects of pairing correlations

closely this reference

making

also use of the corresponding

83

set of Nilsson

parameters.

It is noted, however, that there is one difference concerning the treatment of the mixing of the major oscillator shells NO,, in the stretched coordinates arising from the cranking

term, which in the present

fact, we include

the mixing

to second

paper

is treated

in an approximate

order

following

the perturbative

way. In treatment

described in the appendix. An example of the effect of the NO,,-mixing on J(l) and JC2) calculated with this method is shown in fig. 2 in comparison with the corresponding result of ref. ‘). It is concluded that our perturbative method is, for the present purpose, quite accurate. The Strutinsky renormalized energy is given by E(I)

= E,i,(o,i,)

- E,tru(w,t,“) + EL~(wL~) >

(3.1)

where as usual, E,i,(w,i,) is the energy in the laboratory frame, Eki,-(wmic)y is calculated are obtained by diagonalizing,

= ELic(w,i,)

+ w,icI,

frame. The routhian, by the cranked Nilsson

(3.2)

or the energy single-particle

in the rotating energies which

I e = Lder- W,inJx. The cranking

frequency

W,ic is obtained

(3.3)

from the given value of spin 1, i.e., (3.4)

I

I

0.6 Ro

I

J

0.8

(MeV)

Fig. 2. Influence of IV,,, -mixing on the calculation of the moments of inertia. The calculated values for J(I) and J’*’ are shown by dashed and solid curves, respectively. The deformation parameters are fixed as eZ = 0.582 and s4 = 0.020 appropriate for ‘s2Dy. Three calculations are compared: in the first, labeled “no AN-mixing”, the mixing between oscillator shells in neglected completely; in the second, labeled “full AN-mixing”, the mixing is taken into account exactly; in the third, labeled “eff. AN-mixing”, the mixing is taken into account approximately according to the procedure described in the appendix.

84

ICI?. Shimiru

The same treatment and to the liquid The energy

should be applied drop energy,

E(I)

et af. / Efects

is evaluated

and is then minimized.

It should

ofpairing

to the Strutinsky

correlations

smoothed

energy, Estru(wstru),

&,(~~n). on a mesh in the space of deformation be emphasized

only the spin value but also the microscopic

that it is very important

configuration

parameters to fix not

in the course of minimiz-

ation. Otherwise one cannot obtain a smoothly varying J(‘) as a function of the rotational frequency, as J(*) is very sensitive to the change of the internal structure of the rotational band. The routhian, E~i,( w,iJ, should then be calculated by following the occupation pattern of single-particle states diabatically when changing the parameters of the cranked Nilsson hamiltonian, i.e., the cranking frequency and the deformation parameters. For this purpose, as emphasized in ref. 3), it is convenient to treat the quantum number N,,, as a conserved quantity in the calculation of cranked Nilsson orbits (see the appendix). We have used N,,,= O-8 for both neutrons and protons in the calculation. Once the minimization with respect to defo~ation parameters is carried out the rotational frequency, which is directfy connected to the observed r-ray transition energy, is calculated by the canonical relation w(I)

=g.

(3.5)

From this equation the functional reiationships between I and w, as for example between w and the cranking frequency Wmic, can be obtained. The relation between w and w,ic is approximately linear: w = (1.2-1.3)w,i, due to the well-known overestimate of Strutinsky-smoothed moment of inertia for the Nilsson potential. In the calculation presented in this paper, we have minimized the energy with respect to only Ed and Ed, keeping the potential axially symmetric. This seems to be sufficient for the superdeformed rotational bands in A = 150 region under consideration “). The energy averaging parameter in the Strutinsky procedure is chosen to be 1.27%~~ with hw,= 41/A”3 MeV, which is checked to satisfy the plateau condition.

3.2. TREATMENT

OF THE MONOPOLE

PAIRING

INTERACTION

Since the monopole pairing force is a schematic force used in order to simulate the short-range part of the nucleon-nucleon interaction, it is not realistic to use it in the full model space adopted in the cranked Nilsson-Strutinsky calculation described above. We follow the prescription of ref. “) in the definition of the pairing model space. Introducing the energy cut-off parameter A, the pairing model space, Mp, is chosen as being composed by the time reversal pairs of the non-cranked Nilsson single-particle levels with level number going from N,‘,, = $( (3 N)1’3 - A/‘&,)’

,

Y.R. Shimizu

et al. / EApcts of pairing

correlations

85

to (3.6b) Here N is the proton is based

or neutron

on the semiclassical

number.

This estimate

expressions

of the Fermi

of the pairing energy,

density in the isotropic harmonic-oscillator potential. It is important to choose the force strength, G, consistently space defined above, i.e.,

smoothed

space

with the pairing

model

,

G = 2/(gF sinh-‘(A/d)) where gF is the Strutinsky

model

sr, and of the level

single-particle

(3.7)

state density,

(3.8) evaluated at the Fermi energy, gF= Ed, which we use the standard value,

and i

A = 12/4X Note that we fix the force strength

is the smoothed

energy

gap for

MeV.

at zero rotational

(3.9) frequency,

and do not change

it at finite frequencies. The parameter A is chosen to have the same value as the Strutinsky averaging parameter, i.e. A = 1.2hw,. Once the pairing model space is fixed at zero frequency, the monopole pair transfer operator* is defined in the non-cranked basis (CT, ci) by, (3.10) where

i denotes

monopole

the time

pair transfer

reversal

operator

conjugate

cl =c

I;‘=

orbit

can be expressed

of i. At finite

cp;(o)c:(w)

C M(kt,

frequency,

the

in the cranked basis (cL( w ), ck( w)), (3.11)

w)c~(w)c~(w),

(3.12)

cp;(&p:(w).

(3.13)

k, i

with M(kT,

w) =

C iCMp

The index be noticed

r labels the signature that the pair strength

conjugated state to the cranked orbit 1. It should spreads quickly from the pairing space as defined

* Strictly speaking, ?’ in eq. (3.10) is not the monopole pair operator, because not all the magnetic substates with angular momentum j are necessarily included in the spherical limit, according to the definition (3.6) of the pairing model space.

86

Y.R. Shimizu et al. / Eflects of pairing correlations

at zero frequency, over all available orbits through the rotationaf coupling, which mixes up time reversal pairs or orbits, at finite frequencies (cf. fig. 3). Therefore the set of the cranked orbits (/c, 1) in eqs. (3.12)-(3.13) should be sufficiently large to satisfy the “pairing sum rule” (ref. 6)),

c lM(kl; w)l’ = NC”, - I?,‘,,+ 1.

(3.14)

kr

In the calculations at finite frequencies presented below, we have included enough orbits so as to satisfy the sum rule (3.14) within 0.1%. The rapid spreading of the pair strength as a function of the rotational frequency is evidenced in fig. 3. Finally, the monopole pairing residual interaction is written in terms of the pair transfer operator as ?= -fG(&t&),

(3.15a)

with (3.15b) (3.15c)

00

0.2 I

0.4 1

0.6 I

--,-___

0.8

Fig. 3. Spreading of the pair strength with increasing frequency. The value of the pairing sum rule (3.14), x,rjA4(ki; @)I2 for “‘Dy is shown as a function of the cranking frequency mmic. The different curves result when the summation is extended over all the cranked orbitais that are within +E MeV from the Fermi surface, where E corresponds to the number labeling the curves. The pairing model space defined by eqs. (3.6) at o = 0 spans an interval of approximately *8 MeV, so that at zero frequency all the calculations exhaust the pairing sum rule, except the one having E = 5 MeV. It is seen that in order to satisfy the sum rule at h~,,,~, = 0.8 MeV, E = 50 MeV is needed, so that a very large number of orbitals must be included.

Y.R. Shimizu 3.3. CRANKED

et al. / Effects of pairing correlations

87

BCS AND RPA

As a first step, we can take the pairing correlations into mean-field approximation. Namely based on the hamiltonian, hI&=

if-+AS^+-

the cranked BCS state ICBCS(A, A) > is obtained. are determined by the usual BCS equation, f(A,

account

within

AI?,

(3.16)

The mean-field

parameters

h)=(CBCS[i+ICBCS)-2A/G=O,

g(A, A) = (CBC#jlCBCS)-

(A, A)

(3.17a)

N = 0.

We solve these equations with the two parameter of eqs. (3.17) are given by

the

(3.17b)

newtonian

method.

The gradients

-$=R++(fi=O)-2/G,

(3.18a)

(3.18b)

(3.18~) Here the quantities defined by

R,,

with p, (+ = +, - are elements

of the pairing

response

matrix

(3.19) where

E,( E,) is the cranked

and S*(pF) annihilation

BCS quasiparticle

energy

with signature

r = +i(-i),

is the RPA part (i.e. the terms proportional to the creation atat au of two quasi-particles) of the matrix elements of the operator

and 2,.

The pairing response matrix will be also used below in the RPA calculations. Once the selfconsistent values of (A, A) are obtained the correction to the energy from the static pair correlations is calculated by SE,,, = (CBCSlfi’lCBCS) for each rotational

frequency

- A2/G - Eki,,

(3.20)

O,ic.

In the superdeformed bands discussed in this paper, the static pairing gap A is generally quenched at rather low rotational frequencies because of the large shell gap. Then at higher frequencies and at large deformations the dynamic effects of the pairing correlations are especially important, as has been shown in refs. ‘,‘). Therefore we also calculate the pairing fluctuation energy by means of the RP,4. The dynamic correction to the energy is given by 6Edyn = ;

Cfln-C (E,+E,) n fi”.c

>

,

(3.21)

88

Y.R. Shimizu

et al. / Effects of pairing

where fi, is the RPA eigen-energy. has been developed

In ref. “) a powerful

using the response

function

a7 SE+=

-&

method

technique.

to calculate

SEdrn

The result is given by,

1

da I0

where ?Bi)(*)is obtained

correlations

dh Im(Tr ($‘?Zz”‘(0)x)),

(3.22)

I0

from the 2 x 2 pairing

response

matrix R defined

in eq. (3.19)

by LBcA)= (1- R/ix)-lR,

(3.23)

where x = +G. The A-integration can be performed analytically, see ref. “) for details. In the actual calculations, and in order to perform the numerical integration with respect to 0 we must smoothen out the singularities of the response function. This is done by giving fi an imaginary part, i.e. 0 + 0 + is. The expressions (3.21) and (3.22) coincide in the limit of 6 + 0. In ref. *) we have used a small value of 6 = 80 keV, which causes problems near the critical point of pairing collapse. This is a drawback of the RPA, arising from the small amplitude approximation which is not valid near the critical point. This problem can be solved by going beyond the RPA, e.g. using the boson expansion technique (cf. e.g. ref. “)). In this paper, however, we do not invoke such a sophisticated approach since the critical region corresponds to a rather small interval of rotational frequencies. Consequently, a smooth interpolation is enough to obtain reasonable results, as was also done in ref. *). In keeping with this discussion we found it more convenient to choose a somewhat larger value 6 = 200 keV and to perform the smoothening by Spline polynomials. This procedure may introduce some uncertainty in the calculations, in particular for the evaluation of J@), in cases where the neutron or pairing gap vanishes at a relatively higher frequency. In principle, of pairing

the energy

correlations.

minimization

We have checked

should

be performed

that the resulting

including change

the effects

in the selfcon-

sistent values of deformation parameters is negligibly small. Therefore we used the same deformation parameters as determined in the Strutinsky calculation. from the exchange (Fock) The correlation energy 6Edy, includes the contribution energy

of the pairing

interaction, E exch

=

-+G

which is given by C pF((S+(&+ WC

IS_(pF)l’)

.

(3.24)

The different contributions, 6E,,, , SE+“, Eexch are shown in fig. 4 as a function of the rotational frequency for 152Dy as a typical case. In this case the neutron pairing gap is zero already at zero frequency and the proton gap collapses at Aw,i,= 0.34 MeV. A dip around this frequency is caused by the failure of RPA as discussed above, and will be smoothened out in the calculations of SJFair and SJga!r presented below (cf. eq. (3.29)). It can be seen that E exch shows a very smooth frequency dependence, and its contribution to 8JFJir and 8JrJi, can be estimated to be about 20-40% and lo-20% respectively, in agreement with the results of ref. “).

Y. R. Shimizu et al. / Effects ofpairing correlations

89

hWmic (MeV) Fig. 4. Typical calculation of the pairing correlation energy. The different contributions to the pairing correlation energy SEP,,;, are shown for the case of 15’Dy. We add a constant to the energies, so that they are all equal to zero at zero rotational frequency, and it is easier to compare their frequency dependence. The two solid curves show the total correlation energy SE,,,, and the dynamical correlation energy 6E,,, (cf. eq. (3.21)). Shown by dashed lines are the static pairing energy SE,,, (eq. (3.20)), the exchange energy I?,,,, (eq. (3.24)), and finally the difference SE,,, - Eexch. The static proton pairing gap has vanished for hw,,, > 0.35 MeV (the neutron gap is already zero at w,,,~~= 0), so that the static contribution 6E,,, vanishes and SE,,,, = SE,,, (in fact, due to our normalization, in the figure SE,,,,, and the difference c?E,,~,- ljEdyn reaches a constant value). The absolute values of all the quantities at o,,,~~ = 0 are SEs,,,= -0.427 (0.0, -0.427) MeV, SE,,, = -8.065 (-4.410, -3.655) MeV and Eexch= -5.807 (-3.106, -2.701) MeV where the numbers in parentheses indicate the neutron and proton contributions, respectively. 3.4. CALCULATION

OF JI”’ AND

J(*)

The _I(‘) and J(‘) moments of inertia without pairing correlations are calculated using the canonical I-w relation given by eq. (3.5). The inclusion of pairing correlations leads to corrections to both of them. There arises a subtle problem about how to evaluate the corrections, because the microscopic frequency w,ic, at which the pairing correlations are taken into account, is different from the “real” rotational frequency 60. In this paper we use the following

prescription.

The total energy correction

SE,,i,

arising from both the static and dynamic effects of pairing correlations is given as a function of w,ic by adding the contributions from eqs. (3.20) and (3.22), aEpair(wmic) = 6E,t,(m,i,) The canonical relation should angular momentum is evaluated

always by

+ SEdyn(u,ic)

be satisfied,

.

so that the correction

(3.25) to the

(3.26)

Y.R. Shimizu et al. / Effects of pairing correlations

90

In this way, the “reno~alization”

through

dw,Jdw.

frequency

of the rotational

Thus the total angular

is given simply

frequency

momentum

of the rotational

by (3.27)

,

LJ,(w) = I(w) + %G,(w) leading

is taken into account

as a function

to J,‘U’,= J(i) + &J(i). pa,r

(i=l,2),

(3.28)

with (yJ”‘,

=

paw

_-1$jE w dw

parr,

=_dZSE pa,r .

SJ’? pa3r

dW2

(3.29)

A different prescription might be possible, in which the total, microscopically calculated energy should be used in the Strutinsky procedure, Namely the quantity iE,i,(w,i,) in eq. (3.1) should be replaced by EXc(Grnic)=

where now the rotational

(3.30)

Emic(;mic)+SEpair(;mic),

frequency

(3,,, is fixed by

Imic(;mic)+8~~air(L3mic)

=

(3.31)

I,

with M S1pair(Ljmic)

=

-

$

SEpair(Djmicl

m1c

-

(3.32)

In principle this prescription may be preferable. The problem is that we do not know how to carry out the Strutinsky procedure in the rotating case when the pairing correlations are included by means of the CBCS plus RPA approach. It is certainly not appropriate to perform the St~tinsky smoothening only for the deformed potential without pairing correlations. Because of this conceptual difficulty, we adopt in this paper the prescription described above. From the numerical point of view, however, it is easily seen that both procedures give the same results up to first-order corrections in SIpair or 80,~~ = (&& - w,ic)p so that in practice the calculated moments of inertia agree well at the large angular momenta studied here. This concludes the description of the theory on which our calculations are based. It should be emphasized that the parameters entering into the calculations are fixed once and for all as described above. No changes are made for individual cases. 4. Results and comparisons with experimental data We have calculated the moments of inertia, with and without pairing correlations, for all the nuclei in the A = 150region, for which a superdeformed band has been observed up to now. Furthermore, we have also studied a few adjacent nuclei (cf. fig. 5). The results of the calculations are presented in the rotational frequency

Y.R. Sbimizu et ai. / E#eets of pairing correlations

SUPER - DEFORMED BAND f A

91

- 150 REGION)

Z 68 Er 67

63 Eu 62

Sm I

I

80

I

81

I

I

82

83

I

I

84

I

85

-I

1

1% \ pi /

87

I

88

89

90

N Fig. 5. Nuclei studied in the present work. The proton and neutron numbers of the nuclei in the A = IS0 region are shown. The boxes referring to the nine nuclei studied in this paper are shaded. The six nuclei, for which a superdeformed band has been experimentally observed, are cross-shaded by oblique lines, and the other three single-shaded.

interval 0.3 < hw < 0.8 MeV, namely approximately the starting of the band with observable population

in the range comprised between and the end point at which the

band is suddenly depopulated. In this section, we first discuss the main qualitative features of the results of the detailed calculations, without taking into account pairing correlations. We then briefly describe the main effects caused by pairing correlations. Finally, we present the results of the detailed calculations for each nucleus, comparing them with the experimental data when they are available. 4.1. MAIN

FEATURES

OF THE

RESULTS

WITHOUT

PAIRING

CORRELATIONS

As discussed

in refs. loS1l), the general behaviour of the moments of inertia as a function of neutron and proton number, and of rotational frequency, can be understood in terms of the properties of the single particle, high-j orbitals present in the lowest part of the N = 6 proton shell, and of the iV = 7 neutron shell. This is possible because, for all the nuclei studied in this paper, within the frequency range ACJJ= 0.3-0.8 MeV the static proton and neutron gaps have vanished, or at least are very small. The changes of the equilibrium defo~ation as a function of the proton and neutron number and of angular momentum are shown in fig. 6 (observe the very enlarged scale of the figure). As discussed below, increasing the proton or neutron number generally increases the number of nucleons found in highly alignable orbitals, and they drive the nucleus towards larger quadrupole deformations in a systematic way. On the other hand, angular momentum affects more the hexadecapole than the quadrupole deformation, see ref. “), with the

Y.R. Shimizu et al. / E@cfs ofpairing correlations

92

0.06

I

I

t

I

I

1

0.04 &J* 0.02

0.00

0.54

0.56

0.58

0.60

% Fig. 6. Self-consistent deformations for the different nuclei as a function of spin. The deformations of the nuclei studied in the paper are shown in the +-s4 plane. Gadolinium, terbium and dysprosium isotopes are indicated by squares, triangles, and circtes respectively. For each nucleus, the symbols refer to different spin values: at each step, the spin increases by 10 units in the sense of the arrows, starting from I = 10h and ending at I = 80h.

exception of the lighter gadolinium isotopes. Anyhow, the changes in deformation are not too large, so that we may base the following qualitative discussion on the routhians calculated with the deformation parameters appropriate to ‘z;Dy at w = 0, shown in figs. 7a, b. It should be remarked, however, that the features discussed below may not be valid for other choices of the parameters of the Nilsson potential, or when a different potential is used. For example, calculations performed using a Woods-Saxon potential lead to rather different level schemes and pairing gaps for the nuclei studied here. In particular, the calculations of ref. 13) show band crossings, that are not found in our results, and lead to a different interpretation of the experimental data, giving more importance to deformation changes and to static pairing. Other differences may arise, if our diabatic prescription is not followed in calculating the equilibrium deformations. One can obtain the yrast configurations adding particles on top of the “‘l$m” with the calculations presented in core, as suggested in refs. ‘O*“). In agreement ref. ‘I), one obtains for ‘z$rn a rather constant J”‘=67h2 MeV-‘, and a J’*) which decreases rather slowly from Jf2)= 70h2 MeV’ at hw =0.3 MeV to J”‘= 60tr2 MeV-’ at Rw = 0.8 MeV. The “valence” particles considered for the nuclei studied in this paper are the first four proton orbitals in the N=6 shell and the first four neutron orbitals in the N = 7 shell, cf. figs. 7a, b. These proton and neutron orbitals are “intruder” orbitals, that align very fast with increasing rotational frequency, and contribute a large angular momentum to the nucleus. However, because of the considerable mixing caused by the large deformations, the alignment

Y.R. Shim&

et al. / Effects

of pairing

correlations

93

5.8

.02

DO

.04

.06

Wmic

.08

.I0

.I2

1%

64 6.3 6.2

2

6

,__.-----‘-‘-._,

6.0---------------

\

--T_

.oo

.02

.04

.06

\

l

.08

.I0

.I2

wmic/Wv Fig. 7. Cranked level schemes using the Nilsson potential. (a) The proton singie-panicle routhians are shown as a function of the rotational frequency. Energies and frequencies are plotted in units of the volume-conserving ha~onic-osci~fator frequency w, (this frequency is often written as w,, (sZ. Ed)). The diagram is obtained at the deformation ~*=0.58, c4=0.01, appropriate for the nucleus “*Dy at zero frequency (cf. fig. 6). As discussed in the text, the diagram gives a qualitative indication of the properties of the orbitals that are more relevant in the calculations, but it cannot be used for determining the yrast configurations, since the deformation is kept constant. Orbitals having parity and signature equal to (+, ++), (+,-f), (-, +i) and (-, -1) are shown by solid, short dashed, dash-dotted and short-long dashed lines respectively. We have indicated the oscillator quantum number PI,,,, for some relevant orbitals. A similar plot was shown, for example, in ref. 4). (b) The same as (a), but for the neutron orbitals.

Y.R. Shimizu

94

process

is slow, as compared

a configuration,

indicating

et al. / E#ecfs of pairing correlations

to normally

deformed

for each oscillator

rotating

shell the number

respect to the ‘$Sm core. For example

the configuration

two protons

one neutron

in a pair of N = 6 orbitals,

hole in a N=6

orbital,

is indicated

by ~(6)~,

nuclei.

We will denote

of particles

adopted

or holes,

for ‘z:Gd, having

in a N = 7 orbital and a neutron

v(6)--‘(7)‘.

The routhians calculated with the Nilsson parameters of ref. ‘) at the appropriate equilibrium deformations show large energy gaps for proton numbers 2 = 64 or 2 = 66 and for neutron number N = 86. These gaps lead to negative shell correction energies, and make the superdeformed shapes stable. The relative importance of the Z = 64 and Z = 66 gaps depends strongly on deformation, since changing the values of &2and aq changes the positions of the highly alignable orbitals as compared to the others. Experimentally, until now only a single discrete superdeformed band has been resolved in each nucleus. In selecting the associated configuration in the different nuclei, a difficulty arises. This is because in some cases in the calculations there is not a single configuration, which is yrast in the whole angular range. Experimentally, it is found that the observed superdeformed bands are populated in the frequency range defined by hw = 0.7 MeV and hw = 0.5 MeV, at which frequency they saturate. The bands decay rather abruptly around &J = 0.2.5-0.30 MeV, with the exception of ‘i:Gd, in which the superdeformed band is depopulated already at hw = 0.4 MeV. The mechanisms responsible for the population and decay of the superdeformed bands are at present the object of intense study, but are still rather poorly understood (cf. e.g. refs. 14-16)). We usually select the configuration, which is calculated to be yrast around hw = 0.5-0.6 MeV. In some cases, however, there are a few configurations, having the occupancy

similar energies at this frequency. These configurations may differ in of highly-alignable orbitals, the associated bands having very different

moments of inertia, as well as different energies at low frequencies. In other cases we may find two configurations which are almost degenerate, since they differ only in the occupancy of an orbital of a pair of non-alignable orbits, which remain essentially time-reversed partners up to high frequencies. This is the case of the nucleus ‘$Gd. This result will suggest that two discrete bands should be experimentally observable in this nucleus. The assignment of the intrinsic configurations of different superdeformed bands requires the use of the self-consistent deformation values in the calculation of the routhians, and figs. “/a, b are clearly not appropriate for this purpose. In sect. 4.3, we will show the calculated low-lying configurations for each nucleus. Here instead we briefly describe the resulting assignments, limiting ourselves to the nuclei, in which a superdeformed band has been experimentally observed. Such assignments were also proposed, for example, in ref. “). The yrast configuration for N = 86 and Z = 64 is unambiguously obtained at all frequencies making the first to proton valence orbitals and the first two neutron valence orbitals occupied, and can therefore be indicated as 7r(6)‘, ~(7)~. No

Y. R. Shimizu

crossings should

occur

et al. / Effects

for this configuration,

correspond

to the observed

of pairing correlations

that

has (+, 0) parity

superdeformed

Gd-isotopes studied, ‘z:Gd and ‘$Gd, we adopt 7~(6)~, v(6)-‘(7)’ and ~(6)~, u(7)‘. The configuration excitation

of a N = 6 neutron

and ‘ZgDy, we adopt

95

and

signature,

and

band in ‘ZzGd. For the other two respectively the configurations for ‘i:Gd implies the one hole

of the ‘$Sm core. For the two N = 86 isotones, the configurations ~(6)~, ~(7)~ and ~(6)~,

respectively

‘i:Tb ~(7)~.

Finally for ‘i:Dy we adopt the configuration rr(6)4, v(7)‘. Once the configurations for the nuclei under study have been assigned, one can try to understand qualitatively the behaviour of the moments of inertia from figs. 8a-d (cf. also fig. 13 in ref. 4)), that show the contributions to J(I) and JC2) arising from each valence orbital. The first two N = 6 protons are almost fully aligned at the highest frequencies considered, hwmi, = 0.8 MeV, so that they contribute moderately to J(‘), while they give an important contribution to J(l). The first two N = 7 neutrons similarly give an important contribution to J(l). In this case, due to the fact that these orbitals align less rapidly than their N = 6 counterparts (cf. ref. I’)), they also contribute in a more important way to J(‘) . This lack of complete alignment

\\

6,,,’

PROTON

\.___’

I

-10 ’

0

2

4

6

8

II

IO 0

NEUTRON

I 2

4

6

8

IO

fl WnC

Fig. 8. Single-particle contributions to the moments of inertia. (a) The contributions to the first moment of inertia J(‘) from the two first pairs of “valence” proton orbitals in the N = 6 shell, and from the first pair of proton orbitals in the N = 7 shell are shown as functions of the cranking frequency w,,,,~. We use the same constant deformation parameters as used in fig. 7. The orbitals are drawn according to their parity and signature, as described in the caption to fig. 7. The label 6, indicates the first proton in the N = 6 shell, and so on. (b) The contributions to the second moment of inertia J”’ from the same proton orbitals used in (a) are shown. (c) The contributions to the first moment of inertia J(l) from the two first pairs of neutrons orbitals in the N = 7 shell are shown. (d) The contributions to the second moment of inertia J (‘) from the same neutron orbitals used in (c) are shown.

96

Y.R. Shimizu

at the relevant proton moments

frequencies

is particularly

pair, and is quite relevant of inertia.

angular

momentum

particle

alignment,

et al. / Effects

It somewhat

important

in the case of the second

in order to understand hinders

and the single-particle a distinction

of pairing correlations

the features

a clear distinction angular

which has proved

momentum very useful

N = 6

of the observed

between

a collective

derived

by single-

at smaller

deforma-

tions. In fact, the Coriolis coupling is reduced at these very large deformations, so that the passage from the deformed-aligned to the rotation-aligned regime is made more difficult than in “normal” deformed nuclei. The contribution to J (I’ from the first N = 7 neutron is positive and decreasing with frequency, while the contribution from the second neutron is increasing, going from negative to positive values. Both contributions to Jc2’ are positive, and decreasing in the whole frequency range. Because of these features, the calculations without pairing correlations show an increase in both moments of inertia at large frequencies, going from ‘:iGd or ‘z:Gd to ‘ZiGd. Furthermore, J (” decreases more slowly with frequency, while the opposite is true for J(“. The isotopes ‘i:Dy and ‘ZiDy should also have the third and fourth N = 6 proton orbitals occupied. This pair of orbitals splits at a larger frequency, and even at hw,i, = 0.8 MeV the alignment process is far from being completed, especially for the fourth proton. The contributions to both moments of inertia are positive for the third proton, and increase very rapidly from negative to positive values for the fourth proton. For this reason, the calculations show a reduced dependence on rotational frequency for the Dy-isotopes, and above all for ‘i;Dy, since the increasing contributions from the second pair of N = 7 neutrons counterbalance the decreasing contributions arising from the ‘z$rn core, the first pair of N = 7 neutrons and, especially, the first pair of N = 6 protons. In keeping with the above discussion, one can anticipate the main trends to be expected for J(” and Jc2’ as a function of the detailed calculations presented below, (i) J(“. For all the nuclei considered, decrease, however, is less pronounced for

the rotational frequency, resulting from not including pairing correlations. J(” is a decreasing function of w. The ‘z:Dy and for ‘z:Tb, while J(l) for ‘$Dy

is almost constant. This is due to the contributions to the moments of inertia coming from the second pair of N = 6 protons and from the second N = 7 neutron, that are increasing over most of the frequency range. At large frequencies, increasing the proton or neutron number increases J(l), since all the single-particle contributions to J(” are positive. This is not true at small frequencies, since the contributions from the second pair of N = 6 protons and the second N = 7 neutron are negative in this case. (ii) J (2’. Also J”’ is a decreasing function of w; it decreases faster than J(“, because all the single-particle contributions are decreasing, except for the fourth N = 6 proton, whose effect is evident in the Dy isotopes. Since all the contributions to Jc2’ are positive (except only for the fourth N = 6 proton at small frequencies), increasing the proton or neutron number generally increases J”‘.

Y. R. Shimizu

(iii)

et al. / Effcts

J(l) versus J@). As was already

Consequently

of pairing correlations

remarked,

J(l) is a decreasing

97

function

of w.

(cf. eq. (2.3)) J(1)

_

Jc2)

=

-_w->

dJ”’

0

’

dw

(4.1)

the difference J(l) - Jc2’ calculated without pairing correlations is always positive, for all the nuclei considered at all frequencies. The difference (4.1) and consequently the apparent alignment &,increase with the frequency. The single-particle contributions to J(l)- J(‘) a re positive or close to zero except for the fourth N = 6 proton and for the second N = 7 neutron (especially at low frequency). For this reason J(I) - Jc2’ is especially small for ls2D 66 y at low frequency.

4.2. MAIN

CONSEQUENCES

OF PAIR CORRELATIONS

The pair correlation energy, 6E,,i, is a negative quantity, that is expected to vanish for w -f co. A typical calculation was shown in fig. 4, from which it can be observed that it is generally a monotonic increasing quantity. As a consequence, the contribution of pair correlations to the alignment, SIpair = -dsE,,i,/dw is always negative (dealignment), and it amounts typically to 2-4 units of angular momentum in our calculations. Our value for (Slpair is smaller than that found using the projection method in refs. “,‘*) for the nucleus ‘ZzDy. We have also calculated SIpair in selected cases, making use of number projection techniques, and have found a good agreement with the RPA result (the results of these calculations will be discussed elsewhere). This discrepancy may be due to the use of a different potential (WoodsSaxon potential instead of Nilsson potential), or of different parameters in the two calculations. The first moment

of inertia

becomes

smaller

by the amount

Typical values for SJ~~ir are SJ~~i~^I -12h’ MeV-’ at hw ~0.3 MeV, and SJC,gr-3h’ MeV-’ at hw -0.8 MeV. While 2iJJbfd:r is related to the slope of SEpair, the contribution to J(‘), cifE!r is proportional to the curvature of 6E,,i, as a function of w, SJgJiF = -d*6E,,i,fdw2 (cf. eq. (3.29)). 6E,,i, generally has an inflection point at a frequency w*, that is qualitatively related to the vanishing of the static pairing gaps. For frequencies w > w* the curvature of SE,,i, is negative and SJFJir > 0. For very large frequencies both SJyair and SJFir go to zero, respectively through negative and positive values. The typical trend of the corrections to the energy and to the moments of inertia is shown schematically in fig. 9. For most of the cases studied in this paper, and discussed in the next section, w* lies at the left of the frequency interval considered, so that in these cases we find always positive corrections SJgiire

Y.R. Shimizu

98

et al. f Effects of pairing correlations

1

-10

WW

Fig. 9. Influence of pairing correlations on the moments of inertia. The schematic dependence of the pairing correlation energy SE,,,;, on the rotational frequency is shown, in the lowest part of the figure. It is noted that IYE,,,, has an inflection point at a frequency w*, where d*GE,,,,/dw’=O. From the correlation energy, the typical behaviours of the corrections to the moments of inertia can be deduced; the correction to the first and second moments of inertia, 8JJb’2,rand c?J~!~ are shown in the middle and top part of the figure, respectively.

However, the details of the effects due to pairing fluctuations depend on the behaviour of the pair matrix elements M( kc w) between the various orbitals. Such matrix elements are oscillating refs. x,‘9-22). Moreover, it should J”‘) depend

somewhat

functions of the frequency (cf. the discussion in be recalled that the moments of inertia (especially

on the averaging

parameter

6 and the smoothing

procedure

(cf. sect. 3). Anyway, in the frequency range 0.4 < hw < 0.8 MeV a good description of our results can be obtained writing SE,,i,(w)=A+Bw+Cw2+DW3,

(4.3)

from which one deduces SJ”! pa,r= -B-2C w

-30

w,

SJc2’ pa,r= -2C

- 6Dw.

(4.4)

The values of B, C, D that can be extracted from our calculations for the various nuclei are collected in table 1. It is to be noted that because of time reversal violation at the large frequencies under discussion, the quantity SEpai, can contain both even and odd powers of w.

Y.R. Shimizu et al. / Effects of pairing correlations

99

TABLE 1 The coefficients

to the moments

B, C, D entering in the parametrization (4.3) and (4.4) of the corrections SJrJ,r, SJgir of inertia are shown for different nuclei. The parametrization reproduces well the results shown in sect. 4.3.

Nucleus ‘%d

‘@Gd “‘Gd

B

c

D

4.1 4.5 9.2

-1.6 -1.4 -9.3

0.4 0.0 4.2

Nucleus ‘S’Tb “rDy ‘=Dy

B

C

5.5 6.9 4.1

-1.8 -5.1 -1.2

D -0.4 2.4 0.0

The renormalization of the moments of inertia due to pairing correlations is also clearly evidenced, when the difference J(I) - J’*’ is considered. This is shown in figs. lOa-d, for the nuclei in which the value of J”’ could be estimated experimentally. It is seen that except for the case of 15’ 66Dy, pairing correlations provide an overall account of the data. In particular they are important in reproducing its frequency dependence. The case of ‘2: D y will be discussed below in sect. 4.3. According to these results it is meaningful to refer to i. = o (J(l) - J”‘) as to an apparent alignment. In fact, this quantity can become negative, something that cannot happen to the actual particle alignment. On the other hand, i, reflects the single-particle content

20

(a) _____-------___.__._...._._____-_-_____ 1

>/--Q---------J 9 O S ,’ A= --20 I-

11

14’Gd

15'Dy

Cd)

-201

I

/ 3

4

5

6

a

7 hw

3

4

5

6

7

0

(MeV)

Fig. 10. Difference of the two moment of inertia with and without pairing correlations. (a) The calculated difference between the two moments of inertia, J (I) - J’*’ is shown as a function of the rotational frequency for the nucleus ‘@Gd. The calculations with and without pairing correlations are shown by a solid and a dashed line respectively. The calculations are compared with the experimental data (filled dots). (b) The same as in (a), for the nucleus “OGd. (c) The same as in (a), for the nucleus 15’Dy. (d) The same as in (a), for the nucleus lS2Dy.

100

of total

Y.R. Shimizu ef al. / E@+zts unpairing

angular

momentum,

and

is consequently

correlations

strongly

quenched

by pairing

correlations.

4.3. COMPARISON

WITH

THE

EXPERIMENTAL

DATA

spectra of the lowest superdeformed bands 43.1. The ‘$Gd nucleus. Theoretical in this nucleus are shown in fig. 11. It can be seen that two bands, having (-, 0) and (-, 1) parity and signature are clearly the lowest in the spin range 351i < I < 70h. Both configurations can be indicated as ~(6)~~ v(6)-‘(7)‘, and have one hole in one of the two orbitals of the fourth pair of neutrons in the N = 6 shell. This pair of orbitals is deformed-aligned, and shows a very small signature splitting even at the largest frequencies considered (cf. fig. 7b), so that the two bands are almost degenerate and have practically the same moments of inertia. Experimentally, however, only one band has been observed. The moments of inertia of the (-, 0) band are compared with the data “f in fig. 12. For this nucleus, only J (2) is experimentally available. The calculated JC2) gives

-

IO

30

50

I

70

(h)

Fig. 11. Low-lying superdeformed bands in the nucleus *4sGd. The energies of the lowest superdeformed bands calculated for the nucleus ‘@Gd are shown as functions of spin. For each band, the deformation has been determined selfconsistently as a function of spin without including pairing correlations. The reference moment of inertia is taken to be 90h2/MeV. Ten lowest configurations were selected at spin I = 5Oh. Bands having (+, 0), (+, l), (-, 0) and (--, 1) parity and signature are drawn as solid, dashed, dash-dotted and long-short-dashed curves, respectively. We have labeled some of the low-lying bands; the configuration associated to the (i-, 0) band 1 is n(6)*, while the bands 2 and 3 are the signature partners of m(6)‘, v(6)-‘(7)’ with (-, 0) and (-, 1) parity and signature, respectively. These two bands are discussed in the text.

101

Y. R. Shimizu et al. / Efecrs o~pairing correhtions

1 J

.3

.4

.5

.6

7

.8

moments of inertia of the Fig. 12. Calculated moments of inertia for the nucleus “‘Gd. The calculated low-lying (-, 0) configuration in 14sGd are shown. The first moments of inertia I(‘), calculated with and without pairing correlations, are drawn by a thick and thin dashed line respectively. There was no experimental assignment proposed for J”‘. The second moments of inertia, J”‘, calculated with and without pairing correlations, are drawn by a thick and thin solid line respectively, and are compared with the experimental data (filled dots). The data are taken from ref. 23)_

a good description of the data. The static proton pairing gap collapses at Ro = 0.20 MeV, while the neutron pairing gap vanishes already at zero frequency. The effect of pairing correlations on P’ is not particularly significant in this case, while it is very important for J (I), that is predicted to be almost constant in the whole frequency range. bands 43.2. The ‘z:Gd n~cie~s. Theoretical spectra of the lowest superdefo~ed in this nucleus are shown in fig. 13. As is clear from the figure, one band, having (-, -i) parity and signature, is yrast at all the relevant angular momenta. The calculated configuration is ~(6)~, u(7)‘, so that the same highly alignable orbitals are occupied, as in ‘;fiGd. The moments of inertia for this band are compared with the data 24) in fig. 14. The static proton pairing gap collapses at ho = 0.20 MeV, while the neutron pairing gap vanishes already at zero frequency. The result without pairing fluctuations is similar to that presented in ref. ‘I). The calculated moments of inertia reproduce the overall trend of the data, when pairing fluctuations are included (cf. also fig. 10a). 4.3.3. The *iiGd nucleus. Theoretical spectra of the lowest superdeformed bands in this nucleus are shown in fig. 15. As in the case of ‘$Gd, one band is clearly yrast at all angular momenta. The associated configuration is 7r(6)*, v(7)‘, and has (+, 0) parity and signature. The calcuIated moments of inertia are shown and compared with the experimental data “‘) in fig. 16. The static proton (neutron) pairing gap collapses at hw = 0.17

Y.R. Shimiru et al. / Effects of

102

IO

30

pairing

50

correlations

70

Fig. 13. Low-lying superdeformed bands in the nucleus L49Gd. The same as in fig. 11 but for the nucleus ‘49Gd. Bands having (+, +$), (+, -$), (-, +f), and (-, -f) parity and signature are drawn as solid, dashed, dash-dotted, and long-short-dashed curves, respectively. In this nucleus there is a well separated (-, -$) band with configuration 7r(6)‘, v(7)], labeled by 1 in the figure, which is yrast at all spins. The (+, +i), (+, -$) bands, labeled by 2 and 3, are signature partners of the configuration 7r(6)*, ~(6)’ respectively.

I

I

14'Gd -

100 -

2

=---__

z

---

--- -..

60 I

.3

.4

5

hw

.6

.7

.8

(MeV)

Fig. 14. Calculated moments of inertia for the nucleus ‘49Gd. The calculated moments of inertia of the yrast (-, -f) band in ‘49Gd are shown (cf. fig. 12). The data are taken from ref. 24). In this reference an assignment for J(‘) was proposed, and the experimental data for J(‘) are shown by triangles in the figure.

Y. R. Shimizu et al. / E&c&

IO

ofpairing correlafions

30

103

70

50 I (h)

Fig. 15. Low-lying superdeformed bands in the nucleus *s’Gd. The same as in fig. 11 but for the nucleus ‘s*Gd. In this nucleus there is a well separated (+, 0) band, labeled by 1 in the figure, corresponding to the configuration w(6)‘, v(7)*, which is yrast at all spins. The next lowest bands, labeled by 2 and 3, are signature partners of the configuration a(6)‘, v(5)‘(7) with (+, O), (+, 1) parity and signature.

3 3 60

-

.3

.4

flw Fig. 16. Calculated moments yrast (+, 0) band in “‘Gd

.7

.5

.8

CM,v”,

of inertia for the nucleus “‘Gd . The calculated moments of inertia of the are shown (cf. fig. 12) and compared with the data, taken from ref. 25).

104

Y.R. Shimizu

(0.13) MeV. Although

pairing

et al. / Effects

correlations

of pairing

correlations

bring theory

in better overall

agreement

with the data (cf. also fig. lob), large discrepancies

for fi2’ are found in the frequency

range

calculation,

ho = 0.40-

understand associated

0.45 MeV. Within

the observed

rapid

with the collapse

the present

increase

of the static

we were not able to

of J (*). A possible mechanism may be pairing gaps, although in our present

calculation this occurs at much lower frequencies. However, when higher order correlations are included, the effect of the BCS pair gap collapse on the moment of inertia is more or less smeared out (cf. e.g. refs. 9*26)). It was proposed in ref. “) that the rapid increase in J(*) at low frequencies might be produced by a rather sharp band-crossing. Calculations performed using a Woods-Saxon potential indeed predict proton and neutron alignments for this nucleus 13). In those calculations, the shell gaps at 2 =64 and N=86 seem to be much smaller than ours at the deformations relevant for ‘ZiGd, so that the static pairing gaps may still have large values, and alignments of two quasiparticles may result in large increases in Jc2). It is to be noted, however, that there is no clear indication of band-crossings in the other superdeformed rotational bands observed up to now. In conclusion, the present calculation is not able to reproduce the specific behaviour of the measured Jc2’. 4.3.4. The ‘ii77~ nudeus. Theoretical spectra of the lowest superdeformed bands in this nucleus are shown in fig. 17. One bands turns out to be yrast at all spins. The relevant configuration is 7r(6)3, v(7)*, having (+, +$) parity and signature. The calculated moments of inertia are compared with the data 25) in fig. 18. Only J’*’ is known in this case. Both the static proton and neutron pairing gaps vanish already at zero frequency. Pairing correlations alter considerably the behaviour of J(j). 4.3.5. The ‘ijm nucleus. Theoretical spectra of the lowest superdeformed bands in this nucleus are shown in fig. 19. There is one band, having (-, -f) parity and signature, which is yrast for I <35h and its associated configuration is am, v(7)‘. At larger spins, a few other bands, labeled by 2,3 and 4, become yrast successively. None of these bands can reproduce the large measured value of J(l) [ref. *‘)], when pairing fluctuations are included. The calculations are not able to reproduce the increasing trend of J(‘), either. As an example, we show in fig. 20a the results for J(l) and J(*) calculated for the band 1. In this case, the static proton pairing gap collapses only at hw =0.85 MeV, while the neutron pair gap vanishes already at zero frequency. One should remark that the same con~guration, with one neutron added in the N = 7 orbital, reproduces the experimental J(l) measured in ‘ZzDy quite well (cf. fig. 22). The differences between the experimentally measured moments of inertia in the two isotopes ‘i:Dy and ‘i;Dy are striking. As a matter of fact, both the relative slope and the relative absolute values of J(l) and J(‘) are reversed in the two nuclei. In particular, according to the present experimental assignments, the J(l) value at fiw = 0.3 MeV is larger in ‘i:Dy by about 20 h’/MeV. A reasonable agreement of the measured Jfif with the calculation including the pairing correlations might be obtained for some of the bands, if the experimental assignment were lowered by 4-6 units. A good agreement without changing the

Y.R. Shimizu

et al. / Effects

of pairing

105

correlations

15’Tb

‘\

\._ k.

--__

\>. \:.,

_,44

A.

_=I_--’

---__

--___

/i’

,

/

_--

2 ‘\ ‘\

:

IO

30

--3

70

50 I

(f-l)

Fig. 17. Low-lying superdeformed bands in the nucleus “‘Tb. The same 15rTb. In this nucleus there is clearly one (+, +f) band, labeled by 1 in configuration ~(6)~, v(7)*, which is yrast at all spins. The band labeled this band, and the band 3 has the configuration a(S)‘(6)‘,

as in fig. 13 but for the nucleus the figure, corresponding to the by 2 is the signature partner of v(7)’ with (-, -k).

I

/

I

15’Tb -------------

----__

80

60 .3

1

.4

1

.5

hw Fig. 18. Calculated yrast (+, +i) band

6

I

/

.7

.8

WleW

moments of inertia for the nucleus “rTb. The calculated moments of inertia of the in “‘Tb are shown (cf. fig. 12) and compared with the data, taken from ref. a5). No experimental assignment for J(‘) was given in this case.

106

Y. R. Shimizu 9

et al. / Effects of pairing \

correlations

\

Fig. 19. Low-lying superdeformed bands in the nucleus “‘Dy. The same as in fig. 13 but for the nucleus Is’Dy. The (-, -f) band labeled by 1 has configuration ~(6)~, v(7)’ and is yrast up to 13 35h. After that, the band 3 and 4, which are signature partners of configuration rr(4)‘(6)‘, v(7)’ with (-, -4) and (-, +$) parity and signature, and the (+, -4) band 2, having the configuration ~(5)‘(6)~, u(7)‘, appear to be yrast successively. We especially included the band 5 in addition to the ten low-lying bands, which has a configuration having a N = 7 proton excited, i.e. n(#(7)‘, ~(7)’ with (+, -4) parity and signature. This band is discussed in the text.

experimental assignment is instead obtained, selecting a configuration having the first N = 7 proton occupied. This proton is highly aligned, and yields the angular momentum needed to reproduce the measured value. There is a (+, -4) band, labeled 5 in fig. 19, and corresponding to the configuration ~(6)~(7)‘, v(7)‘, which has a large first moment of inertia, and becomes yrast at very large spins (I > 70h). In fig. 20b we show the calculated

moment

of inertia

for this band.

Both the static

proton and neutron pairing gaps vanish already at zero frequency for this band. It is seen that f(l) IS . in good agreement with the data; but since the fourth N = 6 in disagreement proton is not occupied, J”’ decreases rapidly at large frequencies, with experiment. It should also be noted that, due to its large moment of inertia this band is excited by about 2 MeV above yrast at spin I = 30h. It seems then unlikely, that it may correspond to the observed band. In conclusion, with the present Nilsson parameters there seems to be no calculated band able to reproduce both experimental moments of inertia in ‘i:Dy. In particular, with a configuration similar to the one adopted for ‘i:Dy is in strong disagreement the data when pairing correlations are included. The effect of pairing correlations seems to be important in this nucleus, but at the present stage it is impossible to conclude, whether it improves the agreement with the data, or not.

Y.R. Shimizu

100

et al. / Effects

qf pairing

107

correlations

15'Dy

--rs *

n

L a

----___

d

~

80. ____-----;-;-; 2___ .' l

. .

l

60 -

60

(bl .3

.4

.5

hw

.6

7

.8

(MeV)

Fig. 20. Calculated moments of inertia for the nucleus “‘Dy. The calculated (-, -4) bands labeled 1 and 5 in fig. 19, are shown (cf. fig. 12) and compared ref’“), in (a) and (b), respectively.

moments of inertia for the with the data, taken from

4.3.6. 7’he ‘~~0~ nucleus. This nucleus, in which the first superdeformed band was observed, is the most studied, both experimentally (cf. refs. 2g-31)) and theoretically The theoretical spectrum of the lowest superdeformed bands (cf. refs. ‘oV1s*32-34)). in this nucleus is shown in fig. 21. It is seen that for spin I <45A one band, labeled 1, is yrast.

It has (+, 0) parity

and signature,

~(6)~, ~(7)~. At larger spins, another yrast. This band has the configuration

and the associated

configuration

is

band, with (-, 0) parity and signature, becomes r(5)‘(6)‘, u(7)‘, and differs from the previous

one for having a proton in a moderately aligned N = 5 orbital, rather than in the fourth N=6 orbital. The two configurations have been compared lo,“), and it has been found that the second moment of inertia of the (+, 0) configuration is in much better agreement with the experimental data. In fact, as we discussed in sect. 4.1, the occupation of the fourth N = 6 proton is required, in order to obtain a rather In the (-, 0) band and in constant JC2), such as the one observed experimentally. the other low-lying bands this proton orbital is not occupied, and as a consequence the resulting second moment of inertia decreases too rapidly as a function of the frequency, even when pairing correlations are included. According to this discussion, we have calculated the moments of inertia for the (+, 0) band. However, we may have here an indication that the Nilsson parameters for the N = 6 proton shell

108

Y. R. Shimizu

ofpairing

et al. / Effects

correlations

1 /’

_--

_/ 3

\

L

,,IL 1

50

30 I

(f-l)

70

Fig. 21. Low-lying superdeformed bands in the nucleus “‘Dy. The same as in fig. 11 but for the nucleus “*Dy. For this nucleus we labeled five low-lying bands with numbers going from 1 to 5 in the figure. The configuration related to band 1 is ark, v(7)* with (+, 0) parity and signature; that of band 2 is ~(6)~(5)‘, ~(7)~ with (-, 0) parity and signature; the bands 3 and 4 are signature partner of configuration v(7)* with (+, 0) and (+, 1) parity and signature. Finally, we included the band 5 in addition pa’, to the ten low-lying bands, which has a N = 7 proton excited, having the configuration ~(6)~(7)‘, v(7)* with (-, 0) parity and signature.

should

be modified,

so that the (+, 0) band

stays yrast up to higher

spins.

This

could be obtained increasing the (K, p) Nilsson parameters, as compared to the values suggested in ref. 3), so that the N = 6 orbital would be lowered compared to the N = 4 and N = 5 orbitals, and the 2 = 66 shell gap would become larger. The required change would not alter the moments of inertia considerably. The calculated moments of inertia for the (+, 0) configuration are compared with the data 30) in fig. 22. The calculation without pairing correlations is similar to that of refs. ‘O,“), while a calculation including pairing correlations, but without Strutinsky renormalization, was presented in ref. 34). The effect of the occupation of the fourth N = 6 proton, stabilizing the moments of inertia as functions of the frequency, is apparent comparing fig. 22 with fig. 18, valid for ‘z$b. The static proton pairing gap collapses at ho = 0.42 MeV, while the nuetron pairing gap vanishes already at zero frequency. It can be observed, that the inclusion of pairing correlations improves the agreement with the measured J(‘) substantially, without requiring a different experimental assignment for J (‘I . Pairing correlations also improve the difference J(l) - J(‘) (cf. fig. 10d). However, in this case they worsen the agreement with the data for JC2).

Y.R. Shimizu et al. / Eflects of pairing correlations

ho Fig. 22. Calculated moments (+, 0) band in “‘Dy, labeled

4.4. CALCULATIONS

FOR

109

(MeV)

moments of inertia for the of inertia for the nucleus “‘Dy. The calculated by 1 in the previous figure, are shown (cf. fig. 12) and compared with the data, taken from ref. 30).

OTHER

SUPERDEFORMED

BANDS

We have also calculated low-lying superdeformed bands in a few nuclei, for which no experimental observation has been reported up to now (cf. fig. 5). The calculations for ‘$$b are shown in figs. 23 and 24. The moments of inertia in fig. 24 are calculated for the band labeled 1 in fig. 23, which is yrast in the whole frequency range, and whose configuration is 7r(6)3, v(7)‘, with (-, 0) parity and signature. For this configuration, the static proton pairing gap collapses at hw = 0.42 MeV, while the neutron pairing gap vanishes already at zero frequency. In figs. 25 and 26 we instead show the results for ‘i:Tb. In this case there are two almost degenerate yrast bands, labeled 1 and 2 in fig. 25, with (-, 1) and (-, 0) parity and signature, and corresponding to the configuration ~(6)~, v(5)‘(7)*. The calculated moments of inertia for the (-, 1) band are shown in fig. 26. Both the static proton and neutron pairing gaps vanish already at zero frequency. Finally, we show in figs. 27 and 28 the calculations related to ‘ZiDy. At high spins (I > 45h), two almost degenerate bands, having (+, +G) and (+, -$) parity and signature, and labeled 4 and 5 in fig. 27, are yrast. Their configuration is rr(5)‘(6)‘, v(5)‘(7)*. The moments of inertia related to band 4 are shown in fig. 28b. Both the static proton and neutron pairing gaps vanish already at zero frequency. We also show in fig. 28a the moments of inertia of the band labeled 1 in fig. 27. This band has (-, -$ parity and signature, and its configuration is 7r(6)4, ~(7)~, and therefore can be obtained adding one N = 7 neutron to the configuration adopted for ‘ZiDy. For this configuration, the static proton pairing gap collapses at hw = 0.38 MeV, while the neutron pairing gap vanishes already at zero frequency.

Y.R. Shimizu

110

et al. / Effects of pairing correlations

70

50

30

I

I (6) Fig. 23. Low-lying superdeformed bands in the nucleus ‘50Tb. The same as in fig. 11 but for the nucleus ‘5oTb. In this nucleus we find that a well separated band with (-, 0) parity and signature, having the configuration ~(6)~, u(7)‘, is yrast at all spins. It is labeled by 1. Other two low-lying bands, labeled by signature. 2 and 3, are signature partners of configuration p(6)‘(4)‘, v(7)’ with (-, 1) and (-, 0) parityand

15’Tb

, .3

.4

I

I

1

.5

.6

.7

fro

.8

(MeV)

moments of inertia Fig. 24. Calculated moments of inertia for the nucleus rsOTb. The calculated (-, 0) band in ““Tb, labeled by 1 in the previous figure are shown (cf. fig. 12).

for the

Y.R. Shimizu

et al. / Effects

IO

30

of pairing

111

correlations

70

50 I (i-l)

Fig. 25. Low-lying superdeformed bands in the nucleus ‘?‘b. The same as in fig. 11 but for the nucleus “‘Tb. In this nucleus we find that a pair of signature partner bands with (-, I) and (-, 0) parity and signature, having the configuration ~(6)~, ~(7)~(5)‘, are yrast at all spins. They are labeled by 1 and 2. Another (-, 0) low-lying band, labeled 3, has the configuration ~(6)~, ~(7)~.

,

/

,

I

15*T b 100 :-_ ------------

--------

---___

1

1

80 -

60 -

.3

1

.4

I .7

.8

Fig. 26. Calculated moments of inertia for the nucleus ‘52Tb. The calculated moments of inertia (-, 1) band in “‘Tb, labeled by 1 in the previous figure, are shown (cf. fig. 12).

for the

112

YR.

Shimizu

et al. / Efects

of pairing

\

IO

correlations

‘53Dy \

\

30

\

50

70

I (il) Fig. 27. Low-lying superdeformed bands in the nucleus ‘53Dy. The same as in fig. 13 but for the nucleus ‘s3Dy. In this nucleus several bands appear successively along yrast. First, three almost degenerate bands, labeled by 1, 2 and 3, are yrast up to I=42. The (-, -$) band 1 has configuration ark, ~(7)~ and bands 2 and 3 are signature partners of configuration STY, ~(7)~(5)’ with (-, +$), (-, -i) parity and signature. After that, a pair of signature partner bands, labeled by 4 and 5, come as yrast, whose configuration is ~(6)~(5)‘, v(7)*(5)’ with (+, +f) and (+, -$) parity and signature. As in the other Dy isotopes, we included a (+, -4) band with a N = 7 proton excited, labeled by 6, having the configuration

5. Conclusions We have found that the calculations performed within the cranked Strutinsky method using the Nilsson potential are able to reproduce the main trends observed in the kinematic

and dynamic

moments

of inertia

of the superdeformed

rotational

bands in the A = 150 region. Adding the individual contributions of the valence intruder orbitals in the lowest part of the N = 6 and N = 7 oscillator shells to the “‘46Sm-core”, one can understand in most cases the different behaviours observed in the various nuclei. The first pair of valence orbitals align at rather low frequency and their contributions to both moments of inertia are decreasing in the frequency range in which the superdeformed bands are observed. However, the orbitals of the second pair of N = 6 protons align later and when they are occupied (as in ‘i:Dy) their contributions tend to make JC2) more constant. In fact, the large deformation of intrinsic states on which rotational bands are based implies that alignment is a slower process than in normal deformed nuclei and blurs the clear distinction existing in that case between collective and single-particle angular momentum.

Y.R. Shimizu

et al. f Efiects

of pairing correlations

(a)

.4

.5 hw

113

-

.6

(MeV)

Fig. 28. Calculated moments of inertia for the nucleus ‘53Dy. The calculated moments of inertia for the (-, -f) band labeled by 1 in fig. 27 and for the (+, +f) band labeled by 4 are shown (cf. fig. 12) in (a) and (b) respectively.

For a fixed configuration, the moments of inertia are only slightly affected by the changes of deformation with angular momentum. On the other hand, deformation affects in an essential way the relative energy of the bands, and consequently plays an important role in determining the yrast configuration in the relevant spin range. Although the cranking model can reproduce the main trends of the moments of inertia, at a closer look there appear systematic discrepancies between theory and and the experiment. In general, J(‘) is overestimated, J (*) is underestimated, difference J(l) - I(*) which is proportional to the apparent alignment iO, is always positive, in contradiction to the experimental data. We have found that a better agreement with experiment can be obtained, taking pairing correlations into account. While static pairing correlations do not play a significant role, because in our calculations the BCS pairing gaps are usually quenched at low frequencies, pairing fluctuations affect the moments of inertia in an important way. In fact, typical dealignments introduced by these correlations are of the order of 2-4 units of h, implying reductions of .I(‘) by 5-10h2 MeV-‘, while corrections to J(*) are in general positive and of the order of 5h2 MeV-‘. These two effects lead to a large reduction of io.

Y.R. Shimizu et al. / E&Y%sof pairing correlations

114

Even if the inclusion agreement

the calculations frequencies,

of pairing

with experiment, cannot

correlations

there remain

reproduce

which may indicate

leads in general to a rather satisfactory

a few significant

the abrupt some drastic

‘ZiDy, no choice of the con~guration

increase

discrepancies.

in 5”’ observed

change

of the internal

is able to reproduce

In ‘:tGd, at the lowest structure.

In

both the very large value

for J”’ estimated experimentally and the frequency dependence inadequacy of our model in these cases indicates the need for further and theoretical studies.

of J@). The experimental

We express sincere thanks to Tord Bengtsson for helping us to perform the Strutinsky calculations. This work is executed as a project of the NBI-INS collaboration. One of us (Y.R.S.) is indebted for financial support during his stay at Niels Bohr Institute given by the Grant-in-Aid for Scientific Research of Japan Ministry of Education, Science and Culture under Grant number 63044160.

Appendix The effect of N,,,-mixing caused by the cranking term becomes important for the high frequency rotation and large deformation. As is stressed in sect. 3, the “diabatic” construction of the cranked single-panicle basis is crucial to get the reliable spininterpolation and the selfconsistent determination that it is useful to keep the major shell quantum quantity. In ref. ‘), the cranked harmonic oscillator matrix elements of the Nilsson potential. Here we in a simple manner. The angular momentum operator in the stretched _?,= &I:, + & + /3X$x)

of deformation parameters, so number, N,,,, as a conserved basis is used to evaluate the show how to include the effect coordinates

basis is written

as

(A.11

where & and fX are the spin and the orbital angular momentum operators in the stretched coordinates. The operator fX is responsible for the AN,,, = *2-mixing and is defined, in the stretched coordinates, by

i=*

(A.21

where C,, C, are the annihilation operators of the isotropic harmonic oscillator the stretched coordinates, and the coefficients a, and &. are given by

in

(A.3) By using second-order perturbation in eq. (A.l) with the approximation

theory to evaluate the effect of the last term of taking the unpe~urbed hamiltonian as the

Y.R. Shimizu

isotropic matrix

harmonic element

oscillator

et al. / Effects

of pairing

in the stretched

correlations

coordinates,

115

we obtain

for the energy

in the shell IV,,, = N

AEN,,,,= -

C

P,N'=NIZ

(NCYl(--OPf,)lN’P)(N’Pl(-WPxf,)INa!’) EN' - EN

(NcY([CyCz, c~c~]pva’) )

= -*

(‘4.4)

”

where w, denotes the oscillator frequency unit which preserves the volume conservation condition for given values of deformation parameters. Thus the “effective” cranking term acting on orbits in the same IV,,,-shell is given by

(A.3 with

(A-6) where we have introduced

the spherical CO)=

Im

tensor

J -

4rr

in stretched

coordinates

r* Yl,.

21+1

From this result, we see that the effect of major shell mixing caused by the cranking term appears as an effective change of the monopole and quadrupole deformation terms in the Nilsson potential. Since fy’ is now w-dependent, the angular momentum operator to be used to evaluate the single-particle alignments takes the form, (A.7) where the +( -) sign applies to the orbits with signature the canonical relation, de; -z-i dw between

single-particle

routhians

k,

r = fi (-i),

which guarantees

(A.81

and alignments.

References 1) 2) 3) 4)

A. Bohr and B. Mottelson, Phys. Scripta 24 (1981) 71 J.D. Garrett, J. Phys. Sot. Jap. 54 (suppl.) (1985) 456 T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436 (1985) 14 S. Aberg, T. Bengtsson, 1. Ragnarsson and P. Semmes, in Proc. XXVI Int. Winter Meeting on nuclear physics, Bormio, Italy, 1988, ed. I. Iori (Ricerca Scientifica ed Educazione Permanente, Milano, 1988) 5) M. Brack, J. Damgaard, A.S. Jensen, H. C. Pauli, V.M. Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44(1972)320

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