NEGATIVE-PARITY STATES AND OCTUPOLE COLLECTIVITY OF EVEN GE-ISOTOPES

Negative parity states

and

octupole collectivity

of

even

Ge isotopes

Der-San Chuu

Department

of

Electrophysics, National Chiao Tung Uniuersity, Hsinchu, Taiwan, Republic

of

China

S.

T.

Hsieh and

H.

C.

Chiang

Department

of

Physics, National Tsing Hua Uniuersity, Hsinchu Taiwan, Republic

of

China (Received 11 December 1991)

The negative parity energy levels ofthe even-even Geisotopes with mass number between 64 and 76 are studied systematically by enlarging the model space ofthe interacting boson approximation model to include both collective and noncollective basis states. The basis states consist ofN&

—

1sd-boson plus a f-boson configuration and Ns

—

1 sdboson plus a fermion pair configuration. The fermions are allowed

to occupy the

f,

iz and g9/p single-particle orbitals, respectively. It was found that the negative parity

energy levels of Ge nuclei can be described reasonably well. The intensities of the collective configuration in 3 states increase when going from nucleus Ge tonucleus

'

Geand decrease from

nu-cleus Ge to nucleus Ge. The B(E3;3&~0&+)values are calculated and compared with the available observed data.

PACSnumber(s): 21.10.Re,21.60.Ev,23.20.Lv,27.70.

+

q

I.

INTRODUCTION

In recent years, considerable progress has been made in extending the interacting boson approximation model (IBA) to study the negative parity states

of

even-mass nu-clei and the high spin states anomaly in medium to heavy deformed nuclei [1—

9].

Among these works very few are concentrated on the structure

of

energy levels.

of

medium light nuclei such as the Ge isotopes. During the past few years, the observed nuclear properties

of

the negative parity energy states

of

even-mass Ge isotopes have been accumulated and considerable attention has been attract-ed

[10

—

_{22]. It}

is known that the Ge nuclei are complex nuclear systems with unstable shapes. Both the coex-istence

of

a shape transition from spherical to weakly de-formed and a coexistence

of

different types

of

deforma-tion are expected in these nuclei [23—

25].

Qualitatively these features can be explained with the help

of

the Nilsson model calculation

[26],

shell model calculation

[27],

constrained Hartree-Fock calculation [28], two-quasiparticle plus- rotor calculation

[13],

and

IBA

model calculation

[13].

Substantial experimental evidence suggests that exten-sive regions

of

statically octupole deformed nuclei occur

in some mass regions [29—

33].

The question whether cer-tain nuclei can be octupole unstable has been a subject

of

much experimental and theoretical interest during the past ten years. Nazarewicz and collaborators

[31,

32] have proposed that the nuclei with the strongest octupole correlations (i.

e.

, the best candidates for static octupole deformation) occur when Nand

Z

are equal to 34, 56, 88, and 134. However, Cottle [30]analyzed systematically the behavior

of 3,

states from the available observed data and identified the

X

and

Z

values equal to 40, 64, 88, and 134 for maximum octupole collectivity.

It

is interest-ing to study the problem

of

octupole collectivity from the point

of

view

of

the

IBA

model.

The purpose

of

this work istwofold. First, we want to present a systematic study

of

the negative parity energy

levels

of

even-mass Ge isotopes. Second, we desire

to

in-vestigate the octupole collectivity around the region

of

mass number A

=70

by a hybrid

of

sdf

IBA

and

IBAF

models.

II.

MODEL

The negative parity states

of

even mass Ge isotopes with

Z

=32

and

32(X

~44

will be studied systematical-ly.

For

this mass region, the Ca nucleus or the Ni nu-cleus can be treated as the core. In

IBA

calculation, itis known that the effect

of

using a different core can be ab-sorbed in the interaction strengths. Therefore, we may take the Ca nucleus or Ni nuclei as the core in this work. By assuming Ca nuclei as the core, the boson number for the isotopes Ge and Ge are

%~=12

and 13, respectively.

For

the other isotopes which pass the neutron midshell the neutron boson numbers are counted as one-half

of

the number

of

neutron holes. Thus,

IBA

model assumes valence boson numbers Nz as 13, 12,

11,

10,and 9for the nuclei Ge, Ge,

Ge,

Ge, and Ge, respectively. In this work, the model space isconsidered as the admixture

of

two subspaces: (1)the configuration

of

Ntt

—

1 sd bosons plus one

f

boson; or (2) the configuration

of

N~

—

1 sd bosons plus two fermions which are allowed to distribute in the

f

5iz and g&i2

orbit-als. The former subspace is

of

a more collective behavior

while the later contains some single particle nature.

To

be more specific, the model space is spanned by the hy-brid

of

two types

of

basis states:

in, n&vaL,

f;LTMT)

and ~n,

n&vaLj

ij2(J);LTMT)

where n,+n&

=Nz

—

1,

j

„j2

=5/2

or

9/2,

and

J=2,

3,

. .

.

,

7.

The model Hamiltonian can be expressed as [8]

~here _{H~ is the}

IBA

boson Hamiltonian H~

=aoEd+a,

p

p+a2L

L+a3Q

Q .

The octupole term

T3.

T3 and the hexadecapole term T4 T4 have been omitted in _Hz since they are generally believed to be less important. The fermion Hamiltonian HF 1S

1

m JM

_J1J

where c is the fermion single-particle energy, V 's are the fermion-fermion interactions, a~

(aj

), and

aj

=(

—

1)J

_a,

being the nucleon creation (annihila-tion) operator. Themixing Hamiltonian Vz~ between the sd boson and the fermion is assumed:

V~F=ag

g

(a a )' '

J)J2 where

Q~=(dts+std)'

'

—

_&7/2(d

_d)'

',

and the Hamiltonian related to the f-boson part is

V„=Efnf+yg

(f'f)'"

+g

_{y g'}

[(a'a')'"f+H.

c.

]"'

which includes the f-boson single-particle energy and mixing the Hamiltonian

of

the f-boson with the sd boson and with the fermions. The fermion potential is taken as the Yukawa type with the Rosenfeld mixture. The oscil-lation constant

v=0. 963

' fm with A

=70

is as-sumed. The whole Hamiltonian is then diagonalized in the selected model space. Practically, we first performed a calculation for the positive parity energy levels

of

even mass Ge nuclei with the framework

of

extended

IBA

model [34,

35]. For

the calculation

of

the negative parity energy levels, the interaction strength parameters in Hamiltonian H~ are kept as the same values as those ob-tained from the calculation

of

the positive parity energy

levels

of

Ge nuclei. The mixing parameters

a,

y,

6,

the single f-boson energy

_sf,

and the single-fermion

ener-gies

(j=5/2

and

9/2)

contained in the fermion

Hamil-tonian _{HF, Vz,} and V~F were chosen to reproduce the

spectively. The interaction strengths and single-particle energies for each isotope are allowed to be mass-number dependent.

III.

RESULTAND DISCUSSIQN

Table

I

represents the final chosen values

of

the in-teraction strengths and single-particle energies. The

values

of

_{y which represents} the interaction between the

sd and the f-bosons are in general very small [8]and thus can be set to zero.

It

can be seen from Table

I

that the values

of

the mixing parameters are in general not too large. The smallness

of

the mixing parameters manifests the small mixings between the different configurations.

The single f-boson energy as well as the single fermion

energies has a minimum value around the mass number

A

=68.

This means the effect

of

the

f

boson is most im-portant around the Ge nucleus. One may note that some

of

the effective interaction strength parameters and single particle energies (especially the

_sf

) have a significant change around the nucleus

Ge.

This is be-cause in our work we assume Ca nuclei as the core, the boson number for Ge and Ge are counted as one-half

of

the number

of

nucleons outside the core while other

Ge

isotopes ( Ge) that pass the neutron midshell and the neutron boson numbers are counted as one half

of

the number

of

neutron holes. Therefore, we have particle-particle to particle-hole transitions from nucleus Ge to nucleus

Ge.

And this is the reason that we have an unusual change

of

the interaction parameters in the

A

=68

region. One can also note from Table

I

that the features

of

the variation

of

single fermion energies (s5&z and E9&z) are similar to that

of

single f-boson energy. The values

of

c5&2 and c,_9/2

of

the nuclei Ge and Ge are obtained from the extrapolation.

In the past few years, several observations on the nega-tive parity states

of

Ge nuclei have been performed. de Lima et al.

[13]

studied the low and high spin states

of

Ge through in-beam y-ray spectroscopy via the Ni(' C,2p) Ge, Cu( Li,2n) Ge and Cr('

F,

p2n) Ge reactions. They observed three positive parity bands and two negative parity bands up to two tentative 11 states. Ardouin et

al. [16]

presented some negative parity states data from Ge(p,

t)

Ge reactions. The calculated negative parity energy spectra in this work are compared with the observed ones in Figs. 1 and

2.

The levels marked with asterisks are not included in the

least-TABLE

I.

The interaction parameters (inMev) adopted inthis work.

Nucleus Ge 66Ge e 70Ge 72Ge '4Ge 76Ge ao 0.298 0.254 0.156 0.289 0.289 0.396 0.400 a& -0.220 -0.220 -0.220 -0.155 -0.102 -0.035 -0.025 a2 0.035 0.035 0.023 0.023 0.023 0.023 0.023 Parameter a3 -0.016 -0.008 0.015 -0.001 -0.001 -0.001 -0.001 (MeV) -0.006 -0.661 -0.005 -0.527 -0.527 -0.527 -0.527 -0.4 -0.4 -0.3 -0.3 -0.3 -0.3 -0.3 2.080 1.295 0.836 1.707 2.116 2.760 2.850 ~5/2 0.996 3.245 0.576 3.158 3.974 4.791 5.607 ~9/2 2.235 4.547 1.545 3.903 4.461 5.020 5.578

IO-7

O

6-X

LLJ 5

4-3

2-7 5 64G 9 5

0

3 LLJ

2-

JO-7

0

6

—

X

5-UJ 9 9 7 75- 5~-3 ~Ge -9 9 7

57

-5 3

l-0-

0 Expt Theo. O- o' Expt. Theo.

o-o Expt. Theo.

FIG.

1. The calculated and observed negative parity energy spectra for the nuclei "Ge, Ge,and Ge. The experimental data are taken from Refs.[10—15].

squares fitting. Figure 1shows the calculated and the ob-served negative parity states

of

the three lighter mass nu-clei

Ge.

Figure 2 presents those

of

the three heavier mass nuclei

Ge.

Several levels not yet observed are presented in the figures forfuture reference.

Nazarewicz et al. [32]performed a Strutinsky calcula-tion that included octupole as well as qaudruple and hex-adecapole shape degrees

of

freedom and indicated that nuceli in the light Ge-Se region would be stable but rath-er soft with respect to octupole deformahon. They

pre-dieted that the largest octupole softness would occur for

Ge.

The calculation

of

Gorres et al

[10]

fo.und a mod-est decrease in the

3,

energy in Ge followed by a sharper drop in

Ge.

From a microscopic point

of

view, octupole collectivity originates in the interaction between the unique parity orbit in a major shell and the common parity orbit having both orbital and total angular momentum 3A less than that

of

the unique parity orbit

[24].

The nuclei in which the strongest octupole effects occur are those in which Fermi surfaces

of

both neutrons

TABLE

II.

The relative intensities ofthe (Ns

—

I)-boson plus one f-boson configuration and the

(X&

—

1)-boson plus a fermion pair configuration for negative parity states ofGenuclei. The total

inten-sity ofthese two configurations for each state isnormalized to 1.0. The numbers shown are the intensi-ties forboson plus f-boson intensities.

States 64Ge 66Ge

Nucleus

686.e

"Ge

Ge '4Ge Ge

31 32 33 3g 1i 12 2l 22 23 24 41 42 5i 52 6i 7l 8l 9l 10l 11l 13) 0.799 0.679 0.000 0.000 0.000 0.000 0.919 0.736 0.135 0.029 0.009 0.932 0.804 0.781 0.795 0.000 0.000 0.000 1.000 0.946 0.839 0.732 0.154 0.011 0.022 0.011 0.009 0.953 0.502 0.939 0.781 0.591 0.702 0.780 0.572 0.850 0.786 0.819 0.764 0.882 0.879 0.611 0.848 0.673 0.775 0.810 0.785 0.800 0.947 0.931 0.892 0.925 0.912 0.894 0.899

6-IQ

5-

₈

4-

r

6 5

3

O CD 'K LQ

2

6 -9 7 (3,2) (3,2

!

2 3 2 I 3

"Ge

I I I II II I/' IlI ~/ 3 2 2 3 I

4

0

33 ~2)P2& 3) Ge 35 Bg 3 2 3g 4) 32 2) 8]

o-

o'

Expt. Theo,

o

+

Expt. Theo, Expt Theo.

FIG.

2. The calculated and observed negative parity state energy spectra for the nucleus Ge, Ge,and Ge. The experimental data are taken from Refs. [16—_21].

and protons liebetween the two interacting orbits. Based on the above argument, Nazarewicz et al. proposed that the nuclei with the strongest octupole correlations occur

when N and

Z

are equal to 34, 56, 88, and 134. Cottle [30]performed a systematic observation regarding the en-ergies

of 3,

states which are octupole vibrational states in most nuclei and found that instead

of

those nuclei pro-posed by Nazarewicz et al.

_[32],

the strongest octupole correlations seem to occur at nuclei with N and

Z

being equal to 40, 64, 88,and 134. Our plot

of

3& states

of

Ge

nuclei versus the neutron number N is shown in

Fig.

3.

The octupole deformation behavior can be investigated by the behavior

of

3& states because nuclei which have

3.0— (79 2.

9

2.8 CD 2.7 I 2.6 LQ 2.5

the greatest degree

of

octupole collectivity have the lowest 3& state energies.

It

can be noted from the figure

that the 3& state

of

the Ge nucleus lies at the minimum

position. We also analyzed our calculated wave func-tions. In

Fig.

3,we list the relative wave function intensi-ties

of

the (Nti

—

1)-sd-boson plus af-boson configuration for the lowest 3& states

of Ge

isotopes in the parentheses.

The total intensity

of

the (Nii

—

1)-boson plus one f-boson configuration and the (Nti

—

1)-boson plus two fermions configuration for each state has been normalized to 1000. One can note from the figure that the intensities

of

the

f-boson configuration increase when going from nucleus

Ge

to nucleus Ge and then decrease when going from

Ge

to

Ge.

Our calculation is equivalent to enlarge the model space to consider both collective and noncollective negative parity states. The relative wave-function inten-sities

of

the (Nti

—

1)-sd-boson plus a

f

-boson configuration for the negative parity energy levels

of

Ge nuclei are shown in Table

II.

One can note from Table

II

that, in general, the states with smaller angular momenta

(J

(5

) are dominated by the (N~

—

1)-sd-boson plus a f-boson configuration while the states with higher angu-lar momenta are dominated by the (N~

—

1)-sd-boson plus

TABLE

III.

The calculated and observed 8(E3;0&+~3&)

values. The experimental data are taken from Ref.[33].

8(E3;0)+~3)

)

2.4

l I 1 1 I I I I l I

30 34 38 42 46

N

FIG.

3. The energies of3& states for the Ge nuclei are

plot-ted as a function ofthe neutron number ofGeisotopes.

Nucleus 64Ge 66Ge 68Ge 70Ge 72Ge 74G 76Ge Expt. 35.850 23.751 8.830 8.730 Theo. 31.577 29.762 30.440 23.616 32.028 9.140 5.720

two fermions configuration. The mixings between the two kinds

of

configurations are in general not toolarge.

In order

to

check our wave functions, we calculated the

B(E3;0,

+~3,

) values. There are only a few experi-mental

B(E3;0,

+~3,

) values available

[33].

For

future observation, we present also some theoretical values for which the experimental counter parts are not available now. In our calculation the fermion effective charge e is assumed to be

0.

5e. The average boson effective charge e which is determined by normalizing the calculated

B

(E2)

value [36]

to

the corresponding observed data for the transition

2,

+

—

+0&+ is about

0.

24e.

It

can be noted from Table

III

that our calculated

B

(E3;0,

+

~3,

)values agree satisfactorily with the experimental data, since the

B(E3)

values are very sensitive to the wave functions. The reasonable agreement

of

B

(E3;01+

~3

t ) values

shows that our wave function isalso reasonably good.

IV. SUMMARY

In summary, we have investigated the structure

of

the negative parity energy spectra

of

the

Ge

isotope with mass numbers between 64and

76.

The

IBA

model space is enlarged to consider the collective and noncollective negative parity states.

It

was found that the mixings

of

the configuration

of

(N~

—

I)-sd-boson plus af-boson and the configuration

of

(X~

—

I)-sd-boson plus two fermions in the negative parity states

of Ge

isotopes are in general small. The octupole softening is also analyzed by study-ing the energy values, the wave functions, and the

B

(E3;0,

~3,

)transition rates

of

the

3,

states.

This work was supported by the National Science Council

of ROC

with grant number

NSC81-0208-M009-04.

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