STRUCTURE OF EVEN GE ISOTOPES BY MEANS OF INTERACTING BOSON MODEL WITH A FERMION PAIR MODEL

Structure

of

even

Ge

isotopes

by means

of

interacting boson

model with

a

fermion pair

model

S.

T.

Hsieh and

H.

C.

Chiang

Department

of

Physics, National Tsing Hua University, Hsinchu, Taiwan

Der-San Chuu

Department

of

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

of

China (Received 4December 1991)

The energy levels ofthe even-even Geisotopes with mass number between 64 and 78are studied inthe model ofthe traditional interacting boson approximation. Toaccount forthe multiple band structure of

these isotopes, one boson isallo~ed tobreak and form afermion pair. The two fermions are allowed to

excite to

f,

/2 and g9/2 single-particle orbitals. Itwas found that the energy levels ofthe

"Ge

isotopes

can be reproduced reasonably.

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

+

_q

I.

INTRODUCTION

The nuclear properties

of

nuclei around the N

=40

re-gion and, more particularly,

of

the even-mass Ge isotopes have been investigated by a number

of

experimental and

theoretical works [1

—23].

The Ge nuclei are

character-ized by a complex nuclear system subjected to a variety

of

nuclear interactions which make these nuclei very

un-stable in shape. Hence both the coexistence

of

a shape transition from spherical

to

weakly deformed and a

coex-istence

of

different types

of

deformation occur in these

isotopes. Qualitatively, these features can be explained with the help

of

the Nilsson model

[24]. For

the proton and neutron numbers in this mass region, the Nilsson single-particle energy diagrams display various rather large gaps at different deformations. Thus a competition and coexistence

of

several kinds

of

conGgurations

corre-sponding to various shapes at the low spin region is

ex-pected. Guilbaut et

al. [12]

have presented shell model calculations for

Ge;

however, a satisfactory

reproduc-tion

of

the experimental data was not obtained. Ardouin et

al. [7]

successfully performed constrained

Hartree-Fock

calculations using Skyrme's effective interaction

to

analyze the different structures

of

Ge isotopes in

terms

of

an oblate-to-prolate transition. Petrovici et

al.

[8] studied in detail the shape coexistence phenomena dominating the structure

of

the nucleus Ge by taking into account the dominant correlations on top

of

the symmetry projected quasiparticle mean-Geld solutions. de Lima et

al. [9]

performed two-quasiparticle-plus-rotor calculations [25]and an interacting boson approximation

(IBA) model [26] calculation for the Ge nucleus. The

results obtained can well describe the yrast features

of

the

level scheme. Barclay et

al. [10]

performed a two-quasiparticle-plus-IBA model proposed by Gelberg and Zemel [27] Morrison, Fassler, and Lima [28],and Yoshi-da, Arima, and Qtsuka [29,30] to study the

8

(E2)

_{and g}

factors for the high spin states

of

the

Ge

nucleus. Reasonable agreement between calculated and measured values was obtained.

The purpose

of

this work is twofold.

First,

we want to

present a systematic study

of

the even-mass Ge isotopes.

Second, and most important, we desire

to

investigate to

what extent the observed shape coexistence or multiple band structure

of

these nuclei can be interpreted in terms

of

the interacting-boson-plus-a-fermion pair model. This

model has been successfully applied

to

study the positive and negative parity states and band crossing behavior

of

even-mass deformed nuclei [31

—34].

II.

MODEL

The even-mass Ge isotopes with

Z

=

32 and

32~% ~44

will be studied systematically. Taking the

Ca nucleus as the core, the boson numbers for the

iso-topes

Ge

and

Ge

are

X =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 the

IBA

model assumes valence boson numbers

13,

12,

11,

10,

9,

and 8for the nu-clei Ge, Ge, Ge, Ge, Ge, and Ge, respectively.

In this work it is assumed that one

of

the bosons can be broken to form a fermion pair which may occupy the

f

s/2 org9/2

«bitals.

Our model space includes the

IBA

space with Nbosons and space with N

—

1 bosons plus two fermions. The

model Hamiltonian can be expressed as[32]

H=H~+HF+

V~F ~

where H~ isthe

IBA

boson Hamiltonian

Htt

=aced+a,

p

p+a~L. L+a3Q.

Q

.

The octupole _{term T3 T3 and hexadecapole term T4 T4}

have been omitted in Hz since they are generally believed

tobe less important. The fermion Hamiltonian _HF is

H

=ps.

+2j+1[atXa.

]'

'

1

+

—,'

gV

&2J+1[(atxat)

x(irixa.

₎

]'

',

J,

j

with a being the nucleon creation operator. The mixing Hamiltonian V~Fis assumed tobe

y

QB._Q QB QB where Q

=(d"Xs+s

Xd)'

'

—

(d

Xd)'

',

2 0.4 0.3 0.2— / / / / / / / Qo U(5)

Q=g

+a(a

Xa

)I ' J J

+P[(atXat)'

'Xd

—

d

X(a Xa

)'

']'

' _. J J J J

In the calculation the fermion potential is taken as the Yukawa type with the Rosenfeld mixture. The oscillator

constant

v=0. 963

' fm with A

=70

is assumed. The single-particle energies and interaction strength pa-rameters contained in the boson Harniltonian Hz and

Vz~ were chosen to reproduce the energy level spectra

of

even Ge isotopes with mass number between 64 and

78.

In our calculation the interaction parameters contained in H~ for each nucleus are unified forboth the Npure

bo-son configuration and N

—

1-boson-plus-one-fermion pair configuration. The energy bands with these two kinds

of

configurations are mixed through the diagonalization

of

the energy matrix in the whole model space.

III.

RESULTAND DISCUSSION

The interaction strengths and single-particle energies

for Ge isotopes are allowed to be mass number depen-dent. Table

I

lists the best fitted interaction strengths and single-particle energies for all isotopes. The mixing parameter

_p

can be unified as

p=

—

0.

02 MeV, while the parameter n has a significant change from nucleus Ge to nucleus

Ge.

Since here we have particle-particle to

particle-hole transitions, it is not surprising we have a significant change

of

u at this point.

It

is well known [35] that the four terms

of

Hz relate to the pure symmetries in

the following way: In the U(5) symmetry, only ed and

L L

terms appear; in the SU(3)limit, only

L L

and _{Q Q} terms appear; and in the O(6) limit, only _{p p} and

L L

terms appear.

To

correlate the variation

of

the

interac-tion parameters to the limiting symmetries, the resulting interaction parameters contained in the pure boson Ham-iltonian

of

Ge isotopes as a function

of

mass numbers are plotted in

Fig. 1. To

the far right, we listed the

sym-metries to which each

of

the terms belongs. At the

bot-)

0.)— 0 -0.(— I I / o-———

~

I I 64 66 I / I I / CV 0———-0-—— ',70 72 74 76 78 2 Qa U(5)0(6) SU(3) —A )0()&SU(3~ Qi 0(6) SU(3) U(5) 0(6) 0(6) U(5) U(5)

FIG.

1. Interaction parameters ofHz vs mass number A of Ge isotopes. Indicated on the right-hand side are the sym-metries involved in each term. Indicated in the bottom are the symmetry regions fordifferent Geisotopes.

tom we indicated the possible relevant symmetries along the boson number axis. From

Fig.

1 one can see that

there are possibly symmetry changes from A

=

70to 72,

72to 74, and 76to

78.

The abrupt changes

of

the

single-particle energies

c»2

and _c9/2 for the nuclei Ge, Ge,

and Ge reflect the fact that there are structure changes

in these nuclei. This is consistent with the result ob-tained by Lecomte et al.

[20].

The calculated and observed energy spectra for the Ge

isotopes are shown in Figs. 2

—8.

The levels marked with asterisks are not included in the least-squares fitting.

Fig-ure 2 shows the calculated and observed energy levels

of

the N

=Z

nucleus

Ge.

The structure

of

this very

neu-tron deficient Ge isotope has been investigated recently with the use

of

particle-y coincidence techniques in weak fusion-evaporation channels

[1]

and the evaporation code

CASCADE

_[3]

with the reaction ' C(

Fe,

2n) Ge at 150 MeV. Lister et

al.

[4]investigated the shape changes

of

Ge experimentally and thus provide a direct test

of

a variety

of

nuclear models. Figure 3shows the energy lev-els

of

the Ge nucleus. From Figs. 2and 3, it can noted

that our calculated energy levels

of

the nuclei Ge and

Ge are all in good agreement with their experimental

counterparts. The complex multiple band structures and shape coexistence

of

the nucleus Ge have attracted

TABLE

I.

Interaction parameters (inMeV) adopted inthis work. Parameter (MeV) Nucleus ao a& a2 a3 5/2 ~9/2 64G 66G 0.2978 0.2535

—

_0.22

—

0.22 particle-particle 0.035

—

0.016 0.035

—

0.008 0.03 0.03

—

_0.02

—

_0.02 0._0.249186 1.534 1.488 Ge

"Ge

72G 74G 76G 78G 0.1558 0.2890 0.2890 0.3964 0.4000 0.4300

—

_0.22

—

_0.155

—

0.102

—

_0.035

—

_0.025

—

_0.025 0.023 0.023 0.023 0.023 0.023 0.023 particle-hole 0.015

—

_0.001

—

_0.001

—

_0.001

—

_0.001

—

_0.001

—

0.27

—

0.27

—

_0.27

—

0.27

—

0.27

—

_0.27

—

_0.02

—

_0.02

—

_0.02

—

0.02

—

0.02

—

_0.02 0.111 0.830 1.200 1.200 1.200 1.200 1.080 1.575 1.687 1.687 1.687 1.687

"Ge

e4Ge IO-I4 8— I2 _lp+ x 8 5 8 ₆-IP IP-e' X ₈ 8 4

3—

LLI 4 2' p + 4 + 2—4 2 O-o Expt. Theo. Expt. Theo. Expt. Thep. Expt. Theo. Expt. Theo. + 4 Expt. Theo.

I

-2'

FIG.

4. _{Calculated and observed}

energy spectra for the nu-cleus 6'Ge. The experimental data are taken from Ref.[12].

0-o

Expt. Theo.

FIG.

2. Calculated and observed energy spectra for the

nu-cleus ~Ge. The experimental data are taken from Refs.

[1-5].

vestigated_{mation theory, which is an improvement}the Ge nuclear structure with dynamic

defor-of

the pairing-plus-quadrupole model. Satisfactory results were ob-tained. The spectroscopy

of

the nucleus Ge is

especial-ly interesting because this N

=40

semiclosed shell nu-cleus isone

of

the few even-even nuclei tohave a

0+

state for the first excited state

[14,

17,

18].

Kotlinski et

al. [17]

studied the Coulomb excitation

of

Ge using '

0,

Ni, and Pb targets. They proposed that Oz+ state is an

in-truder state. Our calculated 02+ state has a discrepancy

of

0.

39MeV above the observed value and isin a reversed

order with the calculated 2&+ state. However, the

calcu-lated results in the other energy levels in general agree reasonably with the observed values. The calculated and observed energy levels

of

the nucleus Ge are shown in

Fig.

7.

For

this nucleus only a few levels have been identified experimentally. One can see from the figure

that the agreement between the calculated and observed

levels issatisfactory especially forthose levels which were included in the least-squares fitting. The energy levels

of

the nuclei Ge and Ge are shown in

Fig. 8.

One can

see that the agreement between the theoretical energy

levels and experimental counterparts is quite reasonable.

The analysis

of

the relative wave-function intensities

eeGe + IO 70G 8 + I 8 + a 6 + 6 4 O IJJ x 6 5+ 6 + 4 2+ p+ a 2' 3' Expt. Theo. 2' p+ Expt. Theo. Expt. 2 Theo. Expt. Theo. I 2

0

p' + Q- o

Expt. Theo. Expt, Theo.

FIG.

3. Calculated and observed energy spectra for the nu-cleus 66Ge. The experimental data are taken from Ref.[6].

FIG.

5. Calculated and observed energy spectra for the nu-cleus Ge. The experimental data are taken from Ref.

[13].

much interest recently [7

—

11].

Petrovici et

al.

[8]

inves-tigated the shape coexistence phenomena which dom-inates the structure

of

the nucleus Ge by using an

ap-proach

of

the excited variation after mean-field projection

in a realistic model space. Chaturvedi et al.

_[11]

em-ployed the same approach to study the complex band

structure

of

the

Ge

nucleus and obtained good agree-ment between the theoretical and observed levels. de

Lima et

al.

[9]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 even parity collective bands which can be interpreted fairly well in terms

of

the rotation-aligned and interacting boson models. Our

cal-culated results

of

the nucleus Ge are shown in

Fig. 4.

The different bands are displayed in different columns for clear comparison.

It

can be seen from the figure that the complex multiple bands can be reproduced quite well.

The calculated and observed energy levels

of

the isotopes

in-ve + lo e Ge 3 + I + O+ x Expt. Theo. 5 4 + 2 Expt. Theo. Z LU

2-

4~ 3i I 22 2 ED K 4J + 3g + 23 4, + 22 2' I

0-

o'

Expt. Theo. _{Expt. Theo.} Q

o-Expt. Theo.

FIG.

6. Calculated and observed energy spectra for the nu-cleus 'iGe. The experimental data are taken from Ref.[15].

3

0

2—

0)

X

LLj 4 3 + 2 Expt. Theo. 2'

0-

o+ Expt. Theo.

FIG.

7. Calculated and observed energy spectra for the nu-cleus Ge. The experimental data are taken from Ref.[21]. for the energy levels

of

Ge shows that most

of

the levels

are dominated by the pure boson configuration except for

the

J"=5,

+,

62+, and 7&+ states, which are dominated _by

the configuration

of

N 1boson

—

plus two _f5/2 fermions, and the states

J

=8,

+ and

9,+,

which are dominated by the configuration

of

N

—

1 boson plus two g9/p fermions.

For

the nucleus Ge, most states are dominated by the pure boson configuration except for the states

J

=4&+,

5,

+,

62+, 7&+, and 8&+, which are dominated by the N

—

1

boson plus two

f

_&/2 fermion configuration, and the states

J

=

8&+ and 10&+, which are dominated by the

configuration

of

N

—

1 boson plus two _g9/2 fermions. In our results it was found that the overlapping between different subspaces is very small. Table

II

shows the rela-tive intensities

of

wave functions corresponding to N

bo-son and N

—

1-boson-plus-two-f s/2-or-g9/2-fermions configurations for each state

of

the nuclei Ge, Ge, and

Ge.

The total intensity

of

N boson, N

—

1-boson-plus-two-f

»2-fermions, and N 1-boson-plus— -two-g9/2-fermions configurations for each state is normalized

to

1000. One can see that, in general, the energy levels

of

FIG.

8. Calculated and observed energy spectra for the nu-clei

'

Geand 'Ge. The experimental data are taken from Refs.

[22,23].

these three nuclei are dominated by the pure boson configurations. The N 1-boson—-plus-two-f s/2-fermions configuration is important only in the states

J

=4&+,83+,

103+,and 122+

of

Ge and 42+, 62+, 7j+, and 82+

_of

_Ge,

while the N

—

1-boson-plus-two-g9/2 fermions

configuration is only dominant in the states

of

j

=8,

+,

12&+,and 14&+,

of

Ge,7&+

of

_Ge,and 8&+ and 10&+levels

of

Ge.

If

we increase the _{fs/2 org9/2} single-particle en-ergy so that this orbit becomes effectively irrelevant, then the agreement between the calculated and observed levels will become worse. One can also find that the mixing be-tween difFerent configurations is very small in general.

There are only four states

(J

=42,

+7,+,

82+,g2+) which possess more than

10%

mixing between different kinds

of

configurations.

For

the nuclei Ge, Ge, and Ge, the pure boson configurations are dominant in nearly all

states. Only the states

J

=33+ and 43+

of

_Ge, 44+

of

Ge,

and 43+

of

the nucleus Ge are dominated by the N 1-boson-plus-—two-

f

5/2-fermions configuration.

There are some experimental

B (E2)

values for Ge iso-topes [6,12,13,15,17,21,

22].

The study

of

these values

will give us a good test

of

the model wave functions. The electric quadruple operator can be written as

T(E2)=e

g+e

~(gtg.

)

+,

_peB[(g

tg t)(4)d d't(g _g )(4)](2)

1 1 J J

where _{Q is}taken as

g =(dts+sfd

)"'

—

a(dtd

)"'

In our calculation the fermion effective charge e is as-sumed tobe

0.

5.

It

was found that different values

of

the fermion effective charge cannot yield significant change

in

B(E2)

values. The boson effective charge in the

T(E2)

operator has been determined by normalizing the calculated

B(E2)

value to the corresponding observed

data for the transition 2&~0&. The parameters

a

and

P

are assumed to have the same values as used in the

mix-TABLE

II.

Relative intensities ofthe

X

boson configuration {denoted as 0) and the

X

—

1bosons plus afermion pair occupied in single fermion orbits

f,

/2 (denoted as

f,

/2) org9/i (denoted asg9/2) configurations for 'Ge,

'

Ge,and

'

Geisotopes. The total inten-sity ofconfigurations with and without fermion-pair excitation for each state is normalized to 1000.

Nucleus States Ol 02 03 04 2[ 22 23 24 3] 4) 42 43 6[ 8) 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.994 0.009 0.998 0.989 0.000 68Ge 2

f

s/2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.972 0.002 0.010 0.000 2 g9/2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.019 0.000 0.001 1.000 1.000 1.000 0.999 1.000 1.000 1.000 0.998 0.999 0.999 0.993 0.746 0.958 0.969

0.

000

"Ge

2

f

s/2 0.000 0.000 0.001 0.000 0.000 0.000 0.002 0.001 0.001 0.006 0.246 0.038 0.030 0.000 g9/2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.008 0.004 0.001 1.000 1.000 1.000 0.999 0.998 0.999 0.999 0.998 0.997 0.998 0.996 0.995 0.990 0.981 0.001 72Ge 2

f

s/2 0.000 0.000 0.001 0.002 0.001 0.001 0.002 0.002 0.001 0.004 0.004 0.009 0.016 0.000 g9/2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.000 0.001 0.001 0.003 0.999

Nucleus

_J;

—

+

Jf

Expt. This work Other work'

68Ge

"Ge

2)

—

+Oi

22~0'

4l—+2i 42

—

+2( 42~22 6,

~4,

82~6)

82~62

83~6)

84~6)

10)

~8[

102

—

+83 122

—

+102 14,

~12,

02~2

22~0

22~02

22~2

l

4l~2l

03~22

03~2]

42~22

43~2,

6l~4,

62~42 8,

~6,

82~6(

17.6 0.17 13.9 0.5 0.41 12.0 15.0 12.0 23.0 15.0 3.3 24.0

)

23.0 10.0 4.5 21.0 48.0 1.0 25.0 111.0 24.0 &4.8 &0.14 29.0 2.0 34.0 27.0 6.5 43.0 17.6 16.34 29.2 0.88 0.11 25.28 0.06 8.35 5.83 0.39 23.90 54.45 4.2 11.02 13.45 21.0 50.2 7.94 22.02 35.09 35.04 0.72 0.16 15.11 0.69 42.04 7.43 2.5 44.0 17.6 29.09 33.67 16.07 19.13 9.95 17.60 'Reference [10].

TABLE

III.

Calculated and the experimental

B(E2)

values (in Weisskopf units) for Ge and Ge. The experimental data are adopted from Refs. [12]and

[13].

ing Hamiltonian. The value

of

a ischosen tobe

—

&7//2, which is the generator

of

the SU(3) group.

For

illustra-tion only we list the calculated and experimental

8

(E2)

values for Ge and Ge isotopes in Table

III.

Other theoretical work

[10]

is also presented for comparison. One can note from Table

III

that our calculated values agree reasonably with observed data and other theoretical

values.

IV. SUMMARY

In summary, we have investigated the structure

of

the energy spectra

of

the isotope string

of

Gewith mass

num-ber between 64 and

78.

We extended the

IBA

model to

allow aboson

to

bebroken to form afermion pair which

can occupy the _{fs/2 or g9/2} orbitals. The calculated

en-ergy levels are in satisfactory agreement with the

ob-served values for the whole string

of

Ge isotopes.

The plot

of

the interaction strength versus mass num-ber reveals a transition from the mixture

of

SU(3), O(6),

and U(5) symmetry to O(6) and U(5) mixture and then

finally U(S) symmetry as the mass number increases from

64to

78.

This structure change is apparently manifested in the steep change

of

the Hamiltonian between these two nuclei. We also analyze the relative intensities for

configurations

of

pure 1V bosons and

of

N

—

1-bosons-plus-two-fermions excitation. Our analysis shows that, in general, the mixing s between these two kinds

of

configurations are small.

This work was supported by the National Science

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J.

W. Ar-rison, Phys. Rev. Lett. 58,662(1987).

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