INTERACTING-BOSON-FERMION-APPROXIMATION DESCRIPTION OF NEGATIVE-PARITY STATES IN EVEN-ODD YB ISOTOPES

Interacting-boson-fermion-approximation

description

of

negative-parity

states

in even-odd

Yb isotopes

S.

T.

Hsieh

Department ofPhysics, National Tsing Hua University, Hsinchu, Taiwan, Republic

of

China

D.

S.

Chuu

Department

_of

Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic ofChina (Received 6 March 1991)

The negative-parity energy levels ofthe even-odd Yb isotopes, ' Yb, ' 'Yb, ' Yb, ' 'Yb, and ' Yb, are studied systematically within the framework of the extended interacting-boson-fermion-approximation (IBFA)model inwhich one odd fermion iscouplied with a boson core. The boson coreis

described with the IBAmodel but allowing one boson tobreak into a quasifermion pair. The odd fer-mion is assumed to be in one ofthe f7/2 and p3/2 single-particle orbitals while the fermion pair is al-lowed to occupy the single-particle orbit i&3/2 It was found that the negative-parity energy levels of

these even-odd Yb isotopes and the values oftransition quadrupole moments for the ground-state rota-tional band of' Ybcan be reproduced infairly good agreement with the observed data.

I.

INTRODUCTION

Recently, experimental information on even- and odd-mass Yb isotopes has become more abundant [1—

17].

These nuclei fall in the transitional region and have several interesting properties.

For

example, they can as-sume a variety

of

shapes [8]ranging from oblate to super-deformed prolate, and thus the coexistence

of

different nuclear shapes, collective and noncollective rotational modes, is expected. Second, the ytterbium nuclei in this mass region can bear higher angular momentum [1—3,8,

11,12].

Furthermore, the energy spacings between the negative-parity levels in some even-odd-mass Yb iso-topes have been observed several anomalous phenomena such as the abrupt shrinking

[14,

15]between some adja-cent energy levels and backbendings [3,4]as one plots the moment

of

inertia versus the square

of

the angular veloci-ty for yrast band

of

a nucleus. These phenomena might be a result

of

the complicated interplay between the col-lective and single-particle degrees

of

freedom induced by the Coriolis decoupling

[18].

A generalized calculation within the framework

of

the two-quasiparticle plus rotor band-mixing model

[19]

predicted that the high-spin states are produced by the alignments

of

the angular mo-menta

of

the decoupled quasiparticles along the collective rotation. Most

of

the high-spin data can be understood, at least qualitatively, in terms

of

the cranked shell model [20—22] which assumes aligned high-j nucleons weakly coupled to deformed cores with rather constant deforma-tion parameters. However, the neglect

of

variations in deformation from state to state is dangerous in the

N=88

—90

transition region between rotational and vari-ational nuclei

[9].

Recently, the interacting-boson-approximation (IBA) and interacting-boson-fermion-approximation

(IBFA)

models including fermion pair de-grees

of

freedom [23

—

29] were employed to analyze the positive- and negative-parity high-spin anomalies.

It

was found that high-spin states and the back-bend

phenome-na could be reproduced quite well.

The negative-parity bands

of

even-odd-mass Yb iso-topes, ' Yb ' Yb,

'

Yb, '~'Yb, and ' Yb, are seldom studied in recent years because

of

their complicated band-crossing patterns. In this work we shall employ the extended

IBFA

model

to

study the energy levels

of

the odd-mass Yb isotopes. These odd-mass nuclei are as-sumed to be described by an odd fermion weakly coupled to a boson core. The boson core is studied with the ex-tended

IBA

by allowing one

of

the bosons tobreak into a fermion quasiparticle pair, usually assigned to a unique-parity intruder orbital with spin

j.

In the region

of

well-deformed nuclei, the unique-parity intruder orbital such

as h_&&/2 or _i]3/2 is the most important because both the

Coriolis antipairing and rotation alignment effects in-crease with increasing angular momentum

[18,29].

How-ever, a recent study on the negative-parity high-spin states

of

N=88

isotones [30]manifested that the orbit

i&3/2 is the most important one forthe first back bending.

As a consequence, in this work we restrict the quasiparti-cle pair to be in the _i/3/2 single-particle orbit only. The odd nucleon forour even-odd-mass isotopes isassumed to distribute in the f7/2 or p3/2 single-particle orbital. In the present work, the

IBA-1

basis states are used in the boson core. This is because the IBA-1 has been proven

[31]

to be a valid approximation in transitional regions far from the closed shell.

II.

MODEL

In the calculation

of

the negative-parity energy levels

of

the even-odd-mass Yb isotopes,

Z

=X=

82 is taken as the core. The pure

IBFA

model assumes a valence boson number Nz

=

11,

12,14, 15,and 16and a fermion quasi-particle distributed in the single-particle orbitals _{f7/2 or} p3/p for the five nuclides ' Yb, ' Yb, ' 9Yb, ' 'Yb, and

Yb, respectively. In addition to the pure

IBFA

configuration, we admix the N~

—

1 boson plus 1fermion

pair configuration into the model space.

To

be more specific, the model space is spanned by two types

of

basis states, ln,ndpyL j'~TMT& and l[n,'nd

p'y'L

',

j

' (

J)

]L

',

j;

_JT

_MT

&,

where _n,

+nd=N&,

n,

'+nd=N~

—

1,

j=f7/2

or _p3/2,

j'=

—",_,and

J~4.

The

J

=0

and 2fermion pair states are

excluded to avoid double counting

of

the states. The Hamiltonian consists

of

four parts,

H=H~+HF+

V~F+

V

where the

IBA

boson Hamiltonian H~ can be expressed as

H~=aond+a,

P

_{P+a2L L+a3Q}

_{Q .}

The octupole term T3T3 and hexadecapole term _T4T4 have been omitted in_Hz since they are generally believed tobe less important. The fermion Hamiltonian HF is

HF=QEJa

a~

+

—,' V (a~:a.) (a~a )

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

(a

) is the

nucleon creation (annihilation) operator.

The boson-fermion Hamiltonian V~Fthat describes the interaction between the odd quasinucleon and even-even core nucleus contains, in general, many different terms [32] and is rather complicated. However, it has been shown that V~F may very well be approximated by the following three terms

[33,34]:

V~F=aQ&

QF+

g

A~~':[(aj

Xd)'

'X(d

Xaj')'"']o

':

J,

J,

k

where _{Q~ is}the boson quadrupole operator;

Qz

=(stXd+dtXd)"'+y(dt

Xd)'

',

and QF isthe fermion quadrupole operator,

and

(::)

indicates that the commutation

of

a.

and a is neglected. The SU(3) value

of

y=

—

&7/2

is adopted in this work. The first term in V~F is a quadrupole-quadrupole interaction, while the second term represents an exchange interaction. The origin

of

the exchange term is due to the interchange

of

the odd quasinucleon with one

of

the nucleons that make up the d boson. The 1ast term in V~F is a monopole interaction, which gives rise to arenormalization

of

the boson energy c.

=

_cd

—

4.

The mixing Hamiltonian V between the sd boson and quasifermion pair is assumed

V

=aQ~ (ata

)'

_'+PQ~

[(atat)'

'd

—

_dt(a.

_a )'

']

The radial dependence

of

the fermion potential is taken as the Yukawa type with a Rosenfeld mixture. An oscil-lation constant

v=0. 963

' fm with

3

=160

is as-sumed. The interaction strength

of

the V 's is adjusted by requiring

(

jjl

Vl

jj

& ₂

—

(

jj

Vl

jj

_&J

0=2

MeV. The

whole Hamiltonian is then diagonalized in the selected model space.

The odd-mass Yb isotopes ' Yb, ' Yb, ' Yb, '

'Yb,

and ' Yb are assumed as the system with an odd fermion weakly coupled to the even-even boson cores: ' Yb,

Yb, ' Yb, ' Yb,and '

Yb.

To

describe the even-even boson core, the fermion pair degrees

of

freedom are taken into account in the traditional

IBA

model by allowing a boson to break into one fermion pair which is assumed to distribute in the single-particle orbit i&3/2 In the

practi-cal practi-calculation, the single-particle energy e(i&3/p) and in-teraction parameters contained in the Hamiltonians H& and

H

are first chosen to reproduce the positive-parity energy spectra

of

these five even-even Yb nuclei. After obtaining these parameters, the interaction strengths con-tained in V~F and the single-fermion energy E(p3/2

f7/p)-[the single-particle energy s(f7/2) is set to be zero] were then fitted to the negative-parity energy spectra

of

the even-odd Yb nuclei using the following constrains: (1) The single-particle energy

of

i,

3/2 and interaction

param-eters contained in H~ and

H

are kept at the same value as those obtained from the fitting

of

the positive-parity states

of

even-even Yb nuclei. (2)The parameters a and A are assumed as smooth functions

of

neutron number. (3) The boson energy sd is renormalized as Ed=ed

—

6

due to the strength

of

the monopole interaction. (4)The fermion single-particle energies for each isotope are al-lowed to be mass number dependent and are obtained as aresult

of

fitting.

III.

RESULTS

The searched interaction strengths for the even-even Ybnuclei are listed in Table

I.

Note that the mixing pa-rameter

P

is kept constant for all Yb isotopes. The major effect

of

the mixing parameter

a

is to mix the boson-configuration-dominated low-lying states with the fer-mion pair states. Since the energy-level spacings

of

the low-lying sates

of

these five isotopes are different, conse-quently, the value

of

the mixing parameter

a

for each iso-topes has to vary

to

obtain better agreement between the calculated and observed data. The smallness

of

the mix-ing parameters

a

and

P

manifest in the mixings between the

Xz

pure boson configurations and N~

—

1 boson plus 2 fermion configurations are small, in general, and seem to be consistent with the general analysis

of

the previous work

[19].

The other parameters are all varied monoton-ically with the mass number. The decreasing

of

single-boson energy Ed(ao) and the increasing

of

the strength

of

col-TABLE

I.

Interaction parameters (in MeV) ofthe Hamiltonian for even-even Ybnuclei adopted in this work. ter eV) ao

a,

a2 a3 ~~~13/2) 162Yb 164Yb 166Yb 168Yb »OYb 172Yb 0.614 0.614 0.610 0.610 0.610 0.610 0.047 0.045 0.040 0.033 0.013 0.000 0.0035 0.0035 0.0040 0.0046 0.0057 0.0058

—

_0.0082

—

_0.0082

—

0.0085

—

0.0093

—

_0.0111

—

_0.0120 0.089 0.089 0.089 0.110 0.110 0.110 0.03 0.03 0.03 0.03 0.03 0.03 1.340 1.317 1.223 1.178 1.137 1.075

lective as the boson number increases. This is consistent with the tendency

of

deviating away from U(5) symmetry to SU(3)symmetry. Table

II

lists the interaction parame-ters

of

the total Hamiltonian for the even-odd Yb iso-topes. The single-particle energy

e(i,

3/p) and strengths

of

the

P"

P

term

(a,

),

L L

term (a2), and

Q.

Qterm (a3)are not shown in Table

II

because these values are the same as those used in the corresponding even-even cores. The parameter ao (Ed) varied slightly from those

of

the corre-sponding even-even core. The variation

of

ao comes from the renormalization due

to

the monopole-monopole in-teractions A.

The calculated and observed energy spectra for the even-even nuclei ' Yb, ' Yb, ' Yb, ' Yb, and ' Yb are shown in Figs. 1—5. The observed levels with asterisks in Figs. 1 and 2are not included in the g-squares fitting be-cause their spin

or

parity assignments are not quite cer-tain. One can see from the figures that the agreement be-tween the theoretical values and observed data is quite reasonable in general. Recently, the high-spin states

of

Yb have been studied

[1]

using a ' Sn( Ca,4n) reac-tion with a beam

of

Caions delivered by the tandem ac-celerator. Positive-parity states with spin up to 30 are as-signed tentatively. Our calculated energy levels shown in

Fig.

1 for ' Yb agree fairly well with the observed data and are all in correct order except the 122+state, which is obtained too low.

For

' Yb high-spin states have been populated in the ' Sm('

_0,

4n) and ' Sm('

0,

4n) reac-tions. The ground-state (g.

s.

) band and sideband have

been established to

I

=22+

and

26+.

One can note from

Fig.

2 that the calculated values are, in general, able to reproduce the observed data fairly well. Figure 3shows

the calculated and experimental energy spectra

of

'

Yb.

The g.s. band can be reproduced quite satisfactorily. However, the shrinking

of

the adjacent levels in the _y band cannot obtained reasonably. Especially, the 13&+

state is obtained too low and thus causes reversed order with its neighboring states. Figures 4 and 5 show the comparisons between the theoretical energy spectra and their experimental counterparts for nuclei ' Yb and

Yb.

The fittings

of

these two nuclei are very similar to that

of

' Yb. Their

P

and y bands cannot be reproduced as good as their ground-state bands.

We also analyze the wave function for each state

of

these even-even Yb isotopes to study the relative intensi-ties

of

the Nz-pure boson configuration and the

Xz

—

1

boson plus 1 fermion pair configuration. The analysis

of

the wave functions shows that the states with

I

~

8+

in the g.

s.

band

of

the nucleus ' Yb are dominated by the pure boson configuration. The states with

I

)

16+

in the g.s.band are dominated by the configuration

of

N

—

1

bo-son plus 2

i,

3/2 fermions. The mixing

of

these two

configurations is only observed in the states 10,+ (85.

2%

—14.

8%),

12~ (78.

4%

—

21.6%),

and 142+

(55.

1%

—44.

9%).

For

' Yb the states with

I

(6+

are all dominated

()

94%)

with the pure boson configuration, while the levels with

I

~

16+

in the g.s.band all states in the sideband are pure

Xz

—

1 boson plus 1 fermion pair configuration except

I=142+

(8%

—

_92%)

and 162+ states

(19%

—

_81%).

_For

the g.sband the mixings between these two configurations are only exhibited in the states 8&+

(90%

—

_10%),

10&+

_(85%

—

_15%),

12~+

(78%

—

_22%),

and

14&+

(68%

—

32%).

For

' Yb the pure boson configuration is only dominant in the states with

I

(6+

TABLE

II.

Interaction parameters (inMeV) ofthe total Hamiltonian foreven-even Ybnuclei

adopt-ed inthis work. V) ao &lp3yz

f

7x2 ) 163_Yb 165_Yb 1_69Yb 171Yb 173Yb 0.4738 0.4338 0.4425 0.4101 0.3542

—

_0.1401

—

0.1401

—

_0.1645

—

_0.200

—

_0.2558 0.0311 0.0311 0.0252 0.0199 0.0199 0.2431 0.250 0.531 0.576 0.7556

—

_0.5014

—

_0.4141 2.00 2.00 3.10

IO— 162 38 168 + 28 26

7

24 28+ 26 24' 12

10—

36 $4 32'— CD 20-w 5 18

12—

2—10— +

8—

4 + 2 0 Expt. g.

s.

band Theo. 22' 20—— 18 16' 14 12 10+ 8 6+ Expt. Theo. v' _band 26+

+-5

— ₂₂+ 20 18 16 14 2 12

lp—

8+ 64 4+

0

p+ _{g.s. band} Expt, Theo. 14 13 9~~ yS T6~c 4t 5 2+ Expt. y bond Theo.

FIG.

1. Calculated and observed energy spectra for the ' Yb nucleus. The experimental data are taken from Ref.

[l].

The

levels with asterisks are not included inthe y-squares fitting.

FIG.

3. Calculated and observed energy spectra for the ' Yb nucleus. The experimental data are taken from Ref. [5]. The

levels with asterisks are not included in the y-squares fitting.

5

r-170 164 20

4—

18

6

—

~4 22' 20 16' 14 12 8 6 4+ 2' pV' Expt. Theo g.S. 22_g

20'

18 16 14 12 + 4 2 0 +5 4

Expt. Theo. Fxpt. Theo. Expt. Theo.

s band

p

bond _{y band}

14 2 12- 8'-4 2'.

0

—₀₊ Expt. Theo. g, s. band + lp 6+ 4 2+ 0 Expt. Theo.

P

band + 10 9

e'

7'

6 5 4f 2+ Expt. Theo. band

FIG.

2. Calculated and observed energy spectra for the ' Yb nucleus. The experimental data are taken from Ref. [2]. The

levels with asterisks are not included in the g-squares fitting.

FIG.

4. Calcu1ated and observed energy spectra for the ' Yb nucleus. The experimental data are taken from Ref. [6]. The levels with asterisks are not included in the y-squares fitting.

172

b

TABLE

III.

Relative intensities ofwave functions for energy levels ofthe ' Yband ' Ybnuclei.

Nucleus states 17'_Yb ~2 I 13/2 172Yb I 13/2 14 14 12

O

CD QJ + IO I — + 8 4 0

E~

Theo. g. s.band Expt. Theo.

P

band fxpt. Tho. y bQAd

FIG.

5. Calculated and observed energy spectra for the

"

Yb nucleus. The experimental data are taken from Ref.[7].

in the g.s. band

I

~7+

in _{the y band.} The states with

I~16+

in the g.s. band

I

~ 11+

in _{the y band} are all

dominated with the configuration

of

N

—

1 boson plus 2

i]3/2 fermion. The mixing

of

these two configurations is prominent only in the states 8&+ (86.

7%

—

13.

3%),

8q+

(71.

3%

—

_28.

_7%),

10t+ (78.

9%

—

_21.

_1%),

12~+

(68%

—

32%),

and 14,+

(53%

—

_47%).

_For

the nuclei ' Yb and ' Yb, the relative intensities

of

the pure boson and one-fermion pair excitation configurations

of

the wave function for each state are listed in the Tables

III

and

IV.

One can note from the tables that in terms

of

the relative intensity these states are more dispersive.

The calculated and observed negative-parity energy spectra for odd-mass ' Yb, &6s

Yb &69Yb, '

_'Yb,

_and Yb isotopes are shown in Figs. 6

—

11.

The levels marked with asterisks are not included in the fitting. The dashed levels are the predictions for those yet unobserved levels. There are abundant experimental data observed [6—17]in recent years.

For

example, Becket

al. [3]

used the advent

of

Compton-suppressed multidetector arrays to analyze the triple y-ray coincidence spectra. The negative-parity level scheme

of

' Yb up to

—

", was as-signed. In Figs. 6—9 the different quasibands are displayed in different columns for clear comparison.

For

the nucleus ' Yb,only the yrast band is presented due to scarce and uncertain observed data. One can see from the figures that the calculated energy levels including in the y-squares fitting for odd-mass Yb isotopes agree in general reasonably well with the observed data especially

01 21 41 61 81 101 121 141 161 181 201 02 22 42 62 82 102 122 142 23 31 43 51 63 71 83 91 103 123 143 0.979 0.958 0.944 0.914 0.862 0.782 0.666 0.452 0.001 0.000 0.000 0.959 0.962 0.936 0.862 0.361 0.098 0.000 0.075 0.954 0.942 0.843 0.939 0.194 0.894 0.551 0.007 0.019 0.070 0.064 0.021 0.042 0.056 0.086 0.138 0.218 0.334 0.548 0.999 1.000 1.000 0.041 0.038 0.064 0.138 0.639 0.902 1.000 0.925 0.046 0.058 0.157 0.061 0.806 0.106 0.449 0.993 0.981 0.930 0.936 0.977 0.935 0.936 0.904 0.849 0.768 0.656 0.509 0.006 0.959 0.957 0.918 0.745 0.220 0.099 0.000 0.009 0.952 0.928 0.651 0.921 0.302 0.797 0.122 0.005 0.000 0.000 0.000 0.023 0.047 0.064 0.096 0.151 0.232 0.344 0.491 0.994 0.041 0.043 0.082 0.255 0.780 0.901 1.000 0.991 0.048 0.072 0.349 0.079 0.698 0.203 0.878 0.995 1.000 1.000 1.000

for the g.sband. Figures 6 and 7 show the calculated and observed energy spectra

of

' Yb and '

Yb.

One can note that the calculated levels for these two isotopes are all in correct order and agree reasonably with their exper-imental counterparts, except for very few levels,

e.

g.,the

and —"

,

states in ' Yb and the —"

,

state in '

Yb.

For

these states the theoretical level energies obtained are too low. We also present some experimental unobserved states such as the —", _,

—

",

and

—

_", states in ' Yb and

the —", _, —", _,

—

", _,

—

", _,and

—

", states in ' Yb for future

experimental reference. Figures 8 and 9 present the theoretical and experimental energy levels for ' Yb and '

_'Yb.

_{One can note from the figures that the calculated}

state in ' Yb and '

'Yb

obtained is too high. We need a rather strong exchange interaction in order to lower the calculated excitation energy

of

the —,' state,

which isthe bandhead

of

the ground-state band. Howev-er, the strong exchange interaction will make the agree-ment between the calculated and observed energy spectra become worse. From Figs. 8and 9one can also note that anomalous phenomena such as the abrupt shrinking

of

the energy spacings between two adjacent energy levels (signature properties) observed in the —,

'[521]

band

of

Fig-ure 10 shows the calculated and observed energy

differences between two adjacent levels,

EE(I)

=E(I)

E—

_(I

—

_{1), of}

the ground-state band —,'

[521]

for isotopes ' Yb and '

'Yb.

One can note from the figure that the observed zigzag lines (signature depen-dence) can be fairly well reproduced. Figure 11 shows the calculated and observed energy spectra

of

'

Yb.

Yoshida et

al.

[26]studied the signature dependence

of

the energy levels and

M1

and

E2

transitions in ' Yb in terms

of

the interacting boson-fermion model. This nu-cleus was considered as a system

of

an odd fermion cou-pled with the boson core ' Yb described by

IBA.

In the work

of

Yoshida et al; the fermion pair degrees

of

free-dom are not taken into account for the boson core; hence the high-spin states can only be reproduced qualitatively. On the contrary, one can see from

Fig.

11 that the

high-spin levels can be reproduced reasonably well. In order to investigate the contributions

of

the interaction param-eters Ir _{(Qz Qz term) and A (the exchange term),} we set these two parameters to be zero alternatively. The ener-gy spectra obtained by setting

~=0

or A,

=O

are presented

also in the

Fig.

11. It

can be seen that a zero A lowers the energy levels significantly, while a zero value

of

rccan

make only aslight deviation from the original result. We also calculate the relative intensities

of

the wave functions for each energy level corresponding to

f

_7/2 and p3/2 single-fermion orbitals for the odd-mass Yb isotopes.

It

is found that all states

of

' Yb, ' 'Yb, and ' Yb are dominated by the

f

7i2 single-particle orbital with intensi-ty greater than 96%%uo. However, the intensities

of

the

wave functions for the energy level in the nuclei ' Yb and ' Yb are rather dispersive. Table

III

lists the

rela-TABLE IV. Relative intensities ofwave functions forenergy levels ofthe ' Yband ' Yb nuclei. Nucleus State (—,

')I

(-,

')(

(—,

')I

( & )I 0.435 0.131 0.140 0.489 0.198 0.602 0.238 0.709 0.292 0.832 0.420 0.119 0.103 0.838 0.322 0.103 0.070 0.104 0.105 0.065 0.106 0.108 0.106 0.109 0.111 0.115 0.114 0.210 Yb 0.565 0.869 0.860 0.511 0.802 0.398 0.762 0.291 0.708 0.168 0.580 0.881 0.897 0.162 0.678 0.897 0.930 0.896 0.895 0.935 0.894 0.892 0.894 0.891 0.889 0.885 0.886 0.790 State (—,

')I

(—)I (2)& 0.569 0.194 0.599 0.263 0.688 0.307 0.769 0.368 0.859 0.510 0.119 0.404 0.119 0.417 0.122 0.122 0.125 0.126 0.128 0.135 0.132 0.228 0.135 0.306 0.139 0.333 0.260 0.359 0.272 0.378 0.252 0.378 0.203 0.634 1.000 165Yb 0.431 0.806 0.401 0.737 0.312 0.693 0.231 0.632 0.141 0.490 0.881 0.596 0.881 0.583 0.878 0.878 0.875 0.874 0.872 0.865 0.868 0.772 0.865 0.694 0.861 0.667 0.740 0.641 0.728 0.622 0.748 0.622 0.797 0.366 0.000

5-

47 45 2 43 2

4-

41 2 IB3

II-2 69 2 65-t65Y 67 2 63 2 39 2 37 2 25 M3 2 3l 2 29

2-2

~

2 25 2 21 2 I —17 2

_19:

2 37 2 29 2 25

r

Expt. Theo. 2

2:

O 2~

25

_2~'

i Expt. Theo

FIG.

6. Calculated and observed energy spectra for the ' 'Yb nucleus. The experimental data are taken from Ref.

[11].

The

levels in dashed lines are the predicted energy levels. The levels with asterisks are not included inthe g-squares fitting.

59 2

7

57 2 ₅₅ 2 53 2 51 2 49 Q)

₅

— 2 45 2

4—

2 37 33

2-

'2 25 21 I —2~1 13 2IIy

0

2 = 47 2 43 2 39 r= 35 2 31—

r

27= 2 27

r

--

—--

23:----2 ~l 2 15 9 Theo. 2

Expt. Expt. Theo.

FICx.7. Calculated and observed energy spectra for the ' Yb nucleus. The experimental data are taken from Refs. [12]and [13].The levels in dashed lines are the predicted energy levels.

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

tive intensities

of

the wave functions for these two nuclei. One can see from the table that the mixing

of

_f7/z and p3/p single-particle orbitals is prominent in most

of

the

states in these two nuclei.

For

' Yb there is experimental information on the transition quadrupole moment

[17].

The study

of

these values provides us a good test

of

the model wave func-tions. The electric quadrupole operator can be written as

T(E2)=e

Qs+e

a(a Xa

)' '

+Pe [(a

Xat)'

'Xd

—

dtX(a

Xa.

)'4']'

'.

J J J J kO 2 23 2 21 2 19 CD / / I / / I I I 17 2 IB9

For

the fermion effective charge

e,

an average value

0.

37e

of

the proton and neutron obtained by Alonso, Arias, and Lozano [28] is adopted. The boson efFective charge in the

T(E2)

operator is assumed to be

0.

22e, which isclose to the value used in the previous work

[35].

The

EI=1

and 2 transition quadrupole moments for the ground-state rotational band

of

' Yb are defined as

Q'a

=

_[B(E2;I

~I

—

_KI

₎

_{l[(5/16m )(I2KO~I}

am

)']]'",

-where

IC=

—,

'

was assumed. The parameters

a

and

P

are

assumed the same values as used in the mixing

Hamil-17 2-15 2 15 2 II 2 2 2 9 7 2 2

5'

f523) -2

7p

O— 2 _{a'2 (512)} 2 I//2 _(52f)

Expt. Theo. Expt. Theo. Kxpt Theo

FICi. 8. Calculated and observed energy spectra for the ' Yb nucleus. The experimental data are taken from Ref.[15].

25 2 2I 2 / / / / / / / / / / 25 2 l73YI 2l 2 lgr 2 I3 2 II 2 23 2 2f 2 f9 2 l7 2 l5 2 II 2 l7 2 l5 l3 2 II 2 II/2 _{,

505)

23 2 2I 2 l9 I ₂ 17 2 25 2 27

r

2I

T

9

72

2 ₅ O g-2 2 2 I/2

(52I]

Theo. 9 2 7 2 5//2 _(5I2) Expt. Expt. Theo,

FIG.

9. Calculated and observed energy spectra for the ' 'Yb nucleus. The experimental data are taken from Ref. [16].The

levels in dashed lines are the predicted energy levels.

II 2 9

7:

o

₂ Theo.I KsA 23 ~IT IB ~19

Expt. Theo.2 Theo.3

+

II-K=O

2

FIG.

11~ Calculated and observed energy spectra for the

' _{'Ybnucleus. The experimental data are taken from Ref.[17].}

l2-

~Expt. opresent work xRef.[l2]

04

169 0.2— ~( 7 /. -/ tl / II / / 0.0 (D

&

06-

I7I xTh8o.

l2-UJ &

o~-0.2— 0.

0--0.

2 I I I I I I I I I I eo. 8 CP OJ C3 ~Expt. o Preset cwork x Ref.[Ipj 3 5 7 9 l I I3 I5 l7 l9 2I P$ 25

FIG.

10. Calculated and observed energy difterences AE(I)

=E(I)

—E(I —

1)for the nuclei ' Yband ' 'Yb. The ex-perimental data are taken from Refs. [15]and [16],respectively. The dashed lines represent the observed values, while the solid

lines represent the calculated values inthis work.

I I I I

7 9 II 13 l5 I7 l9 2l 23 25

2 2 2 2 2 2 2

FIG.

12. Calculated and observed transition quadrupole mo-ments for the ' _{'Yb nucleus. The experimental data are taken}

from Ref. [18] and references therein. The dashed curves represent the results obtained in this work, and the solid curve

tonian. Since the structure

of

Ybisclose

to

SU(3) sym-metry, the value

of

_y

is chosen to be

—

&7/2,

which is the value forthe SU(3)group generator. Figure 12 shows the calculated and observed transition quadrupole mo-ments versus the spins

of

the depopulating states. From the figure it can be seen that the parabolic formlike feature

of

Q'

"

from spin —"

,

to —"

,

and the fine vari-ation feature

of

Q' ' _{can be reproduced}

qualitatively, especially when the dip occurs at

I=

—", _, although our

calculated values exaggerate the dip at

I=

—"

,

and yield a somewhat larger Q'

"

value at

I

= —

", . On the con-trary, the previous works

of

Oshima et al.

[17]

with the rotating shell model and Yoshida et

al.

[26] with the

IBFA

model can only obtain nearly Aat curves. IV. SUMMARY AND DISCUSSION

In summary, we have investigated the negative-parity states

of

the odd-mass ytterbium nuclei in the framework

of

the extended interacting boson-fermion model. These nuclei are described by coupling an odd fermion to the even-even core. The core is described by the

IBA

Hamil-tonian with the fermion pair degrees

of

freedom being

taken into account. That is, one

of

the bosons is allowed

to

break into a fermion pair which can occupy the i&3/2

single-fermion orbit. The odd quasinucleon is allowed to occupy the 2p3&2 or If7/2 single-particle orbitals. The parameters contained in the core Hamiltonian are chosen to reproduce the positive-energy spectra

of

the corre-sponding even-even Yb isotopes. %'ith this set

of

param-eters, the strengths

of

odd fermion single-particle ener-gies and the parameters contained in the fermion Hamil-tonian are then chosen to reproduce the energy spectra

of

the even-odd Yb isotopes. The calculated energy levels for the even-even nuclei and even-odd Ybnuclei are all in reasonable agreement with the observed values. Al-though the observed transition quadrupole moments in the odd-mass Yb region are not abundant, our calcula-tion yields good agreement with the observed values. We also analyzed the intensities

of

the wave functions corre-sponding to each single-nucleon orbital for each state.

It

is found that the intensities

of

the wave functions for states

of

' Yband ' Ybisotopes are rather dispersive.

This work is supported by the National Science Coun-cil

of ROC

under the Grant No. NSC80-0208-M009-13.

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