STRUCTURE OF EVEN-EVEN DY NUCLEI IN THE INTERACTING BOSON MODEL WITH 2-QUASIPARTICLE STATES

Structure

of

even-even

Dy nuclei

in

the

interacting

boson

model with two-quasipartic$e

states

D.

S.

Chuu

Department ofElectrophysics, National Chiao Tu-ng University, Hsinchu, Taiwan, Repubiic

of

China

S.

T.

Hsieh

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

(Received 25January 1988)

Theenergy levels ofg.s.bands, _Pbands, and _y bands of

"

' Dy isotopes are studied in the

mod-elofthe traditional interacting boson approximation, allowing one boson tobreak and form a quasi-particle pair. The two quasiquasi-particles are allowed to excite to the ii3/p and h»&2 orbitals. Itwas

found that the energy levels ofthe g.s.,P,and y bands ofeven-even Dy nuclei can be reproduced

very well. The backbends ofthe moment ofinertia can also be reasonably described. The yrast

B

(E2)values are also calculated and compared with the experimental data.

I.

INTRODUCTION

The interacting boson approximation (IBA)model' has been remarkably successful ih the description

of

the low-lying collective states in many medium to heavy even-even nuclei. Recently, alarge amount

of

high-spin states

of

nuclei has been accumulated. Among these data some backbending occurs as one plots the moment

of

inertia versus the square

of

the angular velocity for the yrast band

of

a nucleus. Many efforts ' within the frame-work

of IBA

have been attempted to understand the mechanism

of

the sudden change

of

the moment

of

iner-tia.

It

is believed that the backbending phenomenon is caused by the crossing

of

the ground-state band and a two-quasiparticle band. ' Yoshida and co-workers '

ex-tended the n-p

IBA

(IBA-2) to allow one

of

the bosons to

be changed into apair

of

nucleons and applied this model

to study the Ge isotopes. Alonso et

al.

followed the work

of

Yoshida and applied this model to reproduce the backbending phenomena

of

Dy isotopes. Since the num-ber

of

basic states

of

IBA-2 is tremendously large as the proton number goes away from the closed shell, one therefore needs to employ some kind

of

truncation on the basic states. In order to make the problem manageable, Alonso et

al.

used the weak-coupling technique in their calculation, although their calculation yielded

satisfacto-ry results for the ground-state band, the abundant experi-mental data

of P

and y bands are still impossible to

de-scribe. Recently, Harter et

al.

' investigated the rela-tionship between the IBA-1 and IBA-2 and concluded

that for

Ntr+Nv»

~Ntr Nv ~ the

IBA-l

is

—

_avalid ap-proximation. This, in fact, has been reflected in the suc-cess

of

the semimicroscopic model

of

Morrison et

al.

"

in a boson basis based on the philosophy

of

the

IBF

mod-el.' Hence it should be valuable and interesting to study in more detail whether the structure

of

the whole energy spectrum including g.

s.

,13,and y bands and the observed

backbending phenomena

of

the deformed nuclei can be described in terms

of

all

of

the whole basic states

of

the

IBA-1

plus a two-quasiparticle pair. In this work we shall illustrate the model by taking Dy isotopes as a

test-ing example.

In Sec.

II

we describe the model. In

Sec.

III

we present the results. The Anal section will give the summary and the discussion.

II.

THEMODEL

The Dy isotopes we are interested in have

Z

=66,

and

N lies between 88 and

96.

Thus valence nucleons are in the 50—82 major shell. Taking

Z

=50

and N

=82

as the

core, the traditional

IBA

assumes the valence boson num-bers

11,

12, 13,14, and 15for ' ' Dy isotopes.

It

is as-sumed that one boson could break toform a quasiparticle pair, usually assigned

to

a unique parity intruder orbital with spin

j.

In the region

of

well-deformed nuclei the unique parity intruder orbitals such as h»&2 and

i,

3/2 are

the most important because both the Coriolis antipairing effect and the rotation alignment effect increase with in-creasing angular momentum. In our model the two quasiparticles are allowed to excite to these two orbits. The angular momentum

of

the nucleon pair takes the values 4,6,

.

. .

,

2j

—

1 and is coupled to the rotation

of

the core. The couplings to angular momentum

0

and 2 are excluded in order to avoid the double counting

of

states, because they are included through the s and d

bo-sons, respectively.

Our model space includes the

IBA

space with Nbosons and states with N

—

1 bosons plus two nucleons. The

model Hamiltonian is

H

=H~+HF+

V~F ~

where Hz isthe

IBA

boson Hamiltonian

Htt=

aoed+p

ap+a2L

L+a3Q

Q .

The octopole 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 is

38 STRUCTURE OFEVEN-EVEN Dy NUCLEI IN

THE.

. .

961

with a being the nucleon creation operator. The mixing Hamiltonian VzF isassumed, l aF

=Q'

Q Q—

'

Q'

where

E(Mev)

o ~=ist a

I=16

t x 13/2 ~ _{hl l/2} and Qi'

—

(dtg+gtd

)i '

_(dtd

_)' ' 2

Q=Qs+a(ata

_{J J})

'+p[(a

_{J J}

"at)'

'd

—

dt(a

_Ja_J)'

']'

'

For

the radial dependence

of

the fermion potential, the Yukawa type with Rosenfeld mixture is used and an

os-cillator constant

v=0.

96A ' fm with A

=160

as-sumed. The interaction strength is adjusted so that the

J

=0

state is lower than the

J

=2

state by 2 MeV. The

single-particle energy is obtained as a result

of

fitting.

The other parameters contained in the boson Hamiltoni-an Hz and VzF were chosen to reproduce the energy-level spectra

of

even Dy isotopes. In the calculation the

interaction parameters contained in the boson part for each nuclei are kept

to

be the same values for either the

N boson configuration or the

X

—

1 boson _plus

two-quasiparticle configuration. The interaction strengths and the single-particle energies for each isotope are al-lowed to be mass number dependent.

I I I t I

154 156 I&8 160 I62

FIG.

1. Variation of calculated single-particle energies of

e(h,

»,

)and e(i,3/2) orbits compared with the variation ofthe

energy values of

I

=

16and

I

=

18states for the Dy isotopes.

III.

RESULTS

Table

I

lists the searched interaction strengths and single-particle energies for all isotopes. One can see that

the strength

of

the dboson ao and the mixing parameters

a

and

_p

can be unified for the whole string

of

even-even

Dy isotopes with a mass number between 154 and 162. The magnitudes

of

the pairing term

P

P,

the quadrupole term Q Q, and the term

L L

increase as mass number A

increases, while the single quasinucleon energies E(h»~2)

and e(ii3&2)decrease from nucleus ' Dy tonucleus ' Dy

and then increase for nucleus ' Dy. The variation for

the single-particle energies

of

the quasinucleons coincides approximately with the variational trend

of

the

I

=16

and

I

=

18states through the whole isotope string, as can be seen in

Fig. 1.

The calculated and observed energy spectra for the string

of

Dy isotopes are shown in Figs. 2—

6.

There have been abundant experimental data' ' observed in recent years. The different quasibands are displayed in different columns for clear comparison. The energy levels marked

with asterisks are not included in the fitting.

It

can be seen from the figures that the energy levels can be repro-duced quite well, especially the g.

s.

bands. The

calculat-ed states

of

quasibands are all in correct order and agree reasonably with the observed data, except for very few levels. The relative intensities

of

wave functions for each energy level, corresponding to the N boson and (N

—

1

)-boson plus two

h»

&2 or i&3/2 quasiparticle configurations,

are shown in Tables

II

and

III.

The total intensity

of

the N-boson configuration, the (N

—

1)-boson plus two h»&2 quasinucleons configurations, and the (N

—

l)-boson plus two i&3/2 configuration for each state is normalized to

1.00.

One can see the lower-lying levels

of

the

_p

and the

y bands, and the yrast levels with angular momentum up

to 14 are dominated by the pure boson configuration.

The (N

—

1)-boson plus two h_„&2 quasiparticle configuration becomes dominant from

I

=16,

states for g.

s.

bands. The (N

—

1)-boson plus two

i,

3/2 quasiparticle

configuration becomes more important for the higher-spin states. Therefore,

if

we increase the _h»/2

quasiparti-TABLE

I.

The interaction parameters inMev forIBA-1plus two-quasiparticle configuration model. Parameter (Mev) nuclei 154D 156D 158D 160Dy 162D ao 0.58 0.58 0.58 0.58 0.58 al

—

0.002 0.028 0.048 0.066 0.066 ap 0.0022 0.0032 0.0037 0.0045 0.0045 a3

—

0.009

—

0.009

—

0.009

—

0.008

—

0.005 0.035 0.035 0.035 0.035 0.035 0.025 0.025 0.025 0.025 0.025 611/2 1.16 1.05 0.905 0.865 1.0 613/2 1.505 1.32 1.275 1.25 1.3

E(Me%') E(MeV) +32

7

l 26 156

p

154

p

t g30

6

—24' 28 22' 26 ₂₀

4

24 16 22' 14 +6 20 12t 8 7r 4 8' 6t 2T 0

—

gl8 + 3₂₊ 10 18 Q P 16 8' 6t t 2t4 16 14 14

—

12

Expt. Theo. Expt. Theo. Expt. Theo. Expt. Theo.

10 _f7t 3t4 8 _g.S.

P

_Vl

_Y.

8 6 ₊ t2 gO 4 2' 0

6

_FIG.

4. Calculated and observed energy spectra for

"

Dy.

2'

Ot

1

Expt. Theo. Expt. Theo. Expt. Theo.

g.

s.

p,

Expt.Theo. Expt, Theo.

P,

Y

FIG.

2. Calculated and observed energy spectra for ' Dy.

E(MeV) 156

_p

630 y28 E(MeV)

7

—f26 18 16

_0p

24 16 22 '12 II '1O' 20 14t 20 10 18 18 6t gf 9 8 6 12 16 16 8 14 12t 10 «12 1 1t 10 gt 7t 6+—

4—

3t —14 ~t 3 2'—

.

10' Io 12 8t 6 2' 8 10 ₈ 6f B _4t ~2 ~Q 6 t2r 0

—

-- -. --4 2t 0

0

Expt. Theo.

Expt. Theo. Expt. Theo.

pi

Expt. Theo.

Exp% Theo. Exp t. Theo. Expt. Theo, Expt. Theo,

s

band

g.

s.

g.S.

FIG.

5. Calculated and observed energy spectra for '~Dy.

FIG.

3.Calculated and observed energy spectra for

"

Dy.

cle orbit in energy so that it becomes effectively ir-relevant, then the agreements between the calculated and the observed high spin levels will become worse.

Espe-cially, the good coincidence

of

theoretical and experimen-tal energy levels around the first backbends

of

the mo-ments

of

inertia will get worse. Since Dy isotopes are not

in the region

of

U(5) symmetry, the level spacings are al-most equal. Although the almost equal spacings between the adjacent energy levels around

I

=

14in the string

of

Dy isotopes might be obtained by either increasing the value

of

the parameter

_P

or lowering the

single-38 STRUCTURE OFEVEN-EVEN Dy NUCLEI IN

THE.

. .

963

E(Mev)

l62

D

l4

t

l4 l2 l2 l0 4t

,

2'

0—

l0

8'

0 IO 8 6 St 4 2' 0

Expt. Theo. Expt. Theo. Expt. Theo. Expt, Theo. Expt. Theo.

P~

P2

P~

FIG.6. Calculated and observed energy spectra for ' Dy.

quasiparticle orbit i&3/2 both approaches will certainly

make the agreements between the observed and the

calcu-lated higher-spin levels become worse. This shows the

statistical significance

of

the single-quasiparticle energies e(A ii/2 ) and

e(i,

3/2) listed in the Table

I.

The apparent

variation for the intensities

of

different configurations with the angular momenta shows that two bands

of

different deformation cross between

I

=14

and

I

=

18,

and a rotation aligned band originating from the _h»/2

and _i~3/p nucleon quasiparticle states stems from the

I

=

14state.

The backbendings in the Dy isotopes are commonly in-terpreted as the transition from the ground-state

rota-tional band to the aligned two-quasiparticle

i,

3/2 nucleon band.' Figure 7 shows the results

of

our calculation.

Here we choose the most sensitive expression to plot the conventional

28/R

versus (fico) curves, with

2

4I

—

2

El+a

—

E

and

It

can be seen from

Fig.

7 the agreements between the calculated and the observed curves are very satisfactory.

We also plot the backbending curve for the

_P

band

of

the

Dy nucleus. One can notice that the main feature

of

the observed data can be well reproduced. From the

cal-culated wave-function intensities, as shown in the Tables

II

and

III,

one notices that the configurations, including two _h»/2 quasiparticles and two

i,

3/2 quasiparticles, are

competitive at the point

of

the backbending, being

of

two nucleons h_&&/2 lower in energy.

There are abundant experimental

8

(E2)

values for Dy

isotopes. '

'

' _{The study}

_of

_{these values will give us a}

good test

of

the model wave functions. The electric

quadrupole operator iswritten as

T(E2)=e Q+e

a(a,

a,

)'

+Pe

[(atat}'

'd

—

dt(a

a }'

']'

'

J

j

J J

where Qistaken as

(fico)

=

E~+2

—

_EI

TABLE

II.

The relative intensities ofwave functions forenergy 1evels ofisotopes

"

Dy,

"

Dy, and

'"Dy.

State 0] 2] 4] 6] 8] 10] 12] 14] 16, 18] 20] 22] 24] 26] 02 22 42 62 82 03 23 3] 43 5] 63 7] Nucleus nfiguration 1.00 0.99 0.96 0.93 0.90 0.87 0.84 0.80 0.00 0.00 0.00 0.00 0.00 0.00 0.97 0.98 0.95 0.93 0.88 0.95 0.96 0.97 0.94 0.94 0.01 0.91 154D 2 h»/2 0.00 0.01 0.03 0.05 0.08 0.10 0.12 0.16 0.98 0.97 0.96 0.88 0.04 0.04 0.02 0.02 0.03 0.06 0.10 0.03 0.03 0.02 0.04 0.04 0.91 0.07 2 0.00 0.00 0.01 0.02 0.02 0.03 0.04 0.04 0.02 0.03 0.04 0.12 0.96 0.96 0.01 0.00 0.02 0.01 0.02 0.02 0.01 0.01 0.02 0.02 0.08 0.02 0.97 0.95 0.92 0.89 0.85 0.81 0.77 0.73 0.00 0.00 0.00 0.00 0.00 0.00 0.95 0.96 0.93 0.90 0.83 0.94 0.93 0.94 0.90 0.91 0.02 0.87 156Dy 2 h»/z 0.02 0.03 0.05 0.08 0.11 0.14 0.17 0.21 0.95 0.87 0.06 0.01 0.00 0.00 0.03 0.03 0.05 0.08 0.13 0.04 0.05 0.04 0.07 0.07 0.77 0.10 .2 t]3/Z 0.01 0.02 0.03 0.03 0.04 0.05 0.06 0.06 0.05 0.13 0.94 0.99 1.00 1.00 0.02 0.01 0.02 0.02 0.04

0.

02 0.02 0.02 0.03 0.02 0.21 0.03 0.91 0.90 0.87 0.83 0.79 0.74 0.69 0.63 0.00 0.00 0.00 0.00 0.00 0.00 0.92 0.92 0.88 0.83 0.32 0.91

0.

89 0.89 0.86 0.85 0.04 0.79 158Dy 2 h»/2 0.06 0.07 0.09 0.12 0.16 0.20 0.25 0.31 0.98 0.98 0.98 0.97 0.67 0.01 0.05 0.06 0.09 0.14 0.60 0.07 0.08 0.08 0.11 0.11 0.78 0.17 ~₂ 13/2 0.03 0.03 0.04 0.05 0.05 0.06 0.06 0.06 0.02 0.02 0.02 0.03 0.33 0.99 0.03 0.02 0.03 0.03 0.08 0.02 0.03 0.03 0.03 0.04 0.18 0.04

TABLE

III.

The relative intensities ofwave functions forenergy levels ofisotopes ' Dy and ' Dy. Nucleus configuration 160Dy 2 h /2_13/2 162Dy 2 h /2_13/2 0] 2] 4] 6] 8] 10] 12] 14] 16] 18] Op 22 4q 6~ 82 102 23 3] 43 5] 63 7] 83 9] 103 0.87 0.86 0.84 0.80 0.76 0.71 0.66 0.60 0.00 0.00 0.90 0.87 0.85 0.80 0.55 0.03 0.87 0.86 0.83 0.82 0.04 0.77 0.24 0.56 0.05 0.08 0.09 0.12 0.15 0.19 0.23 0.28 0.34 0.98 0.98 0.07 0.10 0.11 0.16 0.39 0.90 0.10 0.10 0.13 0.14 0.78 0.19 0.63 0.40 0.90 0.05 0.05 0.04 0.05 0.05 0.06 0.06 0.06 0.02 0.02 0.03 0.03 0.04 0.04 0.06 0.07 0.03 0.04 0.04 0.04 0.18 0.04 0.13 0.04 0.50 0.88 0.86 0.84 0.81 0.77 0.73 0.70 0.66 0.62 0.00 0.91 0.89 0.86 0.83 0.78 0.74 0.88 0.87 0.88 0.84 0.83 0.80 0.02 0.76 0.03 0.08 0.09 0.10 0.13 0.16 0.19 0.22 0.25 0.29 0.96

0.

06 0.07 0.09 0.12 0.16 0.20 0.08 0.09 0.08 0.11 0.12 0.14 0.78 0.18 0.85 0.04 0.05 0.06 0.06 0.07 0.08 0.08 0.09 0.09 0.04 0.03 0.04 0.05 0.05 0.06 0.06 0.04 0.04 0.04 0.05 0.05 0.06 0.20 0.06 0.12

38 STRUCTURE OFEVEN-EVEN DyNUCLEI IN

THE.

.

.

965

For

the fermion effective charge

e,

an average value

0.

37

of

the proton and neutron obtained by Alonso et a/. is assumed. The boson effective charge in the

T(E2)

opera-tor has been determined by normalizing the largest

calcu-lated

B(E2)

value for each nucleus tothe corresponding observed data. The parameters

a

and

_P

are assumed to

be the same values as used in the mixing Hamiltonian.

The value

of

a is chosen tobe

—

&7/2

which is the

gen-erator

of

the SU(3)group. Figure 8shows the calculated and observed

B(E2;I~I

—

2) versus the spins

of

the depopulating states. From the figure, it can be seen that

important features

of

the

B (E2)

values are reproduced well. Especially, the decreasing feature at

I

=16

and the increasing feature at

I

=

18

of

nucleus

'

Dy are obtained nicely. However, the interpretation

of

the decreasing feature at the

I

=6

state

of

' Dy isdifficult in our model, since the excitation energy

of

the first

6+

state level is only 770

KeV.

Therefore the mixing

of

the two-quasiparticle configuration is very small and there is no contribution to improve the agreement.

For

the other

nuclei, the

B

(E2;I

~I

—

2)

values for the yrast states are in reasonable agreement with the observed ones.

29~%

(MeV)

80-154py

Yrast

15epy

-

_l40

~

_Expt.

o

_Theo.

~

Expt.

o

_Theo.

0,03

0.

09

O.i5 0,2I

0.

06

O.l2 O,IB P.O —o,oz

0.09

O.t5 f56

Yrast

~

_Expt.

'

_Theo.

0

2I 160

_p

Yrast

0.

03

0.06

_I

0.

09

_I ~

Expt.

'Theo.

0.

~2

!00

-'80

l56

p -band

~

Expt.

o

_Theo.

0.

06

O,l2 I l I

f'

'(M

V')

0.

03

0-06

0-09

4'w'

(MeV')

eo.

olz

-70

I

ISED Y Yrast leoD Yrast CU I I56D Yrast l62D Yrast I i l i I i L i I —

0

lO 20 I58D Yrast ----Theo. Expt. t L a I i I IO 20 30 ANGULAR MOMENTUM

I

IV. SUMMARY AND DISCUSSION

FIG.

8. Calculated and observed

B(E2;I~I

—

2)values for the yrast band vsthe spins ofthe depopulating states.

162. We extend the

IBA-1

model

to

allow a boson to break

to

form a quasiparticle pair which can occupy the

h

_„/2

and

_i,

_3/2 orbitals. The calculated energy levels, in-cluding the ground-state, p,and _y bands are in

satisfacto-ry agreement with the observed values for the whole string

of

Dy isotopes. Backbendings

of

the moment

of

in-ertia

of

the yrast states can be reproduced reasonably.

We also plot the backbending curve

of

the moment

of

in-ertia

of

the

_p

band for ' Dy. The observed data are able

to beexplained. We also calculated

B

(E2)

values for Dy isotopes. Our model yields satisfactory agreement with the observed data.

The effects

of

the two-quasiparticle configuration are manifested in the improvement

of

the energy-level

calcu-lation and the fine variations in the calculated

B(E2)

values. The couplings to angular momenta

J=4,

6, 8,

.

. .

, for the quasiparticles in _h»/2 and

i,

3/2

orbitals might be considered as implicitly including the higher angular momentum _{bosons, such as the g boson} and the

I

boson,

etc.

, and therefore could make the

IBA-1 model space more complete. This is also manifested in

the analysis

of

the wave functions. The high-spin states are usually dominated by the N

—

1 boson plus two-quasiparticle configurations and thus cannot be

repro-duced by the traditional

IBA

model.

Recently, very high spin states up to

I

=40

and a dou-ble backbending have been observed in some nuclei in the rare-earth region. ' These phenomena might hopefully be interpreted by considering two or more bosons

to

break to form more quasiparticle pairs and make more band crossings

to

form the double backbending. In

con-clusion, our calculation suggests afeasible model that can

be extended very easily to handle the recently observed very-high-spin states and the double backbending phe-nomena in some rare-earth nuclei.

In summary we have investigated the structure

of

the energy spectra and the backbending phenomena

of

the isotope string

of

Dy with amass number between 154and

This work is supported by the National Science

Coun-cil

of

the Republic

of

China under Grant No. NSC77-0208-M009-09.

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F.

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