PHYSICAL REVIEW C VOLUME 42, NUMBER 1 _{JULY} 1990

### Effective boson

### number

### calculations near the

### Z

### =64

### subshell

### C.

### S.

Han and### D.

### S.

ChuuDepartment ofElectrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic

### of

China### S.

### T.

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

### of

China (Received 21 November 19&9)The effect ofthe partial subshell closure near Z

### =64

is studied byintroducing the effective bosonnumber in the framework ofthe interacting boson model. The energy spectra and the

### 8

(E2)valuesofthe Sm, Gd, and Dy isotope series are calculated. It isfound that the agreements between the theoretical results and the observed data arevery satisfactory when the partial closure effect is taken into account byasmooth variation ofthe effective proton-boson numbers.

### I.

INTRODUCTIONIn recent years there are considerable interests in

studying the subshell closure effects at

### Z

### =64.

### '

### It

has been shown '_{that}

_{the}

_{Z}

_{=64}

_{subshell}

_{had significant}effect for N

### ~

88 transition nuclei, but disappears asN ap-proaches### 90

due to the increasing importance### of

the n-pinteractions. This subshell closure also introduces significant effects on the calculations using the interacting boson model

### (IBA).

In### IBA,

the number### of

active bosonsis usually determined by counting particles above, or below the nearest spherical-shell closure. Therefore, the counting

### of

the boson numbers becomes ambiguous whensome subshell closure exists. Wolfet

### al.

have calculated the magnetic moment### of

the first### 2+

states### of

the transi-tional nuclei### Ba,

Nd, Sm, and Gd in the framework### of

### IBA-2,

and Gill et### al.

have performed an### IBA

calcula-tion near### Z

### =64

subshell. They made the assumption### of

a drastic change in the proton-boson numbers at N

### =

88, namely, the assumption### of

a### Z

### =64

shell for N 88and a### Z

### =50

shell for N### ~90.

### It

is interesting### to

note that for those nuclei with N 88 the counting schemes from the### Z

### =64

subshell yielded better agreement with the experi-mental data. Scholten has proposed a method to calcu-late the number### of

"effective bosons" in a microscopic model.### It

is interesting to see that when the number### of

effective bosons is calculated, it does yield a minimumvalue at

### Z

### =64.

However, the minimum value is about### 2.

4 instead### of

### 0

as required by a full closure structure. This means that there is considerable washing out### of

the shell closure effect at### Z

### =64.

A similar result has also been obtained by Maino and Venture using the Nilsson model on the basis### of BCS

approximation. Federman and Pittel, Federman et### al.

, and Chen et### al.

studiedI

the shape transition

### of

Zr and Mo isotopes in the frame-work### of

a shell model.### It

was found that the isospin### T

### =0

component### of

the n-p interaction is responsible for the onset### of

deformation.### It

was suggested that the pro-motion### of

the neutron into the h9/2 orbit near### N=90

leads to a concurrent polarization

### of

protons into theh»&2 orbit, via the strong npinter-action between parti-cles in spin-orbit partners orbits, and results in an eradi-cation

### of

the subshell_{gap.}However, it is well known that the shell-model single particle level spacings are dependent on the model space. Therefore, the assump-tion

### of

sudden disappearance### of

the### Z

### =64

subshell at N### =88

made by Wolf et### al.

and Gill et### al.

seems### to

be oversimplified in the sense that itdoes not take the effects### of

partial closure### of

### Z

### =64

and the smooth neutron num-ber dependence into account. In order to investigate these partial subshell closure effects, some preliminary studies'### '"

on Sm and### Er

isotopes with### N=86

### —96

have been done.### It

is found that satisfactory results can be achieved only when smooth variation### of

the proton-boson numbers forN&_{88}

_{are considered.}

In this paper, we present a systematic study on the
148—_{158Sm} 1so—_{1586d, and} 152—_{1MDy}

isotopes

the effects

### of

partial closure and the smooth neutron dependence by considering the effective proton-boson numbers in### IBA.

Aunified Hamiltonian and an### E2

tran-sition operator for each isotope series are used in our cal-culation.### II.

MODELIn the calculation

### of

energy levels, the most general Hamiltonian with nine parameters### of IBA-1

was used### H~=e,

(s### s)+ed(dt

### d)+

### g

—,### '&(2L+1)CLX[[d

### Xd ]

### 'X[d Xd]' 'j'

'L=0,2, 4

### +(

—### ')'

v### [[d

### Xd

### ]'

### 'X[d

### Xs]'

### '+[d

### Xs

]'### 'X[d Xd]' 'j'

'### +(2)

### 0'0[[d

### Xd

### ]'

### 'X[s

### Xs]'

### '+[s

### Xs ]'

### 'X[d Xd]' 'j'

'### +u

_{[[d}

### Xs ]'

### 'X[d Xs]' 'j'

### '+

—,### 'u

### I[s

### Xst]'

### 'X[s

### Xs]'

### 'j'

' .### For

comparison, two calculations were performed. We first consider the conventional### IBA

(hereafter denoted as### MI).

Then the effective proton-boson numbers arein-cluded (hereafter denoted as MII) to investigate the par-tial subshell closure effects

### of

the### Z

### =64

and N### =82.

In both models, the number### of

neutron bosons N is count-ed as usual,### N„=

—,### '(N

### —

82) where N is the number### of

neutrons.

### For

the proton bosons, N is counted from### Z

### =50

closed shell in### MI.

However, in### MII,

we relax allthe shell closure restriction for the proton boson but maintain the only requirement that they must be integers. Therefore, we try different sets

### of

proton-boson numbers for the nuclei in each isotope series in our calculations. Once we chose a set### of

N for the isotope series, a least-squares search for the interaction parameters is then car-ried out in the framework### of

### IBA

to fit the experimental data### of

these nuclei.### It

was found that the best set forthe effective proton-boson numbers for Sm isotopes is N### =2

for ' Sm, N### =4

for ' Sm, and N### =6

for all other Sm isotopes. The best set### of

the effective proton-boson num-bers### of

the Gd isotopes is N### =2

for ' Gd, N### =5

forGd, and N

### =7

for all other Gd isotopes.### For

Dy iso-topes, the best set is### N„=5

for ' Dy, N### =7

for ' Dy, and N### =8

for all other Dy isotopes.### It

is worth noting that for each isotope series, both### MI

and### MII

count the same N for N### 90.

This is consistent with the disap-pearance### of

the### Z

### =64

subshell for N### &90

in this mass region as pointed out in previous works. '_{However, the}

linear variation

### of

N for N 88in### MII

for each isotope series manifests the effects### of

the partial closure and smooth neutron number dependence.In the calculations, 71 reliable energy levels in Sm, 94

levels in Gd, and 100levels in Dy isotopes were included in the least-squares fittings.

### It

is well known that not all the parameters in the Hamiltonian are linearly indepen-dent. Since we are concerned with excitation energies only, the effect_{of e,}

can be absorbed into ed. Also the
parameter uo is kept at zero because it can be absorbed
into other parameters. ' The resulting interaction
pa-rameters and the overall root-mean-square deviations for
Sm, Gd, and Dy isotopes are listed in Table ### I.

### It

isworth noting that unified interaction parameters can be found for each isotope series.### It

isalso seen from the table that the parameters change smoothly from Sm isotopes to Dy isotopes. ~'I+&I 0 ~~ cd ~I&+I V 'a V U' cd V 6 0 0 U V bQ CJ V V 0 V 'a cd Q 0 ~1++I### 4

0 cd CA C@ V C 0 ~& 0 cd### 40%

_{OO}

### 0

O O### %04

OO Ch DO### 000

### 0

O### 0

l OO OO### ~

M CO### ooo ooo

I I I I I I O O### 000

### 000

### 0 0

ev### ooo

### mWt

_{Q}Q

### 000

I I I I I I### o0OO0OO

_{ooo ooo}

### 4N~

### 0oo

I I I### 0 0

### 000

OO cv### ooo

m t_{0 0}

### 000

I I I I I I 40 OO M OO o### ooo ooo

I I I I I I### III.

RESULTS OO DO### 0

H### 0

O Q If) OO OO### 0 0 0

The ground-band levels### of

the Sm, Gd, and Dyiso-topes are shown in

### Fig. 1.

We can see that the energy values for each### J

states change almost linearly for N 88 and become quite flat for### X

### 90.

This is consistent with the onset### of

the### Z

### =64

subshell effect. Furthermore, the linear falling### of

the energy values for each### J

state as N goes from 86to 88 seems tojustify the linear variation### of

the effective proton-boson number### N„

found in### MII

ineach isotope series.

The partial closure effect for

### Z

### =64

subshell can be in-vestigated by plotting the effective proton-boson numbers versus the number### of

protons for each mass number A as### 000

282 C.S.HAN, D. S.CHUU, AND S.

### T.

HSIEH 42 3.4 10 Gd Dy### 0.

### 2'

~ 22 20-W 1S### ~

1.6### X

UJ 1.4~ 1.2~ 1.0 O. S-0.6 OA~ 0.2 I 86 S8 90 92 94 96 N i 86 88 90 92 94 86 88 QO 92 94 96### FIG.

1.The general trend ofthe experimental ground-band level energies ofthe Sm,Gd,and Dy isotopes.shown in

### Fig. 2. For

A### =150,

there is a clear minimumN

### =2

occurring at### Z

### =64.

The nonzero value### of

the minimum indicates that the closure### of

the### Z

### =64

sub-shell is not complete but only partial. This result is con-sistent with that obtained_{by}Scholten as shown in the dashed line

### of

the figure. However, in Scholten's result, there is no calculation to study the way the subshell clo-sure will be decreasing as the neutron number### N,

ap-proaches### 90.

We have also studied this tendency### of

wash-ing out the subshell closure at

### Z

### =64

as N### =90.

This can be clearly seen from the figure that the partial closure effect### of Z

### =64

starts to decrease as A goes from 152to 154and is completely eliminated at A### =156,

where the curve becomes a straight line indicating the spherical clo-sure### of

### Z

### =50

for N### ~90.

The gradual decreasing### of

the partial closure effect in the mass region A### =150

### —

156 manifests the smooth neutron number dependence due to the increasing importance### of

the n-p interaction.The calculated energy spectra

### of

Sm, Gd, and Dy iso-topes compared with the experimental values are shown in Figs. 3—8. In these figures different quasibands are separated in order to have a clear comparison. One can see that the traditional### IBA

(in the column### MB

cannot reproduce well the energy spectra### of

these isotopes. The calculated ground state energies in### MI

in general have much lower values compared### to

the observed data espe-cially for some higher spin states, except for ' Dy and Dy where the calculated values are higher than theex-8~ A=156 A=154 6 ~ - A=152 4 ~ A=150 2 62 64 66

### FIG.

2. The effective proton-boson numbers versus thenum-ber ofprotons for each mass number A. Thy dashed line is

148 Sm 8 4

### —

3 2 0 0 Ol 4J### ~

2 8### —

6### —

3 g 4 LaJ### —

2### —

10### —

4### —

2 8### —

2 4### —

4 4 «3 m2 8 «4 0 3### —

12### —

14' 10 ~### —

12### —

### 8~

### —

5 0### —

p' g 0 r Exp### —

0 9 g r Ml MII 0### —

2### —

0 g Q r_{g}

_{Q}Exp N

### —

2 MII 0" g 0 Exp g 0 r_{g}

_{0}MI MII

### FIG.

3. Experimental and calculated energy spectra for ' 'Sm,### "

Sm, and### '"Sm.

31' 5'S 3 )56' 158 2~ 2

### —

12### —

3 mo### ~

### —

2'### —

12 1p ~ 2' 4### —

12 mo### —

8### —

fp 2'### —

0'### —

2 1~ 8 B 8### —

2. p 0 g 0 r g g r g 0 Exp Ml MII 9 0 r_{9}

_{0}g 8 Exp 0~

### —

4 0 9 Exp Ml MII### FIG.

4. Experimental and calculated energy spectra for### "

Sm, ' Sm, and### "

Sm.152 3' M2

### —

8 W6 4~### —

12 H2 4-3### —

16 MS M6### —

2### —

_{2}-0 ol.

### ~

Exp 4### —

2 4 —2### —

p' 9 0 MI pL g A MII -6 —4### —

3z2. Lal### —

2 4### —

2 2 2### —

2 g 8 r Exp —B 4 —10 —_{6.}-6 -2 g f3 r Ml 4 4 4 —2 0 —4 0 —2 0' g A MII 9 0 Exp wp l2 —6 3

### —

2' 0 g A Ml284 C. S.HAN,

### D.

S.CHUU, AND S.### T.

HSIEH 42 15 3r P### ~4

WO mp### ~2

7 2_{—}

12 ### —

12### —

### e

0### ~o

### —

5' i### —

6### —

10### «3

### —

8### —

2. 1-### —

8 m2'### —

2## ~

~### —

8### —

4### —

_{—}

2
0
### —

5 t### —

5 ~ 4 4 4### ~

### —

2### —

8' 9 g r Exp### —

4 4 2~ 9 8 MI### ~

4 E'xp MII### FIG.

6. Experimental and calculated energy spectra for### "

Gd and### '"Gd.

perimental data. The energy spacings calculated in

### MI

are also in considerable_{disagreement}with the observed data in the

_{P}

band and the _{y}band. Some states are even in reversed order in

_{the y band}

### of

' Dy. Thus the results manifest that the pure### IBA

model is unable to simulate the energy level structure forthe nuclei near### Z

### =64.

The reason forthese discrepancies isthat in the calculation### of

the nuclei near### Z

### =64

with 86 N 96,the introduction### of

the neutron-proton interaction will becomeincreasing-ly important when there are more neutrons outside the N

### =82

shell, and thus tend to eliminate the### Z

### =64

sub-shell closure as mentioned above. Hence, it is reasonable to expect that the calculation which includes the proper number### of

proton bosons will yield better results for the energy level fittings. This is indeed true as shown in thefigures (under the column

### MII)

where the effective156_{D}
&52
+r

### —

12'### ~p

3I M2 3### —

16 W4 M2' M2 5 4 0 LLI MQ### —

8 2 0' g 9 9 Ex& gI MII 4 4### —

7### —

2### —

4### ~+

pi (f 9 Q r G(p### —

s### —

### 7

4### —

2### —

_{0}9 Q r MII Qi Mo'

### —

6 2### -6

### ~4

WO### ~0

~&### —

8 ~Q### —

0' 4### —

_{0}9 8 MII

### FIG.

7. Experimental and calculated energy spectra for### "

Dy,### "

Dy, and ' Dy.1'58 Oy 6-160 5

### —

22### —

20### —

22 4### —

20### —

20 4r m8 4-_{~6}

3 ### —

16### —

16 w6'### x3'

YLJ### —

14 2'_{—}

8### —

8' r 4 7.### 10~ ~o

### ~4

### ~

### —

4 Qe 9 Q r Exp### —

&6### —

14### —

8### —

10### —

8### ~r

r### —

4 o Ml### —

16### —

14 r### —

4 8 r ~### ~

0 9 Q r MII### —

14### )

&2### —

12 hl r### —

10### ~t

4### —

6 9 Q r### —

14### —

12### —

5' 0### ~4

9 g r h4### —

14### —

12### —

10### —

2'. 9 A r MII### FIG.

8. Experimental and calculated energy spectra for### '"Dy

and ' Dy.TABLE

### II.

### 8(E2)

values (in e b )and branching ratios for Sm isotopes. TheoryThis work Previous works Nucleus Sm

### '"Sm

152S J,### ~Jf

### 4)~2)

### 2(~0)

4i### —

+2) 10)### ~8)

4)### ~22

2,### ~0,

### 23~0

22 +2]### 23~2]

23### —

+41 3)### ~2)

/4) 3)### ~2~/2,

### 2)~0)

4i### —

+2i### 6)~4)

### 8)~6)

### 10)~8)

12'### —

+_{10)}

### 23~0)

### 23~2]

### 23~4)

### 23~2)

/0, Expt. 0.151 0.25' 0.274### 053

0435 0447 0.0106b 0.0036 0.0088b 0.27 0.0387 0.0174 0.296b 24+5 0.67 1.017b 1.2' 139' 1 103 1.123" 0.0163d 0.041 7 0.00416 2.44' MI 0.225 0.399 0.355 0.708 0.934 0.70 0.08 0.09 0.0001 0.14 0.005 0.036 0.165 8.54 0.41 0.68 0.88 0.971 0.907 0.826 0.007 0.011 0.002 1.57 MII 0.131 0.20 0.318 0.566 0.609 0.512 0.05 0.021 0.0035 0.11 0.079 0.025 0.342 20.62 0.861 1.197 1.256 1.220 1.131 1.001 0.0102 0.0267 0.0019 2.618 0.275 0.51b 0 139 0.02b 0.02 0.181b 0.024### 109

### 434

0.673### 098

0.41' 0.73' 0.18' 0.008' 0.015' 0.12' 0.029' 0.54' 16.8" 0.75~ 1.0g 0.97g 0.83g286 C. S.HAN,

### D.

S.CHUU, AND S.### T.

HSIEH 42TABLE

### II.

(Continued).Nucleus

_{J;}

### —

_{+Jf}

Expt.
This work
MI ### MII

Theory Previous works 154sm 156Sm 158S### 23~21/4

31 21/41 43~21/41 22~02/21 62~41/61 6~~4~/61 82~61/81 8P~62/81 51 41/61### 21~01

41### —

+21 101~81 121~101 23### —

+01 23 +21### 23~4

### 23~21/0

31~21/41 43—+21/41### 21~0

11.9' 0 95' 0.088' &860' 0.2' 0.08' 54' &0.06' &58' 0.43' O.922' 1.186d 1497 1.538 1.565 0.013 O.O2' 0.0008 1.56" 2.S' o.### oss'

1.22" 1.28" 5.50 0.428 0.0001 50.8 0.081 0.308 128.5 3.59 173.2 0.102 0.657 0.96 1.01 1.05 0.964 0.002 0.012 0.0018 6.0 0.29 0.0013 0.784 0.647 14.28 1.0 0.054 273.7 0.256 0.658 330.8 4.55 175 0.391 0.978 1.364 1.416 1.333 1.212 0.0068 0.018 0.0012 2.65 1.0 0.111 1.089 1.205 'Reference 17. Reference 18. 'Reference 19. Reference 20. 'Reference 21. 'Reference 22. ~Reference 15. "Reference 23.TABLE

### III. B(E2)

values (ine b )forGd isotopes. Theory Nucleus### J;~Jf

Expt. This work MI### MII

Previous work 1526d 1546d### 21~01

41—+21### 61~41

41### —

+2q 22~21 22~01 2p~02 21### —

+01 41 41 81### —

+61 101### ~81

02~21 22~01 2P~21 2P~41 42~21 42 62~41 6q### —

+61 0.33' 0.64' 0.95' 0.096' 0.077' 0.0014' 0.21' 0.773' 1.178' 139' 1.526' 173' 0.258' 0.004 0.033' 0.09' 0003' 0.071' 0.0027'### 0.

033' 0.506 0.803 0.68 0.017### 0.

201### 0.

049 0.085 0.545 1.07 1.16 1.17 1.11 0.03 0.027 0.018### 0.

005 0.009 0.001 7 0.0013 0.009 0.463 0.77 0.84 0.036 0.116 0.027 0.263 0.954 1.132 1.385 1.345 1.235### 0.

02### 0.

017### 0.

017 0.021 0.027 0.018### 0.

047### 0.

019### 033

O.62Ob 0.76 O.O6b o.164b O.O26b 0773' 1.098' 119' 1.20' 1.18' 0.136'### 0.

0168 0.025' 0.087' 0.016' 0.089' 0 013' 0.008 5' 0.33' 0.69' 0.79 O.O11d 0.006' 0.009' 0.87' 1.18' 1.17'Table

### III.

(Continued). Nucleus_{J;~Jg}

Expt.
This work
MI MII
Theory
Previous work
156g,_{d}

### 23~0]

23 +2]### 23~4]

### 31~21

3]### ~4]

### 43~2]

43 4] 43 5]### ~4]

2]### ~0]

4] 4] 10]### ~8]

12,—+10,### 02~2]

22—+0]### 22~2]

### 22~4]

42—+2]### 42~4]

### 42~6]

### 23~2]

### 23~4]

3]### ~2]

### 3]~4]

### 43~2

43### ~4]

5]### ~4]

5]### ~6]

0.00459 0.0 0.001 39 0.0103 0.01 0.001 4 0.01 0.0043 0.0074 0.914' 1.299' 1.470' 1.57' 159' 1.45' 0.029' 0.00316 0.016' 0.018' 0.### 0061'

0.014' 0.009 1' 0.035 5' 0.003 2' 0.0364' 0.028' 0.007 8' 0.046' 0.029 5' 0.041' 0.005 0.071 0.0043 0.011 0.055 0.0013 0.061 0.0039 0.0011 0.86 1.12 1.29 1.28 1.23 1.22 0.197 0.001 0.0028 0.007 9 0.011 0.003 0.012 0.026 0.0014 0.0028 0.005 0.0025 0.0021 0.0001 0.003 0.0067 0.027 0.0012 0.0082 0.017 0.0048 0.028 0.0036 0.0034 1.04 1.45 1.53 1.50 1.42 1.26 0.124 0.002 3 0.015 0.043 0.045 0.035 0.048 0.023 0.0038 0.008 0.015 0.002 6 0,005 0.0006 0.014 0.0052 0.01' 0.00038 0.0176 0.01 0.0019 0.01 0.00081 0.01 0914' 1.286' 1.38' 1.38' 134' 1.26' 0.025' 0.004 0.006' 0.015' 0.0047' 0.004 8' 0.014 8' 0 0333' 0.0023' 0 037' 0019' 0.### 011'

0 039' 0.0289' 0.025' 'Reference 24. bReference 25. 'Reference 26 Reference 15 'Reference 27### 2]~0]

### 23~0]

### 23~2]

### 23~4]

### 43~2]

43### —

+4] 43 61### 22~0

2p 2]### 22~4]

31 21 3] 4] 4, 2, 42 4]### 42~6]

5]### ~4]

5]### ~6]

43~23 42~22 5]### —

+3] 43~22### 43~3]

1.008' 0.0016' 0.001 2' 0.0071' 0.004' 0.0022' 0 0094' 0.016 9' 0.029' 0.0013' 0.0297' 0.0177' 0.006' 0.04' &0.01' 0.023 5' 0.0194' 137e 0.6' 1.01' 0 037' 0.14' 0.75 0.0005 0.0017 0.06 0.008 0.0082 0.0004 0.031 0.03 0.067 0.0017 0.0058 0.014 0.013 0.034 0.002 0.016 0.25 0.57 0.49 0.504 0.637 1.13 0.013 0.0002 0.018 0.007 0.029 0.0024 0.030 0.057 0.077 0.0012 0.006 0.037 0.018 0.07 Q.OQO8 0.014 1.3 0.48 0.66 0.35 0.67 1.008' 00019e 0.000 4' 0.0034' 0004 7' 0.001 5' 0.0056' 0.0236' 0 039' 00019' 0043e 0.021' 0.009 2' 0.04' 0.0028' Q035e 0.028' 1 13' 0.42' 0 70' 0.0207' 0.16'288 C. S.HAN,

### D.

S.CHUU, AND S.### T.

HSIEH 42TABLEIV.

### B(E)

values (in### e'b')

and branching ratios for Dy isotopes. TheoryThis work Previous work Nucleus 154D 156D 158Dy 'Reference 28. Reference 15. 'Reference 25. Reference 29. 'Reference 30.

### J,

### ~Jf

22~0l/2l 42~2~/4l 62 4z/6l### 2l~ol

42~22 6q### ~42

10q~82 12'### ~

102### 142~

122 42 4l### 42~2l

### 62~6l

### 62~4,

### 82~8(

### 102~10)

### 102~8(

### 122~

12l### 122~

10l### 142~

12( 2l### —

+Ol Expt. 0.015' 5.### 1+0.

5' 14+2' 3.79' 1.29 1.42### 149

1.53 1.S6' 1.58d 0.054 0.0024 0.078 O.OO14' o.os4d o.ooo6' 0.077" 0.001" &0.32' o.oo6d &0.0015d 093' 0.095 2.13 1.15 1.04 0.75 1.15 1.19 1.13 0.86 0.68 0.098 0.018 0.53 0.011 0.031 0.0029 0.014 0.0046 0.012 0.0017 0.0033 1.34 0.051 2.92 6.82 1.17 0.792 1.25 1.36 1.32 0.97 0.80 0.09 0.079 0.76 0.0045 0.057 0.0017 0.038 0.002 0.036 0.0053 0.0023 1.40 O.O1lb 1.11b 2.16b 1.O1' O.7S' 1.07 1.O1b 0.81 o.s6' O.27'proton-boson number is taken into account. The results

in

### MII

show that the calculated ground-band energy lev-els are in good agreement with the observed data. The energy spacings### of

the levels in_{p}

_{and y bands improved}

much better. The level ordering in the _{y} band

### of

' Dy is also well reproduced.### To

test the wave function, we also calculate the elec-tromagnetic transitions. The general form### of

the### E2

operator was used,

### T'

### '=a[(d

### Xs+s

### Xd)'

### '+P(dtXd)' ']

(2)The parameters

### a

and_{p}

were determined directly from
least-squares fitting ### to

the observed### 8

### (E2)

values. There are abundant observed### 8

### (E2)

data for the Sm, Gd, and Dy isotopes. We have calculated the### 8(E2)

both in### MI

and in### MII.

In the calculation### of

### MII,

itis interesting to note that a unified parameter### a=0.

165 and### p=

### —

### 0.

105 can be found to reproduce the### 8(E2)

values quite wellfor all the three isotope series. The resulting values

### of

### a

and

_{p}

obtained in our calculation agree approximately
with those obtained in previous works. ' ' A similar
calculation is also performed in ### MI

with unified pararne-ters### a=0.

135and### p=0.

### 115.

In general, the results in### MII

are much better than those in### MI.

Tables### II, III,

and IV show the### 8

### (E2)

values and some branchingra-tios. Some results

### of

the previous works are also included for comparison.### It

can be seen from the tables that our calculated results in### MII

are in better agreement with the observed values than those in### MI,

especially for the tran-sitions within the ground band.### For

the cross band tran-sitions the agreement is also more satisfactory for the### MII

calculation.IV. CONCLUSIONS

This paper presents a scheme tostudy the effect

### of

sub-shell closure at### Z

### =64

for the Sm, Gd, and Dy isotopes in the framework### of IBA.

Wefound that large discrepan-cies occur when### Z

### =

50and N### =

82 is treated as a closed shell in this mass region. But this discrepancy can be im-proved very much when the partial subshell closure effect at### Z

### =64

is taken into account. Therefore, in presenting a scheme for calculating nuclei near### Z

### =64,

it is neces-sary torecognize not only that there is a subshell closure at### Z

### =64,

but also that the neutron-proton interaction among nucleons will gradually wash out the subshell gapwhen there are more neutrons outside the N

### =82

shell. The existence### of

partial subshell closure causes ambigui-ties in counting boson numbers in the IBA-type calcula-tions. Lacking a precise microscopic counting scheme,effective boson numbers in the phenomenological calcula-tions.

### It

is found that the energy spectra and the### B(E2)

values### of

the Sm, Gd, and Dy isotopes can be well ex-plained when the partial subshell closure effect is taken into account by gradually increasing the effective proton bosons asN approaches### 90.

The results### of

our phenome-nological calculations indicate that the effective bosonap-proach in

### IBA

is a rather encouraging approach.### It

may be helpful### to

study the foundation### of

this approach in a more microscopic model, such as the shell model which can manifest the effects### of

the n-p interaction explicitly.This work was supported by the National Science Council

### of

Taiwan, Republic### of

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