ON THE STRUCTURE AND DEFORMATION OF YRAST STATES IN146Gd, 147Gd, AND 152Dy+

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ON THE STRUCTURE AND DEFORMATION OF

YRAST STATES IN146Gd, 147Gd, AND 152Dy+

T. Døssing, K. Neergård, H. Sagawa

To cite this version:

1 4 7 152 ON T H E STRUCTURE AND DEFORMATION OF YRAST STATES I N 1 4 6 ~ d ~ Gdt AND Dy+

**

T. Dsssing, K. ~ e e r ~ ~ r d * and H. Sagawa

.Nordita, BZegdamsvej 17, DK-2100 Kabenhavn 0, Denmark.

Justus-Liebig-Universitat, I n s t i t u t fiir Theoretische Physik I, Heinrich-Buff-Ring 16, **D-6400 Giessen, West Germany.

Kmbenhavns U n i v e r s i t e t , NieZs Bohr I n s t i t u t e t , BZegdamsvej 17, DK-2100 kbbenhuvn 0, Denmark.

Abstract.- The change of shape along the yrast line observed i n l Q 7 G d i s explained in terms of a de- formed independent particle model. Results of calculations for 1 4 6 ~ d and 1 5 2 ~ y are also presented. 1 . I n t r o d u c t i o n smooth i n d e p e n d e n t p a r t i c l e energy of t h e

ground s t a t e c o n f i g u r a t i o n , and ELD i s t h e I n a r e c e n t e x p e r i m e n t 1 ,

the

Sfuadru-

p o l e moment of y r a s t i s o m e r s i n l 4 Gd was l i q u i d d r o p e n e r g y w i t h t h e p a r a m e t e r s of found t o i n c r e a s e s t r o n g l y w i t h t h e a n g u l a r W e r s and s w i a t e c k i 7 . (The p a i r i n g c o n t r i - momentum. I t w i l l b e s e e n t h a t t h i s b u t i o n t o t h e smooth p a r t of ENpBCS can b e phenomenon h a s a s i m p l e e x p l a n a t i o n w i t h i n _{since i t is a constant.)} a deformed i n d e p e n d e n t p a r t i c l e model. The

i n v e s t i g a t i o n of such a model h a s p r e v i o u s - For each c o n f i g u r a t i o n , E i s minimized l y 2 y i e l d e d new i n s i g h t i n t o t h e n a t u r e of w i t h r e s ~ e c t t o t h e common s h a p e o f t h e t h e y r a s t s p e c t r a of n u c l e i i n t h e v i c i - s i n g l e p a r t i c l e p o t e n t i a l and t h e l i q u i d n i t y of 2 0 8 ~ b . d r o p . We c o n s i d e r p u r e l y e l l i p s o i d a l s h a p e s , I n a d d i t i o n t o t h e d i s c u s s i o n of which we d e s c r i b e by t h e p a r a m e t e r l Q 7 ~ d , we p r e s e n t i n t h i s t a l k a l s o t h e r e s u l t s o f c a l c u l a t i o n s o f t h e y r a s t s p e c t r a o f l Q 6 ~ d and 1 5 2 ~ y . ( 3 ) 2. Model S t r u t i n s k y c a l c u l a t i o n s i n t h e f u l l 6-y p l a n e performed w i t h b o t h a Woods- s a x o n 3 and a N i l s s o n 4 p o t e n t i a l i e l d f o r

B

t h e n u c l e i i n t h e v i c i n i t y of l 4 Gd equi- l i b r i u m s h a p e s i n t h e s p i n r a n g e I

2

50 t h a t a r e a x i a l l y symmetric w i t h t h e symme- t r y a x i s a l i g n e d w i t h t h e a n g u l a r momentum v e c t o r . A c c o r d i n g l y , we c o n s i d e r t h e s t a - tionary configurations of independent nucleons

i n a n a x i a l l y symmetric p o t e n t i a l w e l l . The e n e r g y E o f a g i v e n c o n f i g u r a t i o n i s c a l c u l a t e d from t h e e x p r e s s i o n Here, ENpBCS d e n o t e s t h e e x p e c t a t i o n v a l u e of t h e i n d e p e n d e n t p a r t i c l e p l u s p a i r i n g f o r c e Hamiltonian, where q d e n o t e s t h e r a t i o o f t h e d i s t a n c e between t h e p o l e s t o t h e d i a m e t e r of t h e e q u a t o r . A c r u c i a l p o i n t of o u r model a s com- p a r e d t o some r e l a t e d i s t h e u s e of a s i n g l e p a r t i c l e Hamiltonian h which, f o r B = 0

,

r e p r o d u c e s t h e e m p i r i - c a l s i n g l e p a r t i c l e e n e r g i e s i n l 6 ~ d . The Hamiltonian i s written in the form below. We notice that t h i s general expression includes the Nilsson Hamiltonian a s a particular case.

where o 41 2 t ~ N-Z ( , , = - ( I - - 1 A) MeV

,

A' i n t h e c o r r e s p o n d i n g BCS m u l t i - q u a s i p a r - t i c l e s t a t e p r o j e c t e d t o t h e p r o p e r n u c l e o n

numbers. The chemical p o t e n t i a l s A t of t h e

₌

₊

i

P.

,

z

BCS wave f u n c t i o n a r e d e t e r m i n e d s o a s t o

J2Mw,

y i e l d t h e r i g h t n u c l e o n numbers on a v e r a g e

b e f o r e t h e p r o j e c t i o n , and t h e p r o j e c t e d and t h e q u a s i a n g u l a r momentum quantum num- energy i s minimized w i t h r e s p e c t t o t h e g a p b e r s

gt

and jt a r e d e f i n e d i n t e r m s o f p a r a m e t e r s

% .

The p a i r i n g f o r c e c o n s t a n t s ~ i l s s o n ' g s t r e t c h e d o r b i t a l a n g u l a r momentum Gt- a r e g e n e r a t e d a c c o r d i n g t o t h e o p e r a t o r s 0

L '7,

,averaqe gap r e c i p e w i t h 6 , h = 1 4 / 6 MeV.

4:

I i ( b p z

-

b2hy)

e t c . ( 6 )

The second t e r m i n ( l ) , E, i s S t r u t i n s k y ' s

JOURNAL DE PHYSIQUE

& t . t The s p h e r i c a l e n e r g i e s cs h(Nshr , 3 )

a r e determined i n our mode? from em- p i r i c a l d a t a i n a way d e s c r i b e d i n de- t a i l i n s e c t . 3 . For a g i v e n c o n f i g u r a t i o n , we c a l c u - l a t e t h e a n g u l a r momentum I from t h e e x p r e s s i o n where mi i s t h e s i n g l e p a r t i c l e magnetic quantum number. T h i s c o r r e s p o n d s t o a n i n - t e r p r e t a t i o n bf t h e model s t a t e s a s s t a t e s w i t h qood a n g u l a r momentum and maximal magnetic quantum number. I n conse,quence of t h i s i n t e r p r e t a t i o n , a p a r t of t h e model s p a c e i s s p u r i o u s . For s m a l l d e f o r m a t i o n s , t h e s p u r i o u s c o n f i g u - r a t i o n s can i n p r a c t i c e be i d e n t i f i e d by t h e i r quantum numbers. (See t h e more de- t a i l e d d i s c u s s i o n i n r e f . 2.) A t l a r g e r de- f o r m a t i o n s , o u r model s t a t e s can be viewed a s band heads of r o t a t i o n a l band.s i n t h e s t r o n g c o u p l i n g ~ i c t u r e l l , s o t h a t t h e d i s - c u s s i o n of s p u r i o u s s t a t e s r e d u c e s t o t h e f a m i l i a r and y e t unsolved problem of spu- r i o s i t i e s i n t h i s p i c t u r e . 3 . S p h e r i c a l s i n g l e p a r t i c l e e n e r q i e s I n t h e p r o t o n s h e l l NSh = 4

,

we u s e - - - t h e 1 4 6 ~ d l e v e l s o f Waroquier and ~ e y d e l ~ . We n o t i c e t h a t t h e s e l e v e l s have a s e p a r a - t i o n between d and hll,2 e q u a l t o . 5/2 2 . 4 MeV. T h e n e u t r o n l e v e l s i n t h e Nsh = 4 and 5 s h e l l s a r e o b t a i n e d by a n e x t r a - p o l a t i o n of t h e 1 3 t l 4 ( p , d ) and15

(dip)

c e n t r o i d s from t h e even N = 82 n u c l e i w i t h Z = 56-62. I n d e t e r m i n i n g t h e r e l a t i v e p o s i t i o n of t h e p a r t i c l e and h o l e s t a t e s , we make u s e o f t h e v a l u e 1 6 sn(14!jGd)

-

S n ( l k 7 ~ d ) = 3.7 MeV

.

F o r a l l l e v e l s n o t mentioned above, w e t a k e c s p h ( N s h , ~ t , jt) f r m a Woods-Saxon c a l c u - l a t i o n , . Also,' t h e c e n t r o i d of t h e e m p i r i c a l l e v e l s i s f i x e d a t t h e c o r r e s p o n d i n g Woods- Saxon v a l u e . F i n a l l y , s i n c e , e m p i r i c a l l y , t h e s e p a r a t i o n between t h e n e u t r o n NSh = 4 and 5 s h e l l s i s c o n s i d e r a b l y s m a l l e r t h a n i n t h e Woods-Saxon.calculation, it i s ne- c e s s a r y , i n o r d e r t o m a i n t a i n a p l a t e a u i n t h e S t r u t i n s k y c a l c u l a t i o n , t o s h i f t t h e n e u t r o n s h e l l s w i t h Nsh $ 4 o r 5 2.2 MeV towards t h e N = 82 gap. W e have checked t h a t t h e s i n g l e p a r - t i c l e model t h u s o b t a i n e d y i e l d s t h e same g r o s s dependence of t h e s h e l l c o r r e c t i o n energy on t h e nucleon numbers and deforma- t i o n a s t h e Woods-Saxon p o t e n t i a l .

I n a n e a r l y c a l c u l a t i o n , s l i g h t l y d i f f e r e n t p a r a m e t e r s were used, T h e r e f o r e , t h e r e s u l t s g i v e n below show some minor d e v i a t i o n s from p r e v i o u s l y r e p o r t e d ones171 1 8 . 4 . Y r a s t s p e c t r a . o f 146Gd, l 4 7 ~ d and 5 2 ~ y F l g . ? shows t h e c a l c u l a t e d y r a s t l$nes of 1 4 6 ~ d , 147Gd and 1 5 2 ~ y . I n a l l t h r e e cases, t h e energy i n c r e a s e s s t r o n g l y i n t h e b e g i n n i n g a s a f u n c t i o n of I ( I + l ) . T h i s

i s

a r e s u l t of a break-down o f t h e ground s t a t e p a i r c o r r e l a t i o n s d u e t o t h e experiment +

+

r u calculation I 1 t t

J

0 5

10

15 2 0 25 30 35 40

I

F i g . 1. E x c i t a t i o n e n e r g i e s and d e f o r - m a t i o n s of y r a s t s t a t e s i n 1 4 6 ~ d , 1 4 7 ~ d ,

t i o n s ,

dei (:

F )

g=o

6 % -

-

(81

Indeed, configurations w i t h l a r g e I a r e ob- t a i n e d w i t h t h e s m a l l e s t expense of energy by i n v o l v i n g o r b i t a l s w i t h l a r g e magnetic quantum numbers mi. S i n c e , i n t h e s e o r b i - t a l s , t h e v a l u e o f dci/dg i s s t r o n g l y p o s i - t i v e , t h e d e f o r m a t i o n i s d r i v e n t o i n c r e a - s i n g l y n e g a t i v e v a l u e s when t h e s p i n , and t h e r e f o r e t h e number o f s u c h o r b i t a l s i n t h e c o n f i g u r a t i o n , i n c r e a s e s . The i n c r e a s e o f t h e number o f l a r g e - m o r b i t a l s w i t h i n c r e a s i n g 1, i s d i s p l a y e d i i n t h e c o n f i g u r a t i o n s of t r a p s l i s t e d i n t a b l e 1. L i k e i n r e f . 2, we d e f i n e a t r a p a s a c o n f i g u r a t i o n which c a n n o t , due t o t h e e n e r g i e s , decay by a s i n g l e p a r t i c l e t r a n - s i t i o n w i t h A I 2 2

.

i n a d i r e c t manner i n f i g . 2. The s l o p e o f t h e h i g h e r p a r t of t h e y r a s t l i n e of lS2Dy i s s e e n t o b e w e l l r e - produced i n t h e c a l c u l a t i o n . We n o t i c e t h a t t h e s e s t a t e s have B

2

-0.15. Also t h e measured1 2 0 r e l a t i v e p a r i t y o f - a l l s t a t e s w i t h I 2 18 i s r e p r o d u c e d . For t h e r e l a t i v e p a r i t y o f t h e I = 17 and 18 s t a t e s , t h e c a l c u l a t i o n i s i n v a r i a n c e w i t h t h e e x p e r i m e n t a l a s s i g n m e n t . The low- e r p a r t of t h e spectrum h a s i n t h e e x p e r i - ment a c o l l e c t i v e s t r u c t u r e which i s be- yond t h e scope of o u r model.

A s t o t h e a b s o l u t e e n e r g i e s i n t h e h i g h e r p a r t of t h e lS2Dy spectrum, t h e r e i s a 1-2 MeV d i s c r e p a n c y between c a l c u l a - t i o n and e x p e r i m e n t . T h i s d i s c r e p a n c y c o u l d be somewhat reduced by assuming a s t r o n g e r p a i r i n g f o r c e , which would make t h e ground s t a t e more bound r e l a t i v e t o t h e e x c i t e d s t a t e s .

I n 1 4 6 ~ d and 1 4 7 ~ d , whose low-ener- gy s p e c t r a have a l e s s c o l l e c t i v e c h a r a c - ter t h a n t h a t o f l S 2 ~ y , t h e s t a t e s up t o r e s p e c t i v e l y I = 20 and I = 27/2 were T a b l e 1 Asymptotic c o n f i g u r a t i o n s of c a l c u l a t e d t r a p s w i t h I 2 10 Nucleus 1" C o n f i g u r a t i o n The f a c t t h a t l a r e r d e f o r m a t i o n s a r e

S

r e a c h e d i n 1 4 6 t l 4 Gd t h a n i n lS2Dy i s a r e s u l t of n e u t r o n h o l e s b e i n g i n - v o l v e d i n t h e y r a s t c o n f i q u r a - t i o n s w i t h I & 20

.

Indeed, d u e t o t h e l a r g e c o n t r i b u t i o n t o ( d 2 ~ / d B 2 ) _B=o from t h e o r b i t a l d3/2,m=1/2 , ' t h e denominator i n (8) i s d i m i n i s h e d when a h o l e i s made i n t h i s o r b i t a l . T h i s i s r e f l e c t e d a l s o i n t h e v i o l e n t i n c r e a s e o f

I

6

l

i n 6 t 7 ~ d a t I = 2 0 and I = 41/2, r e s p e c t i v e l y . i n t e r p r e t e d 2 1 i n t e r m s o f t h e s p h e r i c a l s h e l l model. Our c o n f i g u r a t i o n s f o r t h e s e s t a t e s a r e i n agreement w i t h t h e r e s u l t o f t h i s a n a l y s i s . 5

.

E l e c t r o m a g n e t i c moments i n l 4 7 ~ d T a b l e 2 shows t h e c a l c u l a t e d and em-

JOURNAL DE PHYSIQUE

Table 2

Electromagnetic moments of observed isomers in 1 4 7 ~ d

Calculation Experiment 1,25 t

1/2

I Q I

9 (ns) (eb)

Fig. 2. Excitation energies of con- figurations in 14'~d plotted as func- tion of the deformation. The minima are indicated by .arrows. Different signatures describe the neutron con- figuration:

-

single particle,

- -

2-particle-l-hole, - *

-

3

-

particle-2;hole etc.

up the contributions of the individual occu- pied orbitals. For the effective charges and single particle gyromagnetic ratios we

-

used the values eeff

-

efree

,

-

-

-

gt,eff gt,free

'

and gs,eff

-

O s 6 gs,free'

For the quadrupole moment, the value, QiSp. thus obtained was renormalized ac-

cording to the formula

M

where Q is Strutinsky's smooth quadrupole moment5, and QLD is the quadrupole moment of a homogeneously charged ellipsoid with the ratio of axes q and the radial para- meter ro = 1.12 fm corresponding to the density p = 0.17fK3. This Strutinsky re- normalization was found, however, to be insig- nificant within the experimental accuracy.

While g is sensitive only to the con- figuration, Q is seen to be in all cases roughly equal to QLD and thus essentially a measure of the deformation B

.

For both quantities calculation and experiment are seen to be in good mutual agreement.

+ Work supported in part by BMFT, Bonn, GSI, Darmstadt, and SNF, Copenhagen.

1

0 . Hzusser et al., Phys. Rev. Lett.

44,

132

(1980)

2

K. Matsuyanagi et al., Nucl. Phys.

W ,

253 (1978)

3 T. Dassing et al., Phys. Rev. Lett.

2,

1395 (1977)

4

C. G. Andersson et al., Nucl. Phys.

m,

141 (1978)

5

M. Brack et al., Rev. Mod. Phys.

44,

320 (L9721

6 ~ . Fkauendorf, Nucl. Phys.

E ,

150 (1976)

-

7 ~ . D. MyerS and W. J. Swiatecki, Aik. Fys.

36, 343 (1967)

-

8

G. Andersson et al., Nucl. Phys. A=, 205 (1976)

9

M. Cerkaski et al., Nucl. Phys.

W ,

269 (1979)

'OS. G. Nilsson, Mat. Fys. Medd.

3,

no 16 (1955)

'

'A. Bohr and B. R. Mottelson, Nuclear Strpc- ture, vol. 2 (Advanced Book Program, W.A. Benjamin, Reading 1975)

1 4

R. K. Jolly and E. Kashy, Phys. Rev.

E,

1398 (1971)

5 ~ Booth et al., Nucl. Phys. .

W ,

301 (1975)'

1 6 ~ . H. Wapstra and K. Bos, Atomic Data

19, 177 (1977)

17

K. Neergsrd, T. DGssing and K. Matsuyanagi, Proc. Symp. High Spin Phenomena in Nuclei, Argonne 1979

1 8 ~ . Dossing et al., contribution to this

conference

19

B. Haas et al., Phys. Lett.

E,

178 (1979)

L I

P. Kleinheinz et al., 2 . Phys.

e,

279 (1979)

*'R. Broda et al., 2. Phys.

E ,

423 (1978)

2 3 ~ . Pedersen et al., priv. com.

2 4

T. Khoo et al., Phys. Rev. Lett.

41,

1027 (1978)

2 5 0 . Hsusser et al., Phys. Rev. Lett.

9,

1451 (1979)