QUASIPARTICLE SPECTRA ABOVE THE YRAST LINE, BAND CROSSINGS AND BACKBENDING PHENOMENA

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QUASIPARTICLE SPECTRA ABOVE THE YRAST LINE, BAND CROSSINGS AND BACKBENDING

PHENOMENA

R. Bengtsson

To cite this version:

R. Bengtsson. QUASIPARTICLE SPECTRA ABOVE THE YRAST LINE, BAND CROSSINGS

AND BACKBENDING PHENOMENA. Journal de Physique Colloques, 1980, 41 (C10), pp.C10-84-

C10-97. �10.1051/jphyscol:19801009�. �jpa-00220628�

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JOURNAL DE PHYSIQUE CoZZoque C I O , suppZJment au n012, Tome 41, de'cembre 1980, page C10-84

QUASIPARTICLE SPECTRA ABOVE THE YRAST LINE, BAND CROSSINGS AND BACKBENDING PHENOMENA

R. Bengtsson.

Centre d 'Etudes NucZdaires de GrenobZe, DRF/CPN, 85X, F-38041 GrenobZe cedes, France.

Abstract.- Properties of low-lying quasiparticle configurations are devided from experimental data and compared with the results of theoretical calculations. A special interest is focused on the defor- mation dependence of e.g. the moment of inertia and the aligned angular momentum.

1. Introduction

particle energies, can be plotted in the Ever since the Nilsson model was first

presented in 1955, it has been customary to display the eigenstates of the single- particle Hamiltonian as single-particle diagrams, in which the single-particle energies are plotted versus some deformation coordinate.

'

⁾These diagrams do not only show the single-particle level order, but give also very important information about the shell structureand its variation with the deformation.

In later applications of the Nilsson model to rotating nuclei an other kind of

single-particle diagrams came in use. 'The single-particle energies (in the rotating frame) are then plotted versus the rotational (cranking) frequency. ^{2 )}

Although the single-particle level diagrams correctly describe the level order near the Fermi surface at the ground state deformation, they cannot be used for a detailed specfroscopic comparison with experimental data. The reason is that the single-particle Hamiltonian does not in- clude the pair interaction. This interaction can however be included in a

straight forward way, leading to the 'HFB- Hamiltonian. Its eigenvalues, the quasi-

same way as the single-particle energies.

The most interesting comparisons with experiment can be made if the quasiparticle energies are plotted versus the rotational frequency like in fig. 1 . For a detailed interpretation of the quasi-

3 ) particle diagrams we refer to ref.

.

In this paper we shall discuss what kind of information we can extract from the quasiparticle energy diagrams. We shall restrict ourselves to well-deformed nuclei in the rare earth and actinide regions, and put a special emphasize on the importance of the systematic changes of the deformation that can be observed within these regions.

2. How to compare the quasiparticle energy diagrams with experimental data.

The quasiparticle energies are calculated as functions of the rotational frequency.

In order to make a direct comparison, the experimental data must be transformed to the rotational frame of reference and expressed as a function of the rotational frequency. This can be done by calculating the rotational frequency

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19801009

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0.20

1 5 8 ~ r

(neutrons),

A

= 6 . 3 8 h u o , E ~0.20, Eq =-0.015, A.0.1rh0,

I

-

1--->2

F i g . 1 Q u a s i p a r t i c l e e n e r g y diagram f o r 1 5 8 ~ r .

where E ( I + I ) and E(1-1) a r e two con-, s e c u t i v e s t a t e s ( o f t h e same s i g n a t u r e ) i n a r o t a t i o n a l band. Ix is t h e a n g u l a r momentum component on t h e r o t a t i o n a l a x i s . I t i s e s t i m a t e d a s

where K i s t h e a n g u l a r momentum of t h e band head. The e n e r g y i n t h e r o t a t i n g frame i s t h e n g i v e n by

E I E + E - I I ~ ( I )

T h i s a l l o w s u s t o d i s p l a y b o t h Ix and E ' a s f u n c t i o n s of w

,

s i n c e t h e f r e - quency c o r r e s p o n d i n g t o a g i v e n a n g u l a r momentum, I

,

can be c a l c u l a t e d from e q . ( l ) .

Both E ' and Ix a r e t o t a l q u a n t i t i e s and c a n n o t be compared t o t h e t h e o r e t i c a l q u a s i p a r t i c l e e n e r g i e s , which o n l y g i v e t h e e x c i t a t i o n e n e r g i e s . We must t h e r e f o r e f i n d a r e f e r e n c e c o n f i g u r a t i o n , r e l a t i v e t o which we c a n c a l c u l a t e e x p e r i m e n t a l e x c i t a t i o n e n e r g i e s . A s a r g u e d i n r e f . 3 t h e most c o n v e n i e n t c h o i c e o f r e f e r e n c e

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JOURNAL DE PHYSIQUE

NEUTRONS, h = 6.51

N 97

1

F i g . 2 T h e o r e t i c a l l y c a l c u l a t e d q u a s i - p a r t i c l e e n e r g i e s f o r n e u t r o n s i n 1 6 7 ~ b

.

c o n f i g u r a t i o n i s t h e g r o u n d c o n f i g u r a t i o n , which f o r even-even n u c l e i c o r r e s p o n d s t o t h e g r o u n d band and i f r e q u i r e d i t s s m o o t h l y e x t r a p o l a t e d p r o l o n g a t i o n . I n odd-A n u c l e i , where t h e ground band i s a l - q.p. e x c i t a t i o n , t h e r e f e r e n c e c o n f i g u r a t i o n h a s t o b e d e t e r m i n e d i n a n o t h e r way., e . g . a s a n a v e r a g e o f t h e

F i g . 3 N e u t r o n q u a s i p a r t i c l e e n e r g i e s i n 1 6 7 ~ b , c a l c u l a t e d from t h e ex- p e r i m e n t a l l y o b s e r v e d r o t a t i o n a l b a n d s .

where t h e i n d e x r d e n o t e s t h e r e f . con- f i g u r a t i o n , e ' ( ~ ) t h e e x c i t a t i o n e n e r g y i n t h e r o t a t i n g f r a m e and i ( w ) t h e a d d i - t i o n a l ( a l i g n e d ) a n g u l a r momentum o f t h e e x c i t e d c o n f i g u r a t i o n r e l a t i v e t o t h e r e f e r e n c e c o n f i g u r a t i o n . F o r t h e g r o u n d band o f a n even-even n u c l e u s b o t h e ' ( w ) and i ( w ) a r e by d e f i n i t i o n z e r o . F o r a l - q . p . e x c i t a t i o n i n a n odd A-nucleus e ' ( w ) c o r r e s p o n d s d i r e c t l y t o a q u a s i - p a r t i c l e l e v e l , w h i l e i ( w ) c o r r e s p o n d s t o t h e s l o p e o f t h e l e v e l ( w i t h r e v e r s e d s i g n ) . T h i s i s i l l u s t r a t e d i n f i g s . 2 and 3 i n which we compare t h e e x p e r i m e n t a l e x c i t a t i o n e n e r g i e s e l ( w ) f o r 6 7 ~ b ground c o n f i g u r a t i o n s o f t h e n e a r l y i n g w i t h t h e c o r r e s p o n d i n g t h e o r e t i c a l q u a s i - 'even-even n u c l e i . We may now d e f i n e p a r t i c l e e n e r g y d i a g r a m . F o r 2 - , 3-,

...

^{q . p .}

t h e q u a n t i t i e s e x c i t a t i o n s e ' ( w ) [ o r i ( w ) ] must b e

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compared t o t h e sum o f t h e e n e r g i e s [ o r s l o p e s ] of t h e t h e o r e t i c a l q u a s i p a r t i c l e l e v e l s f i l l e d by t h e e x c i t a t i o n .

D e t e r m i n a t i o n o f t h e r e f e r e n c e con- f i g u r a t i o n and c a l c u l a t i o n o f t h e a l i g n e d a n g u l a r momentum.

The c a l c u l a t i o n o f t h e e x p e r i m e n t a l e x c i - t a t i o n e n e r g i e s r e q u i r e s t h a t 1:(w) and Ek(w) c a n b e d e t e r m i n e d w i t h r e a s o n a b l e a c c u r a c y . F o r t h i s p u r p o s e one may a d o p t t h e v a r i a b l e moment o f i n e r t i a (VMI) model, a s s u m i n g t h a t t h e e f f e c t i v e moment

of i n e r t i a c a n b e w r i t t e n

where Jo and J, a r e two p a r a m e t e r s , which f o r even-even n u c l e i may b e d e t e r - mined f r o m t h e normal b a c k b e n d i n g p l o t ,

2

J e f f ( w

,

^bymaking a l i n e a r approxima- t i o n i n t h e low f r e q u e n c y r e g i o n p r e c e d - i n g t h e b a c k b e n d i n g f r e q u e n c y . F o r odd- A n u c l e i t h e p a r a m e t e r s c a n b e o b t a i n e d a s , an a v e r a g e v a l u e o f t h e p a r a m e t e r s f o r t h e n e a r l y i n g even-even n u c l e i . One t h e n g e t s

T h i s p r e s c r i p t i o n was u s e d i n r e f . 3 ) w i t h good r e s u l t s f o r n u c l e i i n t h e m i d d l e o f t h e r a r e e a r t h r e g i o n ( A = 160 - 1 7 0 ) . I t h a s a l s o b e e n u s e d f o r c o n s t r u c t i n g f i g . 3.

F i g . 4 The a n g u l a r momentum Ix as a f u n c t i o n of t h e r o t a t i o n a l f r e - quency f o r 6 2 ~ r ( y r a s t b a n d ) , 161

Er ( i l 3 / 2 b a n d ) and ^{( a} p a r t o f t h e a = 1 / 2 i 1 3 / 2 b a n d )

I n f i g . 4 we show I x ( w ) f o r t h e y r a s t c o n f i g u r a t i o n o f 6 2 ~ r

.

The b a c k b e n d i n g i s c l e a r l y v i s i b l e a t

M U

= 0.28 MeV

.

^{F o r}

low f r e q u e n c i e s t h e r e f e r e n c e i s e q u a l t o t h e y r a s t c o n f i g u r a t i o n . I n t h e back- b e n d i n g r e g i o n t h e y r a s t c o n f i g u r a t i o n c h a n g e s i t s c h a r a c t e r f r o m t h a t o f t h e g-band t o t h a t o f t h e S-band, c a r r y i n g a l a r g e a l i g n e d a n g u l a r momentum, i - 7 . 5

Hue.

We a l s o show

I ^X f o r t h e two s i g n a t u r e s o f t h e il 3 , 2 band i n ' 6 1 ~ r

,

c o r r e - s p o n d i n g t o a n e , x c i t a t i o n t o t h e l e v e l a

( a = 1 / 2 ) and b ( a = - 1 / 2 ) r e s p e c t i v e l y . If w e f o r a moment make t h e a p p r o x i m a t i o n t h a t t h e r e f e r e n c e c o n f i g u r a t i o n o f 6 1 ~ r i s t h e same a s t h e one o f 6 2 ~ r

,

we c a n d i r e c t l y d e t e r m i n e t h e a l i g n e d a n g u l a r momentum f o r t h e two s i g n a t u r e s o f t h e

band i n ' 6 1 ~ r ( i x ( 1 / 2 ) and i ( - 7 / 2 1 r e s p e c t i v e l y ) .

X

The S - c o n f i g u r a t i o n o f 1 6 2 ~ r i s a 2 q . p . c o n f i g u r a t i o n i n which t h e l e v e l s a and

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c10-88 JOURNAL DE PHYSIQUE

b a r e f i l l e d . I t s a l i g n e d a n g u l a r momentum i s t h e r e f o r e g i v e n by t h e sum o f t h e c o n t r i b u t i b n t o Ix coming from t h e s e two l e v e l s , and c a n be e s t i m a t e d from t h e I 6 ' E r d a t a t o b e i x ( l / 2 ) + i x ( - 1 / 2 1 , Adding a l s o 1: g i v e s t h e a n g u l a r momentum o f t h e S - c o n f i g u r a t i o n ( t h e d o t - d a s h e d l i n e ) . Although t h e s-band o f I 6 * ~ r i s o n l y known e x p e r i m e n t a l l y above t h e b a c k b e n d i n g , we s e e t h a t i t s a n g u l a r momentum c a n be s m o o t h l y matched t o - g e t h e r w i t h t h e o n e c o n s t r u c t e d f r o m t h e 1 6 1 ~ r - d a t a . T h i s i s one o f t h e most a p p e a l i n g f e a t u r e s o f t h e p u r e q u a s i p a r - t i c l e p i c t u r e , a s d i s p l a y e d by t h e e n e r g y d i a g r a m s : The p r o p e r t i e s o f 2 - , 3- e t c . q u a s i p a r t i c l e c o n f i g u r a t i o n s c a n b e d e t e r - mined by s i m p l y a d d i n g t h e i n d i v i d u a l

c o n t r i b u t i o n s o f t h e e x c i t e d q u a s i p a r - t i c l e s .

4. The l i g h t r a r e e a r t h r e g i o n

I f t h e r e f e r e n c e c o n f i g u r a t i o n i s d e t e r - mined f o r odd-N n u c l e i i n t h e l i g h t r a r e e a r t h r e g i o n , f o l l o w i n g t h e p r e s c r i p t i o n o u t l i n e d i n t h e p r e v i o u s s e c t i o n , one o b t a i n s more o r l e s s u n r e a s o n a b l e r e s u l t s . Thus e . g . t h e a l i g n e d a n g u l a r momentum, i

,

d e c r e a s e s s t r o n g l y when W i n c r e a s e s ,

X

w h i l e t h e q u a s i p a r t i c l e e n e r g y d i a g r a m s p r e d i c t a p r a c t i c a l l y c o n s t a n t v a l u e ( c f . f i g . l ) , a t l e a s t if t h e odd n e u t r o n i s i n a l e v e l o r i g i n a t i n g f r o m t h e K = 1 / 2 o r K = 8 / 2 members o f t h e i 1 3 / 2 s h e l l . One may t h e r e f o r e make t h e a s s u m p t i o n t h a t t h e a n g u l a r momentum o f t h e i 1 3 / 2 band c a n b e w r i t t e n

The c o r r e s p o n d i n g moment o f i n e r t i a becomes

I t g o e s t o i n f i n i t y when w g o e s t o zero, a p r o p e r t y which i s s e e n i n a l l e x p e r i - m e n t a l bands c a r r y i n g a f i n i t e amount o f a l i g n e d a n g u l a r momentum, as shown i n

4 ) r e f .

.

The p a r a m e t e r s J o , J l and ix c a n b e d e t e r m i n e d i n f i t t i n g t h e e x p e r i m e n t a l d a t a . One t h e n f i n d s v a l u e s o f i which

X

a g r e e w e l l w i t h t h e s l o p e s o f t h e t h e o r e - t i c a l q u a s i p a r t i c l e l e v e l s . The v a l u e s of Jo and J1 d e v i a t e , however, system- a t i c a l l y from t h e a v e r a g e v a l u e o f t h e s e p a r a m e t e r s c a l c u l a t e d from t h e n e a r l y i n g even-even n u c l e i . F o r odd-N n u c l e i i s a l w a y s l a r g e r and J l a l w a y s s m a l l e r t h a n e s t i m a t e d from t h e even-even i s o - t o p e s . T h i s i s i l l u s t r a t e d i n f i g . 5.

F i g . 5 The e f f e c t i v e moment o f i n e r t i a as a f u n c t i o n o f t h e s q u a r e d r o t a - t i o n a l f r e q u e n c y . C u r v e A i s c a l - c u l a t e d d i r e c t l y from e x p e r i m e n t a l d a t a and J and J a r e o b t a i n e d by f i t t i n g a 8 t r a i g h i l i n e . Curve B i s a s t r a i g h t l i n e g i v e n by Jo and J1 d e t e r m i n e d f o r 5 9 ~ r .

deff/h2 M

I I I

Average value for 15a Er ond Er A

1

^[&,-l92 M~V-lh2, h = 1 8 0 ~ e V - 3 h ' I

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F i g . 6 The a l i g n e d a n g u l a r momentum i f o r t h e y r a s t band o f some E r i s o t o p e s , For t h e

X

i n t e r p r e t a t i o n of t h e d i f f e r e n t backbendings o r upbendings s e e r e f . 3 ) . For t h e even-even i s o t o p e s i # 0 a t s m a l l f r e q u e n c i e s s i n c e J and J1 have been a d j u s t e d

X 0

t o g i v e a c o n s t a n t v a l u e of ix i n t h e i n t e r v a l between t h e two backbendings. The -3 4

J o - v a l u e s a r e g i v e n i n f i g . 7 . The J 1 - v a l u e s a r e ( i n u n i t s o f MeV

M

) :

1 5 6 ~ r , 1 2 0 ; 1 5 7 ~ r , 1 0 5 ; 1 5 8 ~ r , 9 7 . 5 ; 1 5 9 ~ r , 9 8 ; I 6 ' E r , 9 2 . 0 ; 1 6 3 ~ r , 8 8 .

F i t t i n g J o

,

J 1

,

and f o r odd-N n u c l e i ix

,

i s n e v e r a s t r a i g h t f o r w a r d , t a s k . AS d i s c u s s e d i n r e f . 3 , t h e v a l u e o f 1, may be s t r o n g l y p e r t u r b e d i n c a s e s where t h e i n t e r a c t i o n between t h e g- and t h e S-band i s - s t r o n g . I n odd-N n u c l e i t h i s i n t e r - a c t i o n i s n o t s e e n i n t h e i13,2-band ( c f . f i g . h ) which i n s t e a d i s a f f e c t e d by t h e i n t e r a c t i o n w i t h t h e l o w e s t 3 q . p . con-

f i g u r a t i o n a t a s l i g h t 1 , y h i g h e r f r e q u e n c y (Ha w 0 . 3 5 MeV, s e e f i g . 6 ) . It must a l s o be t a k e n i n t o a c c o u n t t h a t h i g h e r up i n the i 1 3 / 2 s h e l l , t h e l e v e l s a l i g n g r a d u a l l y , and t h e a s s u m p t i o n o f a con- s t a n t v a l u e o f ix becomes q u e s t i o n a b l e . When c o n s t r u c t i n g f i g . 6 w e have t h e r e - f o r e made s m a l l a d j u s t m e n t s of J and J

0 1

t o compensate f o r t h e above e f f e c t s .

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C10-90 JOURNAL DE PHYSIQUE

the levels a and b in fig. 1

.

^{For the}

level a (a = 1/21 i = 5.7M. This is

X

equal to the ayerage value of ix for 1 5 7 ~ r and 1 5 9 ~ r

,

calculated from the data shown in fig. 6.

In fig. 6 we have included the yrast bands

I

I I I I

The values of Jo used for constructing fig. 6 are shown in fig. 7, where also the Jo-values determined for the g-band of the even-even Er-isotopes are shown.

..It. is clearly seen that Jo is larger for the odd-N' isotopes and that the dif- of

'

^{5 6 ~ r}and 1 5 8 ~ r (from refs. 576). The

( M ~ v - ' )

ference increases for smaller N-values.

It should also be observed that the values fitted for the S-band of the even-even isotopes irgremuch better with those of the odd-N isotopes (the i l a f 2 band with

a . = 1/21 than with those of the g-band in the even-even isotopes.

Er

5. The deformation dependence of the moment of inertia

parameters Jo and J 1 ,have been fitted

30 -

for 0.25

5

Mu 5 0.40 (solid lines) where these 'bands have the character of the

20

- -

S-band, and their aligned angular momentum is 10.3

M

and 9.7

M ,

respectively.

The theoretical value for 1 5 8 ~ r is 9 . 9 s 10 ^, which is the sum of the contributions from

he,

^sjrstematic difference in the 'reference moment of inertia between odd-N and even-

Fig. 7 Values of

Jo

for some Er isotopes.

The filled circles indicate the g-band and the open circles the S-band of the even-N isotopes.

The crosses indicate the a = 1/2 branch of the i1 /2 band.

very time consuming calculations. One may.

however get some ideas about the importance of deformation,changes f ~ o m the following simple-minded considerations.

The ground state deformations (at w = 0 ) can easily be calculated and are shown in fig. 8. For N 5 95 we see a systematic displacement of the deformations for the odd-N nuclei compared to what one would expect from the deformation of the even- even nuclei. We do not know whether the same tendency holds also at higher rotational frequencies, but we shall for a

%moment assume that this, is the c,ase, and investigate if deformation changes of this order have a significant influence on the even nuclei, may depend on several factors, moment of inertia. This can be done like

'like

changes in the pair correlations or in fig. 9, where 30 is plotted as a

i f i the deformation caused by the presence function of the deformation parameters

of the odd particle. To make a detailed E and

,

for a number of Er, Dy and theoret i o a l 'inbestigation of this requires Gd isotopes. We see that there is a very

,

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Fig. 8 ~ h e o r e t i c a l ground s t a t e deforma- t i o n s c a l c u l a t e d w i t h t h e m d i f i e d o s c i l l a t o r p o t e n t i a l ( r e f . l 3 ) f o r E r and Dy i s o t o p e s . For odd-N

i s o t o p e s t h e K-value i s g i v e n i n p a r e n t h e s i s i f t h e odd p a r t i c l e i s i n t h e il 3,2 s h e l l .

s t r o n g c o r r e l a t i o n between t h e t h e o r e t i c a l ground s t a t e d e f o r m a t i o n and t h e e x p e r i - mental v a l u e of Jo

.

We a l s o s e e t h a t t h e d i f f e r e n c e i n d e f o r m a t i o n between t h e odd-N and even-even n u c l e i i s l a r g e enough t o g i v e s i g n i f i c a n t changes i n J o ( o f t h e o r d e r 1 0 - 2 0 % i n s e v e r a l . c a s e s ) , and t h a t Jo i s j u s t a s w e l l c o r r e l a t e d w i t h t h e t h e o r e t i c a l d e f o r m a t i o n f o r odd-N a s f o r even-even n u c l e i .

The s i g n i f i c a n c e o f t h e s e o b s e r v a t i o n s can o n l y be confirmed by more e l a b o r a t e d

c a l c u l a t i o n s , b u t t h e y show a t l e a s t t h a t it i s i m p o r t a n t t o t a k e i n t o account a l s o s m a l l d e f o r m a t i o n changes, which e a s i l y can modify t h e r e f e r e n c e c o n f i g u r a t i o n s u f f i c i e n t l y much t o make t h e experimen- t a l l y d e r i v e d r e l a t i v e q u a n t i t i e s e t ( w ) and ix(w) i r r e l e v a n t f o r comparison w i t h t h e t h e o r e t i c a l q u a s i p a r t i c l e l e v e l s . The p a r a m e t e r J 1 depends i n a more complex way on s e v e r a l v a r i a b l e s l i k e d e f o r m a t i o n , p r o t o n and n e u t r o n number. The f i t t e d v a l u e s a r e , however, r e l a t i v e l y u n c e r t a i n and no c l e a r - c u t c o n c l u s i o n s can be drawn.

6 . The d e f o r m a t i o n dependence of t h e a l i g n e d a n g u l a r momentum, t h e backbending f r e q u e n c y and t h e i n t e r a c t i o n between t h e g- and t h e S-bands.

I n t h e p r e v i o u s s e c t i o n we have s e e n t h a t s m a l l d e f o r m a t i o n changes may have a s i g n i f i c a n t i n f l u e n c e on the r e f e r e n c e c o n f i g u r a t i o n . I t i s caused by t h e summed e f f e c t o f s m a l l changes i n many i n d i v i d u a l q u a s i p a r t i c l e l e v e l s and c a n n o t be n e g l e c - t e d a s soon a s it l e a d s t o a change of e . g . t h e r e f e r e n c e energy which i s n o t n e g l i g i b l e compared to t h e e x c i t a t i o n energy e l ( w )

,

i . e . t h e energy o f an i n d i v i d u a l q u a s i p a r t i c l e .

To g e t i m p o r t a n t changes of t h e p r o p e r t i e s of i n d i v i d u a l q u a s i p a r t i c l e l e v e l s r e - q u i r e s much l a r g e r d e f o r m a t i o n changes.

We s e e , however, i f we c o n s i d e r t h e whole r a r e e a r t h r e g ' i o n , t h a t t h e ' g r o u n d s t a t e d e f o r m a t i o n s a r e s p r e a d o u t o v e r a rela- t i v e l y l a r g e r e g i o n of t h e d e f o r m a t i o n

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JOURNAL DE PHYSIQUE

Fig. 9 The moment of inertia parameter So displayed as a function of the deformation parameters E and

.

The dashed lines marked fo = 2 0 , 2 5 etc. are only a tentative estimate based on the fitted values given in parenthesis for each nucleus.

Ground state deformd~ons

4 1 I I I

Fig. 10 Ground state deformations of a number of rare earth nuclei for which high-spin states are ob- served

.

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PROTON LEVELS, RARE EARTHS

Fig. 1 1 Single-proton levels calculated along a path in the deformation plane given by the dot-dashed line in fig. 10.

NEUTRONS, V (A,def.)

.

Scale:

0.0l)i~)o

l l l ~ l J l ~ l l l ~ ~

1 2 3 4 5 6 7

Deformat ion

NEUTRON L E V E L S . RARE EARTHS

r l ' l ' l ' l ' l ' l ' l l

E ,175 ,220 ,260, ,265 ,220 .l73 .l25 E4-.030 -.030 000 ,045 ,060. ,050 ,040 NI* 88 92 96 102 108 112 116

Deformation 1 2 3 4 5 6 , 7

point

Fig. 13 The interaction strength, V

,

^as

a function of the Fermi energy, A

,

and the deformation, here indicated with figures corresponding to the triangles on,the dot-dashed line in fig. 10.

plane as shown in fig. 10. In this region we observe great changes in the single- particle level structure as illustrated

in figs. 1 1 and 12. The most striking feature is the bunching of the levels of the h11/2

'

^h9/2 ^i13/2 and i11/2 shells for positive values of E,,

.

^As

a consequence of this, the position of the neutron Fermi surface relative the Fig. 12 Same as fig. I1,but forneutrons. ' lowest levels of the il 3,2-~he11 ( 1 6 6 0 1 /2 1

The triangles indicate the posi-

tion of the Fermi surface forthe and [ 6 5 1 3/21) is more or less the same neutron nunlbers given below. The

deformation points are those between the neutron numbers 94 and 1 0 8 . indicated in fig. 1 0 .

The degree of alignment of the

i13/2 quasiparticle levels just above the Fermi

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JOURNAL DE PHYSIQUE

NEUTRON NUMBER

Fig. 1 4 The interaction strength, V

,

between the quasiparticle levels a and -b (here multiplied with a factor 2) is shown by the continuous curves. The interaction between the experimental g- and S-band is shown by the various symbols. TY,e limit for backbending (b.b.) is shown by the thin dot-dashed line.

surface (in fig. 2 labelled a and b) is strongly dependent on the position of the 1660 1 / 2 3 and 1651 3/21 levels, and decreases when these levels move away from the Fermi surface. Since, however, the energy of these levels increases with N almost as fast as the Fermi energy, the aligned angular momentum stays relatively constant all the way up to

N ^ry108

.

This is in agreement with 3 ) experiment as can be seen from ref.

.

As a direct cpnsequence of the small variatibn of the alignment, the backbending frequency (in fig. 2 corresponding to the frequency where the levels a and -b (and b and -a) interact) becomes only weakly dependent on the particle number.

An other quantity which depends on the deformation is the interaction strength between the g- and S-band. This interaction can, at least qualitatively, be estimated from the interaction between the quasiparticle levels a and - b y which is shown in fig. 1 3 . We see that when the

levels bunch together, the maximal interaction strength inbetween two zeros decreases, and if we plot the interaction strength as a function of the neutron number, N

,

choosing a representative deformation for each value of N

,

^{we get}

the results shown in fig. 14. We observe that the height of the interaction maxima is only weakly dependent on the neutron number. If on the other hand the deforma-

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tion is kept fixed, like in ref. )

,

^the

height of the maxima increases consider- ably with the neutron number. The agreement between the theoretical curves and the experimental points in fig. 14 is relatively good. It is however only a qualitative agreement, since the theoretical values have been multiplied with a factor 2.

Although we have studied the deformation dependance in a quite rough way by

restricting ourselves to the dot-dashed line in fig. 10, it has clearly shown us the importance of considering the deformation dependence in order to under- stand the experimental results.

PROTONS, A = 6.32h0o ( Z n 911, E ~ 0 . 2 4 ,

7. The actinide region

One may assume that the quasiparticle energy diagrams may be very useful also in other well-deformed mass regions, like the actinide region. Here the neutron Fermi surface is placed in the j15/2 shell and the proton Fermi surface in the i13/2-shell. A couple of representative quasiparticle diagrams are shown in figs. 1 5 and 16. Compared to the rare earth region (cf. ref. 3, ) the major difference is that the frequency of the first interactionbetween levels originating from above and from below the Fermi surface is practically the same for protons and neutrons. The reason is that in the neutron j15,2-shell the K = 7/2 level lies at the Fermi surface

Fig. 15 Quasiparticle energy diagram representative for proton number Z W 91

.

it is the K = 5/2 level. A level with a low K-value aligns faster than one with a high K-value if they are in the same j-shell. On the other hand, a level with a given K-value gets a larger alignment if it belongs to a shell with a higher j-value. In the actinide region these two.

effects compensate each other and we get a similar alignment for neutrons and protons and consequently a similar interaction frequency. This will most certainly make the interpretation of the experimental data more complex.

It is theoretically predicted that also in while in the proton i,3/2-shell

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JOURNAL DE PHYSIQUE

NEUTRONS, h = 7 . 4 2 5 0 o ( N ~ 1 4 3 1 , E = 0.24.

F i g . 1 6 . Q u a s i p a r t i c l e e n e r g y diagram r e p r e s e n t a t i v e f o r n e u t r o n number N w 1 4 3 .

t h e a c t i n i d e r e g i o n t h e i n t e r a c t i o n s t r e n g t h , V

,

between t h e g- and t h e S-band i s an o s c ' i l l a t i n g f u n c t i p n of t h e Fermi e n e r g y a s shown i n f i g . 1 7 . The c o n d i t i o n f o r g e t t i n g backbending ^{3 )} i s t h a t

I v I

^<

& ^.

S i n c e t h e moment o f

C

i n e r t i a J c i s l a r g e i n t h e a c t i n i d e r e g i o n and t h e a l i g n e d , a n g u l a r momentum n o t h i g h e r t h a n i n t h e r a r e e a r t h r e g i o n , t h i s ' l i m i t l i e s r e l a t i v e l y low. If one furtlierniore assumes t h a t t h e t h e o r e t i c a l i n t e r a c t i o n i s t o o low l i k e i n t h e r a r e e a r t h . r e g i o n ( i . e . t h e s o l i d l i n e s i n

. .

f i g . 1 7 have t o be m u l t i p l i e d by r o u g h l y

PROTON NUMBER

NEUTRON NUMBER

F i g . 1 7 The i n t e r a c t i o n s t r e n g t h between t h e q u a s i p a r t i c l e l e v e l s a and -b ( s o l i d l i n e ) and t h e l i m i t f o r backbending ( d o t - d a s h e d l i n e ) .

t i e s f o r g e t t i n g backbending i n t h e a c t i - n i d e r e g i o n a r e q u i t e s m a l l .

I n f i g . 1 8 we show t h e e x p e r i m e n t a l a l i g n e d a n g u l a r momentum, i

,

o f t h e

X

y r a s t band f o r some a c t i n i d e n u c l e i 7

.

We o b s e r v e a pronounced i n c r e a s e o f i.

X

a t a f r e q u e n c y , which a g r e e s w e l l w i t h t h e one p r e g i c t e d from t h e q u a s i - p a r t i c l e e n e r g y diagrams ( i n d i c a t e d w i t h t h e a r r o w s ) . We a l s o s e e t h a t t h e i n c r e a s e o f ix a t wc depends s t r o n g l y on t h e n e u t r o n number. The maximal i n c r e a s e i s a f a c t o r 2 ) we r e a l i z e t h a t t h e p o s s i b i l i -

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C e r t a i n p a r t s of t h i s p a p e r have been s t r o n g l y i n f l u e n c e d by a v e r y f r u i t f u l c o o p e r a t i o n w i t h J.A. P i n s t o n and S. F r a u e n d o r f .

R e f e r e n c e s

1

.

^S^.G.N i l s s o n ,

Mat. Fys. Medd. Dan. V i d . S e l s k . , 29, No. 16, (1955)

-

S.G. N i l s s o n e t a l . ,

Nucl. Phys. (1969) 1

2. G. Andersson e t a l . ,

Nucl. Phys.

A268

(1976) 205

F i g

.

¹⁸ The e x p e r i m e n t a l l y o b s e r v e d a l i g n e d a n g u l a r momentum, ix ^y f o r t h e y r a s t band o f some a c t i n i d e n u c l e i .

however observed f o r N = 146 w h i l e t h e t h e o r y p r e d i c t s N = 144

.

T h i s might p o s s i b l y be e x p l a i n e d by t h e f a c t t h a t t h e d e f o r m a t i o n chosen i n f i g . 17 i s n o t t h e o p t i m a l one. The e x p e r i m e n t a l d a t a a v a i l a b l e i n t h e a c t i n i d e r e g i o n do however n o t p e r m i t u s t o make a more s y s t e m a t i c comparison w i t h t h e t h e o r e - t i c a l p r e d i c t i o n s .

3. R. Bengtsson and S . F r a u e n d o r f , Nucl. Phys. A314 (1979) 27

-

and