A SUM-RULE ESTIMATE OF RPA GROUND-STATE CORRELATIONS IN NUCLEI

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A SUM-RULE ESTIMATE OF RPA GROUND-STATE CORRELATIONS IN NUCLEI

P.-G. Reinhard, J. Friedrich

To cite this version:

P.-G. Reinhard, J. Friedrich. A SUM-RULE ESTIMATE OF RPA GROUND-STATE COR- RELATIONS IN NUCLEI. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-181-C6-190.

�10.1051/jphyscol:1984621�. �jpa-00224222�

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JOURNAL DE PHYSIQUE

Colloque C6, supplément au n06, T o m e 45, juin 1984 page C6-181

A SUM-RULE ESTIMATE OF RPA GROUND-STATE CORRELATIONS I N NUCLEI

P.-G. Reinhard and J. ~riedrich*

I n s t i t u t fi& Theoretische Physik, Universitüt Erlangen, 8520 Erlangen, F.R.G.

* ~ n s t i t u t fi& Kemphysik, Universitat Mains, 6500 Mainz, F . R. G.

Résumé - On a obtenu une description simple des corrélations dans l'état de base utilisant une approximation de règle de somme. Les effets des corrélations sont petits mais non-négligeables.

Abstract - A s i m p l e description o f RPA g r o u n d - s t a t e c o r r e l a t i o n i s obtained by using a s u m - r u l e approximation t o nuclear giant resonances. The correlation effects c o m e out to be s m a l l but non-negligible.

1. I N T R O D U C T I O N

S i n c e the introduction o f density dependent effective f o r c e s ( l i k e the S k y r m e force [l]) Hartree-Fock c a l c u l a t i o n s have found widespread app- l i c a t i o n s i n nuclear physics. B e s i d e s pure ground-state c a l c u l a t i o n s the S k y r m e forces a r e used for the evaluation o f fusion/fission bar- r i e r s or low-energy vibration dynamics, for the d e s c r i p t i o n o f h e a v y - ion d y n a m i c s within the TDHF a p p r o a c h , and for c o n s i s t e n t RPA c a l c u l a - t i o n s on top o f the Hartree-Fock ground s t a t e ( s e e e.g. ref. [ z ] ) or semiclassical approximations t o that. H o w e v e r , t h e r e i s o n e problem in an extended use o f t h e S k y r m e forces: T h e usual p a r a m e t r i s a t i o n s [1,3,

4 1 are designed to be effective forces for nuclear Hartee-Fock c a l c u -

lations and a r e determined (main-ly) by a d j u s t i n g g r o u n d - s t a t e proper- ties; however, i f o n e c a l c u l a t e s RPA-excitations o n e i m p l i e s a n RPA- correlated g r o u n d - s t a t e ; but correlation e f f e c t s a r e n o t fully negli- g i b l e ( s e e e.g. [ 5 , 6 ] ) . Therefore, i n a n RPA c a l c u l a t i o n with S k y r m e - - f o r c e s o n e should u s e parametrisations w h i c h are adjusted t o c o r r e l a - ted ground-state calculations. In order t o s e e whether these theoreti- cal a r g u m e n t s cal1 for practical c o n s e q u e n c e s or not w e h a v e investi- gated the c o r r e l a t i o n e f f e c t s in the context o f Skyrme-Hartree-Fock calculations.

A straightforward evaluation o f the RPA-correlation i s very e x p e n s i v e t h u s prohibiting e x t e n s i v e use in systematic i n v e s t i g a t i o n s over many nuclei and f o r c e s ; i t s inclusion i n t o a fit o f the f o r c e - p a r a m e t r i s a - tion would be completely hopeless. It i s desirable t o h a v e a f a s t e s - t i m a t e o f the RPA ground-state correlations. This can be obtained by exploiting the fact t h a t the dominant c o n t r i b u t i o n s t o the c o r r e l a t i - o n s c o m e from a few collective s t a t e s in the RPA s p e c t r u m , namely the g i a n t r e s o n a n c e s and the low-energy surface vibrations. I n particular for the giant r e s o n a n c e s there exist s i m p l e models i n t e r m s o f s u m - rule or fluid-dynamical a p p r o a c h e s ( w e w i l l not give a long l i s t o f r e f e r e n c e s h e r e , s i n c e much o f the work i n t h i s field w i l l be presen- ted in other c o n t r i b u t i o n s of these proceedings). T h i s h a s motivated u s t o develop a fast estimate o f ground-state c o r r e l a t i o n s making use o f those s i m p l e d e s c r i p t i o n s o f giant resonances. F o r t h e ap r o x i m a - tion s c h e m e w e follow mainly the work o f Brink and Leonardi r i ' ] corn- plemented b y f e a t u r e s o f the fluid-dynamical approach o f Krivine e t

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984621

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C6-182 JOURNAL DE PHYSIQUE

a l [ a ] . A s i m p l e t e c h n i q u e t o e v a l u a t e t h e c o r r e l a t i o n e f f e c t s f o r a g i v e n s t r u c t u r e o f e x c i t a t i o n s i s t a k e n o v e r f r o m a p r e v i o u s s t u d y o f t i m e - d e p e n d e n t c o r r e l a t i o n s [ 9 ] . I n t h e p r e s e n t i n v e s t i g a t i o n , t h e e x - c i t a t i o n s a r e b u i l d o n t o p o f a s p h e r i c a l S k y r m e - H a r t r e e - F o c k s t a t e i n c l u d i n g a s c h e m a t i c p a i r i n g . F o r t h e g i a n t r e s o n a n c e s we c o n s i d e r t h e ( L = O , T = O ) - , ( L = O , T = l ) - , ( L = l , T = l ) - , ( L = 2 , T = O ) - a n d ( L = 2 , T = l ) - modes ( w h e r e L = a n g u l a r momentum a n d T = i s o s p i n ) . The L = 2 c o l l e c - t i v e s u r f a c e mode i s i n c l u d e d i n a s e m i - p h e n o m e n o l o g i c a l way.

Due t o l i m i t e d s p a c e we c a n n o t d e v e l o p t h e t h e o r e t i c a l b a c k g r o u n d i n f u l l l e n g t h . We m e r e l y w i l l p r e s e n t t h e e s s e n t i a l f o r m u l a s w i t h some p l a u s i b i l i t y a r g u m e n t s a n d t h e n p r o c e e d q u i c k l y t o p r e s e n t t h e r e s u l t s . I n s e c t i o n 2 we show how t o e v a l u a t e c o r r e l a t i o n e f f e c t s f o r g i v e n e x - c i t a t i o n . I n s e c t i o n 3 we p r e s e n t t h e d e s c r i p t i o n o f g i a n t r e s o n a n c e s i n t h e s u m - r u l e a p p r o x i m a t i o n . I n s e c t i o n 4 we g i v e t h e r e s u l t s . C o n - c l u s i o n s a r e d r a w n i n s e c t i o n 5 .

2 . CORRELATION EFFECT OF RPA EXCITATION

T h e r e a r e many w a y s t o f o r m u l a t e t h e R P A e q u a t i o n s w h i c h d e t e r m i n e t h e e x c i t a t i o n s p e c t r u m f o r s m a l l o s c i l l a t i o n s a b o u t t h e H a r t r e e - F o c k s t a t e IO>. I n v i e w o f t h e s u m - r u l e a p p r o c h e s we p r e f e r t h e n o t a t i o n i n t e r m s o f c o o r d i n a t e - a n d momentum - l i k e l p h o p e r a t o r s ,ON, a n b PN r e s p e c t i v e l y , w h e r e N l a b e l s t h e e x c i t a t i o n s t a t e s . The Q N a n d PN a r e d e t e r m i n e d b y t h e RPA e q u a t i o n s p l u s c o n j u g a t i o n c o n d i t i o n ,

The e x c i t a t i o n e n e r g i e s a n d w i d t h e s o f t h e e i g e n s t a t e s a r e e v a l u a t e d f r o m m a s s ff a n d s p r i n g - c o n s t a n t I!! i n t h e s t a n d a r d way

N N

hmN = JiN /rZN r A N = /ZN 'PN - ^{( I d )}

I n p r i n c i p l e , o n e s h o u l d u s e t h e R P A - c o r r e l a t e d s t a t e s I $ > i n t h e e q s . ( 1 ) . B u t t h e s e a r e n o t k n o w n b e f o r e t h e c a l c u l a t i o n . T h u s o n e e m p l o y s t h e Q u a s i - B o s o n a p p r o x i m a t i o n a n d r e p l a c e s i n e q . ( 1 ) I $ > b y t h e s i m - p l e H a r t r e e - F o c k s t a t e s ] @ > .

.. -

W i t h t h e Q N a n d PN o n e c a n c o n s t r u c t b y s t a n d a r d o s c i l l a t o r a l g e b r a t h e c r e a t i o n o p e r a t o r f o r s t a t e N

t h e c o r r e s p o n d i n g a n n i h i l a t i o n o p e r a t o r i s t h e n t h e a d j o i n t CN. Now, t h e R P A - c o r r e l a t e d s t a t e i s d e f i n e d t o b e t h e " b o s o n 1 ' - v a c u u m .

One n e e d s n o t t h e know t h e I @ > e x p l i c i t e l y a s a c o h e r e n t sum o f I @ >

p l u s 2 p h - , 3ph-, ... e x c i t a t i o n s . The d e f i n i n g c o n d i t i o n ( 3 ) p l u s t h e a l g e b r a o f t h e Q N a n d P N i s s 2 f f i c i e n t t o d e r i v e t h e c o r r e l a t e d e x p e c - t a t i o n v a l u e o f a n y o p e r a t o r A a s

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The o b v i o u s interpretation o f that expression is: in t h e round brac- k e t s i n t Q e _ s u m s w e h a v e the d i f f e r e n c e o f correlated t o uncorrelated Q N Q M , or P N P M , width. T h i s "correlation width" in each c h a n n e l - i s weighted with the c o u p l i n g o f the N-M-channel t o t h e o p e r a t o r A. From t h i s e x p r e s s i o n it i s k n o w n that i t o v e r e s t i m a t e s the g r o u n d - s t a t e c o r r e l a t i o n s [5]. O n the other h a n d , o n 1 a few c o l l e c t i v e s t a t e s pro- vide the dominating t e r m s in eq. (4) 1 0 ; restricting t h e s u m t o t h e s e few s t a t e s a v o i d s overcounting [lli and g i v e s a s u f f i c i e n t estimate o f t h e c o r r e l a t i o n s a t the s a m e time [10,11]. T h e r e f o r e w e re- strict t h e s u m s in eq. (4) to j u s t one c o l l e c t i v e mode per multipola- r i t y , namely t h e giant resonance. F u r t h e r m o r e , w e a p p r o x i m a t e

t h i s assumption will be valid for Q and P w h i c h d e s c r i b e a c o l l e c t i v e mode (since it i s related t o t h e G a u s s i a n - O v e r l a p - A p p r o x i m a t i o n [12]).

T h e a n l o g u e o f eq. (5) with the s t a t e s I$> i s strictly true d u e t o condition (3).

With that a p p r o x i m a t i o n s , we o b t a i n th? correlated ground s t û t e energy by identifying A = H. t h i s yields

w h e r e the sum r u n s over t h e considered g i a n t r e s o n a n c e s (GR) with mul- tiplicity (2L+1). H o w e v e r , it i s questionable whether t h i s c o r r e c t i o n can be applied a s i t stands. The Çkyrme-force i s s u p p o s e d to be a pa- rametrisation o f the nuclear G-matrix [l]. In t h a t c a s e the RPA-corre- l a t i o n s produce a double-counting, s i n c e the lowest order c o r r e l a t i o n diagram i s already contained i n the Hartree-Fock e n e r g y [5]. T o r e m o v e the double c o u n t i n g , w e subtract the energy o f the l o w e s t order graph w h e r e w e insert (in accordance with the s u m - r u l e approximation) ave- r a g e lph-excitation e n e r g i e s and a v e r a g e f o r c e rnatrix elements. T h e corrected correlated energy becomes

For the c a s e that A i s general one-body operatcr w e a s s u m e further QI@> = i PI@> / ),HF; t h i s approximation i s a t the s a m e level as a p p r o x i - mation (5). T h u s w e o b t a i n

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C6-184 JOURNAL DE PHYSIQUE

A l 1 t h e r e s u l t s o f t h i s s e c t i o n c a n b e d e r i v e d i n d e p e n d e n t l y f r o m t h e G e n e r a t o r - C o o r d i n a t e - M e t h o d . T h i s a d d s some c o n f i d e n c e t o t h e r e s u l t s .

3 . APPROXIMATION FOR GIANT RESONANCES

The c o r r e l a t i o n e f f e c t s a r e s i m p l e t o e v a l u a t e , a s we h a v e s e e n i n t h e p r e v i o u s s e c t i o n . ' t r e m a i n s t h e c u m b e r s o m e t a s k t o s o l v e t h e RPA e q s . ( 1 ) f o r t h e Q a n d P. A f a s t e r a c c e s s t o Q a n d P i s d e s i r a b l e a n d i t i s p o s s i b l e : A n u c l e a r g i a n t r e s o n a n c e a l m o s t e x h a u s t s t h e s t r e n g t h o f i t s c o r r e s p o n d i n g m u l t i p o l e o p e r a t o r ; t h u s i t s h o u l d b e j u s t i f i e d j u s t t o g u e s s t h e c o l l e c t i v e Q t o b e t h i s m u l t i p o l e o p e r a t o r , i . e .

A 1 N-Z ^A T

Q = - 1 ( 1 + -A ~ , ( i ) ) f i ( r i )

i -

w h e r e T = O o r 1 i s t h e - i s o s p i n a n d f~ i s t h e r L Y o o f o r L = O a n d r L Y L ~f o r L 1 ^{O .} W i t h Q g i v e n we c a ~ e v a l u a t e t h e mass M a c c o r d i n g t o e q . ( l a ) ; f o r t h e l o c a l o p e r a t o r Q , a s g i v e n i n e q . ( 9 a ) , a n d w i t h S k y r m e f o r c e s t h i s r e d u c e s t o s i m p l e i n t e g r a l s i n v o l v i n g o n l y t h e d e n - s i t y ~ ( 2 ) . I n o r d e r t o g u e s s t h e c o r r e s o n d i n g P we r e c u r t o a n a l t e r - n a t i v e f o r m u i a t i o n o f e q . ( l a ) , n a m e i y [H,61ph = - i P . We n e g l e c t t h e i n t e r a c t i o n t h e r e i n a n d a p p r o x i m a t e

..

w h e r e T i s t h e k i n e t i c e n e r g y o p e r a t o r . F o r a l o c a l Q, t h i s P i s o f f i r s t o r d e r i n o. ^'t g e n e r a t e s a s i m p l e s c a l i n g t r a n s f o r m a t i o n , _ e . g . f o r T = O we h a v e P 2' C <Li - ^Vif ⁺^{V i f}- ^{F i )}a n d t h e e x p ( - i q P ) j u s t p r o d u c e s a b r e ~ t h i n g - r ⁺e x p ( - - i q ) r: T h i s means t h a t t h e - s r i n g c o n - s t a n t f = < m [ P , [ ~ , P ~ ] I * > = d ~ / d ~ ~ < @ r e x ~ ( - i q P ) H e x p ( - i q P ) P Q > ~ ~ = O c a n b e e v a l u a t e d b y s i m p l e s c a l i n g t e c h n i q u e s . F o r t h e T = 1 mo e s o n e h a s t o b e a b i t c a r e f u l i n a p p l y i n g t h e s c a l i n g a r g u m e n t s , s i n c e p r o t o n s a n d n e u t r o n s s c a l e w i t h o p p o s i t e s i g n .

F o r e v a l u a t i n g t h e c o r r e l a t i o n e f f e c t s o n a n $ o b s e r a b l e A we a'so-em- p l o y t h e s c a l i n g , - u s i n g <QI [ P , [ A , P ] ] I O > = d / d q 2 < a l e x p ( - i q P ) A e x p ( - i q P ) I Q > F o r A b e e i n g t h e c h a r g e d e n s i t y o r a r e l a t e d o b s e r v a b l e , t h i s e x p r e s s i o n i s s i m p l e t o e v a l u a t e .

I n c a s e o f o p e n - s h e l l n u c l e i t h e r e i s a p r o b l e m w i t h t h e u n c o r r e l a t e d w i d t h

The w , va c a n g e t v a l u e s b e t w e e n O a n d 1 a n d t h u s O h - t r a n s i t i o n s ( w i t h y n t h e v a l e n c e s h e l l ) c a n o c c u r . T h e s e Ohw - t r a n s i t i o n s show s t r o n g i s o t o p i c v a r i a t i o n s a n d t h e y c o r r e s p o n d m a i n l y t o t h e l o w - l y - i n g s u r f a c e v i b r a t i o n s . H o w e v e r , w i t h t h e s u m - r u l e a p p r o x i m a t i o n ( 9 ) we m o d e 1 t h e g i a n t r e s o n a n c e s . I n o r d e r t o s u p p r e s s t h e s e O b - t r a n - s i t i o n s we p r e f e r t h e c h o i c e

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U s i n g t h e w i d t h ( 1 1 ) we o b t a i n i n d e e d a s m o o t h A - d e p e n d e n c e o f t h e u n - c o r r e l a t e d w i d t h .

4 . RESULTS

I n t h e f o l l o w i n g we p r e s e n t t h e r e s u l t s f o r g r o u n d - s t a t e c o r r e l a t i o n e f f e c t s i n t h e a p p r o x i m a t i o n o u t l i n e d a b o v e . M o s t c a l c u l a t i o n s h a v e b e e n d o n e w i t h t h e S k y r m e M f o r c e [ 4 ] . I n o r d e r t o s e e t h e s e n s i t i v i t y t o t h e f o r c e , we h a v e a l s o c o n s i d e r e d t h e f o r c e S k y r r n e 3 [l] a n d a s e - r i e s o f f o r c e s w h i c h w a s o b t a i n e d r e c e n t l y b y e x p l i c i t e l y f i t t i n g q r o u n d s t a t e p r o p e r t i e s o f t h e n u c l e i 1 6 0 , 4 0 ~ a , 4 8 ~ a , 5 8 ~ i , 9 0 ~ i ,

1 6 s n , 1 2 4 s n a n d 2 0 8 ~ b . T h e s e f o r c e s a r e d e n o t e d F i t A , F i t 8 , F i t K , F i t L a n d F i t K * i n t h e f o l l o w i n g ; t h e y d i f f e r i n t h e s e l e c t i o n o f g r o u n d - s t a t e p r o p e r t i e s w h i c h h a v e b e e n i n c l u d e d i n t h e f i t ( F i t B a n d L i n c l u d e r a d i u s a n d b i n d i n g e n e r g y ; F i t A , K a!d K * i n -

c l u d e i n a d d i t i o n t h e s u r f a c e t h i c k n e s s ) ; f o r F i t K w a s i n c l u d e d a l s o o n e e x c i t a t i o n p r o p e r t y , t h e ( L = l , T r l ) g i a n t r e s o n a n c e e n e r g y . A l 1 c a l c u l a t i o n s h a v e b e e n d o n e u s i n g w a v e f u n c t i o n s o f a s p h e r i c a l H a r - t r e e - F o c k c a l c u l a t i o n p l u s s c h e m a t i c p a i r i n g .

B e f o r e we s t a r t t o l o o k a t t h e c o r r e l a t i o n s , we j u s t w a n t t o h a v e a s h o r t g l a n c e a t e x c i t a t i o n p r o p e r t i e s . P a r s p r o t o t o , we g i v e i n f i g . 1

Skyrme M

1 ^Ni

-

- F i g . 1

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t h e e x c i t a t i o n e n e r g i e s o f t h e ( L = l , . T = l ) g i a n t r e s o n a n c e f o r f o r c e Ç k y r m e M a n d a v a r i e t y o f i s o t o p i c c h a i n s . T h e A116 r u l e s e e m s t o b e f a i r l y w e l l f u l f i l l e d o v e r t h e A ; b u t t h e r e a r e s t r o n g i s o t o p i c t r e n d s a c r o s s t h e s m o o t h t r e n d w i t h A . T h i s i s a g e n e r a l f e a t u r e o f a l 1 t h e r e s u l t s , f o r e n e r g i e s a n d w i d t h e s o f g i a n t r e s o n a n c e s a s w e l l a s f o r a l 1 c o r r e l a t i o n e f f e c t s . T h e i s o t o p i c t r e n d s a r e n e a r l y l i n e a r . T h u s we p l o t i n t h e f o l l o w i n g f i g u r e s o n l y t h e e n d p o i n t s o f t h e i s o t o p i c c h a i n s a n d c o n n e c t t h e m b y a s t r a i g h t l i n e . F i n a l l y we w a n t t o m e n - t i o n c o n c e r n l n g t h e e x c i t a t i o n e n e r g i e s , t h a t a l 1 T = 1 e n e r g i e s f o l - l o w n e a r l y a n ~ l / ~ - r u l e , b u t a l 1 T = O e n e r g i e s f o l l o w r a t h e r a n A 1 1 3 - r u l e .

T h e r e s u l t s f o r t h e c o r r e l a t i o n e n e r g i e s a r e g i v e n i n f i g . 2 f o r f o r c e S k y r m e M a n d a v a r i e t y o f i s o t o p i c c h a i n s ; t h e f u l l c i r c l e s i n f i g . 2

1

^Skyrme ^M

F i g . 2

s

ioo 200 A

h a v e b e e n e v a l u a t e d w i t h t h e " n a i v e " e x p r e s s i o n ( 6 ) w h e r e a s t h e o p e n c i r c l e s h a v e b e e n e v a l u a t e d w i t h t h e c o r r e c t e d f o r m u l a ( 7 ) . T h e c o r - r e l a t i o n e n e r g i e s a r e f a i r l y l a r g e ( i n a c c o r d a n c e w i t h r e f . [ 5 ] ) ; t h e c o r r e c t e d v a l u e s a r e a p p r o x i r n a t e l y h a l f t h e s i z e ( s t i l l l a r g e ) a n d h a v e o p p o s i t e s i g n . T h i s i s d u e t o t h e d o m i n a n c e o f t h e T = 1 m o d e s

~ h e r e A H F < A . We h a v e i n s e r t e d b o t h c o r r e l a t i o n e n e r g i e s i n a l e a s t - - s q u a r e f i t o f t h e Ç k y r m e f o r c e p a r a m e t e r s a n d f o u n d t h a t t h e c o r r e c - t e d e x p r e s s i o n ( 6 ) f i t s rnuch b e t t e r ; t h i s rnay b e t a k e n a s a p r a c t i c a l

" p r o o f " o f t h e d o u b l e - c o u n t i n g i m p l i c i t i n e q . ( 7 ) .

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In fig. 3 w e s h o w how t h e correlation energy i s added up from t h e five modes under consideration. I n the left part w e s h o w t h e r e s u l t s f o r 208pb only but for a variety o f forces ( a s explained above); i n t h e right part w e s h o w the r e s u l t s for the force F i t K only but for a variety o f isotopes.

Fig. 3

The mode i s denoted in t h e brackets (L,T) i n the m i d d l e o f the figure.

I n any c a s e w e h a v e given the "naive" c o r r e l a t i o n energy (6 ) s i n c e t h e r e a l 1 c o n t r i b u t i o n s have the s a m e sign. It i s o b v i o u s that t h e T = l modes dominate the correlation energy. T h e problem i s t h a t j u s t the T = 1 m o d e s a r e badly determined by c u r r e n t force parmetrisations.

As one s e e s i n the left part o f fig. 3 , the variation with t h e force i s very large. It i s interesting to note that the two forces which fit best the L = l , T = l mode, namely S k y r m e M and F i t K * , give the s m a l - lest correlation effects. That my be the realistic e s t i m a t e , s i n c e w e have made the e x p e r i e n c e that a fit o f t h e force which i n c l u d e s ground s t a t e c o r r e l a t i o n s t e n d s t o make them small.

In fig. 4 w e s h o w the effect o f c o r r e l a t i o n s o n form p a r a m e t e r s o f the charge d i s t r i b u t i o n , namely the diffraction r a d i u s R [13], the r.m.s. r a d i u s r and the surface t h i c k n e s s a [13], for the force S k y r m e M and a variety o f isotopic chains. The effect i s strongest for O ,

moderate f o r r and s m a l l for R . Again w e s e e the t y p i c a l strong isotopic variations. Altbough the e f f e c t s o n R seem t o be s m a l l , they a r e not completely n e g l i g i b l e in view o f t h e precision with which modern Skyrme-Hartree-Fock c a l c u l a t i o n s c a n r e p r o d u c e energy and radius.

Al1 the r e s u l t s a b o v e a r e evaluated only with the g i a n t r e s o n a n c e s a s collective modes. In order t o s e e the e f f e c t o f t h e low energy qua- drupole m o d e w e h a v e included i t phenomenologically by taking hw and X from measured energy and B ( E 2 ) value [ 6 ] . T h e e f f e c t s on the correlation e n e r g y are negligible but there are strong e f f e c t s on the forrn parameters. I n fig. 5 w e s h o w the r e l a t i v e correlation e f f e c t s o n the s u r f a c e t h i c k n e s s O without (full c i r c l e ) and with (open c i r c l e ) the s u r f a c e vibration mode. Obviously the s u r f a c e modes dominate the e f f e c t for open-shell nuclei. They give r i s e to the typical s t r o n g

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C6-188 JOURNAL DE PHYSIQUE

i s o t o p i c v a r i a t i o n s , s m a l l c o r r e c t i o n s t o w a r d s t h e r n a g i c n u c l e i a n d l a r g e c o r r e c t i o n s i n t h e m i d d l e o f t h e v a l e n c e s h e l l .

[ = R d

O I = r

Skyrme M

F i g . 4

5 . C O N C L U S I O N S

T h e s t r o n g e s t c o n t r i b u t i o n s t o t h e R P A g r o u n d - s t a t e c o r r e l a t i o n s a r e e x p e c t e d t o c o m e f r o r n t h e c o l l e c t i v e m o d e s o f t h e s y s t e m . We o b t a i n a f a s t e s t i m a t e o f t h e c o r r e l a t i o n s b y i n v o l v i n g t h e s u m - r u l e a p p r o x i - m a t i o n f o r t h e g i a n t r e s o n a n c e s a n d i n s e r t i n g i t i n t o s i m p l e e x p r e s - s i o n s f o r c o r r e l a t e d e x p e c t a t i o n v a l u e s , o b t a i n e d b y m a n i p u l a t i n g t h e R P A a l g e b r a . T h e s u r f a c e q u a d r u p o l e m o d e i s a d d e d s e m i - p h e n o m e n o - l o g i c a l l y b y t a k i n g i t s w i d t h a n d e n e r g y f r o m e x p e r i r n e n t a l v a l u e s . T h e p r o m i . n e t r e s u l t s a r e :

1 . I n a n y q u a n t i t y we o b s e r v e s t r o n g i s o t o p i c t r e n d s ; t h e c o r r e l a t i o n e f f e c t s i n c r e a s e w i t h a d d i n g n e u t r o n s .

2 . F o r t h e g i a n t r e s o n a n c e s t h e T = l m o d e s d o m i n a t e c l e a r l y t h e c o r r e - l a t i o n e f f e c t s .

3 . S u r f a c e m o d e a n d g i a n t r e s o n a n c e s b e h a v e c o m p l e m e n t a r y . T h e g i a n t r e s o n a n c e s g i v e l a r g e c o r r e l a t i o n e n e r g i e s b u t s r n a l l e f f e c t s o n f o r m p a r a m e t e r s ; w h e r e a s t h e s u r f a c e m o d e h a s n e g l i g i b l e c o r r e l a - t i o n e n e r g y b u t l a r g e f o r m e f f e c t s .

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F i g . 5

A l t o g e t h e r , t h e c o r r e l a t i o n e f f e c t s a r e m o d e r a t e . B u t i n v i e w o f t h e r n a r v e l l o u s p r e c i s i o n o f m o d e r n Ç k y r r n e - H a r t r e e - F o c k c a l c u l a t i o n s , t h e y a r e n o t n e g l i g i b l e . T h u s we w i l l n e e d d i f f e r e n t f o r c e s f o r d i f f e r e n t p u r p o s e s . M e r e g r o u n d s t a t e o r f i s s i o n c a l c u l a t i o n s rnay l i v e w e l l w i t h t h e o l d p a r a r n e t r i s a t i o n s . B u t f o r RPA c a l c u l a t i o n s we n e e d n e w f o r c e s .

REFERENCES

[ l ] V a u t h e r i n , D. a n d B r i n k , D.M., P h y s . R e v . CS ( 1 9 7 2 ) 6 2 6 B e i n e r , M . , F l o c a r d , M . , v a n G i a i , N . a n d Q u e n t i n , P . , N u c l . P h y s . A251 ( 1 9 7 3 ) 1

[ 2 ] K r e w a l d , S . , K l e m t , V . , Ç p e t h , J . a n d F a e s s l e r , A . , N u c l e a r P h y s . A281 ( 1 9 7 7 ) 1 6 6

[ 3 ] K o h l e r , H . S . , N u c l . P h y s . A258 ( 1 9 7 6 ) 3 0 1

[ 4 ] T r e i n e r , I . , K r i v i n e , H . , B o h i g a s , 0 . a n d M a r t o r e l l , K . , N u c l . P h y s . A371 ( 1 9 8 1 ) 2 5 3

B a r t e l , K . , Q u e n t i n , P . , B r a c k , M . G u e t , C . a n d H a k a n s o n , M.-B., N u c l . P h y s . A386 ( 1 9 8 2 ) 7 9

[ 5 ] B o u s s y , A . a n d V i n h M a u , N . , N u c l . P h y s . A229 ( 1 9 7 4 ) 1 [ 6 ] R e i n h a r d , P . - G . a n d D r e c h s e l , D . , Z . P h y s . A290 ( 1 9 7 9 ) 8 5 [ 7 ] B r i n k , D.M. a n d L e o n a r d i , R . , N u c l . P h y s . A258 ( 1 9 7 6 ) 2 8 5

[ a ] K r i v i n e , H . , T r e i n e r , K . a n d B o h i g a s , D . , N u c l . P h y s . A336

( 1 9 8 0 ) 1 5 5

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[ 9 ] R e i n h a r d , P.-G., C u s s o n , R . Y . a n d G o e k e . K . , N u c l . P h y s . A398

( 1 9 8 3 ) 1 4 1

[IO] G o g n y , D., L e c t u r e N o t e s i n P h y s i c s , V o l . 108, p . 8 8 , Ç p r i n g e r V e r l a q , B e r l i n 1 9 7 9

[ll] P a r i k h , I . C . a n d Rowe, D . I . , P h y s . Rev. 175 ( 1 9 6 8 ) 1 2 9 3 [ 1 2 ] R e i n h a r d , P . - G . , N u c l . P h y s . A261 ( 1 9 7 6 ) 2 9 1

[ 1 3 ] F r i e d r i c h , J. a n d V o e g l e r , N., N u c l . P h y s . A373 ( 1 9 8 2 ) 1 9 2