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[ Nuclear Ph ysics A 1 0 2 (1967) 593--601; (~ North-Holland Publishing Co., Amsterdam1 . D . I I Not to be reproduced by photoprint or microfilm without w ritten permission from the publisher

T W O - B O D Y M A T R IX E L E M E N T SF R O M A M O D I F I E D S U R F A C E D E L T A IN T E R A C T I O N

P. W. M. G L A U D E M A N S a n d P. J. B R U S S A A R DFysisch Laboratorium der Rijksuniversiteit, Utrecht, Ne t h e r l a n d s

a n dB. H. WILDENTHAL *

O a k R i d g e N a t i o n a l Laboratory, Oa k Ridge, TennesseeR e c e i v e d 2 5 M a y 1967

A b s t r a c t : T h e a d d i t i o n of a T-dependent but J-independent term t o t h e s u r f a c e d e l t a i n t e r a c t i o ng r e at ly i m p r o v e s t h e a g r e e m e n t b e t w e e n t h e v a l u e s o f t h e t w o - b o d y m a t r i x e l e m e n t s c a l c u la t e dfrom t h i s i n t e r a c t i o n a n d t h e c o r r e s p o n d i n g v a l u e s o b t a i n e d b o t h f r o m e m p i r i c a l f i t s t o l e v e le n e r g i e s a n d from a realistic (Hamada-Johnston) i n t er a c t io n . T h i s i s s h o w n for ld-2s and lf-2pc o n f i g u r a t i o n s . T h e use of this modified surface d e l t a i n t e r a c t i o n as a shell-model residual inter -a c t i o n p r o d u c e s g o o d a g r e e m e n t b e t w e e n t h e c a l c u l a t e d a n d e x p e r i m e n t a l e n e r g i e s of many-nucleon c o n f i g u r a t i o n s . I n p a r t i c u l a r , t o t a l b i n d i n g e n e r g i e s a n d t h e e n e r g y s p a c i n g s b e t w e e nl e v e l s o f d i f f e r e n t i s o s p i n a r e f i t t e d m u c h b e t t e r t h a n i s p o s s i b l e w i t h o u t t h e use of t h e e x t r aT - d e p e n d e n t term.

1 . I n t ro duct i o nS h e l l - m o d e l c a l c u l a t i o n s h a v e b e e n p e r f o r m e d r e c e n t ly w i t h t h e su r f a ce d e l ta i n te r -

a c t i o n 1 ) S D I ) a s a n e f fe c ti v e t w o - b o d y i n t e r a c t io n i n t h e d s sh e l l 2, 3 ) a n d f o r h e a v i ern u c l e i 4 ) . I t w a s f o u n d t h a t t h e e n e r g y s p a c i n g s b e t w e e n l o w - l y i n g l e v e l s o f a g i v e nv a l u e o f t h e m a s s A a n d o f i s o s p i n T c o u l d b e q u i t e w e l l r e p r o d u c e d w i t h t h is in t e r -a c t i o n . I n t h i s p a p e r i t w i l l b e s h o w n t h a t t h e a d d i t i o n o f a T - d e p e n d e n t b u t J - i n -d e p e n d e n t t e r m t o t h e S D I g r e a t ly i m p r o v e s t h e a g r e e m e n t w i t h i ) t h e e x p e r i m e n t a l lyo b s e r v e d b i n d i n g e n e r g i e s a n d e x c i t a t i o n e n e r g i e s , i n c l u d i n g t h o s e o f d i f f e r e n t i s o -s p in , i i) t h e t w o - b o d y m a t r i x e l e m e n t s t h a t ar e o b t a i n e d e m p i r ic a l ly f r o m a l ea s t-s qua res f it t o lev e l energ i e s , a nd i i i) t he t w o - bo dy m a t r i x e lem en t s t ha t a re ca l cu l a t edf r o m a r e a li st ic H a m a d a - J o h n s t o n ) p o t e n t i a l.

T h e S D I i s d e f in e d 4 ) a sV s D , j = - - 4 n A r 6 Q i j ) 6 r i R ) 6 r 1 - R ) , 1 )

whe re ~ 2 i1 i s t he a ng u l a r co o rd i n a t e be t w een t he i n t era c t i ng pa r t i c l e s i a nd j , a nd Rt he nuc l ea r ra d ius . T h e t w o s t reng t hs , A 1 f o r a T = 1 co u pl ed pa i r a nd A o f o r T = 0 ,a r e t h e o n l y p a r a m e t e r s t h a t e n t e r t h e e x p r e s s i o n o f t h e t w o - b o d y m a t r i x e l e m e n t s .

U . S . A . E . C . P o s t d oc t o ra l F e l l o w u n d e r a p p o i n t m e n t f r o m O a k R i d g e A s s o c ia t e d U n i ve r si ti es .593

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594 P . W . M . G L A U D E M A N S e t al .A n a n t i - sy m m e t r i z e d tw o - b o d y m a t r ix e l e m e n t o f t h e S D I c a n b e w r i t t en i n j - j

c o u p l i n g a s_ A r { ( 2 j ~ + l ) ( 2 j b + l ) ( 2 j ~ + l ) ( 2 j d + l ) } ~( L i b I VsDI[L. Id>S . T 2 ( 2 J 1~ (1 -k- 6ab)(1 q-t~cd

X [ ( - l ] l a + l e W ~ b ' W j d / ; l i l l [ ~ \ / ; ; 2 r l d 0 > { l _ ( _ 1 )t~ +ld + ~+ T} . / \ J a 2 - J b - - 2 I v / \ J c 2 J d - -- ( L J b I J 1 ) ( . L j d I J 1 ) { 1 + ( - 1)T}], (2)

w h e r e J a n d T a r e t h e s p i n a n d i s o s p i n o f t h e t w o - p a r t i c l e s y s t e m , Jk a n d 1 t h e s p i na n d t h e o r b i ta l a n g u l a r m o m e n t u m o f a p a r ti c le in o r b it k a n d ( j l m l j z m 2 l J M )a C l e b s c h - G o r d a n c o ef fi ci en t. T h e s m a l l n u m b e r o f a d j u s ta b l e p a r a m e t e r s m a k e s t h eS D I a v e r y c o n v e n i e n t r e s i d u a l in t e r a c t i o n f o r u se in s h e l l- m o d e l c a l c u la t i o n s . H o w -e v e r , t h e p o s i t i o n o f T > T_. s t a t e s w i t h r e s p e c t t o t h e p o s i t i o n o f T = T= s t a t e s c a n n o tb e w e ll r e p r o d u c e d b y t h e S D I g i v e n in e q . ( 1 ), a s w i ll b e s h o w n b e l o w .

2 . M o d i fi c at i on o f t h e S D IC o n s i d e r t w o g r o u p s o f e n e r g y l e v el s o f o n e n u c l e u s, o n e g r o u p w i t h T = T 1 a n d

t h e o t h e r w i t h T = T 2. L e t S 1 2 ( e x p ) b e t h e e x p e r i m e n t a l v a l u e o f th e e n e r g y s p a c i n gb e t w e e n t h e l o w e s t s t a t e w i t h T = T 1 a n d t h e l o w e s t s t a t e w i t h T = T 2 , a n d l e tS ~ 2 ( c al c ) b e t h e th e o r e t i c a l v a l u e o f t h i s s p a c i n g a s r e p r o d u c e d w i t h t h e S D I . L e t u sd e n o t e t h e d i f fe r e n c e b e t w e e n t h e e x p e r i m e n t a l a n d c a l c u l a t e d v a l u e s b y ATIT 2 - -$ 1 2 ( e x p ) - S 12 (ca lc) .

T h e v a l u e s o f S ~ 2 (e x p ) a r e ta k e n f r o m t h e e x p e r i m e n t a l l y d e t e rm i n e d e x c i ta t io ne n e r g ie s o f t h e l o w e s t a n a l o g u e s t at e . W h e n t h e s e e n e r g ie s w e r e n o t k n o w n e x p e r i m e n -t a l l y , t h e y w e r e c a l c u l a t e d w i t h a n e x p r e s s i o n f o r C o u l o m b e n e r g i e s g i v e n b y d e -S h a l i t a n d T a l m i 7). T h i s e x p r e s s i o n h a s b e e n u s e d s u c c e ss f u ll y i n p r e v i o u s c a l c u l a -t i o n s a ) . T h e v a l u e s S ~ 2 ( c a l c ) a r e d e r i v e d f r o m s h e l l - m o d e l c a l c u l a t i o n s i n w h i c h t h es t r e n g t h s A I a n d A 0 (s e e eq . ( 1 )) w e r e d e t e r m i n e d s u c h t h a t t h e c a l c u l a t e d e n e r g ie so f t h e m a n y - n u c l e o n c o n f i g u r a t i o n s y i e l d e d a le a s t - s q u a r e s fi t t o e x p e r i m e n t a l l y o b -s e r v e d e n e r g i e s . O n l y t h e e x c i t a t i o n e n e r g i e s o f l e v e l s o f a g i v e n T w i t h r e s p e c t t o t h el o w e s t s ta t e o f t h e s a m e T w e r e f it te d in m a n y n u c l e i s i m u l t a n e o u s l y a n d a n a v e r a g eo f a b o u t t h r e e l o w - l y in g le v e ls i n e a c h n u c l e u s w a s u s e d. N o a t t e m p t s w e r e m a d e t of it t h e b i n d i n g e n e r g ie s o f t h e g r o u n d s t a te s o f t h e v a r i o u s n u c l ei .

I t i s f o u n d t h a t t h e v a l u e s o f AT~T2 a r e r a th e r i n d e p e n d e n t o f m a s s n u m b e r A . T h i si s i l l u s t r a t e d f o r t h e n u c l e i i n th e m a s s r e g i o n 3 0 < A < 3 8 i n f ig . 1, w h e r e A T IT 2 h a sb e e n p l o t t e d a g a i n s t A . I n ta b l e 1 m e a n v a l u e s o f ATIT2 a r e li st ed . T h e s e m e a n v a l u e sw e r e o b t a i n e d b y a v e r a g i n g AT,T2 o v e r t h e n u c l e i i n t h e t h r e e n o n - o v e r l a p p i n g i n -te r v a ls 18__< A < 22 , 23 < A =< 28 a nd 30 < A < 38 .

I n t h e f ir s t t w o i n t e rv a l s t h e l d ~ a n d 2 s~ su b s h e ll s w e r e t a k e n i n t o a c c o u n t a n d i nt h e t h i r d i n t e r v a l t h e 2s + a n d l d , s u b s h e l ls . T h e r a t h e r d i f f e r e n t v a l u e s o f A f a r 2 i n

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T W O - B O D Y M A T R I X E L E M E N T S 595the second interval are pres umabl y caused by the neglect of the ld~ subshell in themodel calculations.

-- -~ - -- -- - o o A 2 06

>

~ A 3 / 2 1 /22 ~ A I O

0 I i I I I i I i I30 32 34 36 38- - ~ , " - M A S S N U M B E RFig. 1. Mass dependence of the deviations ATIT2 for s~r d k shell nuclei. The definition of AT,T2 isgiven in sect. 2.

TABLE 1Values (in MeV) o f , d T l T ~ and DTIT2 for ds shell nuclei

Mass region Configuration A 1 A 0 Z ] l o Dlo Azo D2o z]k~ D{~i

18 < -- A ~ 22 d{ s { 1 .0 1 . 3 2.6 2.4 7.2 7.2 3.3 3.623 < A ~< 28 d{ s~ 1.1 0.3 4.3 3.9 11.2 11.7 5.9 5.930 ~< A < 38 s~ d{ 1.2 0.4 2.4 2.4 7.3 7.2 3.6 3.6The values of A1 and A0 represent the strengths of the SDI as obtained by fitting excitation energies inthe various mass regions. The meaning of the symbols Z]TIT2 n d DTITE is explained in the text.

Analysis shows that the deviat ions At ,r : ca n be compensated fGr by adding aT-d epe nde nt ter m to the S DI defined in eq. (1). Thus we get

V i j = - - 4 ~ z A r 6 ( Q u ) J ( r i - R ) 6 ( r i - R ) + B T . ( 3 )In t erms of two- body matri x elements this modification amou nts to addin g a term B rto the diagon al matri x elements only. The two parameters B 1 and B o of the modifiedsurface delta interaction (MSDI), eq. (3), do not depend on J but only affect theenergy spac ing between the groups of T = T1 and T = T 2 states. There is a simplerelation between At,r2 and the parameter s BT. I n order to derive this relation, let usconsider a diagonal matrix element of the interaction between k particles occupyingthe shells p ~ . . . P k outside a core. This matrix element can be expressed in terms of

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596 v .w .M . GLAUDEMANS t al.t w o - b o d y m a t r i x e l e m e n t s a s( P I . . . P k I H I p x . . " P k ) J T = Z c ( ,1 )x , ( T ' = 1 ) + ~ c~)Xm(T ' = O)

n m + s i n g l e - p a r t i c l e t e r m s . ( 4 )H e r e x , ( T ' = 1 ) a n d X m ( T ' = 0 ) d e n o t e t h e t w o - p a r t i c l e m a t r i x e l e m e n t s f o r p a i r sc o u p l e d t o T ' = 1 a n d T ' = 0 , r e s p e c t i v e l y . T h e c o e f fi c i en t s c(,1) an d e(,, ) a r e de r iv ed

(1) an di t h s t a n d a r d R a c a h t e c h n i q u e s s ) . T h e s u m s o f t h e s e c o ef f ic i en t s, ~ , c , ~ m C ~ ),g i v e t h e n u m b e r o f p a i r s T ' = 1 a n d T ' = 0 , r e s p e c ti v e l y , t h a t a r e p r e s e n t i n t h eo r ig i na l k - p a r ti c le c o n f ig u r a ti o n [ p ~ . . . P R ) s r . T h e t w o - b o d y o p e r a t o r V i j = T zw i t h T ' = t i + t i h a s e i g e n v a l u e s 1 a n d 0 f o r p a i r s T ' = 1 a n d T ' = 0 , r e s p e ct i v e ly .T h e r e f o r e

kZ c ,1) = P , . . . P k l ~ V , j l p l . . . p ~ ) s rn i j

= ( P l . . . p k I T 2 - - ~ , t 2 + E ( t 2 + t E ) l P l "" " P k )s ri i j

= ~ k ( k - 2 ) + r ( T + 1 ), (5 )w h e r e T = ~ t i .F r o m t h e r e la t io n ~ , C t , 1 ) + ~ r , C ~ 0 ) = ( k ) = k ( k - 1 ) , i t f o ll ow s i m m e d i a t e ly t h a t

Z c~ ) = ~ k ( k + 2 ) - T ( T + 1). (6)mF r o m e q s. ( 4 )- ( 6 ), w e s e e t h a t t h e a d d i t i o n o f t h e t w o t e r m s B 1 a n d B o i n e q . ( 3 )c h a n g e s t h e v a l u e o f t h e d i a g o n a l k - p a r t i c le m a t r i x e l e m e n t s o f is o s p in T b y a n a m o u n to f

[ ] k ( k - 2 ) + T ( T + 1)]B~ + [ ~ k ( k + 2 ) - T ( T + 1 ) ] B o . ( 7 )T h i s v a r i a t i o n i s i n d e p e n d e n t o f t h e s p in J o f t h e k - p a r t i c l e s t at e . T h e o f f - d i a g o n a lk - p a r t i c l e m a t r i x e l e m e n t s a r e n o t a f f e c t e d b y t h e t e r m s B 1 a n d B 0 , s o t h a t a l s o inc a s e o f c o n f i g u r a t i o n m i x i n g t h e B - t e r m s w i ll s h i ft t h e c a l c u l a t e d l e v e l e n e r g ie s i n -d e p e n d e n t l y o f J , a s g i v e n in e q . ( 7 ) . T h i s i m p l i e s t h a t , f o r a f ix e d v a l u e o f k , t h e p o s i -t i o n s o f t h e T = T 1 l e v e ls a r e s h if t e d w i t h re s p e c t t o t h e T = T 2 l e v el s b y a n a m o u n t o f

D r ~ r 2 = [ T ~ ( T ~ + 1 ) - T 2( T2 + 1 ) ]( B t - B o ) , (8)a s f o l l o w s d i r e c t l y f r o m e q . ( 7 ) .

E q . ( 8 ) r e p r o d u c e s t h e d e v i a t i o n A I , T 2 v e r y w e l l w i t h a s in g le v a l u e o f B 1 - B o , a si s d e m o n s t r a t e d f o r t h e m e a n v a l u e s A T ,T 2 i n t a b l e 1 f o r t h r e e c o m b i n a t i o n s o f T 1a n d T 2 [ n o te t h a t B I - - B o = D a o a c c o r d i n g to e q. ( 8 )] . T h u s t h e M S D I n o t o n l yr e n d e r s t h e s p e c t r a o f t h e T = T z a n d T > T z l e v e l s ( a s d i d t h e u n m o d i f i e d S D I ) b u ta l s o t h e c o r r e c t s p a c i n g s o f t h e s e g r o u p s o f le v e ls w i t h r e s p e c t t o e a c h o t h e r.

W e s ti ll h a v e t h e v a l u e o f B~ o r B o (t h e i r d i f fe r e n ce h a s b e e n f i x ed a b o v e ) a t o u r

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5 9 8 P W M G L A U D E M A N S e t al .

3 . Com parison of the M S D I with empirical two-body matrix elementsN u m e r i c a l v a l u e s o f t w o - b o d y m a t r i x e l e m e n t s c a n b e d e r iv e d i n a s h e ll - m o d e l

c a l c u l a t io n f r o m e x p e r i m e n t a l n u c l e a r l ev e l s p e c tr a . T h e t w o - b o d y m a t r i x e l e m e n t sa r e c o n s i d e r e d a s fr e e p a r a m e t e r s t o b e d e t e r m i n e d f r o m a l e a st - sq u a r e s f it ti n g p r o c e -d u r e t o t h e e n e r g y l e v el s.

I t is i n t e r e s t in g t o c o m p a r e s u c h e m p i r i c a l m a t r i x e l e m e n t s w i t h th o s e c a l c u l a t e dw i t h t h e M S D I .3 .1 . M A T R I X E L E M E N T S O F s ~ d k A N D d ~If1_ C O N F I G U R A T I O N S

E m p i r i c a l t w o - b o d y m a t r i x e l e m e n ts o b t a i n e d i n t h e 2 s I d s u b sh e ll s 2 ) a n d i n t h el d l l f ~ s u b sh e ll s s ) a r e c o m p a r e d w i th t h o s e o f t h e M S D I i n f ig . 3 . T h e v a l u es o f t h eM S D I t w o - b o d y m a t r i x e l e m e n t s a r e c a l c u la t e d w i th t h e v a l u e s o f A ~ , A o, B ~ a n d B ot h a t w e r e d e t e r m i n e d i n a l e a s t- s q u a r e s f it t o e x p e r i m e n t a l e n e r g ie s o f s~ d ~ c o n f i g u r a -t i o n s ( s e e s e c t . 2 ) .

>tl)~ - o

. $ 1 / 2 d 3 / 2/ /

o

. - Y = l Y : O , ,, .

M S D Io e m p i r i c a l d 3 / 2 f 7 / 2

t

'gI I ~ I I I ~ I I I I I

9 - T = I , . , - - - T = 0 -

Fig. 3. Values of two -bod y ma trix elements calculated fr om the M SD I with A1 = 1.2 M eV, A0 = 0.4M eV, B1 = 0.7 MeV and B0 ~ --1. 8 M eV are com pare d with matrix elements obtained empirically.T h e n l j values are specified by 1 for 2sx, 3 for ld~ and 7 for lf~.

S o m e a g r e e m e n t b e tw e e n M S D I a n d e m p i ri c al t w o - b o d y m a t r ix e l em e n ts f o r t hes +d ~ c o n f i g u r a t i o n s s h o u l d b e e x p e ct e d , s i nc e b o t h r e p r o d u c e t h e s a m e e x p e r i m e n ta ls p e c t r a. I t is r e m a r k a b l e t o s e e , h o w e v e r , t h a t a l s o f o r d ~f_ ~ c o n f i g u r a t i o n s t h e t w os e ts o f m a t r i x e l e m e n t s a g r e e v e r y w e l l , e s p e c i a l ly s i n c e t h e v a l u e s o f A 1 , A o , B 1 a n dB o a r e t a k e n e x a c t l y t h e s a m e a s t h o s e u s e d f o r t h e s + d~ m a t r i x e l e m e n t s.3.2. M A T R I X E L E M E N T S O F T H E fk P k C O N F I G U R A T I O N

V a l u e s o f th e M S D I p a r a m e t e r s A 1 a n d B 1 c a n b e d e t e r m i n e d b y a l e a s t -s q u a r e sf i tt in g p r o c e d u r e t o l ev e ls o f th e 4 ~ C a t o 4S C a i s o t o p e s w i t h f ~ p ~ c o n f i g u r a t i o n s . T h e

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TWO-BODY MATRIX ELEMENTS 59 9

b e s t fi t t o e i g h t g r o u n d s t a te s a n d 2 0 lo w - l y i n g e x c i te d s t a te s is o b t a i n e d w i t h A ~ = 0 . 6M e V , B ~ = 0 . 2 M e V . T h e a v e r a g e d e v i a t i o n b e t w e e n e x p e r i m e n t a l l y d e t e r m i n e d a n dc a l c u l a t e d e n e r g ie s o f g r o u n d s t at e s a n d l e v e ls w a s f o u n d t o b e 0 .2 M e V .

V a l u e s o f e m p i r i c a l t w o - b o d y m a t r i x e l e m e n t s f o r f k p ~ c o n f ig u r a t i o n s h a v e b e e no b t a i n e d , e .g . b y E n g e l a n d a n d O s n e s 9) a n d b y F e d e r m a n a n d T a l m i x 0 ). I n f ig . 4t h es e e m p i r i c a l m a t r i x e l e m e n t s a re c o m p a r e d w i t h t h o se c a l c u la t e d f r o m t h e M S D If i t . T h e r e i s a g a i n a g o o d c o r r e s p o n d e n c e b e t w e e n t h e M S D I a n d e m p i r i c a l v a l u e s o fa l m o s t a l l m a t r i x e l e m e n t s .

2

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- 2

- 4

MSDIemp i r i ca l

J , \

A~ 3o 2 , 6 2 3 , ~ o 2 ~ 4 o 2 2. . . . . . . . . ' ~' , , .

Fig. 4. Values of the T = 1 two- body matrix elements calculated from MS DI with A x = 0.6 MeVand B1 = 0.2 MeV are co mpar ed with ma trix elements obtain ed empirically. The values indicatedby the triangles are from ref.9), the open circles are from ref.l). The n O values are specified by 7

for Ifk and 3 for lpk.

U n f o r t u n a t e l y n o e m p i r i c a l T = 0 , t w o - b o d y m a t r i x e l e m e n t s a re a v a i l a b l e f o rc o m p a r i s o n , s o th a t w e h a v e n o t a t t e m p t e d t o d e t e r m i n e t h e p a r a m e t e r s A o a n d B of o r f~ p ~ c o n f i g u r a t i o n s .

4 . C o m p a r i s o n o f t h e M S D I w i t h m a t r i x e l e m e n t s f r o m a r e a li s ti c p o t en t i alR e c e n t l y , p r o g r e s s h a s b e e n m a d e i n d e d u c i n g t w o - b o d y m a t r i x e le m e n t s f r o m a

r e a li s ti c t w o - b o d y i n t e r a c t i o n w i th a h a r d c o r e s u c h as t h e H a m a d a - J o h n s t o n p o t e n -t ia l . S u c h r e s u l t s f o r T = 1 t w o - b o d y m a t r i x e l e m e n t s o f p ~ p ~ f ~ c o n f i g u r a t i o n s 6 )a r e c o m p a r e d i n f ig . 5 w i t h t h e M S D I m a t r i x e l e m e n t s , c a l c u l a t e d f o r t h e s a m e v a l u e so f A 1 a n d B 1 a s w e r e g i v e n i n s u b s e c t . 3 .2 .

I t s h o u l d b e n o t e d t h a t t h e q u o t e d n u m e r i c a l v a lu e s o f t h e t w o - b o d y m a t r i x e l e m e n t so f t h e H a m a d a - J o h n s t o n p o t e n t i a l a r e t h e r e su l ts o f a l e n g t h y c a l c u l a t i o n in w h i c hv e r y a p p r e c i a b l e c o r r e c t i o n s d u e t o c o r e e x c i t a t io n s 6) h a d t o b e t a k e n i n t o a c c o u n t .T h e M S D I , b y s h o w i n g a g r e e m e n t w i t h th e se c o r r ec t e d m a t r i x e le m e n ts , a p p e a r s to

8/12/2019 Glaudemans 67 Matrix

8/9

8/12/2019 Glaudemans 67 Matrix

9/9

TWO-BODY MATRIX ELEMENTS 601t e r m B 1 d i m i n i s h e s t h e m a g n i t u d e o f a l m o s t a ll d i a g o n a l T ~- 1 m a t r i x e l e m e n t s .

I t i s w o r t h w h i l e t o r e m a r k t h a t t h e a d d i t i o n o f t h e t e r m B 1 t o t h e p a r t i c le - p a r t i c lei n t e r a c t i o n w i l l s h i ft t h e d i a g o n a l p a r t i c l e - h o l e m a t r i x e l e m e n t s ~Pl P2 1V 121PlP2 ) j rb y t h e a m o u n t -B I + B o ) f o r T = 1 a n d by - 3 B 1 - B o ) f o r T = 0 . I s o sp i n -d e p e n d e n c e o f p a r t i c l e - h o l e i n t e r a c t i o n s h a s b e e n d i s c u s s e d i n r e fs . 1 4 ,1 5) .

T h e a u t h o r s a re v e r y m u c h i n d e b t e d t o D r . E . C . H a l b e rt a n d D r . J . B . M c G r o r yf o r t h ei r p e r m i s s i o n t o m a k e u s e o f t h ei r s h e ll - m o d e l c o m p u t e r c o d e s a n d f o r m a n yh e l p f u l c o n v e r s a t i o n s .

R e f e r e n c e s1) R. Arvieu and S. A. Moszkowski, Phys. Rev. 145 (1966) 8302) P. W. M. Glaudemans, B. H. Wildenthal and J. B. McGrory, Phys. Lett. 21 (1966) 4273) B. H. Wildenthal, P. W. M. Glaudemans, E. C. Halbert and J. B. McGrory, Bull. Am. Phys.

Soc. 12 (1967) 484) A. Plastino, R. Arvieu and S. A. Moszkowski, Phys. Rev. 145 (1966) 8375) F. C. Ern6, Nuc lear Physics 84 (1966) 916) R. D. Lawson, M. H. Macfa rlane and T. T. S. Kuo, Phys. Lett. 22 (1966) 1687) A. de-Shalit and I. Talmi, Nuclear s h e l l t h e o r y (Academic Press, New York, 1963) eq. (30.2)8) P. W. M. Glaudemans, G. Wiechers and P. J. Brussaard, Nuclear Physics 56 (1964) 5299) T. Engeland and E. Osnes, Phys. Lett. 20 (1966) 42410) P. Federman and I. Talmi, Phys. Lett. 22 (1966) 46611) T. T. S. Kuo, p r i va t e c o m m u n i c a t i o n12) S. Cohe n and D. Kurat h, Nuclea r Physics 73 (1965) 1

13) S. Cohe n, R. D. Lawson, M. H. Macfa rlane and M. Soga, Phys. Lett. 10 (1964) 19514) R. K. Bansal and J. B. French, Phys. Let t . 11 (1964) 14515) L. Zamick, Phys. Lett . 19 (1965) 580