Force dependence of 0+ states in 208Bi

Force dependence of 0+ states in 208Bi

Volume 38B, n u m b e r 3 PHYSICS FORCE DEPENDENCE LETTERS OF 0 + 7 F e b r u a r y 1972 STATES IN 208Bi J. RAJEWSKI Deparlmen! of P h y s ...

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Volume 38B, n u m b e r 3

PHYSICS

FORCE

DEPENDENCE

LETTERS

OF

0 +

7 F e b r u a r y 1972

STATES

IN

208Bi

J. RAJEWSKI Deparlmen! of P h y s i c s , Wcizn~a~ln lnsliluA ~ qf Scienee, Rehovol, Israel and DeparlnTen! qf P h y s i e s , Nuclear Reseavch Centre-Negec, P.O.B.9001, B e e r Sheva, Israel

and M. W

KIIISON

Depavl~enl qf P h y s i c s , Wcizn~ann lnslilule of Science, Reh~)vol, Israel Received 2 J a n u a r y 1972

The 0 energy levels of 208Bi are calculated with the reaction m a t r i x deduced from the Sussex and Tabakin potentials. Special attention is paid to the i s o b a r i c analogue of the ground state of 208Pb which comes 4-5 MeV below the e x p e r i m e n t a l value, and is p a r t i c u l a r l y sensitive to the potential.

T h e low l y i n g s t a t e s of 2 0 8 B i c a n b e d e s c r i b e d a s a p r o t o n - p a r t i c l e a n d n e u t r o n - h o l e on a c l o s e d 2 0 8 p b c o r e [ 1 ] T h e n t h e e i g e n v a l u e s a n d e i g e n f u n c t i o n s of 2 0 8 B i c a n b e s i m p l y o b t a i n e d b y d i a g o n a l i z i n g the secular matrix (epl-ehl)512

+ ( P l h l l i Veffip2h211 ~ ,

(1)

w h e r e e p l a n d e h l a r e t h e s i n g l e p a r t i c l e a n d s i n g l e h o l e e n e r g i e s , and Veff i s t h e e f f e c t i v e i n t e r a c t i o n w h i c h c a n b e c a l c u l a t e d f r o m a n u c l e o n - n u c l e o n p o t e n t i a l b a s e d on n u c l e o n - n u c l e o n s c a t t e r i n g d a t a u s i n g t h e B l o c h - H o r o w i t z - B r a n d o w t h e o r y 12]. Kuo p e r f o r m e d a d e t a i l e d c a l c u l a t i o n of 2 0 8 B i e n e r g y l e v e l s [1] with t h e r e a c t i o n m a t r i x e l e m e n t s d e d u c e d f r o m 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 [3]. In t h i s c a l c u l a t i o n the e f f e c t i v e i n t e r a c t i o n i n c l u d e s t h e B r u e c k n e r G - m a t r i x a n d t h e s e c o n d o r d e r d i a g r a m s in B l o c h - H o r o w i t z - B r a n d o w e x p a n s i o n . It a p p e a r s , h o w e v e r , t h a t t h e s e c o n d o r d e r c o r r e c t i o n s to t h e G - m a t r i x a r e g e n e r a l l y u n i m p o r t a n t . T h e fit to t h e e x p e r i m e n t a l e n e r g y l e v e l s i s r e m a r k a b l y good. A r a t h e r s e r i o u s d i s c r e p a n c y w a s f o u n d in K u o ' s c a l c u l a t i o n b e t w e e n t h e t h e o r e t i c a l a n d e x p e r i m e n t a l p o s i t i o n s of t h e i s o b a r i c a n a l o g u e of t h e g r o u n d s t a t e of 2 0 8 P b . A l t h o u g h t h e a d m i x t u r e s ( s q u a r e d a m p l i .-1. s t a t e s ) of t h e c a l c u l a t e d l e v e l c l o s e l y f o l l o w the (2j + 1) p r o p o r t i o n s f o r a p u r e i s o b a r i c t u d e s of I. EJpJh a n a l o g u e s t a t e , t h e c a l c u l a t e d v a l u e of 14.23 M e V f o r t h e e n e r g y of t h i s s t a t e i s a b o u t 4 M e V l o w e r t h a n t h e e x p e r i m e n t a l v a l u e [4,5]. T h e a i m of t h e p r e s e n t w o r k i s to c h e c k t h e d e p e n d e n c e of t h e 0 + e i g e n v a l u e s a n d e i g e n f u n e t i o n s on t h e p a r t i c u l a r n u c l e o n - n u c l e o n p o t e n t i a l used. A c c o r d i n g l y K u o ' s c a l c u l a t i o n is r e p e a t e d for the 0+ s t a t e s with two frequently used potentials: Sussex and Tabakin. T h e 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 ill eq. (1) a r e c a l c u l a t e d b y { ( p l h l 1 ) J T i Veff!(P2h21) J T

J'T'

IJP2Jh2 J'

=

l½ ½

' [(l+Splh2)(l+SP2hl)]l/2{(Plh2)d'T'lVeffi(P2hl)J'T')

(2)

H e r e J = 2 J + l a n d t h e s y m b o l s e n c l o s e d in c u r l y b r a c k e t s a r e t h e s t a n d a r d d - j c o e f f i c i e n t s . We n o t e t h a t p a l w a y s r e f e r s to p r o t o n p a r t i c l e a n d h to n e u t r o n h o l e . B o t h t h e p a r t i c l e - p a r t i c l e a n d p a r t i c l e - h o l e matrix elements are antisymmetrized. T h e s i n g l e p a r t i c l e b a s i s in o u r c a l c u l a t i o n , a s in K u o ' s , i s of h a r m o n i c o s c i l l a t o r (h.o.) w a v e f u n c t i o n s . 162

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LETTERS

7 F e b r u a r y 1972

Thus the particle-particle s t a t e s ] ( a b ) J T ) c a n e a s i l y b e e x p a n d e d a s f i n i t e s u m s of s t a t e s !(nlS)~;N~2, JT', w h e r e l a n d /2 a r e t h e r e l a t i v e o r b i t a l a n g u l a r m o m e n t u m a n d t h e a n g u l a r m o m e n t u m of the c e n t r e - m mass motion respectively, and n and N are the corresponding radial quantum numbers. T h u s t h e p a r t i c l e - p a r t i c l e m a t r i x e l e m e n t s of eq. (2) c a n b e t r a n s f o r m e d i n t o r e l a t i v e - c e n t r e - o f - m a s s

-

<(ab)JTiVeff'~(cd)JT): [(l+5ab)(l+Scd)]-I/2

E A

LL'S

×

~ {nlN~LlnalanblbL)(n'l'N,fJ'L'[nclcnd nln 'l ' N.~ N 'i~'

× G U(J-)lJS;L~) U(J-)'I'JS;L'~) ((nlS)~,NJ2, T

½Jb} A

SJ)

IlL'sJ]

~ Jdl (-i)

l d L ' : (_I)I+I'(I_(_I)I+S+T)

[Veff l(n 'l'S)f?,N'~2'.

T>

(3)

H e r e (nlNl2Lin al a ~ l b L } i s t h e B r o d y - M o s h i n s k y t r a n s f o r m a t i o n b r a c k e t [7], t h e A ' s a r e t h e LS to j j - c o u p l i n g t r a n s f o r m a t i o n c o e f f i c i e n t s [8] a n d U(J21JS: 1_'~) t h e n o r m a l i z e d R a c a h c o e f f i c i e n t s

U(f:~lJS;Z~) =

> -

)2+z+J+s l.t LI ~S J 2 ~

(4)

O u r c a l c u l a t i o n s a r e f i r s t o r d e r in t h e B l o c h - H o r o w i t z - B r a n d o w e x p a n s i o n [2], r e l y i n g on K u o ' s r e s u l t s f o r t h e r e l a t i v e u n i m p o r t a n c e of s e c o n d o r d e r h e r e . T h e s i n g l e - p a r t i c l e a n d s i n g l e - h o l e o r b i t s in w h i c h t h e s e c u l a r m a t r i x (eq. (1)) i s d i a g o n a l i z e d a r e : ( 0 h 9 / 2 , l f T / 2 , l f 5 / 2 , 2P3/2 , 2 p l , / 2 , 0 i 3 / 2 ) " T h e p r o t o n - p a r t i c l e a n d n e u t r o n - h o l e e n e r g i e s u s e d in t h e d i a g o n a l i z a t i o n a r e t h e e x p e r i m e n t a l e n e r g i e s u s e d b y Kuo [1]. T h e S u s s e x g r o u p e x t r a c t e d t h e m a t r i x e l e m e n t s of t h e n u c l e o n - n u c l e o n e f f e c t i v e p o t e n t i a l d i r e c t l y f r o m t h e n u c l e o n - n u c l e o n p h a s e s h i f t s [9]. T h e i n t e r p r e t a t i o n of t h e s e m a t r i x e l e m e n t s f o r a s i n g u l a r p o t e n t i a l i s r e l a t e d to t h e s e p a r a t i o n m e t h o d of S c o t t - M o s z k o w s k i [10] a s a p p l i e d to f i n i t e n u c l e i b y Kuo a n d B r o w n [3], a s s h o w n in r e f . [ l l ] . T h u s t h e y a r e e s s e n t i a l l y e q u i v a l e n t to the B r u e c k n e r G - m a t r i x e l e m e n t s in r e l a t i v e c o o r d i n a t e s p a c e . S u s s e x m a t r i x e l e m e n t s a r e t a b u l a t e d in r e f . [9] f o r s t a t e s w i t h l :~ 4, n < 6 a n d v a r i o u s v a l u e s f o r t h e o s c i l l a t o r l e n g t h p a r a m e t e r b =~/~/'mw. U s i n g K u o ' s //w = 7 M e V we get b = 2.43 fm. In o u r f i r s t c a l c u l a t i o n (to b e d e n o t e d SI) t h e h i g h e r p a r t i a l w a v e s w e r e n e g l e c t e d . In t h e s e c o n d (to b e d e n o t e d SII) t h e y w e r e c a l c u l a t e d u s i n g o n e - p i o n - e x c h a n g e - p o t e n t i a l ( O . P . E . P . ) [12,13]. T h e p a r a m e t e r s f o r t h i s p o t e n t i a l w e r e t a k e n f r o m r e f [13]. T h e T a b a k i n p o t e n t i a l [6,14] i s a s u m of s e p a r a b l e p o t e n t i a l s f o r m i n g a s m o o t h n o n - l o c a l i n t e r a c t i o n . T h e B r u e c k n e r G - m a t r i x ( a n d o u r Vef f) is h e r e a p p r o x i m a t e d b y f i r s t a n d s e c o n d o r d e r l a d d e r d i a g r a m s in t h e p o t e n t i a l . T h i s a m o u n t s to p l u g g i n g into t h e r . h . s , of eq. (3)

{(nlS !p, N f 2 , T l V e f f [ ( n ' l ' S ) p , N ' Z ' ,

V } : [((nlS)~IV](n'l'S)p>

5NN, - B ( n l N . £ J , n ' l ' N ' z ; s r ) ] s z ]

2, ,

(5)

where

B ( n l N . L ), n'I'N'£O;ST) : E {(nlS)p, Nf_2,JT lV[fi}{;3 [VI(n'l'S)9 , N'l.), J r }

e

(6)

-Wo

l

H e r e !;3) a r e s t a t e s in w h i c h two p a r t i c l e s h a v e b e e n e x c i t e d to a n e n e r g y m u c h h i g h e r t h a n t h e s t a t e s a a n d b , W[3 i s t h e e n e r g y of t h e s t a t e ];3) a n d Wo t h e e n e r g y of e i t h e r t h e i n i t i a l o r f i n a l two p a r t i c l e state. T h e f u n c t i o n a l f o r m of V i s g i v e n in r e f s . [6,14]. T h e p a r a m e t e r s of t h e p o t e n t i a l u s e d in o u r c a l c u l a t i o n a r e t h e " n e w " o n e s of [6]. T h e s e p a r a m e t e r s a r e g i v e n t h e r e only f o r m a t r i x e l e m e n t s w i t h l,l'--< 2. In o u r c a l c u l a t i o n ( d e n o t e d TS) we t a k e t h e e l e m e n t s w i t h 2 ~< l ~ 4 f r o m t h e S u s s e x s e t a n d t h o s e w i t h l > 4 f r o m O . P . E . P . , a s in SII. In t h e c a l c u l a t i o n of eq. (5) we u s e o n c e m o r e K u o ' s /iw = 7 MeV. T h e i n t e r m e d i a t e s t a t e s [;3} a r e t h o s e of two p a r t i c l e s in p l a n e w a v e s t a t e s . O n l y s t a t e s w i t h p a r t i c l e m o m e n t a g r e a t e r t h a n t h e F e r m i m o m e n t u m a r e a l l o w e d . T h u s W/3 c o m e s w e l l a b o v e 100 MeV a n d t h e r e s u l t s b e c o m e v e r y i n s e n s i t i v e to 163

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t h e p r e c i s e c h o i c e of Wo. In o u r c a l c u l a t i o n we c h o o s e W o = - 10 M e V w h i c h s e e m s to b e a r e a s o n a b l e a v e r a g e e n e r g y in t h i s m o d e l s p a c e . T h e r e s u l t i n g 0 + e n e r g i e s a r e l i s t e d in t a b l e 1. T h e h i g h e s t 0 + l e v e l a p p e a r s t h e m o s t s e n s i t i v e to v a r i a t i o n s of t h e n u c l e o n - n u c l e o n p o t e n t i a l . C o m p a r i s o n w i t h Kuo [1] s h o w s t h a t t h e S u s s e x p o t e n t i a l i s in s o m e s e n s e w e a k e r t h a n T a b a k i n ' s w h i c h i s i t s e l f w e a k e r t h a n the 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 l o w e s t 0 + s t a t e a l s o a p p e a r s to b e v e r y s e n s i t i v e to the f o r c e u s e d . A c o m p a r i s o n of t h e SI a n d SII n u m b e r s i n d i c a t e s t h a t t h i s i s due a l m o s t e n t i r e l y to t h e O . P . E . P . c o n t r i b u t i o n , a n d i s t h u s not v e r y r e l i a b l e . It s h o u l d b e n o t e d t h a t t h e ene%~)~ of 5.365 M e V r e l a t i v e to t h e 2 0 8 p b g r o u n d s t a t e c o r r e s p o n d s to a n e x c i t a t i o n e n e r g y of 2.495 M e V in Bi. T h i s i s w e l l w i t h i n t h e e x p e r i m e n t a l l y _ o b s e r v e d r e g i o n w h e r e no 0 + s t a t e s o c c u r . T h e s t r u c t u r e of t h e h i g h e s t 0 + s t a t e i s g i v e n in t a b l e 2. All c h o i c e s of t h e n u c l e o n - n u c l e o n p o t e n t i a l g i v e v e r y s i m i l a r r e s u l t s . In all t h e c a l c u l a t i o n s t h e p a r t i c l e - h o l e a m p l i t u d e s c l o s e l y f o l l o w the . ~ j + l p r o p o r t i o n s f o r a p u r e i s o b a r i c a n a l o g u e s t a t e . T h e o v e r l a p s of t h e c a l c u l a t e d s t a t e s with t h e a n a l o g u e a r e v e r y c l o s e to 1. T h i s i s not v e r y s u r p r i s i n g a s it r e f l e c t s t h e f a c t t h a t t h e f o r c e s u s e d in t h e v a r i o u s calculations are isospin invariant. T h e c a l c u l a t e d v a l u e s f o r t h e e n e r g y of t h i s 0 + s t a t e d i f f e r v e r y m u c h f r o m t h e e x p e r i m e n t a l v a l u e of t h e i s o b a r i c a n a l o g u e s t a t e , 18.09 M e V [5], t h e d i s c r e p a n c y b e i n g e v e n l a r g e r f o r S u s s e x a n d T a b a k i n forces than for Kuo's Hamada-Johnston result. We now a t t e m p t to t r a c e t h e s o u r c e of t h e d i s c r e p a n c y b y c o n s i d e r i n g d i f f e r e n t c o n t r i b u t i o n s to t h e a n a l o g u e s t a t e e n e r g y . S i n c e t h e e i g e n s t a t e s f o u n d b y d i a g o n a l i z a t i o n a r e s o c l o s e to t h e a n a l o g u e s t a t e , we s i m p l y u s e t h e a n a l o g u e w a v e f u n c t i o n 1

~--'~J~/j'i(jj-1)j=O r = l }

(7)

J a n d u s i n g eq. (2) we o b t a i n


- 44.1 ~ (-1) Ja--
T'

J'<(jaJb)J'T'lVefflOaJb)J'T'}

(8)

J'T'

H e r e Ja a n d j b a r e t h e s.p. o r b i t s in o u r m o d e l s p a c e . T r a n s f o r m i n g t h e r . h . s , m a t r i x e l e m e n t to r e l a t i v e a n d c e n t r e - o f - m a s s c o o r d i n a t e s u s i n g eq. (3) we g e t e x p l i c i t e x p r e s s i o n s f o r t h e i n d i v i d u a l p o t e n t i a l m a t r i x e l e m e n t c o n t r i b u t i o n s to t h e a n a l o g u e s t a t e e n e r g y . T h e c a l c u l a t i o n w a s p e r f o r m e d f o r SII. T h e t o t a l c o n t r i b u t i o n of T' = 0 m a t r i x e l e m e n t s to (8) i s 8.235 M e V w h i l e t h e T ' = 1 co~ntribution i s - 2.842 M e V (to t h e s e m u s t b e a d d e d t h e s i n g l e p a r t i c l e e n e r g y of t h e p a r t i c l e - h o l e p a i r , r o u g h l y 7 MeV). T h i s s u g g e s t s a p o s s i b l e s t r o n g f o r c e d e p e n d e n c e s i n c e t h e T' = 0 p o t e n t i a l i s l e s s w e l l - k n o w n t h a n T' = 1. T h e l a r g e s t s i n g l e c o n t r i b u t i o n , 7.035 M e V , c o m e s f r o m the c o u p l e d 3S 1 - 3D 1 c h a n n e l s . T h e off d i a g o n a l < 3 S liVI3D1} t e r m s c o n t r i b u t e o n l y - 0 . 0 0 9 MeV. M a n i p u l a t i o n of t h e c e n t r a l / t e n s o r f o r c e r a t i o in t h e s e c h a n n e l s c o u l d a f f e c t the f i n a l a n a l o g u e e n e r g y q u i t e c o n s i d e r a b l y , b u t h a r d l y e n o u g h to a c c o u n t f o r a s i g n i f i c a n t p o r t i o n of t h e o b s e r v e d d i s c r e p a n c y . T h e T = 1 r e s u l t s a r e s i m i l a r l y d o m i n a t e d b y t h e 1S 0 c h a n n e l ( - 2.066 MeV). T h o u g h o t h e r i n d i v i d u a l T' = 1 t e r m s a r e q u i t e l a r g e , t h e y t e n d to c a n c e l o n e a n o t h e r . T h e c o n t r i b u t i o n of t h e m o s t a m b i g u o u s c h a n n e l [13], 1 P 1 , i s - 0 . 9 5 4 M e V w h i c h i s q u i t e big. So d i s c r e p a n c i e s in t h i s p a r t of t h e f o r c e c a n s h o w t h e m s e l v e s s t r o n g l y in t h e e n e r g y of t h e a n a i g o u e s t a t e . Such d i s c r e p a n c i e s a c t u a l l y o c c u r ; f o r i n s t a n c e t h e S = T = 0, n = n' = 0, l = l' = ~ = 1 d i a g o n a l m a t r i x e l e m e n t of K u o ' s G i s 0.606 M e V w h i l e t h e c o r r e s p o n d i n g S u s s e x e l e m e n t i s 0.36 M e V [ a l s o 13]. T h e c o n t r i b u t i o n of < 3 p 2 1 V t 3 p 2 } e l e m e n t s i s - 1.052 M e V a n d t h a t of ( 3 P l l VI3p 1> i s 0.173 MeV. T h e t o t a l 31° d i a g o n a l c o n t r i b u t i o n i s - 0 . 3 0 1 MeV. T h e off d i a g o n a l c o n t r i b u t i o n s a r e g e n e r a l l y s m a l l (<3102!VI3F 2} = - 0 . 0 3 5 MeV). G e n e r a l l y we a r e d e a l i n g w i t h c a n c e l l a t i o n s of q u i t e b i g n u m b e r s w h i c h i s p r o b a b l y r e l a t e d to the e x c e p t i o n a l f o r c e d e p e n d e n c e of t h e h i g h e s t 0 + s t a t e . A s f o r t h e O . P . E . P . c o n t r i b u t i o n to SII it i s of o r d e r of 300 keV. T h i s a g a i n c o m e s f r o m c a n c e l l a t i o n of i n d i v i d u a l c h a n n e l c o n t r i b u t i o n s r a t h e r t h a n f r o m t h e i r s m a l l n e s s . A l t h o u g h t h e s t a n d a r d n u c l e o n n u c l e o n p o t e n t i a l s a r e b u i l t to a p p r o a c h a s y m p t o t i c a l l y t h e O . P . E . P . we a r e not q u i t e c o n f i d e n t u s i n g t h e O.10.E.10 f o r w a v e s a s low a s l = 5. H o w e v e r t h e s e c o n t r i b u t e in SII o n l y - 6 2 1 k e V a n d we do not e x p e c t d r a m a t i c e f f e c t s by t r e a t i n g t h e s e w a v e s m o r e a c c u r a t e l y . 164

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Table 1 + Energy levels of 0 states in 208Bi in MeV. The last column gives Kuo's results [i]. Other headings are expIained in the text. All these are e n e r g i e s relative to the 208pb ground state. SI

SII

TS

Kuo

6.8021 6.8746 7.6369 7.7569 8.4844 12.1654

5.1820 6.8664 7.6411 7.6673 8.4475 12.4636

5.3645 6.9265 7.6695 8.2633 8.8475 12.5744

6.395 6.868 7.429 7.905 9.183 14.228

T h e q u e s t i o n of h i g h e r o r d e r c o r r e c t i o n s in B l o c h - H o r o w i t z - B r a n d o w e x p a n s i o n i s s t i l l o p e n . T h e y c a n c e l , to s e c o n d o r d e r , in K u o ' s c a l c u l a t i o n but t h e i n d i v i d u a l d i a g r a m s a r e q u i t e big and in v i e w of the f o r c e d e p e n d e n c e in t h e f i r s t o r d e r c a l c u l a t i o n we s h o u l d b e r e a d y f o r s u r p r i s e s . H i g h e r o r d e r , c o l l e c t i v e effects could also be important. The calculations involved are rather lengthy as many partial wave matrix elements appear• In k e e p i n g with p r a c t i c e in the p r e c e d i n g w o r k [1] we h a v e u s e d the s a m e l e n g t h p a r a m e t e r f o r p r o t o n s and n e u t r o n s in a l l o u r a b o v e c a l c u l a t i o n s . H o w e v e r if t h e p r o t o n and n e u t r o n d e n s i t i e s h a v e m o r e o r l e s s t h e s a m e r . m . s , r a d i u s , then s i n c e the n e u t r o n s a r e f i l l i n g h i g h e r h.o. l e v e l s t h a n the p r o t o n s , one h a s to u s e h i g h e r //co f o r n e u t r o n s than f o r p r o t o n s [15,16]. U s i n g 5.46 f m a s the r . m . s , r a d i u s f o r b o t h p r o t o n s and n e u t r o n s and f i l l i n g h.o. s t a t e s one o b t a i n s the l e n g t h p a r a m e t e r s bp 2.49 f m and b n = 2.33 fm. T h e r e l a t i v e l e n g t h p a r a m e t e r i s b = 2.41 f m , c l o s e to t h e one we h a v e u s e d f o r e q u a l w e l l s . A c c o r d i n g l y we h a v e r e p e a t e d o u r SII c a l c u l a t i o n with d i f f e r e n t w e l l s f o r p r o t o n s and n e u t r o n s . We h a v e used" t h e s a m e r e l a t i v e m a t r i x e l e m e n t s a s f o r the e q u a l w e l l s c a s e . but a p p r o p r i a t e l y c h a n g e d the B r o d y - M o s h i n s k y t r a n s f o r m a t i o n b r a c k e t s in eq. (3). T h i s , h o w e v e r , d o e s not a c c o u n t f o r t h e d i s c r e p a n c y . T h e h i g h e s t 0 + s t a t e s t i l l h a s 0.994 o v e r l a p with t h e a n a l o g u e s t a t e a n d i t s e n e r g y g o e s down 230 keV c o m p a r e d to the e q u a l w e l l s r e s u l t , s o m e w h a t i n c r e a s i n g the d i s c r e p a n c y . T h e r e i s a n y w a y not m u c h m e a n i n g to t h i s n u m b e r in v i e w of the i n a c c u r a c i e s of t h e S u s s e x m a t r i x e l e m e n t s and the s m a l l d i f f e r e n c e b e t w e e n t h e b we u s e d and the r i g h t one. K a h a n a et al. [15] u s e d W o o d s - S a x o n w e l l s f o r t h e p r o t o n s and n e u t r o n s , c h o o s i n g t h e p a r a m e t e r s s o a s to r e p r o d u c e t h e o b s e r v e d a n a l o g u e s t a t e e n e r g y . T h e i r s i n g l e p a r t i c l e w e l l s a r e b e s t r e p r e s e n t e d by h.o. w e l l s w i t h fiw = 7 MeV f o r p r o t o n p a r t i c l e s a n d he0 = 8.25 MeV f o r n e u t r o n h o l e s . Such l a r g e v a l u e s f o r ha) l e a d to a c o n s i d e r a b l y s m a l l e r r e l a t i v e b , and h e n c e to c o n s i d e r a b l y m o r e a t t r a c t i v e i n t e r a c t i o n m a t r i x e l e m e n t s . A l t h o u g h it i s p o s s i b l e to e l i m i n a t e the d i s c r e p a n c y in t h i s w a y , t h e r e s e e m s to be no a p r i o r i j u s t i f i c a t i o n f o r s u c h a c h o i c e of i/w, w i t h h.o. s i n g l e - p a r t i c l e w e l l s . C o n c l u d i n g , we c a n n o t i n d i c a t e any p a r t i c u l a r r e a s o n f o r t h e a n a l o g u e s t a t e c o m i n g 4-5 MeV l o w e r t h a n t h e e x p e r i m e n t a l v a l u e , a l t h o u g h it i s not e x c l u d e d t h a t a b e t t e r c h o i c e of n u c l e o n - n u c l e o n p o t e n t i a l c o u l d a c c o u n t f o r a l a r g e p a r t of the d i s c r e p a n c y . If t h i s i s s o , t h e n we h a v e , in t h i s u n u s u a l l y s e n s i t i v e s t a t e , a good tool to c h e c k f u t u r e n u c l e o n - n u c l e o n p o t e n t i a l s . Table 2 The wave-functions for the highest 0 state. The last column gives the overlaps of tile calculated wavefunetion with the pure anaiogue state given in the last row. "-2 ........... iriS1

<.pn

ft

9-1

7

7-1

5

5-1

33-1

I 2

0.387

0.342

0.265

: - 1 2

I=! 1:~- I

ovcrlat)

0.487

0.994

calculation"ff'--~ SI

0..507

0.412

SII

0.518

0.394

0.374

0.345

0.263

0.499

0.988

TS

0.527

0.368

0.335

0.331

0.251

0.547

0.993

Kuo

0.513

0.405

0.361

0.322

0.245

0.52,~

0.997

W 2j+ 44 1

0.477

0.426

0.369

0.301

0.213

0.564

1

165

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References [1] [2] [3] [4] [5] [6] [7] [3] [9] [10] [11] [12] [13] [14] [15] [16]

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T. T. S. Kuo, Nucl. Phys. A122 (1968) 325. B. H. Brandow, Rev. Nod. P h y s . 39 (1967) 771. T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40. B. L. A n d e r s e n , J. P. Bondorf and B. S. M a d s e n , P h y s . L e t t e r s 22 (1966) 651. G. H. Lenz and G. M. T e m m e r , Nucl. Phys. Al12 (1968) 625. D. M. Clement and E. U. B a r a n g e r , Nucl. Phys. A108 (1968) 27. T. A. Brody and M. M o s h i n s k y , Tables of t r a n s f o r m a t i o n b r a c k e t s (Monografies del Instituto de F i s i c a , Mexico, 1960). A. de-Shalit and I. T a l m i , Nuclear Shell t h e o r y (Academic P r e s s , New York, 1963). J. P. Eiliott, A. D. J a c k s o n , H. A. M a v r o m a t i s , E A. S a n d e r s o n and B. Singh, Nucl. Phys. A121 (1968) 241. B. L. Scott and S. A. Moszkowski, Nucl. Phys. 29 (1962) 665. H.A. M a v r o m a t i s , E.A. S a n d e r s o n and A . J . J a c k s o n , Nuel. Phys. A124 (1969) 1. A. Bohr and B . R . M o t t e l s o h n , Nuclear S t r u c t u r e , Vol. I (W. A. B e n j a m i n , New Y o r k / A m s t e r d a m , 1969) p. 248. R. V. Reid J r . , Ann. of Phys. 50 (1968) 411. F. Tabakin, Ann. of Phys. 30 (1964) 51. S. Kahana, It. C. Lee and C K. Scott, P h y s . Rev. 180 (1969) 956. T . T . S . Kuo, J. B l o m q v i s t and G.E. Brown, Phys. L e t t e r s 31B (1970) 93.