Charge distribution of Li6

Volume 7, n u m b e r 4

PHYSICS

LETTERS

1December 1963

5) P. Brovetto, V. Canuto and F.Chaos, Physics Letters 5 (1963) 91. 6) V.A.Kravtsov, Soviet Phys. JETP 14 (1962) 1317.

3) H. B. Levy, Phys. Rev. 106 (1957) 1265. 4) S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk., 31, no. 11 (1959). *****

CHARGE

DISTRIBUTION

OF

Li 6 *

E. W. SCHMID

Max-Planck-Institut fiir Physik und Astrophysik, Munich 23, Germany Y. C. TANG

Brookhaven National Laboratory, Upton, Long Island, New York K. WII_DERMUTH

Florida State University, Tallahassee, Florida Received 28 October 1963

A u s e f u l m e t h o d to s t u d y the s t r u c t u r e of light n u c l e i i s the s c a t t e r i n g by h i g h - e n e r g y e l e c t r o n s . F r o m the m e a s u r e d s c a t t e r i n g c r o s s s e c t i o n s , one c a n get i n f o r m a t i o n a b o u t the c h a r g e d i s t r i b u t i o n of the t a r g e t n u c l e i . E x p e r i m e n t s of t h i s kind have b e e n p e r f o r m e d on m a n y n u c l e i b y H o f s t a d t e r a n d c o l l a b o r a t o r s 1). A n a l y s i s of the e x p e r i m e n t a l r e s u l t s showed that IA6 has a l a r g e r m s r a d i u s and a v e r y d i f f u s e s u r f a c e a s c o m p a r e d to o t h e r p - s h e l l n u c l e i 2). Its c h a r g e d i s t r i b u t i o n has a long t a i l which c a n n o t be d e s c r i b e d , for i n s t a n c e , b y u s i n g s i n g l e p a r t i c l e wave f u n c t i o n s in an i n f i n i t e o s c i l l a tor p o t e n t i a l 3). T h i s is in d i r e c t c o n t r a s t to the fact that for a l l o t h e r p - s h e l l n u c l e i , the e x p e r i m e n t a l r e s u l t s c a n b e f i t t t e d f a i r l y w e l l u s i n g wave f u n c t i o n s in s u c h a p o t e n t i a l w e l l 2). As has b e e n p o i n t e d out, the l a r g e d i f f u s e n e s s of IA6 s e e m s to b e a t y p i c a l c l u s t e r i n g effect 4, 5). F o r the g r o u n d s t a t e of IA6, we l e a r n f r o m both e x p e r i m e n t a l e v i d e n c e and t h e o r e t i c a l c a l c u l a t i o n 4) that the p r e d o m i n a n t c l u s t e r s t r u c t u r e i s the a l p h a d e u t e r o n s t r u c t u r e , with o t h e r s t r u c t u r e s s u c h a s H e 3 - t p l a y i n g only a m i n o r r o l e . In the a - d s t r u c t u r e , the two c l u s t e r s a r e b o u n d t o g e t h e r by only 1.45 MeV, which m e a n s that they a r e on the a v e r a g e r a t h e r f a r a p a r t a n d b e h a v e m o r e or l e s s like f r e e p a r t i c l e s . F o r the d e u t e r o n c l u s t e r we c a n thus exp e c t that it w i l l h a v e a long t a i l j u s t a s a f r e e deut e r o n does. Q u a l i t a t i v e l y , we can, t h e r e f o r e , u n d e r s t a n d the f e a t u r e of l a r g e d i f f u s e n e s s and l a r g e r m s r a d i u s of IA6 a s a r i s i n g f r o m the long t a i l of the d e u t e r o n c l u s t e r a n d the fact that the c l u s t e r s a r e widely s e p a r a t e d . To c o n f i r m t h e s e q u a l i t a t i v e a r g u m e n t s we have * Supported by the Office of Naval Research.

c a r r i e d out a v a r i a t i o n a l c a l c u l a t i o n with the following t w o - b o d y p o t e n t i a l : +oo

T''< £C

Vt(r) = { - Vt e x p [ - ~ ( r - r c ) ]

r > rc T ' < ~C

+co

Vs(r; = { - V s e x p [ - B ( r - r c ) ]

(1)

r > rc

a n d Vt = 4 3 4 . 0 M e V . Vs = 2 1 6 . 0 M e V , a = 2 . 4 0 fm -1, t~ = 1.97 fm - l , r c = 0.35 fm. In the a b o v e equation, Vt(r ) and Vs(r ) a r e the t r i p l e t and s i n g l e t i n t e r a c t i o n s in the even o r b i t a l a n g u l a r m o m e n t u m s t a t e s . The i n t e r a c t i o n in the odd o r b i t a l a n g u l a r m o m e n t u m s t a t e s is taken to be z e r o except for a h a r d c o r e of r a d i u s r c. T h i s p o t e n t i a l has b e e n d i s c u s s e d in d e t a i l a s p o t e n t i a l II in an e a r l i e r p a p e r 6). In o r d e r to o b t a i n a r e l i a b l e r m s r a d i u s f r o m a variational calculation, a rather flexible trial f u n c t i o n has to be a s s u m e d . We have c h o s e n the form ¢ ( r l . • • r 6) = A{v~a(rl... r 4 ) ~ d ( r 5 , r6) × (R)

~-~

(2)

f 2(rnm) ¢(~, v)},

n =1,..., 4

m=5, 6 w h e r e the o p e r a t o r A s t a n d s for t o t a l a n t i s y m m e t r i s a t i o n a n d ~(~, ~-) i s a c h a r g e - s p i n f u n c t i o n c h o s e n to g i v e S = 1 a n d T = 0 for the wave f u n c tion ¢. T h e a - c l u s t e r f u n c t i o n is t a k e n a s 263

Volume 7, n u m b e r 4

PHYSICS

4

% -- [ N - Jl(rik)] i >k=l 4 ×{exp[-~al

•

i>k=l

2

,

rikl+a2exp[-~a3

4 2 i>k=l

T h i s f u n c t i o n h a s a l r e a d y b e e n s t u d i e d in the c a s e of a f r e e a - p a r t i c l e 6). T h e d e u t e r o n f u n c t i o n Sd i s g e n e r a t e d b y the Schr6dinger equation h2 d 2 - m drr2 [ r ~ d (r)] + [ Vt(r) - b2] r ~ d ( r ) = 0 between r= 0andr=

b 1. F o r r > b l , the f u n c t i o n

B1 ~ d ( r ) = ~ - [ e x p ( - b 3 r ) - U 2 exp ( - U 3 r ) ] i s u s e d . T h e c o n s t a n t s B1, B 2 and B 3 a r e d e t e r m i n e d b y the c o n d i t i o n that ~d, ~d' a n d ~ d " a r e c o n t i n u o u s at r = b 1. W i t h t h i s t r i a l w a v e f u n c t i o n f o r the d e u t e r o n c l u s t e r , the t a i l c a n b e v a r i e d without a n y v a r i a t i o n in the i n t e r i o r r e g i o n . T h i s s e e m s to b e a d e s i r a b l e f e a t u r e , s i n c e the i n t e r a c t i o n of the c l u s t e r s m i g h t c o n c e i v a b l y a l t e r the t a i l of the d e u t e r o n c l u s t e r f u n c t i o n without having m u c h e f f e c t in the r e g i o n of s t r o n g t w o - b o d y i n t e r a c t i o n . T h e long r a n g e p a r t of the r e l a t i v e m o t i o n f u n c tion i s c h o s e n a s ~((R) = R 2 [ e x p ( - t 1R 2) + t 2 e x p ( - t 3R2)] . T h e i n c l u s i o n of a s e c o n d G a u s s i a n f u n c t i o n s e e m s to b e n e c e s s a r y , s i n c e the b i n d i n g b e t w e e n the two c l u s t e r s i s r a t h e r w e a k . W e f e e l that a s i n g l e G a u s s i a n f u n c t i o n w i l l not b e a b l e to r e p r e s e n t t h e b e h a v i o u r of t h e r e l a t i v e m o t i o n p r o p e r l y . T h e c u t - o f f f u n c t i o n s f l a n d f 2 a r e e q u a l to z e r o within the h a r d - c o r e r e g i o n . O u t s i d e , t h e y a r e g e n e r a t e d b y the d i f f e r e n t i a l e q u a t i o n ~2 d 2 m dr2

Iv/1, 2(r)] + ½~'1, 2 [ Vt(r) + Vs(r)] r / l , 2 (r) = ~ l , 2 r f l , 2(r)

up to r 1 2 w h e r e f l 2 h a s i t s f i r s t m a x i m u m , n o r m a l i z e d ' t o unity. F ~ r v a l u e s of r l a r g e r than rl, 2' t h e f u n c t i o n f l 2 i s t a k e n to b e e q u a l to one. In the above' e q u a t i o n s , the c o o r d i n a t e s rik a r e i n t e r p a r t i c l e d i s t a n c e s and R r e p r e s e n t s t h e d i s t a n c e of s e p a r a t i o n b e t w e e n the two c l u s t e r s . A l s o , we s h o u l d p o i n t out t h a t no t o t a l c e n t r e - o f - m a s s m o t i o n w a v e f u n c t i o n i s c o n t a i n e d in ~. T h e e x p e c t a t i o n v a l u e of the s i x - b o d y H a m i l t o n i a n w a s c a l c u l a t e d b y a M o n t e - C a r l o m e t h o d 7). A b o u t 50 h o u r s of c o m p u t i n g t i m e w a s n e e d e d on t h e I B M - 7 0 9 0 c o m p u t e r . A m i n i m u m w a s found with t h e f o l l o w i n g p a r a m e t e r s : a 1 = 1.17 f m - 2 , a 2 = 0.25, a3=O.52fm -2, b l = l . 3 5 f m , b 2 = - 2 . 2 2 MeV, b3= 264

LETTERS

1 D e c e m b e r 1963

0 . 3 4 f m - 1 , Cl=O.18fm-2 , c2=0.25 , c3=0.065fm-2 , 7 1 = 0 . 6 8 , Y 2 = l . 0 , ~ 1 = 0 , E2=0. T h e p a r a m e t e r s f o r the a - c l u s t e r w a v e f u n c t i o n a r e the s a m e a s t h o s e for the f r e e a - p a r t i c l e 6) * T h i s i n d i c a t e s t h a t the a - c l u s t e r h a s a v e r y s m a l l c o m p r e s s i b i l i t y a s i s to b e e x p e c t e d . T h e d e u t e r o n c l u s t e r h a s a s o m e w h a t s h o r t e r t a i l than a f r e e d e u t e r o n • *. T h e r e l a t i v e m o t i o n f u n c t i o n i s r a t h e r long r a n g e d . T h e e n e r g y at the m i n i m u m i s - 3 1 . 3 + 0 . 8 MeV ( e x p e r i m e n t a l l y - 3 2 . 0 MeV), of w h i c h - 0 . 5 MeV (-1.47 MeV) r e p r e s e n t s the i n t e r a c t i o n e n e r g y b e t w e e n the c l u s t e r s . A l t h o u g h the i n t e r a c t i o n e n e r g y i s r a t h e r s m a l l , the m i n i m t t m i s w e l l d e f i n e d b e c a u s e of the C o u l o m b b a r r i e r w h i c h i s about 1 MeV high. T h e r m s r a d i u s of the c h a r g e d i s t r i b u t i o n i s 2 . 7 3 ± 0 . 1 5 fm, with the e r r o r c o m i n g f r o m the s t a t i s t i c a l u n c e r t a i n t y in the p o s i t i o n of the m i n i m u m . A f i n i t e c h a r g e d i s t r i b u t i o n of the p r o t o n w i t h a G a u s s i a n s h a p e and a r m s r a d i u s of 0.72 fm h a s b e e n i n c l u d e d in t h i s v a l u e . T h e r m s r a d i u s derived from electron scattering experiments is 2 . 7 0 + 0 . 1 5 f m 2). F i g . 1 s h o w s a c o m p a r i s o n b e t w e e n the t h e o r e t i c a l c h a r g e d i s t r i b u t i o n a n d the phenomenological charge distributions derived from electron scattering experiments under various a s s u m p t i o n s 2). A s i s s e e n , the a g r e e m e n t i s q u i t e g o o d in the r e g i o n of the t a i l w h i c h c o n t r i b u t e s m o s t to the r m s r a d i u s . In t h i s r e g i o n , o n l y the c u r v e l a b e l e d " F 6 " s h o w s a r e l a t i v e l y l a r g e d e v i a t i o n f r o m the c a l c u l a t e d one, but t h i s i s b e c a u s e t h i s c u r v e w a s d e r i v e d u n d e r the a s s u m p t i o n of z e r o c h a r g e d e n s i t y b e y o n d 6 fro. In the i n t e r i o r r e g i o n the e x p e r i m e n t s do not g i v e a c c u r a t e i n f o r m a t i o n . T h e a n a l y s i s of M e y e r - B e r k h o u t et al. can only c o n c l u d e t h a t the m a x i m u m c h a r g e d e n s i t y in Li 6 i s 0 . 0 6 4 + 0 . 0 0 6 p r o t o n s / f m 3 2). F r o m t h i s c a l c u l a t i o n , we g e t t h e m a x i m u m c h a r g e d e n s i t y to b e 0.055 p r o t o n s / f m 3 . In c a l c u l a t i n g e x p e c t a t i o n v a l u e s with w a v e funct i o n (2) m o s t of the d i f f i c u l t i e s a r i s e f r o m i t s c u t off p a r t . Many o n e - p a r t i c l e o p e r a t o r s , h o w e v e r , a r e i n s e n s i t i v e to s h o r t r a n g e m a n y - p a r t i c l e c o r r e l a t i o n s and t h e r e f o r e the c u t - o f f p a r t c a n b e d i s r e g a r d e d in c a l c u l a t i n g t h e i r e x p e c t a t i o n v a l u e s . A s an e x a m p l e we h a v e c a l c u l a t e d the p r o t o n d i s t r i b u t i o n of IA6 with and without c u t - o f f ***. T h e * The energy of the c - p a r t i c l e has been recalculated with a higher statistical accuracy. The new value is 28.6+0.2 MeV instead of 29.2+0.6 MeV in ref. 6). ** In the deuteron cluster wave function, B 3 is much l a r g e r than b 3 and B 2 is rather small ; hence, the asymptotic behaviour of this function is determined by b 3. F o r the deuteron cluster, the optimum value of b3 is 0.34, l a r g e r than the corresponding value of 0.23 for a free deuteron. *** In the calculation without cut-off the deuteron function was altered accordingly in its interior region.

Volume 7, number4

PHYSICS LETTERS 007

°°sk

006

007 I

|

oo, L 0o5

r(~(r) l

1 December 1963

/"'" "'",

Fig. I

Fig. 2

005

!y-I-,,.,

":'?F"" \

O~

, , ~

r~(r) O03 002

001

0.01 i

0

""'-.0

1

2

1

4 5 6 r Fig. 1. Theoretical and phenomenological charge d i s t r i butions of Li 6. Solid line : theoretical curve according to the optimum wave function ; a finite charge distribution of the proton is folded in. Broken lines : phenomenological curves, O = o scillator model. F = fermi model. F 6 = Fourier model with cut-off radius equal to 6 fm, see ref. 2). 3

r e s u l t i s s h o wn in fig. 2. A s i s s e e n , t h e c u t - o f f f u n c t i o n s h a v e only m i n o r i n f l u e n c e on the e x p e c t a t i o n v a l u e of the n u c l e o n d e n s i t y o p e r a t o r . We want to thank the A t o m i c E n e r g y C o m m i s s i o n f o r a g e n e r o u s g r a n t of c o m p u t i n g t i m e on t h e i r I B M - 7 0 9 0 c o m p u t e r at t h e N e w Y o r k U n i v e r s i t y .

2

3 r

4

5

6

Fig. 2. Influence of short range correlations on proton distribution. Solid line : proton distribution according to optimum wave function. Broken line : same, without hard-core correlation functions.

References 1) R.Hofstadter, Ann. Rev. Sci. 7 (1957) 231. 2) U. Meyer-Berkhout, K.W. Ford and A. E. S. Green, Ann. Phys. 8 (1959) 119. 3) G.R. Burleson and R. Hofstadter, Phys. Rev. 112 (1958) 1282; D. F. Jackson, Proc. Phys. Soc. (London) 76 (1960) 949; L . R . B . Elton, Nuclear Sizes (Oxford University P r e s s , 1961) p.21. 4) Y. C. Tang, K.Wildermuth and L. D. Pearlstein, Phys. Rev. 123 (1961) 548; Nuclear Phys. 32 (1962) 504. 5) D. F. Jackson, Proc. Phys. Soc. (London) 79 (1962) 1041. 6) E.W. Schmid, Y.C. Tang and R. C. Herndon, Nuclear Phys.42 (1963) 95. 7) E.W.Sehmid, Nuclear Phys.32 (1962) 82.

*****

THE SHAPE OF THE SYMMETRY-TERM IN THE NUCLEAR OPTICAL POTENTIAL T. T E R A S A W A * and G. R. S A T C H L E R

Oak Ridge National Laboratory, Oak Ridge, Tennessee Received 3 November 1963

T h e i s o b a r i c - s p i n d e p e n d e n c e of the o p t i c a l p o t e n t i a l h a s b e e n s t u d i e d by s e v e r a l a u t h o r s 1 - 4 ~ and r e c e n t a n a l y s i s of (p, n) r e a c t i o n s 5) and the * Summer Investigator, 1963 ; from Tokyo University of Education, Tokyo, Japan.

e l a s t i c s c a t t e r i n g of n u c l e o n s 6) h as i m p l i e d the e x i s t e n c e of an a p p r e c i a b l e t e r m p r o p o r t i o n a l to t o • T, w h e r e t o and T a r e t h e / - s p i n o p e r a t o r s f o r p r o j e c t i l e and t a r g e t , r e s p e c t i v e l y . In t h i s n o t e , we g i v e s o m e d i s c u s s i o n of t h e s h a p e of t h i s p o t e n t i a l . Up to t h e p r e s e n t , c a l c u l a t i o n s h a v e 265