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Scattering of high energy electrons and nucleons from 4He and nucleon - nucleon correlations
Volume 25B, Number 5
SCATTERING FROM 4He
P HYS I CS L E T T E RS
18 September 1967
OF HIGH ENERGY ELECTRONS AND NUCLEONS AND NUCLEON - NUCLEON CORRELATIONS W. CZYZ and L. LESNIAK
Instytut Fizyki Jqdrowej. Krak6w. Bronowice. Poland Received 25 July 1967
An attempt is made to give a coherent explanation of the recent experiments on the elastic electron and nucleon scattering from 4He.
T h e r e e x i s t s e v e r a l e x p e r i m e n t s on high e n e r gy s c a t t e r i n g of e l e c t r o n s [1] and nucleons [2, 3] f r o m light nuclei which demand a c o h e r e n t th e o r e t i c a l i n t e r p r e t a t i o n . S o m e a t t e m p t s in this dir e c t i o n have a l r e a d y b e e n made. In r e f s . 4 and 5 a c o h e r e n t i n t e r p r e t a t i o n of the e x p e r i m e n t s of r e f s . 2 and 3 was p r o p o s e d and in ref. 6 of the e x p e r i m e n t s of r e f s . 1 and 2. In this note we p r e sent a d i s c u s s i o n of the influence of the s h o r t range repulsive dynamical nucleon-nucleon corr e l a t i o n s on the m e a s u r e d e l e c t r o n and nucleon e l a s t i c c r o s s s e c t i o n s . We shall l i m i t our d i s c u s sion to the s c a t t e r i n g of e l e c t r o n s and nucleons f r o m 4He, and shall u s e the G l a u b e r m o d e l of the high e n e r g y s c a t t e r i n g throughout this p a p e r [7,4, 5]. T h i s m o d el is e s p e c i a l l y well suited to c o m p a r e the s c a t t e r i n g of e l e c t r o n s with the s c a t t e r ing of s t r o n g l y i n t e r a c t i n g p a r t i c l e s (nucleons, m e s o n s ) . It r e s o l v e s the s c a t t e r i n g amplitude into the s i n g l e - , d o u b l e - , t r i p l e - etc. c o l l i s i o n contributions° Due to the s m a l l n e s s of the e l e c t r o m a g n e t i c coupling constant e l e c t r o n s undergo only single s c a t t e r i n g and p r o v i d e us with the single s c a t t e r i n g c o n t r i b u t i o n of the amplitude of s t r o n g l y i n t e r a c t i n g p a r t i c l e s . This enables us to analyze the e l e c t r o n - , n u c l e o n - , m e s o n - s c a t t e r i n g e x p e r i m e n t s on the s a m e footing and, hopefully, to r e s o l v e c e r t a i n a m b i g u i t i e s of the e l e c t r o n s c a t t e r i n g e x p e r i m e n t s by the nucleon or m e s o n s c a t t e r i n g e x p e r i m e n t s . F i r s t , let us point out that a r e p u l s i v e nucleonnucleon i n t e r a c t i o n m a y , in p r i n c i p l e , be seen in the existing e x p e r i m e n t s . E.g. a hard c o r e r c = = 0.4 fm c o r r e s p o n d s to the m o m e n t u m t r a n s f e r s q u a r e d A2 = 6.25 f m -2 = 0.24 (GeV/c) 2, hence its influence should be s e e n at A2 > 0.25 (GeV/c) 2 which is above the f i r s t m i n i m u m found in ref. 2. In o r d e r to s ee how a r e a s o n a b l e s h o r t r a n g e r e -
p u l s i v e n u c l e o n - n u c l e o n i n t e r a c t i o n influences the e l a s t i c n u cl eo n c r o s s s e c t i o n s let us i n t r o duce the ground st at e nucleon density in the f o r m [8] A
I~ (rl... rA)]2 :
I ~ o ( r l . . " rA) ! 2 l ~ [1-h(rjk)],
j>k
(1)
w h e r e A is the n u m b e r of nucleons, '~o is the unc o r r e l a t e d ground st at e wave function (shell m o d el wave function). The function h(rjk) i n t r o d u c e s c o r r e l a t i o n s of the j , k p a i r at s m a l l d i s t a n c e s rjk = I U - rkl' and s a t i s f i e s the following bounda r c conditions:
h(rjk) ~ O, f o r r j k --* 0% h(Ok ) ~ 1, f o r r j k -* 0. In fact we sh al l a s s u m e that h(rjk > re) ~ 0 and r c << R, w h e r e R in the r a d i u s of the nucleus, and r c p l ay s h e r e the r o l e of a h a r d c o r e r a d i u s . The e l a s t i c s c a t t e r i n g amplitude is [4, 5]:
Tii
6) f d 3 r l . . d 3 r A l ~ ( r l . ,
rA)l 2ip f d 2 b
A × exp(i/I- b) tl-j=l~l [ 1 - 2 - ~ f d26
(2)
× exp {-i(J" ( b - s j ) } f j ( p , 6 ) l l w h e r e fj (p, 5) is the e l a s t i c s c a t t e r i n g amplitude of the incident nucleon (meson, e l e c t r o n ) on the j - t h t a r g e t nucleon (p is the incident m o m e n t u m , 6 the m o m e n t u m t r a n s f e r ) and A is the nucleon (meson, e l e c t r o n ) - nucleus m o m e n t u m t r a n s f e r v e c t o r . 6) is a f a c t o r c o r r e c t i n g f o r the r e c o i l which, f o r a Gaus~ian ground state wave function, amounts to exp (R2LX2/4A). Notice that if A
t~(rl..,
rA)] 2 = ~ [
j=l
p(Irj[)
(3)
319
Volume 25B, n u m b e r 5
PHYSICS
LETTERS
18 September 1967
d~
*He fO"
'He
= -o,; ~. ",~O O2 rr~ ~#'47
53mb
o - s (c,~v/c)-~ e " 17~ev/c
~0 "~
70-T
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'/0 .s 70-3
o
o2
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o6
o8
Fig. 1. Comparison of the nucleon-4He elastic c r o s s sections with the experiment [2] for various models of the ground state wave function. - c o r r e l a t e d ground state wave function: y = 3.4, R = 1.25 fm; . . . . . u n c o r r e l a t e d ground state wave function: y = o% R = 1.25 fro; -. -. - experimentally found Fch given by eq. (6) e m ployed, with c = 0.316 fm and d = 0.681 fro. ~ i s the ratio of the real to the imaginary p a r t s of the nucleonnucleon s c a t t e r i n g amplitude. ~n , I~n the total p r o t o n neutron and proton-proton c r o s s - s e c t i o n . we g e t f r o m eq. (2):
Tii = @ ip f d2b e x p ( i Z l ' b ) A
xIl-j__l~-1 I l - 2 ~ f e x p ( - i ~ ' b )
(4)
fj(p,6)F(5)jl
where F(6) is essentially the charge form factor of t h e t a r g e t n u c l e u s w h i c h i s i d e n t i c a l w i t h t h e charge form factor obtained from the electron e l a s t i c s c a t t e r i n g p r o v i d e d we a s s u m e t h a t t h e neutron and proton densities are the same. W h e n we i n s e r t eq. (1) into eq. (2) a n d e x p a n d t h e p r o d u c t (1) we o b t a i n t h e a m p l i t u d e in f o r m of a s e r i e s of c o n t r i b u t i o n s of d i f f e r e n t n u m b e r s of c o r r e l a t e d p a i r s (no c o r r e l a t e d p a i r s , one c o r r e l a t e d p a i r , two c o r r e l a t e d p a i r s e t c . ) . If r c <
i
fo a'~'v/c)2
i
Fig. 2. Comparison of the F~R(6) measured in ref. 1 and the IFchl calculated to or~d'er from eq. (i) forT, =3.4, R = 1.25fm (B). ~venbyeq. (6) with c = 0.316 fm and d = 0.681 fm is also shown (A).
(re/R)3 IFchl
d i m e n s i o n l e s s p a r a m e t e r (rc/R) 3 << 1, h e n c e it i s e n o u g h to k e e p t h e l o w e s t o r d e r c o r r e c t i o n t o the uncorrelated ground state density. As long as t h e c o n d i t i o n r c << R is f u l f i l l e d t h e r e s u l t i s v e r y i n s e n s i t i v e to t h e p a r t i c u l a r c h o i c e of h(rjk). W e c a l c u l a t e d t h e e l a s t i c n u c l e o n - 4He c r o s s s e c t i o n t o o r d e r (rc/R)3 a n d c o m p a r e d it w i t h t h e B r o o k h a v e n d a t a (fig. 1) f o r t h e G a u s s i a n I~I,o l 2 = : N 2 exp[R - 2 ( ~ + r ~ + r~, + r 2 ) ] ( a s in r e f s . 4 a n d 5) a n d f o r h(r]k ~ = e~xp(-°yr~a/R 2) a n d t h e v a l u e s of t h e p a r a m e t e r s a s i n d i c a t e d in fig. 1. S e v e r a l c o m m e n t s a r e in o r d e r h e r e . 1. T h e s h o r t r a n g e r e p u l s i v e c o r r e l a t i o n s c o r r e c t t h e r e s u l t s g i v e n b y an u n c o r r e l a t e d g r o u n d s t a t e w a v e f u n c t i o n in t h e r i g h t d i r e c t i o n : t h e y m a k e t h e m i n i m u m d e e p e r a n d lift h i g h e r t h e s h o u l d e r of t h e c u r v e on t h e r i g h t h a n d s i d e of t h e m i n i m u m . 2. T h e s e r i e s o n e g e t s by e x p a n d i n g t h e s i n g l e - , d o u b l e - e t c s c a t t e r i n g t e r m s of eqo (2) in p o w e r s of (rc/R)3 h a v e a l t e r n a t i n g s i g n s a n d f a l l off w i t h i n c r e a s i n g A2 s l o w e r a n d s l o w e r a s o n e g o e s to h i g h e r a n d h i g h e r p o w e r s of (rc/R)3. T h i s r e s u l t s in t h e e x i s t e n c e of z e r o s of Tii (of a q u i t e different origin from the zeros which produce m i n i m a f o u n d in t h e B r o o k h a v e n e x p e r i m e n t [2, 4, 5]. In p a r t i c u l a r t h e s i n g l e s c a t t e r i n g c o n t r i b u t i o n
Volume 25B, number 5
PHYSICS
h a s a z e r o , h e n c e the e l e c t r o n c h a r g e f o r m f a c t o r of 4He should h a v e a s h a r p m i n i m u m s o m e w h e r e f o r A2 > 6.25 f m -2, h e n c e f o r A2 l a r g e r than the f i r s t m i n i m u m found in the B r o o k h a v e n e x p e r i m e n t [2]. Such a m i n i m u m has b e e n r e c e n t l y found [1]. F i g . 2 s h o w s I F c h l c a l c u l a t e d (to o r d e r (rc/R)3) and the p o i n t s of the e x p e r i m e n t [1]. 3. T h e p o s i t i o n s of the z e r o s c a u s e d by the c o r r e l a t i o n s m o v e to h i g h e r and h i g h e r A2 as one g o e s f r o m the s i n g l e - , to d o u b l e - , t r i p l e - etc s c a t t e r i n g c o m p o n e n t s of the a m p l i t u d e (2). H e n c e f o r ~ " r e a s o n a b l e " h ~ r d c o r e r a d i u s the only z e r o w h i c h p l a y s an i m p o r t a n t r o l e up to A 2 ~ l(GeV/c) 2 is the f i r s t z e r o of the s i n g l e s c a t t e r i n g c o m p o n e n t and the d e s i r a b l e e f f e c t s d e s c r i b e d in 1. a r e c a u s e d by its e x i s t e n c e . 4. W e did not t r y to find the b e s t f i t s . A l t h o u g h t h e c o r r e l a t i o n c o r r e c t i o n s go in the r i g h t d i r e c t i o n s , the e x p e r i m e n t a l m i n i m u m found in r e f . 1 is s t i l l not w e l l r e p r o d u c e d (fig. 2). It is quite p o s s i b l e that s o m e m o r e s o p h i s t i c a t e d c o r r e l a t i o n s (with m o r e p a r a m e t e r s ) m a y r e p r o d u c e the data still better. 5. One should s t r e s s that if one r e p r o d u c e s I F c h l m e a s u r e d in the e l e c t r o n e x p e r i m e n t [1] i n t r o d u c i n g s h o r t r a n g e c o r r e l a t i o n s one d o e s not prove their existence. The elastic electron scatt e r i n g e x p e r i m e n t a l r e s u l t s can a l w a y s be e x p l a i n e d in t e r m s of an e f f e c t i v e s i n g l e p a r t i c l e d e n s i t y (cf. r e f . 1): p ( r 1) : / d 3 r 2 . . .
d3rA I ~ I ' ( r l . . . rA) i 2
18 September 1967
u r e d Fch [1] f o r the c . m . m o t i o n , the p r o t o n c h a r g e d i s t r i b u t i o n and the m a g n e t i c m o m e n t s c o n t r i b u t i o n s . It t u r n s out h o w e v e r that the f i r s t two c a n c e l e a c h o t h e r to a l a r g e e x t e n t and the t h i r d is a few p e r c e n t c o r r e c t i o n only. In c o n c l u d i n g we would l i k e to s t r e s s that t h e r e is a v e r y good c h a n c e that the m e a s u r e m e n t s of e l a s t i c c r o s s s e c t i o n s of s t r o n g l y i n t e r acting particles scattered from nuclei may indic a t e the p h y s i c a l r e a s o n f o r the e x i s t e n c e of the d i f f r a c t i o n m i n i m u m in the 4He f o r m f a c t o r . Should it t u r n out to be c a u s e d by d y n a m i c a l r e p u l s i v e c o r r e l a t i o n s , the e l a s t i c e l e c t r o n s c a t t e r i n g at l a r g e m o m e n t u m t r a n s f e r s would b e c o m e a v e r y i m p o r t a n t tool in i n v e s t i g a t i n g s u c h c o r r e l a t i o n s in n u c l e i . In o r d e r to a c h i e v e it, we should h a v e a t h e o r y at o u r d i s p o s a l which is r e l i a b l e at l a r g e m o m e n t u m t r a n s f e r s ° It w o u l d s e e m h o w e v e r that a r o u n d t h e m i n i m u m found in r e f . 2, h e n c e f o r r e l a t i v e l y s m a l l m o m e n t u m t r a n s f e r s , w h e r e one can s t i l l t r u s t the m o d e l , our calculations clearly favour rather large valu e s of the r e a l p a r t s of t h e n u c l e o n - n u c l e o n s c a t tering amplitude° We a r e g r a t e f u l to P r o f e s s o r s H. P a l e v s k y and M. R. Y e a r i a n f o r s e n d i n g us t h e i r d a t a b e f o r e p u b l i c a t i o n . We would l i k e a l s o to t h a n k P r o f e s s o r s A. B o h r , B . R . M o t t e l s o n and J. D. W a l e c k a f o r the s t i m u l a t i n g d i s c u s s i o n s , c r i t i c i s m and comments.
(5)
w h i c h m a y o r m a y not c o m e f r o m a c o r r e l a t e d g r o u n d s t a t e w a v e f u n c t i o n . T h i s a m b i g u i t y can p r o b a b l y be r e s o l v e d by m e a s u r e m e n t s of the e l a s t i c c r o s s - s e c t i o n s of s t r o n g l y i n t e r a c t i n g p a r t i c l e s (~ m e s o n s , n u c l e o n s ) at l a r g e m o m e n t u m t r a n s f e r s A2 > I ( G e V / c ) 2 . One m a y e x p e c t it to b e the c a s e b e c a u s e one finds t h a t although the s i n g l e s c a t t e r i n g c o n t r i b u t i o n s to eq. (2) g i v e n by a c o r r e l a t e d w a v e f u n c t i o n l i k e eq. (1) and by the d e n s i t y (5) c a l c u l a t e d f r o m eq. (1) a r e i d e n t i c a l , t h e double s c a t t e r i n g c o n t r i b u t i o n s a r e a l r e a d y quite different. 6. T h e f o r m u l a (4) e n a b l e s us to e m p l o y d i r e c t l y the m e a s u r e d Fch(5) to c a l c u l a t e Cog. the n u c l e o n 4He c r o s s s e c t i o n . One m u s t h o w e v e r k e e p in m i n d that eq. (4) is c o r r e c t u n d e r t h e a s s u m p t i o n (3). W e took Fch(5) = [1-(c252) 6] exp(-d252)
LETTERS
(6)
o b t a i n e d in r e f . 1 and the r e s u l t s a r e s h o w n in f i g . 1. In p r i n c i p l e , in o r d e r to i n s e r t it into eq. (4), one should c o r r e c t the e x p e r i m e n t a l l y m e a s -
References 1. R. F. Frosch, J . S . McCarthy, R.E. Rand and M.R. Yearian, Bull. Am. Phys. Soc. 12 (1967) 16; Stanford preprint HEPL 492 (1967). 2. H. Palevsky, Intern. Conf. on Nuclear physics, Gatlinburg, 1966; Second Intern. Conf. on High energy physics and nuclear structure, Rehovoth 1967, and private communication. 3. G. Bellettini, G. Cocconi, A.N. Diddens, E. Lillethun, G. Matthiae, T. P. Scanlon and A.M.Wetherell, Nucl. Phys. 79 (1966) 609. 4. W. Czy~. and L. Le~niak, Phys. Letters 24B (1967) 227. 5. W. Czy~ and L. Le~niak, Second Intern. Conf. on High energy physics and nuclear structure, Rehovoth, 1967. 6. H. H. Bassel and C.Wilkin, Phys. Rev. Letters 18 (1967) 871. 7. R.J. Glauber, in Lectures in theoretical physics. ed. W. E. Brittin (Interscience Publishers Inc., New York, 1959). 8. R.J. Jastrow, Phys. Rev. 98 (1955) 1479.
321
SCATTERING FROM 4He
P HYS I CS L E T T E RS
18 September 1967
OF HIGH ENERGY ELECTRONS AND NUCLEONS AND NUCLEON - NUCLEON CORRELATIONS W. CZYZ and L. LESNIAK
Instytut Fizyki Jqdrowej. Krak6w. Bronowice. Poland Received 25 July 1967
An attempt is made to give a coherent explanation of the recent experiments on the elastic electron and nucleon scattering from 4He.
T h e r e e x i s t s e v e r a l e x p e r i m e n t s on high e n e r gy s c a t t e r i n g of e l e c t r o n s [1] and nucleons [2, 3] f r o m light nuclei which demand a c o h e r e n t th e o r e t i c a l i n t e r p r e t a t i o n . S o m e a t t e m p t s in this dir e c t i o n have a l r e a d y b e e n made. In r e f s . 4 and 5 a c o h e r e n t i n t e r p r e t a t i o n of the e x p e r i m e n t s of r e f s . 2 and 3 was p r o p o s e d and in ref. 6 of the e x p e r i m e n t s of r e f s . 1 and 2. In this note we p r e sent a d i s c u s s i o n of the influence of the s h o r t range repulsive dynamical nucleon-nucleon corr e l a t i o n s on the m e a s u r e d e l e c t r o n and nucleon e l a s t i c c r o s s s e c t i o n s . We shall l i m i t our d i s c u s sion to the s c a t t e r i n g of e l e c t r o n s and nucleons f r o m 4He, and shall u s e the G l a u b e r m o d e l of the high e n e r g y s c a t t e r i n g throughout this p a p e r [7,4, 5]. T h i s m o d el is e s p e c i a l l y well suited to c o m p a r e the s c a t t e r i n g of e l e c t r o n s with the s c a t t e r ing of s t r o n g l y i n t e r a c t i n g p a r t i c l e s (nucleons, m e s o n s ) . It r e s o l v e s the s c a t t e r i n g amplitude into the s i n g l e - , d o u b l e - , t r i p l e - etc. c o l l i s i o n contributions° Due to the s m a l l n e s s of the e l e c t r o m a g n e t i c coupling constant e l e c t r o n s undergo only single s c a t t e r i n g and p r o v i d e us with the single s c a t t e r i n g c o n t r i b u t i o n of the amplitude of s t r o n g l y i n t e r a c t i n g p a r t i c l e s . This enables us to analyze the e l e c t r o n - , n u c l e o n - , m e s o n - s c a t t e r i n g e x p e r i m e n t s on the s a m e footing and, hopefully, to r e s o l v e c e r t a i n a m b i g u i t i e s of the e l e c t r o n s c a t t e r i n g e x p e r i m e n t s by the nucleon or m e s o n s c a t t e r i n g e x p e r i m e n t s . F i r s t , let us point out that a r e p u l s i v e nucleonnucleon i n t e r a c t i o n m a y , in p r i n c i p l e , be seen in the existing e x p e r i m e n t s . E.g. a hard c o r e r c = = 0.4 fm c o r r e s p o n d s to the m o m e n t u m t r a n s f e r s q u a r e d A2 = 6.25 f m -2 = 0.24 (GeV/c) 2, hence its influence should be s e e n at A2 > 0.25 (GeV/c) 2 which is above the f i r s t m i n i m u m found in ref. 2. In o r d e r to s ee how a r e a s o n a b l e s h o r t r a n g e r e -
p u l s i v e n u c l e o n - n u c l e o n i n t e r a c t i o n influences the e l a s t i c n u cl eo n c r o s s s e c t i o n s let us i n t r o duce the ground st at e nucleon density in the f o r m [8] A
I~ (rl... rA)]2 :
I ~ o ( r l . . " rA) ! 2 l ~ [1-h(rjk)],
j>k
(1)
w h e r e A is the n u m b e r of nucleons, '~o is the unc o r r e l a t e d ground st at e wave function (shell m o d el wave function). The function h(rjk) i n t r o d u c e s c o r r e l a t i o n s of the j , k p a i r at s m a l l d i s t a n c e s rjk = I U - rkl' and s a t i s f i e s the following bounda r c conditions:
h(rjk) ~ O, f o r r j k --* 0% h(Ok ) ~ 1, f o r r j k -* 0. In fact we sh al l a s s u m e that h(rjk > re) ~ 0 and r c << R, w h e r e R in the r a d i u s of the nucleus, and r c p l ay s h e r e the r o l e of a h a r d c o r e r a d i u s . The e l a s t i c s c a t t e r i n g amplitude is [4, 5]:
Tii
6) f d 3 r l . . d 3 r A l ~ ( r l . ,
rA)l 2ip f d 2 b
A × exp(i/I- b) tl-j=l~l [ 1 - 2 - ~ f d26
(2)
× exp {-i(J" ( b - s j ) } f j ( p , 6 ) l l w h e r e fj (p, 5) is the e l a s t i c s c a t t e r i n g amplitude of the incident nucleon (meson, e l e c t r o n ) on the j - t h t a r g e t nucleon (p is the incident m o m e n t u m , 6 the m o m e n t u m t r a n s f e r ) and A is the nucleon (meson, e l e c t r o n ) - nucleus m o m e n t u m t r a n s f e r v e c t o r . 6) is a f a c t o r c o r r e c t i n g f o r the r e c o i l which, f o r a Gaus~ian ground state wave function, amounts to exp (R2LX2/4A). Notice that if A
t~(rl..,
rA)] 2 = ~ [
j=l
p(Irj[)
(3)
319
Volume 25B, n u m b e r 5
PHYSICS
LETTERS
18 September 1967
d~
*He fO"
'He
= -o,; ~. ",~O O2 rr~ ~#'47
53mb
o - s (c,~v/c)-~ e " 17~ev/c
~0 "~
70-T
I0 "J
;0-2 f O "'~
'/0 .s 70-3
o
o2
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o6
o8
Fig. 1. Comparison of the nucleon-4He elastic c r o s s sections with the experiment [2] for various models of the ground state wave function. - c o r r e l a t e d ground state wave function: y = 3.4, R = 1.25 fm; . . . . . u n c o r r e l a t e d ground state wave function: y = o% R = 1.25 fro; -. -. - experimentally found Fch given by eq. (6) e m ployed, with c = 0.316 fm and d = 0.681 fro. ~ i s the ratio of the real to the imaginary p a r t s of the nucleonnucleon s c a t t e r i n g amplitude. ~n , I~n the total p r o t o n neutron and proton-proton c r o s s - s e c t i o n . we g e t f r o m eq. (2):
Tii = @ ip f d2b e x p ( i Z l ' b ) A
xIl-j__l~-1 I l - 2 ~ f e x p ( - i ~ ' b )
(4)
fj(p,6)F(5)jl
where F(6) is essentially the charge form factor of t h e t a r g e t n u c l e u s w h i c h i s i d e n t i c a l w i t h t h e charge form factor obtained from the electron e l a s t i c s c a t t e r i n g p r o v i d e d we a s s u m e t h a t t h e neutron and proton densities are the same. W h e n we i n s e r t eq. (1) into eq. (2) a n d e x p a n d t h e p r o d u c t (1) we o b t a i n t h e a m p l i t u d e in f o r m of a s e r i e s of c o n t r i b u t i o n s of d i f f e r e n t n u m b e r s of c o r r e l a t e d p a i r s (no c o r r e l a t e d p a i r s , one c o r r e l a t e d p a i r , two c o r r e l a t e d p a i r s e t c . ) . If r c <
i
fo a'~'v/c)2
i
Fig. 2. Comparison of the F~R(6) measured in ref. 1 and the IFchl calculated to or~d'er from eq. (i) forT, =3.4, R = 1.25fm (B). ~venbyeq. (6) with c = 0.316 fm and d = 0.681 fm is also shown (A).
(re/R)3 IFchl
d i m e n s i o n l e s s p a r a m e t e r (rc/R) 3 << 1, h e n c e it i s e n o u g h to k e e p t h e l o w e s t o r d e r c o r r e c t i o n t o the uncorrelated ground state density. As long as t h e c o n d i t i o n r c << R is f u l f i l l e d t h e r e s u l t i s v e r y i n s e n s i t i v e to t h e p a r t i c u l a r c h o i c e of h(rjk). W e c a l c u l a t e d t h e e l a s t i c n u c l e o n - 4He c r o s s s e c t i o n t o o r d e r (rc/R)3 a n d c o m p a r e d it w i t h t h e B r o o k h a v e n d a t a (fig. 1) f o r t h e G a u s s i a n I~I,o l 2 = : N 2 exp[R - 2 ( ~ + r ~ + r~, + r 2 ) ] ( a s in r e f s . 4 a n d 5) a n d f o r h(r]k ~ = e~xp(-°yr~a/R 2) a n d t h e v a l u e s of t h e p a r a m e t e r s a s i n d i c a t e d in fig. 1. S e v e r a l c o m m e n t s a r e in o r d e r h e r e . 1. T h e s h o r t r a n g e r e p u l s i v e c o r r e l a t i o n s c o r r e c t t h e r e s u l t s g i v e n b y an u n c o r r e l a t e d g r o u n d s t a t e w a v e f u n c t i o n in t h e r i g h t d i r e c t i o n : t h e y m a k e t h e m i n i m u m d e e p e r a n d lift h i g h e r t h e s h o u l d e r of t h e c u r v e on t h e r i g h t h a n d s i d e of t h e m i n i m u m . 2. T h e s e r i e s o n e g e t s by e x p a n d i n g t h e s i n g l e - , d o u b l e - e t c s c a t t e r i n g t e r m s of eqo (2) in p o w e r s of (rc/R)3 h a v e a l t e r n a t i n g s i g n s a n d f a l l off w i t h i n c r e a s i n g A2 s l o w e r a n d s l o w e r a s o n e g o e s to h i g h e r a n d h i g h e r p o w e r s of (rc/R)3. T h i s r e s u l t s in t h e e x i s t e n c e of z e r o s of Tii (of a q u i t e different origin from the zeros which produce m i n i m a f o u n d in t h e B r o o k h a v e n e x p e r i m e n t [2, 4, 5]. In p a r t i c u l a r t h e s i n g l e s c a t t e r i n g c o n t r i b u t i o n
Volume 25B, number 5
PHYSICS
h a s a z e r o , h e n c e the e l e c t r o n c h a r g e f o r m f a c t o r of 4He should h a v e a s h a r p m i n i m u m s o m e w h e r e f o r A2 > 6.25 f m -2, h e n c e f o r A2 l a r g e r than the f i r s t m i n i m u m found in the B r o o k h a v e n e x p e r i m e n t [2]. Such a m i n i m u m has b e e n r e c e n t l y found [1]. F i g . 2 s h o w s I F c h l c a l c u l a t e d (to o r d e r (rc/R)3) and the p o i n t s of the e x p e r i m e n t [1]. 3. T h e p o s i t i o n s of the z e r o s c a u s e d by the c o r r e l a t i o n s m o v e to h i g h e r and h i g h e r A2 as one g o e s f r o m the s i n g l e - , to d o u b l e - , t r i p l e - etc s c a t t e r i n g c o m p o n e n t s of the a m p l i t u d e (2). H e n c e f o r ~ " r e a s o n a b l e " h ~ r d c o r e r a d i u s the only z e r o w h i c h p l a y s an i m p o r t a n t r o l e up to A 2 ~ l(GeV/c) 2 is the f i r s t z e r o of the s i n g l e s c a t t e r i n g c o m p o n e n t and the d e s i r a b l e e f f e c t s d e s c r i b e d in 1. a r e c a u s e d by its e x i s t e n c e . 4. W e did not t r y to find the b e s t f i t s . A l t h o u g h t h e c o r r e l a t i o n c o r r e c t i o n s go in the r i g h t d i r e c t i o n s , the e x p e r i m e n t a l m i n i m u m found in r e f . 1 is s t i l l not w e l l r e p r o d u c e d (fig. 2). It is quite p o s s i b l e that s o m e m o r e s o p h i s t i c a t e d c o r r e l a t i o n s (with m o r e p a r a m e t e r s ) m a y r e p r o d u c e the data still better. 5. One should s t r e s s that if one r e p r o d u c e s I F c h l m e a s u r e d in the e l e c t r o n e x p e r i m e n t [1] i n t r o d u c i n g s h o r t r a n g e c o r r e l a t i o n s one d o e s not prove their existence. The elastic electron scatt e r i n g e x p e r i m e n t a l r e s u l t s can a l w a y s be e x p l a i n e d in t e r m s of an e f f e c t i v e s i n g l e p a r t i c l e d e n s i t y (cf. r e f . 1): p ( r 1) : / d 3 r 2 . . .
d3rA I ~ I ' ( r l . . . rA) i 2
18 September 1967
u r e d Fch [1] f o r the c . m . m o t i o n , the p r o t o n c h a r g e d i s t r i b u t i o n and the m a g n e t i c m o m e n t s c o n t r i b u t i o n s . It t u r n s out h o w e v e r that the f i r s t two c a n c e l e a c h o t h e r to a l a r g e e x t e n t and the t h i r d is a few p e r c e n t c o r r e c t i o n only. In c o n c l u d i n g we would l i k e to s t r e s s that t h e r e is a v e r y good c h a n c e that the m e a s u r e m e n t s of e l a s t i c c r o s s s e c t i o n s of s t r o n g l y i n t e r acting particles scattered from nuclei may indic a t e the p h y s i c a l r e a s o n f o r the e x i s t e n c e of the d i f f r a c t i o n m i n i m u m in the 4He f o r m f a c t o r . Should it t u r n out to be c a u s e d by d y n a m i c a l r e p u l s i v e c o r r e l a t i o n s , the e l a s t i c e l e c t r o n s c a t t e r i n g at l a r g e m o m e n t u m t r a n s f e r s would b e c o m e a v e r y i m p o r t a n t tool in i n v e s t i g a t i n g s u c h c o r r e l a t i o n s in n u c l e i . In o r d e r to a c h i e v e it, we should h a v e a t h e o r y at o u r d i s p o s a l which is r e l i a b l e at l a r g e m o m e n t u m t r a n s f e r s ° It w o u l d s e e m h o w e v e r that a r o u n d t h e m i n i m u m found in r e f . 2, h e n c e f o r r e l a t i v e l y s m a l l m o m e n t u m t r a n s f e r s , w h e r e one can s t i l l t r u s t the m o d e l , our calculations clearly favour rather large valu e s of the r e a l p a r t s of t h e n u c l e o n - n u c l e o n s c a t tering amplitude° We a r e g r a t e f u l to P r o f e s s o r s H. P a l e v s k y and M. R. Y e a r i a n f o r s e n d i n g us t h e i r d a t a b e f o r e p u b l i c a t i o n . We would l i k e a l s o to t h a n k P r o f e s s o r s A. B o h r , B . R . M o t t e l s o n and J. D. W a l e c k a f o r the s t i m u l a t i n g d i s c u s s i o n s , c r i t i c i s m and comments.
(5)
w h i c h m a y o r m a y not c o m e f r o m a c o r r e l a t e d g r o u n d s t a t e w a v e f u n c t i o n . T h i s a m b i g u i t y can p r o b a b l y be r e s o l v e d by m e a s u r e m e n t s of the e l a s t i c c r o s s - s e c t i o n s of s t r o n g l y i n t e r a c t i n g p a r t i c l e s (~ m e s o n s , n u c l e o n s ) at l a r g e m o m e n t u m t r a n s f e r s A2 > I ( G e V / c ) 2 . One m a y e x p e c t it to b e the c a s e b e c a u s e one finds t h a t although the s i n g l e s c a t t e r i n g c o n t r i b u t i o n s to eq. (2) g i v e n by a c o r r e l a t e d w a v e f u n c t i o n l i k e eq. (1) and by the d e n s i t y (5) c a l c u l a t e d f r o m eq. (1) a r e i d e n t i c a l , t h e double s c a t t e r i n g c o n t r i b u t i o n s a r e a l r e a d y quite different. 6. T h e f o r m u l a (4) e n a b l e s us to e m p l o y d i r e c t l y the m e a s u r e d Fch(5) to c a l c u l a t e Cog. the n u c l e o n 4He c r o s s s e c t i o n . One m u s t h o w e v e r k e e p in m i n d that eq. (4) is c o r r e c t u n d e r t h e a s s u m p t i o n (3). W e took Fch(5) = [1-(c252) 6] exp(-d252)
LETTERS
(6)
o b t a i n e d in r e f . 1 and the r e s u l t s a r e s h o w n in f i g . 1. In p r i n c i p l e , in o r d e r to i n s e r t it into eq. (4), one should c o r r e c t the e x p e r i m e n t a l l y m e a s -
References 1. R. F. Frosch, J . S . McCarthy, R.E. Rand and M.R. Yearian, Bull. Am. Phys. Soc. 12 (1967) 16; Stanford preprint HEPL 492 (1967). 2. H. Palevsky, Intern. Conf. on Nuclear physics, Gatlinburg, 1966; Second Intern. Conf. on High energy physics and nuclear structure, Rehovoth 1967, and private communication. 3. G. Bellettini, G. Cocconi, A.N. Diddens, E. Lillethun, G. Matthiae, T. P. Scanlon and A.M.Wetherell, Nucl. Phys. 79 (1966) 609. 4. W. Czy~. and L. Le~niak, Phys. Letters 24B (1967) 227. 5. W. Czy~ and L. Le~niak, Second Intern. Conf. on High energy physics and nuclear structure, Rehovoth, 1967. 6. H. H. Bassel and C.Wilkin, Phys. Rev. Letters 18 (1967) 871. 7. R.J. Glauber, in Lectures in theoretical physics. ed. W. E. Brittin (Interscience Publishers Inc., New York, 1959). 8. R.J. Jastrow, Phys. Rev. 98 (1955) 1479.
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