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Nucleus-nucleus scattering at high energies
~
Nuclear Physics B28 (1971) 368-380. North-Holland Publishing Company
NUCLEUS-NUCLEUS SCATTERING AT HIGH ENERGIES Alfred MULLENSIE FEN
Institut flit Struktur der Materie der Universitiit Karlsruhe Received 20 July 1970 (Revised manuscript received 8 December 1970)
Abstract: We relate the measured e l e c t r i c nuclear form factor to the opaqueness model of the nucleus. The calculated differential cr o ss section of the proton s c a t tering on 4He, 12C and 160 at 1 GeV is in reasonable agreement with the e x p e r i mental data if the total c r o s s section is taken as a f r e e p a r a m e t e r . Using the r e sults of elastic p a scattering we also get good agreement with the Glauber theory for a a scattering at about 8 G e V /c per nucleon. Our calculations are much more easily handled than those of the Glauber model. They may be considered as an improved optical-limit amplitude. In particular, the c o r r e c t c.m. motion factor differs from that of Czy~. and Maximon for the case of nucleus-nucleus scattering. We calculate d(T/dt for a-12C scattering and estimate the total c r o s s section for the scattering of 4He on C, N and O from which we obtain the mean f r e e path of 4He in air.
I. INTRODUCTION AND CONCLUSION In the o p t i c a l m o d e l t h e s c a t t e r i n g on n u c l e i i s t r e a t e d r e g a r d i n g t h e n u c l e u s as an o p a q u e m e d i u m . R e c e n t l y Chou and Y a n g [1, 2] h a v e d i s c u s s e d d a / d t f o r pp s c a t t e r i n g in the l i m i t of i n f i n i t e l y high e n e r g i e s , t h e i r s u g g e s t i o n b e i n g to r e l a t e the n u c l e o n o p a q u e n e s s to the n u c l e o n e l e c t r o m a g n e t i c f o r m f a c t o r . In t h i s p a p e r w e a p p l y the s a m e i d e a to the 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 and n u c l e i on n u c l e i and thus c o r r e l a t e the m e a s u r e d e l e c t r i c n u c l e a r f o r m f a c t o r w i t h the n u c l e a r o p a q u e n e s s . T h e s t r e n g t h of th e l a t t e r i s d e t e r m i n e d f r o m n u c l e o n - n u c l e u s s c a t t e r i n g . U s i n g only an o p t i c a l m o d e l the a b s o r p t i o n c r o s s s e c t i o n f o r n u c l e i n u c l e i s c a t t e r i n g h a s b e e n d e t e r m i n e d in an e a r l i e r p a p e r [3]. T h e a u t h o r s c h o o s e a s p a r a m e t e r s a m o d i f i e d v a l u e of the t o t a l n u c l e o n - n u c l e o n c r o s s s e c t i o n and the n u c l e a r r a d i i d e t e r m i n e d f r o m n e u t r o n - n u c l e u s s c a t t e r i n g [4]. T h e i r f o r m u l a , h o w e v e r , d o e s not r e d u c e to the r e l a t i o n u s e d in n e u t r o n - n u c l e u s s c a t t e r i n g and thus we b e l i e v e t h a t t h e i r r e s u l t h as to be r e fined. In s e c t . 2 w e s u m m a r i z e the f o r m u l a e and s p e c i f y the a n s a t z f o r the n u c l e a r o p a q u e n e s s P A ( X , y , z ) . T h i s i s e x p r e s s e d in t e r m s of the o p a q u e n e s s of the n u c l e o n and t h e d e n s i t y d i s t r i b u t i o n of the A n u c l e o n s w h i c h f o l l o w s f r o m the m e a s u r e d n u c l e a r e l e c t r i c f o r m f a c t o r . Th e n u c l e o n o p a q u e n e s s i s d e t e r m i n e d f r o m the e l a s t i c n u c l e o n - n u c l e o n s c a t t e r i n g . T h e e l e c t r i c f o r m f a c t o r as well as the o p a q u e n e s s have been taken as a Gauss function o r a
NUCLEUS-NUCLEUS SCATTERING
369
s u m of two G a u s s i a n s . T h e e l a s t i c p r o t o n - n u c l e u s s c a t t e r i n g i s in r e a s o n a b l e a g r e e m e n t w i t h the e x p e r i m e n t a l d a t a . In s e c t . 3 w e c o m p a r e t h i s m o d e l w i t h the o p t i c a l l i m i t of t h e G l a u b e r t h e o r y ( i n f i n i t e l y m a n y s m a l l n u c l e o n s , s e e , e . g . , r e f . [5]). In c o n t r a s t to the o p t i c a l l i m i t c a l c u l a t i o n s of Czy~ and M a x i m o n [5] we g e t r e a s o n a b l e a g r e e m e n t w i t h the m u l t i p l e s c a t t e r i n g t h e o r y in a a s c a t t e r i n g at a b o u t 8 G e V / c p e r n u c l e o n . W e a t t r i b u t e t h i s to o u r m o d i f i e d t r e a t m e n t of t h e c . m . m o t i o n of the n u c l e o n s w i t h i n the n u c l e u s and c o n c l u d e t h a t o u r a m p l i t u d e can b e t a k e n a s an i m p r o v e d o p t i c a l - l i m i t a m p l i t u d e ( s e e e q s . (3.1) and (3.2)). W e h a v e i n t r o d u c e d a l s o a r e a l p a r t of the a m p l i t u d e . W e c a l c u l a t e the a b s o r p t i o n c r o s s s e c t i o n s of a p a r t i c l e s w i t h an e n e r g y of 1 GeV p e r n u c l e o n s c a t t e r e d on 12C, 14N and 160 and o b t a i n a m e a n f r e e p a t h b e t w e e n 44 and 49 g . c m -2 in a i r . The t o t a l aC c r o s s s e c t i o n ~ C i s e x p e c t e d to l i e b e t w e e n 608 and 720 mb. W e c a n n o t g i v e an a c c u r a t e v a l u e s i n c e we h a v e not t r i e d to i m p r o v e the s i m p l e a n s a t z f o r pp s c a t t e r i n g o r to i n c l u d e the e x p e r i m e n t a l e r r o r l i m i t s . It t u r n s out, h o w e v e r , t h a t the t - d e p e n d e n c e of dc~/dt d o e s not c h a n g e d r a s t i c a l l y if ~ ~t i s a l t e r e d . Thus we e x p e c t t h a t o u r s i m p l e m o d e l w i l l y i e l d the c o r r e c t o r - d e r of m a g n i t u d e f o r the e l a s t i c s c a t t e r i n g a n d t h a t o u r r e s u l t s can b e u s e d a s an o r i e n t a t i o n in f u r t h e r c a l c u l a t i o n s and in p l a n n i n g e x p e r i m e n t s w i t h h i g h - e n e r g y a - p a r t i c l e s . Of c o u r s e t h i s m o d e l m a y b e a v e r y c r u d e a p p r o x i m a t i o n , b u t i t s h o u l d a l s o b e p o i n t e d o u t t h a t u s i n g t h e m e a s u r e d e l e c t r i c f o r m f a c t o r a n u m b e r of c o n t r i b u t i o n s (e.g. m e s o n e x c h a n g e c u r r e n t s ) a r e i n c l u d e d to a c e r t a i n d e g r e e w h e r e a s t h e s e e f f e c t s w o u l d be d i f f i c u l t to b u i l d into the G l a u b e r t h e o r y . The s p i n s of the p a r t i c l e s h a v e b e e n n e g l e c t e d t h r o u g h o u t . In a p p e n d i x A we l i s t t h e n e c e s s a r y f o r m u l a e and in a p p e n d i x B we e s t i m a t e the i n f l u e n c e of C o u l o m b s c a t t e r i n g .
2. THE S C A T T E R I N G A M P L I T U D E AND N U C L E A R O P A Q U E N E S S N e g l e c t i n g s p i n e f f e c t s Chou and Y a n g [1] w r i t e f o r t h e s c a t t e r i n g a m plitude 1 a ( u ) = i { s ( x ) - ~.1 s ( u ) ® s ( x ) + ~.v s ( x ) ® s ( u )
®s(u) - +...},
w h e r e ~ =(K..,K~Y') and K2 =K 2 + K2 = - t i s the m o m e n t u m t r a n s f e r . use the definition
f(x)
1 ®g(u) = ~
fff(u
- x')g(u')dx'.
(2.1) They
(2.2)
T h e d i f f e r e n t i a l c r o s s s e c t i o n i s g i v e n by d_~ = ~ l a ( ~ ) f2 dt and s ( x ) i s d e t e r m i n e d in t h e f o l l o w i n g way. F o r an i n c o m i n g p o i n t l i k e p a r t i c l e w i t h an i m p a c t p a r a m e t e r t h e n u c l e u s A a p p e a r s a s a d i s k w i t h an o p a q u e n e s s
(2.3)
b = (x,y)
370
A. MTJLLENSIE
DA(b) = J
FEN
PA(r)dz.
(2.4)
--OO
The e s s e n t i a l a s s u m p t i o n of the Chou-Yang model is that at v e r y high e n e r gies the m a t t e r density P A ( r ) is p r o p o r t i o n a l to the e l e c t r i c density peA(r) , whicl~ in this c a s e is the F o u r i e r t r a n s f o r m of the e l e c t r i c f o r m f a c t o r FA(k~) of the whole nucleus with m a s s n u m b e r A, i.e. P A ( r ) = dAPeA(r),
peA(r) = A f e x p ( - i k r ) F A ( k 2 ) d k .
(2.5)
This a s s u m p t i o n is p r o b a b l y not a v e r y good one f o r a proton and c o m p l e t e l y wrong f o r a neutron, at l e a s t in the m o s t t r i v i a l s e n s e that the neutron is e l e c t r i c a l l y neutral e v e r y w h e r e . H o w e v e r , for nuclei c o m p a r i s o n between e l e c t r o n - n u c l e u s and n u c l e o n - n u c l e u s s c a t t e r i n g shows that eq. (2.5) is indeed a p p r o x i m a t e l y s a t i s f i e d [14]. Speculations exist that the m a t t e r density extends s o m e w h a t f u r t h e r out than the c h a r g e density ("neutron e x c e s s at the n u c l e a r s u r f a c e " ) [15]. In that c a s e d A (which in g e n e r a l is P A ( r ) / p ~ ( r ) ) will be a function of r . H o w e v e r , for the p r e s e n t calculation, we take d A as a constant. It m a y be worth pointing out that the Chou-Yang model has one conceptual advantage o v e r the Glauber model: it does not a s s u m e that the nucleus cons i s t s of protons and neutrons. All hadronic m a t t e r (virtual pions, nucleon i s o b a r s etc.) is included. This could lead to i m p r o v e m e n t s e s p e c i a l l y for strongly bound nuclei. In analogy with the Chou-Yang e x p r e s s i o n f o r the pp s c a t t e r i n g we put (K2 = k 2 + k 2 k z =0) x y' SAB(g 2) = (1 - i a ) K A B d A d B FA(K 2) FB(K2),
(2.61
w h e r e K is d e t e r m i n e d f r o m pp s c a t t e r i n g and d A f r o m e l a s t i c p - n u c l e u s s c a t t e r i n g . With eq. (2.6) we obtain the total c r o s s section (see appendix A) Cr~B = 47r 2 ( f A + fiB) {C+ In x - E i ( - x ) } ,
x = ½KABd A dB(f A + fiB)- 1,
C = 0 . 5 7 7 2 . . . ( E u l e r ' s constant),
(2.7)
(2.8)
w h e r e the e l e c t r i c f o r m f a c t o r has been p a r a m e t r i z e d as FA(k2 ) --- exp ( - f A k 2 ) .
(2.9)
A g e n e r a l i s a t i o n - e s p e c i a l l y to finite e n e r g i e s - is p o s s i b l e if we a s s u m e PA ( r ) = f f f P S (
r - r ' ) W A ( r ') f A ( r ' ) d r ' .
(2.10)
H e r e , P N ( r ) is the nucleon o p a q u e n e s s , which d e s c r i b e s the n u c l e o n - n u cleon s c a t t e r i n g (we do not distinguish between p r o t o n s and neutrons). The
NUCLEUS-NUCLEUS SCATTERING
37I
function WA(r) ( n o r m a l i z e d to A) is the density distribution of the A nucleons calculated f r o m the m e a s u r e d e l e c t r i c f o r m f a c t o r FA(k2) of the nucleus [12]. We take FA(k2) s p h e r i c a l l y s y m m e t r i c s i n c e we neglect spin e f f e c t s , quadrupole m o m e n t s and all o t h e r d i s t o r t i o n m e c h a n i s m s . Thus we have the r e l a t i o n 1
fffwA(r)exp(ikr)dr
=FA(k2)exp(fiN k2) =- GA(k2).
(2.11)
The e x p r e s s i o n exp (/3N t) with fiN = 2.74 (GeV/c) -2 d e s c r i b e s the e l e c t r i c f o r m f a c t o r of the proton and is n e c e s s a r y in eq. (2.11) since GA(k 2) shall r e p r e s e n t the distribution of the nucleons only. Also f A ( r ) = 1 i m p l i e s that the distribution of the nucleons within the nucleus is given by WA(r). Thus the unknown function f A ( r ) d e s c r i b e s the deviation f r o m this s i m p l e model. We a s s u m e that f A ( r ) can be a p p r o x i m a t e d by the constant d A. It can, h o w e v e r , be energy dependent and is d e termined from nucleon-nucleus scattering. F o r an incoming extended p a r t i c l e B the o p a q u e n e s s is a s s u m e d to be of the f o r m (2.12)
D(b) : f f DB(b - b')DA(b')db'.
Within o u r model the amplitude for the e l a s t i c s c a t t e r i n g of B on A is then given by eq. (2.1) taking SAB(~ ) = (1 -
1
ia) ~
fro(b) exp (i ~b) db.
(2.13)
F o r a = 0 the d i f f r a c t i o n m i n i m a in da/dt a r e z e r o and thus the introduction of a r e a l p a r t in the a m p l i t u d e is n e c e s s a r y . We take a as a constant and d e t e r m i n e it e x p e r i m e n t a l l y at t = 0. With 3
(27r) -~
fffPN(t)exp(ikr)dr
= CNON(k),
GN(0) = 1
(2.14)
we get f r o m eqs. (2.10), (2.13) and (2.14)
SAB(kx, ky) = (1 - ia)(2n) 2 c2 dA dBA B G2(kx, ky, O)GA(kx,ky, O)GB(kx,ky, 0).
(2.15)
Here GN(k) is also supposed to be spherically symmetric and thus from eq. (2.15) follows SAB(• 2) = (1 - ia)Kd A dBA
B GA(K 2) GB(K 2) G2(K 2)
using
c(K 2) = ~(k 2) ]kz__0 •
(2.16)
372
A. M O L L E N S I E F E N I
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373
SCATTERING
H e r e , K = (2v) 2 c 2 is e n e r g y dependent and will be d e t e r m i n e d f r o m n u c l e o n nucleon s c a t t e r i n g . The f o r m f a c t o r FA(k2) can be a p p r o x i m a t e d by
(2.17)
FA(k2 ) = c A exp ( - f A k2 ) + (1 - cA)exp (-/~A k2),
as is shown in fig. i for carbon (CA, fA and ~A a r e constants). The o p a q u e n e s s is then
(2.18)
GA(K 2) = c A e x p ( - a A K2) + (1 - c A ) e x p ( - b A n 2 ) ,
w i t h a A = f A - f i N a n d b A = ~ A - f i N . F o r K2 << 1 e v e n c A = 1 i s a g o o d a p proximation. The function GN(K2) is also d e t e r m i n e d f r o m nucleon-nucleon s c a t t e r i n g (A = B =G A =G B = 1). This can, of c o u r s e , be a c o m p l i c a t e function, but fig. 2 shows that the a n s a t z GN(K2) = exp ( - a N K2)
(2.19)
is sufficient for o u r p u r p o s e . The s t r a i g h t line in this f i g u r e r e p r e s e n t s the m e a s u r e d d i f f e r e n t i a l nucleon-nucleon c r o s s section p a r a m e t r i z e d in a f o r m which is e.g. always used in the Glauber theory. With the s i m p l e a n s a t z GA(K 2) = exp ( ' a A K 2 ) ,
GB(K 2) = exp ( - a B K 2 ) ,
(2.20)
we get SAB(K2 ) = (1 - i a ) K d A d B A B
exp[-(aA+aB+2aN)K2
].
(2.21)
Using the optical t h e o r e m and eq. (2.1) the total c r o s s section is now (see appendix A) ~ t B = 47r 2(a A + a B + 2aN) {C + In x - E i ( - x ) } ,
(2.22)
with C = 0 . 5 7 7 2 . . . ( E u l e r ' s constant) and x = ½ K d A d B A B ( a A + aB+ 2aN) -1. If at v e r y high e n e r g i e s a N can be a p p r o x i m a t e d by fiN, then eq. (2.19) is equal to eq. (2.7). Czy~. and Maximon [5] d i s c u s s the aa s c a t t e r i n g at about 8 G e V / c . Taking t h e i r p a r a m e t e r s (see fig. 15 of ref. [5]) the total p a s c a t t e r i n g within the Glauber model is crt a = 139.2 mb. Using this v a l u e , crt a = 375 m b , a N = = 1.8 ( G e V / c ) - 2 (se~ fig. 2) and a a = 9.04 (GeV/c) -2 we get f r o m eq. (2.22) d a = 0.87 and gNN = 40.3 mb which is in r e a s o n a b l e a g r e e m e n t with g t N = = 41 mb used by Czy2 and Maximon as the m e a n value of the total pp and np s c a t t e r i n g (see ref. [19]). We i g n o r e the s m a l l d i f f e r e n c e since we only u s e the a p p r o x i m a t i o n (2.19) f o r GN(K2 ). At 1 GeV we take (~IN = 44 m b , (rt a = 152 m b [11], a a = 9.04 (GeV/c) -2 (ref. [12]) and get the values shown inp table 1. H e r e we see that K d a is n e a r l y constant w h e r e a s d a changes by about 15%. This r e s u l t is p l a u s i b l e
374
A. Mi]LLENSIE FEN Table 1 aN (GeV/c) -2
Kd a (GeV/c) -2
da
1.0
12.20
0.57
0.9
12.33
0.50
s i n c e a s m a l l v a r i a t i o n of a N g i v e s o n l y a s m a l l c h a n g e in c a + 2a N b u t an a p p r e c i a b l e o n e in K in pp s c a t t e r i n g ( s e e eq. (2.22)). W e do not g i v e an a c c u r a t e v a l u e of d a s i n c e w e c o m p l e t e l y i g n o r e the e x p e r i m e n t a l e r r o r l i m i t s and u s e o n l y the s i m p l e a n s a t z (2.19) f o r the n u c l e o n and a - p a r t i c l e o p a q u e n e s s . U s i n g the r e s u l t s of t a b l e 1 we e x p e c t a v a l u e of d a b e t w e e n a b o u t 0.5 and 0.7. In fig. 3 d a / d t c a l c u l a t e d w i t h a N = = 0.9 ( G e V / c ) -2 i s shown f o r p a s c a t t e r i n g . T h e r e s u l t s a r e in good a g r e e m e n t w i t h the m e a s u r e m e n t s and the c a l c u l a t i o n s of t h e G l a u b e r m o d e l (for the 4He f o r m f a c t o r s e e e.g. r e f . [9]). T h e a g r e e m e n t i s a s good f o r p - 1 2 C and p - 1 6 0 s c a t t e r i n g a t 1 GeV if the a p p r o x i m a t i o n (2.17) i s u s e d w i t h c A ¢ 1. In p - 1 6 0 s c a t t e r i n g w e g e t a g r e e m e n t b e y o n d the f i r s t m i n i m u m o n l y if w e t a k e c O ¢ 1. In p - 1 2 C s c a t t e r i n g the c a l c u l a t i o n w i t h c C = I i s s l i g h t l y b e l o w the e x p e r i m e n t a l d a t a w h e r e a s t h a t w i t h c C ¢ 1 i s s l i g h t l y a b o v e t h e m n e a r the f i r s t m a x i m u m .
3. N U C L E U S - N U C L E U S S C A T T E R I N G S i n c e we g e t r e a s o n a b l e r e s u l t s in n u c l e o n - n u c l e u s and aa s c a t t e r i n g we a p p l y t h i s m o d e l to e l a s t i c aC s c a t t e r i n g a s w e l l , and d i s c u s s a l s o the m e a n f r e e p a t h of a - p a r t i c l e s in a i r . W e w a n t f i r s t , h o w e v e r , to c o m p a r e o u r m o d e l in the c a s e of a a s c a t t e r i n g w i t h the o p t i c a l l i m i t r e s u l t w h i c h f o l l o w s f r o m the G l a u b e r t h e o r y and i s g i v e n in r e f . [5]. W e a l s o u s e the p a r t i c l e i n d e p e n d e n t m o d e l f o r 4He and c h o o s e f o r t h e o n e - p a r t i c l e d e n s i t y p(r) = v R 2 e x p ( - r 2 R - 2 ) . T a k i n g into a c count the c . m . c o n s t r a i n t ( s e e e.g. r e f s . [5, 6]) we g e t f o r the o p a q u e n e s s Ga(K2 ) = e x p ( - ~ R2K 2) = e x p ( - a a K 2 ) . F r o m e q s . (2.1) and (2.21) w e o b t a i n f o r the a m p l i t u d e
n=l
nn!
L ~R2+2a N J
e x p { - ( ~ R 2 + 2 a N ) n--}" (3.1)
In the o p t i c a l l i m i t Czy~ and M a x i m o n [5] o b t a i n
a( 2) n=l
nn!
L~(R2 + a). j
w i t h e(g 2) = exp (tR2K2) and d ~ / d t = a2] 1 - i a ] 2 e x p (at)/167r f o r the NN s c a t t e r i n g . A t f i r s t s i g h t t h e r e i s a s t r o n g s i m i l a r i t y b e t w e e n the a m p l i t u d e s
0
I
\
0,2
I
I
l
m m I w
0,1
~
0,3 0,6 O,S OoG -t GeV ~Fig. 3. p-4He s c a t t e r i n g at 1 GeV. The full line has been calculated with a N = 0.9 (GeV/c) -2, f i t = 152 mb, a = -0.33 and Fa(t ) = 1 . 0 2 1 e x p ( l l . 1 7 t ) - 0 . 0 2 1 e x p "( 1- . 2 3 t ) for the 4 He f o r m factor. The c r o s s e s c o r r e s p o n d to Fa(t) = exp (11.78t). The dashed c u r v e (Glauber model) and the e x p e r i m e n t a l data have been taken f r o m ref. [5].
t6
~6 ~
'[
i
I
I
I
I
t
I
-'x% i.%
t
"~~'~.\
%'N,
',,
u s • a"%
"%,
'~. m"%
_J ~'~
s p e c t i v e l y . The c r o s s s e c t i o n d o ' / d t i s 7 . 4 a n d 8.3 b - ( G e V / c ) - 2 at t = 0 .
0 0.! 0,2 0,3 0,4 0,S 0,G {3,7 0,8 0.9 i,O-t GeV:.c-: Fig. 4. ~a s c a t t e r i n g at about 8 G e V / c p e r hucleon. The d a s h - d o t c u r v e (Glauber model) and the dashed c u r v e (optical limit) have been taken f r o m ref. [5]. The full line and the c r o s s e s have been calculated with a** = 1 8 ZGeV/c~ -2 t ~ ~= 375 " m ' b an " d ~400' m b r e a(~ = 9.04 ( G e V / c ) - 2 , o / = - 0 . 3 3 a n d O'qo
~(~z,
1C[3
162
"E 101 ~e~\
•
I0:
f3
!
376
A. MULLENSIE FEN
(3.1) and (3.2). It j u s t i f i e s t h e i n t r o d u c t i o n of a in f o r m u l a (3.1). T h e f a c t o r 0(K2) a r i s e s f r o m the c . m . c o n s t r a i n t of t h e n u c l e o n s w i t h i n t h e n u c l e u s . It i s unity f o r i n f i n i t e l y m a n y n u c l e o n s [5], b u t d a / d t i s too s m a l l in t h i s l i m i t . Czy~ and M a x i m o n m a k e i t p l a u s i b l e to k e e p 8(K 2) in eq. (3.2). T h e d i f f e r e n c e of the n t h p o w e r t e r m b e t w e e n the e x p r e s s i o n s (3.2) and
(3.1)
is 1
- ~ a + 2a N +
(½a+ ½R2)(1
- 1).
(3.3)
Both a and a N d e s c r i b e NN s c a t t e r i n g . Only the t e r m s w i t h n = 1 a r e e q u a l if f o r the m o m e n t w e t a k e ½a = 2a N ( t h i s e q u a l i t y b e i n g o n l y a p p r o x i m a t e l y t r u e ) . A l l the o t h e r t e r m s a r e l a r g e r in the o p t i c a l l i m i t . F o r a a s c a t t e r i n g , fig. 4 s h o w s t h a t a t K2 = 1 ( G e V / c ) 2 the c r o s s s e c t i o n in the o p t i c a l l i m i t i s m o r e than a f a c t o r 102 l a r g e r than the v a l u e o b t a i n e d f r o m the G l a u b e r m o d e l . T h u s eq. (3.2) i s u s e l e s s in n u c l e u s - n u c l e u s s c a t t e r i n g . W h e t h e r the d i s c r e p a n c y b e t w e e n o u r m o d e l and the G l a u b e r m o d e l c a l c u l a t i o n s ( d a s h - d o t c u r v e in fig. 4) can b e r e d u c e d a t h i g h e r m o m e n t u m t r a n s f e r s if a m o r e c o m p l i c a t e d f u n c t i o n f o r GN(K2 ) i s u s e d r e m a i n s to b e i n v e s t i g a t e d . The f i t f o r Ga(K2 ) w i t h c a ¢ 1 g i v e s a l m o s t the s a m e r e s u l t a s t h a t w i t h c a = 1. T h i s d i s c u s s i o n s h o w s t h a t the a m p l i t u d e (3.1) can b e r e g a r d e d a s an i m p r o v e d o p t i c a l l i m i t a m p l i t u d e w h e r e the c . m . m o t i o n of the n u c l e o n s h a s b e e n t a k e n into a c c o u n t in a m o d i f i e d way. We a l s o w a n t to m e n t i o n a p a p e r by K o f o e d - H a n s e n [7] who b r i e f l y d i s c u s s e s aa s c a t t e r i n g in t h e G l a u b e r m o d e l . He i n v e s t i g a t e s the e f f e c t of t r u n c a t i n g the G l a u b e r s e r i e s . F o r aC s c a t t e r i n g w e s t i l l n e e d d C. F r o m ~ t a = 1 5 2 m b , t =370mb ( r e f . [11]) and a C = 23.18 ( G e V / c ) -2 ( r e f . [12]) ~ve c a l c u l a t e the~'"rati~ d c / d a = 0.83 a t 1 GeV f r o m eq. (2.22). U s i n g t h i s v a l u e we o b t a i n a a C = = 839+ 3 3 3 1 n d a (in m b ) , a~C = 608 m b and 720 m b f o r d a = 0.5 and 0.7 r e s p e c t i v e l y . Only In d a a p p e a r s and n o t In d 2 s i n c e the p r o d u c t K d a i s a l r e a d y d e t e r m i n e d f r o m p a s c a t t e r i n g . S e t t i n g d a = d C =1 we can e a s i l y d e t e r m i n e K f r o m n u c l e o n - n u c l e u s s c a t t e r i n g and then c a l c u l a t e a~C. In t h i s w a y we find f r o m p a s c a t t e r i n g atac = 912 m b and f r o m pC s c a t t e r i n g t = 854 mb. aaC T h e d i s c r e p a n c y s h o w s t h a t the c h o i c e d a = d C =1 i s i m p o s s i b l e . In fig. 5 d ~ / d t i s shown f o r aC s c a t t e r i n g a t 1 GeV p e r n u c l e o n f o r a ~ = = 608 m b and 720 mb. T h e t d e p e n d e n c e i s v e r y s i m i l a r in b o t h c a s e s and thus o u r r e s u l t s can b e u s e d a s a f i r s t a p p r o x i m a t i o n to the e l a s t i c aC s c a t t e r i n g . A s in p - 1 2 C s c a t t e r i n g d a / d t i s o n l y s l i g h t l y a l t e r e d if c C ¢ 1 i s u s e d . S i m i l a r l y w e find a s m a l l c h a n g e of the m a x i m a if a = 0 i s c h o s e n . T h u s the t o t a l and a b s o r p t i o n c r o s s s e c t i o n s can w e l l b e c a l c u l a t e d t a k i n g a = 0 and C A = C B = l . S i n c e the p a r a m e t e r s u s e d do not c h a n g e v e r y r a p i d l y w i t h e n e r g y , fig. 5 s h o u l d a l s o be v a l i d a t s l i g h t l y h i g h e r e n e r g i e s . F i n a l l y we e s t i m a t e the m e a n f r e e p a t h of a - p a r t i c l e s in a i r . The a b s o r p tion c r o s s s e c t i o n i s a a = a t - a e, w h e r e a e i s the c o m p l e t e e l a s t i c c r o s s s e c tion
N U C L E U S - N U C L E U S SCATTERING
377
10I
°
t 102
.-.
I
i
101
'~" " ~
'
I0'
1(~2
0
I
I
I
I
i
0.0S
0,1
0,15
0.2
0,25
12
I
[
0,30 0.35 -t GeV 2. c-2
'~s
0.40
2
F i g . 5. ~- C s c a t t e r i n g at 1 GeV p e r nucleon with ~ = -0.33, a N = 0.9 ( G e V / c ) and a a = 9.04 (GeV/c) -2 f o r fftqc = 608 mb (full line and c r o s s e s ) and fftaC = 720 mb (dashed line). The c r o s s e s hav~ been c a l c u l a t e d with F c ( t ) = exp (25.92t~ and the o t h e r c u r v e s with F c ( t ) = 1.83 exp (20.800 - 0 . 8 3 exp (14.67t) f o r the carbon f o r m f a c t o r . The c r o s s s e c t i o n s dcr/dt is 19 and 27 b • ( G e V / c ) - 2 at t = 0.
o"e=
f
o d~
tu
~ t- d t .
(3.4)
H e r e t u i s t h e l o w e r l i m i t of t h e i m p u l s e t r a n s f e r . T h e q u a n t i t y ~e i s e a s i l y c a l c u l a t e d f o r a = 0, c A = c B = 1 and t u = -oo. T h i s a p p r o x i m a t i o n w i l l c e r t a i n l y b e v e r y g o o d s i n c e d(~/dt d e c r e a s e s v e r y r a p i d l y n e a r t = 0 a n d t h u s t h e c o n t r i b u t i o n f r o m t u to i n f i n i t y i s n e g l i g i b l y s m a l l . W e g e t (r~B = 47r(a A + a B + 2 a N ) {C + In ½x - 2Ei(-x) + Ei(-2x)}. The absorption
cross
(3.5)
s e c t i o n ~a i s t h e n
g ~ B = ~ t B _ CrABe = 47r(aA + a B + 2 a N ) {C + In (2x) - Ei(-2x)}.
(3.6)
378
A, M~LLENSIE FEN
A s i m i l a r f o r m u l a h a s b e e n found e a r l i e r [3]. In t h a t p a p e r the n u c l e a r r a d i i a r e t a k e n f r o m an o p t i c a l - m o d e l a n a l y s i s of n e u t r o n - n u c l e u s s c a t t e r i n g [4] but the a b s o r p t i o n c r o s s s e c t i o n d o e s n o t r e d u c e to the f o r m u l a u s e d in n e u t r o n - n u c l e u s a n a l y s i s . T h u s we b e l i e v e t h a t the f o r m u l a in r e f . [3] h a s to b e r e f i n e d . S i n c e E i ( - 2 x ) can b e n e g l e c t e d we g e t w i t h d B = ~d A
~B=47r(aA+aB+2aN){C+In~
ABKd A
= a ~ B ( d A =1) + 4 T r ( a A + a B + 2 a N ) l n d
(3.7)
A .
A t 1 GeV we o b t a i n f o r d A -- 1 ( ~ C = 535 m b and a~O = 656 m b u s i n g a t o = 475 m b and a O = 29.63 ( G e V / c ) -2 ( r e f s . [11, 12]). The r a t i o d o / d a i s 0.82. By l i n e a r e x t r a p o l a t i o n of the c a r b o n and o x y g e n r e s u l t s w e find f o r n i t r o g e n (~aN = 595 m b ( d a = l ) . F o r d a = 0.5 (0.7), a s i s s u g g e s t e d in s e c t . 2, we g e t ~<~ = 420 (476) mb. U s i n g ~ = 44 x 0.95 m b in f o r m u l a (A4.8) of r e f . [3] o n e o b t a i n s oaaC = 472 mb. The mean free path is calculated from Pair l = a a ' ~aO n o + a a N nN
(3.8)
w i t h P a h r = 1.29- 1 0 - 3 g • c m - 3 , nO = 1 . 1 8 . 1 0 1 9 and n N = 4 . 1 9 . 1 0 1 9 p a r t i c l e s p e r c m f o r o x y g e n and n i t r o g e n . W e o b t a i n 39.4 cm_ 2 ; l = 1 + 0.2861n c a g" t h a t i s , l = 49.1 g - c m -2 f o r d a = 0.5 and l = 43.9 g . c m -2 f o r d a = 0.7. U n f o r t u n a t e l y t h e r e a r e no m e a s u r e m e n t s a v a i l a b l e f o r t h e m e a n f r e e p a t h of a - p a r t i c l e s in a i r . C o n s e q u e n t l y i t w o u l d b e i n t e r e s t i n g to a p p l y o u r m e t h o d to the s c a t t e r i n g of 4He on t h o s e n u c l e i w h i c h a r e c o n t a i n e d in n u c l e a r e m u l s i o n s ; s i n c e f o r n u c l e a r e m u l s i o n s the m e a n f r e e p a t h h a s a l ready been measured. I a m v e r y m u c h i n d e b t e d to P r o f . H. P i l k u h n , D r . W. S c h m i d t and D r . D. J u l i u s f o r s t i m u l a t i n g d i s c u s s i o n s and a c r i t i c a l r e a d i n g of the m a n u script.
APPENDIX A In t h i s a p p e n d i x w e s u m m a r i z e -ia
s(K2) = 2 ~ - ~ )
the f o r m u l a e .
For
N
#~=1 c k e x p ( - E k K 2 )
(A.1)
NUCLEUS-NUCLEUS SCATTERING
379
we get N s(g2) ® s ( K 2 ) = 2 ( 1 2 i a ) 2 ~
N ~
CklCk2 exp {- EklEk2 K2 kl=l k2=l Ekl+Ek 2 Ekl+Ek 2 1
(A.2)
with the definition 1
/(~1,K2) ®g(Kl,~2) = ~ ff/(K1 -KI,K 2 -K½)g(~i,K½)dKidK~
(A.3)
The nth convolution is given by s(K2) ® . . . ®
x ~
...
k1=1
s(K2)=2(12i~)
n
N
i
~ Ckl n exp n n K2 kn= l s.~= l Ekl . . EknEk~ . . . i~=l Ekl " EknEk-~
(A.4)
and the amplitude is calculated from eq. (2.1). In the special case N = 1 we get for the nth convolution S(K2) ® . . . ®
( ) E l K2` 1 C~E1)n(1-ia)nexp_---n
S(K2 ) --n 2E
(h.5)
and for the amplitude
"(K2) =i2E 1 ~ (-1)n+l n=l
nn!
[Cl(l+a2)½~neindp L ~
J
(
exp\-
K2/,
(A.6)
with 1
1
sin ~ = -a(1 + a2) -~ ,
cos q5 = (1 + a2) -2 .
The total c r o s s section in this case is given by
~t=4zrIma(O)=4zr2E1 ~
n=l
(A.7)
1
(-1) n+l F C l ( l + a 2 ) ~ l n
nn ! L 2-El
cos n~b.
(A.8)
For ~ = 0 we obtain with C = 0 . 5 7 7 2 . . . (Euler's constant)
I
C a - E l ( - C2~1 l)l
~t=4=2E 1 C+In2~l APPENDIX
"
(A.9)
B
In this appendix we briefly d i s c u s s Coulomb s c a t t e r i n g corrections. Neglecting the real p a r t of the strong interaction we get a rough estimation of this effect f r o m the Coulomb s c a t t e r i n g f o r m u l a (which is still to be multi-
380
A. MIJL LENSIE FEN
p l i e d by the e l e c t r o m a g n e t i c f o r m f a c t o r of the n u c l e i ) : d~
d--[ =
4= ZA ZB [ t a l 2 f ( s , mA, m B) exp ( - ½ ( 4 +
4)
I t]) ,
(A. 10)
w h e r e a A and a B a r e the m e a s u r e d r . m . s , r a d i a of the n u c l e i [11] and
f ( s , m A , mB) i s a known f u n c t i o n ( s e e e.g. r e f . [8]). It i s 1.13 f o r H e - C s c a t t e r i n g a t a b o u t 1 GeV p e r n u c l e o n f o r the i n c o m i n g a - p a r t i c l e . A t t = - 0 . 0 2 5 ( G e V / c ) 2 we o b t a i n da/dt = 68 m b - ( G e V / c ) -2 w i t h o u t the f o r m f a c t o r s and 11 m b . ( G e V / c ) -2 i n c l u d i n g t h e m . S i n c e the C o u l o m b c r o s s s e c t i o n d e c r e a s e s a t l e a s t l i k e t -2 we do not e x p e c t a l a r g e i n f l u e n c e on the s t r o n g a m p l i t u d e n e a r the f i r s t m a x i m u m ( s e e fig. 5). In the r e g i o n of the f i r s t m i n i m u m , h o w e v e r , the C o u l o m b a m p l i t u d e can a l r e a d y b e v e r y i m portant.
REFERENCES [1] T . T . Chou and C.N. Yang, Phys. Rev. 170 (1968) 1591. [2] T. T. Chou and C. N. Yang, Phys. Rev. L e t t e r s 20 (1968) 1213. [3] H. Aizu, Y. Fujimoto, S.Hasegawa, M. Koshiba, I. Mito, J. Nishimura and K. Yokoi, P r o g r . Theor. Phys. Suppl. 16 (1960) 106. [4] T. Coor, D.A. Hill, W. F. Hornyak, L.W. Smith and G. Snow, Phys. Rev. 98 (1955) 1369. [5] W. Czy~ and L. C. Maximon, Ann. of Phys. 52 (1969) 59; Phys. L e t t e r s 27B (1968) 354. [6] S. Gartenhaus and C. Schwartz, Phys. Rev. 108 (1957) 482. [7] D. Kofoed-Hans~n, Nucl. Phys. B l l (1969) 455. [8] H. Pilkuhn, The interactions of hadrons (North-Holland, A m s t e r d a m , 1967). [9] R. F r o s c h , J.S. McCarthy, R . E . Rand and M. R. Yearian, Phys. Rev. 160 (1967) 202. [10] W. Galbraith, E.W. Jenkins, R. F. Kycia, B.A. Leontic, R.H. Phillips, A. L. Read and R.Rubinstein, Phys. Rev. 138 (1965) 913. [11] G. J. Igo, J. L. F r i e d e s , H. Palevsky, R. Sutter, G. Bennett, W.D. Simpson, D. M. Corley, R . L . S t e a r n s , Nucl. Phys. B3 (1967) 181. [12] Landolt-B~rnstein, Zahlenwerte und Funktionen, Neue Serie I/2, Kernradien (Springer-Verlag, 1967). [13] A. Malecki and P. P. Plcchi, Nuovo Cim. L e t t e r s 1 (1969) 81, [14] N. G. Afanasev, N. G. Shevchenko, G. A. Savitskii, I.S. Gulkarov and V. M. Khvastunov, Sov. J. Nucl. Phys. 8 (1969) 646; S. Fernbach, Rev. Mod. Phys. 30 (1958} 414; A. E. Glassgold, Rev. Mod. Phys. 30 (1958) 419; L. R. B. Elton, Nuclear sizes (Oxford, 1961). [15] P . E . N e m i r o v s k i i , Sov. J. Nucl. Phys. 7 (1968) 38.
Nuclear Physics B28 (1971) 368-380. North-Holland Publishing Company
NUCLEUS-NUCLEUS SCATTERING AT HIGH ENERGIES Alfred MULLENSIE FEN
Institut flit Struktur der Materie der Universitiit Karlsruhe Received 20 July 1970 (Revised manuscript received 8 December 1970)
Abstract: We relate the measured e l e c t r i c nuclear form factor to the opaqueness model of the nucleus. The calculated differential cr o ss section of the proton s c a t tering on 4He, 12C and 160 at 1 GeV is in reasonable agreement with the e x p e r i mental data if the total c r o s s section is taken as a f r e e p a r a m e t e r . Using the r e sults of elastic p a scattering we also get good agreement with the Glauber theory for a a scattering at about 8 G e V /c per nucleon. Our calculations are much more easily handled than those of the Glauber model. They may be considered as an improved optical-limit amplitude. In particular, the c o r r e c t c.m. motion factor differs from that of Czy~. and Maximon for the case of nucleus-nucleus scattering. We calculate d(T/dt for a-12C scattering and estimate the total c r o s s section for the scattering of 4He on C, N and O from which we obtain the mean f r e e path of 4He in air.
I. INTRODUCTION AND CONCLUSION In the o p t i c a l m o d e l t h e s c a t t e r i n g on n u c l e i i s t r e a t e d r e g a r d i n g t h e n u c l e u s as an o p a q u e m e d i u m . R e c e n t l y Chou and Y a n g [1, 2] h a v e d i s c u s s e d d a / d t f o r pp s c a t t e r i n g in the l i m i t of i n f i n i t e l y high e n e r g i e s , t h e i r s u g g e s t i o n b e i n g to r e l a t e the n u c l e o n o p a q u e n e s s to the n u c l e o n e l e c t r o m a g n e t i c f o r m f a c t o r . In t h i s p a p e r w e a p p l y the s a m e i d e a to the 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 and n u c l e i on n u c l e i and thus c o r r e l a t e the m e a s u r e d e l e c t r i c n u c l e a r f o r m f a c t o r w i t h the n u c l e a r o p a q u e n e s s . T h e s t r e n g t h of th e l a t t e r i s d e t e r m i n e d f r o m n u c l e o n - n u c l e u s s c a t t e r i n g . U s i n g only an o p t i c a l m o d e l the a b s o r p t i o n c r o s s s e c t i o n f o r n u c l e i n u c l e i s c a t t e r i n g h a s b e e n d e t e r m i n e d in an e a r l i e r p a p e r [3]. T h e a u t h o r s c h o o s e a s p a r a m e t e r s a m o d i f i e d v a l u e of the t o t a l n u c l e o n - n u c l e o n c r o s s s e c t i o n and the n u c l e a r r a d i i d e t e r m i n e d f r o m n e u t r o n - n u c l e u s s c a t t e r i n g [4]. T h e i r f o r m u l a , h o w e v e r , d o e s not r e d u c e to the r e l a t i o n u s e d in n e u t r o n - n u c l e u s s c a t t e r i n g and thus we b e l i e v e t h a t t h e i r r e s u l t h as to be r e fined. In s e c t . 2 w e s u m m a r i z e the f o r m u l a e and s p e c i f y the a n s a t z f o r the n u c l e a r o p a q u e n e s s P A ( X , y , z ) . T h i s i s e x p r e s s e d in t e r m s of the o p a q u e n e s s of the n u c l e o n and t h e d e n s i t y d i s t r i b u t i o n of the A n u c l e o n s w h i c h f o l l o w s f r o m the m e a s u r e d n u c l e a r e l e c t r i c f o r m f a c t o r . Th e n u c l e o n o p a q u e n e s s i s d e t e r m i n e d f r o m the e l a s t i c n u c l e o n - n u c l e o n s c a t t e r i n g . T h e e l e c t r i c f o r m f a c t o r as well as the o p a q u e n e s s have been taken as a Gauss function o r a
NUCLEUS-NUCLEUS SCATTERING
369
s u m of two G a u s s i a n s . T h e e l a s t i c p r o t o n - n u c l e u s s c a t t e r i n g i s in r e a s o n a b l e a g r e e m e n t w i t h the e x p e r i m e n t a l d a t a . In s e c t . 3 w e c o m p a r e t h i s m o d e l w i t h the o p t i c a l l i m i t of t h e G l a u b e r t h e o r y ( i n f i n i t e l y m a n y s m a l l n u c l e o n s , s e e , e . g . , r e f . [5]). In c o n t r a s t to the o p t i c a l l i m i t c a l c u l a t i o n s of Czy~ and M a x i m o n [5] we g e t r e a s o n a b l e a g r e e m e n t w i t h the m u l t i p l e s c a t t e r i n g t h e o r y in a a s c a t t e r i n g at a b o u t 8 G e V / c p e r n u c l e o n . W e a t t r i b u t e t h i s to o u r m o d i f i e d t r e a t m e n t of t h e c . m . m o t i o n of the n u c l e o n s w i t h i n the n u c l e u s and c o n c l u d e t h a t o u r a m p l i t u d e can b e t a k e n a s an i m p r o v e d o p t i c a l - l i m i t a m p l i t u d e ( s e e e q s . (3.1) and (3.2)). W e h a v e i n t r o d u c e d a l s o a r e a l p a r t of the a m p l i t u d e . W e c a l c u l a t e the a b s o r p t i o n c r o s s s e c t i o n s of a p a r t i c l e s w i t h an e n e r g y of 1 GeV p e r n u c l e o n s c a t t e r e d on 12C, 14N and 160 and o b t a i n a m e a n f r e e p a t h b e t w e e n 44 and 49 g . c m -2 in a i r . The t o t a l aC c r o s s s e c t i o n ~ C i s e x p e c t e d to l i e b e t w e e n 608 and 720 mb. W e c a n n o t g i v e an a c c u r a t e v a l u e s i n c e we h a v e not t r i e d to i m p r o v e the s i m p l e a n s a t z f o r pp s c a t t e r i n g o r to i n c l u d e the e x p e r i m e n t a l e r r o r l i m i t s . It t u r n s out, h o w e v e r , t h a t the t - d e p e n d e n c e of dc~/dt d o e s not c h a n g e d r a s t i c a l l y if ~ ~t i s a l t e r e d . Thus we e x p e c t t h a t o u r s i m p l e m o d e l w i l l y i e l d the c o r r e c t o r - d e r of m a g n i t u d e f o r the e l a s t i c s c a t t e r i n g a n d t h a t o u r r e s u l t s can b e u s e d a s an o r i e n t a t i o n in f u r t h e r c a l c u l a t i o n s and in p l a n n i n g e x p e r i m e n t s w i t h h i g h - e n e r g y a - p a r t i c l e s . Of c o u r s e t h i s m o d e l m a y b e a v e r y c r u d e a p p r o x i m a t i o n , b u t i t s h o u l d a l s o b e p o i n t e d o u t t h a t u s i n g t h e m e a s u r e d e l e c t r i c f o r m f a c t o r a n u m b e r of c o n t r i b u t i o n s (e.g. m e s o n e x c h a n g e c u r r e n t s ) a r e i n c l u d e d to a c e r t a i n d e g r e e w h e r e a s t h e s e e f f e c t s w o u l d be d i f f i c u l t to b u i l d into the G l a u b e r t h e o r y . The s p i n s of the p a r t i c l e s h a v e b e e n n e g l e c t e d t h r o u g h o u t . In a p p e n d i x A we l i s t t h e n e c e s s a r y f o r m u l a e and in a p p e n d i x B we e s t i m a t e the i n f l u e n c e of C o u l o m b s c a t t e r i n g .
2. THE S C A T T E R I N G A M P L I T U D E AND N U C L E A R O P A Q U E N E S S N e g l e c t i n g s p i n e f f e c t s Chou and Y a n g [1] w r i t e f o r t h e s c a t t e r i n g a m plitude 1 a ( u ) = i { s ( x ) - ~.1 s ( u ) ® s ( x ) + ~.v s ( x ) ® s ( u )
®s(u) - +...},
w h e r e ~ =(K..,K~Y') and K2 =K 2 + K2 = - t i s the m o m e n t u m t r a n s f e r . use the definition
f(x)
1 ®g(u) = ~
fff(u
- x')g(u')dx'.
(2.1) They
(2.2)
T h e d i f f e r e n t i a l c r o s s s e c t i o n i s g i v e n by d_~ = ~ l a ( ~ ) f2 dt and s ( x ) i s d e t e r m i n e d in t h e f o l l o w i n g way. F o r an i n c o m i n g p o i n t l i k e p a r t i c l e w i t h an i m p a c t p a r a m e t e r t h e n u c l e u s A a p p e a r s a s a d i s k w i t h an o p a q u e n e s s
(2.3)
b = (x,y)
370
A. MTJLLENSIE
DA(b) = J
FEN
PA(r)dz.
(2.4)
--OO
The e s s e n t i a l a s s u m p t i o n of the Chou-Yang model is that at v e r y high e n e r gies the m a t t e r density P A ( r ) is p r o p o r t i o n a l to the e l e c t r i c density peA(r) , whicl~ in this c a s e is the F o u r i e r t r a n s f o r m of the e l e c t r i c f o r m f a c t o r FA(k~) of the whole nucleus with m a s s n u m b e r A, i.e. P A ( r ) = dAPeA(r),
peA(r) = A f e x p ( - i k r ) F A ( k 2 ) d k .
(2.5)
This a s s u m p t i o n is p r o b a b l y not a v e r y good one f o r a proton and c o m p l e t e l y wrong f o r a neutron, at l e a s t in the m o s t t r i v i a l s e n s e that the neutron is e l e c t r i c a l l y neutral e v e r y w h e r e . H o w e v e r , for nuclei c o m p a r i s o n between e l e c t r o n - n u c l e u s and n u c l e o n - n u c l e u s s c a t t e r i n g shows that eq. (2.5) is indeed a p p r o x i m a t e l y s a t i s f i e d [14]. Speculations exist that the m a t t e r density extends s o m e w h a t f u r t h e r out than the c h a r g e density ("neutron e x c e s s at the n u c l e a r s u r f a c e " ) [15]. In that c a s e d A (which in g e n e r a l is P A ( r ) / p ~ ( r ) ) will be a function of r . H o w e v e r , for the p r e s e n t calculation, we take d A as a constant. It m a y be worth pointing out that the Chou-Yang model has one conceptual advantage o v e r the Glauber model: it does not a s s u m e that the nucleus cons i s t s of protons and neutrons. All hadronic m a t t e r (virtual pions, nucleon i s o b a r s etc.) is included. This could lead to i m p r o v e m e n t s e s p e c i a l l y for strongly bound nuclei. In analogy with the Chou-Yang e x p r e s s i o n f o r the pp s c a t t e r i n g we put (K2 = k 2 + k 2 k z =0) x y' SAB(g 2) = (1 - i a ) K A B d A d B FA(K 2) FB(K2),
(2.61
w h e r e K is d e t e r m i n e d f r o m pp s c a t t e r i n g and d A f r o m e l a s t i c p - n u c l e u s s c a t t e r i n g . With eq. (2.6) we obtain the total c r o s s section (see appendix A) Cr~B = 47r 2 ( f A + fiB) {C+ In x - E i ( - x ) } ,
x = ½KABd A dB(f A + fiB)- 1,
C = 0 . 5 7 7 2 . . . ( E u l e r ' s constant),
(2.7)
(2.8)
w h e r e the e l e c t r i c f o r m f a c t o r has been p a r a m e t r i z e d as FA(k2 ) --- exp ( - f A k 2 ) .
(2.9)
A g e n e r a l i s a t i o n - e s p e c i a l l y to finite e n e r g i e s - is p o s s i b l e if we a s s u m e PA ( r ) = f f f P S (
r - r ' ) W A ( r ') f A ( r ' ) d r ' .
(2.10)
H e r e , P N ( r ) is the nucleon o p a q u e n e s s , which d e s c r i b e s the n u c l e o n - n u cleon s c a t t e r i n g (we do not distinguish between p r o t o n s and neutrons). The
NUCLEUS-NUCLEUS SCATTERING
37I
function WA(r) ( n o r m a l i z e d to A) is the density distribution of the A nucleons calculated f r o m the m e a s u r e d e l e c t r i c f o r m f a c t o r FA(k2) of the nucleus [12]. We take FA(k2) s p h e r i c a l l y s y m m e t r i c s i n c e we neglect spin e f f e c t s , quadrupole m o m e n t s and all o t h e r d i s t o r t i o n m e c h a n i s m s . Thus we have the r e l a t i o n 1
fffwA(r)exp(ikr)dr
=FA(k2)exp(fiN k2) =- GA(k2).
(2.11)
The e x p r e s s i o n exp (/3N t) with fiN = 2.74 (GeV/c) -2 d e s c r i b e s the e l e c t r i c f o r m f a c t o r of the proton and is n e c e s s a r y in eq. (2.11) since GA(k 2) shall r e p r e s e n t the distribution of the nucleons only. Also f A ( r ) = 1 i m p l i e s that the distribution of the nucleons within the nucleus is given by WA(r). Thus the unknown function f A ( r ) d e s c r i b e s the deviation f r o m this s i m p l e model. We a s s u m e that f A ( r ) can be a p p r o x i m a t e d by the constant d A. It can, h o w e v e r , be energy dependent and is d e termined from nucleon-nucleus scattering. F o r an incoming extended p a r t i c l e B the o p a q u e n e s s is a s s u m e d to be of the f o r m (2.12)
D(b) : f f DB(b - b')DA(b')db'.
Within o u r model the amplitude for the e l a s t i c s c a t t e r i n g of B on A is then given by eq. (2.1) taking SAB(~ ) = (1 -
1
ia) ~
fro(b) exp (i ~b) db.
(2.13)
F o r a = 0 the d i f f r a c t i o n m i n i m a in da/dt a r e z e r o and thus the introduction of a r e a l p a r t in the a m p l i t u d e is n e c e s s a r y . We take a as a constant and d e t e r m i n e it e x p e r i m e n t a l l y at t = 0. With 3
(27r) -~
fffPN(t)exp(ikr)dr
= CNON(k),
GN(0) = 1
(2.14)
we get f r o m eqs. (2.10), (2.13) and (2.14)
SAB(kx, ky) = (1 - ia)(2n) 2 c2 dA dBA B G2(kx, ky, O)GA(kx,ky, O)GB(kx,ky, 0).
(2.15)
Here GN(k) is also supposed to be spherically symmetric and thus from eq. (2.15) follows SAB(• 2) = (1 - ia)Kd A dBA
B GA(K 2) GB(K 2) G2(K 2)
using
c(K 2) = ~(k 2) ]kz__0 •
(2.16)
372
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SCATTERING
H e r e , K = (2v) 2 c 2 is e n e r g y dependent and will be d e t e r m i n e d f r o m n u c l e o n nucleon s c a t t e r i n g . The f o r m f a c t o r FA(k2) can be a p p r o x i m a t e d by
(2.17)
FA(k2 ) = c A exp ( - f A k2 ) + (1 - cA)exp (-/~A k2),
as is shown in fig. i for carbon (CA, fA and ~A a r e constants). The o p a q u e n e s s is then
(2.18)
GA(K 2) = c A e x p ( - a A K2) + (1 - c A ) e x p ( - b A n 2 ) ,
w i t h a A = f A - f i N a n d b A = ~ A - f i N . F o r K2 << 1 e v e n c A = 1 i s a g o o d a p proximation. The function GN(K2) is also d e t e r m i n e d f r o m nucleon-nucleon s c a t t e r i n g (A = B =G A =G B = 1). This can, of c o u r s e , be a c o m p l i c a t e function, but fig. 2 shows that the a n s a t z GN(K2) = exp ( - a N K2)
(2.19)
is sufficient for o u r p u r p o s e . The s t r a i g h t line in this f i g u r e r e p r e s e n t s the m e a s u r e d d i f f e r e n t i a l nucleon-nucleon c r o s s section p a r a m e t r i z e d in a f o r m which is e.g. always used in the Glauber theory. With the s i m p l e a n s a t z GA(K 2) = exp ( ' a A K 2 ) ,
GB(K 2) = exp ( - a B K 2 ) ,
(2.20)
we get SAB(K2 ) = (1 - i a ) K d A d B A B
exp[-(aA+aB+2aN)K2
].
(2.21)
Using the optical t h e o r e m and eq. (2.1) the total c r o s s section is now (see appendix A) ~ t B = 47r 2(a A + a B + 2aN) {C + In x - E i ( - x ) } ,
(2.22)
with C = 0 . 5 7 7 2 . . . ( E u l e r ' s constant) and x = ½ K d A d B A B ( a A + aB+ 2aN) -1. If at v e r y high e n e r g i e s a N can be a p p r o x i m a t e d by fiN, then eq. (2.19) is equal to eq. (2.7). Czy~. and Maximon [5] d i s c u s s the aa s c a t t e r i n g at about 8 G e V / c . Taking t h e i r p a r a m e t e r s (see fig. 15 of ref. [5]) the total p a s c a t t e r i n g within the Glauber model is crt a = 139.2 mb. Using this v a l u e , crt a = 375 m b , a N = = 1.8 ( G e V / c ) - 2 (se~ fig. 2) and a a = 9.04 (GeV/c) -2 we get f r o m eq. (2.22) d a = 0.87 and gNN = 40.3 mb which is in r e a s o n a b l e a g r e e m e n t with g t N = = 41 mb used by Czy2 and Maximon as the m e a n value of the total pp and np s c a t t e r i n g (see ref. [19]). We i g n o r e the s m a l l d i f f e r e n c e since we only u s e the a p p r o x i m a t i o n (2.19) f o r GN(K2 ). At 1 GeV we take (~IN = 44 m b , (rt a = 152 m b [11], a a = 9.04 (GeV/c) -2 (ref. [12]) and get the values shown inp table 1. H e r e we see that K d a is n e a r l y constant w h e r e a s d a changes by about 15%. This r e s u l t is p l a u s i b l e
374
A. Mi]LLENSIE FEN Table 1 aN (GeV/c) -2
Kd a (GeV/c) -2
da
1.0
12.20
0.57
0.9
12.33
0.50
s i n c e a s m a l l v a r i a t i o n of a N g i v e s o n l y a s m a l l c h a n g e in c a + 2a N b u t an a p p r e c i a b l e o n e in K in pp s c a t t e r i n g ( s e e eq. (2.22)). W e do not g i v e an a c c u r a t e v a l u e of d a s i n c e w e c o m p l e t e l y i g n o r e the e x p e r i m e n t a l e r r o r l i m i t s and u s e o n l y the s i m p l e a n s a t z (2.19) f o r the n u c l e o n and a - p a r t i c l e o p a q u e n e s s . U s i n g the r e s u l t s of t a b l e 1 we e x p e c t a v a l u e of d a b e t w e e n a b o u t 0.5 and 0.7. In fig. 3 d a / d t c a l c u l a t e d w i t h a N = = 0.9 ( G e V / c ) -2 i s shown f o r p a s c a t t e r i n g . T h e r e s u l t s a r e in good a g r e e m e n t w i t h the m e a s u r e m e n t s and the c a l c u l a t i o n s of t h e G l a u b e r m o d e l (for the 4He f o r m f a c t o r s e e e.g. r e f . [9]). T h e a g r e e m e n t i s a s good f o r p - 1 2 C and p - 1 6 0 s c a t t e r i n g a t 1 GeV if the a p p r o x i m a t i o n (2.17) i s u s e d w i t h c A ¢ 1. In p - 1 6 0 s c a t t e r i n g w e g e t a g r e e m e n t b e y o n d the f i r s t m i n i m u m o n l y if w e t a k e c O ¢ 1. In p - 1 2 C s c a t t e r i n g the c a l c u l a t i o n w i t h c C = I i s s l i g h t l y b e l o w the e x p e r i m e n t a l d a t a w h e r e a s t h a t w i t h c C ¢ 1 i s s l i g h t l y a b o v e t h e m n e a r the f i r s t m a x i m u m .
3. N U C L E U S - N U C L E U S S C A T T E R I N G S i n c e we g e t r e a s o n a b l e r e s u l t s in n u c l e o n - n u c l e u s and aa s c a t t e r i n g we a p p l y t h i s m o d e l to e l a s t i c aC s c a t t e r i n g a s w e l l , and d i s c u s s a l s o the m e a n f r e e p a t h of a - p a r t i c l e s in a i r . W e w a n t f i r s t , h o w e v e r , to c o m p a r e o u r m o d e l in the c a s e of a a s c a t t e r i n g w i t h the o p t i c a l l i m i t r e s u l t w h i c h f o l l o w s f r o m the G l a u b e r t h e o r y and i s g i v e n in r e f . [5]. W e a l s o u s e the p a r t i c l e i n d e p e n d e n t m o d e l f o r 4He and c h o o s e f o r t h e o n e - p a r t i c l e d e n s i t y p(r) = v R 2 e x p ( - r 2 R - 2 ) . T a k i n g into a c count the c . m . c o n s t r a i n t ( s e e e.g. r e f s . [5, 6]) we g e t f o r the o p a q u e n e s s Ga(K2 ) = e x p ( - ~ R2K 2) = e x p ( - a a K 2 ) . F r o m e q s . (2.1) and (2.21) w e o b t a i n f o r the a m p l i t u d e
n=l
nn!
L ~R2+2a N J
e x p { - ( ~ R 2 + 2 a N ) n--}" (3.1)
In the o p t i c a l l i m i t Czy~ and M a x i m o n [5] o b t a i n
a( 2) n=l
nn!
L~(R2 + a). j
w i t h e(g 2) = exp (tR2K2) and d ~ / d t = a2] 1 - i a ] 2 e x p (at)/167r f o r the NN s c a t t e r i n g . A t f i r s t s i g h t t h e r e i s a s t r o n g s i m i l a r i t y b e t w e e n the a m p l i t u d e s
0
I
\
0,2
I
I
l
m m I w
0,1
~
0,3 0,6 O,S OoG -t GeV ~Fig. 3. p-4He s c a t t e r i n g at 1 GeV. The full line has been calculated with a N = 0.9 (GeV/c) -2, f i t = 152 mb, a = -0.33 and Fa(t ) = 1 . 0 2 1 e x p ( l l . 1 7 t ) - 0 . 0 2 1 e x p "( 1- . 2 3 t ) for the 4 He f o r m factor. The c r o s s e s c o r r e s p o n d to Fa(t) = exp (11.78t). The dashed c u r v e (Glauber model) and the e x p e r i m e n t a l data have been taken f r o m ref. [5].
t6
~6 ~
'[
i
I
I
I
I
t
I
-'x% i.%
t
"~~'~.\
%'N,
',,
u s • a"%
"%,
'~. m"%
_J ~'~
s p e c t i v e l y . The c r o s s s e c t i o n d o ' / d t i s 7 . 4 a n d 8.3 b - ( G e V / c ) - 2 at t = 0 .
0 0.! 0,2 0,3 0,4 0,S 0,G {3,7 0,8 0.9 i,O-t GeV:.c-: Fig. 4. ~a s c a t t e r i n g at about 8 G e V / c p e r hucleon. The d a s h - d o t c u r v e (Glauber model) and the dashed c u r v e (optical limit) have been taken f r o m ref. [5]. The full line and the c r o s s e s have been calculated with a** = 1 8 ZGeV/c~ -2 t ~ ~= 375 " m ' b an " d ~400' m b r e a(~ = 9.04 ( G e V / c ) - 2 , o / = - 0 . 3 3 a n d O'qo
~(~z,
1C[3
162
"E 101 ~e~\
•
I0:
f3
!
376
A. MULLENSIE FEN
(3.1) and (3.2). It j u s t i f i e s t h e i n t r o d u c t i o n of a in f o r m u l a (3.1). T h e f a c t o r 0(K2) a r i s e s f r o m the c . m . c o n s t r a i n t of t h e n u c l e o n s w i t h i n t h e n u c l e u s . It i s unity f o r i n f i n i t e l y m a n y n u c l e o n s [5], b u t d a / d t i s too s m a l l in t h i s l i m i t . Czy~ and M a x i m o n m a k e i t p l a u s i b l e to k e e p 8(K 2) in eq. (3.2). T h e d i f f e r e n c e of the n t h p o w e r t e r m b e t w e e n the e x p r e s s i o n s (3.2) and
(3.1)
is 1
- ~ a + 2a N +
(½a+ ½R2)(1
- 1).
(3.3)
Both a and a N d e s c r i b e NN s c a t t e r i n g . Only the t e r m s w i t h n = 1 a r e e q u a l if f o r the m o m e n t w e t a k e ½a = 2a N ( t h i s e q u a l i t y b e i n g o n l y a p p r o x i m a t e l y t r u e ) . A l l the o t h e r t e r m s a r e l a r g e r in the o p t i c a l l i m i t . F o r a a s c a t t e r i n g , fig. 4 s h o w s t h a t a t K2 = 1 ( G e V / c ) 2 the c r o s s s e c t i o n in the o p t i c a l l i m i t i s m o r e than a f a c t o r 102 l a r g e r than the v a l u e o b t a i n e d f r o m the G l a u b e r m o d e l . T h u s eq. (3.2) i s u s e l e s s in n u c l e u s - n u c l e u s s c a t t e r i n g . W h e t h e r the d i s c r e p a n c y b e t w e e n o u r m o d e l and the G l a u b e r m o d e l c a l c u l a t i o n s ( d a s h - d o t c u r v e in fig. 4) can b e r e d u c e d a t h i g h e r m o m e n t u m t r a n s f e r s if a m o r e c o m p l i c a t e d f u n c t i o n f o r GN(K2 ) i s u s e d r e m a i n s to b e i n v e s t i g a t e d . The f i t f o r Ga(K2 ) w i t h c a ¢ 1 g i v e s a l m o s t the s a m e r e s u l t a s t h a t w i t h c a = 1. T h i s d i s c u s s i o n s h o w s t h a t the a m p l i t u d e (3.1) can b e r e g a r d e d a s an i m p r o v e d o p t i c a l l i m i t a m p l i t u d e w h e r e the c . m . m o t i o n of the n u c l e o n s h a s b e e n t a k e n into a c c o u n t in a m o d i f i e d way. We a l s o w a n t to m e n t i o n a p a p e r by K o f o e d - H a n s e n [7] who b r i e f l y d i s c u s s e s aa s c a t t e r i n g in t h e G l a u b e r m o d e l . He i n v e s t i g a t e s the e f f e c t of t r u n c a t i n g the G l a u b e r s e r i e s . F o r aC s c a t t e r i n g w e s t i l l n e e d d C. F r o m ~ t a = 1 5 2 m b , t =370mb ( r e f . [11]) and a C = 23.18 ( G e V / c ) -2 ( r e f . [12]) ~ve c a l c u l a t e the~'"rati~ d c / d a = 0.83 a t 1 GeV f r o m eq. (2.22). U s i n g t h i s v a l u e we o b t a i n a a C = = 839+ 3 3 3 1 n d a (in m b ) , a~C = 608 m b and 720 m b f o r d a = 0.5 and 0.7 r e s p e c t i v e l y . Only In d a a p p e a r s and n o t In d 2 s i n c e the p r o d u c t K d a i s a l r e a d y d e t e r m i n e d f r o m p a s c a t t e r i n g . S e t t i n g d a = d C =1 we can e a s i l y d e t e r m i n e K f r o m n u c l e o n - n u c l e u s s c a t t e r i n g and then c a l c u l a t e a~C. In t h i s w a y we find f r o m p a s c a t t e r i n g atac = 912 m b and f r o m pC s c a t t e r i n g t = 854 mb. aaC T h e d i s c r e p a n c y s h o w s t h a t the c h o i c e d a = d C =1 i s i m p o s s i b l e . In fig. 5 d ~ / d t i s shown f o r aC s c a t t e r i n g a t 1 GeV p e r n u c l e o n f o r a ~ = = 608 m b and 720 mb. T h e t d e p e n d e n c e i s v e r y s i m i l a r in b o t h c a s e s and thus o u r r e s u l t s can b e u s e d a s a f i r s t a p p r o x i m a t i o n to the e l a s t i c aC s c a t t e r i n g . A s in p - 1 2 C s c a t t e r i n g d a / d t i s o n l y s l i g h t l y a l t e r e d if c C ¢ 1 i s u s e d . S i m i l a r l y w e find a s m a l l c h a n g e of the m a x i m a if a = 0 i s c h o s e n . T h u s the t o t a l and a b s o r p t i o n c r o s s s e c t i o n s can w e l l b e c a l c u l a t e d t a k i n g a = 0 and C A = C B = l . S i n c e the p a r a m e t e r s u s e d do not c h a n g e v e r y r a p i d l y w i t h e n e r g y , fig. 5 s h o u l d a l s o be v a l i d a t s l i g h t l y h i g h e r e n e r g i e s . F i n a l l y we e s t i m a t e the m e a n f r e e p a t h of a - p a r t i c l e s in a i r . The a b s o r p tion c r o s s s e c t i o n i s a a = a t - a e, w h e r e a e i s the c o m p l e t e e l a s t i c c r o s s s e c tion
N U C L E U S - N U C L E U S SCATTERING
377
10I
°
t 102
.-.
I
i
101
'~" " ~
'
I0'
1(~2
0
I
I
I
I
i
0.0S
0,1
0,15
0.2
0,25
12
I
[
0,30 0.35 -t GeV 2. c-2
'~s
0.40
2
F i g . 5. ~- C s c a t t e r i n g at 1 GeV p e r nucleon with ~ = -0.33, a N = 0.9 ( G e V / c ) and a a = 9.04 (GeV/c) -2 f o r fftqc = 608 mb (full line and c r o s s e s ) and fftaC = 720 mb (dashed line). The c r o s s e s hav~ been c a l c u l a t e d with F c ( t ) = exp (25.92t~ and the o t h e r c u r v e s with F c ( t ) = 1.83 exp (20.800 - 0 . 8 3 exp (14.67t) f o r the carbon f o r m f a c t o r . The c r o s s s e c t i o n s dcr/dt is 19 and 27 b • ( G e V / c ) - 2 at t = 0.
o"e=
f
o d~
tu
~ t- d t .
(3.4)
H e r e t u i s t h e l o w e r l i m i t of t h e i m p u l s e t r a n s f e r . T h e q u a n t i t y ~e i s e a s i l y c a l c u l a t e d f o r a = 0, c A = c B = 1 and t u = -oo. T h i s a p p r o x i m a t i o n w i l l c e r t a i n l y b e v e r y g o o d s i n c e d(~/dt d e c r e a s e s v e r y r a p i d l y n e a r t = 0 a n d t h u s t h e c o n t r i b u t i o n f r o m t u to i n f i n i t y i s n e g l i g i b l y s m a l l . W e g e t (r~B = 47r(a A + a B + 2 a N ) {C + In ½x - 2Ei(-x) + Ei(-2x)}. The absorption
cross
(3.5)
s e c t i o n ~a i s t h e n
g ~ B = ~ t B _ CrABe = 47r(aA + a B + 2 a N ) {C + In (2x) - Ei(-2x)}.
(3.6)
378
A, M~LLENSIE FEN
A s i m i l a r f o r m u l a h a s b e e n found e a r l i e r [3]. In t h a t p a p e r the n u c l e a r r a d i i a r e t a k e n f r o m an o p t i c a l - m o d e l a n a l y s i s of n e u t r o n - n u c l e u s s c a t t e r i n g [4] but the a b s o r p t i o n c r o s s s e c t i o n d o e s n o t r e d u c e to the f o r m u l a u s e d in n e u t r o n - n u c l e u s a n a l y s i s . T h u s we b e l i e v e t h a t the f o r m u l a in r e f . [3] h a s to b e r e f i n e d . S i n c e E i ( - 2 x ) can b e n e g l e c t e d we g e t w i t h d B = ~d A
~B=47r(aA+aB+2aN){C+In~
ABKd A
= a ~ B ( d A =1) + 4 T r ( a A + a B + 2 a N ) l n d
(3.7)
A .
A t 1 GeV we o b t a i n f o r d A -- 1 ( ~ C = 535 m b and a~O = 656 m b u s i n g a t o = 475 m b and a O = 29.63 ( G e V / c ) -2 ( r e f s . [11, 12]). The r a t i o d o / d a i s 0.82. By l i n e a r e x t r a p o l a t i o n of the c a r b o n and o x y g e n r e s u l t s w e find f o r n i t r o g e n (~aN = 595 m b ( d a = l ) . F o r d a = 0.5 (0.7), a s i s s u g g e s t e d in s e c t . 2, we g e t ~<~ = 420 (476) mb. U s i n g ~ = 44 x 0.95 m b in f o r m u l a (A4.8) of r e f . [3] o n e o b t a i n s oaaC = 472 mb. The mean free path is calculated from Pair l = a a ' ~aO n o + a a N nN
(3.8)
w i t h P a h r = 1.29- 1 0 - 3 g • c m - 3 , nO = 1 . 1 8 . 1 0 1 9 and n N = 4 . 1 9 . 1 0 1 9 p a r t i c l e s p e r c m f o r o x y g e n and n i t r o g e n . W e o b t a i n 39.4 cm_ 2 ; l = 1 + 0.2861n c a g" t h a t i s , l = 49.1 g - c m -2 f o r d a = 0.5 and l = 43.9 g . c m -2 f o r d a = 0.7. U n f o r t u n a t e l y t h e r e a r e no m e a s u r e m e n t s a v a i l a b l e f o r t h e m e a n f r e e p a t h of a - p a r t i c l e s in a i r . C o n s e q u e n t l y i t w o u l d b e i n t e r e s t i n g to a p p l y o u r m e t h o d to the s c a t t e r i n g of 4He on t h o s e n u c l e i w h i c h a r e c o n t a i n e d in n u c l e a r e m u l s i o n s ; s i n c e f o r n u c l e a r e m u l s i o n s the m e a n f r e e p a t h h a s a l ready been measured. I a m v e r y m u c h i n d e b t e d to P r o f . H. P i l k u h n , D r . W. S c h m i d t and D r . D. J u l i u s f o r s t i m u l a t i n g d i s c u s s i o n s and a c r i t i c a l r e a d i n g of the m a n u script.
APPENDIX A In t h i s a p p e n d i x w e s u m m a r i z e -ia
s(K2) = 2 ~ - ~ )
the f o r m u l a e .
For
N
#~=1 c k e x p ( - E k K 2 )
(A.1)
NUCLEUS-NUCLEUS SCATTERING
379
we get N s(g2) ® s ( K 2 ) = 2 ( 1 2 i a ) 2 ~
N ~
CklCk2 exp {- EklEk2 K2 kl=l k2=l Ekl+Ek 2 Ekl+Ek 2 1
(A.2)
with the definition 1
/(~1,K2) ®g(Kl,~2) = ~ ff/(K1 -KI,K 2 -K½)g(~i,K½)dKidK~
(A.3)
The nth convolution is given by s(K2) ® . . . ®
x ~
...
k1=1
s(K2)=2(12i~)
n
N
i
~ Ckl n exp n n K2 kn= l s.~= l Ekl . . EknEk~ . . . i~=l Ekl " EknEk-~
(A.4)
and the amplitude is calculated from eq. (2.1). In the special case N = 1 we get for the nth convolution S(K2) ® . . . ®
( ) E l K2` 1 C~E1)n(1-ia)nexp_---n
S(K2 ) --n 2E
(h.5)
and for the amplitude
"(K2) =i2E 1 ~ (-1)n+l n=l
nn!
[Cl(l+a2)½~neindp L ~
J
(
exp\-
K2/,
(A.6)
with 1
1
sin ~ = -a(1 + a2) -~ ,
cos q5 = (1 + a2) -2 .
The total c r o s s section in this case is given by
~t=4zrIma(O)=4zr2E1 ~
n=l
(A.7)
1
(-1) n+l F C l ( l + a 2 ) ~ l n
nn ! L 2-El
cos n~b.
(A.8)
For ~ = 0 we obtain with C = 0 . 5 7 7 2 . . . (Euler's constant)
I
C a - E l ( - C2~1 l)l
~t=4=2E 1 C+In2~l APPENDIX
"
(A.9)
B
In this appendix we briefly d i s c u s s Coulomb s c a t t e r i n g corrections. Neglecting the real p a r t of the strong interaction we get a rough estimation of this effect f r o m the Coulomb s c a t t e r i n g f o r m u l a (which is still to be multi-
380
A. MIJL LENSIE FEN
p l i e d by the e l e c t r o m a g n e t i c f o r m f a c t o r of the n u c l e i ) : d~
d--[ =
4= ZA ZB [ t a l 2 f ( s , mA, m B) exp ( - ½ ( 4 +
4)
I t]) ,
(A. 10)
w h e r e a A and a B a r e the m e a s u r e d r . m . s , r a d i a of the n u c l e i [11] and
f ( s , m A , mB) i s a known f u n c t i o n ( s e e e.g. r e f . [8]). It i s 1.13 f o r H e - C s c a t t e r i n g a t a b o u t 1 GeV p e r n u c l e o n f o r the i n c o m i n g a - p a r t i c l e . A t t = - 0 . 0 2 5 ( G e V / c ) 2 we o b t a i n da/dt = 68 m b - ( G e V / c ) -2 w i t h o u t the f o r m f a c t o r s and 11 m b . ( G e V / c ) -2 i n c l u d i n g t h e m . S i n c e the C o u l o m b c r o s s s e c t i o n d e c r e a s e s a t l e a s t l i k e t -2 we do not e x p e c t a l a r g e i n f l u e n c e on the s t r o n g a m p l i t u d e n e a r the f i r s t m a x i m u m ( s e e fig. 5). In the r e g i o n of the f i r s t m i n i m u m , h o w e v e r , the C o u l o m b a m p l i t u d e can a l r e a d y b e v e r y i m portant.
REFERENCES [1] T . T . Chou and C.N. Yang, Phys. Rev. 170 (1968) 1591. [2] T. T. Chou and C. N. Yang, Phys. Rev. L e t t e r s 20 (1968) 1213. [3] H. Aizu, Y. Fujimoto, S.Hasegawa, M. Koshiba, I. Mito, J. Nishimura and K. Yokoi, P r o g r . Theor. Phys. Suppl. 16 (1960) 106. [4] T. Coor, D.A. Hill, W. F. Hornyak, L.W. Smith and G. Snow, Phys. Rev. 98 (1955) 1369. [5] W. Czy~ and L. C. Maximon, Ann. of Phys. 52 (1969) 59; Phys. L e t t e r s 27B (1968) 354. [6] S. Gartenhaus and C. Schwartz, Phys. Rev. 108 (1957) 482. [7] D. Kofoed-Hans~n, Nucl. Phys. B l l (1969) 455. [8] H. Pilkuhn, The interactions of hadrons (North-Holland, A m s t e r d a m , 1967). [9] R. F r o s c h , J.S. McCarthy, R . E . Rand and M. R. Yearian, Phys. Rev. 160 (1967) 202. [10] W. Galbraith, E.W. Jenkins, R. F. Kycia, B.A. Leontic, R.H. Phillips, A. L. Read and R.Rubinstein, Phys. Rev. 138 (1965) 913. [11] G. J. Igo, J. L. F r i e d e s , H. Palevsky, R. Sutter, G. Bennett, W.D. Simpson, D. M. Corley, R . L . S t e a r n s , Nucl. Phys. B3 (1967) 181. [12] Landolt-B~rnstein, Zahlenwerte und Funktionen, Neue Serie I/2, Kernradien (Springer-Verlag, 1967). [13] A. Malecki and P. P. Plcchi, Nuovo Cim. L e t t e r s 1 (1969) 81, [14] N. G. Afanasev, N. G. Shevchenko, G. A. Savitskii, I.S. Gulkarov and V. M. Khvastunov, Sov. J. Nucl. Phys. 8 (1969) 646; S. Fernbach, Rev. Mod. Phys. 30 (1958} 414; A. E. Glassgold, Rev. Mod. Phys. 30 (1958) 419; L. R. B. Elton, Nuclear sizes (Oxford, 1961). [15] P . E . N e m i r o v s k i i , Sov. J. Nucl. Phys. 7 (1968) 38.