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Elastic and single-charge-exchange scattering of pions by carbon nuclei
I8 •A "83 Nuclear Physics B22 (1970) 179-188
North-Holland Publishing Company
ELASTIC A N D SINGLE-CHARGE-EXCHANGE SCATTERING OF PIONS BY CARBON NUCLEI K. B J g ) R N E N A K , J. F I N J O R D and P. OSLAND F y s i s k I n s t i t u t t , U n i v e r s i t e t e t i Trondheim - N o r g e s LaererhOgskole, N 7000 T r o n d h e i m , Norway
A. R E I T A N F y s i s k I n s t i t u t t , U n i v e r s i t e t e t i Trondheim - Norges t e k n i s k e hpgskole, N 7034 Trondheim - N T H , Norway
Received 25 May 1970
Abstract: The differential c r o s s sections for the reactions 12C(7r-,~-)12C and 13C(~+,y°)13N a r e calculated in the Glauber approximation for pions with kinetic energies in the 180 MeV region. It is found that the f i r s t t e r m s in the Glauber expansion a r e not convergent, due to the large pion-nucleon resonance amplitude. The F e r m i motion of the nucleons in the target nucleus and the exchange t e r m s due to the a n t i - s y m m e t r y of the nuclear wave function have a large influence on the r e s u l t s .
i. INTRODUCTION
T h e G l a u b e r a p p r o x i m a t i o n ( s e e e.g. r e f . [1]) h a s b e e n u s e d in c o n n e c t i o n w i t h p i o n - s c a t t e r i n g p r o b l e m s by s e v e r a l a u t h o r s . S t e r n h e i m [2] t r e a t e d the s c a t t e r i n g of 80 M e V p i o n s by s e v e r a l l i g h t n u c l e i by t h i s m e t h o d a s w e l l a s by o t h e r a p p r o a c h e s to the m u l t i p l e - s c a t t e r i n g p r o b l e m . T h e s c a t t e r i n g of p i o n s by d e u t e r o n s h a s b e e n c o n s i d e r e d by F~ildt and E r i c s o n [3,4] o v e r a l a r g e e n e r g y r e g i o n , and by M i c h a e l and W i l k i n [5] f o r p i o n s in the 4 GeV r e g i o n . T h e d e u t e r o n c a l c u l a t i o n s i n d i c a t e t h a t the c r o s s s e c t i o n i s s e n s i t i v e to d e t a i l s of n u c l e a r s t r u c t u r e , a n d t h a t it i s i m p o r t a n t to t a k e the F e r m i m o t i o n of the s c a t t e r i n g n u c l e o n s into a c c o u n t . Chou [6] h a s c o n s i d e r e d the e l a s t i c s c a t t e r i n g of 8 GeV p i o n s by 4He, and the p r e s e n t a u t h o r s h a v e p r e v i o u s l y u s e d the G l a u b e r e x p a n s i o n to c a l c u l a t e the c r o s s s e c t i o n f o r d o u b l e - c h a r g e - e x c h a n g e s c a t t e r i n g of p i o n s on 51V and 9 0 Z r [7]. T h e G l a u b e r e x p a n s i o n can o n l y be u s e d w i t h c o n f i d e n c e f o r high e n e r g i e s and s m a l l s c a t t e r i n g a n g l e s , but in m a n y c a s e s it s e e m s to w o r k v e r y w e l l a l s o o u t s i d e i t s e x p e c t e d r e g i o n of v a l i d i t y . We h a v e t h e r e f o r e a p p l i e d t h i s a p p r o a c h to the e l a s t i c s c a t t e r i n g of n e g a t i v e p i o n s by 12C and to the s i n g l e - c h a r g e - e x c h a n g e s c a t t e r i n g of p o s i t i v e p i o n s by 13C, f o r pion k i n e t i c e n e r g i e s in the r e g i o n of the 180 MeV p i o n - n u c l e o n r e s o n a n c e . Only t r a n s i tions between nuclear ground states are considered. The differential cross
180
K. BJ(~RNENAK et al.
section for the elastic s c a t t e r i n g has been m e a s u r e d at s e v e r a l e n e r g i e s by Binon et al. [8], and Chivers et al. [9] have m e a s u r e d the total c h a r g e - e x change c r o s s section at 180 MeV.
2. THEORY When the wave function of the s c a t t e r i n g nucleus is taken to be an a n t i s y m m e t r i z e d p r o d u c t of o n e - n u c l e o n wave functions, and the products of the s c a t t e r i n g amplitudes that a p p e a r in the Glauber expansion a r e s y m m e t r i z e d , the m u l t i p l e - s c a t t e r i n g s e r i e s is found to have the f o r m d~ _ [F(q ) [2 df~
A F(q) = ~ Fp(q), ..()=i Fp(q) - (-1;P-1 ik ~lZTr T' ~ f d 2 b exp(iq, b)$`p(b)~Tzu 2~
A $`1 = l~=l gll ,
$'3
A
(i)
A
$`2 = l~= l m = l (JllJmm -Jlm Jml ) '
A A A =~ ~ ~ (JllJmmJnn-JllJmnJnm-JlmJmlJnn l=l m=l n=l +Jlm Jmn Jnl + Jln Jml Jnm - Jln J m m Jnl) ,
etc., w h e r e k is the m o m e n t u m of the pion and q is the m o m e n t u m t r a n s f e r . We use units for which ~ = c = 1. The pion i s o s p i n o r s a r e denoted by }, the z - c o m p o n e n t s being Tz~ and T ~ for the initial and final s t a t e s , r e s p e c tively. The i n t e r a c t i o n i n t e g r a l s a r e
Jlm = (2~ik)-! f d 2 q l exp ( - i q l . b)Klm(ql) , Klm(ql) = f d 3 r l ~ ( 1 ) exp ( i q l . r l ) f l ( k , q l , P, P') qJm(1),
(2)
w h e r e f l is the amplitude for s c a t t e r i n g of the pion on one nucleon, which has initial and final m o m e n t a p and p ' , with m o m e n t u m t r a n s f e r q l . P u r e l s and lp h a r m o n i c - o s c i l l a t o r wave functions
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"Pl(j) = *NLJ~I "~ = R N L ( ~ ) Y L ½ J (3)771 (J)
(3)
a r e u s e d . H e r e R i s the r a d i a l wave f u n c t i o n , Y i s a s p i n - a n g l e f u n c t i o n a n d 7? i s the n u c l e o n i s o s p i n o r . In eq. (2) the f i n a l - s t a t e wave f u n c t i o n has b e e n d e n o t e d by ~'l r a ~ , , , v than by g2l, to i n d i c a t e that f o r the c h a r g e - e x c h a n g e c a s e the n u c l e o n in the ip~ s h e l l i s c h a n g e d f r o m a n e u t r o n to a p r o t o n . F o r a l l o t h e r s t a t e s we have ° ~ l = ~ l - F o r c h a r g e e x c h a n g e the c r o s s s e c t i o n s h o u l d be a v e r a g e d o v e r i n i t i a l a n d s u m m e d o v e r f i n a l v a l u e s of the z - c o m p o n e n t Jz = +½ of a n g u l a r m o m e n t u m . It i s a l s o c l e a r that in this c a s e the a m p l i t u d e F 1 only c o n t a i n s one t e r m , i.e. the o n e w h i c h i s due to the v a l e n c e n u c l e o n in 13C and 13N. T h e f i r s t t e r m ~ J l l J m m . . . in e a c h p a r t i a l a m p l i t u d e Fp is what we below c a l l the d i r e c t t e r m . T h e s e c o n t a i n c o n t r i b u t i o n s f o r w h i c h two o r m o r e i n d i c e s l, rn etc. a r e e q u a l , w h i c h c o r r e s p o n d s to two o r m o r e s c a t t e r i n g s of the pion on the s a m e n u c l e o n . H o w e v e r , t h e s e c o n t r i b u t i o n s a r e c a n c e l l e d by o t h e r s of equal m a g n i t u d e and o p p o s i t e s i g n in the r e m a i n i n g t e r m s , i.e. the e x c h a n g e t e r m s . The c a l c u l a t i o n s a r e p e r f o r m e d for the l a b s y s t e m , w h i c h i s not v e r y d i f f e r e n t f r o m the c . m . s y s t e m f o r the whole s c a t t e r i n g p r o c e s s . It i s , h o w e v e r , i m p o r t a n t to take into c o n s i d e r a t i o n the d i f f e r e n c e b e t w e e n this s y s t e m a n d the c . m . s y s t e m f o r the s i n g l e s c a t t e r i n g p r o c e s s e s 7rN ~ 7rN. T h u s , we s h o u l d not e m p l o y the u s u a l s i m p l e e x p r e s s i o n f o r the 7rN r e s o n a n t s c a t t e r i n g a m p l i t u d e in the c.m° s y s t e m , but r a t h e r the a m p l i t u d e in the o v e r a l l l a b s y s t e m , w h i c h we w r i t e a s
f(k,q,p)
=~
-1
f33(kc.m.)[A(q,p)+i~-2B(q,p)a • (k×q)]v3,
(4)
2
w h e r e ~t i s the pion m a s s a n d O~ i s the p r o j e c t i o n o p e r a t o r f o r the T - 3 • , "2 l s o s p m s t a t e . In eq. (4) we have n e g l e c t e d t e r m s p r o p o r t i o n a l to q . p , k . p , q x p , and k x p . As i n ref. [7] we take the F e r m i m o t i o n of the n u c l e o n s into a c c o u n t by a v e r a g i n g the s c a t t e r i n g a m p l i t u d e o v e r the n u c l e o n m o m e n t u m p , and the t e r m s w h i c h a r e o m i t t e d have c o e f f i c i e n t s w h i c h d e p e n d only w e a k l y on the d i r e c t i o n of p . F u r t h e r m o r e , f o r m o m e n t u m t r a n s f e r s q f o r w h i c h we can hope to get r e l i a b l e r e s u l t s by u s i n g the G l a u b e r e x p a n s i o n , the a m p l i t u d e i s i n d e p e n d e n t of q to w i t h i n a few p e r c e n t . We t h e r e f o r e n e g l e c t the q d e p e n d e n c e of the f u n c t i o n s A and B and u s e t h e i r v a l u e s for q = 0. The c o r r e s p o n d i n g e x p r e s s i o n for A i s g i v e n in ref. [7]. The i m p o r t a n t d i f f e r e n c e b e t w e e n the p r e s e n t c a l c u l a t i o n s a n d t h o s e of ref. [7] i s that we now take into a c c o u n t the e x c h a n g e t e r m s due to the a n t i s y m m e t r i z a t i o n of the n u c l e a r wave f u n c t i o n . On one h a n d , t h i s l e a d s to a c o n t r i b u t i o n f r o m n u c l e a r s p i n - f l i p t r a n s i t i o n s , e v e n f o r the c a s e of e l a s t i c s c a t t e r i n g . In e a c h s c a t t e r i n g o r d e r t h i s c o n t r i b u t i o n i s s m a l l c o m p a r e d with the o t h e r t e r m s , and f o r e l a s t i c s c a t t e r i n g we only take it into a c c o u n t i n the s e c o n d - o r d e r a m p l i t u d e , i.e. the l o w e s t o r d e r in w h i c h e x c h a n g e t e r m s c o n t r i b u t e . F o r c h a r g e - e x c h a n g e s c a t t e r i n g the s p i n - f l i p t e r m s a r e t a k e n into a c c o u n t in the f i r s t - and s e c o n d - o r d e r a m p l i t u d e s . F u r t h e r m o r e , the e x c h a n g e t e r m s l e a d to a c o n t r i b u t i o n f r o m c h a r g e -
182
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MeV
0. . . . .
50
'
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100 '
"
Fig. 1. Theoretical differential c r o s s section for the reaction 12C(~-,~-)12C at 180 MeV. The number of t e r m s in the expansion is indicated. F e r m i motion as well as space and isospin exchange are included in all o r d er s. e x c h a n g e ( i s o s p i n - e x c h a n g e ) i n t e r m e d i a t e s c a t t e r i n g s . In n u c l e i a s l i g h t a s the p r e s e n t o n e s t h e s e g i v e , t o g e t h e r w i t h the s p a c e - e x c h a n g e t e r m s , v e r y l a r g e c o n t r i b u t i o n s to the s c a t t e r i n g a m p l i t u d e . In f a c t , f r o m t h e t h i r d o r d e r o n w a r d s the e x c h a n g e t e r m s a r e a s l a r g e a s o r l a r g e r than the s u m of d i r e c t and e x c h a n g e t e r m s . F o r e l a s t i c s c a t t e r i n g we c a r r y out the e x p a n s i o n to s i x t h o r d e r , i n c l u d i n g e x c h a n g e t e r m s ( e x c e p t spin e x c h a n g e ) in a l l o r d e r s . F o r the c h a r g e e x c h a n g e r e a c t i o n we a l s o i n c l u d e the s e v e n t h - o r d e r t e r m , s i n c e in t h i s c a s e the c o n v e r g e n c e i s p o o r e r . T h e n u c l e a r c . m . m o t i o n i s t r e a t e d by t h e m e t h o d of B a s s e l and W i l k i n [10], and we a l s o u s e t h e i r v a l u e a 2 = 0.401 f m -2 f o r the n u c l e a r s i z e p a r a m e t e r . At s m a l l a n g l e s the C o u l o m b s c a t t e r i n g i s i m p o r t a n t , and we i n c l u d e th e s i n g l e - s c a t t e r i n g C o u l o m b a m p l i t u d e in t h e e l a s t i c c a s e , f o r w h i c h we can c o m p a r e the d i f f e r e n t i a l c r o s s s e c t i o n w i t h e x p e r i m e n t s . When c a l c u l a t i n g the C o u l o m b c o n t r i b u t i o n we n e g l e c t the Fermi motion. 3. N U M E R I C A L R E S U L T S AND D I S C U S S I O N T h e n u m e r i c a l r e s u l t s on e l a s t i c s c a t t e r i n g at 180 M e V a r e d i s p l a y e d in
SCATTERING OF PIONS BY CARBON
183
TK : 180 M eV
10-I
~/~ I
10-;
--
no Fermi motion
---
Fermi m o t i o n
10-3
?
/
10-4
,
o
,
,
,
i
5o
.
.
.
.
l
=_
ei~b l-degl loo
Fig. 2. Theoretical differential cross section for the reaction 12C(~-, ~-)12C at 180 MeV, with and without F e r m i motion. Only direct terms (summed to infinite order) are here considered. figs. 1-3. Fig. 1 s h o w s how the c a l c u l a t e d c r o s s s e c t i o n c h a n g e s as t e r m s of h i g h e r o r d e r a r e i n c l u d e d ; in t h e s e c u r v e s the C o u l o m b c o r r e c t i o n a n d s e c o n d - o r d e r s p i n - f l i p t e r m s h a v e b e e n o m i t t e d , w h e r e a s the F e r m i m o tion is accounted for. It i s s e e n that a l t h o u g h the s e c o n d - o r d e r a m p l i t u d e d o m i n a t e s at the s e c o n d m a x i m u m , the p r o n o u n c e d s h a p e of this m a x i m u m d o e s not a p p e a r u n t i l a l s o the a m p l i t u d e s of h i g h e r o r d e r a r e added. Fig. 2 s h o w s the i n f l u e n c e of the F e r m i m o t i o n on the c r o s s s e c t i o n . When F e r m i m o t i o n is not t a k e n into c o n s i d e r a t i o n , it is not a v e r y good a p p r o x i m a t i o n to n e g l e c t the q - d e p e n d e n c e of the e l e m e n t a r y a m p l i t u d e . To avoid i n t e g r a t i o n o v e r s i x c o u p l e d t w o - d i m e n s i o n a l m o m e n t u m t r a n s f e r s , we have c h o s e n to show the effect of the F e r m i m o t i o n f o r the d i r e c t t e r m s ( e l a s t i c r e s c a t t e r i n g only and no s p i n - o r s p a c e - e x c h a n g e ) s u m m e d to i n f i n i t e o r d e r . C o m p a r i n g figs. 1 and 2 o n e a l s o s e e s that the d i r e c t t e r m s when s u m m e d to i n f i n i t e o r d e r , give a c r o s s s e c t i o n v e r y s i m i l a r to the o n e o b t a i n e d by d i r e c t p l u s e x c h a n g e t e r m s to s i x t h o r d e r , but this i s only due to a c c i d e n t a l c a n c e l l a t i o n s a m o n g the l a r g e e x c h a n g e t e r m s in the v a r i o u s s c a t t e r i n g o r d e r s . The t h e o r e t i c a l d i f f e r e n t i a l c r o s s s e c t i o n is c o m p a r e d with the e x p e r i m e n t a l one [8] in fig. 3, in t h i s c a s e a l s o the C o u l o m b t e r m and s e c o n d - o r d e r s p i n e x -
184
K. BJ(~RNENAK et al.
T~ =180 MeV 10q Theory - - - Experiment -
-
10-2 k ~ ~
10-4
iI
I
II
,
i
i
II
i
50
,
,
!
t i
eta b [Jeg]
I
100
~
F i g . 3. E x p e r i m e n t a l [8] and t h e o r e t i c a l d i f f e r e n t i a ] cross section f o r the r e a c t i o n 12C(?r-,//-)12C at 180 MeV. Coulomb s c a t t e r i n g is inc]uded in f i r s t o r d e r , and spin
exchange in second order. c h a n g e a r e t a k e n into a c c o u n t . At l e a s t at s m a l l a n g l e s t h e r e ks r e a s o n a b l y good a g r e e m e n t with the e x p e r i m e n t s . F i g s . 4 and 5 show the e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s on e l a s t i c s c a t t e r i n g at 120 and 280 MeV. Also in t h e s e c a s e s we have only i n c l u d e d the r e s o n a n c e c o n t r i b u t i o n , and the t h e o r e t i c a l c u r v e s a g a i n c o r r e s p o n d to d i r e c t a s w e l l a s s p a c e - and [ s o s p i n - e x c h a n g e t e r m s up to s i x t h o r d e r , with s p i n e x c h a n g e in s e c o n d o r d e r and C o u l o m b c o r r e c t i o n s in f i r s t o r d e r . T h e t h e o r e t i c a l d i f f e r e n t i a l c r o s s s e c t i o n f o r the c h a r g e - e x c h a n g e r e a c tion at 180 MeV i s shown in fig. 6, in this c a s e w i t h o u t C o u l o m b c o r r e c t i o n s , but i n c l u d i n g f i r s t - and s e c o n d - o r d e r s p i n - f l i p t e r m s a n d o t h e r e x c h a n g e t e r m s up to s e v e n t h o r d e r , a s w e l l a s F e r m i m o t i o n . F o r c o m p a r i s o n we have a l s o shown the c r o s s s e c t i o n w h i c h i s o b t a i n e d when we only c o n s i d e r p r o c e s s e s that do not c h a n g e the v a l u e of Jz for the n u c l e u s . F i n a l l y , in fig. 7 we have shown the o p t i c a l p o t e n t i a l f o r e l a s t i c s c a t t e r i n g at 180 MeV, c o r r e s p o n d i n g to the s i x t h - o r d e r c a l c u l a t i o n s a s e x p l a i n e d a b o v e , but w i t h o u t C o u l o m b a n d s p i n - f l i p c o n t r i b u t i o n s . The p o t e n t i a l i s d e f i n e d a s in ref. [7]. A l t h o u g h the s i x t h - o r d e r a m p l i t u d e h a s l i t t l e e f f e c t on the c r o s s s e c t i o n , e x c e p t at the m i n i m u m and s e c o n d m a x i m u m , it s t i l l has a v e r y l a r g e i n f l u e n c e on the p o t e n t i a l in the i n n e r r e g i o n , and h i g h e r o r d e r s
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PIONS
BY
CARBON
185
I TE =120 MeV
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I
lOO
Fig. 4. Experimental [8] and theoretical differential cross sections for the reaction 12C(~-,Y-)12C at 120 MeV. i n the m u l t i p l e - s c a t t e r i n g e x p a n s i o n a r e e x p e c t e d to give a p p r e c i a b l e c h a n g e s i n the p o t e n t i a l f o r s m a l l r . F o r c h a r g e - e x c h a n g e s c a t t e r i n g t h e c o n v e r g e n c e i s not good e n o u g h to give a m e a n i n g f u l o p t i c a l p o t e n t i a l even when the s e v e n t h - o r d e r t e r m s a r e i n c l u d e d , and t h i s p o t e n t i a l is t h e r e f o r e not shown. Also f o r the c a s e of c h a r g e - e x c h a n g e it i s q u i t e i n s t r u c t i v e to take a look a t the c o n t r i b u t i o n s f r o m the v a r i o u s t e r m s in the e x p a n s i o n . Since h e r e o n l y the t o t a l c r o s s s e c t i o n i s a v a i l a b l e f r o m e x p e r i m e n t s [9] we have in t a b l e 1 shown how the t h e o r e t i c a l v a l u e of the t o t a l c r o s s s e c t i o n c h a n g e s a s m o r e t e r m s a r e i n c l u d e d in the e x p a n s i o n . A l r e a d y the f i r s t t e r m i s in q u a l i t a t i v e a g r e e m e n t with the e x p e r i m e n t a l v a l u e 4.1 ± 1.3 m b , and s i m i l a r r e s u l t s have a l s o b e e n o b t a i n e d by s l i g h t l y d i f f e r e n t f i r s t - o r d e r c a l c u l a t i o n s [11,12]. H o w e v e r , s o m e of the h i g h e r - o r d e r t e r m s a r e l a r g e r than the f i r s t Table 1 Theoretical total cross sections in mb for the reaction 13C(~;+, ~o)13N. The experimental result [9] is 4.1" 1.3 mb. No. of t e r m s Cross section
1
2
3
4
5
6
7
Opt. lim.
11.0
17.0
20.9
6.0
6.7
6.0
6.2
0.16
K. BJ(~RNENAK et al.
186
1 ~
TT~
=280MeV
10-1 - - Theory Experiment
\\ ~ t
10-2
---
10-3
I0 -~
i
i
i
i
50
J
,
J
~
L
(~lob[deg] 100
L
Fig. 5. Experimental [8] and theoretical differential cross section for the reaction 12C(~-,~-)12C at 280 MeV. o r d e r one. By e x p a n d i n g to s e v e n t h - o r d e r we o b t a i n a c r o s s s e c t i o n w h i c h i s in r a t h e r good a g r e e m e n t with the e x p e r i m e n t , w h e r e a s the r e s u l t o b t a i n e d by u s i n g only the n o n - s p i n - f l i p d i r e c t t e r m s to i n f i n i t e o r d e r (the o p t i c a l l i m i t ) i s m u c h too s m a l l . We s h o u l d , h o w e v e r , l i k e to e m p h a s i z e that the good a g r e e m e n t w h i c h i s o b t a i n e d b e t w e e n t h e o r y and e x p e r i m e n t m i g h t s t i l l be p a r t l y a c c i d e n t a l . It s e e m s c l e a r that for the c h a r g e - e x c h a n g e r e a c t i o n s p i n - f l i p t e r m s of t h i r d and h i g h e r o r d e r s h o u l d r e a l l y a l s o h a v e b e e n i n c l u d e d . T h e s e w i l l p r o b a b l y b r i n g the c r o s s s e c t i o n down a f t e r the m i n i m u m in fig. 6. A s m e n t i o n e d a b o v e s p i n - f l i p i s of l i t t l e i m p o r t a n c e when e a c h t e r m of the e x p a n s i o n i s c o n s i d e r e d s e p a r a t e l y , b u t we have e x t e n s i v e c a n c e l l a t i o n s b e t w e e n the l a r g e r c o n t r i b u t i o n s . We r e a l i z e that the p r e s e n t c a l c u l a t i o n s a r e open to c r i t i c i s m a l s o on s o m e o t h e r p o i n t s . A p a r t f r o m the q u e s t i o n of to w h i c h e x t e n t the G l a u b e r e x p a n s i o n i s a p p l i c a b l e to the p r o b l e m in q u e s t i o n , the n u c l e a r p h y s i c s p a r t of the c a l c u l a t i o n s s h o u l d p e r h a p s have b e e n t r e a t e d a b i t m o r e c a r e f u l l y . It d o e s not m a k e m u c h d i f f e r e n c e if the s h e l l - m o d e l p a r a m e t e r a i s v a r i e d w i t h i n r e a s o n a b l e l i m i t s , but a d m i x t u r e s of h i g h e r s t a t e s in the g r o u n d s t a t e w a v e f u n c t i o n s m i g h t h a v e an a p p r e c i a b l e i n f l u e n c e on the r e s u l t s [10, 12], a s w e l l a s i n t e r m e d i a t e t r a n s i t i o n s to e x c i t e d s t a t e s . At l a r g e s c a t t e r i n g angles also s h o r t - r a n g e n u c l e o n - n u c l e o n c o r r e l a t i o n s should
SCATTERING OF PIONS BY CARBON
T11; = 1 8 0
187
MeV
I
10 I|
/
/
/ / iI i
10";
I I I I
/
I
I
10-3 0
so
e,ob .
,oo
Fig. 6. Theoretical differential cross section for the reaction 13C(~+,~°)13N at 180 MeV, to seventh order and including space and isospin exchange. Spin exchange is included in first and second order. The dotted curve does not contain contributions corresponding to a change of the Jz value for the nucleus.
300
200
100
°o
i
2
3
Fig. 7. Optical potential for the reaction 12CUr- , 7r-)12C at 180 MeV. Coulomb scattering and spin-flip are here not considered. h a v e b e e n i n c l u d e d , but t h i s h a r d l y s e e m s w o r t h w h i l e , s i n c e f o r l a r g e m o m e n t u m t r a n s f e r s the g e n e r a l a p p r o a c h i s not e x p e c t e d to b e v e r y t r u s t worthy anyway. In c o n c l u s i o n w e s h o u l d l i k e to e m p h a s i z e that g r e a t c a r e i s n e e d e d in the t h e o r e t i c a l t r e a t m e n t of t h i s t y p e of p r o b l e m s , in the s e n s e that both the F e r m i m o t i o n o f the s c a t t e r e r s and m u l t i p l e s c a t t e r i n g a r e i m p o r t a n t . In p a r t i c u l a r e x c h a n g e c o r r e c t i o n s s h o u l d b e g i v e n s e r i o u s c o n s i d e r a t i o n ,
188
K. BJ~RNENAK
et al.
in spite of the computational difficulties which many scatterings are involved.
they cause
in terms
where
REFERENCES [1] R . J . Glauber, Boulder l e c t u r e s in theoretical physics I (Interscience, New York, 1959) 315; High-energy physics and nuclear st r u ct u r e (North-Holland, A m s t e r d a m , 1967) 311. [2] M . M . S t e r n h e i m , Phys. Rev. 135 (1964) B912. [3] G. F~lldt and T. E. O. E r i c s o n , Nucl. Phys. B8 {1964) 1. [4] G. F~ildt, Nucl. Phys. B10 (1969) 597. [5] C.Michael and C.Wilkin, Nucl. Phys. B l l (1969) 99. [6] T. T. Chou, Phys. Rev. 168 (1968) 1594. [7] K. Bjdrnenak, J. Finjord, P. Osland and A. Reitan, Nucl. Phys. B20 (1970) 327. [8] F. Binon, J. P. Garron, J. GSrres, L. Hugon, R. Meunier, M. Spighel and J. P. Stroot, Nucl. Phys. B17 (1970) 168. [9] D.T. Chivers, J . J . Domingo, E . M . R i m m e r , R.C.Witcomb, B.W. Allardyce and N. W. Tanner, Phys. L e t t e r s 26B (1968) 573. [10] R. H. Bassel and C.Wilkin, Phys. Rev. 174 {1968) 1179. [11] F. H. Bakke and A. Reitan, Nucl. Phys. B10 (1969) 43. [12] Y. Sakamoto, Phys. L e t t e r s 29B (1969) 88.
North-Holland Publishing Company
ELASTIC A N D SINGLE-CHARGE-EXCHANGE SCATTERING OF PIONS BY CARBON NUCLEI K. B J g ) R N E N A K , J. F I N J O R D and P. OSLAND F y s i s k I n s t i t u t t , U n i v e r s i t e t e t i Trondheim - N o r g e s LaererhOgskole, N 7000 T r o n d h e i m , Norway
A. R E I T A N F y s i s k I n s t i t u t t , U n i v e r s i t e t e t i Trondheim - Norges t e k n i s k e hpgskole, N 7034 Trondheim - N T H , Norway
Received 25 May 1970
Abstract: The differential c r o s s sections for the reactions 12C(7r-,~-)12C and 13C(~+,y°)13N a r e calculated in the Glauber approximation for pions with kinetic energies in the 180 MeV region. It is found that the f i r s t t e r m s in the Glauber expansion a r e not convergent, due to the large pion-nucleon resonance amplitude. The F e r m i motion of the nucleons in the target nucleus and the exchange t e r m s due to the a n t i - s y m m e t r y of the nuclear wave function have a large influence on the r e s u l t s .
i. INTRODUCTION
T h e G l a u b e r a p p r o x i m a t i o n ( s e e e.g. r e f . [1]) h a s b e e n u s e d in c o n n e c t i o n w i t h p i o n - s c a t t e r i n g p r o b l e m s by s e v e r a l a u t h o r s . S t e r n h e i m [2] t r e a t e d the s c a t t e r i n g of 80 M e V p i o n s by s e v e r a l l i g h t n u c l e i by t h i s m e t h o d a s w e l l a s by o t h e r a p p r o a c h e s to the m u l t i p l e - s c a t t e r i n g p r o b l e m . T h e s c a t t e r i n g of p i o n s by d e u t e r o n s h a s b e e n c o n s i d e r e d by F~ildt and E r i c s o n [3,4] o v e r a l a r g e e n e r g y r e g i o n , and by M i c h a e l and W i l k i n [5] f o r p i o n s in the 4 GeV r e g i o n . T h e d e u t e r o n c a l c u l a t i o n s i n d i c a t e t h a t the c r o s s s e c t i o n i s s e n s i t i v e to d e t a i l s of n u c l e a r s t r u c t u r e , a n d t h a t it i s i m p o r t a n t to t a k e the F e r m i m o t i o n of the s c a t t e r i n g n u c l e o n s into a c c o u n t . Chou [6] h a s c o n s i d e r e d the e l a s t i c s c a t t e r i n g of 8 GeV p i o n s by 4He, and the p r e s e n t a u t h o r s h a v e p r e v i o u s l y u s e d the G l a u b e r e x p a n s i o n to c a l c u l a t e the c r o s s s e c t i o n f o r d o u b l e - c h a r g e - e x c h a n g e s c a t t e r i n g of p i o n s on 51V and 9 0 Z r [7]. T h e G l a u b e r e x p a n s i o n can o n l y be u s e d w i t h c o n f i d e n c e f o r high e n e r g i e s and s m a l l s c a t t e r i n g a n g l e s , but in m a n y c a s e s it s e e m s to w o r k v e r y w e l l a l s o o u t s i d e i t s e x p e c t e d r e g i o n of v a l i d i t y . We h a v e t h e r e f o r e a p p l i e d t h i s a p p r o a c h to the e l a s t i c s c a t t e r i n g of n e g a t i v e p i o n s by 12C and to the s i n g l e - c h a r g e - e x c h a n g e s c a t t e r i n g of p o s i t i v e p i o n s by 13C, f o r pion k i n e t i c e n e r g i e s in the r e g i o n of the 180 MeV p i o n - n u c l e o n r e s o n a n c e . Only t r a n s i tions between nuclear ground states are considered. The differential cross
180
K. BJ(~RNENAK et al.
section for the elastic s c a t t e r i n g has been m e a s u r e d at s e v e r a l e n e r g i e s by Binon et al. [8], and Chivers et al. [9] have m e a s u r e d the total c h a r g e - e x change c r o s s section at 180 MeV.
2. THEORY When the wave function of the s c a t t e r i n g nucleus is taken to be an a n t i s y m m e t r i z e d p r o d u c t of o n e - n u c l e o n wave functions, and the products of the s c a t t e r i n g amplitudes that a p p e a r in the Glauber expansion a r e s y m m e t r i z e d , the m u l t i p l e - s c a t t e r i n g s e r i e s is found to have the f o r m d~ _ [F(q ) [2 df~
A F(q) = ~ Fp(q), ..()=i Fp(q) - (-1;P-1 ik ~lZTr T' ~ f d 2 b exp(iq, b)$`p(b)~Tzu 2~
A $`1 = l~=l gll ,
$'3
A
(i)
A
$`2 = l~= l m = l (JllJmm -Jlm Jml ) '
A A A =~ ~ ~ (JllJmmJnn-JllJmnJnm-JlmJmlJnn l=l m=l n=l +Jlm Jmn Jnl + Jln Jml Jnm - Jln J m m Jnl) ,
etc., w h e r e k is the m o m e n t u m of the pion and q is the m o m e n t u m t r a n s f e r . We use units for which ~ = c = 1. The pion i s o s p i n o r s a r e denoted by }, the z - c o m p o n e n t s being Tz~ and T ~ for the initial and final s t a t e s , r e s p e c tively. The i n t e r a c t i o n i n t e g r a l s a r e
Jlm = (2~ik)-! f d 2 q l exp ( - i q l . b)Klm(ql) , Klm(ql) = f d 3 r l ~ ( 1 ) exp ( i q l . r l ) f l ( k , q l , P, P') qJm(1),
(2)
w h e r e f l is the amplitude for s c a t t e r i n g of the pion on one nucleon, which has initial and final m o m e n t a p and p ' , with m o m e n t u m t r a n s f e r q l . P u r e l s and lp h a r m o n i c - o s c i l l a t o r wave functions
SCATTERING
.d z
T_
OF
PlONS
BY CARBON
dz.
181
Tz.
"Pl(j) = *NLJ~I "~ = R N L ( ~ ) Y L ½ J (3)771 (J)
(3)
a r e u s e d . H e r e R i s the r a d i a l wave f u n c t i o n , Y i s a s p i n - a n g l e f u n c t i o n a n d 7? i s the n u c l e o n i s o s p i n o r . In eq. (2) the f i n a l - s t a t e wave f u n c t i o n has b e e n d e n o t e d by ~'l r a ~ , , , v than by g2l, to i n d i c a t e that f o r the c h a r g e - e x c h a n g e c a s e the n u c l e o n in the ip~ s h e l l i s c h a n g e d f r o m a n e u t r o n to a p r o t o n . F o r a l l o t h e r s t a t e s we have ° ~ l = ~ l - F o r c h a r g e e x c h a n g e the c r o s s s e c t i o n s h o u l d be a v e r a g e d o v e r i n i t i a l a n d s u m m e d o v e r f i n a l v a l u e s of the z - c o m p o n e n t Jz = +½ of a n g u l a r m o m e n t u m . It i s a l s o c l e a r that in this c a s e the a m p l i t u d e F 1 only c o n t a i n s one t e r m , i.e. the o n e w h i c h i s due to the v a l e n c e n u c l e o n in 13C and 13N. T h e f i r s t t e r m ~ J l l J m m . . . in e a c h p a r t i a l a m p l i t u d e Fp is what we below c a l l the d i r e c t t e r m . T h e s e c o n t a i n c o n t r i b u t i o n s f o r w h i c h two o r m o r e i n d i c e s l, rn etc. a r e e q u a l , w h i c h c o r r e s p o n d s to two o r m o r e s c a t t e r i n g s of the pion on the s a m e n u c l e o n . H o w e v e r , t h e s e c o n t r i b u t i o n s a r e c a n c e l l e d by o t h e r s of equal m a g n i t u d e and o p p o s i t e s i g n in the r e m a i n i n g t e r m s , i.e. the e x c h a n g e t e r m s . The c a l c u l a t i o n s a r e p e r f o r m e d for the l a b s y s t e m , w h i c h i s not v e r y d i f f e r e n t f r o m the c . m . s y s t e m f o r the whole s c a t t e r i n g p r o c e s s . It i s , h o w e v e r , i m p o r t a n t to take into c o n s i d e r a t i o n the d i f f e r e n c e b e t w e e n this s y s t e m a n d the c . m . s y s t e m f o r the s i n g l e s c a t t e r i n g p r o c e s s e s 7rN ~ 7rN. T h u s , we s h o u l d not e m p l o y the u s u a l s i m p l e e x p r e s s i o n f o r the 7rN r e s o n a n t s c a t t e r i n g a m p l i t u d e in the c.m° s y s t e m , but r a t h e r the a m p l i t u d e in the o v e r a l l l a b s y s t e m , w h i c h we w r i t e a s
f(k,q,p)
=~
-1
f33(kc.m.)[A(q,p)+i~-2B(q,p)a • (k×q)]v3,
(4)
2
w h e r e ~t i s the pion m a s s a n d O~ i s the p r o j e c t i o n o p e r a t o r f o r the T - 3 • , "2 l s o s p m s t a t e . In eq. (4) we have n e g l e c t e d t e r m s p r o p o r t i o n a l to q . p , k . p , q x p , and k x p . As i n ref. [7] we take the F e r m i m o t i o n of the n u c l e o n s into a c c o u n t by a v e r a g i n g the s c a t t e r i n g a m p l i t u d e o v e r the n u c l e o n m o m e n t u m p , and the t e r m s w h i c h a r e o m i t t e d have c o e f f i c i e n t s w h i c h d e p e n d only w e a k l y on the d i r e c t i o n of p . F u r t h e r m o r e , f o r m o m e n t u m t r a n s f e r s q f o r w h i c h we can hope to get r e l i a b l e r e s u l t s by u s i n g the G l a u b e r e x p a n s i o n , the a m p l i t u d e i s i n d e p e n d e n t of q to w i t h i n a few p e r c e n t . We t h e r e f o r e n e g l e c t the q d e p e n d e n c e of the f u n c t i o n s A and B and u s e t h e i r v a l u e s for q = 0. The c o r r e s p o n d i n g e x p r e s s i o n for A i s g i v e n in ref. [7]. The i m p o r t a n t d i f f e r e n c e b e t w e e n the p r e s e n t c a l c u l a t i o n s a n d t h o s e of ref. [7] i s that we now take into a c c o u n t the e x c h a n g e t e r m s due to the a n t i s y m m e t r i z a t i o n of the n u c l e a r wave f u n c t i o n . On one h a n d , t h i s l e a d s to a c o n t r i b u t i o n f r o m n u c l e a r s p i n - f l i p t r a n s i t i o n s , e v e n f o r the c a s e of e l a s t i c s c a t t e r i n g . In e a c h s c a t t e r i n g o r d e r t h i s c o n t r i b u t i o n i s s m a l l c o m p a r e d with the o t h e r t e r m s , and f o r e l a s t i c s c a t t e r i n g we only take it into a c c o u n t i n the s e c o n d - o r d e r a m p l i t u d e , i.e. the l o w e s t o r d e r in w h i c h e x c h a n g e t e r m s c o n t r i b u t e . F o r c h a r g e - e x c h a n g e s c a t t e r i n g the s p i n - f l i p t e r m s a r e t a k e n into a c c o u n t in the f i r s t - and s e c o n d - o r d e r a m p l i t u d e s . F u r t h e r m o r e , the e x c h a n g e t e r m s l e a d to a c o n t r i b u t i o n f r o m c h a r g e -
182
K. BJ(~RNENAK e t al.
MeV
0. . . . .
50
'
{~la b'[d e gi
100 '
"
Fig. 1. Theoretical differential c r o s s section for the reaction 12C(~-,~-)12C at 180 MeV. The number of t e r m s in the expansion is indicated. F e r m i motion as well as space and isospin exchange are included in all o r d er s. e x c h a n g e ( i s o s p i n - e x c h a n g e ) i n t e r m e d i a t e s c a t t e r i n g s . In n u c l e i a s l i g h t a s the p r e s e n t o n e s t h e s e g i v e , t o g e t h e r w i t h the s p a c e - e x c h a n g e t e r m s , v e r y l a r g e c o n t r i b u t i o n s to the s c a t t e r i n g a m p l i t u d e . In f a c t , f r o m t h e t h i r d o r d e r o n w a r d s the e x c h a n g e t e r m s a r e a s l a r g e a s o r l a r g e r than the s u m of d i r e c t and e x c h a n g e t e r m s . F o r e l a s t i c s c a t t e r i n g we c a r r y out the e x p a n s i o n to s i x t h o r d e r , i n c l u d i n g e x c h a n g e t e r m s ( e x c e p t spin e x c h a n g e ) in a l l o r d e r s . F o r the c h a r g e e x c h a n g e r e a c t i o n we a l s o i n c l u d e the s e v e n t h - o r d e r t e r m , s i n c e in t h i s c a s e the c o n v e r g e n c e i s p o o r e r . T h e n u c l e a r c . m . m o t i o n i s t r e a t e d by t h e m e t h o d of B a s s e l and W i l k i n [10], and we a l s o u s e t h e i r v a l u e a 2 = 0.401 f m -2 f o r the n u c l e a r s i z e p a r a m e t e r . At s m a l l a n g l e s the C o u l o m b s c a t t e r i n g i s i m p o r t a n t , and we i n c l u d e th e s i n g l e - s c a t t e r i n g C o u l o m b a m p l i t u d e in t h e e l a s t i c c a s e , f o r w h i c h we can c o m p a r e the d i f f e r e n t i a l c r o s s s e c t i o n w i t h e x p e r i m e n t s . When c a l c u l a t i n g the C o u l o m b c o n t r i b u t i o n we n e g l e c t the Fermi motion. 3. N U M E R I C A L R E S U L T S AND D I S C U S S I O N T h e n u m e r i c a l r e s u l t s on e l a s t i c s c a t t e r i n g at 180 M e V a r e d i s p l a y e d in
SCATTERING OF PIONS BY CARBON
183
TK : 180 M eV
10-I
~/~ I
10-;
--
no Fermi motion
---
Fermi m o t i o n
10-3
?
/
10-4
,
o
,
,
,
i
5o
.
.
.
.
l
=_
ei~b l-degl loo
Fig. 2. Theoretical differential cross section for the reaction 12C(~-, ~-)12C at 180 MeV, with and without F e r m i motion. Only direct terms (summed to infinite order) are here considered. figs. 1-3. Fig. 1 s h o w s how the c a l c u l a t e d c r o s s s e c t i o n c h a n g e s as t e r m s of h i g h e r o r d e r a r e i n c l u d e d ; in t h e s e c u r v e s the C o u l o m b c o r r e c t i o n a n d s e c o n d - o r d e r s p i n - f l i p t e r m s h a v e b e e n o m i t t e d , w h e r e a s the F e r m i m o tion is accounted for. It i s s e e n that a l t h o u g h the s e c o n d - o r d e r a m p l i t u d e d o m i n a t e s at the s e c o n d m a x i m u m , the p r o n o u n c e d s h a p e of this m a x i m u m d o e s not a p p e a r u n t i l a l s o the a m p l i t u d e s of h i g h e r o r d e r a r e added. Fig. 2 s h o w s the i n f l u e n c e of the F e r m i m o t i o n on the c r o s s s e c t i o n . When F e r m i m o t i o n is not t a k e n into c o n s i d e r a t i o n , it is not a v e r y good a p p r o x i m a t i o n to n e g l e c t the q - d e p e n d e n c e of the e l e m e n t a r y a m p l i t u d e . To avoid i n t e g r a t i o n o v e r s i x c o u p l e d t w o - d i m e n s i o n a l m o m e n t u m t r a n s f e r s , we have c h o s e n to show the effect of the F e r m i m o t i o n f o r the d i r e c t t e r m s ( e l a s t i c r e s c a t t e r i n g only and no s p i n - o r s p a c e - e x c h a n g e ) s u m m e d to i n f i n i t e o r d e r . C o m p a r i n g figs. 1 and 2 o n e a l s o s e e s that the d i r e c t t e r m s when s u m m e d to i n f i n i t e o r d e r , give a c r o s s s e c t i o n v e r y s i m i l a r to the o n e o b t a i n e d by d i r e c t p l u s e x c h a n g e t e r m s to s i x t h o r d e r , but this i s only due to a c c i d e n t a l c a n c e l l a t i o n s a m o n g the l a r g e e x c h a n g e t e r m s in the v a r i o u s s c a t t e r i n g o r d e r s . The t h e o r e t i c a l d i f f e r e n t i a l c r o s s s e c t i o n is c o m p a r e d with the e x p e r i m e n t a l one [8] in fig. 3, in t h i s c a s e a l s o the C o u l o m b t e r m and s e c o n d - o r d e r s p i n e x -
184
K. BJ(~RNENAK et al.
T~ =180 MeV 10q Theory - - - Experiment -
-
10-2 k ~ ~
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I
II
,
i
i
II
i
50
,
,
!
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I
100
~
F i g . 3. E x p e r i m e n t a l [8] and t h e o r e t i c a l d i f f e r e n t i a ] cross section f o r the r e a c t i o n 12C(?r-,//-)12C at 180 MeV. Coulomb s c a t t e r i n g is inc]uded in f i r s t o r d e r , and spin
exchange in second order. c h a n g e a r e t a k e n into a c c o u n t . At l e a s t at s m a l l a n g l e s t h e r e ks r e a s o n a b l y good a g r e e m e n t with the e x p e r i m e n t s . F i g s . 4 and 5 show the e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s on e l a s t i c s c a t t e r i n g at 120 and 280 MeV. Also in t h e s e c a s e s we have only i n c l u d e d the r e s o n a n c e c o n t r i b u t i o n , and the t h e o r e t i c a l c u r v e s a g a i n c o r r e s p o n d to d i r e c t a s w e l l a s s p a c e - and [ s o s p i n - e x c h a n g e t e r m s up to s i x t h o r d e r , with s p i n e x c h a n g e in s e c o n d o r d e r and C o u l o m b c o r r e c t i o n s in f i r s t o r d e r . T h e t h e o r e t i c a l d i f f e r e n t i a l c r o s s s e c t i o n f o r the c h a r g e - e x c h a n g e r e a c tion at 180 MeV i s shown in fig. 6, in this c a s e w i t h o u t C o u l o m b c o r r e c t i o n s , but i n c l u d i n g f i r s t - and s e c o n d - o r d e r s p i n - f l i p t e r m s a n d o t h e r e x c h a n g e t e r m s up to s e v e n t h o r d e r , a s w e l l a s F e r m i m o t i o n . F o r c o m p a r i s o n we have a l s o shown the c r o s s s e c t i o n w h i c h i s o b t a i n e d when we only c o n s i d e r p r o c e s s e s that do not c h a n g e the v a l u e of Jz for the n u c l e u s . F i n a l l y , in fig. 7 we have shown the o p t i c a l p o t e n t i a l f o r e l a s t i c s c a t t e r i n g at 180 MeV, c o r r e s p o n d i n g to the s i x t h - o r d e r c a l c u l a t i o n s a s e x p l a i n e d a b o v e , but w i t h o u t C o u l o m b a n d s p i n - f l i p c o n t r i b u t i o n s . The p o t e n t i a l i s d e f i n e d a s in ref. [7]. A l t h o u g h the s i x t h - o r d e r a m p l i t u d e h a s l i t t l e e f f e c t on the c r o s s s e c t i o n , e x c e p t at the m i n i m u m and s e c o n d m a x i m u m , it s t i l l has a v e r y l a r g e i n f l u e n c e on the p o t e n t i a l in the i n n e r r e g i o n , and h i g h e r o r d e r s
SCATTERING
OF
PIONS
BY
CARBON
185
I TE =120 MeV
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~~~"~
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10-2
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10-3
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10-41 , . I
I
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Fig. 4. Experimental [8] and theoretical differential cross sections for the reaction 12C(~-,Y-)12C at 120 MeV. i n the m u l t i p l e - s c a t t e r i n g e x p a n s i o n a r e e x p e c t e d to give a p p r e c i a b l e c h a n g e s i n the p o t e n t i a l f o r s m a l l r . F o r c h a r g e - e x c h a n g e s c a t t e r i n g t h e c o n v e r g e n c e i s not good e n o u g h to give a m e a n i n g f u l o p t i c a l p o t e n t i a l even when the s e v e n t h - o r d e r t e r m s a r e i n c l u d e d , and t h i s p o t e n t i a l is t h e r e f o r e not shown. Also f o r the c a s e of c h a r g e - e x c h a n g e it i s q u i t e i n s t r u c t i v e to take a look a t the c o n t r i b u t i o n s f r o m the v a r i o u s t e r m s in the e x p a n s i o n . Since h e r e o n l y the t o t a l c r o s s s e c t i o n i s a v a i l a b l e f r o m e x p e r i m e n t s [9] we have in t a b l e 1 shown how the t h e o r e t i c a l v a l u e of the t o t a l c r o s s s e c t i o n c h a n g e s a s m o r e t e r m s a r e i n c l u d e d in the e x p a n s i o n . A l r e a d y the f i r s t t e r m i s in q u a l i t a t i v e a g r e e m e n t with the e x p e r i m e n t a l v a l u e 4.1 ± 1.3 m b , and s i m i l a r r e s u l t s have a l s o b e e n o b t a i n e d by s l i g h t l y d i f f e r e n t f i r s t - o r d e r c a l c u l a t i o n s [11,12]. H o w e v e r , s o m e of the h i g h e r - o r d e r t e r m s a r e l a r g e r than the f i r s t Table 1 Theoretical total cross sections in mb for the reaction 13C(~;+, ~o)13N. The experimental result [9] is 4.1" 1.3 mb. No. of t e r m s Cross section
1
2
3
4
5
6
7
Opt. lim.
11.0
17.0
20.9
6.0
6.7
6.0
6.2
0.16
K. BJ(~RNENAK et al.
186
1 ~
TT~
=280MeV
10-1 - - Theory Experiment
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10-2
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10-3
I0 -~
i
i
i
i
50
J
,
J
~
L
(~lob[deg] 100
L
Fig. 5. Experimental [8] and theoretical differential cross section for the reaction 12C(~-,~-)12C at 280 MeV. o r d e r one. By e x p a n d i n g to s e v e n t h - o r d e r we o b t a i n a c r o s s s e c t i o n w h i c h i s in r a t h e r good a g r e e m e n t with the e x p e r i m e n t , w h e r e a s the r e s u l t o b t a i n e d by u s i n g only the n o n - s p i n - f l i p d i r e c t t e r m s to i n f i n i t e o r d e r (the o p t i c a l l i m i t ) i s m u c h too s m a l l . We s h o u l d , h o w e v e r , l i k e to e m p h a s i z e that the good a g r e e m e n t w h i c h i s o b t a i n e d b e t w e e n t h e o r y and e x p e r i m e n t m i g h t s t i l l be p a r t l y a c c i d e n t a l . It s e e m s c l e a r that for the c h a r g e - e x c h a n g e r e a c t i o n s p i n - f l i p t e r m s of t h i r d and h i g h e r o r d e r s h o u l d r e a l l y a l s o h a v e b e e n i n c l u d e d . T h e s e w i l l p r o b a b l y b r i n g the c r o s s s e c t i o n down a f t e r the m i n i m u m in fig. 6. A s m e n t i o n e d a b o v e s p i n - f l i p i s of l i t t l e i m p o r t a n c e when e a c h t e r m of the e x p a n s i o n i s c o n s i d e r e d s e p a r a t e l y , b u t we have e x t e n s i v e c a n c e l l a t i o n s b e t w e e n the l a r g e r c o n t r i b u t i o n s . We r e a l i z e that the p r e s e n t c a l c u l a t i o n s a r e open to c r i t i c i s m a l s o on s o m e o t h e r p o i n t s . A p a r t f r o m the q u e s t i o n of to w h i c h e x t e n t the G l a u b e r e x p a n s i o n i s a p p l i c a b l e to the p r o b l e m in q u e s t i o n , the n u c l e a r p h y s i c s p a r t of the c a l c u l a t i o n s s h o u l d p e r h a p s have b e e n t r e a t e d a b i t m o r e c a r e f u l l y . It d o e s not m a k e m u c h d i f f e r e n c e if the s h e l l - m o d e l p a r a m e t e r a i s v a r i e d w i t h i n r e a s o n a b l e l i m i t s , but a d m i x t u r e s of h i g h e r s t a t e s in the g r o u n d s t a t e w a v e f u n c t i o n s m i g h t h a v e an a p p r e c i a b l e i n f l u e n c e on the r e s u l t s [10, 12], a s w e l l a s i n t e r m e d i a t e t r a n s i t i o n s to e x c i t e d s t a t e s . At l a r g e s c a t t e r i n g angles also s h o r t - r a n g e n u c l e o n - n u c l e o n c o r r e l a t i o n s should
SCATTERING OF PIONS BY CARBON
T11; = 1 8 0
187
MeV
I
10 I|
/
/
/ / iI i
10";
I I I I
/
I
I
10-3 0
so
e,ob .
,oo
Fig. 6. Theoretical differential cross section for the reaction 13C(~+,~°)13N at 180 MeV, to seventh order and including space and isospin exchange. Spin exchange is included in first and second order. The dotted curve does not contain contributions corresponding to a change of the Jz value for the nucleus.
300
200
100
°o
i
2
3
Fig. 7. Optical potential for the reaction 12CUr- , 7r-)12C at 180 MeV. Coulomb scattering and spin-flip are here not considered. h a v e b e e n i n c l u d e d , but t h i s h a r d l y s e e m s w o r t h w h i l e , s i n c e f o r l a r g e m o m e n t u m t r a n s f e r s the g e n e r a l a p p r o a c h i s not e x p e c t e d to b e v e r y t r u s t worthy anyway. In c o n c l u s i o n w e s h o u l d l i k e to e m p h a s i z e that g r e a t c a r e i s n e e d e d in the t h e o r e t i c a l t r e a t m e n t of t h i s t y p e of p r o b l e m s , in the s e n s e that both the F e r m i m o t i o n o f the s c a t t e r e r s and m u l t i p l e s c a t t e r i n g a r e i m p o r t a n t . In p a r t i c u l a r e x c h a n g e c o r r e c t i o n s s h o u l d b e g i v e n s e r i o u s c o n s i d e r a t i o n ,
188
K. BJ~RNENAK
et al.
in spite of the computational difficulties which many scatterings are involved.
they cause
in terms
where
REFERENCES [1] R . J . Glauber, Boulder l e c t u r e s in theoretical physics I (Interscience, New York, 1959) 315; High-energy physics and nuclear st r u ct u r e (North-Holland, A m s t e r d a m , 1967) 311. [2] M . M . S t e r n h e i m , Phys. Rev. 135 (1964) B912. [3] G. F~lldt and T. E. O. E r i c s o n , Nucl. Phys. B8 {1964) 1. [4] G. F~ildt, Nucl. Phys. B10 (1969) 597. [5] C.Michael and C.Wilkin, Nucl. Phys. B l l (1969) 99. [6] T. T. Chou, Phys. Rev. 168 (1968) 1594. [7] K. Bjdrnenak, J. Finjord, P. Osland and A. Reitan, Nucl. Phys. B20 (1970) 327. [8] F. Binon, J. P. Garron, J. GSrres, L. Hugon, R. Meunier, M. Spighel and J. P. Stroot, Nucl. Phys. B17 (1970) 168. [9] D.T. Chivers, J . J . Domingo, E . M . R i m m e r , R.C.Witcomb, B.W. Allardyce and N. W. Tanner, Phys. L e t t e r s 26B (1968) 573. [10] R. H. Bassel and C.Wilkin, Phys. Rev. 174 {1968) 1179. [11] F. H. Bakke and A. Reitan, Nucl. Phys. B10 (1969) 43. [12] Y. Sakamoto, Phys. L e t t e r s 29B (1969) 88.