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Rescattering corrections in radiative pion absorption
~
Nuclear Physics B l l (1969) 601-610. North-Holland Publ. Comp., Amsterdam
RESCATTERING CORRECTIONS IN RADIATIVE PION ABSORPTION R e e d GUY a n d J. M. E I S E N B E R G
Department of Physics, University of Virginia, Charlottesville, Virginia* Received 27 May 1969 Abstract: Rescattering corrections to radiative pion absorption in nuclei are considered. The r e s c a t t e r i n g process is known to make important contributions to the nuclear absorption of S orbit pions with two-nucleon ejection. We calculate the r e s c a t t e r i n g corrections both in perturbation theory and with an approach based on the actual pion-nucleus scattering length. For the detailed evaluation of n u clear m a t r i x elements the particle-hole f o r m a l i s m is used, and it is assumed that odd-parity spin-isospin nuclear states are excited by the radiative absorption mechanism. The calculation shows that incoherent r e s c a t t e r i n g effects i n volving an excited nucleus in the intermediate state are of little importance. Retaining only the coherent rescattering, decreases are found in the transition rate of approximately 40% relative to the leading t e r m for 160. The relationship between our calculation and analyses based on the use of an optical potential is presented.
1. I N T R O D U C T I O N T h e n u c l e a r a b s o r p t i o n of a n e g a t i v e p i o n b o u n d i n a n a t o m i c o r b i t a l u s u a l l y l e a d s to the e j e c t i o n of two n u c l e o n s . A s m a l l p e r c e n t a g e of the t i m e i n c o m p l e x n u c l e i , h o w e v e r , a high e n e r g y p h o t o n i s e m i t t e d ; t h i s i s t h e p r o c e s s k n o w n a s r a d i a t i v e p i o n a b s o r p t i o n . I n r e c e n t y e a r s , m u c h of the i n t e r e s t i n t h i s r e a c t i o n h a s c e n t e r e d on i t s c o n n e c t i o n with m u o n c a p t u r e . D e m o n s t r a t i o n s [1, 2] of the u s e f u l n e s s of t h i s r e l a t i o n s h i p f o r c o m p l e x n u c l e i m a d e u s e of t h e i m p u l s e a p p r o x i m a t i o n i n o r d e r to i d e n t i f y t h e r a d i a t i v e a b s o r p t i o n o p e r a t o r with t h e a x i a l - v e c t o r c u r r e n t o p e r a t o r of m u o n c a p t u r e . T h e s e d i s c u s s i o n s e m p h a s i z e d t h a t r a d i a t i v e p i o n a b s o r p t i o n l e a d s to t h e e x c i t a t i o n of s p i n - i s o s p i n c o l l e c t i v e s t a t e s in n u c l e i . Such s t a t e s a r e k n o w n to b e of i m p o r t a n c e i n d e t e r m i n i n g m u o n c a p t u r e r a t e s [3]. E v i d e n c e f o r t h e i r r o l e i n t h i s p r o c e s s , a s w e l l a s the r o l e of t h e i s o s p i n c o l l e c t i v e s t a t e s , h a s a p p e a r e d in m e a s u r e m e n t s of the s p e c t r a of o u t g o i n g n e u t r o n s f o l l o w i n g m u o n c a p t u r e [4]. T h e s t u d y of the s p i n - i s o s p i n s t a t e s c a n b e c a r r i e d out m o r e d i r e c t l y in t h e r a d i a t i v e p i o n a b s o r p t i o n r e a c t i o n , s i n c e t h e r e t h e s p e c t r u m of t h e o u t g o i n g p h o t o n i s e x p e r i m e n t a l l y a c c e s s i b l e [5]. * Work supported in part by the National Science Foundation.
602
R. GUY and J. M. EISENBERG
In the p r e s e n t work, we shall investigate the r e s c a t t e r i n g c o r r e c t i o n s which a r i s e f r o m the situation in which the pion is s c a t t e r e d by a nucleon b e f o r e being r a d i a t i v e l y absorbed. The r e s c a t t e r i n g i n t e r a c t i o n s , which a r e q u a d r a t i c in the pion field, may be seen to e m e r g e f r o m an application of the Dyson t r a n s f o r m a t i o n [6] to the Hamiltonian for p s e u d o s c a l a r m e s o n s with p s e u d o s c a l a r coupling [7]. They w e r e employed by Woodruff [8] in his calculation of the p r o c e s s p+p--d+~+
,
(1)
and l a t e r by Koltun and Reitan [9] who showed that the inclusion of such r e s c a t t e r i n g t e r m s gave a g r e e m e n t with e x p e r i m e n t for the production of swave pions in p r o c e s s (1). In subsequent work [10, 11], t r e a t i n g the a b s o r p tion of S orbit pions followed by the ejection of two nucleons, Koltun and Reitan pointed out that the s - w a v e r e s c a t t e r i n g may play an important p a r t in this reaction. Specifically, these i n t e r a c t i o n s s t r o n g l y enhance the nn/np ratio, p a r t i c u l a r l y f o r l a r g e r e l a t i v e m o m e n t a and opening angles. Calculations of r a d i a t i v e pion a b s o r p t i o n including r e s c a t t e r i n g effects have been c a r r i e d out in d e u t e r i u m [12], and we anticipate that these t e r m s m a y be of c o n s i d e r a b l e significance for radiative a b s o r p t i o n in m o r e c o m plex nuclei. That this is plausible may be seen f r o m a study of the e n e r g y level shifts in pionic a t o m s . It is well known that the s t r o n g interaction between the nucleus and a pion bound in a 1S a t o m i c orbit is repulsive. F r o m the e x p e r i m e n t a l pionic level shifts and widths, one may c o n s t r u c t an optical potential, the r e a l p a r t of which accounts for this r e p u l s i o n and leads to a s u p p r e s s i o n (by about 40%) of the probability of finding the pion in the nucleus [ 13- 15] and a consequent r e d u c t i o n of the radiative a b s o r p t i o n rate. We expect to see a s i m i l a r r e s u l t e m e r g e f r o m a c o n s i d e r a t i o n of the pion s - w a v e r e s c a t t e r i n g i n t e r a c t i o n s . In fact, the calculation of r e s c a t t e r i n g as p r e s e n t e d h e r e will be effectively equivalent to the optical potential app r o a c h p r o v i d e d that i n c o h e r e n t r e s c a t t e r i n g , in which the nucleus is excited during the r e s c a t t e r i n g p r o c e s s , is negligible. We shall t h e r e f o r e exa m i n e the i n c o h e r e n t r e s c a t t e r i n g in s o m e detail, and shall also make explicit the f o r m a l r e l a t i o n s h i p between the r e s c a t t e r i n g and the optical potential a p p r o a c h e s .
2. RESCATTERING FORMALISM We write the i n t e r a c t i o n density * for the leading t e r m in radiative pion a b s o r p t i o n as [7]
w h e r e a j is the spin o p e r a t o r for the jth nucleon at position x j, ~'~ is the nucleon isospin lowering o p e r a t o r , and ~+ is the pion field o p e r a t o r which c r e a t e s a ~+ or annihilates a 7r-. The conventional s p h e r i c a l b a s i s in iso* Units are used such that ~ = c = 1.
603
RESCATTERING CORRECTIONS spin s p a c e has been chosen, so that ~-± = a= ½f2(~-l+iT2), ~b±, M is the nucleon m a s s , and we take g2/(47r) = 15.9 r e s c a t t e r i n g i n t e r a c t i o n densities, whose contributions r e c t i o n s to the leading t e r m a r e taken in the f o r m used tan [9]:
and s i m i l a r l y f o r and e 2 = T~. 1 The will constitute c o r by Koltun and Rei-
H 1(x j) = - 47T;~1 P- 1[ qS_(Xj) dO+(.Xj) + dp+(Xj) el)_( Xj)] ,
(3)
H 2 (x j) = 41Ti;~2 p - 2-tO[4)- (xj)Tr+(Xj) - (b+(xj)~_ (Xj)] .
(4)
and
H e r e /x is the pion m a s s , and ~+(x) a r e the conjugate field o p e r a t o r s , that is ~r± = qS+. The coefficients Xl and ~2 a r e not those that a p p e a r in the Dyson reduction of the p s e u d o s c a l a r i n t e r a c t i o n Hamiltonian. Instead, they a r e obtained f r o m the m e a s u r e d s - w a v e p h a s e shifts for pion-nucleon s c a t t e r i n g and have the values ~1 = 0.005 and ~2 = 0.045. The i n t e r a c t i o n s in eqs. (3) and (4) shall be taken in n o r m a l o r d e r e d f o r m . In H2, we have cons i d e r e d only that p a r t of the i n t e r a c t i o n which does not involve pion c h a r g e exchange. F o r the a b s o r p t i o n of a ~-, the inclusion of such effects would r e q u i r e one to c o n s i d e r the p r o c e s s of r e s c a t t e r i n g followed by radiative a b s o r p t i o n in t h i r d o r d e r . In addition, since the c r o s s section for the photoproduction of ~o n e a r t h r e s h o l d is small, we neglect these t e r m s . In p e r t u r b a t i o n t h e o r y , the amplitude Tfi for the r a d i a t i v e a b s o r p t i o n of a 7r- on a nucleus containing A nucleons, including r e s c a t t e r i n g , is given by A Tfi = ( f ; k p ] ~ i=1
H3(xi)]i;q) A (f;kp f ~
+ ~ ( dq'
j=l
H3(xj) ] n;q')
n a (2~r)3
(f;kp] +
A ~
H1,2(x'O]i;q) i=l E i + wq - E n - wq, + ie
A ~ H1,2(x,i)]n;q,)(n;q' i=1
A ] ~ H3(xj)]i;q) j=l
El+
Wq- (En+ Wq, +wq+k)+i¢
(5)
w h e r e i, n, and f a r e the initial, i n t e r m e d i a t e , and final n u c l e a r states; k is the m o m e n t u m of the photon which has p o l a r i z a t i o n p; and Wq = (q2+p2)½ and ¢Oq, a r e the e n e r g i e s of the initial and i n t e r m e d i a t e negative m e s o n s . The f i r s t - and s e c o n d - o r d e r t e r m s which a p p e a r in eq. (5) a r e shown diag r a m m a t i c a l l y in fig. 1. The t r a n s i t i o n r a t e is given by
w-- 2,, Z; / ] Tfi]2 p(E), spin
(6)
604
R. GUY and J . M. EISENBERG
n
N H1,
~'-
.-"
T'['..,-
.-'1 W'= /
I
P
n
N
N
~
n
H1,
I I ~-"
P
/
N
I-~
-,'"
P
Ca) Cb) Co) F i g . 1. D i a g r a m s f o r the t e r m s c o n s i d e r e d in eq. (5). The s y m b o l N r e f e r s e i t h e r to n e u t r o n (n) o r p r o t o n (p).
w h e r e p(E) is the usual density of plane wave s t a t e s f o r the photon, and the spin s u m includes an a v e r a g e o v e r initial and s u m o v e r final n u c l e a r m a g netic p r o j e c t i o n quantum n u m b e r s , and a s u m o v e r the p o l a r i z a t i o n s t a t e s of the photon. The d i r e c t i o n of the outgoing photon is then, of c o u r s e , i r r e l e v a n t , and is to be i n t e g r a t e d o v e r in eq. (6). We c o n s i d e r that the bound 1S pion is a b s o r b e d f r o m r e s t and neglect the v a r i a t i o n of its wave function within the n u c l e a r volume; f o r it we thus u s e the a p p r o p r i a t e h y d r o g e n i c wave function e v a l u a t e d at the origin. The evaluation of the m a t r i x e l e m e n t s of the pion field o p e r a t o r s in eq. (5) is a c c o m p l i s h e d by f i r s t noting that ( q , ±[ q)+(x) [0) = :F(2eOq)-½ e- i q . x ( q , . l ~ + ( x ) 10) = ~i(½oOq)½ e- i q . x
,
(7)
w h e r e the o p e r a t o r s a r e t a k e n between the pion v a c u u m and a s t a t e of one m e s o n having m o m e n t u m q and c h a r g e +e. T h e s e plane w a v e s , t o g e t h e r with that of the t r a n s v e r s e photon, m a y then be expanded a c c o r d i n g to e~q •
r
= 4~ ~
iZjl(qr)Y~m(~)Ylm(~)--
,
(8)
and ~pe
ik" r
1
= -P(4~r) ~
X^ L M ~ i kj•(kr)C(1XL;pO) D M p ~ k ) T L ; k ( ~ ) . kML
(9)
H e r e A = (2A + 1)½, and the a n g u l a r m o m e n t u m conventions and notations a r e t h o s e of R o s e [16]. Substitution of eqs. (8) and (9), into eq. (5) and the u s e of r e c o u p l i n g t e c h n i q u e s to help identify i r r e d u c i b l e t e n s o r o p e r a t o r s in the n u c l e a r s p a e e l e a d s to an e x p r e s s i o n f o r the t r a n s i t i o n amplitude. Since it will l a t e r t u r n out that the m a i n c o n t r i b u t i o n s to the t r a n s i t i o n r a t e c o m e f r o m d i a g r a m s (a), and (b) with H1, in,fig. 1, only t h e s e t e r m s a r e exhibited h e r e . We then have
RESCATTERING CORRECTIONS
Tfi ~
, (z3 ½
ie ~ M ) (2/z)-2 ~ - ~ J
XML
k/
[-P(4~')2]
L i)~C(1)~L ;pO) D Mp (k) {(f]
A
M
i=l +
605
~
-
jx(kxi)ai" T L;x(xi)'r i Ji)
4~ 1
! ~ ~ "-~*" p~2 l,m .~,f
"ALt(-)L-X+~' -fC 00) (XIl'~;
× C(Llf; Mm)W(1L.C1;~f) ~ f f dq'q'2 n o Wq,[-AEn+tl- ¢Oq,] A M+m . . . . A ~ a i. T;;.~ (xi)3)~(kxi)31(q'xi)ri [ n ) ~ l ~ Ylm(Xj)3l(q*" i=1 j=l
X
" ' 'xj)Ji)} .
(10)
In this equation Z is the a t o m i c n u m b e r , a~ is the Bohr r a d i u s of the pionic a t o m (~ 193 fm), and AE n = IEi- E n ] is the excitation e n e r g y of the nucleus in the i n t e r m e d i a t e s t a t e n. The n u m e r i c a l evaluation of eq. (10) for p a r t i c u l a r n u c l e a r s t a t e s will be c o n s i d e r e d in s e c t . 4, w h e r e we shall s e e t h a t the m a i n contribution to the c o r r e c t i o n t e r m s a r i s e s f r o m c o h e r e n t r e s c a t t e r i n g , n=i, in eq. (10). T h i s m a k e s the r e s c a t t e r i n g a p p r o a c h e s s e n t i a l l y equivalent to that b a s e d on the optical potential (see s e c t . 3). It a l s o allows f o r a c o n s i d e r a b l e s i m p l i f i c a tion in evaluating t h e s e t e r m s since then, at l e a s t f o r ¢0q' = ~, the l a s t m a t r i x e l e m e n t in eq. (10) is s i m p l y r e l a t e d to the s c a t t e r i n g length f o r a pion incident on the c o m p o s i t e nucleus. T h i s in t u r n can be d e t e r m i n e d f r o m the m e a s u r e d e n e r g y shifts of the 1S l e v e l s in pionic a t o m s [14]. Of c o u r s e , one is then f a c e d with a m b i g u i t i e s in extending this r e s u l t off the e n e r g y shell, Wq, > p. T h e r e a r e two n a t u r a l w a y s to r e s o l v e this difficulty. The f i r s t of t h e s e is to n o r m a l i z e the r e s c a t t e r i n g m a t r i x e l e m e n t to the e x p e r i m e n t a l value at ¢Oq, = p and r e t a i n the dependence on q' given by p e r t u r b a tion t h e o r y a s in eq. (10). The second p r o c e d u r e c o n s i s t s in r e p l a c i n g this m a t r i x e l e m e n t with a constant equal to its o b s e r v e d value at z e r o kinetic e n e r g y . As we shall s e e in sect. 4, t h e s e two a l t e r n a t i v e s lead to r a t h e r s i m i l a r r e s u l t s f o r the c o r r e c t i o n t e r m s . Retaining only the dominant c o h e r e n t r e s c a t t e r i n g , eq. (10) s i m p l i f i e s to 1
Tfi ~
-ie ~ M ) ( Z--~33)-~P(2u)(Pk)-½ ~ iX~tC(l~tgf; pO) na~:
× gf
DMfp(k)
{(fl
X A
i=l
3x(kxi)oi'rMf(xi) ~f;
Mf . . . . ai'Tjf,(xi)jx(kxi)Jo(q'xi)~'i i=l •
~
i
_
l i)
A
A
×(fl ~
{i)(il~
+2kl
j=l
~l~ o
dq'q'2
Wq,(p- Wq,)
jo(q'xj')]i)}.
(11)
606
R. GUY and J. M. EISENBERG
In this equation, we note that the H 3 m a t r i x e l e m e n t is the s a m e in the c o r r e c t i o n t e r m a s in the leading t e r m , a p a r t f r o m the f a c t o r jo(q'xi). If this f a c t o r is r e p l a c e d by jo(~'~) we m a y w r i t e eq. (11) as 3 1 i
A i=1
× {1-
2X 1
of
co
A
dq'q'2 ~) (i] ~ jo(q'xj')]i)} . ¢Oq,(Wq,j=l
(12)
H e r e q' ~ 0.85 f m - 1 is the a p p r o x i m a t e p e a k in the q' integrand and ~ is t a k e n to be 1/a ~ 1 . 6 f m , w h e r e a is the u s u a l r a n g e p a r a m e t e r in the shell model h a r m o n i c o s c i l l a t o r r a d i a l wave function. Choosing the f i r s t of the two a l t e r n a t i v e s d e s c r i b e d in the p r e c e d i n g p a r a g r a p h for extending the p i o n - n u c l e u s s c a t t e r i n g length off the e n e r g y shell, the effect of the r e s c a t t e r i n g c o r r e c t i o n s is contained in an a p p r o x i m a t e m u l t i p l i c a t i v e f a c t o r which is to be applied to the leading t e r m in the t r a n s i t i o n amplitude. T h i s f a c t o r has the s i m p l e f o r m
a 1 - ~
? Jo(q'x--)
o
dq, q,2 - -
-
A
Wq,(COq,- p)
w h e r e a is the p i o n - n u c l e u s s c a t t e r i n g c a l c u l a t i o n of the n u c l e a r ground s t a t e s t r a i g h t f o r w a r d . F o r e x a m p l e , taking by c l o s e d s and p shells, we can w r i t e
(i[ E
j=l
jo(q'xj')[i
,
(13)
length. Using the shell model, the expectation value is e x t r e m e l y the 160 ground s t a t e to be d e s c r i b e d this r e s c a t t e r i n g c o r r e c t i o n f a c t o r
as
[ 1 - 0.76 ~ aJo(qX-)] .
(14)
F o r 160, jo(~,/a) z 0.7 and (ref. [14]) a ~ 0 . 4 / p . The s i g n i f i c a n c e of the m i n u s sign in this equation will be d i s c u s s e d in sect. 4.
3. RELATIONSHIP TO O P T I C A L P O T E N T I A L We c o n s i d e r the c a s e of a pion subject to an optical potential Vopt. To f i r s t o r d e r , this potential is given by [17] A (1) = ~
V°pt
Vi "
(15)
i= 1
H e r e Vi is the p i o n - n u c l e o n i n t e r a c t i o n which we take to given by eq. (3) f o r c o n s i s t e n c y with our r e s c a t t e r i n g a p p r o a c h . Then to f i r s t o r d e r , the dist o r t e d pion s t a t e [ ~ ' ) is given in t e r m s of the u n p e r t u r b e d s t a t e I~) by
RESCATTERING CORRECTIONS V(1)
607
,\
°pt]v/ [¢,> = I~) + ~ ( ~ ] x E~ - E x
IX),
(16)
w h e r e IX) denotes an element of the complete set of u n p e r t u r b e d pion states. The amplitude Tfi for r a d i a t i v e pion a b s o r p t i o n may now be written A Tfi = ( f ; k p [ ~ i=1
H3(xi)]i;~')
A
= q;kp] D H3(xi)]i;~) i=1 A (f;kp[ ~
+~
i=1
H3(xi) ] i;~)(i;X I
A ~ i=1
Hl(Xi) I i;~) ,
(17)
E~/- E x
in a notation s i m i l a r to that of eq. (5). Of c o u r s e , in eq. (17) the i n t e r m e diate n u c l e a r state is the ground state, w h e r e a s in our s i n g l e - r e s c a t t e r i n g a p p r o a c h the amplitude of eq. (5) contains a sum over i n t e r m e d i a t e n u c l e a r s t a t e s n. F o r the c a s e in which the i n c o h e r e n t r e s c a t t e r i n g effects a r e negligible, we m a y put n= i in eq. (5), and we thus r e t u r n to the f i r s t - o r d e r optical potential r e s u l t . In c o n s t r a s t to optical models, our r e s c a t t e r i n g a p p r o a c h enables us to evaluate d i r e c t l y the incoherent contributions. Since we u l t i m a t e l y find them to be small, we expect good a g r e e m e n t with the r e suits b a s e d on an optical potential.
4. RESULTS We shall c o n s i d e r as an explicit example the c a s e of pion radiative abs o r p t i o n f r o m the 1S a t o m i c orbit in 160. In deciding which n u c l e a r states to c o n s i d e r , we a r e guided by e a r l i e r work [2, 18], which showed that for 1S pion a b s o r p t i o n the s p i n - i s o s p i n m o d e s of excitation in the giant r e s o nance r e g i o n with J~ = 1-, 2-; T = 1 contribute a p p r o x i m a t e l y 80% to the total radiative a b s o r p t i o n rate. The p a r t i c l e - h o l e f o r m a l i s m of Gillet and Vinh Mau [19] with configuration mixed wave functions was used for the nuc l e a r m a t r i x elements. H a r m o n i c o s c i l l a t o r radial wave functions were e m ployed, with the r a n g e p a r a m e t e r a taken to be 0.629 f m - 1 . The n u c l e a r r a dial i n t e g r a l s in eq. (10) w e r e then evaluated analytically and the r e m a i n i n g i n t e g r a l over the m o m e n t u m of the pion in the i n t e r m e d i a t e state was p e r formed numerically. In spite of the fact that X2 is a p p r o x i m a t e l y ten t i m e s l a r g e r than ~1, H2 does not p r o d u c e l a r g e r r e s c a t t e r i n g effects than H 1. Indeed, employing c l o s u r e f o r the i n t e r m e d i a t e n u c l e a r s t a t e s , the contribution f r o m the nonc h a r g e - e x c h a n g e p a r t of H 2 can be seen to be s m a l l e r by at l e a s t two o r d e r s
608
R. GUY and J. M. EISENBERG
of m a g n i t u d e t h a n t h e t r a n s i t i o n r a t e g i v e n b y t h e l e a d i n g t e r m in eq. (10). A s i m i l a r s i t u a t i o n o b t a i n s if one c o n s i d e r s t h e i n t e r a c t i o n H I f o r t h e c a s e in w h i c h t h e i n t e r m e d i a t e s t a t e c o n t a i n s two p i o n s , c o r r e s p o n d i n g to (c) in fig. 1. In f a c t , t h e m a j o r c o r r e c t i v e e f f e c t c o m e s f r o m t h e c o h e r e n t r e s c a t t e r i n g in w h i c h t h e n u c l e u s r e m a i n s u n e x c i t e d in t h e i n t e r m e d i a t e s t a t e ( n = i ) . T h e e v a l u a t i o n of t h e s e c o n d t e r m in eq. (11), c o r r e s p o n d i n g to (b) w i t h H 1 a c t i n g in fig. 1, l e a d s to a d e c r e a s e in t h e t r a n s i t i o n r a t e of a p p r o x i m a t e l y 15% r e l a t i v e t o t h e f i r s t t e r m in eq. (11) f o r t h e m o s t i m p o r t a n t 1-; T = 1 f i n a l s t a t e . I n v o k i n g c l o s u r e f o r t h e i n t e r m e d i a t e n u c l e a r s t a t e s in eq. (10), w e a r e a b l e t o e s t a b l i s h t h a t t h e c o h e r e n t r e s c a t t e r i n g e f f e c t i s a b o u t one o r d e r of magnitude more significant than the incoherent effects which involve the int e r m e d i a t e n u c l e u s in a l o w - e n e r g y e x c i t a t i o n * . T h e c o n t r i b u t i o n to t h e t r a n s i t i o n r a t e of t h e i n c o h e r e n t r e s c a t t e r i n g in t h e c l o s u r e a p p r o x i m a t i o n i s s h o w n in t a b l e 1 f o r t h e 1- a n d 2 - , T= 1, l e v e l s in t h e A = 16 s y s t e m . The fact that H 2 was found to be insignificant is plausible, since, for 160, t h i s r e s c a t t e r i n g i n t e r a c t i o n c a n n o t a l l o w t h e i n t e r m e d i a t e n u c l e a r s t a t e t o b e e q u a l t o t h e g r o u n d s t a t e d u e t o t h e p r e s e n c e of t h e i s o s p i n o p e r a t o r . If, a s o u t l i n e d in s e c t . 2, w e n o r m a l i z e t h e n u c l e a r r e s c a t t e r i n g m a t r i x e l e m e n t t o o b t a i n a g r e e m e n t a t q ' = 0 w i t h t h e p i o n - 160 s c a t t e r i n g l e n g t h a = 0.56 f m , w e f i n d t h a t t h e c r o s s t e r m in eq. (11) c o n t r i b u t e s a d e c r e a s e of a b o u t 36% r e l a t i v e t o t h e l e a d i n g t e r m , a s w o u l d b e e x p e c t e d f r o m eq. (14). T h i s r e f l e c t s a n i n c r e a s e in t h e n u c l e a r m a t r i x e l e m e n t o v e r t h a t c a l c u l a t e d f r o m eq. (11) in p e r t u r b a t i o n t h e o r y b y a f a c t o r of 2.5. T h i s e f f e c t i s w e l l k n o w n in t h e c o n s t r u c t i o n of p i o n - n u c l e u s o p t i c a l p o t e n t i a l s [13]. It a r i s e s f r o m t h e n e a r - c a n c e l l a t i o n of t h e n e u t r o n a n d p r o t o n r e s c a t t e r i n g c o n t r i b u t i o n s , w h i c h in t u r n m a k e s h i g h e r - o r d e r m u l t i p l e s c a t t e r i n g effects very important. Table 1 Contributions to the transition rate from incoherent r e s c a t t e r i n g for 160. j?T
Energy (MeV)
Leading t e r m (1015 s e c _ l )
Incoherent r e s c a t t e r i n g (1015 sec_l)
11111-
25.4 22.7 19.6 18.1 13.6
150 69 31 47 17
4.0 2.9 0.3 0.8 0.4
22222-
23.7 20.2 19.1 17.7 13.0
81 250 83 2 155
1.0 1.4 7.2 0.2 5.9
The excitation energies a r e given relative to the ground state of 160. The transition r a t e s for the leading t e r m a r e calculated from the interaction in eq. (2), and the r e s c a t t e r i n g is computed after normalization to the observed nuclear s c a t t e r i n g length. * A s i m i l a r r e s u l t has been established for pion-nucleus charge exchange s c a t t e r ing [20].
RESCATTERING CORRECTIONS
609
W e n o t e t h a t w h e n t h e c o m p l e t e r e s c a t t e r i n g m a t r i x e l e m e n t in eq. (11) is replaced by a constant ~--nucleus scattering length, there occurs a dec r e a s e d u e t o H 1 of a p p r o x i m a t e l y 45%. S u c h r e s u l t s b a s e d on eq. (11), f o r t h e 1- ; T = 1 g i a n t r e s o n a n c e f i n a l s t a t e s in t h e A = 16 s y s t e m a r e p r e s e n t e d in fig. 2. T h e s a m e c a l c u l a t i o n s w e r e p e r f o r m e d f o r t h e 2 - ; T = 1 f i n a l s t a t e s a n d t h e s e r e s u l t s a p p e a r in fig. 3. In a l l c a l c u l a t i o n s , o n l y t h e l e a d i n g t e r m a n d t h e c r o s s t e r m in eq. (11) w e r e r e t a i n e d . It i s i m p o r t a n t t o n o t e t h a t t h e r e s c a t t e r i n g i n t e r a c t i o n s do n o t s i g n i f i c a n t l y a l t e r t h e d i s t r i b u t i o n of s t r e n g t h a m o n g t h e v a r i o u s 1- a n d 2- l e v e l s .
"T u
"3" 14
%
CO
~ 0 1,~ ~) "-' 1(:
¢._
._(2
8! L 0
6
<
4
0 in .Q
I I I I
I I
2 0 10
i
ii,
15
ii L ,
20
I I I I
<
II,
I .
25
E(MeV)
Fig. 2. Modifications in the rate of r a diative pion absorption in 160 leading to 1-; T = I final nuclear states of energy E r e l a t i v e to the 160 ground state. The dashed lines r e f e r to the contributions of the leading t e r m as given by eq. (2), and the solid lines a r e the r e s u l t s including the r e s c a t tering c o r r e c t i o n s .
0 10
15
20
25
E(MeV)
Fig. 3. Modifications due to r e s c a t t e r ing c o r r e c t i o n s for 2-; T = 1 levels in the A = 16 s y s t e m (see caption to fig. 2). The level at 19.1MeV is not shown, since its strength is l e s s than 1%.
T h e d e c r e a s e in t h e t r a n s i t i o n r a t e due t o r e s c a t t e r i n g c o r r e s p o n d s to t h e d e c r e a s e in t h e p r o b a b i l i t y of f i n d i n g t h e p i o n in t h e n u c l e u s a s c a l c u l a t e d u s i n g a n o p t i c a l p o t e n t i a l [14, 15]. T h e r e s u l t s of t h e c a l c u l a t i o n p r e s e n t e d h e r e i n d i c a t e t h a t t h e m o d i f i c a t i o n s of t h e r a d i a t i v e p i o n a b s o r p t i o n i n t e r a c t i o n c a l c u l a t e d f r o m r e s c a t t e r i n g a r e in c l o s e a c c o r d w i t h t h e a l t e r native prescription given by the optical potential approach, since incoherent rescattering effects are small.
610
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It i s a p l e a s u r e to a c k n o w l e d g e u s e f u l c o m m u n i c a t i o n s with D r s . M. E r i c s o n , T . E . O . E r i c s o n , L. P. F u l c h e r , D. S. K o l t u n a n d H. P i e t s c h m a n n . A U n i v e r s i t y of V i r g i n i a C o m p u t i n g G r a n t is a l s o g r a t e f u l l y a c k n o w l e d g e d . O n e of u s (R.G.) w o u l d l i k e to e x p r e s s h i s t h a n k s to t h e N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n for fellowship support.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
J. Delorme and T . E . O . Ericson, Phys. Letters 21 (1966) 98. D. K. Anderson and J. M. Eisenberg, Phys. Letters 22 {1966) 164. L. L. Foldy and J. D. Walecka, Nuovo Cimento 34 (1964) 1026. V. Evseyev et al., Phys. Letters 28B (1969) 553. H. Davies, H. Muirhead and J. M. Woulds, Nucl. Phys. 78 (1966) 673. F. J. Dyson, Phys. Rev. 73 (1948) 929. S. D. Drell and E. M. Henley, Phys. Rev. 88 {1952) 1053. A. E. Woodruff, Phys. Rev. 117 {1960) 1113. D.. S. Koltun and A. Reitan, Phys. Rev. 141 {1966} 1413. D.S. Koltun and A. Reitan, Phys. Rev. 155 (1967) 1139. D.S. Koltun and A. Reitan, Nucl, Phys. B4 (1968) 629. A. Reitan, Nucl. Phys. 87 (1966) 232. M. E r i c s o n and T. E. O. Ericson, Ann. of Phys. 36 (1966) 323. R. Seki and A.H. Cromer, Phys. Rev. 156 (1967) 93. L . P . Fulcher, J.M. Eisenberg and J. LeTourneux, Can. J. Phys. 45 {1967) 3313. M . E . R o s e , Multipole fields (John Wiley and Sons, New York, 1955). M. L. Goldberger and K. M. Watson, Collision theory (John Wiley and Sons, New York, 1964). [18] D.K. Anderson, thesis, University of Virginia (1966). [19] V. Gillet and N. Vinh Mau, Nucl° Phys. 54 (1964) 321; 57 {1964) 698. [20] M. Ericson, A. Figureau and A. Molinari, Nucl. Phys. B10 {1969) 501.
Nuclear Physics B l l (1969) 601-610. North-Holland Publ. Comp., Amsterdam
RESCATTERING CORRECTIONS IN RADIATIVE PION ABSORPTION R e e d GUY a n d J. M. E I S E N B E R G
Department of Physics, University of Virginia, Charlottesville, Virginia* Received 27 May 1969 Abstract: Rescattering corrections to radiative pion absorption in nuclei are considered. The r e s c a t t e r i n g process is known to make important contributions to the nuclear absorption of S orbit pions with two-nucleon ejection. We calculate the r e s c a t t e r i n g corrections both in perturbation theory and with an approach based on the actual pion-nucleus scattering length. For the detailed evaluation of n u clear m a t r i x elements the particle-hole f o r m a l i s m is used, and it is assumed that odd-parity spin-isospin nuclear states are excited by the radiative absorption mechanism. The calculation shows that incoherent r e s c a t t e r i n g effects i n volving an excited nucleus in the intermediate state are of little importance. Retaining only the coherent rescattering, decreases are found in the transition rate of approximately 40% relative to the leading t e r m for 160. The relationship between our calculation and analyses based on the use of an optical potential is presented.
1. I N T R O D U C T I O N T h e n u c l e a r a b s o r p t i o n of a n e g a t i v e p i o n b o u n d i n a n a t o m i c o r b i t a l u s u a l l y l e a d s to the e j e c t i o n of two n u c l e o n s . A s m a l l p e r c e n t a g e of the t i m e i n c o m p l e x n u c l e i , h o w e v e r , a high e n e r g y p h o t o n i s e m i t t e d ; t h i s i s t h e p r o c e s s k n o w n a s r a d i a t i v e p i o n a b s o r p t i o n . I n r e c e n t y e a r s , m u c h of the i n t e r e s t i n t h i s r e a c t i o n h a s c e n t e r e d on i t s c o n n e c t i o n with m u o n c a p t u r e . D e m o n s t r a t i o n s [1, 2] of the u s e f u l n e s s of t h i s r e l a t i o n s h i p f o r c o m p l e x n u c l e i m a d e u s e of t h e i m p u l s e a p p r o x i m a t i o n i n o r d e r to i d e n t i f y t h e r a d i a t i v e a b s o r p t i o n o p e r a t o r with t h e a x i a l - v e c t o r c u r r e n t o p e r a t o r of m u o n c a p t u r e . T h e s e d i s c u s s i o n s e m p h a s i z e d t h a t r a d i a t i v e p i o n a b s o r p t i o n l e a d s to t h e e x c i t a t i o n of s p i n - i s o s p i n c o l l e c t i v e s t a t e s in n u c l e i . Such s t a t e s a r e k n o w n to b e of i m p o r t a n c e i n d e t e r m i n i n g m u o n c a p t u r e r a t e s [3]. E v i d e n c e f o r t h e i r r o l e i n t h i s p r o c e s s , a s w e l l a s the r o l e of t h e i s o s p i n c o l l e c t i v e s t a t e s , h a s a p p e a r e d in m e a s u r e m e n t s of the s p e c t r a of o u t g o i n g n e u t r o n s f o l l o w i n g m u o n c a p t u r e [4]. T h e s t u d y of the s p i n - i s o s p i n s t a t e s c a n b e c a r r i e d out m o r e d i r e c t l y in t h e r a d i a t i v e p i o n a b s o r p t i o n r e a c t i o n , s i n c e t h e r e t h e s p e c t r u m of t h e o u t g o i n g p h o t o n i s e x p e r i m e n t a l l y a c c e s s i b l e [5]. * Work supported in part by the National Science Foundation.
602
R. GUY and J. M. EISENBERG
In the p r e s e n t work, we shall investigate the r e s c a t t e r i n g c o r r e c t i o n s which a r i s e f r o m the situation in which the pion is s c a t t e r e d by a nucleon b e f o r e being r a d i a t i v e l y absorbed. The r e s c a t t e r i n g i n t e r a c t i o n s , which a r e q u a d r a t i c in the pion field, may be seen to e m e r g e f r o m an application of the Dyson t r a n s f o r m a t i o n [6] to the Hamiltonian for p s e u d o s c a l a r m e s o n s with p s e u d o s c a l a r coupling [7]. They w e r e employed by Woodruff [8] in his calculation of the p r o c e s s p+p--d+~+
,
(1)
and l a t e r by Koltun and Reitan [9] who showed that the inclusion of such r e s c a t t e r i n g t e r m s gave a g r e e m e n t with e x p e r i m e n t for the production of swave pions in p r o c e s s (1). In subsequent work [10, 11], t r e a t i n g the a b s o r p tion of S orbit pions followed by the ejection of two nucleons, Koltun and Reitan pointed out that the s - w a v e r e s c a t t e r i n g may play an important p a r t in this reaction. Specifically, these i n t e r a c t i o n s s t r o n g l y enhance the nn/np ratio, p a r t i c u l a r l y f o r l a r g e r e l a t i v e m o m e n t a and opening angles. Calculations of r a d i a t i v e pion a b s o r p t i o n including r e s c a t t e r i n g effects have been c a r r i e d out in d e u t e r i u m [12], and we anticipate that these t e r m s m a y be of c o n s i d e r a b l e significance for radiative a b s o r p t i o n in m o r e c o m plex nuclei. That this is plausible may be seen f r o m a study of the e n e r g y level shifts in pionic a t o m s . It is well known that the s t r o n g interaction between the nucleus and a pion bound in a 1S a t o m i c orbit is repulsive. F r o m the e x p e r i m e n t a l pionic level shifts and widths, one may c o n s t r u c t an optical potential, the r e a l p a r t of which accounts for this r e p u l s i o n and leads to a s u p p r e s s i o n (by about 40%) of the probability of finding the pion in the nucleus [ 13- 15] and a consequent r e d u c t i o n of the radiative a b s o r p t i o n rate. We expect to see a s i m i l a r r e s u l t e m e r g e f r o m a c o n s i d e r a t i o n of the pion s - w a v e r e s c a t t e r i n g i n t e r a c t i o n s . In fact, the calculation of r e s c a t t e r i n g as p r e s e n t e d h e r e will be effectively equivalent to the optical potential app r o a c h p r o v i d e d that i n c o h e r e n t r e s c a t t e r i n g , in which the nucleus is excited during the r e s c a t t e r i n g p r o c e s s , is negligible. We shall t h e r e f o r e exa m i n e the i n c o h e r e n t r e s c a t t e r i n g in s o m e detail, and shall also make explicit the f o r m a l r e l a t i o n s h i p between the r e s c a t t e r i n g and the optical potential a p p r o a c h e s .
2. RESCATTERING FORMALISM We write the i n t e r a c t i o n density * for the leading t e r m in radiative pion a b s o r p t i o n as [7]
w h e r e a j is the spin o p e r a t o r for the jth nucleon at position x j, ~'~ is the nucleon isospin lowering o p e r a t o r , and ~+ is the pion field o p e r a t o r which c r e a t e s a ~+ or annihilates a 7r-. The conventional s p h e r i c a l b a s i s in iso* Units are used such that ~ = c = 1.
603
RESCATTERING CORRECTIONS spin s p a c e has been chosen, so that ~-± = a= ½f2(~-l+iT2), ~b±, M is the nucleon m a s s , and we take g2/(47r) = 15.9 r e s c a t t e r i n g i n t e r a c t i o n densities, whose contributions r e c t i o n s to the leading t e r m a r e taken in the f o r m used tan [9]:
and s i m i l a r l y f o r and e 2 = T~. 1 The will constitute c o r by Koltun and Rei-
H 1(x j) = - 47T;~1 P- 1[ qS_(Xj) dO+(.Xj) + dp+(Xj) el)_( Xj)] ,
(3)
H 2 (x j) = 41Ti;~2 p - 2-tO[4)- (xj)Tr+(Xj) - (b+(xj)~_ (Xj)] .
(4)
and
H e r e /x is the pion m a s s , and ~+(x) a r e the conjugate field o p e r a t o r s , that is ~r± = qS+. The coefficients Xl and ~2 a r e not those that a p p e a r in the Dyson reduction of the p s e u d o s c a l a r i n t e r a c t i o n Hamiltonian. Instead, they a r e obtained f r o m the m e a s u r e d s - w a v e p h a s e shifts for pion-nucleon s c a t t e r i n g and have the values ~1 = 0.005 and ~2 = 0.045. The i n t e r a c t i o n s in eqs. (3) and (4) shall be taken in n o r m a l o r d e r e d f o r m . In H2, we have cons i d e r e d only that p a r t of the i n t e r a c t i o n which does not involve pion c h a r g e exchange. F o r the a b s o r p t i o n of a ~-, the inclusion of such effects would r e q u i r e one to c o n s i d e r the p r o c e s s of r e s c a t t e r i n g followed by radiative a b s o r p t i o n in t h i r d o r d e r . In addition, since the c r o s s section for the photoproduction of ~o n e a r t h r e s h o l d is small, we neglect these t e r m s . In p e r t u r b a t i o n t h e o r y , the amplitude Tfi for the r a d i a t i v e a b s o r p t i o n of a 7r- on a nucleus containing A nucleons, including r e s c a t t e r i n g , is given by A Tfi = ( f ; k p ] ~ i=1
H3(xi)]i;q) A (f;kp f ~
+ ~ ( dq'
j=l
H3(xj) ] n;q')
n a (2~r)3
(f;kp] +
A ~
H1,2(x'O]i;q) i=l E i + wq - E n - wq, + ie
A ~ H1,2(x,i)]n;q,)(n;q' i=1
A ] ~ H3(xj)]i;q) j=l
El+
Wq- (En+ Wq, +wq+k)+i¢
(5)
w h e r e i, n, and f a r e the initial, i n t e r m e d i a t e , and final n u c l e a r states; k is the m o m e n t u m of the photon which has p o l a r i z a t i o n p; and Wq = (q2+p2)½ and ¢Oq, a r e the e n e r g i e s of the initial and i n t e r m e d i a t e negative m e s o n s . The f i r s t - and s e c o n d - o r d e r t e r m s which a p p e a r in eq. (5) a r e shown diag r a m m a t i c a l l y in fig. 1. The t r a n s i t i o n r a t e is given by
w-- 2,, Z; / ] Tfi]2 p(E), spin
(6)
604
R. GUY and J . M. EISENBERG
n
N H1,
~'-
.-"
T'['..,-
.-'1 W'= /
I
P
n
N
N
~
n
H1,
I I ~-"
P
/
N
I-~
-,'"
P
Ca) Cb) Co) F i g . 1. D i a g r a m s f o r the t e r m s c o n s i d e r e d in eq. (5). The s y m b o l N r e f e r s e i t h e r to n e u t r o n (n) o r p r o t o n (p).
w h e r e p(E) is the usual density of plane wave s t a t e s f o r the photon, and the spin s u m includes an a v e r a g e o v e r initial and s u m o v e r final n u c l e a r m a g netic p r o j e c t i o n quantum n u m b e r s , and a s u m o v e r the p o l a r i z a t i o n s t a t e s of the photon. The d i r e c t i o n of the outgoing photon is then, of c o u r s e , i r r e l e v a n t , and is to be i n t e g r a t e d o v e r in eq. (6). We c o n s i d e r that the bound 1S pion is a b s o r b e d f r o m r e s t and neglect the v a r i a t i o n of its wave function within the n u c l e a r volume; f o r it we thus u s e the a p p r o p r i a t e h y d r o g e n i c wave function e v a l u a t e d at the origin. The evaluation of the m a t r i x e l e m e n t s of the pion field o p e r a t o r s in eq. (5) is a c c o m p l i s h e d by f i r s t noting that ( q , ±[ q)+(x) [0) = :F(2eOq)-½ e- i q . x ( q , . l ~ + ( x ) 10) = ~i(½oOq)½ e- i q . x
,
(7)
w h e r e the o p e r a t o r s a r e t a k e n between the pion v a c u u m and a s t a t e of one m e s o n having m o m e n t u m q and c h a r g e +e. T h e s e plane w a v e s , t o g e t h e r with that of the t r a n s v e r s e photon, m a y then be expanded a c c o r d i n g to e~q •
r
= 4~ ~
iZjl(qr)Y~m(~)Ylm(~)--
,
(8)
and ~pe
ik" r
1
= -P(4~r) ~
X^ L M ~ i kj•(kr)C(1XL;pO) D M p ~ k ) T L ; k ( ~ ) . kML
(9)
H e r e A = (2A + 1)½, and the a n g u l a r m o m e n t u m conventions and notations a r e t h o s e of R o s e [16]. Substitution of eqs. (8) and (9), into eq. (5) and the u s e of r e c o u p l i n g t e c h n i q u e s to help identify i r r e d u c i b l e t e n s o r o p e r a t o r s in the n u c l e a r s p a e e l e a d s to an e x p r e s s i o n f o r the t r a n s i t i o n amplitude. Since it will l a t e r t u r n out that the m a i n c o n t r i b u t i o n s to the t r a n s i t i o n r a t e c o m e f r o m d i a g r a m s (a), and (b) with H1, in,fig. 1, only t h e s e t e r m s a r e exhibited h e r e . We then have
RESCATTERING CORRECTIONS
Tfi ~
, (z3 ½
ie ~ M ) (2/z)-2 ~ - ~ J
XML
k/
[-P(4~')2]
L i)~C(1)~L ;pO) D Mp (k) {(f]
A
M
i=l +
605
~
-
jx(kxi)ai" T L;x(xi)'r i Ji)
4~ 1
! ~ ~ "-~*" p~2 l,m .~,f
"ALt(-)L-X+~' -fC 00) (XIl'~;
× C(Llf; Mm)W(1L.C1;~f) ~ f f dq'q'2 n o Wq,[-AEn+tl- ¢Oq,] A M+m . . . . A ~ a i. T;;.~ (xi)3)~(kxi)31(q'xi)ri [ n ) ~ l ~ Ylm(Xj)3l(q*" i=1 j=l
X
" ' 'xj)Ji)} .
(10)
In this equation Z is the a t o m i c n u m b e r , a~ is the Bohr r a d i u s of the pionic a t o m (~ 193 fm), and AE n = IEi- E n ] is the excitation e n e r g y of the nucleus in the i n t e r m e d i a t e s t a t e n. The n u m e r i c a l evaluation of eq. (10) for p a r t i c u l a r n u c l e a r s t a t e s will be c o n s i d e r e d in s e c t . 4, w h e r e we shall s e e t h a t the m a i n contribution to the c o r r e c t i o n t e r m s a r i s e s f r o m c o h e r e n t r e s c a t t e r i n g , n=i, in eq. (10). T h i s m a k e s the r e s c a t t e r i n g a p p r o a c h e s s e n t i a l l y equivalent to that b a s e d on the optical potential (see s e c t . 3). It a l s o allows f o r a c o n s i d e r a b l e s i m p l i f i c a tion in evaluating t h e s e t e r m s since then, at l e a s t f o r ¢0q' = ~, the l a s t m a t r i x e l e m e n t in eq. (10) is s i m p l y r e l a t e d to the s c a t t e r i n g length f o r a pion incident on the c o m p o s i t e nucleus. T h i s in t u r n can be d e t e r m i n e d f r o m the m e a s u r e d e n e r g y shifts of the 1S l e v e l s in pionic a t o m s [14]. Of c o u r s e , one is then f a c e d with a m b i g u i t i e s in extending this r e s u l t off the e n e r g y shell, Wq, > p. T h e r e a r e two n a t u r a l w a y s to r e s o l v e this difficulty. The f i r s t of t h e s e is to n o r m a l i z e the r e s c a t t e r i n g m a t r i x e l e m e n t to the e x p e r i m e n t a l value at ¢Oq, = p and r e t a i n the dependence on q' given by p e r t u r b a tion t h e o r y a s in eq. (10). The second p r o c e d u r e c o n s i s t s in r e p l a c i n g this m a t r i x e l e m e n t with a constant equal to its o b s e r v e d value at z e r o kinetic e n e r g y . As we shall s e e in sect. 4, t h e s e two a l t e r n a t i v e s lead to r a t h e r s i m i l a r r e s u l t s f o r the c o r r e c t i o n t e r m s . Retaining only the dominant c o h e r e n t r e s c a t t e r i n g , eq. (10) s i m p l i f i e s to 1
Tfi ~
-ie ~ M ) ( Z--~33)-~P(2u)(Pk)-½ ~ iX~tC(l~tgf; pO) na~:
× gf
DMfp(k)
{(fl
X A
i=l
3x(kxi)oi'rMf(xi) ~f;
Mf . . . . ai'Tjf,(xi)jx(kxi)Jo(q'xi)~'i i=l •
~
i
_
l i)
A
A
×(fl ~
{i)(il~
+2kl
j=l
~l~ o
dq'q'2
Wq,(p- Wq,)
jo(q'xj')]i)}.
(11)
606
R. GUY and J. M. EISENBERG
In this equation, we note that the H 3 m a t r i x e l e m e n t is the s a m e in the c o r r e c t i o n t e r m a s in the leading t e r m , a p a r t f r o m the f a c t o r jo(q'xi). If this f a c t o r is r e p l a c e d by jo(~'~) we m a y w r i t e eq. (11) as 3 1 i
A i=1
× {1-
2X 1
of
co
A
dq'q'2 ~) (i] ~ jo(q'xj')]i)} . ¢Oq,(Wq,j=l
(12)
H e r e q' ~ 0.85 f m - 1 is the a p p r o x i m a t e p e a k in the q' integrand and ~ is t a k e n to be 1/a ~ 1 . 6 f m , w h e r e a is the u s u a l r a n g e p a r a m e t e r in the shell model h a r m o n i c o s c i l l a t o r r a d i a l wave function. Choosing the f i r s t of the two a l t e r n a t i v e s d e s c r i b e d in the p r e c e d i n g p a r a g r a p h for extending the p i o n - n u c l e u s s c a t t e r i n g length off the e n e r g y shell, the effect of the r e s c a t t e r i n g c o r r e c t i o n s is contained in an a p p r o x i m a t e m u l t i p l i c a t i v e f a c t o r which is to be applied to the leading t e r m in the t r a n s i t i o n amplitude. T h i s f a c t o r has the s i m p l e f o r m
a 1 - ~
? Jo(q'x--)
o
dq, q,2 - -
-
A
Wq,(COq,- p)
w h e r e a is the p i o n - n u c l e u s s c a t t e r i n g c a l c u l a t i o n of the n u c l e a r ground s t a t e s t r a i g h t f o r w a r d . F o r e x a m p l e , taking by c l o s e d s and p shells, we can w r i t e
(i[ E
j=l
jo(q'xj')[i
,
(13)
length. Using the shell model, the expectation value is e x t r e m e l y the 160 ground s t a t e to be d e s c r i b e d this r e s c a t t e r i n g c o r r e c t i o n f a c t o r
as
[ 1 - 0.76 ~ aJo(qX-)] .
(14)
F o r 160, jo(~,/a) z 0.7 and (ref. [14]) a ~ 0 . 4 / p . The s i g n i f i c a n c e of the m i n u s sign in this equation will be d i s c u s s e d in sect. 4.
3. RELATIONSHIP TO O P T I C A L P O T E N T I A L We c o n s i d e r the c a s e of a pion subject to an optical potential Vopt. To f i r s t o r d e r , this potential is given by [17] A (1) = ~
V°pt
Vi "
(15)
i= 1
H e r e Vi is the p i o n - n u c l e o n i n t e r a c t i o n which we take to given by eq. (3) f o r c o n s i s t e n c y with our r e s c a t t e r i n g a p p r o a c h . Then to f i r s t o r d e r , the dist o r t e d pion s t a t e [ ~ ' ) is given in t e r m s of the u n p e r t u r b e d s t a t e I~) by
RESCATTERING CORRECTIONS V(1)
607
,\
°pt]v/ [¢,> = I~) + ~ ( ~ ] x E~ - E x
IX),
(16)
w h e r e IX) denotes an element of the complete set of u n p e r t u r b e d pion states. The amplitude Tfi for r a d i a t i v e pion a b s o r p t i o n may now be written A Tfi = ( f ; k p [ ~ i=1
H3(xi)]i;~')
A
= q;kp] D H3(xi)]i;~) i=1 A (f;kp[ ~
+~
i=1
H3(xi) ] i;~)(i;X I
A ~ i=1
Hl(Xi) I i;~) ,
(17)
E~/- E x
in a notation s i m i l a r to that of eq. (5). Of c o u r s e , in eq. (17) the i n t e r m e diate n u c l e a r state is the ground state, w h e r e a s in our s i n g l e - r e s c a t t e r i n g a p p r o a c h the amplitude of eq. (5) contains a sum over i n t e r m e d i a t e n u c l e a r s t a t e s n. F o r the c a s e in which the i n c o h e r e n t r e s c a t t e r i n g effects a r e negligible, we m a y put n= i in eq. (5), and we thus r e t u r n to the f i r s t - o r d e r optical potential r e s u l t . In c o n s t r a s t to optical models, our r e s c a t t e r i n g a p p r o a c h enables us to evaluate d i r e c t l y the incoherent contributions. Since we u l t i m a t e l y find them to be small, we expect good a g r e e m e n t with the r e suits b a s e d on an optical potential.
4. RESULTS We shall c o n s i d e r as an explicit example the c a s e of pion radiative abs o r p t i o n f r o m the 1S a t o m i c orbit in 160. In deciding which n u c l e a r states to c o n s i d e r , we a r e guided by e a r l i e r work [2, 18], which showed that for 1S pion a b s o r p t i o n the s p i n - i s o s p i n m o d e s of excitation in the giant r e s o nance r e g i o n with J~ = 1-, 2-; T = 1 contribute a p p r o x i m a t e l y 80% to the total radiative a b s o r p t i o n rate. The p a r t i c l e - h o l e f o r m a l i s m of Gillet and Vinh Mau [19] with configuration mixed wave functions was used for the nuc l e a r m a t r i x elements. H a r m o n i c o s c i l l a t o r radial wave functions were e m ployed, with the r a n g e p a r a m e t e r a taken to be 0.629 f m - 1 . The n u c l e a r r a dial i n t e g r a l s in eq. (10) w e r e then evaluated analytically and the r e m a i n i n g i n t e g r a l over the m o m e n t u m of the pion in the i n t e r m e d i a t e state was p e r formed numerically. In spite of the fact that X2 is a p p r o x i m a t e l y ten t i m e s l a r g e r than ~1, H2 does not p r o d u c e l a r g e r r e s c a t t e r i n g effects than H 1. Indeed, employing c l o s u r e f o r the i n t e r m e d i a t e n u c l e a r s t a t e s , the contribution f r o m the nonc h a r g e - e x c h a n g e p a r t of H 2 can be seen to be s m a l l e r by at l e a s t two o r d e r s
608
R. GUY and J. M. EISENBERG
of m a g n i t u d e t h a n t h e t r a n s i t i o n r a t e g i v e n b y t h e l e a d i n g t e r m in eq. (10). A s i m i l a r s i t u a t i o n o b t a i n s if one c o n s i d e r s t h e i n t e r a c t i o n H I f o r t h e c a s e in w h i c h t h e i n t e r m e d i a t e s t a t e c o n t a i n s two p i o n s , c o r r e s p o n d i n g to (c) in fig. 1. In f a c t , t h e m a j o r c o r r e c t i v e e f f e c t c o m e s f r o m t h e c o h e r e n t r e s c a t t e r i n g in w h i c h t h e n u c l e u s r e m a i n s u n e x c i t e d in t h e i n t e r m e d i a t e s t a t e ( n = i ) . T h e e v a l u a t i o n of t h e s e c o n d t e r m in eq. (11), c o r r e s p o n d i n g to (b) w i t h H 1 a c t i n g in fig. 1, l e a d s to a d e c r e a s e in t h e t r a n s i t i o n r a t e of a p p r o x i m a t e l y 15% r e l a t i v e t o t h e f i r s t t e r m in eq. (11) f o r t h e m o s t i m p o r t a n t 1-; T = 1 f i n a l s t a t e . I n v o k i n g c l o s u r e f o r t h e i n t e r m e d i a t e n u c l e a r s t a t e s in eq. (10), w e a r e a b l e t o e s t a b l i s h t h a t t h e c o h e r e n t r e s c a t t e r i n g e f f e c t i s a b o u t one o r d e r of magnitude more significant than the incoherent effects which involve the int e r m e d i a t e n u c l e u s in a l o w - e n e r g y e x c i t a t i o n * . T h e c o n t r i b u t i o n to t h e t r a n s i t i o n r a t e of t h e i n c o h e r e n t r e s c a t t e r i n g in t h e c l o s u r e a p p r o x i m a t i o n i s s h o w n in t a b l e 1 f o r t h e 1- a n d 2 - , T= 1, l e v e l s in t h e A = 16 s y s t e m . The fact that H 2 was found to be insignificant is plausible, since, for 160, t h i s r e s c a t t e r i n g i n t e r a c t i o n c a n n o t a l l o w t h e i n t e r m e d i a t e n u c l e a r s t a t e t o b e e q u a l t o t h e g r o u n d s t a t e d u e t o t h e p r e s e n c e of t h e i s o s p i n o p e r a t o r . If, a s o u t l i n e d in s e c t . 2, w e n o r m a l i z e t h e n u c l e a r r e s c a t t e r i n g m a t r i x e l e m e n t t o o b t a i n a g r e e m e n t a t q ' = 0 w i t h t h e p i o n - 160 s c a t t e r i n g l e n g t h a = 0.56 f m , w e f i n d t h a t t h e c r o s s t e r m in eq. (11) c o n t r i b u t e s a d e c r e a s e of a b o u t 36% r e l a t i v e t o t h e l e a d i n g t e r m , a s w o u l d b e e x p e c t e d f r o m eq. (14). T h i s r e f l e c t s a n i n c r e a s e in t h e n u c l e a r m a t r i x e l e m e n t o v e r t h a t c a l c u l a t e d f r o m eq. (11) in p e r t u r b a t i o n t h e o r y b y a f a c t o r of 2.5. T h i s e f f e c t i s w e l l k n o w n in t h e c o n s t r u c t i o n of p i o n - n u c l e u s o p t i c a l p o t e n t i a l s [13]. It a r i s e s f r o m t h e n e a r - c a n c e l l a t i o n of t h e n e u t r o n a n d p r o t o n r e s c a t t e r i n g c o n t r i b u t i o n s , w h i c h in t u r n m a k e s h i g h e r - o r d e r m u l t i p l e s c a t t e r i n g effects very important. Table 1 Contributions to the transition rate from incoherent r e s c a t t e r i n g for 160. j?T
Energy (MeV)
Leading t e r m (1015 s e c _ l )
Incoherent r e s c a t t e r i n g (1015 sec_l)
11111-
25.4 22.7 19.6 18.1 13.6
150 69 31 47 17
4.0 2.9 0.3 0.8 0.4
22222-
23.7 20.2 19.1 17.7 13.0
81 250 83 2 155
1.0 1.4 7.2 0.2 5.9
The excitation energies a r e given relative to the ground state of 160. The transition r a t e s for the leading t e r m a r e calculated from the interaction in eq. (2), and the r e s c a t t e r i n g is computed after normalization to the observed nuclear s c a t t e r i n g length. * A s i m i l a r r e s u l t has been established for pion-nucleus charge exchange s c a t t e r ing [20].
RESCATTERING CORRECTIONS
609
W e n o t e t h a t w h e n t h e c o m p l e t e r e s c a t t e r i n g m a t r i x e l e m e n t in eq. (11) is replaced by a constant ~--nucleus scattering length, there occurs a dec r e a s e d u e t o H 1 of a p p r o x i m a t e l y 45%. S u c h r e s u l t s b a s e d on eq. (11), f o r t h e 1- ; T = 1 g i a n t r e s o n a n c e f i n a l s t a t e s in t h e A = 16 s y s t e m a r e p r e s e n t e d in fig. 2. T h e s a m e c a l c u l a t i o n s w e r e p e r f o r m e d f o r t h e 2 - ; T = 1 f i n a l s t a t e s a n d t h e s e r e s u l t s a p p e a r in fig. 3. In a l l c a l c u l a t i o n s , o n l y t h e l e a d i n g t e r m a n d t h e c r o s s t e r m in eq. (11) w e r e r e t a i n e d . It i s i m p o r t a n t t o n o t e t h a t t h e r e s c a t t e r i n g i n t e r a c t i o n s do n o t s i g n i f i c a n t l y a l t e r t h e d i s t r i b u t i o n of s t r e n g t h a m o n g t h e v a r i o u s 1- a n d 2- l e v e l s .
"T u
"3" 14
%
CO
~ 0 1,~ ~) "-' 1(:
¢._
._(2
8! L 0
6
<
4
0 in .Q
I I I I
I I
2 0 10
i
ii,
15
ii L ,
20
I I I I
<
II,
I .
25
E(MeV)
Fig. 2. Modifications in the rate of r a diative pion absorption in 160 leading to 1-; T = I final nuclear states of energy E r e l a t i v e to the 160 ground state. The dashed lines r e f e r to the contributions of the leading t e r m as given by eq. (2), and the solid lines a r e the r e s u l t s including the r e s c a t tering c o r r e c t i o n s .
0 10
15
20
25
E(MeV)
Fig. 3. Modifications due to r e s c a t t e r ing c o r r e c t i o n s for 2-; T = 1 levels in the A = 16 s y s t e m (see caption to fig. 2). The level at 19.1MeV is not shown, since its strength is l e s s than 1%.
T h e d e c r e a s e in t h e t r a n s i t i o n r a t e due t o r e s c a t t e r i n g c o r r e s p o n d s to t h e d e c r e a s e in t h e p r o b a b i l i t y of f i n d i n g t h e p i o n in t h e n u c l e u s a s c a l c u l a t e d u s i n g a n o p t i c a l p o t e n t i a l [14, 15]. T h e r e s u l t s of t h e c a l c u l a t i o n p r e s e n t e d h e r e i n d i c a t e t h a t t h e m o d i f i c a t i o n s of t h e r a d i a t i v e p i o n a b s o r p t i o n i n t e r a c t i o n c a l c u l a t e d f r o m r e s c a t t e r i n g a r e in c l o s e a c c o r d w i t h t h e a l t e r native prescription given by the optical potential approach, since incoherent rescattering effects are small.
610
R. GUY and J. M. EISENBERG
It i s a p l e a s u r e to a c k n o w l e d g e u s e f u l c o m m u n i c a t i o n s with D r s . M. E r i c s o n , T . E . O . E r i c s o n , L. P. F u l c h e r , D. S. K o l t u n a n d H. P i e t s c h m a n n . A U n i v e r s i t y of V i r g i n i a C o m p u t i n g G r a n t is a l s o g r a t e f u l l y a c k n o w l e d g e d . O n e of u s (R.G.) w o u l d l i k e to e x p r e s s h i s t h a n k s to t h e N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n for fellowship support.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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