- Email: [email protected]
Radiative pion capture in 12C
Volume 36B, number 4
PHYSICS LETTERS
20 September 1971
R A D I A T I V E P I O N C A P T U R E IN 12C S. S K U P S K Y Argonne National Laboratory, Argonne, Illinois 60439, USA * and The University of Chicago, Chicago, Illinois 60637, USA ** Received 28 July 1971
Transition rates for the reaction Ir-(at rest) + 12C(ground) ~ ) ' + 12B (even parity) have been computed in the impulse approximation with Cohen-Kurath wave functions for the nuclear states involved. The two available experiments disagree by a factor of 5 in the ground-state transition rate. The calculated rates agree closely with the results of Bistirlich et al.
R a d i a t i v e pion c a p t u r e is i m p o r t a n t f r o m two points of view: 1) It can g i v e i n f o r m a t i o n about giant dipole and q u a d r u p o l e r e s o n a n c e s in nuclei. 2) It can t e s t the v a l i d i t y of a p a r t i a l c o n s e r v e d a x i a l c u r r e n t (PCAC) by c o m p a r i s o n with muon capture. In o r d e r to study t h e s e two a s p e c t s q u a n t i t a t i v e l y , it is i m p o r t a n t to know how w e l l this m o d e of c a p t u r e can be d e s c r i b e d by e x i s t ing t h e o r i e s . T h e o r e t i c a l tools f o r the study of 12C have been w e l l e s t a b l i s h e d f r o m o t h e r phenomena. It is the p u r p o s e of this p a p e r to apply them to the r e a c t i o n 7t-(at rest) + 12C(ground) -* 7+ 12B(even parity) and to compare the results with experiment. The experimental quantity to be calculated is the ratio R of the number of radiative transitions to the number of stopped pions. In 12C all stopped pions will reach the Is or 2p orbits before absorption. R is related to the l s and 2p radiative capture rates V¢'~and W,, ~ and to the
total capture rates
WanZWby'the relation
where Ps and Pp are percentages of pions in the l s and 2p orbits before absorption. The values used here are Ps = (45.5±3.0)% ,
Pp = (55.5±3.0)% ,
$ Work done on a fellowship from the National Science Foundation. Address after October 1: Weizmann Institute, Rehovoth, Israel. * Work done in part under the auspices of the U.S. Atomic Energy Commission. ** Work done in partial fulfillment of the requirements for the Ph.D. degree from the University of Chicago.
---q' "W~=(4.74+0.24) " x 1018s - I , w T = ( 1 . 5 7 + 0 . 1 9 ) × 1015s - I , w h e r e P s and Pp a r e f r o m the p a p e r by Koch et al. [1] and W Tan d WT a r e f r o m B i s t i r l i c h et al.
[21. It should be e m p h a s i z e d that the e x p e r i m e n t a l quantity R depends on the r a t i o s of r a d i a t i v e c a p t u r e to the total c a p t u r e r a t e s f r o m both the l s and 2p o r b i t s. T h e s e a r e of the s a m e o r d e r of magnitude. H en ce both m u s t be c o n s i d e r e d when computing R, even though Ws >> Wp. C h a r g e d pion photoproduction f r o m nucleons is adequately d e s c r i b e d at low e n e r g y by the CGLN a m p l i t u d e [3]. The i m p u l s e a p p r o x i m a t i o n f o r individual nucleons in a nucleus can be u s e d to c o n s t r u c t an e f f e c t i v e H a m i l t o n i a n H=27r i'r_[A(q)a, e~ + B(q) a . e T t q . k + C(q)a" k e~t" q + D(q) q ' k x
ex] ,
(2)
w h e r e e~t is the photon p o l a r i z a t i o n (;t =± 1), q the pion m o m e n t u m , a the P a u l i spin m a t r i x , and ~-_ the o p e r a t o r that changes a p r o t o n into a neutron. Kawaguchi et al. [4] u s e d the t h r e s h o l d v a l u e s (q = 0) of the c o e f f i c i e n t s and found them to b e A =-0.032 rn~r2 , B = 0 . 0 0 7 5 rn~ 2 C = -0.037 m~ 2 and D = - 0 . 0 1 4 m ~ 2. A s o m e w h a t d i f f e r e n t set was u s e d by D e l o r m e and E r i c s o n [5], ham_ely A = ( - 0 . 0 3 4 ± 0 . 0 0 3 ) m ~ 2, B = 0.019 rn~ 2, C = - 0 .0 1 7 r n ~ 2 and D =0.01 rr~ 2. We do not know the r e a s o n f o r the d i s c r e p a n c y b e tween the two s e t s of c o e f f i c i e n t s , but it is shown below that the d i f f e r e n c e has only a s m a l l effect on the t r a n s i t i o n r a t e s . A l s o below, we a r g u e that t h e s e t h r e s h o l d v a l u e s m ay not be 271
Volume 36B, number 4
PHYSICS LETTERS
a p p r o p r i a t e for c a p t u r e from the p orbit; and a t h i r d set of coefficients, evaluated at q~ my, a r e introduced. The t r a n s i t i o n r a t e is given by W--
(3)
where
¢,(ri) ,
and (Ji' Mi)' (Jr,Mr) and ( L , My) a r e the a n g u l a r m o m e n t u m quantum numbe'~s of the i n i t i a l and final n u c l e a r s t a t e s and the pion state, r e s p e c tively. Units a r e such that ~ = c = p i o n m a s s , mTr --1. The f i r s t t e r m in eq. (2) is the only one that s u r v i v e s in the soft-pion limit. It will be shown below that for i s c a p t u r e this t e r m does indeed account for about 99% of the i n t e r a c t i o n . Howe v e r , for c a p t u r e from the 2p orbit, all four t e r m s a r e l a r g e and must be considered. The pion wave function was calculated by u s e of the K r e l l - E r i c s o n optical potential [6], which has s u c c e s s f u l l y explained pionic e n e r g y - l e v e l shifts. A n u c l e a r density of the h a r m o n i c o s c i l l a t o r type was used, such that ( r 2) 1/2 = 2.40 fro, in a g r e e m e n t with e l e c t r o n - s c a t t e r i n g data. The a b s o r p t i v e p a r t s of the potential w e r e neglected s i n c e t h e i r effect on the wave function is too s m a l l to be significant h e r e [6]. N e u t r o n and proton d e n s i t i e s w e r e a s s u m e d equal. With these specifications, the l s e n e r g y - l e v e l shift obtained is -6.93 keV. The r e p u l s i v e s - w a v e i n t e r a c t i o n r e d u c e s the l s wave function gbs inside the nucleus r e l a t i v e to the nucleus r e l a t i v e to the Bohr wave function ~b~. Defining q~ as the a v e r a g e value of ~b in the n u c l e u s , we obtain [~s/qs~l 2 =0.55. This shows a reduction of about 50°/c. In c o n t r a s t , the p^ i n t e r a c t i o n is a t t r a c t i v e and gives I 4)p/ y ~ P_ z =1.3. F o r the n u c l e a r wave functions of 12C and 12B, the a s s u m p t i o n was made that they c o n s i s t of a closed s shell plus six p - s h e l l nucleons. This alone (i.e., without specifying the d i s t r i b u tion of nucleons among the P l / 2 and P3/2 levels} allows one to make two useful e s t i m a t e ~ 1) The r e l a t i v e o r d e r s of magnitude of the t e r m s in the H a m i l t o n i a n (2) can be calculated. 2) The r a t e of r a d i a t i v e pion c a p t u r e from the I s o r b i t can be d e t e r m i n e d from e x p e r i m e n t a l values for b e t a decay and muon capture. 272
h f R l p 2 ( d e S / d r ) j 1 (kr)r 2 dr
fRlp
× Mi, M~f,X, Myfd~2k] ( jfMflHeffl JiMi)t 2, ri} H(i)
F i r s t , for s capture, the r a t i o of any of the last t h r e e t e r m s in "eq. (2) to the f i r s t t e r m is a p p r o x i m a t e l y the ratio of r a d i a l i n t e g r a l s , i.e., = 0.03.
k 1 1 - (2y52 2Ji+l 2Ly+l
A H etf = ~ exp ( - i k i . i =1
20 September 1971
2¢~sJ o ( k r ) r 2 dr
Here, RID is the l p h a r m o n i c - o s c i l l a t o r wave function, )(kr) is a s p h e r i c a l B e s s e l function, and k is a g a m m a energy of 125MeV. Thus, for s c a p t u r e the last t h r e e t e r m s in eq. (2) should c o n t r i b u t e less than 10%. In the actual c a l c u l a tion for the g r o u n d - s t a t e t r a n s i t i o n , the ratio of the r a t e from the f i r s t t e r m alone to the r a t e from all four t e r m s was 0.99. The situation is v e r y different for p capture, for which the r e l e v a n t ratio of r a d i a l i n t e g r a l s is 0.97. This suggests that all four t e r m s of the H a m i l t o n i a n a r e of the s a m e o r d e r of magnitude and none may be neglected a p r i o r i . W h e r e a s the f i r s t t e r m i n eq. (25 gives a good e s t i m a t e for s capture, it u n d e r e s t i m a t e s the p c o n t r i b u tion by a factor of about 2 in the g r o u n d - s t a t e t r a n s i t i o n . In this p a r t i c u l a r calculation, however, p capture accounts for only 30% of the c o n t r i b u t i o n to R in eq. (15, and t h e r e f o r e the net effect of the m o m e n t u m - d e p e n d e n t t e r m s of H is only 10°/o. T h e r e is another p o s s i b l e s o u r c e of u n c e r tainty in the effective H a m i l t o n i a n for c a p t u r e from the 2p orbit. The r a t i o of r a d i a l i n t e g r a l s u s e d above as an e s t i m a t e of the r e l a t i v e i m p o r t a n c e of the m o m e n t u m - d e p e n d e n t p a r t of the H a m i l t o n i a n has a s i m p l e p h y s i c a l meaning; it is the m e a n m o m e n t u m q (in u n i t s of my) of that p a r t of the pion wave function that lies i n s i d e the n u c l e a r volume. But the c a p t u r e p r o c e s s obviously involves only the p a r t of the pion wave function i n s i d e the nucleus. This s u g g e s t s that the coefficients (A(q), etc.) in the effective H a m i l t o n i a n of eq. (2) should be evaluated at the a p p r o p r i a t e m e a n m o m e n t u m q. F o r l s capture, the ratio of r a d i a l i n t e g r a l s is 0.3, (q ~ 0) so that the t h r e s h o l d coefficients quoted above should be appropriate. F o r 2p c a p t u r e , on the other hand, e] ~ my. Coefficients can then be taken from e x p e r i m e n t s at the a p p r o p r i a t e e n e r gy; i s o v e c t o r t e r m s a r e obtained from ref. [8], and i s o s c a l a r from ref. [9]. The values a r e A =-0.22 rn~ 2 , B-0.0095m~72, C=-0.029m~72 ~. u tJ a n d D =-0.021 rn~ . Of c o u r s e , the whole p r o c e dure for c o n s t r u c t i n g an effective H a m i l t o n i a n via the i m p u l s i v e a p p r o x i m a t i o n and the choice of a s i n g l e m e a n m o m e n t u m ~ at which to e v a l u ate the photoproduction a m p l i t u d e s is u n c e r t a i n . The r e s u l t s obtained should t h e r e f o r e b e viewed
Volume 36B, number 4
PHYSICS L E T T E R S
as a rough e s t i m a t e of a p o s s i b l e s o u r c e of u n c e r t a i n t y in the effective Hamilton±an. If the h i g h e r - m o m e n t u m coefficients a r e u s e d for the 2p contribution, the calculated value of the r a d i a t i v e - c a p t u r e r a t i o R is reduced by about 25% about what is needed to yield c l o s e a g r e e m e n t with the e x p e r i m e n t a l r e s u l t s of B i s t i r l i c h et al.
[2].
The l s c o n t r i b u t i o n should n e v e r t h e l e s s be well d e t e r m i n e d by the threshold value of A alone. To obtain an e s t i m a t e of the l s c o n t r i b u tion, we neglect the L=2 a n g u l a r m o m e n t u m component of the photon wave function. Then eq. (3) b e c o m e s y ~ 2 kA2 12BI . s ,
Ws~ ~
[<
[T-~o(~r)~(r)cr [[12C>12
This is in effect the soft-pion e s t i m a t e . Since only l p nucleons a r e involved, the r a d i a l i n t e g r a l can be factored out and evaluated, the r e sult being W s ~ (0.88× 1016 sec -1) ]<
12B [ [~_~'l ] 12C>12.(4)
The m a t r i x e l e m e n t in eq. (4) is r e d u c e d only in spin and depends on the d i s t r i b u t i o n of p - s h e l l nucleons. The e l e m e n t ](12B[ l•_a' [ I12C)l 2 also app e a r s in s t a n d a r d e x p r e s s i o n s for muon c a p t u r e [10] and beta decay, after the r a d i a l p a r t s a r e f a c t o r e d out. Using a m u o n - c a p t u r e r a t e [11] of +0.3 (6.75_0.75)×103 s-1 and a n f t value [12] of (1.18 + 0.007)× 1 0 4 s j we obtain = .u .. ~. ._ 0+.01. 004 5 s and 1.08 ± 0.007 from the two p r o c e s s e s . Substituting the beta decay value into ea. (4) gives Ws(12C --* 12Bg) =0.95 × 1016 s -1. The c a l c u l a t e d s h e l l - m o d e l value for this r a t e is 1.05 × 1016 s-1. The s m a l l d i f f e r e n c e a r i s e s from n e g l e c t of the L =2 component of the photon wave function. The c l o s e r e l a t i o n exhibited h e r e between g - c a p t u r e and r a d i a t i v e 7r c a p t u r e from the l s o r b i t is of
}<12BI
] 112c>tz
20 September 1971
c o u r s e well known [5]. B e c a u s e of this r e l a t i o n between m a t r i x e l e m e n t s for muon capture, beta decay and r a d i a t i v e pion capture, we cannot hope to p r e d i c t the l a s t if the f i r s t two a r e not d e s c r i b e d c o r rectly. The C o h e n - K u r a t h wave functions [13] a r e able to d e s c r i b e the f i r s t two adequately. The v e r s i o n of the wave functions u s e d h e r e p r e d i c t e d {<12B[ ]~' [ l l 2 c ) [ 2 = 0 . 9 3 , which is about 15% low. They were then modified, following the suggestion of Fuji± et al. [14], by adding to the ground s t a t e of 12C a 4% a d m i x t u r e of the f i r s t excited 0+ state. This s m a l l amount should not affect the energy l e v e l s , but it does r a i s e the value of [(12B] [(7' [ [12C>[ 2 to 1.08, in a g r e e ment with both beta decay and muon capture. T h e s e n u c l e a r wave functions and t h e i r coefficients of f r a c t i o n a l p a r e n t a g e w e r e calculated by u s e of the Argonne DELPHI c o m p u t e r p r o gram. T r a n s i t i o n r a t e s to all t h e J ~ =0 +, 1+, 2+ and 3+ states p r e d i c t e d by the C o h e n - K u r a t h model [12] were calculated. T h e r e w e r e 22 such s t a t e s with e n e r g i e s r a n g i n g up to 25 MeV above the 12B ground state. Of these, only the two listed in t a b l e 1 made a c o n t r i b u t i o n of 0.01% or g r e a t e r to R (the n u m b e r of r a d i a t i v e t r a n s i t i o n s p e r stopped pion). The values of R for each of the t h r e e sets of coefficients a r e tabulated, and an e s t i m a t e d e r r o r of 50~o has been added. This a r i s e s roughly from t h r e e s o u r c e s : 25% from u n c e r t a i n t i e s in the Hamilton±an, 10% from the n u c l e a r and pion wave functions and 15% from the p a r a m e t e r s in eq. (1). C o m p a r i s o n with e x p e r i m e n t a l is h a m p e r e d by lack of a g r e e m e n t between the two published r e s u l t s [2, 15] of work in which s o m e n u c l e a r s t r u c t u r e could be resolved. The energy r a n g e in t a b l e 1 c o r r e s p o n d s to the region of the f i r s t peak of B i s t i r l i c h et al. [2]. (The r e m a i n d e r of
Table 1 Even-parity states in 12B that contribute 0.01% or more to R (the number of radiative transitions per stopped pion) Rs and Rp are the contributions to R from the l s and 2p pionic Bohr orbits. The heading 1, 2 and 3 refer to three alternative sets of coefficients in the Hamilton±an of eq. (2); 1 refers to the coefficients taken directly from ref. [4], 2 to the coefficients of ref. [5] while in 3, the coefficients for 2p capture were evaluated (as described in the text) from pion photoproduetion at q ~ m~. 1
energy (MeV)
J~"
125.9 124.5
1+ 2+
R total
2
Rp
(10 -3 )
(10-3)
(10-3)
(10-3)
(10-3)
(10-3)
0.98 0.14
0.28 0.21
1.01 0.14
0.35 0.21
0.98 0.14
0.14 0.07
1.6 ± 0.8
Rs
3
Rs
Rp
1.7 ± 0.9
Rs
Rp
1.3 ± 0.7 273
Volume 36B, number 4
PHYSICS LETTERS
t h e i r s p e c t r u m m u s t r e s u l t from t r a n s i t i o n s to o d d - p a r i t y s t a t e s and to continuum. ) The e x p e r i m e n t a l r a t i o s in this energy region a r e given as R = (0.97 ± 0.09)x 10 -3 by B i s t i r l i c h et al. [2] and R = (5.8± 1.3)× 10 -3 by H i l s c h e r et al. [15]. All the calculated c a p t u r e r a t i o s in table 1 a r e a g r e e m e n t , to within the stated p r o b a b l e e r r o r s , with the e x p e r i m e n t a l r e s u l t s of B i s t i r l i c h et al. C l o s e s t a g r e e m e n t is obtained u s i n g H a m i l t o n i a n coefficients evaluated at q ~ m~ for c a p t u r e from the 2p orbit. B e s i d e s the difference in e x p e r i m e n t a l n u m b e r s , refs. [2] and [15] also d e t e r m i n e diff e r e n t values of the l s and 2p pion populations P . and P_p from the work of Koch et al. The i n t e r p r e t a t i o n u s e d h e r e was the s a m e as that of H i l s c h e r et al. The a l t e r n a t i v e v a l u e s listed by B i s t i r l i c h et al. w e r e Ps = 12.3 + 1.4% and Pp = 88.7± 1.4%. If t h e s e had been the n u m b e r s u s e d in our calculation, then it would have b e e n the t h r e s h o l d r e s u l t s that gave b e s t a g r e e m e n t with the values of B i s t i r l i c h et al. The high m o m e n tum r e s u l t s would have been about 50% lower but s t i l l in r e a s o n a b l e a g r e e m e n t . Our c o n c l u s i o n is that the g r e a t e s t u n c e r tainty in the t h e o r e t i c a l t r e a t m e n t for r a d i a t i v e pion c a p t u r e 12C lies in the choice of a H a m i l t o n i a n r a t h e r than in the wave functions. By u s e of the above p a r a m e t e r s we find that the b e s t a g r e e m e n t with the e x p e r i m e n t of B i s t i r lich et al. is attained by u s i n g H a m i l t o n i a n coefficients for l a r g e pion m o m e n t u m (~ ~ m~) i n s t e a d of the t h r e s h o l d v a l u e s in calculating c a p t u r e from the p orbit. C l e a r l y , m o r e e x p e r i m e n t s should be done in this r e g i o n to r e s o l v e the e x p e r i m e n t a l d i s c r e p a n c y and to d e t e r m i n e the r a n g e of validity of the t h e o r e t i c a l t r e a t m e n t . In a forthcoming p a p e r [16] these c a l c u l a t i o n s a r e extended to o d d - p a r i t y t r a n s i t i o n s . F o r the wave functions, this p a p e r u s e s a c o r e - c o u p l i n g model b a s e d on the C o h e n - K u r a t h d e s c r i p t i o n of the l p core; this model gives r e m a r k a b l e a g r e e m e n t with e x p e r i m e n t a l data on photoab-
274
20 September 1971
s o r p t i o n and m u o n capture. The r e s u l t s a r e that the positions and heights of the peaks in the u n c o r r e c t e d s p e c t r u m of B i s t i r l i c h et al. a r e well r e p r o d u c e d within the u n c e r t a i n t i e s outlined above. The author is grateful to Dr. M. H. M a c f a r l a n e for m a n y v a l u a b l e d i s c u s s i o n s and for a c r i t i c a l r e a d i n g of this paper. Computations for this work were g r e a t l y s i m p l i f i e d by p r o g r a m m i n g in SPEAKEASY [17].
References [1] H.Koch, M.Krell, Ch. v.d. Malsberg, G.Poelz, H. Schmitt, L. Tauscher, G. Backenstoss, S. Charalambus and H.Daniel, Phys. Letters 29B (1969) 140. [2] J. A. Bistirlich, K.M. Crowe, A. S. L. Parsons, P. Skarek and P. Truoel, Phys. Rev. Letters 25 (1970) 689. [3] G.F.Chew, M.L.Goldberger, F.E.Low and Y. Nambu, Phys. Rev. 106 (1957) 1345, hereafter referred to as CGLN. [4] M. Kawaguehi, H.Ohtsubo and Y. Sumi, Progr. Theoret. Phys. (Kyoto) Suppl. Extra Number (1968} 28. [5] J. Delorme and T. E. O. Ericson, Phys. Letters 21
(1966) 98. [6] M.Krell and T.E.O.Ericson, Nuel. Phys. B l l (1969) 521. [7] As quoted in ref. [2]. [8] P. Noelle, W. Pfeil and D. Schwela, Nucl. Phys. B26 (1971) 461. [9] F.A. Berends. A. Donnachie and D. L. Weaver, Nucl. Phys. B4 (1967) 54. [10] For example, J.R.Luyten, H.P.S.Rood and H.A.Tolhoek, Nucl. Phys. 41 (1963) 236. [11] E.J. Maier, R.M. Edelstein and R. T. Siegel, Phys. Rev° 133 (1964) B663. [12] R.W.Kavanagh, Phys. Rev. 133 (1964) B1504. [13] S. Cohen and D.Kurath, Nucl. Phys. 73 (1965) I. [14] A. Fujii, M. Morita and H.Ohtsubo, Progr. Theoret. Phys. (Kyoto) Suppl. Extra Number (1968) 303. [15] H. Hilscher, W.D. Krebs, G. Sepp and V. Soergel, Nucl. Phys. A158 (1970) 584. [16] S. Skupsky, submitted to Nucl. Phys. [ 17] S. Cohen and C. M. Vincent, Argonne Physics Division Informal Report PHY-1968E (1968).
PHYSICS LETTERS
20 September 1971
R A D I A T I V E P I O N C A P T U R E IN 12C S. S K U P S K Y Argonne National Laboratory, Argonne, Illinois 60439, USA * and The University of Chicago, Chicago, Illinois 60637, USA ** Received 28 July 1971
Transition rates for the reaction Ir-(at rest) + 12C(ground) ~ ) ' + 12B (even parity) have been computed in the impulse approximation with Cohen-Kurath wave functions for the nuclear states involved. The two available experiments disagree by a factor of 5 in the ground-state transition rate. The calculated rates agree closely with the results of Bistirlich et al.
R a d i a t i v e pion c a p t u r e is i m p o r t a n t f r o m two points of view: 1) It can g i v e i n f o r m a t i o n about giant dipole and q u a d r u p o l e r e s o n a n c e s in nuclei. 2) It can t e s t the v a l i d i t y of a p a r t i a l c o n s e r v e d a x i a l c u r r e n t (PCAC) by c o m p a r i s o n with muon capture. In o r d e r to study t h e s e two a s p e c t s q u a n t i t a t i v e l y , it is i m p o r t a n t to know how w e l l this m o d e of c a p t u r e can be d e s c r i b e d by e x i s t ing t h e o r i e s . T h e o r e t i c a l tools f o r the study of 12C have been w e l l e s t a b l i s h e d f r o m o t h e r phenomena. It is the p u r p o s e of this p a p e r to apply them to the r e a c t i o n 7t-(at rest) + 12C(ground) -* 7+ 12B(even parity) and to compare the results with experiment. The experimental quantity to be calculated is the ratio R of the number of radiative transitions to the number of stopped pions. In 12C all stopped pions will reach the Is or 2p orbits before absorption. R is related to the l s and 2p radiative capture rates V¢'~and W,, ~ and to the
total capture rates
WanZWby'the relation
where Ps and Pp are percentages of pions in the l s and 2p orbits before absorption. The values used here are Ps = (45.5±3.0)% ,
Pp = (55.5±3.0)% ,
$ Work done on a fellowship from the National Science Foundation. Address after October 1: Weizmann Institute, Rehovoth, Israel. * Work done in part under the auspices of the U.S. Atomic Energy Commission. ** Work done in partial fulfillment of the requirements for the Ph.D. degree from the University of Chicago.
---q' "W~=(4.74+0.24) " x 1018s - I , w T = ( 1 . 5 7 + 0 . 1 9 ) × 1015s - I , w h e r e P s and Pp a r e f r o m the p a p e r by Koch et al. [1] and W Tan d WT a r e f r o m B i s t i r l i c h et al.
[21. It should be e m p h a s i z e d that the e x p e r i m e n t a l quantity R depends on the r a t i o s of r a d i a t i v e c a p t u r e to the total c a p t u r e r a t e s f r o m both the l s and 2p o r b i t s. T h e s e a r e of the s a m e o r d e r of magnitude. H en ce both m u s t be c o n s i d e r e d when computing R, even though Ws >> Wp. C h a r g e d pion photoproduction f r o m nucleons is adequately d e s c r i b e d at low e n e r g y by the CGLN a m p l i t u d e [3]. The i m p u l s e a p p r o x i m a t i o n f o r individual nucleons in a nucleus can be u s e d to c o n s t r u c t an e f f e c t i v e H a m i l t o n i a n H=27r i'r_[A(q)a, e~ + B(q) a . e T t q . k + C(q)a" k e~t" q + D(q) q ' k x
ex] ,
(2)
w h e r e e~t is the photon p o l a r i z a t i o n (;t =± 1), q the pion m o m e n t u m , a the P a u l i spin m a t r i x , and ~-_ the o p e r a t o r that changes a p r o t o n into a neutron. Kawaguchi et al. [4] u s e d the t h r e s h o l d v a l u e s (q = 0) of the c o e f f i c i e n t s and found them to b e A =-0.032 rn~r2 , B = 0 . 0 0 7 5 rn~ 2 C = -0.037 m~ 2 and D = - 0 . 0 1 4 m ~ 2. A s o m e w h a t d i f f e r e n t set was u s e d by D e l o r m e and E r i c s o n [5], ham_ely A = ( - 0 . 0 3 4 ± 0 . 0 0 3 ) m ~ 2, B = 0.019 rn~ 2, C = - 0 .0 1 7 r n ~ 2 and D =0.01 rr~ 2. We do not know the r e a s o n f o r the d i s c r e p a n c y b e tween the two s e t s of c o e f f i c i e n t s , but it is shown below that the d i f f e r e n c e has only a s m a l l effect on the t r a n s i t i o n r a t e s . A l s o below, we a r g u e that t h e s e t h r e s h o l d v a l u e s m ay not be 271
Volume 36B, number 4
PHYSICS LETTERS
a p p r o p r i a t e for c a p t u r e from the p orbit; and a t h i r d set of coefficients, evaluated at q~ my, a r e introduced. The t r a n s i t i o n r a t e is given by W--
(3)
where
¢,(ri) ,
and (Ji' Mi)' (Jr,Mr) and ( L , My) a r e the a n g u l a r m o m e n t u m quantum numbe'~s of the i n i t i a l and final n u c l e a r s t a t e s and the pion state, r e s p e c tively. Units a r e such that ~ = c = p i o n m a s s , mTr --1. The f i r s t t e r m in eq. (2) is the only one that s u r v i v e s in the soft-pion limit. It will be shown below that for i s c a p t u r e this t e r m does indeed account for about 99% of the i n t e r a c t i o n . Howe v e r , for c a p t u r e from the 2p orbit, all four t e r m s a r e l a r g e and must be considered. The pion wave function was calculated by u s e of the K r e l l - E r i c s o n optical potential [6], which has s u c c e s s f u l l y explained pionic e n e r g y - l e v e l shifts. A n u c l e a r density of the h a r m o n i c o s c i l l a t o r type was used, such that ( r 2) 1/2 = 2.40 fro, in a g r e e m e n t with e l e c t r o n - s c a t t e r i n g data. The a b s o r p t i v e p a r t s of the potential w e r e neglected s i n c e t h e i r effect on the wave function is too s m a l l to be significant h e r e [6]. N e u t r o n and proton d e n s i t i e s w e r e a s s u m e d equal. With these specifications, the l s e n e r g y - l e v e l shift obtained is -6.93 keV. The r e p u l s i v e s - w a v e i n t e r a c t i o n r e d u c e s the l s wave function gbs inside the nucleus r e l a t i v e to the nucleus r e l a t i v e to the Bohr wave function ~b~. Defining q~ as the a v e r a g e value of ~b in the n u c l e u s , we obtain [~s/qs~l 2 =0.55. This shows a reduction of about 50°/c. In c o n t r a s t , the p^ i n t e r a c t i o n is a t t r a c t i v e and gives I 4)p/ y ~ P_ z =1.3. F o r the n u c l e a r wave functions of 12C and 12B, the a s s u m p t i o n was made that they c o n s i s t of a closed s shell plus six p - s h e l l nucleons. This alone (i.e., without specifying the d i s t r i b u tion of nucleons among the P l / 2 and P3/2 levels} allows one to make two useful e s t i m a t e ~ 1) The r e l a t i v e o r d e r s of magnitude of the t e r m s in the H a m i l t o n i a n (2) can be calculated. 2) The r a t e of r a d i a t i v e pion c a p t u r e from the I s o r b i t can be d e t e r m i n e d from e x p e r i m e n t a l values for b e t a decay and muon capture. 272
h f R l p 2 ( d e S / d r ) j 1 (kr)r 2 dr
fRlp
× Mi, M~f,X, Myfd~2k] ( jfMflHeffl JiMi)t 2, ri} H(i)
F i r s t , for s capture, the r a t i o of any of the last t h r e e t e r m s in "eq. (2) to the f i r s t t e r m is a p p r o x i m a t e l y the ratio of r a d i a l i n t e g r a l s , i.e., = 0.03.
k 1 1 - (2y52 2Ji+l 2Ly+l
A H etf = ~ exp ( - i k i . i =1
20 September 1971
2¢~sJ o ( k r ) r 2 dr
Here, RID is the l p h a r m o n i c - o s c i l l a t o r wave function, )(kr) is a s p h e r i c a l B e s s e l function, and k is a g a m m a energy of 125MeV. Thus, for s c a p t u r e the last t h r e e t e r m s in eq. (2) should c o n t r i b u t e less than 10%. In the actual c a l c u l a tion for the g r o u n d - s t a t e t r a n s i t i o n , the ratio of the r a t e from the f i r s t t e r m alone to the r a t e from all four t e r m s was 0.99. The situation is v e r y different for p capture, for which the r e l e v a n t ratio of r a d i a l i n t e g r a l s is 0.97. This suggests that all four t e r m s of the H a m i l t o n i a n a r e of the s a m e o r d e r of magnitude and none may be neglected a p r i o r i . W h e r e a s the f i r s t t e r m i n eq. (25 gives a good e s t i m a t e for s capture, it u n d e r e s t i m a t e s the p c o n t r i b u tion by a factor of about 2 in the g r o u n d - s t a t e t r a n s i t i o n . In this p a r t i c u l a r calculation, however, p capture accounts for only 30% of the c o n t r i b u t i o n to R in eq. (15, and t h e r e f o r e the net effect of the m o m e n t u m - d e p e n d e n t t e r m s of H is only 10°/o. T h e r e is another p o s s i b l e s o u r c e of u n c e r tainty in the effective H a m i l t o n i a n for c a p t u r e from the 2p orbit. The r a t i o of r a d i a l i n t e g r a l s u s e d above as an e s t i m a t e of the r e l a t i v e i m p o r t a n c e of the m o m e n t u m - d e p e n d e n t p a r t of the H a m i l t o n i a n has a s i m p l e p h y s i c a l meaning; it is the m e a n m o m e n t u m q (in u n i t s of my) of that p a r t of the pion wave function that lies i n s i d e the n u c l e a r volume. But the c a p t u r e p r o c e s s obviously involves only the p a r t of the pion wave function i n s i d e the nucleus. This s u g g e s t s that the coefficients (A(q), etc.) in the effective H a m i l t o n i a n of eq. (2) should be evaluated at the a p p r o p r i a t e m e a n m o m e n t u m q. F o r l s capture, the ratio of r a d i a l i n t e g r a l s is 0.3, (q ~ 0) so that the t h r e s h o l d coefficients quoted above should be appropriate. F o r 2p c a p t u r e , on the other hand, e] ~ my. Coefficients can then be taken from e x p e r i m e n t s at the a p p r o p r i a t e e n e r gy; i s o v e c t o r t e r m s a r e obtained from ref. [8], and i s o s c a l a r from ref. [9]. The values a r e A =-0.22 rn~ 2 , B-0.0095m~72, C=-0.029m~72 ~. u tJ a n d D =-0.021 rn~ . Of c o u r s e , the whole p r o c e dure for c o n s t r u c t i n g an effective H a m i l t o n i a n via the i m p u l s i v e a p p r o x i m a t i o n and the choice of a s i n g l e m e a n m o m e n t u m ~ at which to e v a l u ate the photoproduction a m p l i t u d e s is u n c e r t a i n . The r e s u l t s obtained should t h e r e f o r e b e viewed
Volume 36B, number 4
PHYSICS L E T T E R S
as a rough e s t i m a t e of a p o s s i b l e s o u r c e of u n c e r t a i n t y in the effective Hamilton±an. If the h i g h e r - m o m e n t u m coefficients a r e u s e d for the 2p contribution, the calculated value of the r a d i a t i v e - c a p t u r e r a t i o R is reduced by about 25% about what is needed to yield c l o s e a g r e e m e n t with the e x p e r i m e n t a l r e s u l t s of B i s t i r l i c h et al.
[2].
The l s c o n t r i b u t i o n should n e v e r t h e l e s s be well d e t e r m i n e d by the threshold value of A alone. To obtain an e s t i m a t e of the l s c o n t r i b u tion, we neglect the L=2 a n g u l a r m o m e n t u m component of the photon wave function. Then eq. (3) b e c o m e s y ~ 2 kA2 12BI . s ,
Ws~ ~
[<
[T-~o(~r)~(r)cr [[12C>12
This is in effect the soft-pion e s t i m a t e . Since only l p nucleons a r e involved, the r a d i a l i n t e g r a l can be factored out and evaluated, the r e sult being W s ~ (0.88× 1016 sec -1) ]<
12B [ [~_~'l ] 12C>12.(4)
The m a t r i x e l e m e n t in eq. (4) is r e d u c e d only in spin and depends on the d i s t r i b u t i o n of p - s h e l l nucleons. The e l e m e n t ](12B[ l•_a' [ I12C)l 2 also app e a r s in s t a n d a r d e x p r e s s i o n s for muon c a p t u r e [10] and beta decay, after the r a d i a l p a r t s a r e f a c t o r e d out. Using a m u o n - c a p t u r e r a t e [11] of +0.3 (6.75_0.75)×103 s-1 and a n f t value [12] of (1.18 + 0.007)× 1 0 4 s j we obtain = .u .. ~. ._ 0+.01. 004 5 s and 1.08 ± 0.007 from the two p r o c e s s e s . Substituting the beta decay value into ea. (4) gives Ws(12C --* 12Bg) =0.95 × 1016 s -1. The c a l c u l a t e d s h e l l - m o d e l value for this r a t e is 1.05 × 1016 s-1. The s m a l l d i f f e r e n c e a r i s e s from n e g l e c t of the L =2 component of the photon wave function. The c l o s e r e l a t i o n exhibited h e r e between g - c a p t u r e and r a d i a t i v e 7r c a p t u r e from the l s o r b i t is of
}<12BI
] 112c>tz
20 September 1971
c o u r s e well known [5]. B e c a u s e of this r e l a t i o n between m a t r i x e l e m e n t s for muon capture, beta decay and r a d i a t i v e pion capture, we cannot hope to p r e d i c t the l a s t if the f i r s t two a r e not d e s c r i b e d c o r rectly. The C o h e n - K u r a t h wave functions [13] a r e able to d e s c r i b e the f i r s t two adequately. The v e r s i o n of the wave functions u s e d h e r e p r e d i c t e d {<12B[ ]~' [ l l 2 c ) [ 2 = 0 . 9 3 , which is about 15% low. They were then modified, following the suggestion of Fuji± et al. [14], by adding to the ground s t a t e of 12C a 4% a d m i x t u r e of the f i r s t excited 0+ state. This s m a l l amount should not affect the energy l e v e l s , but it does r a i s e the value of [(12B] [(7' [ [12C>[ 2 to 1.08, in a g r e e ment with both beta decay and muon capture. T h e s e n u c l e a r wave functions and t h e i r coefficients of f r a c t i o n a l p a r e n t a g e w e r e calculated by u s e of the Argonne DELPHI c o m p u t e r p r o gram. T r a n s i t i o n r a t e s to all t h e J ~ =0 +, 1+, 2+ and 3+ states p r e d i c t e d by the C o h e n - K u r a t h model [12] were calculated. T h e r e w e r e 22 such s t a t e s with e n e r g i e s r a n g i n g up to 25 MeV above the 12B ground state. Of these, only the two listed in t a b l e 1 made a c o n t r i b u t i o n of 0.01% or g r e a t e r to R (the n u m b e r of r a d i a t i v e t r a n s i t i o n s p e r stopped pion). The values of R for each of the t h r e e sets of coefficients a r e tabulated, and an e s t i m a t e d e r r o r of 50~o has been added. This a r i s e s roughly from t h r e e s o u r c e s : 25% from u n c e r t a i n t i e s in the Hamilton±an, 10% from the n u c l e a r and pion wave functions and 15% from the p a r a m e t e r s in eq. (1). C o m p a r i s o n with e x p e r i m e n t a l is h a m p e r e d by lack of a g r e e m e n t between the two published r e s u l t s [2, 15] of work in which s o m e n u c l e a r s t r u c t u r e could be resolved. The energy r a n g e in t a b l e 1 c o r r e s p o n d s to the region of the f i r s t peak of B i s t i r l i c h et al. [2]. (The r e m a i n d e r of
Table 1 Even-parity states in 12B that contribute 0.01% or more to R (the number of radiative transitions per stopped pion) Rs and Rp are the contributions to R from the l s and 2p pionic Bohr orbits. The heading 1, 2 and 3 refer to three alternative sets of coefficients in the Hamilton±an of eq. (2); 1 refers to the coefficients taken directly from ref. [4], 2 to the coefficients of ref. [5] while in 3, the coefficients for 2p capture were evaluated (as described in the text) from pion photoproduetion at q ~ m~. 1
energy (MeV)
J~"
125.9 124.5
1+ 2+
R total
2
Rp
(10 -3 )
(10-3)
(10-3)
(10-3)
(10-3)
(10-3)
0.98 0.14
0.28 0.21
1.01 0.14
0.35 0.21
0.98 0.14
0.14 0.07
1.6 ± 0.8
Rs
3
Rs
Rp
1.7 ± 0.9
Rs
Rp
1.3 ± 0.7 273
Volume 36B, number 4
PHYSICS LETTERS
t h e i r s p e c t r u m m u s t r e s u l t from t r a n s i t i o n s to o d d - p a r i t y s t a t e s and to continuum. ) The e x p e r i m e n t a l r a t i o s in this energy region a r e given as R = (0.97 ± 0.09)x 10 -3 by B i s t i r l i c h et al. [2] and R = (5.8± 1.3)× 10 -3 by H i l s c h e r et al. [15]. All the calculated c a p t u r e r a t i o s in table 1 a r e a g r e e m e n t , to within the stated p r o b a b l e e r r o r s , with the e x p e r i m e n t a l r e s u l t s of B i s t i r l i c h et al. C l o s e s t a g r e e m e n t is obtained u s i n g H a m i l t o n i a n coefficients evaluated at q ~ m~ for c a p t u r e from the 2p orbit. B e s i d e s the difference in e x p e r i m e n t a l n u m b e r s , refs. [2] and [15] also d e t e r m i n e diff e r e n t values of the l s and 2p pion populations P . and P_p from the work of Koch et al. The i n t e r p r e t a t i o n u s e d h e r e was the s a m e as that of H i l s c h e r et al. The a l t e r n a t i v e v a l u e s listed by B i s t i r l i c h et al. w e r e Ps = 12.3 + 1.4% and Pp = 88.7± 1.4%. If t h e s e had been the n u m b e r s u s e d in our calculation, then it would have b e e n the t h r e s h o l d r e s u l t s that gave b e s t a g r e e m e n t with the values of B i s t i r l i c h et al. The high m o m e n tum r e s u l t s would have been about 50% lower but s t i l l in r e a s o n a b l e a g r e e m e n t . Our c o n c l u s i o n is that the g r e a t e s t u n c e r tainty in the t h e o r e t i c a l t r e a t m e n t for r a d i a t i v e pion c a p t u r e 12C lies in the choice of a H a m i l t o n i a n r a t h e r than in the wave functions. By u s e of the above p a r a m e t e r s we find that the b e s t a g r e e m e n t with the e x p e r i m e n t of B i s t i r lich et al. is attained by u s i n g H a m i l t o n i a n coefficients for l a r g e pion m o m e n t u m (~ ~ m~) i n s t e a d of the t h r e s h o l d v a l u e s in calculating c a p t u r e from the p orbit. C l e a r l y , m o r e e x p e r i m e n t s should be done in this r e g i o n to r e s o l v e the e x p e r i m e n t a l d i s c r e p a n c y and to d e t e r m i n e the r a n g e of validity of the t h e o r e t i c a l t r e a t m e n t . In a forthcoming p a p e r [16] these c a l c u l a t i o n s a r e extended to o d d - p a r i t y t r a n s i t i o n s . F o r the wave functions, this p a p e r u s e s a c o r e - c o u p l i n g model b a s e d on the C o h e n - K u r a t h d e s c r i p t i o n of the l p core; this model gives r e m a r k a b l e a g r e e m e n t with e x p e r i m e n t a l data on photoab-
274
20 September 1971
s o r p t i o n and m u o n capture. The r e s u l t s a r e that the positions and heights of the peaks in the u n c o r r e c t e d s p e c t r u m of B i s t i r l i c h et al. a r e well r e p r o d u c e d within the u n c e r t a i n t i e s outlined above. The author is grateful to Dr. M. H. M a c f a r l a n e for m a n y v a l u a b l e d i s c u s s i o n s and for a c r i t i c a l r e a d i n g of this paper. Computations for this work were g r e a t l y s i m p l i f i e d by p r o g r a m m i n g in SPEAKEASY [17].
References [1] H.Koch, M.Krell, Ch. v.d. Malsberg, G.Poelz, H. Schmitt, L. Tauscher, G. Backenstoss, S. Charalambus and H.Daniel, Phys. Letters 29B (1969) 140. [2] J. A. Bistirlich, K.M. Crowe, A. S. L. Parsons, P. Skarek and P. Truoel, Phys. Rev. Letters 25 (1970) 689. [3] G.F.Chew, M.L.Goldberger, F.E.Low and Y. Nambu, Phys. Rev. 106 (1957) 1345, hereafter referred to as CGLN. [4] M. Kawaguehi, H.Ohtsubo and Y. Sumi, Progr. Theoret. Phys. (Kyoto) Suppl. Extra Number (1968} 28. [5] J. Delorme and T. E. O. Ericson, Phys. Letters 21
(1966) 98. [6] M.Krell and T.E.O.Ericson, Nuel. Phys. B l l (1969) 521. [7] As quoted in ref. [2]. [8] P. Noelle, W. Pfeil and D. Schwela, Nucl. Phys. B26 (1971) 461. [9] F.A. Berends. A. Donnachie and D. L. Weaver, Nucl. Phys. B4 (1967) 54. [10] For example, J.R.Luyten, H.P.S.Rood and H.A.Tolhoek, Nucl. Phys. 41 (1963) 236. [11] E.J. Maier, R.M. Edelstein and R. T. Siegel, Phys. Rev° 133 (1964) B663. [12] R.W.Kavanagh, Phys. Rev. 133 (1964) B1504. [13] S. Cohen and D.Kurath, Nucl. Phys. 73 (1965) I. [14] A. Fujii, M. Morita and H.Ohtsubo, Progr. Theoret. Phys. (Kyoto) Suppl. Extra Number (1968) 303. [15] H. Hilscher, W.D. Krebs, G. Sepp and V. Soergel, Nucl. Phys. A158 (1970) 584. [16] S. Skupsky, submitted to Nucl. Phys. [ 17] S. Cohen and C. M. Vincent, Argonne Physics Division Informal Report PHY-1968E (1968).