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Microscopic form factors for charge exchange reactions to unbound residual states
Volume 40B, number 1
PHYSICS LETTERS
MICROSCOPIC FORM FACTORS REACTIONS TO UNBOUND
12 June 1972
FCR CHARGE EXCHANGE RESIDUAL STATES *
W. R. COKER and G. W. HOFFMANN
Center for Nuclear Studies, Umvers~ty of Texas, Austin, Texas, USA Received 11 April 1972
It is shown that, although final nuclear states polupated in charge exchange reactmns on most nuclei with A > 60 are proton unbound, there is at most a 10% enhancement of the mmroscoplc charge-exchange from factor when the decay of the final state is taken into account.
A n u m b e r of e x p e r i m e n t a l [1-3] and t h e o r e t i cal [4-6] p a p e r s have r e c e n t l y dealt with direct p r o t o n - t r a n s f e r r e a c t i o n s , p a r t i c u l a r l y (d, n) and (3He, d), which populate i s o b a r i c analog r e s o n a n c e s as r e s i d u a l n u c l e a r s t a t e s . It is found both e x p e r i m e n t a l l y and t h e o r e t i c a l l y that the t r a n s f e r r e a c t m n c r o s s s e c t i o n s for population of r e s o n a n c e s a r e often 2 to 5 t i m e s what they would be ff computed by the usual d i s t o r t e d wave Born a p p r o x i m a t i o n (DWBA), with the p a r e n t n e u t r o n bound state function used as form factor [4-7]. Such an e n h a n c e m e n t is expected on s i m p l e phymcal grounds: the e n h a n c e m e n t is g r e a t e s t for t r a n s i t i o n s with the g r e a t e s t a n g u l a r m o m e n tum m i s m a t c h , since these t r a n s l t m n s a r e s e n sitive to the shape of the form factor over a region extending r a t h e r far beyond the n u c l e a r s u r f a c e region [5, 8]. However, it has also been suggested r e c e n t l y that, s i n c e the q u a s i - e l a s t i c (p, n) r e a c t i o n on t a r g e t s with A > 60 also populates unbound r e sidual s t a t e s , "in a heavy n u c l e u s t h e r e may be a f a c t o r - o f - 2 [ e n h a n c e m e n t of the c h a r g e exchange (p, n) c r o s s section] whmh would then d r a s t i c a l l y a l t e r the p a r a m e t e r s of s y m m e t r y p o t e n t i a l s deduced from [such (p, n)] data" [7]. Offhand, one would not think so. C o n s i d e r the way that m i c r o s c o p i c charge exchange form f a c t o r s a r e c u s t o m a r i l y computed [9, 10]. The r a d i a l p a r t of the form factor is given by
I ( r ) = fRp(ri)Rn(ri)go(ri, r) r21 dri
(1)
where Rn, p(r) is the r a d i a l wave function of the i n i t i a l , final single n u c l e a r p a r t i c l e involved m * Supported in part by the U. S Atomic Energy Commmslon under Contract AT-(40-1)-2972.
the reaction. It is common p r a c t i c e to choose the n u c l e o n - n u c l e o n i n t e r a c t i o n to have the radial f o r m g ( r ) = e x p ( - ~ r ) , / ~ r , with ~ about 1 fm -1 [9, 10]. The f u n c t i o n g o ( r i , r ) is the zeroth o r d e r coefficient in the multlpole e x p a n s m n of g ( r - r i), in coordinates r, r i Since such an m t e r a c t l o n g is r e a l i s t i c a l l y s h o r t - r a n g e d , and R n ( r ) is well localized within the n u c l e a r volume, it is difficult to see how Io(r) could be very s e n s i t i v e to the b e h a v i o r of Rp(r) outside the nucleus. To test these c o n s i d e r a t i o n s quantitatively, the m i c r o s c o p i c charge exchange form factor was computed for the r e a c t i o n s 91Zr(p, n)91NbA and 209Bl(p, n)209poA. The form factor for each r e a c t i o n was calculated in two different ways. In the f i r s t , conventional, calculation (A) l d e n t m a l radial^_ wave functions were used for R n ( r ) and R~(r). Thus the r a d i a l shape of the i s o b a r i c analog state (IAS) function was taken to be identical to that of the p a r e n t bound n e u t r o n state function [5]. In the second calculation (R), RR(r) was obtained from a r e s o n a n c e w a v e - f u n c t m n for a proton in the IAS, computed with the s h e l l model theory of r e a c t i o n s developed by Bledsoe and T a m u r a [11]. In such an approach, rRR(r) is given by
rRn(r) +P
f
v2(E') ~p(E ')(E R - E ')-1 dE' 1
+1(2To+ 1)~ ( r - Fp)~p(ER)/2~(ER).
(2)
Here v(E) =
Volume 40B, number 1
PHYSICS
Such an e x p r e s s i o n f o r rRp(r) is the r e s u l t of i n s e r t i n g the full e n e r g y - d e p e n d e n t r e s o n a n c e s t a t e f u n c t i o n [11] into the DWBA e x p r e s s i o n f o r the t h r e e - b o d y (p, n~) c r o s s s e c t i o n [12] and i n t e g r a t i n g to e x t r a c t the (p, n) c r o s s s e c t i o n : d(z _ ; f df~n ~
d3c ~ dEnd~2~ . df~ndf~dE n
LETTERS
Table 1 Volume integral of the microscopic charge exchange form factor, using the state function of the parent analog to describe the residual analog resonance (column A), or using a resonance state function from the shell model theory of reactions of ref. [11] (column R) *. ~ is the reciprocal range of the Yukawa potential in fm -1.
(3)
Overlap H e r e A lS the e x p e r i m e n t a l width of the o b s e r v e d n e u t r o n group. T h e f o r m o b t a i n e d f o r the c r o s s s e c t i o n dcr/df~ n is then i d e n t i c a l to that of the o r d i n a r y DWBA c r o s s s e c t i o n for q u a s i - e l a s t i c (p, .n) to a bound f i n a l s t a t e , with eq. (2) f o r rRA(r). A s d i s c u s s e d in r e f . [5], t h i s r e d u c t i o n zs a c o n v e n i e n t p r o p e r t y of a r e s o n a n c e s t a t e f u n c t m n g e n e r a t e d by a s h e l l - m o d e l r e a c t i o n t h e o r y , and s t e m s f r o m the e x p l i c i t f a c t o r a b l e e n e r g y d e p e n d e n c e of the f u n c t m n [11]. T h e e x p r e s s m n (2) was c o m p u t e d u s i n g the
(2°9poAIVl2°9Bi> I I
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5
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15
r (fm)
Fig. 1. The radial part of the microscopic charge e x change f ~ v fact°r2~Pr°/~riate for 91Zr(p, n)9INb A and for Bi(p, n) Po . In each case the solid line is the form factor computed using the radial wave function of the parent neutron state to describe the analog state. The dashed lines show the real and imaginary parts of the form factor computed using a resonance radial wave function from a shell model theory of reactions [11], The nucleon-nucleon i n t e r a c tion, of Yukawa form, has ~ = 1 0 fm -1 82
12 June 1972
91Zr-91Nb A
209Bi-209poA
/~
A
R
0.5 1.0 1.5 0.5 1.0 1.5
3821.4 478.0 141.8 4933.4 617.0 183.0
3798.6 475.1 140.9 5470.9 684.3 203.0
* Only the real part of the volume integral is gwen in column R. p r o g r a m J P 3 : the only input r e q u i r e d , o t h e r than the p r o t o n o p t i c a l m o d e l p o t e n t i a l and the n e u t r o n s e p a r a t i o n e n e r g y , is the e x p e r i m e n t a l t o t a l width of the IAS w h e n t h e r e a r e m a n y open c h a n n e l s [13]. T h e c a l c u l a t i o n s for Io(r) w e r e m a d e with the C A T H E N A s u b r o u t i n e s f r o m DWUCK [14]. R e s u l t s a r e shown, for p = 1.0 fm -1, in fig. l(a) and (b). T h e f o r m f a c t o r s IoA and I R for 9 1 Z r ( p , n)91Nb a r e e s s e n t i a l l y i d e n t i c a l , although the r e s o n a n c e f o r m f a c t o r IR(r) h a s a s m a l l i m a g i n a r y p a r t due to the fact that rRR(r) is c o m p l e x . T h e r e a l p a r t s of the f o r m f a c t o r s for 209Bi(p, n)209po a r e a l s o s i m i l a r , d i f f e r i n g at p e a k by only 15%, d e s p i t e the g r e a t d i f f e r e n c e m E R , F, and F p r e l a t z v e to the 9 1 Z r c a s e . T h e B e c c h e t t i - G r e e n l e e s o p t i c a l p a r a m e t e r s [15] w e r e u s e d , e x c e p t that WD was set to 4.0 MeV f o r 9 1 Z r [16]. T h e r e s o n a n c e p a r a m e t e r s for the 9 1 Z r c a s e w e r e [13, 1 6 ] E R = 4 . 7 1 M e V , F p = 3 . 1 k e V , F = 19.4 keV: for 209B1, E R = 14.97 M e V , F p = 13.4 keV, F = 300 keV. T h e bound s t a t e f u n c t i o n s w e r e c o m p u t e d with p o t e n t i a l s h a v i n g the Becchettl-Greenlees geometry. T h e s e n s i t i v i t y of the e n h a n c e m e n t to the int e r a c t i o n r a n g e w a s s t u d i e d by v a y i n g p f r o m 0.5 to 1.5 fm -1, s p a n n i n g the r a n g e of v a l u e s c o m m o n l y u s e d in DWBA a n a l y s i s of (p, n) and (I-, t) d a t a with m i c r o s c o p i c f o r m f a c t o r s . It w a s found that, a l t h o u g h the v o l u m e i n t e g r a l of the f o r m f a c t o r I A i n c r e a s e s by a f a c t o r of a l m o s t 30 as one g o e s f r o m ~ = 1.5 to 0.5 fm -1, the e n h a n c e m e n t of the r e a l p a r t of the i n t e g r a l r e m a i n s c o n s t a n t at a r o u n d 10.3% f o r Bi, a s shown m the t a b l e : t h e r e is no e n h a n c e m e n t for Zr. F i n a l l y , the 1.0 fm -1 r a n g e f r o m f a c t o r s I A and Io1~ for 209Bi(p, n ) 2 0 9 p o A w e r e u s e d m a
Volume 40B, n u m b e r 1
PHYSICS
D W B A c a l c u l a t i o n a t a n i n c i d e n t p r o t o n e n e r g y of 35 M e V , w i t h t h e o p t i c a l m o d e l p a r a m e t e r s of Becchetti and Greenlees; the ratio (a(p, n)R/'cr(p, n) A) i s f o u n d to b e 1.25. In s u m m a r y , e v e n f o r a n IAS in t h e l e a d r e g i o n , t h e e n h a n c e m e n t of t h e q u a s i - e l a s t i c (p, n) c r o s s - s e c t i o n in D W B A , d u e to t h e p a r t i c l e u n s t a b l e n a t u r e of t h e r e s i d u a l s t a t e , i s o n l y a r o u n d 20%, p r o b a b l y s o m e w h a t l e s s t h a n t h e a b s o l u t e e r r o r in t h e m a g n i t u d e of t h e e x p e r i m e n t a l (p, n) c r o s s s e c t i o n s to b e fit. T h e p r o g r a m J P 3 w i t h w h i c h rRR(r) w a s c o m p u t e d w a s w r i t t e n b y D r s . T. T a m u r a , H. B l e d s o e , J. W o l t e r a n d W. R. C o k e r . W e a r e g r a t e f u l to P r o f e s s o r T a m u r a f o r i n s t r u c t i v e comments and inspiration.
References [1] R. L. M c G r a t h , N. Cue, W. R. Hering, L. L. Lee, B. L L m b l e r and Z Vager, Phys Rev L e t t e r s 25 (1970) 682. [2] S. A. A. Zaidl, C L. Hollas, J. L. Horton, P. J. Riley, J. L. C. Ford and C. M. J o n e s , Phys. Rev. L e t t e r s 25 (1970) 1503.
LETTERS
12 June 1972
[3] D. H. Youngblood and R L. Kozub, Phys Rev. L e t t e r s 26 (1970) 572. [4] B. J Cole, R. Huby and J R. Mines, Phys. Rev. L e t t e r s 26 (1971) 204. [5] S. A. A. Zaldl and W. R. Coker, Phys. Rev. C4 (1971) 236. [6] H.T Fortune and C M Vincent, Phys. Rev Letters 27 (1971) 1664. [7] U Strobusch, H J Korner, G. C. Morrlson and J P. Schiffer, Phys. Rev. Letters 28 (1972) 47. [8]W.R. Coker and S A A Zaidi, Nucl. Phys. , to be published. [9] M B. Johnson, L W Owen and G R Satcher, Phys Rev. 142 (1966) 748. [10] J.J. Wesolowskl, E H Sehwarez, P. J Roos and C A Ludemann, Phys. Rev. 169 (1968) 878. [11] H Bledsoeand T Tamura, Nuel. Phys. A164 (1971) 191. [12] F. S. Levin, Ann Phys. (N.Y.) 46 (1968) 41. [13] W.R. Coker, H Bledsoeand T. Tamura, to be pubhshed. [14] P. D. Kunz, Univ. of Colorado, private eommumcation. [15] F D Becchettl and G W Greenlees, Phys. Rev. 182 (1969) 1190. [16] W. Kretschmer and G. Graw, Phys. Rev. Letters 27 (1971) 1294
83
PHYSICS LETTERS
MICROSCOPIC FORM FACTORS REACTIONS TO UNBOUND
12 June 1972
FCR CHARGE EXCHANGE RESIDUAL STATES *
W. R. COKER and G. W. HOFFMANN
Center for Nuclear Studies, Umvers~ty of Texas, Austin, Texas, USA Received 11 April 1972
It is shown that, although final nuclear states polupated in charge exchange reactmns on most nuclei with A > 60 are proton unbound, there is at most a 10% enhancement of the mmroscoplc charge-exchange from factor when the decay of the final state is taken into account.
A n u m b e r of e x p e r i m e n t a l [1-3] and t h e o r e t i cal [4-6] p a p e r s have r e c e n t l y dealt with direct p r o t o n - t r a n s f e r r e a c t i o n s , p a r t i c u l a r l y (d, n) and (3He, d), which populate i s o b a r i c analog r e s o n a n c e s as r e s i d u a l n u c l e a r s t a t e s . It is found both e x p e r i m e n t a l l y and t h e o r e t i c a l l y that the t r a n s f e r r e a c t m n c r o s s s e c t i o n s for population of r e s o n a n c e s a r e often 2 to 5 t i m e s what they would be ff computed by the usual d i s t o r t e d wave Born a p p r o x i m a t i o n (DWBA), with the p a r e n t n e u t r o n bound state function used as form factor [4-7]. Such an e n h a n c e m e n t is expected on s i m p l e phymcal grounds: the e n h a n c e m e n t is g r e a t e s t for t r a n s i t i o n s with the g r e a t e s t a n g u l a r m o m e n tum m i s m a t c h , since these t r a n s l t m n s a r e s e n sitive to the shape of the form factor over a region extending r a t h e r far beyond the n u c l e a r s u r f a c e region [5, 8]. However, it has also been suggested r e c e n t l y that, s i n c e the q u a s i - e l a s t i c (p, n) r e a c t i o n on t a r g e t s with A > 60 also populates unbound r e sidual s t a t e s , "in a heavy n u c l e u s t h e r e may be a f a c t o r - o f - 2 [ e n h a n c e m e n t of the c h a r g e exchange (p, n) c r o s s section] whmh would then d r a s t i c a l l y a l t e r the p a r a m e t e r s of s y m m e t r y p o t e n t i a l s deduced from [such (p, n)] data" [7]. Offhand, one would not think so. C o n s i d e r the way that m i c r o s c o p i c charge exchange form f a c t o r s a r e c u s t o m a r i l y computed [9, 10]. The r a d i a l p a r t of the form factor is given by
I ( r ) = fRp(ri)Rn(ri)go(ri, r) r21 dri
(1)
where Rn, p(r) is the r a d i a l wave function of the i n i t i a l , final single n u c l e a r p a r t i c l e involved m * Supported in part by the U. S Atomic Energy Commmslon under Contract AT-(40-1)-2972.
the reaction. It is common p r a c t i c e to choose the n u c l e o n - n u c l e o n i n t e r a c t i o n to have the radial f o r m g ( r ) = e x p ( - ~ r ) , / ~ r , with ~ about 1 fm -1 [9, 10]. The f u n c t i o n g o ( r i , r ) is the zeroth o r d e r coefficient in the multlpole e x p a n s m n of g ( r - r i), in coordinates r, r i Since such an m t e r a c t l o n g is r e a l i s t i c a l l y s h o r t - r a n g e d , and R n ( r ) is well localized within the n u c l e a r volume, it is difficult to see how Io(r) could be very s e n s i t i v e to the b e h a v i o r of Rp(r) outside the nucleus. To test these c o n s i d e r a t i o n s quantitatively, the m i c r o s c o p i c charge exchange form factor was computed for the r e a c t i o n s 91Zr(p, n)91NbA and 209Bl(p, n)209poA. The form factor for each r e a c t i o n was calculated in two different ways. In the f i r s t , conventional, calculation (A) l d e n t m a l radial^_ wave functions were used for R n ( r ) and R~(r). Thus the r a d i a l shape of the i s o b a r i c analog state (IAS) function was taken to be identical to that of the p a r e n t bound n e u t r o n state function [5]. In the second calculation (R), RR(r) was obtained from a r e s o n a n c e w a v e - f u n c t m n for a proton in the IAS, computed with the s h e l l model theory of r e a c t i o n s developed by Bledsoe and T a m u r a [11]. In such an approach, rRR(r) is given by
rRn(r) +P
f
v2(E') ~p(E ')(E R - E ')-1 dE' 1
+1(2To+ 1)~ ( r - Fp)~p(ER)/2~(ER).
(2)
Here v(E) =
Volume 40B, number 1
PHYSICS
Such an e x p r e s s i o n f o r rRp(r) is the r e s u l t of i n s e r t i n g the full e n e r g y - d e p e n d e n t r e s o n a n c e s t a t e f u n c t i o n [11] into the DWBA e x p r e s s i o n f o r the t h r e e - b o d y (p, n~) c r o s s s e c t i o n [12] and i n t e g r a t i n g to e x t r a c t the (p, n) c r o s s s e c t i o n : d(z _ ; f df~n ~
d3c ~ dEnd~2~ . df~ndf~dE n
LETTERS
Table 1 Volume integral of the microscopic charge exchange form factor, using the state function of the parent analog to describe the residual analog resonance (column A), or using a resonance state function from the shell model theory of reactions of ref. [11] (column R) *. ~ is the reciprocal range of the Yukawa potential in fm -1.
(3)
Overlap H e r e A lS the e x p e r i m e n t a l width of the o b s e r v e d n e u t r o n group. T h e f o r m o b t a i n e d f o r the c r o s s s e c t i o n dcr/df~ n is then i d e n t i c a l to that of the o r d i n a r y DWBA c r o s s s e c t i o n for q u a s i - e l a s t i c (p, .n) to a bound f i n a l s t a t e , with eq. (2) f o r rRA(r). A s d i s c u s s e d in r e f . [5], t h i s r e d u c t i o n zs a c o n v e n i e n t p r o p e r t y of a r e s o n a n c e s t a t e f u n c t m n g e n e r a t e d by a s h e l l - m o d e l r e a c t i o n t h e o r y , and s t e m s f r o m the e x p l i c i t f a c t o r a b l e e n e r g y d e p e n d e n c e of the f u n c t m n [11]. T h e e x p r e s s m n (2) was c o m p u t e d u s i n g the
(2°9poAIVl2°9Bi> I I
io
5
4
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4
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Imogmory
._T'n°r'\. ,
5
5
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15
r (fm)
Fig. 1. The radial part of the microscopic charge e x change f ~ v fact°r2~Pr°/~riate for 91Zr(p, n)9INb A and for Bi(p, n) Po . In each case the solid line is the form factor computed using the radial wave function of the parent neutron state to describe the analog state. The dashed lines show the real and imaginary parts of the form factor computed using a resonance radial wave function from a shell model theory of reactions [11], The nucleon-nucleon i n t e r a c tion, of Yukawa form, has ~ = 1 0 fm -1 82
12 June 1972
91Zr-91Nb A
209Bi-209poA
/~
A
R
0.5 1.0 1.5 0.5 1.0 1.5
3821.4 478.0 141.8 4933.4 617.0 183.0
3798.6 475.1 140.9 5470.9 684.3 203.0
* Only the real part of the volume integral is gwen in column R. p r o g r a m J P 3 : the only input r e q u i r e d , o t h e r than the p r o t o n o p t i c a l m o d e l p o t e n t i a l and the n e u t r o n s e p a r a t i o n e n e r g y , is the e x p e r i m e n t a l t o t a l width of the IAS w h e n t h e r e a r e m a n y open c h a n n e l s [13]. T h e c a l c u l a t i o n s for Io(r) w e r e m a d e with the C A T H E N A s u b r o u t i n e s f r o m DWUCK [14]. R e s u l t s a r e shown, for p = 1.0 fm -1, in fig. l(a) and (b). T h e f o r m f a c t o r s IoA and I R for 9 1 Z r ( p , n)91Nb a r e e s s e n t i a l l y i d e n t i c a l , although the r e s o n a n c e f o r m f a c t o r IR(r) h a s a s m a l l i m a g i n a r y p a r t due to the fact that rRR(r) is c o m p l e x . T h e r e a l p a r t s of the f o r m f a c t o r s for 209Bi(p, n)209po a r e a l s o s i m i l a r , d i f f e r i n g at p e a k by only 15%, d e s p i t e the g r e a t d i f f e r e n c e m E R , F, and F p r e l a t z v e to the 9 1 Z r c a s e . T h e B e c c h e t t i - G r e e n l e e s o p t i c a l p a r a m e t e r s [15] w e r e u s e d , e x c e p t that WD was set to 4.0 MeV f o r 9 1 Z r [16]. T h e r e s o n a n c e p a r a m e t e r s for the 9 1 Z r c a s e w e r e [13, 1 6 ] E R = 4 . 7 1 M e V , F p = 3 . 1 k e V , F = 19.4 keV: for 209B1, E R = 14.97 M e V , F p = 13.4 keV, F = 300 keV. T h e bound s t a t e f u n c t i o n s w e r e c o m p u t e d with p o t e n t i a l s h a v i n g the Becchettl-Greenlees geometry. T h e s e n s i t i v i t y of the e n h a n c e m e n t to the int e r a c t i o n r a n g e w a s s t u d i e d by v a y i n g p f r o m 0.5 to 1.5 fm -1, s p a n n i n g the r a n g e of v a l u e s c o m m o n l y u s e d in DWBA a n a l y s i s of (p, n) and (I-, t) d a t a with m i c r o s c o p i c f o r m f a c t o r s . It w a s found that, a l t h o u g h the v o l u m e i n t e g r a l of the f o r m f a c t o r I A i n c r e a s e s by a f a c t o r of a l m o s t 30 as one g o e s f r o m ~ = 1.5 to 0.5 fm -1, the e n h a n c e m e n t of the r e a l p a r t of the i n t e g r a l r e m a i n s c o n s t a n t at a r o u n d 10.3% f o r Bi, a s shown m the t a b l e : t h e r e is no e n h a n c e m e n t for Zr. F i n a l l y , the 1.0 fm -1 r a n g e f r o m f a c t o r s I A and Io1~ for 209Bi(p, n ) 2 0 9 p o A w e r e u s e d m a
Volume 40B, n u m b e r 1
PHYSICS
D W B A c a l c u l a t i o n a t a n i n c i d e n t p r o t o n e n e r g y of 35 M e V , w i t h t h e o p t i c a l m o d e l p a r a m e t e r s of Becchetti and Greenlees; the ratio (a(p, n)R/'cr(p, n) A) i s f o u n d to b e 1.25. In s u m m a r y , e v e n f o r a n IAS in t h e l e a d r e g i o n , t h e e n h a n c e m e n t of t h e q u a s i - e l a s t i c (p, n) c r o s s - s e c t i o n in D W B A , d u e to t h e p a r t i c l e u n s t a b l e n a t u r e of t h e r e s i d u a l s t a t e , i s o n l y a r o u n d 20%, p r o b a b l y s o m e w h a t l e s s t h a n t h e a b s o l u t e e r r o r in t h e m a g n i t u d e of t h e e x p e r i m e n t a l (p, n) c r o s s s e c t i o n s to b e fit. T h e p r o g r a m J P 3 w i t h w h i c h rRR(r) w a s c o m p u t e d w a s w r i t t e n b y D r s . T. T a m u r a , H. B l e d s o e , J. W o l t e r a n d W. R. C o k e r . W e a r e g r a t e f u l to P r o f e s s o r T a m u r a f o r i n s t r u c t i v e comments and inspiration.
References [1] R. L. M c G r a t h , N. Cue, W. R. Hering, L. L. Lee, B. L L m b l e r and Z Vager, Phys Rev L e t t e r s 25 (1970) 682. [2] S. A. A. Zaidl, C L. Hollas, J. L. Horton, P. J. Riley, J. L. C. Ford and C. M. J o n e s , Phys. Rev. L e t t e r s 25 (1970) 1503.
LETTERS
12 June 1972
[3] D. H. Youngblood and R L. Kozub, Phys Rev. L e t t e r s 26 (1970) 572. [4] B. J Cole, R. Huby and J R. Mines, Phys. Rev. L e t t e r s 26 (1971) 204. [5] S. A. A. Zaldl and W. R. Coker, Phys. Rev. C4 (1971) 236. [6] H.T Fortune and C M Vincent, Phys. Rev Letters 27 (1971) 1664. [7] U Strobusch, H J Korner, G. C. Morrlson and J P. Schiffer, Phys. Rev. Letters 28 (1972) 47. [8]W.R. Coker and S A A Zaidi, Nucl. Phys. , to be published. [9] M B. Johnson, L W Owen and G R Satcher, Phys Rev. 142 (1966) 748. [10] J.J. Wesolowskl, E H Sehwarez, P. J Roos and C A Ludemann, Phys. Rev. 169 (1968) 878. [11] H Bledsoeand T Tamura, Nuel. Phys. A164 (1971) 191. [12] F. S. Levin, Ann Phys. (N.Y.) 46 (1968) 41. [13] W.R. Coker, H Bledsoeand T. Tamura, to be pubhshed. [14] P. D. Kunz, Univ. of Colorado, private eommumcation. [15] F D Becchettl and G W Greenlees, Phys. Rev. 182 (1969) 1190. [16] W. Kretschmer and G. Graw, Phys. Rev. Letters 27 (1971) 1294
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