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Role of tensor force in charge exchange reactions
Volume
30B,
number
ROLE
4
OF
PHYSICS
TENSOR
FORCE
LETTERS
13 October
IN C H A R G E
EXCHANGE
1969
REACTIONS
E. ROST and P. D. KUNZ
Department of Physics, University of Colorado, Boulder, Colorado, USA Received 20 August 1969
The discrepancy between theory and experiment in the analysis of the 54Fe(3He,t)54Co reaction is r e moved by including tensor force in the effective interaction.
The (3He, t) r e a c t i o n has been u s e d s u c c e s s fully in the e x c i t a t i o n of analogue s t a t e s [1]. The t h e o r e t i c a l i n t e r p r e t a t i o n of t h e s e e x p e r i m e n t s by a c h a r g e - e x c h a n g e m e c h a n i s m in a conventional DWBA c a l c u l a t i o n g i v e s a c o n s i s t e n t explanation to the data and c o m p a r e s f a v o r a b l y with the a n a l y s i s of (p, n) e x p e r i m e n t s [2]. H o w e v e r , r e c e n t (3He, t) e x p e r i m e n t s which e x c i t e nonanalogue s t a t e s in the final n u c l e u s have been r e p o r t e d [3,4] and indicate a s e r i o u s d e s c r e p a n c y b e t w e e n the t h e o r y and e x p e r i m e n t . T h i s l e t t e r p r o p o s e s an explanation which r e m o v e s the difficulty with the t h e o r y . In the usual a n a l y s i s the c h a r g e exchange p r o c e s s is c a u s e d by an e f f e c t i v e t r a n s i t i o n o p e r a t o r b et ween the point p r o j e c t i l e (o) and a t a r g e t nucleon (k) of the f o r m Vef f : v o. r k (V.r+V(yTao'ak) g(] r o - r k [ ) ,
(1)
w h e r e g(r) is taken to have Yukawa shape with a r a n g e of about 1 fm; crand ~-are t w i c e the spin and i s o b a r i c spin o p e r a t i o n s r e s p e c t i v e l y . The m a t r i x e l e m e n t of Veff ~s t a k e n between the initial and final n u c l e a r wave functions and the r e s u l t i n g function of the p r o j e c t i l e c o o r d i n a t e s is i n s e r t e d into a s t a n d a r d DWBA p r o g r a m [*]. F o r 0 + ~ 1+ t r a n s i t i o n s in the 180(3He, t)18F~reaction [3] and for t r a n s i t i o n s to odd J, even p a r i t y s t a t e s of 54Co in the 54Fe(3He, t)54Co r e a c t i o n only the VaT t e r m in (1) c o n t r i b u t e s ( a s s u m i n g (f~)-2 conf i g u r a t i o n s in the l a t t e r ) . The DWBA t h e o r y p r e dic ts that the c r o s s s e c t i o n is d o m i n a t e d by o r bital a n g u l a r m o m e n t u m t r a n s f e r L = J - 1 with only a s m a l l a d m i x t u r e of L = J + 1. T h e data, Work supported in part by the U. S. Atomic Energy Commission. * If spin-orbit effects in distorted waves can be ignored as is very reasonable for mass 3 projectiles, the sum over L is incoherent.
on the o t h e r hand, a r e b e t t e r d e s c r i b e d by the r e v e r s e situation, i .e., the h i g h e r L fits b e t t e r [3,4]. This r e s u l t is not s e n s i t i v e to c o n f i g u r a t i o n mixing; in p a r t i c u l a r f o r e x c i t a t i o n of the J = 7 st at e in 54Co it is not p o s s i b l e to a c h i e v e the des i r e d L-- 8 a d m i x t u r e within the fp shell. We p r o p o s e that the e f f e c t i v e i n t e r a c t i o n for c h a r g e - exchange t r a n s i t i o n s r e q u i r e s an additional term, /
Veff = v o" v k I(V~- + Vcrl- a o ' a k) g(r) + + VTE N Sok h(r) I Sok = 3 r - 2 ( a o . r ) ( a k . r ) r~
(2)
- ao.a k
ro-r k
the e x t r a t e r m b ei n g a t e n s o r f o r c e . Since an e x t r a an g u l ar m o m e n t u m t r a n s f e r of 2 is a v a i l a b l e through the t e n s o r o p e r a t o r , this f o r c e will allow f o r the e x t r a a n g u l a r m o m e n t u m needed for the J = 7 t r a n s i t i o n in the 54Fe(3He, t)54Co r e a c t i o n . Two functional f o r m s w e r e u s e d and a r e O P E P : g(r) = exp (- p r ) / ( ~ r ) ,
h (r) = g(r) [ l + 3 / ( p r ) + 3/(l~r) 2] Yon- = V T E N ,
~ = 0.7 f m - 1
(3)
E F F : g(r) = e x p ( - p r ) / ( p r ) = h(r) V~z = VTEN ,
P = 1.0 f m - 1
The O P E P f o r m is s t a n d a r d and is obtained by exchanging a p s e u d o s c a l a r pion b et w een two nuc l e o n s [5]. It should be a c c u r a t e at l a r g e s e p a r a tions but it is quite s i n g u l a r and probably a poor r e p r e s e n t a t i o n at s m a l l r e l a t i v e d i s t a n c e s . The e f f e c t i v e E F F f o r m is an e m p i r i c a l i n t e r a c t i o n 231
Volume 30B, n u m b e r 4
k 5C
PHYSICS
LETTERS
•r'~
57.7 MeV OPEP POTENTIAL
1
54Fe(3He,t)54Co
I
54Fe(3He,t)54Co k,/r ~
13 October 1969
i
~
fl~
:57.7 MeV
EFF POTENTIAL
• i i+ Ex=O94MeV 2o~
\1'
~,\"
il 21
Ld t-CO Jl|
I ~Nk
20
+
"10
I0
20
30 40 OC.M.
50
60
70
Fig. 1. Differential c r o s s sections for the 54Fe(3He,t)54Co reaction at 37.5 MeV. The spin assignments were taken from ref. 5 and the 1.82 MeV (3+ ) and 2.10 MeV (5+ ) a s signments are tentative. The solid curves are DW]3A fits using the OPEP effective interaction d e s c r i b e d in the text. The numbers on the right r e f e r to the potential strength used in fitting the curves to the data. c h o s e n to g i v e t h e s a m e r a d i a l d e p e n d e n c e f o r t h e t e n s o r p a r t a s h a s b e e n u s e d in t h e V.r a n d V~-r t e r m s by o t h e r a u t h o r s [3,4]. The theoretical cross sections are compared w i t h t h e d a t a of r e f . [4] in f i g s . 1 a n d 2 u s i n g t h e O P E P and E F F p r e s c r i p t i o n s f o r t h e e f f e c t i v e i n t e r a c t i o n . T h e n u m b e r s on t h e r i g h t of t h e c u r v e s g i v e t h e v a l u e of V(~T = V T E N u s e d in drawing the t h e o r e t i c a l c u r v e s . The fits a r e far b e t t e r t h a n with t h e p u r e c e n t r a l f o r m in r e f . [4]. It i s s e e n t h a t t h e a n g u l a r d i s t r i b u t i o n s f o r t h e J = 7 t r a n s i t i o n a r e p u s h e d out to l a r g e r a n g l e s ; indeed, the O P E P curve contains p e r h a p s too m u c h L = 8. On t h e o t h e r h a n d , t h e J = l fit i s b e t t e r w i t h O P E P a l t h o u g h e a r l i e r w o r k [1] h a s shown that L = 0 angular distributions are very s e n s i t i v e t o d e t a i l s of t h e DWBA c a l c u l a t i o n . Since the n o r m a l i z a t i o n s a r e m o r e constant with E F F it m a y b e p r e f e r a b l e , a l t h o u g h m o r e a n a l y s e s will b e r e q u i r e d t o b e t t e r pin down t h e
232
I0
20
50
40
50
60
70
OC.M. Fig. 2. Same as caption to fig. 1 except that the E F F potential d e s c r i b e d in the text is used. functional form of the effective interaction. However, for non- analogue charge- exchange reactions it is clear that a rather strong t e n s o r force is an essential ingredient.
References 1. A.G. Blair and H. E. Wegner, Phys. Rev. Letters 9 (1962) 168; J. J. Wesolowski, E.H. Schwartz, P. G. Roos and C. A. Ludemann, Phys. Rev. 169 (1968) 878; P. D. Kunz, E. Host, G. D. Jones, R. H. Johnson and S. I. Hayakawa, to be published. 2. V.A. Madsen, Nuel. Phys. 80 (1966) 177; G. R. Satehler, Nucl. Phys. A95 (1967) 1. 3. L. F. Hansen, M. L. Stelts, J. G. Vidal, J. J. Wesolowski and V. A. Madsen, Phys. Rev. 174 (1968) 1155; H. H. Duhm, C. Detraz and R. Finlay, Proc. Conf. on Nuclear Isospin, Asilomar> 1969; P. Kossanyi-Demay, P. Houssel, H. Faraggi and R. Schaeffer, to be published. 4. S. I. Hayakawa, J.J. Kraushaar, P. D. Kunz and E. Rost, Phys. Letters 29B (1969) 327. 5. A. Bohr and ]3. H. Mottelson, Nuclear structure, Vol. I (W. A. ]3enjamin, Inc., New York, 1969). 6. J . J . Schwartz, R. Sherr and T. Bhattia, Bull. Am. Phys. Soc. 13 (1968) 1446.
30B,
number
ROLE
4
OF
PHYSICS
TENSOR
FORCE
LETTERS
13 October
IN C H A R G E
EXCHANGE
1969
REACTIONS
E. ROST and P. D. KUNZ
Department of Physics, University of Colorado, Boulder, Colorado, USA Received 20 August 1969
The discrepancy between theory and experiment in the analysis of the 54Fe(3He,t)54Co reaction is r e moved by including tensor force in the effective interaction.
The (3He, t) r e a c t i o n has been u s e d s u c c e s s fully in the e x c i t a t i o n of analogue s t a t e s [1]. The t h e o r e t i c a l i n t e r p r e t a t i o n of t h e s e e x p e r i m e n t s by a c h a r g e - e x c h a n g e m e c h a n i s m in a conventional DWBA c a l c u l a t i o n g i v e s a c o n s i s t e n t explanation to the data and c o m p a r e s f a v o r a b l y with the a n a l y s i s of (p, n) e x p e r i m e n t s [2]. H o w e v e r , r e c e n t (3He, t) e x p e r i m e n t s which e x c i t e nonanalogue s t a t e s in the final n u c l e u s have been r e p o r t e d [3,4] and indicate a s e r i o u s d e s c r e p a n c y b e t w e e n the t h e o r y and e x p e r i m e n t . T h i s l e t t e r p r o p o s e s an explanation which r e m o v e s the difficulty with the t h e o r y . In the usual a n a l y s i s the c h a r g e exchange p r o c e s s is c a u s e d by an e f f e c t i v e t r a n s i t i o n o p e r a t o r b et ween the point p r o j e c t i l e (o) and a t a r g e t nucleon (k) of the f o r m Vef f : v o. r k (V.r+V(yTao'ak) g(] r o - r k [ ) ,
(1)
w h e r e g(r) is taken to have Yukawa shape with a r a n g e of about 1 fm; crand ~-are t w i c e the spin and i s o b a r i c spin o p e r a t i o n s r e s p e c t i v e l y . The m a t r i x e l e m e n t of Veff ~s t a k e n between the initial and final n u c l e a r wave functions and the r e s u l t i n g function of the p r o j e c t i l e c o o r d i n a t e s is i n s e r t e d into a s t a n d a r d DWBA p r o g r a m [*]. F o r 0 + ~ 1+ t r a n s i t i o n s in the 180(3He, t)18F~reaction [3] and for t r a n s i t i o n s to odd J, even p a r i t y s t a t e s of 54Co in the 54Fe(3He, t)54Co r e a c t i o n only the VaT t e r m in (1) c o n t r i b u t e s ( a s s u m i n g (f~)-2 conf i g u r a t i o n s in the l a t t e r ) . The DWBA t h e o r y p r e dic ts that the c r o s s s e c t i o n is d o m i n a t e d by o r bital a n g u l a r m o m e n t u m t r a n s f e r L = J - 1 with only a s m a l l a d m i x t u r e of L = J + 1. T h e data, Work supported in part by the U. S. Atomic Energy Commission. * If spin-orbit effects in distorted waves can be ignored as is very reasonable for mass 3 projectiles, the sum over L is incoherent.
on the o t h e r hand, a r e b e t t e r d e s c r i b e d by the r e v e r s e situation, i .e., the h i g h e r L fits b e t t e r [3,4]. This r e s u l t is not s e n s i t i v e to c o n f i g u r a t i o n mixing; in p a r t i c u l a r f o r e x c i t a t i o n of the J = 7 st at e in 54Co it is not p o s s i b l e to a c h i e v e the des i r e d L-- 8 a d m i x t u r e within the fp shell. We p r o p o s e that the e f f e c t i v e i n t e r a c t i o n for c h a r g e - exchange t r a n s i t i o n s r e q u i r e s an additional term, /
Veff = v o" v k I(V~- + Vcrl- a o ' a k) g(r) + + VTE N Sok h(r) I Sok = 3 r - 2 ( a o . r ) ( a k . r ) r~
(2)
- ao.a k
ro-r k
the e x t r a t e r m b ei n g a t e n s o r f o r c e . Since an e x t r a an g u l ar m o m e n t u m t r a n s f e r of 2 is a v a i l a b l e through the t e n s o r o p e r a t o r , this f o r c e will allow f o r the e x t r a a n g u l a r m o m e n t u m needed for the J = 7 t r a n s i t i o n in the 54Fe(3He, t)54Co r e a c t i o n . Two functional f o r m s w e r e u s e d and a r e O P E P : g(r) = exp (- p r ) / ( ~ r ) ,
h (r) = g(r) [ l + 3 / ( p r ) + 3/(l~r) 2] Yon- = V T E N ,
~ = 0.7 f m - 1
(3)
E F F : g(r) = e x p ( - p r ) / ( p r ) = h(r) V~z = VTEN ,
P = 1.0 f m - 1
The O P E P f o r m is s t a n d a r d and is obtained by exchanging a p s e u d o s c a l a r pion b et w een two nuc l e o n s [5]. It should be a c c u r a t e at l a r g e s e p a r a tions but it is quite s i n g u l a r and probably a poor r e p r e s e n t a t i o n at s m a l l r e l a t i v e d i s t a n c e s . The e f f e c t i v e E F F f o r m is an e m p i r i c a l i n t e r a c t i o n 231
Volume 30B, n u m b e r 4
k 5C
PHYSICS
LETTERS
•r'~
57.7 MeV OPEP POTENTIAL
1
54Fe(3He,t)54Co
I
54Fe(3He,t)54Co k,/r ~
13 October 1969
i
~
fl~
:57.7 MeV
EFF POTENTIAL
• i i+ Ex=O94MeV 2o~
\1'
~,\"
il 21
Ld t-CO Jl|
I ~Nk
20
+
"10
I0
20
30 40 OC.M.
50
60
70
Fig. 1. Differential c r o s s sections for the 54Fe(3He,t)54Co reaction at 37.5 MeV. The spin assignments were taken from ref. 5 and the 1.82 MeV (3+ ) and 2.10 MeV (5+ ) a s signments are tentative. The solid curves are DW]3A fits using the OPEP effective interaction d e s c r i b e d in the text. The numbers on the right r e f e r to the potential strength used in fitting the curves to the data. c h o s e n to g i v e t h e s a m e r a d i a l d e p e n d e n c e f o r t h e t e n s o r p a r t a s h a s b e e n u s e d in t h e V.r a n d V~-r t e r m s by o t h e r a u t h o r s [3,4]. The theoretical cross sections are compared w i t h t h e d a t a of r e f . [4] in f i g s . 1 a n d 2 u s i n g t h e O P E P and E F F p r e s c r i p t i o n s f o r t h e e f f e c t i v e i n t e r a c t i o n . T h e n u m b e r s on t h e r i g h t of t h e c u r v e s g i v e t h e v a l u e of V(~T = V T E N u s e d in drawing the t h e o r e t i c a l c u r v e s . The fits a r e far b e t t e r t h a n with t h e p u r e c e n t r a l f o r m in r e f . [4]. It i s s e e n t h a t t h e a n g u l a r d i s t r i b u t i o n s f o r t h e J = 7 t r a n s i t i o n a r e p u s h e d out to l a r g e r a n g l e s ; indeed, the O P E P curve contains p e r h a p s too m u c h L = 8. On t h e o t h e r h a n d , t h e J = l fit i s b e t t e r w i t h O P E P a l t h o u g h e a r l i e r w o r k [1] h a s shown that L = 0 angular distributions are very s e n s i t i v e t o d e t a i l s of t h e DWBA c a l c u l a t i o n . Since the n o r m a l i z a t i o n s a r e m o r e constant with E F F it m a y b e p r e f e r a b l e , a l t h o u g h m o r e a n a l y s e s will b e r e q u i r e d t o b e t t e r pin down t h e
232
I0
20
50
40
50
60
70
OC.M. Fig. 2. Same as caption to fig. 1 except that the E F F potential d e s c r i b e d in the text is used. functional form of the effective interaction. However, for non- analogue charge- exchange reactions it is clear that a rather strong t e n s o r force is an essential ingredient.
References 1. A.G. Blair and H. E. Wegner, Phys. Rev. Letters 9 (1962) 168; J. J. Wesolowski, E.H. Schwartz, P. G. Roos and C. A. Ludemann, Phys. Rev. 169 (1968) 878; P. D. Kunz, E. Host, G. D. Jones, R. H. Johnson and S. I. Hayakawa, to be published. 2. V.A. Madsen, Nuel. Phys. 80 (1966) 177; G. R. Satehler, Nucl. Phys. A95 (1967) 1. 3. L. F. Hansen, M. L. Stelts, J. G. Vidal, J. J. Wesolowski and V. A. Madsen, Phys. Rev. 174 (1968) 1155; H. H. Duhm, C. Detraz and R. Finlay, Proc. Conf. on Nuclear Isospin, Asilomar> 1969; P. Kossanyi-Demay, P. Houssel, H. Faraggi and R. Schaeffer, to be published. 4. S. I. Hayakawa, J.J. Kraushaar, P. D. Kunz and E. Rost, Phys. Letters 29B (1969) 327. 5. A. Bohr and ]3. H. Mottelson, Nuclear structure, Vol. I (W. A. ]3enjamin, Inc., New York, 1969). 6. J . J . Schwartz, R. Sherr and T. Bhattia, Bull. Am. Phys. Soc. 13 (1968) 1446.