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Pole dominance in the reaction 6Li(16O, d)20Ne
Volume 30B, number 3
POLE
PHYSICS LETTERS
DOMINANCE
IN THE
REACTION
29 September 1969
6Li(160,d)20Ne
M. J A M E E L *
Internalional Atomic Energy Agency, International Centre for Theoretical Physics, Miramare, Trieste, Italy Received 5 August 1969
A simple pole-dominance model is shown to give a satisfactory explanation of the observed angular distribution in the reaction 6Li(160, d) leading to 20Ne in the ground state.
R e c e n t l y , Cheon and Sen Gupta [1] have p r o pose d a double s c a t t e r i n g m o d e l within the f r a m e w o r k of c l u s t e r i n g t h e o r y in o r d e r to explain the o b s e r v e d a n g u l a r d i s t r i b u t i o n [2] in the 6Li(160,d) r e a c t i o n l ead i n g to 20Ne in the ground state. They obtain the b a c k w a r d peak through the p i c k -up of an alpha p a r t i c l e f r o m the 6Li t a r g e t nuc l e u s (a s e c o n d - o r d e r p r o c e s s ) while the f o r w a r d peak is r e p r o d u c e d by c a l c u l a t i n g a f o u r t h - o r d e r F e y n m a n d i a g r a m in plane wave Born approximation. The p u r p o s e of this note is to show that the e x p e r i m e n t a l d i s t r i b u t i o n can be s a t i s f a c t o r i l y u n d e r s t o o d without having to c o n s i d e r f o u r th - o r d e r p r o c e s s e s , p r o v i d e d one does not take the notion of c l u s t e r i n g too s e r i o u s l y . The p r e sent a p p r o a c h is b a s e d on the concept~ of pole d o m i n a n c e of the r e a c t i o n amplitude. We c o n s i d e r the r e a c t i o n
6Li + 160~ d +20Ne
(i)
to proceed principally through the two modes depicted in fig. 1, viz. the virtual exchange of the nuclei 4He and 14N. We assume further that these nuclides are exchanged in their ground s t a t e s , which is a r e a s o n a b l e a s s u m p t i o n , p a r t i c u l a r l y b e c a u s e we a r e only c o n s i d e r i n g the c a s e in which the final 20Ne nucleus is a l s o in the ground state. Since 4He c a r r i e s no spin and 14N has spin one, the c o n t r i b u t i o n of t h e s e two diag r a m s to the r e a c t i o n a m p l i t u d e can be w r i t t e n :~Permanent address: Pakistan Atomic Energy Com,_mission, Karachi, Pakistan. Considerations of singularities in the amplitude were first introdueed in direet nuclear reaction theory by Amado and Shapiro [3].
He20
d
d
Ne 20
Li6
LI6
Fig. 1. The two diagrams whose contribution to the amplitude has been evaluated. as
A(E) P 0 ( c o s 0 )
T:
u-M
S(E) Pl(COSO)
+
(5)
w h e r e M 1 and M 2 a r e the m a s s e s of 4He and 14N r e s p e c t i v e l y , t and u a r e the u su al M a n d e l s t a m m o m e n t u m t r a n s f e r v a r i a b l e s , 0 is the s c a t t e r i n g angle in the c e n t r e - o f - m a s s s y s t e m and the L e g e n d r e p o l y n o m i a l s r e f l e c t the a n g u l a r m o m e n ta c a r r i e d by the exchanged p a r t i c l e s . The e n e r gy-dependent functions A (E) and B(E) a r e r e l a t e d to f o r m f a c t o r s at the v e r t i c l e s but, si n ce they do not involve the s c a t t e r i n g angle, they a r e not i m p o r t a n t f o r the p r e s e n t a n a l y s i s . In fact, at given r e a c t i o n e n e r g y , they can be t r e a t e d a s constant p a r a m e t e r s . F u r t h e r , as the a m p l i t u d e is anyway u n d e t e r m i n e d to an o v e r - a l l n o r m a l i zation f a c t o r , t h e r e is indeed only one a d j u s t a b l e p a r a m e t e r in this model, which we call CE to u n d e r s c o r e its e n e r g y dependence. The a m p l i t u d e (2) can then be r e w r i t t e n a s 1
T-u-M
C E " cos 0
2+
t-M 2
(3)
The explicit e x p r e s s i o n s f o r l and u a r e given below f o r the sake of c o m p l e t e n e s s :
155
Volume 30B, n u m b e r 3
PHYSICS
/)2
u = (rn a - m d )2 -
p ,2
2pp'
DI 2
YR a l~l c
marn d (P~29+P'2 + 2pp' ~2
LETTERS
29 September 1969
(4) c o s 0)
3O0
c o s 0)
mamd
(5) w h e r e mR, rob, m c a n d rn d a r e t h e m a s s e s , r e s p e c t i v e l y , of 1 6 0 , 6 L i , d a n d 20Ne; p a n d p ' a r e g i v e n i n t e r m s of t h e c e n t r e - o f - m a s s energy E a n d t h e Q - v a l u e of t h e r e a c t i o n a s p2 = 2 m a rnb . m a + rn b ,
p , 2 _ 2 m c rnd rnc + rn d
(E+Q) .
Finally, the reaction cross-section in arbitrary units by the expression
cr(E,O ) = N E ITI 2
(6)
is given (7)
w h e r e N E is a suitable chosen normalization factor which would, in general, vary with energy. T h e c a l c u l a t i o n s h a v e b e e n c a r r i e d out a t E -= 20 M e V , w h i c h i s t h e c e n t r e - o f - m a s s energy of t h e e x p e r i m e n t . T h e a n g u l a r d i s t r i b u t i o n i s f o u n d to b e q u i t e i n s e n s i t i v e to t h e v a l u e of t h e p a r a m e t e r C E s o l o n g a s it i s not c l o s e to z e r o . This agrees with our expectation, as the vertex f u n c t i o n s s h o u l d not a f f e c t t h e a n g u l a r d i s t r i b u t i o n t o o s t r o n g l y . T h e c u r v e s h o w n in fig. 2 h a s b e e n d r a w n w i t h C E ~ 3.5, a v a l u e c h o s e n s o that the experimental points are spread more or less uniformly about the predicted curve. The a g r e e m e n t of t h e o r y w i t h e x p e r i m e n t i s r a t h e r s t r i k i n g i n v i e w of t h e s i m p l i c i t y of t h e m o d e l c h o s e n f o r t h e p r e s e n t c a l c u l a t i o n s . It m a y a l s o be noted that both the theoretical curve and the experimental distribution exhibit similar a s y m m e t r y about 90 ° , although the m i n i m u m of the curve is s o m e w h a t displaced from the lowest dip in the observations. W e can therefore conclude that the pole-dominance model (without any rescattering corrections) gives a very satisfactory explanation of the angular distribution observed in the 6Li(160, d)20Ne reaction.
156
I00
|
30
6o
90 0c.a.a.
12o
15o
in d e g r e e s
Fig. 2. C o m p a r i s o n of the predictions of the p r e s e n t work (solid curve) with the e x p e r i m e n t a l data for the r e a c t i o n 6Li(160, d)20Ne at 20 MeV leading to the ground state. I w o u l d l i k e to a c k n o w l e d g e a n i n t e r e s t i n g d i s c u s s i o n w i t h D r . I1-T. C h e o n . It i s a l s o a p l e a s u r e to t h a n k P r o f e s s o r s A b d u s S a l a m a n d P. Budini as well as the International Atomic Energy Agency for hospitality at the International Centre for Theoretical Physics, Trieste.
References 1. I1-T. Chcon and H. M. Sen Gupta, Phys. L e t t e r s 29B • (1969) 268 and IC]?P, T r i e s t e , p r e p r i n t l C / 6 9 / 1 8 (1969) Nucl. Phys., to be published 2. D. W. Heikkinen and W. Feldman, Nucl. Phys. Al13 (1968) 57. 3. R.D. Amado, Phys. L e t t e r s 2 (1959) 399. I.S. Shapiro, Nucl. Phys. 28 (1961) 244.
POLE
PHYSICS LETTERS
DOMINANCE
IN THE
REACTION
29 September 1969
6Li(160,d)20Ne
M. J A M E E L *
Internalional Atomic Energy Agency, International Centre for Theoretical Physics, Miramare, Trieste, Italy Received 5 August 1969
A simple pole-dominance model is shown to give a satisfactory explanation of the observed angular distribution in the reaction 6Li(160, d) leading to 20Ne in the ground state.
R e c e n t l y , Cheon and Sen Gupta [1] have p r o pose d a double s c a t t e r i n g m o d e l within the f r a m e w o r k of c l u s t e r i n g t h e o r y in o r d e r to explain the o b s e r v e d a n g u l a r d i s t r i b u t i o n [2] in the 6Li(160,d) r e a c t i o n l ead i n g to 20Ne in the ground state. They obtain the b a c k w a r d peak through the p i c k -up of an alpha p a r t i c l e f r o m the 6Li t a r g e t nuc l e u s (a s e c o n d - o r d e r p r o c e s s ) while the f o r w a r d peak is r e p r o d u c e d by c a l c u l a t i n g a f o u r t h - o r d e r F e y n m a n d i a g r a m in plane wave Born approximation. The p u r p o s e of this note is to show that the e x p e r i m e n t a l d i s t r i b u t i o n can be s a t i s f a c t o r i l y u n d e r s t o o d without having to c o n s i d e r f o u r th - o r d e r p r o c e s s e s , p r o v i d e d one does not take the notion of c l u s t e r i n g too s e r i o u s l y . The p r e sent a p p r o a c h is b a s e d on the concept~ of pole d o m i n a n c e of the r e a c t i o n amplitude. We c o n s i d e r the r e a c t i o n
6Li + 160~ d +20Ne
(i)
to proceed principally through the two modes depicted in fig. 1, viz. the virtual exchange of the nuclei 4He and 14N. We assume further that these nuclides are exchanged in their ground s t a t e s , which is a r e a s o n a b l e a s s u m p t i o n , p a r t i c u l a r l y b e c a u s e we a r e only c o n s i d e r i n g the c a s e in which the final 20Ne nucleus is a l s o in the ground state. Since 4He c a r r i e s no spin and 14N has spin one, the c o n t r i b u t i o n of t h e s e two diag r a m s to the r e a c t i o n a m p l i t u d e can be w r i t t e n :~Permanent address: Pakistan Atomic Energy Com,_mission, Karachi, Pakistan. Considerations of singularities in the amplitude were first introdueed in direet nuclear reaction theory by Amado and Shapiro [3].
He20
d
d
Ne 20
Li6
LI6
Fig. 1. The two diagrams whose contribution to the amplitude has been evaluated. as
A(E) P 0 ( c o s 0 )
T:
u-M
S(E) Pl(COSO)
+
(5)
w h e r e M 1 and M 2 a r e the m a s s e s of 4He and 14N r e s p e c t i v e l y , t and u a r e the u su al M a n d e l s t a m m o m e n t u m t r a n s f e r v a r i a b l e s , 0 is the s c a t t e r i n g angle in the c e n t r e - o f - m a s s s y s t e m and the L e g e n d r e p o l y n o m i a l s r e f l e c t the a n g u l a r m o m e n ta c a r r i e d by the exchanged p a r t i c l e s . The e n e r gy-dependent functions A (E) and B(E) a r e r e l a t e d to f o r m f a c t o r s at the v e r t i c l e s but, si n ce they do not involve the s c a t t e r i n g angle, they a r e not i m p o r t a n t f o r the p r e s e n t a n a l y s i s . In fact, at given r e a c t i o n e n e r g y , they can be t r e a t e d a s constant p a r a m e t e r s . F u r t h e r , as the a m p l i t u d e is anyway u n d e t e r m i n e d to an o v e r - a l l n o r m a l i zation f a c t o r , t h e r e is indeed only one a d j u s t a b l e p a r a m e t e r in this model, which we call CE to u n d e r s c o r e its e n e r g y dependence. The a m p l i t u d e (2) can then be r e w r i t t e n a s 1
T-u-M
C E " cos 0
2+
t-M 2
(3)
The explicit e x p r e s s i o n s f o r l and u a r e given below f o r the sake of c o m p l e t e n e s s :
155
Volume 30B, n u m b e r 3
PHYSICS
/)2
u = (rn a - m d )2 -
p ,2
2pp'
DI 2
YR a l~l c
marn d (P~29+P'2 + 2pp' ~2
LETTERS
29 September 1969
(4) c o s 0)
3O0
c o s 0)
mamd
(5) w h e r e mR, rob, m c a n d rn d a r e t h e m a s s e s , r e s p e c t i v e l y , of 1 6 0 , 6 L i , d a n d 20Ne; p a n d p ' a r e g i v e n i n t e r m s of t h e c e n t r e - o f - m a s s energy E a n d t h e Q - v a l u e of t h e r e a c t i o n a s p2 = 2 m a rnb . m a + rn b ,
p , 2 _ 2 m c rnd rnc + rn d
(E+Q) .
Finally, the reaction cross-section in arbitrary units by the expression
cr(E,O ) = N E ITI 2
(6)
is given (7)
w h e r e N E is a suitable chosen normalization factor which would, in general, vary with energy. T h e c a l c u l a t i o n s h a v e b e e n c a r r i e d out a t E -= 20 M e V , w h i c h i s t h e c e n t r e - o f - m a s s energy of t h e e x p e r i m e n t . T h e a n g u l a r d i s t r i b u t i o n i s f o u n d to b e q u i t e i n s e n s i t i v e to t h e v a l u e of t h e p a r a m e t e r C E s o l o n g a s it i s not c l o s e to z e r o . This agrees with our expectation, as the vertex f u n c t i o n s s h o u l d not a f f e c t t h e a n g u l a r d i s t r i b u t i o n t o o s t r o n g l y . T h e c u r v e s h o w n in fig. 2 h a s b e e n d r a w n w i t h C E ~ 3.5, a v a l u e c h o s e n s o that the experimental points are spread more or less uniformly about the predicted curve. The a g r e e m e n t of t h e o r y w i t h e x p e r i m e n t i s r a t h e r s t r i k i n g i n v i e w of t h e s i m p l i c i t y of t h e m o d e l c h o s e n f o r t h e p r e s e n t c a l c u l a t i o n s . It m a y a l s o be noted that both the theoretical curve and the experimental distribution exhibit similar a s y m m e t r y about 90 ° , although the m i n i m u m of the curve is s o m e w h a t displaced from the lowest dip in the observations. W e can therefore conclude that the pole-dominance model (without any rescattering corrections) gives a very satisfactory explanation of the angular distribution observed in the 6Li(160, d)20Ne reaction.
156
I00
|
30
6o
90 0c.a.a.
12o
15o
in d e g r e e s
Fig. 2. C o m p a r i s o n of the predictions of the p r e s e n t work (solid curve) with the e x p e r i m e n t a l data for the r e a c t i o n 6Li(160, d)20Ne at 20 MeV leading to the ground state. I w o u l d l i k e to a c k n o w l e d g e a n i n t e r e s t i n g d i s c u s s i o n w i t h D r . I1-T. C h e o n . It i s a l s o a p l e a s u r e to t h a n k P r o f e s s o r s A b d u s S a l a m a n d P. Budini as well as the International Atomic Energy Agency for hospitality at the International Centre for Theoretical Physics, Trieste.
References 1. I1-T. Chcon and H. M. Sen Gupta, Phys. L e t t e r s 29B • (1969) 268 and IC]?P, T r i e s t e , p r e p r i n t l C / 6 9 / 1 8 (1969) Nucl. Phys., to be published 2. D. W. Heikkinen and W. Feldman, Nucl. Phys. Al13 (1968) 57. 3. R.D. Amado, Phys. L e t t e r s 2 (1959) 399. I.S. Shapiro, Nucl. Phys. 28 (1961) 244.