- Email: [email protected]
Method of pseudo-bound states for stripping to unbound levels
Volume 33B, number 5
METHOD
OF
PHYSICS
PSEUDO-BOUND
STATES
LETTERS
FOR
STRIPPING
9 November
TO
UNBOUND
1970
LEVELS
B. J. C O L E , R. HUBY and J. R. MINES University of Liverpool, Liverpool, UK Received 26 September 1970
A simple approximate method of performing DWBA calculations for stripping to unbound levels is presented together with some applications. The justification of the method is considered in an accompaying letter.
Stripping reactions can lead to states of the residual nucleus which are unstable aginst nucleon decay. For example, recent measurements of the reaction, A(3He, d)B*, include deuteron angular distributions corresponding to the two-stage process A+
3He ~ d+
B* \
p+A. These deuteron angular distributions are similar in character to those for stripping to particle stable states of the residual nucleus B*. The decay of B* is usually unobserved. There is considerable interest in the DWBA analysis of the increasing volume of (3He, d), (d, n) and (d,p) data [1-8] for unbound levels; one wishes to determine the orbital angular momentum l of the captured nucleon and a suitable defined spectroscopic factor 02 . In this letter we present a simplified method of performing the analysis, the justification of which is given by Huby in an accompanying letter [9]. The DWBA theory appropriate to stripping to unbound levels has been investigated previously [i0]. It was found to be closely related to the DWBA for bound levels of the residual nucleus. The essential differences are, firstly, the replacement of the bound state wave-function for the captured nucleon by a wave-function representing the resonant scattering of the nucleon on the target nucleus A (one selects a single partial-wave for a particular unbound level) and, secondly, an integration over the width of the level. A trial calculation was carried out for the 160(d, p) 170 reaction leading to the unbound 5.08 MeV single-particle d3/2 state of 170 [6, 7]. Here the resonant scattering wave-function was derived from a real potential known to correctly 320
r e p r o d u c e the o b s e r v e d 160(n, n) p h a s e shifts. T h e slowly c o n v e r g i n g r a d i a l i n t e g r a l s [10] w e r e e v a l u a t e d u s i n g the d e v i c e of a G a u s s i a n c o n v e r g e n c e f a c t o r . (A b e t t e r p r o c e d u r e , r e c e n t l y p r o p o s e d by F o r t u n e and V i n c e n t [11], i s to u s e the t e c h n i q u e s of c o n t o u r i n t e g r a t i o n . ) The i n t e g r a tion o v e r the width of the l e v e l was p e r f o r m e d n u m e r i c a l l y . T h i s c a l c u l a t i o n and a m o r e r e c e n t one of F o r t u n e and V i n c e n t [11] y i e l d s a t i s f a c t o r y agreement with experiment. B e f o r e m a k i n g a m o r e g e n e r a l a p p l i c a t i o n of the t h e o r y , one h a s to c o n s i d e r how the r e s o n a n t w a v e - f u n c t i o n should be c h o s e n when one no l o n g e r h a s a p u r e s i n g l e - p a r t i c l e r e s o n a n c e . One a p p r o a c h , which h a s d i s t i n c t a d v a n t a g e s , is to e m p l o y W e i n b e r g ' s q u a s i - p a r t i c l e s [12]. T h i s c h o i c e l e a d s n a t u r a l l y to a j u s t i f a e a t i o n of the m e t h o d of p s e u d o - b o u n d s t a t e s o u t l i n e d in the f o l l o w i n g p a r a g r a p h . (Bang and Z i m f i n y i [13] t a k e a " G a m o w " w a v e - f u n c t i o n with an e x p o n e n t i a l l y i n c r e a s i n g tail.) The p s e u d o - b o u n d s t a t e s m e t h o d h a s the m e r i t of s i m p l i c i t y in that it i s a l m o s t i d e n t i c a l to the DWBA f o r bound l e v e l s and a v o i d s the p r o b l e m s a s s o c i a t e d with the r a d i a l and e n e r g y i n t e g r a l s . H o w e v e r , i t s d e r i v a t i o n [9] i n v o l v e s a p p r o x i m a t i o n s r e l a t i n g to a s m o o t h c u t - o f f of the o s c i l l a t o r y t a i l of the s c a t t e r i n g w a v e - f u n c t i o n and the e n e r g y d e p e n d e n c e of the T - m a t r i x e l e m e n t . T h e m e t h o d is the following. The w e l l depth of a r e a l S a x o n - W o o d s p o t e n t i a l i s a d j u s t e d to g i v e a r e s o n a n c e in the lth p a r t i a l w a v e ( c o r r e s p o n d i n g to a n u c l e a r p h a s e shift of at the o b s e r v e d e n e r g y of the l e v e l . (The n u m e r i c a l t e c h n i q u e i s e s s e n t i a l l y the s a m e a s that e m p l o y e d f o r bound s t a t e s . ) F o r a n a r r o w unbound l e v e l the r e s o n a n t s c a t t e r i n g w a v e - f u n c t i o n has a l a r g e a m p l i t u d e i n s i d e the n u c l e u s w h e r e it r e s e m b l e s a bound s t a t e w a v e - f u n c t i o n . We m o d i f y t h i s w a v e - f u n c t i o r
Volume 33B, number 5
PHYSICS
b y c u t t i n g it off s m o o t h l y in t h e r e g i o n of the f i r s t n o d e o u t s i d e t h e n u c l e u s . It i s t h e n n o r m a l i s e d to unity and substituted for the bound state wavef u n c t i o n in t h e D W B A f o r b o u n d l e v e l s . C r o s s s e c t i o n s c o m p u t e d in t h i s way a r e c o m p a r e d w i t h the experimental cross-sections integrated over t h e w i d t h of the l e v e l s a n d s p e c t r o s c o p i c f a c t o r s a r e d e d u c e d in t h e u s u a l way. S u c h c a l c u l a t i o n s should be reliable for states which are only slightly unbound and become more uncertain as the excitation energy increases. Somewhat simi l a r d e v i c e s h a v e b e e n p r o p o s e d e l s e w h e r e [3, 5]. A n u m b e r of c a l c u l a t i o n s h a v e b e e n m a d e u s i n g t h e a b o v e m e t h o d . In f i g s . 1 a n d 2 we g i v e e x a m p l e s of c a l c u l a t i o n s we h a v e m a d e f o r t h e 1 5 N ( ~ H e , d) 1 6 0 , 2 0 N e ( d , n) 2 1 N a a n d 4 2 C a ( d , n) 4 3 C a r e a c t i o n s . S a t i s f a c t o r y f i t s to a n g u l a r d i s tributions have been obtained for levels unbound b y a s m u c h a s 2 ½ M e V . T h e s e n s i t i v i t y of t h e c a l c u l a t i o n s to t h e m e t h o d of c u t t i n g off t h e resonant scattering wave-function has been inv e s t i g a t e d . T h i s w a s f o u n d to b e i n s i g n i f i c a n t f o r l e v e l s u n b o u n d b y up to 1 M e V b u t g i v e s r i s e to u n c e r t a i n t i e s in t h e m a g n i t u d e of t h e p r e d i c t e d c r o s s - s e c t i o n f o r m o r e h i g h l y e x c i t e d l e v e l s . In o r d e r to p r o p e r l y e s t a b l i s h t h e d o m a i n of r e l i a b i l ity of t h e m e t h o d , it i s n e c e s s a r y to m a k e c o m p a r i s o n s w i t h m o r e p r e c i s e c a l c u l a t i o n s , e.g. t h o s e f o r 1 6 0 , w h i c h do n o t i n c l u d e t h e a b o v e m e n t i o n e d a p p r o x i m a t i o n s . T h i s w o r k i s n o w in progress.
+
LETTERS
9 November 1970
15N(3He,d)16 O
20Ne (d, rn)21 N a
unbound
~(e)
unDound
(arb. units) L=2
e e•• ~
30
60
90
Cm
!
L=2
30
60
90
Fig. 2. Results of computations for data f r o m r e f s . [1] and [8]. An interesting comparison can be made between the spectroscopic factors deduced from D W B A c a l c u l a t i o n s f o r s t r i p p i n g to u n b o u n d levels and the same information deduced from t h e a n a l y s i s of t h e c o r r e s p o n d i n g e l a s t i c s c a t tering measurements. A n u m b e r of e x a m p l e s a r e g i v e n i n t a b l e 1. In t h e s e c a s e s r e a s o n a b l e agreement is obtained taking into account all the uncertainties in the data and analysis.
42Ca (d.n)435c
Table 1
0-(8)
d
(arb. units)
30
60
90
~Cm
Levels of 160
30
0
J Y
~
l
Spectrosc£pic f a c t o r s (3He, d ) (p, p)
12.44
1-
½
0
0.80
0.64
12.53
2-
~
2
1.24
0.9
12.97
2-
}
2
1.08
0.75
13.26 21Na
3-
}
2
1.0 (d, n)
0.70 (p, p)
4.29
~5+
~
2
0.15
0.10
4.47
~
}
2
0.27
0.33
3+
9
Fig. 1. Results of computations using the pseudo-bound method for data f r o m ref. [4].
The DWBA s p e c t r o s c o p i c f a c t o r s a r e f r o m our analysis of data f r o m r e f s . [1] and [8]. The (p, p) s p e c troscopic f a c t o r s were obtained by c o m p a r i n g e x p e r i mental widths taken f r o m compilations with computed s i n g l e - p a r t i c l e widths.
321
Volume 33B, number 5
PHYS1CS
The a u t h o r s w o u l d like t o t h a n k D r . J. S c h e e r a n d h i s g r o u p at the H a h n - M e i n e r - I n s t i t u t e , B e r l i n , f o r s t i m u l a t i n g t h i s w o r k . T h e y would a l s o like to t h a n k D r . P. D. F o r s y t h a n d h i s g r o u p f o r p r o v i d i n g d a t a p r i o r to p u b l i c a t i o n a n d f o r v a l u a b l e d i s c u s s i o n s . One of u s ( B . J . C . ) a c k n o w l e d g e s a S.R.C. grant.
References [1] W. Bohne, H. Homeyer, H. Lettau, H. Morgenstern, J. Scheer and F. Siehelsehmidt, Nucl. Phys. A128 (1969} 537. [2] T. B. Grundy, W.J. McDonald, W.K. Dawson and G.C. Neilson, Nucl. Phys. A l l l (1968) 469. [3] D. A. Gedcke, S.T. Lain, S.M. Tang, G.M. Stinson, J. T. Sample, T . B . Grundy, W.J. McDonald, W.K.
322
LETTERS
9 November 1970
Dawson and G . C . N e i l s o n , Nuel. Phys. A134 (1969) 141. [4] H. T, Fortune and C. M. Vincent, Phys. Rev. 185 (1969) 1401. [5] H. T. Fortune, T . J . Gray, W. Trost and N.R. F l e t cher, Phys. Rev. 179 (1969) 1033. [6] J. L. Alty, L. L. Green, R. Huby, G. D. Jones, J. R, Mines and J, F. Sharpey-Schafer, Nucl. Phys. A97 (1967) 541. [7] I. M. Naqib and L . L . Green, Nucl. Phys. A l l 2 (1968) 76. [8] C. J. Oliver, B. C. Walsh, P . D . F o r s y t h and G. Kaye, to be published. [9] R. Huby, P h y s i c s L e t t e r s 33B {1970) [10] R. Huby and J. R. Mines, Rev. Mod. Phys. 37 (1965) 406. [11] C. M. Vincent and H. T. Fortune, to be published. [12] R. Huby, Nucl. Phys. A138 (1969) 442 and to be published. [131 J. Band and J. ZimSnyi, Nuel. Phys. A139 (i969) 534.
METHOD
OF
PHYSICS
PSEUDO-BOUND
STATES
LETTERS
FOR
STRIPPING
9 November
TO
UNBOUND
1970
LEVELS
B. J. C O L E , R. HUBY and J. R. MINES University of Liverpool, Liverpool, UK Received 26 September 1970
A simple approximate method of performing DWBA calculations for stripping to unbound levels is presented together with some applications. The justification of the method is considered in an accompaying letter.
Stripping reactions can lead to states of the residual nucleus which are unstable aginst nucleon decay. For example, recent measurements of the reaction, A(3He, d)B*, include deuteron angular distributions corresponding to the two-stage process A+
3He ~ d+
B* \
p+A. These deuteron angular distributions are similar in character to those for stripping to particle stable states of the residual nucleus B*. The decay of B* is usually unobserved. There is considerable interest in the DWBA analysis of the increasing volume of (3He, d), (d, n) and (d,p) data [1-8] for unbound levels; one wishes to determine the orbital angular momentum l of the captured nucleon and a suitable defined spectroscopic factor 02 . In this letter we present a simplified method of performing the analysis, the justification of which is given by Huby in an accompanying letter [9]. The DWBA theory appropriate to stripping to unbound levels has been investigated previously [i0]. It was found to be closely related to the DWBA for bound levels of the residual nucleus. The essential differences are, firstly, the replacement of the bound state wave-function for the captured nucleon by a wave-function representing the resonant scattering of the nucleon on the target nucleus A (one selects a single partial-wave for a particular unbound level) and, secondly, an integration over the width of the level. A trial calculation was carried out for the 160(d, p) 170 reaction leading to the unbound 5.08 MeV single-particle d3/2 state of 170 [6, 7]. Here the resonant scattering wave-function was derived from a real potential known to correctly 320
r e p r o d u c e the o b s e r v e d 160(n, n) p h a s e shifts. T h e slowly c o n v e r g i n g r a d i a l i n t e g r a l s [10] w e r e e v a l u a t e d u s i n g the d e v i c e of a G a u s s i a n c o n v e r g e n c e f a c t o r . (A b e t t e r p r o c e d u r e , r e c e n t l y p r o p o s e d by F o r t u n e and V i n c e n t [11], i s to u s e the t e c h n i q u e s of c o n t o u r i n t e g r a t i o n . ) The i n t e g r a tion o v e r the width of the l e v e l was p e r f o r m e d n u m e r i c a l l y . T h i s c a l c u l a t i o n and a m o r e r e c e n t one of F o r t u n e and V i n c e n t [11] y i e l d s a t i s f a c t o r y agreement with experiment. B e f o r e m a k i n g a m o r e g e n e r a l a p p l i c a t i o n of the t h e o r y , one h a s to c o n s i d e r how the r e s o n a n t w a v e - f u n c t i o n should be c h o s e n when one no l o n g e r h a s a p u r e s i n g l e - p a r t i c l e r e s o n a n c e . One a p p r o a c h , which h a s d i s t i n c t a d v a n t a g e s , is to e m p l o y W e i n b e r g ' s q u a s i - p a r t i c l e s [12]. T h i s c h o i c e l e a d s n a t u r a l l y to a j u s t i f a e a t i o n of the m e t h o d of p s e u d o - b o u n d s t a t e s o u t l i n e d in the f o l l o w i n g p a r a g r a p h . (Bang and Z i m f i n y i [13] t a k e a " G a m o w " w a v e - f u n c t i o n with an e x p o n e n t i a l l y i n c r e a s i n g tail.) The p s e u d o - b o u n d s t a t e s m e t h o d h a s the m e r i t of s i m p l i c i t y in that it i s a l m o s t i d e n t i c a l to the DWBA f o r bound l e v e l s and a v o i d s the p r o b l e m s a s s o c i a t e d with the r a d i a l and e n e r g y i n t e g r a l s . H o w e v e r , i t s d e r i v a t i o n [9] i n v o l v e s a p p r o x i m a t i o n s r e l a t i n g to a s m o o t h c u t - o f f of the o s c i l l a t o r y t a i l of the s c a t t e r i n g w a v e - f u n c t i o n and the e n e r g y d e p e n d e n c e of the T - m a t r i x e l e m e n t . T h e m e t h o d is the following. The w e l l depth of a r e a l S a x o n - W o o d s p o t e n t i a l i s a d j u s t e d to g i v e a r e s o n a n c e in the lth p a r t i a l w a v e ( c o r r e s p o n d i n g to a n u c l e a r p h a s e shift of at the o b s e r v e d e n e r g y of the l e v e l . (The n u m e r i c a l t e c h n i q u e i s e s s e n t i a l l y the s a m e a s that e m p l o y e d f o r bound s t a t e s . ) F o r a n a r r o w unbound l e v e l the r e s o n a n t s c a t t e r i n g w a v e - f u n c t i o n has a l a r g e a m p l i t u d e i n s i d e the n u c l e u s w h e r e it r e s e m b l e s a bound s t a t e w a v e - f u n c t i o n . We m o d i f y t h i s w a v e - f u n c t i o r
Volume 33B, number 5
PHYSICS
b y c u t t i n g it off s m o o t h l y in t h e r e g i o n of the f i r s t n o d e o u t s i d e t h e n u c l e u s . It i s t h e n n o r m a l i s e d to unity and substituted for the bound state wavef u n c t i o n in t h e D W B A f o r b o u n d l e v e l s . C r o s s s e c t i o n s c o m p u t e d in t h i s way a r e c o m p a r e d w i t h the experimental cross-sections integrated over t h e w i d t h of the l e v e l s a n d s p e c t r o s c o p i c f a c t o r s a r e d e d u c e d in t h e u s u a l way. S u c h c a l c u l a t i o n s should be reliable for states which are only slightly unbound and become more uncertain as the excitation energy increases. Somewhat simi l a r d e v i c e s h a v e b e e n p r o p o s e d e l s e w h e r e [3, 5]. A n u m b e r of c a l c u l a t i o n s h a v e b e e n m a d e u s i n g t h e a b o v e m e t h o d . In f i g s . 1 a n d 2 we g i v e e x a m p l e s of c a l c u l a t i o n s we h a v e m a d e f o r t h e 1 5 N ( ~ H e , d) 1 6 0 , 2 0 N e ( d , n) 2 1 N a a n d 4 2 C a ( d , n) 4 3 C a r e a c t i o n s . S a t i s f a c t o r y f i t s to a n g u l a r d i s tributions have been obtained for levels unbound b y a s m u c h a s 2 ½ M e V . T h e s e n s i t i v i t y of t h e c a l c u l a t i o n s to t h e m e t h o d of c u t t i n g off t h e resonant scattering wave-function has been inv e s t i g a t e d . T h i s w a s f o u n d to b e i n s i g n i f i c a n t f o r l e v e l s u n b o u n d b y up to 1 M e V b u t g i v e s r i s e to u n c e r t a i n t i e s in t h e m a g n i t u d e of t h e p r e d i c t e d c r o s s - s e c t i o n f o r m o r e h i g h l y e x c i t e d l e v e l s . In o r d e r to p r o p e r l y e s t a b l i s h t h e d o m a i n of r e l i a b i l ity of t h e m e t h o d , it i s n e c e s s a r y to m a k e c o m p a r i s o n s w i t h m o r e p r e c i s e c a l c u l a t i o n s , e.g. t h o s e f o r 1 6 0 , w h i c h do n o t i n c l u d e t h e a b o v e m e n t i o n e d a p p r o x i m a t i o n s . T h i s w o r k i s n o w in progress.
+
LETTERS
9 November 1970
15N(3He,d)16 O
20Ne (d, rn)21 N a
unbound
~(e)
unDound
(arb. units) L=2
e e•• ~
30
60
90
Cm
!
L=2
30
60
90
Fig. 2. Results of computations for data f r o m r e f s . [1] and [8]. An interesting comparison can be made between the spectroscopic factors deduced from D W B A c a l c u l a t i o n s f o r s t r i p p i n g to u n b o u n d levels and the same information deduced from t h e a n a l y s i s of t h e c o r r e s p o n d i n g e l a s t i c s c a t tering measurements. A n u m b e r of e x a m p l e s a r e g i v e n i n t a b l e 1. In t h e s e c a s e s r e a s o n a b l e agreement is obtained taking into account all the uncertainties in the data and analysis.
42Ca (d.n)435c
Table 1
0-(8)
d
(arb. units)
30
60
90
~Cm
Levels of 160
30
0
J Y
~
l
Spectrosc£pic f a c t o r s (3He, d ) (p, p)
12.44
1-
½
0
0.80
0.64
12.53
2-
~
2
1.24
0.9
12.97
2-
}
2
1.08
0.75
13.26 21Na
3-
}
2
1.0 (d, n)
0.70 (p, p)
4.29
~5+
~
2
0.15
0.10
4.47
~
}
2
0.27
0.33
3+
9
Fig. 1. Results of computations using the pseudo-bound method for data f r o m ref. [4].
The DWBA s p e c t r o s c o p i c f a c t o r s a r e f r o m our analysis of data f r o m r e f s . [1] and [8]. The (p, p) s p e c troscopic f a c t o r s were obtained by c o m p a r i n g e x p e r i mental widths taken f r o m compilations with computed s i n g l e - p a r t i c l e widths.
321
Volume 33B, number 5
PHYS1CS
The a u t h o r s w o u l d like t o t h a n k D r . J. S c h e e r a n d h i s g r o u p at the H a h n - M e i n e r - I n s t i t u t e , B e r l i n , f o r s t i m u l a t i n g t h i s w o r k . T h e y would a l s o like to t h a n k D r . P. D. F o r s y t h a n d h i s g r o u p f o r p r o v i d i n g d a t a p r i o r to p u b l i c a t i o n a n d f o r v a l u a b l e d i s c u s s i o n s . One of u s ( B . J . C . ) a c k n o w l e d g e s a S.R.C. grant.
References [1] W. Bohne, H. Homeyer, H. Lettau, H. Morgenstern, J. Scheer and F. Siehelsehmidt, Nucl. Phys. A128 (1969} 537. [2] T. B. Grundy, W.J. McDonald, W.K. Dawson and G.C. Neilson, Nucl. Phys. A l l l (1968) 469. [3] D. A. Gedcke, S.T. Lain, S.M. Tang, G.M. Stinson, J. T. Sample, T . B . Grundy, W.J. McDonald, W.K.
322
LETTERS
9 November 1970
Dawson and G . C . N e i l s o n , Nuel. Phys. A134 (1969) 141. [4] H. T, Fortune and C. M. Vincent, Phys. Rev. 185 (1969) 1401. [5] H. T. Fortune, T . J . Gray, W. Trost and N.R. F l e t cher, Phys. Rev. 179 (1969) 1033. [6] J. L. Alty, L. L. Green, R. Huby, G. D. Jones, J. R, Mines and J, F. Sharpey-Schafer, Nucl. Phys. A97 (1967) 541. [7] I. M. Naqib and L . L . Green, Nucl. Phys. A l l 2 (1968) 76. [8] C. J. Oliver, B. C. Walsh, P . D . F o r s y t h and G. Kaye, to be published. [9] R. Huby, P h y s i c s L e t t e r s 33B {1970) [10] R. Huby and J. R. Mines, Rev. Mod. Phys. 37 (1965) 406. [11] C. M. Vincent and H. T. Fortune, to be published. [12] R. Huby, Nucl. Phys. A138 (1969) 442 and to be published. [131 J. Band and J. ZimSnyi, Nuel. Phys. A139 (i969) 534.