- Email: [email protected]
Validity of the dwba for inelastic scattering from nuclei
Volume 5, number 3
VALIDITY
PHYSICS
OF
THE
DWBA
FOR
LETTERS
INELASTIC
SCATTERING
1 July 1963
FROM
NUCLEI
F. P E R E Y and G. R. SATCHLER Oak Ridge National Laboratory, Oak Ridge, Tennessee Received 11 June 1963
Many s u c c e s s f u l a n a l y s e s have been m a d e of the i n e l a s t i c s c a t t e r i n g of alpha p a r t i c l e s 1) and nucleons 2,3) using the d i s t o r t e d - w a v e s B o r n - a p p r o x i m a t i o n (DWBA). R e c e n t l y it has been s u g g e s t e d 4 , 5 ) that the coupling between the ground s t a t e and one or m o r e excited s t a t e s m a y be s u f f i c i e n t l y s t r o n g that this a p p r o x i m a t i o n is p o o r , and one should solve e x a c t l y the s e t of coupled S c h r 0 d i n g e r equations f o r these s t a t e s . In p a r t i c u l a r if the coupling s t r e n g t h be e x p r e s s e d in t e r m s of the c o l l e c t i v e m o d e l it w a s suggested 5) that DWBA was only adequate f o r d e f o r m a t i o n s ~ < 0.2 f o r m e d i u m e n e r g y (10 - 20 MeV) p r o t o n s s c a t t e r i n g f r o m m e d i u m weight nuclei. The p u r p o s e of this note i s to point out that t h e s e r e s u l t s a r e m i s l e a d i n g , and that if the DWBA is applied c o n s i s t e n t l y , t h i s a p p r o x i m a t i o n has a cons i d e r a b l y w i d e r r a n g e of v a l i d i t y . The d i s t o r t e d - w a v e s of the DWBA include the effects of e l a s t i c s c a t t e r i n g b e f o r e and a f t e r the i n e l a s t i c event; in p r a c t i c e they a r e g e n e r a t e d f r o m an o p t i c a l m o d e l p o t e n t i a l which r e p r o d u c e s the o b s e r v e d e l a s t i c s c a t t e r i n g at that e n e r g y f r o m the s a m e nucleus. T h i s m o d e l t a k e s account of r e a c t i o n s a s s i m p l e a b s o r p t i o n , through the i n c l u sion of an i m a g i n a r y t e r m in the p o t e n t i a l . In this way r e a c t i o n s a r e a s s u m e d only to p r o d u c e a d a m p ing of the incident and outgoing w a v e s . H o w e v e r , if the i n e l a s t i c s c a t t e r i n g into n p a r t i c u l a r channels i s s t r o n g , it m a y be n e c e s s a r y to c o n s i d e r in m o r e d e t a i l i t s effects on individual p a r t i a l w a v e s , and the concept of s i m p l e a b s o r p t i o n m a y b e c o m e i n adequate. It is then s i m p l e s t to solve e x a c t l y the coupled Schr0dinger equations f o r t h e s e n c h a n n e l s , while the effects of s c a t t e r i n g into o t h e r , weakly coupled, channels i s s t i l l d e s c r i b e d by a b s o r p t i o n through an i m a g i n a r y p o t e n t i a l . An a n a l y s i s has been given of p r o t o n s c a t t e r i n g f r o m e v e n - e v e n nuclei, solving the coupled equations in a t w o - c h a n n e l a p p r o x i m a t i o n , n a m e l y the 0 + ground state and 2+ f t r s t - e x c i t e d s t a t e , and using an o p t i c a l m o d e l p o t e n t i a l which i s allowed to be n o n - s p h e r i c a l , with a quadrupole d e f o r m a t i o n ~. This w o r k 5) includes a c o m p a r i s o n between the r e s u l t s of t w o - c h a n n e l and DWBA p r e d i c t i o n s for the excitation of a c o l l e c t i v e r o t a t i o n a l or v i b r a tional state. The s a m e o p t i c a l p o t e n t i a l s a r e used 212
in both m e t h o d s of c a l c u l a t i o n , mad it i s then shown that the DWBA o v e r e s t i m a t e s the i n e l a s t i c c r o s s section (for a given ~) f o r d e f o r m a t i o n s g r e a t e r than about 0.2, even though the shape of the a n g u l a r d i s t r i b u t i o n i s v e r y l i t t l e changed. Indeed, while the DWBA c r o s s s e c t i o n i n c r e a s e s p r o p o r t i o n a l to ~2, the coupled equation p r e d i c t i o n s i n c r e a s e m o r e slowly, a p p r o x i m a t e l y like fl f o r ~ > 0.2. However, it should a l s o be noted that f o r t h e s e d e f c r m a t i o n s the e l a s t i c s c a t t e r i n g p r e d i c t i o n s a r e beginning to d e v i a t e f r o m those obtained with no quadrupole coupling; v i r t u a l e x c i t a t i o n of the 2+ s t a t e is b e coming i m p o r t a n t . That i s , the elastic scattering described by the d i s t o r t e d - w a v e s and predicted by the c~epled equations are no longer identical. The coupled equations p r e d i c t i o n s should r e p r o d u c e the o b s e r v e d e l a s t i c s c a t t e r i n g . At the s a m e t i m e , the d i s t o r t e d - w a v e s of the DWBA a r e supposed to be g e n e r a t e d by an opt i c a l p o t e n t i a l which a l s o r e p r o d u c e s the o b s e r v e d e l a s t i c s c a t t e r i n g as c l o s e l y a s p o s s i b l e . C l e a r l y then the o p t i c a l m o d e l p a r a m e t e r s u s e d in DWBA should differ f r o m those used in the coupled equations c a l c u l a t i o n by the a m o u n t s needed to r e p r o duce the effects of v i r t u a l excitation. The effect this h a s on the b e h a v i o u r of o p t i c a l m o d e l p a r a m e t e r s f o r p r o t o n s h a s been d i s c u s s e d in d e t a i l e l s e w h e r e 6). E v i d e n t l y one effect will be to i n c r e a s e the s t r e n g t h of the a b s o r p t i v e p o t e n t i a l W o v e r that used in the coupled equations c a l c u l a t i o n . The i n c r e a s e i s n e c e s s a r y in o r d e r to a c c o u n t f o r the a d d i t i o n a l a b s o r p t i o n f r o m the e l a s t i c channel which is due to e x c i t a t i o n of the 2+ s t a t e , and which i s included e x p l i c i t l y in the coupled equations c a l c u l a tion. T h i s i s the m o s t i m p o r t a n t effect of the coupling in the s c a t t e r i n g of 40 MeV a l p h a p a r t i c l e s 7). However, as v i r t u a l excitation b e c o m e s m o r e i m p o r t a n t one can e x p e c t the effective v a l u e s of o t h e r o p t i c a l m o d e l p a r a m e t e r s to change. The work of r e f . 6) shows that f o r m e d i u m e n e r g y p r o t o n s these effects can be l a r g e l y accounted f o r by a s m a l l i n c r e a s e in the depth V of the r e a l p o t e n t i a l , a s well as that in W. The r e s u l t of i n c r e a s i n g W is to r e d u c e the i n e l a s t i c c r o s s s e c t i o n s p r e d i c t e d in DWBA, and this can be r e a d i l y u n d e r s t o o d b e c a u s e of the a d d i t i o n a l
Volume 5, number 3
PHYSICS
d a m p i n g i m p o s e d on the d i s t o r t e d w a v e s in the v i c i n i t y of the n u c l e a r s u r f a c e . T h e effect i s l a r g e ; f o r m e d i u m e n e r g y p r o t o n s and v a l u e s of W of p h y s i c a l interest the inelastic cross section is roughly prop o r t i o n a l to W-2. In a d d i t i o n , the s h a p e s of the a n gular distributions are very little changed; appar e n t l y the e x t r a d a m p i n g o c c u r s quite u n i f o r m l y o v e r the i m p o r t a n t a r e a s of the n u c l e a r s u r f a c e . Since the i n c r e a s e i n W i s l a r g e r f o r l a r g e r fl, we can u n d e r s t a n d that the DWBA c r o s s s e c t i o n , o b t a i n e d in t h i s s e l f - c o n s i s t e n t way, m a y i n c r e a s e m u c h l e s s r a p i d l y t h a n f12. (The m u c h s m a l l e r i n c r e a s e s i n V which a r e a l s o r e q u i r e d t e n d s to have the o p p o s i t e effect on the i n e l a s t i c p r e d i c t i o n s , and so ~educe s o m e w h a t the effect of i n c r e a s i n g W alone.) To i l l u s t r a t e t h e s e e f f e c t s , a n u m b e r of c a l c u l a t i o n s w e r e m a d e for the s c a t t e r i n g of p r o t o n s of 10, 12, 14, 17 and 22 MeV by Fe56. It was a s s u m e d that a v i b r a t i o n a l 2+ s t a t e of e n e r g y 0.85 MeV was e x c i t e d by i n t e r a c t i o n with a n o n - s p h e r i c a l o p t i c a l potential, as described in detail elsewhere 1,2,5). D e f o r m a b i l i t i e s / 3 = 0.1, 0.2, 0.3 and 0.4 w e r e c o n s i d e r e d ( d e f o r m a b i l i t i e s in e x c e s s of 0.3 have not b e e n found for a c t u a l n u c l e i in t h i s m a s s r e g i o n ) . T h e o p t i c a l p o t e n t i a l u s e d was of the f o r m found to give a good a c c o u n t of the o b s e r v e d e l a s t i c s c a t tering 5,6), U(r) = - V(e x + 1) -1 +
4iWD(d/dx') (ex '
+ 1) -1
+ L . ( T (l~i/m~c) 2 V&) r - 1 ( d / d r ) (eX+ 1) -1 ,
where !
x = (r-
roA3)/a
LETTERS
Table 1 Effective deformations obtained by DWBA calculations. The "true" deformation B is that used in the coupled equations calculations; B1 is required to give the same peak c r o s s s e c t i o n in DWBA using the same optical parameters as with coupled equations, and B2 is required in the selfconsistent DWBA using the adjusted values of V' and Wb in the last two columns which give the same elastic scattering as the coupled equations.
E
B
B1
B2
V'
Wb
10
0.1
0.099
0.102
49.4
12.0
0.2 0.3 0.4
0.188 0.260 0.313
0.213 0.321 0.413
50.2 51.9 54.1
13.4 15.7 18.5
12
0.1 0.2 0.3 0.4
0.099 0.192 0.272 0.337
0.101 0.208 0.315 0.402
48.8 49.4 50.8 52.5
11.9 13.0 14.6 16.0
14
0.1 0.2 0.3 0.4
0.099 0.193 0.277 0.346
0.101 0.209 0.318 0.419
48.2 48.7 49.9 51.5
11.9 13.0 14.6 16.2
17
0.1 0.2 0.3 0.4
0.100 0.194 0.281 0.354
0.101 0.208 0.320 0.413
47.3 47.8 48.9 50.7
11.9 13.0 14.7 16.1
22
0.1 0.2 0.3 0.4
0.099 0.195 0.283 0.361
0.102 0.201 0.302 0.400
46.7 47.1 47.5 47.8
11.7 12.4 13.2 14.2
the s a m e p e a k c r o s s s e c t i o n a s the " d a t a " , i . e . , the coupled e q u a t i o n s p r e d i c t i o n s , w e r e d e t e r m i n e d . T h e s e " e f f e c t i v e " B's a r e c o m p a r e d with the " t r u e " B's of t a b l e 1; a l s o g i v e n a r e the v a l u e s of ~ which
1
,
x' = ( r -
roA3)/a'
404
,
and with r o = 1.25 f m , a = 0.65 f m , a ' = 0.47 f m , VSo = 8 MeV. The C o u l o m b potential, f r o m a u n i f o r m c h a r g e of r a d i u s 1.25 AT w a s a l s o included. The coupled e q u a t i o n s c a l c u l a t i o n s w e r e m a d e with V = 49.2 (10 MeV), 48.6 (12 MeV), 48.0 (14 MeV), 47.1 (17 MeV) and 46.6 (22 MeV), and W D = 11.5 MeV, t h e s e v a l u e s b e i n g c l o s e to t h o s e a c t u a l l y r e q u i r e d 5) to a c c o u n t for the e l a s t i c and i n e l a s t i c s c a t t e r i n g in this m a s s region. The elastic differe n t i a l c r o s s s e c t i o n s p r e d i c t e d by t h e s e coupled e q u a t i o n s c a l c u l a t i o n s w e r e then t r e a t e d a s " e x p e r i m e n t a l " data to which a s p h e r i c a l o p t i c a l m o d e l f i t w a s o b t a i n e d i n the u s u a l way 6). T h e o p t i c a l p o t e n t i a l w a s i d e n t i c a l to that u s e d i n the coupled e q u a t i o n s c a l c u l a t i o n s except f o r the a d j u s t m e n t of V and W D to o b t a i n a b e s t fit. In each c a s e a v e r y s a t i s f a c t o r y f i t to the " d a t a " was o b t a i n e d with the v a l u e s of V' and W~) i n t a b l e 1; a t y p i c a l f i t is shown in fig. 1, w h e r e it i s a l s o c o m p a r e d with the o p t i c a l m o d e l p r e d i c t i o n s b e f o r e V and W D w e r e a d j u s t e d . T h e s e new v a l u e s of V' and W b w e r e t h e n e m p l o y e d in DWBA c a l c u l a t i o n s of the i n e l a s t i c s c a t t e r i n g 8), and the v a l u e s of B n e e d e d to give
1July1963
2 403
~ ~
Fe5Q+ P 40MeV
5 8 2
,I 5 2 t0'
5 2 400 0
25
50
75 400 ec.M.(deg)
425
450
475
Fig. 1. Elastic scattering at 10 MeV with deformation B = 0.3. The circles are the coupled equations p r e d i c tions, the dashed curve is the prediction for an optical potential with the same parameters V and WD as the coupled equations, while the full curve is the optical potential prediction with the adjusted values V' and W~ given in the table.
213
Volume 5. number 3
PHYSICS
would be needed to give the s a m e p e a k c r o s s s e c t i o n in DWBA if the unadjusted value of V and W D had been used. As p r e v i o u s l y noted, }.he l a t t e r p r o c e d u r e a l r e a d y gives a r e a s o n a b l y a c c u r a t e e s t i m a t e of/3 f o r the r a n g e of v a l u e s found in a c t u a l nuclei in this r e g i o n (/3 < 0.3). The s e l f - c o n s i s t e n t p r o c e d u r e , h o w e v e r , l e a d s to e f f e c t i v e / 3 ' s which a r e in much b e t t e r o v e r a l l a g r e e m e n t with the t r u e /3's even f o r /3 = 0.4, although t h e r e i s now s o m e t e n dency to o v e r e s t i m a t e / ~ . T y p i c a l a n g u l a r d i s t r i b u tions a r e shown in fig. 2. At 22 MeV the r e s u l t s of coupled equations and both DWBA c a l c u l a t i o n s have v e r y s i m i l a r s h a p e s even f o r /3 = 0.4, and of c o u r s e the a g r e e m e n t i s even b e t t e r f o r s m a l l e r v a l u e s of /3. At 14 MeV the a g r e e m e n t between coupled equations and both the DWBA p r e d i c t i o n s f o r the m a g n i tude of the c r o s s s e c t i o n s is quite good, although with ~ = 0.3 s o m e c h a r a c t e r i s t i c changes in a n g u l a r d i s t r i b u t i o n a r e beginning to a p p e a r f o r the s e l f c o n s i s t e n t DWBA. T h e s e d i f f e r e n c e s can be t r a c e d to the change in V r a t h e r than in W D. A l s o shown in fig. 2 a r e the DWBA p r e d i c t i o n s at 14 MeV (and /3=0.3) when V and W have the s a m e v a l u e s a s in the coupled equations c a l c u l a t i o n s ; the p e a k c r o s s s e c tion i s now o v e r e s t i m a t e d by r o u g h l y 20%, though the a n g u l a r d i s t r i b u t i o n shape is in somewhat c l o s e r a g r e e m e n t with the coupled equations r e s u l t s . At 10 MeV, and f o r d e f o r m a t i o n s B g r e a t e r than about 0.2, the changes in the a n g u l a r d i s t r i b u t i o n s h a p e s a r e b e c o m i n g m o r e m a r k e d , although the p e a k c r o s s s e c t i o n i s s t i l l in b e t t e r a g r e e m e n t with the coupled equations r e s u l t s than i s the unmodified DWBA (see table 1). Thus we s e e that the u s u a l p r o c e d u r e of obtaining an o p t i c a l m o d e l fit to the o b s e r v e d e l a s t i c s c a t t e r ing and then using t h e s e p a r a m e t e r s in a DWBA c a l culation of the i n e l a s t i c s c a t t e r i n g gives much m o r e r e l i a b l e r e s u l t s than was a p p a r e n t f r o m p r e v i o u s work. (This i s p a r t i c u l a r l y so f o r those e n e r g i e s and coupling s t r e n g t h s which a r e of m o s t p h y s i c a l i n t e r e s t . ) Of c o u r s e t h e r e a r e l i m i t s to t h i s : a s the e n e r g y i s d e c r e a s e d and the n u m b e r of p a r t i a l w a v e s involved i s r e d u c e d , the a p p r o x i m a t i o n b e c o m e s s t e a d i l y p o o r e r 9). In this r e s p e c t , some s h o r t c o m i n g s of the p r e s e n t c a l c u l a t i o n s should be noted. In p a r t i c u l a r , it w a s a s s u m e d that the coupling effects could be accounted f o r s o l e l y by v a r y ing the s t r e n g t h s of the r e a l and i m a g i n a r y p o t e n t i a l s in the o p t i c a l model. It i s p o s s i b l e that s o m e of the d e f i c i e n c i e s in the s h a p e s of the a n g u l a r d i s t r i b u t i o n s at lower e n e r g i e s a r e due to t h i s r e s t r i c t i o n . One might expect other p a r a m e t e r s , f o r e x a m p l e the effective s u r f a c e d i f f u s e n e s s , to be affected a s s t r o n g l y as V o r W. It should a l s o be r e m a r k e d that when the coupling i s s t r o n g the 2channel a p p r o x i m a t i o n for the coupling equations tends to o v e r e m p h a s i s e the r o l e of the 2+ s t a t e ; inclusion of e x p l i c i t coupling to higher e x c i t e d 214
LETTERS
1July 1963
t00 i
20 t0
i
Fes6 (#,p') O - - 0.85 MeV .I'--2
50
j\
\
-- 22 MeV 5 --/9-0.4
-.,w
2 I
5o I
t
20 •
o
.
t0 y
.wj
5
",\,,
- - 14 MeV / 9 l 0.3
I
2
t
IO
5 " ' ~ ' ~.X~" - - I 0 MeV NX
--1 ~ 0.5
\
.8-- 0.3 J.
".~ .
.'
/ 3 - 0.1
~\
OJ
\
\
..~
0.05
"~
0.02 0.0t 0
25
50
75 400 eC.M.(deg)
t25
150
175
Fig. 2. Inelastic scattering cross sections. The full curves are coupled equations predictions, and the dashed curves are the self-consistent DWBA p r e dictions using the ~ and W~ given in table 1. At 14 MeV, the dot-dash curve is the DWBA prediction using the same optical parameters as the coupled equations. The DWBA curves are here normalised with the same deformations B as the coupled equations curves. s t a t e s m a y sufficiently " s m o o t h out" the effects of the 2+ that the o p t i c a l m o d e l w a v e s have a w i d e r r a n g e of v a l i d i t y than i s a p p a r e n t f r o m the r e s u l t s
Volume 5, number 3
PHYSICS
r e p o r t e d h e r e . T h e s e and o t h e r q u e s t i o n s a r e b e i n g s t u d i e d in m o r e d e t a i l . F i n a l l y i t i s i n t e r e s t i n g to s p e c u l a t e t h a t t h e s e c o n s i d e r a t i o n s lend a d d i t i o n a l s u p p o r t to the u s e of t h e s i m p l e DWBA in o t h e r r e a c t i o n s s u c h a s s t r i p p i n g , e v e n when the i n e l a s t i c s c a t t e r i n g i s known to be strong. T h a n k s a r e due to N. A u s t e r n f o r s t i m u l a t i n g t h e p u b l i c a t i o n of t h e s e r e s u l t s , and to R. H. B a s s e l , B. Buck and R. M. D r i s k o f o r h e l p f u l d i s c u s s i o n and f o r m a k i n g a v a i l a b l e the c o m p u t e r c o d e s f o r t h e DWBA and c o u p l e d e q u a t i o n s c a l c u l a t i o n s . 1) E.Rost and N.Austern, Phys. Rev. 120 (1960) 1375. R . H . B a s s e l et al., Phys. Rev. 128 (1962) 2693.
ON
THE IN
IMPORTANCE STRIPPING
OF AND
LETTERS
1July1963
2) C.A. Levinson and M. K. Banerjee, Ann. Phys. 2 (1957) 471. N.K.Glendenning, Phys. Rev. 114 (1959) 1297. F. Perey, l>roc. Conf. on Nuclear Structure, Kingston, Canada, 1960, ed. D. A. Bromley and E. Vogt (University of Toronto P r e s s , Toronto, 1960). 3) G.R. Satchler, to be published. 4) S. Yoshida, Prec. Phys. Soc. (London) A 69 (1956) 668. D. M. Chase, L. Wilets and A.R. Edmonds, Phys. Rev. 110 (1958) 1080. 5) B. Buck, Phys. Rev. 130 (1963) 712. 6) F. Perey, Phys. Rev., to be published. 7) B. Buck and G.R.Satchler, to be published. 8) R.H. Bassel, R. M. Drlsko and G. R. Satchler, Oak Ridge National Laboratory Report ORNL-3240 (1962). 9) N.Austern, lecture notes from Int. Summer School held in Low Tatra Mountains, Czechoslovakia, 1962.
THE RANGE KNOCKOUT
OF INTERACTIONS AMPLITUDES *
F. B. M O R I N I G O
California Institute of Technology, Pasadena, California Received 20 May 1963
In a r e c e n t p a p e r 1) T a n i f u j i d i s c u s s e s the k n o c k o u t m o d e f o r s t r i p p i n g r e a c t i o n s . He c a l c u l a t e s u s i n g t h e c u t - o f f B o r n a p p r o x i m a t i o n and d e l t a - f u n c t i o n i n t e r a c t i o n s and a s s e r t s t h a t t h e k n o c k o u t a m p l i t u d e if f o r w a r d p e a k e d , w h i l e the heavy-particle knockout amplitude is backward p e a k e d . T h i s l e t t e r w i s h e s to p o i n t out t h a t T a n i f u j i ' s r e s u l t s a r e not g e n e r a l but a r e due to h i s c h o i c e of an e x t r e m e l y s h o r t - r a n g e i n t e r a c t i o n . If one c a l c u l a t e s a k n o c k o u t a m p l i t u d e in a B o r n a p proximation without a cut-off, using harmonicoscillator wave functions, together with interact i o n s of g a u s s i a n s h a p e , the r e s u l t s m a y b e q u a i l t a t i v e l y d i f f e r e n t f r o m T a n i f u j i ' s if t h e r a n g e of t h e p o t e n t i a l i s not i n f i n i t e s i m a l . W h a t w a s thought b a c k w a r d p e a k e d b e c o m e s f o r w a r d p e a k e d , and vice versa. T h e e s s e n t i a l p o i n t i s i n d e p e n d e n t of the d e t a i l s of the a n g u l a r - m o m e n t u m c o u p l i n g r u l e s , s o t h a t we s h a l l d i s c u s s h e r e o n l y the c a s e of s p i n l e s s p a r t i c l e s , and bound s t a t e s of z e r o o r b i t a l a n g u l a r m o m e n t u m . T h e g e n e r a l c a s e of a r b i t r a r y s p i n s and h i g h e r a n g u l a r m o m e n t a w i l l be t h e s u b j e c t of a later paper. * Supported by the Office of Naval Research.
F i r s t , we c l a s s i f y the t e r m s in the a m p l i t u d e of a nuclear reaction x + Y-" W + z
(I)
according to the four m o d e s which appear in firstorder theory. The particles x and z are the light particles, for example, deuteron and proton in a typical (d,p) reaction. The four distinct ways in which the process occurs depends on whether the particle z w a s originally a part of x or of Y, and whether the m o m e n t u m exchange which results in the transition involves the particle z or its c o m panion, the captured particle. The four main m o d e s are easily understood by referring to fig. 1. The dashed lines represent m o m e n t u m exchanges. W e shall discuss explicitly the knockout m o d e corresponding to an interaction Vxz; the others are easily done by equally simple methods. W e consider the initial target state Y to be a bound state of z and s o m e other particle b. Similarly, w e consider the final state W to be a bound state of x and b. If w e denote the vector positions of these particles by z, b, x, the wave vector of the initial motion by Ko, and the wave vector of the final motion by A ~ , the initial and final states are the following:
215
VALIDITY
PHYSICS
OF
THE
DWBA
FOR
LETTERS
INELASTIC
SCATTERING
1 July 1963
FROM
NUCLEI
F. P E R E Y and G. R. SATCHLER Oak Ridge National Laboratory, Oak Ridge, Tennessee Received 11 June 1963
Many s u c c e s s f u l a n a l y s e s have been m a d e of the i n e l a s t i c s c a t t e r i n g of alpha p a r t i c l e s 1) and nucleons 2,3) using the d i s t o r t e d - w a v e s B o r n - a p p r o x i m a t i o n (DWBA). R e c e n t l y it has been s u g g e s t e d 4 , 5 ) that the coupling between the ground s t a t e and one or m o r e excited s t a t e s m a y be s u f f i c i e n t l y s t r o n g that this a p p r o x i m a t i o n is p o o r , and one should solve e x a c t l y the s e t of coupled S c h r 0 d i n g e r equations f o r these s t a t e s . In p a r t i c u l a r if the coupling s t r e n g t h be e x p r e s s e d in t e r m s of the c o l l e c t i v e m o d e l it w a s suggested 5) that DWBA was only adequate f o r d e f o r m a t i o n s ~ < 0.2 f o r m e d i u m e n e r g y (10 - 20 MeV) p r o t o n s s c a t t e r i n g f r o m m e d i u m weight nuclei. The p u r p o s e of this note i s to point out that t h e s e r e s u l t s a r e m i s l e a d i n g , and that if the DWBA is applied c o n s i s t e n t l y , t h i s a p p r o x i m a t i o n has a cons i d e r a b l y w i d e r r a n g e of v a l i d i t y . The d i s t o r t e d - w a v e s of the DWBA include the effects of e l a s t i c s c a t t e r i n g b e f o r e and a f t e r the i n e l a s t i c event; in p r a c t i c e they a r e g e n e r a t e d f r o m an o p t i c a l m o d e l p o t e n t i a l which r e p r o d u c e s the o b s e r v e d e l a s t i c s c a t t e r i n g at that e n e r g y f r o m the s a m e nucleus. T h i s m o d e l t a k e s account of r e a c t i o n s a s s i m p l e a b s o r p t i o n , through the i n c l u sion of an i m a g i n a r y t e r m in the p o t e n t i a l . In this way r e a c t i o n s a r e a s s u m e d only to p r o d u c e a d a m p ing of the incident and outgoing w a v e s . H o w e v e r , if the i n e l a s t i c s c a t t e r i n g into n p a r t i c u l a r channels i s s t r o n g , it m a y be n e c e s s a r y to c o n s i d e r in m o r e d e t a i l i t s effects on individual p a r t i a l w a v e s , and the concept of s i m p l e a b s o r p t i o n m a y b e c o m e i n adequate. It is then s i m p l e s t to solve e x a c t l y the coupled Schr0dinger equations f o r t h e s e n c h a n n e l s , while the effects of s c a t t e r i n g into o t h e r , weakly coupled, channels i s s t i l l d e s c r i b e d by a b s o r p t i o n through an i m a g i n a r y p o t e n t i a l . An a n a l y s i s has been given of p r o t o n s c a t t e r i n g f r o m e v e n - e v e n nuclei, solving the coupled equations in a t w o - c h a n n e l a p p r o x i m a t i o n , n a m e l y the 0 + ground state and 2+ f t r s t - e x c i t e d s t a t e , and using an o p t i c a l m o d e l p o t e n t i a l which i s allowed to be n o n - s p h e r i c a l , with a quadrupole d e f o r m a t i o n ~. This w o r k 5) includes a c o m p a r i s o n between the r e s u l t s of t w o - c h a n n e l and DWBA p r e d i c t i o n s for the excitation of a c o l l e c t i v e r o t a t i o n a l or v i b r a tional state. The s a m e o p t i c a l p o t e n t i a l s a r e used 212
in both m e t h o d s of c a l c u l a t i o n , mad it i s then shown that the DWBA o v e r e s t i m a t e s the i n e l a s t i c c r o s s section (for a given ~) f o r d e f o r m a t i o n s g r e a t e r than about 0.2, even though the shape of the a n g u l a r d i s t r i b u t i o n i s v e r y l i t t l e changed. Indeed, while the DWBA c r o s s s e c t i o n i n c r e a s e s p r o p o r t i o n a l to ~2, the coupled equation p r e d i c t i o n s i n c r e a s e m o r e slowly, a p p r o x i m a t e l y like fl f o r ~ > 0.2. However, it should a l s o be noted that f o r t h e s e d e f c r m a t i o n s the e l a s t i c s c a t t e r i n g p r e d i c t i o n s a r e beginning to d e v i a t e f r o m those obtained with no quadrupole coupling; v i r t u a l e x c i t a t i o n of the 2+ s t a t e is b e coming i m p o r t a n t . That i s , the elastic scattering described by the d i s t o r t e d - w a v e s and predicted by the c~epled equations are no longer identical. The coupled equations p r e d i c t i o n s should r e p r o d u c e the o b s e r v e d e l a s t i c s c a t t e r i n g . At the s a m e t i m e , the d i s t o r t e d - w a v e s of the DWBA a r e supposed to be g e n e r a t e d by an opt i c a l p o t e n t i a l which a l s o r e p r o d u c e s the o b s e r v e d e l a s t i c s c a t t e r i n g as c l o s e l y a s p o s s i b l e . C l e a r l y then the o p t i c a l m o d e l p a r a m e t e r s u s e d in DWBA should differ f r o m those used in the coupled equations c a l c u l a t i o n by the a m o u n t s needed to r e p r o duce the effects of v i r t u a l excitation. The effect this h a s on the b e h a v i o u r of o p t i c a l m o d e l p a r a m e t e r s f o r p r o t o n s h a s been d i s c u s s e d in d e t a i l e l s e w h e r e 6). E v i d e n t l y one effect will be to i n c r e a s e the s t r e n g t h of the a b s o r p t i v e p o t e n t i a l W o v e r that used in the coupled equations c a l c u l a t i o n . The i n c r e a s e i s n e c e s s a r y in o r d e r to a c c o u n t f o r the a d d i t i o n a l a b s o r p t i o n f r o m the e l a s t i c channel which is due to e x c i t a t i o n of the 2+ s t a t e , and which i s included e x p l i c i t l y in the coupled equations c a l c u l a tion. T h i s i s the m o s t i m p o r t a n t effect of the coupling in the s c a t t e r i n g of 40 MeV a l p h a p a r t i c l e s 7). However, as v i r t u a l excitation b e c o m e s m o r e i m p o r t a n t one can e x p e c t the effective v a l u e s of o t h e r o p t i c a l m o d e l p a r a m e t e r s to change. The work of r e f . 6) shows that f o r m e d i u m e n e r g y p r o t o n s these effects can be l a r g e l y accounted f o r by a s m a l l i n c r e a s e in the depth V of the r e a l p o t e n t i a l , a s well as that in W. The r e s u l t of i n c r e a s i n g W is to r e d u c e the i n e l a s t i c c r o s s s e c t i o n s p r e d i c t e d in DWBA, and this can be r e a d i l y u n d e r s t o o d b e c a u s e of the a d d i t i o n a l
Volume 5, number 3
PHYSICS
d a m p i n g i m p o s e d on the d i s t o r t e d w a v e s in the v i c i n i t y of the n u c l e a r s u r f a c e . T h e effect i s l a r g e ; f o r m e d i u m e n e r g y p r o t o n s and v a l u e s of W of p h y s i c a l interest the inelastic cross section is roughly prop o r t i o n a l to W-2. In a d d i t i o n , the s h a p e s of the a n gular distributions are very little changed; appar e n t l y the e x t r a d a m p i n g o c c u r s quite u n i f o r m l y o v e r the i m p o r t a n t a r e a s of the n u c l e a r s u r f a c e . Since the i n c r e a s e i n W i s l a r g e r f o r l a r g e r fl, we can u n d e r s t a n d that the DWBA c r o s s s e c t i o n , o b t a i n e d in t h i s s e l f - c o n s i s t e n t way, m a y i n c r e a s e m u c h l e s s r a p i d l y t h a n f12. (The m u c h s m a l l e r i n c r e a s e s i n V which a r e a l s o r e q u i r e d t e n d s to have the o p p o s i t e effect on the i n e l a s t i c p r e d i c t i o n s , and so ~educe s o m e w h a t the effect of i n c r e a s i n g W alone.) To i l l u s t r a t e t h e s e e f f e c t s , a n u m b e r of c a l c u l a t i o n s w e r e m a d e for the s c a t t e r i n g of p r o t o n s of 10, 12, 14, 17 and 22 MeV by Fe56. It was a s s u m e d that a v i b r a t i o n a l 2+ s t a t e of e n e r g y 0.85 MeV was e x c i t e d by i n t e r a c t i o n with a n o n - s p h e r i c a l o p t i c a l potential, as described in detail elsewhere 1,2,5). D e f o r m a b i l i t i e s / 3 = 0.1, 0.2, 0.3 and 0.4 w e r e c o n s i d e r e d ( d e f o r m a b i l i t i e s in e x c e s s of 0.3 have not b e e n found for a c t u a l n u c l e i in t h i s m a s s r e g i o n ) . T h e o p t i c a l p o t e n t i a l u s e d was of the f o r m found to give a good a c c o u n t of the o b s e r v e d e l a s t i c s c a t tering 5,6), U(r) = - V(e x + 1) -1 +
4iWD(d/dx') (ex '
+ 1) -1
+ L . ( T (l~i/m~c) 2 V&) r - 1 ( d / d r ) (eX+ 1) -1 ,
where !
x = (r-
roA3)/a
LETTERS
Table 1 Effective deformations obtained by DWBA calculations. The "true" deformation B is that used in the coupled equations calculations; B1 is required to give the same peak c r o s s s e c t i o n in DWBA using the same optical parameters as with coupled equations, and B2 is required in the selfconsistent DWBA using the adjusted values of V' and Wb in the last two columns which give the same elastic scattering as the coupled equations.
E
B
B1
B2
V'
Wb
10
0.1
0.099
0.102
49.4
12.0
0.2 0.3 0.4
0.188 0.260 0.313
0.213 0.321 0.413
50.2 51.9 54.1
13.4 15.7 18.5
12
0.1 0.2 0.3 0.4
0.099 0.192 0.272 0.337
0.101 0.208 0.315 0.402
48.8 49.4 50.8 52.5
11.9 13.0 14.6 16.0
14
0.1 0.2 0.3 0.4
0.099 0.193 0.277 0.346
0.101 0.209 0.318 0.419
48.2 48.7 49.9 51.5
11.9 13.0 14.6 16.2
17
0.1 0.2 0.3 0.4
0.100 0.194 0.281 0.354
0.101 0.208 0.320 0.413
47.3 47.8 48.9 50.7
11.9 13.0 14.7 16.1
22
0.1 0.2 0.3 0.4
0.099 0.195 0.283 0.361
0.102 0.201 0.302 0.400
46.7 47.1 47.5 47.8
11.7 12.4 13.2 14.2
the s a m e p e a k c r o s s s e c t i o n a s the " d a t a " , i . e . , the coupled e q u a t i o n s p r e d i c t i o n s , w e r e d e t e r m i n e d . T h e s e " e f f e c t i v e " B's a r e c o m p a r e d with the " t r u e " B's of t a b l e 1; a l s o g i v e n a r e the v a l u e s of ~ which
1
,
x' = ( r -
roA3)/a'
404
,
and with r o = 1.25 f m , a = 0.65 f m , a ' = 0.47 f m , VSo = 8 MeV. The C o u l o m b potential, f r o m a u n i f o r m c h a r g e of r a d i u s 1.25 AT w a s a l s o included. The coupled e q u a t i o n s c a l c u l a t i o n s w e r e m a d e with V = 49.2 (10 MeV), 48.6 (12 MeV), 48.0 (14 MeV), 47.1 (17 MeV) and 46.6 (22 MeV), and W D = 11.5 MeV, t h e s e v a l u e s b e i n g c l o s e to t h o s e a c t u a l l y r e q u i r e d 5) to a c c o u n t for the e l a s t i c and i n e l a s t i c s c a t t e r i n g in this m a s s region. The elastic differe n t i a l c r o s s s e c t i o n s p r e d i c t e d by t h e s e coupled e q u a t i o n s c a l c u l a t i o n s w e r e then t r e a t e d a s " e x p e r i m e n t a l " data to which a s p h e r i c a l o p t i c a l m o d e l f i t w a s o b t a i n e d i n the u s u a l way 6). T h e o p t i c a l p o t e n t i a l w a s i d e n t i c a l to that u s e d i n the coupled e q u a t i o n s c a l c u l a t i o n s except f o r the a d j u s t m e n t of V and W D to o b t a i n a b e s t fit. In each c a s e a v e r y s a t i s f a c t o r y f i t to the " d a t a " was o b t a i n e d with the v a l u e s of V' and W~) i n t a b l e 1; a t y p i c a l f i t is shown in fig. 1, w h e r e it i s a l s o c o m p a r e d with the o p t i c a l m o d e l p r e d i c t i o n s b e f o r e V and W D w e r e a d j u s t e d . T h e s e new v a l u e s of V' and W b w e r e t h e n e m p l o y e d in DWBA c a l c u l a t i o n s of the i n e l a s t i c s c a t t e r i n g 8), and the v a l u e s of B n e e d e d to give
1July1963
2 403
~ ~
Fe5Q+ P 40MeV
5 8 2
,I 5 2 t0'
5 2 400 0
25
50
75 400 ec.M.(deg)
425
450
475
Fig. 1. Elastic scattering at 10 MeV with deformation B = 0.3. The circles are the coupled equations p r e d i c tions, the dashed curve is the prediction for an optical potential with the same parameters V and WD as the coupled equations, while the full curve is the optical potential prediction with the adjusted values V' and W~ given in the table.
213
Volume 5. number 3
PHYSICS
would be needed to give the s a m e p e a k c r o s s s e c t i o n in DWBA if the unadjusted value of V and W D had been used. As p r e v i o u s l y noted, }.he l a t t e r p r o c e d u r e a l r e a d y gives a r e a s o n a b l y a c c u r a t e e s t i m a t e of/3 f o r the r a n g e of v a l u e s found in a c t u a l nuclei in this r e g i o n (/3 < 0.3). The s e l f - c o n s i s t e n t p r o c e d u r e , h o w e v e r , l e a d s to e f f e c t i v e / 3 ' s which a r e in much b e t t e r o v e r a l l a g r e e m e n t with the t r u e /3's even f o r /3 = 0.4, although t h e r e i s now s o m e t e n dency to o v e r e s t i m a t e / ~ . T y p i c a l a n g u l a r d i s t r i b u tions a r e shown in fig. 2. At 22 MeV the r e s u l t s of coupled equations and both DWBA c a l c u l a t i o n s have v e r y s i m i l a r s h a p e s even f o r /3 = 0.4, and of c o u r s e the a g r e e m e n t i s even b e t t e r f o r s m a l l e r v a l u e s of /3. At 14 MeV the a g r e e m e n t between coupled equations and both the DWBA p r e d i c t i o n s f o r the m a g n i tude of the c r o s s s e c t i o n s is quite good, although with ~ = 0.3 s o m e c h a r a c t e r i s t i c changes in a n g u l a r d i s t r i b u t i o n a r e beginning to a p p e a r f o r the s e l f c o n s i s t e n t DWBA. T h e s e d i f f e r e n c e s can be t r a c e d to the change in V r a t h e r than in W D. A l s o shown in fig. 2 a r e the DWBA p r e d i c t i o n s at 14 MeV (and /3=0.3) when V and W have the s a m e v a l u e s a s in the coupled equations c a l c u l a t i o n s ; the p e a k c r o s s s e c tion i s now o v e r e s t i m a t e d by r o u g h l y 20%, though the a n g u l a r d i s t r i b u t i o n shape is in somewhat c l o s e r a g r e e m e n t with the coupled equations r e s u l t s . At 10 MeV, and f o r d e f o r m a t i o n s B g r e a t e r than about 0.2, the changes in the a n g u l a r d i s t r i b u t i o n s h a p e s a r e b e c o m i n g m o r e m a r k e d , although the p e a k c r o s s s e c t i o n i s s t i l l in b e t t e r a g r e e m e n t with the coupled equations r e s u l t s than i s the unmodified DWBA (see table 1). Thus we s e e that the u s u a l p r o c e d u r e of obtaining an o p t i c a l m o d e l fit to the o b s e r v e d e l a s t i c s c a t t e r ing and then using t h e s e p a r a m e t e r s in a DWBA c a l culation of the i n e l a s t i c s c a t t e r i n g gives much m o r e r e l i a b l e r e s u l t s than was a p p a r e n t f r o m p r e v i o u s work. (This i s p a r t i c u l a r l y so f o r those e n e r g i e s and coupling s t r e n g t h s which a r e of m o s t p h y s i c a l i n t e r e s t . ) Of c o u r s e t h e r e a r e l i m i t s to t h i s : a s the e n e r g y i s d e c r e a s e d and the n u m b e r of p a r t i a l w a v e s involved i s r e d u c e d , the a p p r o x i m a t i o n b e c o m e s s t e a d i l y p o o r e r 9). In this r e s p e c t , some s h o r t c o m i n g s of the p r e s e n t c a l c u l a t i o n s should be noted. In p a r t i c u l a r , it w a s a s s u m e d that the coupling effects could be accounted f o r s o l e l y by v a r y ing the s t r e n g t h s of the r e a l and i m a g i n a r y p o t e n t i a l s in the o p t i c a l model. It i s p o s s i b l e that s o m e of the d e f i c i e n c i e s in the s h a p e s of the a n g u l a r d i s t r i b u t i o n s at lower e n e r g i e s a r e due to t h i s r e s t r i c t i o n . One might expect other p a r a m e t e r s , f o r e x a m p l e the effective s u r f a c e d i f f u s e n e s s , to be affected a s s t r o n g l y as V o r W. It should a l s o be r e m a r k e d that when the coupling i s s t r o n g the 2channel a p p r o x i m a t i o n for the coupling equations tends to o v e r e m p h a s i s e the r o l e of the 2+ s t a t e ; inclusion of e x p l i c i t coupling to higher e x c i t e d 214
LETTERS
1July 1963
t00 i
20 t0
i
Fes6 (#,p') O - - 0.85 MeV .I'--2
50
j\
\
-- 22 MeV 5 --/9-0.4
-.,w
2 I
5o I
t
20 •
o
.
t0 y
.wj
5
",\,,
- - 14 MeV / 9 l 0.3
I
2
t
IO
5 " ' ~ ' ~.X~" - - I 0 MeV NX
--1 ~ 0.5
\
.8-- 0.3 J.
".~ .
.'
/ 3 - 0.1
~\
OJ
\
\
..~
0.05
"~
0.02 0.0t 0
25
50
75 400 eC.M.(deg)
t25
150
175
Fig. 2. Inelastic scattering cross sections. The full curves are coupled equations predictions, and the dashed curves are the self-consistent DWBA p r e dictions using the ~ and W~ given in table 1. At 14 MeV, the dot-dash curve is the DWBA prediction using the same optical parameters as the coupled equations. The DWBA curves are here normalised with the same deformations B as the coupled equations curves. s t a t e s m a y sufficiently " s m o o t h out" the effects of the 2+ that the o p t i c a l m o d e l w a v e s have a w i d e r r a n g e of v a l i d i t y than i s a p p a r e n t f r o m the r e s u l t s
Volume 5, number 3
PHYSICS
r e p o r t e d h e r e . T h e s e and o t h e r q u e s t i o n s a r e b e i n g s t u d i e d in m o r e d e t a i l . F i n a l l y i t i s i n t e r e s t i n g to s p e c u l a t e t h a t t h e s e c o n s i d e r a t i o n s lend a d d i t i o n a l s u p p o r t to the u s e of t h e s i m p l e DWBA in o t h e r r e a c t i o n s s u c h a s s t r i p p i n g , e v e n when the i n e l a s t i c s c a t t e r i n g i s known to be strong. T h a n k s a r e due to N. A u s t e r n f o r s t i m u l a t i n g t h e p u b l i c a t i o n of t h e s e r e s u l t s , and to R. H. B a s s e l , B. Buck and R. M. D r i s k o f o r h e l p f u l d i s c u s s i o n and f o r m a k i n g a v a i l a b l e the c o m p u t e r c o d e s f o r t h e DWBA and c o u p l e d e q u a t i o n s c a l c u l a t i o n s . 1) E.Rost and N.Austern, Phys. Rev. 120 (1960) 1375. R . H . B a s s e l et al., Phys. Rev. 128 (1962) 2693.
ON
THE IN
IMPORTANCE STRIPPING
OF AND
LETTERS
1July1963
2) C.A. Levinson and M. K. Banerjee, Ann. Phys. 2 (1957) 471. N.K.Glendenning, Phys. Rev. 114 (1959) 1297. F. Perey, l>roc. Conf. on Nuclear Structure, Kingston, Canada, 1960, ed. D. A. Bromley and E. Vogt (University of Toronto P r e s s , Toronto, 1960). 3) G.R. Satchler, to be published. 4) S. Yoshida, Prec. Phys. Soc. (London) A 69 (1956) 668. D. M. Chase, L. Wilets and A.R. Edmonds, Phys. Rev. 110 (1958) 1080. 5) B. Buck, Phys. Rev. 130 (1963) 712. 6) F. Perey, Phys. Rev., to be published. 7) B. Buck and G.R.Satchler, to be published. 8) R.H. Bassel, R. M. Drlsko and G. R. Satchler, Oak Ridge National Laboratory Report ORNL-3240 (1962). 9) N.Austern, lecture notes from Int. Summer School held in Low Tatra Mountains, Czechoslovakia, 1962.
THE RANGE KNOCKOUT
OF INTERACTIONS AMPLITUDES *
F. B. M O R I N I G O
California Institute of Technology, Pasadena, California Received 20 May 1963
In a r e c e n t p a p e r 1) T a n i f u j i d i s c u s s e s the k n o c k o u t m o d e f o r s t r i p p i n g r e a c t i o n s . He c a l c u l a t e s u s i n g t h e c u t - o f f B o r n a p p r o x i m a t i o n and d e l t a - f u n c t i o n i n t e r a c t i o n s and a s s e r t s t h a t t h e k n o c k o u t a m p l i t u d e if f o r w a r d p e a k e d , w h i l e the heavy-particle knockout amplitude is backward p e a k e d . T h i s l e t t e r w i s h e s to p o i n t out t h a t T a n i f u j i ' s r e s u l t s a r e not g e n e r a l but a r e due to h i s c h o i c e of an e x t r e m e l y s h o r t - r a n g e i n t e r a c t i o n . If one c a l c u l a t e s a k n o c k o u t a m p l i t u d e in a B o r n a p proximation without a cut-off, using harmonicoscillator wave functions, together with interact i o n s of g a u s s i a n s h a p e , the r e s u l t s m a y b e q u a i l t a t i v e l y d i f f e r e n t f r o m T a n i f u j i ' s if t h e r a n g e of t h e p o t e n t i a l i s not i n f i n i t e s i m a l . W h a t w a s thought b a c k w a r d p e a k e d b e c o m e s f o r w a r d p e a k e d , and vice versa. T h e e s s e n t i a l p o i n t i s i n d e p e n d e n t of the d e t a i l s of the a n g u l a r - m o m e n t u m c o u p l i n g r u l e s , s o t h a t we s h a l l d i s c u s s h e r e o n l y the c a s e of s p i n l e s s p a r t i c l e s , and bound s t a t e s of z e r o o r b i t a l a n g u l a r m o m e n t u m . T h e g e n e r a l c a s e of a r b i t r a r y s p i n s and h i g h e r a n g u l a r m o m e n t a w i l l be t h e s u b j e c t of a later paper. * Supported by the Office of Naval Research.
F i r s t , we c l a s s i f y the t e r m s in the a m p l i t u d e of a nuclear reaction x + Y-" W + z
(I)
according to the four m o d e s which appear in firstorder theory. The particles x and z are the light particles, for example, deuteron and proton in a typical (d,p) reaction. The four distinct ways in which the process occurs depends on whether the particle z w a s originally a part of x or of Y, and whether the m o m e n t u m exchange which results in the transition involves the particle z or its c o m panion, the captured particle. The four main m o d e s are easily understood by referring to fig. 1. The dashed lines represent m o m e n t u m exchanges. W e shall discuss explicitly the knockout m o d e corresponding to an interaction Vxz; the others are easily done by equally simple methods. W e consider the initial target state Y to be a bound state of z and s o m e other particle b. Similarly, w e consider the final state W to be a bound state of x and b. If w e denote the vector positions of these particles by z, b, x, the wave vector of the initial motion by Ko, and the wave vector of the final motion by A ~ , the initial and final states are the following:
215