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Nonlocal potential barrier and the perey-effect in alpha decay
Volume23, number 12
PHYSICS
LETTERS
19 December 1966
(1) of the s u r f a c e t e n s i o n . T h e i r a p p r o a c h i s d i f f e r e n t f r o m o u r s . T h e s p e c i f i c s u r f a c e e n e r g y ~ = aSA-] can be c o m p a r e d with a v a l u e known f r o m a n a l y s i s of e x p e r i m e n t s ~12[. If one a s s u m e s that the n u c l e u s i s s p h e r i c a l and h a s the s a t u r a t i o n d e n s i t y Po = = (2/37r z -)kf~ ~ A / ~ , t h e s p e c i f i c s u r f a c e e n e r g y i s g i v e n by 7 = ( 3 / P o ) # (41r)~a. We h a v e c a l c u l a t e d 7 f o r d i f f e r e n t ~ a l u e s of the t h i c k n e s s t a = 4a log 3 of the p o t e n t i a l Uo(x) in (5) and f o r d i f f e r e n t v a l u e s of the p a r a m e t e r / 3 in (3). M o r e o v e r , e a c h t i m e we h a v e e v a l u a t e d the t h i c k n e s s t (90%-10% d e f i n i t i o n ) of the d e n s i t y s l o p e in t h e s u r f a c e r e g i o n . A s f i x e d input d a t a we h a v e t a k e n the F e r m i m o m e n t u m kf = 1 . 4 5 f -1 c o r r e s p o n d i n g to 7o = ½ (91r)~kf-1 = 1.05 f and t h e F e r m i e n e r g y [ 1 3 ] ~ f = - 1 5 . 5 M e V . The r e s u l t s a r e p l o t t e d in f i g . 1. It s h o w s t h a t one g e t s the e x p e r i m e n t a l v a l u e s of ~, and t if a = 0 . 6 4 f a n d / 3 = 0 . 8 8 f . F r o m t h e s e v a l u e s we d e r i v e Vo = ( - ~ f + t f ) (1-¼~2kf2} -1 = 99.5 M e V . F o r c o m p a r i s o n we s h o u l d n o t e t h a t f o r e x a m p l e P e r e y and B u c k [10] h a v e o b t a i n e d a = 0 . 6 5 f , fl = 0 . 8 5 f , and Vo = 71 MeV by f i t t i n g the s c a t t e r i n g d a t a of s l o w n e u t r o n s . M e l d n e r et al. [14] h a v e p r o p o s e d a = 0 . 6 5 f , /3 = 0 . 9 0 f , and Vo = 76 M e V f o r c a l c u l a t i o n of n u c l e a r g r o u n d s t a t e e n e r g i e s . T h e i r p a r a m e t e r s r e f e r to a n o n - l o c a l p o t e n t i a l , which i s w e l l a p p r o x i m a t e d by o u r l o c a l e n e r g y - d e p e n d e n t v e r s i o n . F i g . 1 s h o w s t h a t t a k i n g / 3 = 0 one g e t s a n e g a t i v e s u r f a c e t e n s i o n , a s if the n u c l e u s w e r e not s t a b l e . T h e r e f o r e t h e s e r e s u l t s g i v e n e w e v i d e n c e f o r the n e c e s s i t y of i n t r o d u c i n g a v e l o c i t y d e p e n d e n t p o t e n t i a l in s h e l l m o d e l c a l c u l a t i o n s . References 1. C.F.v.Weizs~tcker, Die Atomkerne (Akademische Verlagsgesellschaft m.b.H., Leipzig, 193.7) § 1O. 2. P.Gombas, Die statistische Theorie des Atoms und ihre Anwendungen (Springer Verlag, Wien, 1949). 3. W . J . Swiatecki, P r o c . Phys. Soc. (London) A64 (1951) 226. 4. P . J . H i l l and J . A . W h e e l e r , Phys. Rev. 89 (1953) 1125. 5. G. Lanzl, Dissertation (1962)G~ttingen, Germany. 6. E.Hilf, Diplomarbeit (1963)Frankfurt a.M., Germany. 7. E . H i l f and G.Sflssmarm, Physics Letters 21 (1966) 654. 8. S.A.Guruits, A.A.Migdal and A . M . Polykow, Soviet Phys. J E T P 19 (1964) 149. 9. B . H . J . M c K e l l a r and M.A.Naqvi, Nucl. Phys. 71 (1965) 161. 1O. F. P e r e y and B.Buck, Nucl. Phys. 32 (1962) 353. 11. G.Sflssmann, to be published. 12. R.Hofstadter, Rev. Mod. Phys. 28 (1956) 214. 13. R.Muthukrishnan and M.Baranger, Physics Letters 18 (1965) 160. 14. H.Meldner, G.Sfissmann and W. Ulriei, Z. Naturforscb. 20a (1965) 160. 15. S.Knaak, Diplomarbeit, Frankfurt a . M . , Germany, to be published. ~****
NONLOCAL
POTENTIAL
BARRIER AND IN A L P H A DECAY
THE
PEREY-EFFECT
Gy. B E N C Z E * R e s e a r c h Institute f o r Theoretical P h y s i c s , University o f Helsinki, Helsinki, Finland Received 12 November 1966
It is shown that nonlocality in the alpha-nucleus potential increases the b a r r i e r penetrability. The range of nonlocality of the alpha-nucleus potential is estimated by comparing experimental and theoretical alpha decay rates.
In r e c e n t y e a r s a s a t i s f a c t o r y d e s c r i p t i o n of the r e l a t i v e v a l u e s of a l p h a d e c a y r a t e s h a s b e e n * On leave of absence from Central Research Institute for Physics, Budapest, Hungary.
achieved. However, the absolute values still diff e r c o n s i d e r a b l y f r o m the e x p e r i m e n t a l o n e s . S e v e r a l a r g u m e n t s s u g g e s t that the b a r r i e r p e n e t r a b i l i t i e s a r e r e s p o n s i b l e f o r the d i s c r e p a n c y , s i n c e the p o t e n t i a l b a r r i e r i s r a t h e r v a g u e l y d e f i n e d . In the 713
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PHYSICS LETTERS
usual p r o c e d u r e the potential b a r r i e r i s taken to be the sum of the r e a l p a r t of the a l p h a - n u c l e u s potential and of the Coulomb i n t e r a c t i o n . Howe v e r , due to the well-known a m b i g u i t i e s in the c o m p o s i t e p a r t i c l e o p t i c a l p o t e n t i a l s , t h e r e i s no unique choice f o r the potential b a r r i e r , and s o m e o t h e r a r g u m e n t s have to be invoked. AcCording to Rook [1], the c o m p o s i t e p a r t i c l e o p t i c a l m o d e l p o t e n t i a l s should a l w a y s be at l e a s t a s deep a s the sum of the optical p o t e n t i a l s a c t i n g on the constituent nucleons. In a r e c e n t p a p e r [2] it was shown that indeed the choice of an a l p h a - n u c l e u s o p t i c a l - m o d e l potential of p r o p e r depth and the use of an a c c u r a t e method i n s t e a d of the WKBmethod f o r c a l c u l a t i n g p e n e t r a b i l i t i e s can e x plain the d i s c r e p a n c i e s in the a b s o l u t e v a l u e s . N e v e r t h e l e s s , the o p t i c a l - m o d e l potential i s known to be nonlocal and the nonlocality a l s o ought to be taken into account. T h i s p r o b l e m was studied f i r s t by Chaudhury [3], who i n t r o d u c e d a s m a l l nonlocality into the Igo p o t e n t i a l and c a l culated the b a r r i e r p e n e t r a b i l i t i e s by using the effective m a s s and WKB a p p r o x i m a t i o n s . It was found that the nonlocality i n c r e a s e d the p e n e t r a b i l i t i e s by about 50%. The a i m of the p r e s e n t note i s to point out that this i n c r e a s e i s a d i r e c t consequence of the s o - c a l l e d P e r e y - e f f e c t , and give a s i m p l e e x p r e s s i o n for the b a r r i e r p e n e t r a b i l i t i e s c a l c u l a t e d f r o m the nonlocal potential. F i r s t we r e c a l l the definition of the b a r r i e r p e n e t r a b i l i t y [4] PI(R) : 2kR
__~Pl (¢°)2 ~
(1)
where ¢l i s the channel wave function s a t i s f y i n g p u r e l y outgoing a s y m p t o t i c boundary conditions, l d e n o t e s the o r b i t a l a n g u l a r m o m e n t u m , R i s the channel r a d i u s , and k i s the wave n u m b e r . L e t us a s s u m e that the a l p h a - n u c l e u s p o t e n t i a l can be r e p r e s e n t e d in the f o r m [5]
V(~,9') = V N (~(~+~'))Sf~ (I~-WI) ,
19 December 1966
the r e d u c e d m a s s . Then the wave functions ~ N and ~ L c o r r e s p o n d i n g to the nonlocal and l o c a l equivalent p o t e n t i a l , r e s p e c t i v e l y , have the s a m e asymptotic form. However, Perey's numerical c a l c u l a t i o n s [6] have shown that the nonlocal wave function i s c o n s i s t e n t l y s m a l l e r than the c o r r e s p o n d i n g wave function of the equivalent l o c a l potential in the i n t e r a c t i o n r e g i o n . T h i s s o c a l l e d P e r e y - e f f e c t h a s been d i s c u s s e d by A u s t e r n [7] and quite r e c e n t l y by F i e d e l d e y [8]. In r e f . 8 it i s shown that the following a p p r o x i m a t e r e l a t i o n s h i p h o l d s between the wave f u n c tions ~ L and ~ N : #~N(~) = e x p { ( ~ 2 / 4 / ~ 2) V L ( r ) } . ~L(~).
(5)
It i s then e a s y to d e r i v e the c o r r e s p o n d i n g r e l a t i o n between the b a r r i e r p e n e t r a b i l i t i e s . L e t u s denote the l o c a l and nonlocal channel wave functions by q~lN(r) and ~ / L ( r ) , r e s p e c t i v e l y . Using eqs. (1) and (5) and the f a c t that ~IN(~) = = q~/L(¢¢), one i m m e d i a t e l y g e t s
P/(R) = exp{-(~2/2~2) VL(R)}P/(R),
(6)
where PlN and PI L are the corresponding penetrabilities. Becahse the potential VL(r) is attractive, the exponential factor is greater than unity. Thus the nonlocality increases the penetrability. Since the optical-model analyses use local optical-model potentials, they determine only the local equivalent potential VL(r). However, according to eq. (6) the penetrabilities calculated from VL should always be multiplied by the correction factor due to nonlocality. The alpha decay constant can be expressed in terms of the penetrabilities and reduced widths as
= n jl
~ol(R)
=
(2)
exp{(/zfl2/292) VL(R)]X-,pL(R, 2 (R" where the nonlocality factor is chosen to be
I) exp {- (~_~,)2/~2} 3
= (3)
-
It i s well-known that for a nonlocal potential of type (2) one can a l w a y s find an equivalent l o cal p o t e n t i a l [5], which g i v e s the s a m e s c a t t e r ing phase shifts. The l o c a l equivalent potential of P e r e y and Buck [5] i s given by
VL(r) = VN(r) exp {(#f12/2t/2) [ VL(r) + Vc(r) - E]}(4) where Vc(r) is the Coulomb interaction, and # is 714
I~
-Z.~ l i ) ~ l j l ~ 1. jl
(7)
It can be s e e n f r o m eq. (7) that the n o n l o c a l i t y c o r r e c t i o n i s the s a m e for d i f f e r e n t a l p h a g r o u p s and a f f e c t s only the a b s o l u t e value of the d e c a y constant. The r e l a t i o n (6) m a y have s e v e r a l i n t e r e s t ing a p p l i c a t i o n s . If we a s s u m e that the a l p h a - n u c l e u s i n t e r a c t i o n can be r e p r e s e n t e d by the n o n l o c a l f o r m (2) and if the r e d u c e d widths ~ l j l 2 a r e sufficiently a c c u r a t e l y p r e d i c t e d by n u c l e a r m o del c a l c u l a t i o n s , then the d i s c r e p a n c y in the a b s o l u t e value can be a t t r i b u t e d e n t i r e l y to the
Volume 23, number 12
PHYSICS LETTERS
s e t s of p o t e n t i a l s in table 1. H o w e v e r , f o r c o n s i s t en cy the r e d u c e d widths should then a l s o have been c a l c u l a t e d with d i f f e r e n t v a l u e s of R. As a c o n s e q u e n c e , the channel r a d i u s f o r the 191.9 MeV p o t en t i al is in the r e g i o n of r a p i d v a r i a t i o n and r e s u l t s in g r e a t e r p e n e t r a b i l i t i e s than the d e e p e s t one. If a c c o r d i n g to r e f . 1 we choose the 225.8 MeV potential, we can conclude that the r e a s o n a b l e c h o i c e ~ ~ 0.6 can p r e d i c t the c o r r e c t absolute v a l u e s . T h i s s u g g e s t s that r e d u c e d widths m ay be sufficiently a c c u r a t e l y c a l c u l a t e d f r o m c u r r e n t nuclear models.
Table 1 Potential of ref. 9
U ( M e V ) to(fro )
a(fm)
fi(fm) calculated according to the text
1. 2.
143.3 191.9
1.351 1.336
0.500 0.495
0.96 0.47
3.
225.8
1.304
0.515
0.63
p e n e t r a b i l i t i e s . Then taking VL(r) f r o m the o p t i c a l - m o d e l a n a l y s e s , the r a n g e of nonlocality fl can be d e t e r m i n e d f r o m the r e q u i r e m e n t that eq. (7) give the e x p e r i m e n t a l l y m e a s u r e d value. Or c o n v e r s e l y , if we know fi the s a m e p r o c e d u r e can be u s e d fo r r e s o l v i n g the p o te n t ia l depth a m biguities. As an i l l u s t r a t i o n of this procedure,.~ ..calculations w e r e p e r f o r m e d f o r the 2 ~ p u ~ 2 ~ U alpha decay p r o c e s s . F o r the a - ~°~92U i n t e r a c t i o n w e r e u s e d t h r e e s e t s of optical potentials obtained in a r e c e n t a n a l y s i s by M c F a d d e n and S a t c h l e r [9]. F o r this p a r t i c u l a r p r o c e s s the t h e o r y g i v e s a decay constant w h ic h is about 60 t i m e s s m a l l e r than the e x p e r i m e n t a l one. T h e r e f o r e the r a n g e of nonlocality/3 in eq. (6) was chosen so as to give p e n e t r a b i l i t i e s g r e a t e r by a f a c t o r of 60 than t h o s e in r e f . 10. T h e channel r a d i u s was s et at R = 8.25, as in r e f . 8. The r e s u l t s of the c a l c u l a t i o n s a r e given in table 1. It should be noted, that the e x a c t p e n e t r a b i l i t i e s a r e r a p i d l y v a r y i n g functions of R expect in the b a r r i e r r e g i o n . T h e r e f o r e , the channel r a d i u s h a s to be c a r e f u l l y chosen. In the n u m e r i c a l e x a m p l e s we have u s e d R = 8.25 a c c o r d i n g to r e f . 10, although R should be d i f f e r e n t f o r the t h r e e
19 December 1966
The author w i s h e s to thank P r o f e s s o r K . V . L a u r i k a i n e n f o r the kind h o sp i t al i t y extended to h i m at the R e s e a r c h Institute f o r T h e o r e t i c a l P h y s i c s . He is a l s o indebted to M r s . P i r k k o E s k o l a f o r p e r f o r m i n g the n u m e r i c a l c a l c u l a t i o n s .
References 1. J.R.Rook, Nucl. Phys. 61 (1965) 219. 2. Gy.Bencze and A.Sandulescu, Phys. Letters 22 (1966) 473. 3. M.L.Chaudhury, Phys. Rev. 130 (1963) 2339. 4. H.J. Mang, Ann. Rev. Nucl. Sci. 14 (1964) 1. 5. F . G . P e r e y and B.Buck, Nucl. Phys. 32 (1962) 353. 6. F.G. Perey, in: Direct interactions and nuclear reaction mechanisms, eds. E.Clementel and C. Villi (Gordon and Breach, N e w York 1961) p. 125. 7. N.Austern, Phys. Rev. 137 (1965) B852. 8. H.Fiedeldey, Nuel. Phys. 77 (1966)149. 9. L.McFadden and G.R.Satchler, Nucl. Phys. 84
(1966) 177. lO. I.K. Poggenburg, Ph.D. Thesis, UCRL-16187, 1965.
* * * * *
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PHYSICS
LETTERS
19 December 1966
(1) of the s u r f a c e t e n s i o n . T h e i r a p p r o a c h i s d i f f e r e n t f r o m o u r s . T h e s p e c i f i c s u r f a c e e n e r g y ~ = aSA-] can be c o m p a r e d with a v a l u e known f r o m a n a l y s i s of e x p e r i m e n t s ~12[. If one a s s u m e s that the n u c l e u s i s s p h e r i c a l and h a s the s a t u r a t i o n d e n s i t y Po = = (2/37r z -)kf~ ~ A / ~ , t h e s p e c i f i c s u r f a c e e n e r g y i s g i v e n by 7 = ( 3 / P o ) # (41r)~a. We h a v e c a l c u l a t e d 7 f o r d i f f e r e n t ~ a l u e s of the t h i c k n e s s t a = 4a log 3 of the p o t e n t i a l Uo(x) in (5) and f o r d i f f e r e n t v a l u e s of the p a r a m e t e r / 3 in (3). M o r e o v e r , e a c h t i m e we h a v e e v a l u a t e d the t h i c k n e s s t (90%-10% d e f i n i t i o n ) of the d e n s i t y s l o p e in t h e s u r f a c e r e g i o n . A s f i x e d input d a t a we h a v e t a k e n the F e r m i m o m e n t u m kf = 1 . 4 5 f -1 c o r r e s p o n d i n g to 7o = ½ (91r)~kf-1 = 1.05 f and t h e F e r m i e n e r g y [ 1 3 ] ~ f = - 1 5 . 5 M e V . The r e s u l t s a r e p l o t t e d in f i g . 1. It s h o w s t h a t one g e t s the e x p e r i m e n t a l v a l u e s of ~, and t if a = 0 . 6 4 f a n d / 3 = 0 . 8 8 f . F r o m t h e s e v a l u e s we d e r i v e Vo = ( - ~ f + t f ) (1-¼~2kf2} -1 = 99.5 M e V . F o r c o m p a r i s o n we s h o u l d n o t e t h a t f o r e x a m p l e P e r e y and B u c k [10] h a v e o b t a i n e d a = 0 . 6 5 f , fl = 0 . 8 5 f , and Vo = 71 MeV by f i t t i n g the s c a t t e r i n g d a t a of s l o w n e u t r o n s . M e l d n e r et al. [14] h a v e p r o p o s e d a = 0 . 6 5 f , /3 = 0 . 9 0 f , and Vo = 76 M e V f o r c a l c u l a t i o n of n u c l e a r g r o u n d s t a t e e n e r g i e s . T h e i r p a r a m e t e r s r e f e r to a n o n - l o c a l p o t e n t i a l , which i s w e l l a p p r o x i m a t e d by o u r l o c a l e n e r g y - d e p e n d e n t v e r s i o n . F i g . 1 s h o w s t h a t t a k i n g / 3 = 0 one g e t s a n e g a t i v e s u r f a c e t e n s i o n , a s if the n u c l e u s w e r e not s t a b l e . T h e r e f o r e t h e s e r e s u l t s g i v e n e w e v i d e n c e f o r the n e c e s s i t y of i n t r o d u c i n g a v e l o c i t y d e p e n d e n t p o t e n t i a l in s h e l l m o d e l c a l c u l a t i o n s . References 1. C.F.v.Weizs~tcker, Die Atomkerne (Akademische Verlagsgesellschaft m.b.H., Leipzig, 193.7) § 1O. 2. P.Gombas, Die statistische Theorie des Atoms und ihre Anwendungen (Springer Verlag, Wien, 1949). 3. W . J . Swiatecki, P r o c . Phys. Soc. (London) A64 (1951) 226. 4. P . J . H i l l and J . A . W h e e l e r , Phys. Rev. 89 (1953) 1125. 5. G. Lanzl, Dissertation (1962)G~ttingen, Germany. 6. E.Hilf, Diplomarbeit (1963)Frankfurt a.M., Germany. 7. E . H i l f and G.Sflssmarm, Physics Letters 21 (1966) 654. 8. S.A.Guruits, A.A.Migdal and A . M . Polykow, Soviet Phys. J E T P 19 (1964) 149. 9. B . H . J . M c K e l l a r and M.A.Naqvi, Nucl. Phys. 71 (1965) 161. 1O. F. P e r e y and B.Buck, Nucl. Phys. 32 (1962) 353. 11. G.Sflssmann, to be published. 12. R.Hofstadter, Rev. Mod. Phys. 28 (1956) 214. 13. R.Muthukrishnan and M.Baranger, Physics Letters 18 (1965) 160. 14. H.Meldner, G.Sfissmann and W. Ulriei, Z. Naturforscb. 20a (1965) 160. 15. S.Knaak, Diplomarbeit, Frankfurt a . M . , Germany, to be published. ~****
NONLOCAL
POTENTIAL
BARRIER AND IN A L P H A DECAY
THE
PEREY-EFFECT
Gy. B E N C Z E * R e s e a r c h Institute f o r Theoretical P h y s i c s , University o f Helsinki, Helsinki, Finland Received 12 November 1966
It is shown that nonlocality in the alpha-nucleus potential increases the b a r r i e r penetrability. The range of nonlocality of the alpha-nucleus potential is estimated by comparing experimental and theoretical alpha decay rates.
In r e c e n t y e a r s a s a t i s f a c t o r y d e s c r i p t i o n of the r e l a t i v e v a l u e s of a l p h a d e c a y r a t e s h a s b e e n * On leave of absence from Central Research Institute for Physics, Budapest, Hungary.
achieved. However, the absolute values still diff e r c o n s i d e r a b l y f r o m the e x p e r i m e n t a l o n e s . S e v e r a l a r g u m e n t s s u g g e s t that the b a r r i e r p e n e t r a b i l i t i e s a r e r e s p o n s i b l e f o r the d i s c r e p a n c y , s i n c e the p o t e n t i a l b a r r i e r i s r a t h e r v a g u e l y d e f i n e d . In the 713
Volume 23, number 12
PHYSICS LETTERS
usual p r o c e d u r e the potential b a r r i e r i s taken to be the sum of the r e a l p a r t of the a l p h a - n u c l e u s potential and of the Coulomb i n t e r a c t i o n . Howe v e r , due to the well-known a m b i g u i t i e s in the c o m p o s i t e p a r t i c l e o p t i c a l p o t e n t i a l s , t h e r e i s no unique choice f o r the potential b a r r i e r , and s o m e o t h e r a r g u m e n t s have to be invoked. AcCording to Rook [1], the c o m p o s i t e p a r t i c l e o p t i c a l m o d e l p o t e n t i a l s should a l w a y s be at l e a s t a s deep a s the sum of the optical p o t e n t i a l s a c t i n g on the constituent nucleons. In a r e c e n t p a p e r [2] it was shown that indeed the choice of an a l p h a - n u c l e u s o p t i c a l - m o d e l potential of p r o p e r depth and the use of an a c c u r a t e method i n s t e a d of the WKBmethod f o r c a l c u l a t i n g p e n e t r a b i l i t i e s can e x plain the d i s c r e p a n c i e s in the a b s o l u t e v a l u e s . N e v e r t h e l e s s , the o p t i c a l - m o d e l potential i s known to be nonlocal and the nonlocality a l s o ought to be taken into account. T h i s p r o b l e m was studied f i r s t by Chaudhury [3], who i n t r o d u c e d a s m a l l nonlocality into the Igo p o t e n t i a l and c a l culated the b a r r i e r p e n e t r a b i l i t i e s by using the effective m a s s and WKB a p p r o x i m a t i o n s . It was found that the nonlocality i n c r e a s e d the p e n e t r a b i l i t i e s by about 50%. The a i m of the p r e s e n t note i s to point out that this i n c r e a s e i s a d i r e c t consequence of the s o - c a l l e d P e r e y - e f f e c t , and give a s i m p l e e x p r e s s i o n for the b a r r i e r p e n e t r a b i l i t i e s c a l c u l a t e d f r o m the nonlocal potential. F i r s t we r e c a l l the definition of the b a r r i e r p e n e t r a b i l i t y [4] PI(R) : 2kR
__~Pl (¢°)2 ~
(1)
where ¢l i s the channel wave function s a t i s f y i n g p u r e l y outgoing a s y m p t o t i c boundary conditions, l d e n o t e s the o r b i t a l a n g u l a r m o m e n t u m , R i s the channel r a d i u s , and k i s the wave n u m b e r . L e t us a s s u m e that the a l p h a - n u c l e u s p o t e n t i a l can be r e p r e s e n t e d in the f o r m [5]
V(~,9') = V N (~(~+~'))Sf~ (I~-WI) ,
19 December 1966
the r e d u c e d m a s s . Then the wave functions ~ N and ~ L c o r r e s p o n d i n g to the nonlocal and l o c a l equivalent p o t e n t i a l , r e s p e c t i v e l y , have the s a m e asymptotic form. However, Perey's numerical c a l c u l a t i o n s [6] have shown that the nonlocal wave function i s c o n s i s t e n t l y s m a l l e r than the c o r r e s p o n d i n g wave function of the equivalent l o c a l potential in the i n t e r a c t i o n r e g i o n . T h i s s o c a l l e d P e r e y - e f f e c t h a s been d i s c u s s e d by A u s t e r n [7] and quite r e c e n t l y by F i e d e l d e y [8]. In r e f . 8 it i s shown that the following a p p r o x i m a t e r e l a t i o n s h i p h o l d s between the wave f u n c tions ~ L and ~ N : #~N(~) = e x p { ( ~ 2 / 4 / ~ 2) V L ( r ) } . ~L(~).
(5)
It i s then e a s y to d e r i v e the c o r r e s p o n d i n g r e l a t i o n between the b a r r i e r p e n e t r a b i l i t i e s . L e t u s denote the l o c a l and nonlocal channel wave functions by q~lN(r) and ~ / L ( r ) , r e s p e c t i v e l y . Using eqs. (1) and (5) and the f a c t that ~IN(~) = = q~/L(¢¢), one i m m e d i a t e l y g e t s
P/(R) = exp{-(~2/2~2) VL(R)}P/(R),
(6)
where PlN and PI L are the corresponding penetrabilities. Becahse the potential VL(r) is attractive, the exponential factor is greater than unity. Thus the nonlocality increases the penetrability. Since the optical-model analyses use local optical-model potentials, they determine only the local equivalent potential VL(r). However, according to eq. (6) the penetrabilities calculated from VL should always be multiplied by the correction factor due to nonlocality. The alpha decay constant can be expressed in terms of the penetrabilities and reduced widths as
= n jl
~ol(R)
=
(2)
exp{(/zfl2/292) VL(R)]X-,pL(R, 2 (R" where the nonlocality factor is chosen to be
I) exp {- (~_~,)2/~2} 3
= (3)
-
It i s well-known that for a nonlocal potential of type (2) one can a l w a y s find an equivalent l o cal p o t e n t i a l [5], which g i v e s the s a m e s c a t t e r ing phase shifts. The l o c a l equivalent potential of P e r e y and Buck [5] i s given by
VL(r) = VN(r) exp {(#f12/2t/2) [ VL(r) + Vc(r) - E]}(4) where Vc(r) is the Coulomb interaction, and # is 714
I~
-Z.~ l i ) ~ l j l ~ 1. jl
(7)
It can be s e e n f r o m eq. (7) that the n o n l o c a l i t y c o r r e c t i o n i s the s a m e for d i f f e r e n t a l p h a g r o u p s and a f f e c t s only the a b s o l u t e value of the d e c a y constant. The r e l a t i o n (6) m a y have s e v e r a l i n t e r e s t ing a p p l i c a t i o n s . If we a s s u m e that the a l p h a - n u c l e u s i n t e r a c t i o n can be r e p r e s e n t e d by the n o n l o c a l f o r m (2) and if the r e d u c e d widths ~ l j l 2 a r e sufficiently a c c u r a t e l y p r e d i c t e d by n u c l e a r m o del c a l c u l a t i o n s , then the d i s c r e p a n c y in the a b s o l u t e value can be a t t r i b u t e d e n t i r e l y to the
Volume 23, number 12
PHYSICS LETTERS
s e t s of p o t e n t i a l s in table 1. H o w e v e r , f o r c o n s i s t en cy the r e d u c e d widths should then a l s o have been c a l c u l a t e d with d i f f e r e n t v a l u e s of R. As a c o n s e q u e n c e , the channel r a d i u s f o r the 191.9 MeV p o t en t i al is in the r e g i o n of r a p i d v a r i a t i o n and r e s u l t s in g r e a t e r p e n e t r a b i l i t i e s than the d e e p e s t one. If a c c o r d i n g to r e f . 1 we choose the 225.8 MeV potential, we can conclude that the r e a s o n a b l e c h o i c e ~ ~ 0.6 can p r e d i c t the c o r r e c t absolute v a l u e s . T h i s s u g g e s t s that r e d u c e d widths m ay be sufficiently a c c u r a t e l y c a l c u l a t e d f r o m c u r r e n t nuclear models.
Table 1 Potential of ref. 9
U ( M e V ) to(fro )
a(fm)
fi(fm) calculated according to the text
1. 2.
143.3 191.9
1.351 1.336
0.500 0.495
0.96 0.47
3.
225.8
1.304
0.515
0.63
p e n e t r a b i l i t i e s . Then taking VL(r) f r o m the o p t i c a l - m o d e l a n a l y s e s , the r a n g e of nonlocality fl can be d e t e r m i n e d f r o m the r e q u i r e m e n t that eq. (7) give the e x p e r i m e n t a l l y m e a s u r e d value. Or c o n v e r s e l y , if we know fi the s a m e p r o c e d u r e can be u s e d fo r r e s o l v i n g the p o te n t ia l depth a m biguities. As an i l l u s t r a t i o n of this procedure,.~ ..calculations w e r e p e r f o r m e d f o r the 2 ~ p u ~ 2 ~ U alpha decay p r o c e s s . F o r the a - ~°~92U i n t e r a c t i o n w e r e u s e d t h r e e s e t s of optical potentials obtained in a r e c e n t a n a l y s i s by M c F a d d e n and S a t c h l e r [9]. F o r this p a r t i c u l a r p r o c e s s the t h e o r y g i v e s a decay constant w h ic h is about 60 t i m e s s m a l l e r than the e x p e r i m e n t a l one. T h e r e f o r e the r a n g e of nonlocality/3 in eq. (6) was chosen so as to give p e n e t r a b i l i t i e s g r e a t e r by a f a c t o r of 60 than t h o s e in r e f . 10. T h e channel r a d i u s was s et at R = 8.25, as in r e f . 8. The r e s u l t s of the c a l c u l a t i o n s a r e given in table 1. It should be noted, that the e x a c t p e n e t r a b i l i t i e s a r e r a p i d l y v a r y i n g functions of R expect in the b a r r i e r r e g i o n . T h e r e f o r e , the channel r a d i u s h a s to be c a r e f u l l y chosen. In the n u m e r i c a l e x a m p l e s we have u s e d R = 8.25 a c c o r d i n g to r e f . 10, although R should be d i f f e r e n t f o r the t h r e e
19 December 1966
The author w i s h e s to thank P r o f e s s o r K . V . L a u r i k a i n e n f o r the kind h o sp i t al i t y extended to h i m at the R e s e a r c h Institute f o r T h e o r e t i c a l P h y s i c s . He is a l s o indebted to M r s . P i r k k o E s k o l a f o r p e r f o r m i n g the n u m e r i c a l c a l c u l a t i o n s .
References 1. J.R.Rook, Nucl. Phys. 61 (1965) 219. 2. Gy.Bencze and A.Sandulescu, Phys. Letters 22 (1966) 473. 3. M.L.Chaudhury, Phys. Rev. 130 (1963) 2339. 4. H.J. Mang, Ann. Rev. Nucl. Sci. 14 (1964) 1. 5. F . G . P e r e y and B.Buck, Nucl. Phys. 32 (1962) 353. 6. F.G. Perey, in: Direct interactions and nuclear reaction mechanisms, eds. E.Clementel and C. Villi (Gordon and Breach, N e w York 1961) p. 125. 7. N.Austern, Phys. Rev. 137 (1965) B852. 8. H.Fiedeldey, Nuel. Phys. 77 (1966)149. 9. L.McFadden and G.R.Satchler, Nucl. Phys. 84
(1966) 177. lO. I.K. Poggenburg, Ph.D. Thesis, UCRL-16187, 1965.
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