- Email: [email protected]
DWBA - calculation of the amplitudes of rectangle diagrams in direct nuclear reactions
Volume 31B, n u m b e r 3
OF
PHYSICS
DWBA-CALCULATION RECTANGLE DIAGRAMS
IN
LETTERS
OF THE DIRECT
2 F e b r u a r y 1970
AMPLITUDES NUCLEAR REACTIONS
E. Zh. M A G Z U M O V , V . G . N E U D A T C H I N
Institute of Nuclear P h y s i c s , Moscow State University, Moscow, USSR Received 1 D e c e m b e r 1969
The differential c r o s s - s e c t i o n s of the (t, p) and (p,p') r e a c t i o n s r e a l i z e d on p - s h e l l nuclei and d e s c r i b e d with the r e c t a n g l e d i a g r a m a r e calculated. The DWBA with s u r f a c e i n t e r a c t i o n is used. The r e s u l t s show that the f o r m of a n g u l a r d i s t r i b u t i o n is i n v a r i a n t as c o m p a r e d to the c a s e of the plane waves. T h e a b solute value of the c r o s s - s e c t i o n d e c r e a s e s rapidly with i n c r e a s i n g e n e r g y of the b o m b a r d i n g p a r t i c l e .
As was noted earlier [1,2], in some cases the (t,p) - (3He,p) reaction may take place through the mechanism of independent nucleon capture described by the rectangle diagram. For this process to happen, it is necessary that the difference in the nucleon binding energies should be very substantial in the vertices (A+ I) c ~ A + nI and (A+2) B ~ (A+ I)c + n2, i.e. the difference in the behaviour of radial wave functions of captured nucleons should be substantial in the nuclear surface region. It was indicated that the range of realization of the rectangle diagrams is not limited to only the twonucleon stripping. Under certain circumstances [2,3], such a mechanism may also show up in the r e actions of inelastic scattering of protons and deuterons on light nuclei. A simple oscillator method was proposed [2,3] for calculating the angular distributions in the reaction corresponding to rectangle diagrams. Analysis of some specific examples is indicative of a fairly pronounced forward peak which is a characteristic feature of angular distributions with small angular momentum transfer (L = 0, I, 2) and momenta (q --< 0.4 fm-l). It will be noted that this feature cannot be explained by rotational excitation of either initial or final nucleus [4]. (For example, the (t,p)-reactions, even on such nuclei as 24Mg [5] and 28Si [6], do not show a contribution from the rotational excitation mechanism and preserve the usual picture of stripping). This fact is also difficult to be explained by the influence of distortions in the case of the pole mechanism of stripping [7,8]. Transition to strong distortions [9] results in the fact that the pronounced forward peak should be observed in all, without exception, angular distributions of the (t,p) - (3He,p) reactions including the r e actions with L = 2, and this contradicts the vast volume of experimental data [I]. However, another unsolved problem bearing on the influence of distortions on differential cross-sections already within the framework of rectangle diagram remains, e.e. as to whether the forward peak is actually a feature inherent in the rectangle diagram or a false effect due to excessive roughness of the plane wave approach. Since even in comparatively simple cases of the pole stripping, the distorted wave calculations often do not give good results for absolute cross-sections [8,7] we have calculated the angular distributions using the simplest "surface" version of the distorted wave method [I0]. If the absolute cross-sections are not of interest, this method will give good results, in particular for calculating the relative values of cross-sections for a broad range of various levels of the final nucleus [11]. The problem of estimates of absolute values of the cross-sections will be discussed below. The application of the distorted wave method with surface interaction to the calculation of the (t,p)reaction cross-sections within the framework of the rectangle diagram gives the following expression:
106
Volume 31B, number 3
~
~ ,± ~,± [/1][/2][A][A][A ]~ [ A ] e
LSJA
[S][p] ½ [L] ~
d q_~ ~ B d~
:PH Y S I C S L E T T E R S
U(~:SdS)
2 February 1970
U(A AA A:Lp)(AOA0 [p0)(A'0.&'0 IpOlPp (cos ( k t ' k p ) ) x
h'~. 7i'p
x ~ (A0/~0t~0)(~0Z20IA'0)(-~0/~0I'~0)(~0Z2017V0) U( A'12All :h L) U( X'12 All:~L)R AhA ' R A~ ~, where
11 =l 2 =l ,
[/]=- ( 2 / + 1 ) ,
EC, 1 = E ~ - E t h r .
a r e the d i s t o r t e d r a d i a l wave f u n c t i o n s of t r i t o n , p r o t o n , and i n t e r m e d i a t e d e u t e r o n r e s p e c t i v e l y ; Ull(Ro) and ull(R O) a r e the r a d i a l wave f u n c t i o n s of c a p t u r e d n e u t r o n s in a W o o d s - S a x o n p o t e n t i a l ; R 0 is the r a d i u s of the n u c l e a r s u r f a c e ; 2rod _ E t) K 2 = ~ 2 (¢C,1 ct,1 +
= A,B,C
U(SA ~ S B ~.S d S) U(L A l I L s 12:L C L)(T A MTA ~ - ~I TC M T - )(TC M T C ½ - ~ITB M T B) × x a LASA a LBSB a 2L c S c /ln+2Btln+l C, l) (l n+l C I lnA, l) 12
(the l a s t c o f a c t o r s d e n o t e the f r a c t i o n a l p a r e n t a g e c o e f f i c i e n t s , s e e a l s o ref. 1). A s i m i l a r f o r m u l a has a l s o b e e n o b t a i n e d for the d i f f e r e n t i a l c r o s s - s e c t i o n of the ( p , p ' ) r e a c t i o n [12]. The d i s t o r t e d wave f u n c t i o n with o p t i c a l p o t e n t i a l s in the W o o d s - S a x o n f o r m w e r e c a l c u l a t e d on the b a s i s of the m e t h o d and p r o g r a m p r o p o s e d in ref. 13. The o p t i c a l p a r a m e t e r s for the r e a c t i o n s 10B(t,p)12B and 9 B e ( p , p ' ) 9 B e ( ½ + , E * = 1.75 MeV) had the f o l l o w i n g v a l u e s [8,14]: - Vt = 152 MeV, Wt =
O5
a
\
,
~o3
\,...5/
E~-s,~v
fp=TMeV
Ct = / 0 Mev
O4
~
v
o.2
ot
20
~o
go
do
~o
Fig. 1. Theoretical differential cross-sections of the 10B(t, p)12B reaction.
Fig. 2. Theoretical differential cross-sections of the 9Be(p,p')gBe* (½+) reaction. 107
Volume 3IB, n u m b e r 3
PHYSICS
LETTERS
2 F e b r u a r y 1970
= 14 M e V , r o t = 1.9 fro, a t = 0.5 f m ; - Vd = 81 M e V , - Wd = 18 M e V , r o d = 1.9 f m , a d = 0.5 f m ; - Vp = = 53 M e V , - Wp = 20.5 M e V , t o p = 1.8 f m , a p = 0.48 f m ; (the c o n s i d e r a b l e d e p t h of t h e r e a l p a r t of t h e o p t i c a l p o t e n t i a l c o r r e s p o n d s t o a g r e a t n u m b e r of o s c i l l a t i o n s of t h e w a v e f u n c t i o n of t h e n u c l e o n c l u s t e r m o t i o n in t h e n u c l e u s [15] a s it i s r e q u i r e d i n t h e s h e l l m o d e l ) . F i g s . l a n d 2 s h o w t h e r e s u l t s of c a l c u l a t i o n s f o r d i f f e r e n t e n e r g i e s of b o m b a r d i n g t r i t o n s a n d p r o t o n s . It c a n b e s e e n f r o m t h e f i g u r e s t h a t t h e a n g u l a r d i s t r i b u t i o n s k e e p a p p r o x i m a t e l y t h e i r f o r m i n t h e s m a l l a n g l e r a n g e w i t h c h a n g i n g e n e r g y (they o n l y b e c o m e s l i g h t l y n a r r o w e r ) b u t t h e a b s o l u t e v a l u e s of t h e i n t e g r a l c r o s s - s e c t i o n s a r e s t r o n g l y d e c r e a s e d w i t h i n c r e a s i n g e n e r g y . Such a p r o n o u n c e d d e p e n d e n c e i s p r o b a b l y a v e r y g e n e r a l p r o p e r t y of t h e a m p l i t u d e s of c o m p l e x d i a g r a m s ( s e e a l s o r e f . 16) a n d m a y b e a n a d d i t i o n a l c r i t e r i o n f o r i d e n t i f i c a t i o n of t h e s e d i a g r a m s . It c a n b e i n d i c a t e d , f o r t h e c o m p a r i s o n s a k e , t h a t t h e ( d , p ) - r e a c t i o n c r o s s - s e c t i o n s in t h e s a m e r a n g e of e n e r g i e s a n d n u c l e i a r e d e c r e a s e d m u c h m o r e s l o w l y (for e x a m p l e t h e ! 2 C ( d , p)13C r e a c t i o n c r o s s - s e c t i o n i n t h e 5 - 2 0 M e V e n e r g y r a n g e i s d e c r e a s e d b y a f a c t o r of only 1 . 5 - 2 [17]). T h u s , at s u f f i c i e n t l y h i g h e n e r g i e s t h e r e c t a n g l e m e c h a n i s m of t h e d i r e c t r e a c t i o n s s h o u l d give a v e r y s m a l l c o n t r i b u t i o n to t h e p r o c e s s a s c o m p a r e d to s i m p l e r m e c h a n i s m s . T h e a b o v e a n a l y s i s of t h e i n f l u e n c e of d i s t o r t i o n s on a n g u l a r d i s t r i b u t i o n s h a s s h o w n t h a t t h e p e a k a t z e r o a n g l e i s not s e n s i t i v e to t h e d i s t o r t i o n s a n d i s a c h a r a c t e r i s t i c f e a t u r e of t h e r e c t a n g l e d i a g r a m s . T h i s f a c t j u s t i f i e s t h e a p p l i c a t i o n of t h e p l a n e w a v e a p p r o x i m a t i o n , in t h i s c a s e the a b s o l u t e v a l u e s of t h e c r o s s - s e c t i o n s of t h e r e a c t i o n s (t, p) a n d (p, p) c a n b e e s t i m a t e d u s i n g " r e n o r m a l i z a t i o n " of t h e s i n g l e - p a r t i c l e r e d u c e d w i d t h s c a l c u l a t e d on t h e b a s i s of t h e d a t a on the r e a c t i o n s (t, d) a n d ( d , p ) u s i n g the appropriate experimental and theoretical data and thereby can include sufficiently reliable the inf l u e n c e of t h e d i s t o r t i o n s a l r e a d y on t h e a b s o l u t e c r o s s - s e c t i o n s . F o r t h e 1 0 B ( t , p ) I Z B r e a c t i o n (E t = 10 MeV) s u c h a n a p p r o a c h g a v e t h e v a l u e [d~/d~2 ]8 =0 ° -- 0.7 m b / s t e r a d ( e x p e r i m e n t a l v a l u e i s 1.2 m b / s t e r a d [5]); t h i s v a l u e c a l c u l a t e d in t h e o s c i l l a t o r a p p r o x i m a t i o n [2] w i t h t h e a b o v e m e n t i o n e d r e n o r m a l i z a t i o n i s 0.4 m b / s t e r a d . S i m i l a r c a l c u l a t i o n s of t h e c r o s s - s e c t i o n s of t h e 9 B e ( p , p ' ) g B e ( ½ +) r e a c t i o n g i v e t h e v a l u e I d a / d 2 ] 0 =0 o = 0.2 m b / s t e r a d (the e x p e r i m e n t a l v a l u e i s u n k n o w n ; e x t r a p o l a t i o n to z e r o a n g l e of t h e c r o s s - s e c t i o n s f r o m r e f . 18 m e a s u r e d w i t h l a r g e e r r o r g i v e s t h e v a l u e of ~ 0 . 5 - 3 m b / s t e r a d ) . T h u s , the i m p o r t a n t c o n c l u s i o n m a y b e d r a w n t h a t t h e a b s o l u t e v a l u e s of t h e c r o s s - s e c t i o n s c a n b e o b t a i n e d w i t h i n t h e p r o p e r o r d e r of m a g n i t u d e ; in p a r t i c u l a r , t h e v a l u e of the c r o s s - s e c t i o n of t h e ( t , p ) r e a c t i o n r e a l i z e d t h r o u g h t h e r e c t a n g l e d i a g r a m f o r low t r i t o n e n e r g i e s ( 5 - 1 0 MeV) i s only s l i g h t l y s m a l l e r t h a n t h e u s u a l c r o s s - s e c t i o n s of t h e p o l e ( t , p ) - r e a c t i o n s (1-5 m b / s t e r a d ) . A s w a s n o t e d a b o v e , w i t h i n c r e a s i n g e n e r g y of t h e b o m b a r d i n g p a r t i c l e s , t h e s i t u a t i o n s h o u l d a b r u p t l y c h a n g e . P r o b a b l y t h e r a n g e of c o m p a r a t i v e l y low e n e r g i e s i s of s p e c i a l i n t e r e s t f o r t h e p h y s i c s of d i r e c t p r o c e s s e s on l i g h t nuclei.
References
1. 2. 3. 4.
18.
E. Bang, N.S. Zelenskaya, E. Zh. Magzumov, V. G. Neudatchin, J. Nucl. Phys. (USSH) 4 (1965) 962. V. G. Neudatchin, N.S. Zelenskaya, E. Zh. Magzumov, E.A. Romanovsky, Phys. L e t t e r s 27B (1968) 490. V. G. Neudatchin, E. Zh. Magzumov. E. A. Romanovsky, J. Nucl. Phys. (USSR) 9 (1969) 94. P . J . Iono, S.K. Penny, R. M. Drisko. Nucl. Phys. A127 (1969) 47; B. Kozlowsky. A. De Shalit, Nucl. Phys. 77 (1966) 215; F. S. Levin, Phys. Rev. 147 (1966) 71~. R. Middleton, D.J. Pullea. Nucl. Phvs. 51 (1964) 50, 77. J . C . H a r d y , I.S. Towner, Phys. L e t t e r s 25B (1967) 577. E. M. Henley, D.V.L. Yu. Phys. Rev. 133 (1964) B1445. D. Fleming, J. Cerny, C. C. Maples, N.K. Glendenning, Phys. Rev. 166 (1968) 1012. A. Dar, Nucl. Phys. 55 (1964) 305; Phys. Rev. 137 (1965) B102. N.K. Glendenning, Phys. Hey. 114 (1957) 1297. N.K. Glendenning, Phys. Hey. 137 (1965) B102; Ann. Hey. Nucl. Sci. 13 (1963); Phys. Rev. 156 (1967) 1344. E. Zh. Magzumov, V. G. Neudatchin, M. S. Belkin, J. Nucl. Phys. (USSR), to be published. K. A. Gridnev, L. V. Krasnov et al., P r e p r i n t JINR 2458, Dubna, 1965. L . L . Lee J r . , J. P. Schiffer et al., Phys. Rev. 136 (1964) B971; A. A. Fife, G . C . N e i l s o n , W. K. B e r t r a m , L. J. T a s s i e , Phys. Rev. 156 (1968) 1029. V. G. Neudatchin, Yu. F. Smirnov, Atomic Energy Rev., IAEA, Vienna, v. 3 (1965) 157. P. Beregi, V. G. Neudatchin, Yu. F. Smirnov, Nucl. Phys. 26 (1961) 306. M.H. M c F a r l a n e , J. B. French, Hev. Mod. Phys. 32 (1960) 567; J. N. McGruer, E.K. Warburton, R. S. Bender, Phys. Rev. 100 (1955) 235. R. G. S u m m e r s - G i l l , Phys. Rev. 103 (1958) 1591.
108
*****
5. 6. 7. 8. 9. 10. 11. [2. 13. 14. 15. 16. 17.
OF
PHYSICS
DWBA-CALCULATION RECTANGLE DIAGRAMS
IN
LETTERS
OF THE DIRECT
2 F e b r u a r y 1970
AMPLITUDES NUCLEAR REACTIONS
E. Zh. M A G Z U M O V , V . G . N E U D A T C H I N
Institute of Nuclear P h y s i c s , Moscow State University, Moscow, USSR Received 1 D e c e m b e r 1969
The differential c r o s s - s e c t i o n s of the (t, p) and (p,p') r e a c t i o n s r e a l i z e d on p - s h e l l nuclei and d e s c r i b e d with the r e c t a n g l e d i a g r a m a r e calculated. The DWBA with s u r f a c e i n t e r a c t i o n is used. The r e s u l t s show that the f o r m of a n g u l a r d i s t r i b u t i o n is i n v a r i a n t as c o m p a r e d to the c a s e of the plane waves. T h e a b solute value of the c r o s s - s e c t i o n d e c r e a s e s rapidly with i n c r e a s i n g e n e r g y of the b o m b a r d i n g p a r t i c l e .
As was noted earlier [1,2], in some cases the (t,p) - (3He,p) reaction may take place through the mechanism of independent nucleon capture described by the rectangle diagram. For this process to happen, it is necessary that the difference in the nucleon binding energies should be very substantial in the vertices (A+ I) c ~ A + nI and (A+2) B ~ (A+ I)c + n2, i.e. the difference in the behaviour of radial wave functions of captured nucleons should be substantial in the nuclear surface region. It was indicated that the range of realization of the rectangle diagrams is not limited to only the twonucleon stripping. Under certain circumstances [2,3], such a mechanism may also show up in the r e actions of inelastic scattering of protons and deuterons on light nuclei. A simple oscillator method was proposed [2,3] for calculating the angular distributions in the reaction corresponding to rectangle diagrams. Analysis of some specific examples is indicative of a fairly pronounced forward peak which is a characteristic feature of angular distributions with small angular momentum transfer (L = 0, I, 2) and momenta (q --< 0.4 fm-l). It will be noted that this feature cannot be explained by rotational excitation of either initial or final nucleus [4]. (For example, the (t,p)-reactions, even on such nuclei as 24Mg [5] and 28Si [6], do not show a contribution from the rotational excitation mechanism and preserve the usual picture of stripping). This fact is also difficult to be explained by the influence of distortions in the case of the pole mechanism of stripping [7,8]. Transition to strong distortions [9] results in the fact that the pronounced forward peak should be observed in all, without exception, angular distributions of the (t,p) - (3He,p) reactions including the r e actions with L = 2, and this contradicts the vast volume of experimental data [I]. However, another unsolved problem bearing on the influence of distortions on differential cross-sections already within the framework of rectangle diagram remains, e.e. as to whether the forward peak is actually a feature inherent in the rectangle diagram or a false effect due to excessive roughness of the plane wave approach. Since even in comparatively simple cases of the pole stripping, the distorted wave calculations often do not give good results for absolute cross-sections [8,7] we have calculated the angular distributions using the simplest "surface" version of the distorted wave method [I0]. If the absolute cross-sections are not of interest, this method will give good results, in particular for calculating the relative values of cross-sections for a broad range of various levels of the final nucleus [11]. The problem of estimates of absolute values of the cross-sections will be discussed below. The application of the distorted wave method with surface interaction to the calculation of the (t,p)reaction cross-sections within the framework of the rectangle diagram gives the following expression:
106
Volume 31B, number 3
~
~ ,± ~,± [/1][/2][A][A][A ]~ [ A ] e
LSJA
[S][p] ½ [L] ~
d q_~ ~ B d~
:PH Y S I C S L E T T E R S
U(~:SdS)
2 February 1970
U(A AA A:Lp)(AOA0 [p0)(A'0.&'0 IpOlPp (cos ( k t ' k p ) ) x
h'~. 7i'p
x ~ (A0/~0t~0)(~0Z20IA'0)(-~0/~0I'~0)(~0Z2017V0) U( A'12All :h L) U( X'12 All:~L)R AhA ' R A~ ~, where
11 =l 2 =l ,
[/]=- ( 2 / + 1 ) ,
EC, 1 = E ~ - E t h r .
a r e the d i s t o r t e d r a d i a l wave f u n c t i o n s of t r i t o n , p r o t o n , and i n t e r m e d i a t e d e u t e r o n r e s p e c t i v e l y ; Ull(Ro) and ull(R O) a r e the r a d i a l wave f u n c t i o n s of c a p t u r e d n e u t r o n s in a W o o d s - S a x o n p o t e n t i a l ; R 0 is the r a d i u s of the n u c l e a r s u r f a c e ; 2rod _ E t) K 2 = ~ 2 (¢C,1 ct,1 +
= A,B,C
U(SA ~ S B ~.S d S) U(L A l I L s 12:L C L)(T A MTA ~ - ~I TC M T - )(TC M T C ½ - ~ITB M T B) × x a LASA a LBSB a 2L c S c /ln+2Btln+l C, l) (l n+l C I lnA, l) 12
(the l a s t c o f a c t o r s d e n o t e the f r a c t i o n a l p a r e n t a g e c o e f f i c i e n t s , s e e a l s o ref. 1). A s i m i l a r f o r m u l a has a l s o b e e n o b t a i n e d for the d i f f e r e n t i a l c r o s s - s e c t i o n of the ( p , p ' ) r e a c t i o n [12]. The d i s t o r t e d wave f u n c t i o n with o p t i c a l p o t e n t i a l s in the W o o d s - S a x o n f o r m w e r e c a l c u l a t e d on the b a s i s of the m e t h o d and p r o g r a m p r o p o s e d in ref. 13. The o p t i c a l p a r a m e t e r s for the r e a c t i o n s 10B(t,p)12B and 9 B e ( p , p ' ) 9 B e ( ½ + , E * = 1.75 MeV) had the f o l l o w i n g v a l u e s [8,14]: - Vt = 152 MeV, Wt =
O5
a
\
,
~o3
\,...5/
E~-s,~v
fp=TMeV
Ct = / 0 Mev
O4
~
v
o.2
ot
20
~o
go
do
~o
Fig. 1. Theoretical differential cross-sections of the 10B(t, p)12B reaction.
Fig. 2. Theoretical differential cross-sections of the 9Be(p,p')gBe* (½+) reaction. 107
Volume 3IB, n u m b e r 3
PHYSICS
LETTERS
2 F e b r u a r y 1970
= 14 M e V , r o t = 1.9 fro, a t = 0.5 f m ; - Vd = 81 M e V , - Wd = 18 M e V , r o d = 1.9 f m , a d = 0.5 f m ; - Vp = = 53 M e V , - Wp = 20.5 M e V , t o p = 1.8 f m , a p = 0.48 f m ; (the c o n s i d e r a b l e d e p t h of t h e r e a l p a r t of t h e o p t i c a l p o t e n t i a l c o r r e s p o n d s t o a g r e a t n u m b e r of o s c i l l a t i o n s of t h e w a v e f u n c t i o n of t h e n u c l e o n c l u s t e r m o t i o n in t h e n u c l e u s [15] a s it i s r e q u i r e d i n t h e s h e l l m o d e l ) . F i g s . l a n d 2 s h o w t h e r e s u l t s of c a l c u l a t i o n s f o r d i f f e r e n t e n e r g i e s of b o m b a r d i n g t r i t o n s a n d p r o t o n s . It c a n b e s e e n f r o m t h e f i g u r e s t h a t t h e a n g u l a r d i s t r i b u t i o n s k e e p a p p r o x i m a t e l y t h e i r f o r m i n t h e s m a l l a n g l e r a n g e w i t h c h a n g i n g e n e r g y (they o n l y b e c o m e s l i g h t l y n a r r o w e r ) b u t t h e a b s o l u t e v a l u e s of t h e i n t e g r a l c r o s s - s e c t i o n s a r e s t r o n g l y d e c r e a s e d w i t h i n c r e a s i n g e n e r g y . Such a p r o n o u n c e d d e p e n d e n c e i s p r o b a b l y a v e r y g e n e r a l p r o p e r t y of t h e a m p l i t u d e s of c o m p l e x d i a g r a m s ( s e e a l s o r e f . 16) a n d m a y b e a n a d d i t i o n a l c r i t e r i o n f o r i d e n t i f i c a t i o n of t h e s e d i a g r a m s . It c a n b e i n d i c a t e d , f o r t h e c o m p a r i s o n s a k e , t h a t t h e ( d , p ) - r e a c t i o n c r o s s - s e c t i o n s in t h e s a m e r a n g e of e n e r g i e s a n d n u c l e i a r e d e c r e a s e d m u c h m o r e s l o w l y (for e x a m p l e t h e ! 2 C ( d , p)13C r e a c t i o n c r o s s - s e c t i o n i n t h e 5 - 2 0 M e V e n e r g y r a n g e i s d e c r e a s e d b y a f a c t o r of only 1 . 5 - 2 [17]). T h u s , at s u f f i c i e n t l y h i g h e n e r g i e s t h e r e c t a n g l e m e c h a n i s m of t h e d i r e c t r e a c t i o n s s h o u l d give a v e r y s m a l l c o n t r i b u t i o n to t h e p r o c e s s a s c o m p a r e d to s i m p l e r m e c h a n i s m s . T h e a b o v e a n a l y s i s of t h e i n f l u e n c e of d i s t o r t i o n s on a n g u l a r d i s t r i b u t i o n s h a s s h o w n t h a t t h e p e a k a t z e r o a n g l e i s not s e n s i t i v e to t h e d i s t o r t i o n s a n d i s a c h a r a c t e r i s t i c f e a t u r e of t h e r e c t a n g l e d i a g r a m s . T h i s f a c t j u s t i f i e s t h e a p p l i c a t i o n of t h e p l a n e w a v e a p p r o x i m a t i o n , in t h i s c a s e the a b s o l u t e v a l u e s of t h e c r o s s - s e c t i o n s of t h e r e a c t i o n s (t, p) a n d (p, p) c a n b e e s t i m a t e d u s i n g " r e n o r m a l i z a t i o n " of t h e s i n g l e - p a r t i c l e r e d u c e d w i d t h s c a l c u l a t e d on t h e b a s i s of t h e d a t a on the r e a c t i o n s (t, d) a n d ( d , p ) u s i n g the appropriate experimental and theoretical data and thereby can include sufficiently reliable the inf l u e n c e of t h e d i s t o r t i o n s a l r e a d y on t h e a b s o l u t e c r o s s - s e c t i o n s . F o r t h e 1 0 B ( t , p ) I Z B r e a c t i o n (E t = 10 MeV) s u c h a n a p p r o a c h g a v e t h e v a l u e [d~/d~2 ]8 =0 ° -- 0.7 m b / s t e r a d ( e x p e r i m e n t a l v a l u e i s 1.2 m b / s t e r a d [5]); t h i s v a l u e c a l c u l a t e d in t h e o s c i l l a t o r a p p r o x i m a t i o n [2] w i t h t h e a b o v e m e n t i o n e d r e n o r m a l i z a t i o n i s 0.4 m b / s t e r a d . S i m i l a r c a l c u l a t i o n s of t h e c r o s s - s e c t i o n s of t h e 9 B e ( p , p ' ) g B e ( ½ +) r e a c t i o n g i v e t h e v a l u e I d a / d 2 ] 0 =0 o = 0.2 m b / s t e r a d (the e x p e r i m e n t a l v a l u e i s u n k n o w n ; e x t r a p o l a t i o n to z e r o a n g l e of t h e c r o s s - s e c t i o n s f r o m r e f . 18 m e a s u r e d w i t h l a r g e e r r o r g i v e s t h e v a l u e of ~ 0 . 5 - 3 m b / s t e r a d ) . T h u s , the i m p o r t a n t c o n c l u s i o n m a y b e d r a w n t h a t t h e a b s o l u t e v a l u e s of t h e c r o s s - s e c t i o n s c a n b e o b t a i n e d w i t h i n t h e p r o p e r o r d e r of m a g n i t u d e ; in p a r t i c u l a r , t h e v a l u e of the c r o s s - s e c t i o n of t h e ( t , p ) r e a c t i o n r e a l i z e d t h r o u g h t h e r e c t a n g l e d i a g r a m f o r low t r i t o n e n e r g i e s ( 5 - 1 0 MeV) i s only s l i g h t l y s m a l l e r t h a n t h e u s u a l c r o s s - s e c t i o n s of t h e p o l e ( t , p ) - r e a c t i o n s (1-5 m b / s t e r a d ) . A s w a s n o t e d a b o v e , w i t h i n c r e a s i n g e n e r g y of t h e b o m b a r d i n g p a r t i c l e s , t h e s i t u a t i o n s h o u l d a b r u p t l y c h a n g e . P r o b a b l y t h e r a n g e of c o m p a r a t i v e l y low e n e r g i e s i s of s p e c i a l i n t e r e s t f o r t h e p h y s i c s of d i r e c t p r o c e s s e s on l i g h t nuclei.
References
1. 2. 3. 4.
18.
E. Bang, N.S. Zelenskaya, E. Zh. Magzumov, V. G. Neudatchin, J. Nucl. Phys. (USSH) 4 (1965) 962. V. G. Neudatchin, N.S. Zelenskaya, E. Zh. Magzumov, E.A. Romanovsky, Phys. L e t t e r s 27B (1968) 490. V. G. Neudatchin, E. Zh. Magzumov. E. A. Romanovsky, J. Nucl. Phys. (USSR) 9 (1969) 94. P . J . Iono, S.K. Penny, R. M. Drisko. Nucl. Phys. A127 (1969) 47; B. Kozlowsky. A. De Shalit, Nucl. Phys. 77 (1966) 215; F. S. Levin, Phys. Rev. 147 (1966) 71~. R. Middleton, D.J. Pullea. Nucl. Phvs. 51 (1964) 50, 77. J . C . H a r d y , I.S. Towner, Phys. L e t t e r s 25B (1967) 577. E. M. Henley, D.V.L. Yu. Phys. Rev. 133 (1964) B1445. D. Fleming, J. Cerny, C. C. Maples, N.K. Glendenning, Phys. Rev. 166 (1968) 1012. A. Dar, Nucl. Phys. 55 (1964) 305; Phys. Rev. 137 (1965) B102. N.K. Glendenning, Phys. Hey. 114 (1957) 1297. N.K. Glendenning, Phys. Hey. 137 (1965) B102; Ann. Hey. Nucl. Sci. 13 (1963); Phys. Rev. 156 (1967) 1344. E. Zh. Magzumov, V. G. Neudatchin, M. S. Belkin, J. Nucl. Phys. (USSR), to be published. K. A. Gridnev, L. V. Krasnov et al., P r e p r i n t JINR 2458, Dubna, 1965. L . L . Lee J r . , J. P. Schiffer et al., Phys. Rev. 136 (1964) B971; A. A. Fife, G . C . N e i l s o n , W. K. B e r t r a m , L. J. T a s s i e , Phys. Rev. 156 (1968) 1029. V. G. Neudatchin, Yu. F. Smirnov, Atomic Energy Rev., IAEA, Vienna, v. 3 (1965) 157. P. Beregi, V. G. Neudatchin, Yu. F. Smirnov, Nucl. Phys. 26 (1961) 306. M.H. M c F a r l a n e , J. B. French, Hev. Mod. Phys. 32 (1960) 567; J. N. McGruer, E.K. Warburton, R. S. Bender, Phys. Rev. 100 (1955) 235. R. G. S u m m e r s - G i l l , Phys. Rev. 103 (1958) 1591.
108
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5. 6. 7. 8. 9. 10. 11. [2. 13. 14. 15. 16. 17.