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The dipole photoabsorption in 9Be
Volume
27B, number
PHYSICS
8
THE
Nuclear Research
DIPOLE
Institute
LETTERS
16 September
PHOTOABSORPTION
IN
L. MAJLING of the Czechoslovak Academy of Science,
V. I. KUKULIN and Yu. F. SMIRNOV Nuclear Research Institute, Moscow State University, Received
‘Be
fiez.,
Czechoslowakia
Moscow
15 June 1968
The absorption spectrum for dipole gamma rays with energies particle nuclear shell model
Although the photodistintegration of gBe has been investigated since 1934 [l], the attenuation of theoreticians and experimentalists was mainly devoted to the low-energy region, i.e. to the socalled “pygmy” resonance, related to single particle transitions of the valence neutron. Recently several experimental papers have appeared, in which the photodisintegration of gBe is investigated in more detail in the region of higher energies, when dipole transitions of nucleons in the “core” of 8Be occur [2-51. A theoretical analysis of photoabsorption meets considerable technical difficulties, so that only semiquantitative estimates of the (r,n) and (y,p) reaction cross-sections exist so far [3], based upon Wilkinson’s single-particle model. We consider it desirable to calculate the absorption cross-section and the partial reaction cross-sections in more detail, using a many particle nuclear model [6], although it is necessary to make simplifying assumptions concerning the residual interactions, so as to overcome the computational difficulties, due to the large number of possible configurations. Namely, we assume that it is possible to neglect spin-orbit forces, which really are small in the considered nucleus, as well as other spin-dependent forces, in the residual interaction. In such a case the supermultiplet approximation is valid, [f], L, S, T,J, are good quantum numbers and states with equal values of [f] and L but different values of S, T and J are degenerate. Thus, the total matrix of the Hamiltonian, the order of which exceeds 600, decomposes into a number of submatrices of a considerably lower order, which can be diagonalized separately. Photoprocesses on ‘Be are analyzed in the same
1968
0
below 35 MeV is calculated
IO
20
using the many
M
Ey(MeV
Fig. 1. The y-quantum absorption cross section. The empty: hatched and full columns correspond to the states with [f] = [441], [432] and [4311] respectively. By means of the curves the experimental data on (y,n) and (y, p) reaction according to refs. 4 and 5 are presented in the upper and lower part, respectively.
manner as in ref. 7. We assume that the nucleons of 9Be move in a harmonic oscillator potential that leaves degenerate the states of ls4 lp4( ld-2s) and ls3 Ip6 configurations with the total momentum and parity characteristics J” = i+, i’ and f’ and isospins T = i and z. Next, the residual interaction between all nine nucleons is taken into account. We use Serber forces with the usual parameters, namely V. = 45 MeV, while the radius p of the Gaussian potential V = = V, exp(-r2/p2) has been kept equal to the “oscillator radius” ro = (E/Mw)l’~ = 1.7 fm. In the supermultiplet approximation, taking into account the large statistical weight of the \ls41p5[441] L=l, S=f, T=$, J=%)compo487
Volume 27B. number 8
PHYSICS
n e n t of t h e 9Be g r o u n d s t a t e w a v e f u n c t i o n [8], it i s n e c e s s a r y to c o n s i d e r o n l y t h o s e e x c i t e d l e v e l s of 9Be to w h i c h a E1 t r a n s i t i o n f r o m t h i s c o m p o n e n t i s a l l o w e d by the s e l e c t i o n r u l e s w i t h r e s p e c t to L , [ f ] , S and T; i.e. L = 0, 1 and 2; 3 1 4 4 1 ] ' S = 3~, T = ~~,. [ f ] = [ 4 3 2 ] , S = ~, T = 1 [f] ~ a n d s ; I f ] : [ 4 3 1 1 ] , S =½, T ½ a n d S . It t u r n s out that o n l y 9 m a t r i c e s of an o r d e r not e x c e e d i n g 10 n e e d to be d i a g o n a l i z e d . T h e m a t h e m a t i c a l f o r m a l i s m b a s e d on the R a c a h a l g e b r a of t h e SU(4) g r o u p , e m p l o y e d in c a l c u l a t i n g b o t h r e s i d u a l i n t e r a c t i o n m a t r i x e l e m e n t s and t h e r e d u c e d w i d t h s of s t a t e s with a g i v e n Young t a b l e a u , h a s b e e n d e s c r i b e d s e p a r a t e l y in r e f . 9. We w i s h to point out, t h a t c o n t r a r y to t h e u s u a l p a r t i c l e - h o l e c a l c u l a t i o n s on n o n - m a g i c n u c l e i , we h a v e a l s o i n c l u d e d in o u r c o n s i d e r a t i o n s c o n f i g u r a t i o n s to w h i c h a d i r e c t E1 t r a n s i t i o n f r o m t h e g r o u n d s t a t e i s f o r b i d d e n , e.g. t h e s t a t e lls41p4122] L : 0, ld: [432]L =2, S = i E ½ > . Such states form a giant resonance over the strongly excited levels of the 8Be core. They are excited in the dipole photoabsorption only bec a u s e of the a d m i x t u r e of s t a t e s , to w h i c h E1 t r a n s i t i o n s a r e a l l o w e d . T h e m i x i n g of a l l o w e d and f o r b i d d e n t r a n s i t i o n s i n f l u e n c e s c o n s i d e r a b l y the c a l c u l a t e d width of the g i a n t r e s o n a n c e and i t s d e c a y p r o p e r t i e s . In t h i s way we o b t a i n the r e l a t i v e p o s i t i o n s of d i p o l e l e v e l s . T h e a b s o l u t e v a l u e s of the e n e r g y l e v e l s h a v e to be found by an i n d e p e n d e n t c a l c u l a t i o n of t h e (p, 2p) r e a c t i o n on t h e n e i g h b o u r i n g n u c l e u s 10B in the s a m e m a n n e r a s in r e f . 7. On fig. 1 we p r e s e n t t h e r e s u l t s of a c a l c u l a t i o n of t h e a b s o r p t i o n c r o s s - s e c t i o n . T h e e x p e r i m e n t a l r e s u l t s on (V, n) and (~, p) [5] r e a c t i o n s a r e a l s o given. T h e g e n e r a l a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t i s e v i d e n t . To b e g i n with, t h e r e i s a s t r o n g s p l i t t i n g of the g i a n t r e s o n a n c e w i t h r e s p e c t to t h e p a r t i t i o n I f ] . We h a v e u s e d e m p t y , h a t c h e d and full c o l u m n s f o r t h e s t a t e s w i t h I f ] = [441], [432] and [4311], r e s p e c t i v e l y . S e c o n d l y , the c a l c u l a t e d c r o s s - s e c t i o n p o s s e s s e s a r e m a r k a b l e m a x i m u m in the r e g i o n of 10 MeV and m i n i m u m in t h e r e g i o n f r o m 16 to 18 MeV, t h e b u l k of d i p o l e s t a t e s b e i n g c o n c e n t r a t e d in the r e g i o n f r o m 19 to 28 M e V , in c o m p l e t e a g r e e m e n t with e x p e r i m e n t . A s t r o n g t r a n s i t i o n a b o v e 30 MeV should a l s o be m e n t i o n e d . A s f o r the low l y i n g l e v e l s , we h a v e o b t a i n e d the c a l c u l a t e d l e v e l s t h a t can b e c o n s i d e r e d to c o r r e s p o n d to t h e s e " p y g m y " r e s o n a n c e s . H o w e v e r , t h e i n t e n s i t y of t h e s e t r a n s i t i o n s a p p e a r s to be m u c h l o w e r t h a n t h e e x p e r i m e n t a l one. T h e t o t a l c r o s s s e c t i o n of p h o t o a b s o r p t i o n f o r t h e two l o w e s t p o s i t i v e p a r i t y s t a t e s , the 1.7 MeV and 488
LETTERS
16 September 1968
3.4 MeV i s , a p p r o x i m a t e l y , 20 t i m e s l e s s t h a n t h e e x p e r i m e n t a l one. N e v e r t h e l e s s , it i s e a s y to u n d e r s t a n d t h e r e a s o n f o r t h i s . T h e c a l c u l a t e d w a v e f u n c t i o n s of d i p o l e s t a t e s a r e , in f a c t , c l o s e to t h e f u n c t i o n of t h e SU(3) s c h e m e of E l l i o t t [10], s i n c e t h e c o m p o n e n t with d e f i n i t e v a l u e of (~t, ~) p r e v a i l s in t h e m . F o r the l o w e s t d i p o l e s t a t e s we h a v e (;% ~) = (60) - the o v e r l a p p i n g i n t e g r a l b e i n g e q u a l to 0.97 - and t h e d i p o l e t r a n s i t i o n s f r o m the g r o u n d s t a t e with (~t, ~) = = (31) to t h e s e s t a t e s a r e f o r b i d d e n by t h e s e l e c t i o n r u l e s of the SU(3) s c h e m e . We s u p p o s e t h a t it is i m p o r t a n t to c o n s i d e r the c l u s t e r s t r u c t u r e of t h e c o r e of 8Be. An a n a l y s i s of f a s t e l e c t r o n s c a t t e r i n g on 9Be h a s shown [11] that t h e a - c l u s t e r s in t h i s n u c l e u s a r e m o r e s e p a r a t e d t h e n one w o u l d e x p e c t f r o m the s h e l l m o d e l . T h i s i s why the w a v e f u n c t i o n of c o r e 8Be in the 9Be n u c l e u s d i f f e r s f r o m the s h e l l m o d e l f u n c t i o n with (;% t~) = = (40) and m u s t c o n t a i n a d m i x t u r e s of h i g h e r c o n f i g u r a t i o n s t a t e s w i t h (;% ~) = (60). T h e s e l a s t c o m p o n e n t s of t h e w a v e f u n c t i o n of the c o r e , t o g e t h e r w i t h t h e w a v e f u n c t i o n of the 9th p - n u c l e o n , f o r m a d m i x t u r e s w i t h (k, # ) = (70) and (51) to the w a v e f u n c t i o n of the g r o u n d s t a t e of 9Be. T h e e f f e c t of t h e m g r e a t l y e n h a n c e s the e x c i t a t i o n p r o b a b i l i t y f o r the l o w e s t d i p o l e l e v e l s , s i n c e the t r a n s i t i o n to t h e s t a t e s with (;%/~) = (60) i s now a l l o w e d . N e v e r t h e l e s s , an a c c u r a t e a c c o u n t of t h e c l u s t e r e f f e c t s r e p r e s e n t s a d i f f i c u l t t a s k and we did not intend to r e s o l v e it in t h i s p a p e r . T h e w a v e f u n c t i o n s of d i p o l e s t a t e s , p a r t i a l r e a c t i o n c r o s s - s e c t i o n s and s p e c t r a of p h o t o n u c l e o n s a r e p r e s e n t e d in r e f . 12. We a r e i n d e b t e d to P r o f . V. G. N e u d a t c h i n f o r m a n y f r u i t f u l d i s c u s s i o n s . One of t h e a u t h o r s ( L . M . ) w o u l d l i k e to e x p r e s s h i s g r a t i t u d e to P r o f . F. J a n o u c h and D r . P. W i n t e r n i t z f o r a c r i t i c a l r e a d i n g of the m a n u s c r i p t .
Ref eT'ences 1. W. Gentner, Compt. Rend. 199 (1934) 1211; L. Szilard and I. A. Chalmers, Nature 134 (1934) 494. 2. A . P . Komar and E. D. Makhanovsky, Nuel. Physics 65 (1965) 662: C. Beeehi, L. Meneghetti and M. Sanzone, S. Vitale, Nucl. Physics 59 (1964) 375. 3. B. Cujee, Nucl. Physics 37 (1962) 396. 4. S. Costa, L. Pasqualini, G. Pirogino and L. Roasio, Nuovo Cimento 42 (1966) B306. 5. V . P . Denisov and L. A. Kulchitsky, Yadernaja Fizika 3 (1966) 268. 6. J . P . Elliott and B. H. Flowers, Proc. Roy. Soe. A242 (1957) 57: V. G. Shevchenko and N. P. Yudin, Atomic Energy Review 3 No. 3 (1965).
Volume 27B, n u m b e r 8
PHYSICS
7. I . V . K u r d y u m o v , S . H . E1 S a m a r a i , Yu. F. S m i r n o v and K. V. Shitihova, I z v e s t i a Akad. Nauk SSSR - F i z . 30 (1966) 292. 8. A . N . B o y a r k i n a , I z v e s t i a Akad. Nauk SSSR - F i z . 28 (1964) 337; F . C . B a r k e r , Nucl. P h y s i c s 83 (1966) 418.
LETTERS
16 S e p t e m b e r 1968
9. V . I , Kukulin, Yu. F. S m i r n o v and L. Majling, Nucl. P h y s i c s A103 (1967) 681. 10. J . P . E l l i o t t , P r o c . Roy. Soc. A245 (1958) 128, 562. 11. Y u . A . K u d e y a r o v . V . G . Neudatchin and Yu. F. S m i r nov, I z v e s t i a Akad. Nauk SSSR - Fiz. 30 (1966) 235. 12. L. Mailing, V . I . Kukulin and Yu. F. S m i r n o v , Czech. J . P h y s . , to be published.
489
27B, number
PHYSICS
8
THE
Nuclear Research
DIPOLE
Institute
LETTERS
16 September
PHOTOABSORPTION
IN
L. MAJLING of the Czechoslovak Academy of Science,
V. I. KUKULIN and Yu. F. SMIRNOV Nuclear Research Institute, Moscow State University, Received
‘Be
fiez.,
Czechoslowakia
Moscow
15 June 1968
The absorption spectrum for dipole gamma rays with energies particle nuclear shell model
Although the photodistintegration of gBe has been investigated since 1934 [l], the attenuation of theoreticians and experimentalists was mainly devoted to the low-energy region, i.e. to the socalled “pygmy” resonance, related to single particle transitions of the valence neutron. Recently several experimental papers have appeared, in which the photodisintegration of gBe is investigated in more detail in the region of higher energies, when dipole transitions of nucleons in the “core” of 8Be occur [2-51. A theoretical analysis of photoabsorption meets considerable technical difficulties, so that only semiquantitative estimates of the (r,n) and (y,p) reaction cross-sections exist so far [3], based upon Wilkinson’s single-particle model. We consider it desirable to calculate the absorption cross-section and the partial reaction cross-sections in more detail, using a many particle nuclear model [6], although it is necessary to make simplifying assumptions concerning the residual interactions, so as to overcome the computational difficulties, due to the large number of possible configurations. Namely, we assume that it is possible to neglect spin-orbit forces, which really are small in the considered nucleus, as well as other spin-dependent forces, in the residual interaction. In such a case the supermultiplet approximation is valid, [f], L, S, T,J, are good quantum numbers and states with equal values of [f] and L but different values of S, T and J are degenerate. Thus, the total matrix of the Hamiltonian, the order of which exceeds 600, decomposes into a number of submatrices of a considerably lower order, which can be diagonalized separately. Photoprocesses on ‘Be are analyzed in the same
1968
0
below 35 MeV is calculated
IO
20
using the many
M
Ey(MeV
Fig. 1. The y-quantum absorption cross section. The empty: hatched and full columns correspond to the states with [f] = [441], [432] and [4311] respectively. By means of the curves the experimental data on (y,n) and (y, p) reaction according to refs. 4 and 5 are presented in the upper and lower part, respectively.
manner as in ref. 7. We assume that the nucleons of 9Be move in a harmonic oscillator potential that leaves degenerate the states of ls4 lp4( ld-2s) and ls3 Ip6 configurations with the total momentum and parity characteristics J” = i+, i’ and f’ and isospins T = i and z. Next, the residual interaction between all nine nucleons is taken into account. We use Serber forces with the usual parameters, namely V. = 45 MeV, while the radius p of the Gaussian potential V = = V, exp(-r2/p2) has been kept equal to the “oscillator radius” ro = (E/Mw)l’~ = 1.7 fm. In the supermultiplet approximation, taking into account the large statistical weight of the \ls41p5[441] L=l, S=f, T=$, J=%)compo487
Volume 27B. number 8
PHYSICS
n e n t of t h e 9Be g r o u n d s t a t e w a v e f u n c t i o n [8], it i s n e c e s s a r y to c o n s i d e r o n l y t h o s e e x c i t e d l e v e l s of 9Be to w h i c h a E1 t r a n s i t i o n f r o m t h i s c o m p o n e n t i s a l l o w e d by the s e l e c t i o n r u l e s w i t h r e s p e c t to L , [ f ] , S and T; i.e. L = 0, 1 and 2; 3 1 4 4 1 ] ' S = 3~, T = ~~,. [ f ] = [ 4 3 2 ] , S = ~, T = 1 [f] ~ a n d s ; I f ] : [ 4 3 1 1 ] , S =½, T ½ a n d S . It t u r n s out that o n l y 9 m a t r i c e s of an o r d e r not e x c e e d i n g 10 n e e d to be d i a g o n a l i z e d . T h e m a t h e m a t i c a l f o r m a l i s m b a s e d on the R a c a h a l g e b r a of t h e SU(4) g r o u p , e m p l o y e d in c a l c u l a t i n g b o t h r e s i d u a l i n t e r a c t i o n m a t r i x e l e m e n t s and t h e r e d u c e d w i d t h s of s t a t e s with a g i v e n Young t a b l e a u , h a s b e e n d e s c r i b e d s e p a r a t e l y in r e f . 9. We w i s h to point out, t h a t c o n t r a r y to t h e u s u a l p a r t i c l e - h o l e c a l c u l a t i o n s on n o n - m a g i c n u c l e i , we h a v e a l s o i n c l u d e d in o u r c o n s i d e r a t i o n s c o n f i g u r a t i o n s to w h i c h a d i r e c t E1 t r a n s i t i o n f r o m t h e g r o u n d s t a t e i s f o r b i d d e n , e.g. t h e s t a t e lls41p4122] L : 0, ld: [432]L =2, S = i E ½ > . Such states form a giant resonance over the strongly excited levels of the 8Be core. They are excited in the dipole photoabsorption only bec a u s e of the a d m i x t u r e of s t a t e s , to w h i c h E1 t r a n s i t i o n s a r e a l l o w e d . T h e m i x i n g of a l l o w e d and f o r b i d d e n t r a n s i t i o n s i n f l u e n c e s c o n s i d e r a b l y the c a l c u l a t e d width of the g i a n t r e s o n a n c e and i t s d e c a y p r o p e r t i e s . In t h i s way we o b t a i n the r e l a t i v e p o s i t i o n s of d i p o l e l e v e l s . T h e a b s o l u t e v a l u e s of the e n e r g y l e v e l s h a v e to be found by an i n d e p e n d e n t c a l c u l a t i o n of t h e (p, 2p) r e a c t i o n on t h e n e i g h b o u r i n g n u c l e u s 10B in the s a m e m a n n e r a s in r e f . 7. On fig. 1 we p r e s e n t t h e r e s u l t s of a c a l c u l a t i o n of t h e a b s o r p t i o n c r o s s - s e c t i o n . T h e e x p e r i m e n t a l r e s u l t s on (V, n) and (~, p) [5] r e a c t i o n s a r e a l s o given. T h e g e n e r a l a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t i s e v i d e n t . To b e g i n with, t h e r e i s a s t r o n g s p l i t t i n g of the g i a n t r e s o n a n c e w i t h r e s p e c t to t h e p a r t i t i o n I f ] . We h a v e u s e d e m p t y , h a t c h e d and full c o l u m n s f o r t h e s t a t e s w i t h I f ] = [441], [432] and [4311], r e s p e c t i v e l y . S e c o n d l y , the c a l c u l a t e d c r o s s - s e c t i o n p o s s e s s e s a r e m a r k a b l e m a x i m u m in the r e g i o n of 10 MeV and m i n i m u m in t h e r e g i o n f r o m 16 to 18 MeV, t h e b u l k of d i p o l e s t a t e s b e i n g c o n c e n t r a t e d in the r e g i o n f r o m 19 to 28 M e V , in c o m p l e t e a g r e e m e n t with e x p e r i m e n t . A s t r o n g t r a n s i t i o n a b o v e 30 MeV should a l s o be m e n t i o n e d . A s f o r the low l y i n g l e v e l s , we h a v e o b t a i n e d the c a l c u l a t e d l e v e l s t h a t can b e c o n s i d e r e d to c o r r e s p o n d to t h e s e " p y g m y " r e s o n a n c e s . H o w e v e r , t h e i n t e n s i t y of t h e s e t r a n s i t i o n s a p p e a r s to be m u c h l o w e r t h a n t h e e x p e r i m e n t a l one. T h e t o t a l c r o s s s e c t i o n of p h o t o a b s o r p t i o n f o r t h e two l o w e s t p o s i t i v e p a r i t y s t a t e s , the 1.7 MeV and 488
LETTERS
16 September 1968
3.4 MeV i s , a p p r o x i m a t e l y , 20 t i m e s l e s s t h a n t h e e x p e r i m e n t a l one. N e v e r t h e l e s s , it i s e a s y to u n d e r s t a n d t h e r e a s o n f o r t h i s . T h e c a l c u l a t e d w a v e f u n c t i o n s of d i p o l e s t a t e s a r e , in f a c t , c l o s e to t h e f u n c t i o n of t h e SU(3) s c h e m e of E l l i o t t [10], s i n c e t h e c o m p o n e n t with d e f i n i t e v a l u e of (~t, ~) p r e v a i l s in t h e m . F o r the l o w e s t d i p o l e s t a t e s we h a v e (;% ~) = (60) - the o v e r l a p p i n g i n t e g r a l b e i n g e q u a l to 0.97 - and t h e d i p o l e t r a n s i t i o n s f r o m the g r o u n d s t a t e with (~t, ~) = = (31) to t h e s e s t a t e s a r e f o r b i d d e n by t h e s e l e c t i o n r u l e s of the SU(3) s c h e m e . We s u p p o s e t h a t it is i m p o r t a n t to c o n s i d e r the c l u s t e r s t r u c t u r e of t h e c o r e of 8Be. An a n a l y s i s of f a s t e l e c t r o n s c a t t e r i n g on 9Be h a s shown [11] that t h e a - c l u s t e r s in t h i s n u c l e u s a r e m o r e s e p a r a t e d t h e n one w o u l d e x p e c t f r o m the s h e l l m o d e l . T h i s i s why the w a v e f u n c t i o n of c o r e 8Be in the 9Be n u c l e u s d i f f e r s f r o m the s h e l l m o d e l f u n c t i o n with (;% t~) = = (40) and m u s t c o n t a i n a d m i x t u r e s of h i g h e r c o n f i g u r a t i o n s t a t e s w i t h (;% ~) = (60). T h e s e l a s t c o m p o n e n t s of t h e w a v e f u n c t i o n of the c o r e , t o g e t h e r w i t h t h e w a v e f u n c t i o n of the 9th p - n u c l e o n , f o r m a d m i x t u r e s w i t h (k, # ) = (70) and (51) to the w a v e f u n c t i o n of the g r o u n d s t a t e of 9Be. T h e e f f e c t of t h e m g r e a t l y e n h a n c e s the e x c i t a t i o n p r o b a b i l i t y f o r the l o w e s t d i p o l e l e v e l s , s i n c e the t r a n s i t i o n to t h e s t a t e s with (;%/~) = (60) i s now a l l o w e d . N e v e r t h e l e s s , an a c c u r a t e a c c o u n t of t h e c l u s t e r e f f e c t s r e p r e s e n t s a d i f f i c u l t t a s k and we did not intend to r e s o l v e it in t h i s p a p e r . T h e w a v e f u n c t i o n s of d i p o l e s t a t e s , p a r t i a l r e a c t i o n c r o s s - s e c t i o n s and s p e c t r a of p h o t o n u c l e o n s a r e p r e s e n t e d in r e f . 12. We a r e i n d e b t e d to P r o f . V. G. N e u d a t c h i n f o r m a n y f r u i t f u l d i s c u s s i o n s . One of t h e a u t h o r s ( L . M . ) w o u l d l i k e to e x p r e s s h i s g r a t i t u d e to P r o f . F. J a n o u c h and D r . P. W i n t e r n i t z f o r a c r i t i c a l r e a d i n g of the m a n u s c r i p t .
Ref eT'ences 1. W. Gentner, Compt. Rend. 199 (1934) 1211; L. Szilard and I. A. Chalmers, Nature 134 (1934) 494. 2. A . P . Komar and E. D. Makhanovsky, Nuel. Physics 65 (1965) 662: C. Beeehi, L. Meneghetti and M. Sanzone, S. Vitale, Nucl. Physics 59 (1964) 375. 3. B. Cujee, Nucl. Physics 37 (1962) 396. 4. S. Costa, L. Pasqualini, G. Pirogino and L. Roasio, Nuovo Cimento 42 (1966) B306. 5. V . P . Denisov and L. A. Kulchitsky, Yadernaja Fizika 3 (1966) 268. 6. J . P . Elliott and B. H. Flowers, Proc. Roy. Soe. A242 (1957) 57: V. G. Shevchenko and N. P. Yudin, Atomic Energy Review 3 No. 3 (1965).
Volume 27B, n u m b e r 8
PHYSICS
7. I . V . K u r d y u m o v , S . H . E1 S a m a r a i , Yu. F. S m i r n o v and K. V. Shitihova, I z v e s t i a Akad. Nauk SSSR - F i z . 30 (1966) 292. 8. A . N . B o y a r k i n a , I z v e s t i a Akad. Nauk SSSR - F i z . 28 (1964) 337; F . C . B a r k e r , Nucl. P h y s i c s 83 (1966) 418.
LETTERS
16 S e p t e m b e r 1968
9. V . I , Kukulin, Yu. F. S m i r n o v and L. Majling, Nucl. P h y s i c s A103 (1967) 681. 10. J . P . E l l i o t t , P r o c . Roy. Soc. A245 (1958) 128, 562. 11. Y u . A . K u d e y a r o v . V . G . Neudatchin and Yu. F. S m i r nov, I z v e s t i a Akad. Nauk SSSR - Fiz. 30 (1966) 235. 12. L. Mailing, V . I . Kukulin and Yu. F. S m i r n o v , Czech. J . P h y s . , to be published.
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