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On the projected spectra from a deformed correlated intrinsic state
V o l u m e 36B, n u m b e r 3
P HY SIC S L E T T E R S
ON T H E A DEFORMED
PROJECTED
6 S e p t e m b e r 1971
SPECTRA
CORRELATED
FROM
INTRINSIC
STATE
N. U L L A H and K. K. G U P T A Tata Institute of Fundamental Research, Bombay-5, India R e c e i v e d 25 J u n e 1971
It is argued that the residual interaction being relatively of long range, should produce Random Phase Approximation (RPA) type of correlations in the Hartree-Fock (HF) intrinsic state. A model is described to take these correlations inti account in the intrinsic state. A comparison of the projected spectra from this state with the exact shell model diagonalization for a model problem bears out this point. An a p p l i c a tion of the model in distinguishing two almost degenerate HF solutions for the 2 s - l d shell n u c l e i is m e n tioned.
T h e concept of an i n t r i n s i c s t a t e [1], f r o m which the l o w - l y i n g l e v e l s of a n u c l e u s a r e o b t a i n e d by a n g u l a r m o m e n t u m p r o j e c t i o n , has p r o v e d q u i t e f r u i t f u l in n u c l e a r s t r u c t u r e c a l c u lations. T h i s i n t r i n s i c s t a t e is obtained by p e r f o r m i n g a d e f o r m e d H a r t r e e - F o c k (HF) c a l c u l a tion [2]. In many of t h e s e c a l c u l a t i o n s it has b e e n o b s e r v e d that this p r o c e d u r e does not g i v e fully s a t i s f a c t o r y r e s u l t s f o r the e x c i t e d s t a t e s . T h i s u n s a t i s f a c t o r y s t a t e of a f f a i r a r i s e s due to t he fact that no c o r r e l a t i o n s a r e i n c l u d e d in t h e s e i n t r i n s i c H a r t r e e - F o c k w a v e functions. T h e p u r p o s e of the p r e s e n t note is to s e e what kind of c o r r e l a t i o n s one has to include in t h e s e i n t r i n s i c wave functions and then d e s c r i b e a m o d e l which t a k e s them into account. By c o m p a r i n g the p r o j e c t e d e n e r g i e s f r o m t h e c o r r e l a t e d i n t r i n s i c s t a t e of our m o d e l with the e x a c t s h e l l m o d e l d i a g o n a l i z a t i o n we d e m o n s t r a t e that i n c l u s i o n of such c o r r e l a t i o n s indeed l e a d s to e x c e l l e n t a g r e e m e n t with the e x a c t e n e r g i e s . T h e e f f e c t i v e i n t e r a c t i o n which is u s e d in n u c l e a r s t r u c t u r e c a l c u l a t i o n s i s the one o b t a i n e d f r o m a r e a l i s t i c i n t e r a c t i o n , l i k e the H a m a d a - J o h n s t o n potential. It is g e n e r a t e d by doing a B r u e c k n e r type of K - m a t r i x calculation. S i nc e this p r o c e d u r e t a k e s into a c c o u n t t h e s h o r t r a n g e c o r r e l a t i o n s , th e r e s i d u a l i n t e r a c t i o n m u s t e s s e n t i a l l y be a long r a n g e one. T h e c o r r e l a t i o n s p r o d u c e d by it should then b e d e s c r i b e d by R a n d o m - P h a s e - A p p r o x i m a l i o n (RPA). We next d e s c r i b e a m o d e l to g e n e r a t e the R P A c o r r e l a t e d i n t r i n s i c state. T h i s i n t r i n s i c s t a t e i n s t e a d of the p u r e H F s t a t e will be u s e d f o r 196
p r o j e c t i n g out t h e s t a t e s of good a n g u l a r m o m e n tum. As u s u a l we w r i t e t h e H a m i l t o n i a n as: + +
¼
~ , (I) VA[ ~8>aaa~asa + +
where o~ denotes the set of quantum numbers j m m T and is the a n t i s y m m e t r i z e d two-body m a t r i x e l e m e n t . Th e d e f o r m e d o r b i t a l s b~ a r e obtained by p e r f o r m i n g a d e f o r m e d H a r t r e e - F o c k calculation. If an ax i al s y m m e t r y is a s s u m e d then k stands f o r the i n d i c e s R rr~.. T h e l o w e s t of the d e f o r m e d o r b i t a l s b~ a r e u s e d to c o n s t r u c t the HF s t a t e ]@o): N
i'o>--1-I
Io ,
(2)
k=l w h e r e [0> is the c l o s e d s h e l l s t a t e a n d N is the n u m b e r of nucleons. Th e r e m a i n i n g o r b i t a l s b~, a r e c a l l e d u n o c c u p i e d o r b i t a l s and a r e denoted by putting a p r i m e on k. We take the R P A c o r r e l a t i o n s into account by c o n s t r u c t i n g the o p e r a t o r s O~ of the R P A t h e o r y
[3],
Ok
~?ot - z a ~ )
t3[
,
+
(3)
w h e r e 77a c r e a t e s a d e f o r m e d p a r t i c l e - h o l e state. Th e c o e f f i c i e n t s Y, Z a r e obtained by so l v i n g the R P A equations [3]
(aB w h e r e the m a t r i x e l e m e n t s of A a n d B a r e :
PHYSICS L E T T E R S
Volume 36B, number 3
A
= <*o [
I ¢o>,
BotB = -<~ol [7/or, [H, 7//~]] [ ~bo>.
(5a)
w2=1"8853 ' ' , y ~ ] = \0.7101
(Sb)
E q . (8) d e t e r m i n e s the c o r r e l a t i o n m a t r i x C in t e r m s of the v e c t o r s Y and Z. The p r o j e c t i o n of the s t a t e s J y =0 +, T=I and J~ =2 +, T =1 from the i n t r i n s i c RPA c o r r e l a t e d wave function given by e x p r e s s i o n (7) gives t h e i r e n e r g i e s to be - l l . 1 8 M e V and -9.595MeV r e s pectively, which a r e in excellent a g r e e m e n t with exact r e s u l t s . F o r the m o d e l p r o b l e m , one can also cons t r u c t a BCS type of wave function from b~ and p r o j e c t s t a t e s of definite n u m b e r of p a r t i c l e s and a n g u l a r m o m e n t a from it. We find that it does not i m p r o v e the s i t u a t i o n v e r y much and, t h e r e fore, supports our a r g u m e n t that it is the RPA type r a t h e r than s h o r t r a n g e (BCS) type of c o r r e l a t i o n s , which have to be included in i n t r i n s i c HF wave function. As a f u r t h e r check on our model we have applied it to the exactly solvable L i p k i n - M e s h k o v Glick model [5]. T h e s e r e s u l t s will be published e l s e w h e r e Later. We r e m a r k h e r e that the effect of ground s t a t e c o r r e l a t i o n s have also b e e n studied by the e q u a t i o n s - o f - m o t i o n method [6] p r o p o s e d by Rowe. His method needs a p r i o r knowledge of the ground state, while the method p r o p o s e d by us a i m s at finding f i r s t a c o r r e l a t e d i n t r i n s i c s t a t e from which both the ground and excited s t a t e s a r e obtained by projection. We note by looking at the RPA c o r r e l a t e d ground s t a t e given by e x p r e s s i o n (7) that it is a l i n e a r c o m b i n a t i o n of all 2n p a r t i c l e - 2 n hole d e f o r m e d s t a t e s or d e t e r m i n a n t a l functions which one can c o n s t r u c t from HF state l~o}" The excited i n t r i n s i c s t a t e s a r e obtained by u s i n g the o p e r a t o r O~ on this state. In our model the a n g u l a r m o m e n t u m s t a t e s belonging to v a r i o u s bands a r e [7] p r o j e c t e d from t h e s e i n t r i n s i c states. This is a b e t t e r p r e s c r i p t i o n than the one which was e a r l i e r given by B a s s i c h i s et al. [8] in which s i n g l e p a r t i c l e hole d e t e r m i n a n t a l functions w e r e used. F u r t h e r in t h e i r c a l c u l a t i o n they w e r e only i n t e r e s t e d in the energy of the lowest m e m b e r of the band and so had made f u r t h e r a p p r o x i m a t i o n which does not need the difficult but exact a n g u l a r m o m e n t u m projection. Before we conclude this note, we would like to m e n t i o n an application of our model. It has b e e n known for s o m e t i m e that s o m e of the 2 s - l d s h e l l nuclei, like 24Mg, 28Si, have two a l m o s t d e n e r a t e HF s o l u t i o n s , one of which is p r o l a t e and the other is oblate [9]. The c o n s t r u c t i o n of
The RPA c o r r e l a t e d i n t r i n s i c s t a t e IR> is defined by
o~[R> = 0.
(6)
A method due to Sanderson [4] gives us the following e x p r e s s i o n for this state,
]R > No(exp [-½ ~ C ~ ~a+ ~+] ) I ° o >
(7)
=
w h e r e the m a t r i x C is given by Z : C Y,
(8)
and N O is the n o r m a l i z a t i o n constant. As a model p r o b l e m , we c o n s i d e r the n u c l e u s 180, with 160 b e i n g c o n s i d e r e d as an i n e r t core. To simplify the c a l c u l a t i o n we a s s u m e the two a c t i v e n e u t r o n s to m o v e in ld~ and 2s~_ orbits. The s i n g l e - p a r t i c l e e n e r g i e s ~f t h e s e ~)rbits a r e taken to be e,~i = -4.50 MeV and ~s ! = - 3 . 2 8 MeV. ~ . . 2 The two-body i n t e r a c t i o n is taken to b e a 40MeV Rosenfeld i n t e r a c t i o n . The a x i a l l y - s y m m e t r i c HF c a l c u l a t i o n leads to the following lowest d e f o r m e d o r b i t
bk= ~+ =0.980
a~d~ll -O.199a~s ½½.
(9)
Other o r b i t a l s , v i z . , k = - ~,~ k = ~ ' , k= -~" a r e obtained by orthogonality and t i m e r e v e r s a l operation. The HF s i n g l e p a r t i c l e e n e r g i e s a r e E HF
k =~
=
- 5.420MeV,
E
HF
k =[
= -3.300
MeV.
T h e HF i n t r i n s i c s t a t e is given by
I¢o :
g:l b;:l
116o>.
(10)
On c a r r y i n g out the a n g u l a r m o m e n t u m p r o j e c t i o n from this state, which has no c o r r e l a t i o n s in it, we find the energy o f J ~ =0 +, T = I s t a t e to be -10.184 MeV and that of the J ~ = 2+, T= 1 is -10.247 MeV. The c o r r e s p o n d i n g exact s h e l l model e n e r g i e s for t h e s e two s t a t e s a r e -11.131 MeV and -9.597 MeV r e s p e c t i v e l y . Thus we s e e that p r o j e c t i o n from the i n t r i n s i c HF s t a t e with no c o r r e l a t i o n s p r e s e n t leads to e x t r e m e l y poor r e s u l t s . We next p e r f o r m the RPA c a l c u l a t i o n and find that the e i g e n v a l u e s and the c o r r e s p o n d i n g e i g e n v e c t o r s Y and A a r e given by ( 0.7096~ ( Z I ~ Y~/ = \-0.7096/'\Z~/=
wi=2.0630, (Y~
6 September 1971
(-0.0591~ \ 0.0591/'
2
0.0646) "
197
Volume 36B, n u m b e r 3
PHYSICS
LETTERS
the RPA correlated intrinsic states for these two HF solutions followed by the angular room e n t u m p r o j e c t i o n w o u l d t e l l u s w h i c h of t h e intrinsic HF states is closer to reality. This work which is in progress now will be reported shortly elsewhere.
Refe'Fences [1] M . G . R e d l i c h , Phys. Rev. 110 (1958} 468. [2] I . K e l s o n and C . A . L e v i n s o n , Phys. Rev. 134 (1964) B269. *****
198
6 September 1971
[3] G . E . Brown, Unified theory of n u c l e a r models and f o r c e s (North-Holland, 1967). [4] E . S a n d e r s o n , Phys. L e t t e r s 19 (1965) 141. [5] H. J. Lipkin, N. Meshkov and A . J . G l i c k , Nuc. Phys. 62 (1965) 188. [6] D . J . R o w e , Rev. Mod. P h y s . 40 (1968) 153. [7] D . L . Hill and J . A . W h e e l e r , Phys. Rev. 89 (1953) 1102. [8] W. H. B a s s i c h i s , C.A. Levinson and I. Kelson, Phys. Rev. 136 (1964) B380. [9] G.Ripka, in: Advances in n u c l e a r physics, eds. M. B a r a n g e r and E. Vogt (Plenum P r e s s , N.Y., 1968) Vol. I.
P HY SIC S L E T T E R S
ON T H E A DEFORMED
PROJECTED
6 S e p t e m b e r 1971
SPECTRA
CORRELATED
FROM
INTRINSIC
STATE
N. U L L A H and K. K. G U P T A Tata Institute of Fundamental Research, Bombay-5, India R e c e i v e d 25 J u n e 1971
It is argued that the residual interaction being relatively of long range, should produce Random Phase Approximation (RPA) type of correlations in the Hartree-Fock (HF) intrinsic state. A model is described to take these correlations inti account in the intrinsic state. A comparison of the projected spectra from this state with the exact shell model diagonalization for a model problem bears out this point. An a p p l i c a tion of the model in distinguishing two almost degenerate HF solutions for the 2 s - l d shell n u c l e i is m e n tioned.
T h e concept of an i n t r i n s i c s t a t e [1], f r o m which the l o w - l y i n g l e v e l s of a n u c l e u s a r e o b t a i n e d by a n g u l a r m o m e n t u m p r o j e c t i o n , has p r o v e d q u i t e f r u i t f u l in n u c l e a r s t r u c t u r e c a l c u lations. T h i s i n t r i n s i c s t a t e is obtained by p e r f o r m i n g a d e f o r m e d H a r t r e e - F o c k (HF) c a l c u l a tion [2]. In many of t h e s e c a l c u l a t i o n s it has b e e n o b s e r v e d that this p r o c e d u r e does not g i v e fully s a t i s f a c t o r y r e s u l t s f o r the e x c i t e d s t a t e s . T h i s u n s a t i s f a c t o r y s t a t e of a f f a i r a r i s e s due to t he fact that no c o r r e l a t i o n s a r e i n c l u d e d in t h e s e i n t r i n s i c H a r t r e e - F o c k w a v e functions. T h e p u r p o s e of the p r e s e n t note is to s e e what kind of c o r r e l a t i o n s one has to include in t h e s e i n t r i n s i c wave functions and then d e s c r i b e a m o d e l which t a k e s them into account. By c o m p a r i n g the p r o j e c t e d e n e r g i e s f r o m t h e c o r r e l a t e d i n t r i n s i c s t a t e of our m o d e l with the e x a c t s h e l l m o d e l d i a g o n a l i z a t i o n we d e m o n s t r a t e that i n c l u s i o n of such c o r r e l a t i o n s indeed l e a d s to e x c e l l e n t a g r e e m e n t with the e x a c t e n e r g i e s . T h e e f f e c t i v e i n t e r a c t i o n which is u s e d in n u c l e a r s t r u c t u r e c a l c u l a t i o n s i s the one o b t a i n e d f r o m a r e a l i s t i c i n t e r a c t i o n , l i k e the H a m a d a - J o h n s t o n potential. It is g e n e r a t e d by doing a B r u e c k n e r type of K - m a t r i x calculation. S i nc e this p r o c e d u r e t a k e s into a c c o u n t t h e s h o r t r a n g e c o r r e l a t i o n s , th e r e s i d u a l i n t e r a c t i o n m u s t e s s e n t i a l l y be a long r a n g e one. T h e c o r r e l a t i o n s p r o d u c e d by it should then b e d e s c r i b e d by R a n d o m - P h a s e - A p p r o x i m a l i o n (RPA). We next d e s c r i b e a m o d e l to g e n e r a t e the R P A c o r r e l a t e d i n t r i n s i c state. T h i s i n t r i n s i c s t a t e i n s t e a d of the p u r e H F s t a t e will be u s e d f o r 196
p r o j e c t i n g out t h e s t a t e s of good a n g u l a r m o m e n tum. As u s u a l we w r i t e t h e H a m i l t o n i a n as: + +
¼
~ , (I) VA[ ~8>aaa~asa + +
where o~ denotes the set of quantum numbers j m m T and is the a n t i s y m m e t r i z e d two-body m a t r i x e l e m e n t . Th e d e f o r m e d o r b i t a l s b~ a r e obtained by p e r f o r m i n g a d e f o r m e d H a r t r e e - F o c k calculation. If an ax i al s y m m e t r y is a s s u m e d then k stands f o r the i n d i c e s R rr~.. T h e l o w e s t of the d e f o r m e d o r b i t a l s b~ a r e u s e d to c o n s t r u c t the HF s t a t e ]@o): N
i'o>--1-I
Io ,
(2)
k=l w h e r e [0> is the c l o s e d s h e l l s t a t e a n d N is the n u m b e r of nucleons. Th e r e m a i n i n g o r b i t a l s b~, a r e c a l l e d u n o c c u p i e d o r b i t a l s and a r e denoted by putting a p r i m e on k. We take the R P A c o r r e l a t i o n s into account by c o n s t r u c t i n g the o p e r a t o r s O~ of the R P A t h e o r y
[3],
Ok
~?ot - z a ~ )
t3[
,
+
(3)
w h e r e 77a c r e a t e s a d e f o r m e d p a r t i c l e - h o l e state. Th e c o e f f i c i e n t s Y, Z a r e obtained by so l v i n g the R P A equations [3]
(aB w h e r e the m a t r i x e l e m e n t s of A a n d B a r e :
PHYSICS L E T T E R S
Volume 36B, number 3
A
= <*o [
I ¢o>,
BotB = -<~ol [7/or, [H, 7//~]] [ ~bo>.
(5a)
w2=1"8853 ' ' , y ~ ] = \0.7101
(Sb)
E q . (8) d e t e r m i n e s the c o r r e l a t i o n m a t r i x C in t e r m s of the v e c t o r s Y and Z. The p r o j e c t i o n of the s t a t e s J y =0 +, T=I and J~ =2 +, T =1 from the i n t r i n s i c RPA c o r r e l a t e d wave function given by e x p r e s s i o n (7) gives t h e i r e n e r g i e s to be - l l . 1 8 M e V and -9.595MeV r e s pectively, which a r e in excellent a g r e e m e n t with exact r e s u l t s . F o r the m o d e l p r o b l e m , one can also cons t r u c t a BCS type of wave function from b~ and p r o j e c t s t a t e s of definite n u m b e r of p a r t i c l e s and a n g u l a r m o m e n t a from it. We find that it does not i m p r o v e the s i t u a t i o n v e r y much and, t h e r e fore, supports our a r g u m e n t that it is the RPA type r a t h e r than s h o r t r a n g e (BCS) type of c o r r e l a t i o n s , which have to be included in i n t r i n s i c HF wave function. As a f u r t h e r check on our model we have applied it to the exactly solvable L i p k i n - M e s h k o v Glick model [5]. T h e s e r e s u l t s will be published e l s e w h e r e Later. We r e m a r k h e r e that the effect of ground s t a t e c o r r e l a t i o n s have also b e e n studied by the e q u a t i o n s - o f - m o t i o n method [6] p r o p o s e d by Rowe. His method needs a p r i o r knowledge of the ground state, while the method p r o p o s e d by us a i m s at finding f i r s t a c o r r e l a t e d i n t r i n s i c s t a t e from which both the ground and excited s t a t e s a r e obtained by projection. We note by looking at the RPA c o r r e l a t e d ground s t a t e given by e x p r e s s i o n (7) that it is a l i n e a r c o m b i n a t i o n of all 2n p a r t i c l e - 2 n hole d e f o r m e d s t a t e s or d e t e r m i n a n t a l functions which one can c o n s t r u c t from HF state l~o}" The excited i n t r i n s i c s t a t e s a r e obtained by u s i n g the o p e r a t o r O~ on this state. In our model the a n g u l a r m o m e n t u m s t a t e s belonging to v a r i o u s bands a r e [7] p r o j e c t e d from t h e s e i n t r i n s i c states. This is a b e t t e r p r e s c r i p t i o n than the one which was e a r l i e r given by B a s s i c h i s et al. [8] in which s i n g l e p a r t i c l e hole d e t e r m i n a n t a l functions w e r e used. F u r t h e r in t h e i r c a l c u l a t i o n they w e r e only i n t e r e s t e d in the energy of the lowest m e m b e r of the band and so had made f u r t h e r a p p r o x i m a t i o n which does not need the difficult but exact a n g u l a r m o m e n t u m projection. Before we conclude this note, we would like to m e n t i o n an application of our model. It has b e e n known for s o m e t i m e that s o m e of the 2 s - l d s h e l l nuclei, like 24Mg, 28Si, have two a l m o s t d e n e r a t e HF s o l u t i o n s , one of which is p r o l a t e and the other is oblate [9]. The c o n s t r u c t i o n of
The RPA c o r r e l a t e d i n t r i n s i c s t a t e IR> is defined by
o~[R> = 0.
(6)
A method due to Sanderson [4] gives us the following e x p r e s s i o n for this state,
]R > No(exp [-½ ~ C ~ ~a+ ~+] ) I ° o >
(7)
=
w h e r e the m a t r i x C is given by Z : C Y,
(8)
and N O is the n o r m a l i z a t i o n constant. As a model p r o b l e m , we c o n s i d e r the n u c l e u s 180, with 160 b e i n g c o n s i d e r e d as an i n e r t core. To simplify the c a l c u l a t i o n we a s s u m e the two a c t i v e n e u t r o n s to m o v e in ld~ and 2s~_ orbits. The s i n g l e - p a r t i c l e e n e r g i e s ~f t h e s e ~)rbits a r e taken to be e,~i = -4.50 MeV and ~s ! = - 3 . 2 8 MeV. ~ . . 2 The two-body i n t e r a c t i o n is taken to b e a 40MeV Rosenfeld i n t e r a c t i o n . The a x i a l l y - s y m m e t r i c HF c a l c u l a t i o n leads to the following lowest d e f o r m e d o r b i t
bk= ~+ =0.980
a~d~ll -O.199a~s ½½.
(9)
Other o r b i t a l s , v i z . , k = - ~,~ k = ~ ' , k= -~" a r e obtained by orthogonality and t i m e r e v e r s a l operation. The HF s i n g l e p a r t i c l e e n e r g i e s a r e E HF
k =~
=
- 5.420MeV,
E
HF
k =[
= -3.300
MeV.
T h e HF i n t r i n s i c s t a t e is given by
I¢o :
g:l b;:l
116o>.
(10)
On c a r r y i n g out the a n g u l a r m o m e n t u m p r o j e c t i o n from this state, which has no c o r r e l a t i o n s in it, we find the energy o f J ~ =0 +, T = I s t a t e to be -10.184 MeV and that of the J ~ = 2+, T= 1 is -10.247 MeV. The c o r r e s p o n d i n g exact s h e l l model e n e r g i e s for t h e s e two s t a t e s a r e -11.131 MeV and -9.597 MeV r e s p e c t i v e l y . Thus we s e e that p r o j e c t i o n from the i n t r i n s i c HF s t a t e with no c o r r e l a t i o n s p r e s e n t leads to e x t r e m e l y poor r e s u l t s . We next p e r f o r m the RPA c a l c u l a t i o n and find that the e i g e n v a l u e s and the c o r r e s p o n d i n g e i g e n v e c t o r s Y and A a r e given by ( 0.7096~ ( Z I ~ Y~/ = \-0.7096/'\Z~/=
wi=2.0630, (Y~
6 September 1971
(-0.0591~ \ 0.0591/'
2
0.0646) "
197
Volume 36B, n u m b e r 3
PHYSICS
LETTERS
the RPA correlated intrinsic states for these two HF solutions followed by the angular room e n t u m p r o j e c t i o n w o u l d t e l l u s w h i c h of t h e intrinsic HF states is closer to reality. This work which is in progress now will be reported shortly elsewhere.
Refe'Fences [1] M . G . R e d l i c h , Phys. Rev. 110 (1958} 468. [2] I . K e l s o n and C . A . L e v i n s o n , Phys. Rev. 134 (1964) B269. *****
198
6 September 1971
[3] G . E . Brown, Unified theory of n u c l e a r models and f o r c e s (North-Holland, 1967). [4] E . S a n d e r s o n , Phys. L e t t e r s 19 (1965) 141. [5] H. J. Lipkin, N. Meshkov and A . J . G l i c k , Nuc. Phys. 62 (1965) 188. [6] D . J . R o w e , Rev. Mod. P h y s . 40 (1968) 153. [7] D . L . Hill and J . A . W h e e l e r , Phys. Rev. 89 (1953) 1102. [8] W. H. B a s s i c h i s , C.A. Levinson and I. Kelson, Phys. Rev. 136 (1964) B380. [9] G.Ripka, in: Advances in n u c l e a r physics, eds. M. B a r a n g e r and E. Vogt (Plenum P r e s s , N.Y., 1968) Vol. I.