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The alpha - cluster ground state of 16O
Volume 38B, number 7
PHYSICS
THE
ALPHA-CLUSTER
LETTERS
GROUND
3 April 1972
STATE
OF
160
C. A B U L A F F I O and J. M. IRVINE Department of Theoretical P h y s i c s , University o f Manchester, Manchester, UK Received 12 January 1972
It is shown that there is an or-cluster configuration which is more bound than the optimum regular tetrahedron which is usually assumed to correspond to the ground state of 160. There is no sharp minimum in the energy of the ground state of lffOin the or-cluster model suggesting that the ground state must be a mixture of or-clusters, or that it is important to treat the cluster co-ordinates as dynamical variables. R e c e n t l y , a n u m b e r of p a p e r s [1-4] h a v e r e p o r t e d c a l c u l a t i o n s b a s e d on t h e M a r g e n a u [5] B r i n k [6] a - c l u s t e r m o d e l f o r l i g h t n u c l e i . T h e s e c a l c u l a t i o n s h a v e a s s u m e d that c e r t a i n s t a t e s of t h e l i g h t n u c l e i can b e w e l l r e p r e s e n t e d by s i m p l e p o i n t g r o u p c o n f i g u r a t i o n s of t h e a - c l u s t e r s . In fig. 1 we i n d i c a t e the a s s u m e d f o r m of fhe g r o u n d s t a t e c o n f i g u r a t i o n s f o r s o m e of t h e l i g h t a - p a r ticle nuclei. It h a s r e c e n t l y b e e n d e m o n s t r a t e d [7] that, w h i l e the e q u i l a t e r a l t r i a n g l e is the b e s t r e p r e s e n t a t i o n of the g r o u n d s t a t e of 12C, the 7.65 MeV 0 + l e v e l is not r e p r e s e n t e d by a l i n e a r c h a i n of c l u s t e r s a s had p r e v i o u s l y b e e n a s s u m e d but that the t r u e e n e r g y m i n i m u m o c c u r s f o r an i s o s c e l e s t r i a n g l e of a p e x a n g l e ~ 150 °. We h a v e e x a m i n e d the t e t r a h e d r a l c o n f i g u r a t i o n of 160 to s e e if the r e g u l a r t e t r a h e d r o n is
0
oBe
12c
160
2"M9 Fig. 1. Alpha cluster configurations usually associated with the ground states of some light nuclei. 492
i n d e e d the b e s t r e p r e s e n t a t i o n of the g r o u n d s t a t e . W e h a v e c o n s i d e r e d the c o n f i g u r a t i o n in fig. 2 and t r e a t e d t h e p a r a m e t e r s 5 = h / h ' and d as v a r i a t i o n a l p a r a m e t e r s in m i n i m i s i n g t h e e n e r g y a r i s i n g f r o m the V o l k o v [8] i n t e r a c t i o n
V(r)= (0.35+0.65px)60{exp[-(r/ l.O1)2]-exp[-(r/1.8)2]} (1) w h e r e d i s t a n c e s a r e m e a s u r e d in f m and e n e r g i e s in MeV. In fig. 3 we r e p r o d u c e a c o n t o u r p l o t of o u r r e s u l t i n g b i n d i n g e n e r g y . T h e p o i n t R c o r r e s p o n d s to the m i n i m u m e n e r g y of t h e r e g u l a r t e t r a h e d r o n with a binding e n e r g y of - 126.8 MeV. T h e a b s o l u t e m i n i m u m o c c u r s at t h e p o i n t M c o r r e s p o n d i n g to an e n e r g y of - 127.2 MeV. A l t h o u g h t h e a b s o l u t e v a l u e s of t h e e n e r g i e s and t h e e x a c t l o c a t i o n of the p o i n t s R and M v a r y , the s a m e g e n e r a l c o n t o u r d i a g r a m r e s u l t s f r o m u s i n g the B r i n k and B o e k e r [9] i n t e r a c t i o n . In all o u r c a l c u l a t i o n s we h a v e c a r r i e d out the p r o j e c t i o n of a n g u l a r m o m e n t u m and p a r i t y b e f o r e t h e v a r i a t i o n a l c a l c u l a t i o n . In p r e l i m i n a r y c a l c u l a t i o n s we v a r i e d t h e o s c i l l a t o r s i z e b of t h e a - c l u s t e r s but found t h a t the o p t i m u m v a l u e of b w a s i n s e n s i t i v e to 5 and d. T h u s in c o m p o s i n g fig. 3 we f r o z e b at 1.5 fm. We h a v e not i n c l u d e d t h e C o u l o m b e n e r g y in c o m p u t i n g t h e e n e r g i e s f o r fig. 3 but both t h e C o u l o m b e n e r g y and t h e r . m . s , r a d i u s a r e r a t h e r i n s e n s i t i v e to the e x a c t p o s i t i o n in t h e e q u i l i b r i u m v a l l e y . T h i s is b e c a u s e a s 5 i n c r e a s e s d d e c r e a s e s and the e f f e c t s l a r g e l y c o m p e n s a t e one a n o t h e r . If a n y t h i n g we s h o u l d e x p e c t t h e highly d i s t o r t e d c o n f i g u r a t i o n M to h a v e a s l i g h t l y l o w e r C o u l o m b e n e r g y t h a n t h e r e g u l a r c o n f i g u r a t i o n R and h e n c e the i n c l u s i o n of t h e C o u l o m b f o r c e w i l l l o w e r M r e l a t i v e to R s t i l l f u r t h e r but by only a f r a c t i o n of an MeV.
Volume 38B, number 7
PHYSICS
t
LETTERS
3 April 1972
h" "U
Fig. 2. A right t e t r a h e d r a l configuration of a l p h a - p a r t i c l e c l u s t e r s . The b a s e is an equilateral triangle of side d and the height of the t e t r a hedron is h. The height of the r e g u l a r t e t r a h e d r o n on the s a m e base is h'. In the calculations d and 5 =h/h' a r e t r e a t e d as variational p a r a m e t e r s .
Fig. 3. A contour plot of the calculated binding energy in MeV as a function of d and 5.
In fig. 4 w e p r e s e n t a c r o s s s e c t i o n of t h e c o n t o u r plot c o r r e s p o n d i n g to the e n e r g y m i n i m u m a s a f u n c t i o n of 5. W e s e e t h a t a l o n g t h e v a l l e y in t h e c o n t o u r d i a g r a m t h e r e i s no a - c l u s t e r c o n f i g u r a t i o n c o r r e s p o n d i n g to a sharp minimum and this result is independent of t h e i n t e r a c t i o n e m p l o y e d , w h e t h e r t h e C o u l o m b e n e r g y i s i n c l u d e d o r not, t h e e x a c t v a l u e of b o r t h e c o m p u t a t i o n a l p r e c i s i o n e m p l o y e d . T h u s w e s h o u l d e x p e c t c o n s i d e r a b l e m i x i n g of c o n f i g u r a t i o n s o v e r a c o n s i d e r a b l e r e g i o n of the valley and a w a r n i n g that the c l u s t e r coordinates should be treated as dynamical v a r i a b l e s [ 10]. We have c a l c u l a t e d the o v e r l a p of the e x t r e m e s t a t e s in t h e v a l l e y w h i c h l i e w i t h i n 0.5 MeV of t h e b o t t o m , i . e . t h e s t a t e s w i t h 5 = 2.5, d = 1.25 a n d 5 = 0.5, d = 2.5 a n d t h e m i x i n g of t h e s e s t a t e s p r o d u c e d by t h e V o l k o v f o r c e . T h e r e s u l t s a r e p r e s e n t e d in t a b l e 1.
Tab[e 1 5~1__and ~ 2 are the norma[ised s t a t e s corresponding to 0.5, d = 2.5 fro, 5 = 2.5, d = 1.25 fm r e s p e c t i v e l y and ~ and ~PM are the n o r m a l i s e d states c o r r e s p o n d int to the points R and M in fig. 3. The Ek(i,} ,--) are the energy eigenva[ues obtained when the states ~/~, ~j - are mixed by the Vo[kov force.
1 2 R M
I
2
R
1.000 0.980 0.980 0.990
0.980 1.000 0.985 0.984
M
0.980 0.985 1.00 0.994
i
Ei(1,2)
Ei(1, 2, R, M)
1 2 3
-126.92 - 90.76
-127.19 -104.02 - 93.17
4
0.990 0.984 0.994 1.000
- 88.10
-126-
-127R
~J
6 -126 O.S
O.75 '
Ii O
1.2S '
I.S '
1.75 '
2 0i
2. '2 $
21S
!
!
2.7S
3.0
Fig. 4. The calculated optimum binding energy as a function of 5. 493
Volume 38B, number 7
PHYSICS LETTERS
It is i n t e r e s t i n g to note the e x t r e m e l y l a r g e o v e r l a p b et ween s t a t e s which apparently l i e f a r a p a r t in the a - c l u s t e r c o n f ig u r a ti o n space. We a l s o note that although the two s t a t e s a r e ext r e m e l y c l o s e in t h e i r u n p e r t u r b e d e n e r g y , when they a r e m i x e d one e i g e n v a l u e r e m a i n e x t r e m e l y c l o s e to the e x p e r i m e n t a l e n e r g y while the o t h e r m o v e s to an e x c i t a t i o n e n e r g y of 36 MeV. T h i s highly a s y m m e t r i c b e h a v i o r r e s u l t s f r o m the non orthogonality of our b a s i s functions. Th e m i x i n g of s t a t e s of high d i s t o r t i o n 5 into the ground s t a t e e m p h a s i s e s the i m p o r t a n c e of 4 p a r t i c l e - 4 hole c o n f i g u r a t i o n s in the d e s c r i p tion of this s t at e and s u g g e s t s that the four p a r t i c l e c o n f i g u r a t i o n s need not be l i m i t e d to the s d - s h e l l . The i n cl u s io n of such configurations will r a i s e the i n t r i n s i c m o m e n t of i n e r t i a of the ground s t at e and hence may lead to a l o w e r i n g of the e x c i t a t i o n e n e r g y of the h i g h e r a n g u l a r m o m e n t u m s t a t e s which is n e c e s s a r y in o r d e r to b r i n g t h e m into c l o s e r a g r e e m e n t with e x p e r i ment. We have r e s t r i c t e d o u r s e l v e s to right t e t r a hedra. T h e r e may well be an additional l o w e r i n g of the e n e r g y if this r e s t r i c t i o n is r e l a x e d . In the c a l c u l a t i o n s of ref. [7] on 12C the s t a t e s of good an g u la r m o m e n t u m w e r e p r o j e c t e d a f t e r the v a r i a t i o n a l calculation. We have r e p e a t e d the c a l c u l a t i o n s with p r o j e c t i o n b e f o r e v a r i a t i o n and find that the m i n i m a o c c u r at apex angles of ~ 40 ° and ~ 150 °. Thus once m o r e
494
3 April 1972
t h e r e is a l o w e r m i n i m a than that obtained f o r the m o s t s y m m e t r i c configuration. T h e fact that the l o w e s t e n e r g y c o n f i g u r a t i o n s do not c o r r e s p o n d to the m o s t s y m m e t r i c c l u s t e r c o n f i g u r a t i o n should not be s u r p r i s i n g it is s i m ply an e x a m p l e of the J a h n - T e l l e r [11] effect f a m i l i a r in m o l e c u l a r p h y s i c s . We acknowledge support through SRC g r a n t NG205 during the c o u r s e of this work and c o m putational f a c i l i t i e s p r o v i d e d by the SCR N u c l e a r L a b o r a t o r y , D a r e s b u r y . We should also like to thank M. G. J a m e s and V. F, E. P u c k n e l l f o r t h e i r a s s i s t a n c e with computational p r o b l e m s .
RefeYences [1] D. Brink et al., Phys. Letters 33B (1970) 143. [2] H. Friedrich and A. Weiguny, Phys. Letters 35B (1971) 105. [3] S.B. Khadkikar, Phys. Letters 36B (1971) 451. [4] K. Ananthanarayanan, S. Das Gupta and N. De Takacsy, Phys. Letters 37B (1971) 143. [5] H. Margenau, Phys. Rev. 59 (1941) 37. [6] D. Brink, The alpha-particle model of light nuclei, Int. School of Phys. 'Enrico Fermi' course XXXVI (1965). [71 H. Friedrich, L. Satpathy and A. Weiguny, Phys. Letters 36]3 (1971) 189. [Sl A. B. Volkov, Nucl. Phys. 74 (1965) 33. [91 D. Brink and E. Boecker, Nucl. Phys. A91 (1967) 1. [101 B. Giraud and D. Zaikine, Phys. Letters 37B (1971) 25. [I11 H.A.Jahn and E. Teller, Proc. Roy. Soc. 161A (1937) 220.
PHYSICS
THE
ALPHA-CLUSTER
LETTERS
GROUND
3 April 1972
STATE
OF
160
C. A B U L A F F I O and J. M. IRVINE Department of Theoretical P h y s i c s , University o f Manchester, Manchester, UK Received 12 January 1972
It is shown that there is an or-cluster configuration which is more bound than the optimum regular tetrahedron which is usually assumed to correspond to the ground state of 160. There is no sharp minimum in the energy of the ground state of lffOin the or-cluster model suggesting that the ground state must be a mixture of or-clusters, or that it is important to treat the cluster co-ordinates as dynamical variables. R e c e n t l y , a n u m b e r of p a p e r s [1-4] h a v e r e p o r t e d c a l c u l a t i o n s b a s e d on t h e M a r g e n a u [5] B r i n k [6] a - c l u s t e r m o d e l f o r l i g h t n u c l e i . T h e s e c a l c u l a t i o n s h a v e a s s u m e d that c e r t a i n s t a t e s of t h e l i g h t n u c l e i can b e w e l l r e p r e s e n t e d by s i m p l e p o i n t g r o u p c o n f i g u r a t i o n s of t h e a - c l u s t e r s . In fig. 1 we i n d i c a t e the a s s u m e d f o r m of fhe g r o u n d s t a t e c o n f i g u r a t i o n s f o r s o m e of t h e l i g h t a - p a r ticle nuclei. It h a s r e c e n t l y b e e n d e m o n s t r a t e d [7] that, w h i l e the e q u i l a t e r a l t r i a n g l e is the b e s t r e p r e s e n t a t i o n of the g r o u n d s t a t e of 12C, the 7.65 MeV 0 + l e v e l is not r e p r e s e n t e d by a l i n e a r c h a i n of c l u s t e r s a s had p r e v i o u s l y b e e n a s s u m e d but that the t r u e e n e r g y m i n i m u m o c c u r s f o r an i s o s c e l e s t r i a n g l e of a p e x a n g l e ~ 150 °. We h a v e e x a m i n e d the t e t r a h e d r a l c o n f i g u r a t i o n of 160 to s e e if the r e g u l a r t e t r a h e d r o n is
0
oBe
12c
160
2"M9 Fig. 1. Alpha cluster configurations usually associated with the ground states of some light nuclei. 492
i n d e e d the b e s t r e p r e s e n t a t i o n of the g r o u n d s t a t e . W e h a v e c o n s i d e r e d the c o n f i g u r a t i o n in fig. 2 and t r e a t e d t h e p a r a m e t e r s 5 = h / h ' and d as v a r i a t i o n a l p a r a m e t e r s in m i n i m i s i n g t h e e n e r g y a r i s i n g f r o m the V o l k o v [8] i n t e r a c t i o n
V(r)= (0.35+0.65px)60{exp[-(r/ l.O1)2]-exp[-(r/1.8)2]} (1) w h e r e d i s t a n c e s a r e m e a s u r e d in f m and e n e r g i e s in MeV. In fig. 3 we r e p r o d u c e a c o n t o u r p l o t of o u r r e s u l t i n g b i n d i n g e n e r g y . T h e p o i n t R c o r r e s p o n d s to the m i n i m u m e n e r g y of t h e r e g u l a r t e t r a h e d r o n with a binding e n e r g y of - 126.8 MeV. T h e a b s o l u t e m i n i m u m o c c u r s at t h e p o i n t M c o r r e s p o n d i n g to an e n e r g y of - 127.2 MeV. A l t h o u g h t h e a b s o l u t e v a l u e s of t h e e n e r g i e s and t h e e x a c t l o c a t i o n of the p o i n t s R and M v a r y , the s a m e g e n e r a l c o n t o u r d i a g r a m r e s u l t s f r o m u s i n g the B r i n k and B o e k e r [9] i n t e r a c t i o n . In all o u r c a l c u l a t i o n s we h a v e c a r r i e d out the p r o j e c t i o n of a n g u l a r m o m e n t u m and p a r i t y b e f o r e t h e v a r i a t i o n a l c a l c u l a t i o n . In p r e l i m i n a r y c a l c u l a t i o n s we v a r i e d t h e o s c i l l a t o r s i z e b of t h e a - c l u s t e r s but found t h a t the o p t i m u m v a l u e of b w a s i n s e n s i t i v e to 5 and d. T h u s in c o m p o s i n g fig. 3 we f r o z e b at 1.5 fm. We h a v e not i n c l u d e d t h e C o u l o m b e n e r g y in c o m p u t i n g t h e e n e r g i e s f o r fig. 3 but both t h e C o u l o m b e n e r g y and t h e r . m . s , r a d i u s a r e r a t h e r i n s e n s i t i v e to the e x a c t p o s i t i o n in t h e e q u i l i b r i u m v a l l e y . T h i s is b e c a u s e a s 5 i n c r e a s e s d d e c r e a s e s and the e f f e c t s l a r g e l y c o m p e n s a t e one a n o t h e r . If a n y t h i n g we s h o u l d e x p e c t t h e highly d i s t o r t e d c o n f i g u r a t i o n M to h a v e a s l i g h t l y l o w e r C o u l o m b e n e r g y t h a n t h e r e g u l a r c o n f i g u r a t i o n R and h e n c e the i n c l u s i o n of t h e C o u l o m b f o r c e w i l l l o w e r M r e l a t i v e to R s t i l l f u r t h e r but by only a f r a c t i o n of an MeV.
Volume 38B, number 7
PHYSICS
t
LETTERS
3 April 1972
h" "U
Fig. 2. A right t e t r a h e d r a l configuration of a l p h a - p a r t i c l e c l u s t e r s . The b a s e is an equilateral triangle of side d and the height of the t e t r a hedron is h. The height of the r e g u l a r t e t r a h e d r o n on the s a m e base is h'. In the calculations d and 5 =h/h' a r e t r e a t e d as variational p a r a m e t e r s .
Fig. 3. A contour plot of the calculated binding energy in MeV as a function of d and 5.
In fig. 4 w e p r e s e n t a c r o s s s e c t i o n of t h e c o n t o u r plot c o r r e s p o n d i n g to the e n e r g y m i n i m u m a s a f u n c t i o n of 5. W e s e e t h a t a l o n g t h e v a l l e y in t h e c o n t o u r d i a g r a m t h e r e i s no a - c l u s t e r c o n f i g u r a t i o n c o r r e s p o n d i n g to a sharp minimum and this result is independent of t h e i n t e r a c t i o n e m p l o y e d , w h e t h e r t h e C o u l o m b e n e r g y i s i n c l u d e d o r not, t h e e x a c t v a l u e of b o r t h e c o m p u t a t i o n a l p r e c i s i o n e m p l o y e d . T h u s w e s h o u l d e x p e c t c o n s i d e r a b l e m i x i n g of c o n f i g u r a t i o n s o v e r a c o n s i d e r a b l e r e g i o n of the valley and a w a r n i n g that the c l u s t e r coordinates should be treated as dynamical v a r i a b l e s [ 10]. We have c a l c u l a t e d the o v e r l a p of the e x t r e m e s t a t e s in t h e v a l l e y w h i c h l i e w i t h i n 0.5 MeV of t h e b o t t o m , i . e . t h e s t a t e s w i t h 5 = 2.5, d = 1.25 a n d 5 = 0.5, d = 2.5 a n d t h e m i x i n g of t h e s e s t a t e s p r o d u c e d by t h e V o l k o v f o r c e . T h e r e s u l t s a r e p r e s e n t e d in t a b l e 1.
Tab[e 1 5~1__and ~ 2 are the norma[ised s t a t e s corresponding to 0.5, d = 2.5 fro, 5 = 2.5, d = 1.25 fm r e s p e c t i v e l y and ~ and ~PM are the n o r m a l i s e d states c o r r e s p o n d int to the points R and M in fig. 3. The Ek(i,} ,--) are the energy eigenva[ues obtained when the states ~/~, ~j - are mixed by the Vo[kov force.
1 2 R M
I
2
R
1.000 0.980 0.980 0.990
0.980 1.000 0.985 0.984
M
0.980 0.985 1.00 0.994
i
Ei(1,2)
Ei(1, 2, R, M)
1 2 3
-126.92 - 90.76
-127.19 -104.02 - 93.17
4
0.990 0.984 0.994 1.000
- 88.10
-126-
-127R
~J
6 -126 O.S
O.75 '
Ii O
1.2S '
I.S '
1.75 '
2 0i
2. '2 $
21S
!
!
2.7S
3.0
Fig. 4. The calculated optimum binding energy as a function of 5. 493
Volume 38B, number 7
PHYSICS LETTERS
It is i n t e r e s t i n g to note the e x t r e m e l y l a r g e o v e r l a p b et ween s t a t e s which apparently l i e f a r a p a r t in the a - c l u s t e r c o n f ig u r a ti o n space. We a l s o note that although the two s t a t e s a r e ext r e m e l y c l o s e in t h e i r u n p e r t u r b e d e n e r g y , when they a r e m i x e d one e i g e n v a l u e r e m a i n e x t r e m e l y c l o s e to the e x p e r i m e n t a l e n e r g y while the o t h e r m o v e s to an e x c i t a t i o n e n e r g y of 36 MeV. T h i s highly a s y m m e t r i c b e h a v i o r r e s u l t s f r o m the non orthogonality of our b a s i s functions. Th e m i x i n g of s t a t e s of high d i s t o r t i o n 5 into the ground s t a t e e m p h a s i s e s the i m p o r t a n c e of 4 p a r t i c l e - 4 hole c o n f i g u r a t i o n s in the d e s c r i p tion of this s t at e and s u g g e s t s that the four p a r t i c l e c o n f i g u r a t i o n s need not be l i m i t e d to the s d - s h e l l . The i n cl u s io n of such configurations will r a i s e the i n t r i n s i c m o m e n t of i n e r t i a of the ground s t at e and hence may lead to a l o w e r i n g of the e x c i t a t i o n e n e r g y of the h i g h e r a n g u l a r m o m e n t u m s t a t e s which is n e c e s s a r y in o r d e r to b r i n g t h e m into c l o s e r a g r e e m e n t with e x p e r i ment. We have r e s t r i c t e d o u r s e l v e s to right t e t r a hedra. T h e r e may well be an additional l o w e r i n g of the e n e r g y if this r e s t r i c t i o n is r e l a x e d . In the c a l c u l a t i o n s of ref. [7] on 12C the s t a t e s of good an g u la r m o m e n t u m w e r e p r o j e c t e d a f t e r the v a r i a t i o n a l calculation. We have r e p e a t e d the c a l c u l a t i o n s with p r o j e c t i o n b e f o r e v a r i a t i o n and find that the m i n i m a o c c u r at apex angles of ~ 40 ° and ~ 150 °. Thus once m o r e
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t h e r e is a l o w e r m i n i m a than that obtained f o r the m o s t s y m m e t r i c configuration. T h e fact that the l o w e s t e n e r g y c o n f i g u r a t i o n s do not c o r r e s p o n d to the m o s t s y m m e t r i c c l u s t e r c o n f i g u r a t i o n should not be s u r p r i s i n g it is s i m ply an e x a m p l e of the J a h n - T e l l e r [11] effect f a m i l i a r in m o l e c u l a r p h y s i c s . We acknowledge support through SRC g r a n t NG205 during the c o u r s e of this work and c o m putational f a c i l i t i e s p r o v i d e d by the SCR N u c l e a r L a b o r a t o r y , D a r e s b u r y . We should also like to thank M. G. J a m e s and V. F, E. P u c k n e l l f o r t h e i r a s s i s t a n c e with computational p r o b l e m s .
RefeYences [1] D. Brink et al., Phys. Letters 33B (1970) 143. [2] H. Friedrich and A. Weiguny, Phys. Letters 35B (1971) 105. [3] S.B. Khadkikar, Phys. Letters 36B (1971) 451. [4] K. Ananthanarayanan, S. Das Gupta and N. De Takacsy, Phys. Letters 37B (1971) 143. [5] H. Margenau, Phys. Rev. 59 (1941) 37. [6] D. Brink, The alpha-particle model of light nuclei, Int. School of Phys. 'Enrico Fermi' course XXXVI (1965). [71 H. Friedrich, L. Satpathy and A. Weiguny, Phys. Letters 36]3 (1971) 189. [Sl A. B. Volkov, Nucl. Phys. 74 (1965) 33. [91 D. Brink and E. Boecker, Nucl. Phys. A91 (1967) 1. [101 B. Giraud and D. Zaikine, Phys. Letters 37B (1971) 25. [I11 H.A.Jahn and E. Teller, Proc. Roy. Soc. 161A (1937) 220.