- Email: [email protected]
SU3 symmetry and realistic interactions
Volume 26B, number 2
PHYSICS
LETTERS
e v e r t h i s i s not a l w a y s t r u e ( a s c a n b e s e e n , f o r e x a m p l e , in t h e 8 B e and 12C g r o u n d s t a t e b a n d s [4]). We n o t i c e t h e s t r i k i n g a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l and the calculated s p e c t r u m . This i s a n o t h e r e v i d e n c e t h a t t h e 8Be t y p e l e v e l s a r e p u r e 8p - 8h p r o j e c t e d s t a t e s .
SU 3 SYMMETRY
AND
25 December 1967
1. P, Chevallier, F.Seheibling. G.Goldring. I , P l e s s e r . and M.W. Sachs, Phys, Rev. 160 (1967) 827, 2. Y.Abgrall. E . C a u r i e r a n d G . M o n s o n e g o , Phys. L e t t e r s 24B. (1967) 609, 3. H.Rouhaninejad and J . Y o e c o z , Nucl. Phys. 78 (1966) a5a. 4. G.Baron and H.Rouhaninejad. a . d e Phys.28 (1967)142.
REALISTIC
INTERACTIONS
**
J. F L O R E S * P a l m e r Physical Laboratory Princeton University, Princeton, N.J. and R. PI~REZ Instituto de F[sica Universidad de M~xico, M~xico 20, O . F . , M~xico Received 10 October 1967
The energy levels and wave functions are calculated for s y s t e m s of three and four particles in the 2s - ld shell using an effective interaction determined from the Hamada - Johnston potential and wave functions classified in the SU3 scheme. It is shown that the degree of SU3 s y m m e t r y violation is very low for these e a s e s that c o r r e s p o n d to low isispin values.
Up to t h e p r e s e n t m a n y c a l c u l a t i o n s of n u c l e a r p r o p e r t i e s in t h e 2 s - l d s h e l l h a v e b e e n p e r f o r m e d u s i n g t h e SU 3 s c h e m e ( r e f . 1 and 2). A l l of t h e s e c a l c u l a t i o n s h a v e b e e n d o n e u n d e r two f u n d a m e n t a l a s s u m p t i o n s , 1) t h e u s e of a m o d e l s p a c e with a b a s i s t r u n c a t e d b y SU 3 s y m m e t r y and 2) t h e u s e of d i f f e r e n t t y p e s of p h e n o m e n o l o g i c a l e f f e c t i v e i n t e r a c t i o n s , w h i c h c o n t a i n one or more parameters, whose values are determ i n e d by t h e r e s u l t s of t h e c a l c u l a t i o n s . T h e a i m of t h e p r e s e n t w o r k i s to c a l c u l a t e t h e e n e r g y l e v e l s and w a v e f u n c t i o n s of t h e low l y i n g s t a t e s of s o m e n u c l e i in t h e b e g i n n i n g of t h e 2s - l d s h e l l w i t h o u t u s i n g t h e s e two f u n d a m e n t a l a s s u m p t i o n s . F i r s t o f a l l , the c a l c u l a t i o n will be done u s i n g a c o m p l e t e 2s - l d s h e l l b a s i s and s e c o n d l y the e f f e c t i v e i n t e r a c t i o n , o b t a i n e d by Kuo a n d B r o w n ( r e f . 3) f r o m t h e H a m a d a J o h n s t o n p o t e n t i a l i s u s e d . We c a n t h e n c h e c k h o w g o o d a s s u m p t i o n (1) i s , by t r u n c a t i n g t h e b a s i s a c c o r d ing to the d i f f e r e n t s y m m e t r i e s i n t r o d u c e d in the calculation. T h e s t a t e s w e u s e in t h e c a l c u l a t i o n s a r e r e p r e s e n t e d b y t h e k e t s of t h e f o r m
] [qa(x, ~)~oL;SST,J)
(1)
w h e r e [h] d e n o t e s t h e Young p a r t i t i o n of the u n i t a r y g r o u p in s i x d i m e n s i o n s U6, (X, p ) g i v e s t h e i r r e d u c i b l e r e p r e s e n t a t i o n of t h e s u b g r o u p SU3; L i n d i c a t e s t h e t o t a l o r b i t a l a n g u l a r m o m e n t u m and S and T t h e t o t a l s p i n and i s o s p i n r e s p e c t i v e l y . T h e q u a n t u m n u m b e r s a, co and fi a r e u s e d to c o m p l e t e the c l a s s i f i c a t i o n w h e n e v e r necessary. In o r d e r to p e r f o r m t h e c a l c u l a t i o n , w e h a v e c o n s t r u c t e d t h e s t a t e s by t h e l o w e r i n g o p e r a t o r t e c h n i q u e o u t l i n e d by M o s h i n s k y ( r e f . 4). A c o m p u t e r p r o g r a m f o r t h e IBM 7094 w a s c o n s t r u c t e d to o b t a i n t h e w a v e f u n c t i o n (1) f o r any v a l u e s o f t h e q u a n t u m n u m b e r s t h e r e i n d i c a t e d and t h e m a t r i x e l e m e n t s of an a r b i t r a r y two b o d y o p e r a t o r with r e s p e c t to t h e s e w a v e f u n c t i o n s . A d e t a i l e d * P r e s e n t a d d r e s s : Instituto de Fisfca, Universidad de M~xico. ** This work was supported by the U.S.Atomic Energy Commission and by the Comisi6n National de Energfa Nuclear, M~xico. This work made use of Princeton Computer Facilities, supported in part by the National Science Foundation Grant NSF GP-579. 55
Volume
26B, number
2
PHYSICS
LETTERS
25 December
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.~-"-
G +
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1967
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.--
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-20'
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complete
a
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(x.~,)H3,t)
Exp.
b
c
d
Fig. 1. Energy l e v e l s of 20Ne. d e s c r i p t i o n of t h e p r o g r a m w i l l b e p u b l i s h e d e l s e whe re. In fig. l a we s h o w t h e e n e r g y l e v e l s of 2 0 N e ( T = 0) s t a t e s o b t a i n e d u s i n g t h e p o t e n t i a l of Kuo a n d B r o w n a n d t h e c o m p l e t e SU 3 b a s i s f o r 4 p a r t icles in the 2s - ld shell. As can be seen from the comparison with the experimental spectrum, g i v e n i n fig. l d , t h e f i r s t f o u r l e v e l s a r e r e p r o d u c e d v e r y w e l l , i n c l u d i n g t h e b i n d i n g e n e r g y *. H o w e v e r , a t a b o u t 6 M e V of e x c i t a t i o n e n e r g y t h e r e a p p e a r two e x p e r i m e n t a l J = 2 l e v e l s , of which only one is obtained in our calculation; t h i s c o u l d b e a m a n i f e s t a t i o n of c o r e e x c i t a t i o n , w h i c h we h a v e not i n c l u d e d . Our main interest in this work is, however, to analyze the effect on the spectrum so calcula t e d of t h e b a s i s t r u n c a t i o n . W e p r e s e n t i n fig. l b t h e r e s u l t i n g e n e r g y l e v e l s w h e n we u s e s t a t e s c o r r e s p o n d i n g t o t h e Y o u n g d i a g r a m s [4] a n d [31] of t h e o r b i t a l g r o u p U 6 a n d i n fig. l c , t h e r e s u l t of c o n s i d e r i n g k e t s of t h e t y p e (1) s u c h t h a t i n d i c e s (X, ~ ) c o r r e s p o n d t o w e i g h t s h i g h e r t h a n (ko, iz o) = (3, 1), only. In b o t h c a s e s it c a n b e s e e n f r o m t h i s f i g u r e s t h a t t h e s p e c t r u m i s not affected by the truncation. We consider that this i m p l i e s t h a t t h e d e g r e e of v i o l a t i o n of b o t h o r b i t a l * The binding energy referred to the ground state of 1 7 0 s i n c e t h e v a l u e s E~ = 0 M e V , E± = 0 . 8 7 M e V a n d
E_~ = 5.08 MeV have be~n used for tl~e single p a r t i c l e . s p2 h.t t m g s , throught this work.
56
Fig. 2. Ground state wave function of 20Ne r e p r e s e n t e d by the square of the overlap of the physical wave function with the SU 3 states (solid lines) and j - j coupling states (dotted lines). These states are ordered by inc r e a s i n g energy of a Q2 force and spin - orbit coupling, respectively. a n d SU 3 s y m m e t r y b y a r e a l i s t i c i n t e r a c t i o n i s low i n t h i s c a s e . In f a c t , we h a v e p l o t t e d in fig. 2, t h e g r o u n d s t a t e w a v e f u n c t i o n , r e p r e s e n t i n g it b y t h e s q u a r e of t h e o v e r l a p it h a s w i t h e a c h p o s s i b l e w a v e f u n c t i o n of t y p e (1) w i t h J = O. W e s e e t h a t t h e i n t e n s i t y w i t h t h e l e a d i n g SU 3 s t a t e i s of t h e o r d e r of 81% a n d t h a t t h e c o n t r i b u t i o n f r o m h i g h e r w e i g h t SU3 s t a t e s i s v e r y low. In t h e s a m e g r a p h , a n d w i t h d o t t e d l i n e s , we give t h e s q u a r e of t h e o v e r l a p of t h e g r o u n d s t a t e w a v e function with states classified by j - j coupling; this amplitude is different from zero for many j -j s t a t e s a n d f o r no o n e it e x c e e d s 25%. T h e g r a p h s c o r r e s p o n d i n g to a l l o t h e r l o w lying states are qualitatively the same as the g r o u n d s t a t e ; we c a n u n d e r s t a n d i n t h i s way t h e f a c t t h a t t h e s p e c t r u m i s not a f f e c t e d b y t h e t r u n c a t i o n . T h e s a m e f e a t u r e i s n o t t r u e of j - j c o u p l i n g , s i n c e t h e r e i s no w a y of t r u n c a t i n g t h e b a s i s a n d -1 ZO
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F i g . 3. E n e r g y
l e v e l s of 2 0 0 .
Volume 26B, n u m b e r 2
PHYSICS
s t i l l p r e s e r v i n g t h e m a i n p r o p e r t i e s of t h e w a v e function. We a n a l y z e n o w t h e c a s e of t h e n u c l e u s 2 0 0 ( T = 2). In fig. (3) we p l o t t h e e n e r g y l e v e l s c o m p a r i n g in fig. (3a) a n d fig. (3d) t h e r e s u l t s of t h e complete shell model calculation with the experim e n t a l e n e r g i e s . We o b s e r v e t h a t n o w t h e l e v e l s t r u c t u r e i s not w e l l r e p r o d u c e d , s i n c e t h e s e c o n d O i s p r e d i c t e d t o o low. T h i s c o u l d i n d i c a t e t h a t t h e m o d e l s p a c e i s not l a r g e e n o u g h i n t h i s c a s e , t h e deformed states being important even for very l o w e x c i t a t i o n e n e r g i e s . N e v e r t h e l e s s , we c a n p r o c e e d to t r u n c a t e t h e b a s i s a n d i n f i g s . (3b) we p l o t t h e r e s u l t i n g s p e c t r u m w h e n we i n c l u d e in t h e c a l c u l a t i o n w a v e f u n c t i o n s of o r b i t a l s y m m e t r i e s [22] a n d [211] o n l y a n d in fig. (3c) we g i v e t h e r e s u l t w h e n o n l y t h o s e SU 3 k e t s w i t h w e i g h t h i g h e r t h a n ()to, p o ) = (31) a r e u s e d . T h e t r u n c t i o n p r o d u c e s r a t h e r l a r g e e f f e c t in b o t h c a s e s a n d we c o n s i d e r t h a t it i s n o t j u s t i f i e d f o r t h i s n u c l e u s . If a n i n t e n s i t y g r a p h i s g i v e n f o r t h e e i g e n s t a t e s of 2 0 0 we s e e t h a t t h e r e i s no SU 3 s t a t e t h a t h a s a r e m a r k a b l y l a r g e o v e r l a p with the "physical" wave function. The corresponding intensity graph for j - j coupiing scheme, indicates that, although this coupling scheme is b e t t e r t h a n SU 3 in t h e c a s e of T = 2 s t a t e s , t h e t r u n c a t i o n of t h e b a s i s i s not p o s s i b l e a c c o r d i n g
LETTERS
25 D e c e m b e r 1967
to it, s i n c e , f o r e x a m p l e , t h e da c o m p o n e n t s of the wave function are always important. We h a v e a l s o o b t a i n e d t h e r e s u l t s f o r 1 9 F a n d 1 9 0 . A g a i n t h e s p e c t r u m of 1 9 F i s v e r y w e l l r e p r o d u c e d a n d t h e t r u n c a t i o n a l o n g t h e SU 3 s c h e m e i s p o s s i b l e w i t h no e f f e c t w h a t s o e v e r on t h e r e s u l t s . O n t h e o t h e r h a n d , t h e c a s e of 1 9 0 i s s i m i l a r to t h e 2 0 0 n u c l e u s . F i r s t of a l l , o u r r e s u l t s do not r e p r o d u c e t h e e x p e r i m e n t a l s p e c t r u m a n d s e c o n d l y , no t r u n c a t i o n i s p o s s i b l e , without altering seriously the results. It s e e m s , t h e r e f o r e , t h a t a s s u m p t i o n (1) i s c o r r e c t o n l y f o r low v a l u e s of t o t a l i s o s p i n a n d t h a t in t h e s e c a s e s t h e e x p e r i m e n t a l e n e r g i e s a r e very well reproduced by a realistic interaction. We w o u l d l i k e to t h a n k D r . T. T. S. Kuo f o r providing us with the two-body matrix elements a n d P r o f . M. M o s h i n s k y f o r h e l p f u l d i s c u s s i o n r e g a r d i n g t h e t e c h n i q u e s u s e d . O n e of u s (J. F.) w i s h e s to t h a n k P r o f . G. E. B r o w n f o r h i s h o s pitality at Palmer Physical Laboratory. 1. M. ttarvey and J. P. Elliot. Proe. Roy. Soe.. A272 (1963) 557. 2. M. de Llano. P . A . Mello. E. Chac5n and ,I. Flores. Nucl. Phys. 72 (1965) 379. 3. T . T . S . K u o a n d G . E . B r o w n , Nuel. Phys. 85(1966) 87. 4. M. Moshinsky in Physics of many particle s y s t e m s . e d . E . M e r o n (Gordon and Breach. New York, 1967).
57
PHYSICS
LETTERS
e v e r t h i s i s not a l w a y s t r u e ( a s c a n b e s e e n , f o r e x a m p l e , in t h e 8 B e and 12C g r o u n d s t a t e b a n d s [4]). We n o t i c e t h e s t r i k i n g a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l and the calculated s p e c t r u m . This i s a n o t h e r e v i d e n c e t h a t t h e 8Be t y p e l e v e l s a r e p u r e 8p - 8h p r o j e c t e d s t a t e s .
SU 3 SYMMETRY
AND
25 December 1967
1. P, Chevallier, F.Seheibling. G.Goldring. I , P l e s s e r . and M.W. Sachs, Phys, Rev. 160 (1967) 827, 2. Y.Abgrall. E . C a u r i e r a n d G . M o n s o n e g o , Phys. L e t t e r s 24B. (1967) 609, 3. H.Rouhaninejad and J . Y o e c o z , Nucl. Phys. 78 (1966) a5a. 4. G.Baron and H.Rouhaninejad. a . d e Phys.28 (1967)142.
REALISTIC
INTERACTIONS
**
J. F L O R E S * P a l m e r Physical Laboratory Princeton University, Princeton, N.J. and R. PI~REZ Instituto de F[sica Universidad de M~xico, M~xico 20, O . F . , M~xico Received 10 October 1967
The energy levels and wave functions are calculated for s y s t e m s of three and four particles in the 2s - ld shell using an effective interaction determined from the Hamada - Johnston potential and wave functions classified in the SU3 scheme. It is shown that the degree of SU3 s y m m e t r y violation is very low for these e a s e s that c o r r e s p o n d to low isispin values.
Up to t h e p r e s e n t m a n y c a l c u l a t i o n s of n u c l e a r p r o p e r t i e s in t h e 2 s - l d s h e l l h a v e b e e n p e r f o r m e d u s i n g t h e SU 3 s c h e m e ( r e f . 1 and 2). A l l of t h e s e c a l c u l a t i o n s h a v e b e e n d o n e u n d e r two f u n d a m e n t a l a s s u m p t i o n s , 1) t h e u s e of a m o d e l s p a c e with a b a s i s t r u n c a t e d b y SU 3 s y m m e t r y and 2) t h e u s e of d i f f e r e n t t y p e s of p h e n o m e n o l o g i c a l e f f e c t i v e i n t e r a c t i o n s , w h i c h c o n t a i n one or more parameters, whose values are determ i n e d by t h e r e s u l t s of t h e c a l c u l a t i o n s . T h e a i m of t h e p r e s e n t w o r k i s to c a l c u l a t e t h e e n e r g y l e v e l s and w a v e f u n c t i o n s of t h e low l y i n g s t a t e s of s o m e n u c l e i in t h e b e g i n n i n g of t h e 2s - l d s h e l l w i t h o u t u s i n g t h e s e two f u n d a m e n t a l a s s u m p t i o n s . F i r s t o f a l l , the c a l c u l a t i o n will be done u s i n g a c o m p l e t e 2s - l d s h e l l b a s i s and s e c o n d l y the e f f e c t i v e i n t e r a c t i o n , o b t a i n e d by Kuo a n d B r o w n ( r e f . 3) f r o m t h e H a m a d a J o h n s t o n p o t e n t i a l i s u s e d . We c a n t h e n c h e c k h o w g o o d a s s u m p t i o n (1) i s , by t r u n c a t i n g t h e b a s i s a c c o r d ing to the d i f f e r e n t s y m m e t r i e s i n t r o d u c e d in the calculation. T h e s t a t e s w e u s e in t h e c a l c u l a t i o n s a r e r e p r e s e n t e d b y t h e k e t s of t h e f o r m
] [qa(x, ~)~oL;SST,J)
(1)
w h e r e [h] d e n o t e s t h e Young p a r t i t i o n of the u n i t a r y g r o u p in s i x d i m e n s i o n s U6, (X, p ) g i v e s t h e i r r e d u c i b l e r e p r e s e n t a t i o n of t h e s u b g r o u p SU3; L i n d i c a t e s t h e t o t a l o r b i t a l a n g u l a r m o m e n t u m and S and T t h e t o t a l s p i n and i s o s p i n r e s p e c t i v e l y . T h e q u a n t u m n u m b e r s a, co and fi a r e u s e d to c o m p l e t e the c l a s s i f i c a t i o n w h e n e v e r necessary. In o r d e r to p e r f o r m t h e c a l c u l a t i o n , w e h a v e c o n s t r u c t e d t h e s t a t e s by t h e l o w e r i n g o p e r a t o r t e c h n i q u e o u t l i n e d by M o s h i n s k y ( r e f . 4). A c o m p u t e r p r o g r a m f o r t h e IBM 7094 w a s c o n s t r u c t e d to o b t a i n t h e w a v e f u n c t i o n (1) f o r any v a l u e s o f t h e q u a n t u m n u m b e r s t h e r e i n d i c a t e d and t h e m a t r i x e l e m e n t s of an a r b i t r a r y two b o d y o p e r a t o r with r e s p e c t to t h e s e w a v e f u n c t i o n s . A d e t a i l e d * P r e s e n t a d d r e s s : Instituto de Fisfca, Universidad de M~xico. ** This work was supported by the U.S.Atomic Energy Commission and by the Comisi6n National de Energfa Nuclear, M~xico. This work made use of Princeton Computer Facilities, supported in part by the National Science Foundation Grant NSF GP-579. 55
Volume
26B, number
2
PHYSICS
LETTERS
25 December
N: °
Ne z°
.~-"-
G +
_
G+
_
Z+
2'"
> O÷
1967
-
-
2+ 0+
-
-
0+
J : 0+
- -
.....
0.5,
SU> j-j
.--
%/
'%',,,_
/
,--- -,,
4+,,.
-20'
2+
- _ _ _
2+
-25" 0 + ..
_
complete
a
0+
_
[4][31]
(x.~,)H3,t)
Exp.
b
c
d
Fig. 1. Energy l e v e l s of 20Ne. d e s c r i p t i o n of t h e p r o g r a m w i l l b e p u b l i s h e d e l s e whe re. In fig. l a we s h o w t h e e n e r g y l e v e l s of 2 0 N e ( T = 0) s t a t e s o b t a i n e d u s i n g t h e p o t e n t i a l of Kuo a n d B r o w n a n d t h e c o m p l e t e SU 3 b a s i s f o r 4 p a r t icles in the 2s - ld shell. As can be seen from the comparison with the experimental spectrum, g i v e n i n fig. l d , t h e f i r s t f o u r l e v e l s a r e r e p r o d u c e d v e r y w e l l , i n c l u d i n g t h e b i n d i n g e n e r g y *. H o w e v e r , a t a b o u t 6 M e V of e x c i t a t i o n e n e r g y t h e r e a p p e a r two e x p e r i m e n t a l J = 2 l e v e l s , of which only one is obtained in our calculation; t h i s c o u l d b e a m a n i f e s t a t i o n of c o r e e x c i t a t i o n , w h i c h we h a v e not i n c l u d e d . Our main interest in this work is, however, to analyze the effect on the spectrum so calcula t e d of t h e b a s i s t r u n c a t i o n . W e p r e s e n t i n fig. l b t h e r e s u l t i n g e n e r g y l e v e l s w h e n we u s e s t a t e s c o r r e s p o n d i n g t o t h e Y o u n g d i a g r a m s [4] a n d [31] of t h e o r b i t a l g r o u p U 6 a n d i n fig. l c , t h e r e s u l t of c o n s i d e r i n g k e t s of t h e t y p e (1) s u c h t h a t i n d i c e s (X, ~ ) c o r r e s p o n d t o w e i g h t s h i g h e r t h a n (ko, iz o) = (3, 1), only. In b o t h c a s e s it c a n b e s e e n f r o m t h i s f i g u r e s t h a t t h e s p e c t r u m i s not affected by the truncation. We consider that this i m p l i e s t h a t t h e d e g r e e of v i o l a t i o n of b o t h o r b i t a l * The binding energy referred to the ground state of 1 7 0 s i n c e t h e v a l u e s E~ = 0 M e V , E± = 0 . 8 7 M e V a n d
E_~ = 5.08 MeV have be~n used for tl~e single p a r t i c l e . s p2 h.t t m g s , throught this work.
56
Fig. 2. Ground state wave function of 20Ne r e p r e s e n t e d by the square of the overlap of the physical wave function with the SU 3 states (solid lines) and j - j coupling states (dotted lines). These states are ordered by inc r e a s i n g energy of a Q2 force and spin - orbit coupling, respectively. a n d SU 3 s y m m e t r y b y a r e a l i s t i c i n t e r a c t i o n i s low i n t h i s c a s e . In f a c t , we h a v e p l o t t e d in fig. 2, t h e g r o u n d s t a t e w a v e f u n c t i o n , r e p r e s e n t i n g it b y t h e s q u a r e of t h e o v e r l a p it h a s w i t h e a c h p o s s i b l e w a v e f u n c t i o n of t y p e (1) w i t h J = O. W e s e e t h a t t h e i n t e n s i t y w i t h t h e l e a d i n g SU 3 s t a t e i s of t h e o r d e r of 81% a n d t h a t t h e c o n t r i b u t i o n f r o m h i g h e r w e i g h t SU3 s t a t e s i s v e r y low. In t h e s a m e g r a p h , a n d w i t h d o t t e d l i n e s , we give t h e s q u a r e of t h e o v e r l a p of t h e g r o u n d s t a t e w a v e function with states classified by j - j coupling; this amplitude is different from zero for many j -j s t a t e s a n d f o r no o n e it e x c e e d s 25%. T h e g r a p h s c o r r e s p o n d i n g to a l l o t h e r l o w lying states are qualitatively the same as the g r o u n d s t a t e ; we c a n u n d e r s t a n d i n t h i s way t h e f a c t t h a t t h e s p e c t r u m i s not a f f e c t e d b y t h e t r u n c a t i o n . T h e s a m e f e a t u r e i s n o t t r u e of j - j c o u p l i n g , s i n c e t h e r e i s no w a y of t r u n c a t i n g t h e b a s i s a n d -1 ZO
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Volume 26B, n u m b e r 2
PHYSICS
s t i l l p r e s e r v i n g t h e m a i n p r o p e r t i e s of t h e w a v e function. We a n a l y z e n o w t h e c a s e of t h e n u c l e u s 2 0 0 ( T = 2). In fig. (3) we p l o t t h e e n e r g y l e v e l s c o m p a r i n g in fig. (3a) a n d fig. (3d) t h e r e s u l t s of t h e complete shell model calculation with the experim e n t a l e n e r g i e s . We o b s e r v e t h a t n o w t h e l e v e l s t r u c t u r e i s not w e l l r e p r o d u c e d , s i n c e t h e s e c o n d O i s p r e d i c t e d t o o low. T h i s c o u l d i n d i c a t e t h a t t h e m o d e l s p a c e i s not l a r g e e n o u g h i n t h i s c a s e , t h e deformed states being important even for very l o w e x c i t a t i o n e n e r g i e s . N e v e r t h e l e s s , we c a n p r o c e e d to t r u n c a t e t h e b a s i s a n d i n f i g s . (3b) we p l o t t h e r e s u l t i n g s p e c t r u m w h e n we i n c l u d e in t h e c a l c u l a t i o n w a v e f u n c t i o n s of o r b i t a l s y m m e t r i e s [22] a n d [211] o n l y a n d in fig. (3c) we g i v e t h e r e s u l t w h e n o n l y t h o s e SU 3 k e t s w i t h w e i g h t h i g h e r t h a n ()to, p o ) = (31) a r e u s e d . T h e t r u n c t i o n p r o d u c e s r a t h e r l a r g e e f f e c t in b o t h c a s e s a n d we c o n s i d e r t h a t it i s n o t j u s t i f i e d f o r t h i s n u c l e u s . If a n i n t e n s i t y g r a p h i s g i v e n f o r t h e e i g e n s t a t e s of 2 0 0 we s e e t h a t t h e r e i s no SU 3 s t a t e t h a t h a s a r e m a r k a b l y l a r g e o v e r l a p with the "physical" wave function. The corresponding intensity graph for j - j coupiing scheme, indicates that, although this coupling scheme is b e t t e r t h a n SU 3 in t h e c a s e of T = 2 s t a t e s , t h e t r u n c a t i o n of t h e b a s i s i s not p o s s i b l e a c c o r d i n g
LETTERS
25 D e c e m b e r 1967
to it, s i n c e , f o r e x a m p l e , t h e da c o m p o n e n t s of the wave function are always important. We h a v e a l s o o b t a i n e d t h e r e s u l t s f o r 1 9 F a n d 1 9 0 . A g a i n t h e s p e c t r u m of 1 9 F i s v e r y w e l l r e p r o d u c e d a n d t h e t r u n c a t i o n a l o n g t h e SU 3 s c h e m e i s p o s s i b l e w i t h no e f f e c t w h a t s o e v e r on t h e r e s u l t s . O n t h e o t h e r h a n d , t h e c a s e of 1 9 0 i s s i m i l a r to t h e 2 0 0 n u c l e u s . F i r s t of a l l , o u r r e s u l t s do not r e p r o d u c e t h e e x p e r i m e n t a l s p e c t r u m a n d s e c o n d l y , no t r u n c a t i o n i s p o s s i b l e , without altering seriously the results. It s e e m s , t h e r e f o r e , t h a t a s s u m p t i o n (1) i s c o r r e c t o n l y f o r low v a l u e s of t o t a l i s o s p i n a n d t h a t in t h e s e c a s e s t h e e x p e r i m e n t a l e n e r g i e s a r e very well reproduced by a realistic interaction. We w o u l d l i k e to t h a n k D r . T. T. S. Kuo f o r providing us with the two-body matrix elements a n d P r o f . M. M o s h i n s k y f o r h e l p f u l d i s c u s s i o n r e g a r d i n g t h e t e c h n i q u e s u s e d . O n e of u s (J. F.) w i s h e s to t h a n k P r o f . G. E. B r o w n f o r h i s h o s pitality at Palmer Physical Laboratory. 1. M. ttarvey and J. P. Elliot. Proe. Roy. Soe.. A272 (1963) 557. 2. M. de Llano. P . A . Mello. E. Chac5n and ,I. Flores. Nucl. Phys. 72 (1965) 379. 3. T . T . S . K u o a n d G . E . B r o w n , Nuel. Phys. 85(1966) 87. 4. M. Moshinsky in Physics of many particle s y s t e m s . e d . E . M e r o n (Gordon and Breach. New York, 1967).
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