A study of density and momentum dependent delta function forces

Volume 36B, number 1 A STUDY PHYSICS OF LETTERS DENSITY AND MOMENTUM DELTA FUNCTION FORCES* 9 August 1971 DEPENDENT K. R. LASSEY Department of Physics, McMasler University, Hamillon 16, Ontario, Canada and A. B. V O L K O V * * Department of Physics, State Uni~,ersity of Neu, York, Stony Brook, N.Y. 11790, USA Received 18 June 1971 The suitability of density and momentum dependent delta function forces in nuclear calculations is studied using H a r t r e e - F o c k calculations in the 2 s - l d shell. The effect of the compressibility upon deformation and rms radius is investigated. The t e c h n i c a l d i f f i c u l t i e s a s s o c i a t e d w i t h c a l c u l a t i o n s of the p r o p e r t i e s of h e a v y n u c l e i , e s p e c i a l l y d e f o r m a t i o n e n e r g i e s , s h e l l e f f e c t s in the f i s s i o n p r o c e s s , e t c . , u s i n g two body f o r c e s , r e q u i r e s the u s e of a s i m p l e p h e n o m e n o l o g i c a l but n e v e r - t h e - l e s s " r e a l i s t i c " two body i n t e r a c t i o n . In f a c t t h e r e i s r e a s o n to b e l i e v e [1,2] that the m o s t hopeful p o s s i b i l i t y i s s o m e v a r i a n t of the S k y r m e [2] i n t e r a c t i o n w h i c h i s a m o m e n t u m and d e n s i t y d e p e n d e n t d e l t a f u n c t i o n i n t e r a c t i o n , and in p a r t i c u l a r s o m e f o r m of the M o s z k o w s k i [4] m o d i f i e d d e l t a i n t e r a c t i o n (MDI). The MDI a s p r o p o s e d by M o s z k o w s k i h a s the form V(p, R , r) = - a S ( r ) * ½¢3[p2 5(r) + 5( r)p 21 + + ~Tfi(R) 5(r) (1) w h e r e p , R and r a r e the r e l a t i v e m o m e n t u m , c e n t r e of m a s s c o o r d i n a t e , and r e l a t i v e s e p a r a = tion f o r the i n t e r a c t i n g p a r t i c l e s . The 152 t e r m s , w h i c h s i m u l a t e in p a r t the f i n i t e r a n g e and the r e p u l s i v e c o r e of the f o r c e , and the k 2 ( R ) t e r m , w h i c h s i m u l a t e s m a n y body " d e n s i t y d e p e n d e n t " e f f e c t s , i n c l u d ! n g the r e p l a c e m e n t of the t e n s o r i n t e r a c t i o n by a c e n t r a l i n t e r a c t i o n [5,6]. a r e j o i n t l y r e s p o n s i b l e for the s a t u r a t i o n of n u c l e a r s y s t e m s . The p a r a m e t e r s a , ~ and ), a r e fixed by f i t t i n g n u c l e a r m a t t e r at s a t u r a t i o n , and 160 Work supported in part by the National Research Council of Canada and the United States Atomic Energy Commission. ** Permanent address. Department of Physics, McMaster University, Hamilton 16, Ontario, Canada. binding e n e r g y in a s i m p l e v a r i a t i o n a l c a l c u l a t i o n . T h e s a t u r a t i o n p r o p e r t y of the i n t e r a c t i o n r e p r e s e n t s one of i t s m o s t i m p o r t a n t " r e a l i s t i c " features. Moszkowski has obtained reasonable e s t i m a t e s for the s u r f a c e e n e r g i e s and for l s s i n g l e p a r t i c l e e n e r g i e s for 160 and 2 0 8 P b , etc. H o w e v e r , it i s i m p o r t a n t to e x a m i n e the i n t e r action more critically, especially its propensity to inhibit or e n h a n c e d e f o r m a t i o n of n u c l e i . F o r light n u c l e i it h a s b e e n shown [7] that d e f o r m a t i o n i s s e n s i t i v e to the s a t u r a t i o n m e c h a n i s m of the i n t e r a c t i o n a s e x e m p l i f i e d by the e x change m i x t u r e u s e d and i s p r o b a b l y d o m i n a t e d by k i n e t i c e n e r g y m i n i m i z a t i o n w h i l e m a i n t a i n i n g a s m u c h s p a c i a l s y m m e t r y in the w a v e f u n c t i o n a s p o s s i b l e . The i m p o r t a n c e of the r e p u l s i v e p 2 5 ( r ) + 5(r)p 2 t e r m in the MDI, which w i l l h a v e a k i n e t i c e n e r g y like b e h a v i o u r , s u g g e s t s that the MDI wiI1 tend to e x a g e r a t e d e f o r m a t i o n s , though the p u r e S e r b e r e x c h a n g e c h a r a c t e r ( e v e n s t a t e i n t e r a c t i o n only) w i l l tend to c o m p e n s a t e t h i s e x a g e r a t i o n s o m e w h a t when c o m p a r e d to the e x c h a n g e m i x t u r e s u s e d in finite r a n g e d e n s i t y dependent forces. F u r t h e r m o r e , an e x t e n s i v e study of the p r o p e r t i e s of light n u c l e i u s i n g f i n i t e r a n g e d e n s i t y d e p e n d e n t f o r c e s , i n c l u d i n g the s p e c t r u m of 4He [8,9], the s p e c t r a of the l p s h e l l n u c l e i [9], the l p - l h e x c i t a t i o n s of 1 6 0 [ 9 ] , and H a r t r e e F o c k c a l c u l a t i o n s in the 2 s - l d s h e l l [11,12], p e r f o r m e d in a v a r i a t i o n a l m a n n e r , i n c l u d i n g nuc l e a r s i z e , s e e m s to f a v o u r f o r c e s w h o s e n u c l e a r m a t t e r c o m p r e s s i b i l i t y f a i l s in the 2 0 0 - 2 5 0 MeV r a n g e . H i g h e r c o m p r e s s i b i l i t y f o r c e s with the Volume 36B, number 1 PIIYSICS s a m e e x c h a n g e m i x t u r e s and n u c l e a r m a t t e r s a t u r a t i o n p r o p e r t i e s g e n e r a l l y lead to s m a l l e r r a d i i [8-10] and s m a l l e r binding e n e r g i e s . T h i s i s a c o n s e q u e n c e of the h i g h e r c o m p r e s s i b i l i t y f o r c e i n c r e a s i n g the a v e r a g e d e n s i t y of the finite s y s t e m away f r o m the e x p e r i m e n t a l v a l u e and c l o s e r to the n u c l e a r m a t t e r v a l u e w h i c h both r e d u c e s the r m s r a d i u s of the s y s t e m and o v e r e s t i m a t e s the k i n e t i c e n e r g y . The M D I h a s a c o m p r e s s i b i l i t y K = 306 MeV which is p a r t i a l l y r e s p o n s i b l e for the s m a l l e r r a d i i which a r e found f o r this i n t e r a c t i o n . F o r this r e a s o n we s h a l l a l s o c o n s i d e r a m o d i f i c a t i o n of the MDI which a l l o w s d i f f e r e n t v a l u e s of K (we s h a l l r e f e r to this m o d i f i e d v e r s i o n a s MDIK). In this i n v e s t i g a t i o n H a r t r e e - F o c k c a l c u l a t i o n s a r e p e r f o r m e d for all " a l p h a p a r t i c l e " n u c l e i f r o m 160 to 4 0 C a u s i n g for c o m p a r i s o n p u r p o s e s a f i n i t e - r a n g e d e n s i t y and m o m e n t u m d e p e n d e n t f o r c e , the MDI, and the MDIK which i s c h o s e n to h a v e the s a m e c o m p r e s s i b i l i t y K = 225.6 MeV as the f i n i t e r a n g e f o r c e . T h e f i n i t e r a n g e f o r c e i s of the f o r m 9 August 1971 LETTERS t i o n a l c a l c u l a t i o n of the s p e c t r u m (no s h e l l m o d e l a s s u m p t i o n s ) i s the m o s t i m p o r t a n t of the d) c r i t e r i a . In any e v e n t it is found that i n t e r a c t i o n s of the f o r m of eq. (2) s a t i s f y i n g the b a s i c c r i t e r i a a) - c) and h a v i n g a p p r o x i m a t e l y the s a m e K all lead to c o m p a r a b l e r e s u l t s f o r the d) c r i t e r i a . M o s z k o w s k i ' s MDI can be w r i t t e n a s VMDI( p, l~, r) = -1160 5(r) +227.2(p 2 5(r) + 5( r )p2) + 980.4 02, '3(R)6(r) (3) in u n i t s MeV and fm. The MDIK i n t r o d u c e s an additional density dependent term proportional to p-l/'3 which is s u g g e s t e d by the w o r k of B h a d u r i and W a r k e [6] and which a l l o w s K to have v a l u e s in the r a n g e d e s i r e d . O t h e r than d e m a n d ing that K = 225.6 MeV (the s a m e a s VA) the p a r a m e t e r s of the MDIK a r e d e t e r m i n e d in e x a c t l y the s a m e way a s f o r the MDI (E/'A = -15.7 MeV in n u c l e a r m a t t e r which d i f f e r s s l i g h t l y f r o m the -16 MeV c r i t e r i o n u s e d f o r VA). T h i s then gives VMDIK( P, R , r ) = - 5 3 8 . 3 5 ( r ) + 315.5(p 26( r ) + 5 ( r )p2: VA( r 1 , r 2 , R , te) = [ ( l + c 3 P 1/3 ( r l , r 2 ) ) V a ( r ) + _ 2 2 8 . 6 p - 1 / ' 3 ( R ) 5 ( r ) _ 20.74p2 '3 ( R ) 5 ( r ) (4) + (l+c4p2/3(rl, r 2 ) ) Y ( r , k)] × × (1-m+~nPX+ b P ~ + h p T) (27 w h e r e r = r l - r 2, Px, Pv, PT are the s p a c e , spin and i s o s p i n e x c h a n g e o p e r a t o r s and m , b and h a r e the a p p r o p r i a t e e x c h a n g e p a r a m e t e r s . V a ( r ) is a s i m p l e a t t r a c t i v e g a u s s i a n p o t e n t i a l w h i l e V(r, k) i s a r e p u l s i v e g a u s s i a n pQtentential w h o s e r a n g e d e p e n d s on the a v e r a g e r e l a t i v e m o m e n t u m of the i n t e r a c t i n g p a r t i c l e s . T h i s f o r c e is f o r c e A of Ho and Volkov [8] which i s b a s e d on the r a d i a l p a r a m e t e r s of f o r c e 4 of Manning and V o l k o v [13]. The f o r c e p a r a m e t e r s h a v e been c h o s e n to s a t i s f y the f o l l o w i n g c r i t e r i a a) a fit to the w e i g h t e d a v e r a g e of the 1S and 3S s c a t t e r i n g p h a s e s h i f t s up to 250 MeV in the z e r o d e n s i t y l i m i t , b) an a p p r o p r i a t e fit to n u c l e a r m a t t e r at s a t u r a t i o n (k F = 1 . 3 6 fm -1, E / A = - 1 6 MeV), c) a r e a s o n a b l y c l o s e fit to the binding e n e r g y (inc l u d i n g the two body C o u l o m b i n t e r a c t i o n e n e r g y ) and r m s r a d i u s of 1 6 0 and d) a r e a s o n a b l e fit by an a p p r o p r i a t e v a r i a t i o n a l ( n o n - m o d e l ) c a l c u l a tion of the 4He s p e c t r u m , lp s h e l l s p e c t r a , l p - l h 1 6 0 s p e c t r u m and the H a r t r e e - F o c k p r o p e r t i e s of 2 s - l d s h e l l n u c l e i . The l a s t (d) c r i t e r i a a r e q u a l i t a t i v e f i t s which l e a d to VA a s one of the b e t t e r i n t e r a c t i o n s a m o n g a l a r g e f a m i l y of p o s s i b i l i t i e s s a t i s f y i n g a) - c). The 1 6 0 l p - l h e x c i t a t i o n s i n c l u d i n g all 16 p a r t i c l e s in a v a r i a - for the MDIK i n t e r a c t i o n (in u n i t s MeV and fro). The a t t r a c t i v e n a t u r e of the d e n s i t y d e p e n d e n t t e r m s i s not i n c o n s i s t e n t with the r e s u l t s of B h a d u r i and W a r k e , and the a t t r a c t i o n d e c r e a s e s with i n c r e a s i n g d e n s i t y a s it d o e s with the MDI. S a t u r a t i o n r e s u l t s f r o m the r e l a t i v e S s t a t e n a t u r e of the i n t e r a c t i o n and the s i m u l a t i o n of (a) the r e p u l s i v e c o r e by the p2 t e r m s , and (b) the t e n s o r t e r m by d e n s i t y - d e p e n d e n t t e r m s . The H a r t r e e - F o c k c a l c u l a t i o n s a r e p e r f o r m e d in the f o l l o w i n g m a n n e r . All p a r t i c l e s in the s y s t e m a r e a c t i v e and only the i n t r i n s i c e n e r g y of the s y s t e m is c o n s i d e r e d by s u b t r a c t i n g out the c e n t r e - o f - m a s s k i n e t i c e n e r g y . The H a r t r e e Fock program employs a cylindrical harmonic o s c i l l a t o r b a s i s i n c l u d i n g all s t a t e s for N = 2n+ I n t ] + n z < 3, i . e . , a f o u r s h e l l p r o g r a m including all s t a t e s t h r o u g h the 2 p - l f s h e l l in the s p h e r i c a l l i m i t . B e f o r e p e r f o r m i n g the H a r t r e e F o c k v a r i a t i o n an a t t e m p t i s m a d e to find the opt i m u m set of o s c i l l a t o r p a r a m e t e r s for e v e r y p o s s i b l e s i n g l e p a r t i c l e s t a t e (each c y l i n d r i c a l b a s i s s t a t e h a s two o s c i l l a t o r c o n s t a n t s ) c o m p a t i b l e with o r t h o g o n a l i t y c o n s t r a i n t s , by p e r f o r m ing a v a r i a t i o n a l c a l c u l a t i o n on the d o m i n a n t h a r m o n i c o s c i l l a t o r d e t e r m i n a n t of the s y s t e m . The m a x i m u m n u m b e r of o s c i l l a t o r c o n s t a n t s o b tained in t h i s m a n n e r is 11 for 32S and 36Ar. We l a b e l the r e s u l t s of t h i s v a r i a t i o n a s M I N D E T . Volume 36B, n u m b e r 1 PHYSICS LETTERS 9 August 1971 Table 1 The binding e n e r g i e s and shapes of nuclei for different interactions. R 2 = ~d ~ i (x2"+v2+z~ ~ O = I <Ei(2z2i_x2_.,2,\ i Yi ~/ and ASYM :: ( E (x/2-y2)} / ( , E (x/2 y/2)}. MINDET values arc given in p a r e n t h e s e s . Q/R 2 Rms radius R (fro) ASYIvI Binding energy (MeV) Interaction VA MDI MDIK VA MDI MDIK VA MDI MDIK 0.0 O.O 0.0 160 Ground state 125.6 123.5 129.7 (120.4) (119.2) (126.7) 2.76 2.44 2.56 (2.85) (2.55) (2.63) 160 4p-4h state 109.1 116.7 122.4 (93.3) (83.5) (96.6) 3.25 2.89 3.16 (3.34) (3.02) (3.15) 0.846 0.881 0.80 (0.924) (0.965) (0.961) 3.09 2.77 2.90 (3.21) (2.91) (3.00) 0.532 0.601 0.587 (0.591) (0.640) (0.627) 20Ne 153.3 150.0 158.6 t 188.7 184.8 191.6 (146.6) (140.0) (150.9) 24Mg 3.25 2.92 3,09 0.538 0.58 MD] MDIK 0.0 0.0 0.0 -0.71 -0.78 (3.35) (3.04) (3.13) (0.579) (0.624) (0.615) 28Si Oblate state 229.2 219.0 230.9 (215.6) (197.0) (215.2) 3.38 3.06 3.16 (3.48) (3.18) (3.26) -0.416 -0.445 -0.457 (-0.484) (-0.513) (-0.508) 28Si Prolate state 226.2 212.3 227.8 (222.5) (205.8) (222.6) 3.38 3.06 3.18 (3.47) (3.15) (3.25) 32S Oblate state 270.3 252.4 267.3 (253.5) (223.6) (247.1) 3.45 3.14 3.23 (3.56) (3.22) (3.28) 32S Prolate state 268.9 253.3 266.3 (256.8) (228.2) (248.7) 3.43 3.11 3.20 (3.49) (3.17) (3.26) 36Ar 312.9 281.3 304,4 (306.1) (269.5) (296.0) 3.52 3.22 3.29 (3.56) (3.26) (3.32) 0.635 0.623 (0.623) (0.615) -0°292 -0.302 -0.339 (-0.354) (-0.386) (-0.381) 0.0 0.0 -0.24 -0.1 to -0.4 -0.29 0.0 0.0 0.0 0.0 0.0 0.0 -0.26 -0.23 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.21 0.310 0.339 0.345 (0.324) (0.346) (0.354) | -0.239 -0.246 -0.264 (-0.252) (-0.269) (-0.272) -0.91 0.0 0.60 ((176.5) (162.1) {177.1) 0.573 (0.578) VA i 40Ca 3.60 3.32 3.35 358.1 308.5 3 4 3 . 1 ~ (353.4) (302.8) (339.1)~ (3.60) (3.31) (3.35) On the basis of extensive investigation an appropriate interpolation procedure is used to obtain a near optimum set of oscillator constants for the full Hartree-Fock variational procedure. The Hartree-Fock program includes the possibility of both axially symmetric and non-axially symmetric deformations. A two-body Coulomb interaction is included in all the calculations. A suitable one body density dependent spin-orbit interaction [14,15] for deformed nuclei of the form VSO(R) = 8 v F s • (Vp ( R ) × k ) , (5a) w h e r e s i s tile s p i n and k i s i v , h a s b e e n u s e d in t h e s e c a l c u l a t i o n s . In t h e l i m i t of s p h e r i c a l s y m m e t r y eq. (5a) r e d u c e s to Vso(R) do = 8 v F 1 d ~ 1. s (5b) c o n s i s t e n t w i t h N e g e l e [16] and V a u t h e r i n . T h e v a l u e of F = 6.0 MeV. fro5, d e d u c e d by N e g e l e f o r t h e c a l c i u m r e g i o n , h a s b e e n u s e d (in p r i n ciple the strength should be valid for all nuclei). C a l c u l a t i o n s u s i n g t h e u s u a l s h e l l m o d e l /. s f o r m h a v e a l s o b e e n p e r f o r m e d but g e n e r a l l y (5) seems superior. The results for single particle 0.0 0.0 0.0 l s p l i t t i n g s u s i n g (5) d e p e n d r a t h e r s e n s i t i v e l y on t h e e q u i l i b r i u m r m s r a d i u s of t h e n u c l e u s a n d t h u s s p i n - o r b i t s p l i t t i n g s a r e e x a g e r a t e d f o r the MDI. T h i s s h o u l d be r e m e m b e r e d w h e n c o m p a r ing the splitting between occupied and non-occupied levels for the different interactions. T h e m a i n a p p r o x i m a t i o n m a d e in the c a l c u l a t i o n s i s to r e p l a c e t h e t r u e d e n s i t y f o r t h e s y s t e m at a n y p a r t of t h e c a l c u l a t i o n by a s i n g l e c y l i n d r i c a l g a u s s i a n r e p r e s e n t a t i o n of the d e n sity which has the same z 2 and r2 as the true d e n s i t y a n d w h i c h i s n o r m a l i z e d to g i v e the d e s i r e d n u m b e r of p a r t i c l e s . O t h e r w i s e t h e d e n s i t y is treated self-consistently until density convergence is obtained. Previous investigations have shown that this is a fairly reliable approxim a t i o n f o r a l l but the a s y m m e t r i c ( o r t r i a x i a l ) s t a t e s , e.g. the 4 p - 4 h 1 6 0 e x c i t e d s t a t e , 2 4 M g a n d 32S, w h o s e a s y m m e t r y m a y b e g r e a t l y e x a g gerated. However, there is some compensating cancellation even for these cases since the app r o x i m a t i o n l e a d s to too h i g h a d e n s i t y i n s o m e p a r t s of t h e n u c l e u s a n d too low i n o t h e r p a r t s . In any e v e n t t h e u s e of t h e s a m e a p p r o x i m a t i o n f o r a l l i n t e r a c t i o n s i s s u f f i e n t l y c o n s i s t e n t to a l l o w q u a l i t a t i v e s t a t e m e n t s to be m a d e w i t h some confidence. V o l u m e 36B, n u m b e r 1 PHYSICS LETTERS 9 A u g u s t 1971 Table 2 Some s i n g l e p a r t i c l e e n e r g i e s of n u c l e i f o r d i f f e r e n t i n t e r a c t i o n s . T h e f i r s t e n e r g y is the p r o t o n s i n g l e p a r t i c l e st'~te and the s e c o n d e n e r g y is the n e u t r o n s i n g l e p a r t i c l e s t a t e . F o r the s p h e r i c a l 1 6 0 g r o u n d s t a t e the s i n g l e p a r t i c l e st:ires a r e the l S l / 2 , l p l / 2 and l d 5 / 2 s t a t e s . F o r 40Ca the s t a t e s a r e the l s l / 2 , 2 S l / 2 and l f 7 / 2 s t a t e s . ] Lowest single particle energy (MeV) E n e r g y of l a s t o c c u p i e d s t a t e ] E n e r g y of f i r s t u n o c c u p i e d s t a t e (MeV) (MeV) VA MDI MDIK VA MDI MDIK VA MDI MD]K -39.9 -43.0 -45.8 -49.4 -46.7 -50.2 -17.3 -20.3 -15.2 -1~.6 -16.6 -19.8 - 2.1 - 5.5 - 3.1 7.0 + 0.4 - 3.5 -35.1 -37.9 -40.6 -43.9 -36.9 -39.9 -14.8 -17.5 -17.3 -20.3 -17.6 -20.4 - 5.7 - 8.3 - 4.8 8.3 - 5.9 - 9.2 20Ne -41.0 -44.8 -46.9 -51.2 -47.7 -51.9 -15.6 -19.1 -17.3 -21.2 -16.9 -20.6 - 6.8 -10.6 7.3 -11.6 - 7.7 -11.8 2-tMg -43.4 -47.8 -48.6 -53.6 -49.9 -54.8 -14.9 -19.J -15.0 -19.6 -15.4 -19,7 - '7.3 -11.7 7.2 -12.3 - 7,1 -11,8 2~3Si Oblate s t a t e -45.2 -50.3 -49.8 -55.6 -52.1 -57.8 -16.2 -20.8 -15.9 -21.1 -16.5 -21,6 - ~.6 -12.7 5.6 -11.3 - 6.8 -12.3 2SSi Prolate state 32 S Oblate s t a t e 32 s Prolate state -45.5 -50.6 -50.7 -56.5 -52.6 -58.3 -14.1 -12.2 -1 3 .4 - 8.5 - 8.2 - -18.9 -17.5 -18.5 -13.6 -I3,8 -13.2 -47.6 -53,3 -51.5 -58.0 -54.'7 -61.1 -15.2 -20.6 -13.6 -19.5 -1,t.5 -2003 - 8.7 -14,3 6.2 -12,5 - 7.5 -13.6 -47.4 -53.1 -51.6 -58.1 -54.4 -6o.8 -15.2 -20.5 -13.6 -19.6 -14.4 -20.2 - 8.7 -1-i.4 - 5.9 -12,3 - 6.9 -13.2 36Ar -49.3 -55.7 -52.6 -59.8 -57.0 -64.2 -15.3 -21.3 -11.9 -1~.4 -14.0 -20.4 - 7.5 -13.8 - 7.5 - 14.3 - 6.5 -13.1 40Ca -58.1 -50.9 -54.3 -59.1 i -15.8 -22.6 -11.5 -18.8 -15.0 -22.1 - 5.8 -12.7 - 6.1 -13.6 - 4.0 -11.5 Interaction 160 Ground s t a t e 10 O 4p-4h state -62.3 -67.1 j T h e r e s u l t s o f t h e s e c a l c u l a t i o n s a r e s h o w n in two tables. Table 1 gives the Hartree-Fock binding energies and nuclear shapes of the nuclei studied for the different forces. The charge radii, corrected for finite proton size, are consistently about 0.1 fm larger than the nuclear radii, and Q JR 2 i s v e r y s i m i l a r to t h e n u c l e a r v a l u e . T h e numbers in parenthesis are the corresponding MINDET, single determinant, values. Table 2 lists some of the single particle properties of the different nuclei. The main conclusions are that the MDI gives systematically smaller values for the rms radii a n d i n t h e c a s e of 1 6 0 a n d 4 0 C a , w i t h e x p e r i m e n t a l v a l u e s of 2 . 7 0 f m a n d 3 . 4 9 f m r e s p e c t i v e l y , t h e s e r a d i i a r e t o o s m a l l by 5 - 10%. T h e finite-range V A g i v e s r a d i i w h i c h a r e a t m o s t 4% too large. The MDIK radii are larger than for t h e M D I a n d i n f a c t s o m e w h a t b e t t e r t h a n f o r V A. T h e b i n d i n g e n e r g i e s f o r VA a n d M D I K a r e b e t t e r than for the MDI which are too small. (Q/R 2 and The MDI and MDIK deformations ASYM) are systematically l a r g e r t h a n f o r VA, whose value for 20Ne compares well with other calculations [17-20]. Despite many recent innovations including the reorientation effect in 7.8 coulomb excitation [21], experimental data does not pin down this quadrupole moment very accurately. Recent work, summarized in ref. [22], i n d i c a t e s a q u a d r u p o l e m o m e n t of b e t w e e n 0 . 4 0 and 0.84 barn*, which, assuming R = 2.8 fm corresponds to Q+R 2 i n t h e r a n g e 0 . 2 6 to 0 . 5 4 . In a d d i t i o n , S t a m p ' s e s t i m a t e of Q R 2 b a s e d o n experimental d a t a i s 0 . 8 9 [17]. H e n c e a l t h o u g h MDI and MDIK do have somewhat larger deform a t i o n s t h a n VA t h e y a r e n o t i n s e r i o u s c o n f l i c t with experiment. The asymmetry of (4p-4h) 160 and 24Mg do conflict with other calculations using a cartesian basis [18], but we do not take our values seriously for these cases. The MDI and MDIK single particle energies are systematically d e e p e r t h a n VA b u t t h e v a l u e s are not unreasonable. The gap between occupied a n d u n o c c u p i e d l e v e l s in 1 6 0 a n d 4 0 C a i s s m a l l e r for the MDI than for the MDIK but this is mainly a t t r i b u t a b l e to t h e s m a l l e r r a d i i a n d c o n s e q u e n t l y stronger resulting spin-orbit interaction. It i s i n t e r e s t i n g to o b s e r v e t h a t a l l t h r e e * The quadrupole m o m e n t s quoted h e r e a r e < ~ i ( 3z2 - r ~ ) ) . and d i f f e r by a f a c t o r of A with t h o s e in table 1, Volume 36B, n u m b e r 1 PHYSICS f o r c e s p r o d u c e m o r e b i n d i n g in t h e o b l a t e s t a t e of 28Si t h a n i n t h e p r o l a t e s t a t e . T h e e n e r g y d i f f e r e n c e b e t w e e n the two i s a v e r y s e n s i t i v e f u n c t i o n of t h e s p i n - o r b i t s t r e n g t h , a n d t h e o b l a t e state in particular has both energy and deformat i o n w h i c h i s m o s t s e n s i t i v e to t h e s p i n - o r b i t s t r e n g t h . ( T h i s p o i n t h a s a l s o b e e n n o t e d by R i p k a [23]. ) In f a c t i n t h e a b s e n c e of s p i n - o r b i t , o r w i t h j u s t a p u r e / . s w i t h the s t r e n g t h to s p l i t t h e p - s t a t e s i n 1 6 0 c o r r e c t l y , the p r o l a t e s t a t e c o n t a i n s m o r e b i n d i n g [13]. A s i m i l a r f e a t u r e i s e v i d e n t in 32S w h e r e a slightly triaxial, but predominantly oblate configuration has energy nearly degenerate with an axially-symmetric prolate solution. The singlep a r t i c l e e n e r g i e s of e a c h s o l u t i o n a r e v e r y s i m i l a r , but t h e y a p p e a r n o t to be t h e s a m e i n t r i n s i c s t a t e b e c a u s e of t h e i r v e r y d i f f e r e n t d e f o r m a t i o n s . H o w e v e r , o t h e r a u t h o r s [18] c o n s i d e r , w i t h o u t e x p l a n a t i o n , t h e t r i a x i a l s o l u t i o n to b e s t d e s c r i b e t h e g r o u n d s t a t e of 32S. R i p k a [23] h a s s t u d i e d t h e e f f e c t of s p i n o r b i t ( s i m p l e / . s ) s t r e n g t h on t h e e n e r g i e s of v a r i o u s i n t r i n s i c c o n f i g u r a t i o n s . T h e m a i n c o n c l u s i o n to be d r a w n f r o m t h i s w o r k i s t h a t t h e MDI i s p r o b a b l y u n s u i t a b l e f o r variational calculations in heavy nuclei since it w o u l d l e a d to too s m a l l n u c l e i a n d too l a r g e d e formations. Smaller deformations can probably be a c h i e v e d by r e d u c i n g t h e i m p o r t a n c e of t h e p2 t e r m s a n d a t t r i b u t i n g m o r e of t h e s a t u r a t i o n to d e n s i t y - d e p e n d e n t t e r m s . H o w e v e r , t h e i m p r o v e d r e s u l t s f o r t h e MDIK, w h i c h i s by no means the optimum delta function interaction, i n d i c a t e s t h a t a f o r c e of t h i s t y p e w i t h a p r o p e r compressibility K should be suitable for calculat i o n s of the p r o p e r t i e s of h e a v y n u c l e i , if i n a d dition the force contains spin and isospin exchange mixtures which will allow for correct symmetry energies. The symmetry energy does n o t i n f l u e n c e t h e p r o p e r t i e s of t h e N = Z n u c l e i considered in this work, but would play an ext r e m e l y i m p o r t a n t r o l e i n t h e p r o p e r t i e s of h e a v y nuclei with large neutron excess. LETTERS 9 August 1971 T h e a u t h o r s w o u l d l i k e to t h a n k M r . R o b i n Griffin, who determined the parameters for the MDIK a n d D r . H. C. P a u l i f o r m a n y s t i m u l a t i n g discussions. References [1] R. Griffin, C. Ko, It. C. Pauli and A. B. Voikov. work in p r o c e s s . [2] D. Vautherin and D. M. Brink. Phys. L e t t e r s 32B (1970) 149; D. Vautherin, M. V6n6roni and D. M. Brink, Phys. L e t t e r s 33B (1970) 381. [3] T . H . R . Skyrme, Phil. Mag. 1 (1956) 1043. Nucl. Phys. 6 (1959) 615. ]4] S. A. Moszkowski, Phys. Rev. C 2 (1970) 402. [5] T . T . S . Kuo and G . E . B r o w n , Phys. L e t t e r s 18 (1965) 54. [6] R. K. Bhaduri and C. S. Warke, Phys. Rev. L e t t e r s 20 (1968) 1379. [7] A.B. Vo[kov, Nucl. Phys. A141 (1970) 337. [8] T. H. Ho and A. B. VoIkov, Phys. L e t t e r s 31B (1970) 259. [9] A. B. Volkov, unpublished. [10] D.J, Hughes, Ph.D. Thesis. McMaster University (1970), unpublished. [11] M. R. Manning, Ph.D. Thesis, McMaster University (1967), unpublished. [12] K. H. Lassey, unpublished. [13] M. R. Manning and A.B. Volkov. Phys. L e t t e r s 26B (1967) 60. [14] W.I. van Hij and C . T . H e s s , Nucl. Phys. A142 (1970) 72. [15] R. R. Scheerbaum, Ph. D. Thesis, Cornell University (1969), unpublished. [16] J . W . Negele, Phys. Rev. C 1 (1970) 1260. [17] A . P . Stamp, Phys. L e t t e r s 33B (1970) 257. [18] J. Zofka and G. Ripka, Phys. L e t t e r s 34B (1971) 10; G. Ripka, p r e p r i n t (from "International Centre for T h e o r e t i c a l P h y s i c s " . to be published in the Proe. "Nuclear Structure Symposium of the Thousand Lakes", Jyvaskyia. Finland, 1970). [19] D. G. Vautherin, private communication. [20J S . J . K r e i g e r , Phys. Rev. C 1 (1970) 76. [21] K. Nakai, F.S. Stephens and R. M. Diamond, Nucl. Phys. A 1 5 0 (1970) 114. [22] H. Rebel, G.W. Schweiner, J. Specht, G. Schatz, R. LBhken, D. Habs, G. Hauser and H. KleweNebenius, Phys. Rev. L e t t e r s 26 (1971) 1190. [23] G. Ripka, Advances in Nuclear Physics, Vol. I (Plenum P r e s s , New York, 1968).