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A comment on shell corrections to nuclear binding energies
Number 38B, number 3
A COMMENT
PHYSICS LETTERS
ON S H E L L
CORRECTIONS
TO NUCLEAR
7 February 1972
BINDING
ENERGIES
A. BOTTINO and G. CIOCCHETTI Istituto di Fisica dell' UniversitY, Torino, Italy Istituto Nazionale di Fisica Nucleate, Sezione di Torino Received 26 November 1971
Ane[ementary method to correct the semi-elassical evaluation of nuclear in presented and compared with other approaches.
Th e s m a l l d i f f e r e n c e s b e tw e e n the e x p e r i m e n t a l v a l u e s of n u c l e a r m a s s e s (or, e q u i v alently, n u c l e a r binding e n e r g i e s ) and t h e i r s t a t i s t i c a l evaluation by m e a n s of s t a n d a r d s e m i - e m p i r i c a l mass formulae are apparently due to quantum m e c h a n i c a l e f f e c t s . As a m a t t e r of fact, the s t a t i s t i c a l a p p r o a c h , which is e s s e n t i a l l y s e m i - c l a s s i c a l in c h a r a c t e r , cannot account for the d i s c r e t e n e s s p r o p e r t y of the e n e r g y s p e c t r a of bound s y s t e m s which c a u s e s s h e l l s t r u c t u r e in n u c l e i , but r a t h e r d e s c r i b e s n u c l e a r e n e r g y s p e c t r a as s m o o t h functions. Since the evaluation of n u c l e a r m a s s e s with a m i c r o s c o p i c t h e o r y is a v e r y c o m p l i c a t e d p r o b l e m , a convenient way to p r o c e e d s i m p l y c o n s i s t s in c o r r e c t i n g the s e m i - c l a s s i c a l v a l u e s for the quantum (shell) e f f e c ts [1,2]. It is p o s s i b l e to p r o v e that the c o r r e c t i o n in the tota l n u c l e a r e n e r g y is a p p r o x i m a t e l y given by 5E= E-
Esc
w h e r e E is the sum o v e r the quantum s i n g l e p a r t i c l e e n e r g i e s of a s e l f - c o n s i s t e n t p o t e n t i a l E = , ~ Wn¢n(Wn is the weight f a c t o r of the d e g e n e r a t e n-th l e v e l of e n e r g y en) a n d E s c is the c o r r e s p o n d i n g s e m i - c l a s s i c a l quantity. The m a x i m u m v a l u e of the index n is obviously d e t e r m i n e d by the condition N = ~ n COn, w h e r e N is the total n u m b e r of p a r t i c l e s (for s i m p l i c ity we s h al l only c o n s i d e r one kind of nucleons in what follows). As is c l e a r a nice p r o p e r t y of f o r m u l a (1) is the a b s e n c e of any r e s i d u a l interaction. The p r o b l e m of evaluating 6E is then c o m p l e t e l y r e d u c e d to a p r o p e r c a l c u l a t i o n of Esc. To c a s t the p r o b l e m in a m o r e p r e c i s e way let us r e w r i t e the e x p r e s s i o n of N and E as follows
140
(1)
~max N : f g(¢)d¢, 0
masses for shell effects
£max E : f Eg(¢)dc 0
(2)
w h e r e the density function g(¢ ) ~ d N / d e is g i v en by g(e) = ~n ¢OnS(e - ¢ n )
(3)
and E m a x is any r e a l v al u e g r e a t e r than the F e r m i energy. It is now a p p a r e n t that f o r m u l a e (2) a r e v e r y convenient for a s e m i - c l a s s i c a l c a l c u l a t i o n of N and E; in this c a s e one should u s e a s m o o t h density function gsc(~) to be obtained f r o m g(E) by m e a n s of s o m e s m e a r i n g p r o c e d u r e of the quantum l e v e l s o v e r the energy spectrum. Out of the d i f f er en t s u g g e s t e d a p p r o a c h e s to obtain gsc(E), we wish to c o n s i d e r the following two: 1) Strutinsky method [1-4]. This is the m o s t c o m m o n l y u s e d method; it c o n s i s t s in s m e a r i n g e a c h d el t a d i s t r i b u t i o n of e x p r e s s i o n (3) into a g a u s s i a n function and then sm o o t h i n g out the s h o r t - r a n g e ( r a n g e of the o r d e r of the e n e r g y l e v e l spacing) o s c i l l a t i o n s of the r e s u l t i n g d i s t r i b u t i o n by p r o p e r l y multiplying each g a u s s i a n t e r m by a p o l y n o m i a l f act o r . This a p p r o a c h although r a t h e r e m p i r i c a l p r o v e s v e r y useful for n u m e r i c a l ev al u at i o n s of 5E and has been e x t e n s i v e l y e m p l o y e d for the study of heavy nuclei [1, 3]. 2) B h a d u r i - R o s s method. A c o m p l e t e l y d i f f er en t a p p r o a c h which m ak es u s e of the concept of the n u c l e a r p a r t i t i o n function Z(/3) = 2] n con exp (-/3E n)has been r e c e n t l y p r e s e n t e d by Bh ad u r i and Ross [5]. Since the density function g(a) is s i m p l y the i n v e r s e L a p l a c e t r a n s f o r m of the p a r t i t i o n function
Volume
38B,
number
1
3
PHYSICS
f
n
_oo
exp {ia(¢ - ¢n)}da
E
f
exp
exp(-~en))d ~
(4)
N :- f
_ioo
min
E
¢on
gsc(¢)de'
(5)
rain F r o m this e x p r e s s i o n it is s t r a i g t h f o r w a r d to identify, in the l i m i t (Ae)n ~ 0 (or equivalently -~ 0), gsc(e) with con/(Ae)n where n has to b e c o n s i d e r e d as a function of e
gsc(e) = %/(ae)n
£
egsc(E)de+ £1col
min
m i n = ½(el + e2)
(8)
w h e r e the s u b s c r i p t s 1,2 r e f e r to the f i r s t and the second quantum l e v e l s , r e s p e c t i v e l y . The c o n t r i b u t i o n to N and E due to the f i r s t level could be included in the i n t e g r a l by adding to gsc(E) a p r o p e r delta function. Let us apply now the p r e s e n t method to the potentials c o n s i d e r e d in ref. [5]. 1) Isotropic h a r m o n i c o s c i l l a t o r . Since the e n e r g i e s and the weight factors a r e given by e n = ~¢o(n+ 3/2), wn = (n+ 1)(n+ 2) our s e m i c l a s s i c a l density function is gsc(¢) = e2/(~w) 3 - 1/4~co
(9)
which coincides with that found in ref. [5] **. 2) Q u a s i - r o t a t i o n a l s i n g l e - p a r t i c l e s p e c t r u m . e n = A ( n + ½ ) 2, ¢On=2(2n+1 ) (A = c o n s t ) . In this c a s e our r e c i p e gives
g
max
(AE)n = • f
1972
max
gsc (£)dE+ col' E = f
E
they p r o p o s e to c a l c u l a t e the exact p a r t i t i o n function Z(fl) of the quantum s y s t e m , then to make a s e m i - c l a s s i c a l (t/~ 0) a p p r o x i m a t i o n Zsc(fl) of Z(fl) and finally to get gsc(e) u s i n g again eq. (4) *. By applying this p r o c e d u r e to s o m e p o t e n t i a l s of i n t e r e s t . o n e is able to obtain analytic e x p r e s s i o n s for gsc(E) which a r e in good a g r e e m e n t with the S t r u t i n s k y r e s u l t s . Two s h o r t c o m i n g s of this method should propably be pointed out: a) divergent t e r m s in gsc(~) can appear if one does not p r o p e r l y cut the s e m i c l a s s i c a l expansion of Z(/3); b) for many p o t e n t i a l s it can be very difficult to sum the s e r i e s defining Z(fl). In the p r e s e n t note we wish to p r e s e n t a v e r y s i m p l e - m i n d e d approach which n e v e r t h e l e s s enables one to r e p r o d u c e e s s e n t i a l l y the a n a l y t i c e x p r e s s i o n s of gsc(E) given by Bhaduri and Ross, in an e l e m e n t a r y way. In o r d e r to build up the function gsc(C), let u s r e w r i t e N as follows
N = ~n ( ~ n
£
max
+i~
=~
7 February
to the lowest quantum level, this state has to be taken into account s e p a r a t e l y . Then the final e x p r e s s i o n s for N and E a r e
+oo
g(E) : E ¢on ~
LETTERS
n : n(e)
(6)
This p r o c e d u r e can be r e f o r m u l a t e d as follows: i) one a t t r i b u t e s the ¢on d e g e n e r a t e s t a t e s to an energy i n t e r v a l (A~)n; ii) the h i s t o g r a m so obtained is i n t e r p o l a t e d by s i m p l y e x p r e s s i n g all q u a n t i t i e s in t e r m s of ~n and then letting En be a continous v a r i a b l e E. A convenient choise for (aE)n could be (AE)n = ~(en+ 1 - e n _ l )
(¢) = 2 / A . (10) sc This e x p r e s s i o n differs from that of Bhaduri Ross (eq. (10) of ref. [5]) by a t e r m p r o p o r t i o nal to a delta function 5(e); this d i s a g r e e m e n t s i m p l y r e f l e c t s the inadequacy of any method to t r e a t p r o p e r l y the d i s c r e t e s p e c t r u m in the neighbourhood of the lowest level and is n u m e r i c a l l y i n e s s e n t i a l for p r a c t i c a l n u c l e a r applications (N = 80 to 200). 3) Cubix box. Denoting by L the side of the box, the energy s p e c t r u m is given by e n=Tr2n2 / B , w h e r e B= 2ML 2 / y2, n 2 =n2 + n2y + n 2
Since f o r m u l a (6) cannot obviously be applied
and M is the nucleon m a s s . The m a i n difficulty in this case is to take into p r o p e r account the n u m b e r of d e g e n e r a t e states for each energy level. Since no exact function con e x i s t s , we have tentatively used throughout the whole energy s p e c t r u m a s t a t i s t i c a l f o r m u l a which is known to be c o r r e c t only for l a r g e n . More p r e c i s e l y , s i n c e the total n u m b e r of states with quantum n u m b e r s between 1 and n (n large) is given by [7]
* Analogous methods are used to evaluate nuclear level densities as fucntions of excitation energy and mass number [e. g. 6].
** The same result can be obtained with a ThomasFermi calculation [2].
g s c (e) = 2con/(en + 1 - e n - 1) n =n(E)"
(7)
141
Volume 38B, number 3
PHYSICS LETTERS
=~ },~ 3
C
(11)
n
we h a v e e m p l o y e d f o r con the e x p r e s s i o n
con = 2(C n - C n _ l )
(12)
w h e r e the f a c t o r 2 is due to the spin d e g e n e r a c y . We then obtain 1
3
£ s c (E) = 2u2 B ~ f ~ -
1
1
1
2~ B + ~ B ~ / ~ f ~ .
i) the p r e s e n t method though v e r y e l e m e n t a r y g i v es a c c u r a t e e v a l u a t i o n s of the s e m i - c l a s s i c a l density function as t e s t e d in the p r e v i o u s e x a m p l e s , w h e r e p a r t i c u l a r n a t u r e of the s e l f c o n s i s t e n t p o t e n t i a l s p e r m i t s finding analytic e x p r e s s i o n of gsc(e); ii) as a c o n s e q u e n c e , t h i s a p p r o a c h could p o s s i b l y p r o v e u s e f u l also f o r c a s e s in which a n u m e r i c a l a n a l y s i s is necessary.
(13)
This formula displays the same structure as the corresponding expression [5, eq. (9)], but the coefficients of the second and third terms are sightly different; this is not surprising in view of our approximate estimate of ¢on which affects higher order corrections in a semi-classical expansion. Nevertheless numerical calculations show that the difference between Esc values obtained by eq. (13) and the corresponding values computed with the Bhaduri-Ross formula are of the order of 10% throughout the range N = 80 to 200° We wish to conclude with two final r e m a r k s :
142
7 February 1972
References [1] V.M. Strutinsky, Nuel. Phys. A95 (1967) 420, and Nucl. Phys. A122 (1968) 1. [2] A.S. Tyapin, Soy. J. Nucl. Phys. 11 (1970) 53 and r e f e r e n c e s given therein. [3] S.G. N i l s s o n e t a l . , Nucl. Phys. A131 (1969) 1. [4] Wing-fai Lin, Phys. Rev. C2 (1970) 871. [5] R.K. Bhaduri and C.K. Ross, Phys. Rev. L e t t e r s 27 (1971) 606, [6] A. Bohr and B. Mottelson, Nuclear Structure, vol. 1 (W.A. Benjamin, Inc., New York, 1969) p.281. [7] M. Born, Atomic Physics (Blaekie and Son Limited, London, 1959) p.251.
A COMMENT
PHYSICS LETTERS
ON S H E L L
CORRECTIONS
TO NUCLEAR
7 February 1972
BINDING
ENERGIES
A. BOTTINO and G. CIOCCHETTI Istituto di Fisica dell' UniversitY, Torino, Italy Istituto Nazionale di Fisica Nucleate, Sezione di Torino Received 26 November 1971
Ane[ementary method to correct the semi-elassical evaluation of nuclear in presented and compared with other approaches.
Th e s m a l l d i f f e r e n c e s b e tw e e n the e x p e r i m e n t a l v a l u e s of n u c l e a r m a s s e s (or, e q u i v alently, n u c l e a r binding e n e r g i e s ) and t h e i r s t a t i s t i c a l evaluation by m e a n s of s t a n d a r d s e m i - e m p i r i c a l mass formulae are apparently due to quantum m e c h a n i c a l e f f e c t s . As a m a t t e r of fact, the s t a t i s t i c a l a p p r o a c h , which is e s s e n t i a l l y s e m i - c l a s s i c a l in c h a r a c t e r , cannot account for the d i s c r e t e n e s s p r o p e r t y of the e n e r g y s p e c t r a of bound s y s t e m s which c a u s e s s h e l l s t r u c t u r e in n u c l e i , but r a t h e r d e s c r i b e s n u c l e a r e n e r g y s p e c t r a as s m o o t h functions. Since the evaluation of n u c l e a r m a s s e s with a m i c r o s c o p i c t h e o r y is a v e r y c o m p l i c a t e d p r o b l e m , a convenient way to p r o c e e d s i m p l y c o n s i s t s in c o r r e c t i n g the s e m i - c l a s s i c a l v a l u e s for the quantum (shell) e f f e c ts [1,2]. It is p o s s i b l e to p r o v e that the c o r r e c t i o n in the tota l n u c l e a r e n e r g y is a p p r o x i m a t e l y given by 5E= E-
Esc
w h e r e E is the sum o v e r the quantum s i n g l e p a r t i c l e e n e r g i e s of a s e l f - c o n s i s t e n t p o t e n t i a l E = , ~ Wn¢n(Wn is the weight f a c t o r of the d e g e n e r a t e n-th l e v e l of e n e r g y en) a n d E s c is the c o r r e s p o n d i n g s e m i - c l a s s i c a l quantity. The m a x i m u m v a l u e of the index n is obviously d e t e r m i n e d by the condition N = ~ n COn, w h e r e N is the total n u m b e r of p a r t i c l e s (for s i m p l i c ity we s h al l only c o n s i d e r one kind of nucleons in what follows). As is c l e a r a nice p r o p e r t y of f o r m u l a (1) is the a b s e n c e of any r e s i d u a l interaction. The p r o b l e m of evaluating 6E is then c o m p l e t e l y r e d u c e d to a p r o p e r c a l c u l a t i o n of Esc. To c a s t the p r o b l e m in a m o r e p r e c i s e way let us r e w r i t e the e x p r e s s i o n of N and E as follows
140
(1)
~max N : f g(¢)d¢, 0
masses for shell effects
£max E : f Eg(¢)dc 0
(2)
w h e r e the density function g(¢ ) ~ d N / d e is g i v en by g(e) = ~n ¢OnS(e - ¢ n )
(3)
and E m a x is any r e a l v al u e g r e a t e r than the F e r m i energy. It is now a p p a r e n t that f o r m u l a e (2) a r e v e r y convenient for a s e m i - c l a s s i c a l c a l c u l a t i o n of N and E; in this c a s e one should u s e a s m o o t h density function gsc(~) to be obtained f r o m g(E) by m e a n s of s o m e s m e a r i n g p r o c e d u r e of the quantum l e v e l s o v e r the energy spectrum. Out of the d i f f er en t s u g g e s t e d a p p r o a c h e s to obtain gsc(E), we wish to c o n s i d e r the following two: 1) Strutinsky method [1-4]. This is the m o s t c o m m o n l y u s e d method; it c o n s i s t s in s m e a r i n g e a c h d el t a d i s t r i b u t i o n of e x p r e s s i o n (3) into a g a u s s i a n function and then sm o o t h i n g out the s h o r t - r a n g e ( r a n g e of the o r d e r of the e n e r g y l e v e l spacing) o s c i l l a t i o n s of the r e s u l t i n g d i s t r i b u t i o n by p r o p e r l y multiplying each g a u s s i a n t e r m by a p o l y n o m i a l f act o r . This a p p r o a c h although r a t h e r e m p i r i c a l p r o v e s v e r y useful for n u m e r i c a l ev al u at i o n s of 5E and has been e x t e n s i v e l y e m p l o y e d for the study of heavy nuclei [1, 3]. 2) B h a d u r i - R o s s method. A c o m p l e t e l y d i f f er en t a p p r o a c h which m ak es u s e of the concept of the n u c l e a r p a r t i t i o n function Z(/3) = 2] n con exp (-/3E n)has been r e c e n t l y p r e s e n t e d by Bh ad u r i and Ross [5]. Since the density function g(a) is s i m p l y the i n v e r s e L a p l a c e t r a n s f o r m of the p a r t i t i o n function
Volume
38B,
number
1
3
PHYSICS
f
n
_oo
exp {ia(¢ - ¢n)}da
E
f
exp
exp(-~en))d ~
(4)
N :- f
_ioo
min
E
¢on
gsc(¢)de'
(5)
rain F r o m this e x p r e s s i o n it is s t r a i g t h f o r w a r d to identify, in the l i m i t (Ae)n ~ 0 (or equivalently -~ 0), gsc(e) with con/(Ae)n where n has to b e c o n s i d e r e d as a function of e
gsc(e) = %/(ae)n
£
egsc(E)de+ £1col
min
m i n = ½(el + e2)
(8)
w h e r e the s u b s c r i p t s 1,2 r e f e r to the f i r s t and the second quantum l e v e l s , r e s p e c t i v e l y . The c o n t r i b u t i o n to N and E due to the f i r s t level could be included in the i n t e g r a l by adding to gsc(E) a p r o p e r delta function. Let us apply now the p r e s e n t method to the potentials c o n s i d e r e d in ref. [5]. 1) Isotropic h a r m o n i c o s c i l l a t o r . Since the e n e r g i e s and the weight factors a r e given by e n = ~¢o(n+ 3/2), wn = (n+ 1)(n+ 2) our s e m i c l a s s i c a l density function is gsc(¢) = e2/(~w) 3 - 1/4~co
(9)
which coincides with that found in ref. [5] **. 2) Q u a s i - r o t a t i o n a l s i n g l e - p a r t i c l e s p e c t r u m . e n = A ( n + ½ ) 2, ¢On=2(2n+1 ) (A = c o n s t ) . In this c a s e our r e c i p e gives
g
max
(AE)n = • f
1972
max
gsc (£)dE+ col' E = f
E
they p r o p o s e to c a l c u l a t e the exact p a r t i t i o n function Z(fl) of the quantum s y s t e m , then to make a s e m i - c l a s s i c a l (t/~ 0) a p p r o x i m a t i o n Zsc(fl) of Z(fl) and finally to get gsc(e) u s i n g again eq. (4) *. By applying this p r o c e d u r e to s o m e p o t e n t i a l s of i n t e r e s t . o n e is able to obtain analytic e x p r e s s i o n s for gsc(E) which a r e in good a g r e e m e n t with the S t r u t i n s k y r e s u l t s . Two s h o r t c o m i n g s of this method should propably be pointed out: a) divergent t e r m s in gsc(~) can appear if one does not p r o p e r l y cut the s e m i c l a s s i c a l expansion of Z(/3); b) for many p o t e n t i a l s it can be very difficult to sum the s e r i e s defining Z(fl). In the p r e s e n t note we wish to p r e s e n t a v e r y s i m p l e - m i n d e d approach which n e v e r t h e l e s s enables one to r e p r o d u c e e s s e n t i a l l y the a n a l y t i c e x p r e s s i o n s of gsc(E) given by Bhaduri and Ross, in an e l e m e n t a r y way. In o r d e r to build up the function gsc(C), let u s r e w r i t e N as follows
N = ~n ( ~ n
£
max
+i~
=~
7 February
to the lowest quantum level, this state has to be taken into account s e p a r a t e l y . Then the final e x p r e s s i o n s for N and E a r e
+oo
g(E) : E ¢on ~
LETTERS
n : n(e)
(6)
This p r o c e d u r e can be r e f o r m u l a t e d as follows: i) one a t t r i b u t e s the ¢on d e g e n e r a t e s t a t e s to an energy i n t e r v a l (A~)n; ii) the h i s t o g r a m so obtained is i n t e r p o l a t e d by s i m p l y e x p r e s s i n g all q u a n t i t i e s in t e r m s of ~n and then letting En be a continous v a r i a b l e E. A convenient choise for (aE)n could be (AE)n = ~(en+ 1 - e n _ l )
(¢) = 2 / A . (10) sc This e x p r e s s i o n differs from that of Bhaduri Ross (eq. (10) of ref. [5]) by a t e r m p r o p o r t i o nal to a delta function 5(e); this d i s a g r e e m e n t s i m p l y r e f l e c t s the inadequacy of any method to t r e a t p r o p e r l y the d i s c r e t e s p e c t r u m in the neighbourhood of the lowest level and is n u m e r i c a l l y i n e s s e n t i a l for p r a c t i c a l n u c l e a r applications (N = 80 to 200). 3) Cubix box. Denoting by L the side of the box, the energy s p e c t r u m is given by e n=Tr2n2 / B , w h e r e B= 2ML 2 / y2, n 2 =n2 + n2y + n 2
Since f o r m u l a (6) cannot obviously be applied
and M is the nucleon m a s s . The m a i n difficulty in this case is to take into p r o p e r account the n u m b e r of d e g e n e r a t e states for each energy level. Since no exact function con e x i s t s , we have tentatively used throughout the whole energy s p e c t r u m a s t a t i s t i c a l f o r m u l a which is known to be c o r r e c t only for l a r g e n . More p r e c i s e l y , s i n c e the total n u m b e r of states with quantum n u m b e r s between 1 and n (n large) is given by [7]
* Analogous methods are used to evaluate nuclear level densities as fucntions of excitation energy and mass number [e. g. 6].
** The same result can be obtained with a ThomasFermi calculation [2].
g s c (e) = 2con/(en + 1 - e n - 1) n =n(E)"
(7)
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Volume 38B, number 3
PHYSICS LETTERS
=~ },~ 3
C
(11)
n
we h a v e e m p l o y e d f o r con the e x p r e s s i o n
con = 2(C n - C n _ l )
(12)
w h e r e the f a c t o r 2 is due to the spin d e g e n e r a c y . We then obtain 1
3
£ s c (E) = 2u2 B ~ f ~ -
1
1
1
2~ B + ~ B ~ / ~ f ~ .
i) the p r e s e n t method though v e r y e l e m e n t a r y g i v es a c c u r a t e e v a l u a t i o n s of the s e m i - c l a s s i c a l density function as t e s t e d in the p r e v i o u s e x a m p l e s , w h e r e p a r t i c u l a r n a t u r e of the s e l f c o n s i s t e n t p o t e n t i a l s p e r m i t s finding analytic e x p r e s s i o n of gsc(e); ii) as a c o n s e q u e n c e , t h i s a p p r o a c h could p o s s i b l y p r o v e u s e f u l also f o r c a s e s in which a n u m e r i c a l a n a l y s i s is necessary.
(13)
This formula displays the same structure as the corresponding expression [5, eq. (9)], but the coefficients of the second and third terms are sightly different; this is not surprising in view of our approximate estimate of ¢on which affects higher order corrections in a semi-classical expansion. Nevertheless numerical calculations show that the difference between Esc values obtained by eq. (13) and the corresponding values computed with the Bhaduri-Ross formula are of the order of 10% throughout the range N = 80 to 200° We wish to conclude with two final r e m a r k s :
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7 February 1972
References [1] V.M. Strutinsky, Nuel. Phys. A95 (1967) 420, and Nucl. Phys. A122 (1968) 1. [2] A.S. Tyapin, Soy. J. Nucl. Phys. 11 (1970) 53 and r e f e r e n c e s given therein. [3] S.G. N i l s s o n e t a l . , Nucl. Phys. A131 (1969) 1. [4] Wing-fai Lin, Phys. Rev. C2 (1970) 871. [5] R.K. Bhaduri and C.K. Ross, Phys. Rev. L e t t e r s 27 (1971) 606, [6] A. Bohr and B. Mottelson, Nuclear Structure, vol. 1 (W.A. Benjamin, Inc., New York, 1969) p.281. [7] M. Born, Atomic Physics (Blaekie and Son Limited, London, 1959) p.251.