Pairing forces with neutrons and protons

Volume 19, number 4

PHYSICS LETTERS

E n e r g y L a b o r a t o r y , the continuing i n t e r e s t and a d v i c e of Dr. R. C. Hanna in t h i s w o r k and the a s s i s t a n c e of E. A. M c C l a t c h i e and F. G. Kingston d u r i n g the e x p e r i m e n t a l runs. One of us (J. H. P. C. M.) w i s h e d to acknowledge the f i n a n c i a l a s s i s t a n c e given by the M i n i s t r y of E d u c a t i o n for Northern Ireland.

2, W. Grffbler, W. Haeberli and P. Extermarm, Proc. 3.

4.

5.

6.

Refe~'ences 1. L. Rosen, J . E . Brolley and L. Stewart, Phys. Rev. 121 (1961) 1423.

7. *

PAIRING

FORCES

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1 November 1965

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WITH

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Intern. Conf. on Polarization phenomena of nucleons, Karlsruhe (1965). H.E.Conzett, G.Igo and W.J.Knox, Phys.Rev. Letters 12 (1964) 222. S.J.Hall, W.R.Gibson, A.R.Johsten, R.J.Grtffiths, E. A. McClatchie and K. M. Knight, Compt. Rend. du Congr~s Intern. de Physique nucl~aire, Paris (1964) p.219. H. E. Conzett, H.S. Goldberg, E. Shield, R.J. Slobodrian and S.Yamabe, Physics Letters 11 (1964) 68. S.J.Hall, A.R.Johnston and R.J.Grtffiths, Physics Letters 14 (1965) 212. K. L. Kowalski and D. Feldman, Phys. Rev. 130 (1963) 276.

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NEUTRONS

AND

PROTONS

J. P. E L L I O T T and D. A. L E A School of Physical Sciences, University of Sussex, Brighton, England Received 25 September 1965

The p r o b l e m of extending the s i m p l e B. C. S. t r e a t m e n t of p a i r i n g between l i k e p a r t i c l e s to the fully c h a r g e - i n d e p e n d e n t p a i r i n g f o r c e b e tween nn, pp and np p a i r s h a s i n t e r e s t e d m a n y a u t h o r s [ i - 5 ] . In t h i s note we show how a s o l u tion m a y be d e d u c e d f r o m the u s u a l B. C. S. s o l u t i o n s f o r like p a r t i c l e s by an i s o s p i n p r o j e c t i o n . The p r o c e d u r e i s s i m i l a r to the c o n s t r u c t i o n of r o t a t i o n a l wave functions b y an a n g u l a r m o m e n t u m p r o j e c t i o n f r o m a s i m p l e i n t r i n s i c function. In c a l c u l a t i n g p a i r i n g f o r c e e n e r g i e s , the p r o j e c t i o n m a y be t r e a t e d f o r m a l l y , without any e x p l i c i t c a l c u l a t i o n s . A c o m p a r i s o n with s o m e e x a c t s o l u t i o n s d e r i v e d f o r a s i m p l e s y s t e m by Hecht [6] i n d i c a t e s that the a c c u r a c y of the m e t h o d i s s i m i l a r to that of the u s u a l B. C. S. procedure. We w r i t e the c h a r g e - i n d e p e n d e n t p a i r i n g Hamfltonian a s

t r a n s f o r m a t i o n s s e p a r a t e l y f o r n e u t r o n s and p r o t o n s to p r o v i d e s o l u t i o n s 9 - and 9 - f o r m e a n n u m b e r s n n and np of n e u t r o n s l a n d p r o t o n s . The p r o d u c t ~p = ~Pn~p i s then an eigenfunction of H ~1111 + H__, within the u s u a l B " C " S " a p p r o x i m a ~.l~ tion. It m a y f u r t h e r be shown that the o p e r a t o r I-In P - ½ G ( T 2 - T 2o - ½~), when ~ i s the n u m b e r o p e r a t o r , p o s s e s s e s no " d a n g e r o u s t e r m s " (i.e. of two q - p c r e a t i o n type) and that even the t e r m s which c r e a t e f o u r q - p , o r d e s t r o y one and c r e a t e t h r e e q - p , v a n i s h in the d e g e n e r a t e l i m i t . N e g l e c t i n g t h e s e f o u r q - p o p e r a t o r s , in the s p i r i t of the B. C. S. method, we thus have Hnu ~ ½ G × x ( T 2 - T 2 - ½h). If we now define the i s ~ s u i n p r o j e c t i o ~ q~T = P T ~ , w h e r e P T i s a p r o j e c t i o n o p e r a t o r , we have H~ T = HPT~

= PTH~

~ P T { E n + Ep + ½G(T 2 - T2o - ½ n ) } q ~

(1)

H = Hnn + H p p ÷ Hnp En + Ep + ~ G

w h e r e , in the u s u a l notation,

(j+j' ÷ m + m ' +l+l')

~ (-z)

_

Here, E n and E D are the usual B. C. S. energies

Hnn = j ~ m e j a J m a j m +

-¼c~

-

%m

We f i r s t i n t r o d u c e the u s u a l q u a s i - p a r t i c l e (q-p)

f o r n e u t r o n s a n d p r o t o n s s e p a r a t e l y . In d e r i v i n g (1) we have n e g l e c t e d the f o u r q - p t e r m s and t a k e n the m e a n v a l u e s f o r the n u m b e r o p e r a t o r s and T O = ½(nn - riD)" T h i s l a t t e r a p p r o x i m a t i o n m a y be a v o i d e d by inaking a f o r m a l p r o j e c t i o n with r e s p e c t to the n u m b e r of p a r t i c l e s . With 291

Volume 19, number 4

PHYSICS LETTERS

t h i s m o d i f i c a t i o n , the e n e r g y (1) a g r e e s with the w e l l - k n o w n ex act a n s w e r in the c a s e of d e g e n e r ate s i n g l e - p a r t i c l e l e v e l s . In the d e g e n e r a t e c a s e , the e n e r g y is then independent of M T = ½(nn - rip), as it should be f o r a c h a r g e - i n d e p e n d e n t potential, but in the nond e g e n e r a t e c a s e , the a p p r o x i m a t i o n s m a d e in d e r i v i n g (1) d e s t r o y this p r o p e r t y . T h e r e is, t h e r e f o r e , a f r e e d o m in the c h o i c e of M T which we ma y exploit. The c h o i c e M T = T, i.e. m a x i m u m M T f o r given T, a p p e a r s to give the g r e a t e s t a c c u r a c y s i n c e it tends to e l i m i n a t e f r o m the v a c u u m any c o m p o n e n t s with T l e s s than that r e q u i r e d and a l s o e n s u r e s that the e x a c t a n s w e r is obtained in the l i m i t of v a n i s h i n g p a i r ing f o r c e . C o n s i d e r a nucleus of m a s s A in which we 1 suppose that an equal n u m b e r ~A o of n e u t r o n s and p r o t o n s f o r m an i n e r t c o r e of c l o s e d s h e l l s while c h a r g e - i n d e p e n d e n t p a i r i n g f o r c e s a r e o p e r a t i v e among the r e m a i n i n g n = A - A o nuc l e o n s when they l i e in a s p e c i f i e d s e t of s i n g l e p a r t i c l e l e v e l s . To find the low l e v e l s f o r given 1 1 T we thus c h o o s e n n = T + ~n, np = - T + ~ n and then s o l v e the s e p a r a t e n e u t r o n and p r o t o n gap e qua t i o n s to find the e n e r g i e s E n and Ep. The tota l e n e r g y f r o m (1) is then E n + Ep + ½G(T-½n). It is i m p o r t a n t to notice that, if n n f u r n s out to be even, then En will be the v a c u u m e n e r g y and ~n will have J = 0 while, if n n is odd, E n will contain a q - p e n e r g y and ~n will have the c o r r e sponding value f o r J. Since the e n e r g y g e n e r a l l y i n c r e a s e s with T, the ground s t at e of a n u c le u s with g iv e n N and will g e n e r a l l y have the l o w e s t p o s s i b l e v a l u e f o r T, n a m e l y T = ½(N - Z) and our m e t h o d then i m p l i e s the c h o i c e n n = N - ~Ao, 1 np = Z - ~A 1 o. In o t h e r w o r d s n n and np a r e p r e c i s e l y the n u m b e r s of n e u t r o n s o r p r o t o n s beyond the c o r e , although f o r h i g h e r v a l u e s of T in the s a m e n u c l e u s , diff e r e r e n t v a l u e s f o r n n and np would be needed. F o r an e v e n - e v e n n u c le u s t h e r e f o r e , the ground s t a t e is obtained by taking the v a c u u m f o r both ~on and q~p and p r o j e c t i n g T= ½ ( N - Z). F o r an odd nucl eu s e i t h e r q~n or ~p m u s t contain a q - p

292

1 November 1965

ex ci t at i o n , depending on w h et h er N or Z is odd. F o r an odd-odd nucleus t h e r e is a c o m p e t i t i o n between the s t a t e s with the s m a l l e s t v al u e T = = ½(N - Z) which d e m a n d s a q - p e x c i t a t i o n in both q~n and ~D and the next s m a l l e s t value T = = ~ ( N - Z) + 1 which has the v a c u u m en er g y . Although t h e i r e a r l i e r c a l c u l a t i o n s w e r e conc e r n e d only with s e m i - c l o s e d shell nuclei, the subsequent work of K i s s l i n g e r and S o r e n s o n [7] c o n s i d e r s m o r e g e n e r a l nuclei. They u se a p a i r i n g - p l u s - q u a d r u p o l e H am i l t o n i an but allow p a i r i n g only between like p a r t i c l e s . We have not thoroughly i n v e s t i g a t e d the effect of making t h e i r p a i r i n g f o r c e p r o p e r l y c h a r g e - i n d e p e n d e n t by the method given above but it does a p p e a r that many of t h e i r c o n c l u s i o n s will be unaffected by the change. F o r e x a m p l e our method shows that f o r an odd n u cl eu s the s p e c t r u m is s i m p l y the q - p s p e c t r u m f o r the odd group, (neutrons o r p r o tons). The e x p r e s s i o n s u s e d by K i s s l i n g e r and S o r e n s o n f o r the o d d - e v e n m a s s d i f f e r e n c e s a r e a l s o obtained f r o m (1), the additional T - d e p e n dent t e r m s c a n c e l l i n g out. F r o m a p h y s i c a l point of view the i n cl u si o n of n e u t r o n - p r o t o n p a i r i n g , d e s c r i b e d h e r e , is only p a r t of the s t o r y b e c a u s e , even f o r a s h o r t r an g e f o r c e , t h e r e is s t r o n g i n t e r a c t i o n a l s o in the two-body s t a t e s with T = 0 and odd J, which has not b e e n included in our method. The method is a l s o applicable to the p r o b l e m of p a i r i n g in L - S coupling but although this r e p r e s e n t s a much b e t t e r t r e a t m e n t of the T = 0 i n t e r a c t i o n s , it s u f f e r s f r o m the d i s a d v a n t a g e i n h e r e n t in all L - S coupling a p p r o a c h e s , that it does not include the i m p o r t a n t s p i n - o r b i t f o r c e .

References

I. 2. 3. 4. 5.

V.G. Soloviev, Nuel. Phys. 18 (1960) 161. B. Br~mond and J. G. Valatin, Nucl. Phys. 41 (1963} 40. B.H. Flowers and M. Vuji~ic, Nucl. Phys. 49 (1963) 586. A.Goswami, Nucl. Phys.60 (1964) 228. P.Camiz, A.Covello and M.Jean, Nuovo Cimento 36 ~1965) 6. K.T.Hecht, Phys. Rev. 139 (1965) 794. 7. L.S. Kisslinger and R. A. Sorenson, Rev. Mod. Phys. 35 (1963) 853.