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A connection between the surface delta interaction and the pairing plus quadrupole model in spherical nuclei
Volume 21, number 1
A CONNECTION THE PAIRING
PHYSICS LETTERS
15 April 1966
BETWEEN THE SURFACE DELTA INTERACTION AND PLUS QUADRUPOLE M O D E L IN S P H E R I C A L NUCLEI J . TOUCHARD
Institut de Physique Nucl~aire , Division de Physique Th$orique, Orsay, France Received 10 March 1966
It is shown that the surface delta interaction in single closed shell spherical nuclei leads to a description which is very similar to the usual pairing plus quadrupole model if the recoupling terms are ignored.
It has b e e n suggested r e c e n t l y by G r e e n and Moszkowski [1] that the s u r f a c e delta i n t e r a c t i o n acting as a r e s i d u a l f o r c e between n u c l e o n s in open s h e l l s , could s i m u l a r e in a v e r y s i m p l e way both s h o r t r a n g e effects (e.g. p a i r i n g effects) and long r a n g e p r o p e r t i e s for J = 2 s t a t e s . It has been studied in m o r e detail by A r v i e u and Moszkowski [2], using the g e n e r a l i z e d q u a s i - s p i n f o r m a l i s m and by A r v i e u , P l a s t i n o and Moszkowski [3] for single-closed-shell nuclei. It is i n t e r e s t i n g to c o n s i d e r how we can e s t a b l i s h a connection in the q u a s i - p a r t i c l e s c h e m e , b e t w e e n the s u r f a c e delta i n t e r a c t i o n and the p a i r ing plus quadrupole model, as used for i n s t a n c e by K i s s l i n g e r and S o r e n s e n [4, 5]. The s u r f a c e delta i n t e r a c t i o n is defined as a c t ing only when n u c l e o n s a r e at the n u c l e a r s u r f a c e . In addition, it is s t r i c t l y a contact i n t e r a c t i o n . We complete the definition of the model by taking a constant amplitude u o for the r a d i a l functions at the n u c l e a r s u r f a c e * u n l ( R ) = Uo
•
The p l a u s i b i l i t y of such an idealized model is d i s c u s s e d in r e f . 2 . Our i n t e r a c t i o n is then: G 6(r l - R ) V12 = - (-R-~o)4
6(r2 - R) ~k
Ck(Wl)Ck(W2)
where Ck (°~l)Ck ( w 2 ) = P k ( c ° s O 1 2 ), and G is a coupling constant. C o n s i d e r i n g only identical nucleons in s u b s h e l l s J~n' Jb, e t c . , we r e p l a c e the q u a n t u m n u m b e r s a, la, Ja) by a single s y m b o l a . As a c o n s e q u e n c e of the above a s s u m p t i o n s ,
the g e n e r a l m a t r i x e l e m e n t Gj(abcd) = = r e d u c e s to:
Gj(abcd) = -G ~ (2k + 1)(abJ! P k l cdJ> , k i . e the Slater i n t e g r a l s a r e all the s a m e . and the state dependence of G j i s p u r e l y g e o m e t r i c a l . A s t r a i g h t f o r w a r d c a l c u l a t i o n shows that:
Go(aabb ) = - ½G (-) la+ lb ~t ~ , with ~ = ~ / ( 2 J a + l ) . T h e r e f o r e , the gap equation is the s a m e as for a p a i r i n g force of s t r e n g t h G. On the other hand, the s e l f - e n e r g y t e r m is s t a t e - i n d e p e n d e n t and can be included in the c h e m i c a l potential. Then the exact s u r f a c e delta i n t e r a c t i o n gap equation gives the s a m e r e s u l t as the usual p a i r i n g plus quadrupole model gap equation, where the s e l f - e n e r g y t e r m and the c o n t r i b u t i o n of the P2 f o r c e a r e r e m o v e d . If G is fixed, as in the p a i r i n g f o r c e model, by the binding e n e r g y d i f f e r e n c e s , the s u r f a c e delta i n t e r a c t i o n is completely d e t e r m i n e d . Looking now at the collective excited s t a t e s , we r e m e m b e r that in the p a i r i n g plus quadrupole model the f i r s t 2+ l e v e l s a r e given by the incomplete q u a s i - p a r t i c l e r a n d o m - p h a s e - a p p r o x i m a t i o n where 1) the c o n t r i b u t i o n of the p a i r i n g force is neglected and 2) the ecoupling t e r m s coming f r o m the P2 i n t e r a c t i o n a r e also neglected; this e n s u r e s the s e p a r a b i l i t y of the model and can be c o n s i d e r e d as a p a r t of the model itself [6]. The s i g n i f i c a n t p a r t of the m a t r i x e l e m e n t s of the q u a s i - p a r t i c l e r a n d o m - p h a s e - a p p r o x i m a t i o n have the following s e p a r a b l e f o r m :
• We choose here the phases of radial functions such thai asymptotic values of Unl all have the same sign. 85
Volume 21, number 1
PHYSICS LETTERS
(UaVb +VaUb) (- ) lah 2 (ab)
X (UcVd + VcUv~ (-)lch2(cd) ' ,/(1 + ~cd)
(2)
w h e r e ot stands f o r the two q u a s i - p a r t i c l e s s t a t e (a,b), u a and v a a r e the c o e f f i c i e n t s of the Bogoliubov t r a n s f o r m a t i o n *, and h2(ab) = (a]lC 2]]b).
c ~ =} X x a x~
(ra)u H e r e X a n d ( r ~ u a r e r e s p e c t i v e l y the P2 i n t e r a c tion s t r e n g t h and a s p e c i f i c m e a n - v a l u e of r 2, as in r e f . 4 . Now, with our choice of p h a s e s f o r the r a d i a l functions, all the x~ have the s a m e sign, and it is e a s y to v e r i f y that, inside a given s h e l l , t h e i r absolute value does not d e v i a t e a p p r e c i a b l y f r o m unity. 1 G' = vX. In o r d e r T h e r e f o r e , we have G~fl to c o m p a r e the s u r f a c e d e l ta i n t e r a c t i o n model to the p a i r i n g plus quadrupole model we take the r e coupling t e r m s out of the s u r f a c e delta i n t e r a c t i o n m a t r i x e l e m e n t s of the q u a s i - p a r t i c l e r a n d o m p h a s e - a p p r o x i m a t i o n , which l e a d s to an e x p r e s sion d i r e c t l y c o m p a r a b l e to the p a i r i n g plus quadrupole m o d el eq. (2). This s i m p l i f i e d s u r face delta i n t e r a c t i o n e x p r e s s i o n i s :
R a ft = -G (UaVb+ VaUb) (-)la h2(ab) `/(1 + 5ab)
×
× (UcVd + VcUd) (-)lch2(cd), ~/(1 + 5cd)
86
15 April 1966
which is i d e n t i c a l to R~/~ if G =~X (apart f r o m the slight fluctuations of x~). K i s s l i n g e r and Sor e n s e n , have shown that, in o r d e r to fit the binding e n e r g y d i f f e r e n c e s and the e n e r g i e s of the f i r s t 2 + s t a t e s in s i n g l e - c l o s e d - s h e l l s p h e r i c a l m i cl ei , we must take X ~ 125/.4 and G ~ 25/.4 f o r the p a i r i n g plus quadrupole m o d el (A is the m a s s number), or G ~ -~X. If we r e m e m b e r that the gap equations a r e the s a m e in the two m o d e l s , we get in both c a s e s the s a m e value of G. We can t h e r e f o r e conclude that the p a i r i n g plus quadrupole model with the above e m p i r i c a l values of X and the d e s c r i b e d s e p a r a b l e s u r f a c e delta int e r a c t i o n model give q u a l i t a t i v e l y the s a m e b e h a v i o r f o r the 2 + s t a t e s . H o w e v e r , a c o m p l e t e t r e a t m e n t of the q u a s i - p a r t i c l e r a n d o m - p h a s e a p p r o x i m a t i o n f o r our i d e a l i z e d s u r f a c e i n t e r action would d ev i at e f r o m the c o r r e c t e x p e r i mental energies.
References 1. 2. 3. 4. 5. 6.
I.M.GreenandS.A.Moszkowski, Phys. Rev. 139 (1965) B790. R.Arvieu and S.A. Moszkowski, to be published. R.Arvieu, A. Plastino and S. A. Moszkowski, to be published. L.S. Kisslinger and R.A. Sorensen, Kgl. Dan. Vid. Sels., Mat. Fys. Medd. 32 (1960). L.S.Kisslinger and R.A.Sorensen, Rev. Mod. Phys. 35 (1963) 853, M.Baranger and K.Kumar, Nuclear Physics 62 (1965) 113.
* We have chosen, for convenience, the following definition of the pairing force:
Gj(abcd) = -½G(-) la +Ic ~ ~5jo 5ab 5cd , i.e., the same phase convention as in eq. (1). The sign of ua (or Va) is then state-independent.
A CONNECTION THE PAIRING
PHYSICS LETTERS
15 April 1966
BETWEEN THE SURFACE DELTA INTERACTION AND PLUS QUADRUPOLE M O D E L IN S P H E R I C A L NUCLEI J . TOUCHARD
Institut de Physique Nucl~aire , Division de Physique Th$orique, Orsay, France Received 10 March 1966
It is shown that the surface delta interaction in single closed shell spherical nuclei leads to a description which is very similar to the usual pairing plus quadrupole model if the recoupling terms are ignored.
It has b e e n suggested r e c e n t l y by G r e e n and Moszkowski [1] that the s u r f a c e delta i n t e r a c t i o n acting as a r e s i d u a l f o r c e between n u c l e o n s in open s h e l l s , could s i m u l a r e in a v e r y s i m p l e way both s h o r t r a n g e effects (e.g. p a i r i n g effects) and long r a n g e p r o p e r t i e s for J = 2 s t a t e s . It has been studied in m o r e detail by A r v i e u and Moszkowski [2], using the g e n e r a l i z e d q u a s i - s p i n f o r m a l i s m and by A r v i e u , P l a s t i n o and Moszkowski [3] for single-closed-shell nuclei. It is i n t e r e s t i n g to c o n s i d e r how we can e s t a b l i s h a connection in the q u a s i - p a r t i c l e s c h e m e , b e t w e e n the s u r f a c e delta i n t e r a c t i o n and the p a i r ing plus quadrupole model, as used for i n s t a n c e by K i s s l i n g e r and S o r e n s e n [4, 5]. The s u r f a c e delta i n t e r a c t i o n is defined as a c t ing only when n u c l e o n s a r e at the n u c l e a r s u r f a c e . In addition, it is s t r i c t l y a contact i n t e r a c t i o n . We complete the definition of the model by taking a constant amplitude u o for the r a d i a l functions at the n u c l e a r s u r f a c e * u n l ( R ) = Uo
•
The p l a u s i b i l i t y of such an idealized model is d i s c u s s e d in r e f . 2 . Our i n t e r a c t i o n is then: G 6(r l - R ) V12 = - (-R-~o)4
6(r2 - R) ~k
Ck(Wl)Ck(W2)
where Ck (°~l)Ck ( w 2 ) = P k ( c ° s O 1 2 ), and G is a coupling constant. C o n s i d e r i n g only identical nucleons in s u b s h e l l s J~n' Jb, e t c . , we r e p l a c e the q u a n t u m n u m b e r s a, la, Ja) by a single s y m b o l a . As a c o n s e q u e n c e of the above a s s u m p t i o n s ,
the g e n e r a l m a t r i x e l e m e n t Gj(abcd) = =
Gj(abcd) = -G ~ (2k + 1)(abJ! P k l cdJ> , k i . e the Slater i n t e g r a l s a r e all the s a m e . and the state dependence of G j i s p u r e l y g e o m e t r i c a l . A s t r a i g h t f o r w a r d c a l c u l a t i o n shows that:
Go(aabb ) = - ½G (-) la+ lb ~t ~ , with ~ = ~ / ( 2 J a + l ) . T h e r e f o r e , the gap equation is the s a m e as for a p a i r i n g force of s t r e n g t h G. On the other hand, the s e l f - e n e r g y t e r m is s t a t e - i n d e p e n d e n t and can be included in the c h e m i c a l potential. Then the exact s u r f a c e delta i n t e r a c t i o n gap equation gives the s a m e r e s u l t as the usual p a i r i n g plus quadrupole model gap equation, where the s e l f - e n e r g y t e r m and the c o n t r i b u t i o n of the P2 f o r c e a r e r e m o v e d . If G is fixed, as in the p a i r i n g f o r c e model, by the binding e n e r g y d i f f e r e n c e s , the s u r f a c e delta i n t e r a c t i o n is completely d e t e r m i n e d . Looking now at the collective excited s t a t e s , we r e m e m b e r that in the p a i r i n g plus quadrupole model the f i r s t 2+ l e v e l s a r e given by the incomplete q u a s i - p a r t i c l e r a n d o m - p h a s e - a p p r o x i m a t i o n where 1) the c o n t r i b u t i o n of the p a i r i n g force is neglected and 2) the ecoupling t e r m s coming f r o m the P2 i n t e r a c t i o n a r e also neglected; this e n s u r e s the s e p a r a b i l i t y of the model and can be c o n s i d e r e d as a p a r t of the model itself [6]. The s i g n i f i c a n t p a r t of the m a t r i x e l e m e n t s of the q u a s i - p a r t i c l e r a n d o m - p h a s e - a p p r o x i m a t i o n have the following s e p a r a b l e f o r m :
• We choose here the phases of radial functions such thai asymptotic values of Unl all have the same sign. 85
Volume 21, number 1
PHYSICS LETTERS
(UaVb +VaUb) (- ) lah 2 (ab)
X (UcVd + VcUv~ (-)lch2(cd) ' ,/(1 + ~cd)
(2)
w h e r e ot stands f o r the two q u a s i - p a r t i c l e s s t a t e (a,b), u a and v a a r e the c o e f f i c i e n t s of the Bogoliubov t r a n s f o r m a t i o n *, and h2(ab) = (a]lC 2]]b).
c ~ =} X x a x~
(ra)u H e r e X a n d ( r ~ u a r e r e s p e c t i v e l y the P2 i n t e r a c tion s t r e n g t h and a s p e c i f i c m e a n - v a l u e of r 2, as in r e f . 4 . Now, with our choice of p h a s e s f o r the r a d i a l functions, all the x~ have the s a m e sign, and it is e a s y to v e r i f y that, inside a given s h e l l , t h e i r absolute value does not d e v i a t e a p p r e c i a b l y f r o m unity. 1 G' = vX. In o r d e r T h e r e f o r e , we have G~fl to c o m p a r e the s u r f a c e d e l ta i n t e r a c t i o n model to the p a i r i n g plus quadrupole model we take the r e coupling t e r m s out of the s u r f a c e delta i n t e r a c t i o n m a t r i x e l e m e n t s of the q u a s i - p a r t i c l e r a n d o m p h a s e - a p p r o x i m a t i o n , which l e a d s to an e x p r e s sion d i r e c t l y c o m p a r a b l e to the p a i r i n g plus quadrupole m o d el eq. (2). This s i m p l i f i e d s u r face delta i n t e r a c t i o n e x p r e s s i o n i s :
R a ft = -G (UaVb+ VaUb) (-)la h2(ab) `/(1 + 5ab)
×
× (UcVd + VcUd) (-)lch2(cd), ~/(1 + 5cd)
86
15 April 1966
which is i d e n t i c a l to R~/~ if G =~X (apart f r o m the slight fluctuations of x~). K i s s l i n g e r and Sor e n s e n , have shown that, in o r d e r to fit the binding e n e r g y d i f f e r e n c e s and the e n e r g i e s of the f i r s t 2 + s t a t e s in s i n g l e - c l o s e d - s h e l l s p h e r i c a l m i cl ei , we must take X ~ 125/.4 and G ~ 25/.4 f o r the p a i r i n g plus quadrupole m o d el (A is the m a s s number), or G ~ -~X. If we r e m e m b e r that the gap equations a r e the s a m e in the two m o d e l s , we get in both c a s e s the s a m e value of G. We can t h e r e f o r e conclude that the p a i r i n g plus quadrupole model with the above e m p i r i c a l values of X and the d e s c r i b e d s e p a r a b l e s u r f a c e delta int e r a c t i o n model give q u a l i t a t i v e l y the s a m e b e h a v i o r f o r the 2 + s t a t e s . H o w e v e r , a c o m p l e t e t r e a t m e n t of the q u a s i - p a r t i c l e r a n d o m - p h a s e a p p r o x i m a t i o n f o r our i d e a l i z e d s u r f a c e i n t e r action would d ev i at e f r o m the c o r r e c t e x p e r i mental energies.
References 1. 2. 3. 4. 5. 6.
I.M.GreenandS.A.Moszkowski, Phys. Rev. 139 (1965) B790. R.Arvieu and S.A. Moszkowski, to be published. R.Arvieu, A. Plastino and S. A. Moszkowski, to be published. L.S. Kisslinger and R.A. Sorensen, Kgl. Dan. Vid. Sels., Mat. Fys. Medd. 32 (1960). L.S.Kisslinger and R.A.Sorensen, Rev. Mod. Phys. 35 (1963) 853, M.Baranger and K.Kumar, Nuclear Physics 62 (1965) 113.
* We have chosen, for convenience, the following definition of the pairing force:
Gj(abcd) = -½G(-) la +Ic ~ ~5jo 5ab 5cd , i.e., the same phase convention as in eq. (1). The sign of ua (or Va) is then state-independent.