Effective nuclear force of a realistic potential: A selfconsistent approach

Volume 27B. number 10

EFFECTIVE

PHYSICS LETTERS

NUCLEAR FORCE OF A SELFCONSISTENT

14 October 1968

A REALISTIC POTENTIAL: APPROACH *

M. GMITRO **

International Atomic Energy Agency, International Centre for Theoretical Physics, Trieste, Italy Received 1 July 1968

Core polarization corrections to the Tabakin realistic nucleon-nucleon potential have been calculated by a selfconsistent method. The derived residual interaction is demonstrated by the example of the tin isotope spectroscopy.

In phenomenological s h e l l - m o d e l c a l c u l a t i o n s one c o n s i d e r s the effective n u c l e o n - n u c l e o n f o r ces a l r e a d y r e n o r m a l i z e d for the core effects and for the t r u n c a t i o n of H i l b e r t spaces involved. On the other hand, when m a t r i x e l e m e n t s of a " b a r e " r e a l i s t i c potential a r e u s e d for the r e s i d u a l i n t e r action, only d i a g o n a l i z a t i o n s in H i l b e r t spaces of all p o s s i b l e configurations can be expected to d e s c r i b d e n u c l e a r s p e c t r a c o r r e c t l y . Instead, s e v e r a l a u t h o r s have studied the p o s s i b l e r e n o r m a l i z a t i o n of r e a l i s t i c potentials. Second o r d e r ( 3 - p a r t i c l e and 1-hole in the i n t e r m e d i a t e state) c o r r e c t i o n s proposed by Kuo and Brown [1] have been proved s u c c e s s f u l in a s e r i e s of p a p e r s [1,2] devoted to the s p e c t r o s c o p y of s - d shell n u c l e i and Ni isotopes. Following this s u c c e s s , s i m i l a r c o r r e c t i o n s to the T a b a k i n and Yale potential have been i n v e s t i g a t e d in tin isotopes [3,4] w h e r e q u a s i p a r t i c l e techniques proved adequate. Although g e n e r a l l y fruitful, the p r o c e d u r e s e e m s r a t h e r s e n s i t i v e to the choice of s i n g l e p a r t i c l e e n e r g i e s Es, which a p p e a r as p a r a m e t e r s in the core p o l a r i z a t i o n f o r m u l a s . A m b i g u i t i e s in the " e x p e r i m e n t a l " v a l u e s for these p a r a m e t e r s have been d e m o n s t r a t e d in ref. 3. We shall p e r f o r m a s e l f c o n s i s t e n t calculation of the s i n g l e - p a r t i c l e e n e r g i e s es in analogy with the H a r t r e e - F o c k method [5]. Usually H a r t r e e Fock c a l c u l a t i o n s a r e c a r r i e d out with the a i m of finding the s e l f c o n s i s t e n t set of s i n g l e - p a r t i c l e o r b i t a l s I~,> which lead to m~ energy m i n i m u m for d e t e r m i n a n t a l wave function of A s i n g i e - p a r t i c l e o r b i t a l s . Here we a r e m o r e i n t e r e s t e d in a s e l f c o n s i s t e n t r e n o r m a l i z a t i o n p r o c e d u r e for a r e a l i s * To be submitted for publication. ** On leave of absence from Nuclear Research Institute. ~{e~ (Prague), Czechoslovakia. 616

tic n u c l e o n - n u c l e o n potential to be used in s p e c troscopic calculations. We s h a l l u s e h a r m o n i c o s c i l l a t o r o r b i t a l s in the capacity of a fixed s i n g l e - p a r t i c l e b a s i s , for it has been o b s e r v e d s e v e r a l t i m e s [6] that h a r monic o s c i l l a t o r o r b i t a l s u s u a l l y give good a p p r o x i m a t i o n to the r a d i a l shape of c o r r e s p o n d i n g H a r t r e e - F o c k o r b i t a l s . This choice i s c o n s i s t e n t with the u s e of the h a r m o n i c o s c i l l a t o r b a s i s in the s p e c t r o s c o p i c calculation which we r e p o r t below. Then the f o r m u l a for s i n g l e - p a r t i c l e energy c o r r e c t e d for the s e l f - e n e r g y ~ s can be w r i t t e n in the f o r m [8]

~s = - ~s(~t)

(1)

where the single-particle operator u contains kinetic energy, a spin-orbit term and all the effects of core protons. For the sake of definiteness it is preferable to work with a fixed set of n u m e r i c a l values for . An i n t e r p r e t a t i o n of as eigenvalues of a Woods-Saxon pot e n t i a l s e e m s to be r e a s o n a b l e . We have chosen s i n g l e - p a r t i c l e binding e n e r g i e s of the WoodsSaxon potential calculated by the Bonn group [7]. In the p a r t i c l e - h o l e vector coupling notation

[8]

<~ Iv + Av(~t)17~>

--

= - 2 ~ ~;(acdbJ; Q)(-)Jb-m~ +Jc-rn~ x JM × (JarnevJc - rnT1JM)(Jdm5 Jb - m~[ JM) we have for s e l f e n e r g y c o r r e c t i o n s ~s(~t) the following e x p r e s s i o n :

(2)

Volume 27B, number 10

Us(~t) - ~ 2

~s , ~

PHYSICS LETTERS

~(sss's'J=

14 October 1968

The highest core s u b s h e l l s which we a s s u m e to be involved in the core p o l a r i z a t i o n c o r r e c t i o n s are: lg~, 2P½, lf~ and 2p~. In the s i m p l e v a r i a n t of the h a r m o n i c o s c i l l a t o r model our "core" lg~ and "valence" s - d - g s u b s h e l l s a r e d e g e n e r a t e . F o r this r e a s o n the i n t r o d u c t i o n of s p i n - o r b i t splitting t e r m s into the s i n g l e - p a r t i c l e o p e r a t o r u as d e s c r i b e d above s e e m s to be unavoidable, at l e a s t in our case of the tin isotope s p e c t r o s copy. F r o m our experience, the c o n v e r g e n c e of the above p r o c e d u r e is v e r y quick (3-5 i t e r a t i o n steps) and r e s u l t s do not depend on the i n i t i a l choice of es" In table 1, in the c o l u m n s headed Es, we give our r e s u l t s for Sn isotopes with A = 114, 116, 118 and 120. S i m u l t a n e o u s l y our input p a r a m e t e r s taken f r o m the work of B l e u l e r et al. [7] as eigenvalues of the Woods-Saxon potential a r e listed. In the next step we have used the Tabakin pot e n t i a l r e n o r m a l i z e d by the d e s c r i b e d s e l f c o n s i s tent method in a s p e c t r o s c o p i c calculation. The gap equations [8] with this potential and the c a l culated set of the s i n g l e - p a r t i c l e e n e r g i e s Es have been solved for the s i n g l e - q u a s i p a r t i c l e e n e r g i e s E s = [(E s _~)2 +A2]½. Our r e s u l t s of the t w o - q u a s i p a r t i c l e T a m m - D a n c o f f theory [3] for the lowest s t a t e s of l l 4 S n , l l 6 S n , 118Sn and 120Sn isotopes a r e c o m p a r e d in table 2 with the c o r r e s p o n d i n g e x p e r i m e n t a l data. S u m m i n g up, we should like to s t r e s s two l i m i t a t i o n s of the p r e s e n t approach. F i r s t , our

O" Et) (3)

where the s u m m a t i o n is c a r r i e d out over occupied o r b i t a l s only. F o r the chosen ( r e a l i s t i c ) two-body potential v and an i n i t i a l set of s i n g l e - p a r t i c l e e n e r g i e s es, we can c a l c u l a t e the a b o v e - m e n t i o n e d 3 - p a r t i c l e 1-hole core p o l a r i z a t i o n c o r r e c t i o n s Av(Et) as d e s c r i b e d in refs. 1-3. Then a new set of Es a r i s e s f r o m eqs. (1) and (3), which can be used in the s u b s e q u e n t step for c a l c u l a t i n g c o r r e c t i o n s Av(Et) until s e l f c o n s i s t e n c y is obtained. In the capacity of two-body r e s i d u a l i n t e r a c tion v, the T a b a k i n [9] n o n - l o c a l potential has been chosen. A l m o s t complete equivalence of this potential with the m o r e u s u a l hard core c o n t a i n ing Yale r e a l i s t i c potential in the s p e c t r o s c o p y of tin isotopes has a l r e a d y been d e m o n s t r a t e d [4]. The m a i n advantage of the T a b a k i n potential for our a l m s c o n s i s t s in the p o s s i b i l i t y of avoiding the i n t r o d u c t i o n of the r e a c t i o n m a t r i x , which i s n e c e s s a r y for s i n g u l a r potentials. T h e r e , new p r o b l e m s of s e l f c o n s i s t e n c y in c a l c u l a t i n g the reaction matrix arise. As a n u m e r i c a l example we have chosen the s p e c t r o s c o p y of s e v e r a l isotopes of tin, which a r e u s u a l l y c o n s i d e r e d as r e p r e s e n t a t i v e s of s o called v i b r a t i o n a l n u c l e i . In our model, tin i s o topes c o n s i s t of the doubly magic core (50 n e u t r o n s plus 50 protons) and "valence" n e u t r o n s filling the s u b s h e l l s 2d~, lg½, 3s½, 2d~ and lh,~.

Table 1 Woods-Saxon potential eigenvalues (slu s) from the calculation of Bleuler et al. [9] and our self-consistent values of single-particle energies Es, both in MeV. ll4sn

ll6Sn

ll8Sn

120Sn

ns/sJ s

2d~

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

lg~

1.21

2.31

1.160

2.49

1.11

2.07

1.07

1.66

3s~

2.08

2.92

2.070

2.53

2.06

2.48

2.06

2.43

2d~

2.77

3.94

2.740

4.09

2.72

3.92

2.70

3.76

lh,,2

3.42

2.88

3.36

3.00

3.30

2.97

3.25

2.97

lg~

- 4.74

-6.37

-4.72

- 6.20

- 4.70

- 6.09

- 4.67

- 5.95

lp~

- 6.83

- 8.12

- 6.76

- 7.37

- 6.71

- 7.46

- 6.65

-

lf~

-8.60

-8.39

-8.55

-8.17

- 8.50

- 8.47

- 8.44

- 8.74

lp_~

-8.36

- 9.24

- 8.30

- 9.03

- 8.20

- 8.98

-8.12

-8.93

7.53

617

V o l u m e 2 7 B , n u m b e r 10

PHYSICS

LETTERS

14 October 1968

Table 2 Energy levels (in MeV) of the lowest s t a t e s of even tin isotopes as calculated in the p r e s e n t work are compared with e x p e r i m e n t a l data. ll4Sn

ll6Sn

118Sn

exper,

calc.

exper,

calc.

exper,

O2

1.58

1.88

1.76

1.77

1.76

1.80

+

1.95

3.74

3.55

2.06

3.20

-

1.30

1.52

1.29

1.46

1 °23

1.41

1.17

-

2.77

2.11

2.81

2.65

-

2.19

2.08

2.39

2.02

1.91

2.19

2.49

2.53

2.50

2.66

-

2.2 7

2.54

2.27

2.56

2.68

2.39

2.89

51

2.81

2.21

2.36

2.11

2.05

2.28

1.95

61

-

2.44

2.77

2.34

2.28

-

71

-

2.25

2.91

2.25

2.25

2.48

+

O3 +

22 +

41 +

42

-

u s e of t h e W o o d s - S a x o n p o t e n t i a l i n a d d i t i o n t o the kinetic energy operator, although reasonable, i s n e v e r t h e l e s s a r a t h e r a r b i t r a r y s o l u t i o n of p r o b l e m s c o n n e c t e d w i t h t h e d e g e n e r a c y of t h e (core) subshell lg~ with four valence subshells in the harmonic oscillator model. Second, our picture is clearly far from saturation in the sense of s t a b i l i t y of r e s u l t s a g a i n s t t h e i n c l u s i o n of deeperlying core subshells; ideally one must inc l u d e a l l c o r e s u b s h e l l s in t h e r e n o r m a l i z a t i o n procedure. W e c a n c o n c l u d e t h a t t h e c a l c u l a t e d s e t s of t h e s i n g l e - p a r t i c l e e n e r g i e s ~s f o r f o u r i s o t o p e s of t i n g i v e s t h e f i r s t (to o u r k n o w l e d g e ) s e l f c o n s i s tent information about these parameters which, especially for the core subshells, are treated as p r a c t i c a l l y f r e e i n m o s t of t h e r e c e n t c a l c u l a t i o n s . On t h e o t h e r h a n d , t h e q u e s t i o n of t h e p r o p e r e n e r g y d e n o m i n a t o r s AE i n t h e r e n o r m a l i z a t i o n m e t h o d of Kuo a n d B r o w n [1,2] h a s b e e n i n v e s t i g a t e d . A c c o r d i n g to t h e r e s u l t s of t a b l e 2 t h e suggested self consistent prescription seems to give all the most important corrections leading to a n a t l e a s t s e m i q u a n t i t a t i v e o v e r a l l a g r e e m e n t of t h e c a l c u l a t e d a n d e x p e r i m e n t a l s p e c t r a .

618

120Sn

2.28

2.32

i

2.57

eale.

exper,

calc.

1.87

1.86 2.81

1.29 2.47

1.72 2.96

2.19

2.04

The author would like to thank Professor J. Sawicki and Dr. J. Hendekovi5 cordially for int e r e s t i n g d i s c u s s i o n s . He w o u l d a l s o l i k e t o t h a n k P r o f e s s o r s A. S a l a m , P . B u d i n i a n d t h e IAEA for hospitality at the International Centre for Theoretical Physics, Trieste, and UNESCO for financial support.

1. T . T . S . Kuo a n d G . E . B r o w n , Nucl. Phys. 85 (1966) 40; A92 {1967) 481. 2. T . T . S . K u o , Nucl. Phys. A90 (1967) 199. 3. M . G m i t r o , J. Hendekovi5 and J. Sawicki, Phys. L e t t e r s 26B (1968) 252; Phys. Rev. 169 (1968) 983; M . G m i t r o , A . R i m i n i , T . W e b e r and J. Sawicki, ICTP, T r i e s t e , p r e p r i n t I C / 6 8 / 2 9 . 4. M . G m i t r o and J. Sawicki, Phys. L e t t e r s 26B (1968) 493. 5. G. Ripka, Advances in nuclear physics, eds. M. B a r a n g e r and E.Vogt (Plenum P r e s s , New York, 1968) Vol. 1,p. 183. 6. R . M u t h u k r i s h n a n and M. B a r a n g e r , Phys. L e t t e r s 18 (1965) 160. 7. K . B l e u l e r , M . B e i n e r and R.De Toureil, Nuovo Cimento 52B (1967) 45, 149 and private c o m m u n i c a tion f r o m M. Beiner. 8. M . B a r a n g e r , Phys. Rev. 120 (1960) 957. 9. F. Tabakin, Ann. Phys. (N.Y.) 30 (1964) 51.