Shell-model calculation for nuclei of masses 30 through 33

Volume 27B, number 10

SHELL-MODEL

PHYSICS LETTERS

CALCULATION

FOR

NUCLEI

14 October 1968

OF MASSES

30 T H R O U G H

33 *

B. H. WILDENTHAL **, J . B . McGRORY and E. C. HALBERT

Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA and P. W, M. GLAUDEMANS

Physisch Laboratorium, Utrecht, The Netherlands Received 12 July 1968

Level energies and spectroscopic factors for isotopes of P, Si, and S are calculated with a shell model which includes active particles in the ld~. 2s- and ld~ shells.

This note p r e s e n t s s o m e r e s u l t s of s h e l l model c a l c u l a t i o n s for the low-lying e v e n - p a r i t y s t a t e s of nuclei with m a s s e s 30 --
* Research sponsored in part by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation. ** AEC Postdoctoral Fellow under appointment from Oak Ridge Associated Universities.

fit the o b s e r v e d energy l e v e l s . We make the u s u a l a s s u m p t i o n of a (1 + 2)-body effective H a m i l t o n i a n . In our s - d shell b a s i s , the most g e n e r a l (1 +2)-body Hamiltonian i s specified by 63 two-body m a t r i x e l e m e n t s and 3 s i n g l e - p a r t i c l e e n e r g i e s . However, it is not p r a c t i c a l to vary all of these 66 p a r a m e t e r s independently in a l e a s t - s q u a r e s s e a r c h . Instead, we r e s t r i c t e d the two-body effective i n t e r a c t i o n to have the form of the modified s u r f a c e delta i n t e r a c t i o n

[5-% VT(iJ) = - 4 n A T 6 ( r - r j ) f i

j +B T •

Here T i n d i c a t e s the i s o s p i n (0 or 1) of the two i n t e r a c t i n g p a r t i c l e s ; A T and B T a r e s t r e n g t h s depending only on T; and fij is an o p e r a t o r which has the effect of making

f f r i dr i ~ drj Ra(ri)R b (rj) 5 (ri-r j)fij Re (ri) Rd(rj) equal to unity (where Ra, Rb, Rc, R d a r e the s i n g l e - p a r t i c l e r a d i a l wave functions involved in

). In this way the 63 two-body m a t r i x e l e m e n t s a r e specified by the 4 p a r a m e t e r s A0, B0, A1, B 1 of the modified s u r f a c e delta i n t e r a c t i o n . T h e s e 4 p a r a m e t e r s , together with the 3 s i n g l e p a r t i c l e e n e r g i e s , were d e t e r m i n e d by a d j u s t i n g them so as to give a l e a s t - s q u a r e s fit to the exp e r i m e n t a l data for the 9 g r o u n d - s t a t e binding e n e r g i e s and for 44 excitation e n e r g i e s of 30Si, 30p, 31Si ' 31p, 32Si ' 32p, 32S, 331)and 33S. The l e a s t - s q u a r e s solution is quite stable a g a i n s t i n c l u s i o n or exclusion of any p a r t i c u l a r level in the fitting c r i t e r i o n . (Indeed, a p r e l i m i n a r y fit 611

Volume 27B, number 10

PHYSICS

LETTERS

14 October 1968

Table 1 Experimental and calculated e n e r g i e s for some e v e n - p a r i t y states in 30 --< A -.< 33 nuclei. Nucleus A

T

30 30 30

30

First

Second

J

Eex

0

1

156.22

156.31

0.71

0.93

3.02

0

2

1.45

1.65

2.72

2 . 3 4 [1]

4.34

0

3

1.97

2.19

2.54

2.95

-

0

4

4.18

5.01

-

30

0

5

3.74

5.11

31

1_ 2

£2

3.56

168.50

Eth

168.67

Eex

3.14

Eth

Third Eex

Fourth Eth

Eex

Eth

3.34

-

3.75

4 . 7 5 [1]

-

5.05

3.39

-

4.51

6.08

-

6.44

-

6.01

-

6.92

*5.25

3.91

-

5.16

31

~

32__

1.26

1.17

3.51

4.06

4.26

4.53

-

4.80

31

£2

!2

2.23

2.35

3.29

2.86

4.19

4.75

-

4.98

31

1

2?_

3.41

3.56

4.18

-

5.57

-

5.79

31

!2

-~ 2

-

5.10

5.66

-

6.70

-

7.18

32

0

0

183.56

183.45

32

0

1

4.70

4.76

32

0

2

2.24

1.97

32

0

3

-

5.88

32

0

4

4.47

5.48 [14]

3.78 4.29

3.87

-

7.63

-

8.38

6.61

-

8.22

-

8.44

4.90

5.55

5.63

-

6.83

7.20

-

7.67

-

8.22

6.19

-

6.99

-

8.23

Binding e n e r g i e s ( c o r r e c t e d for Coulomb e n e r g i e s as in ref. 9 are listed for the ground states along with excitation e n e r g i e s for the excited s t a t e s . Levels marked by a s t e r i s k s were not included in the l e a s t - s q u a r e s s e a r c h . Except where noted, experimental e n e r g i e s are taken from ref. 4. w a s m a d e t o e n e r g i e s f o r A = 30 a n d 31 o n l y . T h e resulting Hamiltonian, and the calculated energy l e v e l s up t h r o u g h A = 33, w e r e e s s e n t i a l l y i d e n t i c a l t o t h o s e p r e s e n t e d h e r e [8].) T h e o p t i m i z e d v a l u e s of t h e m o d i f i e d s u r f a c e d e l t a i n t e r a c t i o n s t r e n g t h s obtained f r o m the 30-33 fit a r e A 0 = = 0.892 M e V , A 1 = 0.873 M e V , B 0 = - 1 . 0 7 1 MeV a n d B 1 = 0.708 M e V ; a n d t h e o p t i m i z e d b i n d i n g e n e r g i e s of t h e l d ~ , 2s,:, a n d ld~ n u c l e o n s a r e 7.55, - 5.73, a n d - 3.3~ M e V r e s p e c t i v e l y . T a b l e 1 s h o w s s o m e of t h e s p e c t r a f r o m t h e s e calculations , together with the experimentally determined energy levels. The deviations between 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 e n e r g i e s in t h i s t a b l e a r e t y p i c a l of a l l t h e r e s u l t s . T h e r . m . s . d e v i a t i o n f o r t h e 53 p i e c e s of d a t a in t h e f i t w a s 0.34 M e V , a n d t h e a v e r a g e a b s o l u t e d e v i a t i o n w a s 0.27 M e V . In v i e w of t h e s i m p l e a n d r i g i d n a t u r e of t h e i n t e r a c t i o n , w e f i n d t h e e n e r g y level agreement quite pleasing. Particularly sign i f i c a n t a r e t h e s e v e r a l i n s t a n c e s in w h i c h t h e o b s e r v e d e n e r g y s p a c i n g s of t h r e e o r f o u r l e v e l s of o n e s p i n in a g i v e n n u c l e u s a r e r e p r o d u c e d in t h e m o d e l s p e c t r a . It i s n o t y e t c l e a r w h e t h e r t h e f e w s u b s t a n t i a l d e v i a t i o n s t h a t do o c c u r a r e to b e t r a c e d to d e f i c i e n c i e s in t h e m o d e l s p a c e o r to -

612

the modified surface delta interaction assumption. A t e s t of s o m e i m p o r t a n t f e a t u r e s of t h e m o d e l wave functions obtained from these calculations i s p r o v i d e d by c o m p a r i n g 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 l y o b s e r v e d [11-15] v a l u e s f o r s i n g l e - n u cleon spectroscopic factors. This comparison i s s h o w n in t a b l e 2. A g a i n t h e a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t i s good. In p a r t i c u l a r , t h e r e l a t i v e s t r e n g t h s o b s e r v e d f o r 2s_, a n d ld_~ transfer are well reproduced. This indicates that the theoretical wave functions give the correct r e l a t i v e o c c u p a t i o n p r o b a b i l i t i e s f o r t h e s e two o r b i t s . T h e a b s e n c e of e x p e r i m e n t a l l y o b s e r v e d s t r o n g ld~ s t r i p p i n g t r a n s i t i o n s , w h i c h i m p l i e s t h a t t h e l d ~ s h e l l i s e f f e c t i v e l y f u l l in t h e g r o u n d s t a t e s , i s a l s o c o r r e c t l y r e p r o d u c e d by t h e m o d e l The main discrepancy between the model and exp e r i m e n t s e e m s to b e t h e l a c k of e n o u g h ld~ s i n g l e - h o l e s t r e n g t h in t h e s e c o n d ~+ 2 level calcul a t e d f o r A = 31, T = ½ ( 3 1 p _ 31S)" T h i s w o r k i s b e i n g e x t e n d e d to c o n s i d e r t h e A = 34 a n d 35 n u c l e i , e l e c t r o m a g n e t i c t r a n s i t i o n r a t e s , a n d a l t e r n a t i v e f o r m s of t h e e f f e c t i v e i n t e r a c t i o n . T h e c o m p u t a t i o n a l m e t h o d s u s e d in t h i s w o r k a r e d e s c r i b e d in r e f s . 15 a n d 16.

V o l u m e 27B, n u m b e r 10

PHYSICS

Table 2 Experimental and calculated spectroscopic factors. Final state Transition

Sx

Sth

(j?T, Ex(MeV)) 3C)Si(3He,d)31p [10] *

30Si(d,p)31Si [11]

32S(h,~)31S [12] ** 32S(p, d)31S [13]

32S(d,p)33S [14]

½+, 3+,

0.00

0.68

0.95

1.26

0.67

0.74

5+ ,

2,23

0.065

0,06

I+ ,

3.14

0.02

0.07

5+ , 3+ ,

3.29

0.003

0.01

3.51

0.004

0.14

3 +, 1+ ,

0,00

0.86

0.70

0.75

0.27

0.17

1.70

0.02

0.002

5+ , 3+ ,

2.32

0.96

0.01

_5+ 2 ,

2.79

0.04

0.01

1+, 3+ ,

0.00

1.9

2.12

1,24

2.0

1.66

~÷ 2 , ½+,

2,23

5.5

4.52

3.08

weak

0.02

5 +,

3,29

1.5

0.02

~+,

3,43

weak

0.01

~3 + , 1+ ,

0.00

0.69

0.61

0,84

0.27

0.28

2+, 3+ ,

1.97

weak

0.002

2.31

0.05

0.08

LETTERS

14 O c t o b e r 1968

References I . G . I . H a r r i s and A . K . Hyder J r . , P h y s . Rev. 157 (1967) 958 and p r i v a t e c o m m u n i c a t i o n . 2. A . C . Wolff, M . A . M e y e r and P. M. Endt, Nucl. P h y s . A107 (1968) 332, 3. A . R . Poletti and M . A . G r a c e , Nucl. P h y s . 78 (1966) 319. 4. P . M . Endt and C. Van d e r Leun, Nucl. P h y s . A105 (1967) 1. 5. R.Arvieu and S.A.Moszkowski, Phys. Rev. 145 (1966) 830. 6. P . W . M . G l a u d e m a n s , B. H. Wildenthal and J. B. M e G r o r y , P h y s . L e t t e r s 21 (1966) 427. 7. P . W . M . G l a u d e m a n s , P . J . B r u s s a a r d and B. H. Wildenthal, Nuel. P h y s . A102 (1967) 593. 8. E . C . H a l b e r t , P r o c . T h i r d . I n t e r n . Syrup. on the s t r u c t u r e of low m e d i u m m a s s nuclei, L a w r e n c e , K a n s a s , 1968. 9. P . W . M . G l a u d e m a n s , G . W i e c h e r s and P , J . B r u s s a a r d , Nucl. P h y s . 56 (1964) 529. 10. M . B e t i g e r i et al., Z. N a t u r f o r s c h . 21a (1966) 980. 11. B . H . Wildenthal and P. W. M. G l a u d e m a n s , Nucl. P h y s . A108 (1968) 49. 12. C . M . F o u and R. W. Zurm~ihle, P h y s . Rev. 151 (1966) 927, 13. R. L. Kozub, p r e p r i n t . 14. M . C . M e r m a z , C . A . W h i t t e n J r . and D . A . B r o m l e y , Bull. A m . P h y s . Soc. 13 (1968) 675, and p r i v a t e communication. 15. J . B . F r e n c h , E . C . H a l b e r t , J . B . M c G r o r y and S. S. M. Wong, to be published. 16. P . W . M . G l a u d e r n a n s and B . H . W i l d e n t h a l , to be published.

* With a r b i t r a r y n o r m a l i z a t i o n . ** The v a l u e s Sx l i s t e d h e r e a r e the a v e r a g e s f r o m r e f s . 12 and 13.

613