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 --<A < 33. S e v e r a l
of these nuclei have been i n t e n s i v e l y studied exp e r i m e n t a l l y [1-4], but u n t i l now t h e r e has been
no c o m p r e h e n s i v e and detailed model offered for
the l a r g e set of low-excitation energy data p r e sently a v a i l a b l e .
In our c a l c u l a t i o n s we r e s t r i c t the s h e l l - m o d e l
b a s i s to a s u b s e t of all the P a u l i - a l l o w e d states
f o r m e d by d i s t r i b u t i n g (A-16) p a r t i c l e s in the o r bits ld~, 2sz, and ld~ outside a closed (ls) 4
(lp) 12 Core. ~ Our r e s t r i c t e d b a s i s c o m p r i s e s all
such s t a t e s having at l e a s t ten ld~ p a r t i c l e s ; in
other words we allow no m o r e than two holes in
the ld~ shell. Then for given A, T and J t h e n u m b e r of m a n y - p a r t i c l e b a s i s states is typically b e tween 100 and 300. (In the full s - d - s h e l l v e c t o r
space for these n u c l e i , typical d i m e n s i o n s would
be 1000 to 3000.) Our hope is that the p e r t u r b a tive effects of f u r t h e r ld~ holes and of lp holes,
if p a r t i c l e s , etc. can be adequately t r e a t e d by
use of a s u i t a b l e effective i n t e r a c t i o n . Indeed,
i n s p e c t i o n of the e x p e r i m e n t a l data on p a r i t i e s of
low-lying states does suggest that the neglect of
explicit n o n - s - d - s h e l l configurations is a better
approximation for A = 30-33 than for nuclei near
either end of the s-d shell.
The effective interaction for our model was
obtained by varying its p a r a m e t e r s so as to best
* 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
<abJT ]VT(iJ) IcdJT>).
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
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* 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
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613