27 Toward a Shell-Model Description of Intruder States and the Onset of Deformation 1
1
2
3
K. Heyde, P. Van Isacker, R. F. Casten , and J. L. Wood 1
Institute for Nuclear Physics, Proeftuinstraat 86, B-9000 Gent, Belgium Brookhaven National Laboratory, Upton, NY 11973 School of Physics, Georgia Institute of Technology, Atlanta, GA 30332
2
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3
Basing on the nuclear shell-model and concentrating on the monopole,pairing and quadrupole corrections originating from the nucleon-nucleon force,both the appearance of low-lying 0 intruder states near major closed shells(Z=50, 82)and sub-shell regions (Z=40,64) can be described.Moreover,a number of new facets related to the study of intruder states are presented. +
In a nucleus,the major part of the nucleonic motion is determined by the average single-particle field.This field approximation implies the existence of magic numbers i.e. 20,28,50,82,126 ,determining extra stability for the par ticular nuclei having proton and/or neutron number equal to such a magic num ber.The energy separation between the last filled single-particle orbitals and the lowest,unfilled orbitals varies from ~6 MeV(light nuclei) to~3.5 MeV(Z=82 gap).Besides,a number of subshell closures have by now been established i.e. Z=40,64.In these cases,a much smaller energy gap of 1 and 2.5 MeV,respectively results.It is precisely the existence of particular configurations that makes the study of nuclear structure approximate tractable.Most nucleons can be con sidered to contribute to the "core" nucleus and thus to the average field lea ving only a small number of nucleons,called valence nueleons(particles or holes) outside closed shells(see fig.1).In region I,few valence nucleons are present
Figure 1. Schematic divisions of nuclei in three major regions: (I) region near doubly closed shells; (II) region of strongly deformed nuclei; (III) region of intruder states.
ζ
AO
50 Ν
60
70
0097-6156/86/0324-0184$06.00/0 © 1986 American Chemical Society
In Nuclei Off the Line of Stability; Meyer, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
27.
and t h e n u c l e a r s h e l l - m o d e l
techniques[SHA63Jal
l y i n g nuclear excited s t a t e s . I n cal
nucléons are p r i m o r d i a l
til!
this
region
case,the
(or s p i n - o r b i t
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of the v a r i a t i o n
r
a
detailed
properties
II
study of between
[B0H75]) moving i n
o r b i t a l s [bHA53,FED79]
interaction
of shell
are
i s t h e dominant
1,11 v e r y much r e l i e s
shell configurations.Since
gaps can be shown t o be A d e p e n d e n t ( s e e analysis
o
low-
identi-
identical
present.In
contribution
nuclei.
i n two r e g i o n s
defined closed
(region
partner)
strong proton-neutron
The s e p a r a t i o n
f
and s t r o n g l y c o l l e c t i v e m o t i o n does n o t d e v e l o p u n -
and l e a d s t o p e r m a n e n t l y deformed
a set of well
l o w
I,the pairing
many v a l e n c e p r o t o n s and n e u t r o n s
single-particle
185
Shell-Model Description of Intruder States
H E Y D E ET AL.
sect.2
on t h e e x i s t e n c e
the value o f tne
[GU077,S0R84]) ,a more
gaps as a f u n c t i o n
of A w i l l
of
shell
detailed
have t o be s t u -
died. T h e r e now a l s o e x i s t s
an i d e a l i z e d r e g i o n
o f nucléons i s near a closed s h e l l and t h e o t h e r has a maximal
configuration(very
i n such n u c l e i , t h e
lyinq particle-hole
(p-h)excitations
in detail
f o r odd-A n u c l e i
vered i n d e t a i l . I n
closed shell
nucléons)
result(lp-2h
first,a
or Zp-lh
in
larqer excitation
i n odd-A n u c l e i
"intruder"
studied
regions are co-
even-even
approach f o r d e s c r i b i n g both t h e small
ha-
statesfor
e n e r g y , h a v e been
[HEY83] ,where most c l o s e d - s h e l l
and t h e p a r t i c u l a r Α-dependence o f such i n t r u d e r
situation).It
can be e x c i t e d and l o w -
the present d i s c u s s i o n ( s p e c i f I C f o r
present a shell-model
where one t y p e
few v a l B n c e
l h or Ip outside a closed shell).These
w h i c h one w o u l d e x p e c t , a t
fig.l)
number o f v a l e n c e nue I e o n s ( m i d - s h e l 1
has been shown t h a t
ving o r i q i n a l l y
III(see
nuclei),we
excitation
states throughout
energy
region
III
nuclei. 2.She!1-model
description
In a schematic we t a k e as t h e l o w e s t
of
intruder
presentation(see intruder
state
fig.2)
states of a single-closed shell
(0 state) +
a proton
( π ) 2p-2h
nucleus,
configuration
Figure 2. Schematic representation of a proton (π) intruder 2p-2n 0 configuration where denotes the regular o r b i t a l , the intruder orbital and j the neutron orbitals f i l l e d i n a BCS way ( v ) +
v
#
In Nuclei Off the Line of Stability; Meyer, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
186
NUCLEI OFF THE LINE OF STABILITY
w i t h the valence neutrons tals.The
d i s t r i b u t e d over the a v a i l a b l e s i n g l e - p a r t i c l e
unperturbed energy then
We now s h o r t l y
discuss the d i f f e r e n t
pairing correlations
both
2p-2h c o n f i g u r a t i o n s
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in
BCS o c c u p a t i o n p r o b a b i l i t i e s
energies
= 2(fc = l
of the j
-ΐ
results
)-2(ε
A
Z
effects
become m o d i f i e d due t o t h e
proton-
2 energy[TAL63]
and ν
neutron orbitals.because of
these
energy v a r i a t i o n s , a f i r s t
intruder configurations
ΔΕ
due t o c o r e p o l a r i z a t i o n
t h e f o l l o w i n g way
π ν ν ) the average p r o t o n - n e u t r o n
single-particle
in
binding
correction
The p r o t o n s i n g l e - p a r t i c l e
,j
e n e r g i e s and t h e
+
neutron i n t e r a c t i o n ii
strong
0 core.
2 . 1 Monopole f i e l d
with E(j
due t o o o t n t h e
and t n e p r o t o n - n e u t r o n
the proton s i n g l e - p a r t i c l e
energy o f these i n t r u d e r o f the underlying
energy c o r r e c t i o n s
i n t h e 2p and 2h c o n f i g u r a t i o n s
teraction affecting
orbi
becomes
interaction
correction
to the unperturbed
.
the
proton
as
-ε
)
[E(J;.J )-E(J,.J )] V
(3)
V
ν T h i s monopole c o r r e c t i o n shell
has been s t u d i e d i n some d e t a i l
Z = b 0 , « 2 and t h e s u b s n e l l
Z=40,b4 r e g i o n s [ H E Y 8 5 ] . i t shell
for
both tne
large
t h e r e b y becomes
that f o r the large spherical
single-particle
m o d i f i c a t i o n o f the o r i g i n a l
gap r e s u l t s . O n t h e c o n t r a r y , f o r
the subshell
gions,when Ν i n c r e a s e s , a n almost complete e r a d i c a t i o n o f t h e subshell ( f o r Z=40 n e a r N = b 8 , b 0 ; f o r Z=b4 n e a r IM=88,9U) r e s u l t s c a s e s , w e end up w i t h n u c l e i
similar
to the region
II
and t h u s , i n
field
the l o w e s t - l y i n g states
A E
A
P
E
(similar
=
A E
p a i r W
0
l
e
S
>
=
a r t
-)
+
A E
case.
coupled c o n f i g u r a t i o n s
+
using the residual
pairing(
h o l e s
nucleon-nucleon
p
a
r
n
a r t
- ) ) »where
gain.
as
)
-> | 0 β 0 > + ct|2 fi 2 > + . . . . (6) π ν π ν π ν U s i n g p e r t u r b a t i o n t h e o r y and t h e l o w e s t s e n i o r i t y s h e l l - m o d e l d e s c r i p t i o n o f the 0 p a i r d i s t r i b u t i o n , o n e c a l c u l a t e s a quadrupole p o l a r i z a t i o n energy gain +
+
+
1
+
+
1
+
1
+
ΔΕ where ( K
$ m
Q η
