8 Interacting Boson Model-2 for High-Spin States Raymond A. Sorensen and Kevin Fowler Carnegie-Mellon University, Pittsburgh, PA 15213
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The N-P Interacting Boson Model is extended to include bosons of spins 4, 6, 8,.. in addition to the usual S and D bosons, in order to treat nuclear states of high spin within the IBM formalism. Up u n t i l about t e n years ago, most c a l c u l a t i o n s of n u c l e i assumed them t o be composed of neutrons and protons with t h e i r s p a c i a l and s p i n degrees of freedom, and with two body i n t e r a c t i o n s between them. Recent developments i n d i c a t e a quark substructure to nucléons so there are now attempts t o f i n d and c a l c u l a t e nuclear phenomena r e q u i r i n g these e x t r a degrees of freedom. On the other hand, there are a large number of nuclear c a l c u l a t i o n s being performed today using many fewer degrees of freedom than those represented by the neutrons and protons, namely the i n t e r a c t i n g boson model (IBM) c a l c u l a t i o n s CARI783. The IBM c a l c u l a t i o n s use only two nuclear c o n s t i t u e n t s , i d e n t i c a l S and D bosons, or i n the IBM-2, proton (Ή) and neutron (υ) S and D bosons. While t h i s model cannot contain a l l the p r o p e r t i e s of models with more degrees of freedom, i t has the advantage of s i m p l i c i t y and also seems t o be able to account f o r many d e t a i l s of the quadrupole c o l l e c t i v e motion i n a s i n g l e formalism. Thus one might hope to be able t o use the IBM to extrapolate from known p r o p e r t i e s of n u c l e i near the s t a b i l i t y l i n e to n u c l e i f a r from s t a b i l i t y . The parameters are the t o t a l numbers of Ή and ν bosons, Ν and Ν , and t h e i r one and two boson i n t e r a c t i o n s . The usefulness of t h i s theory f o r e x t r a p o l a t i o n depends on these parameters being independent of Ν and Z, or known functions of N,Z. Ή
υ
Two d e f i c i e n c i e s of the IBM are that the force parameters are i n general not independent of N,Z so that separate f i t s to i n d i v i d u a l n u c l e i are o f t e n r e q u i r e d . And second, the usual theory i s l i m i t e d t o low s p i n s t a t e s . In t h i s paper we describe an extension of the usual IBM-2 designed to be a p p l i c a b l e to deformed n u c l e i i n c l u d i n g the high s p i n s t a t e s CSOR853. The model, which contains bosons with L = 4,6,8...as w e l l as the usual L=0, S bosons and L = 2, D bosons, has many more degrees of freedom than the usual IBM, but remains much simpler than the treatment i n terms of fermions. The higher s p i n bosons are supposed to simulate the e f f e c t s of high s p i n nucléon p a i r s such as the a l i g n e d p a i r s that are important i n backbending n u c l e i . We d i s c u s s general features of the model f o r s t r o n g l y backbending n u c l e i , and then present a f i t to Ι^βγ^. then study to what extent the model with the same force parameters can f i t the s p e c t r a of the neighboring n u c l e i . W
e
The Hamiltonian The boson c r e a t i o n operators are where ρ = π,υ i n d i c a t e s proton or neutron and i = L,M the angular momentum of the b a s i s bosons. We consider even L, even p a r i t y f o r the bosons, which are supposed to represent p a i r s of l i k e p a r t i c l e s (or h o l e s ) . The numbers N = Σ.±ηρ± are assumed to equal 1/2 the number of proton or neutron p a r t i c l e s or holes from the nearest c l o s e d s h e l l , and are thus w e l l d e f i n e d f u n c t i o n s of Ν and Z, to the p
0097-6156/ 86/ 0324-0053S06.00/ 0 © 1986 American Chemical Society
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
NUCLEI OFF THE LINE OF STABILITY
54 extent that s u b s h e l l s are not considered. a
s
The number operators are d e f i n e d
" p i = Y p i YpiFor the Hamiltonian, we use H = E
t
ε
Ή
ΐ
η
Ή
ΐ
+ Σ
±
ε
υ
1
η
υ
1
+ κΟπ-Ο»,
= (N«! Ν ! > ~ (B ) (B A lo>, (3) ° ΤΤΟ \>0 ' where B J i s the condensate boson c r e a t o r , which w i l l depend on the cranking r o t a t i o n a l frequency or angular momentum. 1 / 2
0
r
R
n
T
υ
71
V
For the rare e a r t h n u c l e i the s e l f c o n s i s t e n t f i e l d i s deformed and the cranking procedure (of adding a term o>j t o the Hamiltonian) produces a c o l l e c t i v e r o t a t i o n . T h i s method gives s t a t e s ΙΦ > = GSB (ground s t a t e band), which are not eigenstates o f the angular momentum, but have a predetermined value o f the χ component o f angular momentum. The t o t a l angular momentum I i s i d e n t i f i e d as I = . An i n t e r e s t i n g e f f e c t occurs i n the HB c a l c u l a t i o n s i f the parameters, ε'β and a's, are chosen t o produce a backbending spectrum. The e f f e c t i s that as the angular momentum i s increased, ΔΙ, the angular momentum spread o f the s t a t e ΙΦ > increases r a p i d l y at and above the backbend. T h i s suggests that the s i n g l e boson condensate s t a t e i n t h i s r e g i o n i s not a good approximation t o the exact wave f u n c t i o n , which would be an angular momentum e i g e n s t a t e . One s o l u t i o n t o t h i s problem i s t o consider Bands with one (or a few) R e c i t e d Bosons r e p l a c i n g one or more o f the condensate bosons. Such s t a t e s are l a b e l e d x
0
x
0
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
8.
SORENSEN AND FOWLER
55
High-Spin States
1-BEB, 2-BEB e t c . We f i n d that such s t a t e s have much improved angular momentum p r o p e r t i e s and are not much more d i f f i c u l t to c a l c u l a t e i n the HB approx imation. In the r e g i o n above the backbend, where ΔΙ f o r ΙΦ > 11 i s i n c r e a s i n g r a p i d l y , ΔΙ f o r the s t a t e with an e x c i t e d boson decreases to a minimum. At a s t i l l higher s p i n , the 2-BEB s t a t e has the lowest ΔΙ value, » as seen i n Figure 1. 1 2
0
1 0
Χ
Figure 1. The rms d e v i a t i o n Δ Ι i n the x component of the angular momentum, vs. I f o r a s t r o n g l y backbending nucleus. The three curves are f o r the GSB, 1-BEB, and 2-BEB.
·
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Χ
β
5 Η—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—I t
0
2
4
β
8
10
12
14
If
18
20
I Angular Momentum P r o j e c t i o n The HB method can give good r e s u l t s p a r t i c u l a r l y f o r cases i n which the p r o p e r t i e s of the s t a t e s do not change too r a p i d l y with s p i n . For back bending n u c l e i , f o r which the energy does not vary smoothly with angular momentum, a b e t t e r method i s needed i f q u a n t i t a t i v e r e s u l t s are r e q u i r e d . To t r e a t these cases we have p r o j e c t e d the HB s t a t e s to s t a t e s of good angular momentum. The p r o j e c t i o n of these HB s t a t e s i s somewhat simpler than t h a t f o r fermions i n the HF approximation, and we are able t o p r o j e c t even our non-axial high s p i n s t a t e s to angular momentum eigenstates CSOR773. The r e s u l t i s t h a t a good r o t o r spectrum i n the HB approximation i s not much changed by p r o j e c t i o n , but f o r a back bending nucleus the p r o j e c t e d and unprojected energy s p e c t r a are s i g n i f i c a n t l y d i f f e r e n t . F i g u r e 2 on the next page shows the energy vs. angular momentum p l o t f o r the backbending model of Figure 1; and F i g u r e 3 shows the usual p l o t of moment of i n e r t i a S vs. for the same nucleus. The parameters used are shown i n the t a b l e s below together with those of Yb. 1 6 8
Table 1. L 0 2 4 6 8
ε
7!
0 0.5 3.0
Backbender υ
%
«υ
L
0 O.S 3.0 3.0 O.S
1.0 0.9 0.8
1.0 0.9 0.6 0.6 0.9
0 2 4 6
ε
168
Table 2.
κ = -0.028, Μη = 6, Ny = 8
Y b
ε
ε
υ
«π
«υ
0 1.0 1.6
0 1.0 1.6 1.9
1.0 0.9 0.8
1.0 0.9 0.8 0.8
τι
κ = -0.015, Rn = 6, Ny = 8
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
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56
NUCLEI OFF T H E LINE OF STABILITY
Results o f the C a l c u l a t i o n s In Figure 2, the upper three bands are the GSB, 1-BEB, and 2-BEB resp. and the lowest one i s the p r o j e c t e d band. Note that as I approaches the backbend region, the 1-BEB i s almost equal i n energy t o the GSB. At higher energy s t i l l , a l l three s e l f c o n s i s t e n t bands are n e a r l y degenerate. We i n t e r p r e t t h i s as i n d i c a t i n g that the " e x c i t e d " band at low s p i n i s s i m u l a t i n g an a l i g n e d nucléon p a i r , which at higher s p i n crosses the ground band. T h i s i n t e r p r e t a t i o n CSTE723 i s r e i n f o r c e d by the c a l c u l a t e d wave f u n c t i o n , which shows that the e x c i t e d boson has most of the angular momentum at the backbend. The second c r o s s i n g resembles two a l i g n e d p a i r s . The shape of the y r a s t l i n e q u a l i t a t i v e l y resembles the p r o j e c t e d energy curve, but as seen i n Figure 3, the p r o j e c t e d and unprojected bands d i f f e r s i g n i f i c a n t l y . The angular momentum p r o j e c t i o n i s done from the GSB. From Table 1 i t i s seen that the backbend or band c r o s s i n g i s produced by having a s i n g l e high s p i n boson at low energy. The α parameters are a l s o chosen to enhance the Q-Q i n t e r a c t i o n of that boson. In c o n t r a s t , a boson spectrum which i s more r e g u l a r , with the boson energy i n c r e a s i n g monotonieally with L, such as that of Table 2, leads to a smooth upbending spectrum to which the unprojected HB mean f i e l d approximation i s r a t h e r good. A l s o , i n that case the ΔΙ value i s lowest f o r the GSB and the e x c i t e d 1-BEB does not come so c l o s e to the GSB as the energy i s increased.
F i g u r e 2. P l o t s of Energy vs. Angular Momentum f o r a model backbender with parameters of Table 1.
0.24
F i g u r e 3. Moment of I n e r t i a vs. the square of the angular v e l o c i t y f o r the backbender. The GSB and the p r o j e c t e d bands are shown.
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
8.
High-Spin States
SORENSEN AND FOWLER
57
For the remainder of the paper ve w i l l d i s c u s s the model f i t to the spectrum of Yb, f o r which the HB approximation i s reasonably good. We w i l l show that the same parameter set can give a good f i t to the c o l l e c t i v e character of the neighboring n u c l e i as w e l l . 1 6 8
The energy spectrum i s f i t q u i t e w e l l and only the S vs. ω curve i s shown i n Figure 4 . Note that the small d e v i a t i o n s shown would be b a r e l y seen i n a p l o t of Ε vs. I. With the many ε and α parameters a v a i l a b l e , i t i s not s u r p r i s i n g that a good f i t can be made, and no e f f o r t was made f o r f u r t h e r improvement.
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2
However, the t e s t of the model becomes much more severe i f i t i s r e q u i r e d that t h i s model, with i t s parameters unchanged, f i t the s p e c t r a of neighboring n u c l e i as w e l l . Thus, i n the remaining two f i g u r e s the dependence on Ν and Ζ of the c a l c u l a t e d energy s p e c t r a are compared with the experimental data.
0.12
F i g u r e 4. Moment of i n e r t i a vs. the Angular v e l o c i t y f o r Y b . The l i n e i s the model of Table 2 and the
