10 The Importance of an Accurate Determination of Interacting Boson Model-2 Parameters 1,4
2
3
B. R. Barrett , I. Morrison , and J. G. Zabolitzky 1
Physics Division, National Science Foundation, Washington, DC 20550 School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia Institute for Theoretical Physics, University of Köln, 5 Köln, 41, Federal Republic of Germany
2
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3
First, a brief description of the neutron-proton Interacting Boson Model (IBM-2) is given. Next, this model is applied to experimental data in order to determine its empirical parameters. Finally, we discuss why an accurate determination of these parameters is so important. 1. Introduction The Interacting Boson Model (IBM) of A r i m a and Iachello [ARI76, ARI81, BAR81] has been highly successful in correlating and describing a wide variety of experimental data regarding the collective properties of medium-to-heavy mass nuclei. A s originally formulated [ARI76], it is a purely phenomenological model, whereby the properties of nuclei are described in terms of interacting s (J=0) and d (J=2) bosons, such that the number of bosons is conserved. This original version did not distinguish between proton bosons and neutron bosons and is commonly referred to as the IBM-1. Later, the IBM-2 [ARI77] was developed to treat the neutron-proton interaction [ARI77] and to allow a connection to be established with the microscopic nuclear shell model. We will first outline the IBM-2 formalism and will then discuss problems related to an accurate determination of the IBM-2 parameters. 2. The IBM-2 Formalism The principal idea of the IBM-2 is to exploit the observation of Talmi [TAL82] and others [FED79] that it is the interaction between active protons and neutrons which is mainly responsible for causing nuclei with several valence nucléons to deform. The lowest rank proton-neutron interaction which can produce this effect is a quadrupolequadrupole interaction [TAL71, T A L 8 2 , ARI77]. For this reason, and in order to keep the number of variable parameters small, the usual form taken for the IBM-2 Hamiltonian is [ARI77, IAC79] IBM-2
where
Q
p
+n
(
r i
= (d xs + s x d ) p +
+
2)
) + KQ - Q + M
+V
+V
+ X ( d x d ) p , Ρ = π, ν +
p
(2)
2)
M—% = Majorana term = % (s xd - d x s * / ^ · (s xd^ - d xs n
+
+
+
2
(1)
%
'Permanent address: Department of Physics, University of Arizona, Tucson, A Z 85721 0097-6156/ 86/ 0324-0065$06.00/ 0 © 1986 American Chemical Society
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
66
NUCLEI OFF THE LINE OF STABILITY
V
Q 0 μ
μ
=
Σ |(2L 1) L=0,2,4 +
1 / 2
C
L o
[(d xd )J μ μ +
+
L )
χ (dxd)i ] μ L )
( 0 )
, Ρ = π,ν
(4)
m 3
m = ^
d
-m ·
»
The " · " represent scalar products and the " x " represent tensor products. The purpose of the Majorana term was to remove states which are non-symmetric under interchange of the proton and neutron degrees of freedom by shifting them up in energy. The nonsymmetric neutron-proton states are now of some physical interest [BOH 84]. For simplicity, and to decrease the number of variable parameters, we have taken ε = ε = ε. In general, this will not be true.
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π
ν
3. Application of the IBM-2 Formalism Since the ξ| are usually held constant, there could be ten variable parameters to. be determined for each nucleus studied. This number is usually reduced to six by assuming that only V^viV-mr) contributes to relative splittings in isotopes (isotones) and that the contribution of is negligible. The remaining six parameters are ε, κ, X C o (ρ=π or ν). A f t e r the first isotope (isotone) is described, X ^ ( X ) is determined and is assumed to be the same for all remaining isotopes (isotones), leaving five parameters per nucleus. The goal is to determine empirically the values of these six parameters which yield the best description of the low-lying spectra of medium-to-heavy nuclei and which at the same time vary smoothly with changes in the neutron- and proton-boson numbers. It is important that the set of parameters for one set of isotopes be quite similar to the set of parameters for the neighboring series of isotopes. The values of these parameters should not vary in a random manner, if the IBM-2 is a truly meaningful description of the properties of nuclei in this mass region. So, it is definitely of interest to determine empirically these six parameters as well as possible for medium-to-heavy mass nuclei. Work along this line has been carried out (e.g., [BAR81, SCH80]), but this entire mass region has so far not been investigated within the IBM-2 approach. There is the difficulty that most IBM-2 investigations carried out to date have been done by fitting the model to the data by "eye." That is, one chooses values for the IBM-2 parameters and uses them to obtain eigenenergies and eigenvectors. These eigenenergies are compared with the experimental excitation energies for a given nucleus. New values of the parameters are selected and the process repeated until (i) a "reasonable" fit has been obtained to the experimental energies and (ii) a set of parameters has been obtained which vary smoothly with Ν and Ζ for neighboring isotopes and isotones. This procedure generally yields a satisfactory fit to the experimental data after a small number of iterations (less than 4 or 5). The advantage of this procedure is that (i) it uses less computer time than a least-squares fit to the data and (ii) it allows the person doing the fit to check on the smoothness of the parameter variations with Ν and Z. This latter feature is guided by our microscopic understanding of the IBM-2 in terms of the nuclear shell model [ARI77, B A R 81, TAL82]. To improve on this situation, we have used the Glasgow shell model code, rewritten to treat bosons [MOR80], to perform a least-squares fit to the excitation energies of a single nucleus using the IBM-2 Hamiltonian (1) without the Majorana term. In general, the model is applicable to only 10 to 12 experimentally known energy levels M
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
V
10.
Interacting Boson Model-2 Parameters
BARRETT ET A L .
67
per nucleus. Since the IBM-2 Hamiltonian used contains five or six variable parameters, there are only two or less pieces of data per parameter per nucleus in the least-squares fit. What this means is that it is possible to find several sets of parameter values which fit the experimental data equally well (i.e., have essentially the same χ value). These parameter sets can be rather different, especially in their values for χ and X . When the six parameters ε, κ, χ , χ , C , and C were all allowed to vary in the least-squares-fit procedure, a "minimum" was found in all cases studied by changes only in ε and κ, with χ , Χ , C g , and C essentially unchanged. Typical results are shown in Table 1. ν ν ζ
π
v
ν
π
0
2
v
ν
π
v
2
Table 1. IBM-2 parameter sets producing similar least-squares fits to the excitation energies for 7 | P t l
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1 1 8
Case
e(MeV) ic(MeV)
χ
π
χ
ν
C ^(MeV) C ^(MeV) χ n
2
2
Initial guesses
0.580
-0.180 -0.800
1.050
0.600
0.020
--
Vary all 6
0.568
-0.209 -0.800
1.052
0.600
0.018
3.83
0.580
-0.180
-0.800
1.083
0.600
0.020
4.94
Vary all 6, X (initial)=1.083
0.572
-0.209 -0.800
1.084
0.600
0.019
3.69
Vary only χ and X
0.580
-0.180
1.160
0.600
0.020
2.86
Vary only χ
ν
v
π
-0.853
v
Source: Reproduced with permission from [BAR84]. Copyright 1984 World S c i e n t i f i c P u b l i s h i n g Co. When the eigenvectors from these different fits for a given nucleus were then used to calculate electromagnetic properties, such as transition rates and quadrupole moments, the results were found to agree with one another within 10%. Hence, it was evident that we needed to increase the amount of data included in a given IBM-2 fit in order to " t i e down" the empirical values of the model parameters. We did this by making a least-squares fit of the IBM-2 Hamiltonian simultaneously to the excitation energies of several neighboring nuclei. For our fit we chose the isotopes of X e , B a , C e , N d , and Sm with 66
