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Physical Review C

Comparison of two approaches to microscopy of the interacting boson model

Jan Dobeš, P. Navrátil, Olaf Scholten

Published Jun 1, 1992• 1 citation
Publisher

Abstract

Two approaches to the microscopical derivation of the interacting boson model are examined and compared. The first one is the well-known Otsuka-Arima-Iachello method as extended to a multishell case by using the generalized seniority scheme or exact calculations in the seniority equal to 0 and 2 space. The second one is a procedure based on the similarity transformed Dyson mapping. Close agreement between the two approaches is found.

DOI
10.1103/physrevc.45.2795

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Two approaches to the microscopical derivation of the interacting boson model are examined and compared. The first one is the well-known Otsuka-Arima-Iachello method as extended to a multishell case by using the generalized seniority scheme or exact calculations in the seniority equal to 0 and 2 space. The second one is a procedure based on the similarity transformed Dyson mapping. Close agreement between the two approaches is found.
The microscopic basis of the IBM is discussed in this paper, presenting a derivation of the IBM system from the multi-nucleon system. Although there could be different approaches to this goal as stated in the preface of this supplement, we focus on two approaches. One is the Otsuka-Arima-Iachello mapping, which works for the spherical and near-spherical nuclei. The other is a mapping for deformed nuclei. In this paper, we discuss mapping methods for deriving the Interacting Boson Model (IBM). l) We first present a mapping method for spherical and near-spherical nuclei, usually referred to as the Otsuka-Arima-Iachello (OAI) mapping. 2 ), 3 ) The OAI mapping is based on the seniority scheme as will be discussed in detail. This mapping has been developed so that realistic calculations can be carried out as presented in a subsequent paper. In this paper, the basic concepts of the mapping are discussed rather in detail. Furthermore, the proton-neutron Interacting Boson Model (IBM-2) is discussed as a natural consequence of this mapping. 2 ), 3 ) In the second half of this paper, a mapping method for strongly deformed nuclei 8 ) will be reviewed. Although this mapping is for deformed nuclei, it is somewhat related to the OAI mapping. The work on deformed nuclei is much before any sort of completion, and considerable effort should be made in the future. For this purpose, the Quantum Monte Carlo Diagonalization method proposed recently 4 )- 7 ) may be useful. When the IBM was proposed from the phenomenological viewpoint by Arima and Iachello, the microscopic picture of bosons of this model was not known as pointed out in the preface of this supplement. The microscopic theory has made a crucial contribution even to phenomenological studies by the IBM, also as mentioned in the preface. Before starting rather detailed discussions, it may be useful to overview relevant properties of the effective nucleon-nucleon interactions. The short-range nuclear force favors two nucleons lying close to each other. This means that, if the wave functions of the two neutrons have large spatial overlap, the matrix element of this interaction becomes larger. On the other hand, two identical fermions cannot occupy the same quantum state, and this is the case for two neutrons. The next optimum case for gaining energy is that the two neutrons are moving on the same orbital but in opposite directions. Because the direction is opposite, the quantum states of the two neutrons are different. In this case, the total angular momentum of the two-neutron system is zero, because the rotation is completely cancelled. In fact, the
We show that the S-D subspaces, which are used in the Otsuka-Arima-Iachello microscopic derivation of the interacting-boson model, form a particle-hole-symmetric family. Consequently, there exist particle-hole-symmetric prescriptions for determining the structure of the S and D pairs. This result holds independently of whether the Hamiltonian conserves generalized seniority. Nevertheless, there are deviations from particle-hole symmetry when boson matrix elements involving more than two d bosons are calculated in lowest order using the boson mapping procedure of Otsuka, Arima, and Iachello. These deviations are used to estimate the inaccuracies introduced by the lowest-order mapping.
We discuss the construction of ideal space seniority images of single fermion operators which proceeds via the generalized Dyson mapping and a subsequent similarity transformation. Particular attention is paid to the implications of a consistent treatment of single-particle and collective degrees of freedom. The results are compared with expressions for the mapped single fermion operators previously obtained, respectively, within the Otsuka-Arima-Iachello framework and by using the prescriptions of nuclear field theory. We point out certain inconsistencies in existing constructions of interacting-boson-fermion model operators in general and the interacting-boson-fermion model quadrupole operator in particular.
The qualitative ideas put forward by Geyer and Lee are given quantitative content by constructing a similarity transformation which reexpresses the Dyson boson images of the single-jjj shell fermion operators in terms of seniority bosons. It is shown that the results of Otsuka, Arima, and Iachello, or generalizations thereof which include ggg bosons or even bosons with J>4J>4J>4, can be obtained in an economic and transparent way without resorting to any comparison of matrix elements.
The validity of the essential ingredients of the microscopic formulation of the interacting boson model: (i) truncation of the shell-model or fermion space and (ii) the procedures for mapping the truncated fermion space on to the boson space, is tested over various (spherical, deformed and transitional) regions both in the simplistic single j-shell, identical boson sys­ tems and the more realistic multi j-shell, proton-neutron systems depending on the feasibility of such calculations. In the situations where the truncation and mapping procedures fail to reproduce the exact shell-model results, it is demonstrated that renormalization and higher order mapping procedures cure the associated problems, respectively and provide excellent agreements. In addition, a Hamiltonian of the interacting boson-fermion model which is relevant for a study of odd mass nuclei is derived for a single j-shell case in the shell-model framework, which yields results identical to those in the shell-model for the monopole pairing interaction.