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Journal of Mathematical Physics

Boson basis realization for shell-model fermion problem

Y. K. Gambhir, R. S. Nikam, C. R. Sarma, Javid A. Sheikh

Published Aug 1, 1985• 7 citations
Publisher

Abstract

The boson basis for the fermion shell-model problem has been realized over a representation space Vn(n−1)/2 of the unitary group U(n(n−1)/2). A contraction of this space was found to yield the generator algebra of U(n). Based on this result it is shown that the transformations induced by the Dyson-mapped boson Hamiltonian on the Dyson boson basis are identical to the corresponding transformations by the fermion Hamiltonian on the shell-model basis.

DOI
10.1063/1.526828

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The boson basis for the fermion shell-model problem has been realized over a representation space Vn(n−1)/2 of the unitary group U(n(n−1)/2). A contraction of this space was found to yield the generator algebra of U(n). Based on this result it is shown that the transformations induced by the Dyson-mapped boson Hamiltonian on the Dyson boson basis are identical to the corresponding transformations by the fermion Hamiltonian on the shell-model basis.
This Brief Report deals with the solution of the shell-model Hamiltonian in the boson representation. In the present approach, the fermion Hamiltonian is exactly mapped to the boson space using the Dyson boson mapping. The problem of mapping the basis states is resolved by employing the intrinsic boson basis. The method has been applied to a model Hamiltonian consisting of a monopole pairing and a deformation force. It is observed that only s and d bosons are dominant in the intrinsic wave function for the model case studied. This is synonymous to the interacting boson model approach.
An algorithm for obtaining a recursive algebraic expression for the various expansion coefficients appearing in the physical boson basis states is presented, using the Dyson boson mapping. It is observed that only the first few terms of the physical boson basis states, expressed in the present form, contribute to the matrix element of the Hamiltonian. Explicit numerical calculations have been performed for Ni and Sn isotopes in the obtained physical boson basis states. The pairing Hamiltonian has been used as the model interaction. The results are more than 90% in agreement with exact shell model calculations.
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 systems 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.
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.