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From the Shell Model to the Interacting Boson Model

Joseph N. Ginocchio, Calvin W. Johnson

Published Jul 1, 1994• 0 citations
OpenAlex

Abstract

Starting from a general, microscopic fermion-pair-to-boson mapping of a complete fermion space that preserves Hermitian conjugation, we show that the resulting infinite and non-convergent boson Hamilitonian can be factored into a finite (e.g., a 1 + 2-body fermion Hamiltonian is mapped to a 1 + 2-body boson Hamiltonian) image Hamilitonian times the norm operator, and it is the norm operator that is infinite and non-convergent. We then truncate to a collective boson space and we give conditions under which the exact boson images of finite fermion operators are also finite in the truncated basis.

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Starting from a general, microscopic fermion-pair-to-boson mapping of a complete fermion space that preserves Hermitian conjugation, we show that the resulting infinite and non-convergent boson Hamilitonian can be factored into a finite (e.g., a 1 + 2-body fermion Hamiltonian is mapped to a 1 + 2-body boson Hamiltonian) image Hamilitonian times the norm operator, and it is the norm operator that is infinite and non-convergent. We then truncate to a collective boson space and we give conditions under which the exact boson images of finite fermion operators are also finite in the truncated basis.
Starting from a general, microscopic fermion-to-boson mapping that preserves Hermitian conjugation, we discuss truncations of the boson Fock space basis. We give conditions under which the exact boson images of finite fermion operators are also finite [e.g., a (1+2)-body fermion Hamiltonian is mapped to a (1+2)-body boson Hamiltonian] in the truncated basis. For the most general case, where the image is not necessarily exactly finite, we discuss how to make practical and controlled approximations.
After a brief review of various mappings of fermion pairs to bosons, we rigorously derive a general approach. Following the methods of Marumori and Otsuka, Arima, and Iachello, our approach begins with mapping states and constructs boson representations that preserve fermion matrix elements. In several cases these representations factor into finite, Hermitian boson images times a projection or norm operator that embodies the Pauli principle. We pay particular attention to truncated boson spaces, and describe general methods for constructing Hermitian and approximately finite boson image Hamiltonians. This method is akin to that of Otsuka, Arima, and Iachello introduced in connection with the interacting boson model, but is more rigorous, general, and systematic.
I derive in a novel fashion exact formulas for the calculation of general matrix elements, including the overlap (norm) matrix, between states constructed from fermion pairs. Mapping the fermion pairs to bosons, I show how to construct finite and exact (in the sense of preserving matrix elements) boson representations of the norm operator and one- and two-fermion operators. This may lead to a microscopic basis for the Interacting Boson Model, as well as new truncation schemes for the nuclear shell model.
After a brief review of various mappings of fermion pairs to bosons, we rigorously derive a general approach. Following the methods of Marumori and Otsuka, Arima, and Iachello, our approach begins with mapping states and constructs boson representations that preserve fermion matrix elements. In several cases these representations factor into finite, Hermitian boson images times a projection or norm operator that embodies the Pauli principle. We pay particular attention to truncated boson spaces, and describe general methods for constructing Hermitian and approximately finite boson image Hamiltonians, including effective operator theory to account for excluded states. This method is akin to that of Otsuka, Arima, and Iachello introduced in connection with the Interacting Boson Model, but is more rigorous, general, and systematic.
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.