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Physics Letters B

The one nucleon transfer operator in the microscopic IBM without NOA

José Barea, Clara E. Alonso, J. M. Arias

Published Aug 19, 2014• 7 citations
Publisher

Abstract

The mapping of the single fermion creation operator (ci†) onto the Interacting Boson–Fermion space (IBFM) is revisited within the Generalized Seniority scheme. In the original work the Number Operator Approximation (NOA) was used. Here the exact evaluation of the relevant terms using exact values for the fermion matrix elements in the Generalized Seniority scheme is worked out. This provides a new, improved, single particle transfer operator to be used in IBFM.

DOI
10.1016/j.physletb.2014.08.026

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The mapping of the single fermion creation operator (ci†) onto the Interacting Boson–Fermion space (IBFM) is revisited within the Generalized Seniority scheme. In the original work the Number Operator Approximation (NOA) was used. Here the exact evaluation of the relevant terms using exact values for the fermion matrix elements in the Generalized Seniority scheme is worked out. This provides a new, improved, single particle transfer operator to be used in IBFM.
We derive an approximate boson-quasifermion mapping of a single-j shell-model algebra, applicable to the seniority classification scheme, as an expansion in the ratio of the seniority quantum number to the level degeneracy, with selective summation of higher order terms. The procedure used is an extension of the algebraic technique first applied to the purely bosonic part of the mapping by Bonatsos, Klein, and Li. Only the maps of the generators are needed to derive a Hamiltonian for the interacting boson fermion model (IBFM), and these can be obtained in Hermitian form without having to go through the intermediary of a mapping of the fermion operators. Application of the results is illustrated by a fresh discussion of the so-called exchange interaction of the IBFM. A consistent mapping of the single fermion in terms of bosons and quasifermions is also derived that includes a consideration of the unusual properties of the quasifermions.
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.
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 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 discuss microscopic ground of the IBM from the quasiparticle viewpoint, paying attention to the effects of nucleon-number conservation. The OAI mapping is revisited, showing for a single-j system how the number conservation effects are taken into account. The number-conserving quasiparticle approximation exploited by Li is applied to incorporate the number conservation effects. The significance of the number conservation is confirmed via the tests in the Ginocchio model and in a 2j system. An effective boson number is evaluated based on the number-conserving quasiparticle method. It is shown that the nucleon-number conservation effects are well incorporated in the IBM, even for systems with non-degenerate single-particle orbits.