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Progress of Theoretical Physics Supplement

Microscopic Basis of the Interacting Boson Model

Takaharu Otsuka

Published Jan 1, 1996• 7 citations
Publisher

Abstract

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

DOI
10.1143/ptps.125.5

Related papers

Microscopic calculation for deformed nuclei

Takaharu Otsuka

The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.

•Sep 24, 1984•0 citations

Comparison of two approaches to microscopy of the interacting boson model

Jan Dobeš, P. Navrátil, Olaf Scholten

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.

Physical Review C•Jun 1, 1992•1 citation

Microscopic Foundations of the Interacting Boson Model from the Shell-Model Point of View

Naotaka Yoshinaga, T. Mizusaki, A. Arima +1 more

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.

Progress of Theoretical Physics Supplement•May 16, 2013•1 citation

Microscopic Foundations of the Interacting Boson Model from the Shell-Model Point of View

Naotaka Yoshinaga, T. Mizusaki, Akito Arima +1 more

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.

Progress of Theoretical Physics Supplement•Jan 1, 1996•50 citations

Fermion to boson mappings revisited

Joseph N. Ginocchio, Calvin W. Johnson

We briefly review various mappings of fermion pairs to bosons, including those based on mapping operators, such as Belyaev-Zelevinskii, and those on mapping states, such as Marumori; in particular we consider the work of Otsuka-Arima-Iachello, aimed at deriving the Interacting Boson Model. We then give a rigorous and unified description of state-mapping procedures which allows one to systematically go beyond Otsuka-Arima-Iachello and related approaches, along with several exact results.

arXiv:nucl-th/9408014•Aug 15, 1994•0 citations

Operators

FRANCESCO IACHELLO, P. Van Isacker

Introduction In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons. The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.

Cambridge University Press eBooks•Aug 30, 1991•0 citations
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
The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.
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.
Introduction In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons. The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.