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Physical Review C

Mixing of Collective States in an Exactly Soluble Three-Level Model

N. Meshkov

Published Jun 1, 1971• 14 citations
Publisher

Abstract

A general class of exactly soluble three-level models is presented for an $N$-fermion system. Assuming a monopole-monopole interaction among the fermions allows one to express the model Hamiltonian in terms of the generators of the $S{U}_{3}$ group. A specialized version of this class of models is adapted for the study of mixing two collective states of opposite parity, analogous to the mixing of giant-dipole-resonance and surface quadrupole vibrations in nuclei. Exact results of the model are studied as a function of interaction parameters and compared with approximations including up to three-particle-three-hole excitations. It is shown that in the model many-particle-many-hole states play an important role in the mixing of collective states and cannot be properly neglected. The implications of this result for the study of actual nuclei is discussed.

DOI
10.1103/physrevc.3.2214

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A general class of exactly soluble three-level models is presented for an NNN-fermion system. Assuming a monopole-monopole interaction among the fermions allows one to express the model Hamiltonian in terms of the generators of the SU3S{U}_{3}SU3​ group. A specialized version of this class of models is adapted for the study of mixing two collective states of opposite parity, analogous to the mixing of giant-dipole-resonance and surface quadrupole vibrations in nuclei. Exact results of the model are studied as a function of interaction parameters and compared with approximations including up to three-particle-three-hole excitations. It is shown that in the model many-particle-many-hole states play an important role in the mixing of collective states and cannot be properly neglected. The implications of this result for the study of actual nuclei is discussed.
An exactly soluble three-level model for a system of fermions with Hamiltonian formed from generators of the group SU(3) is studied. A basis for the representation of which the ground state is a member is constructed. It is demonstrated that in this representation, the generators can each be expanded in a series as functions of a pair of ``kinematical'' boson operators; the series are uniquely determined to satisfy the operator algebra and the invariants of the representation by a method of Marumori. It is seen that the lowest anharmonic approximation to the Hamiltonian and other operators yields excellent numerical agreement with exact results for all regimes of interaction strength considered. An alternative description of the system in terms of a dynamically more meaningful boson, called the ``physical'' boson, is shown to be appropriate for relatively weak coupling where one has a near harmonic spectrum.
The present paper attempts to give independent descriptions of the collective degrees of freedom of nuclei such as the monopole and quadrupole ones in order to avoid the seniority mixing problem. This is done by the operatorized Bogoliubov transformation (OBT) which separates the degree of freedom from the rest described by the seniority particle operator which is not coupling to the pairs, and in terms of which the independent pair is composed. The OBT is derived in such a way that it can be generalized easily to include a new kind of “seniority particle operator” neither coupling to the nor to the pairs. In terms of this new particle operator the independent pair or the independent odd particle in odd nuclei is constructed.
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.
We discuss a multistep variational approach to collective excitations. The approach is developed in a boson formalism (bosons representing particle-hole excitations) and based on an iterative sequence of diagonalizations in subspaces of the full boson space. The purpose of these diagonalizations is that of searchingf or the best approximation of the ground state of the system. The procedure also leads us to define a set of excited states and, at the same time, of operators which generate these states as a result of their action on the ground state. We examine the cases in which these operators carry one-particle-one-hole and up to two-particle-two-hole excitations. We also explore the possibility of associating bosons to Tamm-Dancoff excitations and of describing the spectrum in terms of only a selected group of these. Tests within an exactly solvable three-level model are provided.
For pt.I see ibid., vol.14, p.2581 (1981). If one begins with N non-interacting fermions, the Pauli principle can be easily incorporated by the use of Slater determinants. This is not the case for N interacting fermions. The authors consider exactly soluble 'Coulomb-type' quantum systems in three dimensions of N interacting identical spin- 1 / 2 -fermions. A systematic procedure for constructing Pauli antisymmetry adapted wavefunctions is given. The resulting antisymmetric wavefunctions are labelled by conserved 'good' quantum numbers. In particular, for N=2, all the physically acceptable states are obtained. For N=3, they present a class of antisymmetric states which consists of all the ground states, all the first excited states and the states obtained by the hyperradial excitations of these. For N=2 and 3, the ground states of the authors' model systems are found to be 1 S(s 2 ) and 2 P(s 2 p), respectively, in the quantum numbers of the interacting system.
The monopole (L=0) and quadrupole (L=2) strength distributions in normal 3He^{3}\mathrm{He}3He clusters are calculated within the random-phase approximation. We use a phenomenological, zero-range $^{3}\mathrm{\ensuremath{-}}^{3}$He interaction to generate the Hartree-Fock single-particle spectrum and the residual particle-hole interaction. The evolution of the collective modes with the number of atoms in the cluster is discussed.