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Progress of Theoretical Physics

On the “Anharmonic Effects” on the Collective Oscillations in Spherical Even Nuclei. I

Toshio Marumori, Masatoshi Yamamura, Akira Tokunaga

Published Jun 1, 1964• 267 citations
Publisher

Abstract

A new method which enables us to transcribe the dynamics of the system of even number of fermions into that of the boson system is developed, with the purpose of analysing the “anharmonic effects” on the collective oscillations in spherical even nuclei from the standpoint of the microscopic theory of the collective excitations. In the “harmonic approximation”, a pair of bound quasi-particles is replaced by the “phonon” as an ideal boson. This replacement inevitably leads to neglecting two main effects; the dynamical effect due to the residual interactions omitted under the “harmonic approximation” and the effect of the Pauli-principle among the quasi-particles composing the different pairs. According to our theory, the two effects which cause “anharmonic effects” can be evaluated correctly, in principle. It is not our purpose to go into detailed quantitative calculations, but rather to develop the basic idea.

DOI
10.1143/ptp.31.1009

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•Jan 1, 1979•1 citation
A new method which enables us to transcribe the dynamics of the system of even number of fermions into that of the boson system is developed, with the purpose of analysing the “anharmonic effects” on the collective oscillations in spherical even nuclei from the standpoint of the microscopic theory of the collective excitations. In the “harmonic approximation”, a pair of bound quasi-particles is replaced by the “phonon” as an ideal boson. This replacement inevitably leads to neglecting two main effects; the dynamical effect due to the residual interactions omitted under the “harmonic approximation” and the effect of the Pauli-principle among the quasi-particles composing the different pairs. According to our theory, the two effects which cause “anharmonic effects” can be evaluated correctly, in principle. It is not our purpose to go into detailed quantitative calculations, but rather to develop the basic idea.
A new method for systematic treatment of the anharmonicity effect in the many-phonon states in fermion space is proposed. The most important part of the method is a recursion formula for the norm of many-phonon states in fermion space. This recursion formula is applied to the calculation of every matrix element of an arbitrary operator. A successive approximation procedure to calculate the recursion formula is also proposed. This new method is rewritten so as to be suitable for the case of collective-phonon truncation of fermion space. The recursion formula in this case is expressed in a form expanded with respect to to what extent the anharmonicity effect due to the Pauli principle is taken into account. It is shown that the cutoff factor appearing in the boson representation of the SU(6) algebra is obtained by the lowest order of the successive approximation procedure for this expansion and the values of the cutoff factor estimated in this approximation show good fit to those obtained from the exact solution of the many-quasi-particle states which were previously calculated by the present authors. This implies that the new method proposed in this paper gives good results even in the lowest order approximation.
Collective excitations are one of the most common and interesting features of many-body systems. Of particular interest are those collective modes which can be interpreted in terms of vibrations. Nuclei show a large variety of such modes, both low-lying and high-lying. In particular the giant dipole resonance is due to the coherent motion of protons against neutrons. Its analogue in metal clusters is the dipole plasmon excitation, due to the oscillation of electrons against the positive ions. The Random Phase Approximation (RPA) is extensively used as a microscopic theory to study the basic properties of these collective excitations. In the derivation of RPA use is made of the Quasi Boson Approximation (QBA). It consists in replacing the expectation values in the correlated ground state with those in the uncorrelated (Hartree-Fock) one. Strictly related to QBA is the feature of RPA of predicting a harmonic spectrum. On the other hand, the existence of anharmonicities in the multiphonon spectra of nuclei and their relevance in various physical processes are well established. Overcoming QBA has represented the starting point of many attempts aiming at improving RPA. A recently developed approach in this line will be presented and discussed. A very nice property of RPA is to satisfy Energy Weighted Sum Rules (EWSR). However, some violations are present in several extensions of RPA. We will show how to cure this shortcoming.
The quantum theory of third order anharmonic effects in the photon production amplitude developed by V.A. Khodel has been generalized for nuclei with pairing and for a more accurate description of low-energy collective phonons. An expression for the transition amplitude between two- and one-phonon states has been obtained. This expression has been compared to the solution of a similar problem considered within the quasiparticle–phonon model. It has been shown that this approach describes a number of new effects, including three- and four-quasiparticle correlations in the ground state ( backward-going graphs ).
A nonlinear generalization of the quasiparticle random phase approximation is constructed with the aim of describing anharmonic effects in the low-lying spectra of vibrational nuclei. Angular momentum zero pairing correlations of the ground state are described by a generalized BCS calculation based on a variational method. The dynamics of the vibrational states are described by the Heisenberg equations of motion for fermion pair operators; matrix elements of products of pair operators are treated by sum rule methods. Kinematical constraints are expressed by the algebra of the pair operators and by boson-like polynomial expansions for the quasiparticle conserving multipole operators in terms of the quasiparticle pair changing operators, these expansions enforcing the content of the Pauli principle. The physical basis for the choice of approximations is described. The final algorithm is organized so as to involve little more than repeated BCS-like and RPA-like calculations, converging to a self-consistent solution of the full nonlinear scheme. Detailed discussion is carried out for a modified form of the pairing plus quadrupole interaction Hamiltonian, but the method is not confined to this model.NUCLEAR STRUCTURE Approximate many-body methods applied to modified pairing plus quadrupole model. Description of vibrational spectrum of single closed shell nuclei. Equations of motion and algebraic methods.
We propose a new boson expansion theory that does not premise the closed-algebra approximation, indispensable for formulation until now, as an extension of the conventional practical boson expansion methods that have tried to elucidate nuclear collective motion, which is a method that allows the closed-algebra approximation not to be used or to be used appropriately, enables us to obtain the boson expansion easier, and reproduces the fermion subspace onto the boson subspace more faithfully than the conventional practical methods. The two-phonon norm matrix composed of all the phonon excitation modes is investigated in detail, which reveals the mechanism, essential for the boson expansion methods, of how we should construct the fermion subspace to be mapped from the whole fermion space. The conventional practical boson expansion methods have applied the closed-algebra approximation improperly to strengthen the effect of the Pauli principle inappropriately, which should be replaced by those that do not use that approximation or use it properly.
The aim of this series of papers is to propose a microscopic theory to go beyond the situations where collective motions are described by the random phase approximation, i.e., by small amplitude harmonic oscillations about equilibrium. The theory is thus ap­ propriate for the microscopic description of the large amplitude collective motion of soft nuclei. The essential idea is to develop a method to determine the collective subspace (or submanifold) in the many-particle Hilbert space in an optimal way, on the basis of a fundamental princi­ ple called the invariance principle of the Schrodinger equation. By using the principle with­ in the framework of the Hartree-Fock theory, it is shown that the theory can clarify the structure of the so-called phonon-bands by self-consistently deriving the collective Hamil­ tonian where the number of the physical phonon is conserved. The purpose of this paper is not to go into detailed quantitative discussion, but rather to develop the basic idea.