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Progress of Theoretical Physics

Effective Interaction in Nuclear Many-Body Problem

Masaru Yasuno

Published Mar 1, 1961• 1 citation
Publisher

Abstract

General treatment of nuclear many-body problem will be given in terms of Green functions and the correlation property of nucleus will be exhibited by correlation factor. Effective interaction certifying the independent particle motion is introduced and proved to identify with Brueckner's K-matrix. And on the basis of this point, applying the method of the particle-hole pair to the system, a collective mode will be derived and the effective interaction contributing to collective motion will be studied. It will also be shown that this effective interaction has the property very different from electron plasma case.

DOI
10.1143/ptp.25.411

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General treatment of nuclear many-body problem will be given in terms of Green functions and the correlation property of nucleus will be exhibited by correlation factor. Effective interaction certifying the independent particle motion is introduced and proved to identify with Brueckner's K-matrix. And on the basis of this point, applying the method of the particle-hole pair to the system, a collective mode will be derived and the effective interaction contributing to collective motion will be studied. It will also be shown that this effective interaction has the property very different from electron plasma case.
The methods of quantum statistics previously developed by the author are applied to the determination of properties of the ground state of nuclear matter. An expansion in powers of the pair-function of quantum statistics is made and expressions are derived for the momentum distribution, pair-correlation function, binding energy, and effective single-particle energies. The leading terms of these expressions can be interpreted in terms of an effective two-body interaction, and a model of nuclear matter which consists of interacting quasi-particles whose energies are the effective single-particle energies is thereby suggested. The theory is compared with Brueckner's theory and also with Landau's phenomenological theory of the Fermi liquid.
Many-body Green functions are a very efficient formulation of the many-body problem. We review the application of this method to nuclear physics problems. The formulas which can be derived are of general applicability, e.g., in self-consistent as well as in nonself-consistent calculations. With the help of the Landau renormalization, one obtains relations without any approximations. This allows to apply conservation laws which lead to important general relations. We investigate the one-body and two-body Green functions as well as the three-body Green function and discuss their connection to nuclear observables. The generalization to systems with pair correlations are also presented. Numerical examples are compared with experimental data.
The equations which determine the one-particle energy and effective two-body interaction in an interacting Fermi gas are constructed within the approximation which sums up all pair creation-annihilation processes. The equation corresponds to the familiar equation for the KKK matrix which represents the interaction between particles (or holes) and sums up the particle-particle (or hole-hole) scattering processes. The method of the equation of motion is used in this paper. Our result for the one-particle energy is shown to lead to the result previously obtained by Quinn and Ferrell, and by Rockmore for the case of the electron gas with Coulomb interactions, when we replace screened potentials by bare potential in the self-consistent energy equation. For nuclear matter, it is shown that the presence of an attractive interaction in the equation of motion for number density causes an "enhancement" of exchange forces, whereas in the electron gas repulsive Coulomb interactions lead to "screening" of the exchange force. The strength of the isospin density interaction pseudopotential is enhanced by a factor of two when one solves the self-consistent equation; and a simple estimate shows that the Goldhaber-Teller mode lies about 15% higher than the value pFqm\frac{{p}_{F}q}{m}mpF​q​ previously estimated by Glassgold et al. (qqq: momentum of the oscillation, pF{p}_{F}pF​: Fermi momentum).
Abstract The aim of this article is the representation of the theory of nuclear matter with the technique of many‐body Green functions. We have treated the theory of Green functions thoroughly in the first chapters taking support of the functional method, which offers a convenient approach to the construction of the theory. After treatment of the properties of the exact two‐point function and of the effective single‐particle potential, preconditions and properties of the approximate solutions are discussed. Results achieved todate are treated in chapter 6. Brueckner theory and allied methods are not further discussed, since these are sufficiently known. The theory of linear response, being important for the semi‐phenomenological treatment of real atomic nuclei, is presented in appendix A.
In this paper we present the theory of nuclear matter in the framework of relativistic nuclear field theory utilizing the technique of relativistic many-body Green's functions. The analytic properties of the Green's functions and related quantities are given and used to obtain the final set of equations for the nuclear matter problem. In detail, we have investigated theoretically and numerically the so-called ${\ensuremath{\Lambda}}^{00}$ and ${\ensuremath{\Lambda}}^{10}$ approximations. For comparison we also calculated the relativistic Hartree-Fock and the Brueckner-Hartree-Fock approximation, respectively. For masses and coupling strengths we used (phenomenological) parameter sets given by the Bonn and Groningen group. The results of the \ensuremath{\Lambda} approximations are similar to the relativistic Brueckner treatment.