Influence of Correlations on the Dielectric Behavior of an Electron Gas. Nonperturbative Approach to the Many-Fermion System

Abstract

Starting with the equation of motion for the density matrix operator of an electron gas in second quantization, a system of matrix equations is derived for matrix elements connecting states with various numbers of electron-hole pairs. Allowing up to $n$ electron-hole pairs and neglecting all states with more than $n$ pairs yields a set of ${(n+1)}^{2}$ coupled equations for the various amplitudes in close analogy to the $n\mathrm{th}$ order Tamm-Dancoff approximation. The starting point of this approximation, in which all matrix elements connecting the Fermi vacuum with any pair states are neglected (zero-order Tamm-Dancoff approximation), is an equation for the self-consistent field rather than for free particles as in the usual Tamm-Dancoff method. Correlation effects are therefore automatically taken into account when pair states are included. In a certain sense the approximation scheme given here can be thought of as being an expansion in the deviation of the Fermi surface from the sharp edge of the free-particle ground state, the deviation being caused by the Coulomb interaction among the particles. In this paper, we present calculations based on the first Tamm-Dancoff approximation (taking into account only one electron-hole pair). An expression for the dielectric constant will be derived and applied to a determination of the effects of correlations on the wavelength dependence of plasma oscillations. The theory will be compared with the experiments by Watanabe on the energy loss of fast electrons in metal foils and it will be seen that the agreement with experiment is good considering the inherent simplicity of the theory.