Effect of the Lattice on Dielectric Properties of an Electron Gas

Abstract

A system of $N$ electrons in the presence of a rigid periodic background of positive charge is considered. Following Martin and Schwinger, an inverse dielectric operator, ${\mathcal{K}}^{\ensuremath{'}}$, is introduced. An approximate equation which takes into account the long-range nature of the Coulomb field is derived for ${\mathcal{K}}^{\ensuremath{'}}$. A representation is used where ${\mathcal{K}}^{\ensuremath{'}}$ is a matrix with rows and columns labeled by vectors of the reciprocal lattice. Poles and zeros in the dielectric operator are found to be manifestations of Bragg's law. Assuming these to be the major effect of the lattice, the equation for ${\mathcal{K}}^{\ensuremath{'}}$ is solved. The result is examined in the weak-binding limit and seen here, except at the Bragg reflections, to agree with that of Nozi\`eres and Pines. Finally the ground-state energy of the system is exhibited.