Interaction of Charged Particles in a Dielectric

Abstract

This paper is a sequel to an earlier study, from a many-particle standpoint, of the motion of an electron (or hole) in an insulator. We consider an insulator (whose nuclei are taken as fixed), including the electron-electron interaction. It is then shown that the static dielectric constant $\kappa$ defined in terms of the force between two distant point charges immersed in the medium, and the effective dielectric constant ${\kappa }^{*}$, which determines the force between an extra electron (or hole) and a distant point charge, are equal. These results may be summarized by the statement that, if sufficiently distant, external charges in a dielectric interact with each other and with the charge of an extra electron or hole as if all charges were renormalized according to the prescription $Q\to \frac{Q}{{\kappa }^{\frac{1}{2}}}$. The method of proof is to treat the Coulomb interactions between the electrons and between the electrons and external charges by perturbation theory and to establish a correspondence, to all orders, between the Feynman graphs which define $\kappa$ and those which define ${\kappa }^{*}$. The result is therefore exact, at least as long as the perturbation series converge.