Effective Fields in Cubic Lattices with Extended Charges

Abstract

The effective field in cubic lattices is calculated for a simple model in which the electrons have spatially extended charge distributions. For simple cubic, body-centered cubic, and face-centered cubic lattices in which the electrons in each primitive cell are infinitesimally displaced from rigid cores, the effective field can be written ${\mathrm{E}}_{\mathrm{eff}}=\mathrm{E}+(\frac{4\ensuremath{\pi}}{3})\ensuremath{\gamma}\mathrm{P}$, where E is the average electric field in the medium, and P is the polarization. The coefficient $\ensuremath{\gamma}$ varies from zero for very extended electronic charge distributions to 1 for the limit of point charges. Values of $\ensuremath{\gamma}$ for Gaussian distributions of intermediate width are given. Effective fields are also calculated for the rocksalt, zincblende, and cesium chloride structures. These results involve an additional coefficient ${\ensuremath{\gamma}}^{\ensuremath{'}}$ which also varies between 0 and 1. For moderate overlaps between electronic charge distributions of next-nearest neighbors the effective fields differ appreciably from the Lorentz field $E+(\frac{4\ensuremath{\pi}}{3})\mathrm{P}$.