Reaction, cavity, and dipole fields in a discrete lattice

Abstract

Reaction, cavity, and dipole fields are evaluated in a simple model crystal. Aside from possible permanent dipole moments the electrical properties of isotropic molecules located on the sites of a cubic lattice are mimicked by a susceptibility appropriate to a collection of harmonic oscillators. Although qualitatively similar as a function of the dielectric constant ε, the reaction field in a discrete cubic lattice is found to be smaller than in a continuum. The cavity field in a discrete lattice is found to be larger than in a continuum and to diverge linearly in ε for large ε. It is found that the present expression for the reaction field is not in disagreement with data gleaned from studies of the dependence of chemical shifts on ε. Following the classic calculation of Onsager, the reaction and cavity fields are used to determine the dielectric constant of a polar crystal. In body centered cubic and face centered cubic lattices it is found that the expression for ε is singular at a temperature of about 0.30 TD, indicating a phase transition below the Debye temperature TD but nevertheless nonzero. Because of the anomalous dispersion of its energy bands, the simple cubic lattice undergoes a different sort of phase transition at a temperature of about 0.84 TD.