Long-Wave Lattice Dynamics of the Fluorite Structure

Abstract

A simple shell model is used to calculate expressions for the long-wavelength vibrational eigenfrequencies and elastic constants of the fluorite (Ca${\mathrm{F}}_2$ type) lattice. The concept of effective charge is introduced and expressions analogous to the Szigeti relations for lattices with diatomic unit cells are given. These and other relations between the acoustic and optical properties derived for the dipole shell model are in better agreement with the measured properties of the alkaline-earth fluorides than are those derived for a rigid-ion model. The large deviation from the Cauchy relation $C_{12}=C_{44}$ seems to be largely explainable in terms of the shell-model correction to the internal strain.