Order-disorder transitions in ionic crystals containing reorientable dipoles

The dielectric properties of a crystal made of several interpenetrating lattices of ions and reorientable permanent point dipoles are investigated. The ions and dipoles are both assumed to be polarizable. A set of functions ni is introduced to describe the orientational disordering of the permanent dipoles. The electrostatic internal energy is calculated in the mean-field approximation, and from it the free energy is obtained in terms of the ni's. A transcendental system of equations for the latter is derived, from which the equilibrium configuration as a function of T can be found by numerical computation. The model predicts the possible occurence of an ordered phase (which may be ferro-, antiferro- or ferri-electric) and a disordered, para-electric phase. The electric susceptibility, specific heat and other quantities of interest are evaluated as functions of T. The formalism is applied to several simple tetragonal structures. In each case a ferro- or antiferro- to para-electric transition is found, the Curie temperature is evaluated and some interesting features of the model are discussed.