# Phase transitions in a model ferroelectric

## Abstract

A model crystal exhibiting a ferroelectric transition from a NaCl-structure phase to a distorted C3v5 phase is examined within the context of the lowest-order self-consistent phonon approximation. The difficulties associated with the usual harmonic approximation are avoided by solving a set of nonlinear integral equations for the renormalized phonon spectrum. The paraelectric phase admits a single self-consistent solution, whereas the ordered phase yields two solutions one stable and one unstable. From the properties of these solutions several important results are obtained. In particular, for the crystal stabilized by quartic anharmonicity alone the frequency of the soft zone-center TO mode of the paraelectric phase does not vanish at the transition. Indeed, the transition from the ordered to the disordered phase takes place with a discontinuous change in the optic-mode order parameter, so that the resulting transition is distinctly first order. The implications of these results with respect to previous microscopic derivations of the Devonshire free-energy expansion are discussed. © 1972 The American Physical Society.