Statistical Mechanics of a Simple Model of a Displacive Ferroelectric

A displacive ferroelectric is represented by a model in which there is only one species of movable ion. The potential energy is taken to be an arbitrary unstable quadratic form in the ionic displacements, plus a quartic term γΣ(r⃗_i·r⃗_i)², (γ>0). The statistical mechanics is done variationally by means of approximate distribution functions containing parameters which are chosen so as to minimize the free energy. The mean-field approximation uses a distribution function of the form Πg(r⃗_i) and the self-consistent phonon approximation (SPA) uses a Gaussian in the quadratic normal coordinates, the coefficients in the Gaussian being identified with the squares of the temperature-dependent frequencies. In both approximations it is found that the ferroelectric phase transition can be either first or second order, depending on the quadratic potential. The SPA shows "softening" of the q⃗=0 mode in the paraelectric phase. The frequency of this mode vanishes when T=T_c, where T_c is the lowest temperature at which the paraelectric phase exists either stably or metastably. The SPA leads to a Curie-law susceptibility in the paraelectric phase only when the quadratic forces are of sufficiently long range.