Thermodynamics of an eight-site order-disorder model for ferroelectrics

Thermodynamical properties of an order-disorder model of ferroelectrics proposed by Comes, Lambert, and Guinier for the perovskites BaTi${\mathrm{O}}_3$ and KNb${\mathrm{O}}_3$, are discussed. It is shown that in the absence of crystalline distortion induced by the motion of the tunneling ion, the only stable ordered phase is that of symmetry $C_{3v}$. The ferroelectric-paraelectric transition is a "supersingular" point where the paraelectric phase coexists with three ferroelectric phases. The lattice distortion breaks this degeneracy; unless the distortion is too drastic the system then shows three ferroelectric phases of symmetries $C_{3v}$, $C_{2v}$, and $C_{4v}$, appearing in this order for increasing temperature, in complete agreement with BaTi${\mathrm{O}}_3$ and KNb${\mathrm{O}}_3$. The calculated entropy changes associated to the phase transformations are consistently larger than the measured values, showing that a correlation between the configurations of neighboring cells is an essential feature of the model.