Successive Orientational Transitions in Crystals

Abstract

An extended discussion is given of a crystal model showing successive ``rotational'' or orientational transitions. This model consists of an array of classical rotators with next-neighbor coupling, the potential energy of coupling being E(θij)=AP1(cosθij)+BP2(cosθij), where θij is the angle between the molecular axes of symmetry. It is treated by both the internal field approximation (Bragg-Williams approximation) and Chang's modification of Bethe's method. The internal field approximation is applied in two ways, the first involving use of consistency relations, the second involving calculations of the thermodynamic potentials. The consistency relations are shown to be equivalent to the condition that the free energy be stationary, but not necessarily a minimum, with respect to variation of the orientational distribution of the molecules. Depending on the relative values of A and B, the model shows a single second-order transition, a single first-order transition, two first-order transitions, or a second-order transition followed by a first-order transition as T rises. The treatment by Chang's method is not as complete as that by the internal field method, but gives confidence that the latter method is sufficiently accurate to indicate correctly the general behavior of the model. Transition temperatures are computed by both methods, latent heats and specific heats by the internal field method only. The relatively complex behavior of the model can be understood in general terms. In particular, when two transitions occur, the one at lower temperature marks a change from an ordered state in which molecular axes tend to be all parallel to a preferred direction (``ferromagnetic'' case) or alternately parallel and antiparallel (``antiferromagnetic'' case) to another ordered state in which parallel and antiparallel orientations are equally probable for each molecule; the second transition is to a state of complete long-range orientational disorder. There are suggestive similarities between the behavior of this model and the observed properties of the hydrogen and deuterium halides.