Applicability of the cubic model to the critical behavior of real systems

Abstract

We have previously developed a model Hamiltonian called the cubic model to explain the salient features of the critical behavior of a group of cubic rare-earth compounds. There are interaction terms present in real systems that were neglected in this model. Here we consider how they modify the tricritical-like phase transition of the cubic model that was predicted by using the mean-field approximation. We find that for rare-earth compounds with large angular momentum $J$, the neglect of the overlap between the sixfold degenerate states is justified. Crystal fields which make a nonmagnetic state lowest in energy tend to drive the transition first order. These results which are readily obtained only within the mean-field approximation are expected to hold in better statistical approximations. Within both the mean-field and Bethe-Peierls-Weiss approximation quadrupolar pair interactions that favor parallel ordering of the moments drive the transition first order, while those favoring perpendicular ordering of the moments drive the transition second order. As the cubic model has a first-order phase transition in the Bethe-Peierls-Weiss approximation in zero fields, the transition is not tricritical. Therefore we have determined by using this approximation the size of the single-ion anisotropy, or quadrupolar pair interaction, needed to drive the system tricritical. The magnitudes required to achieve this are within the limits estimated from experimental data.