Mean-field study of the degenerate Blume–Emery–Griffiths model in a random crystal field11Work partially supported by the Brazilian agencies CNPq and FINEP.

The degenerate Blume–Emery–Griffiths (DBEG) model has recently been introduced in the study of martensitic transformation problems. This model has the same Hamiltonian as the standard Blume–Emery–Griffiths (BEG) model but, to take into account vibrational effects on the martensitic transition, it is assumed that the states S=0 have a degeneracy p (p=1 corresponds to the usual BEG model). This model was studied by E. Vives et al. for a particular value of Δ, through a mean-field approximation and numerical simulation. When the parameter p increases, the ferromagnetic phase shrinks and the region where the transition is of first order increases. In some materials, however, the transition would be better described by a disordered DBEG model; further, the inclusion of disorder in the DBEG model may be relevant in the study of shape memory alloys. From the theoretical point of view, it would be interesting to study the consequence of conflicting effects: the parameter p, which increases the first-order phase-transition region, and disorder in the crystal field, which tends to diminish this region in three dimensions. In order to study this competition in high-dimensional systems, we apply a mean-field approximation: it is then possible to determine the critical behavior of the random DBEG model for any value of the interaction parameters. Finally, we comment on (preliminary) results obtained for a two-dimensional system, where the randomness in the crystal field has a more drastic effect, when compared to the three-dimensional model.