Statistical mechanics of a spin-one Ising model on a Bethe lattice

Exact expressions for the Curie temperature, magnetization, quadrupolar moment and the susceptibility of a spin-one Ising model on a Bethe lattice are derived. Biquadratic exchange and single-ion anisotropy are included, in addition to the bilinear exchange interactions. The variation of the critical temperature with the relative strengths of these interactions is studied. The expressions for the magnetization and quadrupolar moment are compared, where possible, to those previously reported for Ising systems on a Bethe lattice. The equivalence of our results with those obtained by a generalized constant-coupling approximation applied to a spin-one Ising model on a regular lattice is demonstrated. The magnetization curves and the thermal variation of the quadrupolar moment are studied in depth for a range of interaction parameters where unusual features appear because of the presence of an anti-Curie temperature. The temperature variation of the zero field susceptibility is also studied in this region.