Variational method study of the spin-1 Ising model with biaxial crystal-field anisotropy

The phase diagram of the spin-1 Ising model in the presence of a biaxial crystal-field anisotropy is studied within the framework of a variational approach, based on the Bogolyubov inequality for the free energy. We have investigated the effects of a transverse crystal-field $D_y$ on the phase diagram in the $T-D_x$ plane. Results obtained by using effective-field theory (EFT) on the honeycomb (% $z=3$), square ($z=4$) and simple cubic ($z=6$) lattices ($z$ is the coordination number) show only continuous phase transitions, while the variational approach presents first-order and continuous phase transitions for $D_y=0$% . We have also used the EFT for larger values of $z$ and we observe the presence of tricritical points in the phase diagrams, for $z\ge 7$, in accordance with the variational approach results.