Tri-critical behavior of the Blume Capel model on a diamond lattice

The mean field approximation results are obtained in a five-site cluster on the diamond lattice from the Bogoliubov inequality. Spin correlation identities for the Blume-Capel model on diamond lattice are derived from a five-site cluster and used to obtain an effective field approximation. The free-energy, magnetization, critical frontiers and tricritical points are obtained from the mean field approximation and the effective field approximation and are compared to those obtained by other methods. From the mean-field approximation, we also studied the unstable and metastable states besides the stable states present in the model.