Cluster expansion for the Blume-Capel model

The two-spin cluster approximation in the cluster expansion method is applied for the Blume-Capel model on the simple cubic lattices. Within this framework the phase diagram and the temperature dependence of the magnetization are obtained. We have found in the ordered phase a first-order line ending at an isolated critical point in the midst of an ordered phase (coexistence of two critical phases). The results are in qualitative agreement with the results of the molecular field approximation.