Ordering in fcc lattices with first- and second-neighbor interactions

A prototype phase diagram for the $〈1\frac{1}{2}0〉$ family of the ordered superstructures in fcc lattices is calculated in the tetrahedron-octahedron approximation of the cluster variation method. The tetrahedron-octahedron cluster combination allows for first- and second-neighbor interactions, both essential for the $〈1\frac{1}{2}0〉$ superstructures to be ground states. Calculations are carried out for positive (antiferromagnetic) nearest-neighbor pair interactions and for a ratio of second- to first-neighbor interaction energies of 0.25. Salient features of the resulting phase diagram are tricritical points in the fcc to $A_2{B}_2$ and the fcc to $A_{3}B$ transitions, and the presence of a bicritical point at the junction of the $A_2{B}_2$ and $A_{3}B$ critical lines with the $A_2{B}_2$ to $A_{3}B$ first-order transition line. The $A_{2}B$ phase of the $〈1\frac{1}{2}0〉$ family is found to be stable at relatively low temperatures and in a very narrow composition range. Stability of the disordered state is investigated in $k$ space and the $〈1\frac{1}{2}0〉$ ordering spinodal is determined.