The critical behavior of the mixed ferrimagnetic ternary alloy on the Bethe lattice

The critical behavior of the mixed Ising model of the type AB$_p$C$_{1-p}$ ternary alloy consisting of spins $\sigma=1/2$, $S=1$, and $m=3/2$ is investigated on the Bethe lattice by using the exact recursion relations. The exact expressions for the magnetizations and magnetic susceptibilities are found, and thermal behaviors of the magnetizations and susceptibilities are studied. We also investigate the behavior of magnetizations and susceptibilities as a function of the crystal-field interaction or single-ion anisotropy. We construct the phase diagram and find that the system always undergoes a second-order phase transition for the coordination number $q\le3$ and a second- and first-order phase transitions for $q>3$; hence the system gives a tricritical point. The system also exhibits the reentrant behaviors.