Journal of Statistical Physics, Vol. 54, Nos. 1/2, 1989. p. 191-200.
We study a generalized Ising system consisting of a Bethe lattice on every site of which two spin-l/2 and two isospin-1/2 (or atomic species A) states can be realized, the spin-species interacting with appropriate nearest neighbor couplings. The system is equivalent to that of four states per site and we obtain its exact thermodynamic behavior. The case of a fixed concentration of species is the annealed or liquid magnetic binary alloy. The temperature dependence of the short-range-order (SRO) parameter of such systems is obtained as an
application of the theory and discussed in connection with relevant material from the literature. When fixing both the concentration and the nearest neighbor spatial correlation of the species, we obtain Eggarter's formulas for the "frozen-in" species problem, which therefore are only approximately valid even on the Bethe lattice.