An Ising bilayer system consisting of spin-32 and spin-12 atoms

A bilayer Ising model consisting of two Bethe lattices each with a branching ratio of q Ising spins with one of the layers having only spin-32 and the other having only spin-12 is laid over the top of the other and the two layers are tied together via an interaction between the vertically aligned spins. The problem was studied by using the exact recursion relations in a pairwise approach in terms of the intralayer bilinear interactions J1 and J2 of the upper and lower layers, respectively, and the interlayer bilinear interaction J3. After obtaining the ground state phase diagrams on the (J2/|J1|,J3/q|J1|) plane with either J1>0 or J1<0, the variations of the order-parameters and the free energy were analyzed to obtain the temperature dependent phase diagrams. They were calculated for only the ferromagnetic ordering in each of the layers and ferromagnetic or antiferromagnetic ordering of the adjacent nearest-neighbor (NN) spins of the layers. It was found that the system presents both second- and first-order phase transitions, besides the isolated critical and triple points. The model also presents compensation temperatures when J2 of spin-12 layer can compete with J1 of spin-32 layer.