The crystal field effects for the Ising bilayer system consisting of spin-3/2 and spin-1/2

A model consisting of two Bethe lattices each with a branching ratio of q Ising spins with one layer having spin-3/2 with nearest-neighbor (NN) interaction J(1) and crystal field interaction 1, and the other having spin-1/2 with NN interaction J(2) is considered. The layer with spin-3/2 is laid over the top of the other with spin-1/2 and the two layers are tied together via an interaction between the vertically aligned spins denoted as J(3). After obtaining the ground state (GS) phase diagrams on different possible planes depending on the given system parameters, the variations of the order-parameters and the free energy are investigated using the exact recursion relations in a pairwise approach to obtain the temperature dependent phase diagrams of the model by considering only the ferromagnetic ordering of the layers, i.e. J(1) > 0 and J(2) > 0, and ferromagnetic or antiferromagnetic ordering of the adjacent NN spins of the layers, J(3) > 0 or J(3) > 0, respectively. Besides the second-order phase transition lines with different kinds of behaviors, the first-order phase transition lines either ending at a tricritical point or at an isolated critical point are found. The model presents compensation temperatures when J(2) of the lower layer can compete with J(1) of the upper layer. The paramagnetic phase is also divided into two phases by studying the thermal behaviors of the quadrupolar moment for the layer with spin-3/2.