The critical behavior of the mixed spin-1 and spin-2 Ising ferromagnetic system on the Bethe lattice

The phase diagrams of the mixed spin-1 and spin-2 Blume-Capel Ising ferromagnetic system (J>0) with two crystal-field interactions, DA for spin-1 and DB for spin-2, is studied on the Bethe lattice by using the exact recursion equations. The exact expressions of the order-parameters for the sublattices with spin-1 and spin-2, and the Curie temperature and the free energy of the system are obtained in terms of the recursion relations. The phase diagrams on the (kTc/J,DA/J) and (kTc/J,DB/J) planes are studied for selected values of DB/J and DA/J, respectively, for the coordination numbers q=3,4, and 6. Besides the second-order phase transition lines, the lines of the first-order phase transitions terminating either at a tricritical point or at an isolated critical point are found.