We consider a mixed spin-3/2 and spin-2 Ising model consisting of two sublattices A and B, which are arranged alternately. In this system, the sites of sublattice A are occupied by spin
, which take spin values ±3/2 and ±1/2, while those of the sublattice B are occupied by spins
, which take spin values ±2, ±1 and 0. The Hamiltonian of the system is given by
where 
indicates a summation over all pairs of nearest-neighboring sites and the first summation is carried out only over nearest neighbour pairs of spins on different sublattices and J (J < 0) is the nearest-neighbour exchange parameter, DA is the crystal field interaction constant of spin-2 ions and DB is that of spin-3/2 ions. h is the external magnetic field acting on the lat tice.
In order to treat the model approximately we employ a variational method based on the Bogoliubov inequality for the Gibbs free energy which is given by the inequality, 
, where 
is the true free energy of the model described by the Hamiltonian (1) and 
is given by the relation
is the average free energy of a trial Hamiltonian 
and 
denotes a thermal average over the ensemble defined by
.
As the conventional procedure, the trial Hamiltonian is assumed to be in the form
where 
and 
are the two variational parameters related to the molecular fields acting on the two different sublattices, respectively.
By evaluating Equation (2), it is easy to obtain the expression of the free energy per site in MFA
where
, N is the total number of sites of the lattice and z is the number of the nearest neighbors of every ion in the lattice. mA and mB are the sublattice magnetizations per site which are defined by Equations (5) and (6) below:
Now, by minimizing the free energy in Equation (4) with respect to 
and
, we obtain
The mean field properties of the present system are then given by Equations (5)-(7). As the set of Equations (5)-(7) have in general several solutions for the pair, the
pair chosen is that which minimizes the free energy. given in Equation (5). We are here interested in studying the thermal variation of the sublattice magnetizations and the averaged total magnetization per site which defined as
On the other hand, the sublattice initial susceptibilities
are defined by
.
From which the total initial susceptibility per site is given by
, namely the ordered ferrimagnetic phases
, and selected values of
. Notice that the selection of 
corresponds to the crossover from the O1 to the O2 phase (see the ground state phase diagram 
, the sublattice magnetizations mA and mB have standard characteristic convex shape. When 
(slightly above the boundary between the phase O1 and the phase O3 in the ground state-phase diagram, where
) the sublattice magnetization mA may exhibit a rather rapid decrease from its saturation value mA = 2.0 with the increase of temperature from T = 0 K to a certain temperature T. When 
(at the boundary between the ordered phase O1 and the ordered phase O2 in the ground state phase-diagram), the saturation value of mA is 1.5, which indicates that the half of the spins on the sublattice A are equal to +2 (or −2 as well) and the other half are equal to +1 (or −1 as well). Note that this mixed state persists as long as 
and
. When 
the ground state phase is O2 phase, with mA = 1.0 at T = 0 K. However, in this case the thermal variation of mA exhibits an interesting feature which is the initial rise of mA with the increase of temperature before decreasing to zero value at the critical point Tc.
(slightly below the boundary between the ordered phase O2 and the disordered phase D1 in the ground state
), the sublattice magnetization mA exhibits a rapid decrease before it decreases normally by increasing the value of 
to the critical point Tc. In this case, a large magnetization jump is observed at the critical point, indicating a first-order transition. On the other hand, for all values of 
the sublattice magnetization mB decreases normally by increasing the value of 
to the critical point Tc, even though it is coupled to mA. The previous results for sublattice magnetization are similar to those observed in the Mixed Spin-3/2 and Spin-2 Ising Ferrimagnetic System within the Effective- field Theory [
, when
. In this case, the selection of 
corresponds to the crossover from the O1 to the O3 phase. When
, the sublattice magnetization mA may show normal behaviour. When 
(slightly above the boundary between the ordered phase O1 and the ordered phase O2, where
) the magnetization curve mB may exhibit a rather rapid decrease from its saturation value (mB = −3/2) at T = 0 K, while for the value of 
(slightly below that boundary), there is a rapid increase of mB from the saturation value (mB = −3/2) with the increase in T. When the value of
, the saturation value of the sublattice magnetization mB at T = 0 K is (mB = −1.0). It indicates that at this point, the spin configuration of 
in the ground state consists of the mixed state; half of the spins on the sublattice B are equal to −3/2 (or +3/2 as well) and the other half are equal to −1/2 (or +1/2 as well). It is also seen from 
, the sublattice magnetization mB decreases normally from its saturation value (mB = −1/2) to vanish at the critical tem-