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The magnetic properties of a mixed Ising ferrimagnetic system consisting of spin-3/2 and spin-2 with different single ion anisotropies and under the effect of an applied longitudinal magnetic field are investigated within the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. The ground-state phase diagram is constructed. The thermal behaviours of magnetizations and magnetic susceptibilities are examined in detail. Finally, we find some interesting phenomena in these quantities, due to applied longitudinal magnetic field.

Recently, there have been many theoretical studies of mixed-spin Ising ferrimagnetic systems. These systems have been of interest because they have less translational symmetry than their single-spin counterparts, since they consist of two interpenetrating unequivalent sublattices. For this reason, They are studied not only for purely theoretical interest but also because they have been proposed as possible models to describe a certain type of ferrimagnetic systems such as molecular-based magnetic materials [1-3] which are of current interest. Moreover, the increasing interest in these systems is mainly related to the potential technological applications of these systems in the area of thermomagnetic recording [4,5]. Therefore, the synthesis of new ferrimagnetic material is an active field in material science.

One of these models to be studied was the mixed-spin Ising system consisting of spin-1/2 and spin-S (S > 1/2) in uniaxial crystal field. The model for different values of S (S > 1/2) has been investigated by exact on honeycomb lattice [6-8], as well as on Bethe lattice [9,10], mean-field approximation [

In this paper, our aim is to investigate the magnetic properties of the mixed spin-3/2 and spin-2 Ising system in the presence of longitudinal magnetic field within the framework of the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. The outline of this work is as follows. In Section 2, we define the model and present the mean-field theory based on Bogoliubov inequality for the Gibbs free energy for the mixed-spin system with the applied longitudinal magnetic field. In Section 3, we discuss the temperature dependences of the sublattice and total magnetizations and sublattice and total susceptibilities for selected values of single-ion anisotropies. Finally, In Section 4 we present our conclusions.

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, D_{A} is the crystal field interaction constant of spin-2 ions and D_{B} 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. m_{A} and m_{B} 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

We begin with the ground-state structure of the system. At zero temperature, we find four phases with different values of, namely the ordered ferrimagnetic phases

and disordered phases

where the parameters q_{A} and q_{B} are defined by:

From Hamiltonian (1) and by comparing the groundstate energies of the different phases, the ground-state phase diagram can be determined, and is shown in

In this subsection, let us at first examine the temperature dependence of the sublattice magnetizations m_{A} and m_{B} for the system. The results are depicted in

, and selected values of. Notice that the selection of corresponds to the crossover from the O_{1} to the O_{2} phase (see the ground state phase diagram _{A} and m_{B} have standard characteristic convex shape. When (slightly above the boundary between the phase O_{1} and the phase O_{3} in the ground state-phase diagram, where) the sublattice magnetization m_{A} may exhibit a rather rapid decrease from its saturation value m_{A} = 2.0 with the increase of temperature from T = 0 K to a certain temperature T. When (at the boundary between the ordered phase O_{1} and the ordered phase O_{2} in the ground state phase-diagram), the saturation value of m_{A} 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 O_{2} phase, with m_{A} = 1.0 at T = 0 K. However, in this case the thermal variation of m_{A} exhibits an interesting feature which is the initial rise of m_{A} with the increase of temperature before decreasing to zero value at the critical point T_{c}.

As shown in _{2} and the disordered phase D_{1} in the ground state), the sublattice magnetization m_{A} exhibits a rapid decrease before it decreases normally by increasing the value of to the critical point T_{c}. 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 m_{B} decreases normally by increasing the value of to the critical point T_{c}, even though it is coupled to m_{A}. 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 [

_{1} to the O_{3} phase. When, the sublattice magnetization m_{A} may show normal behaviour. When (slightly above the boundary between the ordered phase O_{1} and the ordered phase O_{2}, where) the magnetization curve m_{B} may exhibit a rather rapid decrease from its saturation value (m_{B} = −3/2) at T = 0 K, while for the value of (slightly below that boundary), there is a rapid increase of m_{B} from the saturation value (m_{B} = −3/2) with the increase in T. When the value of, the saturation value of the sublattice magnetization m_{B} at T = 0 K is (m_{B} = −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 _{B} decreases normally from its saturation value (m_{B} = −1/2) to vanish at the critical tem-