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The mixed spin-2 and spin-3/2 Blume-Emery-Griffiths (BEG) Ising ferrimagnetic system is studied by the Bethe lattice approach. The ground-state phase diagram is constructed. The influence of the crystal-field and the biquadratic interactions among neighboring spins on the thermal behaviors of the system is singled out. The system displays very rich critical behaviors with the existence of tricritical points. Compensation points where the global magnetization of the system vanishes have been detected for appropriate values of the system parameters.

Ising systems have attracted much interest in the three last decades because of their critical behaviors. Mixed Ising systems, beyond their theoretical purposes, have been proposed as possible systems to describe ferrimagnetic materials [

One of the earliest, simplest and the most extensively studied mixed-spin Ising model is the spin-1/2 and spin-1 mixed system. Different approaches have been used: renormalization-group technique [

Recently, these investigations have been extended to high order mixed spin ferrimagnetic systems in order to study their magnetic properties. Bobak et al. [

In this work, we study the mixed spin-2 and spin-3/2 Blume-Emery-Griffiths (BEG) ferrimagnetic system on the Bethe lattice in terms of exact recursion equations to investigate the influence of the crystal-field and biquadratic spin interactions on the critical behaviors of the model. It has been shown that the partition function in the Bethe lattice approach is that of an Ising model in the Bethe-Peierls approximation [

The remainder of this work is organized as follows. In Section 2, a brief formulation of the Bethe lattice approach is given. Section 3 is devoted to the formulation of the critical temperatures of the model. In Section 4, besides the ground-state phase diagram, the thermal properties of the model are presented and discussed in details in the model parameters’ space. Some concluding remarks are given in the last section.

A Bethe lattice is an infinite Cayley tree, i.e. a connected graph without circuits. It consists of a central spin

The Hamiltonian of the system is given by:

where each spin

The partition function of the model reads:

where

where,

Advancing along any branch, we get a site that is next-nearest to the central spin, hence

Let us give some examples of the calculated

In order to find the recursion relations, we introduce the following variables as a ratio of

and for the spin-

The BEG model is characterized by two order parameters, the magnetization M and the quadrupolar moment Q. Four order parameters:

They are easily expressed in terms of the recursion relations, namely Equation (10), and calculated as:

Similarly, we get:

The energy F of the system is defined as

Then, the phase diagrams of the system for a given coordination number q are obtained by studying the thermal variations of the order parameters and the free energy.

In the thermodynamic limit,

Also, in this case, substituting

Usually, multiple solutions of

perature that we take as

The most common phase transitions are of second or first order type for all kind of systems.

The second order phase transition (SOT) temperature

and

At

In order to calculate the first-order phase transition (FOT) temperature, we need an analysis of the free energy expression given above in terms of the recursion relations.

We have also investigated the compensation temperature

The real compensation occurs when

It is instructive to analytically analyze the ground-state phase diagrams from the ground-state energies of the model Hamiltonian. The ground-state configuration is that with the lowest ground state energy. Here, we have six different ground-state configurations as in ref. [

Two disordered phases are obtained

Thermal magnetic properties of the system, namely the sublattice magnetiztions are presented. It’s worthwhile to first mention that the disordered phases

the boundary of phases

In

susceptibility in the low-temperature region originates from the behaviour of the sublattice susceptibility

Now, in order to explain the appearance of the broad maximum in the susceptibility of the sublattice B in the low-temperature region (

In

To explain the physical scenario for the appearance of the divergence of the susceptibility of the sublattice B (

In

existence of a TCP and a compensation line with two end-points indicated by full squares. The ordered domain F is not homogeneous in the sense that it does not consist of only one ferrimagnetic phase. Indeed, one gets three ferrimagnetic phase

In order to check the obtained compensation temperatures, we have illustrated the thermal behavior of the net magnetization

It is important to mention that the model shows interesting numerical behavior when

In summary, the mixed spin-2 and spin-3/2 BEG Ising ferrimagnetic system is studied on the Bethe lattice using exact recursion equations. The ground phase diagram of the model was constructed in (

the magnetizations and susceptibility curves and found interesting behavior results. Finally, the influences of the crystal field and the biquadratic interactions are investigated by obtaining the phase diagrams on the (

M.Karimou,R.Yessoufou,F.Hontinfinde, (2015) Mixed Spin-2 and Spin-3/2 Blume-Emery-Griffiths (BEG) Model on the Bethe Lattice. World Journal of Condensed Matter Physics,05,187-200. doi: 10.4236/wjcmp.2015.53020