Mean-Field Solution of a Mixed Spin-3/2 and Spin-2 Ising Ferrimagnetic System with Different Single-Ion Anisotropies

The mixed spin-3/2 and spin-2 Ising ferrimagnetic system with different single-ion anisotropies in the absence of an external magnetic field is studied within the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. Second-order critical lines are obtained in the temperature-anisotropy plane. Tricritical line separating secondorder and first-order lines is found. Finally, the existence and dependence of a compensation points on single-ion anisotropies is also investigated for the system. As a result, this mixed-spin model exhibits one, two or three compensation temperature depending on the values of the anisotropies.


Introduction
In the last two decades, much attention has been paid to the study of the magnetic properties of two-sublattice mixed-spin ferrimagnetic Ising systems, because they are well adapted to consider some types of ferrimagnetism, namely the molecular-based magnetic materials [1][2][3] which have less translational symmetry than their singlespin counterparts since they consist of two interpenetrating sublattices and have increasing interest.In a ferrimagnetic material, the different temperature dependences of the sublattice magnetizations raise the possibility of the existence of a compensation temperature: a temperature below the critical point where the total magnetization is zero [4].This interesting behaviour has important applications in the field of thermomagnetic recording [5,6].For this reason, in recent years, there have been many theoretical studies on the magnetic properties of systems formed by two sublattices with different spins and with different crystal field interactions.
One of the earliest and simplest of these models to be studied was the mixed-spin Ising system consisting of spin-1/2 and spin-S (S > 1/2) in a uniaxial crystal field.
It should be mentioned that the effects of different sublattice crystal-field interactions on the magnetic properties of the mixed spin-1 and spin-3/2 Ising ferromagnetic system with different single-ion anisotropies have been investigated with the use of an effective field theory [22,23], mean field theory [24], a cluster variational method [25] and Monte Carlo simulation [26].Recently, The attention was devoted to the high order mixed spin ferrimagnetic systems (mixed spin-3/2 and spin-2 ferrimagnetic system mixed spin-2 and spin-5/2 ferrimagnetic system and mixed spin-3/2 and spin-5/2 system) in order to construct their phase diagrams in the temperatureanisotropy plane and to consider their magnetic properties.Bobak and Dely investigated the effect of single-ion anisotropy on the phase diagram of the mixed spin-3/2 and spin-2 Ising system by the use of a mean-field theory based on the Bogoliubov inequality for the free energy F. ABUBRIG 271 [27].Albayrak also studied the mixed spin-3/2 and spin-2 Ising system with two different crystal-field interactions on Bethe lattice by using the exact recursion equations [28].Bayram Deviren et al. have used the effective field theory to study the magnetic properties of the ferrimagnetic mixed spin-3/2 and spin-2 Ising model with crystal field in a longitudinal magnetic field on a honeycomb and a square lattice [29].We should mention that an early attempt to study the mixed-spin-2 and spin-5/2 system on a honeycomb lattice was made by Kaneyoshi and co-workers [30] within the frame work of the EFT.Nakamura [31,32] applied Monte Carlo (MC) simulations to study the magnetic properties of a mixed spin-2 and spin-5/2 system on a honeycomb lattice.Li et al. [33, 34]  studied the magnetic properties of the mixed spin-2 and spin-5/2 system on a layered honeycomb lattice by a multisublattice green-function technique to investigate the magnetic properties of a mixed ( ) ( ) and to consider the compensation behaviour of the system.Wei and co-worker [35] examined the internal energy, specific heat and initial susceptibility of the mixed spin-2 and spin-5/2 ferrimagnetic system with an interlayer coupling by the use of the EFT with correlations.Albayrak [36] studied the critical behaviour of the mixed spin-2 and spin-5/2 Ising ferrimagnetic system on Bethe lattice.And he also examined the critical and the compensation temperatures of the mixed spin-2 and spin-5/2 Ising ferrimagnetic system on Bethe lattice by using the exact recursion equations.Keskin and Ertas [37] investigated the Existence of a dynamic compensation temperature of a mixed spin-2 and spin-5/2 Ising ferrimagnetic system in an oscillating field.
In this paper, we studied the effects of two different single-ion anisotropies in the phase diagram and in the compensation temperature of the mixed spin-2 and spin-5/2 Ising ferrimagnetic system within the theoretical framework of the mean-field theory and we found some outstanding features in the temperature dependences of total and sublattice magnetizations.
The outline of this work is as follows.In Section 2, we define the model and present the mean-field theory based on the Bogoliubov inequality for the Gibbs free energy and then, we describe a Landau expansion of the free energy in the order parameter.In Section 3, we present the results and the discussion about the phase diagrams and compensation temperature for various values of the single-ion anisotropies, as well as the temperature dependences of the magnetizations in some particular cases.Finally, in Section 4, we present our conclusions.

The Model and Calculation
We consider a mixed Ising spin-2 and spin-5/2 system consisting of two sublattices A and B, which are arranged alternately.The sublattice A are occupied by spins i , which take the spin values of , while the sublattice B are occupied by spins S 2, 1, 0 ± ± j S , which take the spin values of 5 2, 3 2,1 2.

± ±
In each site of the lattice, there is a single-ion anisotropy ( A in the sublattices A and D B D in the sublattice B) acting in the spin-2 and spin-5/2.The Hamiltunian of the system according to the mean-field theory is given by , where the first summation is carried out only over nearest-neighbor pairs of spins on different sublattices and J is the nearest-neighbour exchange interaction.
The most direct way of deriving the mean-field theory is to use the variation principle for the Gibbs free energy, ( ) ( ) where is the true free energy described by Hamiltonian given in the relation (1), is the free energy described by the trial Hamiltonian 0 ( ) H which depends on variational parameters and 0 denotes a thermal average over the ensemble defined by  0 H . Depending on the choice of the trial Hamiltonian, one can construct approximate methods of different accuracy.However, owing to the complexity of the problem, we consider in this work the simple choice of 0 H , namely: where A γ and B γ are the two variational parameters related to the molecular fields acting on the two different sublattices, respectively.Through this approach, we found the free energy and the equations of state (sublattice magnetization per site A m F. ABUBRIG 272 where 1 , is the total number of sites of the lattice and z is the coordination number.

The sublattice magnetization per site and
Now, by minimizing the free energy (4) with respect to A γ and B γ , we obtain The mean-field properties of the present model are then given by Equations ( 4)- (7).Since the Equations ( 5)-( 7) have in general several solutions for the pair , the stable phase will be the one which minimizes the free energy.When the system undergoes the second-order transition from an ordered state , to the paramagnetic state , this part of the phase diagram can be determined analytically.

(
) Because the magnetizations A and m B m are very small in the neighborhood of second-order transition point, we can expand Equations ( 4)-( 6) to obtain a Landau-like expansion. ( where the expansion coefficients are given by ( )( ( ) In this way, critical and tricritical points are determined according to the following routine; 1) Second-order transition lines when a = 0 and b > 0; 2) Tricritical points when a = b = 0, and c > 0; 3) The first-order transition lines are determined by comparing the corresponding Gibbs free energies of the various solutions of Equations ( 5) and ( 6) for the pair Even so, we have also checked that c > 0 in all T, D A , D B space.The critical behaviour is the same for both ferromagnetic (J > 0) and ferrimagnetic (J < 0) systems, because the coefficients a, b and c are even 1 functions of J. On the other hand, the total magnetization per site. ( and the signs of sublattice magnetizations m A and m B are different, therefore, a compensation temperature at which the total magnetization is equal to zero may be exist in the system, although and .In our paper we shall prove whether the present mixed-spin system can exhibit a compensation point or not. (

Phase Diagrams
The ground-state phase diagram is easily determined from Hamiltonian (1) by comparing the ground-state energies of the different phases and is shown in Figure 1.
At zero temperature, we find six phases with different values of { } , , , m m q q , namely the ordered ferrimagnetic phases and three disordered phases where the parameters and A q B q are defined by:

Temperature Phase Diagrams
In Figures 2 and 3, the phase diagrams of the mixed spin-2 and spin-5/2 Ising ferrimagnetic system are shown in the ( ) D z J k T z J planes for some selected values of B D z J for spin-5/2 and A D z J for spin-2, respectively.The solid and light dotted lines are used for the second and first-order transition, respectively, the heavy dashed curve represents the positions of tricritical points.The second-order phase transition lines are easily obtained from Equations ( 10) and ( 11) by setting a = 0 and b > 0.
The tricritical points (the critical points at which the phase transitions change from second to first order) are determined from Equations ( 10) and ( 11) by setting a = b = 0, however, the first-order phase transitions must be determined by comparing the corresponding Gibbs free energies of the various solutions of ( 5) and ( 6) for the pair .
( ) m m In Figure 2, we note that the value of the critical temperature increases when B D z J and A D z J increases.Above each second-order lines the system is in the paramagnetic state, while below them is in the ferrimagnetic state.We note that the system gives only second-order phase transitions (solid lines) for all the values of 0.4661 ) .For this rea- son, the coordinates of the tricritical point in the limit of large positive B D z J are five times higher than those for large negative B D z J .

Magnetization Curves
Thermal behaviour of the sublattice magnetizations A and m B m are obtained by solving the coupled Equations.( 5) and ( 6).The results are depicted in Figure 4 for the system with 1.0 Notice that the selection of B D z J corresponds to the crossover from the 1 phase to the 2 phase and from the 2 to the 3 phase (see the ground-state phase diagram in Figure 1).Therefore, the ground state is always ordered and Figure 4 shows that the system undergoes only the second-order phase transition, because the sublattice magnetizations go to zero continuously as the temperature increases.In Figure 5(a), all the curves increase monotonically with B D z J and terminate at the corresponding phase boundaries (solid lines).This behaviour implies the occurrence of one compensation point only.As A D z J is reduced, the range of B D z J over which the compensation points occur gradually becomes small, but the compensation temperature still reaches the corresponding transition line.In the

Conclusion
In this paper, we have determined the global phase diagrams of the mixed spin-2 and spin-5/2 Ising ferrimagnetic system with different single-ion anisotropies acting on the spin-2 and spin-5/2 by using mean-field approximation.In the phase diagrams, the critical temperature lines versus single-ion anisotropies are shown.The system presents tricritical behaviour, i.e., the second-order phase transition line is separated from the first-order transition line by a tricritical point.We also observed that this mixed-spin ferrimagnetic system may exhibit one, two, three or four compensation points.The theoretical prediction of the possibility of compensation points and the design and preparation of materials with such unusual behaviour will certainly open a new area of research on such materials.

Figure 1 .
Figure 1.Ground-state phase diagram of mixed spin-2 and spin-5/2 Ising ferrimagnetic system with the coordination number z and different single-ion anisotropies D A and D B .The nine phases: ordered O 1 , O 2 , O 3 , O 4 , O 5 , O 6 and disordered D 1 , D 2 , D 3 are separated by lines of first-order transitions.

A
D z J > −and the phase diagram is topologically equivalent to that of the spin-5/2 Blume-Capel model which does not include any tricritical point.For the values of 2second-order phase transition lines (solid lines) at higher temperatures, first-order phase transition lines (light dotted lines) at lower temperatures and a curve of tricritical (heavy dashed lines) points separates

Figure 2 .
Figure 2. Phase diagram in the (D B , T) plane for the mixedspin Ising ferrimagnet with the coordination number z, when the value of D B /z|J| is changed.The solid and dotted lines, respectively, indicate second and first-order phase transitions, while the heavy dashed line represents the positions of tricritical points.
gives only first-order phase transition lines.In Figure3, the phase diagrams of ( )B ck T z J versus A D z J are shown for selected values of B D z J From this figure, it is clear that in regions of high temperatures, for all positive or negative values of, and for any value of B D z J , the phase diagram shows only second-order phase transitions.When 1.4650 B D z J ≥ , all the second-order lines end in the same tricritical point given by ≤ − , all the second-order lines end in the same tricritical point given by ( this figure, we also note that for (0.4661,0.2272 − B D z J → +∞ , the mixed spin Ising system behaves like a two-levels system since the spin-5/2 behaves like 5 k T z J of the tricritical point are .( ) 2.3315,1.1360− On the other hand, for B D z J → −∞ , the 5 states are suppressed and the system becomes equivalent to mixed spin-1/2 and spin-2 Ising model with tricritical point located at

5 AS
between the ordered-phase 1 and the ordered phase 2 in the ground-state phase diagram), the temperature dependences of m B may exhibit a rather rapid decrease from its saturation value at T = 0 K.The phenomena is further enhanced when the value of O O B D z J approaches the boundary.At 0.D z J = − and for, the saturation value of m B is , which indicates that in the ground state the spin configuin the system consists of the mixed state; in this state half of the spins on sublattice B are equal to

Figure 3 .
Figure 3. Phase diagram in the (D A , T) plane for the mixedspin Ising ferrimagnet with the coordination number z, when the value of D B /z|J| is changed.The solid and dotted lines, respectively, indicate second and first-order phase transitions, while the heavy dashed line represents the positions of tricritical points.

Figure 4 . 5 B−) 5 B
Figure 4. Thermal variations of sublattice magnetizations m A , m B for the mixed-spin Ising ferrimagnet with the coordination number z, when the value of D B /z|J| is changed for fixed D A /z|J| = 1.0.For one curve (D A /z|J|, D B /z|J|) = (−0.8,−0.3).+5/2 (or −5/2) and the other half are equal to +3/2 (or −3/2).Note that this mixed state persists as long as 0.5 B D z J = − and 0.5 A D z J > − .In this case, the total magnetization for the ferrimagnetic system is at , and hence, there is a compensation point at which the two sublattice magnetization cancel.0 M = 0 K T =

Figure 5 ( 5 BFigure 5 Figure 5 .
Figure 5. Dependence of the compensation temperature (dotted curves) on the single-ion anisotropy.D B /z|J| in a mixed-spin Ising ferrimagnet with coordination number z, when the value of D B /z|J| is changed.(a) The curves show the positions of one compensation points; (b) The curves show the positions of two, three and four compensation points.The solid and dashed curves represent the second and first-order transitions.

Figure 6 .
Figure 6.Thermal variations of the total magnetization M for the mixed-spin Ising ferrimagnet with the coordination number z, when the value of D A /z|J| = −0.498and the value of D B /z|J| = −0.499222.exhibitsome outstanding features.At this point, as the temperature is increased from zero, the sublattice magnetizations m A and m B exhibit four jumps (discontinuity) before the magnetizations vanish, indicating the existence of four first order transitions at the temperature values 0.0797 B k T z J = , 0.1583 and 0.2526 respectively.In the same time, as shown in Figure7(b), the total magnetization exhibits four first order transition points and four compensation temperatures.

Figure 7 .
Figure 7. Thermal variations of (a) The total magnetization M; (b) The sublattice magnetizations m A , m B for the mixedspin Ising ferrimagnet with the coordination number z, when the value of D A /z|J| = −0.4999and the value of D B /z|J| = −0.5. )