Open Journal of Applied Sciences, 2013, 3, 270277 doi:10.4236/ojapps.2013.33034 Published Online July 2013 (http://www.scirp.org/journal/ojapps) MeanField Solution of the Mixed Spin2 and Spin5/2 Ising Ferrimagnetic System with Different SingleIon Anisotropies Fathi Abubrig Department of Physics, Faculty of Science, Elmergeb University, Zliten, Libya Email: dr_fathiomar@yahoo.com Received December 26, 2012; revised February 13, 2013; accepted February 20, 2013 Copyright © 2013 Fathi Abubrig. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The mixed spin2 and spin5/2 Ising ferrimagnetic system with different anisotropies ( A DzJ ) for the spin2 and () for the spin5/2 is studied by the use of the meanfield theory based on the Bogoliubov inequality for the free energy. First, the ground state phase diagram of the system at zero temperature is obtained on the B DzJ () , AB DzJDzJ , different kinds of phase diagrams are achieved by changing the temperature and the values of the single ion anisotropies plane. Topologically A DzJ and B Dz des secr transition lines, first order phase transition lines terminating at tricritical points, are found. The existence and dependence of a compensation temperature on singleion anisotropies is also investigated. J. Besiondorde Keywords: Mixed Spin; Ising Model; Ferrimagnetic; Sublattice Magnetization; Tricritical Points 1. Introduction In the last two decades, much attention has been paid to the study of the magnetic properties of twosublattice mixedspin ferrimagnetic Ising systems, because they are well adapted to consider some types of ferrimagnetism, namely the molecularbased magnetic materials [13] which have less translational symmetry than their single spin counterparts since they consist of two interpenetrat ing sublattices and have increasing interest. In a ferri magnetic material, the different temperature dependences of the sublattice magnetizations raise the possibility of the existence of a compensation temperature: a tempera ture below the critical point where the total magnetiza tion 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 mixedspin Ising system consisting of spin1/2 and spinS (S > 1/2) in a uniaxial crystal field. The model for different values of S (S > 1/2) has been investigated by acting on honeycomb lattice [79], as well as on Bethe lattice [10,11]), mean field approxima tion [12], effective field theory with correlations [1317], cluster variational theory [11], renormalizationgroup tech nique [18] and MonteCarlo simulation [1921]. It should be mentioned that the effects of different sublattice crystalfield interactions on the magnetic prop erties of the mixed spin1 and spin3/2 Ising ferromag netic system with different singleion anisotropies have been investigated with the use of an effective field theory [22,23], mean field theory [24], a cluster variational me thod [25] and Monte Carlo simulation [26]. Recently, The attention was devoted to the high order mixed spin ferrimagnetic systems (mixed spin3/2 and spin2 ferri magnetic system mixed spin2 and spin5/2 ferrimagnetic system and mixed spin3/2 and spin5/2 system) in order to construct their phase diagrams in the temperature anisotropy plane and to consider their magnetic proper ties. Bobak and Dely investigated the effect of singleion anisotropy on the phase diagram of the mixed spin3/2 and spin2 Ising system by the use of a meanfield theory based on the Bogoliubov inequality for the free energy Copyright © 2013 SciRes. OJAppS
F. ABUBRIG 271 [27]. Albayrak also studied the mixed spin3/2 and spin2 Ising system with two different crystalfield 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 ferrimag netic mixed spin3/2 and spin2 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 mixedspin2 and spin5/2 sys tem on a honeycomb lattice was made by Kaneyoshi and coworkers [30] within the frame work of the EFT. Na kamura [31,32] applied Monte Carlo (MC) simulations to study the magnetic properties of a mixed spin2 and spin5/2 system on a honeycomb lattice. Li et al. [33, 34] studied the magnetic properties of the mixed spin2 and spin5/2 system on a layered honeycomb lattice by a multisublattice greenfunction technique to investigate the magnetic properties of a mixed () () 24 21 AFeFeCOANCH,3, 5 nn nn ΙΙ ΙΙΙ + == 3 and to consider the compensation behaviour of the system. Wei and coworker [35] examined the internal energy, spe cific heat and initial susceptibility of the mixed spin2 and spin5/2 ferrimagnetic system with an interlayer coupling by the use of the EFT with correlations. Albay rak [36] studied the critical behaviour of the mixed spin 2 and spin5/2 Ising ferrimagnetic system on Bethe lat tice. And he also examined the critical and the compen sation temperatures of the mixed spin2 and spin5/2 Ising ferrimagnetic system on Bethe lattice by using the exact recursion equations. Keskin and Ertas [37] investi gated the Existence of a dynamic compensation tem perature of a mixed spin2 and spin5/2 Ising ferrimag netic system in an oscillating field. In this paper, we studied the effects of two different singleion anisotropies in the phase diagram and in the compensation temperature of the mixed spin2 and spin 5/2 Ising ferrimagnetic system within the theoretical frame work of the meanfield theory and we found some out standing 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 meanfield 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 singleion anisotropies, as well as the temperature de pendences of the magnetizations in some particular cases. Finally, in Section 4, we present our conclusions. 2. The Model and Calculation We consider a mixed Ising spin2 and spin5/2 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 sublat tice B are occupied by spins S 2,1, 0±± S, which take the spin values of 52, 32,12.±± In each site of the lattice, there is a singleion anisotropy (A in the sublattices A and D D in the sublattice B) acting in the spin2 and spin5/2. The Hamiltunian of the system according to the meanfield theory is given by () () () 22 , AB AB ij AiBj ij HJSSDS DS=− −− , (1) where the first summation is carried out only over near estneighbor pairs of spins on different sublattices and J is the nearestneighbour exchange interaction. The most direct way of deriving the meanfield theory is to use the variation principle for the Gibbs free energy, () () 00 0 0 GHG HHH≤Φ≡+ −, (2) where is the true free energy described by Ham iltonian given in the relation (1), is the free energy described by the trial Hamiltonian 0 () GH () 0 GH which depends on variational parameters and 0 denotes a thermal average over the ensemble defined by 0 . 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 , namely: () () 2 0 2, AA AiA i iA BB BjBj jB HSDS SDS γ γ ∈ ∈ =− + −+ (3) where A and 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 ()() ()() 1ln12exp4cosh2 2exp cosh 2 1255 93 11 ln2expcosh 2expcosh 2expcosh 24 24242 111 , 222 AA AA BB BB B ABAA BB gDD N DDD zJmmmm ββγββγ β ββγ ββγ ββ β γγ Φ− == ++ −++ −++ B γ (4) Copyright © 2013 SciRes. OJAppS
F. ABUBRIG 272 where 1 , B , N NkT β Φ == is the total number of sites of the lattice and z is the coordination number. The sublattice magnetization per site and A m m are defined by 0 A Ai mS= and 0 B Bj mS=, thus ()()() () ()()() 2sinh 2xp3sinh cosh 2exp3cosh0.5exp4 AAA AAAA A D mDD βγβ βγ βγβ βγβ +− =+−+ − e (5) and () () () () 53 5sinh3exp 4sinhexp 6sinh 122 53 2coshexp 4coshexp 6cosh 222 BBBB B BBBB D m DD βγββγβγ βγ βγ ββγ ββγ +− +− = +− +− 1 2 1 B B . (6) Now, by minimizing the free energy (4) with respect to A and , we obtain , ABB zJm zJmA ==. (7) The meanfield 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 secondorder transition from an ordered state , to the paramagnetic state AB , this part of the phase diagram can be determined analytically. () , AB mm () 0≠0, AB mm≠ ) 0= ( 0,mm= Because the magnetizations A and m m are very small in the neighborhood of secondorder transition point, we can expand Equations (4)(6) to obtain a Lan daulike expansion. ( 2468 0AAA A ) gambmcmOm=++ + +, (8) where the expansion coefficients are given by ()( 0 1ln 1 2AABB B gXYXY β =−+++ + ) Z , (9) 22 4 2 12 1 1 24 8 32 tt t aaaa β 1 b =−− , (10) 43 2 2 11212 3 1 2768 19296 tt t baccaa β c =++ , (11) () () 74 8 52 32 5 22 45213214126 216 33 21152018423 768018432 245760 c atctt ccctcaaaaaaa t β =++ −−+−− c (12) with 12 1 21 925 , 1 RR aRR ++ =++ 2 4, 1 AA AA Y aXY + =++ 12 3 12 81625 , 1 RR aRR ++ =++ 4 16 , 1 AA AA Y aXY + =++ 12 5 12 72915625 , 1 RR aRR ++ =++ 6 64 , 1 AA AA Y aXY + =++ 1 25 9, BB BB B YZ bXYZ ++ =++ 2 625 81, BB BB B YZ bXYZ ++ =++ 3 15625 729, BB BB B YZ bXYZ ++ =++ () 3 2 213 3, 2 t caa=− ( 2 22 3131 3412 4 t cbaab=+−− ) 2 , () 2 41212 1530 , 4 t cbbbb=−−− () 3 3 5513 30 15, 4 t caaaa=+− () () 43 66 2 61241612 1530 , 12 t caaaaaaa=−− where () 2exp 4, AA XD β = () 2 exp254, BB XD β = () 2exp , AA YD β = () 2exp 94 , BB YD β = () 2exp4, BB ZD β = , () 12exp 4A RD β =− () 22exp 6A RD β =− . In this way, critical and tricritical points are deter mined according to the following routine; 1) Secondorder transition lines when a = 0 and b > 0; 2) Tricritical points when a = b = 0, and c > 0; 3) The firstorder 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, DA, DB 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 ( , AB mm ) . 1 Copyright © 2013 SciRes. OJAppS
F. ABUBRIG 273 functions of J. On the other hand, the total magnetization per site. ( 1 2AB ) mm=+ (13) and the signs of sublattice magnetizations mA and mB 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 mixedspin system can exhibit a compensation point or not. ( kk c TT T< 0 B m≠ ) 0 A m≠ 3. Results and Discussions 3.1. Phase Diagrams The groundstate phase diagram is easily determined from Hamiltonian (1) by comparing the groundstate energies of the different phases and is shown in Figure 1. At zero temperature, we find six phases with different values of ,,, ABAB mmqq , namely the ordered ferri magnetic phases 1 525 2,,4, 24 O =− , 2 39 2,,4,, 24 O =− 3 11 2,,4,, 24 O =− 4 525 1, ,1,, 24 O =− 5 39 1, ,1,, 24 O =− 6 11 1, ,1, 24 O =− , and three disordered phases 1 25 0, 0,0,4 D = , 2 9 0, 0,0,4 D = , 3 1 0, 0,0,4 D = , where the parameters and A q q are defined by: Figure 1. Groundstate phase diagram of mixed spin2 and spin5/2 Ising ferrimagnetic system with the coordination number z and different singleion anisotropies DA and DB. The nine phases: ordered O1, O2, O3, O4, O5, O6 and disor dered D1, D2, D3 are separated by lines of firstorder transi tions. 2 A Ai qS=, 2 B Bj qS= 3.2. Temperature Phase Diagrams In Figures 2 and 3, the phase diagrams of the mixed spin2 and spin5/2 Ising ferrimagnetic system are shown in the () , ABc DzJkTzJ and () , BBc DzJkTzJ planes for some selected values of B DzJ for spin5/2 and A DzJ for spin2, respectively. The solid and light dotted lines are used for the second and firstorder transition, respectively, the heavy dashed curve repre sents the positions of tricritical points. The secondorder 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 firstorder phase transitions must be determined by comparing the corresponding Gibbs free energies of the various solutions of (5) and (6) for the pair . () , AB mm In Figure 2, we note that the value of the critical tem perature increases when B DzJ and A DzJ in creases. Above each secondorder lines the system is in the paramagnetic state, while below them is in the ferri magnetic state. We note that the system gives only sec ondorder phase transitions (solid lines) for all the values of 0.4661 A DzJ>− and the phase diagram is topo logically equivalent to that of the spin5/2 BlumeCapel model which does not include any tricritical point. For the values of 2.3315 0.4661 A DzJ−≤ ≤− the system includes secondorder phase transition lines (solid lines) at higher temperatures, firstorder phase transition lines (light dotted lines) at lower temperatures and a curve of tricritical (heavy dashed lines) points separates Figure 2. Phase diagram in the (DB, T) plane for the mixed spin Ising ferrimagnet with the coordination number z, when the value of DB/zJ is changed. The solid and dotted lines, respectively, indicate second and firstorder phase transitions, while the heavy dashed line represents the posi tions of tricritical points. Copyright © 2013 SciRes. OJAppS
F. ABUBRIG 274 the second and the firstorder critical lines. When 2.5 2.3315 A DzJ−<<−, the system gives only firstorder phase transition lines. In Figure 3, the phase diagrams of () Bc kTzJ ver sus A DzJ are shown for selected values of B DzJ From this figure, it is clear that in regions of high tem peratures, for all positive or negative values of, and for any value of B DzJ, the phase diagram shows only secondorder phase transitions. When 1.4650 B DzJ≥, all the secondorder lines end in the same tricritical point given by () () 3 , 2.3315,1.1360 ABc DzJkTzJ=− and when 0.8450 B DzJ≤− , all the secondorder lines end in the same tricritical point given by ( 3 , ABc DzJkTzJ= ) ) . From this figure, we also note that for ( 0.4661,0.2272− B DzJ→+∞, the mixed spin Ising system behaves like a twolevels system since the spin5/2 behaves like 52 B j S=± and the coordinates () 3 , ABc DzJkTzJ of the tricritical point are . () 2.3315,1.1 360− On the other hand, for B DzJ→−∞, the 52 B j S=± and 32 B j S=± states are suppressed and the system becomes equivalent to mixed spin1/2 and spin2 Ising model with tricritical point located at () ( 3 , 0.4661,0.2272 ABc DzJkTzJ=− ) . For this rea son, the coordinates of the tricritical point in the limit of large positive B DzJ are five times higher than those for large negative B DzJ. 3.3. Magnetization Curves Thermal behaviour of the sublattice magnetizations A and m m are obtained by solving the coupled Equations. (5) and (6). The results are depicted in Figure 4 for the system with 1.0 A DzJ=, when the value of B DzJ is changed from 0.45 B DzJ=− to −1.05. Notice that the selection of B DzJ corresponds to the crossover from the 1 phase to the 2 phase and from the 2 to the 3 phase (see the groundstate phase diagram in Figure 1). Therefore, the ground state is always ordered and Figure 4 shows that the system undergoes only the secondorder phase transition, because the sublattice mag netizations go to zero continuously as the temperature increases. O OO O As shown in Figure 4, when 0.45 B DzJ=− (close to the boundary between the orderedphase 1 and the ordered phase 2 in the groundstate phase diagram), the temperature dependences of mB 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 DzJ approaches the boundary. At 0.5 A DzJ=− and for , the saturation value of mB is , which indicates that in the ground state the spin configu ration of 0KT=2.0 B m= S in the system consists of the mixed state; in this state half of the spins on sublattice B are equal to Figure 3. Phase diagram in the (DA, T) plane for the mixed spin Ising ferrimagnet with the coordination number z, when the value of DB/zJ is changed. The solid and dotted lines, respectively, indicate second and firstorder phase transitions, while the heavy dashed line represents the posi tions of tricritical points. Figure 4. Thermal variations of sublattice magnetizations mA, mB for the mixedspin Ising ferrimagnet with the coor dination number z, when the value of DB/zJ is changed for fixed DA/zJ = 1.0. For one curve (DA/zJ, DB/zJ) = (−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 DzJ=− and 0.5 A DzJ>− . In this case, the total magnetization for the ferrimag netic system is at , and hence, there is a compensation point at which the two sublattice mag netization cancel. 0M=0KT= By further decreasing B DzJ, the ground state be comes O2, with at T = 0 K. In this region, when 1.5 B m= 0.55=− B (slightly below the boundary between the ordered phases O1 and O2) the thermal varia tion of mB exhibits an interesting feature which is the initial rise of mB with the increase of temperature before decreasing to zero at the critical point. On the other hand, for all values of DzJ B DzJ, even though the sublattice Copyright © 2013 SciRes. OJAppS
F. ABUBRIG 275 magnetization mA may show normal behaviour it is cou pled to mB. When B DzJ has the values −0.95, −1.0 and −1.05 (close to the end at the boundary between the ordered phases O2 and O3 in the groundstate phase diagram), it is clear from Figure 4 that the temperature dependences of mB and mA exhibit similar behaviours to the temperature dependences of mB and mA in the previous case. At the point () () ,0.8, AB DzJDzJ=− −0.3, the system will be in the ordered phase O5 (see the ground state phase diagram in Figure 1). In this case, the satu rated values of are () , AB mm ( 1, 32− ) at T = 0 K. No tice that the sublattice magnetization mB has initial rise with temperature before decreasing to its zero value at the critical point, and the sublattice magnetization mA may show a normal behaviour with temperature. 3.4. Compensation Temperature A Compensation temperature Tk of the system can be evaluated by requiring the condition M = 0; in Equation (13). Figures 5(a) and (b) show the behaviour of k (dot ted lines) in the T ( , BB DzJkTzJ ) plane for different values of A DzJ . As seen from the figures, all Tk curves emerge from 0.5 B DzJ=− at and exhibit some characteristic behaviours when the value of 0KT= A DzJ is changed. In Figure 5(a), all the curves increase monotonically with B DzJ and terminate at the corresponding phase boundaries (solid lines). This behaviour implies the oc currence of one compensation point only. As A DzJ is reduced, the range of B DzJ over which the com pensation points occur gradually becomes small, but the compensation temperature still reaches the corresponding transition line. In the Figure 5(b), and in a restricted re gion of B DzJ, close to 0.5 B DzJ=− , a new type of compensation curves appear and the compensation temperature lines exhibit an interesting features in their behaviours, which implies the occurrence of two, three, or four compensation points. In this figure, for A DzJ, close to 0.5 A DzJ=− , a new type of compensation curves appear: the curves are extended to k T B DzJ→−∞ below the corresponding transition lines. The curve labeled 0.498 A DzJ=− is an example of such behaviour of k T. Finally, a total magnetization curve (which refers to the compensation temperatures presented in Figure 5(b)) when 0.498 A DzJ=− and 0.499222 B with four compensation points are shown in Figure 6. Furthermore, In Figure 7(a), when DzJ=− 0.4999DzJ=− A and 0.5 B DzJ=− (very close to the point () () ,0.5,0.5DzJDzJ=− − AB which is in the boundary between five phases in the ground state phase diagram), the magnetization curves (a) (b) Figure 5. Dependence of the compensation temperature (dotted curves) on the singleion anisotropy. D B/zJ in a mixedspin Ising ferrimagnet with coordination number z, when the value of DB/zJ 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 firstorder transitions. Figure 6. Thermal variations of the total magnetization M for the mixedspin Ising ferrimagnet with the coordination number z, when the value of DA/zJ = −0.498 and the value of DB/zJ = −0.499222. exhibit some outstanding features. At this point, as the temperature is increased from zero, the sublattice mag netizations mA and mB exhibit four jumps (discontinuity) before the magnetizations vanish, indicating the exis tence of four first order transitions at the temperature values 0.0797 B kT zJ=, 0.1583 and 0.2526 respec tively. In the same time, as shown in Figure 7(b), the total magnetization exhibits four first order transition points and four compensation temperatures. 4. Conclusion In this paper, we have determined the global phase dia grams of the mixed spin2 and spin5/2 Ising ferrimag netic system with different singleion anisotropies acting on the spin2 and spin5/2 by using meanfield approxi mation. In the phase diagrams, the critical temperature lines versus singleion anisotropies are shown. The sys tem presents tricritical behaviour, i.e., the secondorder Copyright © 2013 SciRes. OJAppS
F. ABUBRIG 276 (a) (b) Figure 7. Thermal variations of (a) The total magnetization M; (b) The sublattice magnetizations mA, mB for the mixed spin Ising ferrimagnet with the coordination number z, when the value of DA/zJ = −0.4999 and the value of DB/zJ = −0.5. phase transition line is separated from the firstorder transition line by a tricritical point. We also observed that this mixedspin 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. REFERENCES [1] O. Khan, “Molecular Magnetism,” VCH publishers, New York, 1993. [2] T. Kaneyoshi and Y. Nakamura, “A Theoretical Investi gation for LowDimensional MolecularBased Magnetic Materials,” Journal of Physics: Condensed Matter, Vol. 10, No. 13, 1998, p. 3003. doi:10.1088/09538984/10/13/017 [3] T. Kaneyoshi, Y. Nakamura and S. Shin, “A Diluted Mixed Spin2 and Spin5/2 Ferrimagnetic Ising System; a Study of a MolecularBased Magnet,” Journal of Physics: Condensed Matter, Vol. 10, No. 31, 1998, p. 7025. doi:10.1088/09538984/10/31/018 [4] L. Néel, “Propriétées Magnétiques des Ferrites; Férrimag nétisme et Antiferromagnétisme,” Annales de Physique (Paris), Vol. 3, 1948, pp. 137198. [5] M. Mansuripur, “Magnetization Reversal, Coercivity, and the Process of Thermomagnetic Recording in Thin Films of Amorphous Rare Earth Transition Metal Alloys,” Jour nal of Applied Physics, Vol. 61, No. 4, 1987, pp. 1580 1587. doi:10.1063/1.338094 [6] F. Tanaka, S. Tanaka and N. Imamura, “MagnetoOptical Recording Characteristics of TbFeCo Media by Magnetic Field Modulation Method,” Japan Journal of Applied Physics, Vol. 26, 1987, pp. 231235. doi:10.1143/JJAP.26.231 [7] L. L. Goncaloves, “Uniaxial Anisotropy Effects in the Ising Model: An Exactly Soluble Model,” Physica Scripta, Vol. 32, No. 3, 1985, p. 248. doi:10.1088/00318949/32/3/012 [8] L. L. Goncaloves, “Uniaxial Anisotropy Effects in the Ising Model: An Exactly Soluble Model,” Physica Scripta, Vol. 33, No. 2, 1986, p. 192. doi:10.1088/00318949/33/2/018 [9] A. Dakhama and N. Benayad, “On the Existence of Com pensation Temperature in 2d MixedSpin Ising Ferri magnets: An Exactly Solvable Model,” Journal of Mag netism and Magnetic Materials, Vol. 213, No. 12, 2000, pp. 117125. doi:10.1016/S03048853(99)00606X [10] N. R. da Silva and S. R. Salinas, “MixedSpin Ising Model on Beth Lattice,” Physical Review, Vol. 44, No. 2, 1991, pp. 852855. doi:10.1103/PhysRevB.44.852 [11] J. W. Tucker, “The Ferrimagnetic Mixed Spin1/2 and Spin1 Sing System,” Journal of Magnetism and Mag netic Materials, Vol. 195, No. 3, 1999, pp. 733740. doi:10.1016/S03048853(99)003029 [12] T. Kaneyoshi and J. C. Chen, “MeanField Analysis of a Ferrimagnetic Mixed Spin System,” Journal of Magnet ism and Magnetic Materials, Vol. 98, No. 12, 1991, pp. 201204. doi:10.1016/03048853(91)90444F [13] T. Kaneyoshi, “Curie Temperatures and Tricritical Points in Mixed Ising Ferromagnetic Systems,” The Physical So ciety of Japan, Vol. 56, 1987, pp. 26752680. doi:10.1143/JPSJ.56.2675 [14] T. Kaneyoshi, “Phase Transition of the Mixed Spin Sys tem with a Random Crystal Field,” Physica A, Vol. 153, No. 3, 1988, pp. 556566. doi:10.1016/03784371(88)902403 [15] T. Kaneyoshi, M. Jascur and P. Tomczak, “The Ferri magnetic Mixed Spin1/2 and Spin3/2 Ising System,” Journal of Physics: Condensed Matter, Vol. 4, No. 49, 1992, pp. L653L658. doi:10.1088/09538984/4/49/002 [16] T. Kaneyoshi, “Tricritical Behavior of a Mixed Spin1/2 and Spin2 Ising System,” Physica A, Vol. 205, No. 4, 1994, pp. 677686. doi:10.1016/03784371(94)902291 [17] A. Bobak and M. Jurcisin, “Discussion of Critical Behav iour in a MixedSpin Ising Model,” Physica A, Vol. 240, No. 34, 1997, pp. 647656. doi:10.1016/S03784371(97)000447 [18] S. G. A. Quadros and S. R. Salinas, “Renormalization Group Calculations for a MixedSpin Ising Model,” Phy sica A: Statistical Mechanics and Its Applications, Vol. 206, No. 34, 1994, pp. 479496. [19] G.M. Zhang and C.Z. Yang, “Monte Carlo Study of the Copyright © 2013 SciRes. OJAppS
F. ABUBRIG Copyright © 2013 SciRes. OJAppS 277 TwoDimensional Quadratic Ising Ferromagnet with Spins S = 1/2 and S = 1 and with CrystalField Interactions,” Physical Review B, Vol. 48, No. 13, 1993, pp. 94529455. doi:10.1103/PhysRevB.48.9452 [20] G. M. Buendia and M. A. Novotny, “Numerical Study of a Mixed Ising Ferrimagnetic System,” Journal of Physics: Condensed Matter, Vol. 9, No. 27, 1997, pp. 59515964. doi:10.1088/09538984/9/27/021 [21] G. M. Buendia and J. A. Liendo, “Monte Carlo Simula tion of a Mixed Spin1/2 and Spin3/2 Ising Ferrimag netic System,” Journal of Physic s: Condensed Matter, Vol. 9, No. 25, 1997, pp. 54395448. doi:10.1088/09538984/9/25/011 [22] A. Bobak, “The Effect of Anisotropies on the Magnetic Properties of a Mixed Spin1 and Spin3/2 Ising Ferri magnetic System,” Physica A, Vol. 258, No. 12, 1998, pp. 140156. doi:10.1016/S03784371(98)002337 [23] O. F. Bobak and D. H. Abubrig, “An EffectiveField Study of the Mixed Spin1 and Spin3/2 Ising Ferrimag netic System,” Journal of Magnetism and Magnetic Ma terials, Vol. 246, No. 12, 2002, pp. 177183. doi:10.1016/S03048853(02)000483 [24] O. F. Abubrig, D. Horvath, A. Bobak and M. Jascur, “MeanField Solution of the Mixed Spin1 and Spin3/2 Ising System with Different SingleIon Anisotropies,” Phy sica A, Vol. 296, No. 34, 2001, pp. 437450. doi:10.1016/S03784371(01)001765 [25] J. W. Tucker, “Mixed Spin1 and Spin3/2 BlumeCapel Ising Ferromagnet,” Journal of Magnetism and Mgnetic Materials, Vol. 237, No. 2, 2001, pp. 215224. doi:10.1016/S03048853(01)006916 [26] Y. Nakamura and J. W. Tucker, “Monte Carlo Study of a Mixed Spin1 and Spin3/2 Ising Ferromagnet,” IEEE Transactions on Magnetics, Vol. 38, No. 5, 2002, pp. 24062408. doi:10.1109/TMAG.2002.803598 [27] A. Bobak and J. Dely, “Phase Transitions and Multicriti cal Points in the Mixed Spin3/2 and Spin2 Ising System with a SingleIon Anisotropy,” Journal of Magnetism and Magnetic Materials, Vol. 310, No. 2, 2007, pp. 1419 1421. doi:10.1016/j.jmmm.2006.10.427 [28] E. Albayrak, “The Critical and Compensation Tempera tures of the Mixed Spin3/2 and Spin2 Ising Model,” Physica B: Condensed Matter, Vol. 391, No. 1, 2007, pp. 4753. doi:10.1016/j.physb.2006.08.045 [29] B. Deviren, E. Kantar and M. Keskin, “Magnetic Proper ties of a Mixed Spin3/2 and Spin2 Ising Ferrimagnetic System within the EffectiveField Theory,” Journal of the Korean Physical Society, Vol. 56, No. 6, 2010, pp. 1738 1747. doi:10.3938/jkps.56.1738 [30] Y. Nakamura, S. Shin and T. Kaneyoshi, “The Effects of Transverse Field on the Magnetic Properties in a Diluted Mixed Spin2 and Spin5/2 Ising System,” Physica B, Vol. 284288, 2000, pp. 14791480. doi:10.1016/S09214526(99)02668X [31] Y. Nakamura, “Monte Carlo Study of a Mixed Spin2 and Spin5/2 Ising System on a Honeycomb Lattice,” Journal of Physics: Condensed Matter, Vol. 12, No. 17, 2000, pp. 40674074. doi:10.1088/09538984/12/17/312 [32] Y. Nakamura, “Existence of a Compensation Tempera ture of a Mixed Spin2 and Spin5/2 Ising Ferrimagnetic System on a Layered Honeycomb Lattice,” Physical Re view B, Vol. 62, No. 17, 2000, pp. 1174211746. doi:10.1103/PhysRevB.62.11742 [33] J. Li, A. Du and G. Z. Wei, “Green Function Study of a MixedSpin2 and Spin5/2 Heisenberg Ferrimagnetic Sys tem on a Honeycomb Lattice,” Physica Status Solidi (b), Vol. 238, No. 1, 2003, pp. 191197. [34] J. Li, A. Du and G. Z. Wei, “The Compensation Behavior of a MixedSpin2 and Spin5/2 Heisenberg Ferrimag netic System on a Honeycomb Lattice,” Physica B, Vol. 348, No. 14, 2004, pp. 7988. doi:10.1016/j.physb.2003.11.074 [35] G. Wei, Q. Zhang, Z. Xin and Y. Liang, “Internal Energy and Initial Susceptibility of Mixed Spin2 and Spin5/2 Ferrimagnetic Ising System with Interlayer Coupling,” Jour nal of Magnetism and Magnetic Materials, Vol. 277, No 12, 2004, pp. 115. doi:10.1016/j.jmmm.2003.06.001 [36] E. Albayrak, “MixedSpin2 and Spin5/2 BlumeEmery Griffiths Model,” Physica A: Statistical Mechanics and Its Applications, Vol. 375, No. 1, 2007, pp. 174184. [37] M. Keskin and M. Ertas, “Existence of a Dynamic Com pensation Temperature of a Mixed Spin2 and Spin5/2 Ising Ferrimagnetic System in an Oscillating Field,” Phy sical Review E, Vol. 80, No. 6, 2009, Article ID: 061140. doi:10.1103/PhysRevE.80.061140
