Magnetic Properties of a Mixed-Spin-3/2 and Spin-2 Ising Ferrimagnetic System in an Applied Longitudinal Magnetic Field

Fathi Abubrig

Department of Physics, Faculty of Science, Almergeb University, Zliten, Libya.

DOI: 10.4236/wjcmp.2013.32018                

Abstract

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.

Keywords

Mixed-Spin; Ising; Susceptibility

1. Introduction

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 [11], effective-field theory with correlations [12-16], cluster variational theory [10], renormalization-group technique [17] and Monte-Carlo simulation [18-20]. On the other hand, ferromagnetic and ferrimagnetic systems subjected to longitudinal magnetic fields have been investigated for many years [21-23]. The results assured that the longitudinal magnetic field has strong effects on the magnetic properties of these systems. Subsequently, the attention has been devoted to mixed-spin systems in a longitudinal field theoretically. The magnetic properties of the mixed spin-1/2 and spin-1 Ising ferromagnetic system with a crystal-field interaction in the presence of a longitudinal magnetic field by using the cluster variational method was investigated by Ekiz and Keskin [24]. Wei et al. [25] studied the magnetic properties of a mixed spin-1/2 and spin-3/2 Ising model in a longitudinal magnetic field within the framework of EFT with correlations. They examined the thermal behaviours of the magnetizations, susceptibilities and phase diagrams in detail. The magnetic properties of a mixed spin-1/2 and spin-3/2 Ising system in a longitudinal magnetic field on a Bethe lattice were studied by using the recursion relation scheme [26]. The thermodynamic and magnetic properties of a mixed Ising system on a triangular array in the presence of longitudinal magnetic field were investigated by Aouzi et al by using EFT with correlations [27]. Jiang and Bai also studied the influences of an external longitudinal magnetic field on the magnetic properties of mixed spin-1/2 and spin-3/2 Ising ferromagnetic or ferrimagnetic bilayer system [28]. The Magnetic properties of an anti-ferromagnetic and ferrimagnetic mixed spin-1/2 and spin-5/2 Ising model in the longitudinal magnetic field and the Magnetic Properties of a Mixed Spin-3/2 and Spin-2 Ising Ferrimagnetic System within the Effective-field Theory are studied by Bayram Deviren et al by using the effective-field theory with correlations [29,30] and the results show that the longitudinal magnetic field plays an important role in the magnetic properties of the mixed spin Ising systems.

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.

2. Formulation of the Model and Its Mean-Field Solution

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

(1)

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

(2)

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

(3)

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

(4)

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

(7)

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

(5) (6)

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

. (8)

On the other hand, the sublattice initial susceptibilities are defined by

.

From which the total initial susceptibility per site is given by

. (9)

3. Results and Discussions

3.1. Phase Diagrams

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 Figure 1.

3.2. Sublattice Magnetizations m_A and m_B

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 Figure 2 with

Figure 1. Ground-state phase diagram of the mixed spin-3/2 and spin-2 Ising ferrimagnetic system with the coordination number z and different single-ion anisotropies and. The four phases: ordered O₁, O₂, O₃, O₄ and disordered D₁, D₂ are separated by first-order transition lines.

Figure 2. Thermal variation of the sublattice magnetizations m_A, m_B for the mixed-spin Ising ferrimagnet with the coordination number z, when the value of is changed for fixed.

, 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 Figure 1), therefore the ground state is always ordered. When, the sublattice magnetizations m_A and m_B 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 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 O₁ and the ordered phase O₂ 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 O₂ 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 Figure 2, for (slightly below the boundary between the ordered phase O₂ and the disordered phase D₁ 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 [30] and in the mixed-spin-1 and spin-3/2 Ising ferrimagnetic system [31-33].

Figure 3 shows the sublattice magnetization curves as a function of temperature for several values of, when. In this case, the selection of corresponds to the crossover from the O₁ to the O₃ phase. When, the sublattice magnetization m_A may show normal behaviour. When (slightly above the boundary between the ordered phase O₁ and the ordered phase O₂, 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 Figure 3 that when, the sublattice magnetization m_B decreases normally from its saturation value (m_B = −1/2) to vanish at the critical tem-

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	O. Khan, “Molecular Magnetism,” VCH Publishers, New York, 1993.
[2]	T. Kaneyoshi and Y. Nakamura, “A Theoretical Investigation for Low-Dimensional Molecular-based Magnetic Materials,” Journal of Physics: Condensed Matter, Vol. 10, No. 13, 1998, p. 3003. doi:10.1088/0953-8984/10/13/017
[3]	T. Kaneyoshi, Y. Nakamura and S. Shin, “A Diluted Mixed Spin-2 and Spin-5/2 Ferrimagnetic Ising System; a Study of a Molecular-Based Magnet,” Journal of Physics, Condensed Matter, Vol. 10, No. 31, 1998, p. 7025. doi:10.1088/0953-8984/10/31/018
[4]	M. Mansuripur, “Magnetization Reversal, Coercivity, and the Process of Thermomagnetic Recording in Thin Films of Amorphous Rare Earth Transition Metal Alloys,” Journal of Applied Physics, Vol. 61, No. 4, 1987, pp. 1580-1587. doi:10.1063/1.338094
[5]	F. Tanaka, S. Tanaka and N. Imamura, “Magneto-Optical Recording Characteristics of TbFeCo Media by Magnetic Field Modulation Method,” Japan Journal of Applied Physics, Vol. 26, 1987, pp. 231-235. doi:10.1143/JJAP.26.231
[6]	L. L. Goncaloves, ‘‘Uniaxial Anisotropy Effects in the Ising Model: An Exactly Soluble Model,” Physica Scripta, Vol. 32, No. 3, 1985, p. 248. doi:10.1088/0031-8949/32/3/012
[7]	L. L. Goncaloves, “Uniaxial Anisotropy Effects in the Ising Model: An Exactly Soluble Model,” Physica Scripta, Vol. 33, No. 2, 1986, p. 192. doi:10.1088/0031-8949/33/2/018
[8]	A. Dakhama and N. Benayad, “On the Existence of Compensation Temperature in 2d Mixed-Spin Ising Ferrimagnets: An Exactly Solvable Model,” Journal of Magnetism and Magnetic Materials, Vol. 213, No. 1-2, 2000, pp. 117-125. doi:10.1016/S0304-8853(99)00606-X
[9]	N. R. da Silva and S. R. Salinas, “Mixed-Spin Ising Model on Beth Lattice,” Physical Review, Vol. 44, No. 2, 1991, pp. 852-855. doi:10.1103/PhysRevB.44.852
[10]	J. W. Tucker, “The Ferrimagnetic Mixed Spin-1/2 and Spin-1 Sing System,” Journal of Magnetism and Magnetic Materials, Vol. 195, No. 3, 1999, pp. 733-740. doi:10.1016/S0304-8853(99)00302-9
[11]	T. Kaneyoshi and J. C. Chen, “Mean-Field Analysis of a Ferrimagnetic Mixed Spin System,” Journal of Magnetism and Magnetic Materials, Vol. 98, No. 1-2, 1991, pp. 201-204. doi:10.1016/0304-8853(91)90444-F
[12]	T. Kaneyoshi, “Curie Temperatures and Tricritical Points in Mixed Ising Ferromagnetic Systems,” The Physical Society of Japan, Vol. 56, 1987, pp. 2675-2680. doi:10.1143/JPSJ.56.2675
[13]	T. Kaneyoshi, “Phase Transition of the Mixed Spin System with a Random Crystal Field,” Physica A, Vol. 153, No. 3, 1988, pp. 556-566. doi:10.1016/0378-4371(88)90240-3
[14]	T. Kaneyoshi, M. Jascur and P. Tomczak, “The Ferrimagnetic Mixed Spin-1/2 and Spin-3/2 Ising System,” Journal of Physics: Condensed Matter, Vol. 4, No. 49, 1992, pp. L653-L658. doi:10.1088/0953-8984/4/49/002
[15]	T. Kaneyoshi, “Tricritical Behavior of a Mixed Spin-1/2 and Spin-2 Ising System,” Physica A, Vol. 205, No. 4, 1994, pp. 677-686. doi:10.1016/0378-4371(94)90229-1
[16]	A. Bobak and M. Jurcisin, “Discussion of Critical Behaviour in a Mixed-Spin Ising Model,” Physica A, Vol. 240, No. 3-4, 1997, pp. 647-656. doi:10.1016/S0378-4371(97)00044-7
[17]	S. G. A. Quadros and S. R. Salinas, “RenormalizationGroup Calculations for a Mixed-Spin Ising Model,” Physica A: Statistical Mechanics and Its Applications, Vol. 206, No. 3-4, 1994, pp. 479-496.
[18]	G.-M. Zhang and C.-Z. Yang, “Monte Carlo Study of the Two-Dimensional Quadratic Ising Ferromagnet with Spins S=1/2 and S=1 and with Crystal-Field Interactions,” Physical Review B, Vol. 48, No. 13, 1993, pp. 9452-9455. doi:10.1103/PhysRevB.48.9452
[19]	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. 5951-5964. doi:10.1088/0953-8984/9/27/021
[20]	G. M. Buendia and J. A. Liendo, “Monte Carlo Simulation of a Mixed Spin-1/2 and Spin-3/2 Ising Ferrimagnetic System,” Journal of Physics: Condensed Matter, Vol. 9, No. 25, 1997, pp. 5439-5448. doi:10.1088/0953-8984/9/25/011
[21]	M. Doerr, S. Kramp, M. Loewenhaupt, M. Rotter, R. Kratz, H. Krug, D. Eckert, H. Siegel and P. Verges, “Anomalous Magnetic Behaviour of NdCu2 in High Magnetic Fields,” Physica B: Condensed Matter, Vol. 294-295, 2001, pp. 164-167. doi:10.1016/S0921-4526(00)00633-5
[22]	K. Koyama, H. Fujii, T. Goto, H. Fukuda and Y. Janssen, “Magnetic Phase Transitions of Ce2Fe17 under High Pressures and High Magnetic Fields,” Physica B: Condensed Matter, Vol. 294-295, 2001, pp. 168-171. doi:10.1016/S0921-4526(00)00634-7
[23]	F. Albertini, F. Bolzoni, A. Paoluzi, L. Pareti and E. Zannoni, “Magnetic-Field Induced First-Order Transitions in the Intermetallic Compound Pr2Fe17,” Condensed Matter, Vol. 294-295, 2001, pp. 172-176. doi:10.1016/S0921-4526(00)00635-9
[24]	C. Ekiz and M. Keskin, “Magnetic Properties Of The Mixed Spin-1/2 and Spin-1 Ising Ferromagnetic System,” Physica A, Vol. 317, No. 3-4, 2003, pp. 517-534. doi:10.1016/S0378-4371(02)01356-0
[25]	G.-Z. Wei, Y.-Q. Liang, Q. Zhang and Z.-H. Xin, “Magnetic Properties of Mixed-Spin Ising Systems in a Longitudinal Magnetic Field,” Journal of Magnetism and Magnetic Materials, Vol. 271, No. 2-3, 2004, pp. 246-253. doi:10.1016/j.jmmm.2003.09.043
[26]	C. Ekiz, “Mixed Spin-1/2 and Spin-3/2 Ising System in a Longitudinal Magnetic Field,” Journal of Magnetism and Magnetic Materials, Vol. 293, No. 3, 2005, pp. 913-923. doi:10.1016/j.jmmm.2004.12.012
[27]	M. Aouzi, M. El Hafidi and E. M. Sakhaf, “Thermodynamic and Magnetic Properties of a Mixed Ising System on a Triangular Array in Presence of Longitudinal Field,” Physica A, Vol. 345, No. 3-4, 2005, pp. 575-590.
[28]	W. Jiang and B.-D. Bai, “Hysteresis Loops and Susceptibility of Ferromagnetic or Ferrimagnetic Bilayer System,” Physica Status Solidi (b), Vol. 243, No. 12, 2006, pp. 2892-2900. doi:10.1002/pssb.200541244
[29]	B. Deviren, M. Keskinb and O. Cankob, “Magnetic Properties of an Anti-Ferromagnetic and Ferrimagnetic Mixed Spin-1/2 and Spin-5/2 Ising Model in the Longitudinal Magnetic Field within the Effective-Field Approximation,” Physica A, Vol. 388, No. 9, 2009, pp. 1835-1848. doi:10.1016/j.physa.2009.01.032
[30]	B. Deviren, E. Kantar and M. Keskin, “Magnetic Properties of a Mixed Spin-3/2 and Spin-2 Ising Ferrimagnetic System within the Effective-Field Theory,” Journal of the Korean Physical Society, Vol. 56, No. 6, 2010, pp. 1738-1747. doi:10.3938/jkps.56.1738
[31]	O. F. Abubrig, D. Horvath, A. Bobak and M. Jascur, “Mean-Field Solution of the Mixed Spin-1 and Spin-3/2 Ising System with Different Single-Ion Anisotropies,” Physica A, Vol. 296, No. 3-4, 2001, pp. 437-450.
[32]	A. Bobak, O. F. Abubrig and D. Horvath, “An Effective-Field Study of the Mixed Spin-1 and Spin-3/2 Ising Ferrimagnetic System,” Journal of Magnetism and Magnetic Materials, Vol. 246, No. 1-2, 2002, pp. 177-183. doi:10.1016/S0304-8853(02)00048-3
[33]	C. Ekiz, “The Possibility of Two Compensation Points in a Ferrimagnetic Mixed Spin-1 and Spin-3/2 Ising System Using Bethe Lattice Approach,” Journal of Magnetism and Magnetic Materials, Vol. 307, No. 1, 2006, pp. 139-147. doi:10.1016/j.jmmm.2006.03.059
[34]	Y. Nakamura and J. W. Tucker, “Monte Carlo Study of a Mixed Spin-1 and Spin-3/2 Ising Ferromagnet,” IEEE Transactions on Magnetics, Vol. 38, No. 5, 2002, pp. 2406-2408. doi:10.1109/TMAG.2002.803598
[35]	A. Bobak, O. F. Abubrig and D. Horvath, “Magnetic Properties of a Mixed Ferro-Ferrimagnetic Ternary Alloy,” Physica A, Vol. 312, No. 1-2, 2002, pp. 187-207. doi:10.1016/S0378-4371(02)00864-6
[36]	T. Kaneyoshi, I. Tamura and R. Honmura, “Dilute Ising Ferromagnet: Its Physical Proprties,” Physical Review B, Vol. 29, No. 5, 1984, pp. 2769-2776. doi:10.1103/PhysRevB.29.2769
[37]	A. Bobak and M. Jascur, “Correlated Effective-Field Theory of the Site-Diluted Ising Model,” Journal of Magnetism and Magnetic Materials, Vol. 136, No. 1-2. 1994, pp. 105-117. doi:10.1016/0304-8853(94)90454-5
[38]	A. A. da Silva and F. G. Brady Moreira, “Thermodynamic Properties and Phase Transitions of the Site-BondCorrelated Ising Model,” Physical Review B, Vol. 40, No. 16, 1989, pp. 10986-10991. doi:10.1103/PhysRevB.40.10986