Dynamic compensation temperature in kinetic spin-5/2 Ising model on hexagonal lattice

We present a study of the dynamic behavior of a two-sublattice spin-5/2 Ising model with bilinear and crystal-field interactions in the presence of a time-dependent oscillating external magnetic field on alternate layers of a hexagonal lattice by using the Glauber-type stochastic dynamics. The lattice is formed by alternate layers of spins σ=5/2 and S=5/2. We employ the Glauber transition rates to construct the mean-field dynamic equations. First, we investigate the time variations of the average sublattice magnetizations to find the phases in the system and then the thermal behavior of the dynamic sublattice magnetizations to characterize the nature (first- or second-order) of the phase transitions and to obtain the dynamic phase transition (DPT) points. We also study the thermal behavior of the dynamic total magnetization to find the dynamic compensation temperature and to determine the type of the dynamic compensation behavior. We present the dynamic phase diagrams, including the dynamic compensation temperatures, in nine different planes. The phase diagrams contain seven different fundamental phases, thirteen different mixed phases, in which the binary and ternary combination of fundamental phases and the compensation temperature or the L-type behavior strongly depend on the interaction parameters.     

Dynamic compensation temperature in the mixed spin-1 and spin-2 Ising model in an oscillating field on alternate layers of a hexagonal lattice

The dynamic behavior of a mixed spin-1 and spin-2 Ising system with a crystal-field interaction in the presence of a time-dependent oscillating external magnetic field on a hexagonal lattice is studied by using the Glauber-type stochastic dynamics. The lattice is formed by alternate layers of spins σ=1$\sigma =1$ and S=2. The Hamiltonian model includes intersublattice, intrasublattice and crystal-field interactions. The set of mean-field dynamic equations is obtained by employing the Glauber transition rates. Firstly, we study time variations of the average sublattice magnetizations in order to find the phases in the system, and the thermal behavior of the average sublattice magnetizations in a period or the dynamic sublattice magnetizations to obtain the dynamic phase transition points as well as to characterize the nature (continuous and discontinuous) of transitions. Then, the behavior of the dynamic total magnetization as a function of the temperature is investigated to find the dynamic compensation points as well as determine the type of behavior. We also present the dynamic phase diagrams for both presence and absence of the dynamic compensation temperatures in the nine different planes. According to the values of Hamiltonian parameters, besides the paramagnetic (p), antiferromagnetic (af), ferrimagnetic (i) and non-magnetic (nm) fundamental phases, eight different mixed phases and the compensation temperature or L- and N-types behavior in the Néel classification nomenclature exist in the system.     

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

The magnetic properties of an anti-ferromagnetic and ferrimagnetic mixed spin-1/2 and spin-5/2 Ising model with a crystal field in a longitudinal magnetic field on the honeycomb (z=3) and square lattice (z=4) are studied by using the effective-field theory with correlations. The ground state phase diagram of the model is obtained in the longitudinal magnetic field (h) and a single-ion potential or crystal-field interaction (Δ) plane. We also investigate the thermal variations of the sublattice and total magnetizations, and present the phase diagrams in the (Δ/|J|, ) plane. The phase diagrams have one, two or even three compensation temperatures depending on the values of the crystal-field interaction. Moreover, the susceptibility, internal energy and specific heat of the system are numerically examined, and some interesting phenomena in these quantities are found due to the applied longitudinal magnetic field.     

Dynamic compensation temperature in the kinetic spin-1 Ising model in an oscillating external magnetic field on alternate layers of a hexagonal lattice

The dynamic behavior of a two-sublattice spin-1 Ising model with a crystal-field interaction (D) in the presence of a time-varying magnetic field on a hexagonal lattice is studied by using the Glauber-type stochastic dynamics. The lattice is formed by alternate layers of spins σ=1 and S=1. For this spin arrangement, any spin at one lattice site has two nearest-neighbor spins on the same sublattice, and four on the other sublattice. The intersublattice interaction is antiferromagnetic. We employ the Glauber transition rates to construct the mean-field dynamical equations. Firstly, we study time variations of the average magnetizations in order to find the phases in the system, and the temperature dependence of the average magnetizations in a period, which is also called the dynamic magnetizations, to obtain the dynamic phase transition (DPT) points as well as to characterize the nature (continuous and discontinuous) of transitions. Then, the behavior of the total dynamic magnetization as a function of the temperature is investigated to find the types of the compensation behavior. Dynamic phase diagrams are calculated for both DPT points and dynamic compensation effect. Phase diagrams contain the paramagnetic (p) and antiferromagnetic (af) phases, the p+af and nm+p mixed phases, nm is the non-magnetic phase, and the compensation temperature or the L-type behavior that strongly depend on the interaction parameters. For D<2.835 and H₀>3.8275, H₀ is the magnetic field amplitude, the compensation effect does not appear in the system.     

Dynamic phase transition in the kinetic Blume--Emery--Griffiths model: Phase diagrams in the temperature and interaction parameters planes

As a continuation of our previously published work, the dynamic phase transitions are studied further, within a mean-field approach, in the kinetic Blume--Emery--Griffiths model in the presence of a time varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. The dynamic phase transitions are obtained and the phase diagrams are constructed in two different planes, namely in the reduced temperature (T) and biquadratic interaction (k) plane and found eight fundamental types of phase diagrams for various values of reduced crystal-field interaction (d) and magnetic field amplitude (h), and in the (T,?d) plane and obtained six distinct topologies for different values of k and h. Phase diagrams exhibit one or two dynamic tricritical points and a dynamic double critical end point, dynamic triple and quadruple points, and besides disordered and ordered phases, three coexistence phase regions exist in which occurring of these strongly depend on the values of d, k and h.     

Multicritical behaviour of the antiferromagnetic spin- Blume-Capel model

The multicritical behaviour of the spin- Ising model on the square lattice with nearest-neighbour antiferromagnetic exchange interaction (J<0), a crystal-field interaction (D) and an external magnetic field (H) is studied within mean field approximation (M.F.A.). The phase diagram exhibits a rich variety of behaviour: second order, first order and critical points of different order.     

The possibility of two compensation points in a ferrimagnetic mixed spin-1 and spin-3/2 Ising system using Bethe lattice approach

We give an exact formulation of a mixed spin-1 and spin-3/2 Ising model on the Bethe lattice, which shows ferrimagnetism and compensation points. The model incorporates antiferromagnetic nearest-neighbor interaction which is relevant to describe ferrimagnetism. The influence of two sublattice crystal fields, D_A and D_B, on compensation points is studied in detail. For certain crystal-field values, the single or double compensation temperature may occur in the present system.     

The mixed spin-1/2 and spin-1 Ising system on a two-layer Bethe lattice

Two layered magnetic Bethe lattice with varying coordination number q is introduced and numerically studied via exact recursion relations within a pairwise approach. The system is influenced by competing interlayer and intralayer nearest-neighbour (NN) coupling interactions and also by the crystal and external magnetic fields. Cases where both layers are ferromagnetic or one is ferro and the other antiferromagnetic are considered. System configurations’ energy calculations are used to devise some ground state phase diagrams that have proven useful for the investigation of the very low temperature behaviour of the model. Analysis of the thermal behaviours of the total magnetization within the model parameters’ space yield interesting phase diagrams which display fascinating properties, in particular the presence of tricritical points. Increasing negative values of the crystal field strength stabilizes the disordered paramagnetic phase and sometimes gives rise to wavy transition lines.     

The effective-field theory studies of critical phenomena in a mixed spin-1 and spin-2 Ising model on honeycomb and square lattices

An effective-field theory with correlations has been used to study critical behaviors of a mixed spin-1 and spin-2 Ising system on a honeycomb and square lattices in the absence and presence of a longitudinal magnetic field. The ground-state phase diagram of the model is obtained in the longitudinal magnetic field (h) and a single-ion potential or crystal-field interaction (Δ) plane. The thermal behavior of the sublattice magnetizations of the system are investigated to characterize the nature of (continuous and discontinuous) of the phase transitions and obtain the phase transition temperature. The phase diagrams are presented in the (Δ/|J|, k_BT/|J|) plane. The susceptibility, internal energy and specific heat of the system are numerically examined and some interesting phenomena in these quantities are found due to the absence and presence of the applied longitudinal magnetic field. Moreover, the system undergoes second- and first-order phase transition; hence, the system gives a tricritical point. The system also exhibits reentrant behavior.     

Kinetics of a mixed spin-1/2 and spin-3/2 Ising ferrimagnetic model

We present a study, within a mean-field approach, of the kinetics of a mixed ferrimagnetic model on a square lattice in which two interpenetrating square sublattices have spins that can take two values, , alternated with spins that can take the four values, . We use the Glauber-type stochastic dynamics to describe the time evolution of the system with a crystal-field interaction in the presence of a time-dependent oscillating external magnetic field. The nature (continuous and discontinuous) of transition is characterized by studying the thermal behaviors of average order parameters in a period. The dynamic phase transition points are obtained and the phase diagrams are presented in the reduced magnetic field amplitude (h) and reduced temperature (T) plane, and in the reduced temperature and interaction parameter planes, namely in the (h, T) and (d, T) planes, d is the reduced crystal-field interaction. The phase diagrams always exhibit a tricritical point in (h, T) plane, but do not exhibit in the (d, T) plane for low values of h. The dynamic multicritical point or dynamic critical end point exist in the (d, T) plane for low values of h. Moreover, phase diagrams contain paramagnetic (p), ferromagnetic (f), ferrimagnetic (i) phases, two coexistence or mixed phase regions, (f+p) and (i+p), that strongly depend on interaction parameters.     

Dynamic Phase Transitions and Compensation Temperatures in a Mixed Spin-3/2 and Spin-5/2 Ising System

We examine the dynamic phase transitions and the dynamic compensation temperatures, within a mean-field approach, in the mixed spin-3/2 and spin-5/2 Ising system with a crystal-field interaction under a time-varying magnetic field on a hexagonal lattice by using Glauber-type stochastic dynamics. The model system consists of two interpenetrating sublattices with σ=3/2 and S=5/2. The Hamiltonian model includes intersublattice, intrasublattice, and crystal-field interactions. The intersublattice interaction is considered antiferromagnetic and to be a simple but interesting model of a ferrimagnetic system. We employ the Glauber transition rates to construct the mean-field dynamic equations, and we solve these equations in order to find the phases in the system. We also investigate the thermal behavior of the dynamic sublattice magnetizations and the dynamic total magnetization to obtain the dynamic phase transition points and compensation temperatures as well as to characterize the nature (continuous and discontinuous) of transitions. We also calculate the dynamic phase diagrams including the compensation temperatures in five different planes. According to the values of Hamiltonian parameters, five different fundamental phases, three different mixed phases, and six different types of compensation behaviors in the Néel classification nomenclature exist in the system.     

Magnetic properties of the mixed spin-5/2 and spin-3/2 Blume-Capel Ising system on the two-fold Cayley tree

We use exact recursion relations to study the magnetic properties of the half-integer mixed spin-5/2 and spin-3/2 Blume-Capel Ising ferromagnetic system on the two-fold Cayley tree that consists of two sublattices A and B. Two positive crystal-field interactions Δ₁ and Δ₂ are considered for the sublattice with spin-5/2 and spin-3/2 respectively. For different coordination numbers q of the Cayley tree sites, the phase diagrams of the model are presented with a special emphasis on the case q = 3, since other values of q reproduce similar results. First, the T = 0 phase diagram is illustrated in the (D _A = Δ₁/J,D _B = Δ₂/J) plane of reduced crystal-field interactions. This diagram shows triple points and coexistence lines between thermodynamically stable phases. Secondly, the thermal variation of the magnetization belonging to each sublattice for some coordination numbers q are investigated as well as the Helmoltz free energy of the system. First-order and second-order phase transitions are found. The second-order phase transitions become sharper and sharper when D _A or D _B increases. The first-order transitions only exist for some appropriate non-zero values of D _A and/or D _B . The corresponding transition lines never connect to the second-order transition lines. Thus, the non-existence of tricritical points remains one of the key features of the present model. The magnetic exponent β ₀ of the model is estimated and found to be ¼ at small values of D _A = D _B = D and β ₀ = ½ at large values of D. At intermediate values of D, there is a crossover region where the magnetic exponent displays interesting behaviours.     

Dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model in an oscillating field: the effective-field theory based on the Glauber-type stochastic dynamics

Using the effective-field theory based on the Glauber-type stochastic dynamics (DEFT), we investigate dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model under an oscillating magnetic field. We presented the dynamic phase diagrams in (T/J, h₀/J), (D/J, T/J) and (K/J, T/J) planes, where T, h₀, D, K and z are the temperature, magnetic field amplitude, crystal–field interaction, biquadratic interaction and the coordination number. The dynamic phase diagrams exhibit several ordered phases, coexistence phase regions and special critical points, as well as re-entrant behavior depending on interaction parameters. We also compare and discuss the results with the results of the same system within the mean-field theory based on the Glauber-type stochastic dynamics and find that some of the dynamic first-order phase lines and special dynamic critical points disappeared in the DEFT calculation.     

The spin-2 antiferromagnet on the Bethe lattice

The complete phase diagrams of the antiferromagnetic spin-2 Blume-Capel Ising system is studied on the Bethe lattice by the use of exact recursion relations. In order to specify the states of the system, i.e. the different spin configurations, the ground state phase diagram is obtained on the (H/|J|, D/|J|) plane corresponding to the reduced external magnetic and crystal fields, respectively. As a result, the thermal change of the order-parameters, the magnetisations belonging to the two sublattice system, was investigated to obtain the full phase diagrams of the system on the (H/|J|, kT/|J|) planes. The behavior of the order-parameters with respect to the external magnetic field was also studied for the given values of D/|J|. Besides the interesting thermal and external magnetic field change of the sublattice magnetisations, the system also exhibits interesting critical behaviors including first- and second-order phase transitions, therefore, triciritical points and the reentrant behavior. The calculations are carried out for the coordination number q=4, corresponding to the square lattice, only.     

Ground state phase diagrams for the mixed Ising 3/2 and 5/2 spin model

We calculate the ground state phase diagrams of a mixed Ising model on a square lattice where spins S (± 3/2, ± 1/2) in one sublattice are in alternating sites with spins Q (± 5/2, ± 3/2, ± 1/2), located on the other sublattice. The Hamiltonian of the model includes first neighbor interactions between the S and Q spins, next-nearest-neighbor interactions between the S spins, and between the Q spins, and crystal field. The topologies of the phase diagrams depend on the values of the parameters in the Hamiltonian. The diagrams show some key features: coexistence between regions, points where two, three, four, five and six states can coexist. Besides being very useful as a way to check the low temperature limit of the finite-temperature phase diagram, often obtained by mean-field theories, the richness of the ground state diagrams for certain combinations of parameters can be used as a guide to explore interesting regions of the finite-temperature phase diagram of the model.     

A renormalization-group analysis of a spin-1 Ising ferromagnet with competing crystal-field and repulsive biquadratic interactions

Phase diagrams have been produced and critical exponents calculated for a Blume-Emery-Griffiths system with competing biquadratic and crystal-field interactions with uniform ferromagnetic bilinear interactions. This competition directly effects the clustering and density of nonmagnetic impurities. These results have been produced using renormalization-group methods with a hierarchical lattice. A series of planes of constant, repulsive biquadratic coupling have been probed while varying the temperature and concentration of annealed vacancies in the system. The sinks have been analyzed and interpreted, and critical exponents calculated for the higher order transitions.     

Dynamic compensation temperatures in a mixed spin-1 and spin-3/2 Ising system under a time-dependent oscillating magnetic field

We study the existence of dynamic compensation temperatures in the mixed spin-1 and spin-3/2 Ising ferrimagnetic system Hamiltonian with bilinear and crystal-field interactions in the presence of a time-dependent oscillating external magnetic field on a hexagonal lattice. We employ the Glauber transitions rates to construct the mean-field dynamic equations. We investigate the time dependence of an average sublattice magnetizations, the thermal behavior of the dynamic sublattice magnetizations and the total magnetization. From these studies, we find the phases in the system, and characterize the nature (continuous or discontinuous) of transitions as well as obtain the dynamic phase transition (DPT) points and the dynamic compensation temperatures. We also present dynamic phase diagrams, including the compensation temperatures, in the five different planes. A comparison is made with the results of the available mixed spin Ising systems.     

Phase diagrams of the mixed spin-1 and spin-1/2 Ising-Heisenberg model on the diamond-like decorated Bethe lattice

The ground-state and finite-temperature behavior of the mixed spin-1 and spin-1/2 Ising-Heisenberg model on the diamond-like decorated Bethe lattice is investigated within the framework of two rigorous methods: the decoration-iteration transformation and exact recursion relations. The model under consideration describes a hybrid classical-quantum system consisting of the Ising and Heisenberg spins, which interact among themselves either through the Ising or XXZ Heisenberg nearest-neighbor interaction. Both sublattice magnetizations of the Ising and Heisenberg spins are exactly calculated with the aim to examine phase diagrams, thermal variations of the total and sublattice magnetizations. The finite-temperature phase diagrams form continuous (second-order) phase transition lines only, which exhibit a small reentrant region if the diamond-like decorated Bethe lattice with a sufficiently high coordination number is considered.     

The spin-1 Ising model on a two-layer Bethe lattice with FM/AFM interactions

Two-layer Bethe lattice with the Ising spins of the top layer having only ferromagnetic (FM) interactions and the bottom layer having only antiferromagnetic (AFM) interactions are allowed to interact with the interlayer interaction of either FM or AFM type. The model is studied by using the exact recursion relations in a pairwise approach for given coordination numbers q=3, 4 and 6 with equal external magnetic fields acting on the layers. The phase diagrams of the model are obtained on different planes for given system parameters by studying the ground state (GS) phase diagrams and the thermal variations of the order-parameters and the response functions, i.e. the susceptibility and the specific heat, in detail. The model presents second- and first-order phase transitions, and where their lines are combined is the tricritical point. The critical end points also exist. The reentrant behavior is also seen when the model presents two Néel temperatures.     

Dynamic phase transitions in the kinetic spin-1 Blume-Capel model on the Bethe lattice

The stationary states of the kinetic spin-1 Blume-Capel (BC) model on the Bethe lattice are analyzed in detail in terms of recursion relations. The model is described using a Glauber-type stochastic dynamics in the presence of a time-dependent oscillating external magnetic field (h) and crystal field (D) interactions. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. It is found that the magnetization oscillates around nonzero values at low temperatures (T) for the ferromagnetic (F) phase while it only oscillates around zero values at high temperatures for the paramagnetic (P) phase. There are regions of the phase space where the two solutions coexist. The dynamic phase diagrams are obtained on the (kT/J,h/J) and (kT/J,D/J) planes for the coordination number q=4. In addition to second-order and first-order phase transitions, dynamical tricritical points and triple points are also observed.