Dynamic phase transitions and dynamic phase diagrams of the spin-2 Blume-Capel model under an oscillating magnetic field within the effective-field theory

The dynamic phase transitions are studied in the kinetic spin-2 Blume-Capel model under a time-dependent oscillating magnetic field using the effective-field theory with correlations. The effective-field dynamic equation for the average magnetization is derived by employing the Glauber transition rates and the phases in the system are obtained by solving this dynamic equation. The nature (first- or second-order) of the dynamic phase transition is characterized by investigating the thermal behavior of the dynamic magnetization and the dynamic phase transition temperatures are obtained. The dynamic phase diagrams are constructed in the reduced temperature and magnetic field amplitude plane and are of seven fundamental types. Phase diagrams contain the paramagnetic (P), ferromagnetic-2 (F₂) and three coexistence or mixed phase regions, namely the F₂+P, F₁+P and F₂+F₁+P, which strongly depend on the crystal-field interaction (D) parameter. The system also exhibits the dynamic tricritical behavior.     

Dynamic phase transitions and dynamic phase diagrams in the kinetic spin-5/2 Blume-Capel model in an oscillating external magnetic field: Effective-field theory and the Glauber-type stochastic dynamics approach

Using an effective field theory with correlations, we study a kinetic spin-5/2 Blume-Capel model with bilinear exchange interaction and single-ion crystal field on a square lattice. The effective-field dynamic equation is derived by employing the Glauber transition rates. First, the phases in the kinetic system are obtained by solving this dynamic equation. Then, the thermal behavior of the dynamic magnetization, the hysteresis loop area and correlation are investigated in order to characterize the nature of the dynamic transitions and to obtain dynamic phase transition temperatures. Finally, we present the phase diagrams in two planes, namely (T/zJ, h₀/zJ) and (T/zJ, D/zJ), where T absolute temperature, h₀, the amplitude of the oscillating field, D, crystal field interaction or single-ion anisotropy constant and z denotes the nearest-neighbor sites of the central site. The phase diagrams exhibit four fundamental phases and ten mixed phases which are composed of binary, ternary and tetrad combination of fundamental phases, depending on the crystal field interaction parameter. Moreover, the phase diagrams contain a dynamic tricritical point (T), a double critical end point (B), a multicritical point (A) and zero-temperature critical point (Z).     

Dynamic magnetic hysteresis behavior and dynamic phase transition in the spin-1 Blume-Capel model

The nature (time variation) of response magnetization m(wt) of the spin-1 Blume-Capel model in the presence of a periodically varying external magnetic field h(wt) is studied by employing the effective-field theory (EFT) with correlations as well as the Glauber-type stochastic dynamics. We determine the time variations of m(wt) and h(wt) for various temperatures, and investigate the dynamic magnetic hysteresis behavior. We also investigate the temperature dependence of the dynamic magnetization, hysteresis loop area and correlation near the transition point in order to characterize the nature (first- or second-order) of the dynamic transitions as well as obtain the dynamic phase transition temperatures. The hysteresis loops are obtained for different reduced temperatures and we find that the areas of the loops are decreasing with the increasing of the reduced temperatures. We also present the dynamic phase diagrams and compare the results of the EFT with the results of the dynamic mean-field approximation. The phase diagrams exhibit many dynamic critical points, such as tricritical (•), zero-temperature critical (Z), triple (TP) and multicritical (A) points. According to values of Hamiltonian parameters, besides the paramagnetic (P), ferromagnetic (F) fundamental phases, one coexistence or mixed phase region, (F+P) and the reentrant behavior exist in the system. The results are in good agreement with some experimental and theoretical results.     

Dynamic phase transition in the kinetic spin-5/2 Blume–Emery–Griffiths model in an oscillating external magnetic field

Employing a mean-field approach, we study the stationary states of the kinetic spin-5/2 Blume–Emery–Griffiths (BEG) model under the presence of a time-varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. We employ the Glauber transition rates to construct the set of dynamic mean-field equations. We investigate the time variation in average order parameters to find the phases in the system, and the thermal behavior of the dynamic order parameters to characterize the nature (continuous or discontinuous) of the dynamic phase transtions and to the dynamic phase transition temperature. The dynamic phase diagrams are presented in three different planes. The phase diagrams contain the ferromagnetic-5/2, the ferromagnetic-3/2, the ferromagnetic-1/2, the ferroquadrupolar, and disordered fundamental phases. They also include the nine coexisting or mixed phases composed of binary and ternary combinations of fundamental phases that strongly depend on the interaction parameters. The phase diagrams display the critical end point, double critical end point, triple point, quadruple point, and one, two, or three special points and the dynamic tricritical point that depends on the interaction parameters.     

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.     

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.     

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.     

Dynamic phase transitions in the kinetic mixed spin-1/2 and spin-5/2 Ising model under a time-dependent oscillating magnetic field

We calculate the dynamic phase transition (DPT) temperatures and present the dynamic phase diagrams in the kinetic mixed spin-1/2 and spin-5/2 Ising model under the presence of a time-dependent oscillating external magnetic field. We employ the Glauber transition rates to construct the set of mean-field dynamic equations. The time variation of the average magnetizations and the thermal behavior of the dynamic magnetizations are investigated, extensively. The nature (continuous or discontinuous) of the transitions is characterized by studying the thermal behaviors of the dynamic magnetizations. The DPT points are obtained and the phase diagrams are presented in two different planes. Phase diagrams contain four fundamental phases and three coexistence or mixed phases, which strongly depend on interaction parameters. The phase diagrams are discussed and a comparison is made with the results of the other mixed spin Ising systems.     

Dynamic magnetic properties of the kinetic cylindrical Ising nanotube

The nature (time variation) of response magnetizations m(wt) of the kinetic cylindrical Ising nanotube in the presence of a periodically varying external magnetic field h(wt) is studied by employing the effective-field theory (EFT) with correlations as well as the Glauber-type stochastic dynamics. We have determined the time variations of m(wt) and h(wt) for various temperatures, and investigated the dynamic magnetic hysteresis behavior. Temperature dependence of the dynamic magnetizations, hysteresis loop areas and correlations are investigated in order to characterize the nature (first- or second-order) of the dynamic transitions as well as to obtain the dynamic phase transition temperatures. We also present the dynamic phase diagrams in the three different planes and compare the results of the equilibrium and nonequilibrium states. The phase diagrams exhibit dynamic tricritical, isolated critical, multicritical and triple points. The results are in good agreement with some experimental and theoretical results.     

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.     

The ferrimagnetic phase of the Blume-Emery-Griffiths model on the cellular automaton

The Blume-Emery-Griffiths model is simulated using the cooling algorithm which is improved from the Creutz cellular automaton (CCA) under periodic boundary conditions. The simulations are carried out on a simple cubic lattice at K/J = −1.5 in the range of −3.5 < D/J < 0.5, with J and K representing the nearestneighbour bilinear and biquadratic interactions, D being the single-ion anisotropy parameter. The phase diagram characterizing phase transition of the model is obtained. We found different kinds of phase transitions between the ferromagnetic, quadrupolar, staggered quadrupolar and ferrimagnetic phases for K/J = −1.5. In particular, the region of the phase diagram containing a ferrimagnetic phase is explored and compared to those obtained by other methods. The simulations confirm that the ferrimagnetic phase occurs in the narrow interval −3.006 ≤ D/J < −3. This result is in a good agreement with Monte Carlo renormalization group and closer to the cluster variation method result than the mean field approximation result.      

Dynamic phase transitions in a cylindrical Ising nanowire under a time-dependent oscillating magnetic field

The dynamic phase transitions in a cylindrical Ising nanowire system under a time-dependent oscillating external magnetic field for both ferromagnetic and antiferromagnetic interactions are investigated within the effective-field theory with correlations and the Glauber-type stochastic dynamics approach. The effective-field dynamic equations for the average longitudinal magnetizations on the surface shell and core are derived by employing the Glauber transition rates. Temperature dependence of the dynamic magnetizations, the dynamic total magnetization, the hysteresis loop areas and the dynamic correlations are investigated in order to characterize the nature (first- or second-order) of the dynamic transitions as well as the dynamic phase transition temperatures and the compensation behaviors. The system strongly affected by the surface situations. Some characteristic phenomena are found depending on the ratio of the physical parameters in the surface shell and the core. According to the values of Hamiltonian parameters, five different types of compensation behaviors in the Néel classification nomenclature exist in the system. The system also exhibits a reentrant behavior.     

Dynamic phase transition in the kinetic spin-1 Blume-Capel model: Phase diagrams in the temperature and crystal-field interaction plane

Within a mean-field approach, the stationary states of the kinetic spin-1 Blume-Capel model in the presence of a time-dependent oscillating external magnetic field is studied. The Glauber-type stochastic dynamics is used to describe the time evolution of the system and obtain the mean-field dynamic equation of motion. The dynamic phase-transition points are calculated and phase diagrams are presented in the temperature and crystal-field interaction plane. According to the values of the magnetic field amplitude, three fundamental types of phase diagrams are found: One exhibits a dynamic tricritical point, while the other two exhibit a dynamic zero-temperature critical point. The text was submitted by the authors in English.     

Dynamic phase transition and dynamic phase diagrams in the spin-5/2 Blume–Capel model under a time-dependent oscillating external field

We calculate the dynamic phase transition (DPT) temperatures and present the dynamic phase diagrams in the kinetic spin-5/2 Blume–Capel model under the presence of a time-dependent oscillating external magnetic field. First, we employ the Glauber transition rates to construct the mean-field dynamic equation. Then, we study the time variation of the average magnetization to find the phases in the system. We also investigate the behaviour of the dynamic magnetization to characterize the nature (continuous and discontinuous) of transition and to obtain the DPT points. We present the dynamic phase diagrams in two different planes. The phase diagrams include the ferromagnetic-5/2 (f_5/2), the ferromagnetic-1/2 (f_1/2) and paramagnetic (p) fundamental phases. In addition to these fundamental phases, we find 10 mixed phases, depending on the interaction parameters. The phase diagrams display many special points, such as a dynamic tricritical point, a double critical end point, a triple point and a quadruple point.     

Dynamic magnetic properties in the kinetic mixed spin-2 and spin-5/2 Ising model under a time-dependent magnetic field

We extend the recent paper [W. Jiang, V-C. Lo, B-D. Bai, J. Yang, Physica A 389 (2010) 2227-2233] to present a study, within a mean-field approach, the dynamic magnetic properties of the mixed spin-2 and spin-5/2 Ising ferrimagnetic system, which corresponds the molecular-based magnetic materials AFe^IIFe^III(C₂O₄)₃ [ A=N(n-C_nH_2n+1)₄, n=3-5], by using the Glauber-type stochastic dynamics. This mixed Ising ferrimagnetic system is used on a layered honeycomb lattice in which Fe^II (S=5/2) and Fe^III (σ=2) occupy sites. First, we investigate the time variations of average order parameters to find the phases in the system and then the thermal behavior of the dynamic order parameters to obtain the dynamic phase transition (DPT) points as well as to characterize the nature (first-or second-order) phase transitions. We also present the dynamic phase diagrams and study the dynamic magnetic hysteresis loop behaviors of the kinetic mixed spin-2 and spin-5/2 Ising ferrimagnetic system. The results are compared with some experimental and theoretical works and a good overall agreement is found.     

Hysteretic response characteristics and dynamic phase transition via site dilution in the kinetic Ising model

The decay of the hysteresis loop area of the system, which is obeying a site diluted kinetic Ising model, is considered by the disorder parameter using the effective field theory analysis. The exhibition focuses on the understanding of external field frequency, amplitude and the site concentration dependency of the hysteresis loop area for several powerful treatments. Important characteristics of the hysteretic response, such as frequency dispersion, effect of domain nucleation phenomenon on the dynamic process, etc., has been introduced together with well known other characteristics. An attempt has been made to explain the relations between the competing time scales (intrinsic microscopic relaxation time of the system and the time period of the external oscillatory field) and the shape of the response. As a result of the detailed investigations, existence of essentially three, particularly four types of dispersion curves have been propounded.     

Dynamic Dipole and Quadrupole Phase Transitions in the Kinetic Spin-3/2 Model

The dynamic phase transition has been studied, within a mean-field approach, in the kinetic spin-3/2 Ising model Hamiltonian with arbitrary bilinear and biquadratic pair interactions in the presence of a time dependent oscillating magnetic field by using the Glauber-type stochastic dynamics. The nature (first- or second-order) of the transition is characterized by investigating the behavior of the thermal variation of the dynamic order parameters and as well as by using the Liapunov exponents. The dynamic phase transitions (DPTs) are obtained and the phase diagrams are constructed in the temperature and magnetic field amplitude plane and found nine fundamental types of phase diagrams. Phase diagrams exhibit one, two or three dynamic tricritical points, and besides a disordered (D) and the ferromagnetic-3/2 (F_3/2) phases, six coexistence phase regions, namely F _3/2+ F _1/2, F _3/2+ D, F _3/2+ F _1/2+ FQ, F _3/2+ FQ, F _3/2+ FQ + D and FQ + D, exist in which depending on the biquadratic interaction. PACS number(s): 05.50.+q, 05.70.Fh, 64.60.Ht, 75.10.Hk     

Dynamic magnetic behavior of the mixed-spin bilayer system in an oscillating field within the mean-field theory

The dynamic magnetic behavior of the mixed Ising bilayer system (σ=2$\sigma =2$ and S=5/2$S=5/2$), with a crystal-field interaction in an oscillating field are studied, within the mean-field approach, by using the Glauber-type stochastic dynamics for both ferromagnetic/ferromagnetic and antiferromagnetic/ferromagnetic interactions. The time variations of average magnetizations and the temperature dependence of the dynamic magnetizations are investigated. The dynamic phase diagrams are presented in the reduced temperature and magnetic field amplitude plane and they exhibit several ordered phases, coexistence phase regions and critical points as well as a re-entrant behavior depending on interaction parameters.     

Effects of random-crystal-field on the kinetic Blume Capel model

The effects of random crystal field on the stationary states of the kinetic spin-1 Blume Capel model are studied using the Glauber-type stochastic dynamics under the time-dependent oscillating field assumption. Our investigation, based on the equilibrium ground state phase diagram, revealed many interesting phenomena. The known phases, in the equilibrium case, are obtained for high field and are represented by limit cycles. The phase diagram of the pure pure kinetic Ising spin- and spin-1 Blume Capel models are deduced as particular cases. First-order, second-order transition lines, dynamical critical and dynamical double critical end points are also obtained.     

Nonequilibrium behavior of the kinetic metamagnetic spin-5/2 Blume–Capel model

The dynamic magnetic behavior of the kinetic metamagnetic spin-5/2 Blume–Capel model is examined, within a mean-field approach, under a time-dependent oscillating magnetic field. To describe the kinetics of the system, Glaubertype stochastic dynamics has been utilized. The mean-field dynamic equations of the model are obtained from the Master equation. Firstly, these dynamic equations are solved to find the phases in the system. Then, the dynamic phase transition temperatures are obtained by investigating the thermal behavior of dynamic sublattice magnetizations. Moreover, from this investigation, the nature of the phase transitions(first- or second-order) is characterized. Finally, the dynamic phase diagrams are plotted in five different planes. It is found that the dynamic phase diagrams contain the paramagnetic(P),antiferromagnetic(AF5/2, AF3/2, AF1/2) phases and five different mixed phases. The phase diagrams also display many dynamic critical points, such as tricritical point, triple point, quadruple point, double critical end point and separating point.