Path Probability Method Research Articles

Nonequivalent Antiferromagnetically Coupled Sublattices Induce Two-Step Spin-Crossover Transitions: Equilibrium and Nonequilibrium Aspects

As a continuation to the previously published work (Yalçın et al. (2022)), we investigate the equilibrium and nonequilibrium properties of the spin-crossover systems, with a specific focus on the nonequivalent sublattice, and compare these properties with those of the equivalent sublattices. We used the lowest approximation of the cluster variation method (LACVM) to derive the static equations for the order parameters of the two sublattices and determine high-spin fraction in relation to temperature and external magnetic field in a spin-crossover system. At a low temperature, the transition from stable high-spin (HS) state where nHS=1 occurs in the plateau region, where nHS=0.5 for nonequivalent sublattices. The order parameters for non-equivalent sublattices exhibit different states at the transition temperature. Also, we study the nonequilibrium properties of the order parameters and high-spin fraction using the path probability method (PPM). With the current model, we obtain and analyze the relaxation curves for the order parameters Sa, Sb, and high-spin fraction. These curves demonstrate the existence of bistability at low temperatures. At the end of this study, we present the flow diagram that shows the order parameters for different temperature values. The diagram exhibits states that are stable, metastable, and unstable.

Dynamic magnetic hysteresis features of a mixed spin (2, 5/2) Ising system on a hexagonal lattice under an oscillating magnetic field within the path probability method

Recently, we [J. Magn. Magn. Mater. 514 (2020) 167242] presented a study of the dynamic magnetic loop features of molecular-based magnetic materials and ferrimagnetism that were investigated by using a mixed spin 1/2 and spin-1 ferrimagnetic Ising system within the path probability method. We found several interesting dynamic magnetic hysteresis properties and observed that the obtained results are in qualitatively good agreement with some theoretical and experimental studies in magnetic materials. We extended our recent previous published work to study the dynamic magnetic hysteresis properties of molecular-based magnetic materials and ferrimagnetism utilizing the mixed spin (2, 5/2) ferrimagnetic Ising system on a hexagonal lattice employing the path probability method. The spin (2, 5/2) Ising system is one of the most suitable and most used prototypical systems to investigate many molecular-based magnetic materials and ferrimagnetic materials. We found more interesting dynamic magnetic hysteresis properties such as double, triple, quadruple and quintuple loop behaviors for the layer magnetization, and only double and triple loops for the total magnetization. We found that some of our obtained results are in qualitatively good agreement with some theoretical, experimental reported works, and the works on some magnetic materials as obtained by the melt-spinning technique.

Dynamic behavior of the antiferromagnetically coupled bilayer Ising model

Using the path probability and lowest approximation of cluster variation method, we study the dynamic and equilibrium properties of a bilayer magnetic system, consisting of two ferromagnetic monolayers antiferromagnetically coupled for different spins (σ = 1/2 and S = 1). Firstly, numerical results of the monolayer and total magnetizations are presented under the effect of the diverse physical parameters, and the phase diagrams of bilayer system are discussed. Then, since it is well established that the path probability method is an effective method for the existence of metastable states, the time evolution of monolayer- and total magnetizations is investigated.

Dynamic Magnetic Features of a Mixed‐Spin‐2 and Spin‐5/2 Ising Ferrimagnetic System under a Time‐Dependent Oscillating Magnetic Field: Path Probability Method Approach

A study, within the path probability method, of the dynamic magnetic features of the mixed spin (2, 5/2) Ising ferrimagnetic system on a hexagonal lattice under the presence of a sinusoidal external magnetic field is presented. The Hamiltonian of the system contains intersublattice, intrasublattice, and crystal‐field interactions. The intersublattice interaction is taken as antiferromagnetic. The time variations of average dynamic magnetizations and the thermal behavior of the dynamic magnetizations are investigated to obtain the phases in the system and to characterize the nature of the dynamic phase transitions and find the dynamic phase transition temperature points. The dynamic phase diagrams in six different phase planes are presented, in which they contain the paramagnetic (p) phase, three different ferrimagnetic (i 1, i 2, i 3) fundamentals, and the p + i 1, i 1 + i 2, and i 2 + i 3 mixed phases as well as the dynamic tricritical point and the dynamic double critical end point, depending on the interaction parameters. The dynamic compensation behavior is also examined and only N‐type behaviors are observed. Some of the obtained dynamic phase diagrams are in qualitatively good agreement within some works on the dynamics of mixed‐spin Ising system.

Stable, metastable and unstable states of the mixed spin (1, 3/2) Ising model in the absence and presence of constant magnetic fields

Stable, metastable and unstable states of the mixed spin (1, 3/2) Ising system in the absence and presence of constant external magnetic fields (CEMFs) were investigated both by the cluster variation method (CVM) and the path probability method (PPM). First, we examine the thermal behavior of the magnetizations by using the CVM and search the metastable and unstable branches of magnetizations in addition to the stable branches. We present the metastable phase diagrams besides to equilibrium phase diagrams in the absence of the CEMFs. We also constructed only the metastable phase diagram under the presence of CEMFs. Since the metastability is a stable state of a dynamical system, we used the PPM on the model and found the dynamic equations. Dynamic equations were solved by means of the flow diagrams (FDs). The FDs clearly display the stable, metastable and unstable states in which also obtained within the CVM. The FDs display the role of the metastable, unstable states and initial conditions as well as how the system trapped into the metastable state. Some properties of the metastable states were observed from the FDs. These studies were made both for the ferrimagnetic and antiferrimagnetic cases.PACS: 05.50. + q, 05.70.Fh, 05.70.Ln, 64.60.Cn, 64.60. My, 75.10.Hk

Static and dynamic properties of two-sublattice spin-crossover systems

We investigate the static and dynamic properties of spin-crossover systems in the presence of an external magnetic field using two-sublattice Ising-like Hamiltonian. At low temperatures both sublattices are in the LS state, and in the HS state at high temperatures. The new feature of spin-crossover is depicted for our model when we take the non-equivalent sublattices JA≠JB, the two-step spin-crossover phase transition is obtained with the presence of the hysteresis loop. In order to study the properties of two-step spin-crossover, we use the lowest approximation of the cluster variation method for static properties. The dynamic properties of the system with bilinear interactions are studied by the path probability method. We present high-spin state fraction nHS vs temperature and magnetic field variations for constant values of the degeneracy ratio between high-spin and low-spin states. In the present model, we obtained the relaxation curves of the order parameters SA and SB using different temperature values. Finally, the flow diagram of the order parameters is depicted for different values of rate constant k1 and temperature, which displays the stable, metastable, and unstable properties. Static and dynamic self-consistent equations of the system were performed by using Newton-Raphson and Runge-Kutta methods.

Nonequlibrium magnetic features in a mixed spin (2, 5/2) Ising system driven by the external oscillating magnetic field by path probability method

Utilizing the path probability method, we investigated nonequilibrium magnetic features in a mixed spin (2, 5/2) Ising model Hamiltonian composed of bilinear and crystal-field interactions in the presence of an external oscillating magnetic field. We numerically solved the time dependence of average magnetizations to find the phases in the system. We examined the dynamic magnetizations to obtain dynamic phase transition (DPT) temperatures, the nature of the DPTs, and phases in the system. The dynamic phase diagrams (DPDs) were constructed in reduced temperature and the amplitude of oscillating magnetic field plane for various interaction parameters. We observed that the system gives very rich and interesting topological behaviors of DPDs, such as up to two dynamic tricritical points, six critical end points, a double critical end points, a zero-temperature critical point, one inverse critical end point and a quadruple point depending on interaction parameters. The system also exhibits paramagnetic, six distinct ferrimagnetic and three different nonmagnetic phases as well as up to eleven different mixed or hybrid phases. In addition, the system exhibits the reentrant behavior.

Dynamic hysteresis features of a mixed spin (1/2, 3/2) Ising system within the path probability method

ABSTRACT The dynamic hysteresis features (DHFs) of a mixed spin (1/2, 3/2) model on a square lattice was studied within the path probability method (PPM) in the presence of a time-varying (sinusoidal) magnetic field. The PPM contains three rate constants in which one of them (k 2) may be considered as corresponding to the wheel speed in the melt-spinning method (MSM). Influences of the temperature (T), angular frequency (ω), and the rate constant (k2 ) on the DHFs for the attractive bilinear interaction were examined. We also investigated the magnetic coercive fields and remanent magnetizations as functions T, ω, and k 2. Some of our obtained results are in qualitatively good agreement with some theoretical studies, experimental reported works on some magnetic materials, and the hysteresis loops obtained by using the MSM.

Dynamic phase transitions and compensation behaviors in a mixed spin (1/2, 3/2) Ising model on a hexagonal lattice by path probability method

Dynamic phase transitions and compensation behaviors in a mixed spin (1/2, 3/2) Ising model on a hexagonal lattice by path probability method

Dynamic multicritical phase diagrams of the mixed spin (2, 5/2) Blume-Emery-Griffiths model with repulsive biquadratic coupling

We investigated the dynamic phase transitions (DPTs) in the mixed spin (2, 5/2) Blume-Emery Griffiths model with repulsive biquadratic interaction in the presence of a time-varying magnetic field. We used the path probability method to obtain the set of the dynamic equations. We numerically solved these dynamic equations to characterize the nature of first- and second-order phase transitions and to find the DPT temperatures as well as obtain the phases in the system. We constructed the dynamic phase diagrams (DPDs) in reduced temperature and amplitude of oscillating magnetic field plane. We observed that the DPDs display richer, complex and more topological various type of phase diagrams. In particular, DPDs exhibit the disordered phase, antiquadrupolar or staggered phase, six different ferrimagnetic phases, three different nonmagnetic phases, and numerous mixed phases. DPDs also display two dynamic tricritical points for only smaller values of crystal-field interactions, multiple critical end and double critical end points, one zero-temperature critical point, one inverse critical end point, and a quadruple point depending on interaction parameters. The system always shows the reentrant behaviors for the higher values of magnetic field amplitude, but it does not exhibit the dynamic tricritical behavior for higher values of crystal-field parameter.

Influences of the interaction parameters on the dynamic hysteresis of a mixed spin (1/2, 3/2) Ising model under the presence of an oscillating magnetic field

Within the path probability method, the influences of the interaction parameters on the dynamic hysteresis properties of a mixed spin (1/2, 3/2) Ising model driven by a time-varying magnetic field were investigated. First, we examined the effects of the interaction parameters, namely temperature, angular frequency, the rate constant and the exchange interactions on the dynamic hysteresis loops. Then, we investigated the impacts of the interaction parameters on the dynamic coercive fields and dynamic remanent magnetizations. Some of obtained results are in, qualitatively, good agreement with some theoretically reported studies as well as experimental works in some magnetic materials and the hysteresis loops obtained within the melt spinning method.

Magnetic features and compensation behaviors of a mixed spin (1/2, 1) Ising ferrimagnetic system on a hexagonal lattice

We studied dynamic magnetic properties and compensation behaviors of a mixed spin (1/2, 1) Ising ferrimagnetic system on a hexagonal lattice within the path probability method that contains more than one rate constants and the average dynamic order parameters were coupled. We found the phases in the system by solving the average dynamic magnetizations. We examined the thermal features of dynamic magnetizations to characterize the nature of the dynamic phase transitions (DPTs) and obtained the DPT temperatures. We constructed the dynamics phase diagrams in six different planes, which contain the dynamic tricritical, critical end, double critical end, zero-temperature critical and multicritical special points as well as the paramagnetic (P), non-magnetic (NM), ferromagnetic (I) fundamental phases and P+I, NM+I mixed or hybrid phases depending on system parameters. We also investigated the dynamic compensation behaviors, and the N-, P-, Q-, R- and the S-types compensation behaviors were found.

Dynamic magnetic hysteresis loop features of a mixed spin (1/2, 1) Ising system on a hexagonal lattice using path probability method

Batı and Ertaş [Physica B 513 (2017) 40] presented a study of dynamic magnetic hysteresis loop properties (DMHLFs) of a mixed spin (1/2, 1) Ising system (IS) on a hexagonal lattice within the mean-field theory based on the Glauber dynamics approach. They observed a number of interesting DMHLFs. We extended the investigation of DMHLFs of a mixed spin (1/2, 1) IS on a hexagonal lattice using the path probability method. We also examined the influences of the rate constants on DMHLFs as well as magnetic coercive fields and remanent magnetizations. We observed that some of our results are in quantitatively good agreement with some theoretical works on various ISs and experimental reported works on some magnetic materials as well as works within the melt-spinning method. We also found that the effects of rate constants are more effective on DMHLFs of the magnetization of spin-1 particles than loops of the magnetization of spin-1/2 particles.

Dynamic phase of transitions of the mixed spin (1/2, 3/2) Ising model in the presence of a time-varying magnetic field by using the path probability method

Dynamic phase transitions (DPTs) of the mixed spin (1/2, 3/2) Ising ferrimagnetic model on the two interpenetrating square lattice with the bilinear and crystal-field interactions under a time-varying (sinusoidal) magnetic field were examined by using the path probability method (PPM). We studied stationary solutions of dynamic average magnetizations and obtained phases and then we investigated thermal behaviors of dynamic magnetizations to designate the nature of the DPTs and find the DPT temperatures. We also constructed the dynamic phase diagrams (DPDs) in two different planes. DPDs exhibit the paramagnetic (p), ferrimagnetic (i), and mixed phases (i+p), and one or two dynamic tricritical points and dynamic double critical end points. The reentrant behavior is also seen. We observed that the PPM is a more appropriate dynamic method to study the dynamic features of ferrimagnetism.

Determination of the steady-state kinetics in a spin-1 Ising model using path probability method

As a continuation of our previously published work, we propose a theoretical framework for the determination of steady-state kinetics in a spin-1 Ising model by the path probability method. The framework is based on the principles of non-equilibrium statistical physics and is quite different from the phenomenological approach. We construct a set of linear kinetic equations for the order parameters using the non-linear dynamic (or rate) equations in the presence of external magnetic field. From the steady-state solutions of the linear equations, an expression for the complex (or dynamic) magnetic susceptibility [Formula: see text] is derived. The temperature dependence of the magnetic dispersion relation [Formula: see text] and magnetic absorption factor [Formula: see text] has been studied in the ferromagnetic (FM) and paramagnetic (PM) phases as well as near the critical regime.

Dynamic magnetic hysteresis loop features of molecular-based magnetic materials and ferrimagnetic systems by the path probability method

Dynamic magnetic hysteresis loops (DMHL) features of molecular-based magnetic materials and ferrimagnetic systems are investigated by a mixed spin (1/2, 1) model by means of the path probability method (PPM). The PPM contains two rate constants; hence the DMHL features have been studied by using the method with two rate constants. In particular, the effects of the repulsive (J<0) and attractive (J>0) bilinear nearest-neighbor pair interactions, temperature (T), crystal-field interaction (D), angular frequency (ω) of the oscillating magnetic field and especially the rate constant (k2), which corresponds to the wheel speed in the melt spinning technique (MST), on the DMHL are investigated. The shapes and areas of hysteresis loops are affective sensibility by these system parameters. We also investigate the magnetic coercive fields and remanents magnetizations as a function ω. We found that the most of our results are in a good agreement with some theoretical and experimental reported works. We also observe the novel and interesting DMHL behaviors for lower values of T, and higher values of ω, k2 and D.

Dynamic magnetic properties of the mixed spin (1/2, 3/2) Ising system in the presence of magnetic field within the path probability method

The dynamic magnetic properties of the mixed spin (1/2, 3/2) Ising system with bilinear nearest-neighbor exchange interactions and a crystal-field term under the presence of the external longitudinal magnetic field (ELMF) are investigated by the path probability method. First we briefly examine the equilibrium magnetic properties of the system within the cluster variation method in order to clearly understand the dynamic behavior. In particular, we investigate thermal behavior of the magnetization in detail to search whether or not the metastable and unstable branches of magnetizations occur in the system. We find that the metastable and unstable branches of magnetization only exist if the system is in the presence of the ELMF. We also presented the metastable phase diagrams in the ELMF. Then, we apply the path probability method to the system and obtain the dynamic equations. Numerical solutions of these equations are given in the form of the flow diagrams (FDs). FDs explicitly illustrate the stable, metastable and unstable states as well as the role of the unstable states and initial conditions. FDs also display how one can obtain the metastable state more easily.

Relaxation times obtained from the rate equations using path probability method for the spin-1 Ising model

We have presented the time-dependent behaviors of a spin-1 Ising model (with both bilinear and biquadratic interactions) in the neighborhood of the equilibrium states via the path probability method. The rate equations as introduced by Keskin and co-workers (1989) have been linearized to obtain the characteristic relaxation times. The temperature variations of these times were produced in accordance with the Tanaka and Mannari’s work (1976) on the equilibrium properties. Results are compared to one using Onsager’s phenomenological theory and a good agreement is achieved.

Application of Cluster Variation and Path Probability Methods to the Tetragonal–Cubic Phase Transition in ZrO2

Cluster variation method (CVM) and path probability method (PPM) have generally been employed to study replacive phase transitions in alloy systems. Recently, displacive phase transitions have been explored within the realm of replacive phase transition in the CVM theoretical framework by viewing displaced atoms as different atomic species, i.e., by converting a freedom of atomic displacement to a configurational freedom. The same methodology is applied to the PPM calculations in this work, and the kinetics of displacive phase transition from tetragonal to cubic phases in ZrO2 are investigated as well as their equilibrium states.

Atomistic relaxation process in a Ni3Al ordered phase using path probability method with vacancy mechanisms

The path probability method (PPM), which is a natural extension of the cluster variation method (CVM) to a time domain, has been employed in a relaxation process of atomic configurations in alloy systems. Although the vacancy mechanism is the main atomic migration process in an alloy system, most studies of PPM have used the spin flipping mechanism (or the direct exchange mechanism) because of the huge computational burden imposed by the vacancy mechanism. In this paper the computational problem is circumvented by treating various path variables in the PPM as cluster probabilities in the CVM, and the vacancy mechanism is explicitly taken into account in the theoretical framework. The method is employed to explore the relaxation process in a Ni3Al ordered phase within the tetrahedron approximation, and the effect of vacancy concentration is investigated.