Dynamic Behavior of a Spin- 1 Ising Model. I. Relaxation of Order Parameters and the “Flatness” Property of Metastable States (original) (raw)
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Equilibrium properties of a spin-1 Ising system with bilinear, biquadratic and odd interactions
Physica A: Statistical Mechanics and its Applications, 1996
The equilibrium properties of the spin-1 Ising system Harniltonian with arbitrary bilinear (J), biquadratic (K) and odd (L), which is also called dipolar~luadrupolar, interactions is studied for zero magnetic field in the lowest approximation of the cluster variation method. The odd interaction is combined with the bilinear (dipolar) and biquadratic (quadrupolar) exchange interactions by the geometric mean. In this system, phase transitions depend on the ratio of the coupling parameters, ct = J/K; therefore, the dependence of the nature of the phase transition on ~t is investigated extensively and it is found that for ct ~< 1 and ~> 2000 a second-order phase transition occurs, and for 1 < ~t < 2000 a first-order phase transition occurs. The critical temperatures in the case of a second-order phase transition and the upper and lower limits of stability temperature in the case of a first-order phase transition are obtained for different values of ct calculated using the Hessian determinant. The first-order phase transition temperatures are found by using the free energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves and the thermal variations of these solutions as a function of the reduced temperature are investigated. The unstable solutions for the first-order phase transitions are obtained by displaying the free energy surfaces in the form of a contour map. Results are compared with the spin-1 Ising system Hamiltonian with the bilinear and biquadratic interactions and it is found that the odd interaction greatly influences the phase transitions.
Equilibrium and nonequilibrium behavior of the spin-1 Ising model in the quadrupolar phase
Physica A: Statistical Mechanics and its Applications, 2002
Equilibrium properties of the spin-1 Ising model with the arbitrary bilinear (J) and biquadratic (K) pair interactions are studied on a body centered cubic lattice by using the pair approximation of the cluster variation method, which is identical to the Bethe approximation, in the quadrupolar phase. First, the thermal variation of the metastable and unstable dipole and quadrupolar moment order parameters is investigated and the metastable and unstable branches of them are found besides the stable branch of the quadrupolar moment order. In addition, it is found that when the values of = J=K decreases, transition temperatures also decrease and some of metastable and unstable branches of order parameters disappear. On the other hand, nonequilibrium behavior of the model in the quadrupolar phase is studied by using the path probability method and the set of nonlinear di erential equations, which is also called the dynamic or rate equations, is obtained. The solutions of the dynamic equations are expressed by means of ow diagrams near the transition temperatures. The stable, metastable and unstable solutions are shown and the "overshooting" phenomenon is seen in the ow diagrams, explicitly. The role of the unstable points, as separators between the stable and the metastable points, is described and how a system freezes in a metastable state is also investigated, extensively.
Physics Letters A, 2005
The relaxation behavior of the spin-3/2 Ising model Hamiltonian with bilinear and biquadratic interactions near the secondorder phase transition temperature or critical temperature is studied by means of the Onsager's theory of irreversible thermodynamics or the Onsager reciprocity theorem (ORT). First, we give the equilibrium case briefly within the molecular-field approximation in order to study the relaxation behavior by using the ORT. Then, the ORT is applied to the model and the kinetic equations are obtained. By solving these equations, three relaxation times are calculated and examined for temperatures near the second-order phase transition temperature. It is found that one of the relaxation times goes to infinity near the critical temperature on either side, the second relaxation time makes a cusp at the critical temperature and third one behaves very differently in which it terminates at the critical temperature while approaching it, then showing a "flatness" property and then decreases. We also study the influences of the Onsager rate coefficients on the relaxation times. The behavior of these relaxation times is discussed and compared with the spin-1/2 and spin-1 Ising systems.
Metastable states in a two-dimensional Ising model with dipolar interactions
Physica D: Nonlinear Phenomena, 2002
The equilibrium phase diagram of a two-dimensional Ising model with competing exchange and dipolar interactions is analyzed using a Monte Carlo simulation technique. We consider the low temperature region of the (δ, T ) phase diagram (δ being the ratio between the strengths of the exchange and dipolar interactions) for the range of values of δ where striped phases with widths h = 1 and 2 are present. We show that the transition line between both phases is a first order one. We also show that, associated with the first-order phase transition, there appear metastable states of the phase h = 2 in the region where the phase h = 1 is the thermodynamically stable one and vice versa.
Critical properties of a spin- Ising model with bilinear and biquadratic interactions
Physica A: Statistical Mechanics and its Applications, 2003
We have studied the temperature dependences of the magnetization and the quadrupolar moment of a spin-3 2 Ising model Hamiltonian with bilinear and biquadratic interactions using the molecular ÿeld approximation. A number of characteristic behaviors for the thermal variations of the order parameters are investigated and besides the stable solutions, metastable and unstable solutions are found. These solutions are classiÿed by the means of free energy surfaces based on the form of a contour map. The ÿrst-and second-order phase transition temperatures are found by using free energy values while increasing and decreasing the temperature, and the Hessian determinant, respectively. The metastable phase diagram in addition to the equilibrium phase diagram is presented in (T=K; J=K) plane in detail and it exhibits zero-temperature critical point, critical points, triple point, tricritical points and multicritical point.
Non-equilibrium phase transitions in one-dimensional kinetic Ising models
Journal of Physics A: Mathematical and General, 1995
A family of nonequilibrium kinetic Ising models, introduced earlier, evolving under the competing effect of spin flips at zero temperature and nearest neighbour random spin exchanges is further investigated here. By increasing the range of spin exchanges and/or their strength the nature of the phase transition 'Ising-to-active' becomes of (dynamic) mean-field type and a first order tricitical point is located at the Glauber (δ = 0) limit. Corrections to mean-field theory are evaluated up to sixth order in a cluster approximation and found to give good results concerning the phase boundary and the critical exponent β of the order parameter which is obtained as β ≃ 1.0.
Phase diagrams of a spin-1 Ising system with competing short- and long-range interactions
Physical review. E, Statistical, nonlinear, and soft matter physics, 2015
We have studied the phase diagrams of the one-dimensional spin-1 Blume-Capel model with anisotropy constant D, in which equivalent-neighbor ferromagnetic interactions of strength -J are superimposed on nearest-neighbor antiferromagnetic interactions of strength K. A rich critical behavior is found due to the competing interactions. At zero temperature two ordered phases exist in the D/J-K/J plane, namely the ferromagnetic (F) and the antiferromagnetic one (AF). For lower values of D/J(D/J<0.25) these two ordered phases are separated by the point K_{c}=0.25J. For 0.25<D/J≤0.50, the paramagnetic phase P emerges in a region separated between the lines determined by D/J=0.5-K/J and D/J=K/J. For D/J>0.5, only phases AF and F exist and are separated by a line given by D/J=K/J. At finite temperatures, we found that the ferromagnetic region of the phase diagram in the k_{B}T/J-D/J plane is enriched by another ferromagnetic phase F^{^{'}} above a first-order line for 0.195<K/...
Critical properties of a spin- 3 2 Ising model with bilinear and biquadratic interactions
Physica a, 2003
We have studied the temperature dependences of the magnetization and the quadrupolar moment of a spin-3 2 Ising model Hamiltonian with bilinear and biquadratic interactions using the molecular ÿeld approximation. A number of characteristic behaviors for the thermal variations of the order parameters are investigated and besides the stable solutions, metastable and unstable solutions are found. These solutions are classiÿed by the means of free energy surfaces based on the form of a contour map. The ÿrst-and second-order phase transition temperatures are found by using free energy values while increasing and decreasing the temperature, and the Hessian determinant, respectively. The metastable phase diagram in addition to the equilibrium phase diagram is presented in (T=K; J=K) plane in detail and it exhibits zero-temperature critical point, critical points, triple point, tricritical points and multicritical point.
The European Physical Journal B, 2000
We investigate the dynamical properties of the 1-D Ising-like Hamiltonian taking into account short and long range interactions, in order to predict the static and dynamic behavior of spin crossover systems. The stochastic treatment is carried out within the frame of the local equilibrium method [1]. The calculations yield, at thermodynamic equilibrium, the exact analytic expression previously obtained by the transfer matrix technique [2]. We mainly discuss the shape of the relaxation curves: (i) for large (positive) values of the short range interaction parameter, a saturation of the relaxation curves is observed, reminiscent of the behavior of the width of the static hysteresis loop [3]; (ii) a sigmoidal (self-accelerated) behavior is obtained for large enough interactions of any type; (iii) the relaxation curves exhibit a sizeable tail (with respect to the mean-field curves) which is clearly associated with the transient onset of firstneighbor correlations in the system, due to the effect of short-range interactions. The case of negative short-range interaction is briefly discussed in terms of two-step properties.