Narrow Results

The two-dimensional antiferromagnetic spin-1 Ising model with positive biquadratic interaction is simulated on a cellular automaton which based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transition of the model are presented for a comparison with those obtained from other calculations. We confirm the existence of the intermediate phase observed in previous works for some values of J/K and D/K. The values of the static critical exponents (β, γ and ν) are estimated within the framework of the finite-size scaling theory for D/K<2J/K. Although the results are compatible with the universal Ising critical behavior in the region of D/K<2J/K-4, the model does not exhibit any universal behavior in the interval 2J/K-4<D/K<2J/K.

The spin-1 Ising model with the dipole–quadrupole interaction (ℓ = L/J) has been simulated using a cellular automaton (CA) algorithm improved from the Creutz cellular automaton (CCA) for a face-centered cubic (fcc) lattice. The simulations have been made for different ℓ values at the reentrant phase transition and the special points such as the tricritical point (k = K/J = 0, d = D/J = 5.7) and the critical end point (k = -0.9, d = 0.7). The simulation results show that the model has the dense ferromagnetic (df, df(+), df(-)) and the ferromagnetic (F, F(+), F(-)) phases with the dipole–quadrupole interaction. The type and the order of the phase transitions change for the nonzero values of ℓ on the special points. Furthermore, the effect of ℓ is similar with the effect of the external magnetic field (h).

The equilibrium properties of the spin-1 Blume-Capel model are studied by the lowest approximation of the cluster-variation method which is identical to the mean-field approximation. The lattice is divided into two sublattices A and B, and different bilinear interaction parameters J_AB and J_BA between the nearest-neighbor spins are assumed. The temperature variations of the order-parameters are studied, therefore, the phase diagrams are obtained on the (J_AB, T) planes for given values of J_BA and crystal fields D. It was found that the model yields both second- and first-order phase transitions at higher positive D values and the lines of which combine at tricritical points. Negative values of D lead only second-order phase transitions. The phase diagrams are symmetric under the exchange of J_AB with J_BA as expected.

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.

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 $(\chi ={\chi }_1-i{\chi }_2)$ is derived. The temperature dependence of the magnetic dispersion relation $({\chi }_1)$ and magnetic absorption factor $({\chi }_2)$ has been studied in the ferromagnetic (FM) and paramagnetic (PM) phases as well as near the critical regime.