Stable, metastable and unstable solutions of a spin-1 Ising system obtained by the molecular-field approximation and the path probability method

Abstract

The spin-1 Ising model Hamiltonian with arbitrary bilinear ( J) and biquadratic ( K) pair interactions has been studied for zero field using the lowest approximation of the cluster variation method. The spin-1 or three-state system will undergo a first- or second-order phase transition depending on the ratio of the coupling parameters α = J/ K. The critical temperatures and, in the case of the first-order phase transition, the limit of stability temperatures are obtained for different values of α calculating by 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. The dynamics of the same model is studied by means of the path probability method and the resulting set of equations is solved at temperatures below the critical temperatures below the critical temperature for three different cases: a system with second-order transition, a system with a first-order transition both in the quadrupole moment and the spin moment, at a temperature in between the upper and lower stability points, and a system that undergoes a first-order transition in the quadrupole moment only. The stable, metastable and unstable solutions are shown explicitly in the flow diagrams. The unstable solutions for the first-order phase transitions are found by solving the dynamic equations and the role of these solutions in the phase diagram is also described. Finally a discussion is given of how metastable phase could be obtained.