Cooperative Phenomena in the Triangular Lattice

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Abstract
A three-point triangle cluster and a W-shaped five-point cluster are used as the bases of the cluster-variation method in the triangular lattice, and the results are used to predict the properties to be obtained by the rigorous calculation. The following systems are treated using the same mathematical formulations: (a) the lattice-gas—liquid system, (b) the phase-separating binary alloy, and (c) the Ising model. In the pressure-versus-density isotherm of (a), the van der Waals-type loop shrinks as the approximation is improved and shows a tendency to approach the flat coexisting line. The central hump of the excess free energy in (b) lowers as the approximation is improved, and tends toward the flat line connecting the minima of the free-energy curve. The free energy of the Ising model (c) as a function of the long-range order parameter behaves in the same manner. When the binary alloy system (b) is quenched, the central hump remains in the free-energy curve (i.e., it does not become flat) in the rigorous treatment. It is pointed out that this nonequilibrium free energy is the quantity to be used in the treatment of the spinodal decomposition.