New approximate solutions to the Ising problem in three dimensions

A new approximate theory is proposed to treat the three-dimensional Ising model. The theory is based on a generalization to three dimensions of the classical Ono-Kondo equations for the density profile near a surface. Results are obtained in closed form. The accuracy of the proposed model is demonstrated by comparison with Monte Carlo simulation data for a hexagonal lattice gas. In addition, this approach represents a significant advance (compared to the Bragg-Williams and Bethe-Guggenheim approximations) in that it takes into account the lattice structure, not just the coordination number. Therefore, it can predict differences in energies between different lattices that have the same coordination number. This new model can be considered as an alternative approach to the cluster variation method (CVM). We compare the model predicted here with CVM, and it is shown that the two models are in close agreement. In addition, it is shown that CVM can be used to include larger cluster sizes in the framework of the Ono-Kondo approach.