Application of the Mayer Method to the Melting Problem

Abstract

The antiferromagnetic Ising model can also describe a gas with purely repulsive intermolecular forces, and a comparison of the two is made, the transition of such a `gas' to a solid-like structure being the analogue of the appearance of an antiferromagnetic structure. Three simple lemmas enable the asymptotic behaviour of the Mayer cluster sums to be related to other combinatorial problems on the lattice, such as that studied by Hammersley and Broadbent (1957). Some information on these problems is already available, but a detailed discussion is reserved to another paper. The Mayer formalism can be generalized to deal with the antiferromagnetic problem, with the conclusion that the transition is probably first-order and associated with a singularity of the Mayer b-series on the positive real axis of the z variable, but that there is a closer one on the negative real axis. The Mayer β-series can probably be extended beyond the transition and corresponds to a metastable liquid. Its divergence probably represents the absolute limit of the metastable state. The symmetry of the particular model used implies a second transition at a still higher density which may be likened to a change of crystal structure. (The gas of completely rigid molecules is a limiting case, and has only one transition, the transition curve then being a linear relation between P and T.) These considerations agree well with other work on the antiferromagnetic Ising model. A `solid-gas' critical temperature seems most unlikely for any realistic interaction, through it could occur for a sufficiently `soft' one.