Statistical Mechanics of Nearest Neighbor Systems

In many solids the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The thermodynamic properties of a solid with respect to a given coordinate, α, (for example, in a ferromagnetic system α might represent the excess unpaired electron spin at a given lattice point; or in a substitutional binary alloy α might denote which of the two possible kinds of atoms are at a given lattice point) can be found from the factor of the partition function which averages over all possible configurations of α at all lattice points. Here the evaluation of such a factor of the partition function is reduced to the calculation of the largest characteristic value of a linear homogeneous operator equation involving the potential energy between two adjacent layers of lattice points in the solid. By assuming that all possible configurations of a layer are equally probable, a lower bound for the partition function for a simple cubic lattice is found in terms of the partition function of a cube of eight molecules. The operator equation for a real crystal becomes quite complex, but in some problems it is possible to obtain characteristic vectors and characteristic values (and thus the partition function) at high and low temperatures. For intermediate temperatures a theory of resonance between the low temperature ``almost ordered'' characteristic vectors and the high temperature ``almost disordered'' characteristic vectors is developed. In cases where the operator equation can be solved for microcrystals a few layers thick, a perturbation method is discussed in which interactions between the individual microcrystals are treated as perturbations. Some remarks are made concerning phase transitions and finally the general theory is applied to a superficial treatment of two-dimensional ferromagnetic plates.