Two Problems in the Statistical Mechanics of Crystals. I. The Propagation of Order in Crystal Lattices. I. The Statistics of Two-Dimensional Lattices with Four Components.

We have considered a two-dimensional square net consisting of four kinds of atoms supposing that only nearest neighbors interact and that there are only two distinct potential energies of interaction, one between like and one between unlike atoms. In extension of a method due to Onsager it is found that for the case where like atoms attract one another a simple "reciprocity" relation exists between the partition functions at pairs of temperatures "reciprocally" related to one another. As one temperature $T$ tends to zero, the other ${T}^{*}$ tends to infinity. If one further assumes that only one "Curie" transition point exists, the relation between $T$ and ${T}^{*}$ enables one to locate the Curie temperature. Predictions can be made concerning the nature of the transition point with results similar to those of Kramers and Wannier. The reciprocity relation for the case of attraction between like atoms is found to be not valid for the case where unlike atoms attract one another.