Order-disorder statistics. II. A two-dimensional model

The present paper is concerned with the detailed calculation of the partition functions for a two-dimensional quadratic lattice from the matrix derived previously. A brief discussion is given of the general methods of calculation available. The method here employed is the expansion of the eigenvector and eigenvalue as power series about zero temperature. This is readily applicable to finite matrices of quite high order, and, when a suitable notation has been introduced, to the infinite case. It is found more convenient to deal with the general problem of an unsymmetric net with different interactions in the two directions; for a ferromagnetic in the absence of a magnetic field a symmetry relation observed empirically enables terms to be derived successively, and it is assumed that this solution is equivalent to Onsager’s (1944). The method is applied in the presence of a magnetic field, and several terms of a generalized series are deduced. Several terms are hence obtained of a series for the spontaneous magnetization. An inspection of the generalized series leads one to conjecture that the specific heat curve becomes continuous in the presence of a magnetic field. A rearrangement of the terms of the generalized series enables several terms of the high-temperature expansion to be deduced. Finally, the results are applied to the theory of binary solid solutions; the solubility curve for the two substances is formally closely related to the spontaneous magnetization. The separation into two phases is established, and the corresponding specific heat singularities are analyzed.