Crystal Statistics. III. Short-Range Order in a Binary Ising Lattice

The degree of order in a binary lattice is described in terms of a family of "correlation" functions. The correlation function for two given lattice sites states what is the probability that the spins of the two sites are the same; this probability is, of course, a function of temperature, as well as of the distance and orientation of the atoms in the pair. It is shown that each correlation function is given by the trace of a corresponding $2^n$-dimensional matrix. To evaluate this trace, we make use of the apparatus of spinor analysis, which was employed in a previous paper to evaluate the partition function for the lattice. The trace is found in terms of certain functions of temperature, ${\Sigma }_a$, and these are then calculated with the aid of an elliptic substitution.

Correlations for the five shortest distances (without restriction as to the orientation of the pair within the plane) are plotted as functions of temperature. In addition, the correlation for sites lying within the same row is given to any distance. For the critical temperature this correlation is plotted as a function of distance. It is shown that the correlation tends to zero as the distance increases, that is to say: there is no long-range order at the critical temperature.