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.