Order-disorder statistics. III. The antiferromagnetic and order-disorder transitions

Abstract The method of the previous paper is applied to a two-dimensional model of an antiferromagnetic. An alternative notation is developed, and this shows that in the absence of a magnetic field the antiferromagnetic is effectively identical with the ferromagnetic, a result first demonstrated by Kramers & Wannier (1941). In the presence of a magnetic field a number of terms of a series expansion are obtained, and these are used in conjunction with the corresponding high-temperature ferromagnetic expansions to derive a number of qualitative features of an antiferromagnetic. High- and low-temperature series for the magnetic susceptibility in zero field are deduced, and the results are compared with standard approximations. The theory of order-disorder transitions with constituent ratios differing from unity is discussed, and it is shown that for concentrations of one constituent less than 0.226 no long-range order can exist, and there is no singularity. The application of the results to adsorption theory is discussed. The method of Ashkin & Lamb (1943) is generalized to derive a series for long-range order when the constituent ratio differs from unity.