Some Remarks on Correlations in Classical Order-Disorder Phenomena

Abstract

By means of a spin-magnitude variation parameter, equations are derived for spin-pair correlation functions in classical lattice theories. Specifically, the case of two spin states present in equal average numbers is considered, but with arbitrary scalar inter-action between pairs of spin sites. It is pointed out that in this case the spin-triplet correlation functions which occur in the theory, as well as all higher odd-order correlation functions, may be reduced exactly to linear combinations of lower-order correlation functions. The nature of spin-quadruplet correlations, which are also required in this method for rigorous determination of the pair functions, unfortunately have no corresponding reduction, but their character is discussed qualitatively. The approximate theory resulting from use of a simple closure relation for quadruplets is examined, with particular attention to the position of the critical temperature for ferromagnetic coupling of varying range. It is remarked, on the basis of the approximate theory, that spin-pair correlations at large distance tend to become independent of direction (relative to the fundamental axes of the lattice) as the critical temperature is approached, when the interaction satisfies a certain second-moment isotropy condition.