Curie Temperatures for Layer Structures

Abstract

The problem of calculating Curie (or Néel) temperatures for layer structures is discussed by considering a simple example—that of a simple-cubic lattice of spins in which the (Heisenberg) exchange interactions $J$ (or $-J$) within a set of parallel planes is allowed to differ from the interactions $K$ (or $-K$) between them. Ferromagnetism and antiferromagnetism are both considered, and particular attention is paid to the cases where $\frac{K}{J\ll 1}$. Most of the well-tried methods for obtaining transition temperatures are discussed, and it is shown that the molecular-field theory, the Opechowski high-temperature expansion method, the constant-coupling treatment, and the cluster methods of Oguchi and of Bethe-Peierls-Weiss are all unable to give results which are even qualitatively satisfactory for the weakly interacting layer problem, if we accept the spin-wave conditions for the existence or nonexistence of long-range order at low temperatures. The breakdown of these methods is shown to be particularly serious in the antiferromagnetic case. Finally, the method of Green functions is used and is shown to give acceptable approximations for both ferromagnetism and antiferromagnetism.