Use of Green Functions in the Theory of Ferromagnetism. I. General Discussion of the Spin-SCase

Abstract

The theory of ferromagnetism developed by Tyablikov for the case of spin-½ particles is extended to higher spin values. The chain of equations for the various Green functions is made finite by a decoupling which is completely analogous to the one used by Tyablikov and our results reduce to his for the case $S=\frac{1}{2}$. This decoupling procedure seems, however, to be better for larger $S$ values. Explicit expressions are given for the magnetization for $S=\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, \mathrm{and} 3$ and those expressions are used to obtain series expansions at temperatures well below the Curie temperature, just below the Curie temperature, and well above the Curie temperature. The results obtained are compared with those of Dyson for the spin-wave region, with those of Lax obtained via the spherical model near the Curie point, and with those of the molecular-field theory at high temperatures. It is interesting to note that we obtain the same Curie temperatures as those following from the spherical model. The results obtained here seem to be reasonable. In the appendix we examine critically similar work by Izyumov and Yakovlev and by Kawasaki and Mori.