Some recent progress in the theory of magnetism for non-migratory models

The present paper is concerned entirely with models in which the magnetic electrons do not migrate from atom to atom. Part I reviews the progress made since 1950 by various workers in calculating the Curie point for the Heisenberg model and the behavior of the susceptibility above it. The calculations have been greatly improved in recent years by including more terms in the series method, and in extending both this method and that of Bethe-Peierls-Weiss to higher values of the spins. Furthermore, the B-P-W theory has been applied to ferrimagnetism, and especially to antiferrimagnetism where it gives both the Néel point and the corresponding maximum value of the susceptibility. The so-called constant coupling approximation of Kasteleijn and Van Kranendonk is a relatively simple method which gives surprisingly good results for three-dimensional lattices. Part II reviews recent developments in the calculation of the magnetization at very low temperatures by the method of spin waves. Part III is concerned with ferromagnetic anisotropy. Here it is necessary to generalize the Heisenberg model by including spin-orbit interaction, since otherwise no anisotropy results. A general proof is given that with quadrupole type coupling, the cubic anisotropy constant K1 should vary as the tenth power of the magnetization a low temperatures. In ferrites a " one-atom " model of the anisotropy can be used, but in ferromagnetic materials where the spins of the individual atoms are 3/2 or less it, is necessary to include coupling between atoms to obtain appreciable anisotropy. The temperature variation of the magnetic anisotropy of nickel is still a mystery. On the other hand, crystalline field theory furnishes an explanation of the unusually large anisotropy of dilute cobalt ferrites.