Application of the Bethe-Weiss Method to Ferrimagnetism

The Bethe-Weiss theory of ferromagnetism is extended and applied to systems containing two nonequivalent sets of sites, designated by A and B, for the magnetic atoms. Each B atom has nₐ nearest A neighbors and each A atom has nb nearest B neighbors. In the theoretical development, the following restrictions are imposed: the spin per atom is {} and only nearest neighbor A-B interactions are considered (α=β=0, in N\'eel's notation). The A-B interaction J may be either positive or negative, however, so that the sublattice magnetizations below the Curie temperature may be either parallel or antiparallel, respectively. Expressions are derived for the Curie temperature and for the susceptibility above the Curie temperature. If the two sublattices are made equivalent, our results for positive J reduce to Weiss' equations for the ferromagnetic case and our results for negative J reduce to Li's equations for the antiferromagnetic case.A table of values of the ratio kTc|J| as computed by the N\'eel, Ising-Bethe, and Bethe-Weiss theories is given for various values of nₐ and nb. For a given magnitude of J, the Bethe-Weiss Curie temperature for negative J is always greater than that for positive J. An empirical formula which approximates the Bethe-Weiss criterion for the existence of a ferrimagnetic transition is nₐnb>5(nₐ+nb)2.The N\'eel and Bethe-Weiss susceptibilities are calculated and compared for a ferrimagnet with nₐ=4, nb=8. The two general expressions for the susceptibility are found to agree quantitatively at extremely high temperatures. It is shown that for a ferrimagnet with $J<0$ the sublattice magnetizations in an external field become antiparallel not at the Curie temperature but considerably above.