Application of the Bethe-Weiss Method to the Theory of Antiferromagnetism

The P. R. Weiss method developed in the theory of ferromagnetism is applied to antiferromagnetism by introducing sublattices. Atomic lattices (spin ½ per atom) with negative Heisenberg exchange coupling $J$ between the nearest neighbors are investigated. It has been found that two-dimensional lattices cannot sustain antiferromagnetic order. The Curie temperatures of the simple cubic and b.c.c. structure are, respectively, $\frac{2.004\left|J\right|\mathrm{}}{k}$ and $\frac{3.18\left|J\right|\mathrm{}}{k}$. That the f.c.c. lattice cannot be ordered by the interactions among the nearest neighbors is deduced from the disorder of a quadratic layer and the ineffectiveness of the interactions between layers in a f.c.c. lattice in producing order. This helps to understand why the ordering pattern of spins in Mn ions in MnO should be such as observed by Shull. Curves are obtained for reciprocal susceptibility and for short-range order vs temperature above the Curie point $T_c$. The experimental formula $\chi =\frac{\mathrm{const}}{(T+\theta )}$ is compared with our theory. We obtain for the simple cubic and b.c.c. lattice $\theta =1.5\mathrm{}{T}_c \mathrm{and} 1.25T_c$ respectively if we extrapolate our theoretical curve from extremely high temperatures, and $\theta$ is slightly higher than these values if we extrapolate from the temperature range at which experimental readings are taken. This compares more favorably with the experimental data than the prediction $\theta =T_c$ of the molecular field theory. The general validity of our theory and its failure in the range of low temperatures are discussed.