Anisotropic Superexchange Interaction and Weak Ferromagnetism

A theory of anisotropic superexchange interaction is developed by extending the Anderson theory of superexchange to include spin-orbit coupling. The antisymmetric spin coupling suggested by Dzialoshinski from purely symmetry grounds and the symmetric pseudodipolar interaction are derived. Their orders of magnitudes are estimated to be ($\frac{\ensuremath{\Delta}g}{g}$) and ${(\frac{\ensuremath{\Delta}g}{g})}^{2}$ times the isotropic superexchange energy, respectively. Higher order spin couplings are also discussed. As an example of antisymmetric spin coupling the case of Cu${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O is illustrated. In Cu${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O, a spin arrangement which is different from one accepted so far is proposed. This antisymmetric interaction is shown to be responsible for weak ferromagnetism in $\ensuremath{\alpha}$-${\mathrm{Fe}}_{2}$${\mathrm{O}}_{3}$, MnC${\mathrm{O}}_{3}$, and Cr${\mathrm{F}}_{3}$. The paramagnetic susceptibility perpendicular to the trigonal axis is expected to increase very sharply near the N\'eel temperature as the temperature is lowered, as was actually observed in Cr${\mathrm{F}}_{3}$.