Magnetic Properties of Magnetite

Below the Verwey transition the ${\mathrm{Fe}}^{2+}$ and ${\mathrm{Fe}}^{3+}$ ions on B sites in magnetite are magnetically coupled to $A$-site ions by different exchange constants. We calculate the six-sublattice spin-wave spectrum of this ordered orthorhombic structure, for arbitrary spin quantum numbers and for arbitrary ferrimagnetic nearest-neighbor exchange interactions coupling $A$ ions with two different kinds of $B$-site ions. Inelastic neutron scattering by Verwey-ordered magnetite, which has not been studied, will allow evaluation of the two exchange constants and of the dynamics of their averaging through the transition. In spite of differences of the exchange constants and of the numbers of nearest neighbors (an $A$-site ion has twice as many nearest neighbors as does a $B$-site ion), we show that all sublattices and the net moment have the same temperature dependence, in the spin-wave region, apart from their proportionality to their 0°K value. This holds also if intrasublattice exchange, which we ignore, is included, and for other symmetries. Besides sublattice magnetizations we also calculate the ferrimagnetic-resonance $g$ factor and specific heat, both above and below the Verwey transition. The observed Mössbauer spectrum is explained, but many other experiments remain unexplained.