Theory of Verwey and Charge-Density-Wave-State Ordering in Magnetite

The Cullen-Callen Hartree-approximation band model of magnetite is solved self-consistently, and it is found that if one starts with the three-order-parameter state of Cullen and Callen, the solution iterates self-consistently to the Verwey ordered state. A charge-density-wave-state ordering is proposed to explain recent neutron- and electron-diffraction, magnetic-resonance, and M\"ossbauer-effect experiments on magnetite, which suggest a larger unit cell than occurs in the Verwey ordering. It is argued that the Verwey ordering could very easily be unstable to the formation of such a state. A discussion is also given of small polarons in a degenerate electron system and applied to magnetite.