Title:

A computational study of the domain states for magnetite

A threedimensional micromagnetic model is described which predicts the equilibrium magnetisation states of magnetite, one of the most common natural magnetic minerals. Solutions are presented for cubic, octahedral and irregularly shaped grains and their stability to external magnetic fields studied. Two aspects of the model make it more powerful than previous models. Firstly, a Fourier transform algorithm has been developed which reduces the number of calculations from O(N^{2}) to O(N log N) where N is the number of elementary magnetisations vectors. Secondly, the model is implemented on a parallel computer which reduces the computation time by a factor of approximately 1/(4N_{p}), where N_{p} = 16000 is the number of processors. Cubic and octahedral grains smaller than d = 0.07μm, where d is the width of grain, occupy uniform magnetisation states. However Néel's relaxation theory predicts that these singledomain (SD) states are only stable to thermal fluctuations in grains larger than d ≃ 0.045μm. Therefore SD states are only stable over a narrow size range suggesting that a significant percentage of remanence in magnetic minerals resides in grains larger than 0.07μm. Between 0.07μm and 0.1μm grains can occupy either flower states or vortex states. The magnetisation in flower states is mostly uniform but with some deflection at the corners of the grain. Vortex states are characterised by a circular magnetisation pattern which rotates around a line running through the centre of the grain. For cubic grains, vortex states are lower energy states than SD states for all grain sizes. However, for octahedral grains, vortex states are energetically favourable only in grains larger than 0.11μm. Hysteresis simulations show that vortex states can predict remanences which are ¼ of those predicted by SD theory and coercivities which are 2/3. These results are important in showing how a gradual transition in the bulk magnetic properties of magnitite, from SD states to multidomain states can occur. Flower states are unstable in grains larger than 0.1μm and grains between 0.2μm and 1μm can occupy one of several different vortex states, each one of which is a separate local energy minimum state. The model predicts that the high values of saturation remanence and coercivity measured experimentally in grains in this size range are due to grains occupying high energy vortex states with an associated high magnetic moment. There is a gradual transition from vortex states in grains below 1μm to multidomain states in grains larger than 1μm. During this transition vortices become localised features which can nucleate domain walls.
