On heating, the thermodynamics of many minerals is affected by cations becoming progressively disordered over their original sites. This may have a significant stabilising effect with respect to other phases and thus must be understood in order to correctly predict the dominant phases in the Earth's mantle. Experimental studies are often limited in temperature range or are not sufficiently sensitive.
Some studies of order-disorder processes have already been performed, aimed at determination of ordering energies (36). The only serious restriction on such studies are the sizes of the supercells required for long-range interactions to be investigated, but with carefully chosen configurations progress is possible. We have recently performed calculations on MgAl2O4 spinel, in which Mg in tetrahedral sites exchange with Al in octahedra, to deduce effective cluster interactions with which to perform Monte Carlo simluations. This approach aims to overcome the size limitation on ab initio calculations, which prohibit the very large calculations necessary for accurate simulations of statistical processes such as disorder.
We used a supercell of eight formula units for CETEP calculations, and made ten sets of different Mg Al exchanges. The energies obtained were fitted with a variety of schemes, with terms including a baseline energy; a `chemical potential' term, representing the energy difference between the two types of site; two different nearest-neighbour pair exchanges; and two different three-body cluster terms, for three sites sharing a single oxygen. The last two terms arise mainly from elastic interactions, but the second term is instead mostly chemical and is hence very hard to obtain with empirical potentials. First principles calculations are thus essential to obtain good values for these interactions, despite the limitations of such a simple parametrisation scheme.
We found that satisfactory representation of the ab initio data could be made with only four of these terms -- one of each sort -- and that over the limited range of neutron scattering experimental data, good agreement was found with the Monte Carlo simulations using these four terms. This demonstrates that finding simple effective interactions with quantum mechanical methods can open the way for much larger simulations which are currently out of the reach of electronic structure simulations.