Omphacite, a clinopyroxene mineral with two distinct crystallographic sites, M1 and M2, and composition intermediate between diopside and jadeite, is abundant throughout the Earth's upper mantle, and is the dominant mineral in subducted oceanic crust. Unlike the end-members, omphacite exists in two distinct phases, a P2/n ordered phase at low temperature and a high-temperature C2/c disordered phase. The crystal structure and full elastic constants tensor of ordered P2/n omphacite have been calculated to 15 GPa using plane-wave density functional theory. Our results show that several of the elastic constants, notably C11, C12, and C13 deviate from linear- mixing between diopside and jadeite. The anisotropy of omphacite decreases with increasing pressure and, at 10 GPa, is lower than that of either diopside or jadeite. The effect of cation disorder is investigated through force-field calculations of the elastic constants of Special Quasirandom Structures supercells with simulated disorder over the M2 sites only, and over both cation sites. These show that cation order influences the elasticity, with some components displaying particular sensitivity to order on a specific cation site. C11, C12, and C66 are sensitive to disorder on M1, while C22 is softened substantially by disorder on M2, but insensitive to disorder on M1. This shows that the elasticity of omphacite is sensitive to the degree of disorder, and hence the temperature. We expect these results to be relevant to other minerals with order-disorder phase transitions, implying that care must be taken when considering the effects of composition on seismic anisotropy
