Calculations of the elastic constants of crystals as functions of pressure with applications to quartz and cristobalite

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Virginia Tech


Assuming that its deformation is both static and homogeneous, a method was devised within the framework of finite strain theory to calculate the elastic constants of a crystal at zero pressure as well as functions of pressure. As this method can be used for a crystal of any symmetry with a variety of potentials, elastic constant calculations were completed for both quartz and cristobalite using the potentials derived by Lasaga and Gibbs (1987, 1988), Tsuneyuki et al. (1988), van Beest et al. (1990) and Boisen and Gibbs (1993). Among the four theoretical potentials derived from force fields calculated for molecule H₆Si₂Q₇, two potentials associated with the SiO bond stretching as well as the OSiO and the SiOSi bond angle bending terms yield reasonable agreements with the experimental data, supporting the assumption that the binding forces in crystals are similar to those in small molecules (Gibbs 1982). Using the semi-empirical potentials, including the SQLOO potential derived with the covalent model (Boisen and Gibbs 1993) and two derived with the ionic model (Tsuneyuki et al. 1988; van Beest et al. 1990), calculations of the elastic constants reproduce the experimental results for both quartz and cristobalite. The pressure derivatives of the elastic constants calculated with these potentials also agree with the experimental results measured for quartz at low pressures and yield pressure derivatives of the bulk modulus that are in close agreement with that observed for quartz. Using the SQLOO potential, although the results of the calculations do not agree with the experimental data as well as those calculated using the two ionic potentials, the agreement of the calculations made with the theoretical potentials was improved significantly. Finally, using the three semi-empirical potentials, the elastic constants were calculated as functions of pressure for quartz and cristobalite up to their transitional pressures. Calculated for both quartz and cristobalite, different behaviors of the elastic constants were found using different potentials. For cristobalite, the variations of its elastic constants and bulk modulus are better modeled by the SQLOO potential as its structural behaviors calculated with the SQLOO potential are consistent with the X-ray diffraction studies.