Wave velocities of MgSiO 3 akimotoite

Brillouin spectroscopy measurements were collected at eleven pressure points for DAC1 and DAC2 up to a maximum pressure of 24.9(1) GPa using the Brillouin setup described in section 2.7.5. An example of dispersion curves of vP, vS1 and vS2 collected for crystal X1 at the maximum pressure reached in this study is given in Figure 3-10. For the last two pressure points, an overlapping of the diamond vS (red circles in Figure 3-10) with the akimotoite vP occurred for different orientation directions; however, the number of observable, i.e. wave velocities, was still sufficient to constrain all elastic stiffness coefficients using the Christoffel equation as described in section 2.7.5.

Figure 3-10: Dispersion curves of vP (black), vS1 (blue) and vS2 (green) of crystal X1 at 24.9(1) GPa. With increasing pressure, some of the vP signals of the sample were masked by the vS signals of the diamond (red) reducing the amount of signals to be fitted with the Christoffel equation.

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The resulting cij coefficients obtained at each pressure point are reported in Table 3-4. The uncertainties of the fitted cij’s (Table 3-4) are below 1 - 2.5 % for c11, c33, c44, c12, c13 and around 1-2 GPa for the smallest cij’s, c14 and c25, that range between values of -39 to 23 GPa. Most of the coefficients increase monotonously with increasing density (Figure 3-11), except for c14 and c25

which decrease with pressure. The cij’s at ambient pressure are in good agreement with the cij’s determined by Weidner and Ito (1985) (orange triangles in Figure 3-11) who have investigated a single-crystal of MgSiO3 akimotoite at 16 different orientations. The c14 reported in this study is negative likely due to a different setting for the transformation from the trigonal to the orthogonal coordinates. The large differences between all stiffness coefficients (Figure 3-11) is stressing the fact that akimotoite is a highly anisotropic mineral.

Figure 3-11: Variation with pressure of the stiffness coefficients cij of akimotoite. The large anisotropy of MgSiO3 akimotoite can be seen in the large difference between all stiffness coefficients. The black solid lines represent BM3 EoS fits through the stiffness coefficients. The uncertainties are smaller than the symbol size.

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Table 3-4: Single-crystal stiffness coefficients cij’s, absolute pressure (Pabs), pressure calculated using the ruby calibration reported by Dewaele et al. (2004) and density measured by single-crystal X-ray diffraction. The RVH average bulk (isothermal KT and adiabatic KS) and shear moduli are reported, as well as the aggregate velocities vP and vS. KT was calculated using the relation: 𝑲_𝑺𝟎= 𝑲_𝑻𝟎∗ (𝟏 + 𝜶𝜸𝑻)

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The Reuss and Voigt bounds of the bulk and shear moduli were calculated using the cij’s and the sij’s obtained inverting the elastic tensor (see Section 2.7.3) and are shown in Figure 3-12 as a function of density. As proposed by Watt et al. (1976), the arithmetic average has been used to calculate the Reuss-Voigt-Hill average which represents the aggregate bulk and shear moduli of a composite material. The resulting Reuss-Voigt-Hill average bulk and shear moduli obtained from the Brillouin spectroscopy measurements are shown in Figure 3-12 (black symbols). The adiabatic bulk modulus of 208(1) GPa obtained from Brillouin spectroscopy is slightly smaller than the recalculated adiabatic bulk modulus obtained from EoSfit of 211(2) GPa.

Figure 3-12: Elastic moduli KT and G versus density. The black circles are the RVH average isothermal bulk and shear moduli determined experimentally. The solid lines represent BM3 EoS fits through the RVH average (black), the upper Voigt bound (red) and the lower Reuss bound (blue). The uncertainties are smaller than the symbol size.

The aggregate compressional and shear wave velocities, were calculated following equations (17) and (18), respectively, by using the elastic moduli determined with the Reuss-Voigt-Hill average and density measured using single-crystal X-ray diffraction on the same crystals (Figures 3-13 and 3-14). The difference between the Reuss and Voigt bounds for both the longitudinal and shear wave velocities are ~0.3 km/s and decrease slightly with increasing pressure indicating a decrease in anisotropy (Figures 3-13 and 3-14).

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Figure 3-13: Pressure dependence of the longitudinal aggregate velocities vP. The Voigt (red) and Reuss (blue) bounds were plotted to show the range of possible longitudinal wave velocities through MgSiO3 akimotoite. The black circles represent the experimentally determined aggregate vP calculated from the RVH-average versus the absolute pressure and the black line correspond to a BM3 EoS fit through this data set. The uncertainties are smaller than the symbol size.

Figure 3-14: Pressure dependence of the shear aggregate velocities vS. The Voigt (red) and Reuss (blue) bounds were plotted to show the range of possible shear wave velocities through MgSiO3 akimotoite. The black circles represent the experimentally determined aggregate vS

calculated from the RVH-average versus the absolute pressure and the black line correspond to a BM3 EoS fit through this data set.

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A comparison between the absolute pressures calculated using equation (28) and the pressures determined by the ruby fluorescence shift using the calibration reported by Dewaele et al. (2004) is plotted in Figure 3-15. Both pressures are in agreement within 2.5 % uncertainty (black solid lines) of the ruby pressure scale up to 25 GPa. The ruby pressure may deviate from the absolute pressure due to non-hydrostatic stresses. Note that the absolute pressure appears smaller than that measured with ruby when the broadening of the reflections during single-crystal X-ray diffraction were observed due to stresses in the DAC (e.g. between 13-17 GPa and > 22 GPa).

Figure 3-15: Comparison of the absolute pressure calculated using the bulk modulus KT, K’ and the density determined during the Brillouin experiment with the ruby pressure calculated using the ruby fluorescence pressure calibration reported by Dewaele et al. (2004). The black lines in the plot on the right side represent the uncertainty (2.5 %) of the ruby pressure scale.