all show significant impact on computed wave velocities. Both P‐ and S‐wave velocities of the minerals included in Figures 3.1 and 3.2 appear to be sensitive to variations in the Grüneisen parameter γ0, in particular at low pressures and high temperatures. In general, the isotropic volume strain derivative q0 = ηV0/γ0 of the Grüneisen parameter has more impact on P‐wave velocities while the isotropic shear strain derivative ηS0 mostly affects S‐wave velocities. The S‐wave velocities of transition zone minerals seem to be particularly sensitive to ηS0. This sensitivity arises from comparatively large uncertainties on ηS0 for these minerals and from relatively high temperatures in the transition zone at comparatively small finite strains. The exothermic phase transitions from olivine to wadsleyite and from wadsleyite to ringwoodite are expected to raise the temperatures in the transition zone in addition to adiabatic compression (Katsura et al., 2010). For olivine and bridgmanite, adiabatic compression over extended pressure ranges somewhat mitigates the influence of thermoelastic parameters as compressional contributions to elastic moduli increase at the expense of thermal contributions.
A complete analysis of uncertainties on computed P‐ and S‐wave velocities would include correlations between finite‐strain parameters that are, however, not regularly reported. The derivation of an internally consistent matrix of covariances between finite‐strain parameters requires coherent data sets of elastic properties at high pressures and high temperatures, i.e., data sets than can be simultaneously inverted for all relevant finite‐strain parameters. Due to experimental challenges in performing measurements of sound wave velocities at combined high pressures and high temperatures, however, finite‐strain parameters for most mantle minerals have been derived from separate but complementary data sets. Elastic moduli and their pressure derivatives are often obtained from sound wave velocity measurements at high pressures but ambient temperatures. Thermoelastic parameters are then independently derived from a thermal EOS, from the results of separate measurements at high temperatures, or from a computational study. This approach does not always generate data sets that can be combined and jointly inverted for complete and consistent sets of finite‐strain parameters and their covariances. As a consequence, the results of different studies are combined in terms of finite‐strain parameters that have been derived from separate data sets. As an example, the inversion of the combined data set of elastic stiffness tensors and unit cell volumes of San Carlos olivine that have been determined in separate studies at high pressures (Zha et al., 1998) and at combined high pressures and high temperatures (Mao et al., 2015; Zhang and Bass, 2016) required fixing the components of the Grüneisen tensor γii0 and the Debye temperature θ0 in order to stabilize the inversion results. Correlations between finite‐strain parameters can be reduced by optimizing the sampling of the relevant volume–temperature space. Future studies that provide consistent elasticity data sets at combined high pressures and high temperatures will allow for analyzing parameter correlations and help to reduce uncertainties in computed P‐ and S‐wave velocities by integrating covariances into the propagation of uncertainties.
Figure 3.1 Variations in P‐wave velocities for isotropic polycrystalline aggregates of olivine (1st column), wadsleyite (2nd column), ringwoodite (3rd column), and bridgmanite (4th column) that result from propagating uncertainties on individual finite‐strain parameters. Respective parameters and uncertainties are given in each panel. Bold black curves show adiabatic compression paths separated by temperature intervals of 500 K at the lowest pressure for each mineral. See text for references.
Figure 3.2 Variations in S‐wave velocities for isotropic polycrystalline aggregates of olivine (1st column), wadsleyite (2nd column), ringwoodite (3rd column), and bridgmanite (4th column) that result from propagating uncertainties on individual finite‐strain parameters. Respective parameters and uncertainties are given in each panel. Bold black curves show adiabatic compression paths separated by temperature intervals of 500 K at the lowest pressure for each mineral. See text for references.
3.6 ELASTIC PROPERTIES OF SOLID SOLUTIONS
Most minerals form solid solutions spanned by two or more end members. When the molar or unit cell volumes Vi of the end members are known, a complex mineral composition given in terms of molar fractions xi of the end members is readily converted to volume fractions vi. Assuming ideal mixing behavior for volumes, the volume of the solid solution is then given by:
Based on the volume fractions vi = xiVi/V, the elastic properties of end members can then be combined according to one of the averaging schemes introduced in Section 3.2 to approximate the elastic behavior of the solid solution. Mineral‐physical databases compile elastic and thermodynamic properties for many end members of mantle minerals (Holland et al., 2013; Stixrude & Lithgow‐Bertelloni, 2011). However, the physical properties of some critical end members remain unknown, either because they have not been determined at relevant pressures and temperatures or because the end members are not stable as pure compounds.
Bridgmanite is believed to be the most abundant mineral in the lower mantle and adopts a perovskite crystal structure. Most bridgmanite compositions can be expressed as solid solutions of the end members MgSiO3, FeSiO3, Al2O3, and FeAlO3. Of these end members, only MgSiO3 is known to have a stable perovskite‐structured polymorph at pressures and temperatures of the lower mantle. FeSiO3 perovskite is unstable with respect to the post‐perovskite form of FeSiO3 or a mixture of the oxides FeO and SiO2 (Caracas and Cohen, 2005; Fujino et al., 2009). Al2O3 transforms from corundum to a Rh2O3 (II) structured polymorph instead of adopting a perovskite structure (Funamori and Jeanloz, 1997; Kato et al., 2013; Lin et al., 2004). For FeAlO3, both perovskite and Rh2O3 (II) structures have been proposed (Caracas, 2010; Nagai et al., 2005). Although first‐principle calculations can access the elastic properties of compounds in thermodynamically unstable structural configurations, for example for FeSiO3 and Al2O3 in perovskite structures (Caracas & Cohen, 2005; Muir & Brodholt, 2015a; Stackhouse et al., 2005a, 2006), it is unclear to which extent the results are useful for modeling the elastic properties of complex solid solutions that do not extend towards those end member compositions and might be affected by deviations from ideal mixing behavior at intermediate compositions.
When the physical properties