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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 x_i of the end members is readily converted to volume fractions v_i. Assuming ideal mixing behavior for volumes, the volume of the solid solution is then given by:

Based on the volume fractions v_i = x_iV_i/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 of end members are unknown, it might still be possible to approximate the elastic properties of a solid solution as long as the effects of different chemical substitutions on elastic properties are captured by available experiments or computations on intermediate compositions of the solid solution. Let x_im be the molar fractions of end member i for a set of intermediate compositions for which volumes and elastic properties are known. For each composition m of this set, the molar fractions x_im of all end members form a composition vector x_m of dimension n equal to the total number of end members. If the set of compositions x_m forms a vector basis of ℝⁿ, it is possible to construct a matrix M <span class="dbond"></span> {x_im} that contains the vectors x_m as columns. Any composition vector x of the solid solution can then be transformed into a vector y by using the inverse matrix: y = M⁻¹x. The components of the vector y express the composition x in terms of molar fractions y_m of the intermediate compositions x_m. In this way, the compositions x_m are combined to match the required composition, and their volumes and elastic properties can be combined according to the mixing laws introduced above. It is important to note that this type of mixing is strictly valid only within the compositional limits defined by the compositions x_m that may not cover the full compositional space as spanned by the end members. When volumes can be assumed to mix linearly across the entire solid solution, however, it is possible to extrapolate volumes beyond these compositional limits. The extrapolation of elastic properties requires special caution since negative molar fractions y_m < 0 can lead to unwanted effects when computing bounds on elastic moduli.

Figure 3.3 illustrates the uncertainties that arise from mixing the elastic properties of bridgmanite compositions. P‐ and S‐wave velocities were calculated for bridgmanite solid solutions in the systems MgSiO₃‐Al₂O₃‐FeAlO₃ and MgSiO₃‐Al₂O₃‐FeSiO₃ at 40 GPa and 2000 K using available high‐pressure experimental data on different bridgmanite compositions (Chantel et al., 2012; Fu et al., 2018; Kurnosov et al., 2017; Murakami et al., 2012, 2007) that provided the basis of composition vectors in the approach outlined in the previous paragraph. For all compositions, I adopted the thermoelastic properties given by Zhang et al. (2013). Uncertainties on P‐ and S‐wave velocities due to mixing were estimated as the differences that arise from combining the elastic properties of bridgmanite compositions according to either the Voigt or the Reuss bound relative to velocities of the Voigt‐Reuss‐Hill average, i.e., dlnv = (v^V − v^R)/v^VRH. For bridgmanite compositions that are similar to the compositions studied in experiments and used here to compute P‐ and S‐wave velocities, the uncertainties remain below 0.5%. When extrapolating elastic properties beyond the compositional limits defined by available experimental data, however, uncertainties rise substantially. Note that bridgmanite compositions falling outside the compositional range as delimited by experiments imply negative molar fractions in terms of the experimental compositions that form the basis of composition vectors. As a result, the Reuss bound may exceed the Voigt bound and dlnv < 0. As mentioned above, such extrapolations may exert strong leverages on sound wave velocities and need to be restricted to compositions that remain close to the compositional limits defined by available data.

Figure 3.3 Variations in P-wave (a) and S-wave (b) velocities of bridgmanite solid solutions at 40 GPa and 2000 K that reflect the differences between the Voigt and Reuss bounds when combining the elastic properties of bridgmanite compositions (triangles). Each large ternary diagram spans the section marked black in the small full ternary diagram next to each large ternary diagram. See Table 3.1 for references to finite‐strain parameters for bridgmanite compositions (triangles) and Figure 3.6 for references to mineral compositions observed in experiments on different bulk rock compositions (circles and diamonds).

In addition to mapping uncertainties on modeled sound wave velocities, Figure 3.3 summarizes bridgmanite compositions observed in experiments on bulk rock compositions of interest for Earth’s lower mantle. Many of these compositions, in particular for metabasaltic rocks, fall outside the compositional limits defined by bridgmanite compositions for which elastic properties have been determined in experiments. This highlights the need for further studies on the elastic properties of bridgmanite solid solutions to expand compositional limits, to establish reliable trends for individual substitution mechanisms, and to resolve inconsistencies between experiments and computations. Although I focused on bridgmanite solid solutions as they are of highest relevance for the lower mantle, there is a similar need to systematically analyze and describe the effect of chemical composition on the elastic properties of other chemically complex mantle minerals such as garnets, pyroxenes, and the calcium ferrite‐type aluminous phase. Internally consistent thermodynamic databases on mantle minerals provide highly valuable resources for end member properties (Holland et al., 2013; Komabayashi & Omori, 2006; Stixrude & Lithgow‐Bertelloni, 2011, 2005). To date, however, these databases do not include all relevant end members and/or do not include shear properties. Internally consistent thermodynamic properties of hydrous high‐pressure phases, for example, have been derived for magnesium end members only (Komabayashi and Omori, 2006). Recent experimental results on the elastic properties of wadsleyite and ringwoodite allowed for modeling the combined effects of iron and hydrogen on sound wave velocities in the mantle transition zone (Buchen et al., 2018b; Schulze et al., 2018). When sufficient data on intermediate compositions of a complex solid solution are available, it might be possible to determine end member properties by making assumptions about how end member elastic properties combine to those of intermediate compositions. Although restricted to the equation of state, Buchen et al. (2017) devised models for wadsleyite solid solutions in the system Mg₂SiO₄‐Fe₂SiO₄‐MgSiO₂(OH)₂‐Fe₃O₄ by assuming solid solutions between end members to follow either the Reuss or the Voigt bound when mixing the properties of end members. In principle, such an approach can be extended to shear properties and other minerals once sufficient data become available.