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Most mantle rocks are composed of several mineral phases, each of which has different elastic properties. For a polymineralic rock composed of different mineral phases with volume fractions vi, the bulk and shear moduli Mi of individual mineral phases are then combined to the Voigt bound as (Watt et al., 1976):
and to the Reuss bound as (Watt et al., 1976):
While the Voigt and Reuss averages provide bounds to the elastic behavior of a crystalline aggregate, the Voigt‐Reuss‐Hill averages are often used as an approximation to the actual elastic response (Chung & Buessem, 1967; Hill, 1952; Thomsen, 1972b):
Note that the calculation of the Voigt and Reuss bounds in their strictest sense would require the elastic stiffness and compliance tensors of all minerals in the rock at the respective pressure and temperature. For many minerals, however, bulk and shear moduli have so far only been determined on polycrystalline aggregates and are themselves averages over the elastic properties of many individual crystals. For polycrystalline aggregates, the changes of the aggregate bulk and shear moduli of a mineral that result from compression and heating can be described by combining the finite‐strain expressions for individual components cijkl to finite‐strain expressions for the isotropic bulk and shear moduli (Birch, 1947; Davies, 1974; Stixrude & Lithgow‐Bertelloni, 2005). Under the assumption of isotropic finite strain, the expressions would resemble those for the Voigt bound. In analogy to the elastic moduli, the components of the tensor ηijkl are combined to the isotropic parameters ηV and ηS with respect to volume and shear strain, respectively (Stixrude & Lithgow‐Bertelloni, 2005). When determined on a polycrystalline sample, however, aggregate moduli are not readily identified with either of the bounds or with a simple average as the elastic response of the aggregate falls somewhere in between the Voigt and Reuss bounds and may even fall outside the bounds if a CPO is present.
Once the elastic properties of individual minerals have been combined to bulk and shear moduli of an isotropic polymineralic rock, the propagation velocities v of longitudinal waves (P) and transverse or shear waves (S) can be calculated by:
The density ρ of the rock is computed by combining the molar masses and volumes of all minerals according to their abundances. Note that wave velocities discussed throughout this chapter are based on the elastic response of minerals and rocks only and do not include corrections for anelastic behavior that might affect wave velocities at seismic frequencies and at high temperatures (Jackson, 2015; Karato, 1993).
3.3 EXPERIMENTS
Finite‐strain theory describes the elastic stiffnesses or moduli as functions of volume and temperature. The parameters needed for this description are therefore best determined from data sets that relate elastic stiffnesses or moduli with the corresponding volume (or density) and temperature. Since sound wave velocities are combinations of elastic moduli and density, they contain information on the elastic properties that is best retrieved when volume and temperature are determined along with sound wave velocities. Alternatively, volume can be inferred from pressure measurements via an equation of state (EOS) that relates pressure to volume and temperature. More information on theoretical and experimental aspects of high‐pressure equations of state for solids has been summarized in recent review articles (Angel, 2000; Angel et al., 2014; Holzapfel, 2009; Stacey & Davis, 2004). Here, I give a brief overview of methods that constrain elastic properties other than the EOS. More detailed introductions into these and other methods to characterize the elastic properties of minerals can be found in Schreuer and Haussühl (2005) and in Angel et al. (2009).
The elastic stiffness tensor completely describes the elastic anisotropy of a crystal and hence also determines the velocities of sound waves travelling in different crystallographic directions. The components of the elastic stiffness tensor can therefore be derived by measuring the velocities of sound waves that propagate through a single crystal along a set of different directions selected so as to adequately sample the elastic anisotropy of the crystal. The minimum number of individual measurements depends on the number of independent components cijkl and hence on the crystal symmetry. At high pressures, sound wave velocities can be determined using light scattering techniques on single crystals contained in diamond anvil cells (DAC). Both Brillouin spectroscopy and impulsive stimulated scattering (ISS) are based on inelastic scattering of light by sound waves (Cummins & Schoen, 1972; Dil, 1982; Fayer, 1982). The applications of Brillouin spectroscopy and ISS to determine elastic properties at high pressures have been reviewed by Speziale et al. (2014) and by Abramson et al. (1999), respectively. Light scattering experiments on single crystals compressed in DACs have proven successful in deriving full elastic stiffness tensors of mantle minerals up to pressures of the lower mantle (Crowhurst et al., 2008; Kurnosov et al., 2017; Marquardt et al., 2009c; Yang et al., 2015; Zhang et al., 2021). At combined high pressures and high temperatures, elastic stiffness tensors have been determined using Brillouin spectroscopy by heating single crystals inside DACs using resistive heaters or infrared lasers (Li et al., 2016; Mao et al., 2015; Yang et al., 2016; Zhang & Bass, 2016).
Resistive heaters for DACs commonly consist of platinum wires coiled on a ceramic ring that is placed around the opposing diamond anvils with the pressure chamber between their tips (Kantor et al., 2012; Sinogeikin et al., 2006). With this configuration, the pressure chamber containing the sample is heated from the outside together with the diamond anvils and the gasket. Heating and oxidation of diamond anvils and metallic gaskets can destabilize the pressure chamber and even cause it to collapse. Therefore, measurements of elastic properties using resistive heating of samples contained in DACs have so far been limited to temperatures of about 900 K. Such experiments require purging of DAC environments with inert or reducing gases, typically mixtures of argon and hydrogen, to prevent oxidation of diamonds and gaskets. Higher temperatures can be achieved with graphite heaters and by surrounding the DAC with a vacuum chamber (Immoor et al., 2020; Liermann et al., 2009). Setups with graphite heaters have been developed for X‐ray diffraction experiments and could potentially be combined with measurements of elastic properties. When heating DACs with external heaters, temperatures are typically measured with thermocouples that are placed close to but outside of the pressure chamber. Besides those limitations, resistive heaters create a temperature field that can be assumed to be nearly homogeneous across the pressure chamber and can be held stable for long enough periods of time to perform light scattering and X‐ray diffraction experiments on samples inside DACs.
Substantially higher temperatures can be achieved by using infrared (IR) lasers to directly heat a sample inside the pressure chamber of a DAC through the absorption of IR radiation. For a given material, the strength of absorption and hence the heating efficiency depend on the wavelength of the IR laser radiation. The IR radiation emitted by common rare‐earth‐element (REE) lasers, e.g., lasers based on REE‐doped host crystals or optical fibers, is centered at wavelengths between
