Millimeter‐sized polycrystalline samples can be compressed and heated in multi‐anvil presses while the travel times of ultrasonic waves through the sample are being measured by an interference technique (Li et al., 2004; Li and Liebermann, 2014). When the experiment is conducted at a synchrotron X‐ray source, the sample length can be monitored by X‐ray radiography, which requires the intense X‐rays generated by the synchrotron. Otherwise the sample length can be inferred from an equation of state or solved for iteratively. Sound wave velocities can then be calculated from combinations of travel times and sample length. The stability of modern multi‐anvil presses facilitates sound wave velocity measurements on samples held at pressures and temperatures that exceed those of the transition zone in Earth’s mantle (Gréaux et al., 2019, 2016). By combining sample synthesis and ultrasonic interferometry in the same experiment, Gréaux et al. (2019) and Thomson et al. (2019) were able to determine the sound wave velocities of the unquenchable cubic polymorph of calcium silicate perovskite, CaSiO3. After synthesis at high pressure and high temperature, cubic calcium silicate perovskite cannot be recovered at ambient conditions as its crystal structure instantaneously distorts from cubic to tetragonal symmetry below a threshold temperature (Shim et al., 2002; Stixrude et al., 2007). A similar situation is encountered for stishovite, SiO2, which reversibly distorts from tetragonal to orthorhombic symmetry upon compression (Andrault et al., 1998; Karki et al., 1997b). Such displacive phase transitions can substantially change the elastic properties of materials and illustrate the need to determine elastic properties at relevant pressures and temperatures.
The wavelengths of sound waves used to derive the elastic moduli may also affect how the elastic properties of individual grains in a polycrystalline material are averaged by the measurement. At ultrasonic frequencies, sound waves travel with wavelengths between 10 μm and 10 mm that are long enough to probe the collective elastic response of fine‐grained polycrystalline samples. Sound waves probed by light scattering techniques, however, typically have wavelengths on the order of 100 nm to 10 μm (Cummins & Schoen, 1972; Fayer, 1982), which is similar to typical grain sizes in polycrystalline samples. When the wavelength is similar to or smaller than the grain size, the measured sound wave velocity may be dominated by the elastic response of individual crystals or of the assembly of only a few crystals. When light is scattered by these single‐ or oligo‐crystal sound waves, the measurement on a polycrystal takes an average over sound wave velocities within single crystals rather than averaging over the elastic properties of a sufficiently large collection of randomly oriented crystals that determine the aggregate sound wave velocities at longer wavelengths. The intensity of the scattered light also depends on the orientations of the individual crystals via the photoelastic coupling that can enhance light scattering for some orientations and emphasize their sound wave velocities over others (Marquardt et al., 2009a; Speziale et al., 2014). Nevertheless, light scattering experiments on polycrystalline samples have contributed substantially in characterizing the elastic properties of mantle minerals at high pressures (Fu et al., 2018; Murakami et al., 2009b) and at high pressures and high temperatures (Murakami et al., 2012).
Synchrotron X‐rays can be used to probe the lattice vibrations of crystalline materials. Inelastic X‐ray scattering (IXS) combines the scattering geometry of the X‐ray–lattice momentum transfer with measurements of minute energy shifts in scattered X‐rays that result from interactions with collective thermal motions of atoms in a crystalline material (Burkel, 2000). At low vibrational frequencies, these collective motions are called acoustic phonons and resemble sound waves. Their velocities can be derived from an IXS experiment by setting the scattering geometry to sample acoustic phonons that propagate along a defined direction and with a defined wavelength and calculating their frequencies from the measured energy shifts of inelastically scattered X‐rays. The energy distribution of lattice vibrations, the phonon density of states, can be studied by exciting the atomic nuclei of suitable isotopes and counting the reemitted X‐rays (Sturhahn, 2004). Because the atomic nuclei are coupled to lattice vibrations, a small fraction of them absorbs X‐rays at energies that are modulated away from the nuclear resonant energy reflecting the energy distribution of phonons that involve motions of the resonant isotope. 57Fe is by far the most important isotope in geophysical applications of nuclear resonant inelastic X‐ray scattering (NRIXS). Both IXS and NRIXS can be performed on samples compressed in diamond anvil cells to constrain the elastic properties of single crystals or polycrystalline materials (Fiquet et al., 2004; Sturhahn & Jackson, 2007). Given the low efficiency of inelastic X‐ray scattering in general and the selective sensitivity of NRIXS to Mössbauer‐active isotopes such as 57Fe, many IXS and NRIXS studies focused on iron‐bearing materials, including potential alloys of Earth’s core (Antonangeli et al., 2012; Badro et al., 2007; Fiquet et al., 2001) and minerals relevant to Earth’s lower mantle (Antonangeli et al., 2011; Finkelstein et al., 2018; Lin et al., 2006; Wicks et al., 2017, 2010).
As mentioned at the beginning of this section, experimentally determined sound wave velocities are ideally combined with measurements of density or volume strain at the same pressure and temperature. Densities are routinely determined by X‐ray diffraction, and all methods outlined above can in principle be combined with X‐ray diffraction experiments. X‐ray diffraction at high pressures has been treated in numerous review articles (Angel et al., 2000; Boffa Ballaran et al., 2013; Miletich et al., 2005; Norby and Schwarz, 2008) and is used to determine the volume of the crystallographic unit cell of a crystalline material. Particularly instructive examples of combining measurements of elastic properties with measurements of unit cell volumes were given by Zha et al. (1998, 2000), who determined bulk moduli as a function of volume by combining single‐crystal Brillouin spectroscopy and X‐ray diffraction. Combinations of bulk moduli K and unit cell volumes V at different compression states define the function K(V) that, upon integration, gives a direct measure for pressure:
Sample sizes in diamond anvil cells are very small, and samples are surrounded by heater and thermal insulation materials in multi‐anvil presses so that, in most cases, intense and focused X‐rays from synchrotron sources are required to generate diffraction patterns of suitable quality. Converting the unit cell volume to density requires information on the atomic content of the unit cell and hence on the chemical composition of the material. Uncertainties on molar masses of chemically complex materials typically subvert the high precision on unit cell volumes achievable with modern X‐ray diffraction techniques. Note also that densities based on X‐ray diffraction do not capture amorphous materials or porosity that might be present along grain boundaries or cracks in polycrystalline materials. A rather new technique to determine the bulk modulus uses synchrotron X‐ray diffraction to capture the elastic response of a polycrystalline sample that is subjected to cyclic loading at seismic frequencies. For this type of experiment, a DAC is attached to a piezoelectric actuator that generates small pressure oscillations. The resulting oscillations in unit cell volume can be measured by recording the time-resolved diffraction of intense X-rays with sufficiently fast and sensitive detectors and be analyzed to constrain the bulk modulus. Marquardt et al. (2018) successfully used this technique to probe the softening of the bulk modulus across the spin transition in ferropericlase at seismic frequencies. When combined with resistive heating, piezo‐driven DACs may facilitate cyclic loading experiments at combined high pressures and high temperatures (Méndez et al., 2020).
3.4 COMPUTATIONS
Quantum‐mechanical computations and molecular dynamic simulations have evolved to powerful tools in predicting the elastic properties of minerals at high pressures and high temperatures and complement experiments for conditions that are not readily accessible with current experimental methods (Karki et al., 2001a; Stixrude et al., 1998). First‐principle calculations are based on the Schrödinger equation:
that yields the total energy E of a system of electrons and atomic nuclei as the eigenvalue of the Hamiltonian operator