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 1 μm and 3 μm and efficiently absorbed by metals and by most opaque or iron‐rich minerals. Light scattering experiments, however, are commonly performed on optically transparent materials, such as oxides and silicates with lower iron contents, that may not absorb IR radiation at wavelengths around 1–3 μm efficiently enough for uniform and steady heating. CO2 gas lasers emit IR radiation with wavelengths of about 10 μm that is absorbed even by many optically transparent materials. As a consequence, CO2 lasers have been used to heat transparent mineral samples while probing their elastic properties with light scattering (Kurnosov et al., 2019; Murakami et al., 2009a; Sinogeikin et al., 2004; Zhang et al., 2015).
Requirements of uniform heating as well as stabilization and accurate assessment of temperatures impose particular challenges on laser‐heating experiments. Typical sample sizes for light scattering experiments on the order of several tens to hundred micrometers may exceed the sizes of hot spots generated by IR lasers. As a result, samples may not be heated uniformly, and the resulting thermal gradients can bias the measurements of both temperature and elastic properties. During laser‐heating experiments, temperatures are determined by analyzing the thermal emission spectrum of the hot sample. Modern optical instrumentation allows combining spectral with spatial information of the hot spot to generate temperature maps that reveal thermal gradients and facilitate more accurate temperature measurements (Campbell, 2008; Kavner & Nugent, 2008; Rainey and Kavner, 2014). Analyses of laser‐heated hot spots indicate that temperatures may vary by several hundreds of kelvins over a few tens of micrometers across the hot spot. The interaction of the sample with the IR laser often changes in the course of a laser‐heating experiment and may lead to temporal temperature fluctuations in addition to thermal gradients. As a consequence, uncertainties of temperature measurements on laser‐heated samples tend to be on the order of several hundred kelvins. The combination of sound wave velocity measurements on samples held at high pressures inside DACs with laser heating remains one of the major experimental challenges in mineral physics. The potential to determine elastic properties at pressures and temperatures that resemble those predicted to prevail throughout Earth’s mantle has motivated first efforts to combine Brillouin spectroscopy with laser heating (Kurnosov et al., 2019; Murakami et al., 2009a; Sinogeikin et al., 2004; Zhang et al., 2015) and led to successful measurements of sound wave velocities at combined high pressures and high temperatures (Murakami et al., 2012; Zhang & Bass, 2016).
When large enough crystal specimens are available, sound wave velocities can be derived by measuring the travel time of ultrasonic waves through single crystals. Initially developed for centimeter‐sized samples and using frequencies in the megahertz range (e.g., Spetzler, 1970), this technique can be adopted to micrometer‐sized samples contained in DACs by raising the frequencies of ultrasonic waves into the gigahertz range (Bassett et al., 2000; Reichmann et al., 1998; Spetzler et al., 1996). While first ultrasonic experiments in DACs were restricted to P‐wave velocity measurements, Jacobsen et al. (2004, 2002) designed P‐to‐S wave converters to generate S waves with frequencies up to about 2 GHz and to enable the measurement of S‐wave velocities on thin single crystals in DACs up to pressures of about 10 GPa (Jacobsen et al., 2004; Reichmann and Jacobsen, 2004). The relatively young techniques of picosecond acoustics and phonon imaging use ultrashort laser pulses to excite sound waves and to measure their travel times on the order of several hundred picoseconds, allowing to further reduce sample thickness. These techniques have been implemented with DACs to study elastic properties at high pressures and bear the potential to derive full elastic stiffness tensors (Decremps et al., 2014, 2010, 2008). At ambient pressure, the ultrasonic resonance frequencies of a specimen with a well‐defined shape can be measured as a function of temperature and then be inverted for the elastic properties (Schreuer & Haussühl, 2005). The technique of resonant ultrasound spectroscopy (RUS) has been used, for example, to trace the elastic stiffness tensors of olivine single crystals up to temperatures relevant for the upper mantle (Isaak, 1992; Isaak et al., 1989).
The elastic response of a perfectly isotropic polycrystalline aggregate is, in principle, fully captured by the isotropic bulk and shear moduli. These moduli can be determined from the velocities of sound waves propagating in any direction through an isotropic polycrystal. Consequently, the elastic properties of many minerals have been studied on polycrystalline samples or powders, especially when single crystals of suitable size and quality were not available. Elastic moduli determined on polycrystalline samples, however, do not allow computing absolute bounds on the elastic response nor can they capture the elastic anisotropy of the individual crystals that compose the polycrystal. Polycrystalline samples need to be thoroughly characterized to exclude bias by an undetected CPO or by small amounts of secondary phases that might have segregated along grain boundaries during sample synthesis or sintering. Microcracks or vanishingly small fractions of porosity might also alter the overall