Читать книгу Mantle Convection and Surface Expressions - Группа авторов - Страница 62

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 c_ijkl 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 elastic response of a polycrystal (Gwanmesia et al., 1990; Speziale et al., 2014).

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).