Sound velocities of bridgmanite from density of states determined by nuclear inelastic scattering and first-principles calculations

Insight into the accretion of the Earth and its subsequent differentiation can be gained through knowledge of the present-day Earth composition. The lower mantle in particular has been the focus of much attention as well as controversy, and questions such as how closely it approximates a chondritic composition have occupied geochemists for decades. Comparison of laboratory measurements of elastic wave velocities of lower mantle minerals with seismic data has played a crucial role in the discussion starting with Birch (1952), and experimental techniques such as inelastic X-ray scattering (e.g., Fiquet et al. 2004), Brillouin scattering (e.g., Sinogeikin and Bass 2002; Jackson et al. 2005; Murakami et al. 2007; Murakami et al. 2008), ultrasonics (e.g., Irifune et al. 2008; Gwanmesia et al. 2009; Liu et al. 2015), and impulsive stimulated scattering (e.g., Crowhurst et al. 2008) have provided important results on transition zone and lower mantle minerals. In situ measurements are important, because some transitions (for example, spin transitions) are not quenchable, and may influence the elastic properties of iron-containing minerals.

Nuclear inelastic scattering (NIS) offers the attractive possibility to determine elastic wave velocities of iron-containing minerals in the laser-heated diamond anvil cell through direct measurement of the partial density of states (e.g., Sturhahn and Jackson 2007), and results for mantle minerals have already been reported for ferropericlase (Lin et al. 2006; Wicks et al. 2010; Chen et al. 2012; Sinmyo et al. 2014), orthoenstatite (Jackson et al. 2009) and post-perovskite (Mao et al. 2006). So far, however, no investigations using NIS have been carried out on the Earth’s most abundant mineral that constitutes the bulk of the lower mantle, namely bridgmanite.

Computational studies provide an important complement to experimental measurements and allow a more extensive probe of parameter space. Methods have advanced substantially in recent years to enable densities and elastic wave velocities of candidate lower mantle assemblages to be calculated for the appropriate compositions at relevant pressure and temperature compositions (e.g., Wang et al. 2015 and references therein). Electronic transitions in iron present a challenge, however, and the apparent inconsistency between experimental and computational results on the stability of the intermediate spin state of Fe²⁺ in bridgmanite remains controversial (e.g., Stackhouse 2008; McCammon et al. 2013; Shukla et al. 2015). Nevertheless, a strong advantage of computations is the capability to calculate a wide range of properties including experimental observables based on vibrational density of states (DOS), for example infrared spectra of hydrous post-perovskite (Townsend et al. 2015). The combination of theoretical DOS calculations with experimental determinations using NIS provides a powerful new tool to understand the vibrational properties of iron-containing phases.

In this work, we present an experimental study using NIS to determine sound velocities of five different compositions of (Mg,Fe)(Si,Al)O₃ bridgmanite at pressures between 0 and 89 GPa at room temperature, including a comparison with results for (Mg,Fe)(Si,Al)O₃ majorite with the same composition. We present a parallel theoretical study to calculate the DOS of two bridgmanite end-members (MgSiO₃ and FeSiO₃) at comparable pressures that provides insight into the anomalous experimental results.

NIS measurements were made using panoramic DACs that were designed and constructed at Bayerisches Geoinstitut. For each experiment, a starting material in the form of powder was loaded into a Be gasket together with small ruby chips for pressure determination using the ruby scale of Mao et al. (1986). Generally, LiF or NaCl was used as a pressure medium. The diameters of the diamond culets and opening holes in the gaskets varied depending on the pressure range of the experiment. The samples were laser-heated in the DAC using a portable system mounted on the beamline incorporating either single-sided (Dubrovinsky et al. 2009) or double-sided (Kupenko et al. 2012) laser heating.

The NIS data were collected at ambient conditions on the nuclear resonance beamline ID18 at the European Synchrotron Radiation Facility (Rüffer and Chumakov 1996) between 2008 and 2010. Further details of the NIS setup and data analysis are given in Glazyrin et al. (2013) and references therein. Nuclear forward scattering (NFS) data were collected at the same time as the NIS data to monitor the electronic state of iron, and X-ray diffraction was used to confirm the identity of bridgmanite off-line after laser heating. The NIS data were collected typically over a range of −80 to 80 meV around the ⁵⁷Fe nuclear resonance energy of 14.4 keV in steps of 0.2 meV. The energy resolution of the X-ray beam was 1 meV. Debye sound velocities (V _D) were determined from the reduced partial density of states (pDOS) using the “homogeneous model” described in Sinmyo et al. (2014). The densities required for the calculation of V _D of bridgmanite were determined using a third-order Birch-Murnaghan equation of state based on volumes and elastic parameters reported in Boffa Ballaran et al. (2012) (sample S4850) and Glazyrin et al. (2014) (samples U1219, SL16, SL18, S4883), where the generalized model in the latter work was used to calculate values relevant to the different compositions. Equations of state parameters for majorite were taken from McCammon and Ross (2003) (cell volume) and Kavner et al. (2000) (elastic parameters). The adiabatic bulk modulus can be used to calculate longitudinal wave (V_P) and transverse wave (V_S) velocities from V _D and vice versa (e.g., Sturhahn and Jackson 2007).

First-principles calculations were performed based on the local density approximation of DFT to determine the DOS for MgSiO₃ bridgmanite at 60 GPa and antiferromagnetic FeSiO₃ bridgmanite at 20, 40, and 60 GPa. We computed the dynamical matrices on a regular grid of 4 × 4 × 4 special q points (Monkhorst and Pack 1976). We used Fourier interpolation techniques to obtain the interatomic force constants on a dense grid in the reciprocal space, from which we obtained the phonon DOS (Gonze et al. 2005). We determined both the total and partial DOS for all the elements and calculated V _D using the same approach as for the experimental data. The equation of state parameters were taken from Caracas and Cohen (2005). We also determined seismic wave velocities from the full elastic tensor where elastic constants were obtained from the linear relation between stresses and strains. Uniaxial and/or pure shear strains on the order of ±1 % and ±2 % were applied, and then the positions of the atoms were relaxed. The elastic constants were determined from the residual stresses divided by the strains. Voigt-Reuss-Hill schemes were employed to average the elastic tensor, and seismic wave velocities were determined based on homogeneous aggregates.