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

Sound velocities of bridgmanite measured in the laboratory are a key to deciphering the composition of the lower mantle. Here, we report Debye sound velocities determined using nuclear inelastic scattering (NIS) for one majorite composition (Mg0.82Fe0.18SiO3) and five bridgmanite compositions (Mg0.82Fe0.18SiO3, Mg0.86Fe0.14Si0.98Al0.02O3, Mg0.88Fe0.12SiO3, Mg0.6Fe0.4Si0.63Al0.37O3, Mg0.83Fe0.15Si0.98Al0.04O3) measured in a diamond anvil cell at pressures up to 89 GPa at room temperature. Debye sound velocities for majorite determined from NIS are consistent with literature data from Brillouin scattering and ultrasonics, while Debye sound velocities for bridgmanite are significantly lower than literature values from the same methods. We calculated partial and total density of states (DOS) for MgSiO3 and FeSiO3 bridgmanite using density functional theory and demonstrate that Debye sound velocities calculated from the reduced DOS using the same approach as for the experimental data (i.e., the limit of D(E)/E2 as energy goes to zero) give the same sound velocities for each phase irrespective of which partial DOS is used. In addition, we show that Debye sound velocities calculated using this approach are consistent with values obtained from the calculation of the full elastic tensor. Comparison of the calculated DOS with the one obtained from NIS indicates that the experimental DOS has enhanced intensity at low energies that leads to a different slope of the DOS and hence a lower sound velocity. This effect is present in all of the bridgmanite samples examined in this study.


Background
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 Fe 2+ 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)O 3 bridgmanite at pressures between 0 and 89 GPa at room temperature, including a comparison with results for (Mg,Fe)(Si,Al)O 3 majorite with the same composition. We present a parallel theoretical study to calculate the DOS of two bridgmanite end-members (MgSiO 3 and FeSiO 3 ) at comparable pressures that provides insight into the anomalous experimental results.

Methods
The starting materials for the DAC experiments are listed in Table 1. Sample S4883 was synthesized from orthopyroxene loaded into a Re capsule in a Kawai-type multianvil press with a LaCrO 3 heater assembly and W 75 Re 25 /W 97 Re 03 thermocouple at 26 GPa and 2000°C for 40 min (Keppler and Frost 2005). The compositions of the starting materials were determined using the electron microprobe and conventional (radioactive point source) Mössbauer spectroscopy. 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 doublesided (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 57 Fe 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) 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 MgSiO 3 bridgmanite at 60 GPa and antiferromagnetic FeSiO 3 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 Fig. 4 Influence of pressure on S4883 bridgmanite. a NIS spectra. b pDOS (upper) and reduced pDOS (lower). The sample was loaded as bridgmanite into the DAC, and data were collected on compression during the same beamtime experiment Fig. 5 Influence of pressure on S4850 bridgmanite. a NIS spectra. b pDOS (upper) and reduced pDOS (lower). The sample was loaded as bridgmanite into the DAC, and data were collected during two different beamtime experiments: #1-50 GPa and #2-53 and 72 GPa. The sample was laser-heated at 50 GPa, and data were collected both before and after laser heating Fig. 6 Comparison of majorite and bridgmanite sample data at 0 GPa. a NIS spectra. b pDOS (upper) and reduced pDOS (lower). Majorite was measured as the starting material in air while bridgmanite samples were measured after decompression 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.

Results and discussion
We measured room temperature NIS spectra of one majorite composition and five bridgmanite compositions as a function of pressure. In the case of majorite, sample U1219 was first measured at 33 GPa and then laserheated to produce bridgmanite and measured again without changing pressure. The NIS spectra show a clear change between majorite and bridgmanite (Fig. 1a) that leads to different pDOS (Fig. 1b). Subsequent compression of the U1219 bridgmanite sample produced systematic changes in the NIS spectra (Fig. 2a) that are reflected in the pDOS (Fig. 2b). A similar behavior on compression was observed in bridgmanite with lower iron content (SL18, Fig. 3) and Al-containing bridgmanite with lower (S4883) and higher (S4850) amounts of Fe 3+ (Figs. 4 and 5, respectively). Finally, we collected data for the five different samples at ambient conditions for comparison (Fig. 6). Debye sound velocities (V D ) were calculated from the reduced DOS in the limit as energy goes to zero by averaging D(E)/E 2 in the low-energy range as described in Sinmyo et al. (2014). V D values for majorite (Table 2) (Fig. 7). These lower values are consistent with the higher iron content of our majorite (Mg 0.82 Fe 0.18 SiO 3 ). In contrast, the V D values for bridgmanite (Table 2) are unrealistically low compared to the results from Brillouin scattering (Fig. 8). We can rule out the influence of local clustering as a significant factor that lowers velocities since the effect is too small to account for the large discrepancy (e.g., Sinmyo et al. 2014).
For a better understanding of the DOS for end-member bridgmanite, we calculated the total DOS for MgSiO 3 bridgmanite and antiferromagnetic FeSiO 3 bridgmanite as well as the pDOS for each element at 60 GPa (Figs. 9 and 10). For consistency with previous finite difference computations (Caracas and Cohen 2005), we calculated the full elastic tensor to confirm that we obtained the same elastic velocities within error. We then applied the same approach used to obtain V D from the experimental DOS obtained from NIS data to the theoretical DOS shown in Figs. 9 and 10. We found that the V D values calculated from each pDOS as well as the total DOS for each phase were identical within error, i.e., the same result is obtained irrespective of which element pDOS is used. In addition, we found that the V D values calculated using this approach (Table 2) are consistent with values that we obtained from the full elastic tensor (8.23 and 6.78 km/s for MgSiO 3 and FeSiO 3 , respectively).
A Birch plot for bridgmanite data from both calculations and experiments highlights the anomalous behavior of sound velocities determined using NIS (Fig. 11). To obtain further insight into the origin of the low velocities of bridgmanite, we compare the pDOS for experiments and calculations. The breadth of vibrational states is smaller for bridgmanite calculated at 0 K compared to the experimental data at room temperature, but it is similar for the experimental data for both bridgmanite and majorite at room temperature (Fig. 12, upper). In the reduced pDOS, the enhanced intensity at low energies in the experimental bridgmanite data leads to a significantly higher intercept and hence lower velocity compared to both calculated bridgmanite and experimental majorite (Fig. 12, lower). While particle size (e.g., Marquardt et al. 2011) or anisotropy ) could be a factor, we note that this effect is present in all the bridgmanite compositions that we measured using NIS, irrespective of sample synthesis method or history (i.e., whether they were  synthesized in the multianvil press or by laser heating in the DAC). While it is clear from our results that sound velocities for bridgmanite determined using NIS cannot be used for comparison with seismic data, there is clearly scope for further work to identify the contributions to the vibrational DOS derived from NIS data in order to more fully understand the lattice vibrational properties of bridgmanite.

Conclusions
Our work has presented an extensive dataset for bridgmanite with different compositions based on NIS measurements in a DAC as well as first-principles calculations: (1) NIS measurements of majorite give sound velocities that are consistent with literature data from other methods, while NIS measurements of bridgmanite give sound velocities that are significantly lower than literature data.
(2) Sound velocities of bridgmanite determined from NIS data are consistently low, irrespective of composition or sample history.
(3) Treatment of the DOS calculated using DFT for MgSiO 3 and FeSiO 3 bridgmanite using the same approach as for the NIS data gives sound velocities that are consistent with each other for each phase irrespective of which partial DOS is used. (4) Treatment of the DOS calculated using DFT for MgSiO 3 and FeSiO 3 bridgmanite using the same approach as for the NIS data gives sound velocities that are consistent with the values obtained from the calculation of the full elastic tensor. (5)Comparison of the calculated and experimental DOS shows enhanced intensity at low energies in the latter that leads to the lower sound velocities.