Single-Crystal Elasticity of MgSiO 3 Bridgmanite to Mid-Lower Mantle Pressure

The combination of seismic observations and mineral physics data represents a unique tool to understand the structure and evolution of the deep Earth's interior. However, to date, elasticity data on both compressional ( v P ) and shear ( v S ) wave velocities of MgSiO 3 bridgmanite are limited to shallow mantle conditions, hampering the resolution of mineral physics models. Here, we report the first single-crystal measurements of v P and v S of MgSiO 3 bridgmanite up to 79 GPa using high-pressure Brillouin scattering and single-crystal X-ray diffraction in a diamond anvil cell. At shallow lower mantle pressures, the elastic anisotropy of MgSiO 3 bridgmanite was found to be similar, albeit smaller than that of Fe,Al-bearing bridgmanite of Kurnosov et al. (2017) but differed significantly from that proposed in the recent study of Fu et al. (2019). Using the elastic stiffness coefficients of bridgmanite obtained in this study at different pressures, we calculate the pressure dependence of the adiabatic bulk modulus, K S0 = 257.1(6) GPa, K' S0 = 3.71(4), and of the shear modulus, G 0 = 175.6(2) GPa, G' 0 = 1.86(1). These elastic parameters are included in a self-consistent thermodynamic model to calculate seismic wave velocities along a lower mantle adiabat

is different, however, from that measured by Kurnosov et al. (2017), who performed simultaneous Brillouin scattering and single crystal X-ray diffraction measurements on Fe-and Al-bearing bridgmanite up to ∼40 GPa. To date, no direct experimental constraints exist on v P of bridgmanite at pressures of the mid lower mantle and although a number of computational studies have been performed, they partially disagree with one another due to different choices in the potentials and approximations used (Oganov et al., 2001;Wentzcovitch et al., 1998Wentzcovitch et al., , 2004Zhang et al., 2013). For instance, both compressional and shear wave velocities calculated at high pressure and room temperature by Oganov et al. (2001) are 2%-3% faster than those reported by Wentzcovitch et al., (2004). Results from ultrasonic measurements by Chantel et al. (2012) at low pressure seem to agree with the v P of Wentzcovitch et al. (2004) and the v S of Oganov et al. (2001), while the v S of Murakami et al. (2007) are in good agreement with those of Wentzcovitch et al. (2004).
In this study, we show that single-crystal DAC elasticity measurements can provide some constraints on the v P of MgSiO 3 bridgmanite even at shallow lower mantle pressures, if the in-plane orientation of the crystal is correctly chosen. Because both the diamond anvils and the sample are single crystals, their sound wave velocities depend on the wave propagation direction. If the minimum of the v P of the sample matches the maximum of the v S of the diamond, it is possible to observe a portion of the dispersion curve of bridgmanite even at shallow lower mantle pressure (∼30 GPa). At higher pressure, the v P overlaps completely with the diamond v S , as the sample has higher pressure derivatives of the elastic moduli than diamond. At deep lower mantle pressures, however, a crossover occurs, and the v P of bridgmanite appears at higher frequency (i.e., velocity) than the v S of the diamond in the Brillouin spectrum and, therefore, can be observed again. In this study, we have conducted single-crystal X-ray diffraction and Brillouin scattering measurements on four crystals of MgSiO 3 bridgmanite in the DAC up to pressures of the middle lower mantle. Combining all elasticity and density measurements in a global fit (Buchen, 2018), we self-consistently determine, for the first time, the full elastic tensor of MgSiO 3 bridgmanite at mid lower mantle pressures. The newly determined values of the elastic moduli and their pressure derivatives are used to update existing mineral physics models and estimate seismic velocities for a lower mantle with a pyrolite bulk composition.

Materials and Methods
Single crystals of MgSiO 3 bridgmanite were synthesized at 25 GPa and 1573 K (Boffa Ballaran et al., 2012) in a multianvil large volume press (LVP). Inclusion free bridgmanite crystals were selected for which sharp X-ray diffraction peak profiles (full width half maxima < 0.12°) were obtained in omega-scan rotations. Suitable crystals were then oriented and polished on both sides to platelets with a thickness of 8-10 μm. High-pressure single-crystal X-ray diffraction and Brillouin spectroscopy measurements were performed in a BX90 piston-cylinder-type DAC (Kantor et al., 2012). (100)-oriented Boehler-Almax type Ia diamond anvils were combined with Boehler-Almax seats to achieve opening angles larger than 80° (Boehler & De Hantsetters, 2004). The diameter of the diamond culets was 400 or 350 μm for measurements up to ∼35 GPa, and 250 μm for measurements at higher pressure. Before gluing the diamond anvils, Brillouin scattering measurements were performed on the two diamonds to identify the v S dispersion curves relative to the orientation of each diamond. By rotating one of the two diamonds with respect to the other, it was possible to align their dispersion curves and, therefore, minimize the effective width of the diamond v S peak in the Brillouin spectra. Re gaskets were indented to a thickness of ∼60 and ∼35 μm for target pressures of 35 and 80 GPa, respectively. Ne or He were loaded as quasi-hydrostatic pressure transmitting media using a gas loading apparatus (Kurnosov et al., 2008).
High-pressure Brillouin scattering experiments conducted in platelet geometry (Speziale et al., 2014;Whitfield et al., 1976) require at least two different platelet orientations to solve for the nine independent elastic stiffness components, c ij , of an orthorhombic crystal such as bridgmanite. This is achieved by combining experimentally measured density and sound wave velocities to fit the Christoffel's equation (e.g., Every, 1980): where c ijkl are the elastic stiffness components (tensorial notation, see supporting information S1), q j and q l are direction vectors, ρ is density, v i are the sound wave velocities and  ik is the Kronecker delta. Three CRINITI ET AL.

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sets of high-pressure experiments (runs) were performed employing two different sets of bridgmanite single-crystal platelets with different orientations. In the first run, crystals with Cartesian indices (0.81, 0.49, 0.31) and (0.35, 0.16, 0.92), were selected based on the low extent of c ij correlation at ambient conditions (for a more detailed description, see supporting information S1). These crystals were loaded into two separate DACs and Brillouin scattering and X-ray diffraction measurements were performed up to ∼35 GPa. At this pressure, even the slowest bridgmanite v P values overlapped with the v S of the diamond anvils, leaving many c ij values unconstrained. To overcome this limitation two further runs were performed using a second set of single crystal platelets with orientations chosen to maximize the appearance of the v P signals on either side of the diamond v S peak. The platelet orientations were first selected by simulating the velocities for different crystal planes using published c ij values (Sinogeikin et al., 2004). To simulate realistic datasets, synthetic velocities were scattered by randomly adding up to ± 40 m/s to v S and 70 m/s to v P , which are values comparable to our estimated experimental uncertainties. Platelets with Cartesian indices (0.00, 0.80, 0.55) and (1.00, 0.00, 0.00), which correspond to the (011) and (100) crystallographic planes, were found to be optimal based on the low correlation between the c ij s (supporting information S1). These orientations also have relatively slow and fast v P respectively, such that the direction of the v P maximum in the (100) plane could be aligned with the diamond v S minimum, while the v P minimum in the (011) plane could be aligned with the diamond v S maximum, to maximize the pressure interval over which these signals could be visible ( Figure 1a). To load two crystals into the same DAC, while also maximizing their dimensions with respect to the gasket hole dimension, it was necessary to cut them into semicircular-shaped platelets, which meant that both crystals had to be correctly aligned relative to the diamonds and to each other before cutting. Two double-sided-polished but unshaped platelets in the selected planes were, therefore, first placed on a diamond anvil ( Figure 1b) and the desired orientations relative to the diamonds were then determined for each crystal using single-crystal X-ray diffraction. The two single-crystal platelets were then cut into semi-circles for loading into the DAC ( Figure 1b) using a FEI Scios focused ion beam (Schulze et al., 2017), while maintaining the required orientations relative to the straight edges. Ion beam cutting was performed using a beam current of 3 nA at an acceleration voltage of 30 kV. According to the target pressure, one or two platelets were loaded into the DAC sample chamber together with a ruby chip for preliminary pressure determination (Figures 1c-1e). Before gas loading, the bridgmanite crystals were rotated on the diamond culet using a needle, until the desired orientations (determined by means of XRD) relative to the diamond CRINITI ET AL.

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3 of 17  (100) (purple) and (011) (blue) relative to each other and to the diamond v S (red) at room pressure. The shaded pink area represents the predicted full width of the diamond peak at high pressure (see supporting information S1), where the v P of the samples cannot be resolved. Note that the diamonds have been aligned in such a way to produce only one v S signal at each orientation. The difference in rotation angle (Δχ 0 ) between the g 0 vectors of the two crystal platelets was used together with the UB matrix of the crystals to calculate the desired FIB cutting pattern. (b) Photograph of the two platelets 100 and 011 before FIB cutting in a test DAC (middle circular inset) and secondary electron image of the same platelets after FIB cutting.
(c-e) Photographs of platelets 100 and 011 inside the relevant sample chambers at high pressure, loaded with Ne (c) or He (d and e) gas.
anvils were achieved. The two crystals were loaded together in the same DAC for measurements up to ∼30 GPa (second run), and in two separate DACs for measurements above 50 GPa (third run).
Brillouin scattering measurements were performed using the system installed at the BGI (Trots et al., 2013).
The system employs a Coherent Verdi V2 solid-state Nd:YVO 4 laser with a 532 nm single wavelength output and a six-pass Sandercock-type piezoelectrically driven scanning tandem Fabry-Pérot interferometer (Sandercock, 1982) equipped with a single pixel photon counter detector (Hamamatsu C11202-50). A laser power of 100-200 mW was used, which corresponds to 40-80 mW before entering the DAC. The scattering angle between the incident and the analyzed laser beam, equal to 80°, was calibrated using a fused silica glass standard. Further details concerning the system alignment and calibration, as well as a discussion of potential errors related to misalignment, are discussed in supporting information S1. Simultaneous density measurements were performed on a Huber Eulerian single-crystal diffractometer coupled with an ultra-high intensity rotating anode X-ray source FR-E + SuperBright from Rigaku equipped with multilayer VaryMax™ focusing optics (MoKα radiation) (Trots et al., 2011). The rotating anode was operated at 45 kV and 55 mA, and the diffractometer was driven by the SINGLE software (Angel & Finger, 2011). Cell parameters were determined and refined with the vector least-squares method (Ralph & Finger, 1982), using up to 15 unique reflections centered in eight positions (King & Finger, 1979) to eliminate the effect of crystal offsets and diffractometer aberrations. To reduce the very long counting times required when using a point detector on small crystals at very high pressures, density measurements above 50 GPa were conducted on a Kappa-geometry Oxford Diffraction single-crystal X-ray diffractometer (Xcalibur2) equipped with a Sapphire 2 CCD area detector. The diffractometer uses graphite-monochromatized MoKα radiation (λ = 0.70937 Å) and was operated at 50 kV and 40 mA. At each pressure point, intensity data collections of ∼24 h with 60 s of exposure time were performed, and lattice parameters were determined from at least 100 reflections.

Sound Wave Velocities of MgSiO 3 Bridgmanite at High Pressure
Sound wave velocities of MgSiO 3 bridgmanite were measured at 14 pressure points, ranging from ambient pressure to 78.8(5) GPa (Table 1) over the three different sets of runs. Typical Brillouin spectra in the lowand high-pressure range (Figure 2a) show the progressive overlap between the v P of bridgmanite and v S of diamond, with a crossover taking place at pressures above 50 GPa. To our knowledge, these are the first direct measurements of P-wave velocities of MgSiO 3 bridgmanite at deep lower mantle pressures, albeit that the v P of bridgmanite could be observed only in certain ranges of rotation angle (χ) at such high pressures. Usually Equation 1 is solved using the appropriate number of acoustic velocities collected at each individual pressure point, which in the case of an orthorhombic crystal, such as bridgmanite, requires the measurement at the same conditions of at least two differently oriented crystals. This procedure is hereinafter referred to as an individual fit.
a Combined velocities from pressure points at 29.37 GPa (set 2) and 31.20 GPa (set 1).

Table 1 Elastic Stiffness Coefficients (c ij ) and Elastic Moduli From Individual Fits and Global Fit, Reported in GPa
where c ijkl is the elastic stiffness component (tensorial notation, see supporting information S1) at a certain is the finite Eulerian strain, c ijkl,0 is the elastic stiffness component at ambient conditions and c′ ijkl,0 its pressure derivative, K 0 is the bulk modulus and K′ 0 its pressure derivative, and  ij kl is −3 for c 1111 , c 2222 , and c 3333 and −1 for the other six independent components of the tensor. The coefficients were fitted to the complete velocity dataset using a script implemented in the software package Origin 2019 (OriginLab corporation, Northampton, MA, USA) and described more in detail in Buchen (2018). In our fitting procedure, c ij,0 were initially fixed to the values determined independently at ambient conditions, while c′ ij,0 were refined. Only in the last cycle, the c ij,0 were allowed to vary and were therefore refined together with the c′ ij,0 . Further details are provided in the supplementary information (supporting information S1). This approach (hereinafter referred to as a global fit) has the advantage that all velocity data from all pressure points can be used to constrain the c ij , also at those pressures where data are scarce due to the challenging conditions. For instance, c 22 is not well constrained at 30 GPa, due to the overlap of the v P of bridgmanite with the v S of the diamond. However, its pressure evolution is well constrained by the measurements conducted above 50 GPa, i.e., at pressures where v P can be observed again, at least in some orientations CRINITI ET AL.  on the other side of the diamond v S . On the other hand, c 11 is poorly constrained above 50 GPa, but its high-pressure evolution is very well constrained by measurements on 4 different platelets up to 35 GPa. This approach reduces the number of independent variables from 117 (9 c ij × 13 pressure points) down to 18 (9 c ij,0 + 9 pressure derivatives) while having the same number of observables. In this way, it was possible to obtain the elastic tensor of bridgmanite, within reasonable uncertainties, at all pressure points. Measured velocities in the low-pressure range (i.e., less than 30 GPa) constrain well the zero-pressure values of the c ij (i.e., c ijkl,0 ). If we considered only velocities measured in the low-pressure range, data scattering could still bias the third-order fit of the c ij , even when a global fit is employed. This happens because the higher the pressure at which wave velocities are measured, the larger is the constraint they provide on the pressure derivative of the c ij (i.e., c' ij,0 ). Therefore, despite being unable to perform individual fits at pressure points higher than 30 GPa, the measured velocities at mid-lower mantle conditions are of extreme importance in the global fit procedure. Furthermore, we stress that virtually no discrepancies between the c ij calculated from the finite strain EOS and the values determined using the individual fits can be observed, with values typically identical within two standard deviations (Table 1, Figure S1).
A potential downside of the global fit procedure is that we are of course assuming that the c ij s follow a third-order finite strain equation, but this should be a safe assumption given the known high-pressure behavior of MgSiO 3 bridgmanite (Boffa Ballaran et al., 2012) where no phase transitions either structural or electronic occur in the pressure range investigated. Although at the highest pressure some c ij s are not as well constrained (e.g., c 11 ) as others in the global fit, the constraints provided by the lower pressure data leave very little room for uncertainty, unless significant anomalous compression behavior occurs. As described in supporting information S1, we also tested global fitting using fourth-order EOSs. Although convergence in the refinement was not possible for all c ij s, this led to a slightly poorer agreement with the high-pressure shear velocity data, as shown in Figure S10 and a third-order fit was, therefore, retained for all c ij s. A detailed assessment of possible errors arising from the deviation of the c ij of bridgmanite from the third-order EOS fit is further discussed in supporting information S1.

Aggregate Elastic Properties of MgSiO 3 Bridgmanite
At each experimental pressure point, the adiabatic bulk (K S ) and shear moduli (G) have been calculated from the c ij values in the Reuss and Voigt bounds (supporting information S1) and are reported in Table 1. The elastic moduli were then fitted with a third-order finite strain EOS (Stixrude & Lithgow-Bertelloni, 2005) to obtain K S0 , G 0 , K' S0 , and G' 0 (Table 2). While the third-order fits of the Voigt bulk and shear moduli, as well as the Reuss bulk modulus, have only small residuals relative to experimental data points (i.e., typically less than 0.5 GPa), there is a change in the slope of the Reuss bound of the shear modulus (G R ) at ρ ∼ 4.54 g cc −1 (i.e., 35 GPa). This is apparent from the systematic deviations in the fitting residual at both low and high densities (i.e., pressures) ( Figure S2a, inset). This subtle change in the slope of the Reuss shear modulus occurs in the pressure interval where the shear wave anisotropy of MgSiO 3 bridgmanite shows a minimum, as already pointed out by Karki et al. (1997), and G V and G R tend to converge. At densities higher than 4.54 g cc −1 , the two bounds start to diverge again ( Figure S2c) as the shear wave anisotropy of bridgmanite increases. One possible explanation for this behavior is that it results from a change in the response of octahedral tilting and polyhedral distortion of the bridgmanite structure with pressure, i.e., under an CRINITI ET AL.
10.1029/2020JB020967 7 of 17 Values of M″ 0 in italics are implied from third-order truncations.

Table 2 EOS Parameters for K and G in the Voigt and Reuss Bound
isotropic stress environment (Reuss bound), for which corroborating evidence can be found in the evolution of lattice strain components for MgSiO 3 bridgmanite. Lattice strain components for the orthorhombic perovskite-type structure of bridgmanite can be simply calculated from unit cell parameters and describe the extent of lattice distortion with respect to a cubic perovskite structure, as described in previous studies (Boffa Ballaran et al., 2012;Carpenter, 2007;Carpenter et al., 2001). Of the three strain components, the tetragonal strain (e tx ), which is sensitive to both the degree of octahedral tilting and A site distortion, has also been shown to change in slope at ∼35 GPa (Boffa Ballaran et al., 2012) as shown in Figure S3. We argue that the change in slope of G R may, therefore, be detected from the change in the distortion mechanism of the perovskite structure with pressure relative to an undeformed cubic perovskite. It should be noted that this behavior of G R results from a combination of pressure dependencies of the individual c ij s, through the matrix inversion needed to obtain the compliances (s ij ) to calculate the Reuss bound, and does not mean that these dependences are not well described by a third-order finite stain equation.
To better describe the change in slope of G R , we fitted a fourth-order EOS to our shear moduli. To do so, we fixed the value of K S0 , K' S0 , as well as K'' S0 (value implied by a third-order truncation of K, see supporting information S1) which can be determined independently from K-V data, and refined only the shear modulus G 0 and its derivatives G' 0 and G'' 0 . A similar situation, where a higher truncation order for G is required with respect to K, was observed before in MgO (Zha et al., 2000). Alternatively, it is possible to fit two separate third-order EOS below 30 GPa and above 50 GPa to the same shear moduli dataset ( Figure S2b), assuming two distinct shear mechanisms, as justified by the change in the slope of e tx with pressure ( Figure S3). The two fitting strategies are indistinguishable within uncertainty ( Figure S2b). Fitting parameters of third-and fourth-order EOS are reported in Table 2.

Absolute Pressure Determination
One of the advantages of conducting elasticity and density measurements simultaneously on the same sample is that pressure can be determined without using any secondary pressure scale (e.g., ruby or Sm:YAG). The absolute pressure ( abs P ) can be determined by integration of the isothermal bulk modulus in the Reuss bound and the volume over a certain volume range: where K TR is the Reuss isothermal bulk modulus, K TR0 is its value at ambient conditions and K' TR0 its pressure derivative. Using the relations K SR = K TR (1 + αγT) and α = γC V /K TR V, we converted our experimentally determined K SR into K TR and calculated K TR0 and K' TR0 by fitting a third-order Birch-Munraghan EOS to our K T -V data. Thermoelastic parameters used for the conversion of K SR to K TR at high pressure are reported in Table S2. The absolute pressure was then determined using experimentally measured volumes, and our K TR0 = 254.5(4) GPa and K' TR0 = 3.73(2). The deviation of abs P from the ruby pressure scale is virtually zero at pressures below 40 GPa and increases to 1-5 GPa at the highest pressure achieved ( Figure S4) depending on the employed pressure calibration for the ruby fluorescence (Dewaele et al., 2004;Jacobsen et al., 2008;Mao et al., 1986). The best agreement is found with the pressure scale determination of Jacobsen et al., (2008). For comparison, we also fitted a third-order Birch-Munraghan EOS to the experimental bridgmanite unit cell volumes using pressures determined from the ruby pressure scale of Jacobsen et al. (2008), and the EOSFit software (Angel et al., 2014;Gonzalez-Platas et al., 2016). The obtained fitting parameters of K T0 = 255(10) GPa and K' T0 = 3.8(2) are in good agreement with the values determined in the K T -V fit reported above. Pressures calculated with our P-V EOS are in broad agreement, within uncertainty, with those calculated from single-crystal synchrotron X-ray diffraction measurements by Boffa Ballaran et al. (2012), powder diffraction measurements in the LVP by Katsura et al. (2009) and the recent computational study of Zhang et al. (2013) (Figure S5). On the other hand, powder X-ray diffraction data by Fiquet et al. (1998) and Tange et al. (2012) tend to overestimate pressures by more than 2 GPa already at 50 GPa. Tange et al. (2012) pointed out that the wide range of reported MgSiO 3 bridgmanite EOS parameters results from the use of different pressure scales and experimental techniques. Their MgSiO 3 bridgmanite EOS employed a primary MgO pressure scale calibrated with data sets where absolute pressure measurements were made (Tange, Nishi-hara, & Tsuchiya, 2009). However, the accuracy of such EOS measurements is reduced by both the error on the primary pressure scale and those on the volume measurements of the pressure marker. Our abs P , on the other hand, is a primary pressure scale and has been derived from volume and elasticity measurements on single-crystals in a quasi-hydrostatic environment, which reduces the sources of systematic errors.

Discussion
Aggregate velocities of MgSiO 3 bridgmanite are computed from the Voigt-Reuss-Hill (VRH) average of the bulk and shear moduli (Hill, 1963) and show excellent agreement between individual fits and the global fit procedure (Figure 3a). A comparison of the global fit with previous experimental and computational elasticity studies is shown in Figure 3b. Ultrasonic interferometry data collected by Chantel et al. (2012) in a LVP are in good agreement with this study at pressures up to the top of the lower mantle, although the values of K S0 and K' 0 , 247(4) GPa and 4.5(2) respectively, would lead to an overestimation of v P at deeper lower mantle conditions. Shear wave velocity data reported by Murakami et al. (2007) plot below the values determined in this study throughout the whole pressure range investigated. To make a fairer comparison, the v S data of Murakami et al. (2007) were refitted using a consistent third-order truncation of the shear modulus EOS, as proposed by Chantel et al. (2016), rather than that derived from the fourth-order EOS of Davies and Dziewonski (1975)  sizes can potentially trigger nonnegligible scattering from grain boundaries, causing sound velocity softening even at high pressure (Marquardt et al., 2011). This effect was reported to be particularly pronounced on the shear modulus of the polycrystalline aggregate, which, in the case of MgO, can be reduced by 1%-2% even when the crystal size is 500 nm (Marquardt et al., 2011). A grain size of 100 nm, for example, could therefore potentially explain the 3%-4% difference in shear wave velocities observed between our data set and that of Murakami et al. (2007).
Longitudinal wave velocities obtained in the computational studies of Oganov et al. (2001) and Wentzcovitch et al. (2004) are in reasonable agreement with our data at deep mantle conditions, as well as the shear wave velocities reported by Oganov et al. (2001). On the other hand, shear wave velocities reported by Wentzcovitch et al. (2004) are smaller by more than 3% at deep lower mantle pressure and close to the experimental results of Murakami et al. (2007). Based on their velocity data and previous static compression experiments Murakami et al. (2007Murakami et al. ( , 2012 argued that a perovskite-rich (at least 93 volume %) lower mantle is required to match 1D seismic velocity models. Our elasticity data, instead, suggest that the shear modulus of bridgmanite is 3% stiffer than the values reported by Murakami et al. (2007) at mid lower mantle pressures. Therefore, the addition of softer materials (i.e., ferropericlase) is required to explain the slower v S of the lower mantle assemblage, pointing toward a bulk composition that is more compatible with that of the upper mantle.

Effect of Chemistry on the Elasticity of Bridgmanite
The effect of Fe and Al on the compressional and shear wave velocities of bridgmanite has been investigated in several previous studies by means of Brillouin scattering, ISLS and ultrasonic interferometry (Chantel et al., 2012;Fu et al., 2018Fu et al., , 2019Jackson et al., 2005;Kurnosov et al., 2017). A comparison between velocity data for MgSiO 3 bridgmanite from this study with selected previous studies, summarized in Table S1, is shown in Figure 3b. Chantel et al. (2012) reported that the incorporation of Fe in bridgmanite decreases its sound velocities relative to MgSiO 3 . Despite showing much larger K' 0 (4.7) and lower G' 0 (1.56) than values reported here, sound velocities measured by Chantel et al. (2012)  The v P of Fe-bearing bridgmanite is subparallel to the MgSiO 3 endmember presented in this study, except between 40 and 60 GPa, where a spin crossover of Fe 3+ in the B site of bridgmanite is proposed to take place causing a drop in v P . Note that at the onset of the spin crossover, the slope of the v S increases and becomes slightly larger than that of MgSiO 3 . Jackson et al. (2005) studied the effect of the AlAlO 3 component on the aggregate acoustic velocities of bridgmanite using Brillouin scattering in a DAC. Incorporation of 0.05 Al atoms in both A and B sites of the bridgmanite structure seems to decrease both v P and v S (Jackson et al., 2005), but does not appear to affect their pressure dependence compared to the MgSiO 3 endmember reported in this study. On the other hand, the incorporation of the FeAlO 3 component decreases acoustic velocities at ambient conditions and causes an increase in their pressure derivatives (Kurnosov et al., 2017). In particular, the slope of the shear wave velocity for 10 mol% FeAlO 3 bridgmanite is considerably higher than that measured for MgSiO 3 (Kurnosov et al., 2017). The decrease of both v P and v S in Fe-bearing Al-free bridgmanite at ambient conditions can be mainly attributed to the increase in density due to incorporation of Fe in the structure . On the other hand, Al incorporation in the B-site of bridgmanite increases the volume of the octahedral site causing an increase in its compressibility, which results in a lower bulk elastic moduli for Fe, Al-bearing bridgmanite (Kurnosov et al., 2017).

Elastic Anisotropy
The elastic properties of MgSiO 3 bridgmanite determined in this study at ambient conditions are in overall good agreement with previously reported values by Sinogeikin et al. (2004) in terms of both elastic anisotropy ( Figure S6) and average properties (Table S1). The stiffest elastic component of MgSiO 3 bridgmanite at all pressures investigated in this study is the c 22 , while the c 11 and c 33 show a crossover at ∼15 GPa (Figure S1) with c 33 becoming stiffer than c 11 . The off-diagonal components c 13 and c 23 have very similar slopes throughout the whole pressure range investigated in this study, while the c 12 is much steeper ( Figure S1). A similar behavior was observed for FeAlO 3 -bearing bridgmanite by Kurnosov et al. (2017). Among the shear components, the stiffest is the c 44 , while the observed crossover in the c 55 and c 66 might be related to changes in the compression mechanism of individual Mg-O bonds with increasing pressure, affecting the shear mechanism of the A site of bridgmanite. While the c 44 and c 55 of FeAlO 3 -bearing bridgmanite measured by Kurnosov et al. (2017) are subparallel to MgSiO 3 bridgmanite, the slope of c 66 is considerably higher. Once again, this is probably related to the higher degree of distortion of FeAlO 3 -bearing bridgmanite relative to the MgSiO 3 end member, which is described by the lattice strain term e tx ( Figure S3). Several first principle studies have also calculated the anisotropic elastic properties of MgSiO 3 bridgmanite at high pressure (Karki et al., 1997;Oganov et al., 2001;Wentzcovitch et al., 1998Wentzcovitch et al., , 2004 and, in general, they show fairly good agreement with our results. While the diagonal components predicted by computational studies match well the values obtained in this study, the off-diagonal components tend to be more than 10% stiffer than our experimental values at mid-lower mantle pressures. There are a number of inconsistencies that arise when the elastic stiffness components determined in this study and those determined by Kurnosov et al. (2017) Karki et al. (1997), who clearly report a c 22 larger than c 33 at all pressures. The trends as a function of FeAlO 3 component reported by Fu et al. (2019) appear therefore inconsistent when compared to experimental and theoretical studies reported in the literature (Karki et al., 1997;Kurnosov et al., 2017;Oganov et al., 2001;Wentzcovitch et al., 1998Wentzcovitch et al., , 2004) ( Figure S7). Moreover, the v P anisotropy determined by Fu et al. (2019) for the Fe6Al4 and Fe12Al11 compositions at 25 GPa is quite different, even though the compositional change is relatively small (Figure 4). The v P anisotropy determined by Kurnosov et al. (2017), on the other hand follows the same general trend as that of the MgSiO 3 end member measured in this study and deviates quite strongly from the results of Fu et al. (2019). At 35 GPa, the v S anisotropy displayed by the Fe6Al4 and Fe12Al11 samples of Fu et al. (2019) becomes more similar ( Figure S8), because of a major change in the anisotropy of the Fe6Al4 sample. This would imply a significant change in the compressibility mechanism for these samples over a pressure interval of only 10 GPa, which seems unlikely. Although the three studies employed Brillouin scattering to probe the v S of the bridgmanite samples, the v S anisotropy displayed by the CRINITI ET AL.
10.1029/2020JB020967 11 of 17 Fe6Al4 and Fe12Al11 samples of Fu et al. (2019) is not always in agreement with that of MgSiO 3 bridgmanite and FeAl-bearing bridgmanite samples measured in this study and by Kurnosov et al. (2017) Figure 4, suggesting orientation problems. Note that the XRD, ISLS and Brillouin scattering measurements reported by Fu et al. (2019) were performed on three different setups and combined only afterward. On the other hand, XRD and Brillouin scattering measurements performed in this study and by Kurnosov et al. (2017) were performed on the same setup, without moving the sample. Further investigations on bridgmanite samples with intermediate compositions are therefore required to unequivocally solve this issue.
Using our high-pressure elastic stiffness components, the axial compressibility of MgSiO 3 bridgmanite can also be calculated for comparison purposes. The same relation between axial compressibilities β a > β c > β b (or axial moduli, M b > M c > M a ) can be observed in MgSiO 3 and FeAlO 3 -bridgmanite through both elasticity and X-ray diffraction measurements independently (Boffa Ballaran et al., 2012;Kurnosov et al., 2017). We have used third-order linear EOS (Angel et al., 2014) to fit the unit cell parameters reported by Boffa Ballaran et al. (2012) for MgSiO 3 bridgmanite using the EosFit7-GUI software (Gonzalez-Platas et al., 2016). The axial moduli are calculated from the compliance components, s ij , of MgSiO 3 bridgmanite obtained by inverting at each pressure the elastic stiffness tensor reported in this study. The axial moduli calculated in this way are in qualitative agreement with those calculated from the compression data reported in Boffa Ballaran et al. (2012), despite showing a less anisotropic behavior at high pressure and a smaller M b ( Figure S9). The axial moduli calculated from our elasticity measurements are also in fair agreement with computational studies (Oganov et al., 2001;Wentzcovitch et al., 1998Wentzcovitch et al., , 2004 which reproduce well the axial compressibility of Boffa Ballaran et al. (2012) (Figure S9). Larger values of M b observed at high pressure by Boffa Ballaran et al. (2012) may be explained by the larger calculated K' T0 relative to this study, which is most likely due to the different pressure scales employed.

A Mineral Physics Model for the Lower Mantle
Using previously published parameters for the thermal EOS of MgSiO 3 bridgmanite along with elasticity data for other components of the lower mantle, a mineral physical model can be constructed to predict seismic wave velocities for a given bulk mantle composition, which can then be compared with 1D seismic reference models (Dziewonski & Anderson, 1981;Kennett et al., 1995). Currently such a model can only be preliminary as the phase equilibria and EOS data required to model a complex mantle composition throughout the lower mantle are not available. Without more accurate knowledge, for example, of how the proportions and site occupancies of Fe 3+ and Al in bridgmanite change with pressure, their inclusion in such a model introduces more uncertainty than they likely resolve. However, as uncertainties rise with the complexity of the model, a comparison with simpler chemical models allows factors to be identified that may be important in a suitably realistic model of the mantle.
Seismic wave velocities were computed as a function of depth for a primitive mantle composition (McDonough & Rudnick, 1998) consisting of bridgmanite, ferropericlase and Ca-perovskite in the CaO-MgO-SiO 2 (CMS) and CaO-FeO-MgO-SiO 2 (CFMS) systems. Mass fractions of Al 2 O 3 , Cr 2 O 3 , and TiO 2 were added to SiO 2 . Calculations were performed using the formalism of Stixrude & Lithgow-Bertelloni (2011 to determine mineral compositions (in the CFMS system), seismic velocities and the adiabatic temperature profile, assuming a temperature of 1,960 K at 24 GPa (Katsura et al., 2010 where X is the mole fraction of either bridgmanite (Brg) or ferropericlase (Fp) components. To model the effect of Fe-Mg nonideality of mixing, we used the parameters given by Nakajima et al. (2012). However, we calculate the standard state Gibbs free energy change for the corresponding exchange equilibrium (Equation 4) self consistently using the model of Stixrude & Lithgow-Bertelloni (2005. The model results in values of K D which are in excellent agreement with recent multianvil studies performed up to ∼60 GPa Tange, Takahashi, et al., 2009) and in good agreement with some studies performed in the diamond anvil cell up to ∼100 GPa (e.g., Auzende et al., 2008;Sakai et al., 2009), even though the Fe-Mg partitioning model does not explicitly account for the effect of an electronic spin transition of Fe 2+ in ferropericlase. Using appropriate mineral component averaging schemes (Cottaar et al., 2014) the aggregate wave velocities in the Voigt and Reuss bound, as well as in Voigt-Reuss-Hill average, were then calculated. We make no allowance for the effects of attenuation in the calculation, which may further lower the calculated velocities (Cottaar et al., 2014) and the effects of iron spin transitions on the elastic properties have not been explicitly considered.
In Figure 5 velocities calculated along a mantle adiabat for a primitive mantle composition (McDonough & Rudnick, 1998) are compared in different systems with 1D seismic models (Dziewonski & Anderson, 1981;Kennett et al., 1995). Velocities for single-phase MgSiO 3 bridgmanite and the CMS model are faster than the seismic models; moreover, the CMS velocities also have distinctly steeper slopes, particularly for v P ( Figure 5). This arises from the slightly higher K' 0 of MgO (Kono et al., 2010)   v S and the seismic models throughout the entire lower mantle, with residuals never exceeding 0.6%. Even more striking agreement exists for v P , which perfectly overlaps with PREM along the entire modeled depth interval within the Voigt and Reuss bounds. The improved agreement in the slope of v P compared with the CMS model arises mainly because the Fe 2+ -Mg K D decreases with pressure, in line with experimental observations (Nakajima et al., 2012). As the proportion of the FeO endmember of ferropericlase increases, both v P and v S decrease. Despite the excellent match between the slope of v P for the CMFS phase assemblage and that of seismic models, it should be noted that the v S slope is not in perfect agreement. This difference cannot be attributed to temperature, as it would similarly displace both v P and v S . This likely arises because the model does not include potential effects of Al or the spin crossover transition of Fe 2+ in ferropericlase, which would need to be considered in any meaningful comparison. As far as the spin crossover of Fe 2+ in ferropericlase is concerned (Kantor et al., 2006;Lin et al., 2005;Marquardt et al., 2018;Valencia-Cardona et al., 2017, and references therein), its effect on the elastic properties has not been considered at high pressure and temperature, but the CMFS model should at least capture any resulting effects on the Fe-Mg K D because the available experimental data at pressure above the spin crossover transition are relatively well reproduced by our model. The incorporation of Al in bridgmanite is known to also raise the bridgmanite Fe 3+ /∑Fe ratio to levels of ∼0.5 (Frost et al., 2004), at least at conditions of the top of the lower mantle, due to the favorable substitution of an FeAlO 3 component. Fe 3+ is likely produced in the lower mantle through charge disproportionation of Fe 2+ that also leads to precipitation of Fe-rich alloy. As Al is added to bridgmanite half of the Fe 2+ , therefore, becomes Fe 3+ . The effect of adding an FeAlO 3 component to bridgmanite is twofold; on the one hand the incorporation of an FeAlO 3 component will lower the bulk and shear modulus of bridgmanite (Fu et al., 2019;Kurnosov et al., 2017) to some extent, while on the other hand the replacement of Fe 2+ for Fe 3+ will reduce the amount of Fe 2+ that partitions into ferropericlase, which will tend to raise the bulk and shear modulus. These combined effects may, therefore, tend to cancel out to some extent. However, quantifying this effect by incorporating it into a model for the entire bridgmanite stability field is not currently possible, because the definitive experiments to determine how the proportion of the bridgmanite FeAlO 3 component is likely to vary with depth in a primitive mantle composition have not yet been performed. It is quite possible, for example, that at higher pressures the proportion of the FeAlO 3 component decreases either due to changes in the Fe 2+ disproportionation reaction, as proposed recently for silica saturated assemblages (Shim et al., 2017) or due to a raised preference for the AlAlO 3 and FeFeO 3 bridgmanite components. In addition to these problems, however, there is also a lack of high-pressure elasticity data on the effects of all bridgmanite components other than MgSiO 3 , as elasticity measurements performed so far on FeSiO 3 -and AlAlO 3 -bearing bridgmanite, for example, were conducted only on polycrystalline samples (Chantel et al., 2012;Fu et al., 2018;Jackson et al., 2005;Murakami et al., 2012).
In this study, we have shown that self-consistent measurements of X-ray diffraction and Brillouin scattering of single crystals can provide accurate and precise elastic parameters of bridgmanite up to pressures of the mid lower mantle. To improve the resolution of mineral physics models, more self-consistent single-crystal elasticity measurements at high-pressure and high-temperature are required to better constrain the effects of these bridgmanite components. Assumed thermal parameters and the effects of attenuation also need to be addressed. Reducing the chemical and thermal uncertainties should eventually allow more reliable complex mineral physics models to be developed through which the average bulk composition of the lower mantle, in addition to the nature of large-scale apparent heterogeneities, can ultimately be constrained.

Data Availability Statement
Sound velocity data used for c ij calculation are available at: https://figshare.com/articles/dataset/ sound_velocity_data_xlsx/14265176.