Atomistic origins of pressure-induced changes in the O $K$-edge x-ray Raman scattering features of $\mathrm{Si}{\mathrm{O}}_2$ and $\mathrm{MgSi}{\mathrm{O}}_3$ polymorphs: Insights from ab initio calculations

Despite its fundamental importance in condensed matter physics and geophysical implications, establishing the systematic and direct link between the pressure-induced structural changes in crystalline and noncrystalline low-$z$ oxides and their corresponding evolution in O $K$-edge core-electron excitation features under extreme compression has been challenging. Here we calculated the site-resolved partial density of states and O $K$-edge x-ray Raman scattering (XRS) spectra for two of the important oxide phases in the Earth's lower mantle, $\mathrm{MgSi}{\mathrm{O}}_3$ bridgmanite and post-bridgmanite, up to 120 GPa using ab initio calculations, revealing the electronic origins of the O $K$-edge features for oxides under compression. The absorption threshold $\left(E_A\right)$ and band gap increase linearly with a decrease in the O-O distance in diverse $\mathrm{Si}{\mathrm{O}}_2$ and $\mathrm{MgSi}{\mathrm{O}}_3$ high-pressure phases $[E_A\left(\mathrm{eV}\right)\approx -10.9d_{\text{O-O}}\left(\AA \right)+34.4]$, providing a predictive relationship between the $E_A$ and the O-O distances in the oxide at high pressure. Despite densification, upon isobaric phase transition from bridgmanite to post-bridgmanite at 120 GPa, a decrease in band gap results in a decrease in edge energy because of an increase in O-O distance. The oxygen proximity is a useful structural proxy of oxide densification upon compression, as it explains the pressure-induced changes in O $K$-edge XRS features of crystalline and amorphous $\mathrm{Si}{\mathrm{O}}_2$ and $\mathrm{MgSi}{\mathrm{O}}_3$ at high pressures. These results can be applied to studies of the pressure-bonding transitions in a wide range of oxides under extreme compression.

Figure 1

Crystal structures of (a) bridgmanite at 25 GPa and (b) post-bridgmanite at 120 GP. Crystallographically inequivalent oxygen atoms in the unit cells are labeled O1 and O2. There are two crystallographically distinct oxygen sites in $\mathrm{MgSi}{\mathrm{O}}_3$ bridgmanite at 25 GPa: an apical oxygen atom, ${}^{\left[6\right]}\mathrm{Si}$-O1-${}^{\left[6\right]}\mathrm{Si}$ (O1, ${}^{\left[6\right]}\mathrm{Si}$-O1: 1.7725 Å, ${}^{\left[6\right]}\mathrm{Si}$-O1-${}^{\left[6\right]}\mathrm{Si}$ angle: $146.{14}^{\circ }$, average Mg-O1: 2.0382 Å), and a planar oxygen atom, ${}^{\left[6\right]}\mathrm{Si}$-O2-${}^{\left[6\right]}\mathrm{Si}$ (O2, average ${}^{\left[6\right]}\mathrm{Si}$-O2: 1.7681 Å, ${}^{\left[6\right]}\mathrm{Si}$-O2-${}^{\left[6\right]}\mathrm{Si}$ angle: $146.{36}^{\circ }$, average O2-O2 distance: 2.5005 Å, average Mg-O2: 2.2098 Å). The oxygen environments in post-bridgmanite at 120 GPa include a corner-sharing oxygen atom, ${}^{\left[6\right]}\mathrm{Si}$-O1-${}^{\left[6\right]}\mathrm{Si}$ (O1, ${}^{\left[6\right]}\mathrm{Si}$-O1: 1.6312 Å, ${}^{\left[6\right]}\mathrm{Si}$-O1-${}^{\left[6\right]}\mathrm{Si}$ angle: $138.{37}^{\circ }$, Mg-O1: 1.8686 Å) and an edge-sharing oxygen atom, ${}^{\left[6\right]}\mathrm{Si}$-O2-$2{}^{\left[6\right]}\mathrm{Si}$ (O2, average ${}^{\left[6\right]}\mathrm{Si}$-O2: 1.7123 Å, average O2-O2 distance: 2.4213 Å).

Figure 2

Crystal densities (i.e., the total atomic mass divided by unit cell volume) of the $\mathrm{Si}{\mathrm{O}}_2$ (α-quartz, coesite, and stishovite) and $\mathrm{MgSi}{\mathrm{O}}_3$ [enstatite, ilmenite-type $\mathrm{MgSi}{\mathrm{O}}_3$, bridgmanite (BR), and post-bridgmanite (PBR)] high-pressure phases with respect to average Si-O bond lengths (a) and average O-O distances (b); average O-O distances (c), and average Si-O bond lengths (d) of $\mathrm{Si}{\mathrm{O}}_2$ and $\mathrm{MgSi}{\mathrm{O}}_3$ high-pressure phases with increasing pressure. The black line refers to the trend for all the phases studied here, whereas a blue trend line is for the bridgmanite phase only. The linear relationships between crystal densities (ρ) and average O-O distances $\left(d_{\text{O-O}}\right)$ are described as $\rho (\mathrm{amu}/{\AA }^3)\approx -5.1d_{\text{O-O}}\left(\AA \right)+15.3(R^2=0.9)$ for all the phases and $\rho (\mathrm{amu}/{\AA }^3)\approx -4.0d_{\text{O-O}}\left(\AA \right)+12.6(R^2=1.0)$ for the bridgmanite phase only. The linear relationship between average O-O distances and pressures for all the phases is described as $d_{\text{O-O}}\left(\AA \right)\approx -0.002P\left(\mathrm{GPa}\right)+2.6(R^2=0.8)$. The linear relationship between average Si-O bond lengths and pressures for the bridgmanite phase is described as $d_{\text{Si-O}}\left(\AA \right)\approx -0.001P\left(\mathrm{GPa}\right)+1.8(R^2=1.0)$.

Figure 3

$l$-resolved oxygen PDOS (for unoccupied states) of O1 (a) and O2 (b) for $\mathrm{MgSi}{\mathrm{O}}_3$ bridgmanite (BR) and post-bridgmanite (PBR) with varying pressure. The Gaussian-broadening FWHM for PDOS is 0.02 Ry (red solid, oxygen $s$ state; blue solid, oxygen $p$ state; green solid, oxygen $d$ state; black solid, total DOS of oxygen).

Figure 4

DOS threshold energy for oxygen $s$ state (red), oxygen $p$ state (blue), and oxygen $d$ state (green) for O1 (a) and O2 sites (b) in $\mathrm{MgSi}{\mathrm{O}}_3$ bridgmanite (BR; closed circle) and post-bridgmanite (PBR; open circle), with varying average O-O distance. The linear relationships between DOS thresholds $\left(E_{\mathrm{DOS}}\right)$ of and average O-O distances $\left(d_{\text{O-O}}\right)$ for the bridgmanite structures are described as $E_{\mathrm{DOS}}\left(\mathrm{O}1\text{−}s\right)\left(\mathrm{eV}\right)\approx -20.8d_{\text{O-O}}\left(\AA \right)+58.3(R^2=1.0)$ for the $\mathrm{O}1 s$ state, $E_{\mathrm{DOS}}\left(\mathrm{O}1\text{−}p\right)\left(\mathrm{eV}\right)\approx -21.1d_{\text{O-O}}\left(\AA \right)+59.7(R^2=1.0)$ for the $\mathrm{O}1 p$ state, $E_{\mathrm{DOS}}\left(\mathrm{O}1\text{−}d\right)\left(\mathrm{eV}\right)\approx -19.9d_{\text{O-O}}\left(\AA \right)+56.5(R^2=1.0)$ for the $\mathrm{O}1 d$ state, $E_{DOS}\left(\mathrm{O}2\text{−}s\right)\left(\mathrm{eV}\right)\approx -22.8d_{\text{O-O}}\left(\AA \right)+63.3(R^2=1.0)$ for the $\mathrm{O}2 s$ state, $E_{\mathrm{DOS}}\left(\mathrm{O}2\text{−}p\right)\left(\mathrm{eV}\right)\approx -22.1d_{\text{O-O}}\left(\AA \right)+61.8(R^2=1.0)$ for the $\mathrm{O}2 p$ state, and $E_{\mathrm{DOS}}\left(\mathrm{O}2\text{−}d\right)\left(\mathrm{eV}\right)\approx -21.2d_{\text{O-O}}\left(\AA \right)+59.4(R^2=1.0)$ for the $\mathrm{O}2 d$ state, respectively.

Figure 5

Band gap of the $\mathrm{Si}{\mathrm{O}}_2$ (α-quartz, coesite, and stishovite) and $\mathrm{MgSi}{\mathrm{O}}_3$ [enstatite, ilmenite-type $\mathrm{MgSi}{\mathrm{O}}_3$, bridgmanite (BR), and post-bridgmanite (PBR)] high-pressure phases, with varying average O-O distance (a) and pressure (b). The linear relationship between band gaps and average O-O distances is described as $E_G\left(\mathrm{eV}\right)\approx -14.1d_{\text{O-O}}\left(\AA \right)+42.6(R^2=0.8)$ for all the phases and $E_G\left(\mathrm{eV}\right)\approx -22.7d_{\text{O-O}}\left(\AA \right)+63.9(R^2=1.0)$ for the bridgmanite phase only.

Figure 6

(a) Site-resolved and total O $K$-edge XRS spectra [blue solid, O1, apical corner-sharing oxygen; red solid, O2, planar corner-sharing oxygen; black solid, total $(\mathrm{O}1:\mathrm{O}2=1:2)$] for bridgmanite (BR) at 25 GPa. (b) $l$-resolved O DOSs (for unoccupied states) of each orbital (red solid, $s$ state; blue solid, $p$ state; green solid, $d$ state; black solid, total O DOS) for bridgmanite at 25 GPa. (c) Site-resolved and total O $K$-edge XRS spectra [blue solid, O1, apical corner-sharing oxygen; red solid, O2, planar corner-sharing oxygen; black solid, total $(\mathrm{O}1:\mathrm{O}2=1:2)$] for bridgmanite at 120 GPa. (D) $l$-resolved O DOSs (for unoccupied states) of each orbital (red solid, $s$ state; blue solid, $p$ state; green solid, $d$ state; black solid, total O DOS) for bridgmanite at 120 GPa. The Gaussian-broadening FWHM for PDOSs and O $K$-edge XRS spectra are 0.02 Ry and 0.5 eV, respectively. The experimental O $K$-edge XRS spectra of bridgmanite (open black circles with a black solid line) [25] were compared with the calculated O $K$-edge XRS spectra for the bridgmanite at 25 GPa.

Figure 7

Site-resolved (a) and total (b) O $K$-edge XRS spectra of high-pressure $\mathrm{MgSi}{\mathrm{O}}_3$ [enstatite, ilmenite-type $\mathrm{MgSi}{\mathrm{O}}_3$, bridgmanite (BR), and post-bridgmanite (PBR)] polymorphs. Those for crystallographically inequivalent oxygen sites [O1 (black), O2 (red), and O3 (blue)] are shown. The spectra of the enstatite and ilmenite-type $\mathrm{MgSi}{\mathrm{O}}_3$ are from a previous study [24]. The Gaussian-broadening FWHM for O $K$-edge XRS spectra is 0.5 eV. (C) Direct comparison of the calculated O $K$-edge XRS pattern for bridgmanite and post-bridgmanite at 120 GPa.

Figure 8

Absorption threshold energies in the O $K$-edge XRS of $\mathrm{Si}{\mathrm{O}}_2$ phases (α-quartz, coesite, and stishovite) and $\mathrm{MgSi}{\mathrm{O}}_3$ [enstatite, ilmenite-type $\mathrm{MgSi}{\mathrm{O}}_3$, bridgmanite (BR), and post-bridgmanite (PBR)] polymorphs with varying crystal density (a), average Si-O bond lengths (b), and average O-O distance (c). The black line refers to the trend for all the phases studied here, whereas a blue trend line is for the bridgmanite phase only. The absorption threshold for $\mathrm{Si}{\mathrm{O}}_2$ phases, enstatite, and ilmenite-type $\mathrm{MgSi}{\mathrm{O}}_3$ phases were obtained from a previous study [24] and that for the result of $\mathrm{Si}{\mathrm{O}}_2$ coesite will be presented in a future contribution. The linear relationships between absorption threshold energies $\left(E_A\right)$ and average O-O distances $\left(d_{\text{O-O}}\right)$ are described as $E_A\left(\mathrm{eV}\right)\approx -10.5d_{\text{O-O}}\left(\AA \right)+33.7(R^2=0.7)$ for all the phases and $E_A\left(\mathrm{eV}\right)\approx -21.5d_{\text{O-O}}\left(\AA \right)+60.4(R^2=1.0)$ for the bridgmanite structures, respectively.

Figure 9

Calculated O $K$-edge XRS spectra for the bridgmanite (BR) structures with the O $K$-edge XRS features of bridgmanite and $\mathrm{MgSi}{\mathrm{O}}_3$ glasses [25]. The black and red solid lines with open circles indicate the experimental O $K$-edge XRS features of the $\mathrm{MgSi}{\mathrm{O}}_3$ glasses at 1 atm and 39 GPa, respectively, and the blue solid line with open circles indicates the O $K$-edge XRS features of the bridgmanite. The black, blue, and red solid lines at the bottom of the figure refer to the calculated O $K$-edge XRS spectra for the bridgmanite structures at 25, 79, and 120 GPa, respectively.