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American Mineralogist; July 2008; v. 93; no. 7; p. 965-976; DOI: 10.2138/am.2008.2691
© 2008 Mineralogical Society of America
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Rietveld structure refinement of MgGeO3 post-perovskite phase to 1 Mbar

Atsushi Kubo1,*, Boris Kiefer2, Sang-Heon Shim3, Guoyin Shen4,{dagger}, Vitali B. Prakapenka4 and Thomas S. Duffy1

1 Department of Geosciences, Princeton University, Princeton, New Jersey 08544, U.S.A.
2 Department of Physics, New Mexico State University, Las Cruces, New Mexico 88003, U.S.A.
3 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02319, U.S.A.
4 Center for Advanced Radiation Sources, University of Chicago, Chicago, Illinois 60637, U.S.A.


   ABSTRACT
Top
 Abstract
Introduction
 Methods
 Results and discussion
 Acknowledgments
 References cited
 
Using the CaIrO3-type structure model (space group Cmcm), lattice parameters and atomic positions of the MgGeO3 post-perovskite (pPv) phase were determined based on Rietveld refinements at 78–109 GPa and first-principles calculations based on density functional theory. The reproducibility of structural parameters obtained for different samples, consistency with theoretical calculations, and good agreement with expected bond lengths based on structurally similar materials all provide evidence for both validity of CaIrO3-type structure model for the pPv phase in MgGeO3 exceeding 1 Mbar and reliability of structural parameters obtained by Rietveld refinements approaching 1 Mbar. The MgGeO3 pPv phase exhibits strong anisotropy in axial compressibility, with the b-axis being most compressible. The polyhedral bulk modulus for the GeO6 octahedron is 1.9x larger than that for the MgO8 hendecahedron. Examination of neighboring O-O distances shows that the O-O distance aligned along the a direction is one of the longest and that aligned along c is one of the shortest, and these may be related to the lower compressibility along c compared with a. Comparison of structural features of MgGeO3 pPv with those for MgSiO3, NaMgF3, and CaIrO3 pPv show that MgSiO3 pPv has more similarity with NaMgF3 and MgGeO3 pPv than with CaIrO3 pPv in such parameters as degree of octahedral distortion, implying that both NaMgF3 and MgGeO3 pPv are better analogs to MgSiO3 pPv than CaIrO3 pPv.

Key Words: Post-perovskite • MgGeO3 • Rietveld refinement • high-pressure experiment • first-principles calculation • laser-heated diamond anvil cell • density functional theory • polycrystalline X-ray diffraction


   INTRODUCTION
Top
 Abstract
 Introduction
Methods
 Results and discussion
 Acknowledgments
 References cited
 
The post-perovskite (pPv) phase, with CaIrO3-type structure (space group Cmcm) (Rodi and Babel 1965) (Fig. 1Go) discovered in MgSiO3 at 125 GPa and 2500 K (Murakami et al. 2004; Oganov and Ono 2004; Iitaka et al. 2004), has been extensively studied due to its geophysical importance (e.g., Ono and Oganov 2006; Hirose 2006; Merkel et al. 2007). Due to the experimental difficulties in the synthesis of silicate pPv phase at >1 Mbar and high temperature (e.g., Shim et al. 2004; Mao et al. 2004; Shieh et al. 2006), theoretical calculations have played an important role in predicting stability and physical properties of the pPv phase (e.g., Oganov and Ono 2004; Tsuchiya et al. 2004; Oganov et al. 2005; Wentzcovitch et al. 2006). Experimental studies have also focused on the CaIrO3-type pPv phase in analog materials to silicates such as MgGeO3 (Hirose et al. 2005; Kubo et al. 2006), MnGeO3 (Tateno et al. 2006), and NaMgF3 (Liu et al. 2005; Martin et al. 2006a). For example, MgGeO3 pPv phase can be synthesized as low as ~70 GPa at 2000 K (Hirose et al. 2005; Runge et al. 2006), and the pPv phase of CaIrO3 is stable at ambient conditions up to ~1650 K (Hirose and Fujita 2005; Kojitani et al. 2007). By using these analog materials, predictions for the behavior of the silicate pPv phase can be obtained experimentally (e.g., Merkel et al. 2006; Miyagi et al. 2008; Shim et al. 2007; Walte et al. 2007). It should also be noted that the CaIrO3-type phase also has been discovered in several sesquioxide compounds at high pressures: Al2O3 (Ono et al. 2006a), Fe2O3 (Ono and Ohishi 2005), and Mn2O3 (Santillán et al. 2006).


Figure 1
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  		FIGURE 1. Crystalg structure of CaIrO3-type post-perovskite phase in MgGeO3. (a) MgO8 and GeO 6 polyhedra are shown with blue and orange, respectively. (b) Spheres show atoms: red = Mg; blue = Ge; yellow = O1 (corner-shared oxygen); green = O2 (edge-shared oxygen). White and black bars indicate Mg-O and Ge-O bonds, respectively.

 
At high pressures where single-crystal diffraction is not accessible, the Rietveld method has been applied to examine detailed structures of polycrystalline materials (e.g., Fiquet et al. 2002). As pressure increases, the use of this technique becomes more challenging because diffraction data may suffer from effects of differential stress, preferred orientation, and poorer crystal statistics. Although differential stress in the sample can be reduced by annealing (e.g., Fiquet et al. 2002), it is generally difficult to eliminate preferred orientation in the sample. If preferred orientation exists in the sample, refined parameters that depend on diffraction intensities such as atomic positions could be affected (Martin et al. 2006b), even if lattice parameters can be reliably refined. Theoretical calculations based on density functional theory also can predict structure parameters as a function of pressure (e.g., Kubo et al. 2006). Due to the complementary nature of these two methods, use of both Rietveld and theory is desirable to obtain robust structure parameters at very high pressures.

So far, the Rietveld method has been applied to the pPv phase in MgGeO3 (Hirose et al. 2005, Kubo et al. 2006), Fe2O3 (Ono and Ohishi 2005), MnGeO3 (Tateno et al. 2006), Mn2O3 (Santillán et al. 2006), and NaMgF3 (Martin et al. 2006a) mainly to confirm that diffraction data are consistent with the CaIrO3-type structure. An important goal of our study is to provide a test of the reliability of Rietveld refinement methods at megabar pressures where effects such as poor crystal statistics, limited 2{theta} range, preferred orientation, and differential stress may limit refinement quality. Here we examine pressure and temperature dependencies of structure parameters of the pPv phase in MgGeO3 by conducting both Rietveld refinements and first-principles calculations. The results are compared with structure parameters predicted for MgSiO3 pPv at 120 GPa, NaMgF3 pPv at 30 GPa, and experimentally determined for CaIrO3 pPv at ambient conditions. This study presents further analysis of data reported by Kubo et al. (2006).


   METHODS
Top
 Abstract
 Introduction
 Methods
Results and discussion
 Acknowledgments
 References cited
 
High-pressure experiments
The pPv phase was synthesized from MgGeO3 orthopyroxene using laser-heated diamond cells. Powder X-ray diffraction of the starting material showed orthoenstatite with minor amounts of excess GeO2, as reported in Kubo et al. (2006). Unit-cell parameters of the orthopyroxene at ambient conditions were determined to be a = 18.8088(9) Å, b = 8.9490(4) Å, and c = 5.3431(2) Å from Le Bail refinement. These unit-cell parameters are in very good agreement with Ozima and Akimoto (1983). This starting material was mixed with 20 wt% Pt powder that served as both a pressure standard and laser absorber. Experiments were conducted using either Ar (run 1) or NaCl (run 2) as pressure media and thermal insulation. The sample was heated using a Nd:YLF laser (TEM00 mode) with double-sided heating (Shen et al. 2001).

Angle dispersive X-ray diffraction experiments were conducted at beamline 13-ID-D of the GSECARS sector at the Advanced Photon Source (APS) (Shen et al. 2005). The incident monochromatic X-ray beam of 0.3344 Å wavelength was focused to a size of ~6 x 6 µm2. An angle dispersive geometry with an image plate (Mar 345) was used. The detector was calibrated using CeO2. Pressure was determined from the equation of state of platinum (Holmes et al. 1989). Further details can be found in Kubo et al. (2006).

In run 1, the pPv phase was synthesized by heating the starting material to 1600–1900 K at 83–87 GPa for 10 min. We then compressed the sample to 99 GPa at room temperature and heated to ~1600 K. After quenching to room temperature, we further compressed the sample to 109 GPa at room temperature, but diffraction peaks of Pt and Ar completely overlapped at this pressure. The sample was then decompressed without further heating. Diffraction data were collected at room temperature at various pressures during both compression and decompression. In run 2, MgGeO3 pPv phase was synthesized during laser heating at 92–94 GPa and 1400–1700 K. Diffraction data were also collected in situ during laser heating. Subsequently, the sample was decompressed without heating, and diffraction data were obtained at various pressures.

Rietveld refinements
Two-dimensional (2D) diffraction images obtained after laser heating contained continuous Debye rings from all phases in the diamond cell, and few diffraction spots were present (Fig. 2Go). These two factors are essential to conduct reliable Rietveld refinement (cf. McCusker et al. 1999). To reduce the intensity of some strong diffraction spots originating from diamond in run 2, we tilted the diamond cell 1° away from the normal. For this reason, the outermost Debye rings were only partially recorded. To obtain one-dimensional (1D) data with the maximum 2{theta} range and reliable intensities, we restricted the region of integration to a ~60° slice where the outermost Debye ring was recorded (Fig. 2Go). Most of the observable diffraction spots in this region were masked, and then using Fit2d (Hammersley et al. 1996), the 2D data were integrated to obtain 1D diffraction patterns that were used for the Rietveld refinements.


Figure 2
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  		FIGURE 2. Polycrystalline diffraction data of MgGeO3 pPv phase at 88 GPa and room temperature from run 2 (B019).

 
We estimated the standard deviation ({sigma}) of diffraction intensity at each 2{theta} based on the equation (von Dreele, unpubl.): {sigma} = (F{checkmark}I)/tan2{theta}, where F is the parameter that represents intrinsic noise level of each diffraction datum and I is the diffraction intensity including background. F was assumed to be constant across the 2{theta} range for each pattern. To determine F, we sampled the noise level ({sigma}) and background intensity I at 2{theta} = 5.2 ± 0.4° where no diffraction peaks were observed.

Prior to Rietveld refinements, we conducted Le Bail refinements using GSAS/ EXPGUI (Larson and von Dreele 2004; Toby 2001) to refine lattice parameters and peak shape parameters (profile terms "GW and LX" in the constant wavelength profile function 2 in GSAS) to be used as initial values in Rietveld refinement. Here GW and LX are the basic profile terms that define pseudo-voigt peak shape. It is known empirically that Le Bail refinement yields the best fitting results (with best reliability factor) achievable in Rietveld refinement. This in turn means that the diffraction data that cannot be well fitted by Le Bail refinement are unlikely to be successfully refined by Rietveld refinement.

After successful Le Bail refinement, we conducted Rietveld refinement using GSAS/EXPGUI by the following three steps. For each step, parameters for the pPv phase, Pt, and either Ar or NaCl were separately refined initially, but eventually refined simultaneously. Before starting the refinement, a background was manually defined using a Chebyshev polynomial (typically 10 terms). In the first step, we refined only phase fractions and atomic positions assuming no preferred orientation, with lattice parameters and profile terms being fixed. Subsequently, we fixed all the parameters refined in the first step, and then refined only spherical harmonic terms for preferred orientation correction. We assumed cylindrical sample symmetry (fiber texture). The spherical harmonic order used for refinements was 2 (run 1) or 4 (run 2) for the pPv phase, and 6 for Pt, NaCl, and Ar. In the third step, we refined all the parameters including lattice parameters and profile terms simultaneously to complete the refinement and to obtain correct estimated standard deviations {sigma} (McCusker et al. 1999). For data obtained at 105 and 109 GPa in run 1, we ignored the presence of Ar in the diffraction data because peak overlap between Pt and Ar was nearly complete and diffraction peaks from Ar seemed weak and broad.

It is very challenging to refine isotropic displacement parameter Uiso [=B/(8{pi}2), where B is Debye parameter] from diffraction data obtained under pressure with a limited 2{theta} range. We concluded that our diffraction data do not have enough quality to refine displacement parameters because negative displacement parameters that are physically meaningless were typically obtained. Sugahara et al. (2006) demonstrated in a single-crystal X-ray diffraction study up to 15 GPa that equivalent isotropic displacement parameters Biso for perovskite-type MgSiO3 do not change strongly as a function of pressure at room temperature. Therefore, we fixed Uiso to the following values: 0.005 Å2 for all atoms in the pPv phase, 0.004 Å2 for Pt, 0.04 Å2 for Ar, and 0.02 Å2 for all atoms in B2 phase of NaCl. These Uiso values are based on equivalent isotropic displacement parameters of MgSiO3 perovskite (Sugahara et al. 2006) and Debye parameters listed in International Tables for X-ray Crystallography (Ibers et al. 1968) for Pt (293 K), krypton (93 K), and B1 phase of NaCl (293 K), respectively.

We tried to reduce the number of refined parameters as much as possible to avoid mathematically better but physically meaningless fitting ("overfitting"). For this purpose, we did not refine the background and Uiso, and we limited the number of peak shape parameters and spherical harmonic order for the preferred orientation correction to be as small as possible. In fact, Kubo et al. (2006) presented atomic positions for MgGeO3 pPv phase with some differences from those reported here because more parameters were refined for each phase including Uiso and one more Gaussian-related peak shape parameter in that study. The atomic positions reported by Kubo et al. (2006) are superseded by those obtained here.

Theoretical calculations
We carried out first-principles calculations based on density functional theory (Hohenberg and Kohn 1964) to calculate atomic positions and lattice parameters of the pPv phase as a function of pressure at 0 K as described in Kubo et al. (2006). These calculations were carried out with the software package vASP (Kresse and Hafner 1993, 1994; Kresse and Furthmüller 1996) using the projector-augmented-wave (PAW) method (Blöchl 1994; Kresse and Joubert 1999). Electronic correlations were treated within the local density approximation (LDA) in the parameterization of Perdew and Zunger (1981). The reliability of the predicted structure was verified by alternate calculations within the general gradient approximation (GGA) in the parameterization by Perdew-Burke-and Ernzerhof (PBE, Perdew et al. 1996). We used PAW-LDA potentials with core region cut-off radii of 2.0 a.u. for Mg (valence configuration 2p63s2), 1.9 a.u. for Ge (valence configuration 3d104s24p2), and 1.52 a.u. for O (valence configuration 2s22p4). The core cut-off radii and valence configurations in the PAW-PBE calculations were the same with the exception of the cut-off radius for Ge that was 2.3 a.u. Tests showed that converged solutions to the Kohn-Sham equations (Kohn and Sham 1965) could be obtained with an energy-cutoff of 600 ev and a 6 x 4 x 6 k-point grid. Total energies are converged to better than 2.3 mev/atom, and stresses due to the incompleteness of the basis-set are <0.5 GPa and 0.6 GPa in the LDA and GGA calculations, respectively.

We optimized the lattice parameters and atomic positions in the CaIrO3-type structure for volumes between 108 and 180 Å3 by LDA and between 110 and 195 Å3 by GGA to determine the groundstate of MgGeO3 at these volumes. The pressure and all structural parameters of MgGeO3 pPv were obtained from the relaxed configurations for eleven volumes in the pressure range of –2 to 298 GPa by LDA and twelve volumes between –4 and 298 GPa by GGA. It is noted that the pPv phase is metastable below ~47 GPa at 0 K according to the pPv phase transition boundary determined by Hirose et al. (2005), and Kubo et al. (2006) have confirmed the stability of the pPv phase up to 2 Mbar at ~1600 K. The results are shown in Table 1Go.


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  		TABLE 1. Lattice parameters and atomic positions for MgGeO3 post-perovskite phase at 0 K predicted by theoretical calculations using local density approximation (LDA) and generalized gradient approximation (GGA)
 

   RESULTS AND DISCUSSION
Top
 Abstract
 Introduction
 Methods
 Results and discussion
Acknowledgments
 References cited
 
Using the CaIrO3-type structure model, we could successfully conduct Rietveld refinements for three diffraction spectra in run 1 in the pressure range of 100–109 GPa and seven diffraction patterns including two data at high temperatures in run 2 at 78–94 GPa. Although we had diffraction data in a wide pressure range down to 7 GPa, we could not successfully refine lower pressure data probably due to higher differential stresses and metastability of the pPv phase at pressures < ~45 GPa (Kubo et al. 2006). Peak widths for diffraction data below 45 GPa are much broader than higher pressure data. Figure 3Go shows a selected Rietveld refinement result (see online supplementary materials1 for complete Rietveld results). It is clear from Figure 3Go that diffraction peaks of the pPv phase are well fitted. Together with the reasonable atomic positions and lattice parameters listed in Table 2Go, this observation validates the CaIrO3-type structure model for the pPv phase in MgGeO3 at 78–109 GPa. However, misfit can be seen for the most intense diffraction peak of Pt 200 and peaks from B2 phase of NaCl due to differential stress in the sample chamber, as discussed later. To evaluate the effect of misfit of Pt on refined atomic positions of the pPv phase, we conducted another set of Rietveld refinements using the diffraction data with the Pt 200 peak manually subtracted. In these refinements, the atomic positions were mostly reproduced within 1{sigma} deviation from original results, implying that effects of differential stress in Pt are not significant within 1{sigma}. However, for patterns A038 and A042 (Table 2Go and online supplementary items1), atomic positions were not reproducible within 3{sigma}. Therefore, despite better R-factor in run 1 than in run 2 (Table 2Go), Rietveld results from run 2 appear to be more reliable than those from run 1, which is supported by the larger 2{theta} range, weaker Pt intensity, and less peak overlap of Pt with pressure medium in run 2 (cf. Toby 2006). Note that the uncertainty shown in Table 2Go and all the figures in this paper are 1{sigma} obtained from least squares calculations in Rietveld refinements, and this represents the precision of the recovered parameters.


Figure 3
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  		FIGURE 3. An example of Rietveld refinement result at 88 GPa and room temperature from run 2. The crosses represent observed intensities (Iobs). The solid line is the calculated intensities (Icalc). The tick marks show the positions of calculated diffraction lines: upper ticks = NaCl (B2-phase); middle ticks = Pt; lower ticks = pPv. The line below the tick marks is the intensity difference curve (Iobs – Icalc).

 

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  		TABLE 2. Lattice parameters and atomic positions for MgGeO3 pPv phase determined by Rietveld refinements
 

Lattice parameters, axial ratios, and unit-cell volumes
Lattice parameters, axial ratios, and unit-cell volume of the pPv phase determined in this study are listed in Tables 1Go and 2Go. These results are also shown in Figure 4Go together with Rietveld results by Hirose et al. (2005) for MgGeO3 pPv at 78 GPa and room temperature, theoretical predictions for MgSiO3 pPv at 120 GPa and 0 K by Tsuchiya et al. (2004) and NaMgF3 pPv at 30 GPa and 0 K by Umemoto et al. (2006), and single-crystal X-ray diffraction results for CaIrO3 at ambient conditions by Rodi and Babel (1965). Note that theoretical results for MgSiO3 pPv by Tsuchiya et al. (2004) and Iitaka et al. (2004) using LDA are similar to within 0.2% for atomic positions and 0.3% for lattice parameters, and the differences between theoretical results by GGA (Oganov and Ono 2004) and LDA (Tsuchiya et al. 2004; Iitaka et al. 2004) are within 0.2% for atomic positions and 0.9% for lattice parameters. Although Rietveld results are available for both MgSiO3 pPv (Ono et al. 2006b) and NaMgF3 pPv (Martin et al. 2006a), we use theoretical results for these phases because, unlike for MgGeO3, theory and experiment are not currently in good agreement in terms of atomic positions. In this and subsequent figures, open symbols refer to data at room temperature (Rietveld) or 0 K (theory), and filled symbols indicate high-temperature data. Also, in the following discussion, we compare structure parameters in MgGeO3 at ~70 GPa with those in MgSiO3, NaMgF3, and CaIrO3 pPv phases at 120, 30, and 0 GPa, respectively, as these pressures are close to lowest stability limit of these pPv phases at high temperatures.


Figure 4
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  		FIGURE 4. (a) Lattice parameters, (b) axial ratios, and (c) unit-cell volume of the pPv phase. Open symbols indicate data at room temperature (Rietveld) or 0 K (theory), and filled symbols indicate data at high temperature from the present study. Data from other studies are triangle = Rietveld refinement by Hirose et al. (2006) for MgGeO 3 pPv at 78 GPa and 300 K; plus sign (+) = theoretical calculation by Tsuchiya et al. (2004) for MgSiO3 pPv at 120 GPa and 0 K (LDA); asterisk (*) = theoretical calculation by Umemoto et al. (2006) for NaMgF3 pPv at 30 GPa and 0 K (LDA); cross (x) = single-crystal X-ray diffraction by Rodi and Babel (1965) for CaIrO3 pPv at ambient conditions. Dashed and dotted lines represent LDA and GGA results, respectively. that of MgGeO3, NaMgF3, and CaIrO3 pPv phases, respectively.

 
Almost all the data points at room temperature in Figure 4aGo determined here by Rietveld refinement fall on the compression curves previously reported by Kubo et al. (2006) that were based on individual peak fitting using five diffraction peaks of the pPv phase, indicating that peak fitting can yield lattice parameters that are as reliable as those determined by Rietveld method. Lattice parameters predicted by LDA (GGA) are slightly smaller (larger) than those observed in our experiments, which are typical for these calculations.

Figure 4bGo shows ratios of unit-cell axes. Pressure dependence of b/a ratio shows that the b-axis is more compressible than a- and c-axes whose axial compressibilities are similar as manifested by less pressure dependence of c/a ratio. This finding is consistent with experimental observations for the MgGeO3 pPv phase (Hirose et al. 2005; Merkel et al. 2006; Kubo et al. 2006) and the MgSiO3 pPv phase (e.g., Murakami et al. 2004). While theory predicts a comparatively strong pressure dependence of b/a, experimental results show a small pressure dependence at 80–110 GPa, which is also reported in Kubo et al. (2006). On the other hand, c/a at 80–110 GPa shows a mild increase with pressure, in excellent agreement between experiments and theory. At 1700 K, both b/a and c/a become slightly smaller. The b/a ratio of MgSiO3 pPv is expected to be 0.8, 4.2, and 4.2% larger than while the c/a ratio of MgSiO3 pPv is 0.8, 0, and 6.5% larger than that of MgGeO3, NaMgF3, and CaIrO3 pPv phases, respectively, showing remarkable similarity of MgGeO3 pPv with MgSiO3 pPv in terms of axial ratios.

Figure 4cGo shows unit-cell volumes determined by Rietveld refinement and theory. All data points determined by Rietveld refinement fall on the compression curve of the pPv phase experimentally determined by Kubo et al. (2006), again validating consistency between peak fitting and Rietveld methods. It is noted that a previous Rietveld refinement by Hirose et al. (2005) is also in excellent agreement with our Rietveld results in spite of serious inconsistency in atomic positions between our results and Hirose et al. (2005) as discussed later. By using a third-order Birch-Murnaghan equation of state, LDA results give zero-pressure isothermal bulk modulus (K0) of 205 GPa with pressure derivative (K0 ’) of 4.28, while GGA results yield K0 of 174 GPa with K0 ’ of 4.25. Our GGA results agree well with theoretical work by Fang and Ahuja (2006) using GGA. Since LDA results are more consistent with experimentally determined K0 of 207(5) GPa and K0 ’ of 4.4 by Kubo et al. (2006), we mainly use LDA results for comparison with Rietveld results hereafter.

Preferred orientation and differential stress
Figure 5aGo shows values of texture indices from the Rietveld refinements, parameters that indicate the magnitude of preferred orientation, with J = 1 if there is no preferred orientation, otherwise J > 1 and J = {infty} for a single crystal (Von Dreele 1997). All materials in the sample chamber exhibit evidence for preferred orientation. In all diffraction patterns, the relative intensity of pPv diffraction peaks 020 and 110 are stronger and weaker than expected from the ideal intensity relationship, respectively. All the spherical harmonic coefficients for the pPv phase are shown in Supplementary Table 11, from which one can obtain pole figures. Our pole figure analysis showed that the b-axis tends to align perpendicular to the compression direction of the diamond cell while {100}, {110}, {101}, and {111} tend to align almost parallel to the compression axis. These observations are consistent with preferred orientation of MgGeO3 pPv observed by Merkel et al. (2006) and theoretical predictions by Oganov et al. (2005) for MgSiO3 pPv. Similar experimental observations have been reported for both MgSiO3 and Mn2O3 pPv by Murakami et al. (2004) and Santillán et al. (2006), respectively.


Figure 5
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  		FIGURE 5. (a) Texture index obtained from Rietveld refinements. (b) Differential stress in Pt estimated from 111 and 200 diffraction peaks.

 
We estimated differential stress in Pt from d-values of Pt111 and Pt200 based on Singh’s (1993) method, assuming an axially symmetric stress field in the sample, iso-stress model (Reuss limit), and elastic anisotropy of Pt (Kavner and Duffy 2003). Details of our calculations can be found in Runge et al. (2006). The results, shown in Figure 5bGo, indicate that the magnitude of differential stress in our samples (MgGeO3 mixed with Pt) is ~0.3–1.0 GPa. There is no discernible correlation between the magnitude of differential stress and type of pressure medium (Ar or NaCl). At high temperature, differential stress tends to be somewhat reduced.

Atomic positions
There are four variable atomic positions (fractional coordinates) in the CaIrO3-type structure (Fig. 1Go). In the present study, we represent atomic positions of the ABX3 pPv phase as follows; A (0, y, 1/4), B (0, 0, 0), X1 (0, y, 1/4), X2 (0, y, z), where X1 and X2 correspond to corner and edge shared anions of the BX6 octahedron (Fig. 1Go), respectively. Note that Rodi and Babel (1965) contains typographical errors in the table of atomic positions for CaIrO3.

Atomic positions from this study are shown in Tables 1Go–2Go, and Figure 6Go. Results by LDA and GGA calculations are in good agreement. Rietveld results are consistent with theoretical results within 3{sigma}, although both considerable data scatter and the limited pressure range make it difficult to constrain the pressure dependence of atomic positions solely from Rietveld results. Our theoretical calculations predict small changes of the atomic positions at pressures between ~50 and 300 GPa, implying no change in compression mechanism of the pPv phase at this pressure range. However, Mg y and O2 positions change much more between 0 and ~50 GPa, implying different compression behavior in the low-pressure metastable region. Within an uncertainty of 3{sigma}, we do not find any consistent temperature dependence of the atomic positions from Rietveld results, and thermal effects may be below our level of resolution.


Figure 6
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  		FIGURE 6. Atomic positions (fractional coordinates) of ABX3 pPv phase. X1 and X2 are corner- and edge-shared anions in BX6 octahedra, respectively. See Figure 4Go for definitions of symbols and references.

 
Hirose et al. (2005) reported Rietveld refinement from a single diffraction spectrum of MgGeO3 pPv at 78 GPa and 300 K. Hirose et al.’s results for Mg y and O2 z are consistent with our results, but their results for O1 y and O2 y are quite inconsistent beyond 3{sigma}. The diffraction data obtained by Hirose et al. (2005) exhibits a markedly different intensity relationship compared with our data in Figure 3Go. According to our Rietveld refinement, the texture index of the pPv phase is always <1.5 (Fig. 5aGo), showing that preferred orientation of our samples is minimal. Also, our observed diffraction intensities are similar to expected intensities based on the structure. Therefore, we infer that Hirose et al.’s sample had a significant degree of preferred orientation, and their atomic positions may be affected by overfitting in the preferred orientation correction. Overall, the A position in these ABX3 pPv phases is similar, but X positions are generally different. In particular, O1 y position in MgGeO3 is remarkably smaller than those in MgSiO3, NaMgF3, and CaIrO3. It is noted that all the atomic positions in MgSiO3 are very close to those in NaMgF3.

Interatomic distances
CaIrO3-type ABX3 pPv phase has two coordination polyhedra in its structure, namely AX8 bicapped trigonal prism (hendecahedron) and BX6 octahedron (Fig. 1Go) (Ijjaali et al. 2004). The BX6 octahedra share X1 corners along the c-axis and X2-X2 edges along the a axis to form an infinite sheet. The AX8 hendecahedra share both faces on trigonal prisms in b-c plane and edges on pyramid caps to form a layer that separates the sheets of BX6 octahedra. Figure 7aGo shows averaged lengths of A–X and B–X bonds in AX8 hendecahedron and BX6 octahedron, respectively. Averaged distances of Mg-O in MgSiO3 perovskite up to 12.6 GPa (Ross and Hazen 1990) and Ge-O in CaGeO3 perovskite at room pressure (Sasaki et al. 1983) determined by single-crystal X-ray diffraction at room temperature are compared in this figure. Also shown are Rietveld results of averaged Mg-O distance in MgSiO3 perovskite at 79.7 GPa and 1681 K (Fiquet et al. 2000) and averaged Ge-O distances in {alpha}-PbO2 type GeO2 at room temperature at 60 GPa (Prakapenka et al. 2003) and 70.7 GPa (Shiraki et al. 2003). Here all Mg and Ge atoms have coordination number of 8 and 6, respectively. Since LDA and GGA results for lattice parameters and atomic positions are consistent, we show only LDA results in figures hereafter for clarity. Theory predicts averaged Mg-O and Ge-O lengths that are in excellent agreement with experimental results using single crystals, validating reliability of theoretical results at low pressures. Previous Rietveld results for averaged Mg-O and Ge-O lengths are also in good agreement with theoretical results, which also support reliability of theoretical results at high pressures. Due to large uncertainty in Rietveld refinement results at high temperature, however, it is difficult to discuss thermal expansion of bond lengths from Rietveld results. The average Mg-O bond length in MgSiO3 pPv is 0.9% smaller than that in MgGeO3 pPv at 120 GPa. Ratio of average bond lengths A-X/B-X in ABX3 pPv is 1.13 for MgGeO3, 1.17 for MgSiO3, 1.16 for NaMgF3, and 1.21 for CaIrO3.


Figure 7
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  		FIGURE 7. (a) Average bond lengths, and (b) individual bond lengths. Single-crystal results are by Ross and Hazen (1990) for MgSiO3 perovskite up to 12.6 GPa and by Sasaki et al. (1983) for CaGeO3 perovskite at room pressure. Rietveld results are by Fiquet et al. (2000) for MgSiO3 perovskite at 79.7 GPa and 1681 K, and by Prakapenka et al. (2003) and Shiraki et al. (2003) for {alpha}-PbO2 type GeO2 at room temperature at 60 and 70.7 GPa, respectively. See Figure 4Go for the other references.

 
Figure 7bGo shows individual bond lengths in the pPv phase. Our Rietveld results from run 1 and run 2 are consistent within uncertainty, and they are also consistent with our theoretical results. Theory predicts enhanced compressibility of Mg-O2 (x4) bond between 0 and ~50 GPa, which is closely related to rapid change of Mg y and O2 positions predicted between 0 and ~50 GPa (Fig. 6aGo). This implies a change of the compression behavior in MgO8 hendecahedral layer around 50 GPa, which might induce both unusual volume expansion below ~45 GPa and breakdown of the pPv phase below 7 GPa during decompression reported by Kubo et al. (2006). At high temperatures, it is likely that both Ge-O1 and Ge-O2 expand in the GeO6 octahedron, while Mg-O2(4) is the only bond in MgO8 hendecahedron that shows evidence for thermal expansion. In MgSiO3, MgGeO3, NaMgF3, and CaIrO3 pPv phases, bond lengths relationships are always B-X1 < B-X2 and A-X1 < A-X2 (x4) < A-X2 (x2).

Distances of seven different neighboring X-X anions in ABX3 pPv phases are shown in Supplementary Figure 21. Rietveld results, which are generally consistent with theoretical predictions, show that compressibilities of O-O pairs in MgO8 hendecahedron are generally greater than those in GeO6 octahedron. The most incompressible O-O pair locates in octahedron (X2-X2_o in Supplementary Fig. 2), while the most compressible pair locates in hendecahedron (both X2-X2_h2 and X1-X2_h in Supplementary Fig. 2). Theoretical calculations showed that compressibilities of O-O pairs in hendecahedron are significantly different from each other compared with those in octahedron. The O2-O2 pair that aligns in the c-direction (X2-X2_h1 in Supplementary Fig. 2) is one of the closest O-O pairs, while the O2-O2 pair that aligns in the a-direction (X2-X2_oh in Supplementary Fig. 2) is one of the longest O-O pairs in both MgGeO3 and MgSiO3 pPv. These facts may explain the reason for the less axial compressibility of the c-axis than the a-axis in both MgGeO3 and MgSiO3 pPv (Kubo et al. 2006; Guignot et al. 2007) due to expected higher O-O repulsion in c-direction.

Polyhedral volumes
Figure 8aGo shows the volume change of GeO6 octahedron and MgO8 hendecahedron as a function of pressure and temperature. Polyhedral volumes can be calculated from unit-cell volume Vcell and the atomic positions using the following equations: VGeO6 = (8 x O2 z x O1 y – 8 x O2 z – 4 x O1 y + 2 x O2 y + 3) xVcell/6, and VMgO8 = (–4 x O2 z x O1 y + 12 x O2 z + 5 x O1 y – O2 y – 12 x O2 y x O2 z – 3) x Vcell/12. It is noted that due to existence of void spaces in the crystal structure (Fig. 1aGo), the unit-cell volume of the pPv phase is given by 4VGeO6 + 4VMgO8 + Vvoid. Our Rietveld results are in agreement with our theoretical prediction within 2{sigma}.


Figure 8
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  		FIGURE 8. (a) Polyhedral volumes and (b) volume occupancy of polyhedra and void space in ABX3 pPv phase. Results for the second-order Birch-Murnaghan equation of state fitting using Rietveld results are shown with dotted lines in a. See Figure 4Go for references.

 
Theory predicts the following polyhedral compression parameters at 0 K based on third-order Birch-Murnaghan equation of state: polyhedral bulk modulus of 161 GPa for MgO8 and 301 GPa for GeO6, pressure derivative of 3.94 for MgO8 and 4.91 for GeO6, and zero-pressure polyhedral volume of 18.919 Å3 for MgO8 and 9.069 Å3 for GeO6. Rietveld results yield the following parameters at room temperature based on second-order Birch-Murnaghan equation of state: polyhedral bulk modulus of 174(3) GPa for MgO8 and 333(11) GPa for GeO6, with fixed zero-pressure polyhedral volume of 18.919 Å3 for MgO8 and 9.069 Å3 for GeO6. As shown here, polyhedral bulk modulus for GeO6 is 1.9x larger than that for MgO8. The compression curves fitted to the Rietveld data are also shown in Figure 8aGo. It is noted that polyhedral volume of MgO8 in MgSiO3 pPv is similar to that in MgGeO3 despite significant difference of octahedral volume between these two pPv phases.

Figure 8bGo shows volume occupancy of MgO8 hendecahedra, GeO6 octahedra, and void space. Both Rietveld results and theoretical prediction are consistent with each other within 3{sigma}, indicating that volume proportion of the MgO8 hendecahedral layer significantly decreases with pressure. We do not find any temperature dependence of volume ratio of MgO8 and GeO6 polyhedra within uncertainty, implying similar thermal expansivity for MgO8 and GeO6 polyhedral layers. Polyhedral volume ratio (VAX8/VBX6) is 1.89 for MgGeO3, 2.08 for MgSiO3, 2.02 for NaMgF3, and 2.34 for CaIrO3. volume occupancy of the void space is insensitive to pressure, and is similar among the pPv phases.

Octahedral tilting and distortion
The pPv phase has fewer degrees of freedom than the perovskite phase in terms of both tilting and distortion of the octahedra. Because the octahedra in the perovskite phase share corners along all directions, both tilting and symmetrical distortion of octahedra are allowed in all the directions. On the other hand, since the octahedra in the pPv phase share X2-X2 edges along the a-axis and share corners in c-direction (Fig. 1Go), tilting of octahedra is allowed only around the a-axis, and the X2-X2-X2-X2 plane in an octahedron must have a square or rectangular shape (Fig. 1Go). Therefore, the shape of octahedral layer in ABX3 pPv phase can be fully described by specifying both tilting of octahedra around the a-axis (B-X1-B angle in b-c plane) and the distortion of an octahedron that can be fully described by following three variables: X2-B-X2 angle that represents the deviation of the X2-X2-X2-X2 plane from square to rectangular shape, the angle between X2-X2-X2-X2 plane normal and the B-X1 vector in the b-c plane (angle q), and B-X1/B-X2 lengths ratio (Fig. 1Go). For an ideal octahedron these three variables are 90°, 0°, and 1°, respectively.

As shown below, Rietveld results at 78–109 GPa are consistent with theoretical predictions within 3{sigma}, but considerable data scatter and the limited pressure and temperature range make it difficult to constrain the pressure and temperature dependencies of octahedral tilting and distortion solely from Rietveld results. Therefore, we use theoretical results to compare octahedral tilting and distortion in MgGeO3, MgSiO3, NaMgF3, and CaIrO3 pPv phases.

Tilting of octahedra.
Figure 9Go shows B-X1-B angle in the b-c plane. Theoretical results show a subtle monotonic increase of this angle from 131.3 to 132.7° at 0–300 GPa, indicating that octahedral tilting is insensitive to pressure. Octahedral tilting is ~132° for MgGeO3, ~138° for MgSiO3, and ~140° for both NaMgF3 and CaIrO3, indicating a greater degree of bending in the connection of octahedra in MgGeO3 than the other pPv. The order of octahedral tilting among these pPv can be qualitatively understood by polyhedral volume ratio VAX8/VBX6: A larger (smaller) octahedral tilting makes the length of octahedral layer in c-direction shorter (longer). Since MgGeO3 (CaIrO3) has the smallest (largest) VAX8/VBX6 among these pPv, octahedra in MgGeO3 (CaIrO3) need to be more (less) bent than the other pPv to share the a-c plane with the relatively small (large) hen-decahedral layer.


Figure 9
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  		FIGURE 9. Tilting angle of octahedra around the a-axis (= B-X1-B angle between two octahedra in b-c plane) in ABX3 pPv phase. See Figure 4Go for references.

 
Octahedral distortion.
At first, we survey the overall octahedral distortion using angular variance (Fig. 10aGo), which has been used to show variations in bond angle in an octahedron (Robinson et al. 1971). A regular octahedron has a value of 0°2. Theoretical results show a mild decrease of angular variance from 12 to 6°2 by compression from 50 to 300 GPa, predicting that basic evolution of angular variance with pressure is to decrease distortion over a wide pressure range. However, the pressure change of angular variance is somewhat larger below 50 GPa (from 12 to 18°2 by decompression to 0 GPa), consistent with different compression behavior below 50 GPa. Angular variance of the pPv phase is greater than that of MgSiO3 perovskite phase (1.7°2 at ambient conditions by Ross and Hazen 1990; 2.4°2 at 79.7 GPa, 1681 K by Fiquet et al. 2000), and is ~11°2 for MgGeO3, ~7°2 for both MgSiO3 and NaMgF3, and ~38°2 for CaIrO3, showing that MgSiO3, MgGeO3, and NaMgF3 are similarly less distorted than CaIrO3.


Figure 10
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  		FIGURE 10. Distortion parameters for BX6 octahedron in ABX3 pPv phase. (a) Angular variance, (b) X2-B-X2 angle (where X2-X2 vector is parallel to a direction), (c) angle q (the angle between X2-X2-X2-X2 plane normal and B-X1 vector in b-c plane), and (d) B-X1/B-X2 lengths ratio. X1 and X2 are corner- and edge-shared oxygen in BX6 octahedra, respectively. See Figure 4Go for references.

 
Figure 10bGo shows X2-B-X2 angle in a BX6 octahedron. We define this angle so that the X2-X2 vector is parallel to the a-direction. Theoretical calculations predict a modest decrease of this angle with pressure, which contributes to decrease of octahedral distortion with pressure. Results from Rietveld refinements are consistent with theory within 3{sigma}, but Rietveld results also imply that this angle may change more than predicted (~92° at 90 GPa to ~94° at 80 GPa during decompression). Since faster increase of this angle during decompression can result in rapid expansion of a-axis, we speculate that the greater change of this angle with pressure may explain the anomaly in b/a ratio reported by Kubo et al. (2006). This angle is 92.1° in MgGeO3, 93.4° in MgSiO3, 94.4° in NaMgF3, and 99.2° in CaIrO3, showing significant distortion of CaIrO3. The order of this angle among these pPv can be qualitatively understood by considering the VAO8/VBO6 polyhedral volume ratio following the same logic put forward in the discussion of the octahedral tilting among these pPv.

Figure 10cGo shows pressure dependence of the angle between X2-X2-X2-X2 plane normal and B-X1 vector in b-c plane (angle q). Theory predicts slight monotonic decrease of this angle by compression, which also contributes to decrease of octahedral distortion with pressure. This angle is ~3° in MgSiO3, ~1° in NaMgF3, and ~5° in both MgGeO3 pPv and CaIrO3, showing that NaMgF3 is least distorted.

Figure 10dGo shows the ratio of B-X1 and B-X2 lengths. Theory predicts deviation of this ratio from 1 with pressure with relatively rapid change below 50 GPa and relatively mild change above 50 GPa, contributing to an increase of octahedral distortion with pressure. This ratio is 0.971 in MgGeO3, 0.967 in both MgSiO3 and NaMgF3, and 0.939 in CaIrO3, showing that MgSiO3, NaMgF3, and MgGeO3 are similarly less distorted than CaIrO3.

Comparison of MgGeO3, MgSiO3, NaMgF3, and CaIrO3 pPv
There are some similarities between MgSiO3 and MgGeO3 pPv represented by axial ratios, averaged Mg-O distance, MgO8 polyhedral volume, and degree of octahedral distortion as well as consistency in elastic systematics and generally similar behavior in axial compressibilities reported by Kubo et al. (2006). On the other hand, NaMgF3 pPv phase shows more similarities in structural parameters with MgSiO3 pPv than MgGeO3 pPv, while the similarity between MgSiO3 and CaIrO3 pPv is limited to octahedral tilt angle. Therefore, we infer that NaMgF3 may be a good analog material to MgSiO3 pPv, and among oxide pPv, MgGeO3 is a better analog than CaIrO3. It is noted that Lindsay-Scott et al. (2007) have also inferred that CaIrO3 may not be a good analog for MgSiO3 pPv based on comparison of axial incompressibility ratios for CaIrO3 and MgSiO3 pPv. It should also be noted that MgGeO3 and CaIrO3 pPv phases tend to show qualitatively opposite character with the relationship to MgSiO3 pPv phase, such as axial ratios, atomic positions, ratio of average bond lengths (A-O/B-O), variation of bond lengths in AO8 and BO6 polyhedra (Fig. 7bGo), polyhedral volume ratio, and octahedral tilting angle. This implies that physical properties of MgSiO3 pPv (such as elasticity and rheology) might be intermediate between the properties of the two analog materials.

Reliability of Rietveld refinements to Mbar pressures
Many previous high-pressure studies report only a single optimum Rietveld refinement and this leaves open significant questions about the robustness of the reported results. In addition, there is a large disagreement between the Rietveld results reported here and that of Hirose et al. (2005) on the same material at similar conditions. The existence of such discrepancies, despite each providing an overall reasonable fit to the diffraction data, could raise questions about the reliability of the Rietveld method at extreme conditions.

The present results show that we can obtain consistent Rietveld refinements for experiments involving separate samples with different pressure transmitting media that cover a similar pressure range. Furthermore, these refinement results are in good agreement with our independent first-principles calculations. Consistent results are also obtained for individual and averaged bond lengths between theory and experiment. Finally, the averaged Mg-O and Ge-O bond lengths found here are consistent with previous experimental studies (single-crystal and polycrystalline) of other materials with similar structural elements at high pressures. These consistencies all suggest that Rietveld refinement can provide reasonable structural parameters at pressures approaching 1 Mbar.


   ACKNOWLEDGMENTS
Top
 Abstract
 Introduction
 Methods
 Results and discussion
 Acknowledgments
References cited
 
This work was supported by NSF and the Carnegie-DOE Alliance Center. A.K. benefited greatly from the powder X-ray diffraction workshop held at National Synchrotron Light Source in January 2005 organized by John Parise, Peter Stephens, and Patrick Woodward. We thank Robert J. Cava for his help with sample synthesis and Sean R. Shieh for experimental assistance. Preliminary experiments were conducted at HPCAT, APS, Argonne National Laboratory with the help of Yue Meng. Comments by Leonid Dubrovinsky and anonymous reviewers improved the manuscript. Discussions with Haozhe Liu and Przemyslaw Dera were also helpful. This work was performed at GSECARS, APS, Argonne National Laboratory. GSECARS is supported by the NSF, Earth Sciences (EAR-0622171) and DOE-Geosciences (DE-FG02-94ER14466). Use of the Advanced Photon Source is supported by DOE Office of Basic Energy Sciences under contract no. DE-AC02-06CH11357.


   Footnotes
 
* Present address: Center for Advanced Radiation Sources, University of Chicago, Chicago, Illinois 60637, U.S.A. E-mail: akubo{at}cars.uchicago.edu Back

{dagger} Present address: High Pressure Collaborative Access Team, Carnegie Institution of Washington, Argonne National Laboratory, Argonne, Illinois 60439, U.S.A. Back

MANUSCRIPT HANDLED BY ARTEM OGANOV

NOTE ADDED IN PROOF
Martin et al. (2008) also recently reported structure refinements of MgGeO3 post-perovskite at 84–89 GPa.

1 Deposit item AM-08-032, Supplementary Table 1 and Supplementary Figures 1 and 2 (complete Rietveld results). Deposit items are available two ways: For a paper copy contact the Business Office of the Mineralogical Society of America (see inside front cover of recent issue) for price information. For an electronic copy visit the MSA web site at http://www.minsocam.org, go to the American Mineralogist Contents, find the table of contents for the specific volume/issue wanted, and then click on the deposit link there. Back

Revision received May 9, 2007. MANUSCRIPT ACCEPTED January 30, 2008


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 Results and discussion
 Acknowledgments
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