High-pressure phase transition of olivine-type Mg₂GeO₄ to a metastable forsterite-III type structure and their equations of state

(Mg,Fe)₂SiO₄ olivine is the most abundant mineral in the Earth’s upper mantle. The major seismic discontinuities (410, 520, and 660 km) in the upper mantle and transition zone can be attributed to pressure-induced phase transitions in Mg-rich olivine to β-olivine (wadsleyite), γ-olivine (ringwoodite), and (Mg,Fe)SiO₃ perovskite (bridgmanite, Pv) + (Mg,Fe)O magnesiowüstite (Ringwood 1991). The D″ layer, located in the lowermost ~250 km of the mantle is characterized by a transition from bridgmanite to post-perovskite (pPv; 125 GPa and 2500 K) (Murakami et al. 2004; Oganov and Ono 2004; Tsuchiya et al. 2004). Post-perovskite (Mg,Fe)SiO₃ is expected to be the highest-pressure silicate phase in the Earth. However, in the case of terrestrial super-Earth planets, where the pressure-temperature conditions at the core-mantle boundary can be substantially higher (e.g., >1600 GPa and ~6500 K for a planet with a mass equivalent to that of 10 Earths) (van den Berg et al. 2019), additional transitions are possible. At ~500 GPa, pPv + MgO is expected to recombine into a tetragonal I42d or cubic I43d Mg₂SiO₄ phase (Umemoto et al. 2017; Dutta et al. 2023), followed by a dissociation into the binary oxides at ~3000 GPa. However, all the post-pPv transitions have only been computationally predicted and not observed experimentally because of the extreme pressure-temperature conditions, which are beyond the limits of conventional experimental techniques. As an alternative, silicate analogs like germanates (Ringwood and Seabrook 1963; Umemoto and Wentzcovitch 2019; Dutta et al. 2018, 2022) and fluorides (Grocholski et al. 2010; Dutta et al. 2019) can be used in high-pressure experiments as they undergo similar phase transitions, but at significantly lower pressures, e.g., the Pv-pPv phase transition, which occurs at 125 GPa in MgSiO₃ is observed at 65 GPa in the germanate (Hirose et al. 2005). Additionally, the I42d/I43d phase in Mg₂GeO₄ has been reported at pressures >170 GPa from experiments (Dutta et al. 2022) and computational studies (Umemoto and Wentzcovitch 2019, 2021) in comparison to the theoretical prediction of 500 GPa in the silicate (Umemoto et al. 2017).

There is considerable interest in understanding the 300 K compression behavior of both the silicate and germanate olivine as well. Knowledge of the metastable transitions in olivine can help in understanding mineral phases formed at impact sites (Van de Moortèle et al. 2007). It is potentially useful in inferring phase transitions in laboratory shock experiments, where the short timescale may prevent the formation of stable assemblages (Kim et al. 2021). In Mg₂SiO₄, existing studies have reported pressure-induced amorphization (Guyot and Reynard 1992; Andrault et al. 1995), change in compression mechanism (Rouquette et al. 2008) or a transition to forsterite-II and forsterite-III (Finkelstein et al. 2014; referred to as Fo-II and Fo-III after this) structures. In Mg₂GeO₄, the stable phase at ambient pressure and low temperatures is the spinel-type structure (Ross and Navrotsky 1987). The high-temperature phase, olivine-type Mg₂GeO₄ reverts to the spinel-type phase at 1083 K (Dachille and Roy 1960) and persists on quenching to ambient temperature. On compressing the germanate olivine at room temperatures, it has been reported to stay stable up to 13 GPa, after which new diffraction peaks were observed (Petit et al. 1996) and could not be resolved. Pressure-induced amorphization has been reported above 22–25 GPa (Petit et al. 1996; Nagai et al. 1994). High-pressure Raman spectroscopic studies have observed the appearance of new modes at ~11 GPa, followed by a sharp decrease in its intensity at ~25 GPa (Reynard et al. 1994). In this work, we aim to resolve the post-olivine structure(s) under compression by studying forsterite-type Mg₂GeO₄ to 105 GPa at both room and high-temperature using laser-heated diamond-anvil cells (LH-DAC) and density functional theory (DFT) based computations.

The starting material, olivine-type Mg₂GeO₄ was synthesized by heating high-purity MgO and GeO₂ at 1473 K for 5 days (Ross and Navrotsky 1987; Dutta et al. 2022) and confirmed using Raman spectroscopy and X-ray diffraction. The synthesized sample was ground with 10 wt% gold, which acts as the laser absorber and pressure marker during the high-pressure experiments. The samples were compressed using symmetric diamond-anvil cells with 300–150 μm diameter culets. Rhenium gaskets were drilled to form the sample chamber. The samples were loaded inside the sample cavities (200–80 μm), and gas loaded with Ne to provide a quasi-hydrostatic environment. In situ X-ray diffraction (XRD) was carried out at sectors 13-ID-D and 16-ID-B of the Advanced Photon Source using a monochromatic beam with wavelengths of 0.2952 and 0.4066 Å, respectively. The two-dimensional X-ray images were radially integrated to the one-dimensional patterns using DIOPTAS (Prescher and Prakapenka 2015). Double-sided laser heating was used to produce the high temperatures. Temperatures were increased in small steps of ~100 K and measured using spectroradiometry (Jephcoat and Besedin 1996; Shen et al. 2001). The (111) Au peak was used to calculate the pressures (Fei et al. 2007) using the Birch Murnaghan equation of state (EOS). The lattice parameters were calculated using least-squares refinement of the peak positions (Holland and Redfern 1997) fitted to Voigt line shapes or whole profile Le Bail refinement as implemented in the GSAS-II package (Toby and Von Dreele 2013). The background was fitted with a sixth-order Chebyschev polynomial. The unit-cell dimensions, instrumental and sample broadening parameters were initially refined separately and then together.

All computations were performed using the plane wave implementation of density functional theory through the Quantum Espresso package (Giannozzi et al. 2009). The generalized gradient approximation (GGA-PBE, Perdew et al. 1996) was used to treat the exchange and correlation functional. We have used a plane wave basis set with a cutoff of 40 Ry and a Monkhorst-Pack (Monkhorst and Pack 1976) k-point grid of 6 × 6 × 6 for all the considered structures. Ultrasoft pseudopotentials (Vanderbilt 1990) were used to describe the electron-ion interactions. The geometry optimizations were carried out using the BFGS algorithm (Broyden 1970) by relaxing the lattice parameters and atomic positions at each pressure step. The structural relaxations were considered complete when the forces on atoms were less than 1 × 10^–4 Ry/Bohr and total energies converged to 1 × 10^–6 Ry.

In three separate experimental runs, the germanate olivine samples were compressed to peak pressures of 26, 54, and 105 GPa at room temperature (Fig. 1). The diffraction patterns up to 30 GPa can be indexed using the ambient-pressure olivine structure, suggesting a metastable persistence. As an example, Online Materials¹ Table S1 shows the observed and calculated d-spacings for forsterite-type Mg₂GeO₄ at 14.6 GPa. The difference between the two values is <0.002 Å, suggesting a good fit of the olivine structure to the observed pattern. This is also reflected in the whole profile Le Bail refinement of the measured pattern at 26 GPa (Fig. 2). In contrast to previous studies (Nagai et al. 1994; Petit et al. 1996), we did not find any evidence for amorphization. The lattice parameters of olivine-type Mg₂GeO₄ at 26 GPa are a = 4.7573 Å, b = 9.6574 Å, and c = 5.7064 Å. Figure 3 and Online Materials¹ Table S2 show the change in the unit-cell dimensions as a function of pressure. Although our work extends to higher pressures, it agrees with existing experimental studies, especially at lower pressures. At higher pressure, the discrepancy possibly arises from the non-hydrostatic conditions (Klotz et al. 2009) inside the DAC in the previous work. The linear compressibilities (×10^–3 GPa^–1) of the axes from the experimental (computational) results are β_a = 1.21 (1.15), β_b = 2.29 (2.35), and β_c = 1.98 (1.96). Despite the GGA’s tendency to overestimate the lattice parameters, the remarkable concurrence of experimental and computed linear compressibilities emphasizes their strong agreement. The order of the axial compressibilities, i.e., β_b > β_c > β_a, also agrees with that of Mg₂SiO₄ forsterite (Zhang 1998; Finkelstein et al. 2014).

Upon further compression to 40 GPa (Fig. 1), new diffraction peaks were observed, which were retained up to the peak pressure of 105 GPa. To understand the structure of the new phase, we computed the enthalpies (Fig. 4) of spinel and post-spinel structures reported in Mg₂SiO₄ and its analogous systems within the germanate framework. The considered structures are: Fo-II, Fo-III, Fo-IV, CaTi₂O₄-, CaFe₂O₄-, and Ca₂IrO₄-type Mg₂GeO₄ (Decker and Kasper 1957; Babel et al. 1966; Yamanaka et al. 2013; Finkelstein et al. 2014; Bouibes and Zaoui 2020; Dutta et al. 2022); Pv-MgGeO₃ (Leinenweber et al. 1994) + B1-MgO and pPv-MgGeO₃ (Hirose et al. 2005) + B1-MgO. It can be seen the Pv-MgGeO₃ + MgO assemblage becomes more stable (lower enthalpy) with respect to the forsterite-type Mg₂GeO₄ structure at ~12 GPa, which then transforms into the pPv-MgGeO₃ + MgO assemblage at ~50 GPa. Taking into account the tendency of the GGA-PBE functional to underestimate transition pressures, these results can be viewed as reasonably consistent with experimental findings (Liu 1977; Hirose et al. 2005). The XRD patterns at P > 40 GPa are not consistent with any of these phases, suggesting the presence of a metastable phase. This can be attributed to the experimental conditions being at room temperature, which creates a kinetic barrier that prevents the transition to the more stable assemblage. Besides Pv-MgGeO₃ + MgO and pPv-MgGeO₃ + MgO, the candidate phases with low enthalpies are the Fo-II type, Fo-III type, and CaTi₂O₄-type Mg₂GeO₄ structures. Figure 5 compares the observed XRD pattern at 61 GPa with the simulated diffraction pattern of these three phases. In agreement with a previous theoretical study (Bouibes and Zaoui 2020) on Mg₂SiO₄, the triclinic Fo-II structure (Finkelstein et al. 2014) was neither energetically favored computationally, nor did it match the XRD data. The closest match to the observed patterns was the ordered Pmma CaTi₂O₄-type phase and the Fo-III type phase. Although the simulated patterns for the two are similar, the Fo-III type structure (CIF¹, optimized DFT structure at 60 GPa) is a better match (fewer peaks) and has a comparatively lower enthalpy.

Post-spinel (e.g., CaMn₂O₄-, CaFe₂O₄-, and CaTi₂O₄-type) structures (Yamanaka et al. 2008) generally feature chains of octahedra that share edges and corners, forming channels that align parallel to the c-axis. The Fo-III structure (Fig. 6) is analogous to an inverse spinel structure. It is related to the non-centrosymmetric variant of the Cmcm CaTi₂O₄ post-spinel structure (Yamanaka et al. 2013) in which half of Mg atoms are situated in the larger trigonal prismatic site (Mg2), while the other half occupy the octahedral (Mg1) site (Finkelstein et al. 2014). This is substantially different from the olivine structure, where both the Mg1 and Mg2 sites are octahedral, with one being more distorted than the other. The Fo-III type structure also marks an increase in the Ge-coordination from 4 (as in olivine) to 6, providing a pathway to the stable six-coordinated Pv and pPv structures. The structural parameters of Fo- and Fo-III type Mg₂GeO₄ are listed in Table 1. Figure 7 shows a Le Bail refinement of the measured diffraction pattern of Mg₂GeO₄ at 74 GPa. The difference between the calculated and observed d-spacings is less than <0.006 Å (Online Materials¹ Table S3, 68 GPa), again suggesting a good fit of the measured diffraction patterns with the Fo-III structure. Figure 8 and Online Materials¹ Table S4 show the variation in lattice parameters of Fo-III type Mg₂GeO₄ with increasing pressure. The experimental a, b, and c parameters are found to decrease by 2.9, 2.7, and 2.7%, respectively, over the pressure range (40.4–73.8 GPa) considered. In agreement with the experiments, the theoretical axial parameters decrease by 3.3, 3.7, and 3.3%, respectively, between 40 and 80 GPa. No further transitions were observed up to the peak pressure of 105 GPa at room temperature.

The pressure-volume data for both the Fo- and Fo-III type Mg₂GeO₄ phases (Fig. 9) were fitted to an isothermal third-order Birch Murnaghan (BM) equation of state. Table 2 presents the EOS parameters for these phases and includes a comparison with the existing studies on the same structures in Mg₂GeO₄ (Weidner and Hamaya 1983; Nagai et al. 1994; Petit et al. 1996) and Mg₂SiO₄ (Andrault et al. 1995; Downs et al. 1996; Zhang 1998; Finkelstein et al. 2014; Zhang et al. 2019; Bouibes and Zaoui 2020). For the germanate olivine, the EOS parameters for the computed data are V₀ = 316.8(3) ų, K₀ = 112.2(13) GPa, and $K0′$ = 3.86(5), where V₀, K₀, and $K0′$ are the unit-cell volume, bulk modulus, and its pressure derivative at ambient pressure, respectively. In case of the experimentally obtained values, $K0′$ was fixed to the theoretical value of 3.86. This yielded V₀ = 305.1(3) and K₀ = 124.6(14) GPa, which are in excellent agreement with existing ultrasonic (K₀ = 120 GPa) (Soga 1971) and Brillouin spectroscopic measurements (K₀ = 120 GPa) (Weidner and Hamaya 1983). However, the obtained bulk modulus is significantly less than that obtained from previous DAC studies [K₀ = 166(15) at fixed $K0′$ = 4] (Petit et al. 1996). The discrepancy likely stems from the limited pressure range (0–10 GPa in the earlier study compared to pressures above 30 GPa in this study). Additionally, the previous work used silicone oil, whereas neon was used in the current study. Silicone oil is known to offer limited hydrostaticity at high pressures (Klotz et al. 2009). The EOS parameters are also in good agreement with those of silicate olivine [K₀ = 130.0(9) GPa and $K0′$ = 4.12(7)] (Finkelstein et al. 2014). The transition from forsterite- to Fo-III type Mg₂GeO₄ is expected to have a substantial volume change of 9.53% at 35 GPa, which agrees with its silicate counterpart (8.3% at 58 GPa). In the case of Fo-III type Mg₂GeO₄, the EOS parameters for the theoretical data are V₀ = 271.8(9) ų, K₀ = 162.9(5) GPa, and $K0′$ = 4.19(1). The fit to the experimental data yielded V₀ = 263.5(15) ų, K₀ = 175(7) GPa, with $K0′$ fixed to the computed value (4.19). These values are consistent with the theoretical EOS parameters for Mg₂SiO₄ (V₀ = 247.4517 ų, K₀ = 197.12 GPa, and $K0′$ = 3.4) (Bouibes and Zaoui 2020).

On laser-heating the Mg₂GeO₄ sample to 2331 ± 148 K for 2–5 min at 26 GPa, we observed new diffraction peaks that could not be explained using forsterite-, spinel-, or forsterite-III type Mg₂GeO₄ structures. The XRD peaks were instead consistent with an assemblage of Pv-MgGeO₃ + B1-MgO. This is in agreement with our computations, which predict a transition from the olivine-type structure to Pv-MgGeO₃ + MgO at 12 GPa and existing experimental studies with an olivine-type starting material (26 GPa) (Liu 1977) as well as a MgGeO₃ pyroxene starting material (25 GPa) (Runge et al. 2006). The sample was further compressed to 54 GPa at room temperature, followed by heating a fresh spot to a peak temperature of 2463 ± 112 K in small steps of 200 K. The observed diffraction pattern could still be indexed using MgGeO₃-Pv + B1-MgO. Figure 10 shows a Le Bail refinement of the XRD pattern at 65 GPa. The lattice parameters obtained from the refinement (a = 4.584 Å, b = 4.858 Å, c = 6.727 Å) are in excellent agreement with previous studies (a = 4.587 Å, b = 4.860 Å, c = 6.721 Å at 65.7 GPa, Runge et al. 2006). In the experiment where a fresh sample was compressed to 105 GPa at room temperature and subsequently heated to 2280 ± 46 K, the diffraction pattern could be explained using a mixture of CaIrO₃-type post-perovskite MgGeO₃ + B1-MgO (Fig. 11). This is consistent with the reported Pv to pPv transition pressure of 63 GPa with an orthoenstatite starting material (Hirose et al. 2005). The lattice parameters obtained from the Le Bail refinement at 110 GPa, 2300 K (a = 2.567 Å, b = 8.301 Å, c = 6.351 Å) are in agreement with existing experimental work (a = 2.575 Å, b = 8.324 Å, c = 6.349 Å at 107 GPa and 300 K) (Kubo et al. 2006).

Knowledge of metastable phases is important for understanding the mineralogy of planetary impact sites and meteorites, e.g., martian meteorites NWA 2737 and NWA 1950 (Van de Moortèle et al. 2007). The ultrafast timescales of dynamic compression experiments are often not enough to stabilize the equilibrium stable structures, leading to the formation of metastable phases. The metastable olivine wedge hypothesis (Soga 1971; Däßler and Yuen 1996) has commonly been used to explain the stagnation of subducting slabs and the origin of deep-focus earthquakes. The P-T conditions in the cold subducting slabs may also stabilize metastable phases like Fo-III and thereby contribute to the high seismic velocities observed near the 660 km discontinuity (Zhang et al. 2019). The occurrence of the same metastable phases in the germanate analog at significantly lower pressures compared to the silicate (for example, the Fo-III type structure appears at pressures >30 GPa in Mg₂GeO₄ vs. 58 GPa in Mg₂SiO₄) enables the use of a broader range of experimental techniques (e.g., Burnley 1990; Burnley et al. 1991). This allows for experiments under more controlled conditions and with larger sample sizes.

Recent laser-based shock compression experiments (Kim et al. 2021) on forsterite Mg₂SiO₄ have shown the presence of a metastable Fo-III phase instead of the stable assemblage i.e., bridgmanite + MgO at pressures >33 GPa. Mg₂GeO₄ olivine is a widely used analog for forsterite Mg₂SiO₄ and is expected to show similar phase transitions but at lower pressures. The high-pressure data on the germanate olivine is limited to pressures <35 GPa and suggests a pressure-induced amorphization under compression at room temperature (Nagai et al. 1994; Petit et al. 1996). Using synchrotron X-ray diffraction measurements and density functional computations, we have shown that olivine-type Mg₂GeO₄ persists metastably up to 30 GPa. It then undergoes a pressure-induced phase transition to a metastable forsterite-III type structure. The Fo-III type phase stays stable up to the peak pressure of 105 GPa (at 300 K), with no evidence of the Fo-II (Finkelstein et al. 2014) type phase seen in the silicate or pressure-induced amorphization. We have also obtained the equation of state parameters of both the Fo- and Fo-III type phases. Although our bulk modulus value for forsterite-type Mg₂GeO₄ (K₀ = 124.6 GPa) is lower than previous high-pressure studies (e.g., K₀ = 166 GPa) (Petit et al. 1996), it is in excellent agreement with ultrasonic (K₀ = 120 GPa) (Soga 1971) and Brillouin spectroscopic measurements (K₀ = 120 GPa) (Weidner and Hamaya 1983). The enhanced quality of our calculated EOS parameters can be attributed to the utilization of a wider data range and the incorporation of a more hydrostatic pressure medium (Ne). To the best of our knowledge, there is no available data for Fo-III type Mg₂GeO₄.

The Fo-III type phase has now been reported in laser (~10 ns timescale) (Kim et al. 2021) and gas gun (approximately hundreds of nanoseconds) (Newman et al. 2018) based shock compression studies as well as static compression experiments in both silicates (Finkelstein et al. 2014) and germanates (this study). This suggests it may be an important transition pathway to stable higher-coordination structures at higher temperatures. On laser-heating at 26 and 54 GPa, a partial dissociation into perovskite MgGeO₃ + B1-MgO was observed. At 105 GPa, post-perovskite MgGeO₃ was observed instead of perovskite. The presence of both the perovskite and post-perovskite structures at high pressures and temperatures in Mg₂GeO₄ makes it an excellent low-pressure analog of Mg₂SiO₄.

We thank F. Miozzi and J. Yang for their help with experiments. We acknowledge the support of GeoSoilEnviroCARS (Sector 13), which is supported by the NSF-Earth Sciences (EAR-1128799) and the U.S. Department of Energy (DOE), Geosciences (DE-FG02-94ER14466). Portions of this work were performed at HPCAT (Sector 16), Advanced Photon Source (APS), Argonne National Laboratory. HPCAT operations are supported by DOE-NNSA’s Office of Experimental Sciences. This research used resources of the Advanced Photon Source; a DOE Office of Science User Facility operated by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. R.D. is grateful to SERB-Department of Science and Technology, India, for financial support. R.V.D. thanks CSIR-India for providing the research fellowship. R.D. and R.E.C. gratefully acknowledge the Gauss Centre for Supercomputing e.V. (https://www.gauss-centre.eu/) for funding this project by providing computing time on the GCS Supercomputer SuperMUCNG at Leibniz Supercomputing Centre (LRZ, http://www.lrz.de).