Computer simulation of the MgSiO3 polymorphs

Six polymorphs of MgSiO₃ have been studied using molecular dynamic (MD) simulation techniques, based on the empirical potential (MAMOK), which is composed of terms to describe pairwise additive Coulomb, van der Waals attraction, and repulsive interactions. Crystal structures, bulk moduli, volume thermal expansivities, and enthalpies were simulated for the known MgSiO₃ polymorphs; orthoenstatite, clinoenstatite, protoenstatite, garnet, ilmenite, and perovskite. The simulated values compare very well with the available experimental data, and the results are quite satisfactory in view of the diversity of the crystal structures of the six polymorphs, the wide range of simulated properties, and the simplicity of the MAMOK potential. MD simulation was further successfully used to study the possibile existence of a post-protoenstatite phase at high temperature, and a C2/c phase at high pressure, both phases being suggested or inferred previously from experimental works.     

CdGeO3-phase transformations at high pressure and temperature and structural refinement of the perovskite polymorph

Four polymorphs of CdGeO₃ were synthesized at high temperatures (600 ～ 1200° C) and high pressures up to 12 GPa. The pyroxenoid phase synthesized under ambient pressure transforms to garnet, ilmenite and perovskite phases with increasing pressure. The phase boundary of ilmenite-perovskite had a slightly negative P-T slope in contrast to the positive P-T slopes of the pyroxenoid-garnet and garnet-ilmenite transition boundaries. CdGeO₃III has the ilmenite structure with hexagonal lattice parameters, a=5.098 Å and c =14.883 Å. The c/a ratio of 2.919 is greater than that of any other ilmenite. CdGeO₃IV has a distorted perovskite structure with orthorhombic lattice parameters a = 5.209 Å, b = 5.253 Å and c = 7.434 Å. Synthesis of a CdGeO₃IV single crystal was successful and structural refinement revealed that the structure is isomorphic to GdFeO₃ with the space group Pbnm. The increase of density with the CdGeO₃III→CdGeO₃IV transformation is the largest (9.8%) for any ilmenite-perovskite transition studied so far.     

Determination of the phase boundary between ilmenite and perovskite in MgSiO3 by in situ X-ray diffraction and quench experiments

Determination of the phase boundary between ilmenite and perovskite structures in MgSiO₃ has been made at pressures between 18 and 24 GPa and temperatures up to 2000 °C by in situ X-ray diffraction measurements using synchrotron radiation and quench experiments. It was difficult to precisely define the phase boundary by the present in situ X-ray observations, because the grain growth of ilmenite hindered the estimation of relative abundances of these phases. Moreover, the slow reaction kinetics between these two phases made it difficult to determine the phase boundary by changing pressure and temperature conditions during in situ X-ray diffraction measurements. Nevertheless, the phase boundary was well constrained by quench method with a pressure calibration based on the spinel-postspinel boundary of Mg₂SiO₄ determined by in situ X-ray experiments. This yielded the ilmenite-perovskite phase boundary of P (GPa) = 25.0 (±0.2) – 0.003 T (°C) for a temperature range of 1200–1800 °C, which is generally consistent with the results of the present in situ X-ray diffraction measurements within the uncertainty of ∼±0.5 GPa. The phase boundary thus determined between ilmenite and perovskite phases in MgSiO₃ is slightly (∼0.5 GPa) lower than that of the spinel-postspinel transformation in Mg₂SiO₄. Received: 19 May 1999 / Accepted: 21 March 2000     

High-pressure transformations of ilmenite to perovskite, and lithium niobate to perovskite in zinc germanate

In-situ X-ray powder diffraction measurements conducted under high pressure confirmed the existence of an unquenchable orthorhombic perovskite in ZnGeO₃. ZnGeO₃ ilmenite transformed into perovskite at 30.0 GPa and 1300±150 K in a laser-heated diamond anvil cell. After releasing the pressure, the lithium niobate phase was recovered as a quenched product. The perovskite was also obtained by recompression of the lithium niobate phase at room temperature under a lower pressure than the equilibrium phase boundary of the ilmenite–perovskite transition. Bulk moduli of ilmenite, lithium niobate, and perovskite phases were calculated on the basis of the refined X-ray diffraction data. The structural relations among these phases are considered in terms of the rotation of GeO₆ octahedra. A slight rotation of the octahedra plays an important role for the transition from lithium niobate to perovskite at ambient temperature. On the other hand, high temperature is needed to rearrange GeO₆ octahedra in the ilmenite–perovskite transition. The correlation of quenchability with rotation angle of GeO₆ octahedra for other germanate perovskites is also discussed.     

The extrapolation of elastic moduli to high pressure and temperature

The elastic constants of a crystal under stress, defined as the second derivative of the crystal free energy with respect to strain, require a correction related to the static pressure at non-zero pressures. The corrections required for the elastic constants calculated by the free energy minimisation code PARAPOCS are described and tested by comparison with the elastic constants calculated numerically by applying small stresses in the appropriate orientations to simulated crystals of fluorite, forsterite, α-quartz and albite. The corrected elastic constants are then used to investigate the extrapolation of the bulk and shear moduli (and hence also the seismic wave velocities V _p and V _s) of β-spinel and forsterite to upper mantle pressures. A Murnaghan equation, thirdorder Eulerian finite strain equation, second order polynomial equation and a logistic equation were all fitted to the simulated bulk and shear moduli between 0 and 3 GPa pressure. The parameters derived for these equations are used to extrapolate the bulk and shear moduli to 14 GPa and the results are compared to the simulated high pressure moduli. Over this pressure range, the second order polynomial provides the best extrapolation of the bulk modulus, but the use of the logistic equation results in the best extrapolation of the shear modulus.     

The single crystal elastic moduli of neighborite

The adiabatic elastic moduli of a single crystal of Neighborite (NaMgF ₃ perovskite) have been measured at ambient conditions using Brillouin spectroscopy. The adiabatic aggregate (Voight-Reuss-Hill) bulk modulus is K = 75.6 GPa, and shear modulus is = 46.7 GPa. The experimental results show the ratio of linear compressibilities _b / _a = 0.80 for neighborite. These ratios reflect the different amounts of tilting freedom of the octahedral framework along each lattice axis of the perovskite structure. It is understood that the elastic compliance S _ij of the crystal can directly sense the behavior of the octahedral tilting in the structural distortion of NaMgF₃ perovskite. The octahedral tilting angles are considered to be the order parameters of the ferroelastic phase transition in the perovskite structure. Single crystal elasticity data provide a basis for understanding the role of octahedral tilting in the ferroelasticity of perovskite. Together with high pressure compressional data, one can thus elucidate the relationship between crystal structure and physical properties of perovskite. A detailed assessment indicates that the dominant compression mechanism for NaMgF₃ perovskite is shortening of the octahedral [MgF] bond, which is also true for orthorhombically distorted MgSiO₃ perovskite.     

Quantum-mechanical Hartree-Fock study of calcite (CaCO3) at variable pressure,and comparison with magnesite (MgCO3)

The static crystal energy of calcite and its structure configuration as functions of pressure were determined by ab initio all-electron periodic Hartree-Fock calculations (CRYSTAL code). Ca, O and C atoms were represented by 22, 18 and 14 atomic orbitals, respectively, in form of contracted Gaussian-type functions. Comparison between theoretical and experimental data was performed for binding energy, equilibrium unit-cell and bond lengths, bulk modulus and C ₃₃ and C ₁₁ + C ₁₂ elastic constants, and vibrational frequency of the symmetrical C-O stretching mode. The agreement is generally satisfactory. A larger compressibility is observed for structural parameters of calcite than for those of magnesite coming from a similar calculation. The Ca-O and C-O chemical bonding was characterized by electron density maps and by Mulliken atomic charges; these are discussed and compared to values determined by empirical fitting of Born-type interatomic potentials.     

Calculation of the equation of state and elastic moduli of MgO using molecular orbital theory

The success of molecular orbital theory in calculating crystal properties such as bond lengths and atomic force constants has been well documented in the literature. Calculations can be extended to crystals under simulated compression and strain to determine elastic moduli and their pressure derivatives. Comparison of the molecular orbital results with both experimental values and results obtained by calculations such as the potential induced breathing model provides insight into the nature of chemical bonding in MgO. In this study, several molecular clusters were investigated as possible models for MgO; the cluster Mg₄O₄H₂₄ was chosen as the best model. Molecular energies were calculated with respect to bond length for both compression and expansion based on clusters that had been optimized for minimum energy. The resulting energy-volume curve was fitted to a recently derived equation of state (Brown, in preparation) to derive the values of K ₀ and dK₀/dP and the individual elastic moduli and their pressure derivatives were calculated by applying strain to the molecular cluster at both zero and elevated pressures. Agreement between theory and experiment varies between parameters, but the overall trend is encouraging. Since the molecular orbital model includes only short range interactions, its ability to approximating model the elastic moduli of MgO suggests a strong contribution to the elastic energy from short range interactions.     

An interatomic potential model for carbonates allowing for polarization effects

An empirical model for investigating the behaviour of CaCO₃ polymorphs incorporating a shell model for oxygen has been created. The model was constructed by fitting to: the structure of aragonite and calcite; their elastic, static and high-frequency dielectric constants; phonon frequencies at the wave vectors [&frac; 0 2] and [0 0 0] of calcite; and vibrational frequencies of the carbonate deformation modes of calcite. The high-pressure phase transition between calcite I and II is observed. The potentials for the CO₃ group were transferred to other carbonates, by refitting the interaction between CO₃ and the cation to both the experimental structures and their bulk modulus, creating a set of potentials for calculating the properties of a wide range of carbonate materials. Defect energies of substitutional cation defects were analyzed for calcite and aragonite phases. The results were rationalized by studying the structure of calcite and aragonite in greater detail.     

Oxide perovskites: pressure derivatives of the bulk and shear moduli

The pressure derivatives of elastic moduli (∂M/∂P; M=K_S and G) for a suite of polycrystalline oxide perovskites (2 titanates, 1 stannate and 2 aluminates) have been measured up to 3 GPa using the ultrasonic interferometry method combined with a buffer rod technique. Two empirical systematic relationships (∂G/∂P vs K_S/G and ∂K_S/∂P vs K_S (/ρ)^1/3) have been used to investigate the elasticity systematics of this suite of perovskites and to estimate ∂M/∂P of MgSiO₃ perovskite. The pressure derivatives ∂G/∂P and ∂K_S/∂P for this suite of perovskites scatter between well-defined linear trends for the rutile, rocksalt and spinel structures. The more diffuse trends observed for the perovskites might reflect greater flexibility in the response of its corner-connected octahedral framework structure to changing pressure. The pressure derivatives of the elastic moduli for MgSiO₃ perovskite estimated by the “perovskite bands” are ∂G/∂P=1.6–2.2 and ∂K_S/∂P=3.9–4.2. Received: 13 November 1997 / Revised, accepted: 31 August 1998     

Heat capacity calculations for Al2O3 corundum and MgSiO3 ilmenite

We have used Kieffer's vibrational model to calculate heat capacities and entropies for Al₂O₃ corundum and MgSiO₃ ilmenite, using available vibrational and elastic data for these phases. The calculated heat capacity for corundum is within 1–2 percent of the experimental values between 100 K and 1,800 K, while that for MgSiO₃ ilmenite is within 1–2 percent of the experimental data between 350 K and 500 K. We have calculated the heat capacity for MgSiO₃ ilmenite from 50 K to 1,800 K, which extends the range of available heat capacity data for this phase. The results of this calculation suggest that there may be differences in the vibrational properties of corundum and MgSiO₃ ilmenite. Finally, we have used the results of our calculation to obtain a transition entropy of near -18.8 J/mol.K for the MgSiO₃ pyroxene-ilmenite reaction.     

Thermodynamic data for phases in the FeO-MgO-SiO2 system and phase relations in the mantle transition zone

Based on the available experimental data on phase equilibria in the FeO -MgO -SiO₂ system the mixing properties of the solid solutions (olivine, β- and γ-spinel, pyroxene, majorite, ilmenite and perovskite and magnesiowustite), the enthalpies of FeO and fictive FeSiO₃ phases with ilmenite and majorite structures have been assessed. The entropies, temperature dependance of heat capacities for fictive FeSiO₃ end-members were estimated from structural analogies. The calculated phase diagrams for Mg₂SiO₄-Fe₂SiO₄ and MgSiO₃ — FeSiO₃ systems at pressures up to 30 GPa and temperatures between 1000 and 2100 K are quite consistent with the available experimental determinations except for the fine features of the phase diagram at 2073 K.     

Calorimetric study of high pressure polymorphism in FeTiO3: Stability of the perovskite phase

A calorimetric study of the ilmenite and lithium niobate polymorphs of FeTiO₃ was undertaken to assess the high-pressure stabilities of these phases. Ilmenite is known to be the stable phase at ambient pressure, but the lithium niobate form may be a quench phase from a perovskite form which has been previously observed in situ at high pressure.In this study, the lithium niobate phase of FeTiO₃ was synthesized from an ilmenite starting material at 15– 16 GPa and 1473 K, using a uniaxial split-sphere high-pressure apparatus (USSA 2000). The energetics of the ilmenite to lithium niobate transformation were investigated through transposed-temperature drop calorimetry. The heat of back-transformation of lithium niobate to ilmenite was measured by dropping the sample in argon from ambient conditions to a temperature where the transformation occurs spontaneously. In drops made at 977 K, an intermediate x-ray amorphous phase was encountered. At 1273 K, the transformation went to completion. A value of -13.5±1.2 kJ/mol was obtained for the heat of transformation.     

Phase boundary between ilmenite and perovskite structures in MnGeO3 determined by in situ X-ray diffraction measurements

In situ X-ray observations of the phase transition from ilmenite to perovskite structure in MnGeO₃ were carried out in a Kawai-type high-pressure apparatus interfaced with synchrotron radiation. The phase boundary between the ilmenite and perovskite structures in the temperature range of 700–1,400°C was determined to be P (GPa) = 16.5(±0.6) − 0.0034(±0.0006)T (°C) based on Anderson’s gold pressure scale. The Clapeyron slope, dP/dT, determined in this study is consistent with that for the transition boundary between the ilmenite and the perovskite structure in MgSiO₃.     

Calorimetric study of high-pressure phase transitions among the CdGeO3 polymorphs (pyroxenoid,garnet, ilmenite,and perovskite structures)

At high pressures, CdGeO₃ pyroxenoid transforms to garnet, then to ilmenite, and finally to perovskite. Enthalpies of transition among the four phases were measured by high temperature calorimetry. The entropies of transition and slopes of the boundaries were calculated using the measured enthalpies and free energies calculated from the phase equilibrium data. Pyroxenoid and garnet are very similar energetically. However garnet is a high pressure phase because of its lower entropy and smaller volume. The pyroxenoid-garnet transition has a small positiveP-T slope. Ilmenite is intermediate in enthalpy between garnet and perovskite, but is lower in entropy than both phases. Therefore the garnet-ilmenite transition has a positivedP/dT, while a negativedP/dT is calculated for the ilmenite-perovskite transition. The thermochemical data for the CdGeO₃ phases are generally consistent with the observed high pressure phase relations. The high entropy of perovskite relative to ilmenite, observed in several ABO₃ comounds including CdGeO₃, is related to the structural features of perovskite, in which relatively small divalent cations occupy the large sites of 8–12 fold coordination. The thermochemistry of the CdGeO₃ polymorphs shows several similarities to that of the CaGeO₃ system.     

Interatomic potentials for CaCO3 polymorphs (calcite and aragonite), fitted to elastic and vibrational data

Calcite and aragonite have been modeled using rigid-ion, two-body Born-type potentials, supplemented by O-C-O angular terms inside the CO₃ groups. A shell model has also been developed for calcite. Atomic charges, repulsive parameters and force constants have been optimized to reproduce the equilibrium crystal structures, the elastic constants and the Raman and infrared vibrational frequencies. The rigid-ion potential RIM (atomic charges:z _O= -0.995e,z _C = 0.985e,z _Ca = 2.0e) fitted to calcite properties is able to account for those of aragonite as well. Experimental unit-cell edges, elastic constants, internal and lattice frequencies are reproduced with average relative errors of 2.1, 5.5, 2.4, 15.1% for calcite and of 0.2, 19.4, 2.5, 11.8% for aragonite, respectively. The RIM potential is suitable for thermodynamic and phase diagram simulations in the CaCO₃ system, and is discussed and compared to other potentials.     

High-temperature properties of geikielite (MgTiO3-ilmenite) from high-temperature high-pressure Raman spectroscopy — Some implications for MgSiO3-ilmenite

The Raman spectra of geikielite (MgTiO₃-ilmenite) have been recorded at high pressure (up to 27 GPa) and at high temperature (up to 1820 K). No phase transitions could be evidenced in both cases. In particular, no cation disordering can be evidenced from the high temperature spectra. The observed Raman wavenumber shifts with pressure and with temperature are used to calculate the intrinsic mode anharmonic parameters. The low absolute values of these parameters indicate that geikielite has a nearly quasi-harmonic behaviour, at least to moderate temperatures. However, systematics and the temperature evolution of Raman linewidths suggest that the absolute values of the anharmonic parameters increase at high temperatures. Anharmonic corrections are applied to Kieffer modelling of the constant volume heat capacity of geikielite. They amount to +4 J · mol^-1 · K^-1 at 1800 K, i.e. are much lower than those inferred for, for instance, olivine and garnet structures. These results are used to discuss some implications on the phase relations of the high-pressure MgSiO₃-ilmenite, and the factors controlling the occurence of order-disorder transitions in ilmenite structures.     

The transition of pyrope to perovskite

The stability field of Mg₃Al₂Si₃O₁₂-pyrope was examined for the first time under hydrostatic pressure conditions in a CO₂-laser heated diamond cell in the pressure range 21–30 GPa between 2300 and 3200 K. The phases were characterized using Raman and fluorescence spectroscopy. With increasing pressure pyrope transforms to an ilmenite phase above ∼21.5 GPa, to perovskite plus ilmenite above ∼24 GPa, and to perovskite above 29 GPa. The pressures of the first occurrence of perovskite in this study are about 2 GPa above the corresponding phase boundary between end-member MgSiO₃-ilmenite and perovskite. A small amount of Al₂O₃ coexists with perovskite up to 43 GPa, as evident from fluorescence spectra resembling those of ruby, but above 43 GPa the entire Al₂O₃ content of the pyrope starting material is accommodated in the perovskite structure. Received: 6 March 1997 / Revised, accepted: 23 July 1997     

The nonlinear elasticity and related properties of anhydrous and hydrous γ-Mg2SiO4: a theoretical study

The second-order elastic constants up to 30 GPa, which encompass the stability field of the spinel forms, their pressure derivatives and the third-order elastic constants of both hydrous and anhydrous -Mg₂SiO₄ have been obtained theoretically. A combination of deformation theory and finite strain elasticity theory has been employed to arrive at the expressions for second-order and third-order elastic constants from the strain energy of the lattice. The strain energy is calculated by taking into account the interactions up to second nearest neighbours in the -Mg₂SiO₄ lattice. This is then compared with the strain-dependent lattice energy from continuum model approximation to obtain the expression of elastic constants. The second-order elastic constants C_ij compare favourably with the measurements in the case of anhydrous as well as hydrous -Mg₂SiO₄ and with other calculations on the anhydrous phase. All the third-order elastic constants of both the compounds are negative. The third-order elastic constant C₁₄₄(–52.41 and –45.07 GPa for anhydrous and hydrous -Mg₂SiO₄, respectively) representing the anisotropy of shear mode has a smaller value than C₁₁₁ (–2443.94 and –2101.25 GPa for anhydrous and hydrous phases, respectively), which corresponds to the longitudinal mode. The pressure-induced variations in the longitudinal elastic constants (i.e.,dC₁₁/dp) are relatively large (4.08 and 4.09 for dry and hydrous ringwoodite, respectively) compared with those for the shear (0.22 and 0.32 for dry and hydrous ringwoodite, respectively) and off-diagonal constants (1.40 and 1.41 for dry and hydrous ringwoodite, respectively). The variation of the shear moduli C_s and anisotropy factor A with pressure have also been studied. The average value of elastic anisotropy is 0.835 in the case of anhydrous -Mg₂SiO₄ and 0.830 in the hydrous phase. The reversal of sign of the Cauchy pressure C₁₂ – C₄₄, which describes the angular character of atomic bonding in metals and other compounds, at around 21 GPa for both the compounds may be a precursor to the phase transition from ringwoodite to periclase and perovskite at an elevated temperature. The aggregate elastic properties like the adiabatic bulk modulus K (175.4 and 150.2 GPa for anhydrous and hydrous phases, respectively), and the isotropic compressional (P) and shear (S) wave velocities were calculated and the mode Grüneisen Parameters (GPs) of the acoustic waves were determined based on the quasi-harmonic approximation. The low temperature limit of both hydrous and anhydrous phases of -Mg₂SiO₄ are positive (1.69 and 1.78, respectively, for hydrous and anhydrous phases) and hence we expect the thermal expansion to be positive down to absolute zero. The Anderson–Grüneisen parameter obtained for hydrous as well as anhydrous phases of -Mg₂SiO₄ from the second-order and third-order elastic constants are 2.30 and 2.29, respectively.