Elasticity of the pyrope (Mg3Al2Si3O12)-majorite (MgSiO3) garnets solid solution

Dense isotropic polycrystalline specimens of majorite-rich garnets (Py₁₀₀, Py₆₂Mj₃₈, Py₅₀Mj₅₀, Py₂₁Mj₇₉ and Mj₁₀₀) along the pyrope (Mg₃Al₂Si₃O₁₂ = Py₁₀₀)-majorite (MgSiO₃ = Mj₁₀₀) join were fabricated in a 2000-ton uniaxial split-sphere anvil apparatus (USSA-2000) at pressures from 10 to 18.5 GPa and temperatures from 1200 to 1850 °C, within their stability fields in runs of 2–4-h duration, using hot-pressing techniques developed by Gwanmesia et al. (1993). These specimens are single-phased, fine-grained (≤5 mm), free of microcracks, and have bulk densities greater than 99% of the corresponding single-crystal X-ray density. Elastic compressional (P) and shear (S) wave velocities were determined at room pressure and temperature for these polycrystalline garnet specimens by phase comparison ultrasonic interferometry. For Mj₁₀₀, the P and S wave velocities are within 1% of the Hashin-Shtrikman averages calculated from the single crystal elastic moduli measured by Brillouin spectroscopy. Both the elastic bulk modulus (K) and the shear modulus (G) decrease continuously with increasing majorite content from pyrope garnet (Py₁₀₀) to pure majorite garnet (Mj₁₀₀). The compositional dependence of K and G are given by K = 172.3 (40) − 0.085X, and G = 91.6 (10) − 0.038X, where X = mol% majorite), respectively, indicating that substitution of Si for Mg and Al decreases both K and G by about 5% along the solid solution series. Received: 25 March 1999 / Accepted: 12 July 1999     

The high-pressure and temperature equation of state of a majorite solid solution in the system of Mg4Si4O12-Mg3Al2Si3O12

The high-pressure and temperature equation of state of majorite solid solution, Mj_0.8Py_0.2, was determined up to 23 GPa and 773 K with energy-dispersive synchrotron X-ray diffraction at high pressure and high temperature using the single- and double-stage configurations of the multianvil apparatuses, MAX80 and 90. The X-ray diffraction data of the majorite sample were analyzed using the WPPD (whole-powder-pattern decomposition) method to obtain the lattice parameters. A least-squares fitting using the third-order Birch-Murnaghan equation of state yields the isothermal bulk modulus, K _T0  = 156 GPa, its pressure derivative, K′ = 4.4(±0.3), and temperature derivative (∂K _T /∂T) _P = −1.9(±0.3)× 10⁻² GPa/K, assuming that the thermal expansion coefficient is similar to that of pyrope-almandine solid solution. Received: 5 October 1998 / Revised, accepted: 24 June 1999     

The cubic-tetragonal phase transition in the system majorite (Mg4Si4O12) – pyrope (Mg3Al2Si3O12), and garnet symmetry in the Earth's transition zone

 Garnets along the join Mg₄Si₄O₁₂ (majorite end member) – Mg₃Al₂Si₃O₁₂ (pyrope) synthesized at 2000 °C, 19 GPa are, after quench, tetragonal in the compositional range up to 20 mol% pyrope, but cubic at higher Al contents. Lattice constants a _tet and a _tet in the tetragonal compositional range converge with increasing pyrope contents towards the lattice constant of the cubic garnets. The elastic strain and the intensity of the (222) reflection as a function of composition indicate a second-order phase transition near 20 mol% pyrope. From the wedge-like shape of pseudomerohedral twins and their interaction near 90° twin-boundary corners, as well as from the absence of growth-induced dislocations, it is concluded that the Al-poor garnets are also cubic at synthesis conditions but invert by (Mg,Si) ordering on the octahedral sites into tetragonal phases of space group I4₁/a upon quench. This implies that the cubic-to-tetragonal phase transition in Mg₄Si₄O₁₂ garnet occurs below 2000 °C at 19 GPa and at even lower temperatures in more aluminous compositions. A composition-dependent Landau model is consistent with a direct transformation from Ia3d to I4₁/a. Comparison of the T-X stability field of majorite-pyrope garnets with the chemistry of majorite-rich garnets expected to occur in the Earth's transition zone shows that the latter will be cubic under all conditions. Softening of elastic constants, which commonly accompanies ferroelastic phase transitions, may affect the seismic velocities of garnets in the deeper transition zone where majorite contents are highest. Received July 5, 1996 / Revised, accepted September 24, 1996     

Cation disorder in garnets along the Mg3Al2Si3O12-Mg4Si4O12 join: an infrared,Raman and NMR study

We have obtained infrared and Raman spectra for garnets synthesized at high (static) pressures and temperatures along the join Mg₃Al₂Si₃O₁₂ (pyrope) — Mg₄Si₄O₁₂ (magnesium majorite). The vibrational spectra of Mg-majorite show a large number of additional weak peaks compared with the spectra of cubic pyrope garnet, consistent with tetragonal symmetry for the MgSiO₃ garnet phase. The Raman bands for this phase show no evidence for line broadening, suggesting that Mg and Si are ordered on octahedral sites in the garnet. The bands for the intermediate garnet compositions are significantly broadened compared with the end-members pyrope and Mg-majorite, indicating cation disorder in the intermediate phases. Solid state ²⁷Al NMR spectroscopy for pyrope and two intermediate compositions show that Al is present only on octahedral sites, so the cation disorder is most likely confined to Mg-Al-Si mixing on the octahedral sites. We have also obtained a Raman spectrum for a natural, shock-produced (Fe,Mg) majorite garnet. The sharp Raman peaks suggest little or no cation disorder in this sample.     

Elastic properties of pyrope

Brillouin spectroscopy was used to measure the single crystal elastic properties of a pure synthetic pyrope and a natural garnet containing 89.9 mol% of the pyrope end member (Mg₃Al₂Si₃O₁₂). The elastic moduli, c _ij , of the two samples are entirely consistent and agree with previous estimates of the elastic properties of pyrope based upon the moduli of solid solutions. Our results indicate that the elastic moduli of pyrope end-member are c ₁₁=296.2±0.5, c ₁₂=111.1±0.6, c ₄₄=91.6±0.3, K_s=172.8±0.3, =92.0±0.2, all in units of GPa. These results differ by several percent from those reported previously for synthetic pyrope, but are based upon a much larger data set. Although the hydrous components of the two samples from the present study are substantially different, representing both dry and saturated samples, we find no discernable effect of structurally bound water on the elastic properties. This is due to the small absolute solubility of water in pyrope, as compared with other garnets such as grossular.     

Single crystal Raman spectrum of uvarovite, Ca3Cr2Si3O12

Single-crystal Raman spectra of synthetic end-member uvarovite (Ca₃Cr₂Si₃O₁₂) and of a binary solution (59% uvarovite, 41% andradite) have been measured using single crystal techniques. For each of these garnets, 22 and 21 of the 25 Raman modes were located, respectively. The spectra for uvarovite garnets closely resemble those of the other calcic garnets, grossular, and andradite. The modes for uvarovites do not fit into the same trends as established by the other five anhydrous end-member garnets: the high energy “internal” Si–O modes do not depend on lattice constant in uvarovite. They exceed frequencies for both andradite and grossular. This is likely due to the large crystal field stabilization energy of trivalent chromium. The low energy and midrange modes are at similar frequencies to the other calcic garnets.     

Local structural heterogeneity in garnet solid solutions

Local heterogeneities in pyrope-almandine, almandine-grossular and pyrope-grossular solid solutions have been investigated using IR-powder absorption spectroscopy. Correlations of the wavenumber shifts and line broadening systematics with the thermodynamic mixing properties were found. Wavenumber shifts of the highest energy modes correlate closely with the Si-O bond distances and give an indirect view of the average distortions across the three solid solutions. They have a linear behaviour for Py-Alm, but show positive variations from linearity for Alm-Gr and Py-Gr systems. An effective line width (Δ_corr) of the absorption bands over a given wavenumber interval was obtained using the autocorrelation function. Line broadening is associated with local heterogeneities arising from cation substitution in the structure of samples at intermediate compositions. Non-linearities of the line broadening were found for Alm-Gr and Py-Gr and have a shape similar to the enthalpy of mixing, ΔH_mix. An empirical analysis was therefore carried out to compare ΔH_mix and Δ_corr quantitatively. Low-temperature far-IR spectra were recorded for the end-members pyrope, almandine and grossular and far-IR and mid-IR low-temperature spectra for Py₆₀Gr₄₀ in the temperature range 292–44 K. Softening of the lowest energy band with decreasing temperature was observed in the spectrum of pyrope and more enhanced in the spectrum of Py₆₀Gr₄₀. The same softening occurs by substitution of grossular component into pyrope. High energy modes of Py₆₀Gr₄₀ show the effect of saturation below 110–130 K, which correlates with the volume saturation at low temperature. This could provide an alternative explanation for the heat capacity anomaly found for Py-Gr solid solution at low-temperatures.     

High-pressure behaviour of Ca-rich C 2 /c clinopyroxenes along the join diopside-enstatite (CaMgSi2O6-Mg2Si2O6)

 An in situ high-pressure (HP) X-ray diffraction investigation of synthetic diopside and of the Ca_0.8Mg_1.2Si₂O₆ clinopyroxene (Di₈₀En₂₀) was performed up to respectively P=40.8 and 15.1 GPa, using high brilliance synchrotron radiation. The compression of the cell parameters is markedly anisotropic, with β_b ⋙ β_c > β_a > β_asinβ for any pressure range and for both diopside and Di₈₀En₂₀. The compressibility along the crystallographic axes decreases significantly with pressure and is higher in Di₈₀En₂₀ than in diopside. The β cell parameter decreases as well with pressure, at a higher rate in Di₈₀En₂₀. The cell volume decreases at almost the same rate for the two compositions, since in diopside a higher compression along a* occurs. A change in the mechanism of deformation at P higher than about 5–10 GPa is suggested for both compositions from the analysis of the strain induced by compression. In diopside at lower pressures, the deformation mainly occurs, at a similar rate, along the b axis and at a direction 145° from the c axis on the (0 1 0) plane. At higher pressures, instead, the deformation occurs mostly along the b axis. In Di₈₀En₂₀ the orientation of the strain axes is the same as in diopside. The substitution of Ca with Mg in the M2 site induces at a given pressure a higher deformation on (0 1 0) with respect to diopside, but a similar change in the compressional behaviour is found. Changes in the M2 polyhedron with pressure can explain the above compressional behaviour. A third-order Birch-Murnaghan equation of state was fit to the retrieved volumes, with K=105.1(9) GPa, K′=6.8(1) for diopside and K=107.3(1.4) GPa, K′=5.7(3) for Di₈₀En₂₀; the same equation can be applied for any pressure range. The elasticity of diopside is therefore not significantly affected by Mg substitution into the M2 site, in contrast to the significant stiffening occurring for Ca substitution into Mg-rich orthopyroxenes. Received: 3 January 2000 / Accepted: 21 May 2000     

Trace-element partitioning between garnet,clinopyroxene and Fe-rich picritic melts at 3 to 7 GPa

We have determined mineral-melt partition coefficients (D values) for 20 trace elements in garnet-pyroxenite run products, generated in 3 to 7 GPa, 1,425–1,750°C experiments on a high-Fe mantle melt (97SB68) from the Paraná-Etendeka continental-flood-basalt (CFB) province. D values for both garnet (∼Py₆₃Al₂₅Gr₁₂) and clinopyroxene (∼Ca_0.2Mg_0.6Fe_0.2Si₂O₆) show a large variation with temperature but are less dependent on pressure. At 3 GPa, D ^cpx/liq values for pyroxenes in garnet-pyroxenite run products are generally lower than those reported from Ca-rich pyroxenes generated in melting experiments on eclogites and basalts (∼Ca_0.3–0.5Mg_0.3–0.6Fe_0.07–0.2Si₂O₆) but higher than those for Ca-poor pyroxenes from peridotites (∼Ca_0.2Mg_0.7Fe_0.1Si₂O₆). D ^grt/liq values for light and heavy rare-earth elements are ≤0.07 and >0.8, respectively, and are similar to those for peridotitic garnets that have comparable grossular but higher pyrope contents (Py_70–88Al_l7–20Gr_8–14). 97SB68 D _LREE^grt/liq values are higher and D _HREE^grt/liq values lower than those for eclogitic garnets which generally have higher grossular contents but lower pyrope contents (Py_20–70Al_10–50Gr_10–55). D values agree with those predicted by lattice strain modelling and suggest that equilibrium was closely approached for all of our experimental runs. Correlations of D values with lattice-strain parameters and major-element contents suggest that the wollastonite component and pyrope:grossular ratio exert major controls on 97SB68 clinopyroxene and garnet partitioning, respectively. These are controlled by the prevailing pressure and temperature conditions for a given bulk-composition. The composition of co-existing melt was found to have a relatively minor effect on 97SB68 D values. The variations in D values displayed by different mantle lithologies are subtle and our study confirms previous investigations which have suggested that the modal proportions of garnet and clinopyroxene are by far the most influential factor in determining incompatible trace-element concentrations in mantle melts. The trace-element partition coefficients we have determined may be used to place high-pressure constraints on garnet-pyroxenite melting models.     

Monte Carlo simulation of mixing in Ca3Fe2Ge3O12–Ca4Ge4O12 garnets and implications for the thermodynamic stability of pyrope–majorite solid solution

Static lattice energy calculations, based on empirical pair potentials, were performed for a large set of structures differing in the arrangement of octahedral cations within the garnet 2 × 2 × 2 supercell. The compositions of these structures varied between Ca₃Fe₂Ge₃O₁₂ and Ca₄Ge₄O₁₂. The energies were cluster expanded using pair and quaternary terms. The derived ordering constants were used to constrain Monte Carlo simulations of temperature-dependent mixing properties in the ranges of 1,073–3,673 K and 0–10 GPa. The free energies of mixing were calculated using the method of thermodynamic integration. The calculations predict a wide miscibility gap between Fe-rich (cubic) and Fe-pure (tetragonal) garnets consistent with recent experimental observations of Iezzi et al. (Phys Chem Miner 32:197–207, 2005). It is shown that the miscibility gap arises due to a very strong cation ordering at the Fe-pure composition, driven by the charge difference between Ca²⁺ and Ge⁴⁺ cations. The structural and thermodynamic analogies between Ca–Ge and Mg–Si systems suggest that a similar miscibility gap should exist between pyrope and Mg–Si-majorite.     

The influence of low aluminum concentrations on the composition and conditions of crystallization of majorite–knorringite garnets: Experiment at 7.0 GPa and 1500–1700°C

Crystallization of garnet in high-chromium restite formed under the conditions of partial melting in the spinel facies and subsequently subducted into the garnet depth facies was studied experimentally in the MgO–Al₂O₃–Cr₂O₃–SiO₂ system. The crystallization of garnet and the dependence of its composition on the temperature and bulk composition of the system with low Al concentration were studied as well. Experiments in the knorringite–majorite–pyrope system with 5, 10, and 20 mol % Prp were carried out at 7 GPa. The phase associations for the starting composition of pure knorringite Mg₃Cr₂Si₃O₁₂ included chromiumbearing enstatite MgSiO₃ (up to 3.2 wt % Cr₂O₃) and eskolaite Cr₂O₃. Addition of Al resulted in crystallization of high-chromium majoritic garnet. The portion of garnet in the samples always exceeded the concentration of pyrope in the starting composition owing to the formation of the complex majorite–knorringite–pyrope series of solid solutions. With increasing content of pyrope (from 5 to 20 mol %) and increasing temperature, the modal concentration of garnet increased significantly (from 6–12 to 22–37%). The garnet was characterized by high concentrations of the pyrope (23–80 mol %) and knorringite (22–70 mol %) components. The excess of Si (>3 f.u.) with decreasing Cr concentration provided evidence for the contribution of the majorite–knorringite trend to the variation in garnet composition. On the basis of the natural data, most of the garnets composing xenoliths of ultrabasic rocks in kimberlites and occurring as inclusions in diamonds are low-chromium; i.e., their protolith was not subjected to partial melting, at least in the spinel depth facies.     

A high-temperature and high-pressure Raman spectroscopic study of CaGeO3 garnet

 High-pressure and high-temperature Raman spectra of CaGeO₃ tetragonal garnet have been collected to 11.5 GPa and 1225 K, respectively, in order to investigate possible intrinsic anharmonic behaviour in this phase. The Raman peak positions were observed to vary linearly with pressure and temperature within the ranges studied, with the higher-energy peaks showing larger P- and T-induced shifts than the low energy modes. The observed T-induced shifts are similar to those reported for grossular and andradite, while the observed P-induced shifts are generally larger than those of aluminosilicate and MgSiO₃ majorite garnets (Gillet et al. 1992; Rauch et al. 1996) due to the larger bulk modulus of CaGeO₃ garnet. The observed mode shifts of CaGeO₃ garnet were used to determine the isothermal and isobaric mode Grüneisen parameters for this phase. These parameters are similar in value to those reported previously for grossular and andradite (Gillet et al. 1992). The calculated intrinsic anharmonic parameters, a _i , for CaGeO₃ garnet were determined to be nonzero, indicating significant anharmonic behaviour for this phase. These values, which range from −3.8 × 10⁻⁵ K⁻¹ to −1.3 × 10⁻⁵ K⁻¹, are also similar to those reported for andradite and grossular, but smaller than those determined for pyrope (Gillet et al. 1992). Hence, we expect MgSiO₃ majorite to show greater anharmonicity than the germanate analogue studied by us. The anharmonic parameters determined for CaGeO₃ tetragonal garnet may now be introduced into quasiharmonic vibrational heat capacity models to account for the observed anharmonic behaviour. Received: 21 April 1999 / Revised, accepted: 11 September 1999     

Trace element incorporation into pyrope–grossular solid solutions: an atomistic simulation study

 We have performed atomistic computer simulations on trace element incorporation into the divalent dodecahedral X-sites of pyrope (Py — Mg₃Al₂Si₃O₁₂) – grossular (Gr — Ca₃Al₂Si₃O₁₂) solid solutions. An ionic model and the Mott–Littleton two-region approach to defect energies were used to calculate the energetics of substitution by a range of divalent trace-elements and of charge-balanced substitution by trivalent ions in the static limit. Results are compared with experimental high-temperature, high-pressure garnet-melt trace element partitioning data obtained for the same garnet solid solution to refine our understanding of the factors controlling element partitioning into solid solutions. Defect energies (U _def,f), relaxation (lattice strain) energies (U _rel), and solution energies (U _sol) were derived using two different approaches. One approach assumes the presence of one type of hybrid X-site with properties intermediate between pure Mg and Ca sites, and the other assumes discrete Mg and Ca X-sites, and thus two distinct cation sublattices. The hybrid model is shown to be inadequate, since it averages out local distortions in the garnet structure. The discrete model results suggest trace elements are more soluble in Py₅₀Gy₅₀ than in either end-member compound. Physically this is due to small changes in size of the X-sites and the removal of unfavourable interactions between third nearest neighbours of the same size. Surprisingly, depending on the local order, large trace element cations may substitute for Mg²⁺ and small trace elements for Ca²⁺ in Py₅₀Gr₅₀. These solubilities provide an explanation for the anomalous trace-element partitioning behaviour along the pyrope–grossular join observed experimentally. Received: 27 January 2000 / Accepted: 14 February 2003     

Structural and magnetic properties of Co2Al4Si5O18 and Mn2Al4Si5O18 cordierite

The crystal structures, lattice dynamics and magnetic properties of synthetic Co-cordierite, Co₂Al₄Si₅O₁₈, and Mn-cordierite, Mn₂Al₄Si₅O₁₈ have been studied by neutron powder diffraction, infrared spectroscopy and magnetisation measurements. Due to different synthesis conditions, the Co-cordierite used here crystallised in the hexagonal α-cordierite structure with a disordered Si/Al distribution in the framework, while for the Mn-cordierite the orthorhombic β-structure has been determined. The experimentally determined paramagnetic moments, _exp (Mn) = 5.47(6) _B and _exp (Co) = 3.88(4) _B , are in good agreement with theoretical predictions for octahedrally coordinated Mn²⁺ and Co²⁺, respectively. In both compounds there is no magnetic long-range order down to at least 1.5 K. However, the onset of an anti-ferromagnetic short-range correlation of magnetic moments along [001] has been observed for Mn-cordierite by magnetisation and neutron diffraction measurements. This short-range magnetic correlation becomes evident from diffuse scattering observed at 2 K. The diffuse scattering has been interpreted in terms of a Blech-Averbach function. Received: 30 June 1998 / Revised, accepted: 3 March 1999     

The effect of chemical composition and oxygen fugacity on the electrical conductivity of dry and hydrous garnet at high temperatures and pressures

The in situ electrical conductivity of hydrous garnet samples (Py₂₀Alm₇₆Grs₄–Py₇₃Alm₁₄Grs₁₃) was determined at pressures of 1.0–4.0 GPa and temperatures of 873–1273 K in the YJ-3000t apparatus using a Solartron-1260 impedance/gain-phase analyzer for various chemical compositions and oxygen fugacities. The oxygen fugacity was controlled by five solid-state oxygen buffers (Fe₂O₃ + Fe₃O₄, Ni + NiO, Fe + Fe₃O₄, Fe + FeO, and Mo + MoO₂). Experimental results indicate that within a frequency range from 10⁻² to 10⁶ Hz, electrical conductivity is strongly dependent on signal frequency. Electrical conductivity shows an Arrhenius increase with temperature. At 2.0 GPa, the electrical conductivity of anhydrous garnet single crystals with various chemical compositions (Py₂₀Alm₇₆Grs₄, Py₃₀Alm₆₇Grs₃, Py₅₆Alm₄₃Grs₁, and Py₇₃Alm₁₄Grs₁₃) decreases with increasing pyrope component (Py). With increasing oxygen fugacity, the electrical conductivity of dry Py₇₃Alm₁₄Grs₁₃ garnet single crystal shows an increase, whereas that of a hydrous sample with 465 ppm water shows a decrease, both following a power law (exponents of 0.061 and −0.071, respectively). With increasing pressure, the electrical conductivity of this hydrous garnet increases, along with the pre-exponential factors, and the activation energy and activation volume of hydrous samples are 0.7731 ± 0.0041 eV and −1.4 ± 0.15 cm³/mol, respectively. The results show that small hopping polarons ( \textFe_\textMg ^· ) \left( {{\text{Fe}}_{\text{Mg}}^{ \cdot } } \right) and protons ( \textH ^· {\text{H}}^{ \cdot } ) are the dominant conduction mechanisms for dry and wet garnet single crystals, respectively. Based on these results and the effective medium theory, we established the electrical conductivity of an eclogite model with different mineral contents at high temperatures and high pressures, thereby providing constraints on the inversion of field magnetotelluric sounding results in future studies.     

Elasticity of single-crystal MgAl2O4 spinel up to 1273 K by Brillouin spectroscopy

The adiabatic single-crystal elastic constants, C _ij , of stoichiometric magnesium aluminate spinel (MgAl₂O₄) have been measured up to 1273 K by highresolution Brillouin spectroscopy, using a 6-pass tandem Fabry-Pérot interferometer and an argon ion laser (514.5 nm). Two platelet samples were employed for probing the acoustic phonons along [100] and [110] directions by platelet and backscattering geometries. The measured temperature dependences of the elastic moduli show a distinct anomaly at 923 K in the shear modulus C _s = (C₁₁-C₁₂)/2 (along [110] direction) and the longitudinal modulus C₁₁ (along [100] direction). This anomaly is consistent with the order-disorder phase transition, resulting from the atomic exchange between Mg at the tetrahedral site and Al at the octahedral site, which has been well documented recently (Peterson et al. 1991; Millard et al. 1992) by neutron powder diffraction and ²⁷Al magic-angle spinning NMR. The values of the temperature derivatives of v _p , v _s , and K _s , in the temperature range 300–923 K, calculated by the Voigt-Reuss-Hill approximation are -0.40ms^?1 K^?1, -0.26ms^?1 K^?1, and -1.89 x 10^?2GPaK^?1.     

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     

Hydrostatic compression and crystal structure of pyrope to 33 GPa

The equation of state and crystal structure of pyrope were determined by single crystal X-ray diffraction under hydrostatic conditions to 33 GPa, a pressure that corresponds to a depth of about 900 km in the lower mantle. The bulk modulus K _T0 and its pressure derivative K ^' _T0 were determined simultaneously from an unweighted fit of the volume data at different pressures to a third order Birch-Murnaghan equation of state. They are 171(2) GPa and 4.4(2), respectively. Over the whole pressure range, MgO₈ polyhedra showed the largest compression of 18.10(8)%, followed by AlO₆ and SiO₄ polyhedra, with compression of 11.7(1)% and 4.6(1)%, respectively. The polyhedral bulk moduli for MgO₈, AlO₆ and SiO₄ are 107(1), 211(11) and 580(24) GPa, respectively, with K ^' _T0 fixed to 4. Significant compression of up to 1.8(1)% in the very rigid Si−O bonding in pyrope could be detected to 33 GPa. Changes in the degree of polyhedral distortion for all three types of polyhedra could also be observed. These changes could be found for the first time for AlO₆ and SiO₄ in pyrope. It seems that the compression of pyrope crystal structure is governed by the kinking of the Al−O−Si angle between the octahedra and tetrahedra. No phase transition could be detected to 33 GPa. Received: 24 March 1997 / Revised, accepted: 29 July 1997     

In search of the mixed derivative ?2 M /? P ? T ( M = G , K ): joint analysis of ultrasonic data for polycrystalline pyrope from gas- and solid-medium apparatus

Elastic wave velocities for dense (99.8% of theoretical density) isotropic polycrystalline specimens of synthetic pyrope (Mg₃Al₂Si₃O₁₂) were measured to 1,000 K at 300 MPa by the phase comparison method of ultrasonic interferometry in an internally heated gas-medium apparatus. The temperature derivatives of the elastic moduli [(∂Ks/∂T) _P = −19.3(4); (∂G/∂T) _P = −10.4(2) MPa K⁻¹] measured in this study are consistent with previous acoustic measurements on both synthetic polycrystalline pyrope in a DIA-type cubic anvil apparatus (Gwanmesia et al. in Phys Earth Planet Inter 155:179–190, 2006) and on a natural single crystal by the rectangular parallelepiped resonance (RPR; Suzuki and Anderson in J Phys Earth 31:125–138, 1983) method but |(∂Ks/∂T) _P | is significantly larger than from a Brillouin spectroscopy study of single-crystal pyrope (Sinogeikin and Bass in Phys Earth Planet Inter 203:549–555, 2002). Alternative approaches to the retrieval of mixed derivatives of the elastic moduli from joint analysis of data from this study and from the solid-medium data of Gwanmesia et al. in Phys Earth Planet Inter 155:179–190 (2006) yield ∂² G/∂P∂T = [0.07(12), 0.20(14)] × 10⁻³ K⁻¹ and ∂² K _S /∂P∂T = [−0.20(24), 0.22(26)] × 10⁻³ K⁻¹, both of order 10⁻⁴ K⁻¹ and not significantly different from zero. More robust inference of the mixed derivatives will require solid-medium acoustic measurements of precision significantly better than 1%.     

Quantum-mechanical modeling of minerals at high pressures. The role of the Hamiltonian in a case study: the beryl (Al4Be6Si12O36)

Ab initio calculations of the beryl structure at room and higher pressures, and of its equation of state, have been performed both at the Hartree–Fock (HF) and density functional (DFT) level. In the latter case, three different hybrid HF/DFT Hamiltonians have been employed, in which the exact non-local exchange contribution increases from 20% (B3LYP Hamiltonian, indicated with the symbol F20) to 50% (F50) and 60% (F60). Within the DFT series, the equilibrium volume (V ₀) increases as the HF exchange contribution decreases; with respect to the experimental datum, F20 overestimates V ₀ by 2.9%, whereas F60 underestimates it by 0.9%; F50 (and HF) volume is very close to the experimental datum (error less than −0.1%). All four Hamiltonians overestimate the bulk modulus (K ₀); with respect to the experimental datum (obtained in the present work) [K ₀=179(1) GPa], the F20 Hamiltonian leads to the smallest error (+2.7%); the corresponding errors for the F50, F60 and HF Hamiltonians are +13.2, +16.2 and +16.3%, respectively. In the case of F20, in spite of the small error in K ₀, the relatively large error in V ₀ leads to an incorrect P(V) equation of state, which significantly overestimates the pressure at a given volume, compared to the experimental one at the same volume; the maximum error in the pressure range investigated is at the largest pressure (P _max=26.4 GPa) and amounts to +34.8%. The corresponding errors for the F50, F60 and HF Hamiltonians are +12.9, +5.7 and +15.5%.