Bose-Einstein Approximation in Equations of State of Minerals

Peter I. Dorogokupets (dor@gpg.crust.irk.ru) & Elena A. Melechova

Institute of the Earth's Crust, Irkutsk, 664033, Russia

A simple numerical models for simultaneous optimization of heat capacity at constant pressure, C_P, heat capacity at constant volume, C_V, volume, V, thermal expansion coefficient,(alpha), isothermal, K_T, and adiabatic, K_S, bulk moduli at zero pressure, and PVT data for minerals have been discussed in (Dorogokupets, 2000). In a new version of these models the free energy, F, and heat capacity at constant volume expressed through the Bose-Einstein function (Kut'in, Pyadushkin, 1998): F=mRT_Bb/(1+T_B/Td)+m₁RT_E1b_E1+m₂RT_E2b_E2, C_V=mR[T_B/(T+T_B/d)]²b(1/d+1+b)+m₁R(T_E1/T)²b_E1(1+b_E1)+m₂R(T_E2/T)²b_E2(1+b_E2), where b=1/[exp(g)-1], g=d ln(1+T_B/Td), b_E1=1/[exp(T_E1/T)-1], b_E2=1/[exp(T_E2/T)-1], m+m₁+m₂=>3n, n is number of atoms, T_B, T_E1, T_E2 are characteristic temperatures, d is degree parameter. Three additional empirical parameters are included in the expression for energy, which take into account anharmonicity, premelting and other effects for real minerals. The volume vs. energy dependence is calculated on the basis of either the Wachtman et al. (1962) or the Suzuki (1975) model or their linear combination. Volume, V₂₉₈, characteristic temperatures, T_B, T_E1, T_E2, parameters m, m₁, m₂, d, isothermal bulk modulus, K_T298, its pressure derivative, K', Gruneisen parameter, (gamma), isothermal Anderson-Gruneisen parameter, _T, and three empirical parameters, a, b, c, which can be equal to zero for Debye-like solids, are fitting parameters of the model. The proposed model enables one to calculate thermodynamic functions of simple substances, oxides and minerals over a temperature range from 0 K up to the melting temperature with a deviation within the scatter of experimental data. Correlation of the proposed model with PVT data is considered. It is shown that the isothermal equation of state results in an unsatisfactory extrapolation of volume in extreme regions. The Wachtman et al. (1962) and the Suzuki (1975) models of the volume vs. energy are extended to high pressure. The high-pressure Wachtman et al. (1962) and Suzuki (1975) model is a version of the Mie-Gruneisen equation of state and allows temperature dependence of thermodynamic functions for any isobar to be easily calculated. This and the classical Mie-Gruneisen model are found to be equivalent at q=1. The model has been tested using rock salt, corundum and lawsonite samples. For corundum we used experimental data for C_P, V, (alpha), K_T, K_S at zero pressure quoted in (Dorogokupets, 2000) and PVT data from (Richet et al. 1988; Dubrovinsky et al., 1998). The derived parameters for corundum are V₂₉₈=25.615 cm³, T_B=1467 K, T_E1=856 K, T_E2=429 K, m=1.225, m₁=9.452, m₂=4.868, d=0.89, g₂₉₈=1.384, K_T298=2.523 GPa, K'=5.02, q=-1.645. Standard deviations of calculated C_P from experimental data are 0.082 J/(mol K) for temperature interval from 0 to 1500 K and 0.195 J/(mol K) for temperature interval from 0 to 2250 K, for calculated adiabatic bulk modulus 0.005 GPa and for calculated pressure 0.31 GPa. The study was supported by the Russian Foundation for Basic Research, grant # 99-05-64891.

Dorogokupets PI, American Mineralogist, 85, (2000).

Dubrovinsky LS, Saxena SK & Lazor P, Physics and Chemistry of Minerals, 25, 434-441, (1998).

Kut'in AM & Pyadushkin DV, Russian Journal of Physical Chemistry, 72, 1567-1572, (1998).

Richet P, Xu JA & Mao HK, Physics and Chemistry of Minerals, 16, 207-211, (1988).

Suzuki I, Journal of Physic Earth, 23, 145-159, (1975).

Wachtman JB, Scuderi TG & Gleek GW, Journal of American Ceramic Society, 45, 310-323, (1962).

Experimental and Theoretical Study of High-Pressure Transformations in Silica

Natalia Dubrovinskaia (natalia.dubrovinskaia @geo.uu.se), Leonid Dubrovinsky, Rajeev Ahuja & Surendra Saxena

Uppsala University, S 75236, Uppsala, Sweden

Properties and behaviour of silicon dioxide SiO₂ at high pressures and temperatures are of great interest due to its wide ranging implications in fundamental physics, geophysics and material sciences. Last decade studies revealed number of enigmatic phenomenon associated with high-pressure silica polymorphs - formation of unidentified phases on the compression of (alpha)-cristobalite, controversial theoretical and experimental information on the possible post-stishovite phases, discovery of new dense natural silica polymorph in the Shergotty meteorite in the mineralogical environment which is unlikely for post-stishovite phase.

New experiments on compression of synthetic (alpha)-cristobalite and natural o-tridymite shows that at pressure above 40 GPa both materials transforms to (alpha)-PbO₂-like silica phase. Rietveld refinement of diffraction patterns collected at pressures between 20 GPa and 70 GPa indicate that correct space group for (alpha)-PbO₂-like silica is Pnc2. We demonstrate that Cristobalite XII and Cristobalite XIII phases discovered by Tsuchida and Yagi (1990), and natural silica polymorph recently found in the SNC meteorite Shergotty (Sharp et al., 1999) are (alpha)-PbO₂-like silica phase. Experiments with electrically and laser heated DAC show that at 80 GPa -PbO₂-like silica stable up to at least 2000 K, but at 60 GPa it transforms to stishovite even at 1000 K.

Stability of Magnesiowüstite in the Lower Mantle

Leonid Dubrovinsky (leonid.dubrovinsky @geo.uu.se)¹, Natalia Dubrovinskaia¹, Surendra Saxena¹, Hans Annersten¹, Elke Hålenius & Tristan Le Bihan²

¹ Uppsala University, S-75236, Uppsala, Sweden

² ESRF, Grenoble, France

Magnesiowüstite (Mg,Fe)O and (Mg,Fe)SiO₃-perovskite are considered to form the bulk of the Earth's lower mantle. At ambient condition, the end members of the MgO-FeO solid solution - periclase (MgO) and wüstite (FeO), - have the same halite NaCl (B1) structures and they form a complete solid solution. However, at pressures above 17 GPa and ambient temperature, wüstite transforms to a phase with rhombohedral structure. On increasing pressure above 100 GPa at 300K it transforms to NiAs (B8) or anti-NiAs (a-B8) structure. Periclase remains in the NaCl structure at least to pressures as high as 227 GPa. The structure of NaCl is based on cubic closed packing of anions but B8 (or a-B8) is formed by hexagonal closed packing of anions (or cations). The topological difference between B8 and B1 structures at high pressure could lead to immiscibility gap at some compositions in the MgO-FeO solid solution. We conducted an in situ high-P,T study of magnesiowüstites of intermediate compositions (Mg_0.5Fe_0.5)O, (Mg_0.6Fe_0.4)O, and (Mg_0.8Fe_0.2)O. We heated samples of magnesiowüstites to a temperature of 1000 K at a pressure above 90 GPa to study the stability of the solid solution at physical conditions of Earth's lower mantle. The in situ x-ray study of the externally heated sample in a Mao-Bell type diamond-anvil cell shows that at pressure above 80 GPa magnesiowüstites (Mg_0.5Fe_0.5)O and (Mg_0.6Fe_0.4)O and at pressure above 90 GPa dissociate in to magnesium and ferrous rich oxide components. The result is important in causing dynamic effects leading to mantle heterogeneity.