Volume 85, Issue 2

Thermal expansivities for the MgSiO 3 phases of orthoenstatite, high clinoenstatite, ilmenite, and majorite; and for stishovite were estimated using the thermodynamic Maxwell relation (∂S/∂P)T = -(∂V/∂T) P where the entropies at high pressures were derived using a statistical method and spectroscopic data. The spectroscopically determined thermal expansivities for all minerals are in excellent agreement with previously determined volumetric data, where available. A value of 3.25(10) × 10 -5 K -1 for orthoenstatite at room temperature was obtained; this value is situated in the middle of the large spread of reported values and is in excellent agreement with the two latest volumetric determinations. For high clinoenstatite, α at room T is estimated as 2.56(9) × 10 -5 K -1 . This method provides good high temperature estimates of α for the high-pressure polymorphs, where data are scanty or unavailable. Included in this report are previous data for the Mg 2 SiO 4 phases and MgO for completeness. The following equations may be used to extrapolate a to higher temperatures at 1 atm in 10 -5 K -1 : a(majorite) = 2.95 + 0.000521x; α(γ-Mg 2 SiO 4 ) = 2.70 + 0.000648x; α(ilmenite) = 2.64+ 0.000537x; α(perovskite) = 2.51 + 0.000805x; and α(stishovite) = 2.19 + 0.000485x, where x is (T/K-750)

The differential laser interferometry method and its application for measuring the thermal expansion of samples is described. The thermal expansion of single-crystal Al 2 O 3 (corundum) was measured up to 1000 K with the differential laser-interferometer. The changes in the distance between two faces of a stepped shape sample are measured. The main feature of the apparatus is simultaneously monitoring two fringe signals that are 90° out of phase. This improvement of the basic laserinterferometer allows us to remotely measure the thermal expansion of minerals with high precision. The overall sensitivity with which we can detect changes in length is about 1/100 of the wavelength of the He-Ne laser (1/100 of 0.6328 × 10 -6 m). After adjusting for errors caused by heat, the thermal expansion data measured with the differential laser-interferometer are in good agreement with previous reports. Use of this apparatus allows the investigation of the high-temperature thermodynamic properties of important rock-forming minerals.

Two new diamond anvil cells have been designed for ultrasonic and X-ray diffraction measurements on a single crystal sample up to 6 GPa and 250 °C. Advances in the generation and transmission of coherent GHz ultrasonic signals with wavelengths of the order of micrometers now make it practical to measure elastic properties of samples small enough to be subjected to pressure and temperature in diamond anvil cells. The signal is carried from a thin transducer through a sapphire buffer rod coupled to one of the diamond anvils by means of force. The signal traverses the diamond anvil and enters the single crystal sample which is coupled to the anvil face by cement, adhesion, or by a normal force. Interference of superimposed waves reflected from the near and far faces of the single crystal is used to measure travel time of the sound waves in the sample. One of the diamond anvil cells employs the conventional geometry in which access for the X-rays is through the diamond anvils. The other provides access for X-rays at high angles to the load axis so that they do not need to pass through the diamond anvils and can therefore have access to the sample while the buffer rod is in place. Both diamond anvil cells make it possible to measure d-spacings at several different orientations using a four-circle goniometer. This capability is used for detecting and correcting displacement of the sample from the center of the goniometer. Measurement of travel times and lattice parameters at the same pressure-temperature conditions allows conversion of travel times to velocities and can provide simultaneous equations of state, which EOS can then be used to make an independent determination of pressure vs. lattice parameter. This provides a primary pressure scale

The purpose of this paper is to demonstrate that resonant ultrasound spectroscopy (RUS) is applicable to conditions of gas pressure. A sphere of silica glass was measured under pressure of helium gas with a sample-gas-cavity container assembly. The pressure-induced shifts of frequency of six toroidal and two spheroidal modes were observed up to 200 MPa (2 kbar). The sample-gas-container system was analyzed as a three-layered shell structure, and the pressure derivatives of rigidity and bulk moduli; G 0 ' = -3.5 and K' 50 = -6.0, respectively, were obtained. These results are in good agreement with previous data

We measured the sound velocity of wadsleyite b-(Mg 0.88 Fe 0.12 ) 2 SiO 4 to 9.6 GPa at room temperature using ultrasonic techniques in a uniaxial split-cylinder apparatus using a polycrystalline specimen hot-pressed at 14 GPa and 1200°C. Bench-top velocity measurements yielded V P = 9.33(3) km/s and V S = 5.43(2) km/s; the calculated bulk modulus (K) and shear modulus (G) are 172(2) GPa and 106(1) GPa, respectively. These K and G values are indistinguishable from those for single crystal and polycrystalline specimens of Mg 2 SiO 4 -wadsleyite. Pressure derivatives of the bulk and shear moduli have been obtained by fitting the current experimental data to 9.6 GPa using third-order, finite strains equations, yielding K' S0 = 4.6(1) and G' 0 = 1.5(1). From comparison with previous data for similar specimens and techniques, we observe no effect of iron content on the pressure derivatives of either the bulk or shear moduli, within the limited compositional range investigated

The sound velocities and single-crystal elastic moduli of spinel-structured g-Mg 2 SiO 4 were measured to 873 K by Brillouin spectroscopy using a new high-temperature cell designed for singlecrystal measurements. These are the first reported acoustic measurements of g-Mg 2 SiO 4 elasticity at high temperatures. A linear decrease of elastic moduli and sound velocities with temperature adequately describes the data. The adiabatic bulk modulus, KS, shear modulus, m, and respective temperature derivatives for γ-Mg 2 SiO 4 are: KS = 185(3) GPa, μ = 120.4(2.0) GPa, (∂KS/∂T) P = -0.024(3) GPa/K and (∂m/∂T) P = -0.015(2) GPa/K. Extrapolation of our data to transition zone pressures and temperatures indicates that the shear and compressional impedance contrasts associated with β- → ∂-(Mg,Fe) 2 SiO 4 transition are sufficient to produce an observable discontinuity at 520 km depth, even with a moderate (30-50%) amount of olivine

The resonant sphere technique, RST, was applied to measure the elastic moduli of spinel. Resonant frequencies of 23 modes were measured between 293 and 1167 K. Results are similar to previous measurements by two of the authors (IS and OLA) using a rectangular prism specimen in that resonant frequencies of all modes decrease with temperature, and some modes show discontinuous change of slopes near 904 K. In the present work, inversion calculations of the frequency data to obtain elastic moduli was repeated until the standard error, s, was minimized so that s = 0.16 kHz (0.012%). Elastic moduli and their probable errors in GPa are: C 11 = 281.310 ± 0.014, C 12 = 155.437 ± 0.013, C 44 = 154.587 ± 0.007, and C s = 62.936 ± 0.003, where density p = 3.5846 g/cm 3 at 293 K. The anisotropy factor is A = C 44 /C s = 2.46 and is much larger than that of other cubic crystals. Isotropic properties are: bulk moduli, K S = 197.39 ± 0.01 GPa; isothermal bulk modulus, K T = 196.20 GPa; and rigidity modulus, μ = 107.81 GPa (the Hill average). Temperature dependence was clarified for elastic moduli in which one of shear moduli, C s [= (C 11 - C 12 )/2] shows a distinctive bend at T c = 904 K. The bend in K S [ = (C 11 + 2C 12 )/3] at T c is less pronounced because of opposite changes of slopes for C 11 and C 12 at 904 K. Combining elasticity and thermal expansivity data, we evaluate anharmonic parameters, resulting in the Grüneisen parameter, γ = 1.17, and the Anderson-Grüneisen parameters, δ S = 2.98 and δ T = 4.72 at T = 300 K; whereas, these are parameters are 1.10, 4.46, and 6.37, respectively, at T = 1200 K. The Grüneisen parameter is about 0.1 lower on the high-temperature side of T c compared to its value on the lowtemperature side

Molecular dynamics (MD) simulation is used to calculate the elastic constants and their temperature and pressure derivatives, and the T-P-V equation of state of MgO. The interionic potential is taken to be the sum of pairwise additive Coulomb, van der Waals attraction, and repulsive interactions. In addition, to account for the observed large Cauchy violation of the elastic constants of MgO, the breathing shell model (BSM) is introduced in MD simulation, in which the repulsive radii of O ions are allowed to deform isotropically under the effects of other ions in the crystal. Quantum correction to the MD pressure is made using the Wigner-Kirkwood expansion of the free energy. Required energy parameters, including oxygen breathing parameters, were derived empirically to reproduce the observed molar volume and elastic constants of MgO, and their measured temperature and pressure derivatives as accurately as possible. The MD simulation with BSM is found to be very successful in reproducing accurately the measured molar volumes and individual elastic constants of MgO over a wide temperature and pressure range. The errors in the simulated molar volumes are within 0.3% over the temperature range between 300 and 3000 K at 0 GPa, and within 0.1% over the pressure range from 0 up to 50 GPa at 300 K. The simulated bulk modulus is found to be correct to within 0.7% between 300 and 1800 K at 0 GPa. Here we present the MD simulated T-P-V equation of state of MgO as an accurate internal pressure calibration standard at high temperatures and high pressures

The structural properties of MgSiO 3 -ilmenite at high pressures are determined using Ab initio variable cell-shape molecular dynamics. Our athermal results at zero pressure are in excellent agreement with single-crystal measurements. The predicted lattice constants compare favorably with powder X-ray diffraction data. The internal parameters are shown to vary only slightly with pressure. The c axis is considerably more compressible than the a axis and our results suggest that this anisotropic behavior arises in the relatively larger compressibility of MgO 6 with respect to SiO 6 octahedra. Both octahedral types remain highly distorted with the degree of distortion decreasing (more rapidly in MgO 6 ) under compression. By comparing free energies, it is shown that MgSiO 3 should transform from the ilmenite to the perovskite structure at 30 GPa (for static lattice). At the transition, the density and elastic moduli increase substantially

By assuming an ideal two-component mixture of (Mg,Fe)SiO 3 perovskite (MgPv) and (Mg,Fe)O magnesiowüstite (Mw), and by using a thermoelastic model for mantle minerals developed previously, we can reproduce the PREM values of density and velocities v P and v S of compressional and shear waves of the lower mantle within ±0.12%, ±0.28%, and ±0.56% except for the transition layers at the both boundaries. The molar fractions and atomic fractions of iron for MgPv and Mw were adjusted to reproduce the PREM values of r, v P , and v S above the point of z = 871 km (which is slightly inside the lower mantle) under constant-entropy condition. This depth avoids the boundary effect. The adiabatic bulk and shear moduli of the mixture are calculated by the Hashin-Shtrikman method for MgPv and Mw and then arithmetically averaged. The temperature profile was calculated assuming that the lower mantle is adiabatic and T(670 km) = 1873 K. The temperature at the top of D'' becomes 2444 K. Being added the temperature increment of 840 K over D'' (z = 2741-2891 km) estimated by Stacey and Loper (1983) to our value, the temperature at the core-mantle boundary (CMB) becomes 3284 K in agreement with T(CMB) of 3300 ± 500 °C by Brown and McQueen. The molar ratios of Fe/(Mg + Fe) and (Mg + Fe)/Si become 0.12 and 2.10. The calculated thermal expansivity, a, of the mixture under lower mantle conditions is in agreement with a of the lower mantle calculated directly from PEM data by Brown and Shankland, and Anderson. For the addition of 5 mol% of CaSiO 3 perovskite to our model, the essential feature of the result is unchanged and the wt% of SiO 2 , MgO, FeO, and CaO become 40.7, 44.6, 11.0, and 3.7

A simple numerical model for simultaneous optimization of heat capacity at constant pressure, C P , heat capacity at constant volume, C V , volume, V, thermal expansion coefficient, α, isothermal, K T , and adiabatic, K S , bulk moduli at zero pressure, and PVT data for minerals has been developed. The basic function is the Debye energy, expressed through the Nernst-Lindemann energy function for T > 0.2Θ. 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, V 298 , Nernst-Lindemann characteristic temperature, Q, isothermal bulk modulus, K T298 , its pressure derivative, K', Grüneisen parameter, g, isothermal Anderson- Grüneisen parameter, δ 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 0.2Θ 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) models are versions of the Mie-Grüneisen equation of state and allow temperature dependencies of thermodynamic functions for any isobar to be easily calculated. The model described here and the classical Mie-Grüneisen model are found to be equivalent at q ≈ 1. The model is tested using rock salt, corundum, and lawsonite

The accuracy of equation-of-state formulations is compared for theoretical total energies or experimental pressure-volume measurements for H 2 , Ne, Pt, and Ta. This spans the entire range of compression found for minerals and volatiles in the Earth. The Vinet equation is found to be most accurate. The origin of the behavior of different equation-of-state formulations is discussed. It is shown that subtle phase transitions can be detected by examining the residuals from an equation-ofstate fit. A change in the electronic structure of Ta is found at high pressures using this procedure, and a possible new transition in H 2

The approximation, (∂K' s /∂T) P=0 ≃ 0, where K' s = (∂Ks/∂P) S , K S is the adiabatic bulk modulus, and P, T, and S represent pressure, temperature, and entropy, is often used, or implicitly assumed, to simplify the interpretation of lower mantle properties when determining plausible mineralogy of that region. This approximation is now found to be unsatisfactory, and a more general treatment gives a positive value which increases strongly with K's. We find that (∂K' s /∂T) P=0 is (6 to 23) × 10 -5 K -1 for the lower mantle. The behavior of this higher order derivative necessitates a reconsideration of the lower mantle equation-of-state fitting by requiring a higher value of K' S0 (K' S at P = 0) than usually considered. The value of K' S0 (T 0 ) where T 0 is the potential temperature, cannot reasonably be less than about 4.2 and may be as high as 4.6, depending on the uncertain value of K' S0 (290 K) for silicate perovskite. There is some trade-off between T 0 and composition, but if a simple mixture of perovskite and magnesiowüstite is assumed, the total iron content is found to be Fe/(Fe + Mg) = 0.22, independent of both T 0 and uncertainty in K' S0 for perovskite. This implausibly high iron content can be attributed to neglect of the presence of Ca perovskite. We therefore conclude that Ca perovskite is an important constituent of the lower mantle and that it is seismologically conspicuous

Equation of state fits to experimental P,V,T data were examined by the inversion of synthetic data sets using the thermoelastic parameters of MgSiO 3 perovskite. Our results show that by extending the pressure and temperature range to 130 GPa and 2500 K, the volume dependence of the Grüneisen parameter, q (∂lng/∂lnV), could be resolved to ~10% under the best circumstances. However, simulations also showed strong correlation between the bulk modulus, K T0 , and its pressure derivative, K' T0 , and q within the currently accepted uncertainty of elastic parameters for MgSiO 3 perovskite. We considered the effect of random error based on the reported uncertainty for different measurement techniques. Even though the laser heated diamond-anvil cell (LHDAC) technique has significantly larger temperature uncertainty, the ability to extend the pressure and temperature ranges allows for improved resolution of higher order thermodynamic parameters. However, systematic error from temperature inhomogeneity in the LHDAC sample could result in overestimation of q. We also performed Birch-Murhanghan-Debye (BMD) equation of state (EOS) fits for currently available data sets. Consistent with the simulation results, combining recent LHDAC (Fiquet et al. 1998) and resistance heated diamond-anvil cell (RHDAC) (Saxena et al. 1999) with lower P-T measurements (Ross and Hazen 1989; Wang et al. 1994; Utsumi et al. 1995; Funamori et al. 1996) we obtained q = 2.0(3) and γ 0 = 1.42(4). The difference between q = 2.0(3) and the normally assumed value of q =1 strongly affects calculated values for higher order thermoelastic parameters [e.g., a, (∂K T /∂T) P ] as well as first order parameters, such as density and bulk modulus at lower mantle conditions. However, possible systematic error sources need to be further investigated and measurements at higher P-T conditions promise to yield better constraints on the thermoelastic parameters of MgSiO 3 perovskite

New synchrotron X-ray diffraction data confirm our previous report of the transformation of the hexagonal close-packed (hcp) phase of iron to the Pbcm orthorhombic lattice (β-iron) at high P and T. The volume differences between the e and β, and the β and γ polymorphs are determined as 1.4 and 1.8%, respectively, indicating positive Clapeyron slopes between these polymorphs in the P-T phase diagram. All three polymorphs have a similar bulk modulus between 30 and 60 GPa. The Pbcm-polymorph can be observed in a metastable state as quenched from high T at high P and also at high T for P lower than 35 GPa where b-iron is not a stable phase. Metastability is possible because the gliding of the same dense atomic layers is involved in both T-induced ε-hcp to γ-fcc and ε-hcp to β-Pbcm transformations. These observations explain why a controversy exists on the structure and P-T stability field of β-iron. From our set of experiments, we estimate that Pbcmiron is stable above 35 GPa and 1500 K, and that the (γ, β, liquid-iron) triple point is located at about 55 GPa and 2400 K

Based on an in situ X-ray study of Fe and a review of the available data, we propose the following triple-points in the Fe phase diagram-the hexagonal closest packed (HCP), face-centered cubic (FCC), and the b phase: P = 40 (4) GPa at T = 1550 (100) K, the b-phase-FCC-melt; P = 60 (10) GPa at T = 2600 (100) K. We define the stability of b phase from a combination of new X-ray results on externally heated Fe between pressures of 37 to 300 GPa, as well as the previous data on externally heated and laser-heated samples. X-ray data on externally heated Fe, without any pressure medium, confirms the double hexagonal closest packed (DHCP) structure for the b phase. The HCP-b phase boundary has a very small negative dP/dT indicating the similarity of physical properties (molar volume, thermal expansion, and bulk modulus) between the two phrases, but a higher entropy and enthalpy for the β phase

We present a method by which the melting curves and corresponding densities along the liquidus of iron phases are calculated at high pressure. The melting curve of the e(hcp) iron phase is calculated from P = 0 to P = 330 GPa. The melting curve of the γ i (fcc) iron phase is calculated from P = 0 to P = 100 GPa. The point where these curves cross, near P = 50 GPa, is the location of the upper triple point on the melting curve. Our method combines the Lindemann melting equation with the Vinet isothermal equation of state. These equations are coupled by means of the Grüneisen parameter, γ, which is given by K' T (the pressure derivative of the isothermal bulk modulus, K T ) of the equation of state. By this thermodynamic formalism, we find values of volume, V; temperature, T; and the Grüneisen parameter, g, along the melting curve. This calculation requires the following thermal parameters: thermal expansivity, aV, at high T; (∂K T /∂T) P at high T; the melting temperature, T m , at P = 0; q = (∂ln g/∂ln r) T at high T; and g. The method also requires zero-pressure equation-of-state parameters: density, r, at T = 300 K; K T at T = 300 K; and (∂K T /∂P) T at T = 300 K. Our results do not confirm or deny the existence of the recently proposed b phase. We demonstrate that with our existing knowledge the physical parameters of the b and e phases may be so close to each other that these two phases need not be considered separately in discussing the physics of iron at core P,T conditions. We find that one set of reasonable thermoelastic parameters of the e phase reproduces the melting temperature and density at pressures of 50 and 240 GPa. Further, these parameters give agreement with previous estimates of the melting temperature and density at 330 GPa

We show that high-quality powder X-ray diffraction data, collected in diamond anvil cells, provide sufficient information for Rietveld refinement and determination of temperature factors. For the first time using a new method based on combination of thermal equation of state and measured mean-square atomic vibrations of high-pressure e-Fe phase, we determine Debye temperatures at pressure up to 300 GPa and temperature over 1000 K. We found that the Grüneisen parameter of e- iron could be described by Anderson’s (1967) equation with parameters γ 0 = 1.78(6), q = 0.69(10) with fixed V 0 = 6.73 cm3/mol

The Grüneisen parameter (γ) is of considerable importance to Earth scientists because it sets limitations on the thermoelastic properties of the lower mantle and core. However, there are several formulations of the Grüneisen parameter in frequent use which not only give different values for γ at ambient pressure but also predict a varying dependence of γ as a function of compression. The Grüneisen parameter is directly related to the equation of state (EOS), yet it is often the case that both the form of γ and the EOS are chosen independently of each other and somewhat arbitrarily. In this paper we have assessed some of the more common definitions of the Grüneisen parameter and the EOS, and have applied them to a test material. Of the EOS considered, when compared against ab initio compressional data for hcp-Fe as our exemplar, we find that the fourth order logarithmic and Vinet relations describe the material with the highest accuracy. Of the expressions for γ considered, it has been suggested, on theoretical grounds, that the modified free-volume formulation should be expected to give the most realistic description of the thermoelastic behavior of a material. However, when we use the fourth order logarithmic EOS to obtain the compressional behavior of the various Grüneisen parameters, we find that there is, in fact, poor agreement between the modifiedfree- volume formulation and the Mie-Grüneisen parameter obtained directly from ab initio free energy calculations on hcp-Fe. We conclude that none of the analytical forms of gamma are sufficiently sophisticated to describe the thermoelastic behavior of real materials with great accuracy, and care must therefore be taken when attempting to model the thermoelastic behavior of solids to ensure that the appropriate γ (ideally obtained from experiments or ab initio calculations) and equations of state are used