Pressure-volume data for epsilon iron and platinum to earth-core pressures and above from one-parameter universal equation of state
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Grüneisen parameter
Outer core
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The first comprehensive PVT description of ε-Fe is presented by collecting the complete data set of previous experiments on ε iron, covering a pressure and temperature range of 6-306 GPa and 293-2255 K, respectively. A single set of equation of state parameters, which is able to cover the entire pressure and temperature range has been determined. The root mean square misfit of the residuals is 1.17 GPa for the PVT version of the Birch-Murnaghan equation of state. The equation of state uses an absolute reference frame (T = 0 K and P = 0 GPa) and follows a path, which eliminates the pressure effect on the volume coefficient of thermal expansion and the temperature effect on the pressure derivative of the bulk modulus.
Atmospheric temperature range
Root mean square
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Isobaric process
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We report here the longitudinal and shear sound velocities on polycrystalline cerium under hydrostatic pressure across the iso-structural γ-α phase transition up to 4.4 GPa. Comparing with previous methods, the pressure-density relation of Ce has been calculated by integrating with the initial travel time and pressure without any fitting. The pressure correction of the Gruneisen parameter and linear expansion coefficient are taken into account during the integration process. The sound velocities, bulk modulus, shear modulus, Debye temperature, and vibrational entropy are achieved and have been compared with previous results. The bulk modulus of cerium in α phase agrees with the previous results determined by neutron and x-ray diffraction. The Debye temperature above and below the phase transition are and , respectively. The difference of the Debye temperature from respective experiment is found and has been expounded. We consider that the vibrational entropy change per atom of 0.44 k B as the Kondo collapse of 17% volume change, and 0.70 k B as the total change from γ phase to complete α phase.
Hydrostatic pressure
Elasticity
Hydrostatic equilibrium
Speed of Sound
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Outer core
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Iron is the main constituent in Earth’s core, along with ~5 to 10 wt% Ni and some light elements (e.g., H, C, O, Si, S). This thesis explores the vibrational thermodynamic and thermoelastic properties of pure hexagonal close-packed iron (e-Fe), in an effort to improve our understanding of the properties of a significant fraction of this remote region of the deep Earth and in turn, better constrain its composition. In order to access the vibrational properties of pure e-Fe, we directly probed its total phonon density of states (DOS) by performing nuclear resonant inelastic x-ray scattering (NRIXS) and in situ x-ray diffraction (XRD) experiments at Sector 3-ID-B of the Advanced Photon Source (APS) at Argonne National Laboratory. NRIXS and in situ XRD were collected over the course of ~14 days at eleven compression points between 30 and 171 GPa, and at 300 K. Our in situ XRD measurements probed the sample volume at each compression point, and our long NRIXS data-collection times and high-energy resolution resulted in the highest statistical quality dataset of this type for e-Fe to outer core pressures. Hydrostatic conditions were achieved in the sample chamber for our experiments at smaller compressions (P ≤ 69 GPa) via the loading of a neon pressure transmitting medium at the GeoSoilEnviroCARS (GSECARS) sector of the APS. For measurements made at P > 69 GPa, the sample was fully embedded in boron epoxy, which served as the pressure transmitting medium. From each measured phonon DOS and thermodynamic definitions, we determined a wide range of vibrational thermodynamic and thermoelastic parameters, including the Lamb-Mossbauer factor; vibrational components of the specific heat capacity, free energy, entropy, internal energy, and kinetic energy; and the Debye sound velocity. Together with our in situ measured volumes, the shape of the total phonon DOS and these parameters gave rise to a number of important properties for e-Fe at Earth’s core conditions. For example, we determined the Debye sound velocity (vD) at each of our compression points from the low-energy region of the phonon DOS and our in situ measured volumes. In turn, vD is related to the compressional and shear sound velocities via our determined densities and the adiabatic bulk modulus. Our high-statistical quality dataset places a new tight constraint on the density dependence of e-Fe’s sound velocities to outer core pressures. Via comparison with existing data for iron alloys, we investigate how nickel and candidate light elements for the core affect the thermoelastic properties of iron. In addition, we explore the effects of temperature on e-Fe’s sound velocities by applying pressure- and temperature-dependent elastic moduli from theoretical calculations to a finite-strain model. Such models allow for direct comparisons with one-dimensional seismic models of Earth’s solid inner core (e.g., the Preliminary Reference Earth Model). Next, the volume dependence of the vibrational free energy is directly related to the vibrational thermal pressure, which we combine with previously reported theoretical values for the electronic and anharmonic thermal pressures to find the total thermal pressure of e-Fe. In addition, we found a steady increase in the Lamb-Mossbauer factor with compression, which suggests restricted thermal atomic motions at outer core pressures. This behavior is related to the high-pressure melting behavior of e-Fe via Gilvarry’s reformulation of Lindemann’s melting criterion, which we used to obtain the shape of e-Fe’s melting curve up to 171 GPa. By anchoring our melting curve shape with experimentally determined melting points and considering thermal pressure and anharmonic effects, we investigated e-Fe’s melting temperature at the pressure of the inner–core boundary (ICB, P = 330 GPa), where Earth’s solid inner core and liquid outer core are in contact. Then, combining this temperature constraint with our thermal pressure, we determined the density of e-Fe under ICB conditions, which offers information about the composition of Earth’s core via the seismically inferred density at the ICB. In addition, the shape of the phonon DOS remained similar at all compression points, while the maximum (cutoff) energy increased regularly with decreasing volume. As a result, we were able to describe the volume dependence of e-Fe’s total phonon DOS with a generalized scaling law and, in turn, constrain the ambient temperature vibrational Gruneisen parameter. We also used the volume dependence of our previously mentioned vD to determine the commonly discussed Debye Gruneisen parameter (γD), which we found to be ~10% smaller than our vibrational Gruneisen parameter at any given volume. Finally, applying our determined vibrational Gruneisen parameter to a Mie-Gruneisen type relationship and an approximate form of the empirical Lindemann melting criterion, we predict the vibrational thermal pressure and estimate the high-pressure melting behavior of e-Fe at Earth’s core pressures, which can be directly compared with our previous results. Finally, we use our measured vibrational kinetic energy and entropy to approximate e-Fe’s vibrational thermodynamic properties to outer core pressures. In particular, the vibrational kinetic energy is related to the pressure- and temperature-dependent reduced isotopic partition function ratios (β-factors) of e-Fe and in turn, provide information about the partitioning behavior of solid iron in equilibrium processes. In addition, the volume dependence of vibrational entropy is directly related to the product of e-Fe’s vibrational component of the thermal expansion coefficient and the isothermal bulk modulus, which we find to be independent of pressure (volume) at 300 K. In turn, this product gives rise to the volume-dependent thermal expansion coefficient of e-Fe at 300 K via established EOS parameters, and the vibrational Gruneisen parameter and temperature dependence of the vibrational thermal pressure via thermodynamic definition.
Hydrostatic equilibrium
Diamond anvil cell
Hydrostatic pressure
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Shock‐wave (Hugoniot) data for initially porous and nonporous samples of iron are inverted to yield values of the Grüneisen parameter (γ), adiabatic bulk modulus ( K s ), and coefficient of thermal expansion (α) along the Hugoniot to pressures of about 150 GPa (1.5 Mbar). This represents the first reliable estimate of thermal properties (e.g., α) to such high pressures based directly on experimental data. The values of γ fit the conventional function γ = γ 0 ( V / V 0 ) n but with n significantly larger than 1, while extrapolation formulas suggested to date for α appear not to provide the best fit to the data. The present analysis yields values of γ between about 1.4 and 1.0 (essentially temperature independent) and α between about 1.0 and 0.4×10 −5 K −1 throughout the earth's core, therefore implying that simple dynamical models of the core are quite viable. These results provide experimental confirmation of the Vaschenko‐Zubarev/Irvine‐Stacey (or ‘free‐volume’) formulation for γ of iron at high pressures, as well as support for Stacey's recent model of the thermal state of the core. The data on density, sound speed, and bulk modulus of iron are extrapolated and corrected to pressures and temperatures existing throughout the earth's core, and compared with current seismological information. This leads to the following conclusions: (1) both densities and bulk moduli in the outer core are less than those of Fe under equivalent conditions (by about 10 and 12%, respectively) but (2) their gradients through the outer core are consistent with gross chemical homogeneity (i.e., uniform intermixing of Fe and a lighter, more compressible element or compound); (3) both densities and bulk moduli for the inner core are compatible with those of iron, suggesting that (4) the inner core‐outer core boundary is likely to be a compositional as well as a phase boundary. Assuming that the outer core consists of Fe and a lighter element or compound, X , the constraints on density, bulk modulus and mass fration of X which must be simultaneously satisfied are given.
Grüneisen parameter
Speed of Sound
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The isothermal equation of state (EOS) for gold has been determined by powder x-ray diffraction experiments up to 123 GPa at room temperature. We have performed experiments independently in two institutions to check the consistency of the results. A He-pressure medium was used to minimize the effect of uniaxial stress on the sample volume and ruby pressures. The stress state in the He-pressure medium gradually becomes nonhydrostatic above about 30 GPa, with the magnitude of the uniaxial stress largely depending on experiments. Since the measured lattice spacings deviate under different stress states, it is a likely cause of the disagreement of the EOS parameters found in the literature. The lattice spacing ${d}_{111}$ for the 111 reflection is least affected by the uniaxial stress in the case of gold. Hence we have calculated the sample volume from ${d}_{111}$ and fitted the obtained pressure-volume data to the Vinet form of EOS. The bulk modulus ${B}_{0}$ at atmospheric pressure was fixed to 167 GPa, a value well established by ultrasonic measurements. The fit gives the pressure derivative of the bulk modulus at atmospheric pressure as ${B}_{0}^{\ensuremath{'}}=5.5(1)$ for the current ruby pressure scale after Zha et al. [Proc. Natl. Acad. Sci. U.S.A. 97, 13494 (2000)]. Alternatively, if we use a different calibration of this standard [Phys. Rev. B 75, 024115 (2007)], we obtain ${B}_{0}^{\ensuremath{'}}=5.9(1)$, which is in excellent agreement with the ultrasonic measurements and first-principles calculations. Discussions are given to the use of gold as a pressure standard and the hydrostaticity of the He-pressure medium.
Isothermal process
Lattice (music)
Lattice constant
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The all-electron, full-potential linear combinations of Gaussian type orbitals--fitting function technique has been used to perform high-precision total-energy calculations on \ensuremath{\alpha}-alumina (corundum). The calculations yield zero-pressure lattice parameters that are in 0.3% agreement with experiment and symmetry-preserving elastic constants that agree with experiment to within 5%. The bulk modulus and pressure derivatives of the lattice parameters are also in good agreement with existing data. The calculated energies have been used to generate an analytical equation of state (EOS) for corundum that should be valid up to at least 250 GPa. The fitted EOS agrees with room temperature diamond anvil cell data up to 175 GPa to within the known limitations of the experimental data. The c/a ratio, band gap, and tetragonal shear modulus have been determined for pressures up to 250 GPa. The c/a ratio varies by less than 3% over the entire pressure range. For pressures above 150 GPa, the band gap changes from direct to indirect and the tetragonal shear modulus softens. The linear pressure coefficient of the band gap is estimated to be 5.1 meV/kbar at zero pressure.
Tetragonal crystal system
Shear modulus
Lattice constant
Corundum
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A revised pressure scale for NaCl is proposed as an update for the 30-year-old work of Decker. An alternative approach to the analysis is utilized in conjunction with more recent data. The zero-Kelvin compression curve is parameterized using local basis functions (splines) and constrained by accurate pressure-volume-temperature data. Thermal pressures are estimated within a quasiharmonic framework using a volume-dependent Grüneisen parameter and the Debye thermal energy. In the pressure regime extending to 5 GPa uncertainties in pressure (based on measured volumes) are estimated to be less than 1%. Uncertainty increases to 1.5% at 10 GPa and 3% at 25 GPa. The largest contribution to systematic uncertainty at the highest pressures is the lack of knowledge of the volume dependence of the Grüneisen parameter. Misfit of other calculated thermodynamic properties with respect to data is relatively small. On the basis of the current analysis, pressures determined using the older Decker calibration are low. Along the 300 K isotherm, apparent errors in the Decker scale are as large as −3% (−0.3 GPa at 10 GPa, −0.47 GPa near 18 GPa, and −0.37 GPa at 25 GPa). At higher temperatures the apparent errors are smaller. At 1100 K and 20 GPa the error is −0.2 GPa.
Debye model
Grüneisen parameter
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Equation of state of gold (Au) is revised using the remeasured shock compression data at pressures up to 580 GPa with including the electronic free-energy contribution. The model, even though determined only using pressure-scale-free data, can reproduce not only the shock compression data but also the zero-pressure thermodynamic properties with remarkable accuracy. Previous models for the EOS of copper, silver, and MgO that were constructed using as a basis old shock compression data are found to underestimate the pressure seriously (up to $\ensuremath{\sim}12%$ at 200 GPa and 300 K). Moreover, we redetermine the EOS model of platinum (Pt) through the same procedure. The determined models of Au and Pt are found mutually highly consistent, which provide quite comparable pressure values in extensive $P$, $T$ range. These are expected to be more reliable primary pressure standards than previous models.
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