Thermoelasticity of Hexagonal Close-Packed Iron from the Phonon Density of States

Citation

Murphy, Caitlin Anne (2012) Thermoelasticity of Hexagonal Close-Packed Iron from the Phonon Density of States. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/GQ17-GV90. https://resolver.caltech.edu/CaltechTHESIS:02162012-075245736

Abstract

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 (ε-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 ε-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 ε-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-Mössbauer 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 ε-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 ε-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 ε-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 ε-Fe. In addition, we found a steady increase in the Lamb-Mössbauer factor with compression, which suggests restricted thermal atomic motions at outer core pressures. This behavior is related to the high-pressure melting behavior of ε-Fe via Gilvarry’s reformulation of Lindemann’s melting criterion, which we used to obtain the shape of ε-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 ε-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 ε-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 ε-Fe’s total phonon DOS with a generalized scaling law and, in turn, constrain the ambient temperature vibrational Grüneisen parameter. We also used the volume dependence of our previously mentioned vD to determine the commonly discussed Debye Grüneisen parameter (γD), which we found to be ~10% smaller than our vibrational Grüneisen parameter at any given volume. Finally, applying our determined vibrational Grüneisen parameter to a Mie-Grüneisen 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 ε-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 ε-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 ε-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 ε-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 ε-Fe at 300 K via established EOS parameters, and the vibrational Grüneisen parameter and temperature dependence of the vibrational thermal pressure via thermodynamic definition.

Thesis Files