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# Sound Velocities of Hot Dense Iron: Birch's Law Revisited

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Science  24 Jun 2005:
Vol. 308, Issue 5730, pp. 1892-1894
DOI: 10.1126/science.1111724

## Abstract

Sound velocities of hexagonal close-packed iron (hcp-Fe) were measured at pressures up to 73 gigapascals and at temperatures up to 1700 kelvin with nuclear inelastic x-ray scattering in a laser-heated diamond anvil cell. The compressional-wave velocities (VP) and shear-wave velocities (VS) of hcp-Fe decreased significantly with increasing temperature under moderately high pressures. VP and VS under high pressures and temperatures thus cannot be fitted to a linear relation, Birch's law, which has been used to extrapolate measured sound velocities to densities of iron in Earth's interior. This result means that there are more light elements in Earth's core than have been inferred from linear extrapolation at room temperature.

The properties of Earth's iron-rich core have been inferred from estimates of iron density at high pressures and temperatures and from measurements of compressional-wave (VP) and shear-wave (VS) velocities passing through the core (113). These data have indicated that Earth's core is less dense than pure iron by approximately 10% for the outer core and 3% for the inner core, suggesting the existence of light elements in the core. On the other hand, Birch's law, a linear sound velocity–density relation (2, 14, 15), has also been used to extrapolate measured sound velocities at high pressures and room temperatures to inner core conditions without considering the temperature effect (9, 12). This linear extrapolation has suggested that the inner core is mainly made of Fe-Ni alloy. The nuclear-resonant inelastic x-ray scattering (NRIXS) technique provides a direct probe of the phonon density of states (DOS) of the resonant isotope (1618) using the 14.4125-keV transition of 57Fe. VP and VS of hexagonal close-packed (hcp) Fe have been measured up to 153 GPa at 300 K (10, 19). However, the effect of temperature on the sound velocity measurements of Fe in static studies is not well understood. Here we report the static NRIXS study of the sound velocities of hcp-Fe up to 73 GPa and 1700 K in a laser-heated diamond anvil cell (LHDAC), and we discuss the temperature effect on the sound velocities and Birch's law.

We conducted NRIXS experiments in an LHDAC at Sector 3 of the Advanced Photon Source (APS) at Argonne National Laboratory (20, 21). Energy spectra were obtained by tuning the x-ray energy (±70 meV) around the nuclear transition energy of 14.4125 keV and collecting the Fe K-fluorescence (the emission of an x-ray photon via the transition of an atomic electron into an unoccupied 1s state) radiation that was emitted with time delay relative to the incident x-ray pulses. We used a quasiharmonic model to extract the phonon DOS from the NRIXS spectra (Fig. 1) according to the procedure described in (1618). With the NRIXS technique, we measured the spectrum of the self-correlation function of the position of the Fe atoms (17). In the model, the atomic motions relative to the temperature-dependent averaged position are assumed to be harmonic under the given conditions of pressure, temperature, and other parameters. Thermal effects, such as expansion and change of force constants with atomic distances, were allowed to change but the vibrations were still assumed to occur in a harmonic potential. The average kinetic energy and force constant independently derived from the moments of the measured spectra were consistent with the values evaluated from the quasiharmonic model (17), indicating the validity of the model to our high-pressure/temperature data (22). The Debye sound velocity (VD) was derived from parabolic fitting of the low-energy regime of the DOS (1618), and the vibrational, elastic, and thermodynamic parameters were obtained by the integration of the DOS. We then calculated the thermal equation-of-state (EOS) parameters of hcp-Fe using the thermal EOS from previous studies (2325) and the Birch-Murnaghan EOS (26). The adiabatic bulk modulus at zero pressure (K0S) is $Math$(1) where K0T is the isothermal bulk modulus at zero pressure (23), α is the thermal expansion coefficient (23), γ is the Grüneisen parameter (γ = 1.78) (25), and T is the temperature. The Birch-Murnaghan EOS is used to calculate the isothermal bulk modulus at high pressures (KT) and the adiabatic bulk modulus at high pressures (KS). The KS, density (ρ), and VD are used to solve for the aggregate VP, VS, and shear modulus G by the following equations (10) $Math$(2) $Math$(3) $Math$(4) where Vϕ is the bulk sound velocity calculated from the thermal EOS parameters of KS and ρ. The derivation of VS is relatively insensitive to the differences in the EOS data (27). Our results at high pressures and room temperatures are consistent with those of a previous study (10). At high temperatures, the bulk sound velocity (Vϕ) followed Birch's law (Vϕ is linearly related to the density and mean atomic weight; dVϕ/dT = 0) (14), whereas VP, VS, and G did not (Fig. 2). At a pressure of ∼54 GPa, VP decreased by ∼7%, VS decreased by ∼14%, and G decreased by ∼28%, with a temperature increase of 1000 K. The effect of temperature on the sound velocities at constant density is smaller than the effect at constant pressure; i.e., at a density of ∼10.25 g cm–3, VP decreased at a rate of 0.00035 km s–1 K–1 (dVP/dT), VS decreased by 0.00046 km s–1 K–1 (dVS/dT), and G decreased by 0.035 GPa K–1 (dG/dT). These values are in general agreement with the Vϕ-density linear relation; if Vϕ is linearly related to the density without temperature effect, then VP(dVP/dT) – 4/3VS(dVS/dT) = 0 (from Eq. 2). X-ray diffraction spectra showed that the samples after laser heating were in the polycrystalline hcp structure at high pressures without significant preferred orientation, suggesting that the strong effect of temperature on the sound velocities cannot be explained simply by the elastic anisotropy in highly textured hcp-Fe, which can account for a few percent of the difference in VP (12). Different thermal pressure conditions varying from no thermal pressure effect to full thermal pressure effect (28) have been used to test the systematic errors in the temperature effect on the sound velocities. We found that the uncertainties in the thermal pressure alone were too small to result in a significant temperature effect on the sound velocities, in particular, the temperature effect on VS.

Extrapolated sound velocities of hcp-Fe at 3000 and 6000 K, obtained by combining our study at moderate pressure and temperature with a previous NRIXS study at high pressures and 300 K and with shock-wave data at high pressure and high temperature, show that the effect of temperature on the sound velocities of Fe is significant at moderate pressures, but weakens under inner core pressures because a highly compressed Fe has a smaller thermal expansion (Fig. 3) (3, 9, 10, 12, 2931). Because the temperature of the inner core is believed to be close to the melting curve of Fe, the extrapolated VS of hcp-Fe in the inner core should be corrected to even lower values. The small deviation in VP between shock-wave data (3) and the previous NRIXS study (10) at high pressures and room temperature also suggests that the temperature effect on VP is suppressed under inner core conditions, whereas the large difference in the VS indicates that temperature has a strong effect on VS even under core pressures (Fig. 3). Theoretical calculations on the elasticity of hcp-Fe predicted that VS and G would decrease with increasing temperature at a constant density of 13.04 g cm–3 (11).

Birch pointed out the likely temperature effect on the sound velocities in his original paper in 1961 (2). Our results confirm this idea. It has been shown that the addition of a light element such as Si or S into Fe increases VP and VS under high pressures (32, 33). Considering the temperature effect on VP and VS of hcp-Fe at inner core pressures and 6000 K (20), a few percent of light elements alloyed with Fe are still needed in the inner core to increase VP to match seismic models (Fig. 3). This results in more light elements in Earth's inner core than has been suggested from the linearly extrapolated VP of hcp-Fe at high pressures and room temperature (12).

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Materials and Methods

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Table S1

References

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