The starting material, Fe₃C, was synthesized from a mixture of Fe and graphite powders by heating at 3 GPa and 1273 K for 12 h using a Kawai-type multianvil apparatus. The starting material of Fe₃C was confirmed to be a single phase of cementite by X-ray diffraction (XRD) and scanning electron microscope (SEM) observations, i.e. it was a single phase of cementite without chemical zoning and interstitial phases.

High pressure was generated using a symmetric-type DAC. Various culet sizes of the diamond anvils from 200 to 350 μm were used, depending on the required experimental pressure conditions. A tungsten gasket was preindented to a thickness of 50 − 70 μm, and an 80 − 120 μm-diameter hole was drilled into the gasket to form the sample chamber. A thin foil of the starting material was made by compressing the Fe₃C chip at room temperature by using opposite anvils (a cold compression technique) and by polishing it to the desired thickness. The XRD pattern of the sample at high pressure shows a clear ring pattern indicating no significant texture effect in the present measurements. A typical two-dimensional (2D) pattern of the sample at high pressure is shown in Fig. 1.

The Fe₃C foil was sandwiched between NaCl layers and was placed in the sample chamber. The NaCl layers served as a pressure-transmitting medium and as a pressure standard for the experiments.

The sound velocity of Fe₃C was measured by the IXS method at SPring-8 (Baron, 2010; Baron, 2016; Baron et al., 2000) using a Si (11 11 11) and Si (9 9 9) configuration. The Si (11 11 11) reflection at 21.747 keV provided a resolution of 1.4 meV in a 40 μm × 55 μm beam size. The momentum transfer, Q = 2 k₀ sin(2θ/2), where k₀ is the wave number of the incident photons and 2θ is the scattering angle, was selected by rotating the spectrometer arm in the horizontal plane. In this setup, IXS spectra were collected in the range of Q = 4.04–11.50 nm⁻¹ at each experimental pressure.

The Si (9 9 9) configuration provided an incident photon energy of 17.794 keV with an energy resolution of 2.8 meV full width at half-maximum (FWHM). The X-ray beam size was focused to 16 μm × 16 μm by a Kirkpatrick–Baez (KB) mirror pair. In this configuration, the IXS spectra were obtained in the range of Q = 7.0–10.3 nm⁻¹ at each experimental pressure. The momentum resolution was set to about 0.4 nm⁻¹ full width. two room temperature and two high temperature runs made at 1400 K and 2300 K were conducted on this configuration. We used a portable double-sided laser heating and temperature measurement system (Fukui et al., 2013) combined with a symmetrical DAC for the determination of the sound velocity at high pressure and high temperature at BL35XU. The laser beam size was 20–30 μm in diameter, which was larger than the X-ray beam size (16 μm × 16 μm). We applied a flattop laser beam using a beam shaper.

Spectra were measured for about 8–12 h at room temperature and 6–8 h at high temperature. The shorter time at high temperature was due to the higher intensity of IXS signals at higher temperature. We monitored the laser heating spot by a charge coupled device (CCD) camera and confirmed that the heating spot was in the correct place of the sample. The position of the laser spot and the heating temperature were controlled during heating and the temperature was recorded every 15 min during the IXS measurements at high temperatures. The fluctuation in temperature was ±200 K during the heating experiments, whereas the temperature gradient at the heating spot was very small by using a flat top beam shaper (Sakamaki et al., 2016). Therefore, the temperature uncertainty was ±200 K at high temperature in the present experiments.

For Fe₃C density determination, XRD patterns of samples were collected using a flat panel (FP; C9732DK, Hamamatsu Photonics K.K.) installed in the IXS optical systems on both setups. The camera length between the sample and the FP was calibrated by a standard crystal of CeO₂.

IXS measurements were carried out at pressures from 33 to 86 GPa. Typical IXS spectra obtained at 44.4 GPa and 300 K, 40.4 GPa and 1400 K, and 67.4 GPa and 300 K are shown in Fig. 2. The spectrum is characterized by an elastic contribution centered at zero energy and inelastic contributions from Fe₃C and diamond derived from the diamond anvils. The peak of the longitudinal acoustic (LA) phonons of Fe₃C is clearly visible between the peak of the elastic scattering of the sample and the transverse acoustic (TA) phonon of diamond. The LA phonons of diamond would be observed at the higher energy position (e.g., Shibazaki et al., 2012). The energy positions of phonons were extracted by fitting the data with a set of Lorentzian functions. To determine V_P, the obtained phonon dispersion relations of the energy position and momentum transfers were fitted using a sine function as shown below (e.g., Fiquet et al., 2004):

The V_P of Fe₃C as a function of the density is shown in Fig. 4. Birch's law of Fe₃C at 300 K was found to be:

We also plotted the sound velocity and density values at high temperatures in Fig. 4. The temperature dependency of Birch's law is not clearly observed and we can ignore it, which is consistent with previous works (e.g., Gao et al., 2011). All data sets of V_P and density relations including 300 K and high temperature were fitted by the following equation:

The V_S of Fe₃C was calculated using the V_P valued determined here. V_S is expressed as follows by using V_P:

Nakajima et al. (2009) reported the phase relationships in the Fe–C system up to 30 GPa based on in situ XRD experiments using a Kawai-type multianvil apparatus. They found that Fe₃C is stable up to 30 GPa at low temperatures and that it melts incongruently to form Fe₇C₃ and liquid at higher temperatures. Lord et al. (2009) determined the Fe–Fe₃C eutectic temperature and the melting temperatures of Fe₃C and Fe₇C₃ up to 70 GPa. There were obvious discrepancies with the melting curves of Fe₃C and Fe₇C₃ reported by Nakajima et al. (2009) and Lord et al. (2009). In addition, Lord et al. (2009) predicted that Fe₃C is not stable above about 80 GPa and decomposes into Fe and Fe₇C₃, and suggested that the inner core is composed of Fe + Fe₇C₃. Sata et al. (2010) reported that Fe₃C is stable up to at least 187 GPa at 300 K. Liu et al. (2016) reported that Fe₃C is stable up to about 140 GPa and it decomposes to Fe and Fe₇C₃ at 157 GPa and 2422 K. In contrast, Tateno et al. (2010) and our recent studies (Takahashi, 2014; Takahashi et al., 2013) confirmed the stable existence of Fe₃C at least up to 192 GPa and 4320 K. A recent study on melting experiments in the Fe–Fe₃C system revealed that Fe₃C is present at least up to 150 GPa (Morard et al., 2017). Thus, Fe₃C might exist under the inner core conditions.

Various studies have been conducted on the magnetic transition of Fe₃C. Lin et al. (2004) reported a paramagnetic-to-nonmagnetic transition at around 25 GPa based on X-ray emission spectroscopy (XES). Prescher et al. (2012) reconcile previous investigations and propose that there is a ferromagnetic (fm) to paramagnetic (pm) transition at 10 GPa and a pm to nonmagnetic (nm) transition around 22 GPa. As our IXS measurements cover the pressure range from 32.8 to 84.3 GPa, and Fe₃C is considered to be nm, we can safely extrapolate our results to higher pressures. Based on XES, Chen et al. (2018) claim that the pm–nm transition is completed only at around 50 GPa. Therefore, a low-pressure region in our measurement may contain the effects of pm Fe₃C. Because we assumed the Fe₃C sample is nm in our IXS measurements, our extrapolation to higher pressures could contain an uncertainty.

Fig. 5 shows the comparison of Birch's law of Fe₃C between this and previous studies (Fiquet et al., 2009; Gao et al., 2008; Gao et al., 2011). The slope of our Birch's law is close to that determined by NRIXS (Gao et al., 2011). Although our Birch's law apparently shows a discrepancy from that reported by Fiquet et al. (2009), our data are consistent with theirs, except for the data measured at 68 GPa, which might include an uncertainty. We conducted two measurements of the sample at 67.4 GPa and 68.9 GPa. The dispersion curves are given in Fig. 3a and b. Our sound velocity is smaller than that reported by Fiquet et al. (2009). The X-ray diffraction profile of the sample (Fig. 1) indicates that there is almost no preferred orientation of the Fe₃C sample during compression. Therefore, we cannot specify the cause of difference between our results and those by Fiquet et al. at around 68 GPa. Fig. 3c shows our dispersion curve determined at 84.3 GPa, where Fiquet et al. (2009) reported softening of the dispersion curve. We could not observe such softening of the dispersion curve at this pressure.

V_P under the ICB condition was determined by Eq. (3). The density of Fe₃C at the inner core condition was calculated by the EoS of Fe₃C (Sata et al., 2010). Here, we ignored the effect of temperature on Birch's law of ρ vs. V_P as our preliminary measurement at high temperature was consistent with that under room temperature conditions. A negligible temperature effect of Fe₃C is consistent with the results by Gao et al. (2011) and Wood et al. (2013), and we used Eqs. (3) and (4) for the ρ–V_P and ρ–V_S relations under the inner core conditions. V_P was plotted as a function of ρ up to the pressure of the center of the Earth's core in Fig. 5. If the inner core is composed of the Fe–Fe₃C system, ρ–V_P should be placed in the shaded area in Fig. 5 assuming that the inner core is the ideal mixture of hcp-Fe and Fe₃C (e.g., Badro et al., 2007). There is no overlap of our extrapolated Birch's law for the V_P data with that of PREM. Thus, it is difficult to explain that the Earth's inner core is composed only of the Fe–Fe₃C mixture.

Fig. 6 shows Birch's law for V_P and V_S of Fe₃C, Eqs. (3) and (4), extrapolated to the inner core boundary (ICB) and to the center of the core (CC), together with those of PREM at ICB and CC. Gao et al. (2011) suggested that the deviation from room temperature Birch's law at high temperature may range from negligible (minimum) to 30%. Our high temperature results shown in Fig. 6 indicate that the deviation at high temperature can be negligible. Present estimations of V_P and V_S of Fe₃C are 12% and 48% faster than those of the PREM inner core at the ICB condition, assuming that ICB is 329 GPa and 5000 K (e.g., Terasaki et al., 2011). The ab initio calculations by Martorell et al. (2013) suggested that V_P and V_S of hcp-Fe drop at temperatures close to the melting point, which was called “the premelting effect”.

Morard et al. (2017) determined the eutectic temperature of Fe–Fe₃C at the core–mantle boundary (CMB) to be 2950 K, whereas Liu et al. (2016) reported that the solidus temperature at CMB is 3200 K. Extrapolation of both results to ICB conditions by Simon's equation indicates that the solidus temperatures are 3700 and 5000 K, respectively. Thus, the temperature of 5000 K at ICB is close to the Fe–Fe₃C eutectic temperature extrapolated to the ICB pressure. The inner core temperature is close to or exceeds the melting temperature of Fe₃C. Thus, the slower V_P and V_S values of the PREM inner core at ICB compared with those for Fe₃C could be attributed to the premelting effect of Fe₃C by analogy to pure Fe (Martorell et al., 2013), or to the effect of partial melting in the Fe–Fe₃C system. The separation of the partial melt might not be so effective because of its Fe-rich nature compared with that of the solid Fe carbides, although we need a quantitative evaluation of the density difference between the solid and the partial melt in the inner core in future.