1 Introduction
Seismic velocities such as the compressional velocity (VP) and shear velocity (VS) are the most important physical properties of the Earth's interior. The profile of the density and seismic velocity of the Earth's interior have been modeled by seismological observations such as the preliminary Earth reference model, PREM (Dziewonski and Anderson, 1981). Sound velocity measurements of Fe and Fe–light element alloys have been performed under high pressure conditions using various methods, such as shockwave experiments (e.g., Brown and McQueen, 1986), inelastic X-ray scattering (IXS) (e.g., Antonangeli et al., 2004; Mao et al., 2012; Ohtani et al., 2013; Sakamaki et al., 2016), nuclear resonance inelastic X-ray scattering (NRIXS or NIS) (e.g., Lin et al., 2005), impulsive stimulated light scattering (e.g., Crowhurst et al., 2004), and picosecond acoustics (e.g., Decremps et al., 2014). The Earth's inner core is considered to be mainly composed of an Fe alloy with some amounts of light elements to account for the core density deficit (Birch, 1964).
The previous data by NRIXS (Gao et al., 2011) and IXS (Fiquet et al., 2009) are generally consistent with each other up to 50 GPa. However, the measurements by Fiquet et al. (2009) indicate that the density–VP relation has an inflection at pressures above 67.4 GPa and that a softening of the dispersion curve appears at 83 GPa. Fiquet et al. (2009) considered that these features might be caused by the magnetic transition at 60 GPa (Vocadlo et al., 2002). However, this transition was reported by Lin et al. (2004) to occur at 25 GPa, and we may need to find another reason for the inflection in Birch's relation. Therefore, it is essential to perform IXS measurements at pressures above 60 GPa and to confirm whether the inflection in the density–VP curve and magnetic softening exists at around 60–80 GPa. In this paper, we determined the sound velocity values of Fe3C based on IXS measurements up to 86 GPa and high temperature, and extrapolated the data to the Earth core conditions.
2 Experimental procedure
2.1 Sample preparation
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.
![](/geoscience/article/CRGEOS_2019__351_2-3_190_0/jpg/src/tex/figures/gr1.jpg)
X-ray diffraction pattern of the sample IXS_Fe3C_002 (83 GPa and room temperature). (a) One-dimensional and (b) two-dimensional diffraction patterns.
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.
2.2 IXS at SPring-8
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.
3 Results
3.1 Measurements of IXS of Fe3C
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 VP, 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):
(1) |
![](/geoscience/article/CRGEOS_2019__351_2-3_190_0/jpg/src/tex/figures/gr2.jpg)
Examples of the IXS spectra collected at 44.4 GPa and 300 K (a), 40.4 GPa and 1400 K (b), and 67.4 GPa and 300 K (c). LA phonon peaks of Fe3C (red diamonds), TA phonon peaks of diamond, and elastic peaks are observed. The data are fitted with a set of Lorentzian functions.
![](/geoscience/article/CRGEOS_2019__351_2-3_190_0/jpg/src/tex/figures/gr3.jpg)
Dispersion curves obtained at (a) 67.4 GPa, 300 K, (b) 68.9 GPa, 2300 K, and (c) 84.3 GPa, 300 K.
Experimental conditions, density, Qmax, VP, and VS.
Run # | Pressure (GPa) | Error | Temperature (K) | Density (g/cm3) | Error | Qmax (nm−1) | Error | VP (km/s) | Error | VS (km/s) | Error |
(11 11 11)/1.5 meV | |||||||||||
IXS_Fe3C_002 | 84.3 | 1.5 | 300 | 9.75 | 0.02 | 14.73 | 0.72 | 8.73 | 0.15 | 3.73 | 0.27 |
IXS_Fe3C_003 | 44.4 | 1.2 | 300 | 9.02 | 0.02 | 13.65 | 0.61 | 7.91 | 0.11 | 3.30 | 0.21 |
IXS_Fe3C_004 | 67.4 | 0.8 | 300 | 9.45 | 0.01 | 14.98 | 0.81 | 8.61 | 0.15 | 3.91 | 0.26 |
(999)/3meV | |||||||||||
IXS103 | 68.9 | 0.7 | 2300 | 9.29 | 0.01 | 13.80 | 0.65 | 8.36 | 0.14 | 3.42 | 0.45 |
IXS107 | 32.8 | 0.9 | 300 | 8.78 | 0.02 | 13.33 | 0.4 | 7.65 | 0.08 | 3.20 | 0.19 |
IXS108 | 40.4 | 1.2 | 1400 | 8.77 | 0.03 | 12.98 | 0.36 | 7.86 | 0.08 | 3.42 | 0.26 |
IXS115 | 49.9 | 1.0 | 300 | 9.13 | 0.02 | 13.55 | 0.81 | 8.25 | 0.22 | 3.74 | 0.37 |
3.2 Birch's law of Fe3C
The VP 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:
(2) |
![](/geoscience/article/CRGEOS_2019__351_2-3_190_0/jpg/src/tex/figures/gr4.jpg)
Density and VP data determined in this work. Solid circles show the data obtained at 300 K at the Si (11 11 11) and Si (9 9 9) configurations. Open circles are the data determined at high temperatures determined at the (9 9 9) configuration. Thick and thin lines show Birch's law for all data and for 300 K only, respectively.
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 VP and density relations including 300 K and high temperature were fitted by the following equation:
(3) |
The VS of Fe3C was calculated using the VP valued determined here. VS is expressed as follows by using VP:
(4) |
4 Discussion
4.1 The stability of Fe3C in the inner core
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.
4.2 Birch's law of Fe3C and the sound velocity of Fe3C 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.
![](/geoscience/article/CRGEOS_2019__351_2-3_190_0/jpg/src/tex/figures/gr5.jpg)
Comparison of Birch's law of Fe3C between this study and previous studies (Fiquet et al., 2009; Gao et al., 2008; Gao et al., 2011). Red solid and open circles show the density and VP profiles by this study at 300 K and high temperatures, respectively. Blue triangles show the results of IXS measurements by Fiquet et al. (2009). Green solid and open diamonds show the results of NRIXS measurements by Gao et al. (2008, 2011). Crosses,× are the plots for the PREM inner core. A red dashed line shows Birch's law of Fe3C at 300 K, a red line shows Birch's law for the entire data set including those at high temperatures. A brown dashed line is the Birch's law of hcp-Fe at 300 K, a brown line is that extrapolated to 5000 K. Red stars are the values for Fe3 C at ICB (329 GPa) and at the center of the core (CC, 364 GPa), whereas brown stars are those for hcp-Fe extrapolated to ICB and CC of the Earth. The hatched area represents the data for the Vp and density values for the Fe–Fe3C mixture.
VP 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. VP 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–VP and ρ–VS relations under the inner core conditions. VP 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–VP 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 VP 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 VP and VS 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 VP and VS 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 VP and VS of hcp-Fe drop at temperatures close to the melting point, which was called “the premelting effect”.
![](/geoscience/article/CRGEOS_2019__351_2-3_190_0/jpg/src/tex/figures/gr6.jpg)
Birch's law for VP and VS of Fe3C at the ICB and at CC conditions together with those of the PREM inner core at ICB and CC. Present estimations of VP and VS of Fe3C are by 12% and 48% faster than those of the PREM inner core at the ICB, assuming that ICB conditions are 329 GPa and 5000 K (e.g., Terasaki et al., 2011). Red solid circles are the present Birch's law for VP at room temperature, whereas red open circles are that at high temperature. Blue solid circles are that for the present VS data at room temperature, whereas blue open circles are that at high temperature. Eqs. (3) and (4) are shown as red and blue lines, respectively. Red triangles are VP and blue triangles are VS of the PREM inner core. The blue and red stars are VP and VS of Fe3C at ICB and CC conditions, respectively.
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 VP and VS 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.
Acknowledgement
This work was supported by JSPS KAKENHI Grant No. 22000002 and No. 15H05748 to E.O. The synchrotron radiation experiments were performed under contracts of the SPring-8 Proposal Nos. 2014B1269, 2014B1465, 2015A1539, 2015A1627, 2015B1202, 2015B1334, 2016A1171, 2016B1419, 2017B1214, 20160096, and 20170051.