In order to determine the P-V-T equation of state of ε-iron, in situ X-ray observations were carried out at pressures up to 80 GPa and temperatures up to 1900 K using the Kawai-type high pressure apparatus equipped with sintered diamond anvils which was interfaced with synchrotron radiation. The present results indicate the unit cell volume at ambient conditionsV0 = 22.15(5) Å3, the isothermal bulk modulus KT0 = 202(7) GPa and its pressure derivative K′T0 = 4.5(2), the Debye temperature θ0 = 1173(62) K, Grüneisen parameter at ambient pressure γ0 = 3.2(2), and its logarithmic volume dependence q = 0.8(3). Furthermore, thermal expansion coefficient at ambient pressure was determined to be α0(K−1) = 3.7(2) × 10−5 + 7.2(6) × 10−8(T-300) and Anderson-Grüneisen parameterδT = 6.2(3). Using these parameters, we have estimated the density of ε-iron at the inner core conditions to be ∼3% denser than the value inferred from seismological observation. This result indicates that certain amount of light elements should be contained in the inner core as well as in the outer core but in definitely smaller amount.
 The earth's core is considered to be predominantly composed of iron alloy [e.g., Birch, 1952]. Although, it was clarified that the inner core is solid from the seismological observation in early 20th century, the crystal structure of iron in the inner core condition has long been an open question. However, recent progress in high pressure experiment using the diamond anvil cell (DAC) have revealed that the iron and its Ni-alloy assumes the hexagonal closed packed structure (hcp,ε-phase) under inner core conditions [Tateno et al., 2010, 2012]. Therefore, the P-V-T equation of state of ε-iron is crucial information to understand the nature of the inner core.
 Although a large number of volume measurements of ε-iron at high pressure have been carried out by means of in situ X-ray diffraction using DAC [Mao and Bell, 1979; Jephcoat et al., 1986; Huang et al., 1987; Mao et al., 1990; Dubrovinsky et al., 1998, 2000; Dewaele et al., 2006; Boehler et al., 2008; Ono et al., 2010] and the Kawai-type multianvil apparatus (KMA) [Funamori et al., 1996; Uchida et al., 2001], P-V-T equation of state for ε-iron has not been constructed in precision high enough to make reliable discussion on state of the inner core. These situations are due to inherent limitations of the high pressure technologies adopted. DAC experiments were mostly conducted at room temperature or up to 750 K at most due to difficulty to realize stable and uniform heating of sample.Dubrovinsky et al. [1998, 2000]expanded the experimental conditions to 1370 K and 68 GPa by adopting the wire-heating technique, and up to 1350 K and 300 GPa using externally heated DAC, respectively. In the conventional KMA, on the other hand, as the maximum pressure was to 30 GPa, the temperature range to observeε-iron was limited to ∼1300 K by the stability ofε-iron [Funamori et al., 1996; Uchida et al., 2001]. Nevertheless, stable and uniform heating of sample and precise measurement of volume from large sample are essential advantages of the KMA in determination of P-V-T relation.
 In the present study, we adopted the KMA equipped with sintered diamond (SD) anvil which now makes it possible to ordinarily generate pressure up to ∼100 GPa [Yamazaki, 2011]. We obtained volume data of ε-iron up to 80 GPa and 1900 K by means of in situ X-ray diffraction over ad-value range from 0.85 to 3 Å. We derived a set of thermoelastic parameters forε-iron, from which we estimated a density ofε-iron under inner core conditions. We also defined the phase boundary betweenγ- andε-iron under pressures from 20 to 50 GPa.
 The high-pressure and high-temperature in situ X-ray diffraction experiment was conducted at the white X-ray beam line BL04B1 in the synchrotron radiation facility SPring-8. The energy dispersive powder X-ray diffraction was adopted at a diffraction angle of ∼6° together with a germanium solid-state detector. A polychromatic X-ray beam collimated to the dimensions of 50μm horizontally × 100 μm vertically was irradiated to the sample through the anvil gaps of the Kawai-type cell. A multi-channel analyzer was used to acquire photons in a range of 20–140 keV, which was calibrated using characteristic fluorescence X-ray lines of Cu, Mo, Ag, Ta, Pt, Ag and Pb. The precision of the energy measurements was approximately 30–100 eV per channel. In order to acquire an averaged diffraction profile minimizing the effect of grain growth, we oscillated the press between 0° and 6° with respect to the direction of incident X-ray [Katsura et al., 2004].
 The Kawai-type cell composed of eight SD cubic anvils with edge length of 14 mm and truncated edge length (TEL) of 1.0 mm was set in the DIA type press (SPEED-Mk.II/Madonna) [Katsura et al., 2004]. Octahedron of MgO + 5% Cr2O3 with edge length of 4.1 mm as pressure medium and cylindrical TiB2 + BN + AlN heater were used (Figure 1). Their low absorption of X-rays make it possible to observe X-ray diffraction of the sample and a radiographic image of the cell as well. Temperature was monitored by a W97%Re3%-W75%Re25% thermocouple with a diameter of 0.05 mm. The pressure effect on EMF of the thermocouple was ignored. Fluctuation of temperature was within ±2 K throughout all the runs. Six runs of volume measurement were conducted using the slightly different octahedral specimen configurations shown in Figure 1. In the first three runs (M1067, M1079 and M1081), we used a gold foil (thickness ∼20 μm) as a pressure marker and sintered iron aggregates from sponge iron (purity: 99.9%) as a sample which was surrounded by thin layer (∼5 μm) of MgO formed by sputtering for electrical insulation from the heater (Figures 1a and 1b). In the other runs (M1154, M1211 and M1230), we prepared the powdered mixture of Au + MgO as the pressure marker and the pre-sintered iron + MgO mixture as the sample (Figures 1c and 1d). The MgO power mixed prevents the grain growth of Au and iron at high temperatures. Sample and pressure marker were separated by using Al2O3 capsules. In two runs (M1079 and M1081), samples were located at the center of the heater (Figure 1b). In three runs (M1067, M1154 and M1211), pressure marker was located at the center of heater (Figures 1a and 1c). In run M1230, the thermocouple junction was located at center of the heater (Figure 1d). From these various setting of the junction, we can evaluate the effect of temperature gradient through the heater on volume measurement.
 Pressure was determined from the volume of Au obtained from (111), (200), (220), (311), (222) ,(400), (331), and (420) diffraction peaks based on the equation of state of Au proposed by Tsuchiya . In some cases, we could not use one or two of them by overlapping with diffraction peaks from the other materials. The volume of ε-iron was determined using eight diffraction peaks: i.e., (100), (002), (101), (102), (110), (103), (112), and (201). The diffraction patterns of the sample and pressure marker were separately taken along the cooling path after heated up to 900–1900 K to minimize the effect of the deviatoric stress in every 200 K step at constant press load.
3. Results and Discussion
 We examined iron grains in the recovered specimen (M1211) by using a field emission scanning electron microscope (FE-SEM) with an energy dispersive spectrometer (EDS). The grain size was a few micron meters. The chemical analysis of the iron grains showed that the grains are free from the contamination by the other element (less than 0.5 wt. %).
3.1. Phase Boundary
 The iron transformed from α (bcc) to ε-iron on the initial compression. No phases other thanε- andγ-iron were observed in all experiments at higher than 10 GPa.Figure 2 represents example of the diffraction patterns of γ-iron (a) at 45.9 GPa and 1700 K andε-iron (b) at 73.8 GPa and 1700 K. We often observed appearance of peaks ofγ-iron on increasing temperature and reversely relative growth of the peaks ofε-iron on the subsequent cooling. These observations were used as criteria to determine the stable phase. Moreover,ε-iron can be regarded as stable phase underP-T conditions at which only ε-iron was present at temperatures higher than 1300 K because the appearance and disappearance of the phases was quick (<1 min) at these temperatures. Thus, we determined the phase boundary betweenγ- andε-iron to be linear equation expressed as,P (GPa) = 0.043 × T (K) −22.7 (Figure 3a).
 In all experiments, the volume measurements were carried out in the stability field of ε-iron. Although small peaks from the Al2O3 capsule, TiB2 + BN + AlN heater, and MgO pressure medium were recognized in the profile, the eight peaks of ε-iron were mostly free from overlapping with these peaks (Figure 2b). The lattice constants and volumes for ε-iron determined in the present study are summarized in Table S1 in theauxiliary material and plotted on the P-T planes in Figure 3. We could not recognize significant systematic difference in the lattice constants due to difference in cell assemblies shown in Figure 1.
 The P-V-T data set of ε-iron was fitted to a single EOS model based on the Mie-Grüneisen equation of state [e.g.,Poirier, 2000] as
where total pressure P(V,T) was expressed by the sum of pressure at a standard temperature of T0 = 300 K, PT0(V), and thermal pressure, ΔPth(V,T) = Pth(V,T)-Pth(V, T0). For the T0-isotherm, we adopted a third-order Birch-Murnaghan equation;
and Vinet equation;
where V0 is the volume under ambient conditions, and KT0 and K′0 are the isothermal bulk modulus and its pressure derivative at zero pressure. The thermal pressure was calculated following Dewaele et al. ;
where R is the gas constant, γ is the Grüneisen parameter expressed as γ = γ0(VP,T0/V0)q, and θ is the Debye temperature as a function of volume as θ = θ0exp[(γ0-γ)/q], where γ0 and θ0are Grüneisen parameter and Debye temperature at ambient pressure and ambient temperature, respectively. The first term of the right-hand side ofequation (4) is the quasiharmonic Debye thermal pressure, which represents the main part of Pth. The second and third terms are the anharmonic and electronic thermal pressures, respectively. In metal, especially ε-iron, large anharmonic and electronic effects are expected [Belonoshko et al., 2008]. We used the parameters, a0 = 3.7×10−5 K−1, m = 1.87, e0 = 1.95×10−4 K−1 and g = 1.339 as fixed values obtained from ab initio anharmonic and electronic thermal pressures [Alfe et al., 2001]. The obtained parameters are listed in Table 1 and the corresponding isothermal equation of states are drawn in Figures 3c and 3d. The T0-isotherms of third-order Birch-Murnaghan and Vinet equations resulted in virtually identical lines in the experimental pressure range (Figures 3c and 3d). The results of fitting the present data to thermal pressure of equation of state (equation (4)) are shown in Figure 4, indicating that thermal pressure depends not only on temperature but also on volume. At 80 GPa and 2000 K, the thermal pressure, ΔPth, amounts to 18.6 GPa and 12% of which is contributed by the anharmonic and electronic thermal pressures. The γ0 and θ0 are slightly changed from 3.2 to 3.5 and from 1168 to 1174 K, respectively, whereas q is significantly changed from 0.8 to 0.4, when the anharmonic and electronic thermal pressure terms in equation (4) were ignored in fitting.
Table 1. Parameters for Equation of States for ε-Iron Determined in the Present Studya
 Bulk modulus (196 and 202 GPa) determined in the present study are relatively high among values reported by previous workers which are bounded by 120 GPa [Jephcoat et al., 1986] and 212 GPa [Huang et al., 1987]. K0 and K0′ for α-iron are well established to be 166 GPa and 5.29 [Guinan and Beshers, 1968], respectively, indicating K = 205 GPa at 7.4 GPa at which α-ε transition occurs at ambient temperature [Akimoto et al., 1987]. It should be noted that the values of KT0 = 202 GPa and K′T0 = 4.5 obtained in the present study yield KT = 235 GPa at 7.4 GPa, which is reasonably larger than that for α-iron, corresponding to higher density ofε-iron [cf.Wang, 1970]. Present values for θ0 and q agree with those of Uchida et al.  (θ0 = 998(85) K and q = 0.91(7)) within uncertainty, whereas γ0 in the present study is larger than Uchida's value of γ0 = 1.36(8). On the other hand, γ0 = 2.8 and q = 1.9, and γ0 = 2.2 and q = 1.3 were estimated from shock experiments [Brown, 2001].
 The change in the thermal expansion coefficient with pressure can be expressed as
where α0 is the thermal expansion coefficient at ambient pressure and δTis the Anderson-Grüneisen parameter. Here we assumed linear temperature dependence ofα0, α0 = a1 + a2(T-300), wherea1 is thermal expansion coefficient at 300 K and 0 GPa and a2 is its temperature derivative. In the present study, a1 and a2 were determined to be 3.7(2) × 10−5 K−1 and 7.2(6) × 10−8 K−2, respectively, by fitting the volume to equation (5) (Figure 3d). These values are consistent with those by Uchida et al. , but slightly lower than that by Dubrovinsky et al. . The δT value of 6.2(3) is in good agreement with 6.1 by Uchida et al. .
 Density of the inner core at around the inner and outer core boundary region (∼330 GPa) is estimated to be 12.76 g/cm3 from seismic observation (e.g., PREM [Dziewonski and Anderson, 1981]). Using thermoelastic parameters listed in Table 1, we calculated the density of iron at 330 GPa and 6000 K to be 13.12(1.67) and 13.18(1.68) g/cm3using a third-order Birch-Murnaghan equation and Vinet equation, respectively, corresponding to the density deficits of 2.7 and 3.1%, respectively. These values are close to those byDubrovinsky et al. , Uchida et al.  and Dewaele et al. . Because Ni-alloying intoε-iron is expected to increase density by approximately 0.4% [Mao et al., 1990], it is highly likely that the inner core contains certain amounts of light elements such as Si, C, O, S, and H, as similar situation as in the outer core but in definitely smaller amounts.
 We thank Y. Tange for discussion and F. Xu for experimental support, and anonymous reviewers to improve the paper. X-ray diffraction measurements were conducted at BL04B1, SPring-8, Japan (proposals 2011B1452, 2012A1702).
 The Editor thanks Denis Andrault and an anonymous reviewer for their assistance in evaluating this paper.