Thermal Equation of State of Fe₃C to 327 GPa and Carbon in the Core

Abstract

The density and sound velocity structure of the Earth’s interior is modeled on seismological observations and is known as the preliminary reference Earth model (PREM). The density of the core is lower than that of pure Fe, which suggests that the Earth’s core contains light elements. Carbon is one plausible light element that may exist in the core. We determined the equation of state (EOS) of Fe₃C based on in situ high-pressure and high-temperature X-ray diffraction experiments using a diamond anvil cell. We obtained the P–V data of Fe₃C up to 327 GPa at 300 K and 70–180 GPa up to around 2300 K. The EOS of nonmagnetic (NM) Fe₃C was expressed by two models using two different pressure scales and the third-order Birch–Murnaghan EOS at 300 K with the Mie–Grüneisen–Debye EOS under high-temperature conditions. The EOS can be expressed with parameters of V₀ = 148.8(±1.0) ų, K₀ = 311.1(±17.1) GPa, K₀^′ = 3.40(±0.1), γ₀ = 1.06(±0.42), and q = 1.92(±1.73), with a fixed value of θ₀ = 314 K using the KBr pressure scale (Model 1), and V₀ = 147.3(±1.0) ų, K₀ = 323.0(±16.6) GPa, K₀^′ = 3.43(±0.09), γ₀ = 1.37(±0.33), and q = 0.98(±1.01), with a fixed value of θ₀ = 314 K using the MgO pressure scale (Model 2). The density of Fe₃C under inner core conditions (assuming P = 329 GPa and T = 5000 K) calculated from the EOS is compatible with the PREM inner core.

1. Introduction

The density of the Earth’s interior has been determined using the preliminary reference Earth model (PREM) [1]. The Earth’s core is composed mainly of iron. The density of iron at specific pressure and temperature conditions has been determined in high-pressure and high-temperature experiments and from theoretical calculations (e.g., [2,3]). These studies implied that the density of pure hcp-Fe is higher than that obtained from seismological models such as PREM [1]. The density deficit is estimated to be ~10% for the outer core [4,5,6,7] and 2–5% for the inner core [8,9]. A recent study estimated the density of hcp-Fe to be 13.8–14.0 g/cm³ [10,11] under inner core conditions, which corresponds to a density deficit of the inner core of 3.5–5.1%. Therefore, light elements may be contained within the Earth’s inner core. Carbon is one candidate for these light elements, and Fe₃C is thought to be a constituent of the inner core (e.g., [12]). There are several works on the phase relations of the Fe–C system. Liu et al. [13] reported that Fe₃C is stable up to 150 GPa and that it melts incongruently into Fe₇C₃ and liquid at high temperatures in this pressure range and decomposes to Fe₇C₃ and Fe at higher pressures and temperatures. However, recent experiments report contradicting results. For example, Mashino et al. [14] reported Fe₃C is stable up to 250 GPa, and they did not find any evidence for decomposition of Fe₃C at around 150 GPa, as was reported by Liu et al. [13]. More recently, Takahashi et al. [15] studied the stability of Fe₃C up to and above 300 GPa and showed that it was stable up to this pressure, whereas it melted incongruently into Fe₇C₃ and liquid under the pressure and temperature conditions studied. The stability of Fe₃C is also supported by a recent shock experiment by Hu et al. [16]. They reported no evidence for decomposition of Fe₃C to Fe₇C₃ and Fe in the pressure range from 80 GPa to 248 GPa along the Hugoniot, which is consistent with the results of Mashino et al. [14] and Takahashi et al. [15]. Therefore, Fe₃C is an important potential constituent of the inner core.

The compression behavior of Fe and Fe alloys is related to their magnetic properties. The transition of magnetic Fe₃C to a nonmagnetic (NM) phase is estimated to occur at 70 GPa [17]. However, based on X-ray emission spectroscopy data [18] and the Mössbauer spectroscopy data [19], it has been revealed that this magnetic transition finishes at 20–25 GPa. Although there is a discrepancy in the absolute pressure of this transition to an NM phase, Fe₃C will be NM under inner core conditions. The compression behavior of NM Fe₃C up to core pressure conditions was reported by Sata et al. [20], but these experiments were only carried out at room temperature. Litasov et al. [21] determined the compression behavior of ferromagnetic (FM) Fe₃C up to about 30 GPa and estimated the density of NM Fe₃C under inner core conditions by assuming that the compression behavior of NM Fe₃C is similar to that of FM Fe₃C, which they determined. Very recently, Hu et al. [16] conducted shock experiments of Fe₃C and estimated the density of Fe₃C under inner core conditions. However, it is not well resolved whether their data correspond to NM Fe₃C. Presently, there are no high-temperature static compression experiments on NM Fe₃C.

To precisely estimate the density of NM Fe₃C in the inner core, it is necessary to understand the compression behavior of NM Fe₃C under high-pressure and high-temperature conditions. In this study, we measured the compression behavior of NM Fe₃C using in situ X-ray diffraction (XRD) experiments in a double-sided, laser-heated diamond anvil cell and determined the equation of state (EOS) to estimate the amount of carbon in the inner core.

2. Materials and Methods

2.1. Sample Preparation

The starting material was a powdered sample of Fe₃C, which was synthesized from a mixture of iron and graphite powders that was heated to 1273 K at 3 GPa for 12 h using a 3000 ton, Kawai-type multianvil apparatus installed at Tohoku University, Japan. The high pressure was generated using a symmetric-type diamond anvil cell and a membrane diamond anvil cell. The culet sizes of the diamond anvils were 100 and 150 µm, respectively. A tungsten gasket was indented to a thickness of 25–40 µm, and a hole with a diameter of 30–50 µm was drilled into the indented gasket to form a sample chamber. A foil from the synthesized Fe₃C sample was made using a cold-compression technique. KBr and MgO were used as the pressure marker [22,23] and the thermal insulator/pressure medium, respectively, for the high-pressure and high-temperature experiments up to 181.7 GPa. The pressure marker of KBr was sandwiched by the Fe₃C layers, and then the KBr and Fe₃C layers were sandwiched between MgO. We sandwiched the Fe₃C foil between the SiO₂ powder of the thermal insulator/pressure medium for the both the experiments above 298.9 GPa and those at room temperature.

2.2. In Situ XRD Experiments at the BL10XU Beamline at Spring-8

The in situ XRD experiments were conducted at the BL10XU beamline at the Spring-8 facility (Hyogo, Japan) [24]. The sample was heated using a double-sided laser heating technique [25] employing an SPI fiber laser (λ = 1.070 µm). The shape of the fiber laser beam was adjusted to a flattop beam using the beam-shaping system at the BL10XU beamline. This technique enabled us to decrease the temperature gradient across the sample and achieve a homogenous heated area with a diameter of approximately 30 µm. The temperature was determined by fitting the emission spectra from the surface of the heated sample to Planck’s radiation law as a grey body formula using the typical wavelength range between 600 and 800 nm. The wavelength of the monochromatic X-ray beam and the distance between the sample and the X-ray detector were calibrated using XRD patterns from CeO₂ based on the double-cassette method. The typical wavelength of the X-rays was 0.4142(2)–0.4157(1) Å. The X-ray beam was collimated to a diameter of 15–20 µm. An imaging plate (IP) detector (RAXIS-IV; Rigaku, Tokyo, Japan) was used to collect the angle-dispersive XRD data. The exposure times for the IP were either 1 or 3 min. Each integrated XRD pattern, along with the 2θ angle (i.e., a one-dimensional XRD profile), was analyzed using an IP analyzer and PD Indexer software package [26].

The experimental pressures were determined using the EOS of KBr (B2) [22] and that of MgO [23] for the high-pressure and high-temperature experiments, whereas the compression curve of pyrite SiO₂ at 300 K was used for determination of pressure for the experiments exceeding 298.9 GPa [27]. The temperature distribution in the pressure medium in the double-heated diamond anvil cell (DAC) was evaluated from 3-dimensional numerical modeling [28]. This model calculation indicates that with a thickness of 1–2 µm, the temperature difference between the inner and anvil surfaces in the MgO layer is around 100–200 K at 2000 K, which is equivalent to the temperature uncertainty in the present experiment. Therefore, the temperature of the MgO layer in the DAC was considered to be the same as the sample temperature. The pressure and temperature conditions for the present experiments, based on the KBr pressure scale [22] and MgO pressure scale [23], are shown in Table S1 in Supplementary Materials.

3. Results

3.1. X-ray Diffraction Patterns of Fe₃C

In situ X-ray diffraction experiments were carried out from 71 GPa up to 182 GPa, and samples were heated up to around 2000 K at each pressure condition. The compression behavior of Fe₃C above 298.9 GPa at 300 K was obtained after quenching the sample from high temperatures above 1500 K, releasing stress during solid state compression. Its experimental pressures were determined by the equation of state of pyrite-type SiO₂ [27]. Previous studies show that the magnetic transition of Fe₃C to a nonmagnetic phase occurs at 20–25 GPa [13,19]. Therefore, Fe₃C is considered to be nonmagnetic under the conditions of the present experiments. Our results are summarized in Table S1. XRD patterns at pressures of 113–116 GPa and high temperatures are shown in Figure 1. XRD patterns of Fe₃C, KBr, and MgO were observed at all present experimental conditions. Fe₃C with the cementite structure was stable at all present pressure–temperature conditions.

3.2. Compression Behavior and the Thermal Equation of State of Fe₃C

The compressional data at 300 K was fitted to the third-order Birch–Murnaghan equation of state (3BM EOS) as follows:

$$P\left(V, 300\right)=\frac{3}{2}{K}_0\left[{\left(\frac{V_0}{V}\right)}^{\frac{7}{3}}-{\left(\frac{V_0}{V}\right)}^{\frac{5}{3}}\right]\left\{1-\frac{3}{4}\left(4-K_0^{\prime }\right)\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right]\right\}$$

(1)

where, V₀, K₀, and K₀^′ are the zero-pressure volume, the bulk modulus, and its pressure derivative, respectively. To investigate the thermoelastic parameters of Fe₃C, the Mie–Grüneisen–Debye (MGD) EOS can be fitted to P–V–T data obtained at 71–326 GPa and 300–2450 K. The MGD EOS is represented as follows (e.g., [29]):

$$P\left(V, T\right)=P\left(V, 300\right)+\Delta P_{th}$$

(2)

where the first and second terms denote the room temperature 3BM EOS (Equation (1)) pressure and the thermal pressure, respectively. The thermal pressure is expressed by the following expressions:

ΔP_th = (γ/V) [E(T, θ_D) − E(300, θ_D)]

(3)

$$E=9nRT{\left(\frac{T}{{\theta }_D}\right)}^3{{\displaystyle \int }}_0^{\frac{{\theta }_D}{T}}\frac{t^3}{e^t-1}dt$$

(4)

where E is internal thermal energy, $\gamma$ is the Grüneisen parameter, θ_D is the Debye temperature, n is the number of atoms per formula unit, and R is the gas constant. The Debye temperature and the Grüneisen parameter are expressed as a function of volume:

$$\gamma ={\gamma }_0{\left(\frac{V}{V_0}\right)}^q$$

(5)

$${\theta }_D={\theta }_0\exp \left(\frac{{\gamma }_0-\gamma }{q}\right)$$

(6)

where q is the volume dependence on the Grüneisen parameter. By fitting five parameters (V₀, K₀, K₀^′, γ₀ and $q$) with the Debye temperature θ₀ = 314 K [21], the following parameters were determined for NM-Fe₃C for the model using the pressure scale of KBr: V₀ = 148.8(1.0) ų, K₀ = 311.1(±17.1) GPa, K₀^′ = 3.40(±0.10), γ₀ = 1.06(±0.42), and $q$ = 1.92(±1.73) [22] (Model 1). Our compression curve of Fe₃C at 300 K and high temperatures (Model 1) is shown in Figure 2a,b. Using the MgO pressure scale [23], we obtained the following parameters for NM Fe₃C: V₀ = 147.3(±1.0) ų, K₀ = 323(±16.6) GPa, K₀^′ = 3.43(±0.09), γ₀ = 1.37(±0.33), and q = 0.98(±1.01), with the Debye temperature of 314 K (Model 2). The compression curve of Fe₃C at 300 K and high temperatures (Model 2) is shown in Figure 2c,d.

The values of K₀ and K₀^′ obtained in this study are shown in

Table 1. Fitting parameters of thermal equations of state of Fe₃C.

V0 (Å3)	K0 (GPa)	K0′	EOS	Phase	Reference
148.8 ± 1.0	311.1 ± 17.1	3.40 ± 0.10	3BM	NM	Model 1 (based on KBr EOS)
147.3 ± 1.0	323 ± 16.6	3.43 ± 0.09	3BM	NM	Model 2 (based on MgO EOS)
149.46	290	3.76	3BM	NM	Sata et al. [20]
148.9	317	4.3	3BM	NM	Vočadlo et al. [17]
155.3	174	4.8	3BM	-	Li et al. [30]
155.2	175	5	Vinet	PM	Litasov et al. [21]
154.42	194	4.6	Vinet	PM	Litasov et al., MGD [21]
155.26	175.4	5.1	3BM	FM	Scott et al. [31]
155.4	167	6.7	Vinet	FM	Ono and Mibe [32]

, together with those of the previous works. These values are close to the NM Fe₃C reported by Sata et al. [20]. On the other hand, the K₀ of NM Fe₃C is higher than that of ferromagnetic and paramagnetic (PM) Fe₃C, and the K₀^′ of the NM Fe₃C is lower than that of FM and PM Fe₃C [17,20,21,30,31]. Ono and Mibe [32] reported a significant reduction in b-axis associated to the phase transition from FM to NM Fe₃C at 55 GPa, although this discontinuity was assigned to the transition from PM to NM Fe₃C by Litasov et al. [21]. However, since our experiments were performed at pressures above 69 GPa (at which NM-Fe₃C is stable) and no magnetic transition exists in the present pressure range, we did not observe the discontinuity in any axis at high pressure. In this work, the MGD EOS parameters of our models (Models 1 and 2), together with those determined by Litasov et al. [21], are shown in

Table 2. Fitting parameters of MGD EOS for Fe₃C.

θ0 (K)	γ0	q	Phase	Reference
314 *	1.06 ± 0.42	1.92 ± 1.73	NM	Model 1 (based on KBr EOS)
314 *	1.37 ± 0.33	0.98 ± 1.01	NM	Model 2 (based on MgO EOS)
314	2.15	−0.03	PM	Litasov et al. [21]

* Fixed following Litasov et al. [21].

.

4. Discussion

Although there are ambiguities concerning the thermal state of the core, the temperatures at the inner core boundary (ICB) and at the center of the Earth are in the range of 5000–6000 K (e.g., [33]). The density of Fe₃C at the pressure of the ICB (329 GPa) has been calculated to be 12.83(±0.67) g/cm³ at 5000 K and 12.79(±0.73) g/cm³ at 6000 K, respectively. The density of Fe₃C at the pressure of the center of the core (364 GPa) has been calculated to be 13.21(±0.70) g/cm³ at 5000 K and 13.18(±0.76) g/cm³ at 6000 K, respectively. PREM data give a density of 12.76 g/cm³ and 13.09 g/cm³ at the ICB (329 GPa) and the center of the core (364 GPa), respectively.

Figure 3a shows a comparison of the density of Fe [10], NM Fe₃C at 300 K and 5000 K, and the PREM density [1] as a function of pressure. The thermal expansion of NM Fe₃C with temperature is small compared to hcp-Fe. Our results indicate that the density of Fe₃C is compatible with seismological data on the inner core, assuming T = ~5000 K at the ICB pressure. The present density values for NM Fe₃C at 5000 K are different from those estimated by Litasov et al. [21] based on FM Fe₃C. The estimation of the high-temperature density based on FM Fe₃C could be different from that of the real NM Fe₃C, although the recent density estimation of Fe₃C based on shock compression [16] is close to the estimation by Litasov et al. [21]. In Figure 3b, we show the density of NM Fe₇C₃ at 5000 K, as described by Chen et al. [34] using thermal parameters for the PM phase [35]. The extrapolated densities of NM Fe₃C are slightly greater than those of Fe₇C₃. However, we cannot quantitatively evaluate the difference in the present accuracy of the experiments.

We concluded that the density of the Earth’s inner core can be explained both by the Fe₃C and Fe₇C₃ density data at 5000~6000 K. Fe and Ni are thought to be major components in the Earth’s core; however, the Earth’s core may also contain carbon and other elements. We need to investigate the effects of Ni and the other elements on the density and sound velocity to form a conclusive argument regarding the composition of the inner core.