Equation of State of Liquid Iron under Extreme Conditions

The density of liquid iron has been determined up to 116 GPa and 4350 K via static compression experiments following an innovative analysis of diffuse scattering from liquid. The longitudinal sound velocity was also obtained to 45 GPa and 2700 K based on inelastic x-ray scattering measurements. Combining these results with previous shock-wave data, we determine a thermal equation of state for liquid iron. It indicates that Earth’s outer core exhibits 7.5%–7.6% density deficit, 3.7%–4.4% velocity excess, and an almost identical adiabatic bulk modulus, with respect to liquid iron.

Figure 1

Structural analyses of liquid iron at high pressures via in situ XRD measurements. (a) Structure factor $S(Q)$ of liquid iron up to 116.1 GPa determined from XRD measurements in this study, showing peak shifts to larger $Q$ values due to the compression of liquid iron. (b) Corresponding radial distribution functions, $g(r)$, determined in this study. Vertical scales are offset for clarity for both $S(Q)$ and $g(r)$ plots.

Figure 2

Distribution function $F(r)$ from run No. 1 calculated based on the present analytical method (red), in which the extension of $S(Q)$ and parameters of $s$, ${\alpha }_N$, and $\rho$ are determined simultaneously, demonstrating that our new method successfully reduces the oscillations at $r<r_{\min }$ and gives a precise liquid density from the slope. $F(r)$ calculated without extension of $S(Q)$ with assuming $s=1$ is shown by the blue line for comparison.

Figure 3

High-pressure inelastic x-ray scattering (IXS) measurements of liquid iron. (a) Typical IXS spectrum of liquid iron collected at 44.9 GPa and 2700 K at momentum transfer $Q=3.0{\mathrm{nm}}^{-1}$. The spectra include three components: a quasielastic peak near zero energy transfer (blue), longitudinal acoustic (LA) phonon mode of liquid Fe (red), and transverse acoustic (TA) phonon mode of diamond (green). The vertical axis is plotted on a logarithmic scale. (b) Longitudinal acoustic phonon dispersion of liquid iron at pressures from 16.0 to 44.9 GPa.

Figure 4

Density ($\rho$), $P$-wave velocity ($V_P$), and adiabatic bulk modulus ($K_S$) of liquid iron. (a),(b) Isothermal $P-\rho$ and $P-V_P$ relations calculated from our EOS for 2000 (blue), 3000 (green), 4000 (yellow), and 5000 K (red) (Table S II [10]). Dashed lines, 2000 (red) and 4000 K (yellow), are from shock-compression study [8]. Red symbols represent experimental data (circles, this study; squares, shock experiments [8]; crosses, multi-anvil (KMA) experiments [5]; diamonds, 1 bar data at 1811 K [39]). Consistency between the red and blue (fit results) symbols indicates that our EOS well reproduces all experimental data points. (c) Calculated isentropic temperature profiles with $T_{\mathrm{ICB}}=5800$ (red), 5400 (green), and 5000 K (blue) (Table S III [10]) (see Supplemental Material [10], Sec. IV). Dashed lines are those proposed by a previous study with a different Grüneisen parameter [8]. (d),(e),(f) Comparison of seismic observations (black circles, PREM [40]) with the $\rho$, $V_P$, and $K_S$ of liquid Fe under core pressures along the isentropic temperature profiles in (c). Uncertainties in the present estimates of $\rho$ and $V_P$ are $\sim 1\%$ (see the uncertainty band around each solid curve and Supplemental Material [10], Sec. IV for details). Dashed lines represent those proposed on the basis of earlier shock-wave data [8].