[S-IT18] Planetary cores: Structure, formation, and evolution

[SIT18-03] Impurity resistivity of the Earth’s inner core

*Hitoshi Gomi^1,2, Kei Hirose^1,2 (1.The University of Tokyo, 2.Tokyo Institute of Technology)

Keywords:Inner core, Impurity resistivity, KKR-CPA

The Earth's inner core has various seismological features (anisotropy, hemispherical asymmetry, inner-inner core). To consider the origin of these features, the possibility of thermal convection, and hence the thermal conductivity, is important. The thermal conductivity of metals can be estimated from their electrical resistivity using the Wiedemann-Franz law. The impurity resistivity of hexagonal close-packed iron (hcp Fe) has been determined for substitutional alloys up to ternary systems by the first principles Korringa-Kohn-Rostoker (KKR) method combined with the coherent potential approximation (CPA). In this study, we extend the method by Gomi and Yoshino (2018) to calculate the impurity resistivity of alloys containing both substitutional and interstitial impurities. By using the Kubo-Greenwood formula, we computed the electrical resistivity of hcp Fe_1-x-yNi^s_xL^s_yLⁱ_z (L^s = Si, C, N, O, P, S, or H, Lⁱ = C, N, O, or H), where the superscript s represents the substitution site and the superscript i represents the octahedral interstitial site. The impurity concentration was set to 0 < x < 0.15, 0 < y < 0.3, and 0 < z < 0.5. Linear regression was performed on the electrical resistivity of the resultant 6105 alloys. We set the concentration of each impurity element as the explanatory variable, considering Mathiesen's rule. The results of the substitutional hcp Fe_0.9-yNi^s_0.1L^s_y (L= Si, C, N, O, P, S, or H) ternary alloy were compared with a previous study (Zidane et al. 2020) with the same composition, which shows that the previous calculation systematically overestimated the impurity resistivity of the inner core. The previous study also showed that hydrogen has a higher impurity resistivity than other impurities, but no such feature was found in the present study. The results for hcp FeHⁱ_z, an interstitial alloy, showed that hydrogen in the interstitial sites hardly contributes to the electrical resistivity, consistent with previous experiments (Ohta et al. 2019). A linear regression on hcp Fe_1-x-yNi^s_xL^s_yLⁱ_z (L^s = Si, C, N, O, P, S, or H, Lⁱ = C, N, O, or H) quaternary alloy with both substitutional and intrusive impurities shows that the constant term in the regression is 33 μΩcm. As a result, the prediction performance was significantly different at around ~40 μΩcm. This behavior may be due to the resistivity saturation. Furthermore, we performed calculations for a six-component alloys of Fe_1-x-yNi^s_x(Si,S)^s_y(H,C)ⁱ_z (0 < x < 0.05, 0 < y < 0.3, 0 < z < 0.5), which have two light elements each at the substitutional and interstitial sites. The results confirmed that the linear regression model of the quaternary alloy correctly predicted the results of the six-component alloys. Adding the contribution of electrical resistivity due to lattice vibrations to the impurity resistivity obtained from first-principles calculations, the thermal conductivity of the Fe_1-x-yNi^s_x(Si,S)^s_y(H,C)ⁱ_z alloys, which satisfies the density deficit in the inner core, is estimated to be in the range 150-257 W/m/K. The temperature profile due to thermal conduction inside the inner core is determined by the thermal conductivity and the age of the inner core (e.g., Buffett 2009). If this thermal conduction profile is lower than the adiabatic temperature profile, thermal convection cannot occur in the inner core. For example, suppose the thermal conductivity of the inner core is 160 W/m/K, and the age of the inner core is 1 billion years. In that case, the inner core temperature due to thermal conduction is ~100 K lower than the adiabatic temperature, making thermal convection impossible. For the estimated range of thermal conductivity in this study > 150 W/m/K, the present inner core is thermally stable even if the inner core age is very young (e.g., 0.5 billion years old).