Abstract
It is widely known that the Earth's Fe dominant core contains a certain amount of light elements such as H, C, N, O, Si, and S. We report the results of first-principles calculations on the band structure and the impurity resistivity of substitutionally disordered hcp and fcc Fe based alloys. The calculation was conducted by using the AkaiKKR (machikaneyama) package, which employed the Korringa-Kohn-Rostoker (KKR) method with the atomic sphere approximation (ASA). The local density approximation (LDA) was adopted for the exchange-correlation potential. The coherent potential approximation (CPA) was used to treat substitutional disorder effect. The impurity resistivity is calculated from the Kubo-Greenwood formula with the vertex correction. In dilute alloys with 1 at. % impurity concentration, calculated impurity resistivities of C, N, O, S are comparable to that of Si. On the other hand, in concentrated alloys up to 30 at. %, Si impurity resistivity is the highest followed by C impurity resistivity. Ni impurity resistivity is the smallest. N, O, and S impurity resistivities lie between Si and Ni. Impurity resistivities of hcp-based alloys show systematically higher values than fcc alloys. We also calculated the electronic specific heat from the density of states (DOS). For pure Fe, the results show the deviation from the Sommerfeld value at high temperature, which is consistent with previous calculation. However, the degree of deviation becomes smaller with increasing impurity concentration. The violation of the Sommerfeld expansion is one of the possible sources of the violation of the Wiedemann-Franz law, but the present results could not resolve the inconsistency between recent electrical resistivity and thermal conductivity measurements. Based on the present thermal conductivity model, we calculated the conductive heat flux at the top of terrestrial cores, which is comparable to the heat flux across the thermal boundary layer at the bottom of the mantle. This indicates that the thermal stratification may develop at the top of the liquid core of super-Earths, and hence, chemical buoyancies associated with the inner core growth and/or precipitations are required to generate the global magnetic field through the geodynamo.
Highlights
Because the electrical current and the heat are mainly transported by mobile electrons in the metallic core, it is important to understand the electron scattering mechanisms in Fe-based alloys at high pressure and temperature to estimate the thermal conductivity and the electrical resistivity. Gomi et al (2013) proposed the core resistivity model that the resistivity saturation was firstly taken into account
Instead of the Linde’s rule, we will show the relative importance of the impurity resistivity of light alloying elements in fcc and hcp Fe by means of the Korringa-Kohn-Rostoker method combined with the coherent potential approximation (KKR-CPA) (Oshita et al, 2009; Kou and Akai, 2018), which successfully reproduces the impurity resistivities of Si and Ni in hcp Fe (Gomi et al, 2016)
We conducted KKR-CPA-DFT calculations of impurity resistivity of Fe-based light elements (C, N, O, Si, S) or Ni alloys, which is consistent with recent diamond-anvil cell experiments (Gomi and Hirose, 2015; Gomi et al, 2016; Suehiro et al, 2017; Zhang et al, 2018)
Summary
Because the electrical current and the heat are mainly transported by mobile electrons in the metallic core, it is important to understand the electron scattering mechanisms in Fe-based alloys at high pressure and temperature to estimate the thermal conductivity and the electrical resistivity. Gomi et al (2013) proposed the core resistivity model that the resistivity saturation was firstly taken into account. The experimentally determined Lorenz number of hcp Fe, which is calculated from recent laser heated diamond-anvil cell (LHDAC) measurements on the electrical resistivity (Ohta et al, 2016) and the thermal conductivity (Konôpková et al, 2016), exhibit substantially smaller than the Sommerfeld value Even though these LHDAC results may have large uncertainty (Dobson, 2016), this fact suggests potential violation of the Widemann-Franz law. The violation of the Sommerfeld expansion may cause 2– 43% deviation from the Sommerfeld value of the Lorenz number, if we adopt the calculated electron density of states (DOS) of fcc and hcp Fe reported by Boness et al (1986) This argument is limited to pure Fe. As well as the impurity resistivity calculation, the KKR-CPA method can simulate the DOS of disordered alloys (Gomi et al, 2016, 2018). The model was applied to the planetary cores with various planetary mass from 0.1 to 10 times Earth mass (section Heat Flux at the CMB of SuperEarths)
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