Review of Electrical Resistivity Measurements and Calculations of Fe and Fe-Alloys Relating to Planetary Cores

There is a considerable amount of literature on the electrical resistivity of iron at Earth’s core conditions, while only few studies have considered iron and iron-alloys at other planetary core conditions. Much of the total work has been carried out in the past decade and a review to collect data is timely. High pressures and temperatures can be achieved with direct measurements using a diamond-anvil cell, a multi-anvil press or shock compression methods. The results of direct measurements can be used in combination with first-principle calculations to extrapolate from laboratory temperature and pressure to the relevant planetary conditions. This review points out some discrepancies in the electrical resistivity values between theoretical and experimental studies, while highlighting the negligible differences arising from the selection of pressure and temperature values at planetary core conditions. Also, conversions of the reported electrical resistivity values to thermal conductivity via the Wiedemann-Franz law do not seem to vary significantly even when the Sommerfeld value of the Lorenz number is used in the conversion. A comparison of the rich literature of electrical resistivity values of pure Fe at Earth’s core-mantle boundary and inner-core boundary conditions with alloys of Fe and light elements (Si, S, O) does not reveal dramatic differences. The scarce literature on the electrical resistivity at the lunar core suggests the effect of P on a wt% basis is negligible when compared to that of Si and S. On the contrary, studies at Mercury’s core conditions suggest two distinct groups of electrical resistivity values but only a few studies apply to the inner-core boundary. The electrical resistivity values at the Martian core-mantle boundary conditions suggest a negligible contribution of Si, S and O. In contrast, Fe-S compositions at Ganymede’s core-mantle boundary conditions result in large deviations in electrical resistivity values compared to pure Fe. Contour maps of the reported values illustrate ρ(P, T) for pure Fe and its alloys with Ni, O and Si/S and allow for estimates of electrical resistivity at the core-mantle boundary and inner-core boundary conditions for the cores of terrestrial-like planetary bodies.


INTRODUCTION
The current interest in direct measurements and modelling of electrical resistivity (ρ) originates mainly from an interest in heat flow modelling of planetary cores. For terrestrial-like planetary bodies that contain predominantly Fe cores, the applications are thermal evolution of the core, which includes freezing of an inner core and sustenance of a dynamo. To model core thermal evolution, the adiabatic heat flow is needed and is normally calculated via the thermal conductivity (k). Although ρ and k of metals are directly related through electron transfer of charge and energy, respectively, the following quote indicates the tolerance for variation in each of these properties in relation to our understanding of core processes. "A factor of two or so uncertainty in ρ does not appear critical to dynamo theory but it has a strong impact on calculations of the thermal regime of the core." (Stacey and Anderson, 2001). The literature on ρ of pure Fe is rich and the values are scattered while the reported data on Fealloys is more scarce but less dispersed. The inconsistencies in measurements and modelling may be the result of different techniques in addition to the range of pressures and temperatures attributed to planetary cores. Much of the total work has been carried out in the past decade and Figure 1 shows the cumulative number of papers published on ρ of pure Fe and Fe-alloys during the past half-century. At the time of writing this article, the four studies labelled on Figure 1 combined for more than 1,000 citations and are viewed as responsible for the increased rate of activity following their publication. This review attempts to summarize both older and recent results to identify a range of reliable and representative values for ρ of the cores of terrestrial-like planetary bodies composed of pure Fe or Fe-alloys.
A general formulation of the core adiabatic heat flow is described as: where k c is k of the core, α is thermal expansion coefficient, g is gravitational acceleration, T is temperature at the top of the core, and C p is heat capacity at constant pressure (P). Direct measurements of k are difficult to make at core-relevant P and T. In contrast, direct measurements of electrical conductivity (σ), which is inversely proportional to ρ, are achievable with relatively high accuracy. The two variables may be related through the electronic component of k (k e ) with the Wiedemann-Franz Law (WFL), where L is the Lorenz number: Thermal conductivity is controlled by electrons and phonons, but the phonon contribution is negligible in metals and metallic alloys (Klemens and Williams, 1986). The appropriate values of the Lorenz number for specific compositions at relevant T are not well constrained. The theoretical value, the Sommerfeld value (L 0 2.44·10 −8 W·Ω·K −2 ), has been shown to account for more than 99% of k e for Fe, suggesting that its use at high T and P < 6 GPa is reasonable (Secco, 2017). While L > L 0 for Fe-Si alloys at high T and low P (Secco, 2017), calculations at Earth core P, T conditions have shown that L < L 0 (de Koker et al., 2012;Xu et al., 2018). The following relationship was developed from measurements of the Seebeck coefficient of Fe up to 6 GPa and 2,100 K (Secco, 2017): where K T is isothermal bulk modulus. For metals in general, ρ is governed by the scattering rate of conduction electrons. Similarly, the Lorenz number has been shown to be both lower and higher than L 0 depending on the state and composition of the system (Pozzo et al., 2012;Pozzo et al., 2013;Pozzo et al., 2014;Pozzo and Alfè, 2016a;Pozzo and Alfè, 2016b;Pourovskii et al., 2020). The scattering rate of conduction electrons is affected by electronphonon, electron-magnon, and electron-electron interactions. Electrical resistivity is proportional to the inverse of the electron mean free path (d), which is proportional to the amplitudes of atomic vibrations (A) and thus proportional to T: Electron-phonon interactions are electron scattering caused by lattice vibrations and are relatively negligible at low T. At high T, the occupation of phonon density of states shifts toward higher energy states which increases the frequency of collisions with conduction electrons. Electron-magnon interactions, or spindisorder scattering, is only relevant in ferromagnetic metals such as Fe and their ferromagnetic alloys. This interaction increases scattering as a function of T 2 up to the Curie T (T c ) and dominates ρ up to approximately 300 K. In Fe-alloys, the interaction of electron-lattice defects (including impurities) also affects the scattering rate of conduction electrons. The increased lattice defects cause electron structure perturbations and shorter electron mean free paths, which results in a larger ρ. The interactions between electrons and lattice defects dominate over the electron-phonon, electron-magnon and electronelectron interactions at low T. Overall, the net effect of T is to increase ρ, while ρ decreases with P as the decreased amplitude of lattice vibrations increases the electron mean free path. The Ioffe-Regel criterion argues that the growth of resistivity is reduced with T increase, i.e. saturates, as the electron mean free path approaches the interatomic distance (Mooij, 1973;Wiesmann et al., 1977;Gurvitch, 1981). Bohnenkamp et al. (2002) estimated a saturation value of 1.68 μΩm for Fe at 1 atm and up to 1,663 K, while there are variations in the saturation value of Fe-Si alloys at high P (Kiarasi and Secco, 2015;Gomi et al., 2016). Gomi et al. (2013) were the first to propose the idea of resistivity saturation for Fe at Earth core conditions however recent work (Zhang et al., 2020) provides contradictory results and suggests resistivity saturation behavior was an experimental artifact.

METHODS OF ELECTRICAL RESISTIVITY DETERMINATION Theory: First-Principles Calculations
Initially meant to describe the diffusion of gases in the atmosphere, Boltzmann (1894) developed the following equation that considers the electronic band structure, phonon dispersion and electron-phonon interactions: where N(ε F ) is the electronic density of states per atom per spin at the Fermi energy (ε F ) level, k B is the Boltzmann constant, Ω is the unit cell volume, V F is the Fermi velocity, and λ tr is the transport coefficient. This work was followed by the introduction of the Free Electron model by Sommerfeld (1928), based on the classical Drude model of electrical conduction (Drude, 1900a;Drude, 1900b) and Fermi-Dirac statistics describing the distribution of particles over energy states. The model predicts σ from the electron density, the mean free time (time between collisions), and the electron charge. The model includes many relations, including the Wiedemann-Franz law, the Seebeck coefficient of the thermoelectric effect and the shape of the electronic density of states function. In their analysis of Earth's magnetic field variations, Elsasser (1946) estimated the ρ of a pure Fe core from the theory of electronic conductivity, which states that σ is inversely proportional to the absolute T and directly proportional to the square of the Debye temperature (Θ D ).
The Debye temperature is proportional to the sound velocity and the acoustic phonon cut-off frequency and inversely proportional to volume (Kittel, 2005). Ziman (1960) later described the Nearly-Free Electron model, which is based on the behavior of electrons, ions, and holes. In the Free Electron model, all energy states from 0 to ∞ are allowed, whereas the Nearly-Free Electron model allows for weak perturbations of electrons by periodic potential ions. Following this idea, Ziman (1961) developed a theory for the behavior of liquid metals. This model was modified by Evans et al. (1971) to include a transition matrix term, introducing a more complex method of calculating ρ of a metallic liquid for metals that have empty d-band states in to which s-conduction electrons may jump (s-d transitions): where e is the electron charge, K F is the radius of the Fermi surface, q K − K' relates to the scattering angle of the Fermi surface, Ω 0 is the atomic volume of the liquid, a(q) is the structure factor, Z is the modified Planck constant, and t(K, K ' ) is the transition matrix that describes the scattering cross-section related to s-electron to d-band (s-d) scattering. While this formulation focuses on the liquid state of a metal, Mott (1972) and Mott (1980) derived a relationship for the solid state of a metal. Mott's theory considers the thermal and impurity contributions, as well as a magnetic contribution in ferromagnetic metals. The theory describes the relationship between ρ and the area of the Fermi surface (S F ). The relationship can be written as follows: The more complex, and commonly used, Kubo-Greenwood formula (Kubo, 1957;Greenwood, 1958) describes the frequency dependent electron conductivity and yields the linear contribution to the current response. The T dependence of ρ, which is related to electron-phonon interactions, is described by the Bloch-Grüneisen equation. Combined with the Bloch-Grüneisen equation for ρ as a function of volume (V) and T, the Kubo-Greenwood equation can be written as follows: where a and b are constants relating to the volume dependence of the vibrational frequencies, the subscript R refers to the reference state, and ρ 0R and ρ 1R are composition-dependent model parameters (de Koker et al., 2012). The electronic structure of a system can be investigated via Density-Functional Theory (DFT), which mainly describes the potential energies of a system. The DFT approach investigates an approximate solution to the Schrödinger equation in 3D based on the electron density, which results in solutions for ion-electron, ion-ion, and electron-electron potential energies (Argaman and Makov, 2000). Often used in combination with DFT, the dynamical mean field theory (DMFT) is a powerful numerical approximation of the potential of a many-particle system, such as electrons in solids. The complex many-particle system is studied by reducing it to a simpler single-particle system with an external mean field. The external field essentially accounts for the other particles in addition to the interactions and local quantum fluctuations that would occur in the many-particle system (Vollhardt et al., 2012). Drchal et al. (2019) and Korell et al. (2019) investigated the effects of magnetism on the electronic properties of metals via the Kubo-Greenwood equation along with DFT and the Tight-Binding Linear Muffin-Tin Orbital method (a method of calculating short-range transitions between electrons in s-, d-and f-orbitals), respectively. Korell et al. (2019) suggested that spin polarization must be accounted for in order to reconcile the first-principle calculations of ρ with measurements. In addition to drawing a similar conclusion, Drchal et al. (2019) suggested that the contributions of the various scattering mechanisms (caused by electron-phonon interactions, electron-magnon interactions, electron-electron interactions, and electron-lattice defects) are comparable but not additive. Through these first-principle calculations which require values of the thermal properties at a reference P and T, theoretical studies are able to estimate ρ of any metal or metalalloys.

Static: Multi-Anvil Press
Static experiments are typically carried out in a multi-anvil press. A typical multi-anvil press uses hydraulic pumps to drive uniaxially a hydraulic ram to compress a pressure cell located at the center of an arrangement of tungsten carbide (WC) anvils.
In the 3000-ton press at the University of Western Ontario for example, there are three steel wedges on the bottom and three wedges on the top part of the pressure module. The wedges provide a nest for eight truncated WC cubes that house and converge on an octahedral pressure cell assembly. The octahedral cell may vary in size depending on the truncation edge length (TEL) of the WC cubes. The TEL may vary from 3 to 25 mm, and the smallest sizes are used to reach higher P, albeit on smaller sample volumes. The octahedral pressure cells are usually composed of Magnesium Oxide (MgO) which provides a balance between machinability, low k and hardness that allows for efficient conversion of applied force to the sample inside. With compression, the octahedral cell extrudes out between the WC cubes creating gaskets. Similarly, other multiple anvil apparatus designs such as the cubic anvil press, which consists of six WC anvils that compress a cubic pressure cell assembly made typically of the mineral pyrophyllite, creates gaskets as the cell extrudes out between the anvils. The insulating material, furnace, sample, and electrodes are placed inside the cell, and arranged co-axially in a hole connecting two opposite faces. Elevated temperatures are generated by Joule heating of an electrically resistant furnace, typically made of graphite, lanthanum chromite or rhenium foil, surrounding the sample. The electrical resistance (R) of the sample is measured with a four-wire method, where a pair of thermocouple wires is placed at each end of the sample. Each pair acts as a temperature sensor, while opposite pairs are assigned as potential and current leads in a switchable circuit configuration. The electrical resistivity is then calculated using the following equation which combines Ohm's law and Pouillet's law: where ΔE is the voltage drop across the sample, I is the input current, A is the cross-sectional area of a wire-shaped sample and l the length of the sample. The general experimental methods for multi-anvil apparatuses are described in more detail by Liebermann (2011) and Ito (2015).

Static: Diamond-Anvil Cell Experiments
A Diamond-Anvil Cell (DAC) consists of two opposing gemquality diamonds that enable the compression of small (∼10 −4 mm 3 scale) samples placed between the two diamond culets. Applying moderate force generates high pressures due to the small diameter, typically 50-250 µm, of the culet. Considering that P is applied uniaxially, a metal gasket with a hole drilled to form the sample chamber provides lateral support. A pressuretransmitting medium such as a gas (Ne, Ar, He, N 2 ) in the sample chamber must be added to obtain hydrostatic pressure but for high T experiments a solid medium (SiO 2 , KCl, Al 2 O 3 , KBr, NaCl) is often used. The system is therefore translucent and allows for the use of XRD to determine the crystallographic structure of the sample. The two main pressure scales are the equation of state of a reference material and the shift in ruby fluorescence lines, although the latter is less reliable at high T. The equation of state describes the relationship between P, T, and V. Measurements of changes in V are compared with a known V-P relation in order to determine the experimental P. The second pressure scale relies on the P-dependent shift in fluorescence wavelength of an irradiated ruby included in the sample chamber. The sample is insulated from the gasket by adding a layer of insulating material, such as Al 2 O 3 . High temperatures can be achieved with single or double-sided laser heating or by internal resistive heating of a metal element, which may be the sample itself. Resistive heating allows for the precise measurement of T with thermocouples but is limited by the oxidization of diamond in air at approximately 973 K. Laser heating can achieve much higher T (up to 7000 K), although large T gradients can exist and care is needed to locate the sample measurements in a relatively uniform T-field. A common method for measuring the sample resistance in DAC experiments is the Van der Pauw method. This method is used for samples of arbitrary shape, as long as the sample is approximately two-dimensional such as a sheet or foil. Four electrodes are connected to the perimeter of the sample. Two opposite electrodes serve as potential leads, which can be machined within the same foil as the sample or carefully connected manually and fixed by applying pressure. The ρ of the sample at a particular P can be calculated as follows: where the subscript zero refers to the ambient conditions. The experimental methods for DAC are described in more detail by Mao and Mao (2007) and Anzellini and Boccato (2020).

Dynamic: Shock Compression Experiments
Shock compression is a dynamic method of generating high pressures which can be used in conjunction with an experimental configuration that allows determination of the physical properties of a sample, usually conducted in a vacuum. This apparatus consists of using compressed gas and/ or gunpowder to launch a metallic projectile onto a stationary sample. The collision generates a strong shock wave that results in simultaneous high P and high T across the sample. The projectile, typically referred to as a flyer plate and made of W or Fe, accelerates to the desired velocities on the order of a few km/s (Bi et al., 2002). A high P, T conductivity zone propagates through the sample under shock compression. The σ of the sample can be measured by contact or contactless methods. The contactless method requires the remote probing of the sample by optical and electromagnetic field sources, which results in values of the impact velocity of the flyer and shock velocity. The arrival of the shock waves generates two electromotive forces that relate to the product of σ and thickness of sample (Gilev, 2011). The contact method requires typically four electrodes placed in contact with the sample. Two electrodes provide the voltage measurements while the other two are used to input constant current. The interface temperatures are usually measured with an optical pyrometer (Gilev, 2011). The most fundamental shock wave characteristic of the sample is its principal Hugoniot curve, which is formed by the locus of shock states along a compression path from initial P, T, and density. The P, T and density properties at the final state are obtained using a standard impedance matching method (Bi et al., 2002). In both methods, σ can be calculated from: where the subscripts 0 and F refer to the initial and final states, respectively, and V is the specific volume. The experimental methods for shock compression experiments are described in more detail by Ahrens (2007) and Asimow (2015).

ELECTRICAL RESISTIVITY OF THE CORES OF TERRESTRIAL-TYPE BODIES Earth
It is widely accepted that the Earth's core composition must be composed of Fe, Ni and some light elements in order to satisfy the core density (Birch, 1961;Birch, 1964;Jeanloz, 1979;Mao et al., 1990). Approximately 10 wt% of light elements is expected in the outer core (Poirier, 1994;Litasov and Shatskiy, 2016). In the inner core, approximately 3-8 wt% of light elements has been suggested (Alfè et al., 2007;Badro et al., 2007;Mao et al., 2012). In addition to approximately 5-10 wt% Ni in the core, light elements include C, H, Si, O and S (Poirier, 1994;Stixrude et al., 1997;Li and Fei, 2003;Alfè et al., 2007). Accordingly, the ρ of multiple Fe binary, ternary and even quaternary alloys must be investigated. Gardiner and Stacey (1971) used available ρ measurements of liquid Fe from the literature to investigate the effects of T, P, and composition. They reported ρ of 0.25 µΩm at 2,500 K and 3,000 km, which corresponds to the Core-Mantle Boundary (CMB). Similarly, they reported 0.69 µΩm at 5,500 K and 5,000 km, which corresponds to the Inner-Core boundary (ICB). These extrapolations were achieved by applying a P-dependence of ρ of solid Fe (Bridgman, 1957) to measurements of liquid Fe at 1 atm (Baum et al., 1967), as shown below: where the subscript P refers to the final pressure, the subscript 0 refers to zero pressure, and D is density. Shock wave experiments by Keeler and Royce (1971) reported a similar value of 0.67 µΩm at 140 GPa, which corresponds to CMB conditions. However, it has been suggested by Bi et al. (2002) that the previous shock compression experiments above 50 GPa underestimated ρ due to the shunting effect of epoxy, which is used to fill gaps between the sample and insulator. Bi et al. (2002) conducted shock compression measurements of σ of Fe up to 208 GPa, corrected for the shunting effect, and suggested a value for ρ of 0.69 µΩm at conditions near the CMB (2,010 K, 101 GPa) and 1.31 µΩm for conditions near the ICB (5,220 K, 208 GPa).
Although these values represent a correction to previously underestimated values, they remain lower than the values reported by Jain and Evans (1972). Jain and Evans (1972) carried out first-principle calculations using the Nearly-Free Electron model and reported ρ of a pure Fe core between 1.00 and 2.00 µΩm. These higher values are in agreement with direct measurements by Secco and Schloessin (1989), who used a largevolume press to measure ρ of Fe up to 7 GPa and above the melting T. The similarities in their measurements with calculations of the density of state functions at low and high P suggest ρ of an Fe solvent is probably between 1.2 and1.5 µΩm at outer core conditions. A theory was developed by Stacey and Anderson (2001) suggesting a constant ρ along the melting boundary of pure metals such as Fe, based on a cancelling of the decreasing effects of P on ρ by the increasing effects of T on ρ. The description of their analysis, rooted mainly in thermo-elastic considerations, begins with the theory of electronic conductivity, as mentioned earlier by Elsasser (1946), while also considering the electron energy. According to the Free-Electron model and the approximation of the electron energy at the Fermi surface, the total energy varies with volume as V −2/3 . This term is multiplied by the inverse square of the Debye T in Equation 6 to calculate ρ. Stacey and Anderson (2001) then extrapolated ρ of Fe at 140 GPa (Matassov, 1977) and 2,180 K (Anderson, 1998) to core temperatures and reported values of 1.22 µΩm at CMB and 1.12 µΩm at ICB, in agreement with earlier estimates (Jain and Evans, 1972;Secco and Schloessin, 1989;Stacey and Anderson, 2001). Stacey and Loper (2007) revised the theory of constant ρ along the melting boundary and suggested that the constant behavior only applies to electronically simple metals with filled d-bands such as Cu but not Fe. They reported a revised ρ of Fe of 3.62 µΩm at the CMB and 4.65 µΩm at the ICB, higher than the values reported thus far. Experiments on several simple metals showed the ρ invariance prediction of Stacey and Loper (2007) to be over-simplified as ρ of Cu (Ezenwa et al., 2017), Ag , Au (Berrada et al., 2018), all decrease on their pressure-dependent melting boundaries whereas Zn (Ezenwa and Secco, 2017a), Co (Ezenwa and Secco, 2017b), Ni (Silber et al., 2017) and Fe (Silber et al., 2018;Yong et al., 2019) all show invariant ρ along the melting boundary. Similarly, Davies (2007) revised the results of Stacey and Anderson (2001) and Bi et al. (2002) by correcting for T at the top of the core (∼4,023 K, 135 GPa), which suggested a ρ of 1.25-1.9 µΩm for Fe at CMB conditions.
First-principle calculations using the Boltzmann equation suggested ρ of 0.75 µΩm for solid ε-Fe at ICB conditions (Sha and Cohen, 2011), while using the Kubo-Greenwood equation via the Bloch-Grüneisen equation on liquid Fe suggested a ρ of ∼0.61-0.69 µΩm at core conditions (de Koker et al., 2012). These results are comparable with first-principle calculations of transport properties based on DFT on liquid Fe alloy mixtures by Pozzo et al. (2012) who reported a value of 0.64 µΩm at the ICB, and 0.73-0.74 µΩm at the CMB. Soon after this work, Pozzo et al. (2013) used first-principle simulations from DFT calculations with the Kubo-Greenwood relation to obtain k while ρ was independently calculated from σ. The appropriate Lorenz number for liquid Fe (2.47-2.51·10 −8 WΩ K −2 ) was then calculated using the Wiedemann-Franz law. Their results also showed 0.64 µΩm for Fe at the ICB, and 0.747 µΩm at the CMB. First-principle calculations combined with molecular dynamic simulations on solid Fe by Pozzo et al. (2014) suggested lower values of 0.50-0.53 µΩm at ICB conditions. Pozzo and Alfè (2016a) extended the set of calculations of core ρ of ε-Fe by Pozzo et al. (2014) to lower temperatures in order to investigate the T dependence of ρ. Their results suggested that ρ increases linearly with T and eventually saturates at high T, implying ρ of Fe of ∼0.72 µΩm not far from CMB conditions (4,350 K, 97 GPa) and ∼0.54 µΩm at Earth's center (6,350 K, 365 GPa), comparable to previous results mentioned above. This study is also in agreement with DAC experiments up to 70 GPa and 300 K by Gomi and Hirose (2015), who suggested values of 0.537 (+0.049/ −0.077) µΩm and 0.431 (+0.058/−0.095) µΩm at CMB (3,750 K, 135 GPa) and ICB (4971 K, 330 GPa) conditions respectively. In contrast, extrapolations of DAC measurements from 26 to 51 GPa up to 2,880 K, assuming resistivity saturation suggested the ρ of Fe is 0.404 (+0.065/−0.097) µΩm at CMB conditions (Ohta et al., 2016). However, in the first study to measure k of Fe at CMB conditions, Konôpková et al. (2016) measured the propagation of heat pulses across Fe foils in a DAC, up to 130 GPa and 3,000 K, and then modelled the T and P dependences of ρ. They derive ρ of Fe of 3.7 ± 1.5 µΩm at the outer core. Although in agreement with the high values reported by Stacey and Loper (2007), this value is higher than the saturation resistivity of 1.43 µΩm suggested by Xu et al. (2018). Xu et al. (2018) computed the electron-phonon and electron-electron contributions to the ρ of solid ε-Fe, from first-principle calculations and molecular dynamics. They suggested values of 0.998 µΩm and 1.008 µΩm at ICB (6,000 K, 330 GPa) and CMB (4,000 K, 136 GPa) respectively. Their results also suggested that previous DAC data (Ohta et al., 2016) may have overestimated the saturation effect. Zhang et al. (2020) proposed that the saturation effect observed at high T by Ohta et al. (2016) should be considered an experimental artifact due to the incorrect positioning of the laser over the sample during heating and inaccurate geometries of the four-probe method. First, the misalignment of the laser during heating generates a large T gradient across the sample. The ρ measurements at high T are then dominated by the colder regions, leading to lower ρ values. Secondly, the measurement uncertainties are expected to be significantly larger than the reported values considering the small size of the sample compared to the location of the electrodes. The geometries of the assembly essentially result in a two-probe method. While considering these key factors, Zhang et al. (2020) used firstprinciple calculations to compare with their ρ measurements of ε-Fe up to ∼170 GPa and ∼3,000 K in a laser heated DAC using the Van der Pauw method. Their analysis suggested a ρ of 0.80 ± 0.05 µΩm at CMB conditions (4,000 K, 136 GPa). Values obtained from DAC measurements extrapolated based on the resistivity saturation model (Gomi and Hirose, 2015;Ohta et al., 2016) are thus expected to be almost doubled according to the results of Zhang et al. (2020). However, these results remain lower than those extrapolated from multi-anvil press measurements (Secco and Schloessin, 1989;Yong et al., 2019). For example, the reported value from Yong et al. (2019) is 1.6× higher than that from Zhang et al. (2020). While Yong et al. (2019) reported measured values of ρ of Fe on the liquid side of the melting boundary, their reported value for ρ at the CMB was linearly extrapolated to 200 GPa based on the P-dependency observed between 14-24 GPa. Zhang et al. (2020) assumed a 10% increase resistivity on melting but only extrapolated from 133 to 136 GPa. In general, the differences in values for ρ at the CMB reported from multi-anvil and DAC experimental studies may arise from uncertainties from large extrapolations, from predispositions specific to each method such as errors in sample geometries, temperature homogeneity of heated sample region, lack of sample symmetry, possible thermoelectric effects and other parasitic voltages, etc. Silber et al. (2018) reported an invariant behavior of Fe ρ at ∼1.2 µΩm, from direct measurements in a multi-anvil press up to 12 GPa and at liquid T. A similar method was used by Yong et al. (2019), who carried out static experiments in a 3000-ton multianvil press from 14-24 GPa into the liquid. Their results suggested an invariant ρ of liquid Fe of 1.20 ± 0.02 µΩm along the melting boundary. However, a statistical linear regression of their measurements, in order to account for the effect of P, suggested a slight deviation from invariant behavior to 1.28 ± 0.09 µΩm along the melting boundary of Fe at 200 GPa.  focused on the behavior of liquid Fe from first-principle calculations based on Ziman's theory on liquid metals. Combining their ICB value of 0.58 µΩm for ε-Fe with the literature values of liquid Fe-Si/S alloys (de Koker et al., 2012;Pozzo et al., 2014), ρ of outer core is expected to be up to 36% larger than that of the inner core. In contrast, estimations of pure Fe from DFT and molecular dynamics suggested almost no increase between ICB and CMB conditions, from 0.60 ± 0.27 µΩm to 0.67 ± 0.27 respectively (Wagle et al., 2019). Recent DFT simulations by Pourovskii et al. (2020) accounted for electron-electron interactions at high T and suggested an inner core ρ of 0.637 µΩm for pure Fe, in general agreement with the lower values mentioned above. Pourovskii et al. (2020) applied DFT and DMFT to Fe and concluded that thermal disorder suppresses the non-Fermi-liquid behavior of bcc Fe which reduces electron-electron scattering at high T. The variations among pure Fe results seem to range between 0.25 µΩm (Gardiner and Stacey, 1971) and 3.7 ± 1.5 µΩm (Konôpková et al., 2016), and may not simply be attributed to the method used.
The results of Gomi and Yoshino (2018) also suggested values ranging from 0.71-0.88 µΩm for Fe-(1.6 to 11.7 wt%)O at CMB conditions. Similarly, de Koker et al. (2012)  GPa and up to liquid T and postulated that the ρ of liquid Fe alloyed with light elements remains unchanged from that of Fe at inner core conditions, but the variation in T at the CMB suggested that ρ could increase up to ∼1.50 µΩm. In only the second study to measure experimentally k of a composition in the Fe system at Earth's core conditions, Hsieh et al. (2020) used a pulsed laser method with a DAC to measure k of Fe-Si alloys up to 144 GPa and 3,300 K. Calculations of ρ using the Wiedemann-Franz law, with the Sommerfeld value of the Lorenz number, and extrapolations to high T suggested approximately 4.6 µΩm (or 20 Wm −1 K −1 ) at ∼136 GPa and 3,750 K for ε-Fe8.7Si (or Fe 0.85 Si 0.15 ) (Hsieh et al., 2020). Hsieh et al. (2020) note that the discrepancy with previously mentioned results may be caused by the assumptions made by studies that did not directly measure k. Both the modeled T-dependence of ρ and the use of the Sommerfeld value of the Lorenz number at high P and T lead to underestimates of k at Earth's core conditions. Shock compression experiments by Matassov (1977) on Fe-Si alloys up to 140 GPa and 2,700 K suggested a ρ of 1.12 µΩm for an Fe33.5Si core, which is within previously reported values from shock wave measurements. As mentioned earlier, this value may be underestimated due to the shunting effect of epoxy above 50 GPa (Bi et al., 2002). The results of de Koker et al. (2012) also suggested a lower range of 0.74-1.03 µΩm for Fe-Si alloys at core conditions. Similar to the results of de Koker et al. (2012), DAC measurements up to 60 GPa and 300 K indicate ρ values for Fe9Si of 0.6-1.3 µΩm at the CMB . Their measurements were extrapolated to CMB conditions using a model of ρ as a function of T, V, and Debye temperature. These results are in agreement with measurements in a cubic-anvil press up to 5 GPa and 2,200 K, indicating ρ of Fe17Si in the range of 0.90-0.94 µΩm at outer core conditions (Kiarasi, 2013). In comparison, Suehiro et al. (2017) reported CMB values of approximately 0.699 µΩm for Fe12.8S (or Fe 80.8 S 19.2 ), 0.741 µΩm for Fe6.1Si6.7S (or Fe 79.7 Si 10.3 S 10 ), and 0.784 µΩm for Fe13.5Si (or Fe 77.5 Si 22.5 ). Suehiro et al. (2017) carried out measurements in a laser-heated DAC up to 110 GPa and 300 K and used Matthiessen's Rule and the saturation resistivity value of 1.68 µΩm, to obtain ρ of Fe-Si-S alloys at core conditions. Thus, the contribution of S to the ρ of Fe-alloys is reported to be weaker than that of Si. Gomi and Hirose (2015) proposed higher ρ values for Fe13.5Si of 1.02 (+0.04/−0.13) µΩm and 0.820 (+0.055/−0.130) µΩm at the CMB and ICB, respectively. These higher values are in agreement with Zhang et al. (2018), who applied their model of k to Fe13.5Si and reported ρ values of 0.92 µΩm at CMB conditions (4,050 K, 136 GPa). Recently, the saturation resistivity of ε-Fe-Si alloys was further investigated in an internally heated DAC up to 117 GPa and 3,120 K (Inoue et al., 2020). Results show that the saturation resistivity of ε-Fe-Si alloys is comparable to that of pure Fe at ∼100 GPa (Inoue et al., 2020). They obtained ρ values for ε-Fe12.7Si of 1.040 (+0.126/−0.212) µΩm and 0.775 (+0.150/-0.231) µΩm at ICB (3,760 K, 135 GPa) and CMB (5,120 K, 330 GPa) respectively. All values discussed above are summarized in Tables 1-4. The variations in ρ, organized in terms of composition, are visualized in Figure 2.
A large quantity of theoretically-and experimentally-based estimates are considered for the ρ of a terrestrial core of pure Fe. In general, ρ values of Fe are centered about 1.05 and 1.08 µΩm at the CMB and ICB, respectively, without considering the high values (>3 µΩm) from Stacey and Loper (2007) and Konôpková et al. (2016). The high values reported in the theoretical work of Stacey and Loper (2007) and very challenging experimental work of Konôpková et al. (2016) can hardly be explained by the selection of P and T values since they are similar to those reported by the studies reporting lower ρ values. However, the values reported by Konôpková et al. (2016) are direct measurements of k, which include the phonon contribution. Their values are thus expected to be higher than measurements or calculations of k e , although such a large discrepancy (3.5× higher than the average) cannot be explained by currently understood physics of the relative contributions of electron and phonon components of k in metals. The average ρ of Fe-Ni alloys is  Gardiner and Stacey (1971) at ICB conditions. The theoretical work of Gardiner and Stacey (1971) reported the highest values corresponding to the estimated ρ of Fe-25Ni,S,Si,MgO. The reported data on Fe-Ni alloys are scattered and a clear relationship between light element content and ρ, or even between theoretical and experimental methods, cannot be readily seen. The average ρ of Fe-O alloys is 0.80 and 0.74 µΩm at the CMB and ICB respectively. Theoretical and experimental ρ values of Fe-Si/S alloys are centered about 1.08 and 0.83 µΩm at the CMB and ICB respectively, without considering the high value from Hsieh et al. (2020) at ICB conditions. Indeed, the ρ value for Fe8.7Si (Hsieh et al., 2020) is expected to be higher than that of Fe7.6Si (Wagle et al., 2019) but lower than that of Fe9Si  and higher Si content alloys. The averages  of all studies reporting values for Earth are 1.07 and 0.97 µΩm at CMB and ICB conditions, respectively. Higher P and lower light element content generally result in a lower ρ for a given core composition, while high T acts to increase ρ. Yet, the consistently greater averages at CMB conditions, which consider a higher light element content and lower P, suggested that the effects of light element content and P dominate over the effects of T.

Moon
A combination of the lunar seismic profiles (Weber et al., 2011) and sound wave velocity measurements of Fe-alloys  (Antonangeli et al., 2015) indicates that the Moon is currently composed of a liquid outer core and solid inner core. The exact amount and nature of light element(s) in the core is not constrained but S, C, and Ni are expected (Wieczorek et al., 2006;Steenstra and van Westrenen, 2017). The presence of Si in the lunar core is also considered possible since the lunar Si isotope composition suggests a mantle composition similar to Earth's (Armytage et al., 2012;Nazarov et al., 2012;de Meijer et al., 2013;Zambardi et al., 2013). Szurgot (2017) reported at least 2.7 wt% Si in the lunar core when considering the uncompressed density of the Moon.
The calculated k of Fe (25-50 W m −1 K −1 ) by Anderson (1998) and Stacey and Anderson (2001) have been commonly used in thermal evolution models of the lunar dynamo (Stegman et al., 2003;Zhang et al., 2013;Evans et al., 2014;Laneuville et al., 2014;Scheinberg et al., 2015;Laneuville et al., 2018). However, the corresponding ρ via the Wiedemann-Franz law varies from study to study due to the large variations in T assigned to the lunar CMB. Direct resistivity measurements of Fe in a multi-anvil press from 3-12 GPa and into the liquid state suggested ρ of 1.23-1.31 µΩm on the core side of the CMB (1,687-1,800 K, 4.9 GPa) (Silber et al., 2018). Using a similar method, Pommier (2018) measured ρ in a multi-anvil apparatus up to 10 GPa and over a wide range of T. Their results suggested a value of 0.66 µΩm, or slightly more than half of the value measured by Silber et al. (2018). Also, Pommier (2018) reported a ρ of 2.15 µΩm for Fe5S at 4.5 GPa and 1,700 K. According to their results, the effect of S, even for such a small wt%, is not negligeable. Recently, Yin et al. (2019) reported measurements of ρ of Fe35.7P (or FeP), Fe21.7P (or Fe 2 P) and Fe15.6P (or Fe 3 P) in a multi-anvil press at 3.2 GPa. These authors include P as a possible light element in the lunar core considering its large solubility in liquid Fe and Fe-S alloys (Zaitsev et al., 1995;Stewart and Schmidt, 2007). Their results indicate that ρ of Fe35.7P is approximately four times that of Fe. Yin et al. (2019) conclude that ρ is expected to be lower than 1.65 µΩm for a lunar core with Fe15.6P. In addition, the direct measurements of Berrada et al. (2020) constrained the ρ of Fe-Si alloys (2, 8.5, 17 wt% Si) between 1.17-1.66 µΩm at the top of the lunar outer core (1,600 K, 5-7 GPa). These studies have all used a multi-anvil apparatus with a four-wire measurement method to measure ρ. The values discussed in this section are summarized in Table 5 and the variations in ρ are visualized in Figure 3.
Although the exact identity and amount of light elements in the lunar core is not constrained, to our knowledge, only few studies have reported ρ values to the relevant P and T conditions of the lunar core. At CMB conditions, the measured ρ of Fe15.6P is lower than that reported for Fe-Si alloys (up to 17 wt% Si) and Fe5S, suggesting the effect of P on the core ρ is relatively negligible. The reported Fe-alloys show greater ρ than that of Fe, as expected considering the additional scattering mechanism caused by electron-impurity interactions. The average of all studies reporting values at CMB conditions is 1.40 µΩm.

Mercury
Our understanding of the light element content of Mercury is mainly based on solar-system chemical abundances, models based on the compositions of Earth and the Moon, and data returned from the MESSENGER X-ray spectrometer (Harder and Schubert, 2001;McCubbin et al., 2012). Early estimates of the FexSi (x 2-17) 1.17-1.66 1,600 K, 5-7 GPa Multi-anvil press (ρ) Berrada et al. (2020) The ρ values associated with studies that have reported k values are obtained via the Wiedemann-Franz law with the Sommerfeld value for the Lorenz number. Values specific to solid Fe are in blue, while unspecified values are in black. Frontiers in Earth Science | www.frontiersin.org September 2021 | Volume 9 | Article 732289 core composition of Mercury suggest ∼99 wt% is metal (Fe, Ni, Co) and ∼1 wt% is Fe36.5S (or FeS) (Morgan and Anders, 1980). Nittler et al. (2017) has argued a Si-bearing core with perhaps some small amounts of S and C, while Rivoldini et al. (2009) suggest a minimum of 5 wt% S. Various thermal evolution models consider an Fe-Si core with uncertainty on the exact Si composition (Knibbe and van Westrenen, 2017;Knibbe and van Westrenen, 2018). Estimates of Si content range between 5 and 25 wt% Si (Malavergne et al., 2010;Chabot et al., 2014), although recent studies propose that 10.5 wt% Si provides the best modelling results of the core's elastic property and geodesy data (Terasaki et al., 2019;Steinbrügge et al., 2021). Early models of Mercury's core formation generally consider k of 40-43 W m −1 K −1 , which correspond to 1.06-1.15 µΩm, at 1,880 K for a core of mainly Fe with 1-5 wt% S (Stevenson et al., 1983;Christensen, 2006;Tosi et al., 2013). Thermal evolution models by Knibbe and van Westrenen (2018) considered a range of k from 30 to 60 W m −1 K −1 for an Fe-Si core with T between 1,800-2,200 K, which corresponds to 0.73-1.79 µΩm via the Wiedemann-Franz law. In contrast, direct measurements of ρ of γ-Fe at 5, 7 and 15 GPa into the liquid state in a multi-anvil press suggested values of ∼0.36 µΩm and ∼0.44 µΩm at ICB (2,200-2500 K, 36 GPa) and CMB (1800-2000 K, 5-7 GPa) conditions respectively (Deng et al., 2013). The direct measurements of ρ for γ-Fe at Mercury core temperatures in a laser-heated DAC suggested values of 1.08-2.44 µΩm (or 35 ± 10 W m −1 K −1 ) at 2,200-2,500 K and ∼40 GPa (Konôpková et al., 2016). Considering the variations in the Lorenz number with T, Secco (2017) reported ρ for Fe of 1.044 µΩm at 1,823 K and 5.5 GPa. Silber et al. (2018) used direct ρ measurements of Fe from 3-12 GPa and up to liquid T and suggested a value of 0.87 ± 0.10 µΩm for Fe at CMB conditions (1,900 K, 5 GPa). Ezenwa and Secco (2019) revised previous measurements of ρ of Fe at Mercury's CMB conditions (1,880 K, 5 GPa) to 1.18 µΩm on the solid side just before melting and 1.24 µΩm on the liquid side. Berrada et al. (2021) reported higher ρ values of Fe8.5Si between 1.40-1.44 µΩm at CMB conditions (1,600-2,100 K, 5-7 GPa) from measurements in a multi-anvil press up to 24 GPa and above the liquid T. Similarly, Pommier et al. (2019) conducted direct measurements in a multi-anvil apparatus up to 10 GPa and over a wide range of T. Their results suggested approximately 0.35 µΩm for Fe10Si, 0.49 µΩm for Fe33.5Si, and 0.33 µΩm for Fe8Si3S at CMB conditions (2,000 K, 6 GPa). In contrast, Manthilake et al. (2019) reported σ data from resistance measurements of Fe36.5S, which convert to 8.01 µΩm near CMB conditions (1,300 K, 8 GPa). This result is significantly larger that of any composition reported in the literature thus far at similar P and T conditions. The values discussed in this section are summarized in Table 6 and the variations in ρ are visualized in Figure 4. Reported theoretical and experimental values of the ρ of Fe applied to Mercury's CMB and ICB conditions show an important disagreement. Deng et al. (2013) reported lower values (∼35% less) than those reported by Konôpková et al. (2016), Secco (2017), Silber et al. (2018), and Ezenwa and Secco (2019). Such a discrepancy is not negligible when calculating q ad at the top of the outer core. The direct k measurements by Konôpková et al. (2016) convert to similar ρ values to the direct ρ measurements by Secco (2017), Silber et al. (2018) and Ezenwa and Secco (2019). The values reported for Fe-Si/S alloys also show two distinct trends. Berrada et al. (2021) reported higher values than those by Pommier et al. (2019), although Pommier (2018) considered a higher light element content (up to 33.5 wt% Si). Indeed, the light element content is expected to increase the scattering contribution and thus the measured ρ. The values reported by Pommier (2018) are consistent with the lower ρ values for Fe reported by Deng et al. (2013) and Konôpková et al. (2016). Similarly, the values reported by Berrada et al. (2021) are consistent with the high ρ of Fe values reported by Secco (2017), Silber et al. (2018),and Ezenwa and Secco (2019). The averages of all studies reporting values for Mercury are 0.88 and 1.12 µΩm at CMB and ICB conditions, respectively, without considering the high values (>3 µΩm) from Manthilake et al. (2019). Although all the reported data at the ICB consist only of pure Fe measurements, the average ρ at ICB conditions is greater than that at CMB conditions which considers Fe-alloys. This contrast with the observations at Earth's core conditions could be explained by the strong effect of T over the effects of light elements and P at Mercury's core conditions, although further studies are necessary to draw reliable conclusions.

Mars
The chemical composition of the Martian core is presumed to be Fe14.2S based on analyses of Martian meteorites (Wänke et al., 1988). Laser-heated DAC experiments and in-situ X-ray diffraction confirm the phase stability of Fe36.5S at core conditions (Kavner et al., 2001). Thermal evolution models also consider a core primarily composed of Fe with 6-8 wt% Ni, and 10-17 wt% S, in addition to containing lower amounts of O, C, H and P (Rivoldini et al., 2011). Stevenson et al. (1983) investigated core evolution models of Mars by combining theories of geomagnetism and fluid dynamics. In their model, the value of k of Fe used for Earth, Mercury, Mars and Venus is 40 W m −1 K −1 , which corresponds to 1.15 µΩm at 1,880 K (Stevenson et al., 1983). Thermal evolution models based on measurements by Anderson (1998) suggested a range of 43-88 W m −1 K −1 at Martian CMB conditions (25 GPa, 1,800 K) (Nimmo and Stevenson, 2000). Using the Wiedemann-Franz law, this suggested ρ values for Fe ranging from 0.50-1.02 µΩm, which are lower than previously mentioned. In fact, Deng et al. (2013) estimated ρ of γ-Fe to be at most 0.40 µΩm at the outermost part of the Martian core (2,000 K, 24 GPa) based on their measurements at 7 GPa. Recent measurements by Silber et al. (2018) reported ρ of Fe of 1.7 µΩm at CMB conditions (1,770 K, 23 GPa), higher than previously reported. Similar experiments by Ezenwa and Yoshino (2021) in a multi-anvil apparatus from 14 to 22.5 GPa and above the liquid T, estimated ρ of 0.68 ± 0.03 µΩm at CMB conditions (2,106 K, 23 GPa). The measurements by Ezenwa and Yoshino (2021) are in agreement with first-principle calculations of pure Fe reporting values from 0.74 ± 0.29 µΩm to 0.75 ± 0.29 at the ICB and CMB respectively (Wagle et al., 2019). Wagle et al. (2019) also calculated the ρ of Fe3. 9O,Fe8.7O,Fe7.6S,Fe16.1S,Fe6.7Si,and Fe14.4Si at the Martian ICB and CMB conditions, as reported in Table 7. Direct measurements of ρ up to 110 GPa at 300 K suggested values from 1.064 µΩm at the CMB to 0.952 µΩm at the center of an Fe15S (or Fe 77.7 S 22.3 ) core (Suehiro et al., 2017). The values discussed in this section are summarized in Table 7 and the variations in ρ are visualized in Figure 5.
The literature on ρ values at Martian core conditions is insufficient to determine a reasonable value at CMB and ICB conditions. While both Deng et al. (2013) and Silber et al. (2018) reported values for solid Fe, the value reported by Deng et al. (2013) is in best agreement with the reported values of liquid Fe (Wagle et al., 2019;Ezenwa and Yoshino, 2021). However, Silber et al. (2018) considered lower T values at the CMB than Ezenwa and Yoshino (2021). Their measured ρ is therefore expected to be lower than that of Ezenwa and Yoshino (2021), although the opposite is observed. Within the results of Wagle et al. (2019), the effect of O in increasing ρ seems to be less than that of Si and S, although all alloys seems to be in agreement within uncertainty. The values for an Fe15S (Suehiro et al., 2017) are in agreement with those of Fe16.1S (Wagle et al., 2019). The average of all studies reporting values at CMB conditions is 0.95 µΩm, while that at ICB conditions is 0.93 µΩm. The greater average at CMB conditions relative to ICB conditions is comparable to that observed at Earth's core conditions.

Ganymede
The presence of Fe36.5S in meteorites indicates that S is probably a major light element in Ganymede's core (Krot, 2005). Hydrogen is also a light element candidate considering that a layer of H 2 O in Ganymede's interior has been proposed (Anderson et al., 1996).
Although the exact composition is not constrained, the light element content is greatly dependent on the core size and oxidation state of the interior during differentiation. The CMB conditions are expected to be near 2,000 K and 7 GPa in the case of a low (1 wt%) S content, while near 1,400 K and 5 GPa in the case of a near eutectic (36.5 wt%) S content (Hauck et al., 2006;Bland et al., 2008;Kimura et al., 2009). Silber et al. (2018) measured ρ of Fe at 1.20 µΩm at 1,500 K (Shibazaki et al., 2011) and 5.9 GPa (Hussmann et al., 2007), while a considerably lower value of 0.59 µΩm is proposed by  In contrast, direct ρ measurements of Fe36.5S, from 2 to 5 GPa and up to 1,785 K in a multi-anvil press, suggested a value of 4.13 ± 0.07 µΩm at CMB conditions (1,411 K, 5 GPa) (Littleton et al., 2021). The values discussed in this section are summarized in Table 8 and the variations in ρ are visualized in Figure 6.
As with the other small planetary bodies reviewed, the available literature is scarce and insufficient to distinguish a reasonable value for ρ at core conditions. Overall, the addition of S impurity scattering increases ρ as expected. However, ρ of Fe36.5S as measured by Littleton et al. (2021) is greater than that of Fe20S (Pommier, 2018). Pressure is expected to decrease ρ, yet the ρ of Fe36.5S from Manthilake et al. (2019) at 8 GPa is almost double that from Littleton et al. (2021) at 5 GPa. Although total k measurements include the phonon contribution to conductivity, it likely does not explain the extent of the disagreement between these two studies. The average of all studies reporting values at CMB conditions is 1.90 µΩm, larger than that at the other planetary bodies reviewed, while the average at ICB conditions is 1.28 µΩm. Figure 7 illustrates contour maps generated from the available literature on ρ for Fe, Fe-Ni alloys, Fe-O alloys and Fe-Si/S alloys, without excluding high values (>3 µΩm). The contour maps suggest ρ(P, T) is unique for each composition, and can hardly be defined by a linear function of P and T. The rich literature on Fe at Earth's and Mercury's CMB and ICB conditions produce an accurate contour map at the relevant P and T. Similarly, the contour map results of the ρ of Fe at the lunar and Martian CMB consist of the mean of the few reported values. However, Fe-Ni alloys were only reported at P and T conditions relevant to Earth's core. Thus, the contour map results for Fe-Ni alloys for the Moon, Mercury, Mars, and Ganymede's cores are unreliable extrapolations. The contour map of Fe-Ni alloys suggest ρ at low P and T is considerably higher than the values reported at high P and T. This would mean the effect of decreasing P, which acts to increase ρ, dominates over the effect of decreasing T, which acts to decrease ρ. In the same idea, ρ of Fe-O alloys was only reported at Earth's and Mars' core conditions, which implies the most reliable areas of the contour map are within those P and T boundaries. The contour map of Fe-Si/S alloys is in agreement with the reported data at the planetary core conditions of Earth, Moon, Mercury, Mars and Ganymede. The advantage of the contour maps is that they take into account the various P and T values used for the planetary core conditions and allow for a general estimate at a specific P and T combination. For example, the contour map of Fe-Si/S shows ρ of ∼1.70 µΩm at the lunar CMB (at 1,750 K which is the mid-T range, and 4.5 GPa), while the average of reported values is 1.66 µΩm without considering the differences in P and T conditions.

DISCUSSION
The interest in determining q ad is namely motivated by thermal evolution modelling. Direct measurements of thermal properties, and in particular k, of metals and metal-alloys at planetary core conditions are challenging and extrapolations from relatively low P and T conditions are not always consistent. Experimental progress made so far in measuring k at core conditions is very promising and future studies are needed to establish conclusively this important core property. The available literature on ρ at Earth's core conditions is focused on pure Fe, while a few studies have considered various light elements and a range of concentrations. Results suggest that the ρ of Fe-alloys at Earth's CMB and ICB does not significantly deviate from that of pure Fe. The scarce literature on ρ at the lunar core suggest the effect of P is negligible when compared to that of Si and S at similar wt%. On the contrary, the effect of Si and S at Mercury's core conditions remains unclear as the literature seems to be divided into two distinct groups of ρ, with ∼35% difference. At the Martian core, the ρ of Fe-alloys are within the variations of ρ for pure Fe. In contrast, the reported ρ of Fe-S alloys at Ganymede's CMB conditions show a great deviation from pure Fe measurements. Overall, the calculated ρ from the reported k values via the Wiedemann-Franz law, with the Sommerfeld value for L, do not significantly vary from direct measurements of ρ.
Results from first-principle calculations are within the variations of those from direct measurements from multi-anvil press, DAC and shock compression experiments. Although ρ values depend on the P and T, the variations in the selection of P and T at the planetary core conditions among the literature seem to have a negligible effect on the average ρ values. The contour maps provide an interpolation of ρ as a function of P and T within the boundaries of the reported values.
Further experimental research on ρ should focus on different light element compositions, particularly in Fe with multi-light element alloys, in order to constrain reliably ρ for FIGURE 7 | 3D visualization of the reported ρ with P and T of (A) Fe, (B) Fe-Ni alloys, (C) Fe-O alloys, and (D) Fe-Si/S alloys. Studies that did not state the exact P, T conditions for Earth's CMB and ICB are placed at 4,000 K and 136 GPa, and 5,000 K and 330 GPa, respectively. Similarly, the lunar CMB is placed at 1,750 K (which is the mid-T range of 1,687-1,800 K) and 4.5 GPa. Mercury's CMB and ICB are placed at 1,900 K and 5 GPa, and 2,350 K (which is the mid-T range of 2,200-2,500 K) and 36 GPa, respectively. The Martian CMB and ICB are placed at 1,770 K and 23 GPa, and 2,000 K and 40 GPa, respectively. Ganymede's CMB is located at 1,700 K (which is the mid-T range of 1,400-2,000 K) and 6 GPa (which is the mid-P range of 5-7 GPa), while the ICB is at 2,200 K and 9 GPa. The areas encompassing the reported data are in full colour, while the faded colors are areas of extrapolations.
Frontiers in Earth Science | www.frontiersin.org September 2021 | Volume 9 | Article 732289 likely core compositions at planetary core conditions. A larger data set on measurements of k will be needed to verify both theoretically determined values of the Lorenz number at extreme conditions of P and T for pure Fe and its many alloys as well as to substantiate the use of ρ data to calculate k. Calculations that account for spin polarization and the effects of magnetism on the electron scattering have recently been developed and further progress in this area to quantify this important contribution to electrical and thermal properties at planetary core conditions is likely. It is clear there is much to be done in the area of ρ and k property determination for application to thermal modelling of terrestrial-like planetary bodies in our Solar System. However, an even greater challenge appearing on the horizon is for similar studies to be carried out at the even greater P, T conditions of terrestrial-type exoplanets where the internal pressures are an order of magnitude higher than in Earth. Studies of those systems, which are currently in the nascent stages of characterizing equation of state and other structure-related characteristics of Fe  and Fe alloys , will then turn to interior modelling of heat flow and core dynamics to understand dynamos in super-Earths, which will require knowledge of ρ and k behavior over much greater P, T space.

AUTHOR CONTRIBUTIONS
MB is responsible for the formal analysis, writing and editing. RS is responsible for supervision, reviewing, editing, and funding.

FUNDING
This work was supported by funds to RS from the Natural Sciences and Engineering Research Council of Canada (grant number 2018-05021).