Presence of primordial Mg can explain the seismic low-velocity layer in the Earth’s outermost outer core

Domain: earth_science
Published: 2026-01-20
Editorial outcome: major_revision

Abstract
The composition of Earth’s Fe-rich liquid outer core has long been debated. Available models incorporating light elements, such as Si, O, C, S, and H, cannot explain the seismically low velocity layer in the uppermost outer core (E′ layer). Here we employ first-principles molecular dynamics simulations to determine the density and sound velocity (V_P) of Fe-Mg liquids under outer core conditions, which were unknown previously. Results show that the presence of Mg slightly decreases the V_P of liquid Fe, in contrast to the enhancing effects of other light elements. Our modeling suggests that 0.5-1.79 wt% Mg is required to match seismically observed core densities and velocities. Such amount of Mg could have entered the outermost outer core following the Moon-forming giant impact, thereby providing a viable explanation for the formation of the E′ layer and contributing to the slight Mg depletion in the bulk silicate Earth relative to chondritic meteorites.

Introduction
The composition of Earth’s core is crucial to understanding several fundamental processes of its evolution, including the early core-mantle differentiation1,2, inner core crystallization and growth3,4, and thermal-chemical convection in the core that powers the geodynamo5,6. Comparing seismically observed density and compressional wave velocity (V_P) of the outer core to those of pure liquid Fe has led to the identification of Si, O, C, S, and H7–10 as candidate light elements in varying concentrations in the outer core, depending on the specific composition model. In most studies, the Preliminary Reference Earth Model (PREM)11 has served as the standard seismic model for such comparison12–14, however, recent seismic models, including ek13715, EPOC16, KHOMC17, and CCMOC18, reveal the presence of a low velocity (up to 1% lower than that of PREM) layer of ~150-700 km thick in the outermost outer core, known as the E′ layer. The existence of the E′ layer has often been attributed to the presence of a chemically stratified layer in the outermost outer core19–24, with supporting evidence from geomagnetic studies17,25–27 and fluid dynamical considerations28. However, the composition of the E′ layer presents a significant paradox. Previous studies have shown that all commonly considered light elements (Si, O, C, S, and H) increase the seismic velocity of the outer core13,14. Thus, reducing the concentrations of light elements in this layer may decrease seismic velocity but would inevitably increase density, causing gravitational instability that would destruct stratification. Although some studies proposed that incomplete mixing22,28 and reactions21,24,29 between the core and lower mantle (or subducted slabs) could form a light and slow E′ layer, it has not been quantitatively evaluated whether such E′ layer compositions would satisfy the density and velocity constraints from seismic models. Furthermore, recent computations of core properties using mineral physics data12 reveal that even for the PREM model, the observed V_P in the outermost few hundred kilometers of the outer core is ~0.5-2.5% lower than those computed for the preferred compositional models, amplifying the mismatch between theoretically predicted and observed seismic velocities for the E′ layer.

Comparison of Mg/Al and Mg/Ca ratios in the bulk silicate Earth (BSE) and CI carbonaceous chondrites suggests nontrivial Mg depletion in the silicate Earth relative to refractory lithophile elements30. Such depletion maybe attributed to the slight volatility of Mg during Earth’s accretion31. Alternatively, high-pressure metal-silicate partitioning experiments5,32,33 suggest that a considerable amount of primordial Mg could have been incorporated into the Earth’s core, facilitated by the enhanced affinity of Mg to liquid Fe under the extreme temperatures associated with the Moon-forming giant impact. Although the subsequent cooling of the core could exsolve some Mg, thereby driving the early geodynamo before inner core crystallization5,33, a substantial quantity of residual Mg, potentially up to ~2 wt%5,32–36 (Supplementary Fig. S1), may remain sequestered in the core. However, the quantitative amount of Mg in the present outer core remains unclear, largely due to the lack of knowledge on the effects of Mg on density and V_P of liquid Fe under outer core condition, in contrast to the relatively well-determined effects of Mg on the properties of solid Fe-Mg phases under inner core conditions37,38.

In this study, we performed first-principles molecular dynamics (FPMD) simulations to determine the density and compressional wave velocity of Fe-Mg alloying liquids as functions of Mg content under Earth’s outer core conditions. Integrating calculated properties of Fe-alloying liquids with geophysical constraints on the outer and inner cores and the chemical equilibrium of light elements at the inner core boundary, we evaluated the potential Mg content in the outermost layer of the core. Our results underscore the previously underestimated role of primordial Mg in understanding the composition and chemical evolution of Earth’s core.

Results
Effects of Mg on the density and V_P of liquid Fe
FPMD simulations were conducted for pure liquid Fe and six different Fe-Mg liquid alloys with Mg contents ranging from 0–4.27 wt% under pressure and temperature conditions of 121–340 GPa and 6000–7500 K (Methods). The Mie–Grüneisen equation of state (EOS) was used to fit the calculated pressure-density-temperature (P-ρ-T) results, with best-fit EOS parameters listed in Supplementary Table S1. In general, our calculated EOS results for pure liquid Fe are consistent with those of previous computational studies14 under outer core conditions, particularly at 4000–6000 K (Supplementary Fig. S5). Following previous studies13,39, we applied a pressure correction to account for the pressure underestimation caused by the use of generalized gradient approximation (GGA) in FPMD simulations. We compared the pressures obtained from our FPMD simulations for pure liquid Fe with those along the adiabatic temperature profile in the outer core from shock-wave experimental data40 and obtained pressure corrections ~10 GPa under CMB conditions (135 GPa, 4000 K) and ~8 GPa at ICB conditions (330 GPa, 6000 K), with values at intermediate conditions obtained by linear interpolation.

As shown in Fig. 1, the density of Fe-Mg liquids increases upon compression but decreases with increasing temperature or Mg content. Using our EOS results, we calculated the density and adiabatic bulk modulus (K_S) of liquid Fe and Fe-Mg alloys under outer core conditions, from which the V_P of Fe-Mg liquids was obtained (Methods). The V_P of Fe-Mg liquids increases with pressure but is nearly independent of temperature (Supplementary Fig. S7a), consistent with the negligible temperature effects previously observed for the V_P of pure liquid Fe14. The effect of Mg on V_P is weak at all pressures and temperatures explored in this study (Supplementary Fig. S7b). Figure 2 shows the calculated density and V_P of Fe-Mg liquids as functions of Mg content at pressures and temperatures corresponding to the CMB and ICB, using reference temperatures of 5400 K and 6000 K at the ICB (Methods). Both ρ and V_P of Fe-Mg liquids decrease linearly with Mg content. Performing the pressure correction decreases the absolute values of density and velocity by 1.6–1.8% and 2.3–2.6%, respectively, at the CMB, and by 0.6–0.7% and 1.2%, respectively, at the ICB, but does not affect the compositional dependences. The V_P values as functions of Mg molar fraction (X_Mg) can be fitted linearly as V_P^CMB (km/s) = 7.80(0.02) − 0.71(0.37) × X_Mg and V_P^ICB (km/s) = 9.93(0.01) − 0.32(0.05) × X_Mg, respectively, confirming that the decreasing trend of V_P with increasing Mg content is significant given the uncertainties at both CMB and ICB conditions. The reduction of V_P in Fe-Mg liquids relative to those containing other light elements likely originates from the weakened metallic bonding between Mg and Fe, in contrast to stronger covalent interactions exhibited partially between light elements such as Si, S, C, O, and H with Fe (Supplementary Fig. S10 and Text S3). Compared with the effects of other light elements, the effect of Mg on reducing density is comparable to that of Si7. Assuming Mg is the sole light element in the outer core, an Mg content of 10.8–11.5 mol%, corresponding to ~5.0–5.3 wt%, is required to explain the outer core density deficit at the CMB conditions, while 13.5-14.1 mol% (6.4–6.7 wt%) of Mg is required to match the ICB density deficit. On the other hand, Mg moderately lowers V_P, in contrast to the increasing effects of other light elements. Thus, although the presence of Mg alone in the outer core is not consistent with the elevated velocity of the outer core compared to pure liquid Fe7,13,14,41, the combined effects of Mg and other light elements (Si, S, O, C, and H) may provide a viable explanation for the reduced velocity of the E′ layer in the outer core.

Mg content in the Earth’s outer core
Here we quantitatively evaluate the Mg content in the outer core by comparing the density and V_P of Mg-bearing Fe-rich liquids with those of seismic models. We consider core compositions of Fe alloyed with Ni, Mg, and other light elements among Si, O, C, S, and H. Mineral physics results from this study for Fe-Mg liquids and those of ref. 14 for Fe-X liquids (X=Si, O, C, S, and H) were used to calculate the density and V_P of the outer core, assuming the ideal mixing of Fe and light elements. As shown in previous studies12,14, the presence of Ni only slightly changes the outer core properties. We thus fixed the core Ni content at 5 wt% according to cosmochemical constraints7 and made corrections to the mixing results for density and V_P (Methods). Following the core composition optimization approach developed in ref. 12, we first searched for the outer core composition that minimizes a reduced misfit function Δ (Methods) that measures the difference between our model predicted outer and inner core densities and outer core velocities and the corresponding values from PREM. Equilibrium partitioning of light elements12,42–45 between the inner and outer cores at ICB conditions were incorporated in the modeling. Two types of models were developed for the outer core: a homogeneous outer core model (one-layer model) and a stratified core model consisting of two homogeneous layers (two-layer model), with the upper layer being ~600 km thick (from 135 to 200 GPa). In latter case, the thickness of the upper layer best resembles the seismically observed E′ layer15–18, although thinner stratified layers are suggested by geomagnetic studies46 to account for the morphology of geomagnetic field at the CMB. In both models, the optimization of a homogenous layer was performed in the Fe-5wt%Ni-Mg-X-Y systems, where X and Y represent any two light elements among Si, O, C, S, and H in addition to Mg. Thus, a total of ten systems were explored for each model (Supplementary Data 1).

For the homogeneous outer core model, the Fe-5wt%Ni-Mg-C-O, Fe-5wt%Ni-Mg-C-H, and Fe-5wt%Ni-Mg-H-O systems yield no solutions with Δ < 2 (Supplementary Data 1) because the preferred partitioning of C, O, and H would lead to larger density jumps at the ICB than seismically observed values. Although the Fe-5wt%Ni-Mg-S-H, Fe-5wt%Ni-Mg-S-O, Fe-5wt%Ni-Mg-S-C, and Fe-5wt%Ni-Mg-S-Si systems (Supplementary Data 1 and Fig. S11) result in Δ values of ~0.10–0.15, the best-fit S contents in these models considerably exceed the cosmochemical limit of 1.7 wt%47. Similarly, Fe-5wt%Ni-Mg-Si-C system with a Δ of 0.04 results in a best-fit C content of ~1.16 wt%, much higher than the core C content of 0.2 wt% allowed by the metal-silicate partition during core-formation48. Consequently, the Fe-5wt%Ni-Mg-Si-H and Fe-5wt%Ni-Mg-Si-O systems, with Δ values of 0.07–0.11, emerge as the most likely solutions for the composition of a homogeneous core. The best-fit outer core composition contains either 0.84±0.35 wt% Mg, 4.45±1.06 wt% Si, and 0.28±0.13 wt% H, or 1.06±0.35 wt% Mg, 5.04±0.91 wt% Si, and 1.71±0.86 wt% O. In both scenarios, Mg is present as a key light element in the core. Compared to previous Mg-free Fe-Ni-Si-O7,13,14,41 or Fe-Ni-Si-H12 core models, Mg primarily substitutes either O or H in the core, thereby reducing the V_P of the outer core while simultaneously maintaining the density deficit. The interchangeable nature of Si and S in modifying outer core properties suggests that a small amount of S may also be present in the core, as long as its content does not exceed that allowed by cosmochemical constraints.

For the stratified (two-layer) core model, the composition of the lower outer core layer was determined using a similar approach to that employed in the one-layer model, assuming no Mg is present. Specifically, chemical equilibrium between the lower outer core and the inner core was included as a constraint, in addition to core density and velocity. The composition of the Mg-bearing upper outer core was, on the other hand, determined using density and velocity constraints only, assuming no chemical equilibrium with either the lower outer core or the lower mantle. Our optimization results suggest that only the Fe-5wt%Ni-Mg-Si-H system yields solutions with overall Δ values lower than 0.25 (Supplementary Data 1 and Fig. S12). The best-fit composition contains 1.15±0.21 wt% Mg, 5.78±1.29 wt% Si, and 0.12±0.12 wt% H for the upper outer core layer, and 4.30±0.51 wt% Si and 0.43±0.06 wt% H for the lower outer core layer, consistent with our previous results using Mg-free one-layer models12.

Most recent seismic models including EPOC16, ek13715, KHOMC17, and CCMOC18 show lower V_P values in the outermost outer core than the PREM model. Here we use the EPOC model16 as a representative model to examine the impact of seismic models on our findings. Using the core density and velocity from EPOC, we performed similar optimizations for both one-layer and two-layer models. We found that for the homogeneous outer core, the Fe-5wt%Ni-Mg-Si-C, Fe-5wt%Ni-Mg-Si-H, Fe-5wt%Ni-Mg-Si-O, and Fe-5wt%Ni-Mg-Si-S systems exhibit minimum Δ values (0.26–0.30) near 0.25, but no model can yield a Δ value lower than 0.25 (Supplementary Data 1). For the two-layer outer core, only the Fe-5wt%Ni-Mg-Si-H system yields solutions with Δ < 0.25 (Supplementary Data 1 and Fig. S13). The best-fit compositions contain 1.31±0.27 wt% Mg, 4.13±1.72 wt% Si, and 0.16±0.16 wt% H in the upper outer core, and 5.03±0.63 wt% Si and 0.23±0.07 wt% H in the lower outer core. Equilibrium partitioning at the ICB42–44 further indicates that the inner core contains 5.03±0.63 wt% Si and 0.07±0.02 wt% H. Compared to PREM, using EPOC as constraints requires more Mg in the upper outer core to further reduce the velocity of Fe-rich liquid.

Our study reveals the critical role played by Mg in reducing velocity in Earth’s outer core. Across all optimized composition models with Δ < 0.25, the best-fit Mg contents range from 0.50 to 1.79 wt% (Fig. 3), suggesting the presence of Mg in the outer core, particularly in the outermost a few hundred kilometers. Using Mg/Al ratios in the BSE and CI carbonaceous chondrites30, our two-layer core model using EPOC density and velocity suggest that about 8% of the missing Mg in the BSE may be sequestered the core. A larger fraction of the missing Mg would reside in the core if the one-layer core model and the PREM density and V_P were used. Regardless of geophysical or compositional models, the inferred upper outer core likely contains Mg as a major light element.

Implications for the formation of the E′ layer
Experimental studies on high-pressure metal-silicate partitioning of Mg5,35,49 suggest that the dissolution of MgO into metal is dependent on temperature. Although some studies32,36 argue that Mg dissolution is strongly affected by the O content in the metal, recent results from ref. 50 incorporating self-consistent Mg-Si-O interactions confirm the notion that temperature is the controlling factor for Mg partitioning as both O and Si contents (two Mg-attracting elements) in the metal increase with temperature. Given the relatively modest temperatures (~2500-3500 K) at the base of the early magma ocean, where metal-silicate equilibration occurred during core formation, the Mg content in the metal was estimated to be 0.2-0.5 wt%5,49,50, significantly lower than our estimated Mg content of 1.31±0.27 wt% in the upper outer core (Fig. 3), when the EPOC model is used for optimization. Thus, the Moon-forming giant impact was necessary to generate the extreme temperatures required to produce the elevated Mg content in the E′ layer. Figure 4e–h schematically illustrate the inferred formation process of the E′ layer, highlighting key stages for the exchange and differentiation of light elements. At first, a proto-core of Fe-Si-H composition (lower outer core of the two-layer model) had formed possibly through the accretion and merge of a serious of relatively small impactor cores equilibrated with magma oceans of various depths51–53 before the Moon-forming giant impact event (Fig. 4e). The extremely high temperatures during the Moon-forming giant impact likely produced an Mg-rich Fe-Si-H-O-Mg metal diapir that had lower density than the proto-core and formed a buoyant layer surrounding the proto-core similar to the process described in ref. 28 (Fig. 4f, g). The subsequent secular cooling of the core could have exsolved some amounts of light elements in the forms of SiO249,54,55, H2O, FeO49,55, and/or MgO5,33,49,55 (Fig. 4h), resulting in an O-depleted E′ layer with an Fe-Si-H-Mg composition. Such exsolution processes are likely determined by the temperature-dependent solubilities of various components in the liquid core, which reflect chemical equilibrium between the core and the exsolved phases, although other solid components in the lower mantle, if in contact with the liquid outer core, may also affect the exsolution processes.

Figure 4a–d compare the differences in elemental concentrations for Si, H, Fe, and Mg between the inner and outer cores, reflecting the combined effects of the giant impact and subsequent light elements exsolution. The net results are such that some Mg from the mantle was incorporated into the upper outer core, while Si, H, Fe originally present in the upper outer core were transported into the mantle. Some O may have dissolved into the upper outer core along with other elements right after the giant impact but became depleted during the cooling process of the core.

The exsolution of Si, Fe, or H into the mantle has been proposed as a potential mechanism for driving the ancient dynamo49,54–56. In addition, some Mg exsolution is also expected5,33, as it is estimated that the giant impact could deliver 1.5–3.3 wt% Mg to the core49, a range surpassing our estimate of 1.31±0.27 wt% Mg in the present-day E′ layer. Exsolution of these primordial elements from the E′ layer to the mantle could have contributed to primordial isotopic signatures observed in some ocean island basalts (OIBs). For example, the anomalously low D/H ratios in Baffin Island and Icelandic lavas57 may reflect core-derived primordial materials brought up to the surface by mantle plumes. Our results highlight the complex Mg-Si, Mg-Fe, and Mg-H exchanges between the core and mantle, primarily driven by giant impacts and core cooling, providing insights into the coupled evolution of Earth’s deep interior.

Methods
First-principles molecular dynamics
First-principles molecular dynamics (FPMD) simulations were performed using the Vienna Ab initio Simulation Package (VASP)58 with the projector augmented wave (PAW) method59. The valence configurations for Fe and Mg in the PAW potentials are 3p63d74s1 and 3s2, respectively. The Perdew-Wang type generalized gradient approximation (GGA) was utilized for the exchange-correlation functional60. The plane wave cutoff energy was set at 450 eV and the Γ point was used to sample the Brillouin zone. Fermi-Dirac distributions were used for the single-particle orbitals. Simulations were carried out in the canonical ensemble (NVT) using a Nosé-Hoover thermostat for temperature control. To determine the effects of Mg on liquid Fe properties, seven supercells—Fe80, Fe78Mg1, Fe78Mg2, Fe78Mg3, Fe78Mg4, Fe78Mg6, and Fe78Mg8—were investigated, corresponding to Mg contents of 0.0 wt%, 0.55 wt%, 1.10 wt%, 1.65 wt%, 2.18 wt%, 3.24 and 4.27 wt%, respectively. The simulations were conducted over a pressure range of 121–340 GPa and a temperature range of 6000–7500 K. Each simulation has a timestep of 1.0 fs and a duration of about 14–24 ps. Initial temperature was set to 10000 K in the first 2 ps and melting of a system was verified by the radial distribution function (RDF) and mean squared displacements (MSD) of atoms. The structures were then quenched to the target temperatures (6000 to 7500 K) over a period of 2 ps, followed by an equilibration phase of 10–20 ps to determine pressure and energy. Different pressures were accessed by varying the volume of a supercell. The Pulay stress was determined to be less than 0.2 GPa using simulations employing a cutoff energy of 800 eV and a 2×2×2 k-point mesh. The effects of simulation cell size and time duration were evaluated by performing test simulations with a larger cell of Fe156Mg16 and/or a longer duration of up to 50 ps. These simulations showed negligible differences in pressure (<0.25 GPa), internal energy (<9.00 meV/atom), and derived compressional wave velocity (<0.3%) (Supplementary Figs. S2-3), indicating that the present system size (~80 atoms) and duration (~10 ps) are sufficient for the calculations of the compressional properties of liquid Fe-Mg alloys (Supplementary Text S1). Additional simulations were performed to determine the EOS of hcp-structured Fe60Mg4 under inner core conditions (Supplementary Table S3). These results were utilized to estimate the volume of Mg at the inner core boundary and to calculate the density of Fe-Mg alloys.

Equation of state fitting
The Mie–Grüneisen equation of state (EOS) was used to describe the pressure-density-temperature (P-ρ-T) data for liquid Fe and Fe-Mg alloys:

P(ρ,T) = 3K_T0 / K_0' * [(ρ/ρ_0)^(K_0'+1)/3 - (ρ/ρ_0)^(5/3)] * {1 + [3(K_0' - 4) / 2] * [(ρ/ρ_0)^(2/3) - 1]} + B(ρ)(T - T_ref),

where ρ_0, K_T0, and K_0' are, respectively, the density, isothermal bulk modulus, and pressure derivative of K_T0 at zero pressure and reference temperature T_ref; B(ρ) = (a + bρ/ρ_0) accounts for the temperature dependence of thermal pressure61,62. The EOS parameters (ρ_0, K_T0, K_0', a, and b) for all liquids were determined by least-squares fitting the P-ρ-T data from simulations to Eq. (1) with a T_ref of 6000 K. The best-fit EOS parameters are presented in Supplementary Table S1.

Calculations of the compressional wave velocity
The compressional wave velocities (V_P) of liquid Fe and Fe-Mg alloys were calculated from the adiabatic bulk modulus (K_S) and density, both of which were obtained using the EOS:

V_P = sqrt(K_S / ρ).

The adiabatic bulk modulus (K_S) is related to the isothermal bulk modulus (K_T), via

K_S = K_T + γT * (dP/dT)_V,

γ = (V/C_V) * (dP/dT)_V,

where γ is the Grüneisen parameter; C_V is the isochoric heat capacity; (dP/dT)_V is the temperature dependence of thermal pressure and is equal to B(ρ) in Eq. (1); K_T can be obtained from the EOS by taking derivative of Eq. (1) as,

K_T = ρ * dP/dρ.

According to the previous studies, e.g. refs. 61,63, γ is independent of temperature but depends on density as

γ = γ_0 * (ρ_0 / ρ)^c,

where γ_0 is the zero-pressure Grüneisen parameter and c is a constant. The best-fit results are listed in Supplementary Table S2. The Grüneisen parameter is related to the density derivative of temperature at constant entropy (S):

γ = (∂logT / ∂logρ)_S.

Integration of Eq. (7) yields the adiabatic temperature profiles in the outer core. Using 5400 K and 6000 K as two endmember temperatures for the inner core boundary (ICB), the estimated temperatures at the core-mantle boundary are about 4000 K and 4500 K, respectively, consistent with those used in previous studies61,64.

Calculations of core properties and optimization of core composition
In this model, the density, adiabatic bulk modulus, and V_P of Fe-alloying liquids are determined by the following equations:

ρ = M / V = (Σ X_i M_i) / (Σ X_i V_i),

K_S = Σ X_i K_S,i,

V_P = sqrt(K_S / ρ),

where M, V, ρ, K_S, and V_P correspond to the molar mass, molar volume, density, adiabatic bulk modulus, and compressional wave velocity of the alloy; X_i, M_i, V_i, and K_S,i represent the atomic fraction, atomic mass, partial atomic volume, and adiabatic bulk modulus of the i-th component in the Fe-alloy, respectively. The components considered include Fe, Si, O, C, S, H, and Mg. For the values of V_i and K_S,i, we utilized data derived from prior FPMD calculations14 that used similar exchange-correlation pseudopotentials and pressure corrections to this study. To account for the presence of 5 wt% Ni in the core, a positive density correction of 0.18% and a negative velocity correction of −0.19% were applied to the calculated properties of the Fe-rich liquids14. To evaluate the validity of the ideal mixing model for estimating density and V_P, we performed additional FPMD simulations for the Fe101Ni5Si11Mg3 composition, corresponding to 4.6 wt% Ni, 4.9 wt% Si, and 1.2 wt% Mg (Supplementary Text S2). This composition closely approximates the best-fit estimate for the E′ layer derived from the EPOC model. The calculated density (Supplementary Fig. S8) and V_P from these simulations are consistent with those predicted by the ideal mixing model within uncertainty, confirming that the ideal mixing model provides a reliable description of the density and V_P of Fe alloys under outer-core conditions.

We employed a simulated annealing algorithm66 to determine the optimal combination of light elements (Si, S, O, C, H, Mg) in the core, simultaneously satisfying seismically observed core properties and the chemical equilibrium at the ICB. The density and V_P of the outer core from both the PREM and EPOC models were used as observational constraints. The reduced misfit function for the optimization process was defined as:

Δ = 1 / (N-M) * [Σ_i ((ρ_Calc,i - ρ_Obs,i) / σ_ρ,i)^2 + Σ_i ((V_P_Calc,i - V_P_Obs,i) / σ_V_P,i)^2 + ((ρ_IC_Calc - ρ_IC_Obs) / σ_ρ,IC)^2],

where ρ_i and V_P,i represent the density and V_P of the outer core (OC) at one of the nine selected depths corresponding to pressures of 135, 150, 175, 200, 225, 250, 275, 300, and 325 GPa. The term ρ_IC denotes the density of the inner core at the ICB. Superscripts “Calc” and “Obs” indicate calculated and observed values, respectively. The degrees of freedom for fitting are given by N-M, where N and M are the number of data points and the number of free parameters, respectively. Uncertainties in the density and V_P are estimated to be ~0.58% and ~0.82%, respectively. The assessment of uncertainties demonstrates that variations in σ exert a negligible effect on the inferred Mg content and do not alter the compositional contrasts between the upper and lower outer core layers for Si, H, Fe, and Mg (Supplementary Text S4 and Figs. S14–17).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript is focused on the hypothesis that a significant concentration of Mg in the outer core might be responsible for the well-known seismic anomalies observed in the outer layers of the core, and in particular in the D' layer. To support this hypothesis the authors perform ab-initio calculations of Fe-Mg mixtures at pressures and temperatures compatible with the outer core conditions. Indeed, the simulations show that the presence of Mg can slightly decrease the velocity Vp, at variance with other light elements, which instead increase it.\nThis is an interesting result, but I feel that the evidence presented in the manuscript is not sufficiently strong to draw robust geophysical conclusions.\n1) Ab initio simulations are performed with a relatively small cells of 80 atoms, and for only a few ps. In a high-quality paper proposing a geophysical hypothesis based on simulations I would have expected the usage of neural network potentials to improve the statistics and check the convergence of the results with respect to system size and simulation time. These potentials can be fine-tuned to the specific conditions of the simulations, starting from one of the available machine learning models. As an alternative, I would suggest performing a careful size and time scale analysis, aimed at demonstrating that the estimate of Vp are meaningful.\n2) According to the estimates present in the vast literature on the subjects, and confirmed in the manuscript it is pretty evident that other light elements beside Mg are present in the outer core. A plausible composition reported at line 164, is ~5 % Ni, ~5 Si, ~ 1% Mg. It is plausible that the presence of a significant fraction of Si might alter the properties of the system at a level which is not capture by the mixture model of eq 8,9 and 10. I suggest performing ab initio or neural network simulations with this composition, estimating directly the density and the Vp value.", "reviewer_2": "The authors use first-principles molecular dynamics (FPMD) simulations to evaluate the equation of state and sound speed of Fe-Mg alloys. The results are significant because the addition of Mg into liquid Fe causes the sound speed to decrease. By comparison all other candidates for the light alloying element in the Earth’s core (e.g. Si, O, C, H, etc) cause the sound speed to increase. To account for the low seismic velocities detected near the top of Earth’s core you need to lower the local concentration of the “usual” light elements. Unfortunately, this change increases the fluid density near the boundary, and it makes the layer convectively unstable. In other words the layer needed to explain the seismic velocities could not survive. The computed properties of Mg-Fe alloys appear to resolve this difficulty.\nA natural question is why Mg behaves differently than other alloying elements. The authors do not offer much insight here. They note that their calculations for pure Fe reproduce previous results. How should readers access the validity of the calculations with increasing concentrations of Mg? Perhaps other reviewers with experience using FPMD simulations could comment on this point. If the results are reliable then the implications are important.\nThe authors use prior results on the partitioning of Mg between silicate melts and Fe metal to access the amount of Mg that could be added to the core during the formation of the planet. Separation of silicate and metal under moderate temperature and pressure would not transfer enough Mg to the core to account for the observed seismic velocities. Instead the authors appeal to the moon-forming impact to generate high temperatures, enabling a sufficient transfer of Mg into the core. The resulting metal should have low density (due to the high concentration of Mg), so it would pond at the top of the core. This configuration motivates a two-layer model with a low Mg concentration in the lower region and a high concentration of Mg in the upper layer. The authors show that this configuration can match the seismic observations across the entire core. Subsequent cooling causes the transfer of O into the base of the mantle (see Fig 4h). While this final step is not essential to the authors argument, it does raise questions about the thermodynamic state on the silicate side of the core-mantle boundary. What is the mantle composition that would allow transfer to of SiO2 or FeO into the mantle? If the authors are going to make the claim then they should include arguments to support this claim.\nThe upper layer of the two-layer model is taken to have a thickness of 600 km. This choice is motivated by several of the seismic velocity models cited in the manuscript. However, there is general consensus in the geomagnetic community that such a thick layer would negatively alter the morphology and time dependence of the Earth’s magnetic field (e.g. Gastine et al., GJI, vol. 222, 2020). This potential concern should be acknowledged in the manuscript.\nA few minor things\nLine 31: convention -> convection\nLine 307: the computed density and compressional velocities are calculated using ideal mixing. There is support for this approximation in prior studies. However, the description included in the manuscript differs from the approach in cited literature (ref. 14). In this study the adiabatic bulk modulus is computed by weighted average of the adiabatic bulk modulus of the individual components. By comparison, ref. 14 averages the isothermal bulk modulus. A standard correction is then applied to convert the isothermal bulk modulus to an adiabatic bulk modulus (which is the one needed for comparison with seismic observation). Have the authors actually converted each component to an adiabatic bulk modulus before computing the weight average? What is the justification for the authors approach?", "reviewer_3": "The manuscript by Liu and Jing reports results from first-principles molecular dynamics (FPMP) simulations concerning the equation of state and sound velocity of Fe-Mg alloy liquids under conditions in the Earth's outer core. The computed properties of the Fe-Mg alloys are employed to elucidate the characteristics of the low-velocity layer observed in the uppermost section of the outer core, referred to as the E' layer. It is hypothesized that primordial magnesium could have been incorporated into the outermost outer core via the Moon-forming giant impact, occurring under extreme high-pressure and high-temperature conditions. I have outlined specific areas below that need clarification and revision to meet publication standards.\n(1) The manuscript's focus on first-principles molecular dynamics (FPMD) calculations falls outside my area of expertise. A review by another expert familiar with this methodology would be useful to confirm if these calculations have been executed correctly. The manuscript mentions (Lines 84-90 and Figures S2-S3) that the generalized gradient approximation (GGA) utilized in the FPMD simulations requires pressure correction. It raises a question whether this correction is also temperature-dependent, potentially increasing with higher temperatures. Additionally, it would be beneficial for the authors to provide a detailed explanation of how the pressure correction was determined under the conditions present at CMB and ICB from 4000 K to 7000 K.\n(2) Fig. 2: The representation of literature data on pure iron from Bargain et al. (2018), Anderson and Ahrens (1994), and Kuwayama et al. (2020) could benefit from clearer plotting. The manuscript extensively discusses the impact of primordial magnesium on the Vp in the Earth's outermost core. More comprehensive statistical and sensitivity analyses are necessary to robustly validate the findings. While the decrease in density with the incorporation of magnesium in iron liquids is evident, it is less apparent that this will also reduce Vp. This uncertainty stems in part from the complexities associated with fitting the equation of state, highlighting the need for more precise analysis to clarify these effects.\n(3) Mg content in the Earth’s outer core: Because of the concerns raised in (2), how would the errors in determining Vp from EOS fittings affect the results on Mg content in the Earth’s outer core. For reproducibility and open science, it would be beneficial to include the codes for the modeling along with the manuscript as datasets. The determination of magnesium content in the Earth’s outer core is critically dependent on the accurate determination of Vp, which is derived from the equation of state (EOS) fittings. The errors inherent in these fittings and calculations based on bulk modulus and density, as outlined in the concerns raised in (2), could significantly impact the inferred content of Mg in Earth’s outer core. This introduces a potential source of uncertainty in our understanding of the outer core's composition. To enhance the reliability and reproducibility of the findings, and in line with principles of open science, it is advisable for the authors to make the computational codes used for modeling available along with the manuscript. Providing these codes as supplementary datasets would allow other researchers to verify the results, potentially replicate the study, and extend the analysis further.\n(4) Fig S4: What are the uncertainties of Vp determined for 4000 K, 5000 K, 6000 K, and 7000 K from low to high pressures?"}, "author_response": "Dear Editor,\nWe are thankful to your handling of our manuscript and the valuable comments from\nthe reviewers. We have revised our manuscript based on these comments. Below are\nour responses to the comments, with the original comments from the reviewers in\nblack, our responses in blue, and the revisions made to the manuscript in orange.\nThank you for considering our manuscript.\nSincerely,\nTao Liu and Zhicheng Jing\nReviewer #1 (Remarks to the Author):\nThe manuscript is focused on the hypothesis that a significant concentration of Mg in\nthe outer core might be responsible for the well-known seismic anomalies observed in\nthe outer layers of the core, and in particular in the D' layer. To support this\nhypothesis the authors perform ab-initio calculations of Fe-Mg mixtures at pressures\nand temperatures compatible with the outer core conditions. Indeed, the simulations\nshow that the presence of Mg can slightly decrease the velocity Vp, at variance with\nother light elements, which instead increase it.\nThis is an interesting result, but I feel that the evidence presented in the manuscript is\nnot sufficiently strong to draw robust geophysical conclusions.\nThank you for your valuable comments. In the revised manuscript, we have\nstrengthened our conclusions by performing additional test simulations and analyses,\nincluding system size tests and time-scale assessments, as well as their respective\ninfluences on V . These simulations and analyses confirm the robustness of our\nP\nresults on the equation of state and sound velocity of Fe-Mg liquids and provide\nstrong support to the derived seismic properties.\n1) Ab initio simulations are performed with a relatively small cells of 80 atoms, and\nfor only a few ps. In a high-quality paper proposing a geophysical hypothesis based\non simulations I would have expected the usage of neural network potentials to\nimprove the statistics and check the convergence of the results with respect to system\nsize and simulation time. These potentials can be fine-tuned to the specific conditions\nof the simulations, starting from one of the available machine learning models. As an\nalternative, I would suggest performing a careful size and time scale analysis, aimed\nat demonstrating that the estimate of Vp are meaningful.\nThank you for your constructive suggestions. While most previous FPMD studies of\nliquid Fe have employed simulation cells containing on the order of 64 atoms (He et\nal., 2022; Li et al., 2020; Luo et al., 2024; Umemoto et al., 2014), we fully agree that\nsystematic convergence tests are essential to assess the robustness of the calculated\nproperties. Although neural network potentials, as you suggested, can in principle\nextend simulations to substantially longer timescales and larger system sizes, they\nalso introduce additional uncertainties arising from the training process, in addition to\nthose inherent in FPMD calculations.\nTo directly address this concern, we conducted benchmark calculations for liquid\nFe Mg by doubling the simulation cell to Fe Mg and extending the simulation\n78 8 156 16\ntime duration to 50 ps in FPMD simulations. As shown in Supplementary Fig. S2-3,\nthe differences in average pressure and internal energy between the systems of\ndifferent sizes at 6000-7500 K remains below 0.25 GPa and 9.00 meV/atom,\nrespectively. These results confirm that the pressures and internal energies obtained\nfrom the larger-cell, longer-time simulations are in excellent agreement with those\nreported in the manuscript, thereby validating the robustness of our original results.\nFurthermore, to evaluate the effect of cell size on compressional velocity (V ),\nP\nwe performed additional EOS simulations (Supplementary Fig. S4) for the large cell\nFe Mg using durations of 14 ps. All simulation results are provided in the\n156 16\nSupplementary Dataset for the revised manuscript. At two representative conditions\ncorresponding to the core-mantle boundary (CMB) and inner-core boundary (ICB),\nthe calculated V values for Fe Mg are 7.79±0.08 km/s and 9.91±0.05 km/s,\nP 156 16\nrespectively. These values are virtually identical to those for Fe Mg in the original\n78 8\nmanuscript (7.77±0.11 km/s at the CMB and 9.90±0.08 km/s at the ICB), with\ndifferences smaller than 0.3%. These results demonstrate that variations in cell size\nand simulation duration have negligible effects on the derived properties, thereby\nconfirming our original results and conclusions.\nIn Supplementary Information we added:\nSupplementary Fig. S2 Convergence assessment of the effects of simulation cell size\nand time duration on the calculated pressure at various temperatures. The P and P\nb r\ndenote the mean pressures obtained from the simulations of the original manuscript\n(blue) and from the benchmark calculations using an enlarged cell with an extended\nsimulation duration (red), respectively.\nSupplementary Fig. S3 Convergence assessment of the effects of simulation cell size\nand time duration on the calculated internal energy at different temperatures. The U\nb\nand U represent the mean internal energies for Fe Mg obtained from the simulations\nr 78 8\nof the original manuscript and from the test simulations with the enlarged simulation\ncell (values normalized to Fe Mg ) with an extended simulation duration,\n78 8\nrespectively.\nSupplementary Fig. S4 Equation of state of liquid Fe Mg obtained under\n156 16\ndifference temperature and pressure conditions. These results are nearly identical to\nthose for liquid Fe Mg in Fig. 1.\n78 8\nOn Lines 283-288 of the revised manuscript, we wrote:\n“\nThe effects of simulation cell size and time duration were evaluated by performing\ntest simulations with a larger cell of Fe Mg and/or a longer duration of up to 50 ps.\n156 16\nThese simulations showed negligible differences in pressure (0.25 GPa), internal\nenergy (9.00 meV/atom), and derived compressional wave velocity (<0.3%)\n(Supplementary Figs. S2-3), indicating that the present system size and duration are\nsufficient for the calculations of the compressional properties of liquid Fe-Mg alloys\n(Supplementary Text S1).\n”\nOn Lines 35-49 of the Supplementary information, we added:\n“\nText S1 Effects of simulation cell size and time duration\nTo assess the influences of simulation cell size and time duration on the calculated\nequation of state and compressional wave velocity (V ), benchmark FPMD\nP\ncalculations were performed for liquid Fe Mg by doubling the simulation cell to\n78 8\nFe Mg and extending the simulation duration to 50 ps. As shown in\n156 16\nSupplementary Figs. S2-3, the differences in average pressure and internal energy\nbetween the systems of different sizes at 6000-7500 K remains below 0.25 GPa and\n9.00 meV/atom, respectively. Furthermore, to evaluate the effect of cell size on V ,\nP\nwe conducted additional EOS simulations (Supplementary Fig. S4) for Fe Mg\n156 16\nusing durations of 14 ps. All simulation results are provided in the Supplementary\nDataset. At two representative conditions corresponding to the core-mantle boundary\n(CMB) and inner-core boundary (ICB), the calculated V values for Fe Mg are\nP 156 16\n7.79±0.08 km/s and 9.91±0.05 km/s, respectively. These values are virtually identical\nto those for Fe Mg (7.77±0.11 km/s at the CMB and 9.90±0.08 km/s at the ICB),\n78 8\nwith differences smaller than 0.3%. In summary, these test simulations confirm that a\nsimulation cell containing ~80 atoms and a time duration of ~10 ps are sufficient to\nachieve reliable convergence of pressure, internal energy, and V in liquid Fe-Mg\nP\nalloys under core conditions.\n”\n2) According to the estimates present in the vast literature on the subjects, and\nconfirmed in the manuscript it is pretty evident that other light elements beside Mg\nare present in the outer core. A plausible composition reported at line 164, is ~5 % Ni,\n~5 Si, ~ 1% Mg. It is plausible that the presence of a significant fraction of Si might\nalter the properties of the system at a level which is not capture by the mixture model\nof eq 8,9 and 10. I suggest performing ab initio or neural network simulations with\nthis composition, estimating directly the density and the Vp value.\nThank you for your constructive suggestions. To further evaluate the robustness of\nour conclusions regarding the composition of the E layer (135-200 GPa), which was\nmodeled as a liquid Fe alloy containing ~5% Ni, ~5% Si, ~1% Mg, we conducted\nadditional FPMD simulations on a representative composition, Fe Ni Si Mg ,\n101 5 11 3\ncorresponding to 4.6 wt% Ni, 4.9 wt% Si, and 1.2 wt% Mg. Although this\ncomposition closely approximates the best-fit model, slight deviations remain due to\ncomputational constraints that limit the use of larger simulation cells required for\nprecise compositional refinement. The resulting EOS for liquid Fe Ni Si Mg is\n101 5 11 3\npresented in Supplementary Fig. S8. The calculated densities and V values of\nP\noutermost outer core are 10.03 ± 0.02 g/cm3 and 7.95 ± 0.09 km/s at 4540 K and 135\nGPa, 10.26 ± 0.02 g/cm3 and 8.16 ± 0.08 km/s at 4693 K and 150 GPa, 10.62 ± 0.02\ng/cm3 and 8.49 ± 0.08 km/s at 4937 K and 175 GPa, and 10.95 ± 0.02 g/cm3 and 8.79\n± 0.07 km/s at 5167 K and 200 GPa, respectively. For comparison, the ideal mixture\nmodel yields densities and V values of 10.09 ± 0.06 g/cm3 and 8.02 ± 0.07 km/s at\nP\nand 8.59 ± 0.07 km/s at 175 GPa, and 10.99 ± 0.06 g/cm3 and 8.90 ± 0.07 km/s at 200\nGPa, respectively. Thus, the directly calculated densities and velocities are within\nuncertainty consistent with those calculated from the ideal mixing model.\nIn Supplementary Information we added:\nSupplementary Fig. S8 Equation of state of liquid Fe Ni Si Mg determined under\n101 5 11 3\nvarying temperatures and pressures.\nOn Lines 342-349 of the revised manuscript, we wrote:\n“\nTo evaluate the validity of the ideal mixing model for estimating density and V ,\nwe\nP\nperformed additional FPMD simulations for the Fe Ni Si Mg composition,\n101 5 11 3\ncorresponding to 4.6 wt% Ni, 4.9 wt% Si, and 1.2 wt% Mg (Supplementary Text S2).\nThis composition closely approximates the best-fit estimate for the E layer derived\nfrom the EPOC model. The calculated density (Supplementary Fig. S8) and V from\nP\nthese simulations are consistent with those predicted by the ideal mixing model within\nuncertainty, confirming that the ideal mixing model provides a reliable description of\nthe density and V of Fe alloys under outer-core conditions.\nP\n”\nOn Lines 71-85 of the Supplementary information, we added:\n“Text S2 Test the effect of the ideal mixing model on density and V\nP\nTo evaluate the validity of the ideal mixing model, we conducted additional FPMD\nsimulations on a representative composition, Fe Ni Si Mg , corresponding to 4.6\n101 5 11 3\nwt% Ni, 4.9 wt% Si, and 1.2 wt% Mg. Although this composition closely\napproximates the best-fit model of the E layer (135-200 GPa), slight deviations\nremain due to computational constraints that limit the use of larger simulation cells\nrequired for precise compositional refinement. The resulting EOS for liquid\nFe Ni Si Mg is presented in Supplementary Fig. S8. The calculated densities and\n101 5 11 3\nV values of the outermost outer core are 10.03 ± 0.02 g/cm3 and 7.95 ± 0.09 km/s at\nP\nGPa, 10.62 ± 0.02 g/cm3 and 8.49 ± 0.08 km/s at 4937 K and 175 GPa, and 10.95 ±\n0.02 g/cm3 and 8.79 ± 0.07 km/s at 5167 K and 200 GPa, respectively. For\ncomparison, the ideal mixing model yields densities and V values of 10.09 ± 0.06\nP\ng/cm3 and 8.02 ± 0.07 km/s at 135 GPa, 10.32 ± 0.06 g/cm3 and 8.24 ± 0.07 km/s at\ng/cm3 and 8.90 ± 0.07 km/s at 200 GPa, respectively. Thus, the directly calculated\ndensities and velocities using FPMD simulations are within uncertainty consistent\nwith those calculated from the ideal mixing model.”\nReviewer #2 (Remarks to the Author):\nThe authors use first-principles molecular dynamics (FPMD) simulations to evaluate\nthe equation of state and sound speed of Fe-Mg alloys. The results are significant\nbecause the addition of Mg into liquid Fe causes the sound speed to decrease. By\ncomparison all other candidates for the light alloying element in the Earth’s core (e.g.\nSi, O, C, H, etc) cause the sound speed to increase. To account for the low seismic\nvelocities detected near the top of Earth’s core you need to lower the local\nconcentration of the “usual” light elements. Unfortunately, this change increases the\nfluid density near the boundary, and it makes the layer convectively unstable. In other\nwords the layer needed to explain the seismic velocities could not survive. The\ncomputed properties of Mg-Fe alloys appear to resolve this difficulty.\nThank you for your supportive comments.\nA natural question is why Mg behaves differently than other alloying elements. The\nauthors do not offer much insight here. They note that their calculations for pure Fe\nreproduce previous results. How should readers access the validity of the calculations\nwith increasing concentrations of Mg? Perhaps other reviewers with experience using\nFPMD simulations could comment on this point. If the results are reliable then the\nimplications are important.\nThank you for the insightful comment and the recognition of the significance of our\nwork.\n(1) To understand the influence of Mg on V relative to other light elements, we\nP\ncomputed the radial distribution functions (RDFs; Supplementary Fig. S9) and\nperformed electron localization function (ELF) analyses (Savin et al., 1997)\n(Supplementary Fig. S10) for Fe X liquids (X = Mg, Si, S, O, C, and H) at 6000 K\n156 16\nand V=1226.52 Å3. The RDFs reveal that the positions and amplitudes of the first\npeaks for Fe–Fe, Fe–Mg, and Mg–Mg pairs in liquid Fe–Mg are nearly identical, in\ncontrast to the distinct peak separations between Fe-Fe and Fe-X peaks observed in\nother Fe–light element alloying liquids. This structural similarity indicates that Mg\nforms metallic bonds with Fe that are nearly identical to Fe-Fe bonds. This is\nconsistent with the absence of pronounced Fe–Mg orbital hybridization in ELF\ncalculations (Supplementary Fig. S10c), with ELF values of ~0.5. However, unlike\nthe delocalization of 3d electrons in Fe contributes to the cohesion of metallic bonds,\nMg lacks d electrons and contributes only s electrons to the conduction band, leading\nto a lower electronic density of states near the Fermi level and weaker metallic\nbonding. Thus, replacing Fe with Mg atoms effectively decreases the strength of\nmetallic bonds and in turn decreases the bulk modulus of the liquid. Although the\nsmaller atomic mass of Mg than Fe decreases the density, the overall effect is that the\nsound velocity decreases slightly with Mg content. On the other hand, other light\nelements including Si, S, C, O, and H exhibit partial covalent interactions with Fe, as\nevidenced by shorter Fe-X distances in RDFs and their distinct hybridization features\nin ELF calculations (with ELF values close to 1.0) (Supplementary Fig. S10). The\ncovalent nature and shorter length of Fe–X bonds enhance local stiffness and\ntherefore increase the bulk modulus and sound velocity of Fe-alloying liquids, in\ncontrast to the effect of Mg.\n(2) Regarding the reliability of our FPMD results at elevated Mg concentrations, we\nnote that our simulations for pure Fe liquid reproduce published data with excellent\nagreement, establishing a solid methodological benchmark. The computational\nprotocol for Fe-Mg alloys is identical to that used for pure Fe, ensuring consistency.\nTo further assess convergence, we conducted additional FPMD simulations using\nsupercells twice (Fe Mg ) the original size and extended simulation time duration\n156 16\nup to 50 ps––five times longer than in the original calculations, as recommended by\nReviewer 1. These tests revealed very minor differences in the calculated pressures\n(less than 0.25 GP) and internal energies (less than 9.00 meV/atom) (see our\nresponses to the comments of Reviewer 1 and Supplementary Figs. S2-3). For direct\ncomparison, two representative thermodynamic states corresponding to the CMB and\nICB were examined. The calculated V values for Fe Mg at these conditions are\nP 156 16\n7.79±0.08 km/s and 9.91±0.05 km/s, respectively, in excellent agreement with the\nvalues reported for Fe Mg in the original manuscript (7.77±0.11 at the CMB and\n78 8\n9.90±0.08 at the ICB), with deviations of less than 0.3%. This confirms the robustness\nof our computations.\nIn Supplementary Information we added:\nSupplementary Fig. S9 RDFs of liquid Fe X alloys (X = C, H, Mg, O, S, and Si)\n156 16\ncalculated at 6000 K and V=1226.52 Å3.\nSupplementary Fig. S10 ELF maps of liquid Fe X alloys (X = C, H, Mg, O, S, and\n156 16\nSi) calculated at 6000 K and V= 1226.52 Å3. The results reveal that light elements such\nas C, H, O, S, and Si form strong covalent-like bonds with Fe (ELF values close to 1),\nwhereas Mg interacts with Fe primarily through weak metallic bonding (ELF values\nclose to 0.5).\nOn Lines 108-111 of the revised manuscript, we added:\n“\nThe reduction of V in Fe-Mg liquids relative to those containing other light elements\nP\nlikely originates from the weakened metallic bonding between Mg and Fe, in contrast\nto stronger covalent interactions exhibited partially between light elements such as Si,\nS, C, O, and H with Fe (Supplementary Fig. S10 and Text S3).\n”\nOn Lines 107-127 of the Supplementary information, we added:\n“\nText S3 Radial distribution and electron localization functions of liquid Fe-X alloys\nTo understand the influence of Mg on V relative to other light elements, we\nP\ncomputed the radial distribution functions (RDFs; Supplementary Fig. S9) and\nperformed electron localization function (ELF) analyses (Supplementary Fig. S10) for\nFe X liquids (X = Mg, Si, S, O, C, and H) at 6000 K and V=1226.52 Å3. The RDFs\n78 8\n(Supplementary Fig. S9) reveal that the positions and amplitudes of the first peaks for\nFe–Fe, Fe–Mg, and Mg–Mg pairs in liquid Fe–Mg are nearly identical, in contrast to\nthe distinct peak separations between Fe-Fe and Fe-X peaks observed in other Fe–\nlight element alloying liquids. This structural similarity indicates that Mg forms\nmetallic bonds with Fe that are nearly identical to Fe-Fe bonds. This is consistent with\nthe absence of pronounced Fe–Mg orbital hybridization in ELF calculations\n(Supplementary Fig. S10c), with ELF values of ~0.5. However, unlike the\ndelocalization of 3d electrons in Fe contributes to the cohesion of metallic bonds, Mg\nlacks d electrons and contributes only s electrons to the conduction band, leading to a\nlower electronic density of states near the Fermi level and weaker metallic bonding.\nThus, replacing Fe with Mg atoms effectively decreases the strength of metallic bonds\nand in turn decreases the bulk modulus of the liquid. Although the smaller atomic\nmass of Mg than Fe decreases the density, the overall effect is that the sound velocity\ndecreases slightly with Mg content. On the other hand, other light elements including\nSi, S, C, O, and H exhibit partial covalent interactions with Fe, as evidenced by\nshorter Fe-X distances in RDFs and their distinct hybridization features in ELF\ncalculations (with ELF values close to 1.0) (Supplementary Fig. S10). The covalent\nnature and shorter length of Fe–X bonds enhance local stiffness and therefore increase\nthe bulk modulus and sound velocity of Fe-alloying liquids, in contrast to the effect of\nMg.\n”\nThe authors use prior results on the partitioning of Mg between silicate melts and Fe\nmetal to access the amount of Mg that could be added to the core during the formation\nof the planet. Separation of silicate and metal under moderate temperature and\npressure would not transfer enough Mg to the core to account for the observed seismic\nvelocities. Instead the authors appeal to the moon-forming impact to generate high\ntemperatures, enabling a sufficient transfer of Mg into the core. The resulting metal\nshould have low density (due to the high concentration of Mg), so it would pond at\nthe top of the core. This configuration motivates a two-layer model with a low Mg\nconcentration in the lower region and a high concentration of Mg in the upper layer.\nThe authors show that this configuration can match the seismic observations across\nthe entire core. Subsequent cooling causes the transfer of O into the base of the mantle\n(see Fig 4h). While this final step is not essential to the authors argument, it does raise\nquestions about the thermodynamic state on the silicate side of the core-mantle\nboundary. What is the mantle composition that would allow transfer to of SiO2 or\nFeO into the mantle? If the authors are going to make the claim then they should\ninclude arguments to support this claim.\nThank you for raising this point. The core-forming metallic liquids were in partial\nequilibrium with silicate melts in the early magma ocean produced by the Moon-\nforming giant impact. However, once the metallic liquids penetrated the solid lower\nmantle through diapirism and formed the liquid core, there is no longer large-scale\nchemical equilibrium between the solid lower mantle and liquid outer core due to the\nslow diffusion rates in solids (Karato, 2023). The exsolution of components such as\nSiO , FeO, and MgO from the liquid outer core is mainly due to the temperature\n2\ndependence of solubility of these components in liquid Fe, but not the equilibrium\npartitioning of these elements between metal and silicate. Thus, the thermodynamic\nstate of the silicate mantle does not have a strong control on the exsolution processes.\nAs demonstrated in previous studies, the secular cooling of the core could lead to the\nexsolution of components such as SiO (Helffrich et al., 2020; Hirose et al., 2017;\nO’Rourke and Stevenson, 2016), and FeO (Deng, 2024; O’Rourke and Stevenson,\n2016), which may form sedimentary layers at the core-mantle boundary or be\nentrained to the mantle through convection.\nOn Lines 222-224 of the revised manuscript, we added:\n“\nSuch exsolution process is mainly determined by the temperature-dependent\nsolubilities of various components in the liquid core, rather than equilibrium\npartitioning between the core and solid lower mantle.\n”\nThe upper layer of the two-layer model is taken to have a thickness of 600 km. This\nchoice is motivated by several of the seismic velocity models cited in the manuscript.\nHowever, there is general consensus in the geomagnetic community that such a thick\nlayer would negatively alter the morphology and time dependence of the Earth’s\nmagnetic field (e.g. Gastine et al., GJI, vol. 222, 2020). This potential concern should\nbe acknowledged in the manuscript.\nThank you for your insightful comment. In response, we have revised the manuscript\nto acknowledge this point.\nOn Lines 138-141 of the revised manuscript, we wrote:\n“\nIn latter case, the thickness of the upper layer best resembles the seismically observed\nE layer (Irving et al., 2018; Kaneshima, 2018; Kennett, 2020; Ma and Tkalčić, 2023),\nalthough thinner stratified layers are suggested by geomagnetic studies (Gastine et al.,\n2020) to account for the morphology of geomagnetic field at the CMB.\n”\nA few minor things\nLine 31: convention -> convection\nThis has been corrected.\nLine 307: the computed density and compressional velocities are calculated using\nideal mixing. There is support for this approximation in prior studies. However, the\ndescription included in the manuscript differs from the approach in cited literature\n(ref. 14). In this study the adiabatic bulk modulus is computed by weighted average of\nthe adiabatic bulk modulus of the individual components. By comparison, ref. 14\naverages the isothermal bulk modulus. A standard correction is then applied to\nconvert the isothermal bulk modulus to an adiabatic bulk modulus (which is the one\nneeded for comparison with seismic observation). Have the authors actually converted\neach component to an adiabatic bulk modulus before computing the weight average?\nWhat is the justification for the authors approach?\nThanks for pointing this out. Our statement in the original manuscript was not clear.\nThe mixing model for density in our study is the same as that from ref. 14 (Umemoto\nand Hirose, 2020), but the model for V is based on the adiabatic bulk modulus (K )\nP S\ninstead of the isothermal bulk modulus (K ). There are two reasons for this. First, we\nT\nused values of V and  for Fe–S, Fe–H, Fe–O, and Fe–C liquids from Umemoto and\nP\nHirose (2020), to compute K , but the values of K and associated thermodynamic\nS T\nparameters (thermal expansion coefficients and Grüneisen parameters) were not\nprovided in Umemoto and Hirose (2020), preventing conversion between K and K .\nT S\nSecond, and more importantly, most previous studies (Badro et al., 2014; Deng et al.,\n2023; Wang et al., 2020) actually applied mixing models to K other than K to\nS T\ncalculate V . Therefore, our use of K is consistent with the established practice in the\nP S\nfield. We have modified the cited references in the revised manuscript to reflect this.\nOn Lines 327-329 of the revised manuscript, we wrote:\n“\nThe density and compressional wave velocity of the Earth’s outer core were\ncalculated using the ideal mixing model of Fe-alloying liquids, similar to the approach\nused in ref.(Deng et al., 2023; Umemoto and Hirose, 2020).\n”\nReviewer #3 (Remarks to the Author):\nThe manuscript by Liu and Jing reports results from first-principles molecular\ndynamics (FPMP) simulations concerning the equation of state and sound velocity of\nFe-Mg alloy liquids under conditions in the Earth's outer core. The computed\nproperties of the Fe-Mg alloys are employed to elucidate the characteristics of the\nlow-velocity layer observed in the uppermost section of the outer core, referred to as\nthe E' layer. It is hypothesized that primordial magnesium could have been\nincorporated into the outermost outer core via the Moon-forming giant impact,\noccurring under extreme high-pressure and high-temperature conditions. I have\noutlined specific areas below that need clarification and revision to meet publication\nstandards.\nThank you for your supportive comments.\n(1) The manuscript's focus on first-principles molecular dynamics (FPMD)\ncalculations falls outside my area of expertise. A review by another expert familiar\nwith this methodology would be useful to confirm if these calculations have been\nexecuted correctly. The manuscript mentions (Lines 84-90 and Figures S2-S3) that the\ngeneralized gradient approximation (GGA) utilized in the FPMD simulations requires\npressure correction. It raises a question whether this correction is also temperature-\ndependent, potentially increasing with higher temperatures. Additionally, it would be\nbeneficial for the authors to provide a detailed explanation of how the pressure\ncorrection was determined under the conditions present at CMB and ICB from 4000\nK to 7000 K.\nThank you for your insightful comment.\nRegarding to the methodology concerns, we have performed additional test\nsimulations and analyses, including system size tests and time-scale assessments, as\nwell as their respective influences on V to respond to the comments raised by\nP\nReviewer #1.\nRegarding the temperature dependence of the GGA pressure correction, we agree this\nis potentially an important issue for the accuracy of our FPMD simulations under core\nconditions. In fact, the temperature dependence of pressure correction remains a hotly\ndebated topic in the literature. While most studies express it as a function of pressure\nor density (Alfè et al., 2002; Badro et al., 2014; Li et al., 2022), some have suggested\na combined pressure and temperature dependence (Umemoto and Hirose, 2020). In\nour study, we follow the outer-core adiabatic temperature profile and effectively\nconsider the pressure and temperature effects on pressure correction simultaneously.\nWe applied the pressure correction by comparing the pressures obtained from our\nFPMD simulations for liquid Fe along the outer-core adiabat with those from shock-\nwave experimental data (Anderson and Ahrens, 1994). The resulting correction\npressures are ~10 GPa under CMB conditions (135 GPa, ~4000 K) and ~8 GPa at\nICB conditions (330 GPa, ~6000 K), with values at intermediate conditions obtained\nby linear interpolation. These corrections are in close agreement with those reported\nin earlier studies (Alfè et al., 2002; Badro et al., 2014), and ensure that the calculated\nV values along the outer-core adiabat are robust and inherently accounts for the\nP\ntemperature effect on the pressure correction.\nOn Lines 84-90 of the revised manuscript, we wrote:\n“\nFollowing previous studies (Alfè et al., 2002; Badro et al., 2014), we applied a\npressure correction to account for the pressure underestimation caused by the use of\ngeneralized gradient approximation (GGA) in FPMD simulations. We compared the\npressures obtained from our FPMD simulations for pure liquid Fe with those along\nthe adiabatic temperature profile in the outer core from shock-wave experimental data\n(Anderson and Ahrens, 1994) and obtained pressure corrections ~10 GPa under CMB\nconditions (135 GPa, 4000 K) and ~8 GPa at ICB conditions (330 GPa, 6000 K), with\nvalues at intermediate conditions obtained by linear interpolation.\n”\n(2) Fig. 2: The representation of literature data on pure iron from Bargain et al.\n(2018), Anderson and Ahrens (1994), and Kuwayama et al. (2020) could benefit from\nclearer plotting. The manuscript extensively discusses the impact of primordial\nmagnesium on the Vp in the Earth's outermost core. More comprehensive statistical\nand sensitivity analyses are necessary to robustly validate the findings. While the\ndecrease in density with the incorporation of magnesium in iron liquids is evident, it\nis less apparent that this will also reduce Vp. This uncertainty stems in part from the\ncomplexities associated with fitting the equation of state, highlighting the need for\nmore precise analysis to clarify these effects.\nFollowing the suggestion, we have revised Fig. 2 to improve its clarity.\nTo assess the robustness of our results, we performed additional FPMD simulations\nusing supercells twice as large (Fe Mg ). The calculated V values for Fe Mg\n156 16 P 156 16\nare 7.79±0.08 km/s under CMB conditions and 9.91±0.05 km/s under ICB conditions,\nin excellent agreement with the corresponding values obtained for Fe Mg in the\n78 8\noriginal manuscript (7.77±0.11 at the CMB and 9.90±0.08 at the ICB), differing by\nless than 0.3%. Although the use of large supercells considerably reduces the\nuncertainty in V , further enlargement of the simulation cell is constrained by\nP\ncomputational limitations. Nevertheless, the close agreement between independent\nsimulations confirms that the mean V values are both robust and reliable. In addition,\nP\nthe fitted results indicate that the slopes of V with respect to Mg content at the CMB\nP\nand ICB conditions are -0.71 ± 0.37 and -0.32 ± 0.05 km/s/mol, respectively. Thus, in\nboth cases, the negative values of the slope are significant given the uncertainties,\nconfirming that V decreases systematically with increasing Mg concentration.\nP\nIn Supplementary Information we added:\nFig. 2\nEffects of the Mg content on the density () and compressional wave velocity (V ) of\nP\nliquid Fe and Fe-Mg alloys at (a) the core-mantle boundary (CMB) and (b) the inner-\nouter core boundary (ICB) conditions. The blue and red lines (or symbols) represent\nproperties calculated at ICB temperatures of 5400 K and 6000 K, respectively. The\ntemperatures at the CMB were calculated using the adiabatic temperature gradient and\nthe ICB temperatures. The circles and triangles represent the results obtained from the\noriginal pressures in FPMD simulations and pressures with corrections for the GGA\napproximation. The magenta squares, yellow hexagons, and cyan diamonds represent\nprevious results from FPMD simulations (T =6000 K)(Bajgain et al., 2019), shock\nICB\nwave experiments (T =5400 K)(Anderson and Ahrens, 1994), and inelastic x-ray\nICB\nscattering measurements (T =5400 K)(Kuwayama et al., 2020), respectively. The\nICB\nhorizontal green dash-dot lines represent the seismically observed density and V of\nP\nthe outer core from the PREM model.\nOn Lines 104-108 of the revised manuscript, we wrote:\n“\nThe V values as functions of Mg molar fraction (X ) can be fitted linearly as\nP Mg\n𝑉𝐶𝑀𝐵(𝑘𝑚⁄𝑠) = 7.80(0.02)−0.71(0.37)×𝑋 and 𝑉𝐼𝐶𝐵(𝑘𝑚⁄𝑠) = 9.93(0.01)−\n𝑃 𝑀𝑔 𝑃\n0.32(0.05)×𝑋 , respectively, confirming that the decreasing trend of V with\n𝑀𝑔 P\nincreasing Mg content is significant given the uncertainties at both CMB and ICB\nconditions.\n”\nOn Lines 283-288 of the revised manuscript, we wrote:\n“\nThe effects of simulation cell size and time duration were evaluated by performing\ntest simulations with a larger cell of Fe Mg and/or a longer duration of up to 50 ps.\n156 16\nThese simulations showed negligible differences in pressure (0.25 GPa), internal\nenergy (9.00 meV/atom), and derived compressional wave velocity (<0.3%)\n(Supplementary Figs. S2-3), indicating that the present system size (~80 atoms) and\nduration (~10 ps) are sufficient for the calculations of the compressional properties of\nliquid Fe-Mg alloys (Supplementary Text S1).\n”\n(3) Mg content in the Earth’s outer core: Because of the concerns raised in (2), how\nwould the errors in determining Vp from EOS fittings affect the results on Mg content\nin the Earth’s outer core. For reproducibility and open science, it would be beneficial\nto include the codes for the modeling along with the manuscript as datasets. The\ndetermination of magnesium content in the Earth’s outer core is critically dependent\non the accurate determination of Vp, which is derived from the equation of state\n(EOS) fittings. The errors inherent in these fittings and calculations based on bulk\nmodulus and density, as outlined in the concerns raised in (2), could significantly\nimpact the inferred content of Mg in Earth’s outer core. This introduces a potential\nsource of uncertainty in our understanding of the outer core's composition. To\nenhance the reliability and reproducibility of the findings, and in line with principles\nof open science, it is advisable for the authors to make the computational codes used\nfor modeling available along with the manuscript. Providing these codes as\nsupplementary datasets would allow other researchers to verify the results, potentially\nreplicate the study, and extend the analysis further.\nThank for your comments. We have carefully revised the manuscript to address these\npoints in detail.\n(1) The uncertainties in the calculated V values were quantified using a Monte Carlo\nP\nerror-propagation method that explicitly accounts for the uncertainties in the fitted\nEOS parameters. The propagated uncertainty in V , denoted as is 𝜎 , was\nP 𝑉𝑃,𝑖\nincorporated into Equation (11):\nΔ=\n1\n{∑ [(\n𝜌\n𝑖\n𝐶𝑎𝑙𝑐−𝜌\n𝑖\n𝑂𝑏𝑠\n)\n2\n+(\n𝑉𝑃𝑖 𝐶𝑎𝑙𝑐−𝑉𝑃𝑖 𝑂𝑏𝑠\n)\n2\n] +(\n𝜌\n𝐼\n𝐶\n𝐶\n𝑎𝑙𝑐−𝜌\n𝐼\n𝑂\n𝐶\n𝑏𝑠\n)\n2\n} (11)\n𝑖\n𝑁−𝑀 𝜎𝜌,𝑖 𝜎𝑉𝑃,𝑖\n𝑂𝐶\n𝜎𝜌,𝐼𝐶\nwhere 𝜎 represents the cumulative uncertainty associated with all light-element\n𝑉𝑃,𝑖\ncontributions to V . A representative uncertainty 𝜎 of ~0.86% was adopted, and\nP 𝑉𝑃,𝑖\nadditional sensitivity tests using uncertainties of 0.7% and 1.1% were performed to\nevaluate their effects on the best-fit composition of the Fe-Ni-Mg-Si-H system for a\ntwo-layer outer core model. The best-fit compositions of the upper outer core\nobtained with uncertainty 𝜎 of 0.7% and 1.1% are 1.33 wt% Mg, 5.40 wt% Si,\n𝑉𝑃,𝑖\nand 0.002 wt% H and 5 wt% Ni, and 1.35 wt% Mg, 5.23 wt% Si, and 0.013 wt% H\nand 5 wt% Ni, respectively. These results are in excellent agreement with those\nderived using 𝜎 =0.86% (1.33 wt% Mg, 5.35 wt% Si, and 0.008 wt% H and 5 wt%\n𝑉𝑃,𝑖\nNi). To further assess the influence of V uncertainty, we examined the compositional\nP\ncontrast between the upper and lower outer core (Supplementary Figs. S14-17). The\nstatistical distributions of Si, H, Fe, and Mg concentrations indicate that variations in\n𝜎 have a negligible influence on the relative differences in Si, H, Fe, and Mg\n𝑉𝑃,𝑖\ncontents between the upper and lower outer core layers. Collectively, these results\nconfirm that our primary conclusions remain robust against reasonable variations in\nthe assumed uncertainty of V .\nP\n(2) We fully agree that sharing our computational codes is vital for ensuring the\ntransparency and reproducibility. We now provide the Mendeley Data repository\n(http:/dx.doi.org/10.17632/vncnpgy22g.1) link to the relevant codes in the Code\nAvailability Section. Specifically, we have included (1) a Python script for EOS\nfitting and V determination, and (2) a Fortran package developed to search the best-\nP\nfit composition of the Earth’s core.\nIn Supplementary Information we added:\nSupplementary Fig. S14 Optimized upper outer core and lower outer core\ncompositions for the two-layer model with reduced misfit ∆< 0.25 using the EPOC\nmodel and a V uncertainty of 0.7% as constraints. The black circles represent the best-\nP\nfit of composition. The color bars indicate the Mg content in upper outer core in wt%\nand the ∆ values for lower outer core, respectively. The solid black circles represent the\nbest-fit compositions.\nSupplementary Fig. S15 The differences in concentrations of key elements including\nSi, H, Fe, and Mg between the upper outer core (E layer) and the lower outer core\nusing the EPOC model and a V uncertainty of 0.7% as constraints.\nP\nSupplementary Fig. S16 Optimized upper outer core and lower outer core\ncompositions for the two-layer model with reduced misfit ∆< 0.25 using the EPOC\nmodel and a V uncertainty of 1.1% as constraints. The black circles represent the best-\nP\nfit of composition. The color bars indicate the Mg content in upper outer core in wt%\nand the ∆ values for lower outer core, respectively. The solid black circles represent the\nbest-fit compositions.\nSupplementary Fig. S17 The differences in concentrations of key elements including\nSi, H, Fe, and Mg between the upper outer core (E layer) and the lower outer core\nusing the EPOC model and a V uncertainty of 1.1% as constraints.\nP\nOn Lines 364-367 of the revised manuscript, we wrote:\n“\nThe assessment of uncertainties in V demonstrates that variations in 𝜎 exert a\nP 𝑉𝑃,𝑖\nnegligible effect on the inferred Mg content and do not alter the compositional\ncontrasts between the upper and lower outer core layers for Si, H, Fe, and Mg\n(Supplementary Text S4 and Figs. S14-17).\n”\nOn Lines 243-246 of the revised manuscript, we wrote:\n“\nCode availability\nThe software package (VASP) used in this study is commercially available at\nhttps://www.vasp.at/. Data and codes are publicly accessible through the Mendeley\nData repository at http:/dx.doi.org/10.17632/vncnpgy22g.1.\n”\nOn Lines 143-155 of the Supplementary Text S4, we wrote:\n“\nText S4 Effects of uncertainties in V on the core composition\nP\nDue to uncertainties in V , we performed additional sensitivity tests using\nP\nuncertainties of 0.7% and 1.1% in V to evaluate their effects on the best-fit\nP\ncomposition of the Fe-Ni-Mg-Si-H system for a two-layer outer core model. The best-\nfit compositions of the upper outer core obtained with uncertainty 𝜎 of 0.7% and\n𝑉𝑃,𝑖\n1.1% are 1.33 wt% Mg, 5.40 wt% Si, and 0.002 wt% H and 5 wt% Ni, and 1.35 wt%\nMg, 5.23 wt% Si, and 0.013 wt% H and 5 wt% Ni, respectively. These results are in\nexcellent agreement with those derived using 𝜎 =0.86% (1.33 wt% Mg, 5.35 wt%\n𝑉𝑃,𝑖\nSi, and 0.008 wt% H and 5 wt% Ni).To further assess the influence of V uncertainty,\nP\nwe examined the compositional contrast between the upper and lower outer core\n(Supplementary Figs. S14-17). The statistical distributions of Si, H, Fe, and Mg\nconcentrations indicate that variations in 𝜎 have a negligible influence on the\n𝑉𝑃,𝑖\nrelative differences in Si, H, Fe, and Mg contents between the upper and lower outer\ncore layers. Collectively, these results confirm that our primary conclusions remain\nrobust against reasonable variations in the assumed uncertainty of V .\nP\n”\n(4) Fig S4: What are the uncertainties of Vp determined for 4000 K, 5000 K, 6000 K,\nand 7000 K from low to high pressures?\nWe have quantified and included the uncertainties of V at 4000 K, 5000 K, 6000 K,\nP\nand 7000 K, and at both 135 GP and 330 GPa, as shown in Supplementary Fig. S7.\nThe results show that the uncertainties in V exhibit a modest decrease with pressure.\nP\nSpecifically, the uncertainties at 4000 K range from 0.96% (135 GPa) to 0.40% (330\nGPa); from 1.23% (135 GPa) to 0.50% (330 GPa) at 5000 K; from 0.96% (135 GPa)\nto 0.40% (330 GPa) at 6000 K; and from 1.80% (135 GPa) to 0.72% (330 GPa) at\nP\nGPa and ~4000 K) and ~0.40% at the ICB (330 GPa and ~6000 K).\nIn Supplementary Information we added:\nSupplementary Fig. S7 Compressional wave velocity (V ) of pure Fe liquid and Fe-\nP\nMg alloying liquids under Earth’s core conditions. (a) The V of pure Fe liquid at\nP\ntemperatures of 4000 K (green), 5000 K (blue), 6000 K (purple), and 7000 K (red).\nThe dots with error bars denote uncertainties in V at 135 GPa and 330 GPa over the\nP\ntemperature range of 4000-7000 K. Fe means that a supercell of 80 Fe atoms was\n80\nused in FPMD simulations. (b) The V of pure Fe liquid (red and green lines) and Fe-\nP\nMg alloying liquids (blue and purple lines) at 4000 K, 5000 K, 6000 K, and 7000 K.\nResults of six Fe-Mg alloys with super cells of Fe Mg , Fe Mg , Fe Mg , Fe Mg ,\n78 1 78 2 78 3 78 4\nFe Mg , and Fe Mg are plotted. Both originally computed pressures from FPMD\n78 6 78 8\nsimulations (red and blue lines for pure Fe and Fe-Mg alloys, respectively) and\ncorrected pressures for the effect of GGA approximation (green and purple lines for\npure Fe and Fe-Mg alloys, respectively) are used for plotting V . As both temperature\nP\nand Mg content show limited effects on V , results at different temperatures and Mg\nP\ncontents are plotted using same colors in (b).\nOn Lines 283-284 of the revised Supplementary manuscript, we added:\n“\nThe dots with error bars denote uncertainties in V at 135 GPa and 330 GPa over the\nP\ntemperature range of 4000-7000 K.\n”"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have correctly addressed my concerns, providing evidence that the errors induced by simulation box size and the short time scale are under control, and the linearity assumptions implicit in their mixing model are approximately correct.", "reviewer_2": "The authors have done a good job responding to the points raised in my review. Brief statements have been included in the manuscript and additional details are presented in the supplement. I think the authors have made their case.\nI would add one final comment on the authors' statement about the solubility of Mg in liquid iron (see line 216-217). The view expressed here is perfectly reasonable. Exsolution describes the transfer of dissolved Mg to new phase (say solid MgO). In other words the solubility represents the concentration of dissolved Mg in equilibrium with solid MgO. Of course, the core liquid is also in contact with other solid components in the lower mantle. It is not entirely obvious that the equilibrium between the core liquid and these other solid components are unimportant, particularly if the exsolution occurs close to the core-mantle boundary.", "reviewer_3": "I have carefully reviewed the revised manuscript and the authors' response to my previous comments. The authors have addressed all of my concerns thoroughly and satisfactorily.\nI have no further comments on the manuscript and recommend acceptance."}, "author_response": "Dear Editor,\nWe very much appreciate your handling of our manuscript and the valuable comments\nfrom the reviewers. We have revised our manuscript based on these comments. Below\nare our responses to the comments, with the original comments from the reviewers in\nblack, our responses in blue. Thank you.\nSincerely,\nTao Liu and Zhicheng Jing\nResponse to Reviewer #1\nThe authors have correctly addressed my concerns, providing evidence that the errors\ninduced by simulation box size and the short time scale are under control, and the\nlinearity assumptions implicit in their mixing model are approximately correct.\nReply: Thank you for your supportive comments.\nResponse to Reviewer #2\nThe authors have done a good job responding to the points raised in my review. Brief\nstatements have been included in the manuscript and additional details are presented\nin the supplement. I think the authors have made their case.\nI would add one final comment on the authors' statement about the solubility of Mg in\nliquid iron (see line 216-217). The view expressed here is perfectly reasonable.\nExsolution describes the transfer of dissolved Mg to new phase (say solid MgO). In\nother words the solubility represents the concentration of dissolved Mg in equilibrium\nwith solid MgO. Of course, the core liquid is also in contact with other solid\ncomponents in the lower mantle. It is not entirely obvious that the equilibrium\nbetween the core liquid and these other solid components are unimportant,\nparticularly if the exsolution occurs close to the core-mantle boundary.\nReply: Thank you for your supportive comments. We agree with these comments and\nhave revised the text in the manuscript as “Such exsolution processes are likely\ndetermined by the temperature-dependent solubilities of various components in the\nliquid core, which reflect chemical equilibrium between the core and the exsolved\nphases, although other solid components in the lower mantle, if in contact with the\nliquid outer core, may also affect the exsolution processes.”\nResponse to Reviewer #3\nI have carefully reviewed the revised manuscript and the authors' response to my\nprevious comments. The authors have addressed all of my concerns thoroughly and\nsatisfactorily.\nI have no further comments on the manuscript and recommend acceptance.\nReply: Thank you for your supportive comments."}]

Decision letter