Radial gradient of superionic hydrogen in Earth's inner core

Hydrogen is considered a major light element in Earth’s core, yet the thermodynamics of its superionic phase and its distribution in the inner core remain unclear. Here, we compute ab initio Gibbs free energies for liquid and superionic hcp and bcc Fe-H phases and construct the superionic-liquid phase diagram over pressure-temperature conditions relevant to the Earth’s inner core. We find that phase diagrams at different inner-core pressures collapse when temperatures are scaled by the melting temperature of pure iron, indicating that solid-liquid partitioning is controlled primarily by a reduced temperature relative to iron melting and is weakly sensitive to pressure. This scaling relation further reconciles previously reported discrepancies in partition coefficients among theoretical studies and yields good agreement with available experimental data at low pressures. By applying thermochemical constraints, our free-energy results reveal a radial hydrogen gradient within the inner core. These results demonstrate that compositional gradients of superionic hydrogen in the inner core emerge naturally from equilibrium thermodynamics and suggest a general mechanism governing the depth-dependent distribution of light elements within Earth’s inner core.

💡 Research Summary

The authors address the long‑standing question of how hydrogen, a leading candidate for the light element in Earth’s core, is partitioned between the solid inner core (IC) and the liquid outer core (OC) under inner‑core conditions. Using a recently developed first‑principles framework, they calculate Gibbs free energies for liquid Fe‑H as well as for two superionic solid phases: hexagonal close‑packed (hcp) and body‑centered cubic (bcc). Ab initio molecular dynamics (AIMD) shows that hydrogen behaves superionically—diffusing like a liquid while the iron lattice remains crystalline—in both hcp and bcc structures. A machine‑learning interatomic potential, trained on high‑pressure DFT data that includes Fe 3s and 3p electrons, enables large‑scale coexistence simulations and thermodynamic integration to obtain accurate free‑energy curves over a wide pressure‑temperature‑composition (P‑T‑x) space (323–360 GPa, 5000–6800 K, up to ~20 at.% H).

From the free‑energy data the authors construct solidus and liquidus curves by common‑tangent constructions. The bcc superionic phase is consistently higher in free energy than hcp, making hcp the only solid phase that can coexist with the liquid under inner‑core conditions. The resulting superionic‑liquid phase diagram shows a markedly higher hydrogen solubility (up to ~10 at.% at 5500 K) than previously reported for superionic oxygen.

A key insight emerges when temperatures are normalized by the melting temperature of pure iron at each pressure, defining a reduced temperature Θ = T/TmFe(P). When plotted versus Θ, the solidus and liquidus lines for 323 GPa and 360 GPa collapse onto nearly identical curves. This scaling indicates that pressure mainly rescales the absolute free‑energy scale of Fe, while the thermodynamics of hydrogen mixing in the superionic and liquid phases are essentially pressure‑independent. The scaling also reconciles previously reported discrepancies in the solid‑liquid hydrogen partition coefficient K_H^{s/l} among different theoretical studies.

Applying the equilibrium condition μ_H^{solid}=μ_H^{liquid} and assuming the inner core is nearly isothermal (≈5500 K, consistent with high thermal conductivity), the authors compute the hydrogen chemical potential as a function of pressure. Because the chemical potential must be spatially uniform throughout the IC, the hydrogen concentration must vary with depth to satisfy μ_H(P, x)=constant. The resulting radial profile shows a decrease from ~16 at.% at the inner‑core boundary (ICB, ~330 GPa) to ~12 at.% at the core’s centre, a total drop of about 7 at.%. Sensitivity tests at 5000 K and 6000 K give gradients of ~10 at.% and ~3 at.% respectively, confirming that a compositional gradient is robust across plausible thermal models. Including the modest gravitational potential energy difference across the IC slightly perturbs the concentrations but does not eliminate the gradient, because the chemical‑potential variation dominates.

The calculated partition coefficients K_H^{s/l} range from 0.70 at 5000 K to 0.57 at 6000 K for 323 GPa, and show a nearly linear temperature dependence. These values agree well with low‑pressure experimental measurements (K≈0.7–0.8 at 1900 K, 45 GPa) and with recent high‑pressure simulations that employed similar DFT settings. The authors argue that differences among earlier studies arise mainly from the treatment of Fe’s inner‑shell electrons and electronic temperature in the underlying DFT calculations, which significantly affect Fe’s thermodynamic properties.

In summary, the paper provides the first comprehensive ab initio free‑energy database for superionic Fe‑H, demonstrates a universal reduced‑temperature scaling of the superionic‑liquid phase diagram, and shows that equilibrium thermodynamics naturally produce a radial hydrogen gradient in Earth’s inner core. This gradient offers a plausible explanation for seismic anisotropy and heterogeneity observed in the inner core and establishes a general mechanism by which light‑element concentrations can vary with depth in planetary interiors.