Radial gradient of superionic hydrogen in the Earth’s inner core

Hydrogen is considered a key 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.