Anharmonicity is essential to the prediction of phase diagram of crystals, but a quantitative understanding of each-order anharmonicity is a long-standing challenge in solid-state physics. Here, we propose a rigorous, systematic, and efficient approach to solve this problem. Through the combination of self-consistent phonon theory (SCPH), improved SCPH, and machine-learning-potential-assisted thermodynamic integration, we investigate each-order anharmonic effects on the free energy and phase diagram of uranium (U) up to 100 GPa from first principles. The experimentally observed phase boundary of the orthorhombic (α) and body-centered cubic (γ) phases is reproduced well. By decomposing the contributions of third-order, fourth-order, and higher-order anharmonicity to the free energy, we demonstrate the competing contributions of different-order anharmonicity in α-U, while observing consistent and stronger anharmonic effects in γ-U. In particular, higher-order anharmonicity, beyond the third and fourth orders, exhibits significant contributions to the free energy.

uranium,phase diagram,first principle,Machine learning techniques