Accurate Prediction of the $α\to β$ Phase Transformation Temperature in Tin via Full Anharmonic Treatment

Abstract

Predicting the $α\to β$ (grey-to-white) transition temperature in tin presents a longstanding challenge for atomistic simulations, with existing theoretical approaches over- or underestimating the experimental boundary (286 K) by up to several hundred Kelvin. In this work, we construct an Atomic Cluster Expansion (ACE) potential trained on density functional theory data to evaluate the finite-temperature free energies of both phases. Evaluated on the same potential energy surface, the quasi-harmonic approximation predicts a transformation temperature of 377 K, whereas full thermodynamic integration, which accounts for explicit vibrational anharmonicity, yields 288 K. This shift directly quantifies the explicit anharmonic free energy, which is substantial for metallic $β$-Sn but negligible for semiconducting $α$-Sn. The anisotropic anharmonicity in $β$-Sn is corroborated by its excess heat capacity, temperature-driven renormalization of its vibrational spectrum, and deviations of its atomic forces and displacements from the harmonic reference. Our results demonstrate that capturing full lattice anharmonicity is essential for predicting the phase stability of tin, while the absolute transition temperature remains limited by the accuracy of the underlying 0 K energetics.