α↔β phase transition in tin: A theoretical study based on density-functional perturbation theory

The free energies of the α and β phases of tin are calculated in the harmonic approximation using density-functional theory and density-functional perturbation theory, within the local-density approximation. At T=0 K the free energy of the β phase lies ≈359cal/mole above that of the α structure. The narrower frequency range spanned by the vibrational band in the β phase makes its entropy larger at high temperature. As a consequence, the free energies of the two phases equal each other at a temperature of 38 ^∘{}C, in close agreement with the observed transition temperature Tc≈13^∘C.