Elastic, vibrational, and thermodynamic properties of Sr10(PO4)6F2 and Ca10(PO4)6F2 from first principles

The electronic, elastic, vibrational, and thermodynamic properties of Sr10(PO4)6F2(Sr-FAP) and Ca10(PO4)6F2(CaFAP) are systematically investigated by the first-principles calculations. The calculated electronic band structure indicates that the Sr-FAP and Ca-FAP are insulator materials with the indirect band gap of 5.273 eV and 5.592 eV, respectively. The elastic constants are obtained by the "stress–strain" method, and elastic modulus are further evaluated and discussed. The vibrational properties, including the phonon dispersion curves, the phonon density of states, the Born effective charge, and associated longitudinal optical and transverse optical(LO–TO) splitting of optical modes, as well as the phonon frequencies at zone-center are obtained within the linear-response approach. Substitution of Ca by Sr causes phonon frequencies to shift to lower values as expected due to the mass effect. Additionally, some phonon-related thermodynamic properties, such as Helmholtz free energy F, internal energy E, entropy S, and specific heat CVof Sr-FAP and Ca-FAP are predicted with the harmonic approximation. The present calculated results of two apatites are consistent with the reported experimental and theoretical results.