Calculation of Stable Isotope (O,C, Ca) Fractionation Factors for Apatite

The isotope fractionation β-factors have been determined for hydroxyapatite, fluorapatite, and carbon–apatite upon ¹⁸O/¹⁶O, ¹³C/¹²C, and ⁴⁴Ca/⁴⁰Ca substitutions depending on the temperature. Calculations were performed based on density functional theory (DFT). The results are expressed in terms of cubic polynomials against x = 10⁶/T(K)² and can be utilized as geothermometers if combined with β-factors of the coexisting phases. The new values of β-factors completely supersede the conventional semi-empirical (estimated by the method of increments) calibrations of isotopic equilibria involving apatite.