First-principles thermodynamics of point defects and off-stoichiometry in \textit{$\beta $}-Mg$_{17}$Al$_{12}$

The mechanical strength of Mg-Al alloys may be enhanced by a fine spatial dispersion of \textit{$\beta $}-Mg$_{17}$Al$_{12}$ precipitates. Native point defects, i.e. vacancies and anti-sites, in Mg$_{17}$Al$_{12 }$are important for understanding the phase stability and unusually asymmetric observed off-stoichiometry in this precipitate phase. In an effort to provide a quantitative picture of the phase stability of this system, we have performed a series of first-principles density functional theory calculations of bulk and defect properties of Mg$_{17}$Al$_{12}$. We consider not only the T=0K static energetics, but also key entropic terms such as the configurational and vibrational entropies. The vibrational entropies are calculated from DFT via the direct force-constant approach using the quasiharmonic approximation. We investigate the effect of atomic vibrations on native point defect free energies of Mg$_{17}$Al$_{12 }$and combine the entropic contributions with the point defect formation energies to evaluate the thermodynamics of off-stoichiometry in this phase. We find there is a large vibrational entropy difference between Mg-rich and Mg-deficient defects in Mg$_{17}$Al$_{12}$, consistent with the strong asymmetry in the observed Mg-Al phase diagram.