Intermetallics in theMg−Ca−Snternary system: Structural, vibrational, and thermodynamic properties from first principles

Abstract

A comprehensive analysis of the structural, vibrational, and thermodynamic properties of the intermetallic compounds in the $\mathrm{Mg}\text{\ensuremath{-}}\mathrm{Ca}\text{\ensuremath{-}}\mathrm{Sn}$ system has been performed via first-principles calculations. The enthalpies of formation at $0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for all the known structures in this ternary system are calculated and the results are favorably compared---within $\ensuremath{\sim}5\phantom{\rule{0.3em}{0ex}}\mathrm{kJ}∕\mathrm{mol}\text{\ensuremath{-}}\mathrm{atom}$ in most cases---to the available experimental data. The vibrational contributions to the thermodynamic properties of fcc Ca, hcp Mg, $\ensuremath{\beta}\ensuremath{-}\mathrm{Sn}$, ${\mathrm{Mg}}_{2}\mathrm{Ca}$, ${\mathrm{Ca}}_{2}\mathrm{Sn}$, $\mathrm{CaSn}$, ${\mathrm{Ca}}_{5}{\mathrm{Sn}}_{3}$, ${\mathrm{CaSn}}_{3}$, ${\mathrm{Mg}}_{2}\mathrm{Sn}$, and $\mathrm{MgCaSn}$ are calculated using the supercell method. In all cases, bond stiffening resulting from compound formation results in upward frequency shifts in the phonon density of states, yielding in turn negative entropies of formation. The effects of volume expansion on the vibrational properties were considered through the quasiharmonic approximation. Thermal electronic contributions were also calculated from the electronic density of states. The electronic degrees of freedom were found to be less important than volume expansion at determining the high temperature thermodynamic properties. The predicted thermodynamic properties of the structures agreed satisfactorily with the experimental data available. The relative importance of these two nonharmonic corrections is reversed when analyzing the formation properties. In all compounds, except for ${\mathrm{CaSn}}_{3}$, it was found that the variation of both the formation enthalpies and entropies with temperature is negative. This results in a destabilization of the compounds with respect to their constituent elements as the temperature is increased.