Vacancies, Interstitials, and Rare Gases in Fluorite Structures

We have calculated the ${\mathrm{Ca}}^{++}$-${\mathrm{F}}^{-}$, ${\mathrm{Ba}}^{++}$-${\mathrm{F}}^{-}$, ${\mathrm{Ca}}^{++}$-${\mathrm{Ca}}^{++}$, ${\mathrm{Ba}}^{++}$-${\mathrm{Ba}}^{++}$, and ${\mathrm{F}}^{-}$-${\mathrm{F}}^{-}$ interatomic potentials using a semiclassical method. Using these potentials, the anion vacancy and interstitial formation and activation energies were determined. Franklin's potentials were also employed for comparison. The results are in excellent agreement with experiment for Ba${\mathrm{F}}_2$ and for most of the Ca${\mathrm{F}}_2$ results. The unusually wide range (0.53-1.65 eV) of experimental values for interstitial-related activation energies makes this direct comparison in Ca${\mathrm{F}}_2$ difficult, but our results clearly indicate the interstitialcy mechanism of anion interstitial migration to be dominant in both Ca${\mathrm{F}}_2$ and Ba${\mathrm{F}}_2$. Using the same method, the energies of formation of He, Ne, and Ar at interstitial sites in Ca${\mathrm{F}}_2$ were determined to be 0.49, 1.59, and 3.08 eV, respectively. The activation energy for interstitial motion of these atoms was found to be ∼ 1.4 eV for all three gases. Furthermore, we find it takes 0.25, 0.95, and 2.30 eV to place a He, Ne, or Ar gas atom, respectively, in an existing anion vacancy. The substitutional detrapping energy, the energy required to move a rare-gas atom from an anion vacancy to its nearest-neighbor stable interstitial position, was found to be 1.8, 2.1, and 2.5 eV for He, Ne, and Ar, respectively. The explicit atomistic calculation is done for many regions of varying size surrounding each defect so that it is possible to say, at least semiquantitatively, that the energy vs radius of the region $r_A$ varies as $\frac{1}{r_A}$ for charged defects and as $\frac{1}{r_A^2}$ for neutral defects.