This paper is concerned with a theoretical examination of the stability of small clusters of substitutional trivalent cation impurities and interstitial fluoride ions in the three fluorites Ca${\mathrm{F}}_2$, Sr${\mathrm{F}}_2$, and Ba${\mathrm{F}}_2$. The energies of a variety of clusters have been calculated and the stability of the clusters examined; of particular interest is their ability to trap or lose interstitials and their ability to change their orientation. We find that in Ca${\mathrm{F}}_2$ the nearest-neighbor (NN) $C_{4v}$ complex $M_s^{}{}_{}{}^{3+}\mathrm{F}_i^{}{}_{}{}^{-}$($s$ denotes substitutional, $i$ denotes interstitial) is more stable, whereas in Ba${\mathrm{F}}_2$ it is the next-NN (NNN) $C_{3v}$ complex that is more stable; in Sr${\mathrm{F}}_2$ the NN and NNN complexes have comparable stability. The stabilization of NN dimer clusters containing two $M_s^{}{}_{}{}^{3+}$ and two $\mathrm{F}_i^{}{}_{}{}^{-}$ by the relaxation in opposite $〈111〉$ directions of NN lattice ${\mathrm{F}}^{-}$ ions is confirmed. Dimers in which the $\mathrm{F}_i^{}{}_{}{}^{-}$ ions are in NNN rather than NN positions have comparable stability to the NN dimers and so are presumably formed when it is the NNN monomer that is favored. Like the monomers, the dimers can lose $\mathrm{F}_i^{}{}_{}{}^{-}$ by dissociation and so may contribute to charge-transport processes. The NN dimer can trap free ${\mathrm{F}}^{-}$ interstitials with remarkable facility; in Ca${\mathrm{F}}_2$ this is so even when these interstitials must come from the dimers themselves. In both Ca${\mathrm{F}}_2$ and Sr${\mathrm{F}}_2$ the NN dimers can take $\mathrm{F}_i^{}{}_{}{}^{-}$ from NN monomers. Wherever possible our theoretical results have been compared with experimental data. Our results on the stability of clusters are in general qualitative agreement with experiment in every case where data are available. It is perhaps surprising, therefore, that the calculated activation energies for dipolar orientation are generally too high by about 0.2 eV and we have been unable to find the origin of this discrepancy; it seems, however, not to lie with the $M_s^{}{}_{}{}^{3+}\mathrm{F}_i^{}{}_{}{}^{-}$ potentials.

• Received 4 December 1981

DOI:https://doi.org/10.1103/PhysRevB.25.6425