Maxwell-Boltzmann Distribution ofM2+: Vacancy Pairs in Compensated fcc Lattice

Coulomb interactions in the face-centered cubic (fcc) lattice are exemplified by the interactions of $M^{2+}$ impurity ions with their charge compensation vacancies in alkali halides. At low temperatures and high dilution, the canonical ensemble partition function leads to a Maxwell-Boltzmann distribution in the distance of separations of the $M^{2+}$: vacancy pairs, which can be calculated in terms of the geometric restrictions of the fcc lattice. The most predominant pairs, in the order of their relative importance, are $C_{4v}$ (2,0,0), $C_s$ (2,1,1,), $C_{2v}$ (1,1,0), $C_{2v}$ (2,2,0), $C_s$ (3,1,0), $C_1$ (3,2,1), $C_{3v}$ (2,2,2), and $C_{4v}$ (4,0,0). The calculated distribution is compared with optical Zeeman and spin-resonance spectroscopic observations in the KCl: ${\mathrm{Sm}}^{2+}$ and KCl: ${\mathrm{Mn}}^{2+}$ systems.