Determination of the Effective Force Constants between a Substitutional Impurity and Its Nearest Neighbors in an Alkali Halide Crystal

The forces between a substitutional impurity and its nearest neighbors in an alkali halide crystal with rocksalt structure are determined in the harmonic approximation. The deformation dipole model is used for the lattice dynamics of the perfect crystal. Two independent force constants are involved in the problem which is treated in two ways. First, the two force constants are obtained from the equation of motion for the local mode of vibration using experimental values of the local-mode frequencies due to ${\mathrm{H}}^{-}$ and ${\mathrm{D}}^{-}$ impurities in NaCl and KCl, and assuming that the force constants are the same for different isotopes of the impurity. This method yields no real solutions for the force constants for $U$ centers in KCl. The second method assumes that the force constants for the impurity (say ${\mathrm{H}}^{-}$) are the same as those in an alkali hydride crystal having the same lattice spacing as the alkali halide crystal. A Born-Mayer interatomic potential is used for this calculation. The first method gives 45% softening of the force constants for $U$ centers in NaCl; the second method gives 70% softening. The results are compared with those of previous theories and with experimental results on the KBr: ${\mathrm{Li}}^{+}$ system.