Renormalized tunneling of off-center impurities in alkali halides

The renormalization of tunneling matrix elements of off-center impurities in alkali halides due to interaction with lattice displacements is studied. The possible forms of interaction between nearest-neighbor lattice ions and impurities oriented along [100], [110], or [111] directions are enumerated. The renormalization factors are expressed in terms of sums over normal modes of the perfect lattice and reduced coupling coefficients describing the impurity-lattice interaction. The lattice sums are tabulated for nine alkali halides and the coupling coefficients are expressed in terms of measurable quantities: the impurity electric dipole moment and stress splitting factors. The results are applied to several experiments: the 90° reorientation of [100]-directed O${\mathrm{H}}^{-}$ dipoles in several alkali halides, the 60° and 90° reorientation of [110]-directed ${\mathrm{Ag}}^{+}$ in RbCl and RbBr (see previous paper), and the reorientation of $\mathrm{O}_2^{}{}_{}{}^{-}$ and $\mathrm{N}_2^{}{}_{}{}^{-}$ in KCl and KI. It is found that in general the relaxation rate can be qualitatively related to the reduction in matrix element due to renormalization.