Defect-induced resonance modes in the asymptotic limit of low frequencies: Isotope effects and amplitude patterns

Results are presented of a theoretical study of the harmonic properties of defect-induced low-frequency resonance modes associated with decreased force constants. Some apparently contradictory results of previous authors are resolved. General relations, model-independent and exact in the limit of zero frequency, are established connecting resonance-mode amplitude patterns and frequency shifts accompanying isotopic substitutions for the defect or host atoms. The largest defect isotope shifts turn out to occur when the impurity and host are completely decoupled, in which case the defect behaves as a simple Einstein oscillator. It is shown that the measured shift for the system KBR:${\mathrm{Li}}^{+}$ is incompatible with any harmonic model. The general results are illustrated for a specific model of an isoelectronic substitutional defect in an alkali-halide crystal. The model assumes weakened nearest-neighbor longitudinal force constants at the impurity, and realistic host-crystal phonons are used to numerically compute isotope shifts and amplitude patterns near the defect for a number of systems. The calculated amplitude patterns show clearly that decoupling does not occur for this model. Estimates are made of the computational uncertainties in the calculated defect isotope shifts, and these shifts are compared with the experimental values. For NaCl:${\mathrm{Cu}}^{+}$ and KI:${\mathrm{Ag}}^{+}$, the calculated and observed shifts agree to within experimental error, but it is shown that this error should be reduced before the correctness of the model for these systems can be claimed. The large observed isotope shift for NaI:${\mathrm{Cl}}^{-}$ is underestimated by the calculations, and realistic extensions of the model are proposed. The possible role played by anharmonicity for specific systems is briefly discussed.