Lattice dynamics of impurity clusters. Application to pairs

A general solution is obtained for the lattice dynamics of a cluster ofn-impurity atoms using the double-time Green’s function formalism. The cluster is characterized byn-mass defect andm-force constant change parameters. It is shown that this general solution for the Green’s function for then-impurity cluster can also be expressed in terms of the Green’s function for the (n−1)-impurity cluster. As an application, the cluster impurity modes for a pair are calculated using the Debye model for the host lattice dynamics. The splitting of the high frequency local modes and nearly zero frequency resonant modes due to pairs show an oscillatory behaviour on varying the distance of separation between the two impurity atoms. These oscillations are most prominent for two similar impurities and get damped for two dissimilar impurities or if one of the impurities produces a force constant change. The predictions of the calculation provide qualitative explanation of the data obtained from the infrared measurements of the resonant modes in mixed crystal system of KBr_1−c Cl _c : Li⁺ and KBr_1−c I _c : Li⁺.