Lattice Response Functions of Imperfect Crystals: Effects Due to a Local Change of Mass and Short-Range Interaction

Lattice response functions, such as the thermal conductivity and dielectric susceptibility of an imperfect crystal with rocksalt structure, are evaluated in terms of the irreducible $T$ matrix accounting for the phonon scattering. It is shown that the effect of defects on thermal conductivity and dielectric susceptibility can be accounted for by expressions which have essentially the same structure. The $T$ matrix for a defect which affects both the mass and the short-range interaction is analyzed according to the irreducible representations of the point group which pertains to the perturbation, and the resonance conditions for ${\Gamma }_1$, ${\Gamma }_{12}$, and ${\Gamma }_{15}$ irreducible representations are considered in detail for any positive impurity in KBr crystals. Hardy's deformation-dipole (DD) model is employed for the description of the host-lattice dynamics. A comparison is made with simplified models, such as diatomic linear chains with nearest-neighbor interaction; it is shown that in polar crystals an effective-force constant has to be used in order to give a reliable description of the short-range interaction between the impurity and the host lattice. An attempt is made to define such effective force constants in the framework of the DD model. The numerical calculations concern positive monovalent impurities in KBr crystals. ${\Gamma }_1$, ${\Gamma }_{12}$, and ${\Gamma }_{15}$ resonance frequencies are evaluated as a function of the change of mass and nearest-neighbor force constant. For KBr:${\mathrm{Li}}^{+}$ and KBr:${\mathrm{Ag}}^{+}$ we also evaluate the band shape of the absorption spectrum at infrared frequencies; good agreement is found between the theoretical prediction and the experimental data on KBr:${\mathrm{Li}}^{+}$. It is shown that some structures actually observed in the spectrum are due to peaks in the projected density of states of the host lattice, and have nothing to do with resonance scattering. Good agreement is found between the impurity-host-lattice interaction as estimated from a priori calculations and as deduced by fitting the ${\Gamma }_{15}$ resonance frequency to the experimental data. A simple explanation of the off-center position of small ions is also suggested. Finally, concentration and stress effects on the absorption coefficient are briefly discussed.