Description of resonant and localized defect vibrations

The vibrational behavior of crystals with point defects is discussed with particular emphasis on defect-induced resonant and localized modes. A method due to Krumhansl and Matthew is applied to obtain a direct description of the local vibrational properties of the defect. For low-frequency resonances an analytic expression of the Green's function of the defect is derived which has the form of the Green's function of the one-dimensional oscillator with an effective force constant $f^{\mathrm{eff}}$, an effective mass $M^{\mathrm{eff}}$, and a velocity-proportional damping $\gamma$. An exact expression for $\gamma$ in terms of $f^{\mathrm{eff}}$ and $M^{\mathrm{eff}}$ is given. Simple approximations for $f^{\mathrm{eff}}$ and $M^{\mathrm{eff}}$ are discussed, which are not based on the calculation of the perfect lattice Green's function. The method is also applied to resonances just below the maximum frequency, which are connected with the formation of localized modes. Furthermore, a simple derivation of upper and lower bounds for the frequencies of localized modes is presented.