Effective Force Constant and Impurity Mode in the Low- and High-Frequency Limits

Abstract

By employing an effective force constant introduced earlier in the framework of a rigid-ion model for the lattice dynamics we have discussed the asymptotic behavior of the high-frequency localized modes and the low-frequency resonance modes due to substitutional impurities in realistic cubic lattices. Simple expressions for the frequencies of these impurity modes and the linewidths of the resonance modes have been obtained. In the asymptotic description the motion of an impurity may be simulated by an Einstein oscillator which is vibrating in the rest of the host crystal assumed to be static. However, the dynamical correlation of the impurity with the other ions of the host lattice is found to be significant especially in the case of an impurity strongly coupled to the lattice. Numerical calculations have been made for CsI crystals doped with impurity ions of different species in the breathing-shell model. A good agreement between the theoretical and experimental results is observed. The values of the frequencies of the impurity modes and the linewidths of the resonant modes calculated in the asymptotic limits compare well with the experimental values. The discrepancy between the calculated and the experimental values in the majority of cases lie within 20%. In both the low- and high-frequency regions, the value of the effective force constant is seen to be independent of frequency. However, the value of the effective force constant in the high-frequency region is smaller in magnitude than that in the low-frequency region. Further, the value of the effective force constant does not depend on the kind of the ion in a unit cell in ionic crystal, i.e., the value is found to be practically the same whether we consider a positively charged impurity ion or a negatively charged impurity ion. The implication of these results is that one may employ a simple rigid-ion model for a realistic crystal lattice with central interactions for understanding the defect properties of solids in the low- and the very high-frequency regions.