Influence of Localized Modes on Thermal Conductivity

The curves of the thermal conductivity versus temperature of some insulating crystals (alkali halides) which are doped with certain impurity centers exhibit a distinct indentation. This suggests a relaxation time of some resonance nature, which is independent of the temperature for molecular impurity centers (KCl: KN${\mathrm{O}}_2$, measured by Pohl) and temperature-dependent for monatomic disturbances (KCl: NaCl, etc., measured by Walker and Pohl). In this paper a theory is given for the influence of inelastic scattering of phonons at localized modes outside the band and at quasilocalized modes inside. The calculated relaxation time for the third-order process $\left(\mathrm{k}\lambda \right)+\left({\mathrm{k}}^{\prime }{\lambda }^{\prime }\right)\rightleftarrows \mathrm{}\left(s\right)\mathrm{}$, describing phonons and localized quanta, respectively, has been tested on the experimental data in the preceding paper by Walker and Pohl; from the very good agreement one may conclude that the indentation effect is quantitatively due to this scattering process. The analysis is also done for the scattering of phonons by internal modes of molecular perturbations. The relaxation time for the second-order process $\left(\mathrm{k}\lambda \right)\rightleftarrows \left(\sigma \right)$, $\left(\sigma \right)=\mathrm{internal}\mathrm{quantum}$, is independent of temperature and of resonance form as observed for KCl: KN${\mathrm{O}}_2$.