Resonant relaxation time for impurity electrons and localized phonons (wave packets) interaction: Application to the thermal conductivity of MgO:Fe2+, ZnS:Fe2+, CdTe:Fe2+, and MgO:Cr2+

For a relaxation time which is a rapidly varying function of $\omega$, phonons are more properly described by wave packets than by plane waves. Approximating the line shape of a wave packet by a Lorentzian function, the effect of localized phonons can be taken into account implicitly in the derivation of the resonant relaxation time $\tau (\omega ,T)$ for impurity electron-phonon interaction. In this phenomenological model the energy spread $\hslash \delta \omega$ of the wave packet simply adds to the width $\hslash \Gamma \mathrm{}\left(T\right)\mathrm{}$ of the expression of $\tau (\omega ,T)$ obtained for plane waves. The results for the calculated thermal conductivity agree with the experimental data, and at low temperature, where localized-phonon effects are not negligible, a quantitative agreement with Casimir's theory is also obtained.