Temperature Dependence of Phonon-Induced Electronic Transitions in Insulating Solids

The temperature dependence of the oscillator strength of a forbidden electronic transition at an impurity in an insulating solid is calculated, assuming that the transition is induced by a local linear electron-phonon interaction. The electronic states are allowed to couple, not merely to one configuration coordinate, but to all the modes of vibration of the imperfect crystal. The frequencies of these modes and their relative contributions to the oscillator strength are calculated using Green's-function techniques. In particular, the $4d^{10}$ to $4d^{9}5s$ transition of ${\mathrm{Ag}}^{+}$ in NaCl is considered. A model which represents the host ions as point charges is shown to be inconsistent with the experimentally determined temperature-dependent oscillator strength. Good agreement with experiment is obtained if one assumes that the short-range interaction between the ${\mathrm{Ag}}^{+}$ impurity and the host ions is important.