Shapes of Impurity Absorption Bands in Solids

Optical-absorption line shapes arising from interaction of the electronic states of impurities with the vibrational modes of the crystal are considered. It is shown that this line shape may be expressed as an $N$-fold convolution integral, where each element of the convolution is the line shape due to a single vibrational mode of the ground state of the system. Two types of vibrational modes, linear and quadratic, are considered in detail in both the semiclassical approximation and quantum mechanically. It is shown that the semiclassical approximation, although often of use for linear interactions, has limited validity for quadratic modes. Formal methods are developed for performing the convolution integral to obtain the line shape due to a number of vibrational modes. Explicit numerical examples are presented.