An approximate treatment of localised vibrational modes

The 'rigid-well' approximation for the local vibrational mode frequency of a light impurity in a crystal in which only the impurity itself is regarded as moving may be regarded as the leading term in an expansion in powers of the ratio of the impurity mass to the host mass or masses. It is found that the retention of only one further term provides an analytic approximation that is quantitatively useful over a much wider range of cases of interest. This approximation assumes a particularly simple form if the contributions from the motion of second and further neighbours of the impurity are negligible. Various applications are presented.