Energy gap law for vibrational relaxation of a molecule in a dense medium

In this paper we advance a model Hamiltonian for direct vibrational relaxation of a guest molecule in a host lattice. General expressions for the vibrational relaxation rate constant are derived utilizing the generating function method, elucidating the gross features of this class of multiphonon relaxation phenomena. Explicit theoretical results were obtained for the vibrational relaxation rate in two temperature regions: for moderately low temperatures below the characteristic Debye temperature and in the high temperature limit. In the low temperature range the vibrational relaxation rate exhibits a nearly exponential variation with the order of the multiphonon processes, revealing the energy gap law, an appreciable isotope effect, and a strong temperature dependence. The predicted energy gap law is compatible with the available experimental data.