Time‐dependent quantum theory III. Model for vibrational relaxation in crystals

A one‐dimensional model consisting of a “diatomic” spring attached on one side to a rigid wall and on the other side to a linear array of mass‐springs is proposed as a model for the vibrational relaxation of small solute molecules in host lattices. A modification allowing a change in the equilibrium internuclear extension of the diatomic spring is also incorporated. The Hamiltonian divides naturally into pure diatomic, pure linear crystal, and the two mixed perturbation terms, one giving rise to stepwise vibrational cascade damping accompanied by phonon emission, and the other process, lattice relaxation, giving rise to phonon emission without any change of the quantum number of the diatomic spring. The cascade damping rate for a diatomic spring with a frequency less than the the maximum frequency of the linear crystal is calculated to second‐order, and it is shown that the perturbation series converges in this range. An upper bound to the cascade damping rate for a diatomic spring with a frequency greater (i.e., 4.5 ×) than the cut‐off frequency of the linear crystal is determined to be very small, λ ≦ 10 4 sec −;1 . The rate for the lattice relaxation process corresponds to a line‐width λ = 6 cm −1 at 0 K. An explanation for the thermal quenching of the low‐temperature luminescence of SO 2 is based upon induced cascade‐phonon emission.