Inelastic Molecular Collisions with a Maxwellian Interaction Energy

The radial Schrödinger equation for arbitrary values of the angular momentum is solved exactly with a repulsive R—4 potential energy, the solutions being Mathieu functions whose arguments are linear functions of logR. These solutions are then applied to the problem of vibrational relaxation in the distorted wave approximation. Emphasis is given to the case in which the energy which is exchanged in the inelastic collision, ε0, is very much greater than the smaller of the energies of the incident and emergent waves. It is found that in this case the energy exchange probability per collision is proportional to a sinusoidal factor which may have a value anywhere between 0 and 1, depending on the precise magnitude of ε0; to a factor which decreases as the exponential of ε0¼; and to a factor which depends on the angular momentum quantum number, l, and on the smaller of the incident and emergent wave numbers, k. This last factor is evaluated for fixed l with k—1 much less than the effective range of molecular interaction; for fixed l with k—1 much greater than the effective range; and for l large with k fixed. The relaxation time is obtained and both its magnitude and temperature dependence are discussed. Over an appropriate range of temperature it is found to decrease as the exponential of —T⅓, which is not the same as the Landau-Teller exponential of T—⅓. The temperature dependence found in the present work is in much better agreement with the observed high-temperature vibrational relaxation of oxygen, and still better agreement is obtained when the present approximation methods are applied with an exponential interaction energy. The magnitude of the relaxation time, however, is not predicted nearly so well as is its temperature dependence.