Vibrational Energy Transfer in a System of Radiating Oscillators

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Abstract
The distribution function and the total band intensity for a system of excited oscillators that can both exchange vibrational energy among themselves through collisions and emit their energy through radiative transitions has been obtained for several pressure conditions. The nonequilibrium solution for an initial Boltzmann distribution is not a time‐dependent Boltzmann distribution as has been obtained for non‐radiating models. This solution is investigated for (1) very low pressures in which there are no interactions between the oscillators and (2) pressures high enough that collisional exchanges are much more rapid than the radiative transitions but not so high as to involve collisional deactivation or reabsorption. Both of these cases yield identical solutions, namely, a Boltzmann distribution with a time‐dependent temperature. The solution for the second condition is also obtained by assuming a Boltzmann distribution is maintained at all times and solving for the temperature as a function of time. Although the non‐equilibrium solution is not Boltzmann, it is very close, deviating at most 2% for CO. The solution for an initial delta‐function distribution is also non‐Boltzmann under nonequilibrium conditions, being given by a hypergeometric function. The total band intensity which is proportional to the first moment is exponential with respect to time and is independent of the initial distribution and the pressure. Also, the effect of first overtone transitions on both the distribution function and the total band intensity is bound to be negligible, the deviation for CO being less than 1%.