Time-Dependent Solutions for De-excitation Rates of Anharmonic Oscillators

Exact numerical solutions are obtained for level populations of a 45-level anharmonic oscillator model of N2 in an expanding relaxation process with initial Tv=2500°K and T=500°K. Both vibration-vibration and vibration-translation energy transfers are included. An implicit numerical method similar to Lomax's is developed for integrating the stiff master relaxation equations. The de-excitation rate is found to decrease rapidly from an extremely high value at the initial stage to the excitation rate at the corresponding heat bath temperature in the secular adjustment period. Laboratory observations on this de-excitation rate will depend on the instant when it is observed. In the initial stage, the intermediate levels are rapidly overpopulated to form a plateau while the lower-level populations adjust to a Treanor distribution and the upper levels to a Boltzmann distribution of T. This plateau descends gradually to form a ∼ shape and eventually will become a straight line indicating Boltzmann equilibrium. For accurate determination of de-excitation rates in the early stages of relaxation, it is important to include more upper levels in the model. However, these upper levels contribute insignificantly in the later stages of secular adjustments. The first-moment method, with a Treanor distribution for a truncated model, accurately predicts the lower-level populations as well as the de-excitation rates in the secular adjustment period. But this method underestimates the rates in the early stages because of neglect of significant VT transitions from higher levels.