Approach to Simple Exponential Decay in Vibrational Relaxation

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Abstract
When diatomic oscillators are dilutely dispersed in an inert heat bath, populations of the various vibrational levels in an initial nonequilibrium distribution relax toward thermal equilibrium with a time dependence that is a sum of terms like exp (λmt), where λ0=0, all other λmm+1<λm. −1/λ1 ordinarily corresponds to the experimental relaxation time τ, since this term will dominate at long times. This work investigates cases in which the coefficient of the λ1 term, depending on the initial vibrational distribution and the heat‐bath temperature, may vanish or be so small that the final simple exponential decay corresponding to λ1 is never reached under experimental conditions. For harmonic oscillators, if the initial distribution has its first n moments equal to the corresponding moments of the equilibrium distribution, then the first n coefficients in the sum of exponentials will be zero. Furthermore, there will be one or more values of the heat‐bath temperature, independent of the initial distribution, for which the normally dominant exponential decay will be missing in the expression for the time dependence of population in any particular level. Some of these results can be extended qualitatively to anharmonic oscillators and more general systems.