Exchange of Translational and Vibrational Energy VT Process

Quantitative calculations of the rate constants ky y for vibrational transitions are relatively easy if the diatomic molecules are simulated by a harmonic oscillator interacting isotropically with the impinging atom (the so-called breathing sphere or the SSH (Schwartz-Slawsky-Herzfeld) model [3, 192, 339, 395] based on the one-dimensional Landau-Teller model [261]). Then the mean transition probability Py y per one gas-kinetic collision calculated to the first order of the semiclassical perturbation treatment is 

Here co is the oscillator frequency, [jl is the reduced mass of colliding partners, and a characterizes the steepness of the intermolecular potential U(R). Compared to the exponential function the pre-exponential factor B only weakly depends on T, CO, (X, a. (For its explicit form see, e.g. [192, 339].) 

The basic condition for the validity of Eq. (14.1) is that the probability (Pv,v+i) must be much lower than unity. This is usually the case when (07T) / 1. Eq. (14.2) shows that the probability increases with the steepness of U(R). This implies that it is the repulsive part of U(R) which is mostly responsible for inducing the transition v - v i 1. For this reason, the attractive part of the potential may often be neglected in VT processes and U(R) can be approximated by the simple exponential U(R) exp (—aR). 

The following specific features of vibrational relaxation stem from Eq. (14.2) known as the Landau-Teller formula  

if anharmonicity is taken into account, (Pv,v+i) increases with v not linearly, as in Eq. (14.2), but faster because the frequency co decreases with increasing v. Assuming, in the usual way, a linear dependence 

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