The high energy asymptotic behavior of line shape cross sections and detailed balance

Abstract

Although line shape relaxation cross sections in the ‘‘impact approximation’’ do not obey detailed balance except in the very high temperature limit, they can be shown, in the Born approximation, to satisfy a somewhat similar asymptotic relation. In this empiric relation, first observed by Boissoles et al. [J. Chem. Phys. 90, 5392 (1989)], the complex conjugate of the cross section for the radiative transition if→i′f′ is asymptotically equal to the cross section for the inverse transition i′f′→if. It is then shown that although cross sections calculated with the presumably more correct Fano collision operator also obey this relation, due to time reversal symmetry they also obey detailed balance. These two observations taken together imply that the imaginary part of line shape relaxation cross sections in the more exact theory decay much more rapidly with energy than the real parts and suggest a quick fix for relaxation cross sections calculated in the ‘‘impact approximation,’’ or as it is sometimes called, the Shafer–Gordon formalism. Numerical calculations of inelastic off-the-energy shell scattering of simple rigid spherelike models support these results and suggest that off-the-energy shell calculations with realistic models are not only desirable but also necessary for computing line shapes of partially overlapping lines.