Scattering Theory of Absorption-Line Profiles and Refractivity

Abstract

Attenuation processes are examined from the viewpoint of collision theory, and the optical theorem is used to connect attenuation cross section $\ensuremath{\sigma}$ and refractive index $n$ with diagonal elements of the $T$ matrix. This approach provides resonance profiles (for "natural" lines shapes) of the form $\ensuremath{\sigma}(\ensuremath{\omega})=C+\frac{(\frac{\ensuremath{\Gamma}}{2})B+(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})A}{{(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}^{2}+{(\frac{\ensuremath{\Gamma}}{2})}^{2}},$ $n(\ensuremath{\omega})\ensuremath{-}1=\frac{\mathrm{Nc}}{2\ensuremath{\omega}}\left[\frac{(\frac{\ensuremath{\Gamma}}{2})A\ensuremath{-}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})B}{{(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}^{2}+{(\frac{\ensuremath{\Gamma}}{2})}^{2}}\ensuremath{-}D\right],$ where the profile parameters $A$, $B$, $C$, $D$, $\ensuremath{\Gamma}$, ${\ensuremath{\omega}}_{0}$ are given in terms of atomic matrix elements.Part I reviews the notion of resonances. Part II summarizes the relevant results of collision theory, stressing physical interpretation, and gives a definition for excited (or resonance) states based on a simple partition of basis states into two classes. Part III applies perturbation theory to the calculation of resonance profiles. Part IV applies these results specifically to the attenuation and refraction of photons by tenuous gases, with particular attention paid to the profiles of autoionizing lines. The effects of degeneracy (the extension of the bound-state ${Z}^{\ensuremath{-}1}$ expansion theory) are noted.