Effect of Reflecting Boundaries on the Transport of Resonance Radiation

Abstract

The Holstein-Biberman theory of the transport of resonance radiation in gases is generalized to include the case in which the boundary is partially reflecting. The problem is formulated in a manner which follows conventional transport theory somewhat more closely than the earlier treatments of Holstein and Biberman. Using the incoherent scattering approximation, we first write down the two coupled Boltzmann equations describing the mutual transport of excited atoms and resonance photons. The photon equation is then solved, with (diffuse) reflecting boundary conditions, by a slight extension of the interreflection method, and inserting the result into the excited atom equation leads at once to the appropriate generalization of the Holstein-Biberman transport equation. The kernel of this integrodifferential equation, G(r,r′), represents the mean probability that a resonance quantum emitted at r′ is absorbed at r, taking into account the contribution of paths which strike the boundary an arbitrary number of times; G(r,r′) is expressed quite generally in terms of the resolvent kernel of the interreflection equation. The theory is used to calculate the effect of a slightly reflecting boundary on the imprisonment time in a gas discharge.