Explicit Form of the Radiative and Collisional Branching Ratios in Polarized Radiation Transport with Coherent Scattering

Abstract

We consider the vector emissivity of the polarized radiation transfer in a $\Lambda$-type atomic transition, which we recently proposed in order to account for both CRD and PRD contributions to the scattered radiation. This expression can concisely be written as $$\varepsilon=\left(\varepsilon^{(1)} -\varepsilon^{(2)}_{\rm f.s.} \right) +\varepsilon^{(2)}\;$$ where $\varepsilon^{(1)}$ and $\varepsilon^{(2)}$ are the emissivity terms describing, respectively, one-photon and two-photon processes in a $\Lambda$-type atom, %the fully and partially redistributed radiations, and where "f.s." means that the corresponding term must be evaluated assuming an appropriate "flat spectrum" average of the incident radiation across the spectral line. In this follow up study, we explicitly consider the expressions of these various terms for the case of a polarized multi-term atom, in order to derive the algebraic forms of the branching ratios between the CRD and PRD contributions to the emissivity. In the limit of a two-term atom with non-coherent lower-term, our results are shown to be in full agreement with those recently derived by Bommier (2017)