X-ray spectroscopy in the microcalorimeter era III: Mathematical Framework of Case C to B transition in iron

Lines of one and two electron species are formed under Case A, Case B, or Case C condition. Case A occurs when the optical depth of the Lyman lines is very small, and no external radiation strikes the cloud. Lines escape the cloud without any scattering. Case B occurs when the Lyman line optical depths are large enough for photons to undergo multiple scatterings and so are converted into Balmer and Ly-α photons. If a broadband continuum source strikes an optically thin cloud, the Lyman lines are enhanced by induced radiative excitation of the atoms/ions by continuum photons, also known as continuum pumping. This situation is originally defined as “Case C” in the literature. Cases A, B, and C have been known from the 1930s for hydrogen and helium. The same has to occur in the X-ray band for H- and He-like systems. In this work, we look into the mathematical framework of Case C to Case B transition in an iron-only model. Such analysis will be important with the advent of future microcalorimeter X-ray observations in the presence of a photoionizing source, for example, any cloud near AGN.