@Kaastra:AAS:1993. Similarly, there are no experimental data at the energies relevant for $K$-shell IA, so we also estimate those cross sections in the same way.

Double-ionization cross sections for Fe$^{1+}$ and Fe$^{3+}$–Fe$^{6+}$ were measured by @Stenke:JPhysB:1999. Fits to these data were given by @Shevelko:JPhysB:2006 using Equations (\[eq:shev6a\]) and (\[eq:shev6b\]). We use their fits here. The existing experimental measurements do not extend high enough to benchmark $K$-shell IA, so here we have used the Lotz formula and the branching ratios given by @Kaastra:AAS:1993 to incorporate those processes. Similarly, for higher order EIMI of these systems we assume that the dominant process is IA and estimate the cross section using the Lotz formula and the branching ratios.

There are no experimental data for double ionization of Fe$^{2+}$. However, Ni$^{4+}$ is isoelectronic and double ionization of this ion was measured by @Stenke:NIMB:1995 and was the basis for a semiempirical fit reported in @Shevelko:JPhysB:2006. We find that the Ni$^{4+}$ double-ionization cross section can be reproduced well using Equation (\[eq:shevtar\]) for the direct ionization contribution plus scaled Lotz cross sections for the indirect contributions. Thus, we have applied the same formulae to the Fe$^{2+}$ double-ionization cross section, with appropriate modifications for the different energy thresholds 