estimate the double-ionization cross section using the semiempirical formulae of @Shevelko:JPhysB:2005, i.e., Equation (\[eq:shev5a\]) for direct ionization and Equation (\[eq:shev5b\]) for the $K$-shell IA. For these charge states, the branching ratios indicate the EIMI of higher order than double ionization is negligible.

Charge State Distribution {#sec:csd}
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The ion abundance $y_{i}$ of charge state $i$ as a function of time is described by $$\frac{\mathrm{d}y_{i}}{\mathrm{d}t}=n_{\mathrm{e}}
\left[I^{j}_{i-j}y_{i-j}+\ldots+I^{1}_{i-1}y_{i-1}-\left(I^{1}_{i}+\ldots+I^{k}_{i}+R_{i}\right)y_{i}+R_{i+1}y_{i}\right], 
\label{eq:dydt}$$ where $I^{j}_{i}$ is the rate coefficient for $j$-times ionization from charge state $i$ to $i+j$ and $R_{i}$ is the recombination rate coefficient from $i$ to $i-1$. The terms on the right represent, from left to right, ionization from lower charge states into $i$, ionization and recombination out of $i$ to other charge states, and recombination from $i+1$ into $i$. For most astrophysical plasmas, the density is low so that multiple recombination is extremely unlikely. Note also that in this expression, the rate coefficients $I$ and $R$ are functions of temperature. In a dynamic plasma, the temperature and density vary in time.

The rate coefficients for Equation (\[eq:dydt\]) were derived from several sources. The EISI rate coefficients come from the recommended data of @Dere:AA:2007. We have found these data to be in reasonable agreement with 