experiment for single ionization from Fe$^{7+}$ and Fe$^{9+}$–Fe$^{17+}$ [@Hahn:JPCS:2014 and references therein]. The radiative and dielectronic recombination rate coefficients are the ones compiled in the CHIANTI atomic database [@Dere:AAS:1997; @Landi:ApJ:2013]. For iron, many of these recombination rate coefficients are based on the calculations of @Badnell:AA:2003 and @Badnell:ApJ:2006[^1]. The dielectronic recombination data have been experimentally benchmarked by ion storage ring experiments [@Schippers:IRAMP:2010 and references therein].

The EIMI rate coefficients have been derived from the formulae given in Section \[sec:cross\] by numerically convolving the cross sections with a Maxwellian electron energy distribution. In dynamic or tenuous plasmas it is possible that the electron energy distribution is non-Maxwellian. Equilibrium-ionization-balance calculations for non-Maxwellian distributions have been given by @Dzifcakova:ApJS:2013, albeit without considering EIMI. Here, all of our calculations are based on Maxwellian distributions.

Equilibrium Ionization Balance {#sec:equilib}
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Collisional ionization equilibrium (CIE) occurs when the left hand side of Equation (\[eq:dydt\]) is zero. In this case the density is a constant factor and plays no role in the solution. For a given temperature, we have a system of algebraic equations. It is easy to see that Equation (\[eq:dydt\]) can be written as a matrix: $$\mathbf{A}\vec{y}=0, 
\label{eq:ay0}$$ where $\mathbf{A}$ is the matrix of the rate 