presented a more accurate formula for double-ionization cross sections. . Here, the cross section is the sum of a direct ionization cross section and several possible indirect ionization channels. In this scheme the direct cross section is given by $$\sigma_{\mathrm{D}}=1-\mathrm{e}^{-3\left(u-1\right)} \frac{p_0}{E_{\mathrm{th}}^3}\left[\frac{u-1}{\left(u+0.5\right)^2}\right]\left[1+0.1\ln{(4u+1)}\right] 
\times10^{-13} \, \mathrm{cm^{2}},
\label{eq:shev6a}$$ where $p_0$ is a fit parameter. The individual indirect ionization cross sections sections, due to IA, are given by $$\sigma_{\mathrm{IA}}=\frac{p_0}{E_{\mathrm{th}}^2}\frac{u-1}{u \left(u+5.0 \right)} \left[1+\frac{0.3}{p_{1}}\ln{(4u+1)}\right]
\times10^{-13} \, \mathrm{cm^{2}},
\label{eq:shev6b}$$ where $p_0$ is a fit parameter and $p_1$ is the principal quantum number of the core electron that is directly ionized. Unlike Equations (\[eq:shev5a\]) and (\[eq:shev5b\]), for Equations (\[eq:shev6a\]) and (\[eq:shev6b\]) the parameters do not have a tabulated dependence on an isoelectronic sequence, but rather are determined by fits to experimental measurements. Thus, the formulae should be accurate, but it is not possible to extrapolate them to systems that have not been measured.

For highly charged ions, the indirect IA process is the dominant contribution to the EIMI cross section. In such cases, the cross section is often well described by multiplying the @Lotz:ZPhys:1969 formula for single ionization of the core electron by the branching ratio $f$ for autoionization of the intermediate state, giving $$\sigma_{\mathrm{IA}}=4.5 f p_{0}\frac{\ln{(u)}}{E_{\mathrm{th}}^{2}u}
\times10^{-14} 