$n_{\rm{e}}$. Because of this alternative reionization pathway at finite densities, the [*effective*]{} DR rate coefficient $\alpha^{\rm{eff}}_{\rm{DR}}(n_{\rm{e}},T)$ is thus [*suppressed*]{} from the zero-density value $\alpha_{\rm{DR}}(T)$ by a density-dependent suppression factor $S(n_{\rm{e}},T)$: $$\begin{aligned}
\alpha^{\rm{eff}}_{\rm{DR}}(n_{\rm{e}},T) & \equiv & S(n_{\rm{e}},T)\alpha_{\rm{DR}}(T)\ .
\label{eqsuppress}\end{aligned}$$

From the earlier detailed studies of @davidson and @badnell, the suppression factor is found to remain unity, corresponding to zero suppression, at lower densities until a certain [*activation*]{} density $n_{\rm{e},\rm{a}}$ is reached, beyond which this factor decreases exponentially from unity with increasing density. We have found that this suppression factor, as a function of the dimensionless log density parameter $x=\log_{10}n_{\rm{e}}$, can be modeled quite effectively by a pseudo-Voigt profile [@Wertheim:1974] — a weighted mixture $\mu$ of Lorentzian and Gaussian profiles of widths $w$ for densities above the activation density $x_{\rm{a}}=\log_{10}n_{\rm{e},\rm{a}}$: $$S(x;x_{\rm{a}}) = \left\{
    \begin{matrix}
    1  &  x \le x_{\rm{a}} \\
    \mu \left[\frac{1}{1+(\frac{x-x_{\rm{a}}}{w})^2}\right]
    + (1 - \mu)\left[{\rm e}^{-(\frac{x-x_{\rm{a}}}{w/\sqrt{\ln 2}})^{2}}\right]  &  x \geq x_{\rm{a}}
    \end{matrix}\right. \ .
    \label{eqsuppression}$$ Fitting this expression to the suppression factor of @badnell for (which was computed at $T=10^5$ K) yielded the values $\mu=0.372$, $w=4.969$, and $x_{\rm{a}}=0.608$, and this parameterization formula is found to be accurate to within 5% for all densities considered (see Fig. \[fig1\]).

![\[fig1\] Pseudo-Voigt fit of 