(b) in Fig.\[fig:auger\_1\] in which the hole is captured by a defect can be calculated in the same way. The result is, $$\begin{aligned}
& & R_{hc} \approx \frac{2\pi}{\hbar} n_{d}f_{d} \sum_{s,s',\alpha} D_{c,s}(q_{\alpha}) |\chi_{d}(q_{\alpha})|^{2}  |b_{v,s'}|^{2}   \nonumber \\
& & \times \left| \frac{1}{A} \sum_{\vec{k}_{r}} V(q_{\alpha}\hat{x}-\vec{k}_{r})  \phi_{\alpha}(\vec{k}_{r}) \right|^{2} \left[ n_{s,s',\alpha} \right. \nonumber \\
& & \left. + \frac{1}{A^{2}} \sum_{\vec{k},\vec{Q}} |\phi_{\alpha}(\vec{k})|^{2} f_{c,s}(\vec{k} + \frac{m_{e}}{m_{ex}} \vec{Q}) \left[ 1 - f_{v,s'}(\vec{k} -  \frac{m_{h}}{m_{ex}} \vec{Q}) \right] \right] \nonumber \\ \label{eq:Rhc_0}\end{aligned}$$ where now $q_{\alpha}$ is approximately given by the relation, $E_{c,s}(q_{\alpha}) - E_{c,s}(0) = E_{d} - E_{\alpha}$. And, as before, if $q_{\alpha} >> k_{r}$ for all values of $k_{r}$ for which $\phi_{\alpha}(\vec{k}_{r})$ is significant, then the above expression reduces to, $$\begin{aligned}
R_{hc} & = & \frac{2 \pi}{\hbar} n_{d} f_{d} \sum_{s,s',\alpha} D_{c,s}(q_{\alpha}) |V(q_{\alpha})|^{2} |\chi_{d}(q_{\alpha})|^{2}  \nonumber \\
& &  \times |b_{v,s'}|^{2} \left[ |\phi_{\alpha}(\vec{r}=0)|^{2} n_{s,s',\alpha} + G_{\alpha} n_{s}p_{s'} \right]       \label{eq:Rhc_1}\end{aligned}$$

Coulomb Correlations and Enhancement of the Auger Capture Rates {#sec:corr}
---------------------------------------------------------------

Equation (\[eq:Rec\_1\]) for the electron capture rate can also be written as, $$\begin{aligned}
R_{ec} & = & \frac{2\pi}{\hbar} n_{d} \left( 1-f_{d} \right) \sum_{s,s',\alpha}  D_{v,s'}(q_{\alpha}) |V(q_{\alpha})|^{2} |\chi_{d}(q_{\alpha})|^{2}  \nonumber \\
& &  \times |b_{c,s}|^{2}  n_{s}p_{s'} \left[  G_{\alpha} + g_{s,s',\alpha}(\vec{r}=0) \right]    \label{eq:Rec_2}\end{aligned}$$ where, $g_{s,s',\alpha}(\vec{r}=0) = |\phi_{\alpha}(\vec{r}=0)|^{2} n_{s,s',\alpha}/(n_{s}p_{s'})$. The quantity inside the square brackets in 