the calculations are given in the Appendix. The final result is, $$\begin{aligned}
& & R_{ec} \approx \frac{2\pi}{\hbar} n_{d}(1-f_{d}) \sum_{s,s',\alpha} D_{v,s'}(q_{\alpha}) |\chi_{d}(q_{\alpha})|^{2}  |b_{c,s}|^{2}   \nonumber \\ 
& & \times \left| \frac{1}{A} \sum_{\vec{k}_{r}} V(q_{\alpha}\hat{x}-\vec{k}_{r})  \phi_{\alpha}(\vec{k}_{r}) \right|^{2} \left[ n_{s,s',\alpha} \right. \nonumber \\
& & \left. + \frac{1}{A^{2}} \sum_{\vec{k},\vec{Q}} |\phi_{\alpha}(\vec{k})|^{2} f_{c,s}(\vec{k} + \frac{m_{e}}{m_{ex}} \vec{Q}) \left[ 1 - f_{v,s'}(\vec{k} -  \frac{m_{h}}{m_{ex}} \vec{Q}) \right] \right] \nonumber \label{eq:Rec_0} \\ \end{aligned}$$ Here, $D_{v,s'}$ is the valence band density of states (per valley per spin) evaluated at the energy of the scattered hole whose wavevector is $q_{\alpha}$. $q_{\alpha}$ is approximately given by the relation, $E_{v,s'}(0) - E_{v,s'}(q_{\alpha}) = E_{g} - E_{\alpha} - E_{d}$. Note that none of the phase factors appear in the above result. The exciton density $n_{s,s',\alpha}$ is, $$n_{s,s',\alpha} = \int \frac{d^{2}\vec{Q}}{(2\pi)^{2}} \, n_{s,s',\alpha}(\vec{Q})$$ 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, $$\begin{aligned}
R_{ec} & = & \frac{2\pi}{\hbar} n_{d} \left( 1-f_{d} \right) \sum_{s,s',\alpha} D_{v,s'}(q_{\alpha}) |V(q_{\alpha})|^{2} |\chi_{d}(q_{\alpha})|^{2}  \nonumber \\
& &  \times |b_{c,s}|^{2} \left[ |\phi_{\alpha}(\vec{r}=0)|^{2} n_{s,s',\alpha} + G_{\alpha} n_{s}p_{s'} \right]         \label{eq:Rec_1}\end{aligned}$$ Expression for $G_{\alpha}$ is given in the Appendix. $G_{\alpha}$ is significant for only the lowest few exciton states.

Hole Capture Rate
-----------------

The rate for process 