Since the excitons are not exact bosons, the value of $\langle B^{\dagger}_{s,s',\alpha}(\vec{Q})  B_{s,s',\alpha}(\vec{Q}) \rangle$ is not just equal to the exciton occupation $n_{s,s',\alpha}(\vec{Q})$. Using the cluster expansion to evaluate $\langle B^{\dagger}_{s,s',\alpha}(\vec{Q})  B_{s,s',\alpha}(\vec{Q}) \rangle$ results in the additional Hartree-Fock term shown above [@Kira06; @Koch06]. The same extra term also shows up in the luminescence spectra of excitons [@Kira12], and, as discussed below, this term results in a quadratic dependence of the capture rate on the exciton density at large exciton densities.

We assume that the electron and hole densities for different spins/valleys (including both free carriers and bound excitons) are $n_{s}$ and $p_{s'}$, respectively, and the defect density is $n_{d}$. The initial ensemble consists of states that are approximate eigenstates of $H_{o} + H_{eh}$ but not of $H_{o} + H_{eh} + H_{hc} + H_{ec}$. Therefore, we consider $H_{hc}$ and $H_{ec}$ as perturbations.

Electron Capture Rate
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We first consider process (a) in Fig.\[fig:auger\_1\] in which the electron is captured by a defect. The average electron capture rate $R_{ec}$ (units: per unit area per second) can be calculated from the first order perturbation theory using the exciton basis described in Section \[sec:ex\_basis\] and the average values given in (\[eq:ex\_avg\]). The details of 