deletions | additions       diff --git a/methods.tex b/methods.tex  index fa94600..d49322b 100644  --- a/methods.tex  +++ b/methods.tex 
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\begin{equation}  \label{eq:pop_inverse_full}  \Delta n \equiv \frac{n_u}{g_u} - \frac{n_l}{g_l} \\  = \frac{P_u/g_u - P_l/g_l}{\Gamma + \left( 1 + g_u/g_l \right)\gamma_{ul}n_{\mathrm{tot}} \right)\gamma_{ul} n_{\mathrm{tot}}}  \end{equation}  This difference, from the Einstein theory, is the criterion for stimulated action \citep{Gray_2009}. As is done in \citet{stahler_palla_2004}, I define the population inversion due to only collisional processes (ie. $\bar{J}=0$), as $\Delta n^{\circ}$. Equation \ref{eq:pop_inverse_full} can then be expressed as:  \begin{equation}