Electron-density diagnostic from metastable populations

In a two-level ground system, balancing collisional upward excitation against radiative and collisional downward loss gives $n_2/n_1=n_e{q}_{12}/(A_{21}+n_e{q}_{21})$. Excited lines fed in different proportions by these levels have a density-sensitive ratio. At low density the population ratio is proportional to density; at high density it saturates at $q_{12}/q_{21}$. A reliable calibration uses a full multilevel model when upper-state branching and cascades matter.

For optically thin lines with local ratio $R(n_e,T)$, their integrated observed ratio is $\int j_1(s)R(n_e(s),T(s))ds/\int j_1(s)ds$. Inverting a uniform-plasma calibration yields an effective diagnostic density. It is generally not the volume mean, nor the density obtained by simply averaging the local densities, because emissivity weighting and the ratio response are nonlinear.