Density sensitivity of a metastable doublet ($R(n)=R_0(1+n/n_{c,3})/(1+n/n_{c,2})$)

For two independently excited optically thin upper levels sharing a lower state, statistical equilibrium gives the displayed ratio with critical densities $n_{c,j}=A_{j1}/C_{j1}$. When $n_{c,2}<n_{c,3}$, the fractional sensitivity $|d\ln R/d\ln n|$ is greatest at $n=\sqrt{n_{c,2}{n}_{c,3}}$. With constant absolute ratio error, the largest $|dR/d\ln n|$ occurs instead at $n=n_{c,2}$. Thus a density diagnostic's optimum requires both an atomic model and a measurement-error convention. Additional collisional coupling changes the intermediate-density curve.