Same-upper-level extinction diagnostic ($j_{ji}/j_{jk}=A_{ji}{\nu }_{ji}/(A_{jk}{\nu }_{jk})$)

Two optically thin lines from the same upper level have an intrinsic energy-emissivity ratio fixed by their radiative branching rates and photon energies. The upper population cancels, so a common electron-temperature or abundance change cannot alter that ratio. Comparing an optical branch with a weakly attenuated infrared branch can determine differential dust extinction, subject to matched emitting volumes and calibrated emissivities.