= Solution
A <spontaneous emission> transition has probability per unit time $A_{ji}$ and emits a photon of <energy> $h\nu_{ji}$. For an optically thin gas, its angle-integrated <emissivity> is $j_{ji}=n_jA_{ji}h\nu_{ji}$. Electron collisions populate excited levels at rates $n_en_iq_{ij}$; they also remove excitation collisionally at rates $n_en_jq_{ji}$, returning the <energy> to the gas rather than producing the line photon. A collisionally excited line is usually emitted by a subsequent spontaneous radiative decay; <collisional de-excitation> instead quenches it.
The <critical density of an atomic transition> compares radiative loss from an upper level with its collisional loss. For a two-level model, $n_{\mathrm{crit}}=A_{ji}/q_{ji}$; for several lower or connected levels use the relevant sums. Below this <density>, excitation is commonly followed by a photon and the line cools the gas. Above it, collisions increasingly suppress radiative emission per excitation and drive the level populations toward Boltzmann ratios.
Optical <forbidden emission lines> have small $A$ because an electric-dipole transition is forbidden by a selection rule. Metastable levels survive long enough to radiate in dilute ionized nebulae, although collisions quench them in dense gas. Ratios from closely spaced levels with different critical densities are electron-density diagnostics; ratios involving different excitation energies probe the <electron temperature> through Boltzmann factors. Combining line emissivities with <electron density>, <temperature> and an <ionization> model estimates ionic abundances.
Infrared <fine-structure emission lines> connect closely spaced sublevels. Their small excitation energies make them useful over <temperature> ranges where optical excitation is exponentially suppressed, and dust extinction is usually weaker in the far infrared. They are important coolants and <density> diagnostics in ionized gas, with the choice of ion specifying the <ionization> zone. Neither an optical nor an infrared ionic abundance automatically equals the total elemental abundance; unobserved <ionization> stages require an <ionization> correction.
Back to article page
