Critical density of an atomic transition ($n_{\mathrm{crit}}=A_{ul}/q_{ul}$)

The critical density of an atomic transition is the electron density at which its collisional de-excitation rate equals its spontaneous radiative decay rate. A multilevel atom requires summed competing rates rather than a single-transition ratio. This atomic density scale is distinct from cosmological critical density.

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.

At fixed density below critical density, optically thin two-level cooling scales as $\mathcal{C}\propto T^{-1/2}{e}^{-\theta /T}$, with $\theta =\Delta E/k_B$. Under temperature-independent heating, $d\ln \mathcal{C}/d\ln T=-1/2+\theta /T$ gives restoring thermal balance for $T<2\theta$ and instability for $T>2\theta$. This boundary assumes fixed ionic abundance and does not describe a constant-pressure perturbation.

The stability boundary in isochoric thermal stability of a collisionally excited line assumes that the abundance of the emitting ion stays fixed during the temperature perturbation. At fixed relevant densities, the cooling factor $T^{-1/2}{e}^{-\theta /T}$ then has logarithmic slope $-1/2+\theta /T$.

If the ion fraction $x(T)$ adjusts, the cooling instead contains that additional factor and its slope gains $d\ln x/d\ln T$. Changes in the electron abundance or heating law introduce further terms. Thus the quoted $T=2\theta$ boundary belongs to the frozen-abundance model; comparing ionization and thermal relaxation times determines whether that assumption fits a particular perturbation.

At the maximum of a cooling curve with constant heating set to that maximum, the linear restoring term vanishes. Because cooling falls on both sides of the maximum, the equilibrium is not generally stable to perturbations of both signs even though a first-order eigenvalue is zero.

At fixed pressure and fixed composition, density varies as $T^{-1}$. With temperature-independent heating per particle, low-density cooling per particle scales as $T^{-3/2}{e}^{-\theta /T}$ and is restoring for $T<2\theta /3$. For constant heating per volume instead, the cooling scales as $T^{-5/2}{e}^{-\theta /T}$ and the threshold is $2\theta /5$. The heating normalization and perturbation constraint must be specified.

When collisions thermalize an optically thin two-level atom, the upper fraction is $f_u=g{e}^{-\theta /T}/(1+g{e}^{-\theta /T})$ and the emitted power per atom is $A_{21}\Delta E{f}_u$. Its logarithmic temperature slope is $(\theta /T)(1-f_u)>0$. Radiative cooling still occurs, but collision frequency no longer sets the emitted power as in the low-density limit.

A schematic statistical equilibrium is $n_l{n}_e{q}_{lu}=n_u(A_{ul}+n_e{q}_{ul})$. The radiative yield per excitation is $A_{ul}/(A_{ul}+n_e{q}_{ul})$. It is near unity below critical density $A_{ul}/q_{ul}$ and decreases above it. Quenching suppresses emission relative to naive low-density excitation scaling; it need not make the absolute emissivity decrease when density rises.