When Nλ³/64 > 1
The cooperative-emission threshold for dense atomic vapor
When the number density of resonant atoms in a quarter-wavelength cube exceeds one, standard radiation-trapping treatments break down. The medium is denser than the photon mode density at the resonance line. What follows is cooperative.
When Nλ³/64 > 1
The load-bearing claim: the medium is denser than the photon mode density. Holstein's eigenmode expansion no longer carries; escape is not a probability but a coherence.
For Na D₂ at 589 nm: - λ = 5.89 × 10⁻⁵ cm - λ³ = 2.04 × 10⁻¹³ cm³ - n_crit = 64 / λ³ ≈ 3.13 × 10¹⁴ cm⁻³
Saturated Na vapor reaches this density around 700-800 K. The cooperative regime is much more accessible than the Hg-isotope industrial literature implies.
## See also - 02-holstein-radiation-trapping - 03-dicke-superradiance - archive/qed.princeton.edu/main/PlasmaWiki/Mean_free_path
Embedded artifact demo
Explorables
Drag T: saturated Na vapor density n(T) climbs the Nesmeyanov curve and crosses the cooperative threshold n_crit = 64/λ³. Watch the regime flip from Holstein-trapped to cooperative to superradiant.