Einstein coefficients and oscillator strengths

Put one atom in a cavity at temperature T. Absorption drains level 1 at B₁₂u_νN₁ and spontaneous emission refills it at A₂₁N₂. In steady state those two alone give u_ν = (A₂₁/B₁₂)(g₂/g₁)e^{(−hν/k_BT)}, which is Wien's law — right in the far tail, catastrophically wrong at low frequency, where Planck's spectrum grows as ν². Einstein's repair was a third channel proportional to the field but driven out of the upper level, B₂₁u_νN₂. Steady state now gives u_ν = A₂₁ / [B₁₂(g₁/g₂)e^(hν/k_BT) − B₂₁], and this matches (8πhν³/c³)/(e^(hν/k_BT) − 1) only if two conditions hold at once: g₁B₁₂ = g₂B₂₁, and A₂₁/B₂₁ = 8πhν³/c³. Nothing about the atom was assumed beyond two levels and a Boltzmann ratio, so the relations bind every atom — and for exactly that reason they cannot deliver a single number. Equilibrium constrains ratios; it does not compute matrix elements.

Divide the stimulated emission rate by the spontaneous one: B₂₁u_ν/A₂₁ = u_νc³/8πhν³, and for a thermal field that is exactly n̄ = 1/(e^(hν/k_BT) − 1), the mean photon number per mode. Which channel wins is therefore a question about the field, not the atom. At 300 K on the sodium D₂ line hν/k_BT = 81.4, so n̄ ≈ 4 × 10⁻³⁶: in a warm room essentially every optical photon an atom emits is spontaneous, and an optical amplifier must manufacture its own mode occupation by pumping a population inversion. At 1.42 GHz the same temperature gives hν/k_BT = 2.27 × 10⁻⁴ and n̄ ≈ 4.4 × 10³, so stimulated emission beats spontaneous by four thousand to one with nobody doing anything. That is why the maser preceded the laser by seven years. The 21 cm hydrogen line, with a spontaneous rate of 2.9 × 10⁻¹⁵ s⁻¹ — a lifetime of eleven million years — shows the same imbalance in space: even the 2.7 K cosmic background makes its stimulated rate some forty times the spontaneous one.

Einstein stopped at ratios; first-order time-dependent perturbation theory supplies the scale. For an electric-dipole transition the spontaneous rate is A₂₁ = ω³Σ_f|⟨f|d|i⟩|²/3πε₀ħc³, with d = −er, summed over the lower sublevels and averaged over the g₂ upper ones. Alongside A, spectroscopists tabulate the oscillator strength f₁₂ = (2mₑω₂₁/3ħ)Σ_f|⟨f|r|i⟩|², which measures the transition against one classical electron bound on a spring at the same frequency. It is dimensionless, usually between 10⁻³ and 1, and for exact eigenfunctions it comes out the same whether you evaluate the matrix element in the length gauge or the velocity gauge — so the gap between those two values is a standard diagnostic of an approximate wavefunction. The bridge back to a rate is A₂₁ = (2πe²ν²/ε₀mₑc³)(g₁/g₂)f₁₂. For Lyman-α, |⟨2p₀|z|1s⟩| = 0.7449a₀, the three 2p sublevels give Σ|⟨f|r|i⟩|² = 1.665a₀², f = 0.416, and A₂₁ = 6.26 × 10⁸ s⁻¹ — a 1.60 ns lifetime.

How do you know a computed set of oscillator strengths is not simply too big? Take the double commutator [x,[H, x]] for H = p²/2m + V(r). Since [H, x] = −iħp/m, the whole object collapses to the pure number ħ²/m, with no reference to V at all. Sandwich it in the ground state and expand over a complete set: 2Σₙ(Eₙ − E₀)|⟨n|x|0⟩|² = ħ²/m. Divide through by ħ²/m and every term becomes an oscillator strength, giving Σₙ fₙ₀ = 1 for one electron and Nₑ for Nₑ of them. A strong line has to be paid for somewhere. The sum runs over the continuum as well, and in hydrogen that is no footnote: the entire bound Lyman series sums to 0.565, so 43.5% of the sum rule sits in photoionisation above 13.6 eV. Use it as a test on any numerical dipole calculation — add your computed f values, subtract from Nₑ, and if what is left for the continuum comes out negative, the calculation is wrong.

A spectrum reaches the atoms only through the optical depth. Write σ(ν) = (e²/4ε₀mₑc) f₁₂ φ(ν) with ∫φ dν = 1; then I(ν) = I₀ exp[−σ(ν)nL], and the frequency-integrated optical depth, ∫σ(ν)nL dν = (e²/4ε₀mₑc) f₁₂N, depends on f and the column density N = nL alone, never on the broadening. Depth is a different story. A Doppler profile has Δν_D = (ν₀/c)√(2k_BT/M) and peak φ(ν₀) = 1/(√π Δν_D), so heating the gas at fixed N flattens the line while its area holds. For Na D₂ at 350 K, Δν_D = 854 MHz and σ₀ = 1.12 × 10⁻¹⁵ m²; 20% transmission through a 7.5 cm cell means τ₀ = ln 5 = 1.61 and n = 1.9 × 10¹⁶ m⁻³. A resolved laser scan reads τ₀ this way for as long as the centre still transmits measurably. A spectrometer that cannot resolve the line sees only its equivalent width, and that tracks N linearly only while τ₀ ≲ 1: push further and the centre saturates, the equivalent width barely grows, and only the damping wings — growing as √N — still count atoms. Reading that flat middle of the curve of growth as if the line were thin is the standard way to underestimate a column density tenfold.

Everything above is a rate equation for populations, and populations are only the diagonal of the density matrix. The optical Bloch equations also carry the coherence ρ₁₂, and eliminating it is legitimate only when it relaxes far faster than the populations do — dephasing rate γ much larger than the Rabi frequency Ω = |d⋅E|/ħ. A broadband thermal field satisfies that easily, its correlation time being femtoseconds; a narrowband laser does not, and the atom then Rabi-flops at Ω instead of relaxing towards a steady state, a regime in which no B coefficient appears at all. Even inside the rate-equation regime the two-level system has a hard ceiling: setting dN₂/dt = 0 under strong driving brings N₂/g₂ up to N₁/g₁ at best — N₂/(N₁ + N₂) → ½ for non-degenerate levels — so a two-level atom can be bleached but never inverted. For Na D₂ the saturation intensity Iₛₐₜ = πhc/3λ³τ is 6.3 mW cm⁻². Real lasers need a third or fourth level precisely because Einstein's own equations forbid the two-level shortcut.