Oscillator Strengths and Einstein Coefficients

Every spectral line an astronomer measures — the dark Fraunhofer bands in a stellar photosphere, the green glow of a planetary nebula, the radio hiss of interstellar hydrogen — is bookkeeping on three microscopic events inside atoms. An atom in an upper level can decay on its own and emit a photon (spontaneous emission); a photon can be swallowed and lift the atom up (absorption); and, most subtly, a passing photon can trigger the atom to decay and emit a second photon locked in step with the first (stimulated emission). Einstein introduced all three in a single 1917 paper, assigned each a rate coefficient — $A_{21}$ , $B_{12}$ , $B_{21}$ — and then, with almost no machinery, showed that if a gas of such atoms is to sit in equilibrium with a Planck radiation field the three coefficients cannot be independent. Detailed balance forces $g_1 B_{12}=g_2 B_{21}$ and $A_{21}=(8\pi h\nu^3/c^3)\,B_{21}$ . These relations are the load-bearing beams of radiative astrophysics. They predate quantum electrodynamics, yet fix the ratio of spontaneous to stimulated emission exactly and prove that stimulated emission must exist — drop it and you cannot build the Planck spectrum. The companion oscillator strength $f_{12}$ ties these quantum rates back to the classical bound electron and is the number tabulated in every atomic-line database. Together they are the dictionary that turns a measured line into a temperature, a density, or an abundance.

Key takeaways

Three processes govern how atoms trade energy with radiation: spontaneous emission ( $A_{21}$ ), absorption ( $B_{12}\bar u_\nu$ ), and stimulated emission ( $B_{21}\bar u_\nu$ ).
Detailed balance with a Planck field forces two universal relations, $g_1 B_{12}=g_2 B_{21}$ and $A_{21}=(8\pi h\nu^3/c^3)\,B_{21}$ — properties of the atom, valid in any radiation field.
Stimulated emission is not optional: without the $B_{21}$ term you recover the Wien tail, not the full Planck spectrum, so its existence is mandatory.
The oscillator strength $f_{12}$ measures a real transition against one classical Lorentz oscillator, whose integrated cross-section is the fixed number $\pi e^2/(m_e c)=0.0265\ \mathrm{cm^2\,Hz}$ .
Coefficients plus populations build the opacity $\alpha_\nu$ , emissivity $j_\nu$ , and source function $S_\nu$ ; in LTE these collapse to Kirchhoff's law $S_\nu=B_\nu(T)$ .
Key idea: Detailed balance with the Planck field forces $g_1 B_{12}=g_2 B_{21}$ and $A_{21}=(8\pi h\nu^3/c^3)B_{21}$ ; these are intrinsic atomic relations valid in any radiation field, and the oscillator strength $f_{12}$ ties them to the classical electron cross-section $\pi e^2/(m_e c)$ .

Where this lives today

Einstein's coefficients and oscillator strengths underpin nearly every quantitative measurement modern astrophysics makes. Stellar chemical abundances — the Sun, ancient metal-poor stars, exoplanet host stars — are read off line strengths through the curve of growth, and the dominant systematic is often not the spectrum but the laboratory $gf$ -value: databases such as NIST ASD, VALD, and Kurucz are constantly revised because a $0.1$ dex error in $\log gf$ propagates straight into the abundance. Astrophysical masers, the runaway limit of the stimulated term, are precision tools: water megamasers around active galactic nuclei let VLBI weigh central black holes and, in NGC 4258, deliver a geometric Hubble constant. Forbidden lines with tiny oscillator strengths — the nebular [O III] $\lambda\lambda4959,5007$ doublet, [N II], [S II] — are the workhorse density and temperature diagnostics of HII regions and emission-line galaxies. And in non-LTE codes the Einstein coefficients populate the statistical-equilibrium matrix, the machinery behind every synthetic spectrum fit to JWST and ELT data.

The physical picture

Picture a sealed box of two-level atoms bathed in radiation. Spontaneous decays fire at a fixed rate $A_{21}$ set by the atom alone, like radioactive decay — indifferent to how much light is around. Absorption and stimulated emission are driven: each is proportional to the number of photons present, so both scale with the energy density $\bar u_\nu$ . In equilibrium the total up-flow must exactly cancel the total down-flow, and it is this accounting — up-arrows balancing down-arrows — that welds the three coefficients into fixed ratios. The first figure lays the three processes side by side with the two relations balance forces on them.

Figure

The three radiative processes between two atomic levels. Absorption lifts the atom at rate B₁₂ uν, spontaneous emission drops it at rate A₂₁, and stimulated emission drops it at rate B₂₁ uν while cloning the incident photon. Detailed balance forces g₁B₁₂ = g₂B₂₁ and A₂₁ = 8πhν³ c⁻³ B₂₁

Stimulated emission is the deep surprise: an incoming photon can clone itself off an excited atom, producing a twin identical in frequency, phase, and direction. Einstein realized that balancing only absorption against spontaneous emission gives $u_\nu\propto e^{-h\nu/k_BT}$ — the Wien tail, missing the crucial $-1$ in the Planck denominator that the stimulated term supplies. The second figure makes this quantitative: the ratio of stimulated to spontaneous emission is exactly the photon occupation number $\bar n=1/(e^{h\nu/k_BT}-1)$ , so the two trade dominance at $h\nu=k_BT\ln 2$ . Optical lines live to the right, where spontaneous emission rules; radio lines live to the left, where a single mode holds thousands of photons and stimulated processes take over.

Figure

Stimulated versus spontaneous emission set by the photon occupation number n̄ = 1 divided by exp(hν/kT) − 1. The rates cross at hν/kT = ln 2 ≈ 0.693. The solar Na D line at hν/kT = 4.2 sits in the spontaneous regime with n̄ = 0.015, while the 21 cm line at hν/kT ≈ 7×10⁻⁴ is deep in the stimulated regime with n̄ ≈ 1470

The oscillator strength adds a third face: a "reality factor" for the classical model. A nineteenth-century physicist modelling the bound electron as a little antenna would predict a fixed integrated absorption of $\pi e^2/(m_e c)$ per electron; quantum mechanics says a real transition absorbs $f_{12}$ times that. A strong line such as Lyman- $\alpha$ has $f_{12}\approx0.42$ , close to a full classical oscillator; a forbidden line has $f_{12}\lesssim10^{-6}$ , a whisper of one.

The core equations

The starting point is pure bookkeeping. Absorption depletes the lower level in proportion to the field; the upper level empties by spontaneous decay plus stimulated decay, the latter again proportional to the field.

\frac{\mathrm{d}n_1}{\mathrm{d}t}\bigg|_{\text{abs}} = B_{12}\,n_1\,\bar{u}_\nu,\qquad \frac{\mathrm{d}n_2}{\mathrm{d}t}\bigg|_{\text{down}} = \left(A_{21}+B_{21}\bar{u}_\nu\right)n_2 \qquad (1)

where: $n_1,\,n_2$ — number densities in the lower and upper levels; $A_{21}$ — spontaneous emission coefficient ( $\mathrm{s^{-1}}$ ); $B_{12},\,B_{21}$ — absorption and stimulated-emission coefficients; $\bar u_\nu$ — line-profile-averaged radiation energy density per unit frequency at the transition. Reading: the driven terms scale with the field; the spontaneous term does not. Balancing the up-rate against the down-rate and demanding a Planckian field at every temperature splits into two conditions. The first ties the two $B$ coefficients through the level degeneracies.

g_1 B_{12} = g_2 B_{21} \qquad (2)

where: $g_1,\,g_2$ — statistical weights (degeneracies) of levels 1 and 2. Reading: the shortcut $B_{12}=B_{21}$ holds only when $g_1=g_2$ . The second condition fixes the ratio of spontaneous to stimulated emission — the same $8\pi h\nu^3/c^3$ that sits in front of the Planck function.

A_{21} = \frac{8\pi h\nu^3}{c^3}\,B_{21} \qquad (3)

where: $\nu$ — transition frequency; $h,\,c$ — Planck constant and speed of light. Reading: because $A_{21}/B_{21}\propto\nu^3$ , spontaneous emission overwhelms stimulated emission in the optical and ultraviolet and reverses toward the radio. To connect these rates to a measurable absorption, write the per-atom cross-section as a fixed classical scale times the oscillator strength times the normalized line shape.

\sigma_\nu = \frac{\pi e^2}{m_e c}\,f_{12}\,\phi(\nu) \qquad (4)

where: $\sigma_\nu$ — absorption cross-section per atom ( $\mathrm{cm^2}$ ); $e,\,m_e$ — electron charge and mass (Gaussian units); $f_{12}$ — absorption oscillator strength; $\phi(\nu)$ — normalized profile with $\int\phi\,\mathrm{d}\nu=1$ . Reading: integrated over the line, $\int\sigma_\nu\,\mathrm{d}\nu=(\pi e^2/m_e c)f_{12}$ , so $f_{12}$ counts classical oscillators. Equating the frequency-integrated photon absorption rate to $B_{12}\bar u_\nu$ gives $B_{12}$ in terms of $f_{12}$ . The form below defines $B$ against the energy density $\bar u_\nu$ , matching equation (1); texts using the mean intensity $\bar J_\nu=(c/4\pi)\bar u_\nu$ carry an extra factor $4\pi/c$ .

B_{12} = \frac{\pi e^2}{m_e\,h\nu}\,f_{12} \qquad (5)

where: $f_{12}$ — oscillator strength bridging classical and quantum absorption. Reading: substituting (5) into (2) and (3) reproduces the $A$ – $f$ relation below — a strong internal consistency check. Combining (2), (3), and (5) expresses the spontaneous rate through the line-strength product $g_1 f_{12}$ , the convention-independent quantity a spectroscopist publishes.

g_2\, A_{21} = \frac{8\pi^2 e^2 \nu^2}{m_e c^3}\,g_1 f_{12} \qquad (6)

where: $g_1 f_{12}$ — the $gf$ -value; $\nu$ — transition frequency. Reading: a laboratory $gf$ gives the spontaneous decay rate with no free parameters. Feeding coefficients and populations into radiative transfer gives the line opacity, with stimulated emission entering as negative absorption; the factored form shows the opacity switching sign when the upper level is over-populated.

\alpha_\nu = \frac{h\nu}{4\pi}\,\phi(\nu)\left(n_1 B_{12}-n_2 B_{21}\right) = \frac{h\nu}{4\pi}\,\phi(\nu)\,n_1 B_{12}\left(1-\frac{g_1 n_2}{g_2 n_1}\right) \qquad (7)

where: $\alpha_\nu$ — absorption coefficient per unit length ( $\mathrm{cm^{-1}}$ ). Reading: the bracket is the stimulated-emission correction, which reverses sign at population inversion. Emission is built from $A_{21}$ , and the ratio of emissivity to opacity defines the source function, which in equilibrium becomes Planckian.

j_\nu = \frac{h\nu}{4\pi}\,\phi(\nu)\,n_2 A_{21},\qquad S_\nu=\frac{j_\nu}{\alpha_\nu} \qquad (8)

where: $j_\nu$ — emission coefficient (energy per volume, time, frequency, solid angle); $S_\nu$ — line source function. Reading: $\phi$ and $h\nu/4\pi$ cancel in the ratio, leaving only populations and the Einstein relations. Finally, oscillator strengths from a level to all others obey a conservation law constraining atomic-structure calculations.

\sum_k f_{ik} = N \qquad (9)

where: $f_{ik}$ — oscillator strengths from level $i$ to all levels $k$ (continuum included; negative for downward transitions); $N$ — number of electrons. Reading: the Thomas–Reiche–Kuhn sum rule fixes an atom's total absorbing power — a benchmark every $f$ -value table must respect.

Derivation AThe Einstein relations from detailed balance

The goal is to show that a gas of two-level atoms in equilibrium inside a Planck radiation field cannot have independent $A$ and $B$ coefficients: demanding that the balance of up- and down-transitions reproduce the Planck energy density at every temperature forces both $g_1 B_{12}=g_2 B_{21}$ and $A_{21}=(8\pi h\nu^3/c^3)B_{21}$ . We write the steady-state balance, solve for the field the atoms demand, insert Boltzmann populations, and compare term by term against the known Planck spectrum — the comparison is what pins the coefficients.

Step 1. In statistical equilibrium the up-rate equals the down-rate, so $n_1 B_{12}\,u_\nu = n_2 A_{21} + n_2 B_{21}\,u_\nu$ . Why: equilibrium means no net traffic between the levels — the single assumption the argument rests on.

Step 2. Solve for the energy density the atoms are in balance with, $u_\nu = A_{21}\big/\big[(n_1/n_2)B_{12}-B_{21}\big]$ . Why: isolating $u_\nu$ lets us match it against the known Planck form, turning a statement about atoms into a constraint on coefficients.

Step 3. The populations follow Boltzmann, $n_1/n_2=(g_1/g_2)\,e^{h\nu/k_BT}$ , so $u_\nu = (A_{21}/B_{21})\big/\big[(g_1 B_{12}/g_2 B_{21})\,e^{h\nu/k_BT}-1\big]$ . Why: thermodynamics dictates the level ratio, and substituting it exposes the temperature dependence the comparison exploits.

Step 4. The Planck energy density is $u_\nu=(8\pi h\nu^3/c^3)\big/(e^{h\nu/k_BT}-1)$ , and the two expressions must agree for all $T$ . Matching the exponential in the denominator forces $g_1 B_{12}=g_2 B_{21}$ ; matching the surviving prefactor forces $A_{21}=(8\pi h\nu^3/c^3)B_{21}$ . Why: an identity in constants that must hold at every temperature can only be satisfied term by term.

Step 5. Since $A_{21}$ , $B_{12}$ , $B_{21}$ are intrinsic atomic constants, the relations derived for a blackbody bath hold in any radiation field, thermal or not. Why: equilibrium was only scaffolding — the relations describe the atom itself, so they survive when it is removed. *Note: the prefactor $8\pi h\nu^3/c^3$ is the photon-mode energy density, and dropping $B_{21}$ collapses the result to the Wien form.*

Go deeper · Derivation B · 1 minThe classical integrated cross-section and the oscillator strength

Derivation BThe classical integrated cross-section and the oscillator strength

The aim is to compute the frequency-integrated absorption cross-section of a single classical bound electron — a damped harmonic oscillator driven by the wave's electric field — and show it equals the universal constant $\pi e^2/(m_e c)$ , independent of the damping. The factor by which a real quantum transition departs from this value is, by definition, the oscillator strength $f_{12}$ . We set up the equation of motion, extract the absorbed power, take the sharp-resonance limit, integrate, and graft on the quantum correction.

Step 1. An electron of charge $-e$ bound with natural frequency $\omega_0$ and radiative damping $\gamma$ obeys $\ddot{x}+\gamma\dot{x}+\omega_0^2 x=-(e/m_e)E_0 e^{-i\omega t}$ . Why: this Lorentz oscillator is the simplest object that both stores energy at a resonance and dissipates it — the two ingredients any absorber needs.

Step 2. The steady-state amplitude gives a time-averaged absorbed power that, divided by the incident flux, yields $\sigma(\omega)=(4\pi e^2/m_e c)\,\gamma\omega^2\big/\big[(\omega_0^2-\omega^2)^2+\gamma^2\omega^2\big]$ . Why: absorption is energy fed to the damping term, and dividing by flux turns power into area.

Step 3. Near resonance ( $\omega\approx\omega_0$ , $\gamma\ll\omega_0$ ) this becomes a Lorentzian, $\sigma(\nu)=(\pi e^2/m_e c)\,(\gamma/4\pi^2)\big/\big[(\nu-\nu_0)^2+(\gamma/4\pi)^2\big]$ . Why: real lines are sharp, so the wings contribute negligibly and the response collapses to a clean peaked profile.

Step 4. Integrating over frequency, the Lorentzian contributes unity and $\int_0^\infty\sigma(\nu)\,\mathrm{d}\nu=\pi e^2/(m_e c)$ — independent of $\gamma$ . Why: the total absorbing strength is a conserved area set only by the electron's charge and mass; damping sets the line's width, not its area.

Step 5. Quantum mechanics keeps this form but scales it by $f_{12}$ : $\int\sigma_\nu\,\mathrm{d}\nu=(\pi e^2/m_e c)\,f_{12}$ , so $f_{12}$ counts how many classical oscillators the real transition is worth. Why: this defines the oscillator strength operationally, packaging the transition dipole into one tabulated number. *Note: $\pi e^2/(m_e c)=0.0265\ \mathrm{cm^2\,Hz}$ in Gaussian units; the integrated area is independent of $\gamma$ , and the sum rule $\sum_k f_{ik}=N$ follows from the same commutator.*

Go deeper · Derivation C · 1 minWhy the line source function reduces to the Planck function

Derivation CWhy the line source function reduces to the Planck function

The purpose is to show that the source function $S_\nu=j_\nu/\alpha_\nu$ , assembled from the Einstein coefficients, collapses to the Planck function $B_\nu(T)$ the moment the level populations take their Boltzmann values — recovering Kirchhoff's law directly from atomic bookkeeping. We form the ratio, cancel shared factors, insert the Einstein relations, and impose Boltzmann populations, watching the atomic constants assemble into the thermal spectrum.

Step 1. From equations (7) and (8), $S_\nu=n_2 A_{21}\big/(n_1 B_{12}-n_2 B_{21})$ ; the profile $\phi(\nu)$ and $h\nu/4\pi$ appear in both emissivity and opacity and cancel. Why: the source function is a ratio, so everything common to emission and absorption drops out.

Step 2. Divide by $n_2 B_{21}$ and insert the Einstein relations: $B_{12}/B_{21}=g_2/g_1$ and, in the transfer (mean-intensity) convention, $A_{21}/B_{21}=2h\nu^3/c^2$ . This gives $S_\nu=(2h\nu^3/c^2)\big/\big[(g_2 n_1)/(g_1 n_2)-1\big]$ . Why: the relations replace the raw coefficients with the Planck prefactor $2h\nu^3/c^2$ , leaving the population ratio as the only variable.

Step 3. Impose Boltzmann populations, $n_1/n_2=(g_1/g_2)e^{h\nu/k_BT}$ , so $(g_2 n_1)/(g_1 n_2)=e^{h\nu/k_BT}$ and $S_\nu=(2h\nu^3/c^2)/(e^{h\nu/k_BT}-1)=B_\nu(T)$ . Why: this is Kirchhoff's law — in LTE a good absorber is an equally good emitter, and $S_\nu$ is the blackbody function at the local temperature.

Step 4. When the populations are not Boltzmann — a pumped maser, a non-LTE atmosphere — $S_\nu$ departs from $B_\nu(T)$ and can diverge or go negative, the formal signature of amplification. Why: the same algebra that gives Kirchhoff's law in LTE quantifies how line formation breaks away from it. *Note: the result reduces to $B_\nu(T)$ in LTE; the $2h\nu^3/c^2$ prefactor is the intensity Planck function, related to the energy-density prefactor by $c/4\pi$ .*

Limiting cases

The three processes trade dominance according to frequency, temperature, and the level populations:

Case	Result	Meaning
High frequency (Wien)	$h\nu\gg k_BT\;\Rightarrow\;A_{21}\gg B_{21}\bar u_\nu$	Spontaneous emission dominates; stimulated emission is negligible. The regime of optical and ultraviolet lines in stellar photospheres.
Low frequency (Rayleigh–Jeans)	$h\nu\ll k_BT\;\Rightarrow\;B_{21}\bar u_\nu\gg A_{21}$	Stimulated processes dominate; many photons per mode. The regime of radio recombination lines and the 21 cm line.
Population inversion	$\dfrac{n_2}{g_2}>\dfrac{n_1}{g_1}\;\Rightarrow\;\alpha_\nu<0$	Net negative absorption: stimulated emission outpaces absorption and radiation is amplified — the maser/laser condition, impossible in LTE.
Allowed versus forbidden	$f_{12}\sim1$ versus $f_{12}\lesssim10^{-6}$	Electric-dipole-allowed lines carry $f\sim0.1$ – $1$ ; forbidden lines carry tiny $f$ yet shine in low-density nebulae where collisions cannot quench them.

Common trapStimulated emission is a small correction you can drop

It is not, and dropping it corrupts both the opacity and the source function. Stimulated emission enters equation (7) as negative absorption, the factor $(1-g_1 n_2/g_2 n_1)$ ; omit it and you overestimate the opacity, most severely at low frequency where $\bar n$ is large — the 21 cm line's net absorption is entirely a small residual between absorption and stimulated emission, so neglecting the latter is a sign-level blunder, not a rounding error. Two companion traps: (i) writing $B_{12}=B_{21}$ is valid only when $g_1=g_2$ ; and (ii) the $B$ coefficients depend on whether the rate is defined against energy density $\bar u_\nu$ or mean intensity $\bar J_\nu=(c/4\pi)\bar u_\nu$ — mixing conventions inserts a spurious $4\pi/c$ . The convention-free anchor is always the $gf$ -value in equation (6).

Worked examples

1 · The Einstein A coefficient of Lyman-alpha (fully worked)

Hydrogen Lyman- $\alpha$ is the $1s\to2p$ transition at $\lambda=1215.67\ \text{Å}$ , with $f_{12}=0.4162$ , $g_1=2$ ( $1s$ ), $g_2=6$ ( $2p$ ). Find the spontaneous decay rate $A_{21}$ and the radiative lifetime of the $2p$ state.

Step 1. Convert to frequency: $\nu=c/\lambda=(2.998\times10^{10})/(1.21567\times10^{-5})=\mathbf{2.466\times10^{15}\ Hz}$ , a vacuum-ultraviolet line deep in the Wien regime.

Step 2. The convention-free constant in equation (6), $8\pi^2 e^2/(m_e c^3)$ , with $e=4.803\times10^{-10}\ \mathrm{esu}$ , $m_e=9.109\times10^{-28}\ \mathrm{g}$ , $c=2.998\times10^{10}\ \mathrm{cm\,s^{-1}}$ , evaluates to $7.42\times10^{-22}$ in cgs, so $g_2 A_{21}=7.42\times10^{-22}\,\nu^2\,g_1 f_{12}$ .

Step 3. Insert $\nu^2=6.08\times10^{30}\ \mathrm{Hz^2}$ and $g_1 f_{12}=2\times0.4162=0.8324$ : $g_2 A_{21}=7.42\times10^{-22}\times6.08\times10^{30}\times0.8324=3.76\times10^{9}\ \mathrm{s^{-1}}$ .

Step 4. Divide by $g_2=6$ : $A_{21}=3.76\times10^{9}/6=\mathbf{6.27\times10^{8}\ s^{-1}}$ , matching the NIST value $6.265\times10^{8}\ \mathrm{s^{-1}}$ . The $2p$ radiative lifetime is $\tau=1/A_{21}=\mathbf{1.60\ ns}$ — one of the strongest lines in astrophysics, which is why Lyman- $\alpha$ dominates the ultraviolet spectra of galaxies and the intergalactic medium.

2 · Spontaneous versus stimulated emission: an optical line and a radio line

Compare stimulated emission for two thermal lines: the solar sodium D line at $\lambda=589\ \mathrm{nm}$ ( $T=5800\ \mathrm{K}$ ), and the neutral-hydrogen 21 cm line at $\nu=1.42\ \mathrm{GHz}$ in a cloud at spin temperature $T=100\ \mathrm{K}$ . Use the occupation number $\bar n=B_{21}\bar u_\nu/A_{21}=1/(e^{h\nu/k_BT}-1)$ .

Step 1. Na D: $\nu=c/\lambda=5.09\times10^{14}\ \mathrm{Hz}$ , so $h\nu/k_BT=(6.626\times10^{-27}\times5.09\times10^{14})/(1.381\times10^{-16}\times5800)=\mathbf{4.21}$ . Then $\bar n=1/(e^{4.21}-1)=\mathbf{0.015}$ : about 66 spontaneous emissions per stimulated one. The opacity correction $1-e^{-h\nu/k_BT}=0.985$ is a mere $1.5\%$ nudge.

Step 2. 21 cm: $h\nu/k_BT=(6.626\times10^{-27}\times1.42\times10^{9})/(1.381\times10^{-16}\times100)=\mathbf{6.8\times10^{-4}}$ , deep in Rayleigh–Jeans. Then $\bar n=1/(e^{6.8\times10^{-4}}-1)\approx\mathbf{1470}$ : stimulated outnumbers spontaneous by more than a thousand to one.

Step 3. The 21 cm stimulated correction is $1-e^{-h\nu/k_BT}\approx h\nu/k_BT=6.8\times10^{-4}$ : the net absorption is a tiny residual between two nearly equal giant terms. This is why the 21 cm line is optically thin and its opacity scales as $1/T_{\rm spin}$ , the cornerstone of Galactic HI surveys. The same algebra that makes stimulated emission a $1.5\%$ afterthought in the optical makes it the dominant actor in the radio.

Physics of the portrait

The whole edifice traces to a single short paper written before quantum mechanics existed — thermodynamic reasoning so sturdy that the full quantum theory, when it arrived, could only confirm it.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · the 1917 quantum theory of radiation

In 1917, a decade after the light-quantum and eight years before matrix mechanics, Einstein published "Zur Quantentheorie der Strahlung" and rederived the Planck law from a startlingly simple premise: atoms with discrete energy levels exchanging quanta with a radiation field. To make the bookkeeping balance he was forced to invent a third process no one had considered — stimulated emission — because absorption and spontaneous emission alone yielded the Wien law, not Planck's. From detailed balance he extracted $g_1 B_{12}=g_2 B_{21}$ and $A_{21}=(8\pi h\nu^3/c^3)B_{21}$ without ever computing a matrix element, and in the same paper noted that each emitted quantum carries momentum $h\nu/c$ — the seed of radiation pressure and, eventually, laser cooling. The argument's power is that it is thermodynamic: it constrains the atomic constants without knowing what the atom is. Dirac's 1927 quantum electrodynamics later computed $A_{21}$ from first principles and confirmed Einstein's ratios exactly, while the maser (1954) and laser (1960) turned the once-abstract stimulated term into the defining technology of the photonic age.

History

1900

— Planck's radiation law introduces the quantum of action

1913

— Bohr's model fixes discrete atomic energy levels

1917

— Einstein derives the A and B coefficients from detailed balance

1925

— Thomas, Reiche and Kuhn state the oscillator-strength sum rule

1927

— Dirac's quantum electrodynamics computes A₂₁ from first principles

1965

— the first astrophysical OH maser is detected, confirming stimulated emission in space

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 9 (Stellar Atmospheres) · Rybicki and Lightman, Radiative Processes in Astrophysics, ch. 1 and ch. 10 · Mihalas, Stellar Atmospheres, 2nd ed., ch. 4 · Dodelson, Modern Cosmology, 2nd ed. (radiation–matter coupling context) · Mo, van den Bosch and White, Galaxy Formation and Evolution, ch. B (radiative processes appendix) · Binney and Tremaine, Galactic Dynamics, 2nd ed. (line-emission diagnostics) · Einstein, A. (1917), Zur Quantentheorie der Strahlung, Phys. Z. 18, 121. Key checks: (1) $g_1 B_{12}=g_2 B_{21}$ and $A_{21}=(8\pi h\nu^3/c^3)B_{21}$ follow from matching the balance solution to $u_\nu=(8\pi h\nu^3/c^3)/(e^{h\nu/k_BT}-1)$ ; (2) the classical integrated cross-section is $\pi e^2/(m_e c)=0.0265\ \mathrm{cm^2\,Hz}$ (Gaussian units); (3) the $A$ – $f$ relation (6) reproduces the NIST Lyman- $\alpha$ value $A_{21}=6.27\times10^{8}\ \mathrm{s^{-1}}$ from $f_{12}=0.4162$ , $g_1=2$ , $g_2=6$ , $\lambda=1215.67\ \text{Å}$ ; (4) the source function reduces to $B_\nu(T)$ in LTE. The $B_{12}$ form in equation (5) uses the energy-density convention consistent with equation (1); the mean-intensity convention differs by $4\pi/c$ . All formulae are in Gaussian (cgs) units as standard in stellar-atmosphere texts.

Prerequisites. Blackbody Radiation and the Planck Law · The Boltzmann Excitation Equation · Specific Intensity and Radiative Flux. Leads to. Opacity and Absorption Coefficients · Emission and the Source Function · Spectral Line Formation · Local Thermodynamic Equilibrium.

Ask the tutor: Ask me to walk through the detailed-balance derivation line by line, or to compute the $A$ coefficient and lifetime of any transition from its $gf$ -value.

Why is the photon from stimulated emission "coherent" with the incident one?

Because stimulated emission is the same quantum-electrodynamic matrix element as absorption, run in reverse and driven by an already-populated mode. The emitted photon is created in that same mode, so it shares the incident photon's frequency, phase, direction, and polarization — the field simply gains one more quantum in an existing state. This is the root of laser and maser light: an avalanche of stimulated emissions pouring into one mode makes an intense, phase-coherent beam. Spontaneous emission seeds a random empty mode, so its photons fly off with random direction and phase — the difference between a laser pointer and a light bulb.

Are the Einstein relations only valid in thermal equilibrium?

No — that is the elegant twist. The relations are derived using a blackbody bath as scaffolding, but the coefficients $A_{21}$ , $B_{12}$ , $B_{21}$ are intrinsic constants of the atom, fixed by its wavefunctions. Once shown in equilibrium, those same constants describe the atom everywhere: a stellar wind, a maser, a laboratory discharge, the early universe. Detailed balance is a tool for finding the relations, not a condition for their validity. What changes out of equilibrium is the populations $n_1,n_2$ , not the coefficients — which is why non-LTE modelling solves for populations while holding the Einstein coefficients fixed.

What is the difference between the f-value and the gf-value, and which should I use?

The oscillator strength $f_{12}$ belongs to a transition; the $gf$ -value multiplies it by the lower level's statistical weight, $g_1 f_{12}$ . Databases tabulate the $gf$ -value because it is the combination appearing in the line opacity and the spontaneous rate (equation 6) with no ambiguity about splitting degeneracies. When computing a line's strength or comparing to a catalogue, use $g_1 f_{12}$ directly and you sidestep bookkeeping errors. The bare $f_{12}$ is more natural for a single sublevel or the Thomas–Reiche–Kuhn sum rule, where oscillator strengths from a level add up to the electron count.

Why do forbidden lines appear in nebulae but essentially never in the laboratory or in dense stars?

Forbidden lines come from magnetic-dipole or electric-quadrupole transitions with minuscule oscillator strengths, hence tiny $A_{21}$ and long upper-level lifetimes — seconds to hours, versus nanoseconds for an allowed line. In dense gas an atom in such a metastable level is knocked out by a collision long before it can radiate, so the line is collisionally quenched. Only at very low density — a planetary nebula, an HII region, the diffuse interstellar medium — does the atom survive long enough to emit. That is why [O III] $\lambda5007$ blazes in a nebula but never in a laboratory cell or a stellar photosphere: those environments are too crowded.

How does the oscillator strength connect to the transition dipole matrix element?

Quantum mechanically, $f_{12}=(2m_e\omega_{21}/3\hbar g_1)\,|\langle 2\|\,\mathbf{d}\,\|1\rangle|^2/e^2$ — up to constants, the squared transition dipole matrix element weighted by frequency and summed over sublevels. A strongly allowed electric-dipole transition has a large dipole and $f\sim1$ ; a forbidden transition has a vanishing electric-dipole term (zero by symmetry) and survives only through weaker magnetic-dipole or quadrupole terms, giving $f\lesssim10^{-6}$ . So the oscillator strength is the transition dipole in disguise, repackaged as a dimensionless number that plugs into the classical cross-section $\pi e^2/(m_e c)$ .

The B coefficient seems to have different values in different books — why?

Because there are two conventions for the driving field. Some texts (and this article's equation 1) define the rate against the energy density $\bar u_\nu$ , giving $B_{12}=\pi e^2/(m_e h\nu)\,f_{12}$ ; others define it against the mean intensity $\bar J_\nu=(c/4\pi)\bar u_\nu$ , absorbing a factor $4\pi/c$ into $B$ . Both are correct — they describe the rate per unit of different quantities. Check the field variable before importing a numerical $B$ , and better still work with $A_{21}$ or the $gf$ -value, which are convention-independent. Equation (6) is the anchor precisely because it involves no $B$ at all.