Einstein Coefficients and Spontaneous Emission

Derivation

Einstein Coefficients and Spontaneous Emission

D-309 Home PU-306 Threads light · energy · chance · waves Depends on Fermi's Golden Rule, planck-radiation-law, Electric-Dipole Selection Rules

Statement

For a pair of non-degenerate atomic levels \(1\) (lower) and \(2\) (upper) separated by \(E_2-E_1=h\nu=\hbar\omega\), the three radiative processes are governed by the Einstein coefficients \(A_{21}\) (spontaneous emission), \(B_{21}\) (stimulated emission) and \(B_{12}\) (absorption). Detailed balance against the Planck spectrum forces the coefficients to satisfy \(g_1 B_{12}=g_2 B_{21}\) and \(A_{21}/B_{21}=8\pi h\nu^3/c^3\); combining this with the stimulated rate \(B_{21}\) obtained from Fermi's golden rule in the electric-dipole approximation yields the spontaneous-emission rate \(\displaystyle A_{21}=\frac{\omega^3\,|\mathbf{d}_{21}|^2}{3\pi\varepsilon_0\hbar c^3}\).

Why it matters

Einstein's 1917 argument is the earliest place quantum jumps, thermal radiation and probability are welded together: without ever solving the atom, it proves that spontaneous emission must exist and fixes its rate relative to the stimulated processes. That single ratio is the seed of the laser (population inversion beats spontaneous loss), of radiative lifetimes and natural linewidths, and of the physics of every emission line in astrophysics.

The result also demonstrates a deep structural point: \(A_{21}\), \(B_{12}\), \(B_{21}\) are intrinsic atomic constants, independent of temperature or the radiation field. Thermal equilibrium is merely the scaffolding used to relate them; once related, the relations hold in any field, including the vacuum.

Assumptions

Two-level system.If dropped, other levels feed and drain populations, so the steady-state bookkeeping needs the full rate matrix; the pairwise relations between \(A_{21},B_{12},B_{21}\) still hold, but the equilibrium populations no longer follow the simple two-level ratio.

Coefficients are intrinsic atomic constants, independent of the field and temperature.This is the crux of the argument: it lets relations forced in thermal equilibrium be exported to arbitrary fields. If \(A,B\) depended on \(T\), matching to Planck at every temperature would be impossible and the relations would be meaningless.

Isotropic, unpolarised, broadband (incoherent) radiation.The golden-rule rate is proportional to spectral energy density only for broadband light; for a coherent monochromatic drive the atom undergoes Rabi oscillations and the constant-rate description fails. Isotropy also supplies the orientational average \(\langle|\hat{\boldsymbol\varepsilon}\cdot\mathbf{d}|^2\rangle=|\mathbf{d}|^2/3\).

Electric-dipole approximation, \(\lambda\gg a_0\).If dropped (forbidden transitions, X-ray wavelengths, tight confinement) the leading term vanishes or higher multipoles (M1, E2) dominate, changing the numerical coefficient and selection rules, though the \(A/B\) relation from detailed balance is untouched.

Weak field / first-order perturbation theory.If dropped the transition saturates, the two populations equalise, and stimulated emission and absorption cancel; the linear rate \(B\rho\) is no longer valid.

Derivation

\[ \dot N_2\big|_{\text{gain}} = N_1 B_{12}\,\rho(\nu), \qquad \dot N_2\big|_{\text{loss}} = N_2\big(A_{21}+B_{21}\rho(\nu)\big) \]

Postulate the three processes: absorption \(\propto\rho\), stimulated emission \(\propto\rho\), spontaneous emission field-independent. \(\rho(\nu)\) is the spectral energy density. A

\[ N_1 B_{12}\,\rho(\nu) = N_2 A_{21} + N_2 B_{21}\,\rho(\nu) \]

Impose steady state \(\dot N_2=0\) in thermal equilibrium: gain balances loss for each level pair (detailed balance). A

\[ \rho(\nu) = \frac{A_{21}/B_{21}}{\dfrac{N_1}{N_2}\dfrac{B_{12}}{B_{21}} - 1} \]

Solve algebraically for the field the atoms must be sitting in. Divide through by \(N_2 B_{21}\rho\) and rearrange. B

\[ \frac{N_1}{N_2} = \frac{g_1}{g_2}\,e^{+h\nu/k_BT} \qquad\Longrightarrow\qquad \rho(\nu)=\frac{A_{21}/B_{21}}{\dfrac{g_1 B_{12}}{g_2 B_{21}}\,e^{h\nu/k_BT}-1} \]

Insert Boltzmann populations for the equilibrium ratio; \(g_i\) are the degeneracies. A

\[ \rho(\nu)=\frac{8\pi h\nu^3}{c^3}\,\frac{1}{e^{h\nu/k_BT}-1} \quad\Rightarrow\quad \boxed{\;g_1 B_{12}=g_2 B_{21}\;},\qquad \frac{A_{21}}{B_{21}}=\frac{8\pi h\nu^3}{c^3} \]

Demand agreement with the Planck law at every \(T\). The exponential prefactor must be unity (fixing \(g_1B_{12}=g_2B_{21}\)) and the \(T\)-independent prefactor must match (fixing the \(A/B\) ratio). Because \(A,B\) are field-independent constants, these relations now hold universally. B

\[ W_{1\to2} = \frac{2\pi}{\hbar}\,\big|\langle 2|{-}\hat{\mathbf d}\cdot\mathbf E|1\rangle\big|^2 \;\xrightarrow{\;\text{broadband, }u=\tfrac12\varepsilon_0 E_0^2\;}\; \frac{\pi}{3\varepsilon_0\hbar^2}\,|\mathbf d_{21}|^2\,\rho(\omega) \]

Fermi's golden rule with the dipole interaction \(H'=-\hat{\mathbf d}\cdot\mathbf E\). Summing the incoherent spectral components converts \(|\mathbf E|^2\) to the energy density \(\rho(\omega)\); the isotropic average \(\langle|\hat{\boldsymbol\varepsilon}\cdot\mathbf d_{21}|^2\rangle=|\mathbf d_{21}|^2/3\) supplies the factor \(1/3\). This identifies \(B_{21}^{(\omega)}=\pi|\mathbf d_{21}|^2/3\varepsilon_0\hbar^2\). C

\[ \rho(\nu)\,d\nu=\rho(\omega)\,d\omega,\;\; \omega=2\pi\nu \;\Rightarrow\; \rho(\nu)=2\pi\,\rho(\omega) \;\Rightarrow\; B_{21}\equiv B_{21}^{(\nu)}=\frac{B_{21}^{(\omega)}}{2\pi}=\frac{2\pi^2}{3\varepsilon_0 h^2}\,|\mathbf d_{21}|^2 \]

Convert the golden-rule coefficient (defined per unit angular frequency) to the energy-density-per-\(\nu\) convention used in Step 5, using \(W=B^{(\omega)}\rho(\omega)=B^{(\nu)}\rho(\nu)\) and \(\hbar=h/2\pi\). B

\[ A_{21}=\frac{8\pi h\nu^3}{c^3}\,B_{21}=\frac{8\pi h\nu^3}{c^3}\cdot\frac{2\pi^2|\mathbf d_{21}|^2}{3\varepsilon_0 h^2}=\frac{16\pi^3\nu^3|\mathbf d_{21}|^2}{3\varepsilon_0 h c^3}=\frac{\omega^3|\mathbf d_{21}|^2}{3\pi\varepsilon_0\hbar c^3} \]

Multiply the golden-rule \(B_{21}\) by the detailed-balance ratio from Step 5, then re-express with \(\omega=2\pi\nu\), \(\hbar=h/2\pi\). B

Result

\[ g_1 B_{12}=g_2 B_{21},\qquad \frac{A_{21}}{B_{21}}=\frac{8\pi h\nu^3}{c^3},\qquad A_{21}=\frac{\omega^3\,|\mathbf d_{21}|^2}{3\pi\varepsilon_0\hbar c^3} \]

Reading. Absorption and stimulated emission are the same intrinsic strength (up to degeneracy), and the "extra" process Einstein was forced to add — spontaneous emission — is not free: its rate is locked to the stimulated rate by \(8\pi h\nu^3/c^3\), the density of electromagnetic modes times \(h\nu\). The last equality shows that this rate is nothing but the transition dipole squared, driven by the \(\omega^3\) mode density; a strong dipole and a high frequency both make the excited state short-lived. Spontaneous emission is stimulated emission driven by the vacuum's zero-point modes.

Units check. \(A_{21}\): \([\omega^3]=\mathrm{s^{-3}}\), \([|\mathbf d|^2]=\mathrm{C^2m^2}\); denominator \([\varepsilon_0\hbar c^3]=(\mathrm{C^2J^{-1}m^{-1}})(\mathrm{J\,s})(\mathrm{m^3 s^{-3}})=\mathrm{C^2 m^2 s^{-2}}\). Hence \([A_{21}]=\mathrm{s^{-3}C^2m^2/(C^2m^2s^{-2})}=\mathrm{s^{-1}}\), a rate. And \([8\pi h\nu^3/c^3]=(\mathrm{J\,s})(\mathrm{s^{-3}})/(\mathrm{m^3 s^{-3}})=\mathrm{J\,s\,m^{-3}}\), exactly the units of spectral energy density \(\rho(\nu)\), so \(A/B\) is dimensionally consistent.

Limiting cases

High frequency / optical (\(h\nu\gg k_BT\)): \(\rho\) is exponentially small, so \(B_{21}\rho\ll A_{21}\) — spontaneous emission dominates. This is why room-temperature optical sources are incoherent and why lasers need external pumping.
Low frequency / microwave (\(h\nu\ll k_BT\)): the photon occupation \(\bar n=(e^{h\nu/k_BT}-1)^{-1}\to k_BT/h\nu\gg1\); stimulated processes swamp spontaneous emission, the Rayleigh–Jeans regime, and masers become natural.
Equal populations, \(g_1B_{12}=g_2B_{21}\): when \(N_1/g_1=N_2/g_2\) stimulated emission exactly cancels absorption — the medium is transparent (saturation), the boundary between absorber and amplifier.
Degenerate levels: setting \(g_1=g_2\) recovers \(B_{12}=B_{21}\), Einstein's original symmetric statement of absorption/emission reciprocity.

Breaks when

Coherent monochromatic driving (laser field, resonant pulse): the radiation is not broadband, so the golden-rule rate equation is invalid; the atom Rabi-oscillates coherently between \(|1\rangle\) and \(|2\rangle\) and populations are not described by constant \(B\rho\) rates.
Strong fields / saturation: first-order perturbation theory breaks; the upper population cannot exceed the lower (for \(g_1=g_2\)), the linear-in-\(\rho\) rates fail, and power broadening sets in.
Dipole-forbidden transitions: when \(\mathbf d_{21}=0\) by selection rules, the electric-dipole \(A_{21}\) formula gives zero; the real (much slower) decay proceeds by magnetic-dipole or electric-quadrupole terms not captured here.
Non-equilibrium or non-thermal spectra during derivation: the detailed-balance step assumes a Planck field at a well-defined \(T\); it cannot be run with an arbitrary \(\rho(\nu)\) — though, crucially, the relations it extracts then apply to any field.

Failure modes

Dropping the \(A\) term. Balancing only absorption against stimulated emission (\(N_1B_{12}\rho=N_2B_{21}\rho\)) forces \(N_1/N_2=\)const, contradicting Boltzmann. The whole point is that spontaneous emission is required for consistency.
Convention confusion \(\rho(\nu)\) vs \(\rho(\omega)\). Forgetting the factor \(2\pi\) between \(B^{(\nu)}\) and \(B^{(\omega)}\) (Step 7) gives a spurious \(2\pi\) in \(A_{21}\). Always state which spectral variable \(\rho\) is per.
Omitting the \(1/3\) orientational average. Using \(|\mathbf d_{21}|^2\) instead of \(|\mathbf d_{21}|^2/3\) triples the stimulated rate; the \(1/3\) comes from averaging \(|\hat{\boldsymbol\varepsilon}\cdot\mathbf d|^2\) over isotropic radiation.
Degeneracy sign/placement errors. Writing \(g_2B_{12}=g_1B_{21}\) (indices swapped) inverts the reciprocity; the correct statement is \(g_1B_{12}=g_2B_{21}\) (lower-level weight on the absorption coefficient).
Treating \(\bar n\) as the field intensity. Confusing photon occupation number \(\bar n\) (dimensionless) with \(\rho(\nu)\) (energy density) when computing stimulated-to-spontaneous ratios.

Discussion

The most striking feature of the argument is its economy: Einstein never solved the Schrödinger equation — quantum mechanics did not yet exist — yet by insisting that a gas of atoms and a Planck radiation field coexist in equilibrium he deduced a process, spontaneous emission, that classical electrodynamics has no room for. The ratio \(A_{21}/B_{21}=8\pi h\nu^3/c^3\) is exactly \(h\nu\) times the electromagnetic mode density \(8\pi\nu^2/c^3\). Read forward, this says spontaneous emission is stimulated emission driven by one photon per mode — the vacuum fluctuations. This is the earliest fingerprint of quantum electrodynamics, later made rigorous by quantising the field.

Structurally, detailed balance does only half the work: it ties \(A\) to \(B\) and fixes the degeneracy weighting, but it cannot give the absolute scale of either coefficient. That absolute scale is set by the atom's internal structure through the transition dipole \(\mathbf d_{21}\), which is where Fermi's golden rule and the dipole selection rules enter. The two ingredients are complementary: thermodynamics fixes ratios, quantum dynamics fixes magnitudes.

The result underlies the laser threshold condition. Because \(A_{21}\) scales as \(\nu^3\), maintaining a population inversion against spontaneous loss becomes progressively harder at higher frequency — the reason X-ray lasers are so much more difficult than microwave masers. The same \(\nu^3\) makes ultraviolet atomic transitions have nanosecond lifetimes while forbidden or low-frequency transitions can live for seconds to hours.

A subtlety worth internalising: \(A_{21}\) here is the total spontaneous rate summed over all photon directions and polarisations, which is why the mode-density factor \(8\pi\nu^2/c^3\) (both polarisations, all angles) appears. In a cavity or structured medium the local mode density is altered, and the spontaneous rate changes accordingly — the Purcell effect. Thus \(A_{21}\) is not, strictly, an immutable property of the atom alone but of the atom plus its electromagnetic environment; free space is simply the default environment. The detailed-balance relation \(A/B=(\text{mode density})\times h\nu\) survives, but with the local mode density substituted.

Common misconceptions. (i) Stimulated and spontaneous emission are not "two kinds of light" — the emitted photon is identical; stimulated emission is coherent with the driving field, spontaneous is random-phase. (ii) \(B_{12}=B_{21}\) only for equal degeneracies; the general statement carries the \(g\)'s. (iii) Spontaneous emission is not a classical instability of the excited state; it is fundamentally a quantum, vacuum-driven process.

Worked examples

\[ A_{21}=\frac{\omega^3\,|\mathbf d_{21}|^2}{3\pi\varepsilon_0\hbar c^3},\qquad \tau=\frac{1}{A_{21}} \]

Radiative lifetime of an allowed optical transition. Take \(\lambda=500\ \mathrm{nm}\) (\(\nu=6.0\times10^{14}\ \mathrm{Hz}\), \(\omega=3.77\times10^{15}\ \mathrm{s^{-1}}\)) and a dipole of order \(|\mathbf d_{21}|=ea_0=8.46\times10^{-30}\ \mathrm{C\,m}\). Symbols first, then numbers. B

\[ \omega^3=(3.77\times10^{15})^3=5.36\times10^{46}\ \mathrm{s^{-3}},\quad |\mathbf d_{21}|^2=7.16\times10^{-59}\ \mathrm{C^2m^2} \]

Evaluate numerator pieces. A

\[ 3\pi\varepsilon_0\hbar c^3=9.42\times(8.854\times10^{-12})(1.055\times10^{-34})(2.70\times10^{25})=2.37\times10^{-19} \]

Evaluate denominator (SI units \(\mathrm{C^2m^2s^{-2}}\)). A

\[ A_{21}=\frac{(5.36\times10^{46})(7.16\times10^{-59})}{2.37\times10^{-19}}=1.6\times10^{7}\ \mathrm{s^{-1}},\quad \tau\approx 62\ \mathrm{ns} \]

Reading. An order-unity dipole at optical frequency gives a lifetime of tens of nanoseconds — precisely the observed scale for strong allowed lines (e.g. sodium D at 16 ns). Units: \(\mathrm{s^{-1}}\), confirmed above.

\[ \frac{\text{stimulated}}{\text{spontaneous}}=\frac{B_{21}\rho(\nu)}{A_{21}}=\frac{1}{e^{h\nu/k_BT}-1}=\bar n \]

Ratio of stimulated to spontaneous emission in a blackbody equals the photon occupation number. Using \(A/B=8\pi h\nu^3/c^3\) and the Planck \(\rho(\nu)\), the prefactors cancel. Compare optical vs microwave at \(T=300\ \mathrm{K}\). B

\[ \text{optical: } \frac{h\nu}{k_BT}=\frac{(6.626\times10^{-34})(6.0\times10^{14})}{(1.381\times10^{-23})(300)}=96 \;\Rightarrow\; \bar n=e^{-96}\approx2\times10^{-42} \]

Evaluate the exponent, then \(\bar n\). A

\[ \text{microwave (10 GHz): } \frac{h\nu}{k_BT}=\frac{(6.626\times10^{-34})(1.0\times10^{10})}{(1.381\times10^{-23})(300)}=1.6\times10^{-3}\;\Rightarrow\;\bar n\approx\frac{1}{1.6\times10^{-3}}\approx6.3\times10^{2} \]

Same formula, low-frequency regime; expand \(e^x-1\approx x\). A

\[ \bar n_{\text{opt}}\sim10^{-42},\qquad \bar n_{\mu\text{w}}\sim6\times10^{2} \]

Reading. At room temperature optical spontaneous emission utterly dominates (hence thermal light is incoherent and lasers need pumping), while at microwave frequencies stimulated emission outweighs spontaneous by hundreds to one — the physical reason masers preceded lasers. Units: \(\bar n\) is dimensionless.

Problems

The hydrogen Lyman-\(\alpha\) transition (\(2p\to1s\)) has \(A_{21}=6.27\times10^{8}\ \mathrm{s^{-1}}\). Find the radiative lifetime of the \(2p\) state.
Solution
\(\tau=1/A_{21}=1/(6.27\times10^{8}\ \mathrm{s^{-1}})=1.60\times10^{-9}\ \mathrm{s}=1.6\ \mathrm{ns}\). This matches the measured natural lifetime and gives a natural linewidth \(\Delta\nu=1/(2\pi\tau)\approx99\ \mathrm{MHz}\).
Two levels have degeneracies \(g_1=1\), \(g_2=3\). If the absorption coefficient is \(B_{12}=4.2\times10^{20}\ \mathrm{m^3 J^{-1}s^{-2}}\), find \(B_{21}\).
Solution
Detailed balance gives \(g_1B_{12}=g_2B_{21}\), so \(B_{21}=(g_1/g_2)B_{12}=(1/3)(4.2\times10^{20})=1.4\times10^{20}\ \mathrm{m^3 J^{-1}s^{-2}}\). Stimulated emission per atom is weaker than absorption per atom because the upper level is threefold degenerate.
Compute the ratio \(A_{21}/B_{21}=8\pi h\nu^3/c^3\) for \(\nu=5.0\times10^{14}\ \mathrm{Hz}\), and state its units.
Solution
\(\nu^3=1.25\times10^{44}\ \mathrm{s^{-3}}\). \(8\pi h=8\pi(6.626\times10^{-34})=1.665\times10^{-32}\). Numerator \(=1.665\times10^{-32}\times1.25\times10^{44}=2.08\times10^{12}\). Divide by \(c^3=2.695\times10^{25}\ \mathrm{m^3 s^{-3}}\): \(A/B=7.7\times10^{-13}\ \mathrm{J\,s\,m^{-3}}\). The units are those of spectral energy density per unit frequency, as required so that \(B\times(A/B)\) has units of \(A\), namely \(\mathrm{s^{-1}}\).
At what temperature does stimulated emission equal spontaneous emission (\(\bar n=1\)) for a \(1\ \mathrm{THz}\) transition?
Solution
\(\bar n=1\Rightarrow e^{h\nu/k_BT}-1=1\Rightarrow h\nu/k_BT=\ln2\). Thus \(T=\dfrac{h\nu}{k_B\ln2}=\dfrac{(6.626\times10^{-34})(1.0\times10^{12})}{(1.381\times10^{-23})(0.693)}=\dfrac{6.626\times10^{-22}}{9.57\times10^{-24}}=69\ \mathrm{K}\). Below this the transition is spontaneous-dominated; above it, stimulated-dominated.
An allowed optical line at \(\lambda=589\ \mathrm{nm}\) is measured to have \(A_{21}=6.2\times10^{7}\ \mathrm{s^{-1}}\). Invert the spontaneous-emission formula to find the transition dipole \(|\mathbf d_{21}|\), and express it in units of \(ea_0\) and in debye.
Solution
\(\omega=2\pi c/\lambda=2\pi(2.998\times10^8)/(589\times10^{-9})=3.20\times10^{15}\ \mathrm{s^{-1}}\), so \(\omega^3=3.27\times10^{46}\ \mathrm{s^{-3}}\). From \(A_{21}=\omega^3|\mathbf d|^2/(3\pi\varepsilon_0\hbar c^3)\), \(|\mathbf d|^2=A_{21}\,3\pi\varepsilon_0\hbar c^3/\omega^3\). The denominator group \(3\pi\varepsilon_0\hbar c^3=2.37\times10^{-19}\). Thus \(|\mathbf d|^2=(6.2\times10^{7})(2.37\times10^{-19})/(3.27\times10^{46})=4.49\times10^{-58}\ \mathrm{C^2m^2}\), giving \(|\mathbf d_{21}|=2.12\times10^{-29}\ \mathrm{C\,m}\). Since \(ea_0=8.46\times10^{-30}\ \mathrm{C\,m}\), this is \(2.5\,ea_0\); and with \(1\ \mathrm{D}=3.336\times10^{-30}\ \mathrm{C\,m}\), \(|\mathbf d_{21}|=6.4\ \mathrm{D}\) — a strong dipole, consistent with the bright sodium D line.