Einstein Coefficient Reference

Einstein coefficients describe absorption, stimulated emission, and spontaneous emission between two declared levels in a weak-field rate model. For a lower level $1$, an upper level $2$, and a narrow transition at $\omega_0=(E_2-E_1)/\hbar>0$,

$$\begin{aligned} R_{1\rightarrow2}^{(\mathrm{abs})} &= B_{12}^{(\omega)} u_\omega(\omega_0), \\ R_{2\rightarrow1}^{(\mathrm{down})} &= A_{21} + B_{21}^{(\omega)} u_\omega(\omega_0). \end{aligned}$$

Here $u_\omega$ is the total isotropic radiation energy density per unit angular frequency. The superscript on $B$ is essential: a $B$ coefficient has a numerical value only relative to the spectral density, directional average, polarization sum, and line-profile convention that it multiplies.

This page is a lookup and audit sheet. The detailed-balance derivation and physical discussion live in Einstein Coefficients.

Canonical Scope

This page owns:

• operational definitions and units of $A_{21}$, $B_{12}^{(\omega)}$, and $B_{21}^{(\omega)}$;
• the degeneracy and Planck-density relations among the three coefficients;
• conversion among angular-frequency, ordinary-frequency, wavenumber, and wavelength $B$ conventions;
• radiative widths, lifetimes, branching fractions, observed lifetimes, and quantum yields;
• E1 conversion to transition dipoles, reduced line strengths, and oscillator strengths;
• compact atomic and molecular database checks; and
• applicability tests for rate equations.

It does not own:

• the historical or modern derivation of Planck’s radiation law;
• mode-quantization derivations of the $n$ and $n+1$ factors;
• Wigner–Weisskopf decay and photon wavepackets;
• optical Bloch dynamics under coherent driving;
• detailed line-shape theory;
• Purcell enhancement and Green-tensor calculations; or
• the derivation of forbidden-multipole line-strength formulas.

The reference formulas are valid only with the conventions stated beside them.

Convention Ledger

Unless a section says otherwise:

• $E_2>E_1$ and $\omega_0=(E_2-E_1)/\hbar$;
• $g_1$ and $g_2$ are the total degeneracies of the lower and upper objects;
• populations are uniform within each degenerate level;
• $A_{21}$ describes one declared upper-to-lower radiative channel;
• $u_\omega(\omega)$ is total isotropic vacuum radiation energy per volume per angular-frequency interval, summed over directions and both transverse polarizations;
• line profiles are normalized to unit area in the variable printed in their subscript;
• the narrow-line approximation is used only when the radiation density is nearly constant across the profile; and
• microscopic matrix-element formulas refer to an electric-dipole, or E1, transition in free space.

The three coefficients are intrinsic to the declared transition model. A rate additionally requires a radiation field or a prepared upper-state population.

Core Definitions

Quantity	Operational definition	SI units	Common database meaning
$A_{21}$	spontaneous downward rate per system in level $2$	$\mathrm{s^{-1}}$	emission transition probability per unit time
$B_{12}^{(\omega)}$	absorption rate divided by matching $u_\omega$	$\mathrm{m^3\,J^{-1}\,s^{-2}}$	usually derived rather than tabulated
$B_{21}^{(\omega)}$	stimulated-emission rate divided by matching $u_\omega$	$\mathrm{m^3\,J^{-1}\,s^{-2}}$	convention dependent
$g_2A_{21}$	upper-weighted spontaneous rate	$\mathrm{s^{-1}}$	a distinct column from $A_{21}$
$\sum_\ell A_{2\ell}$	total radiative decay rate out of level $2$	$\mathrm{s^{-1}}$	inverse radiative lifetime

Spontaneous coefficient

For a system prepared in level $2$,

$$A_{21} = \left. \frac{dP_{2\rightarrow1}}{dt} \right|_{\mathrm{sp}}.$$

For $N_2$ independent emitters, and before collection or detector losses,

$$\dot N_{\gamma,21} = N_2A_{21},$$

and the emitted optical power in that channel is

$$\mathcal P_{21} = N_2\hbar\omega_0A_{21}.$$

Atomic databases often call $A_{21}$ a transition probability. Its units show that it is a probability rate, not a dimensionless finite-time probability.

Stimulated coefficients

The absorption and stimulated-emission rates per system are

$$R_{1\rightarrow2}^{(\mathrm{abs})} = B_{12}^{(\omega)}u_\omega(\omega_0),$$ $$R_{2\rightarrow1}^{(\mathrm{stim})} = B_{21}^{(\omega)}u_\omega(\omega_0).$$

With

$$[u_\omega] = \mathrm{J\,m^{-3}} \left(\mathrm{rad\,s^{-1}}\right)^{-1},$$

and radians treated as dimensionless,

$$[B^{(\omega)}] = \mathrm{m^3\,J^{-1}\,s^{-2}}.$$

A $B$ coefficient alone is not a rate. It becomes one only after multiplication by the matching spectral energy density.

Finite Profiles

Let the absorption profile be normalized by

$$\int_0^\infty \phi_{12}^{(\omega)}(\omega)\,d\omega = 1.$$

The weak-field rate is

$$R_{1\rightarrow2}^{(\mathrm{abs})} = B_{12}^{(\omega)} \int_0^\infty u_\omega(\omega) \phi_{12}^{(\omega)}(\omega)\,d\omega.$$

Only when $u_\omega$ changes negligibly across the line may one write

$$R_{1\rightarrow2}^{(\mathrm{abs})} \simeq B_{12}^{(\omega)}u_\omega(\omega_0).$$

The spontaneous coefficient may likewise be distributed over a normalized emission profile:

$$\frac{dA_{21}}{d\omega} = A_{21}\phi_{21}^{(\omega)}(\omega), \qquad \int_0^\infty \frac{dA_{21}}{d\omega}\,d\omega = A_{21}.$$

The integrated $A$ and the line profile are separate objects. Broad vibronic bands, overlapping transitions, or rapidly varying radiation fields require the integral rather than a line-center shortcut.

Spectral-Density Conventions

Spectral energy is invariant under a coordinate change:

$$u_x(x)\,dx = u_y(y)\,dy.$$

The numerical $B$ must change inversely so that the physical rate remains invariant:

$$B^{(x)}u_x = B^{(y)}u_y.$$

Angular frequency and ordinary frequency

Because

$$\omega=2\pi\nu, \qquad u_\nu(\nu)=2\pi u_\omega(\omega),$$

the coefficients satisfy

$$B_{ij}^{(\nu)} = \frac{1}{2\pi} B_{ij}^{(\omega)}.$$

The two conventions have the same reduced SI dimensions when radians are dimensionless, so dimensional analysis does not reveal a missing $2\pi$.

Spectroscopic wavenumber

For $\widetilde\nu=\nu/c$ measured in $\mathrm{m^{-1}}$,

$$u_{\widetilde\nu} = c\,u_\nu, \qquad B_{ij}^{(\widetilde\nu)} = \frac{1}{c} B_{ij}^{(\nu)}.$$

If wavenumber is measured in $\mathrm{cm^{-1}}$, then

$$B_{ij}^{(\mathrm{cm}^{-1})} = \frac{1}{100c} B_{ij}^{(\nu)}.$$

The unit attached to the wavenumber variable is part of the coefficient definition.

Wavelength

Since $\nu=c/\lambda$,

$$u_\lambda(\lambda) = \frac{c}{\lambda^2} u_\nu(\nu).$$

At a narrow line centered on $\lambda_0$,

$$B_{ij}^{(\lambda)} = \frac{\lambda_0^2}{c} B_{ij}^{(\nu)}.$$

For a broad band, the Jacobian varies across the profile. A single line-center factor is then insufficient.

Conversion table

Density used in the rate	Coordinate units	Coefficient relative to $B^{(\omega)}$
$u_\omega$	$\mathrm{rad\,s^{-1}}$	$B^{(\omega)}$
$u_\nu$	$\mathrm{Hz}$	$B^{(\nu)}=B^{(\omega)}/(2\pi)$
$u_{\widetilde\nu}$	$\mathrm{m^{-1}}$	$B^{(\widetilde\nu)}=B^{(\omega)}/(2\pi c)$
$u_{\widetilde\nu}$	$\mathrm{cm^{-1}}$	$B^{(\mathrm{cm}^{-1})}=B^{(\omega)}/(200\pi c)$
$u_\lambda$ near $\lambda_0$	$\mathrm m$	$B^{(\lambda)}=\lambda_0^2B^{(\omega)}/(2\pi c)$

Relations Among the Coefficients

For sharp levels in free space and the conventions above, thermal detailed balance gives

$$g_1B_{12}^{(\omega)} = g_2B_{21}^{(\omega)}.$$

Thus

$$B_{12}^{(\omega)} = \frac{g_2}{g_1} B_{21}^{(\omega)}.$$

The two level-averaged $B$ coefficients are equal only when the degeneracies are equal. State-to-state coefficients for matched resolved substates can be equal even when the level-averaged coefficients are not.

The spontaneous-to-stimulated relation is

$$\frac{A_{21}} {B_{21}^{(\omega)}} = \frac{\hbar\omega_0^3} {\pi^2c^3}.$$

In ordinary-frequency notation,

$$\frac{A_{21}} {B_{21}^{(\nu)}} = \frac{8\pi h\nu_0^3} {c^3}.$$

These are the same physical relation expressed with different spectral densities. Thermodynamic balance fixes ratios among coefficients; it does not determine their absolute scale. The transition matrix element supplies that scale.

Thermal Radiation Lookup

The mean occupation of a thermal photon mode is

$$\overline n(\omega_0,T) = \frac{1} { \exp\!\left( \dfrac{\hbar\omega_0}{k_{\mathrm B}T} \right)-1 }.$$

Planck’s energy density per angular frequency is

$$u_\omega(\omega_0,T) = \frac{\hbar\omega_0^3} {\pi^2c^3} \overline n.$$

Using the Einstein relations gives the compact rates

$$R_{2\rightarrow1}^{(\mathrm{th})} = A_{21}(\overline n+1),$$ $$R_{1\rightarrow2}^{(\mathrm{th})} = \frac{g_2}{g_1} A_{21}\overline n.$$

The $+1$ is the spontaneous contribution. The thermal population ratio

$$\frac{N_2}{N_1} = \frac{g_2}{g_1} \exp\!\left( -\frac{\hbar\omega_0}{k_{\mathrm B}T} \right)$$

ensures

$$N_1R_{1\rightarrow2}^{(\mathrm{th})} = N_2R_{2\rightarrow1}^{(\mathrm{th})}.$$

For optical transitions at room temperature, $\overline n$ is usually negligible. For microwave and radio-frequency transitions it can be much larger than one.

Lifetimes and Branching

Total radiative width

If level $2$ has several radiative channels,

$$\Gamma_{\mathrm{rad},2} = \sum_\ell A_{2\ell}.$$

The radiative lifetime is

$$\tau_{\mathrm{rad},2} = \frac{1} {\Gamma_{\mathrm{rad},2}}.$$

One channel coefficient equals the inverse radiative lifetime only when it is the sole radiative decay channel.

Branching fractions

The radiative branching fraction into channel $2\rightarrow\ell$ is

$$b_{2\ell}^{(\mathrm{rad})} = \frac{A_{2\ell}} {\sum_jA_{2j}}, \qquad \sum_\ell b_{2\ell}^{(\mathrm{rad})} = 1.$$

Conversely, if a radiative lifetime and complete radiative branching fractions are known,

$$A_{2\ell} = \frac{ b_{2\ell}^{(\mathrm{rad})} }{ \tau_{\mathrm{rad},2} }.$$

Observed lifetime and quantum yield

With additional first-order channels,

$$\Gamma_{\mathrm{tot},2} = \Gamma_{\mathrm{rad},2} + \Gamma_{\mathrm{nr},2} + \Gamma_{\mathrm{coll},2} + \cdots,$$

and

$$\tau_{\mathrm{obs},2} = \frac{1} {\Gamma_{\mathrm{tot},2}}.$$

The radiative quantum yield is

$$\Phi_{\mathrm{rad}} = \frac{\Gamma_{\mathrm{rad},2}} {\Gamma_{\mathrm{tot},2}} = \frac{\tau_{\mathrm{obs},2}} {\tau_{\mathrm{rad},2}}.$$

The last equality assumes the same state and first-order kinetics. In multiexponential, transfer-coupled, diffusion-limited, or time-dependent environments, no single reciprocal rate need describe the measured trace.

Natural Width

For a stable lower state, no pure dephasing, and exponential upper-state decay,

$$\Gamma_1 = \sum_\ell A_{2\ell}, \qquad \Gamma_2 = \frac{\Gamma_1}{2}.$$

The Lorentzian full width at half maximum is then

$$\Delta\omega_{\mathrm{FWHM}} = \Gamma_1,$$

or

$$\Delta\nu_{\mathrm{FWHM}} = \frac{\Gamma_1}{2\pi}.$$

More generally,

$$\Gamma_2 = \frac{\Gamma_{\mathrm{upper}} +\Gamma_{\mathrm{lower}}}{2} + \gamma_\phi,$$

so

$$\Delta\omega_{\mathrm{FWHM}} = 2\Gamma_2.$$

The natural-width conversion therefore uses the sum of all relevant $A$ channels and any lower-state width or pure dephasing. The complete width ledger belongs to Line Shape Reference.

Electric-Dipole Conversions

Nondegenerate transition dipole

For

$$\mathbf d_{21} = \langle2| \widehat{\mathbf D} |1\rangle,$$

the free-space E1 coefficients are

$$A_{21} = \frac{\omega_0^3} {3\pi\epsilon_0\hbar c^3} \left|\mathbf d_{21}\right|^2,$$ $$B_{21}^{(\omega)} = \frac{\pi} {3\epsilon_0\hbar^2} \left|\mathbf d_{21}\right|^2.$$

For nondegenerate endpoints,

$$B_{12}^{(\omega)} = B_{21}^{(\omega)}.$$

Degenerate levels and reduced line strength

For a reduced E1 line strength

$$S_{21} = \left| \left\langle \gamma_2J_2 \left\| \widehat D^{(1)} \right\| \gamma_1J_1 \right\rangle \right|^2,$$

the level-averaged coefficients are

$$\begin{aligned} A_{21} &= \frac{\omega_0^3} {3\pi\epsilon_0\hbar c^3g_2} S_{21}, \\ B_{21}^{(\omega)} &= \frac{\pi} {3\epsilon_0\hbar^2g_2} S_{21}, \\ B_{12}^{(\omega)} &= \frac{\pi} {3\epsilon_0\hbar^2g_1} S_{21}. \end{aligned}$$

The spontaneous and stimulated-emission coefficients average over upper substates; the absorption coefficient averages over lower substates.

Relation to oscillator strength

For an E1 absorption oscillator strength $f_{12}$,

$$A_{21} = \frac{e^2\omega_0^2} {2\pi\epsilon_0m_ec^3} \frac{g_1}{g_2} f_{12}.$$

The angular-frequency stimulated coefficients are

$$B_{12}^{(\omega)} = \frac{\pi e^2} {2\epsilon_0m_e\hbar\omega_0} f_{12},$$ $$B_{21}^{(\omega)} = \frac{g_1}{g_2} B_{12}^{(\omega)}.$$

For vacuum wavelength in ångströms and $A_{21}$ in $\mathrm{s^{-1}}$,

$$f_{12} = 1.49919\times10^{-16} \frac{g_2}{g_1} \lambda_{\mathring{\mathrm A}}^{\,2} A_{21}.$$

For $S_{21}$ in $e^2a_0^2$,

$$A_{21} = \frac{2.02613\times10^{18}} {g_2\lambda_{\mathring{\mathrm A}}^{\,3}} S_{21}.$$

The complete $f$, $gf$, $\log(gf)$, and $S$ ledger is in Oscillator Strength Reference.

Fixed-strength frequency scaling

For a fixed E1 line strength,

$$A_{21}\propto\omega_0^3, \qquad B_{21}^{(\omega)} \propto S_{21}.$$

For a fixed oscillator strength, however,

$$A_{21}\propto\omega_0^2, \qquad B_{12}^{(\omega)} \propto\omega_0^{-1}.$$

Always state which structural quantity is being held fixed before quoting a frequency scaling.

Beyond Free-Space E1

Einstein $A$ values are also used for M1, E2, and higher-multipole transitions. Their microscopic operators, dimensions, angular factors, and frequency powers differ from the E1 formulas above. The transition type must be read before converting a database line strength.

In a cavity, waveguide, dielectric, photonic crystal, near a surface, or in another structured reservoir, spontaneous emission depends on the local electromagnetic density of states and mode functions. A vacuum $A$ can be enhanced, suppressed, redirected, or split among channels. Use a mode-resolved calculation or electromagnetic Green tensor rather than silently applying the free-space formula.

The equilibrium relations among upward and downward processes also assume a specified reservoir and consistent mode normalization. They are not a universal recipe for replacing an environment-dependent decay rate while leaving every other coefficient unchanged.

Atomic Database Notation

Atomic line lists commonly use lower level $i$ and upper level $k$:

$$i\rightarrow k \quad\text{for absorption}, \qquad k\rightarrow i \quad\text{for emission}.$$

Column	Meaning	Required check
$A_{ki}$	spontaneous upper-to-lower rate	$\mathrm{s^{-1}}$ or units of $10^8\,\mathrm{s^{-1}}$
$g_kA_{ki}$	upper-weighted rate	use $g_k=2J_k+1$ for a fine-structure level
accuracy code	assessed uncertainty category	retain the cited transition-probability reference
type	E1, M1, E2, mixed, induced, or other	choose the matching line-strength formula
relative intensity	source-dependent observed strength	do not equate with $A_{ki}$

The lifetime of upper level $k$ requires all radiative channels:

$$\tau_{\mathrm{rad},k}^{-1} = \sum_i A_{ki}.$$

A search window, wavelength table, or detector band may omit important branches. Sum over the level’s complete transition set, not merely the lines visible in one query.

Molecular Database Notation

Molecular $A_{ji}$ values refer to fully or partially resolved rovibronic transitions. Statistical weights can include rotational, electronic, nuclear-spin, parity, and symmetry factors. The same convention must be used in the line list and its partition function.

HITRAN, for example, stores a temperature-independent $A_{ji}$ in $\mathrm{s^{-1}}$, while its line intensity is temperature dependent. In HITRAN’s wavenumber and cgs-style convention,

$$\begin{aligned} S_{ij}^{(\mathrm{HIT})}(T) &= I_{\mathrm a} \frac{A_{ji}} {8\pi c\widetilde\nu_{ij}^{\,2}} \frac{ g_j \exp(-c_2E_i/T) }{ Q(T) } \\ &\quad\times \left[ 1-\exp(-c_2\widetilde\nu_{ij}/T) \right], \end{aligned}$$

where:

• $S_{ij}^{(\mathrm{HIT})}$ has units $\mathrm{cm^{-1}/(molecule\,cm^{-2})}$, equivalently $\mathrm{cm\,molecule^{-1}}$;
• $\widetilde\nu_{ij}$ and $E_i$ are in $\mathrm{cm^{-1}}$;
• $c$ is in $\mathrm{cm\,s^{-1}}$;
• $c_2=hc/k_{\mathrm B}$ is in $\mathrm{cm\,K}$;
• $g_j$ is the upper-state statistical weight;
• $Q(T)$ uses the matching weight convention; and
• $I_{\mathrm a}$ is the terrestrial isotopologue abundance factor used by the database.

The database line intensity $S_{ij}^{(\mathrm{HIT})}$ is not the reduced matrix-element line strength $S_{21}$ used in the E1 formulas. The shared letter $S$ does not make them the same physical quantity.

Rate-Model Applicability

The Einstein rate model is appropriate when:

• radiation is weak enough for induced rates to remain linear in spectral energy density;
• phase coherence between the two levels is irrelevant or rapidly lost;
• populations within a degenerate level follow the assumed distribution;
• memory effects and strong system–reservoir coupling are negligible;
• line overlap can be handled by normalized profile integrals; and
• the reservoir convention matches the coefficient definition.

Use a coherent or mode-resolved description when:

• a narrow laser establishes a definite phase and Rabi oscillations;
• power broadening or saturation matters;
• short pulses prepare coherent superpositions;
• interference among pathways is observable;
• a cavity or nanophotonic environment reshapes the mode density; or
• non-Markovian decay invalidates a constant $A$.

A collimated laser irradiance is not the isotropic $u_\omega$ of Einstein’s thermal argument. Converting irradiance into a field amplitude and using a Rabi frequency is usually the cleanest route.

Common Mistakes

Calling B a rate

$B$ becomes a rate only after multiplication by its matching spectral energy density or after integration over a line profile.

Omitting the spectral coordinate

$B^{(\omega)}$, $B^{(\nu)}$, $B^{(\widetilde\nu)}$, and $B^{(\lambda)}$ are numerically different. The omission is especially dangerous for $\omega$ versus $\nu$ because their reduced SI dimensions are the same.

Setting the two B coefficients equal

For level-averaged coefficients,

$$g_1B_{12}=g_2B_{21}.$$

Equality requires matched degeneracies or resolved state-to-state coefficients.

Equating one A with an inverse lifetime

The radiative inverse lifetime is $\sum_\ell A_{2\ell}$. Nonradiative and collisional rates further shorten an observed lifetime.

Confusing A with observed intensity

Emission also depends on upper-state population, photon energy, reabsorption, collection geometry, branching, and detector response.

Making A temperature dependent

For a fixed free-space transition model, $A$ is intrinsic. Thermal populations and stimulated processes change line intensity and total transition rates, not the stored spontaneous coefficient.

Using E1 formulas for every database type

M1, E2, mixed, and induced transitions require their own microscopic strength conventions.

Applying rate equations to coherent strong driving

Rabi dynamics, optical coherences, saturation, and dressed states require optical Bloch or fuller quantum-optical equations.

Reporting Template

A reproducible statement should specify:

1. lower and upper state or level labels, including all resolved quantum numbers;
2. $g_1$ and $g_2$ and what each degeneracy counts;
3. whether $A_{21}$ is one channel or a sum;
4. transition frequency and vacuum, air, or medium wavelength convention;
5. whether $B$ multiplies $u_\omega$, $u_\nu$, wavenumber density, wavelength density, radiance, irradiance, or another field quantity;
6. line-profile normalization and whether the narrow-line approximation was used;
7. transition multipole and environment;
8. radiative and nonradiative channels used in a lifetime;
9. uncertainty, accuracy code, and bibliographic provenance; and
10. temperature, population, partition-function, and isotopic-abundance conventions for molecular intensities.

Exercises

Exercise 1: Lifetime, branching, and quantum yield

An upper level has radiative channels $A_{21}=3.0\times10^7\,\mathrm{s^{-1}}$ and $A_{20}=1.0\times10^7\,\mathrm{s^{-1}}$, plus a nonradiative rate $\Gamma_{\mathrm{nr}}=1.0\times10^7\,\mathrm{s^{-1}}$. Find the radiative lifetime, radiative branching fraction into level $1$, observed lifetime, and radiative quantum yield.

Solution

The radiative rate is

$$\Gamma_{\mathrm{rad}} = 3.0\times10^7+1.0\times10^7 = 4.0\times10^7\ \mathrm{s^{-1}}.$$

Therefore

$$\tau_{\mathrm{rad}} = \frac{1}{4.0\times10^7} = 25\ \mathrm{ns}.$$

The radiative branching fraction is

$$b_{21}^{(\mathrm{rad})} = \frac{3.0}{4.0} = 0.75.$$

Including the nonradiative channel,

$$\Gamma_{\mathrm{tot}} = 5.0\times10^7\ \mathrm{s^{-1}}, \qquad \tau_{\mathrm{obs}} = 20\ \mathrm{ns}.$$

Finally,

$$\Phi_{\mathrm{rad}} = \frac{4.0}{5.0} = 0.80.$$

The single channel $A_{21}$ is therefore not the inverse lifetime in this example.

Exercise 2: Recover channel coefficients

A level has radiative lifetime $\tau_{\mathrm{rad}}=40.0\,\mathrm{ns}$ and complete radiative branching fractions $0.70$, $0.20$, and $0.10$. Find the three $A$ coefficients.

Solution

The total radiative rate is

$$\Gamma_{\mathrm{rad}} = \frac{1}{40.0\times10^{-9}\,\mathrm s} = 2.50\times10^7\ \mathrm{s^{-1}}.$$

Multiplying by each branching fraction gives

$$\begin{aligned} A_1 &= 1.75\times10^7\ \mathrm{s^{-1}}, \\ A_2 &= 5.00\times10^6\ \mathrm{s^{-1}}, \\ A_3 &= 2.50\times10^6\ \mathrm{s^{-1}}. \end{aligned}$$

Their sum recovers $2.50\times10^7\,\mathrm{s^{-1}}$.

Exercise 3: Degeneracy and spectral convention

Suppose $g_1=2$, $g_2=6$, and $B_{21}^{(\omega)}=4.00\times10^{20}$ in the matching SI units. Find $B_{12}^{(\omega)}$ and $B_{12}^{(\nu)}$.

Solution

Detailed balance gives

$$B_{12}^{(\omega)} = \frac{g_2}{g_1} B_{21}^{(\omega)} = 3(4.00\times10^{20}) = 1.20\times10^{21}.$$

Changing from angular to ordinary frequency,

$$\begin{aligned} B_{12}^{(\nu)} &= \frac{B_{12}^{(\omega)}}{2\pi} \\ &= 1.91\times10^{20}. \end{aligned}$$

The degeneracy factor and the $2\pi$ Jacobian answer different questions and must both be applied.

Exercise 4: Oscillator strength to A

An E1 transition has $\lambda=5000.0\,\mathring{\mathrm A}$, $f_{12}=0.200$, $g_1=2$, and $g_2=4$. Find $A_{21}$. If it is the sole decay channel, find the radiative lifetime.

Solution

Rearrange the numerical conversion:

$$A_{21} = \frac{f_{12}} { 1.49919\times10^{-16} (g_2/g_1) \lambda_{\mathring{\mathrm A}}^{\,2} }.$$

Substitution gives

$$A_{21} = 2.668\times10^7\ \mathrm{s^{-1}}.$$

For a single radiative channel,

$$\tau_{\mathrm{rad}} = \frac{1}{A_{21}} = 37.48\ \mathrm{ns}.$$

If other channels exist, this last step is invalid until their $A$ values are included.

Exercise 5: Thermal detailed balance

For a transition with $A_{21}=1.00\times10^7\,\mathrm{s^{-1}}$, $g_2/g_1=3$, and thermal occupation $\overline n=0.20$, find the upward and downward rates per system. Verify event balance using the equilibrium population ratio.

Solution

The downward rate per upper-state system is

$$R_{2\rightarrow1} = A_{21}(\overline n+1) = 1.20\times10^7\ \mathrm{s^{-1}}.$$

The upward rate per lower-state system is

$$R_{1\rightarrow2} = \frac{g_2}{g_1} A_{21}\overline n = 6.00\times10^6\ \mathrm{s^{-1}}.$$

The Bose factor implies

$$e^{-\hbar\omega_0/(k_{\mathrm B}T)} = \frac{\overline n}{\overline n+1} = \frac{1}{6}.$$

Hence

$$\frac{N_2}{N_1} = 3\left(\frac{1}{6}\right) = \frac{1}{2}.$$

Therefore

$$N_1(6.00\times10^6) = N_2(1.20\times10^7),$$

as required.

Exercise 6: Microwave thermal occupation

Estimate the thermal photon occupation at $\nu_0=10.0\,\mathrm{GHz}$ and $T=300\,\mathrm K$. Is spontaneous emission the dominant downward contribution?

Solution

The dimensionless ratio is

$$\frac{h\nu_0}{k_{\mathrm B}T} = 1.600\times10^{-3}.$$

Thus

$$\overline n = \frac{1} {\exp(1.600\times10^{-3})-1} = 624.6.$$

The stimulated downward contribution is $A_{21}\overline n$, whereas the spontaneous contribution is $A_{21}$. Thermal stimulated emission is therefore about $625$ times larger than the extra spontaneous term. Room-temperature microwave experiments must account for thermal photons.

Exercise 7: Natural linewidth

An upper level decays through channels with $A_1=6.0\times10^7\,\mathrm{s^{-1}}$ and $A_2=2.0\times10^7\,\mathrm{s^{-1}}$. The lower level is stable and pure dephasing is negligible. Find the lifetime and ordinary-frequency FWHM.

Solution

The total population-decay rate is

$$\Gamma_1 = A_1+A_2 = 8.0\times10^7\ \mathrm{s^{-1}}.$$

Hence

$$\tau = \frac{1}{\Gamma_1} = 12.5\ \mathrm{ns}.$$

The natural Lorentzian width is

$$\begin{aligned} \Delta\nu_{\mathrm{FWHM}} &= \frac{\Gamma_1}{2\pi} \\ &= 1.273\times10^7\ \mathrm{Hz} \\ &= 12.73\ \mathrm{MHz}. \end{aligned}$$

Using only $A_1$ would omit the coherence loss caused by the second decay channel.

Exercise 8: Audit a molecular laser claim

A report says: “The HITRAN Einstein $A$ coefficient doubles from 200 K to 400 K, and the stimulated-emission rate is $BI$ for laser irradiance $I$.” Identify the two convention errors.

Solution

First, a stored free-space $A_{ji}$ for a fixed molecular transition is temperature independent. The HITRAN line intensity changes with temperature because level populations, the partition function, stimulated-emission correction, and possibly isotopic conventions enter. That does not make $A_{ji}$ temperature dependent.

Second, the Einstein $B$ used here multiplies a specified isotropic spectral energy density, not an unlabeled irradiance. A directional coherent laser also selects propagation direction and polarization and can preserve optical phase. One must either convert the laser spectrum and geometry to the exact field convention used to define $B$, or, more transparently, use the field amplitude, transition dipole, Rabi frequency, and optical Bloch equations.

Cross-Links

• Einstein Coefficients
• Oscillator Strength Reference
• Line Shape Reference
• Constants and Conversions
• Transition Rates
• Absorption and Emission
• Fluorescence and Phosphorescence
• Spontaneous Emission
• Stimulated Emission
• Optical Bloch Equations
• Planck’s Radiation Law
• Blackbody Radiation
• Transition Rates in Light–Matter Interaction
• Quantum Optical Master Equation

References

• A. Einstein, “Zur Quantentheorie der Strahlung,” Physikalische Zeitschrift 18, 121–128 (1917); English translation, “On the Quantum Theory of Radiation,” in D. ter Haar, ed., The Old Quantum Theory, Pergamon, 1967; CERN Document Server record.
• P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proceedings of the Royal Society A 114, 243–265 (1927), doi:10.1098/rspa.1927.0039.
• R. C. Hilborn, “Einstein coefficients, cross sections, $f$ values, dipole moments, and all that,” American Journal of Physics 50, 982–986 (1982), doi:10.1119/1.12937; revised version; erratum.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley, 1992.
• I. I. Sobelman, Atomic Spectra and Radiative Transitions, 2nd ed., Springer, 1992, doi:10.1007/978-3-642-76907-8.
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