---
title: The Dipole Approximation and Einstein Coefficients
module: Radiative Transitions and Spectral Lines
moduleNumber: 7
lessonNumber: 2
order: 702
summary: >
  The coupling between an atom and light is the interaction of the electron with
  the electromagnetic field. Because an optical wavelength dwarfs the atom, the
  spatial variation of the field across the atom can be dropped, leaving the
  electric-dipole interaction and its matrix element. That matrix element defines
  the oscillator strength, which obeys the Thomas-Reiche-Kuhn sum rule. Einstein's
  three rate coefficients (absorption, stimulated emission, spontaneous emission)
  follow from detailed balance with thermal radiation, fixing the ratio of
  spontaneous to stimulated rates and its steep growth with frequency.
topics: [Radiative Transitions and Spectral Lines]
sources:
  - book: Foot
    ref: "Ch. 7 — The Interaction of Atoms with Radiation; §7.3–7.5 Dipole Approximation, Oscillator Strength, Einstein Coefficients"
  - book: Bransden & Joachain
    ref: "Ch. 4 — Interaction of One-Electron Atoms with EM Radiation; §4.6; Ch. 11 Radiative Transitions"
  - book: Griffiths & Schroeter
    ref: "Ch. 11 — Quantum Dynamics; §11.2–11.3 Emission and Absorption, Spontaneous Emission"
draft: false
---

The [golden rule](/atomic-physics/radiative-transitions-and-line-shapes/time-dependent-perturbation-golden-rule)
delivers a transition rate once the coupling operator $V$ between the atom and the
perturbing field is known. That operator is the interaction of the atomic electron
with an electromagnetic wave. This lesson derives it, reduces it to the
electric-dipole form that dominates optical transitions, and extracts the two
quantities every measured line reports: the **oscillator strength** and the
**Einstein coefficients**. The oscillator strength packages the dipole matrix
element into a dimensionless number obeying a sum rule; the Einstein coefficients
package the same physics into rates and connect spontaneous emission to stimulated
emission through thermodynamics.

## The atom-field interaction

A charged particle in an electromagnetic field described by vector potential
$\vec A$ and scalar potential $\phi$ has the Hamiltonian obtained by minimal
coupling, $\vec p \to \vec p + e\vec A$ for an electron of charge $-e$:

$$
H = \frac{1}{2m}\big(\vec p + e\vec A\big)^2 - e\phi + V_{\text{atom}}(r).
$$

Work in the Coulomb gauge, $\nabla\cdot\vec A = 0$ and $\phi=0$ for the radiation
field, so $\vec A$ commutes with $\vec p$. Expanding the square and dropping the
term quadratic in $\vec A$ (second order in the field, negligible for weak light
and irrelevant to single-photon transitions) leaves

$$
H = H_0 + H'(t),
\qquad
H'(t) = \frac{e}{m}\,\vec A(\vec r,t)\cdot\vec p,
$$

with $H_0$ the atomic Hamiltonian. For a plane wave of angular frequency $\omega$,
wavevector $\vec k$, and polarization $\hat\varepsilon$,

$$
\vec A(\vec r,t) = A_0\,\hat\varepsilon\,
\cos(\vec k\cdot\vec r - \omega t),
$$

the perturbation oscillates harmonically, and the golden-rule matrix element is
$\bra{f}(e/m)A_0\,e^{i\vec k\cdot\vec r}\,\hat\varepsilon\cdot\vec p\ket{i}$.

## The long-wavelength approximation

The matrix element carries the factor $e^{i\vec k\cdot\vec r}$, the spatial phase
of the wave across the atom. Its size is set by $k\,a$, where $a\sim a_0$ is the
atomic radius and $k = 2\pi/\lambda$. For an optical transition $\lambda\sim
500\ \mathrm{nm}$ and $a_0 \sim 0.05\ \mathrm{nm}$, so

$$
k a_0 = \frac{2\pi a_0}{\lambda} \sim \frac{2\pi (0.05)}{500} \approx 6\times 10^{-4}.
$$

The wave is essentially uniform across the atom. Expanding

$$
e^{i\vec k\cdot\vec r} = 1 + i\,\vec k\cdot\vec r + \cdots
$$

and keeping the leading term is the **electric-dipole (E1) approximation**. The
neglected term $i\vec k\cdot\vec r$ is smaller by $\sim k a_0 \sim \alpha Z$ (the
next lesson shows it generates the magnetic-dipole and electric-quadrupole
transitions, weaker by $\sim 10^{-5}$ in rate).

$$
% caption: An optical wavelength (about 500 nm) is roughly ten thousand times the
% Bohr radius, so the field is uniform across the atom and only its value at the
% nucleus matters. The dipole approximation replaces exp(i k . r) by 1.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% long wave
\draw[acc, very thick] plot[smooth] coordinates{
 (-0.3,1.4) (0.6,2.4) (1.5,1.4) (2.4,0.4) (3.3,1.4)
 (4.2,2.4) (5.1,1.4) (6.0,0.4) (6.9,1.4) (7.8,2.4) (8.4,1.9)};
\node[acc, anchor=south] at (4.05,2.42) {field wavelength (about 500 nm)};
% tiny atom
\fill[black, draw=black!70, very thick] (3.9,1.4) circle (0.22);
\fill[black!70] (3.9,1.4) circle (1.5pt);
\draw[black!70] (3.9,1.4) -- (4.6,1.9);
\node[black!70, anchor=west] at (4.62,1.95) {atom (about 0.1 nm)};
% scale bracket by hand
\draw[black] (0.6,-0.15) -- (0.6,0.15);
\draw[black] (4.2,-0.15) -- (4.2,0.15);
\draw[black, <->] (0.6,0) -- (4.2,0);
\node[anchor=north] at (2.4,-0.2) {one wavelength};
\end{tikzpicture}
$$

In the dipole approximation the interaction can be rewritten in a more transparent
form. The momentum matrix element relates to the position matrix element through
the commutator $[\vec r, H_0] = i\hbar\vec p/m$, so

$$
\bra{f}\vec p\ket{i} = \frac{m}{i\hbar}\bra{f}[\vec r,H_0]\ket{i}
= i m\,\omega_{fi}\,\bra{f}\vec r\ket{i}.
$$

Substituting turns the $\vec A\cdot\vec p$ coupling into the equivalent
**length-gauge** dipole interaction

$$
H' = -\vec d\cdot\vec E(t),
\qquad
\vec d = -e\vec r,
$$

where $\vec d$ is the electron's electric-dipole moment and $\vec E$ is the
electric field of the wave. The transition is governed by the **dipole matrix
element** $\vec d_{fi} = \bra{f}\vec d\ket{i} = -e\bra{f}\vec r\ket{i}$. When this
vector vanishes, the E1 transition is forbidden, the subject of the
[selection-rule lesson](/atomic-physics/radiative-transitions-and-line-shapes/selection-rules-forbidden-transitions).

> **Definition (Electric-dipole approximation).** Replacing $e^{i\vec k\cdot\vec r}$
> by $1$ in the atom-field matrix element, valid when the wavelength greatly
> exceeds the atomic size ($ka_0\ll 1$). The interaction reduces to
> $H' = -\vec d\cdot\vec E$, and the transition rate is controlled by the dipole
> matrix element $\vec d_{fi}=-e\bra{f}\vec r\ket{i}$.

## Oscillator strength and the sum rule

The dipole matrix element carries dimensions and depends on the pair of states. It
is conventional to fold it into a dimensionless **oscillator strength**, which
compares the quantum transition with a classical charged oscillator of the same
frequency. For a transition $i\to f$,

$$
f_{fi} = \frac{2 m\,\omega_{fi}}{3\hbar}\,\abs{\bra{f}\vec r\ket{i}}^2,
$$

the factor $\tfrac13$ averaging over the three Cartesian directions of an
unpolarized or randomly oriented sample. Absorption gives $f_{fi}>0$; emission
($\omega_{fi}<0$) gives $f_{fi}<0$. The oscillator strength is what spectroscopic
tables report, because a strong line has $f$ of order unity and a weak one has $f$
orders of magnitude smaller.

The oscillator strengths from any fixed level obey an exact constraint.

> **Theorem (Thomas-Reiche-Kuhn sum rule).** The oscillator strengths for
> transitions from a given state $\ket{i}$ to all other states $\ket{f}$ of a
> one-electron atom sum to one:
> $$
> \sum_f f_{fi} = 1.
> $$
> For an $N$-electron atom the sum equals $N$.

> **Proof.** Start from the double commutator identity. For any two components
> $x_a, x_b$ of position and $H_0 = p^2/2m + V(r)$,
> $$
> \big[x_a,[H_0,x_b]\big] = \frac{\hbar^2}{m}\,\delta_{ab},
> $$
> because $V$ commutes with position and $[x_a,[p^2,x_b]] = 2\hbar^2\delta_{ab}$.
> Take the diagonal matrix element $\bra{i}\cdot\ket{i}$ and insert a complete set
> of states $\sum_f\ket{f}\bra{f}$ between the operators. Each commutator produces
> a factor $E_f-E_i = \hbar\omega_{fi}$, and the two orderings combine so that the
> cross terms add:
> $$
> \frac{\hbar^2}{m}
> = \sum_f 2\hbar\,\omega_{fi}\,\abs{\bra{f}x\ket{i}}^2 .
> $$
> Summing over the three directions and dividing by $\hbar^2/m$ reproduces
> $\sum_f (2m\omega_{fi}/3\hbar)\abs{\bra{f}\vec r\ket{i}}^2 = \sum_f f_{fi} = 1$.
> The generalization to $N$ electrons follows by summing the identity over all
> electron coordinates, giving $N$ on the right.

The sum rule is a conservation law for spectral weight: an atom has a fixed budget
of oscillator strength, shared among all its transitions (including those into the
continuum). A few strong resonance lines exhaust most of the budget; higher members
of a series get progressively less.

$$
% caption: The oscillator-strength budget of a level distributed over its
% transitions. The strong resonance line takes most of the total; higher series
% members and the continuum share the remainder, and the signed sum is fixed at one.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% baseline
\draw[black] (-0.3,0) -- (9.0,0);
\node[anchor=east] at (-0.35,0) {$0$};
% bars
\fill[acc!18, draw=acc, very thick] (0.3,0) rectangle (1.5,3.0);
\node[anchor=south, align=center] at (0.9,3.03) {resonance\\line};
\fill[acc!18, draw=acc, very thick] (2.1,0) rectangle (3.3,0.9);
\node[anchor=south] at (2.7,0.95) {2nd};
\fill[acc!18, draw=acc, very thick] (3.9,0) rectangle (5.1,0.36);
\node[anchor=south] at (4.5,0.42) {3rd};
\fill[acc!18, draw=acc, very thick] (5.7,0) rectangle (6.9,0.18);
\node[anchor=south] at (6.3,0.26) {higher};
\fill[black, draw=black, very thick] (7.5,0) rectangle (8.7,0.5);
\node[anchor=south, align=center] at (8.1,0.55) {continuum};
% sum annotation
\draw[black, dashed] (-0.3,3.4) -- (9.0,3.4);
\node[anchor=west] at (7.0,3.6) {total $= 1$};
\end{tikzpicture}
$$

## Einstein's three coefficients

Einstein derived the relations among absorption, stimulated emission, and
spontaneous emission in 1917 from thermodynamics alone, before quantum mechanics
could compute any of them.[^einstein] Consider two levels, lower $\ket{1}$ and
upper $\ket{2}$ with $E_2-E_1 = \hbar\omega_0$, in equilibrium with blackbody
radiation of spectral energy density $\rho(\omega)$. Three processes change the
populations $N_1, N_2$:

- **Absorption**, rate per lower atom $B_{12}\,\rho(\omega_0)$: the atom absorbs a
  photon and climbs.
- **Stimulated emission**, rate per upper atom $B_{21}\,\rho(\omega_0)$: the field
  drives the atom down, adding a photon coherent with the field.
- **Spontaneous emission**, rate per upper atom $A_{21}$: the atom decays with no
  field present, emitting into a random mode.

The absorption and stimulated rates are proportional to the radiation density and
follow directly from the golden rule; spontaneous emission has no classical driving
term and is the process that requires the quantized field.

$$
% caption: The three radiative processes between two levels bathed in radiation of
% density rho. Absorption and stimulated emission scale with rho; spontaneous
% emission proceeds at the rate A even in the dark.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% absorption panel
\draw[black, very thick] (0,0) -- (1.6,0) node[right]{$1$};
\draw[black, very thick] (0,2.4) -- (1.6,2.4) node[right]{$2$};
\draw[acc, very thick, ->] (0.6,0) -- (0.6,2.4);
\node[acc, anchor=north, align=center] at (0.6,-0.1) {absorption\\$B_{12}$};
% stimulated
\begin{scope}[xshift=4.2cm]
\draw[black, very thick] (0,0) -- (1.6,0) node[right]{$1$};
\draw[black, very thick] (0,2.4) -- (1.6,2.4) node[right]{$2$};
\draw[acc, very thick, ->] (0.6,2.4) -- (0.6,0);
\node[acc, anchor=north, align=center] at (0.6,-0.1) {stimulated\\$B_{21}$};
\end{scope}
% spontaneous
\begin{scope}[xshift=8.4cm]
\draw[black, very thick] (0,0) -- (1.6,0) node[right]{$1$};
\draw[black, very thick] (0,2.4) -- (1.6,2.4) node[right]{$2$};
\draw[black, very thick, ->] (0.6,2.4) -- (0.6,0);
\node[black, anchor=north, align=center] at (0.6,-0.1) {spontaneous\\$A_{21}$};
\end{scope}
\end{tikzpicture}
$$

## Detailed balance fixes the ratios

In equilibrium the upward and downward transition rates balance:

$$
N_1\,B_{12}\,\rho(\omega_0) = N_2\big(A_{21} + B_{21}\,\rho(\omega_0)\big).
$$

Solve for the radiation density,

$$
\rho(\omega_0)
= \frac{A_{21}/B_{21}}{(N_1 B_{12})/(N_2 B_{21}) - 1}.
$$

The populations in thermal equilibrium follow the Boltzmann distribution,
$N_2/N_1 = (g_2/g_1)\,e^{-\hbar\omega_0/k_B T}$ for levels of degeneracy $g_1,g_2$,
so $N_1 B_{12}/(N_2 B_{21}) = (g_1 B_{12}/g_2 B_{21})\,e^{\hbar\omega_0/k_B T}$.
This must reproduce the Planck spectrum

$$
\rho(\omega) = \frac{\hbar\omega^3}{\pi^2 c^3}\,
\frac{1}{e^{\hbar\omega/k_B T}-1}
$$

at every temperature. Matching term by term forces two relations that hold
independently of temperature.

> **Theorem (Einstein relations).** The two absorption/stimulated coefficients and
> the spontaneous coefficient are locked together by detailed balance with the
> Planck spectrum:
> $$
> g_1 B_{12} = g_2 B_{21},
> \qquad
> \frac{A_{21}}{B_{21}} = \frac{\hbar\omega_0^3}{\pi^2 c^3}.
> $$
> The stimulated-emission and absorption coefficients are equal up to degeneracy;
> the spontaneous rate is the stimulated rate multiplied by the mode density of the
> vacuum, $\hbar\omega^3/\pi^2 c^3$.

Two consequences follow immediately. First, $B_{12}=B_{21}$ for non-degenerate
levels: absorption and stimulated emission are the same process run in opposite
directions, exactly as the two counter-rotating terms in the harmonic perturbation
predicted. Second, the ratio $A/B$ grows as $\omega^3$. The mode density of the
electromagnetic field rises steeply with frequency, so spontaneous emission
dominates in the optical and ultraviolet while stimulated processes dominate in the
microwave. This $\omega^3$ scaling is why building a laser (which needs stimulated
emission to win) grows harder toward short wavelengths, and why an X-ray laser is a
far greater engineering feat than a microwave maser.

$$
% caption: The ratio of spontaneous to stimulated emission grows as the cube of the
% frequency. Spontaneous decay dominates in the optical and ultraviolet; stimulated
% processes dominate at microwave frequencies where the mode density is low.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->] (-0.3,0) -- (8.0,0) node[right, black] {frequency};
\draw[->] (0,-0.3) -- (0,3.6) node[above, black] {$\frac{A}{B}$};
% cubic growth
\draw[acc, very thick] plot[smooth] coordinates{
 (0.2,0.02) (1,0.05) (2,0.24) (3,0.72) (4,1.4) (5,2.1) (6,2.7) (6.6,3.2)};
\node[acc, anchor=south west] at (4.2,1.4) {grows as frequency cubed};
% regime marks
\draw[black, dashed] (1.4,0) -- (1.4,0.9);
\node[black, anchor=south, align=center] at (1.4,0.9) {microwave};
\draw[black, dashed] (6.0,0) -- (6.0,2.7);
\node[black, anchor=south, align=center] at (6.0,2.72) {optical / UV};
\end{tikzpicture}
$$

## From matrix elements to the spontaneous rate

Combining the golden rule with the Einstein relations gives the spontaneous
emission rate in terms of the dipole matrix element. The absorption coefficient
computed from $H' = -\vec d\cdot\vec E$ and averaged over polarizations is

$$
B_{12} = \frac{\pi}{3\varepsilon_0\hbar^2}\,\abs{\vec d_{21}}^2,
$$

and multiplying by $A/B = \hbar\omega_0^3/\pi^2 c^3$ gives the spontaneous rate

$$
A_{21} = \frac{\omega_0^3}{3\pi\varepsilon_0\hbar c^3}\,\abs{\vec d_{21}}^2
= \frac{\omega_0^3\,e^2}{3\pi\varepsilon_0\hbar c^3}\,\abs{\bra{2}\vec r\ket{1}}^2.
$$

This is the master formula of radiative decay. It has two ingredients: the
$\omega_0^3$ factor from the vacuum mode density, and the squared dipole matrix
element that carries all the atomic structure. Written through the oscillator
strength,

$$
A_{21} = \frac{e^2\,\omega_0^2}{2\pi\varepsilon_0\, m c^3}\,\abs{f_{12}},
$$

so a transition with $f\sim 1$ and $\omega_0$ in the visible has $A_{21}\sim
10^{8}\ \mathrm{s^{-1}}$, a lifetime of a few nanoseconds — the order of magnitude
of a strong allowed line. The reciprocal of $A_{21}$ is the natural lifetime that
sets the [natural line
width](/atomic-physics/radiative-transitions-and-line-shapes/lifetimes-and-line-shapes).

| Quantity | Symbol | Scaling | Typical value (strong optical line) |
| --- | --- | --- | --- |
| Dipole matrix element | $\abs{\vec d_{21}}$ | $\sim e a_0$ | $\sim 8\times 10^{-30}\ \mathrm{C\,m}$ |
| Oscillator strength | $f_{12}$ | $\lesssim 1$ | $\sim 0.1$–$1$ |
| Spontaneous rate | $A_{21}$ | $\propto \omega_0^3\abs{\vec d_{21}}^2$ | $\sim 10^{8}\ \mathrm{s^{-1}}$ |
| Natural lifetime | $\tau = 1/A_{21}$ | $\propto \omega_0^{-3}$ | $\sim 1$–$100\ \mathrm{ns}$ |

> **Worked example (The lifetime of hydrogen $2p$).** The Lyman-$\alpha$
> transition $2p\to1s$ has $\hbar\omega_0 = \tfrac34\times 13.6\ \mathrm{eV}
> = 10.2\ \mathrm{eV}$, so $\omega_0 = 1.55\times 10^{16}\ \mathrm{s^{-1}}$. The
> hydrogenic radial integral is
> $$
> \bra{R_{10}}\,r\,\ket{R_{21}} = \frac{128\sqrt2}{243}\,a_0 \approx 0.745\,a_0,
> $$
> and assembling the angular factors gives an effective
> $\abs{\bra{1s}\vec r\ket{2p}}^2$ that, inserted into
> $A_{21} = \omega_0^3 e^2\,\abs{\bra{2}\vec r\ket{1}}^2 / 3\pi\varepsilon_0\hbar c^3$,
> yields
> $$
> A_{21} \approx 6.27\times 10^{8}\ \mathrm{s^{-1}},
> \qquad
> \tau = \frac{1}{A_{21}} \approx 1.6\ \mathrm{ns},
> \qquad
> f_{12} \approx 0.42.
> $$
> The oscillator strength near one half confirms Lyman-$\alpha$ is a strong allowed
> line, and the nanosecond lifetime is the natural scale a strong optical or
> ultraviolet transition sets.

### Oscillator strength and the absorption cross section

The oscillator strength also fixes how strongly an atom absorbs. Integrating the
absorption cross section $\sigma(\omega)$ over the whole line gives a result that
depends only on $f_{12}$ and universal constants,

$$
\int \sigma(\omega)\,\d\omega = \frac{\pi e^2}{2\varepsilon_0 m c}\,f_{12}
= 2\pi^2 c\, r_e\, f_{12},
$$

with $r_e = e^2/4\pi\varepsilon_0 m c^2$ the classical electron radius. The
integrated absorption is independent of the line shape, so it is unchanged by
Doppler or pressure broadening: broadening redistributes the same total absorption
over a wider frequency range, lowering the peak but conserving the area. This is
why the equivalent width of an absorption line, not its peak depth, measures the
column density of atoms, and it is the basis of quantitative astrophysical and
laboratory absorption spectroscopy.

The three coefficients and the dipole matrix element are the working currency of
atomic radiation. Absorption and stimulated emission share one coefficient $B$ and
underlie [spectroscopy and
lasers](/atomic-physics/lasers-and-spectroscopy/laser-principles); spontaneous
emission, fixed by $A = (\hbar\omega^3/\pi^2 c^3)B$, sets natural lifetimes and
line widths. The next lesson asks the sharper question the matrix element $\vec
d_{21}=-e\bra{2}\vec r\ket{1}$ raises: for which pairs of states does it vanish, and
what happens to a transition when it does.

[^einstein]: Einstein, A. (1917), "Zur Quantentheorie der Strahlung," _Physikalische Zeitschrift_ **18**, 121. The $A$ and $B$ coefficients and their detailed-balance relations predate the quantum-mechanical calculation of either. The derivation here follows **Foot**, _Atomic Physics_, §7.5, and the matrix-element formulae **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §4.6. The Thomas-Reiche-Kuhn sum rule: **Bransden & Joachain**, §4.6; **Foot**, §7.4.