---
title: Selection Rules and Forbidden Transitions
module: Radiative Transitions and Spectral Lines
moduleNumber: 7
lessonNumber: 3
order: 703
summary: >
  The dipole matrix element vanishes for most pairs of states, and the pattern of
  which survive is the set of selection rules. Parity forces the orbital angular
  momentum to change by one; the angular integral of three spherical harmonics
  restricts the magnetic quantum number to change by zero or one; the photon's spin
  restricts the total angular momentum. When the dipole element vanishes, higher
  multipoles (magnetic dipole and electric quadrupole) can still drive the
  transition at rates smaller by powers of the fine-structure constant, and states
  with no allowed decay become metastable.
topics: [Radiative Transitions and Spectral Lines]
sources:
  - book: Foot
    ref: "Ch. 7 — The Interaction of Atoms with Radiation; §7.4–7.6 Selection Rules, Higher Multipoles"
  - book: Bransden & Joachain
    ref: "Ch. 4 — Interaction of One-Electron Atoms with EM Radiation; §4.6 Selection Rules"
  - book: Demtröder
    ref: "Ch. 7 — Emission and Absorption of Electromagnetic Radiation; §7.2 Selection Rules"
draft: false
---

The [dipole matrix element](/atomic-physics/radiative-transitions-and-line-shapes/dipole-approximation-einstein-coefficients)
$\vec d_{fi} = -e\bra{f}\vec r\ket{i}$ controls whether a transition radiates.
For most pairs of atomic states it is exactly zero, and the transition is
**forbidden** in the electric-dipole approximation. The pattern of nonzero matrix
elements is the set of **selection rules**, and each rule traces to a symmetry:
parity, rotational invariance, and the independence of the dipole operator from
spin. This lesson derives each rule from the integral it enforces, then examines
what drives a transition when the dipole element vanishes: the higher multipoles,
weaker by powers of $\alpha$, and the long-lived metastable states they leave
behind.

## Parity and the change in orbital angular momentum

The position operator $\vec r$ is odd under the parity transformation $\vec r \to
-\vec r$. A hydrogenic state $\ket{n\ell m}$ has definite parity $(-1)^\ell$,
because the spherical harmonic $Y_\ell^m$ satisfies $Y_\ell^m(-\hat r) =
(-1)^\ell Y_\ell^m(\hat r)$. The matrix element $\bra{f}\vec r\ket{i}$ integrates
the product of three factors: the parity of $\ket{i}$, the odd operator $\vec r$,
and the parity of $\ket{f}$. Under $\vec r\to-\vec r$ the integrand picks up
$(-1)^{\ell_f}\cdot(-1)\cdot(-1)^{\ell_i}$. If this sign is $-1$ the integral
equals its own negative and must vanish. A nonzero dipole element therefore
requires

$$
(-1)^{\ell_f + \ell_i + 1} = +1
\quad\Longleftrightarrow\quad
\ell_f + \ell_i \ \text{odd}.
$$

The initial and final orbital angular momenta must have **opposite parity**. This
is the parity selection rule, and it holds for any system with a definite-parity
Hamiltonian, not just hydrogen.

$$
% caption: Parity gates the dipole integral. The operator r is odd; if the two
% states have the same parity the integrand is odd overall and integrates to zero,
% so only states of opposite parity radiate by the dipole.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% allowed row
\node[anchor=west] at (-0.2,2.4) {allowed:};
\node[draw, acc, minimum width=11mm, minimum height=7mm] (a1) at (2.2,2.4) {even};
\node at (3.4,2.4) {$\times$};
\node[draw, black, minimum width=11mm, minimum height=7mm] (a2) at (4.6,2.4) {odd};
\node at (5.8,2.4) {$\times$};
\node[draw, black, minimum width=11mm, minimum height=7mm] (a3) at (7.0,2.4) {odd};
\node at (8.2,2.4) {$=$};
\node[draw, acc, minimum width=13mm, minimum height=7mm] (a4) at (9.6,2.4) {even};
\node[acc, anchor=west] at (10.5,2.4) {nonzero};
% forbidden row
\node[anchor=west] at (-0.2,0.6) {forbidden:};
\node[draw, acc, minimum width=11mm, minimum height=7mm] (b1) at (2.2,0.6) {even};
\node at (3.4,0.6) {$\times$};
\node[draw, black, minimum width=11mm, minimum height=7mm] (b2) at (4.6,0.6) {odd};
\node at (5.8,0.6) {$\times$};
\node[draw, acc, minimum width=11mm, minimum height=7mm] (b3) at (7.0,0.6) {even};
\node at (8.2,0.6) {$=$};
\node[draw, black, minimum width=13mm, minimum height=7mm] (b4) at (9.6,0.6) {odd};
\node[black, anchor=west] at (10.5,0.6) {zero};
% labels
\node[anchor=north, black] at (2.2,2.05) {state $f$};
\node[anchor=north, black] at (4.6,2.05) {operator};
\node[anchor=north, black] at (7.0,2.05) {state $i$};
\end{tikzpicture}
$$

Parity alone allows $\ell_f - \ell_i = \pm 1, \pm 3, \dots$. The angular integral
sharpens this to $\pm 1$.

## The angular integral: the change in the magnetic quantum number

The Cartesian components of $\vec r$ are proportional to the $q=0,\pm1$ spherical
harmonics of rank one:

$$
z = r\sqrt{\tfrac{4\pi}{3}}\,Y_1^0,
\qquad
x \pm iy = \mp r\sqrt{\tfrac{8\pi}{3}}\,Y_1^{\pm 1}.
$$

The dipole matrix element then contains an integral of three spherical harmonics,
the **Gaunt integral**,

$$
\int Y_{\ell_f}^{m_f *}\,Y_1^{q}\,Y_{\ell_i}^{m_i}\,\d\Omega,
$$

which is nonzero only when two conditions hold. The azimuthal integral over the
$e^{i(m_i + q - m_f)\phi}$ phase forces $m_f = m_i + q$, and the triangle
condition on the three ranks forces $\abs{\ell_f-\ell_i}\le 1\le \ell_f+\ell_i$.
Combined with parity, which already excluded $\ell_f=\ell_i$, the surviving cases
are

$$
\Delta\ell = \ell_f - \ell_i = \pm 1,
\qquad
\Delta m = m_f - m_i = 0,\ \pm 1.
$$

The three values of $\Delta m$ correspond to the three polarizations of the
emitted or absorbed photon. The $q=0$ component ($z$, hence $\Delta m=0$) radiates
linearly polarized light along the quantization axis, the $\pi$ component. The
$q=\pm 1$ components ($x\pm iy$, hence $\Delta m=\pm 1$) radiate circularly
polarized light, the $\sigma^{\pm}$ components. This polarization structure is
exactly what the [Zeeman effect](/atomic-physics/atoms-in-external-fields/zeeman-effect)
resolves when a magnetic field splits the $m$ sublevels.

> **Definition (Electric-dipole selection rules).** A one-electron E1 transition
> has a nonzero dipole matrix element only if
> $$
> \Delta\ell = \pm 1,
> \qquad
> \Delta m = 0,\ \pm 1,
> \qquad
> \Delta s = 0,
> \qquad
> \Delta m_s = 0,
> $$
> with a required change of parity. The spin rules follow because the operator
> $\vec r$ does not act on spin, so the spin state must be unchanged for the
> spin overlap $\braket{s_f m_{s,f}}{s_i m_{s,i}}$ to survive.

## Total angular momentum and the photon's spin

An emitted photon carries one unit of angular momentum. Conservation of total
angular momentum between the atom-plus-photon initial and final states restricts
how the atom's total angular momentum $j$ can change. Adding the photon's spin-1 to
the initial $j_i$ by the triangle rule gives

$$
\Delta j = 0,\ \pm 1,
\qquad
j_i = 0 \not\to j_f = 0,
$$

the $0\to 0$ case excluded because a single spin-1 photon cannot connect two
scalar states (there is no way to carry off one unit of angular momentum). The
same argument applied to the projection gives $\Delta m_j = 0,\pm1$. In
[LS-coupled](/atomic-physics/many-electron-atoms/ls-jj-coupling-term-symbols)
many-electron atoms the rules extend to the term symbols: because the dipole
operator is a sum of one-electron position operators, a single electron changes its
orbital by one and the spin is a spectator, giving

$$
\Delta L = 0,\ \pm 1\ (L=0\not\to0),
\qquad
\Delta S = 0,
\qquad
\Delta J = 0,\ \pm 1\ (J=0\not\to0),
$$

with a parity change and $\Delta\ell = \pm 1$ for the jumping electron. The
$\Delta S = 0$ rule forbids **intercombination** lines between different spin
multiplicities, which is why singlet-triplet transitions in helium are so weak,
and it fails only when spin-orbit coupling mixes the multiplicities (progressively
for heavier atoms).

| Rule | Quantity | Allowed change | Origin |
| --- | --- | --- | --- |
| Parity | $\pi = (-1)^{\sum\ell}$ | must change | $\vec r$ is odd |
| Orbital | $\Delta\ell$ | $\pm 1$ | Gaunt integral, parity |
| Magnetic | $\Delta m,\ \Delta m_j$ | $0,\pm 1$ | rank-1 azimuthal integral |
| Total $J$ | $\Delta J$ | $0,\pm 1$ ($0\not\to0$) | photon spin 1 |
| Total $L$ | $\Delta L$ | $0,\pm 1$ ($0\not\to0$) | one electron jumps |
| Spin | $\Delta S$ | $0$ | $\vec r$ spin-independent |

A Grotrian diagram makes the rules visible: allowed lines connect adjacent orbital
columns, and no allowed line runs vertically within a single column.

$$
% caption: A Grotrian diagram. Solid arrows are allowed E1 transitions between
% adjacent orbital columns (Delta-ell = one); the dashed vertical transition within
% the s column violates the parity and orbital rules and is dipole-forbidden.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% column headers
\node[anchor=south] at (1,4.2) {$s$};
\node[anchor=south] at (4,4.2) {$p$};
\node[anchor=south] at (7,4.2) {$d$};
% s levels
\draw[black, very thick] (0.2,0.8) -- (1.8,0.8) node[right, black]{$1s$};
\draw[black, very thick] (0.2,2.6) -- (1.8,2.6) node[right, black]{$2s$};
\draw[black, very thick] (0.2,3.6) -- (1.8,3.6) node[right, black]{$3s$};
% p levels
\draw[black, very thick] (3.2,2.0) -- (4.8,2.0) node[right, black]{$2p$};
\draw[black, very thick] (3.2,3.3) -- (4.8,3.3) node[right, black]{$3p$};
% d levels
\draw[black, very thick] (6.2,3.5) -- (7.8,3.5) node[right, black]{$3d$};
% allowed transitions (adjacent columns)
\draw[acc, very thick, ->] (3.6,2.0) -- (1.4,0.8);
\draw[acc, very thick, ->] (3.6,3.3) -- (1.4,2.6);
\draw[acc, very thick, ->] (6.6,3.5) -- (4.4,2.0);
\node[acc, anchor=south, rotate=20] at (2.6,1.5) {allowed};
% forbidden within-column
\draw[black, very thick, dashed, ->] (1.0,2.6) -- (1.0,0.9);
\node[black, anchor=west] at (1.05,1.7) {forbidden};
\end{tikzpicture}
$$

> **Worked example (Decay channels of hydrogen $3d$).** The $3d$ state has
> $\ell=2$. The parity and orbital rules require $\Delta\ell = \pm1$, so the only
> lower states it can reach by an electric dipole are those with $\ell=1$: here
> $2p$. Decay to $2s$ or $1s$ ($\ell=0$) has $\Delta\ell=-2$, forbidden by both
> parity and the orbital rule; there is no $\ell=3$ state below $3d$ to take the
> $\Delta\ell=+1$ branch. The single allowed E1 channel is therefore
> $$
> 3d \to 2p,
> $$
> after which $2p\to1s$ carries the atom to the ground state. An electron excited
> to $3d$ cannot fall directly to the ground state; it must cascade through $2p$,
> a rule that shapes the observed intensity ratios of a hydrogen spectrum.

### The Wigner-Eckart theorem and hyperfine rules

The angular selection rules are one instance of the **Wigner-Eckart theorem**: a
rank-$k$ spherical tensor operator has matrix elements between angular-momentum
eigenstates that factor into a geometric Clebsch-Gordan coefficient and a
state-independent reduced matrix element. The dipole operator is a rank-1 tensor,
so its Clebsch-Gordan factor vanishes unless the triangle rule $\abs{j_f-j_i}\le 1
\le j_f+j_i$ and $\Delta m_j = 0,\pm1$ hold — exactly the $\Delta j$ and $\Delta
m_j$ rules. The same theorem applied to the total angular momentum $F = I + J$ of a
[hyperfine](/atomic-physics/qed-corrections-and-hyperfine-structure/hyperfine-structure-21cm)
level gives the further rule

$$
\Delta F = 0,\ \pm 1,
\qquad
F=0 \not\to F=0,
$$

for an E1 transition, because the photon still carries one unit of angular
momentum regardless of how the nuclear spin is coupled in. The reduced matrix
element carries all the radial physics; the geometry is universal.

## Higher multipoles

The selection rules are rules for the leading E1 term. When the dipole element
vanishes, the neglected term $i\vec k\cdot\vec r$ in the expansion of $e^{i\vec
k\cdot\vec r}$ can still drive the transition. That term splits into two pieces
with distinct symmetry:

- **Magnetic dipole (M1).** The antisymmetric part couples to the magnetic moment
  operator $-\mu_B(\vec L + 2\vec S)/\hbar$. It has **even** parity, so it connects
  states of the **same** parity ($\Delta\ell = 0$), with $\Delta j = 0,\pm1$ and
  $\Delta m = 0,\pm1$. It drives transitions within a fine-structure or hyperfine
  multiplet, including the [21 cm
  line](/atomic-physics/qed-corrections-and-hyperfine-structure/hyperfine-structure-21cm).
- **Electric quadrupole (E2).** The symmetric part couples to the quadrupole
  moment $\sim r^2 Y_2^q$. It also has even parity ($\Delta\ell = 0,\pm 2$, but
  $0\not\to0$ and $\ell=0\not\to\ell=0$), with $\Delta j = 0,\pm1,\pm2$ (and
  $0\not\to0,1$).

Both are suppressed relative to E1 by the small factor that measures the field's
variation across the atom. In rate the suppression is of order

$$
\frac{A_{\text{M1}}}{A_{\text{E1}}} \sim \alpha^2 \sim 5\times10^{-5},
\qquad
\frac{A_{\text{E2}}}{A_{\text{E1}}} \sim (k a_0)^2 \sim (\alpha Z)^2,
$$

because each higher multipole carries an extra factor of $ka_0 \sim \alpha Z$ in
amplitude, hence its square in rate. A "forbidden" line is not strictly forbidden;
it is weaker by four or five orders of magnitude, and it appears when the geometry
suppresses competing allowed decays (in dilute astrophysical plasmas, or in
trapped single ions where an E2 or M1 clock transition is a feature rather than a
defect).

$$
% caption: Radiative-rate hierarchy. Each successive multipole (E1, M1, E2) is
% suppressed by roughly the square of the fine-structure constant, spanning about
% ten orders of magnitude in transition rate.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black] (-0.3,0) -- (9.2,0);
% E1
\fill[acc!18, draw=acc, very thick] (0.4,0) rectangle (1.8,3.4);
\node[anchor=south] at (1.1,3.43) {E1};
\node[anchor=north, black] at (1.1,-0.1) {$10^8$ per s};
% M1
\fill[black, draw=black, very thick] (3.4,0) rectangle (4.8,1.7);
\node[anchor=south] at (4.1,1.73) {M1};
\node[anchor=north, black] at (4.1,-0.1) {$10^3$ per s};
% E2
\fill[black, draw=black, very thick] (6.4,0) rectangle (7.8,1.0);
\node[anchor=south] at (7.1,1.03) {E2};
\node[anchor=north, black] at (7.1,-0.1) {$1$ to $10^2$ per s};
% suppression arrows
\draw[black, ->] (1.9,2.6) .. controls (2.6,2.9) and (3.0,2.4) .. (3.4,1.5);
\node[black, anchor=south] at (2.7,2.75) {weaker};
\draw[black, ->] (4.9,1.3) .. controls (5.6,1.6) and (6.0,1.2) .. (6.4,0.9);
\node[black, anchor=south] at (5.7,1.5) {weaker};
\end{tikzpicture}
$$

## Metastable states

A state with no allowed E1 decay to any lower level cannot radiate at the strong
$10^8\ \mathrm{s^{-1}}$ rate. Its only decay channels are the suppressed multipoles
or multi-photon processes, so it lives orders of magnitude longer than an ordinary
excited state. Such a state is **metastable**.

The canonical example is the hydrogen $2s_{1/2}$ level. The only lower state is
$1s_{1/2}$, and the $2s\to1s$ transition has $\Delta\ell = 0$: dipole-forbidden by
parity. Nor can it proceed by M1 or E2 to any useful rate (the M1 element between
$2s$ and $1s$ nearly vanishes because the radial wavefunctions are nearly
orthogonal to the operator). The dominant decay is **two-photon emission**,
$2s\to1s + 2\gamma$, with a rate of about $8\ \mathrm{s^{-1}}$, giving a lifetime of
$0.12\ \mathrm{s}$ — roughly a hundred million times the few-nanosecond lifetime of
the neighboring $2p$ state.

$$
% caption: A metastable level. The 2s state cannot decay to 1s by an electric
% dipole (Delta-ell = 0, parity forbidden); the 2p state decays promptly in
% nanoseconds, while 2s survives about 0.1 s via two-photon emission.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% levels
\draw[black, very thick] (0.2,0.4) -- (2.0,0.4) node[right, black]{$1s$};
\draw[black!70, very thick] (0.2,3.2) -- (2.0,3.2) node[right, black]{$2s$ (metastable)};
\draw[black, very thick] (4.6,3.2) -- (6.4,3.2) node[right, black]{$2p$};
% forbidden 2s -> 1s
\draw[black, very thick, dashed, ->] (0.7,3.2) -- (0.7,0.55);
\node[black, anchor=east] at (0.55,1.9) {forbidden};
% two-photon decay
\draw[black, very thick, ->] (1.5,3.2) .. controls (2.6,2.2) and (2.6,1.4) .. (1.5,0.5);
% prompt 2p decay
\draw[acc, very thick, ->] (5.1,3.2) -- (1.9,0.5);
\node[acc, anchor=south, rotate=-32] at (3.7,2.1) {allowed, ns};
% two-photon label below the ground level
\node[black, anchor=north] at (3.2,-0.1) {two photons, about 0.1 s};
\end{tikzpicture}
$$

Metastable states are the raw material of much of atomic physics. Population
accumulates in them because it cannot leak away quickly, and that is the
condition for a
[population inversion](/atomic-physics/lasers-and-spectroscopy/laser-principles) in
a laser. The metastable $2^3S$ level of helium stores energy in discharge lamps
and is the reservoir for the helium-neon laser; metastable levels of trapped ions,
connected to the ground state by an ultra-narrow E2 transition, are the reference
oscillators of [optical
clocks](/atomic-physics/modern-atomic-physics/optical-clocks-precision). The long
lifetime that makes a transition spectroscopically faint also makes its natural
line width extraordinarily narrow, the theme of the final lesson.

The selection rules are the grammar of atomic spectra. A term diagram with its
levels drawn is not yet a spectrum; the selection rules decide which of the many
conceivable transitions actually appear, how strong each is, and how it is
polarized. What they leave over — the forbidden lines, the metastable reservoirs —
is not an afterthought but the working substance of lasers, of the interstellar
medium's emission, and of the most precise clocks yet built. The last lesson turns
from which lines appear to the shape each line has once it does.