Atomic Selection Rules

An atomic selection rule is a statement that a transition matrix element vanishes under specified symmetries, state labels, and interaction operators. It is not a rule about energy differences, and it is not a declaration that a transition can never occur.

The qualification matters. A line may be electric-dipole forbidden but magnetic-dipole allowed, electric-quadrupole allowed, two-photon allowed, or weakly opened by state mixing. Conversely, satisfying the familiar changes in angular momentum and parity does not guarantee a bright line: the reduced matrix element can be small or zero, populations can be absent, and destructive interference can suppress the amplitude.

Required background. Atomic orbitals, fine structure, and hyperfine structure define the state labels. The Wigner–Eckart theorem and dipole transitions give the tensor algebra used below.

Transition Matrix Elements

In zero external field, an atomic level may be labeled schematically as

$$|\gamma J M_J\pi\rangle,$$

where $\gamma$ collects configuration, radial, term, and other labels, $J$ is total electronic angular momentum, $M_J$ its laboratory-axis projection, and $\pi=\pm1$ the parity. If hyperfine coupling is resolved and weak fields preserve $F$ as a useful label, write

$$|\gamma J,I;F M_F,\pi\rangle, \qquad \mathbf F=\mathbf J+\mathbf I.$$

These labels are not decorations. A rule can use only quantum numbers that are good in the Hamiltonian being applied. In a magnetic-field crossover, for example, $F$ may be mixed even though the projection $M_F$ remains conserved.

Let $T_q^{(k)}$ be the $q$ component of an irreducible tensor operator of rank $k$. With the Wigner–Eckart convention used here,

$$\begin{aligned} \mathcal M_{fi}^{(kq)} \equiv{}& \langle \gamma_fJ_fM_f| T_q^{(k)}|\gamma_iJ_iM_i\rangle, \\ \mathcal M_{fi}^{(kq)} ={}& (-1)^{J_f-M_f} \begin{pmatrix} J_f & k & J_i\\ -M_f & q & M_i \end{pmatrix} \times \langle \gamma_fJ_f \|T^{(k)}\| \gamma_iJ_i\rangle . \end{aligned}$$

The two factors answer different questions:

• the $3j$ symbol contains the projection rule, triangle condition, and sublevel-dependent angular strength;
• the reduced matrix element contains radial overlap, configuration composition, spin and relativistic structure, and any additional dynamical zeros.

The projection condition is

$$M_f=M_i+q,$$

while the full triangle condition is

$$|J_f-J_i| \leq k \leq J_f+J_i.$$

The shorthand $|\Delta J|\leq k$ omits the second inequality and can fail for small $J$. For example, a rank-$2$ operator cannot connect $J_i=1/2$ to $J_f=1/2$, even though $\Delta J=0$.

For an electric-dipole transition, the spontaneous-emission rate averaged over an unpolarized upper manifold and summed over lower substates and photon polarizations is

$$A_{i\to f}^{(E1)} = \frac{\omega_{if}^3} {3\pi\epsilon_0\hbar c^3} \frac{ \left| \langle \gamma_fJ_f \|d^{(1)}\| \gamma_iJ_i\rangle \right|^2 }{2J_i+1}.$$

This formula makes two limitations visible. Selection rules test whether the matrix element vanishes, but the rate also depends on its magnitude and on $\omega_{if}^3$. Measured line intensity additionally depends on state populations, branching, optical depth, driving strength, and detection geometry.

A Hierarchy of Rules

Atomic rule tables combine statements with different logical status. Keeping them separated prevents an approximate coupling label from being mistaken for an exact symmetry.

Rule type	Typical labels	Status
rotational rank	$J,M_J$ or $F,M_F$	exact when the stated rotational symmetry and tensor operator apply
parity	$\pi$	exact for a parity-invariant atomic Hamiltonian and an operator of definite parity
hyperfine recoupling	$F,M_F$	exact within a fixed $J,I$ manifold when $F$ is a good label
$LS$-coupling rules	$L,S$	approximate when levels are close to pure $LS$ terms
one-electron orbital rule	$\ell$ of an active electron	model-dependent independent-particle statement
configuration rule	occupied orbitals	approximate; configuration interaction mixes configurations
field-free rule	$J,F,\pi$	may change when external fields mix the corresponding sectors

Apply the exact rank, projection, and parity tests first; use one-electron and LS rules only within their stated approximation. A surviving angular factor still requires a reduced matrix element.

Electric-Dipole Transitions

In the long-wavelength approximation, the leading interaction with an electric field is

$$V_{E1}(t) = -\mathbf d\mathbin{\cdot}\mathbf E(t), \qquad \mathbf d=\sum_a e_a\mathbf r_a.$$

The electric dipole is a rank-$1$ polar tensor and is odd under inversion:

$$\mathsf P\mathbf d\mathsf P^{-1} = -\mathbf d.$$

For levels labeled by total electronic angular momentum, the rigorous field-free $E1$ rules are

$$\pi_f=-\pi_i,$$ $$|J_f-J_i| \leq1 \leq J_f+J_i,$$

and

$$M_f=M_i+q, \qquad q=0,\pm1.$$

The triangle condition is often summarized as

$$\Delta J=0,\pm1, \qquad J=0\not\leftrightarrow J=0.$$

In a pure $LS$-coupling description, the leading nonrelativistic dipole operator does not act on spin. Within that idealization the additional rules are exact; they become approximate for physical levels with spin–orbit admixture:

$$\Delta S=0,$$

and

$$\Delta L=0,\pm1, \qquad L=0\not\leftrightarrow L=0.$$

For one active electron in central-field orbitals, combining rank and parity gives

$$\Delta\ell=\pm1,$$

while $\Delta n$ is unrestricted by angular symmetry. The radial integral may nevertheless be small or vanish through a Cooper minimum or another cancellation. In a many-electron atom, “one electron changes orbital with $\Delta\ell=\pm1$” is an independent-particle rule for a one-body operator, not an exact statement about correlated eigenstates. LS Coupling derives the corresponding pure-term rules and shows how same-$J^\pi$ mixing opens intercombination strength.

Common examples are:

Transition class	$E1$ status	Reason
hydrogen $1s\leftrightarrow2p$	allowed	opposite parity and $\Delta\ell=1$
hydrogen $1s\leftrightarrow2s$	forbidden	same parity and $\Delta\ell=0$
alkali $nS_{1/2}\leftrightarrow n'P_{1/2,3/2}$	allowed	D-line electric-dipole transitions
$J=0\leftrightarrow J'=0$	forbidden	rank-$1$ triangle condition
$S\leftrightarrow D$ in a one-electron picture	forbidden at $E1$ order	same parity and $\Delta\ell=2$; $E2$ may contribute

An $E1$-allowed classification still leaves the reduced matrix element, branching ratios, and experimental preparation to be determined.

Hyperfine-Resolved Electric-Dipole Rules

The electronic dipole operator does not act directly on nuclear spin. For a fixed isotope and ordinary electronic transition,

$$\Delta I=0.$$

When $F$ is a good quantum number, the rank-$1$ rule recouples to

$$\Delta F=0,\pm1, \qquad F=0\not\leftrightarrow F'=0,$$

and

$$\Delta M_F=q.$$

These hyperfine rules do not override an electronic zero. If the reduced electronic $J=0\leftrightarrow J'=0$ matrix element vanishes, merely coupling both states to $I\ne0$ does not make the leading unmixed $E1$ matrix element nonzero.

For the same nuclear spin $I$, the relative hyperfine line strengths follow from a $6j$ coefficient. Up to the reduced electronic matrix element,

$$\left| \langle \gamma_fJ_f,I;F_f \|d^{(1)}\| \gamma_iJ_i,I;F_i\rangle \right|^2 = (2F_f+1)(2F_i+1) \begin{Bmatrix} J_f & F_f & I\\ F_i & J_i & 1 \end{Bmatrix}^{\!2} \times \left| \langle \gamma_fJ_f \|d^{(1)}\| \gamma_iJ_i\rangle \right|^2 .$$

The $6j$ factor redistributes one electronic line among hyperfine components. It does not determine the electronic radial matrix element. Hyperfine Structure develops the coupling hierarchy, while Alkali Atoms places these components in the D-line structure.

Parity Rules

For a parity-conserving atomic Hamiltonian, every stationary level can be assigned $\pi=\pm1$. In a configuration description,

$$\pi_{\mathrm{config}} = (-1)^{\sum_a\ell_a}.$$

Configuration interaction generated by parity-conserving interactions mixes only configurations of the same total parity. A state can therefore be strongly configuration mixed while retaining an exact parity label.

If an operator $T$ has parity $\eta_T$,

$$\mathsf PT\mathsf P^{-1} = \eta_TT,$$

then a necessary condition for a nonzero matrix element is

$$\pi_f\pi_i=\eta_T.$$

The leading radiative cases are:

Process	Rank	Operator parity	Level parity relation
$E1$	$1$	odd	opposite parity
$M1$	$1$	even	same parity
$E2$	$2$	even	same parity
$M2$	$2$	odd	opposite parity

A magnetic field is an axial vector, so the ordinary Zeeman interaction preserves parity even while it mixes $J$ or $F$. A static electric field couples through the odd dipole operator and mixes opposite parity. The appropriate rule must therefore be applied to the field-dressed eigenstates, not to labels that ceased to be exact. Stark Effect in Atoms develops that mixing.

Magnetic-Dipole and Electric-Quadrupole Transitions

If the $E1$ amplitude vanishes, the next one-photon channels are often $M1$ or $E2$. Their symmetry labels are general; their detailed operators and normalizations are treated in Multipole Operators.

Magnetic dipole

The leading nonrelativistic electronic magnetic moment is schematically

$$\boldsymbol\mu \simeq -\mu_B \frac{\mathbf L+g_S\mathbf S}{\hbar},$$

with nuclear and relativistic terms added as required. It is a rank-$1$ axial tensor and has even parity. Thus $M1$ transitions obey

$$\pi_f=\pi_i,$$ $$|J_f-J_i| \leq1 \leq J_f+J_i,$$

and

$$M_f=M_i+q, \qquad q=0,\pm1.$$

In a pure nonrelativistic $LS$ basis, the leading long-wavelength operator preserves $L$ and $S$ and does not change radial occupations. Relativistic corrections, finite wavelength, and mixed configurations can alter these extra restrictions. Consequently, strong $M1$ lines commonly occur within fine-structure or hyperfine manifolds. The hydrogen $21\ \mathrm{cm}$ hyperfine line is an $M1$ transition; its very low frequency further suppresses its spontaneous rate through the photon phase space.

Electric quadrupole

A convenient electric-quadrupole tensor is

$$Q_q^{(2)} = \sum_a e_a r_a^2 C_q^{(2)}(\hat{\mathbf r}_a).$$

It has rank $2$ and even parity. Its rigorous angular rules are

$$\pi_f=\pi_i,$$ $$|J_f-J_i| \leq2 \leq J_f+J_i,$$

and

$$M_f=M_i+q, \qquad q=0,\pm1,\pm2.$$

The full triangle condition automatically excludes $0\leftrightarrow0$, $0\leftrightarrow1$, and $1/2\leftrightarrow1/2$ at $E2$ order. In a one-electron central-field model, parity and rank permit $\Delta\ell=0,\pm2$, subject again to the full triangle condition. The $S_{1/2}\leftrightarrow D_{5/2}$ clock transition in ions such as $\mathrm{Ca}^+$ is a standard $E2$ example.

For atomic size $a$ and photon wavenumber $k_\gamma$, the long-wavelength expansion has

$$k_\gamma a\ll1.$$

Higher electric multipoles carry additional powers of $k_\gamma a$ in their amplitudes, while a typical $M1$ amplitude is relativistically small compared with an ordinary allowed $E1$ amplitude. These are parametric expectations, not rules. Once $E1$ is zero, a nominally weaker channel can control the lifetime.

“Forbidden” always refers to a specified operator and model. An $E1$ line connects opposite parity at leading order; $M1$ and $E2$ can connect same-parity levels; and a perturbation can admix an opposite-parity component with amplitude $\varepsilon$, producing an $E1$ rate that is typically proportional to $|\varepsilon|^2$.

Polarization and Magnetic-Sublevel Rules

Choose a quantization axis and define spherical basis vectors by

$$\mathbf e_0=\mathbf e_z, \qquad \mathbf e_{\pm1} = \mp\frac{ \mathbf e_x\pm i\mathbf e_y }{\sqrt2}.$$

The scalar product of two vectors is

$$\mathbf d\mathbin{\cdot}\boldsymbol\epsilon = \sum_{q=-1}^{1} (-1)^q d_q\epsilon_{-q}.$$

For a transition driven by the operator component $T_q^{(k)}$,

$$\Delta M=q.$$

In the common atomic convention:

Operator component	Polarization label	Projection change
$q=0$	$\pi$	$\Delta M=0$
$q=+1$	$\sigma^+$	$\Delta M=+1$
$q=-1$	$\sigma^-$	$\Delta M=-1$

The association between $\sigma^\pm$ and photon helicity depends on propagation direction, absorption versus emission, and active-versus-passive conventions. The invariant statement is the value of $q$ used in the matrix element.

Linear polarization parallel to the quantization axis is pure $\pi$. Linear polarization perpendicular to that axis is a coherent superposition of $\sigma^+$ and $\sigma^-$. Circular polarization about the quantization axis selects one spherical component, subject to the convention just stated.

For fixed $M_i,M_f,q$, the relative angular strength is

$$S_q(M_i,M_f) \propto{} \left| \begin{pmatrix} J_f & k & J_i\\ -M_f & q & M_i \end{pmatrix} \right|^2 \times \left| \langle \gamma_fJ_f \|T^{(k)}\| \gamma_iJ_i\rangle \right|^2 .$$

The rule $\Delta M=q$ is necessary but not sufficient: an individual $3j$ coefficient can still vanish. For a rank-$1$ transition with $J_f=J_i$, for example, the $M=0\to M'=0$ component vanishes.

Polarization can also create nearly closed cycling transitions. Repeated $\sigma^+$ absorption on a suitable $F\to F+1$ line optically pumps population toward a stretched state. Closure is only as good as the level model: off-resonant hyperfine excitation, polarization impurity, magnetic-field misalignment, collisions, or decay to other manifolds creates leakage.

Forbidden Transitions and Metastable States

A state is metastable when every energetically open decay channel is suppressed enough to produce a long lifetime. For independent decay channels, its total population-decay rate $\Gamma_i$ (in $\mathrm{s}^{-1}$) is a sum over radiative and nonradiative contributions:

$$\begin{aligned} \Gamma_i ={}& \Gamma_i^{\mathrm{rad}} +\Gamma_i^{\mathrm{coll}} +\Gamma_i^{\mathrm{field}} +\cdots, \\ \Gamma_i^{\mathrm{rad}} ={}& \sum_f A_{if}^{E1} +\sum_f A_{if}^{M1} +\sum_f A_{if}^{E2} +\cdots . \end{aligned}$$

and

$$\tau_i=\Gamma_i^{-1}.$$

No single selection rule fixes $\tau_i$. A channel that is tiny compared with an allowed optical $E1$ rate may still dominate when all faster channels vanish.

Three examples expose different mechanisms:

• Hydrogen $2s\to1s$: one-photon $E1$ decay is forbidden because both states have even parity and $\Delta\ell=0$. The level decays mainly through a second-order two-photon $E1$–$E1$ process, giving a lifetime of about $0.12\ \mathrm{s}$ in isolated hydrogen.
• Ionic $S\leftrightarrow D$ clock lines: the same-parity $E1$ amplitude vanishes, but an $E2$ matrix element can drive the transition and set the radiative lifetime.
• Alkaline-earth-like $^1S_0\leftrightarrow{}^3P_0$ clock lines: the leading $E1$ amplitude is blocked by $J=0\leftrightarrow0$ and, in a pure $LS$ picture, by $\Delta S\ne0$. In fermionic isotopes, hyperfine mixing weakly opens the clock transition; in bosonic isotopes, controlled magnetic-field mixing can be used.

Metastability is environment dependent. Collisions can quench a state, blackbody radiation can transfer population, electric fields can mix parity, and a probe can power broaden or depopulate the level. A vacuum lifetime and an observed trap lifetime need not be the same.

How Perturbations Open a Forbidden Line

Suppose a perturbation $W$ admixes other field-free states into the initial level:

$$|\widetilde i\rangle = |i\rangle + \sum_{a\ne i} \frac{ \langle a|W|i\rangle }{ E_i-E_a } |a\rangle +\cdots.$$

A corresponding expression holds for $|\widetilde f\rangle$. The dressed transition amplitude becomes

$$\langle\widetilde f|T|\widetilde i\rangle \simeq{} \langle f|T|i\rangle + \sum_a \frac{ \langle f|T|a\rangle \langle a|W|i\rangle }{ E_i-E_a } + \sum_b \frac{ \langle f|W|b\rangle \langle b|T|i\rangle }{ E_f-E_b }.$$

If the unperturbed amplitude is zero but one admixture has size $\varepsilon$ and connects through an ordinary allowed matrix element, then

$$\mathcal M_{\mathrm{borrowed}} \sim \varepsilon\mathcal M_{\mathrm{allowed}},$$

For comparable transition frequencies, the borrowed rate is typically

$$\Gamma_{\mathrm{borrowed}} \sim |\varepsilon|^2 \Gamma_{\mathrm{allowed}}.$$

When the borrowed and reference optical frequencies differ, the E1 rate ratio also contains $(\omega_{\mathrm{borrowed}}/\omega_{\mathrm{allowed}})^3$. Small energy denominators can make the mixing much larger than a naive perturbation estimate. In that regime, diagonalize the coupled subspace and compute matrix elements between the actual eigenvectors.

Important mechanisms include:

Mixing or new process	What changes	Typical consequence
spin–orbit coupling	mixes nominal spin terms with the same $J,\pi$	intercombination lines acquire $E1$ strength
electrostatic configuration interaction	mixes configurations with common $L,S,J,\pi$ in a spin-independent model	changes radial amplitudes, but cannot by itself violate $\Delta S=0$
hyperfine interaction	mixes electronic $J$ character while coupling to $I$	hyperfine-induced clock transitions
static electric field	mixes opposite parity	parity-forbidden $E1$ amplitudes appear
magnetic field	mixes $J$ or $F$ but preserves parity	angular rules based on unmixed labels weaken
collisions or anisotropic environment	reduces isolated-atom symmetry and adds decay	quenching and pressure-induced lines
finite-wavelength terms	introduces $M1$, $E2$, and higher multipoles	one-photon decay beyond $E1$
two-photon coupling	uses a second-order composite amplitude	transitions such as hydrogen $2s\to1s$

The rule has not mysteriously failed. Either the eigenstates, the interaction operator, or the symmetry of the full problem has changed.

The NIST Atomic Spectra Database provides evaluated energies and transition probabilities where available. A Ritz wavelength inferred from level energies locates a possible line; it does not predict its intensity. For molecular and multiphoton processes, Selection Rules in Spectroscopy extends the operator-based reasoning.

Exercises

Exercise 1: Name the mechanism

Classify each transition at leading one-photon order: hydrogen $1s\to2p$, hydrogen $1s\to2s$, an atomic $J=0\to J'=0$ opposite-parity line, and a one-electron $S_{1/2}\to D_{5/2}$ same-parity line. State whether $E1$, $M1$, or $E2$ is allowed by rank and parity.

Solution

For hydrogen $1s\to2p$, parity changes and $\Delta\ell=1$, so $E1$ is allowed.

For hydrogen $1s\to2s$, the states have the same parity and $\Delta\ell=0$. $E1$ is forbidden. Rank and parity allow $M1$ between their $J=1/2$ levels, but its leading nonrelativistic long-wavelength matrix element vanishes by radial orthogonality. $E2$ is excluded by $J_i+J_f=1<2$. The observed decay proceeds mainly by two-photon emission.

For opposite-parity $J=0\to J'=0$, parity is suitable for $E1$, but a rank-$1$ operator cannot connect two $J=0$ states. $M1$ has the same rank problem and also the wrong parity. $E2$ has the wrong parity and its rank-$2$ triangle condition also excludes $0\to0$.

For $S_{1/2}\to D_{5/2}$, the states have the same parity. $E1$ is forbidden. $M1$ has even parity but rank $1$, so $J=1/2\to5/2$ is excluded. $E2$ has even parity and satisfies

$$\left|\frac52-\frac12\right| =2 \leq2 \leq3,$$

so it is allowed by rank and parity.

Exercise 2: Polarization from a scalar initial state

An $E1$ transition starts from $J_i=0,M_i=0$ and ends in $J_f=1$. Which final $M_f$ is driven by each operator component $q=0,+1,-1$? Are the three components distinguished by reduced atomic dynamics?

Solution

The projection rule gives

$$M_f=M_i+q=q.$$

Thus $q=0$ drives $M_f=0$, $q=+1$ drives $M_f=+1$, and $q=-1$ drives $M_f=-1$. All three share the same reduced matrix element

$$\langle \gamma_f,J_f=1 \|d^{(1)}\| \gamma_i,J_i=0\rangle.$$

Rotational symmetry makes their reduced dynamics identical. Their laboratory excitation strengths differ only through the available field polarization and geometry; with equal pure spherical-component amplitudes, the three $3j$ coefficients have equal magnitude.

Exercise 3: Hyperfine branches

An isotope has $I=3/2$ and an electronic transition $J_i=1/2\to J_f=1/2$. The initial hyperfine level is $F_i=1$. List the $F_f$ values that exist and identify which satisfy the $E1$ hyperfine rule.

Solution

Coupling $J_f=1/2$ to $I=3/2$ gives

$$F_f=|I-J_f|,\ldots,I+J_f=1,2.$$

From $F_i=1$, the rank-$1$ rule permits $\Delta F=0,\pm1$, excluding only $0\leftrightarrow0$. Both existing final values,

$$\begin{gathered} F_i=1\to F_f=1,\\ F_i=1\to F_f=2. \end{gathered}$$

are therefore allowed by hyperfine angular momentum. Their relative strengths require the relevant $6j$ factors and the common reduced electronic matrix element.

Exercise 4: Intensity borrowing

A perturbation admixes an $E1$-allowed opposite-parity state into a nominally forbidden upper state with amplitude $\varepsilon=2\times10^{-3}$. Assume comparable photon frequencies and neglect interference and other channels. Estimate the borrowed rate as a fraction of the fully allowed rate.

Solution

The borrowed amplitude is approximately

$$\mathcal M_{\mathrm b} \simeq \varepsilon\mathcal M_{\mathrm a}.$$

Rates are proportional to squared amplitudes, so

$$\frac{\Gamma_{\mathrm b}} {\Gamma_{\mathrm a}} \simeq |\varepsilon|^2 = 4\times10^{-6}.$$

A part-per-thousand state admixture can therefore create a part-per-million radiative rate relative to an otherwise comparable allowed line.

Exercise 5: A quenched metastable lifetime

An isolated metastable state has an $E2$ decay rate $0.80\ \mathrm{s}^{-1}$ and all other vacuum channels total $0.05\ \mathrm{s}^{-1}$. In a buffer gas, collisions add a quenching rate $0.15\ \mathrm{s}^{-1}$. Find the vacuum and buffer-gas lifetimes.

Solution

In vacuum,

$$\Gamma_{\mathrm{vac}} = 0.80+0.05 = 0.85\ \mathrm{s}^{-1},$$

so

$$\tau_{\mathrm{vac}} = \frac{1}{0.85\ \mathrm{s}^{-1}} \simeq1.18\ \mathrm{s}.$$

With collisional quenching,

$$\Gamma_{\mathrm{gas}} = 0.85+0.15 = 1.00\ \mathrm{s}^{-1},$$

and $\tau_{\mathrm{gas}}=1.00\ \mathrm{s}$. Selection-rule suppression of fast radiative channels does not protect the state from environmental decay.

Exercise 6: Use the full rank-2 triangle

For an $E2$ operator, classify the angular-momentum pairs $1/2\to1/2$, $1/2\to3/2$, $0\to1$, and $0\to2$ using the full triangle condition.

Solution

For rank $k=2$, require

$$|J_f-J_i| \leq2 \leq J_f+J_i.$$

For $1/2\to1/2$, the upper bound is $J_f+J_i=1$, so the transition is excluded. For $1/2\to3/2$, the bounds are $1\leq2\leq2$, so it is allowed by angular momentum. For $0\to1$, the upper bound is $1$, so it is excluded. For $0\to2$, both bounds equal $2$, so it is allowed.

This exercise shows why the shorthand $\Delta J=0,\pm1,\pm2$ is unsafe at small $J$.

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