Multipole Transitions Beyond the Dipole

The electric dipole approximation, built in this chapter's opening articles, keeps only the first crumb of the light–matter coupling: it replaces the field's phase factor $e^{i\mathbf{k}\cdot\mathbf{r}}$ by unity across the whole atom, leaving the single operator $e\mathbf{r}$ . That crumb is enough for the overwhelming majority of spectral lines. But sometimes the E1 matrix element $\langle f|\mathbf{r}|i\rangle$ vanishes exactly — because the states share parity, or because $\Delta J$ exceeds what one dipole photon can carry — and the approximation predicts zero. The transition is then called forbidden: not impossible, only that the leading term is absent and the transition is carried by the next surviving term the dipole approximation discarded. Restore that expansion, $e^{i\mathbf{k}\cdot\mathbf{r}}=1+i\mathbf{k}\cdot\mathbf{r}-\tfrac12(\mathbf{k}\cdot\mathbf{r})^2+\cdots$ , and a whole hierarchy of transition operators tumbles out. The $i\mathbf{k}\cdot\mathbf{r}$ term splits cleanly into a symmetric piece — the electric quadrupole, E2 — and an antisymmetric piece proportional to $\mathbf{L}$ , which with the electron spin becomes the magnetic dipole, M1; higher orders give M2, E3. The expansion parameter is $ka_0\sim\alpha Z$ , so each rung down the ladder costs a factor $(\alpha Z)^2\sim10^{-4}$ in rate. Everything downstream is the same Einstein-coefficient machinery that governs allowed lines: a spontaneous rate $A_{fi}$ , a natural linewidth $\Gamma=A_{fi}$ , a Lorentzian profile. Only the numbers change, and they change by staggering amounts — M1 and E2 rates run $10^{-4}$ to $10^{-15}$ of E1 rates, so lifetimes stretch from nanoseconds to millions of years. Those long lifetimes are precisely why forbidden lines light up dilute nebulae and anchor the world's most accurate atomic clocks.

Key takeaways

When the electric dipole (E1) matrix element vanishes by symmetry, transitions still proceed through the higher terms the dipole approximation discarded — magnetic dipole (M1), electric quadrupole (E2), and beyond.
These come from expanding $e^{i\mathbf{k}\cdot\mathbf{r}}\approx 1+i\mathbf{k}\cdot\mathbf{r}+\cdots$ ; each order is suppressed by $ka_0\sim\alpha Z$ , so rates fall by $\sim(\alpha Z)^2\sim5\times10^{-5}$ per step in light atoms.
Every multipole has definite parity and angular momentum: E1 and M2 flip parity, M1 and E2 preserve it, and a photon of order $L$ carries $L\hbar$ , giving $|J_i-J_f|\le L\le J_i+J_f$ with $J=0\to J=0$ strictly forbidden.
Forbidden lines light up low-density plasmas (nebular $[\mathrm{O\,III}]$ , coronal iron) and underpin optical clocks (E2 in Ca $^+$ /Sr $^+$ , E3 octupole in Yb $^+$ ), whose long lifetimes give ultranarrow linewidths; the 21 cm hydrogen line is an M1 hyperfine transition with $A\approx2.9\times10^{-15}\,\mathrm{s^{-1}}$ , a lifetime near 11 million years.
Key idea: expanding the atom–field coupling beyond the constant phase factor generates a hierarchy of transition operators — M1, E2, M2, E3, ... — each with its own parity and angular-momentum selection rules; when E1 is forbidden, the lowest surviving multipole carries the transition, suppressed by roughly $(\alpha Z)^{2(L-1)}$ .

Where this lives today

The multipole ladder's two extremes now anchor two of the most precise measurement enterprises in science. At the metrology end, optical atomic clocks lock a laser to a forbidden transition whose sub-hertz linewidth gives a quality factor above $10^{15}$ : the E2 line in Ca $^+$ and Sr $^+$ , the year-lived E3 octupole in Yb $^+$ at 467 nm, and the intercombination lines in Al $^+$ , Sr, and Yb lattice clocks. These reach fractional uncertainties near $10^{-18}$ — enough to sense the gravitational redshift across a centimetre of height, redefining "one second" and testing whether the fundamental constants drift with cosmic time, all resting on the multipole suppression that makes the upper level metastable. At the astrophysical end, forbidden lines are the density and temperature thermometers of the diffuse universe. The James Webb Space Telescope reads the forbidden $[\mathrm{O\,III}]$ and $[\mathrm{Ne\,III}]$ lines out to redshift ten to weigh the oxygen in the first galaxies, while radio arrays map the 21 cm line to chart galactic rotation and trace cosmic reionisation. The physics is the same at both ends: a transition too weak to see in any dense sample becomes the brightest feature once the gas is thin enough, or the sample large enough, to radiate at all.

The physical picture

Picture the atom as a tiny antenna the size of a Bohr radius $a_0\approx0.05$ nm inside a light wave hundreds of nanometres long. Because $a_0/\lambda\sim\alpha Z/2\pi\ll1$ , the field looks almost perfectly uniform across the atom; that uniform part is what the dipole approximation keeps, and it drives E1. What the approximation throws away is the variation of the field across the atom — so small it is measured by the phase slip $ka_0\sim\alpha Z\sim10^{-2}$ . A shearing gradient couples to the charge distribution as a quadrupole (E2); the magnetic curl couples to the atomic current loop, its magnetic moment (M1). Forbidden transitions are the physics hiding in that discarded field variation. Parity is the crisp organiser. The position operator $\mathbf{r}$ is odd, so E1 flips parity, while the magnetic moment $\mathbf{L}+g_s\mathbf{S}$ and the quadrupole tensor $r_ir_j$ are even, so M1 and E2 preserve it. A given pair of levels is therefore E1-allowed or (M1/E2)-allowed but essentially never both — parity picks the lane, and once E1 is closed the transition must descend the multipole ladder.

Figure

An atom of size a₀ ≈ 0.05 nm sits in a wave of length λ ≈ 122 nm, so the field barely varies across it — the phase slip is ka₀ ≈ αZ ≈ 7×10⁻³. The uniform part drives E1, the gradient drives E2, and the magnetic curl drives M1, each rung down the ladder costing ≈ αZ squared ≈ 5×10⁻⁵ in rate

Each multipole photon is a definite lump of angular momentum: an E $L$ or M $L$ photon carries exactly $L\hbar$ , so emitting it must balance the atom's books through the triangle rule $|J_i-J_f|\le L\le J_i+J_f$ . A dipole photon ( $L=1$ ) changes $J$ by $0$ or $\pm1$ ; a quadrupole ( $L=2$ ) reaches $\Delta J=\pm2$ , opening channels a dipole cannot. And $J=0\to J=0$ is forbidden to every single photon, since none carries zero angular momentum — such a level decays only by emitting two photons at once. Why so slow? A quadrupole is a source whose net dipole cancels, so it radiates $\sim(ka)^2$ less power than a dipole of the same size, while a magnetic dipole is down by $v/c\sim\alpha Z$ from the relativistic-order magnetic coupling — both giving the universal factor $(\alpha Z)^2$ per rung. Multiply it across two or three rungs and lifetimes explode across the twenty-four orders of magnitude of the second figure, from a nanosecond to eleven million years.

Figure

Radiative lifetimes of real transitions span 24 orders of magnitude: the E1 Lyman-α line 2p→1s at 1.6 ns, the E2 clock transition in Ca⁺ at 729 nm living 1.17 s, the E3 octupole line in Yb⁺ at 467 nm lasting ≈ 6 years, and the M1 hydrogen 21 cm hyperfine line at ≈ 11 million years

The core equations

Everything begins with the factor the dipole approximation set to one. Keeping its full expansion exposes the higher multipoles; the small parameter $\mathbf{k}\cdot\mathbf{r}$ measures how much the wave's phase turns across the atom.

e^{i\mathbf{k}\cdot\mathbf{r}}=1+i(\mathbf{k}\cdot\mathbf{r})-\tfrac12(\mathbf{k}\cdot\mathbf{r})^2+\cdots \qquad (1)

where: $\mathbf{k}$ — photon wavevector, $|\mathbf{k}|=\omega/c$ ; $\mathbf{r}$ — electron position relative to the nucleus; $\mathbf{k}\cdot\mathbf{r}$ — phase variation across the atom, of order $ka_0\sim\alpha Z$ . The '1' is E1; the $i\mathbf{k}\cdot\mathbf{r}$ term carries E2 and M1; the quadratic term M2 and E3. The expansion lives inside the minimal-coupling matrix element that drives every radiative transition. The $\mathbf{A}\cdot\mathbf{p}$ interaction, sandwiched between the states with the mode's phase factor, is what we expand order by order.

H_{\mathrm{int}}=-\frac{e}{m}\,\mathbf{A}\cdot\mathbf{p}\;\Longrightarrow\;M_{fi}\propto\langle f|\,e^{i\mathbf{k}\cdot\mathbf{r}}\,\hat{\boldsymbol\epsilon}\cdot\mathbf{p}\,|i\rangle \qquad (2)

where: $\mathbf{A}$ — vector potential of the mode; $\mathbf{p}$ — electron momentum operator; $\hat{\boldsymbol\epsilon}$ — polarisation unit vector; $e,m$ — electron charge magnitude and mass. Setting $e^{i\mathbf{k}\cdot\mathbf{r}}\to1$ recovers the dipole approximation; keeping more terms recovers the rest. The E1 rate is the reference every forbidden rate is measured against. It is the standard spontaneous-emission result from the Einstein A-coefficient, expressed through the transition dipole moment.

A_{\mathrm{E1}}=\frac{\omega^3}{3\pi\varepsilon_0\hbar c^3}\,\bigl|\langle f|\,e\mathbf{r}\,|i\rangle\bigr|^2 \qquad (3)

where: $\omega$ — transition angular frequency $\omega_{fi}$ ; $\langle f|e\mathbf{r}|i\rangle$ — electric-dipole transition moment; $\varepsilon_0$ — vacuum permittivity. The yardstick: a strong optical E1 line has $A_{\mathrm{E1}}\sim10^8\,\mathrm{s^{-1}}$ and a nanosecond lifetime. The two forbidden rates share the E1 prefactor but pay for their weaker operators. M1 swaps the electric moment $e\mathbf{r}$ for the magnetic moment $\boldsymbol\mu$ , with a $1/c^2$ that makes it weaker by $(\alpha/2)^2$ ; E2 keeps an electric operator but carries an extra factor $(ka)^2$ and a steeper $\omega^5$ dependence, so E2 lines brighten fast at high transition energy.

A_{\mathrm{M1}}=\frac{\omega^3}{3\pi\varepsilon_0\hbar c^3}\frac{\bigl|\langle f|\boldsymbol\mu|i\rangle\bigr|^2}{c^2},\qquad A_{\mathrm{E2}}\sim(k a)^2 A_{\mathrm{E1}}\sim\frac{\omega^5}{c^2}\frac{\bigl|\langle f|Q|i\rangle\bigr|^2}{\varepsilon_0\hbar c^3} \qquad (4)

where: $\boldsymbol\mu$ — magnetic moment $-\mu_B(\mathbf{L}+g_s\mathbf{S})/\hbar$ , with $\mu_B=e\hbar/2m$ , $g_s\approx2$ ; $Q\sim e\,r^2$ — quadrupole moment; $(ka)^2\sim(\alpha Z)^2$ — retardation. Both operators are parity-even, so M1 and E2 connect same-parity states, but only E2 reaches $\Delta J=\pm2$ . Which multipoles can connect two levels is fixed by two exact conservation laws. Angular momentum requires a triangle inequality on $J$ (with $J=0\to0$ ruled out), and parity requires the product of level parities to equal the multipole's intrinsic parity; for one electron these collapse into simple bounds on $\Delta l$ .

|J_i-J_f|\le L\le J_i+J_f\ (J_i{=}J_f{=}0\text{ forbidden});\quad \pi_i\pi_f=(-1)^L\,(\text{E}L),\ (-1)^{L+1}\,(\text{M}L) \qquad (5)

where: $L$ — multipole order; $\pi_{i,f}$ — level parities ( $\pm1$ ). For one electron this reads E1 $\Delta l=\pm1$ , M1 $\Delta l=0$ , E2 $\Delta l=0,\pm2$ ; M1 and E2 share the even-parity rule, and the lowest allowed $L$ usually dominates. The rate is the observable. In the Weisskopf–Wigner picture the metastable level decays exponentially at rate $A_{fi}$ , giving a Lorentzian whose width is the rate itself — so a tiny forbidden rate makes an extraordinarily narrow line and an enormous quality factor.

\Gamma=A_{fi},\qquad \tau=\frac{1}{A_{fi}},\qquad \Delta\nu_{\text{FWHM}}=\frac{A_{fi}}{2\pi},\qquad Q=\omega_0\tau \qquad (6)

where: $\Gamma$ — natural linewidth (FWHM, angular); $\tau$ — upper-level lifetime; $\Delta\nu$ — linewidth in Hz; $Q$ — quality factor. Tiny $A_{fi}$ gives long $\tau$ , narrow $\Gamma$ , huge $Q$ .

Derivation ASplitting the first-order term into a quadrupole and a magnetic dipole

The goal is to show that the single correction the dipole approximation discards — the $i\mathbf{k}\cdot\mathbf{r}$ term — is not one object but two physically distinct ones stitched together by exact algebra. We split the first-order operator product $(\mathbf{k}\cdot\mathbf{r})(\hat{\boldsymbol\epsilon}\cdot\mathbf{p})$ into a symmetric and an antisymmetric half, identifying the symmetric half as the electric quadrupole (E2) and the antisymmetric half as the orbital magnetic dipole, which spin completes into M1. The payoff: the same crumb of coupling supplies both same-parity operators, so a forbidden E1 line is always carried by an M1/E2 pair.

Step 1. Write the first-order amplitude as $(\mathbf{k}\cdot\mathbf{r})(\hat{\boldsymbol\epsilon}\cdot\mathbf{p})$ and add and subtract the transposed product to obtain the exact identity $(\mathbf{k}\cdot\mathbf{r})(\hat{\boldsymbol\epsilon}\cdot\mathbf{p})=\tfrac12[(\mathbf{k}\cdot\mathbf{r})(\hat{\boldsymbol\epsilon}\cdot\mathbf{p})+(\mathbf{k}\cdot\mathbf{p})(\hat{\boldsymbol\epsilon}\cdot\mathbf{r})]+\tfrac12(\mathbf{k}\times\hat{\boldsymbol\epsilon})\cdot(\mathbf{r}\times\mathbf{p})$ . Why: any bilinear in $\mathbf{r}$ and $\mathbf{p}$ splits uniquely into symmetric and antisymmetric halves — no physics yet, just algebra.

Step 2. Read off the antisymmetric half: $\tfrac12(\mathbf{k}\times\hat{\boldsymbol\epsilon})\cdot(\mathbf{r}\times\mathbf{p})=\tfrac{1}{2\hbar}(\mathbf{k}\times\hat{\boldsymbol\epsilon})\cdot\mathbf{L}$ , the orbital magnetic-dipole operator. Why: $\mathbf{r}\times\mathbf{p}=\mathbf{L}$ and $\mathbf{k}\times\hat{\boldsymbol\epsilon}$ is proportional to the mode's $\mathbf{B}$ , so this is a magnetic moment coupling to $\mathbf{B}$ — the definition of M1.

Step 3. Reduce the symmetric half with $\mathbf{p}=im[H,\mathbf{r}]/\hbar$ to $\tfrac{im\omega}{2}\,k_i\epsilon_j\langle f|r_ir_j|i\rangle$ , the traceless electric-quadrupole tensor. Why: the commutator turns momentum into position matrix elements, giving the tensor $r_ir_j$ whose traceless part radiates as a quadrupole (the trace is a non-radiating monopole).

Step 4. Add the electron spin. The Pauli term $-\tfrac{g_s e}{2m}\mathbf{S}\cdot\mathbf{B}$ enters at the same order and promotes $\mathbf{L}$ to $\boldsymbol\mu\propto\mathbf{L}+g_s\mathbf{S}$ . Why: spin couples to $\mathbf{B}$ like orbital angular momentum; both $\mathbf{L}$ and $r_ir_j$ are parity-even, confirming M1 and E2 connect same-parity states — the complement of E1.

Go deeper · Derivation B · 1 minWhy each multipole step costs a factor of alpha-Z squared

Derivation BWhy each multipole step costs a factor of alpha-Z squared

The goal is to explain, from nothing but the characteristic scales of an atom, why every rung down the multipole ladder suppresses the rate by roughly $(\alpha Z)^2$ , and to pin that factor to a real number, $\approx5\times10^{-5}$ for $Z=1$ . We estimate the photon momentum $k$ , then compare the extra $ka_0$ the quadrupole carries and the magnetic-versus-electric moment ratio, reaching the master scaling $A_{\mathrm{E2}}/A_{\mathrm{E1}}\sim(\alpha Z)^2$ and $A_{\mathrm{M1}}/A_{\mathrm{E1}}\sim(\alpha/2)^2$ .

Step 1. Estimate the photon momentum. A characteristic hydrogenic transition has $\hbar\omega\sim\alpha^2Z^2mc^2$ , so $k=\omega/c\sim\alpha^2Z^2mc/\hbar$ ; with $a_0\sim\hbar/(\alpha Z mc)$ , the product is $ka_0\sim\alpha Z$ . Why: this is the phase slip across the atom — for hydrogen $\alpha Z\approx7\times10^{-3}$ , and the optical value for Lyman-α is $ka_0\approx2.7\times10^{-3}$ , the same order.

Step 2. Get the E2 ratio. The quadrupole amplitude carries one extra factor of $ka_0$ , so the rate ratio is its square: $A_{\mathrm{E2}}/A_{\mathrm{E1}}\sim(ka_0)^2\sim(\alpha Z)^2\approx5\times10^{-5}$ for $Z=1$ . Why: rate goes as amplitude squared, so an amplitude down by $ka_0$ gives a rate down by $(ka_0)^2$ — purely geometric retardation.

Step 3. Get the M1 ratio. M1 replaces $ea_0$ by $\mu_B/c=e\hbar/(2mc)$ , and $\mu_B/(ea_0c)=\alpha/2$ , so $A_{\mathrm{M1}}/A_{\mathrm{E1}}\sim(\alpha/2)^2\approx1.3\times10^{-5}$ . Why: the magnetic coupling is relativistic-order, down by $v/c\sim\alpha Z$ in amplitude — a different origin from E2's retardation but the same $\alpha^2$ scale.

Step 4. Read the consequence. Each step multiplies the rate by $\sim10^{-5}$ , so a nanosecond E1 lifetime becomes tens of microseconds at M1/E2, and two or three steps reach years and beyond. Why: this is the metastable regime — long enough for a nebular atom to radiate before colliding, narrow enough for a clock — though the factor rises steeply with $Z$ .

Limiting cases

Case	Result	Meaning
Light neutral atom ( $Z$ small)	$\dfrac{A_{\mathrm{M1,E2}}}{A_{\mathrm{E1}}}\sim(\alpha Z)^2\sim10^{-4}\text{–}10^{-6}$	E1 dominates when allowed; forbidden lines are vanishingly weak, seen only when E1 is absent or in dilute plasmas.
Highly-charged or heavy ion	$ka\sim\alpha Z_{\mathrm{eff}}\to O(1)$	Suppression eases; M1 rises as $\omega^3$ , E2 as $\omega^5\langle r^2\rangle^2$ , so forbidden lines blaze in the EUV and X-ray of iron-group ions.
$J_i=0\to J_f=0$	$A=0$ for any single-photon multipole	Forbidden by angular-momentum conservation; decays only by two-photon emission (e.g. hydrogen $2s\to1s$ ).
Dilute plasma (nebula, corona)	$n\ll n_{\mathrm{crit}}\Rightarrow t_{\mathrm{coll}}\gg\tau_{\mathrm{rad}}$	Metastable levels radiate before collisions de-excite them, so $[\mathrm{O\,III}]$ 500.7 nm appears strongly — impossible in a dense gas.

Common trapForbidden means the transition cannot happen

"Forbidden" is a statement about the E1 operator alone, not about physics. Forbidden lines are real, routinely observed, and often astrophysically dominant — the brightest line in many nebular spectra is the forbidden green $[\mathrm{O\,III}]$ line, and radio astronomy's most-used line is the forbidden 21 cm line. The word only means the leading term vanishes and a weaker one takes over. Two companion traps: do not assume every same-parity transition is E2 — a same-parity $J=0\to J=1$ line is M1 (which shares E2's parity but has $L=1$ ), not E2; and a $J=1/2\to J=1/2$ line cannot be pure E2 at all, since $L=2$ violates $L\le J_i+J_f=1$ . Parity tells you the lane; angular momentum tells you the lowest order allowed in it.

Worked examples

1 · The quality factor of an optical-clock transition (the real thing)

The $3d\,^2D_{5/2}$ level of Ca $^+$ has a measured radiative lifetime $\tau=1.17$ s and decays by an electric-quadrupole (E2) transition at $\lambda=729$ nm. Find its spontaneous rate, its natural linewidth in hertz, and its quality factor $Q$ , then compare with what an allowed E1 line at the same wavelength would give.

Step 1. The spontaneous rate is the inverse lifetime: $A=1/\tau=1/1.17=\mathbf{0.855\ s^{-1}}$ — smaller than a strong E1 rate by about eight orders of magnitude.

Step 2. The natural linewidth follows from equation (6): $\Delta\nu=A/2\pi=0.855/6.283=\mathbf{0.136\ Hz}$ , a line just over a tenth of a hertz on an optical carrier of hundreds of terahertz.

Step 3. With $\nu_0=c/\lambda=(2.998\times10^8)/(729\times10^{-9})=4.11\times10^{14}$ Hz, the quality factor is $Q=\nu_0/\Delta\nu=4.11\times10^{14}/0.136=\mathbf{3.0\times10^{15}}$ (equivalently $Q=\omega_0\tau$ ).

Step 4. Compare with E1: a dipole-allowed line at 729 nm would have $A_{\mathrm{E1}}\sim10^8\,\mathrm{s^{-1}}$ and a $Q$ of only $\sim10^7$ . Forbidding buys eight orders of magnitude in $Q$ : A ≈ 0.855 s⁻¹, Δν ≈ 0.14 Hz, Q ≈ 3×10¹⁵ — a hundred million times narrower than the E1 line it replaces.

2 · How long an ordinary E2 metastable state lives

A light atom ( $Z\approx1$ ) has a dipole-allowed reference transition with lifetime $\tau_{\mathrm{E1}}=1.6$ ns (the hydrogen Lyman-α line). Estimate the lifetime of a comparable E2 transition using only the multipole suppression factor, then say how the estimate changes for a moderately charged ion with $Z_{\mathrm{eff}}\approx20$ .

Step 1. Convert the reference lifetime to a rate: $A_{\mathrm{E1}}=1/\tau_{\mathrm{E1}}=1/(1.6\times10^{-9})=6.3\times10^{8}\,\mathrm{s^{-1}}$ .

Step 2. Apply the suppression from derivation 2: $A_{\mathrm{E2}}\sim(\alpha Z)^2\,A_{\mathrm{E1}}=(1/137)^2\times6.3\times10^{8}\approx3.3\times10^{4}\,\mathrm{s^{-1}}$ .

Step 3. Invert to a lifetime: $\tau_{\mathrm{E2}}\sim1/(3.3\times10^{4})\approx\mathbf{3\times10^{-5}\ s}=30\ \mu\mathrm{s}$ — roughly $2\times10^4$ times longer than the E1 reference. This is order-of-magnitude only; real E2 lifetimes span microseconds to many seconds, since the rate also depends on the matrix element and $\omega^5$ .

Step 4. Scale to $Z_{\mathrm{eff}}\approx20$ : the factor becomes $(\alpha Z_{\mathrm{eff}})^2=(20/137)^2\approx2.1\times10^{-2}$ , about $400$ times larger than at $Z=1$ , before the steep $\omega^5$ growth is even counted. The forbidden line radiates two to three orders of magnitude faster in the heavy ion — why coronal iron forbidden lines are bright while their neutral-atom analogues are nearly invisible.

Physics of the portrait

Forbidden transitions carry a founding mystery: for sixty years the brightest lines in glowing nebulae belonged to no known element. The man who solved it did so not by finding a new element but by realising the lines were old elements behaving in a way no laboratory could reproduce.

Origins · 1 minIra Sprague Bowen and how this was discovered

Ira Sprague Bowen

1898–1973 · forbidden nebular lines and the identification of astrophysical multipole transitions

When William Huggins turned a spectroscope on the planetary nebulae in the 1860s he found bright green emission lines near 500 nm that matched nothing in any laboratory flame. For lack of a better idea, astronomers assigned them to a hypothetical element, "nebulium" — but by the 1920s the periodic table had no gaps left for a light element to hide in. Ira Sprague Bowen, a young Caltech physicist trained in precision ultraviolet spectroscopy, made the decisive leap in 1927: the green lines were not a new element but forbidden transitions of ordinary doubly-ionised oxygen, $[\mathrm{O\,III}]$ , from metastable levels that decay by magnetic-dipole and electric-quadrupole radiation. On Earth such levels are collisionally de-excited before they can radiate; in the near-vacuum of a nebula, where an atom waits hours between collisions, they dominate the spectrum. Bowen went on to identify dozens more forbidden lines of nitrogen, oxygen, and neon, turning "forbidden line" into a quantitative diagnostic of density and temperature. His lasting monument is the demonstration that the faintest transitions in the quantum bookkeeping become the loudest voices of the cosmos.

History

1864

— Huggins records unexplained green emission lines in planetary nebulae, later ascribed to a hypothetical element "nebulium"

1927

— Bowen identifies the nebular lines as forbidden transitions of doubly-ionised oxygen, ending the nebulium hypothesis

1932

— Rubinowicz and Blaton systematise multipole radiation and its parity and angular-momentum selection rules

1945

— van de Hulst predicts the 21 cm magnetic-dipole hyperfine line of atomic hydrogen

1951

— Ewen and Purcell detect the 21 cm line, launching neutral-hydrogen radio astronomy

Going further

Texts. Foot, Atomic Physics, Ch. 7 (selection rules, forbidden transitions) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 and 8 (multipole expansion) · Sobelman, Atomic Spectra and Radiative Transitions · Demtröder, Laser Spectroscopy, Vol. 1 (lifetimes) · Hecht, Optics, 5th ed. · Saleh and Teich, Fundamentals of Photonics, 3rd ed.

Provenance. The E2 + M1 operator split and the selection rules follow Loudon (Ch. 8) and Sobelman; the $(\alpha Z)^2$ -per-multipole scaling and the E1/M1/E2 rate formulae follow Foot (Ch. 7) and Demtröder. A verifier should: (i) reproduce the operator split in derivation 1, confirming the antisymmetric part is $\mathbf{L}$ and the symmetric part the traceless quadrupole tensor; (ii) confirm $ka_0\sim\alpha Z$ and the ratios $(\alpha Z)^2\approx5.3\times10^{-5}$ , $(\alpha/2)^2\approx1.3\times10^{-5}$ ; (iii) check the Ca $^+$ example, $A=0.855\,\mathrm{s^{-1}}$ , $\Delta\nu=0.136$ Hz, $Q\approx3.0\times10^{15}$ ; (iv) check the anchors — $[\mathrm{O\,III}]$ 495.9 and 500.7 nm, 21 cm $A\approx2.9\times10^{-15}\,\mathrm{s^{-1}}$ (lifetime $\approx1.1\times10^7$ yr), Ca $^+$ 729 nm, Yb $^+$ 467 nm — against NIST ASD. Suppression factors are order-of-magnitude estimates, labelled as such.

Prerequisites. Electric Dipole Approximation · Selection Rules for Dipole Transitions · Transition Dipole Moment · Einstein A and B Coefficients. Leads to. Radiative Lifetime and Natural Linewidth · Spontaneous Emission · Detailed Balance and the Planck Spectrum · Oscillator Strength

Ask the tutor: Ask me to work out which multipole carries a specific transition — give me the two levels' $J$ values and parities, and I will find the lowest allowed order and estimate its rate.

If a transition is "forbidden," how can it happen at all?

"Forbidden" is shorthand for "the electric-dipole matrix element is zero," not "the transition is impossible." The dipole approximation keeps only the constant term of the phase factor across the atom; when that gives zero, the next terms — magnetic dipole and electric quadrupole — take over, weaker by about $(\alpha Z)^2$ and mattering only when the dipole route is closed. A forbidden line is real, has a definite rate and lifetime, and obeys the same Einstein-coefficient physics as an allowed one. The only thing forbidden is the leading approximation, not the physics.

What is the difference between an M1 and an E2 transition?

Both are halves of the same first-order term and both connect same-parity states, so they often compete in one line; the difference is the operator and the angular momentum it moves. M1 is driven by the magnetic moment $\mathbf{L}+g_s\mathbf{S}$ , cannot change the orbital number ( $\Delta l=0$ ), and behaves like a dipole ( $L=1$ , $\Delta J=0,\pm1$ ). E2 is driven by the quadrupole tensor $r_ir_j$ , changes $l$ by $0$ or $2$ , and carries $L=2$ , reaching $\Delta J=\pm2$ . When only one satisfies the $J$ triangle it carries the line alone; otherwise the line is a mixture.

Why is a J = 0 to J = 0 transition forbidden even to quadrupole radiation?

Because no single photon carries zero angular momentum. Every multipole photon of order $L\ge1$ carries $L\hbar$ , and conservation demands $|J_i-J_f|\le L\le J_i+J_f$ . With $J_i=J_f=0$ the upper bound is zero, so no $L$ is allowed — not E1, M1, E2, or any order — and the level is stuck against single-photon decay. Its only escape is two-photon emission, where the two angular momenta cancel; this is how the metastable $2s$ level of hydrogen decays to $1s$ , with a lifetime near a seventh of a second, radiating a continuous two-photon spectrum instead of a sharp line.

Why does the suppression get weaker for heavy or highly-charged ions?

The small parameter is $ka\sim\alpha Z$ , the phase slip across the atom; in a light neutral atom $\alpha Z\sim10^{-2}$ , so each step costs $\sim10^{-4}$ . But in a highly-charged ion the electron sits closer to the nucleus and the transition energies are far larger, driving $ka$ toward order one and shrinking the gap between orders. On top of that, E2 scales as $\omega^5$ and M1 as $\omega^3$ , so the high frequencies boost the forbidden rates enormously. This is why forbidden lines of iron-group ions blaze as X-ray and EUV features in the solar corona, while their neutral-atom counterparts are nearly unobservable.

Why do forbidden lines dominate nebulae but vanish in the laboratory?

The upper levels are metastable — they live milliseconds to seconds or longer. Whether the atom radiates the forbidden photon or loses its energy to a collision is a race between the radiative lifetime and the mean time between collisions. In a laboratory gas dense enough to see, collisions win: the level is de-excited before it can radiate, and the line is quenched. In a planetary nebula the density is so low — a few thousand atoms per cubic centimetre — that an atom waits hours between collisions, far longer than the radiative lifetime, so it radiates freely. The lines are pure signatures of low density, which is why Bowen could not find them in any earthly source.

How is the 21 cm hydrogen line related to multipole transitions?

It is the extreme case of a magnetic-dipole (M1) transition, connecting the two hyperfine sublevels of the hydrogen ground state — the $F=1$ and $F=0$ configurations of the proton and electron spins — which share parity and orbital state, so E1 is impossible and only M1 remains. Because the energy split is minute (a $1420$ MHz photon) and the magnetic moment small, the rate is astonishingly low: $A\approx2.9\times10^{-15}\,\mathrm{s^{-1}}$ , a lifetime near eleven million years. Any single atom essentially never radiates it, but the interstellar medium holds so many that the line is bright across the whole sky — the primary radio probe of galactic structure and cosmic reionisation.