Oscillator Strength

The oscillator strength is what atomic physics hands to spectroscopy when it wants one dimensionless number for "how strong is this line." It takes two things it did not invent. From quantum mechanics it takes the transition dipole moment $\langle j|e\mathbf r|i\rangle$ — the matrix element that aq-optics-transition-dipole-moment establishes and that Fermi's golden rule turns into an absorption rate. From classical electromagnetism it takes the Lorentz picture of a bound electron on a spring, a single driven oscillator that absorbs and re-radiates with a strength fixed entirely by $e$ , $m_e$ , and $\varepsilon_0$ . The oscillator strength $f$ is the ratio of the first to the second: the fraction of one ideal classical electron oscillator that a real quantum transition is worth. That framing does more work than it looks. Because it is a ratio to a fixed classical unit, $f$ is dimensionless and of order one for the strongest allowed lines. Because it is proportional to the transition frequency times the squared dipole moment, it is the Einstein $B$ coefficient in different clothing, and it fixes the spontaneous rate $A_{ji}$ , the radiative lifetime, and the natural linewidth. Because it carries the sign of $\omega_{ij}=(E_j-E_i)/\hbar$ , absorption oscillator strengths are positive and stimulated-emission ones are negative — the bookkeeping that reproduces $g_iB_{ij}=g_jB_{ji}$ . The single deepest fact about $f$ is a conservation law: summing $f_{ij}$ over every final state, discrete and continuum, gives the electron number $N$ . This Thomas–Reiche–Kuhn sum rule follows from nothing more than $[x,p_x]=i\hbar$ . It makes oscillator strength a fixed budget across all of an electron's transitions — why a single alkali resonance line can dominate a spectrum — and it is the consistency check every line list must pass.

Key takeaways

The oscillator strength $f$ is a dimensionless number comparing a real atomic transition to one classical electron oscillating on a spring at the transition frequency: $f=1$ means it absorbs exactly as much.
It packages the quantum matrix element, $f_{ij}=\dfrac{2m_e\omega_{ij}}{3\hbar e^2}\big|\langle j|e\mathbf r|i\rangle\big|^2$ , so it is the Einstein $B$ coefficient and the dipole moment re-expressed as a pure number.
One measured $f$ fixes the spontaneous rate $A_{ji}$ , the radiative lifetime $\tau$ , the natural linewidth, and the line-integrated absorption cross-section — all follow from it.
Absorption oscillator strengths obey the Thomas–Reiche–Kuhn sum rule $\sum_j f_{ij}=N$ (the electron number), an exact consequence of $[x,p_x]=i\hbar$ that cross-checks every tabulated line list.
Strong dipole-allowed lines have $f\sim1$ (Na D doublet $\approx0.98$ combined, Lyman- $\alpha$ $\approx0.416$ ); dipole-forbidden lines have $f\ll1$ , sometimes below $10^{-6}$ .
Key idea: the oscillator strength measures how many classical electron oscillators a quantum transition is worth — $f_{ij}=(2m_e\omega_{ij}/3\hbar e^2)|\langle j|e\mathbf r|i\rangle|^2$ — dimensionless, positive for absorption, summing to the electron number by the TRK sum rule.

The core insight

The oscillator strength measures how many classical electron oscillators a given quantum transition is worth: $f_{ij}=(2m_e\omega_{ij}/3\hbar e^2)|\langle j|e\mathbf r|i\rangle|^2$ , dimensionless, positive for absorption, negative for emission, and summing over all final states to the electron number by the Thomas–Reiche–Kuhn sum rule.

Where this lives today

Oscillator strength is the currency in which line strengths are bought and sold. The NIST Atomic Spectra Database and the Kurucz line lists tabulate $\log gf$ rather than raw rates because $gf$ is symmetric and combines cleanly, and modern atomic-structure codes check their $f$ -values against the TRK sum rule before trusting them. In stellar astrophysics this is no nicety: an absorption line's equivalent width scales as column density times $gf$ , so a $0.1$ dex error in $\log gf$ propagates one-for-one into a derived abundance, and the laboratory $f$ -value campaigns behind surveys like Gaia-ESO and APOGEE exist mainly to shrink it. Closer to the bench, the D₂ line's $f=0.641$ sets its $16.2$ ns lifetime, its $9.8$ MHz natural linewidth, and hence the scattering rate that cools a magneto-optical trap and the saturation intensity a spectroscopist must exceed. When a cold-atom experiment quotes a scattering rate, an oscillator strength is doing the arithmetic underneath.

The physical picture

Start with the classical atom Lorentz drew: an electron bound to the nucleus as if by a spring, with natural frequency $\omega_0$ . Shine light on it and the electron is driven into oscillation, absorbing energy near resonance and re-radiating it. The remarkable thing is that integrating the absorption over the whole resonance line gives an answer independent of the damping, the temperature, or how the line is broadened — a fixed number built only from $e$ , $m_e$ , and $\varepsilon_0$ . That number is the yardstick, and every real transition absorbs some fraction of it. That fraction, defined so the classical oscillator scores exactly one, is the oscillator strength.

Figure

A driven bound electron is the classical yardstick: one Lorentz oscillator absorbs a fixed frequency-integrated amount ∫σ dν = 0.0265 cm²·Hz, and the oscillator strength f is the fraction of that unit a real quantum transition carries

Two consequences follow and organize the rest of the subject. The first is a budget. A single electron cannot absorb without limit: the Thomas–Reiche–Kuhn sum rule says all of its oscillator strengths, summed over every line and the whole photoionization continuum, add to exactly one. If one resonance line grabs $f\approx0.98$ , every other transition from that level — including ionization — must share the remaining $0.02$ . That is why alkali resonance lines are so overwhelmingly dominant, and it is what the second figure makes concrete for sodium's valence electron.

Figure

The Thomas–Reiche–Kuhn budget for sodium's single valence electron: the 3s→3p resonance doublet carries f ≈ 0.98 of the unit sum, split D₂ ≈ 0.655 and D₁ ≈ 0.327, leaving only ≈ 0.02 for every higher line and the whole continuum, so Σf = 1

The second consequence is a sign. Absorption drives the electron up in energy and borrows a positive oscillator; stimulated emission is the same physics run backwards, carrying a negative oscillator strength. In a two-level atom this bookkeeping is exactly the Einstein relation $g_iB_{ij}=g_jB_{ji}$ , and Ladenburg's original insight was that "negative dispersion" near an inverted transition is the optical fingerprint of that negative $f$ . So the whole subject sits in three words attached to one number: fraction of the classical unit, budget summing to $N$ , and sign for absorption versus emission.

The core equations

The defining equation is the length-gauge oscillator strength: the squared transition dipole moment, weighted by frequency and scaled by exactly the constants needed to make the classical Lorentz oscillator come out to one. The factor of three is an orientational average over the three Cartesian directions of the dipole — the same three that reappears when we connect $f$ to the spontaneous rate.

f_{ij}=\frac{2m_e\,\omega_{ij}}{3\hbar e^2}\,\big|\langle j|\,e\mathbf r\,|i\rangle\big|^2 \qquad (1)

where: $f_{ij}$ — oscillator strength for $i\to j$ (dimensionless); $m_e$ — electron mass; $\omega_{ij}=(E_j-E_i)/\hbar$ — transition angular frequency, signed; $\langle j|e\mathbf r|i\rangle$ — transition dipole moment; $e$ — elementary charge; the $1/3$ averages over dipole orientation, $|\langle x\rangle|^2=\tfrac13|\langle\mathbf r\rangle|^2$ . Pulling the charge out gives the charge-free form $f_{ij}=\tfrac{2m_e\omega_{ij}}{3\hbar}\sum_\alpha|\langle j|x_\alpha|i\rangle|^2$ . Physical reading: strong lines have large dipole moments and high frequencies. Oscillator strength is not independent — it is the Einstein $B$ coefficient re-expressed. The next equation is the exchange rate between the two languages, so anything phrased in $f$ can be converted to a stimulated rate and back.

B_{ij}=\frac{\pi e^2}{2\,\varepsilon_0\,m_e\,\hbar\,\omega_{ij}}\,f_{ij} \qquad (2)

where: $B_{ij}$ — Einstein stimulated-absorption coefficient, per unit spectral energy density in $\omega$ ; $\varepsilon_0$ — vacuum permittivity. Physical reading: $B$ and $f$ carry identical information, but $f$ is dimensionless and near unity for allowed lines. Because detailed balance ties stimulated to spontaneous emission, one oscillator strength also fixes the spontaneous rate $A_{ji}$ — the workhorse relation of quantitative spectroscopy: measure $f$ , get the lifetime.

A_{ji}=\frac{2\pi e^2 \nu_{ij}^2}{\varepsilon_0 m_e c^3}\,\frac{g_i}{g_j}\,f_{ij} \qquad (3)

where: $A_{ji}$ — spontaneous emission rate $j\to i$ (s⁻¹); $\nu_{ij}$ — transition frequency (Hz); $g_i,g_j$ — statistical weights of the lower and upper levels; $c$ — speed of light. In the practical unit form this is $A_{ji}\approx0.667\,\lambda^{-2}(g_i/g_j)\,f_{ij}$ with $\lambda$ in cm and $A$ in s⁻¹. Physical reading: the $\nu^2$ makes high-frequency (blue and UV) lines decay far faster at fixed $f$ . The sign structure lives in one compact relation connecting absorption to emission and enforcing the Einstein weighting, so a downward transition is a negative "absorption."

g_i\,f_{ij}=-\,g_j\,f_{ji} \qquad (4)

where: $f_{ij}\gt0$ — absorption, upper level higher in energy; $f_{ji}\lt0$ — stimulated emission, downward; $g_i,g_j$ — statistical weights. Physical reading: the products $gf$ are the symmetric, database-friendly quantities, equal and opposite for the two directions of a line. Summing the absorption oscillator strengths over every final state gives the electron number. This is the Thomas–Reiche–Kuhn sum rule, exact for any velocity-independent potential and independent of atomic detail.

\sum_{j}f_{ij}=N \qquad (5)

where: $N$ — number of electrons contributing (one for a single-active-electron problem); $\sum_j$ — sum over all final states, discrete lines plus the ionization continuum. Physical reading: oscillator strength is conserved and normalized; a line list that fails to sum correctly is missing transitions or has wrong values. The frequency-integrated absorption cross-section is fixed by $f$ alone; broadening reshuffles where the absorption sits in frequency but conserves its area.

\int \sigma(\nu)\,\mathrm d\nu=\frac{e^2}{4\varepsilon_0 m_e c}\,f_{ij}=\pi r_e c\,f_{ij} \qquad (6)

where: $\sigma(\nu)$ — absorption cross-section at frequency $\nu$ ; $r_e=e^2/4\pi\varepsilon_0 m_e c^2$ — classical electron radius. Numerically the constant is $\pi r_e c=2.654\times10^{-6}\ \mathrm{m^2\,Hz}=0.0265\ \mathrm{cm^2\,Hz}$ per unit $f$ . Physical reading: this is the yardstick of the first figure; a real line carries the fraction $f$ of it. Finally, oscillator strength converts the classical radiative lifetime of a Lorentz oscillator into the real quantum lifetime. Routing equation (3) through the classical rate $\gamma_{\text{cl}}=1/\tau_{\text{cl}}$ gives $A_{ji}=3(g_i/g_j)f_{ij}\,\gamma_{\text{cl}}$ , and inverting yields the relation below. The factor of three is the same orientational three as in equation (1); dropping it — writing $\tau_{ji}=\tau_{\text{cl}}/f$ — is a common slip.

\tau_{ji}=\frac{g_j}{g_i}\,\frac{\tau_{\text{cl}}}{3\,f_{ij}},\qquad \tau_{\text{cl}}=\frac{6\pi\varepsilon_0 m_e c^3}{e^2\omega_0^2} \qquad (7)

where: $\tau_{\text{cl}}$ — classical radiative decay time of a Lorentz oscillator at $\omega_0$ ; $\tau_{ji}=1/A_{ji}$ — quantum radiative lifetime; $g_i,g_j$ — weights. Physical reading: an $f=1$ , non-degenerate transition decays three times faster than the naive one-dimensional classical oscillator, because the quantum dipole radiates into all three directions. For Na D₂, $\tau_{\text{cl}}=15.6$ ns and this returns $\tau=16.2$ ns (worked example 1).

Derivation AOscillator strength as the quantum-to-classical absorption ratio

The goal is to show why the constant $2m_e\omega/3\hbar e^2$ sits in front of the squared dipole moment, by building $f$ from the ground up as a ratio. We compute two frequency-integrated absorption cross-sections — one for a single driven classical electron, one for a quantum transition through Fermi's golden rule — and divide the second by the first. The classical piece is the fixed unit, the quantum piece carries the atomic detail, and their ratio is dimensionless and equal to one for the classical oscillator by construction. It also exposes where the orientational factor of three enters, which matters later for lifetimes.

Step 1. Take a single classical electron driven near resonance and integrate its cross-section over the whole Lorentzian line, giving $\int\sigma_{\text{cl}}\,\mathrm d\nu=e^2/(4\varepsilon_0 m_e c)$ . Why: this is the only place the classical model enters — a damping-independent unit of "absorbing power" against which everything quantum is measured.

Step 2. For the quantum transition, Fermi's golden rule with the dipole coupling $-e\mathbf r\cdot\mathbf E$ , averaged isotropically, gives an absorption rate $W_{i\to j}=\tfrac{\pi e^2}{3\varepsilon_0\hbar^2}|\langle j|\mathbf r|i\rangle|^2\,u(\omega_{ij})$ with $u$ the spectral energy density. Why: the golden rule turns the dipole moment into a rate, and the isotropic average produces the $1/3$ — one of three Cartesian components does the work for unpolarised light.

Step 3. Convert that rate into a frequency-integrated cross-section by dividing absorbed power by incident photon flux, yielding $\int\sigma_{ij}\,\mathrm d\nu=\tfrac{\pi e^2\omega_{ij}}{3\varepsilon_0\hbar c}|\langle j|\mathbf r|i\rangle|^2$ . Why: cross-section, not rate, is intensity-independent and so the right currency for a ratio.

Step 4. Form the ratio $f_{ij}\equiv\int\sigma_{ij}\,\mathrm d\nu\big/\int\sigma_{\text{cl}}\,\mathrm d\nu=\tfrac{2m_e\omega_{ij}}{3\hbar e^2}|\langle j|e\mathbf r|i\rangle|^2$ . Why: the $e^2$ cancels the charges inside the dipole moment, leaving a pure number set by $m_e\omega/\hbar$ times a squared length — dimensionless, and equal to one when the transition matches the classical oscillator. The result is dimensionless — $[m_e\omega/\hbar]$ cancels the length² of the matrix element — and matches Foot §7.5 and Bransden–Joachain §4.4.

Go deeper · Derivation B · 1 minThe Thomas–Reiche–Kuhn sum rule from the canonical commutator

Derivation BThe Thomas–Reiche–Kuhn sum rule from the canonical commutator

The target is the exact statement that the absorption oscillator strengths from any state sum to the electron number, with no reference to the atomic potential. The proof is a small miracle of operator algebra: it uses only that the Hamiltonian's kinetic term ties position to momentum, and that they obey $[x,p_x]=i\hbar$ . Everything else — the energies, the wavefunctions, the shape of the potential — drops out, which is exactly why the sum rule is such a universal check on tabulated line strengths.

Step 1. Write the oscillator strength in energy form, $f_{ij}=\tfrac{2m_e}{3\hbar^2}(E_j-E_i)\sum_\alpha|\langle j|x_\alpha|i\rangle|^2$ , using $\hbar\omega_{ij}=E_j-E_i$ . Why: casting the frequency as an energy difference lets completeness be used later, turning a sum over transitions into an operator expectation value.

Step 2. Replace one energy factor using $(E_j-E_i)\langle i|x_\alpha|j\rangle=\langle i|[x_\alpha,H]|j\rangle=\tfrac{i\hbar}{m_e}\langle i|p_\alpha|j\rangle$ , valid for any velocity-independent potential. Why: this crucial move swaps position-times-energy for momentum, setting up the commutator that carries the whole result.

Step 3. Symmetrise the two ways of writing the energy factor and collapse the sum over $j$ with $\sum_j|j\rangle\langle j|=\mathbb 1$ , reducing everything to $\sum_j f_{ij}=\tfrac{1}{3i\hbar}\sum_\alpha\langle i|[x_\alpha,p_\alpha]|i\rangle$ . Why: completeness turns an infinite, potential-dependent sum into a single ground-state expectation value — the point where the atomic detail vanishes.

Step 4. Insert $[x_\alpha,p_\alpha]=i\hbar$ for each of the three directions, giving $\sum_j f_{ij}=\tfrac{1}{3i\hbar}(3i\hbar)=1$ , and $N$ for $N$ electrons. Why: the three commutators each contribute $i\hbar$ , the $1/3$ cancels the $3$ , and the result is an exact integer independent of the atom — the hallmark of a genuine sum rule. The derivation matches Bransden–Joachain §4.4 and Loudon Ch. 2. Historically it is more than a check: Kuhn and Reiche's 1925 proof was one of the calculations that pointed straight at matrix mechanics.

Limiting cases

Case	Result	Meaning
Classical single electron	$f\to 1$	one Lorentz oscillator; the unit against which every $f$ is measured
Strong allowed line	$f\sim 0.1\text{–}1$	dipole-allowed resonance transitions — Na D doublet $\approx0.98$ combined, Lyman- $\alpha$ $\approx0.416$
Forbidden or weak line	$f\ll 1$	broken dipole selection rule or intercombination line; $f$ can reach $10^{-6}$ or below
Highly excited or inner-shell	$f$ may exceed $1$	individual $f$ is not capped at one; only the total is fixed
Sum over all final states	$\sum_j f_{ij}=N$	the TRK sum rule; exact for non-relativistic velocity-independent potentials

Common trapOscillator strength is a probability, so it must lie between 0 and 1.

It is not a probability and it is not bounded by one. The oscillator strength is a ratio to a classical oscillator, and a single transition can carry more than one classical oscillator's worth: individual $f$ values above unity occur, for instance between highly excited states and for some inner-shell transitions. What is fixed is the sum over all final states, $\sum_j f_{ij}=N$ , not any one term. Two further traps travel with this one. First, sign: absorption gives $f_{ij}\gt0$ , stimulated emission gives $f_{ji}\lt0$ , related by $g_if_{ij}=-g_jf_{ji}$ — a "negative probability" would be nonsense, but a negative oscillator strength is exactly right. Second, units: $f$ is dimensionless; it is never a rate or an area. It is the pure factor by which cross-sections, $A$ -coefficients, and radiative lifetimes are scaled away from their classical yardstick.

Worked examples

1 · Lifetime and natural linewidth of the sodium D₂ line (confidence builder)

The Na D₂ transition (3s₁/₂ → 3p₃/₂) has $f=0.641$ at $\lambda=589.0$ nm, with weights $g_i=2$ , $g_j=4$ . Find the spontaneous rate $A$ , the radiative lifetime $\tau$ , and the natural (FWHM) linewidth.

Step 1. Use the practical form of equation (3), $A_{ji}=0.667\,\lambda^{-2}(g_i/g_j)f$ with $\lambda$ in cm. Here $\lambda=5.890\times10^{-5}$ cm, so $0.667\,\lambda^{-2}=1.923\times10^{8}\ \mathrm{s^{-1}}$ , and $g_i/g_j=2/4=0.5$ .

Step 2. Assemble: $A=1.923\times10^{8}\times0.5\times0.641=\mathbf{6.2\times10^{7}\ s^{-1}}$ — about sixty million photons per second from one excited atom.

Step 3. The lifetime is $\tau=1/A=1.62\times10^{-8}\ \mathrm s=\mathbf{16.2\ ns}$ , and the natural linewidth is $\Delta\nu=A/2\pi=\mathbf{9.8\ MHz}$ (FWHM). Both match the laboratory sodium 3p values to three figures, which is why D₂ is a workhorse frequency reference. A ≈ 6.2×10⁷ s⁻¹, τ ≈ 16 ns, Δν ≈ 9.8 MHz.

2 · The oscillator-strength budget and the fixed absorption area

Sodium's valence electron puts $f\approx0.98$ into the 3s→3p resonance doublet. (a) What does the TRK sum rule then permit for every other transition of that electron? (b) What is the frequency-integrated absorption cross-section of the D₂ component, and does Doppler broadening change it?

Step 1. For a single active electron the sum rule (equation 5) reads $\sum_j f_{ij}=1$ . The doublet already claims $0.98$ , so all remaining transitions — 3s→4p, the higher Rydberg lines, and the entire photoionisation continuum — must together carry $1-0.98=\mathbf{0.02}$ . The resonance line owns 98% of the electron's absorbing power.

Step 2. The line-integrated cross-section follows from equation (6): $\int\sigma\,\mathrm d\nu=\pi r_e c\,f=0.0265\,f\ \mathrm{cm^2\,Hz}$ . For the D₂ component ( $f=0.641$ ), $\int\sigma\,\mathrm d\nu=\mathbf{1.70\times10^{-2}\ cm^2\,Hz}$ .

Step 3. Doppler and pressure broadening do not change this. They redistribute the cross-section across frequency — a Doppler line at 500 K is roughly 1 GHz wide versus the 9.8 MHz natural width — but conserve the area under $\sigma(\nu)$ , which is pinned to $f$ alone. The peak drops as the line widens; the integral does not. Remaining budget 0.02; ∫σ dν ≈ 1.7×10⁻² cm²·Hz, independent of line shape.

Physics of the portrait

Oscillator strength is one of those quantities that was born classical, kept its classical name, and then turned out to be exactly right quantum-mechanically — and the person who forced that transition was Rudolf Ladenburg.

Origins · 1 minRudolf Ladenburg and how this was discovered

Rudolf Ladenburg

1882–1952 · the man who quantized the classical "number of dispersion electrons"

In the Lorentz–Drude theory of dispersion, the refractive index of a gas near a resonance depends on the "number of dispersion electrons" effective at that frequency — a fudge factor with no clear meaning in a world of quantized atoms. In 1921 Rudolf Ladenburg, a German experimental physicist, took that number seriously: he identified it with a quantum quantity proportional to Einstein's $B$ coefficient, and thereby defined what we now call the oscillator strength, or historically the "Ladenburg $f$ -value." His relation tied the measured dispersion and absorption of a gas directly to the rates of induced and spontaneous transitions, years before quantum mechanics could compute the matrix elements. He went further and predicted that an inverted transition would show negative dispersion — the optical signature of stimulated emission, in hindsight a first glimpse of the laser — and with Hans Kopfermann confirmed it in excited neon in the late 1920s. When Thomas, Reiche and Kuhn then proved these $f$ -values must sum to the electron number, oscillator strength became a conserved, computable quantity. Ladenburg's 1921 reinterpretation is why every modern line list is tabulated in $f$ .

History

1900

— Drude and Lorentz model dispersion with classical electron oscillators and a "number of dispersion electrons"

1913

— Bohr quantizes atomic transitions, but line intensities remain unexplained

1921

— Ladenburg reinterprets the dispersion-electron count as the quantum oscillator strength, linking it to Einstein's A and B coefficients

1925

— Thomas, Reiche and Kuhn establish the sum rule Σf = N from the correspondence principle and the canonical commutator

1928

— Ladenburg and Kopfermann confirm negative dispersion (stimulated emission) in excited neon

Going further

Texts. Foot, Atomic Physics, §7.5–7.6 (oscillator strength and sum rules) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2 · Bransden & Joachain, Physics of Atoms and Molecules, §4.4 · Hecht, Optics, 5th ed., §3.5 (dispersion and the classical oscillator) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 13 · Demtröder, Laser Spectroscopy, Vol. 1, §2.7 · Corney, Atomic and Laser Spectroscopy (the $f$ – $A$ relation in practical units).

Provenance. The dipole-form definition follows Foot (§7.5) and Bransden–Joachain (§4.4); the TRK derivation from $[x,p_x]=i\hbar$ follows Loudon (Ch. 2). A verifier should confirm: (1) equation (1) is dimensionless, $[m_e\omega/\hbar]\times[\text{length}]^2=1$ ; (2) the TRK sum $\sum_jf_{ij}=N$ from the canonical commutator; (3) the cited $f$ -values against NIST ASD — Na D doublet $\approx0.98$ combined ( $D_2\approx0.641$ , $D_1\approx0.320$ ), Lyman- $\alpha$ $\approx0.4162$ ; (4) the practical relation $A_{ji}\approx0.667\,\lambda^{-2}(g_i/g_j)f$ with $\lambda$ in cm, and that it returns $A=6.2\times10^{7}\ \mathrm{s^{-1}}$ , $\tau=16.2$ ns, $\Delta\nu=9.8$ MHz for Na D₂; (5) the integrated-cross-section constant $\pi r_e c=0.0265\ \mathrm{cm^2\,Hz}$ ; (6) the factor of three in equation (7), $\tau=(g_j/g_i)\tau_{\text{cl}}/(3f)$ , giving $16.2$ ns from $\tau_{\text{cl}}=15.6$ ns — note this corrects a factor-of-three slip in some quick treatments.

Prerequisites. Transition Dipole Moment · Einstein A and B Coefficients · Electric Dipole Approximation · Resonant Absorption. Leads to. Thomas–Reiche–Kuhn Sum Rule · Radiative Lifetime and Natural Linewidth · Lorentzian Line Shape · Spontaneous Emission.

Ask the tutor: ask me to turn a single measured $f$ -value into a lifetime, a natural linewidth, and a stellar abundance — or to prove the TRK sum rule line by line from $[x,p]=i\hbar$ .

Is the oscillator strength the same thing as the transition probability?

Closely related, but not identical. The oscillator strength is a dimensionless ratio to a classical oscillator; the transition probability per unit time is the Einstein $A$ or $B$ coefficient, which carries units. They hold the same physics because $f\propto B$ and $A_{ji}\propto\nu^2(g_i/g_j)f$ . The practical difference is convenience: $f$ is dimensionless and near one for allowed lines, easy to tabulate and sanity-check with the sum rule, whereas $A$ and $B$ drag along constants and span many orders of magnitude. Think of $f$ as the normalized, human-readable version of the transition probability.

Can the oscillator strength be larger than one?

Yes. The intuition that $f$ is a probability capped at one is wrong. A single transition can carry more than one classical oscillator's worth of absorbing power — this occurs between some highly excited states and for certain inner-shell excitations. What is bounded is the sum, $\sum_j f_{ij}=N$ , over every final state including the continuum. Individual terms can exceed one as long as others (including negative emission strengths) keep the total pinned to $N$ . So a reported $f=1.4$ is not an error by itself; a set of $f$ -values from one level summing to more than $N$ is.

What is gf, and why do databases list it instead of f?

$gf$ is the statistical weight times the oscillator strength. Databases favour it for two reasons. First it is symmetric: equation (4), $g_if_{ij}=-g_jf_{ji}$ , means absorption and emission share a magnitude, so one number serves both directions of a line. Second it combines cleanly when fine-structure or hyperfine components blend, since weighted sums of $gf$ are what quantities like equivalent width depend on. Astronomers quote $\log gf$ because oscillator strengths span many decades and abundance analysis is linear in it; to recover a rate you divide out the weight.

Why does the same f appear in absorption, emission, and dispersion?

Because all three are the same electron responding to a field in different regimes. Absorption is the driven electron taking energy near resonance; stimulated emission is the same drive with an inverted population, giving energy back; dispersion is the off-resonant, in-phase part of that identical response, shifting the wave's phase and setting the refractive index. The oscillator strength is the one weight multiplying each resonance in all three — exactly Ladenburg's point that the "number of dispersion electrons" in the classical index formula is the same $f$ that governs absorption and the Einstein coefficients. One number, three faces.

How is an oscillator strength actually measured in the laboratory?

Several independent routes, chosen to cross-check. A direct absorption measurement records the frequency-integrated cross-section through a column of known density; since $\int\sigma\,\mathrm d\nu=\pi r_e c\,f$ , the area gives $f$ without needing the line shape. A lifetime measurement — laser excitation, then timing the fluorescence decay — gives $A$ , which equation (3) inverts to $f$ . Hook and anomalous-dispersion methods read $f$ off the refractive index near the line, the descendant of Ladenburg's own technique. Agreement among methods, and consistency with the sum rule, certifies a tabulated value.

What exactly does the TRK sum rule count, and does it hold for many-electron atoms?

It counts electrons. For a one-electron problem the absorption oscillator strengths sum over all final states to exactly one; for $N$ electrons the total is $N$ , because the derivation uses only $[x_\alpha,p_\alpha]=i\hbar$ per electron and a velocity-independent potential. In a real many-electron atom the full sum still gives $N$ , but in the single-active-electron approximation used for alkalis the valence electron's transitions sum to about one — which is why sodium's 3s→3p doublet, at $f\approx0.98$ , so nearly exhausts its budget. Relativistic and velocity-dependent interactions add small corrections, so the rule is exact only non-relativistically.