University Quantum Mechanics II · Emission and Absorption of Radiation · 13.7

Natural linewidth and the Lorentzian line shape

Every excited level has a finite life, so the light it emits spontaneously has a finite spread in frequency. Here 13.04's 1.6 ns lifetime becomes the 100 MHz Lorentzian it implies; then moving and colliding atoms bury that width under gigahertz, and we see what it took to dig the Lamb shift out from under them.

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Build the model

Connect the measurement to the mechanism.

Unit 12 quoted Weisskopf and Wigner: an excited level's amplitude decays as e^(−Γt/2), with Γ = 1/τ. Put that decaying amplitude into the first-order emission amplitude and the time integral becomes a Fourier transform, 1/[Γ/2 − i(ω − ω₀)], whose squared modulus is a Lorentzian of full width Γ in angular frequency, Γ/2π = 1/(2πτ) in hertz. For Lyman-α that is 99.7 MHz, a fractional width of 4 × 10⁻⁸, small because Γ/ω₀ is of order α³.

The shape is only as good as the exponential behind it: Weisskopf and Wigner froze a coupling that really grows as ω³ at its value on resonance, which is harmless within thousands of widths and wrong in the far wings. A lower level that also decays adds its rate to the width, a result also quoted. And in a real gas this natural width is buried.

Doppler shifts spread the line centres into a Gaussian 300 times wider for Lyman-α at 300 K, and collisions add a Lorentzian of their own that grows with pressure. The natural line survives in the far wings, and in experiments built to escape Doppler broadening: Lamb and Retherford's microwaves, and Doppler-free laser spectroscopy.

Simple definition: The natural linewidth is the spread in frequency a spectral line has because its upper level decays: an amplitude falling as e^(−t/2τ) radiates a Lorentzian whose full width at half maximum is Δω = 1/τ, or Δν = 1/(2πτ).
Example: Hydrogen's 2p level lives τ = 1.596 ns, so Lyman-α has Δν = 1/(2π × 1.596 ns) = 99.7 MHz, an energy width ħ/τ = 4.12 × 10⁻⁷ eV, and a fractional width of 4.0 × 10⁻⁸ of its 2.466 × 10¹⁵ Hz frequency.

Lorentzian line shape, unit areag(ω) = (Γ/2π) / [(ω − ω₀)² + (Γ/2)²]

Derived: |∫₀^∞ e^(−Γt/2) e^{(i(ω − ω₀)t)} dt|² = 1/[(ω − ω₀)² + (Γ/2)²]. As Γ → 0 at fixed area it becomes δ(ω − ω₀), the golden rule's sharp line.

ω, ω₀ = (Eᵤ − Eₗ)/ħ and Γ = 1/τ in s⁻¹; g in s, so that ∫g dω = 1; peak value 2/(πΓ)

Natural width from the lifetimeΔνₙₐₜ = Γ/2π = 1/(2πτ)

Derived, and exact for this shape: the angular FWHM times τ is 1. It is not ħ/2 over Δt; that inequality bounds standard deviations, and a Lorentzian has none.

τ the upper-level lifetime in s, Δν in Hz; Lyman-α: τ = 1.596 ns gives 99.7 MHz, and ħΓ = 4.12 × 10⁻⁷ eV

Power within ±Δ of line centreP(|ν − ν₀| ≤ Δ) = (2/π) arctan(2Δ/Δν)

Derived by integrating g. A Gaussian of the same FWHM holds 76% and 98% at the first two; the Lorentzian's slow Δ⁻² tail is what lets its wings outlast a far wider Doppler core.

Δ and the FWHM Δν in Hz; P = 50% at Δ = Δν/2, 70% at Δ = Δν, and only 94% at Δ = 5Δν

A lower level that decays tooΔω = Γᵤ + Γₗ

Quoted from Weisskopf and Wigner, not derived here. Balmer-α's 3s → 2p: (6.32 × 10⁶ + 6.265 × 10⁸ s⁻¹)/2π = 100.7 MHz, a hundred times 3s's own 1.0 MHz.

Γᵤ and Γₗ the total decay rates of the upper and lower level in s⁻¹; Γₗ = 0 for a ground state

Doppler width, a GaussianΔν_D = ν₀ √(8 ln2 k_B T / Mc²)

Derived from the Maxwell distribution of v_z with ν = ν₀(1 + v_z/c). It scales with ν₀, so at 1.06 GHz, the Lamb-shift frequency, it is only 13 kHz.

ν₀ in Hz, T in K, M the atom's mass in kg; hydrogen at 300 K: 30.5 GHz for Lyman-α, 5.64 GHz for Balmer-α

Collisional width, impact modelΔν_c = 1/(πτ_c), with 1/τ_c = nσv̄

Derived for random phase resets; σ is estimated. It adds to Γ/2π. With σ = 5 × 10⁻¹⁹ m² and v̄ = 2.5 × 10³ m s⁻¹ at 300 K: 13 MHz at 133 Pa, 9.7 GHz at 101 kPa.

n the perturber density in m⁻³, σ the phase-breaking cross-section in m², v̄ the mean relative speed in m s⁻¹

From a decaying amplitude to a Lorentzian

Unit 12 quoted the Weisskopf-Wigner result: an atom placed in the upper level u at t = 0 keeps an amplitude cᵤ(t) = e^(−Γt/2), with Γ = 1/τ its total decay rate. Feed that into Unit 12's first-order amplitude for ending in the lower level l with one photon of angular frequency ω. It is proportional to ∫₀^∞ cᵤ(t) e^{(i(ω − ω₀)t)} dt, with ω₀ = (Eᵤ − Eₗ)/ħ, and the integral is 1/[Γ/2 − i(ω − ω₀)]; the rotating-wave approximation has already dropped a partner term in ω + ω₀, smaller by about Γ/ω₀. The emitted spectrum is the squared modulus, ∝ 1/[(ω − ω₀)² + (Γ/2)²]: a Lorentzian that falls to half its peak at ω − ω₀ = ±Γ/2, so its full width at half maximum is Γ. A classical wave train e^(−Γt/2) cos ω₀t, started at t = 0, has the same spectrum. Checks: Γ and ω are both in s⁻¹, so the unit-area form g(ω) carries seconds, as a density per unit ω must; and as Γ → 0 at fixed area g becomes δ(ω − ω₀), the infinitely sharp line the golden rule assumed.

Lyman-α's 100 MHz, checked

Take 13.04's computed lifetime, τ = 1.596 ns, so Γ = 6.265 × 10⁸ s⁻¹. That is the width in angular frequency; in hertz it is Γ/2π = 99.7 MHz, and in energy ħΓ = 4.12 × 10⁻⁷ eV. Beside the line's 2.466 × 10¹⁵ Hz the fractional width is 4.0 × 10⁻⁸, and the order in α explains why: 13.04's Γ = (2/3)⁸α⁵mc²/ħ over ω₀ = (3/8)α²mc²/ħ gives Γ/ω₀ = (2/3)⁸(8/3)α³ = 4.0 × 10⁻⁸. Measurement backs it: timed decays of 2p give 1.60 ns, and the microwave resonances Lamb and Retherford drove out of 2s were about 100 MHz wide, set by this lifetime. Mind the tails. Integrating g, the power within ±Δ of centre is (2/π) arctan(2Δ/Γ): 50% inside the FWHM, 70% within one full FWHM either side, and still only 94% within five, where a Gaussian keeps 76% and 98% at the first two. Most of a Gaussian's light sits in its core; a Lorentzian spends much of its light far out.

Two quoted inputs, and where they bound the result

The exponential is quoted, and the Lorentzian inherits its limits. Weisskopf and Wigner froze the product of |⟨l|d|u⟩|² and the photon density of states, which grows as ω³, at its value at ω₀. Inside the line that is harmless: 10 GHz from Lyman-α's centre, a hundred widths out, ω³ has changed by 1 part in 10⁵. Far out it fails. The Lorentzian puts light at negative frequencies, has no finite variance, and at a detuning of 0.1ω₀, where it predicts 4 × 10⁻¹⁴ of the peak, a factor varying as ω³ has already moved by 30%. The second quoted input is what a decaying lower level does: the width becomes Γᵤ + Γₗ. That is plausible, since the line is carried by the coherence ρᵤₗ, which decays at (Γᵤ + Γₗ)/2, and that half-sum takes the place of Γ/2. With tabulated rates, Balmer-α's 3p → 2s component is 30.2 MHz wide, 2s being metastable, while 3s → 2p is 100.7 MHz, a hundred times what 3s's own 158 ns allows, because 2p empties in 1.6 ns.

Doppler: a Gaussian three hundred times wider

An atom moving toward the observer with velocity component v_z emits at ν = ν₀(1 + v_z/c), to first order in v_z/c. In a gas at temperature T the Maxwell distribution makes v_z Gaussian, ∝ exp(−Mv_z²/2k_BT), so the line centres spread as exp[−Mc²(ν − ν₀)²/(2k_BTν₀²)], which falls to half where (ν − ν₀)² = 2 ln2 k_BTν₀²/Mc². The full width is Δν_D = ν₀√(8 ln2 k_BT/Mc²), derived. For hydrogen at 300 K the root is 1.236 × 10⁻⁵, so Lyman-α's Doppler width is 30.5 GHz, 306 times its natural width, and Balmer-α's is 5.64 GHz. Checks: the root is energy over energy; the width vanishes as T → 0 and falls as M^(−1/2), so deuterium is √2 narrower; and it carries no α, only v/c. The broadening is inhomogeneous. Each atom still emits the whole natural Lorentzian, centred on its own shifted frequency, and the observed Voigt profile is the convolution of the two. A Gaussian dies as exp(−Δ²) and a Lorentzian only as Δ⁻², so the far wings are Lorentzian however wide the core: for Lyman-α at 300 K, beyond about 54 GHz.

Collisions add a Lorentzian of their own

A collision that jolts the emitting atom resets the phase of its wave train at random. If phase-breaking collisions arrive at rate 1/τ_c, the chance that the phase survives an interval s is e^(−s/τ_c), and it multiplies the e^(−Γs/2) of the decaying amplitude in the field's correlation function. Fourier transforming the product gives another Lorentzian, of angular width Γ + 2/τ_c, so collisions add Δν_c = 1/(πτ_c) to the natural width: derived in this impact model, which takes each collision to be brief and to scramble the phase completely. The rate is 1/τ_c = nσv̄. Take a buffer gas at 300 K, an estimated σ = 5 × 10⁻¹⁹ m², and v̄ = 2.5 × 10³ m s⁻¹, the mean relative speed of a hydrogen atom and a heavy partner. At 133 Pa, n = 3.2 × 10²² m⁻³ and Δν_c = 13 MHz, below Lyman-α's natural 100 MHz; at 101 kPa it is 9.7 GHz, a third of the Doppler width. The collisional width grows in proportion to pressure, so wherever a width matters the gas is kept thin.

The Lamb shift, hidden in the Doppler width

Unit 8's Lamb shift puts 2s₁/₂ 1057.8 MHz above 2p₁/₂, a value quoted from quantum electrodynamics and measured. It is about ten times the natural width of Balmer-α's broadest components, 100 to 110 MHz, but a fifth of the line's 5.64 GHz Doppler width at 300 K, and even deuterium cooled to 77 K leaves 2.0 GHz, so optical spectroscopy before 1947 saw only a disputed hint of it. Lamb and Retherford (1947) escaped Doppler broadening instead of fighting it: they drove microwave transitions from metastable 2s₁/₂ to the 2p levels, at 1 to 10 GHz, where the Doppler width, proportional to ν₀, is well under 1 MHz, and their resonances were about 100 MHz wide, the 2p natural width. Optical methods came with lasers that select atoms by velocity. Saturation spectroscopy sends pump and probe beams in opposite directions, so both act on the same atoms only at v_z = 0; it resolved the Lamb shift in Balmer-α in 1972 (Hänsch, Shahin and Schawlow). Doppler-free two-photon spectroscopy of 1s–2s, where opposed photons' shifts cancel, now works against a natural width of 1.3 Hz, set by 2s's quoted two-photon rate of 8.2 s⁻¹.

Change one variable at a time

Make the relationship visible.

Interactive model

Upper-level lifetime τ1.6 ns

Gas temperature T300 K

Buffer-gas pressure p0 kPa

At p = 0 and 300 K, find where the dashed wing climbs above the solid Gaussian, near ±54 GHz: beyond it the line is natural again. Lengthen τ and the wing sinks; raise p and collisions widen the Lorentzian until, at 100 kPa, it is a third of the Doppler width at 300 K and wider than it at 50 K.

NATURAL FWHM 1/(2πτ)99.5 MHz

COLLISIONAL FWHM0 MHz

DOPPLER FWHM30.5 GHz

DOPPLER ÷ LORENTZIAN FWHM306.3

Live interpretationNATURAL FWHM 1/(2πτ): 99.5 MHz. COLLISIONAL FWHM: 0 MHz. DOPPLER FWHM: 30.5 GHz. DOPPLER ÷ LORENTZIAN FWHM: 306.3

Catch the common trap

Explain before calculating.

Misconception check

A measured linewidth is not a lifetime. It is tempting to take Balmer-α's width from a room-temperature lamp, about 5.6 GHz, and invert Δν = 1/(2πτ) to get τ ≈ 28 ps, nearly 200 times shorter than 3p's 5.27 ns, the shortest lifetime at n = 3. The inversion fails twice. The 5.6 GHz is Doppler broadening, which is inhomogeneous: every atom still emits a line only 30 to 110 MHz wide, centred on its own shifted frequency, so the Gaussian envelope records the temperature, not the decay. And even a purely natural width returns Γᵤ + Γₗ, the summed rates of both levels, not the upper lifetime: 3s → 2p is 100.7 MHz wide although 3s lives 158 ns. A width yields a lifetime only when the profile is a Lorentzian of homogeneous origin, with Doppler and collisional broadening shown to be negligible, and even then it yields the summed rate.

Hydrogen's 3s level lives 158 ns and decays only to 2p, which lives 1.60 ns. With Doppler and collisional broadening removed, what is the full width at half maximum, in hertz, of the 3s → 2p line?

Choose an answer to test the model.

Practice & worked examples

Reason from the model, then test the result.

EasyHydrogen's 3p level decays to 1s at 1.6725 × 10⁸ s⁻¹ and to 2s at 2.245 × 10⁷ s⁻¹ (tabulated rates). Ignoring Doppler and collisional broadening, find the natural width of Lyman-β (3p → 1s, λ = 102.57 nm) as an angular frequency, a frequency, an energy and a wavelength, and compare it with Lyman-α's 99.7 MHz.

The lower level, 1s, is stable, so only the upper level's total rate enters, and both branches empty 3p, including the one that emits Balmer-α instead: Γ = 1.6725 × 10⁸ + 0.2245 × 10⁸ = 1.897 × 10⁸ s⁻¹, a lifetime of 5.27 ns.
The Lorentzian's full width in angular frequency is Γ itself, Δω = 1.90 × 10⁸ s⁻¹; in frequency Δν = Γ/2π = 1.897 × 10⁸/6.283 = 3.019 × 10⁷ Hz = 30.2 MHz.
In energy, ħΓ = 6.582 × 10⁻¹⁶ eV s × 1.897 × 10⁸ s⁻¹ = 1.25 × 10⁻⁷ eV.
The line frequency is ν = c/λ = 2.998 × 10⁸ m s⁻¹ / 1.0257 × 10⁻⁷ m = 2.923 × 10¹⁵ Hz, so Δν/ν = 1.03 × 10⁻⁸ and Δλ = λΔν/ν = 102.57 nm × 1.033 × 10⁻⁸ = 1.06 × 10⁻¹⁵ m.
Check: 3p lives 5.27/1.596 = 3.30 times as long as 2p, and 99.7 MHz/30.2 MHz = 3.30, as widths inversely proportional to lifetime must.

AnswerΔω = 1.90 × 10⁸ s⁻¹, Δν = 30.2 MHz, ħΓ = 1.25 × 10⁻⁷ eV and Δλ = 1.06 fm, a fractional width of 1.0 × 10⁻⁸: 3.30 times narrower than Lyman-α.

MediumDeuterium's Balmer-α (ν₀ = 4.568 × 10¹⁴ Hz, atomic rest energy Mc² = 1876 MeV) was the best optical hope for the 1.058 GHz Lamb shift. Find its Doppler width in a discharge at 77 K, the temperature at which that width would fall to 1.058 GHz, and compare both with the natural width of the 3d → 2p component, using Γ₃d = 6.47 × 10⁷ s⁻¹ and Γ₂p = 6.26 × 10⁸ s⁻¹.

k_BT = 8.617 × 10⁻⁵ eV K⁻¹ × 77 K = 6.635 × 10⁻³ eV, so 8 ln2 k_BT/Mc² = 5.545 × 6.635 × 10⁻³ eV / 1.876 × 10⁹ eV = 1.961 × 10⁻¹¹, dimensionless as it must be.
Δν_D = ν₀ √(1.961 × 10⁻¹¹) = 4.568 × 10¹⁴ Hz × 4.429 × 10⁻⁶ = 2.02 × 10⁹ Hz: nearly twice the Lamb shift even after cooling and doubling the mass, against 5.64 GHz for hydrogen at 300 K.
The width scales as √T, so reaching 1.058 GHz needs T = 77 K × (1.058/2.023)² = 21 K.
The 3d → 2p natural width, with both levels decaying (the quoted rule), is (6.47 × 10⁷ + 6.26 × 10⁸ s⁻¹)/2π = 6.907 × 10⁸/6.283 = 1.10 × 10⁸ Hz = 110 MHz, a tenth of the shift.
So the natural width never hid the Lamb shift; the Doppler width did. And a Gaussian width equal to the splitting leaves only a 7% dip between two equal components, and none when one is twice the other, so even 21 K would not have been enough.

AnswerΔν_D = 2.0 GHz at 77 K; it falls to 1.058 GHz only near 21 K; the natural width of the 3d → 2p component, 110 MHz, is ten times smaller than the shift.

HardFor hydrogen Lyman-α at 300 K the natural width is Δν_N = 99.7 MHz and the Doppler width Δν_D = 30.47 GHz. Model the observed profile as the unit-area Doppler Gaussian in the core plus the natural Lorentzian's wing, (Δν_N/2π)/Δ², far out. At what detuning Δ does the wing overtake the Gaussian, how bright is the line there relative to its peak, and is the Weisskopf-Wigner Lorentzian still trustworthy there?

The unit-area Gaussian is G(Δ) = (2/Δν_D)√(ln2/π) exp(−4 ln2 Δ²/Δν_D²). Setting it equal to (Δν_N/2π)/Δ² and writing x = Δ/Δν_D gives x² exp(−4 ln2 x²) = (Δν_N/Δν_D)/(4π√(ln2/π)).
Numbers: Δν_N/Δν_D = 0.0997 GHz/30.47 GHz = 3.272 × 10⁻³ and 4π√(ln2/π) = 5.903, so the right side is 5.54 × 10⁻⁴.
Iterate x² = ln(x²/5.54 × 10⁻⁴)/(4 ln2), which converges fast because the logarithm varies slowly: from x² = 3 it gives 3.100, 3.112, 3.114, so x = 1.765.
Δ = 1.765 × 30.47 GHz = 53.8 GHz, where G/G(0) = exp(−4 ln2 × 3.114) = 1.78 × 10⁻⁴. Below about 2 × 10⁻⁴ of the peak the profile is natural again, and its level measures Δν_N whatever the temperature.
Checks: Δ/Δν_N = 540, so the wing form (Δν_N/2π)/Δ² is fair; and Δ/ν₀ = 53.8 GHz/2.466 × 10¹⁵ Hz = 2.2 × 10⁻⁵, so the ω³ factor the Lorentzian freezes has changed by only 3 × 2.2 × 10⁻⁵ = 6.5 × 10⁻⁵.

AnswerThe Lorentzian wing takes over at about 54 GHz, 1.76 Doppler widths from centre, where the line is 1.8 × 10⁻⁴ of its peak; the Weisskopf-Wigner shape is still good there to better than 10⁻⁴.