Doppler Broadening and Gaussian Line Shape

Take the spectral line of a single, motionless atom — sharp, symmetric, set only by the radiative lifetime — and hand it to statistical mechanics. A real gas is an ensemble drawn from the one-dimensional Maxwell–Boltzmann velocity distribution, and that distribution is what this article consumes from thermal physics. Each atom carries its own line-of-sight velocity $v_z$ and sees the light Doppler-shifted: one sprinting toward the observer resonates below the rest frequency, one receding above it. The observed line is the sum of all these displaced narrow responses, and because $v_z$ is Gaussian-distributed while the shift is linear in $v_z$ , the sum is a Gaussian. That single fact — a Gaussian velocity law pushed through a linear map stays Gaussian — is the whole content of Doppler broadening, and it makes the phenomenon the textbook case of inhomogeneous broadening: the width belongs not to any atom but to the spread among atoms. That is the sharpest contrast with the Lorentzian natural or pressure line, which broadens every atom identically — and it is exactly what lets a monochromatic laser pick out one velocity class and burn a natural-linewidth hole into a broad Doppler line, the trick behind every saturated-absorption lock and optical clock. The width scales as $\nu_0\sqrt{T/M}$ , so optical lines smear a thousandfold more than microwave ones; where both mechanisms act the true profile is a Voigt convolution — Gaussian core reading temperature, Lorentzian wings reading collisions.

Key takeaways

Atoms in a warm gas follow a one-dimensional Maxwell–Boltzmann distribution; the first-order Doppler effect shifts each atom's resonance by $\Delta\nu=\nu_0 v_z/c$ , so the ensemble absorbs over a spread of frequencies.
The line shape is Gaussian, FWHM $\Delta\nu_D=\nu_0\sqrt{8\ln2\,k_BT/(Mc^2)}$ — proportional to $\nu_0$ and $\sqrt{T}$ , inversely to $\sqrt{M}$ .
The broadening is inhomogeneous: distinct velocity classes absorb at distinct frequencies, so the line is a superposition of narrow homogeneous responses, not one width shared by every atom.
For visible transitions at room temperature $\Delta\nu_D\approx1\,\mathrm{GHz}$ , about a hundred times the natural linewidth, so gas-cell spectra are Doppler-limited.
The full profile is a Voigt function — Gaussian (Doppler) convolved with Lorentzian (natural and collisional) — collapsing to a pure Gaussian only when Doppler broadening dominates.
Sub-Doppler methods (saturated absorption, two-photon spectroscopy, laser cooling) defeat the limit by removing the velocity spread, not by narrowing any single atom.
Key idea: A linear velocity-to-frequency map applied to a Gaussian velocity distribution yields a Gaussian line shape of width $\nu_0\sqrt{8\ln2\,k_BT/Mc^2}$ ; the broadening is inhomogeneous because each atom's shift is set by its own velocity.

The core insight

A linear velocity-to-frequency map applied to a Gaussian velocity distribution yields a Gaussian line shape whose FWHM is $\nu_0\sqrt{8\ln2\,k_BT/Mc^2}$ ; the broadening is inhomogeneous because each atom's shift depends on its own velocity, not on a width common to all atoms.

Where this lives today

The most striking modern use inverts the problem: instead of fighting the Doppler width, metrologists measure it to weigh temperature and even fix a constant. In Doppler-broadening thermometry, the Gaussian width of an isolated molecular line in a cell at the triple point of water is fit to extract $k_BT/Mc^2$ ; with $M$ and $T$ known it returns $k_B$ itself, feeding the 2019 SI redefinition of the kelvin. In plasma physics the same $\Delta\nu_D\propto\sqrt{T/M}$ is the standard non-invasive ion thermometer of tokamaks — the Doppler width of an impurity line gives the local ion temperature with no probe inserted, exactly Worked Example 2. Just as often the width is the enemy to be defeated. Saturated-absorption spectroscopy of rubidium and caesium vapour cells is the everyday workhorse that locks diode lasers to atomic transitions in cold-atom, quantum-computing and clock labs — scan the Doppler-broadened absorption, find the natural-width Lamb dips, servo to a dip. Doppler-free two-photon spectroscopy of hydrogen $1S$ – $2S$ underpins the most precise tests of QED, and optical lattice clocks finish the job by trapping atoms so tightly the first-order shift vanishes, reaching accuracies near $10^{-18}$ .

The physical picture

Stand at a spectrometer and watch a warm vapour. The atoms stream in every direction at hundreds of metres per second, but only their component along your line of sight, $v_z$ , matters. An atom moving toward you meets the light blueshifted and resonates below $\nu_0$ ; one moving away resonates above it; one moving sideways sits at $\nu_0$ . Nobody's line is broadened — each atom keeps its razor-thin response. What is broad is the crowd: the absorption is painted across a band by the many velocity classes, a smear whose width tracks the typical thermal speed. Why a Gaussian and not some other bell? Because the projection of an isotropic thermal cloud onto one axis is exactly a Gaussian in $v_z$ (the Maxwell–Boltzmann law), and the Doppler shift is linear in $v_z$ . A linear change of variable rescales and recentres a Gaussian but cannot change its shape — which is why the Doppler profile is Gaussian while the natural line, born from exponential dipole decay, is Lorentzian. The first figure makes this chain — velocity Gaussian in, frequency Gaussian out, through one linear map — a single picture.

Figure

Gaussian in, Gaussian out: the Maxwell–Boltzmann spread of line-of-sight velocity v_z is carried through the linear Doppler map ν = ν₀·(1 + v_z/c) into a Gaussian frequency spread. For sodium at 300 K a velocity width σ_v ≈ 330 m/s becomes σ_ν ≈ 560 MHz

Because each atom's shift is fixed by its own velocity, you can address one velocity class with a monochromatic laser — a labelling by speed no homogeneous mechanism permits. This is the basis of Doppler-free spectroscopy: a narrow laser talks only to the atoms shifted into resonance, so a counter-propagating beam pair digs a natural-width Lamb dip into the broad line's centre. The scalings then hand you the physics for free — width grows as $\sqrt{T}$ , falls as $1/\sqrt{M}$ , rises linearly with $\nu_0$ — so an optical line smears thousands of times more than a microwave hyperfine line. The second figure is the concrete object: the sodium D line at 300 K, its gigahertz width against the hundredfold-narrower natural line underneath.

Figure

The Doppler profile of the sodium D line at T = 300 K: a Gaussian of FWHM Δν_D ≈ 1.32 GHz centred on ν₀. The natural linewidth, about 10 MHz, is roughly 130 times narrower — the green spike drawn to scale — so the gas-cell line is inhomogeneous and Doppler-limited

The core equations

Everything begins with the first-order Doppler shift. An atom whose line-of-sight velocity is $v_z$ sees a laboratory source displaced in frequency, and in the non-relativistic limit ( $v_z\ll c$ ) that displacement is exactly linear in the velocity — the bridge between mechanics and the line shape.

\Delta\nu=\nu_0\,\frac{v_z}{c} \qquad (1)

where: $\nu_0$ — rest-frame transition frequency; $v_z$ — line-of-sight velocity, positive toward the observer; $c$ — speed of light; $\Delta\nu$ — frequency shift. A modest thermal speed makes a tiny fractional shift ( $v_z\sim330$ m/s gives $\Delta\nu/\nu_0\sim10^{-6}$ ), but on an optical $\nu_0\sim5\times10^{14}$ Hz that is hundreds of megahertz. Thermal equilibrium fixes the velocity statistics. Projecting the isotropic three-dimensional Maxwell–Boltzmann distribution onto the observation axis leaves a Gaussian in $v_z$ — the object we transform, and the reason the 3-D speed distribution is not what enters.

P(v_z)\,\mathrm{d}v_z=\sqrt{\frac{M}{2\pi k_BT}}\,\exp\!\left(-\frac{Mv_z^2}{2k_BT}\right)\mathrm{d}v_z \qquad (2)

where: $M$ — atomic mass; $k_B$ — Boltzmann constant; $T$ — absolute temperature; $P(v_z)$ — probability density of $v_z$ . Its standard deviation $\sqrt{k_BT/M}$ is the one-axis thermal speed, setting the scale of every width below. Feed the linear map (1) into the Gaussian (2) by conservation of probability and the line shape drops out. With the intrinsic line taken infinitely narrow, the profile is purely kinematic — the velocity distribution on a frequency axis.

g_D(\nu)=\frac{c}{\nu_0}\sqrt{\frac{M}{2\pi k_BT}}\,\exp\!\left[-\frac{Mc^2(\nu-\nu_0)^2}{2k_BT\,\nu_0^2}\right] \qquad (3)

where: $g_D(\nu)$ — normalized Doppler line shape, $\int g_D\,\mathrm{d}\nu=1$ ; $\nu$ — observation frequency. A Gaussian centred at $\nu_0$ with $\sigma_\nu=(\nu_0/c)\sqrt{k_BT/M}$ ; the prefactor $c/\nu_0$ is the map's Jacobian, keeping the area at one. The standard deviation is natural for a theorist, but spectroscopists quote the full width at half maximum. Their ratio is a fixed factor for any Gaussian, giving the most useful equation here.

\Delta\nu_D=\nu_0\sqrt{\frac{8\ln2\,k_BT}{Mc^2}} \qquad (4)

where: $\Delta\nu_D$ — FWHM of the line; $8\ln2\approx5.545$ converts $\sigma_\nu$ to FWHM via $\Delta\nu_D=2\sqrt{2\ln2}\,\sigma_\nu\approx2.355\,\sigma_\nu$ . Three knobs: the width climbs as $\sqrt{T}$ , falls as $1/\sqrt{M}$ , and rises linearly with $\nu_0$ — so the colour of the transition, not just the temperature, decides the smearing. For quick estimates it helps to fold the constants together and switch to molar units. Writing $k_B/M=R/A$ with $A$ the molar mass in g/mol gives a working formula from a wavelength, a temperature, and a mass number.

\Delta\nu_D=\nu_0\sqrt{\frac{8\ln2\,R\,T}{A\,c^2}}=7.16\times10^{-7}\,\nu_0\sqrt{\frac{T[\mathrm{K}]}{A}} \qquad (5)

where: $R$ — molar gas constant; $A$ — molar mass in g/mol; $7.16\times10^{-7}=\sqrt{8\ln2\,R/c^2}$ with mass in g/mol. Multiply by $\nu_0$ in hertz and the FWHM is in hertz. In any real cell the atoms also decay and collide, so each velocity class carries a Lorentzian, not a delta. Summing those shifted Lorentzians over the Gaussian velocity distribution is a convolution — the Voigt profile a fitting routine uses.

g_V(\nu)=\int_{-\infty}^{\infty} g_D(\nu')\,g_L(\nu-\nu')\,\mathrm{d}\nu' \qquad (6)

where: $g_V$ — observed Voigt shape; $g_D$ — Gaussian (Doppler, inhomogeneous); $g_L$ — Lorentzian (natural plus collisional, homogeneous). The Gaussian core reports temperature, the Lorentzian wings the collision rate, so one spectrum yields both. The transition's oscillator strength fixes the area under the line, so a wider profile must be a lower one. Peak absorption therefore falls as the width grows — why cooling a vapour both narrows and deepens its line centre.

\alpha(\nu_0)\propto \frac{1}{\Delta\nu_D} \qquad (7)

where: $\alpha(\nu_0)$ — absorption coefficient at line centre. Quadrupling $T$ doubles the width and halves the peak, the conserved area spread over twice the span.

Derivation AThe Gaussian line shape from the velocity distribution

The goal is to obtain the full Doppler line-shape function $g_D(\nu)$ directly, without approximation, by transforming the one-dimensional Maxwell–Boltzmann distribution of $v_z$ through the first-order Doppler relation. We treat the intrinsic transition as a delta function so the profile is purely kinematic, and lean on the single fact that the velocity-to-frequency map is linear — which guarantees a constant Jacobian and hence a shape-preserving transformation from bell to bell.

Step 1. Write the kinematics: an atom with line-of-sight velocity $v_z$ absorbs or emits at the laboratory frequency $\nu=\nu_0(1+v_z/c)$ . Why: this is the non-relativistic first-order Doppler formula, the only place the physics of motion enters, and it is linear in $v_z$ — the property everything else rides on.

Step 2. Invert the map: $v_z=c(\nu-\nu_0)/\nu_0$ , so $\mathrm{d}v_z=(c/\nu_0)\,\mathrm{d}\nu$ . Why: the map is affine, so its Jacobian $c/\nu_0$ is a constant; a constant Jacobian rescales a Gaussian without distorting it, which is why the output stays Gaussian.

Step 3. Conserve probability: the fraction of atoms in $[\nu,\nu+\mathrm{d}\nu]$ equals the fraction in the matching velocity slice, so $g_D(\nu)\,\mathrm{d}\nu=P(v_z)\,\mathrm{d}v_z$ . Why: line shape is just a relabelling of the same atoms onto a new axis — none are created or destroyed.

Step 4. Substitute (2) and the Jacobian: $g_D(\nu)=(c/\nu_0)\sqrt{M/2\pi k_BT}\,\exp[-Mc^2(\nu-\nu_0)^2/(2k_BT\nu_0^2)]$ , a normalized Gaussian with $\sigma_\nu=(\nu_0/c)\sqrt{k_BT/M}$ . Why: grouping the constants into $\sigma_\nu$ gives the standard form $g_D=(\sigma_\nu\sqrt{2\pi})^{-1}\exp[-(\nu-\nu_0)^2/2\sigma_\nu^2]$ , whose integral is one — normalization is automatic.

Go deeper · Derivation B · 1 minFWHM and the practical scaling formula

Derivation BFWHM and the practical scaling formula

The aim is to turn the standard deviation $\sigma_\nu$ into the full width at half maximum an experimentalist reads off a spectrum, then compress the constants into a single working number so a Doppler width follows from a wavelength, a temperature, and a mass in seconds. The conversion factor is universal to all Gaussians, fixing the $2\sqrt{2\ln2}$ that appears whenever a width is quoted as a FWHM rather than a sigma.

Step 1. Impose the half-maximum condition on the normalized Gaussian: $\exp[-(\nu-\nu_0)^2/2\sigma_\nu^2]=\tfrac12$ . Why: the FWHM is defined by the two detunings at which the profile drops to half its peak, and only the exponential carries $\nu$ , so the prefactor cancels out of the condition.

Step 2. Solve: taking logarithms gives $(\nu-\nu_0)_{1/2}=\pm\sigma_\nu\sqrt{2\ln2}$ , so the full width is twice this, $\Delta\nu_D=2\sqrt{2\ln2}\,\sigma_\nu\approx2.355\,\sigma_\nu$ . Why: the $\pm$ pair are the symmetric half-power points, and $2\sqrt{2\ln2}$ is the fixed sigma-to-FWHM factor for any Gaussian.

Step 3. Insert $\sigma_\nu=(\nu_0/c)\sqrt{k_BT/M}$ to reach $\Delta\nu_D=\nu_0\sqrt{8\ln2\,k_BT/Mc^2}$ , since $(2\sqrt{2\ln2})^2=8\ln2$ . Why: this is the boxed result; squaring the factor into the radical makes the $\sqrt{T}$ , $1/\sqrt{M}$ , and $\propto\nu_0$ dependences read off directly.

Step 4. Convert to molar units with $k_B/M=R/A$ : $\Delta\nu_D\approx7.16\times10^{-7}\,\nu_0\sqrt{T/A}$ ( $T$ in K, $A$ in g/mol). Why: $\sqrt{8\ln2\,R/c^2}=7.16\times10^{-7}$ with mass in g/mol, a one-line estimator reproducing the gigahertz optical widths of warm vapours.

Go deeper · Derivation C · 1 minWhy the profile is inhomogeneous and the total shape is a Voigt convolution

Derivation CWhy the profile is inhomogeneous and the total shape is a Voigt convolution

The purpose here is to make precise the claim that Doppler broadening is inhomogeneous, and to show that combining it with the homogeneous natural width is not an addition of widths but a convolution of shapes. The distinction is physical, not bookkeeping: it decides whether a laser can burn a spectral hole, and it is why the Voigt profile can be disentangled to read temperature and collision rate separately.

Step 1. Recognize that no single atom has the Doppler width: an atom with definite $v_z$ still radiates a narrow Lorentzian, merely centred at its shifted frequency $\nu_0(1+v_z/c)$ . Why: this is the definition of inhomogeneous broadening — the width lives in the ensemble spread, so a narrow probe resonant with one velocity class leaves the others untouched.

Step 2. Weight each shifted Lorentzian $g_L(\nu-\nu')$ by the Gaussian probability $g_D(\nu')$ of that shifted centre and sum over classes: $g_V(\nu)=\int g_D(\nu')\,g_L(\nu-\nu')\,\mathrm{d}\nu'$ . Why: summing shifted copies of one shape weighted by the distribution of shifts is the definition of a convolution — the Voigt profile is forced, not assumed.

Step 3. Read the limits: when $\Delta\nu_D\gg\Delta\nu_L$ the broad Gaussian dominates and $g_V\to g_D$ ; when $\Delta\nu_L\gg\Delta\nu_D$ the Gaussian acts like a delta and $g_V\to g_L$ . Why: a convolution is controlled by whichever factor is wider, so one integral interpolates the warm-optical and cold-or-dense regimes with no new physics.

Limiting cases

Case	Result	Meaning
Low frequency (microwave / RF)	$\Delta\nu_D\propto\nu_0$	Microwave hyperfine and Zeeman transitions broaden a thousandfold less than optical lines and resolve without cooling — the basis of the caesium clock.
Doppler-dominated ( $\Delta\nu_D\gg\Delta\nu_L$ )	$g(\nu)\to g_{\mathrm{Gauss}}(\nu)$	The generic case for optical lines in a warm dilute gas; the Voigt profile collapses to a pure Gaussian, the Lorentzian surviving only in the far wings.
Lorentzian-dominated ( $\Delta\nu_L\gg\Delta\nu_D$ )	$g(\nu)\to g_{\mathrm{Lorentz}}(\nu)$	Reached at very low $T$ or high pressure; the Gaussian narrows to a delta-like weight and the homogeneous width sets core and wings.
Cold-atom limit ( $T\to0$ )	$\Delta\nu_D\to0$	Laser cooling to microkelvin drops the Doppler width below the natural linewidth, uncovering the homogeneous line and enabling optical lattice clocks.

Common trapThe Doppler width comes from the atoms' full thermal speed, and you get the total linewidth by simply adding the Doppler and natural widths

Two errors travel together. First, the width is set by the one-dimensional line-of-sight velocity $v_z$ , not the full three-dimensional speed — you use $P(v_z)$ with standard deviation $\sqrt{k_BT/M}$ , not the Maxwell speed distribution whose mean is larger; an atom moving fast but transverse contributes no shift at all. Second, the Doppler (Gaussian) and natural (Lorentzian) widths do not add linearly. Line shapes combine by convolution — the Voigt profile — and a linear sum of FWHMs has no rigorous meaning. Only when the Gaussian dominates is the rough rule "widths add in quadrature" a fair approximation, and even then it is a fit to the Voigt result. When the two are comparable you must convolve; when the Lorentzian dominates the profile is essentially Lorentzian whatever a quadrature sum suggests.

Worked examples

1 · The Doppler width of the sodium D line (the flagship number)

Sodium vapour ( $A=23$ ) absorbs its D lines at $\lambda\approx589\,\mathrm{nm}$ , so $\nu_0\approx5.1\times10^{14}\,\mathrm{Hz}$ . Find the Doppler FWHM at $T=300\,\mathrm{K}$ and compare it to the natural linewidth of about $10\,\mathrm{MHz}$ .

Step 1. Apply the practical formula (5): $\Delta\nu_D=7.16\times10^{-7}\,\nu_0\sqrt{T/A}=7.16\times10^{-7}\times(5.1\times10^{14})\times\sqrt{300/23}$ , with radical $\sqrt{13.04}=3.61$ . This isolates the temperature-and-mass factor before touching the big optical frequency.

Step 2. Multiply: $7.16\times10^{-7}\times5.1\times10^{14}=3.65\times10^{8}$ , then $\times3.61=1.32\times10^{9}\,\mathrm{Hz}$ , so Δν_D ≈ 1.32 GHz. Cross-check: $\sigma_v=\sqrt{k_BT/M}=329\,\mathrm{m/s}$ gives $\sigma_\nu=(\nu_0/c)\sigma_v=560\,\mathrm{MHz}$ and $2.355\,\sigma_\nu=1.32\,\mathrm{GHz}$ — the two routes agree.

Step 3. Compare to the natural width: $\Delta\nu_D/\Delta\nu_L\approx1.32\times10^{9}/1.0\times10^{7}\approx\mathbf{132}$ . Doppler broadening beats natural broadening by two orders of magnitude, so the sodium cell is deeply Doppler-limited and the hyperfine structure (order 1 GHz) stays buried under the Gaussian until a sub-Doppler method is used. Gaussian, 1.32 GHz wide, about 130× broader than the natural line it hides.

2 · Kinetic temperature from a measured line width (spectroscopic thermometry)

A hydrogen Balmer-α line ( $A=1$ , $\lambda=656.3\,\mathrm{nm}$ , $\nu_0=4.57\times10^{14}\,\mathrm{Hz}$ ) from a laboratory plasma has Gaussian FWHM $\Delta\lambda=0.050\,\mathrm{nm}$ . Assuming the width is purely Doppler, find the ion temperature.

Step 1. Convert the wavelength width to frequency with $\Delta\nu=c\,\Delta\lambda/\lambda^2=(3.00\times10^{8})(5.0\times10^{-11})/(656.3\times10^{-9})^2=3.48\times10^{10}\,\mathrm{Hz}\approx34.8\,\mathrm{GHz}$ . The fractional width $\Delta\nu/\nu=\Delta\lambda/\lambda$ is the same either way; the $\lambda^2$ factor converts the span.

Step 2. Invert (5) for temperature: $\sqrt{T/A}=\Delta\nu_D/(7.16\times10^{-7}\,\nu_0)$ . The denominator is $7.16\times10^{-7}\times4.57\times10^{14}=3.27\times10^{8}$ , so $\sqrt{T/A}=3.48\times10^{10}/3.27\times10^{8}=106$ . With $A=1$ for hydrogen the mass factor drops out.

Step 3. Square: $T=106^{2}\times A=1.13\times10^{4}\,\mathrm{K}$ , so the plasma runs near T ≈ 11,000 K. This is the workhorse of Doppler thermometry — the Gaussian core reads the kinetic temperature independent of density. T ≈ 1.1×10⁴ K, from a 0.05 nm line width with no other diagnostic.

3 · How cold must you be to reach the natural-width floor?

For that same sodium line, at what temperature does the Doppler FWHM fall to the natural linewidth of $10\,\mathrm{MHz}$ ?

Step 1. Set $\Delta\nu_D=10\,\mathrm{MHz}$ in (5): $\sqrt{T/A}=1.0\times10^{7}/(7.16\times10^{-7}\times5.1\times10^{14})=1.0\times10^{7}/3.65\times10^{8}=2.74\times10^{-2}$ . We solve the same formula for $T$ instead of $\Delta\nu_D$ .

Step 2. Square and multiply by $A=23$ : $T=(2.74\times10^{-2})^{2}\times23=1.7\times10^{-2}\,\mathrm{K}$ . So you need T ≈ 17 mK — far below any cryostat, which is why laser cooling to microkelvin, not refrigeration, is what finally uncovers the homogeneous line. About 17 millikelvin; ordinary cooling cannot do it, laser cooling can.

Physics of the portrait

Doppler broadening sits where three subjects meet — the kinematics of a moving source, the statistical mechanics of a warm gas, and the quantum optics of a radiating atom — and its history is the story of those threads braided together, then a century later cut.

Origins · 1 minDoppler broadening and how this was discovered

Doppler broadening

thermal Gaussian broadening of spectral lines · atomic and laser spectroscopy

The idea that motion colours light is older than the atom. Christian Doppler argued in 1842 that a moving source must shift the frequency an observer measures, and Hippolyte Fizeau in 1848 transferred the principle to the spectral lines of stars. But Doppler broadening, as opposed to a bulk shift, needed the kinetic theory of gases: a thermal swarm does not displace a line, it smears it, and the smear is Gaussian because Maxwell's velocity law is Gaussian and the shift is linear in velocity. Albert Michelson's interferometric line-visibility measurements in the 1890s made the thermal width visible for the first time and pinned it, correctly, on molecular motion. Once both mechanisms were understood, Woldemar Voigt in 1912 wrote the profile that carries his name — the convolution of the Gaussian thermal core with the Lorentzian radiative wings. For fifty years the Doppler width was the wall precision spectroscopy could not climb, setting the resolution of every gas-cell measurement and the gain bandwidth of every gas laser. The wall came down in the 1970s: saturated-absorption spectroscopy, demonstrated on hydrogen Balmer-α by Hänsch, Shahin and Schawlow in 1971, turned the line's inhomogeneous character against itself — counter-propagating beams talk only to the $v_z=0$ class and carve a natural-width Lamb dip into the Gaussian's centre. Doppler-free two-photon spectroscopy then cancelled the first-order shift outright, and laser cooling, proposed in 1975, removed the velocity spread at its source. What began as an unavoidable consequence of heat became a limit engineered away, and atoms once too warm to measure now tick as clocks.

History

1842

— Doppler predicts that a source's motion shifts the frequency an observer measures

1848

— Fizeau extends the principle to the spectral lines of stars

1892

— Michelson's interferometry resolves thermally broadened optical line widths

1912

— Voigt derives the Gaussian–Lorentzian convolution line profile

1971

— Hänsch, Shahin and Schawlow burn sub-Doppler Lamb dips into the Balmer-α line

Going further

Texts. Foot, Atomic Physics (2005), §8.1 — Doppler broadening and the Gaussian line shape · Demtröder, Laser Spectroscopy Vol. 1 (2008), §3.2 and Ch. 2 — Voigt profiles and sub-Doppler methods · Loudon, The Quantum Theory of Light, 3rd ed. (2000), §2.10 · Saleh & Teich, Fundamentals of Photonics, 2nd ed. (2007), §13.3D · Hecht, Optics, 5th ed. (2017), §7.4.

Provenance. The line shape is derived by transforming the one-dimensional Maxwell–Boltzmann distribution through the first-order Doppler relation $\nu=\nu_0(1+v_z/c)$ ; the FWHM follows from the half-maximum condition with the universal factor $2\sqrt{2\ln2}$ . A verifier should recompute: (1) the constant $7.16\times10^{-7}=\sqrt{8\ln2\,R/c^2}$ with mass in g/mol; (2) the sodium D-line width $\Delta\nu_D\approx1.32\,\mathrm{GHz}$ at 300 K and its ratio $\approx132$ to a 10 MHz natural line; (3) the thermometry inversion $T\approx1.1\times10^{4}\,\mathrm{K}$ from a 0.05 nm Balmer-α width; (4) the 17 mK cross-over where the sodium Doppler width meets the natural width. Cross-check against Foot §8.1, Demtröder §3.2, and Loudon §2.10.

Builds on. Radiative Lifetime and Natural Linewidth · Lorentzian Line Shape · Resonant Absorption. Leads to. Einstein A and B Coefficients · Detailed Balance and the Planck Spectrum.

Ask the tutor: Ask me to work out the Doppler width of any line you name — give a wavelength, a temperature, and an atom, and we will get the gigahertz and check whether it swamps the natural width.

Why does only the line-of-sight velocity matter, not the full speed?

Because the first-order shift depends only on the projection of velocity onto the observation direction, $\Delta\nu=\nu_0 v_z/c$ . An atom moving at 500 m/s straight across your line of sight has $v_z=0$ and sits at $\nu_0$ ; only its toward-or-away component shifts the frequency. Projecting the isotropic Maxwell–Boltzmann cloud onto one axis leaves a Gaussian in $v_z$ with standard deviation $\sqrt{k_BT/M}$ — the distribution the line inherits. Using the full 3-D speed distribution would overcount the width, since that distribution is not even Gaussian.

Why is the profile Gaussian when the natural line is Lorentzian?

They come from different physics. The Lorentzian is the spectrum of a single atom's exponentially decaying dipole — a homogeneous width every atom shares, set by the radiative lifetime. The Gaussian is a statistical envelope no atom has: the spread of centre frequencies across the ensemble, and it stays Gaussian because a constant-Jacobian linear map cannot deform a bell. So the two coexist for one line — a Gaussian distribution of narrow Lorentzians, convolving into the Voigt profile.

What is inhomogeneous broadening, and why does it matter?

Inhomogeneous broadening is a width that lives in the spread among emitters rather than in any single one — here, different atoms absorb at different frequencies because they move at different velocities. It matters because it can be selectively addressed: a narrow laser resonant with one velocity class leaves the rest untouched, so you can burn a spectral hole or Lamb dip a homogeneous width would never permit — the mechanism behind saturated-absorption laser locking and Doppler-free spectroscopy. Homogeneous broadening (natural, collisional) has no such structure: every atom responds identically, so none can be singled out.

How does cooling actually help, given the weak square-root law?

Because $\Delta\nu_D\propto\sqrt{T}$ , ordinary cooling is weak: dropping from 300 K to 3 K narrows the line only tenfold, and Worked Example 3 shows sodium needs about 17 mK to reach a 10 MHz floor. That is why the answer is not refrigeration but laser cooling, which reaches microkelvin and drives the Doppler width below the natural linewidth — or, better, tight trapping (optical lattices, ion traps) that suppresses the first-order shift by confining the atom to less than a wavelength (the Lamb–Dicke regime).

Why is a microwave clock transition barely Doppler-broadened while an optical one is not?

Because the Doppler width is proportional to the transition frequency, $\Delta\nu_D\propto\nu_0$ . The caesium hyperfine transition sits near 9.2 GHz, about $10^{5}$ times lower than an optical line, so at the same temperature its Doppler width is $10^{5}$ times smaller — narrow enough to resolve in a warm atomic beam without cooling. The same fractional velocity spread $v_z/c\sim10^{-6}$ makes a shift proportional to $\nu_0$ : a hertz-scale nuisance in the microwave, a gigahertz-scale wall in the visible. This is a major reason the first atomic clocks were microwave devices.

How do astronomers and plasma physicists get a temperature from a line width?

They fit the line to a Voigt profile and read the Gaussian core. The Doppler FWHM depends only on $T$ , $M$ , and $\nu_0$ ; all but $T$ are known for a chosen line, so inverting $\Delta\nu_D=\nu_0\sqrt{8\ln2\,k_BT/Mc^2}$ returns the kinetic temperature — independent of density, as Worked Example 2 shows. The Lorentzian wings of the same fit encode pressure, so one spectrum yields both temperature and density. The caution: bulk flow and turbulence also broaden the Gaussian, so a measured width is an upper bound on the thermal temperature unless non-thermal motions are modelled separately.