# Doppler broadening **Doppler broadening** is the widening of a [[Spectral_line|spectral line]] that arises because the particles absorbing or emitting the light are moving. Each particle carries its own [[Doppler_effect|Doppler shift]], so the line the instrument records is not one particle's line but the sum of many displaced copies of it. In a gas in [[Thermodynamic_equilibrium|thermal equilibrium]] the velocities follow a [[Maxwell–Boltzmann_distribution|Maxwell–Boltzmann distribution]], and the sum takes on that distribution's shape: a bell whose width reports the [[Temperature|temperature]] of the gas and the mass of the particle, and not the properties of the transition at all. The width belongs to the gas, not to the [[Atom|atom]]. In the microsim below the reader builds such a line one atom at a time. A single control sets N, the number of atoms drawn, on a logarithmic scale from 3 to 100,000. Each draw takes a velocity component v∥ along the beam from the one-dimensional Maxwell–Boltzmann distribution, displaces that atom's 5 MHz [[Spectral_line_shape|Lorentzian]] by `df_D = -v_par/lambda`, and adds it to a running sum on a fixed ±1,000 MHz axis. At N = 3 the display is three spikes; by N = 100,000 it is the smooth 1.22 GHz bell given by `df_FWHM = (2.355/lambda)*sqrt(k_B*T/m)`, drawn against the dashed Gaussian for comparison, while every constituent line is still only 5 MHz wide.[^raven-fig47][^raven-width] On the [[Physics]] flagship this article serves Part IV — Branches and fields, section *Atomic, molecular, and optical* (row P57), where it sits beside the natural-linewidth and saturation sims built from the same two-level atom. Its Wikitube siblings [[Spectral_line_shape]] and [[Atomic_physics]] carry the line shapes that Doppler broadening hides. ## Derivation (non-relativistic case) An atom moving with velocity component v∥ along the direction of a beam of light of wavelength λ does not see the laboratory frequency f_L. To first order in v/c the frequency in the atom's own frame is `f_atom = fL*(1 - v_par/c)`, and since f_L = c/λ the change can be written without the [[Speed_of_light|speed of light]] at all: `df_D = -v_par/lambda`.[^raven-shift] The convention matters more than the algebra. In the Portal Book's sign convention v∥ is negative for motion *toward* the source, so an approaching atom has Δf_D > 0 and sees the light shifted up in frequency; reversing the sign of v∥ is, in the book's words, "a very common mistake".[^raven-shift] For motion at an angle θ to the beam only the projection counts, `df_D = -(v/lambda)*cos(theta)`, which is the form that matters for atomic beams and is used again below.[^raven-costheta] The shift is small in relative terms and large in absolute ones. At λ = 940 nm an atom moving at 141 m/s along the beam shifts its line by 150 MHz (derived), a fraction 4.7 × 10⁻⁷ of the optical frequency but thirty times the 5 MHz natural width of the transition.[^raven-threeatoms][^derived] That mismatch of scales is why Doppler broadening dominates ordinary gas-phase [[Spectroscopy|spectroscopy]]: a shift far too small to change the colour of the light is far too large to ignore when the line is being resolved. #### Three atoms, three lines The Portal Book introduces the effect with the smallest interesting case: three atoms, one at rest and two moving with equal and opposite v∥ large enough to give |Δf_D| = 150 MHz. The recorded absorption is then not one line but three, at −150, 0 and +150 MHz, each with the natural width the transition would have if the atom were held still.[^raven-threeatoms] Nothing has happened to the atoms; the spectrometer is simply looking at three different [[Absorption_spectroscopy|absorbers]] whose resonances have been moved apart. Setting the microsim's control to N = 3 reproduces this panel, and it is worth dwelling on, because the whole of Doppler broadening is contained in it. Everything that follows is the same construction with the three velocities replaced by a distribution and the three spikes replaced by their sum. ### Thermal Doppler broadening In a gas at temperature T the component of velocity along any one axis is distributed as `f(v_par) = (m/(2*pi*k_B*T))^(1/2)*exp(-m*v_par^2/(2*k_B*T))`, a Gaussian of standard deviation `sigma = sqrt(k_B*T/m)` centred on zero, with [[Boltzmann_constant|k_B]] = 1.38 × 10⁻²³ J/K.[^raven-mb][^raven-kb] This is the one-dimensional velocity law of the [[Kinetic_theory_of_gases|kinetic theory of gases]], not the three-dimensional speed law, which carries an extra v² factor and a 3/2 power in its normalisation; using the speed law here is the second classic error in the derivation.[^raven-mb] A Gaussian's full width at half maximum is 2.355 times its standard deviation, so the spread of velocities along the beam has `dv_FWHM = 2.355*sqrt(k_B*T/m)`, and mapping it through Δf_D = −v∥/λ turns it into the Doppler width of the line,[^raven-width] `df_FWHM = (2.355/lambda)*sqrt(k_B*T/m)`. Three things are absent from that expression, and their absence is the physics. The lifetime of the excited state does not appear, so the natural width is irrelevant to it. The intensity of the light does not appear, so saturation cannot change it. The identity of the transition enters only through λ. What remains is a thermometer: the width measures T, given the mass, and measures the mass, given T. Rearranged, `T = m*lambda^2*df_FWHM^2/(2.355^2*k_B)` (derived), which is how the ion temperature of a hot [[Plasma_(physics)|plasma]] or the kinetic temperature of an [[Interstellar_medium|interstellar]] cloud is read off a line profile.[^derived] #### The worked case: nitrogen at 400 K The microsim is built on the parameters of the Portal Book's own simulation: a [[Nitrogen|nitrogen]] atom of mass m = 2.33 × 10⁻²⁶ kg at T = 400 K, probed at λ = 940 nm on a transition of natural width γ = 5 MHz, displayed on an axis of ±1,000 MHz.[^raven-fig47] Those numbers give σ = 486.7 m/s for the velocity spread, Δv_FWHM = 1,146 m/s, and a Doppler width of 1.22 GHz — about 244 natural linewidths (derived).[^derived] The ratio is the number that decides how the experiment must be done. A spectrometer that resolves 5 MHz will still see a 1.22 GHz line, and the 244 individual resonances hidden inside it are not merely blurred but genuinely superposed, since each atom absorbs at a frequency no other atom shares. The scaling is easy to carry away. Width goes as √T, so raising the gas from 400 K to 10,000 K widens the line five-fold, to 6.1 GHz (derived). Width goes as 1/√m, so a [[Hydrogen|hydrogen]] atom, fourteen times lighter than nitrogen, gives a line √14 ≈ 3.7 times wider at the same temperature and wavelength (derived). And width goes as 1/λ, so the same gas broadens far less in the infrared than in the ultraviolet. Light, hot and blue all mean wide.[^derived] #### Building the bell by Monte Carlo The sim does not evaluate the Gaussian; it draws atoms. Each of the N velocities is a seeded pseudorandom sample from the one-dimensional distribution, and the displayed curve is the normalised running sum of N Lorentzians of the form `L(f) = A/(1 + 4*(f - fr)^2/gamma^2)`, one per atom.[^raven-lorentz] This is a [[Monte_Carlo_method|Monte Carlo]] construction of a line shape, and its convergence is the thing to watch: the sum is ragged at small N because the tallest spikes come from whichever few atoms happened to be drawn near a given detuning, and the raggedness falls only as 1/√N, so the three decades between N = 100 and N = 100,000 buy a factor of about thirty in smoothness. The Portal Book prints panels at N = 3, 10, 100, 1,000, 10,000 and 100,000, and the sim places checkpoints at the last three so the reader can compare directly.[^raven-fig47] The book's own counting example makes the sampling concrete: of 5,000 atoms drawn from such a distribution, fractions of 0.726, 0.061 and 0.347 correspond to 3,630, 305 and 1,735 atoms falling in the chosen velocity intervals.[^raven-counts] Two properties survive the whole build, and the sim marks both. The constituent lines never widen — each is 5 MHz across at N = 3 and at N = 100,000 alike — and the envelope is never exactly a Gaussian, because a sum of Lorentzians weighted by a Gaussian is a [[Spectral_line_shape|Voigt profile]], whose wings fall off as 1/Δf² rather than exponentially. The Portal Book puts it plainly: the profile is only "pretty close to" Gaussian, and the dashed comparison curve in the sim is there to show where the two part company.[^raven-fig47] ## Applications and caveats Doppler broadening is exploited as a measurement and defeated as a nuisance, often in the same laboratory. As a measurement it is the standard non-contact thermometer of hot and remote gases: the inverted width formula turns a recorded profile into a temperature for a [[Nuclear_fusion|fusion]] plasma, a flame, a discharge lamp, or the atmosphere of a [[Star|star]]. Because the same formula contains the mass, a profile that is too wide for the measured temperature is evidence of turbulence or of a lighter species, and the line shape of the [[Sun|Sun]]'s photospheric lines is read this way. The effect is a first-order Doppler shift summed over a population, and it is distinct from the systematic [[Redshift|redshift]] of a receding source, which moves the whole line without broadening it. #### Getting rid of it The angular form of the shift is the escape route. Since `df_D = -(v/lambda)*cos(theta)`, an atomic or molecular beam observed at right angles to its own direction of travel has cos θ = 0 and shows no first-order broadening at all.[^raven-costheta] The residual is set by how well the right angle is held: at θ = 90° ± 1 mrad, an atom at the 487 m/s of the worked case above still contributes a shift of about 520 kHz (derived), three orders of magnitude below the 1.22 GHz it would have contributed along the beam.[^derived] Cooling attacks the same term from the other side, since the width falls as √T, and the saturation behaviour of the two-level atom — the scattering rate `r = pi*gamma*s/(1 + s + 4*delta^2/gamma^2)`, which selects only the velocity class in resonance with a detuned [[Laser|laser]] — is what makes laser cooling and Doppler-free saturated absorption possible in the first place.[^raven-saturation] These are the techniques behind precision [[Atomic,_molecular,_and_optical_physics|atomic, molecular and optical]] measurement, and they exist because the width of a room-temperature line is a property of the container rather than of the [[Photon|photon]]. #### What the treatment leaves out Four caveats attach to the derivation as it stands. It is non-relativistic: the expansion keeps only the first power of v/c, which is ample for thermal atoms and fails for relativistic beams.[^raven-nonrel] It assumes equilibrium, so that the velocity law applies at all; the equipartition result `(1/2)*m*<v^2> = (3/2)*k_B*T` is the statement being leaned on, and a gas with a directed flow or a non-thermal tail has a profile that no single temperature describes.[^raven-equipartition] It ignores every other broadening mechanism: the natural width set by the excited-state lifetime through `tau = 1/(2*pi*gamma)` — a 6.25 ns lifetime gives 25 MHz (derived), and the [[Caesium|caesium]] line the Portal Book works has γ = 5.22 MHz — the power broadening `gamma_s = gamma*sqrt(1 + s)` that grows without limit as the light intensity rises, and collisional or pressure broadening, which the Portal Book explicitly does not cover.[^raven-lorentz][^raven-saturation][^derived] All of these add to the observed profile, and only the Doppler term carries the temperature. Finally, the figure the sim reproduces is itself a simulation and not measured data, a point the book makes about its own Fig. 4.7; the sim inherits that status, and the comparison Gaussian is an ILLUSTRATIVE reference curve rather than a fit to an experiment.[^raven-fig47] The relation to the [[Uncertainty_principle|energy–time uncertainty relation]] is worth stating precisely, because it is easy to confuse the two widths. ΔE·Δt ≈ ħ fixes the natural width from the lifetime and is a property of the transition; the Doppler width is a property of an ensemble and would vanish for a single atom held at rest. A gas of long-lived atoms has narrow lines that are nonetheless broad in the spectrometer, and cooling — not a longer lifetime — is what recovers them. ## See also - [[Spectral_line_shape]] - [[Atomic,_molecular,_and_optical_physics]] - [[Atomic_physics]] - [[Maxwell–Boltzmann_distribution]] - [[Doppler_effect]] - [[Spectral_line]] - [[Quantum_optics]] ## References [^raven-shift]: Raven, Will (2025). *Atomic Physics for Everyone: An Introduction to Atomic Physics, Quantum Mechanics, and Precision Spectroscopy with No College-Level Prerequisites*. Part I, "Atom–Light Interactions," Ch. 4 "Atoms in Motion," pp. 78–79 (the first-order shift f_atom = f_L(1 − v∥/c) = f_L − v∥/λ and the sign convention for v∥). Portal Book 046. https://open.umn.edu/opentextbooks/textbooks/atomic-physics-for-everyone-an-introduction-to-atomic-physics-quantum-mechanics-and-precision-spectroscopy-with-no-college-level-prerequisites [^raven-costheta]: Raven, *Atomic Physics for Everyone* (2025), p. 79 (the general form Δf_D = −(v/λ)·cos θ). Portal Book 046. [^raven-threeatoms]: Raven, *Atomic Physics for Everyone* (2025), p. 81 (three atoms with |Δf_D| = 150 MHz give features at −150, 0 and +150 MHz). Portal Book 046. [^raven-mb]: Raven, *Atomic Physics for Everyone* (2025), pp. 83–84 (the one-dimensional Maxwell–Boltzmann velocity distribution, σ = √(k_B·T/m), Δv_FWHM = 2.355·σ, and the distinction from the three-dimensional speed law). Portal Book 046. [^raven-kb]: Raven, *Atomic Physics for Everyone* (2025), p. 83 (k_B = 1.38 × 10⁻²³ J/K). Portal Book 046. [^raven-width]: Raven, *Atomic Physics for Everyone* (2025), p. 86 (the Doppler width Δf_FWHM = (2.355/λ)·√(k_B·T/m)). Portal Book 046. [^raven-fig47]: Raven, *Atomic Physics for Everyone* (2025), pp. 85–86 (Fig. 4.7: m = 2.33 × 10⁻²⁶ kg, T = 400 K, λ = 940 nm, γ = 5 MHz, an axis of ±1,000 MHz, and panels at N = 3, 10, 100, 1,000, 10,000 and 100,000 randomly drawn atoms; the profile is "pretty close to" Gaussian; the figure is a simulation, not measured data). Portal Book 046. [^raven-counts]: Raven, *Atomic Physics for Everyone* (2025), p. 83 (counting example on 5,000 atoms: fractions 0.726 → 3,630, 0.061 → 305, 0.347 → 1,735). Portal Book 046. [^raven-lorentz]: Raven, *Atomic Physics for Everyone* (2025), pp. 55–60 (the low-power Lorentzian L(f) = A/(1 + 4(f − f_r)²/γ²); Γ = 2πγ; τ = 1/Γ = 1/(2πγ); the 6.25 ns excited-state lifetime). Portal Book 046. [^raven-saturation]: Raven, *Atomic Physics for Everyone* (2025), pp. 63–67 (saturation parameter s = I/I_s, the scattering rate r = πγs/(1 + s + 4δ²/γ²), the power-broadened width γ_s = γ√(1 + s), the caesium line γ = 5.22 MHz at 852 nm, and the statement that pressure broadening is not covered). Portal Book 046. [^raven-equipartition]: Raven, *Atomic Physics for Everyone* (2025), p. 87 (equipartition: ½·m·⟨v²⟩ = (3/2)·k_B·T). Portal Book 046. [^raven-nonrel]: Raven, *Atomic Physics for Everyone* (2025), p. 90 (the treatment of moving atoms is non-relativistic). Portal Book 046. [^derived]: Values marked *derived* on this page were computed for Wikitube from the Portal Book's own equations and printed parameters, and are not printed in the book. From Fig. 4.7's m = 2.33 × 10⁻²⁶ kg, T = 400 K and λ = 940 nm: σ = √(k_B·T/m) = 486.7 m/s; Δv_FWHM = 2.355·σ = 1,146 m/s; Δf_FWHM = Δv_FWHM/λ = 1.22 GHz, which is 244 times the 5 MHz natural width. A 150 MHz shift at 940 nm corresponds to v∥ = 141 m/s, and 150 MHz is 4.7 × 10⁻⁷ of the 319 THz optical frequency. Width scales as √(T/m)/λ, giving 6.1 GHz at 10,000 K and a factor √14 ≈ 3.7 between atomic hydrogen and atomic nitrogen at equal T and λ. Transverse geometry: at θ = 90° ± 1 mrad an atom at 486.7 m/s contributes |Δf_D| = 486.7 × 10⁻³/940 × 10⁻⁹ ≈ 5.2 × 10⁵ Hz. Inverting the width formula gives T = m·λ²·Δf_FWHM²/(2.355²·k_B). A 6.25 ns lifetime gives γ = 1/(2π·6.25 × 10⁻⁹) = 25.5 MHz. <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Doppler_broadening.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Doppler broadening* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Doppler_broadening.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Doppler_broadening.html" data-title="Doppler broadening"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Doppler_broadening) : [Wikitube](https://en.wikitube.io/wiki/Doppler_broadening) · pinned revision [1358645647](https://en.wikipedia.org/w/index.php?oldid=1358645647) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P57 · sim pending (matter/Doppler_broadening).*