Collisional and Pressure Broadening

An excited atom does not radiate forever. The preceding articles of this chapter establish that spontaneous emission alone damps the atomic dipole over the radiative lifetime, and that the Fourier transform of an exponentially decaying wavetrain is a Lorentzian of width set by that lifetime — the natural linewidth. This article takes those owned results, the natural Lorentzian from aq-optics-radiative-lifetime-natural-linewidth and the correlation-to-spectrum machinery behind aq-optics-lorentzian-line-shape, and asks what happens when the atom is not alone. In a gas the radiator is surrounded by perturbers, and every so often one passes close enough to jolt its energy levels and scramble the optical phase it has been accumulating. Each collision truncates the coherent wavetrain and starts a new one with an unrelated phase. The physics is entirely a statement about coherence time. A wavetrain chopped every few tens of picoseconds carries a broader Fourier spectrum than one that decays over nanoseconds, so collisions broaden the line — and because the collision rate is proportional to the perturber density, the width grows linearly with pressure. This is pressure broadening, the dominant homogeneous broadening of optical and microwave transitions in any gas above a fraction of an atmosphere. Collisions also do more than randomize phase: the transient level shift during each encounter, if it does not average away, drags the line centre to the red or blue, a fingerprint of the interatomic potential. At atmospheric pressure the effect is enormous — gigahertz of width where the natural line is only megahertz.

Key takeaways

Collisions interrupt the phase of the radiating dipole; each interruption truncates the wavetrain, and the finite coherence time broadens the line into a Lorentzian of FWHM $\Delta\nu_{\mathrm{col}}=1/(\pi\tau_{\mathrm{col}})$ , with $\tau_{\mathrm{col}}$ the mean time between collisions.
The collision rate is $1/\tau_{\mathrm{col}}=n\,\sigma\,\bar v_r$ , so the width grows linearly with density and hence with pressure — pressure broadening.
In the impact (Lorentz) approximation collisions are instantaneous events on top of the radiative decay; both parts are Lorentzian, so their half-widths add linearly, never in quadrature.
Elastic collisions that transiently shift the levels also shift the line centre; the shift-to-broadening ratio fingerprints the interatomic potential (attractive → red, repulsive → blue).
Collisional broadening is homogeneous — every atom sees the same statistics — so unlike the Doppler width it cannot be defeated by Doppler-free spectroscopy.
Key idea: a collision that scrambles the optical phase truncates the coherent emission after a mean time $\tau_{\mathrm{col}}$ ; the finite coherence produces a homogeneous Lorentzian of FWHM $\Delta\nu_{\mathrm{col}}=1/(\pi\tau_{\mathrm{col}})=n\,\sigma\,\bar v_r/\pi$ , linear in density and therefore in pressure.

The core insight

A collision that scrambles the optical phase truncates the coherent emission after a mean time $\tau_{\mathrm{col}}$ ; the finite coherence produces a Lorentzian of FWHM $\Delta\nu_{\mathrm{col}}=1/(\pi\tau_{\mathrm{col}})=n\,\sigma\,\bar v_r/\pi$ , linear in number density and hence in pressure.

Where this lives today

Pressure broadening is the everyday enemy and diagnostic of gas-phase spectroscopy. In tunable-diode-laser absorption spectroscopy (TDLAS), engineers scan a distributed-feedback laser across a single line of water, CO, or CO2 in a combustor or engine exhaust and fit the Voigt profile: its Gaussian part gives the temperature through the Doppler width, its Lorentzian part the pressure and concentration through the collisional width and shift. Those coefficients fill the HITRAN and HITEMP databases behind every atmospheric and combustion radiative-transfer model, and the quasi-static wings of CO2 and water lines set the greenhouse forcing of a planetary atmosphere. At the opposite extreme of precision, the collisional shift is the villain in vapour-cell atomic clocks. A rubidium or caesium clock runs in a buffer gas that slows atomic diffusion to the walls, but every buffer-gas collision shifts the clock transition; because attractive and repulsive perturbers shift it oppositely, builders blend gases (say nitrogen and argon) in a ratio that nulls the net shift, and even tune the temperature to its turning point. The broadening that ruins resolution at high pressure is thus, read in reverse, a remote thermometer, barometer, and probe of the interatomic potential.

The physical picture

Picture an oscillator humming at a fixed frequency that at random moments takes a sharp kick jumping its phase to a new, unrelated value. Listen a long time and you never hear a pure tone; you hear a smear whose frequency width is roughly the inverse of the average time between kicks — the Fourier uncertainty principle, since a signal coherent only for a time $\tau$ cannot be sharper than $1/\tau$ . The radiating atom is exactly this oscillator, and the kicks are collisions. The first figure makes the truncation literal. Undisturbed, an atom radiates a long, phase-continuous wavetrain that decays only over its radiative lifetime — nanoseconds for an allowed transition. Add a gas and the same atom is struck every few tens of picoseconds, each strike resetting the phase and dicing the train into stubby, incoherent segments. It is the length of the coherent segment, not the total emission time, that sets the linewidth.

Figure

A radiating atom's wavetrain without collisions advances in phase smoothly and decays only over the nanosecond natural lifetime (top); with collisions each encounter at the red marks resets the optical phase (bottom), and at 1 atm these arrive every τ_col ≈ 62 ps, chopping the coherent train into segments far shorter than the natural decay

Collisions come in two flavours. Weak, distant encounters nudge the phase (dephasing — broadening, plus a shift if the mean phase is nonzero); strong, close encounters quench the atom outright, ending the emission. Both truncate the coherent train, but only phase-shifting collisions move the line centre. Pressure enters only as a proxy for collision rate. Double the density and you double how often the radiator is struck, halving the coherence time and doubling the linewidth — a strictly linear pressure dependence at fixed temperature, the signature separating collisional from Doppler broadening. And because every atom samples the same statistics, the broadening is homogeneous: you cannot burn a spectral hole in it. The second figure shows the payoff — the width climbing straight through the fixed Doppler width.

Figure

Collisional width grows linearly with pressure while the Doppler width stays fixed: for the sodium D line perturbed by argon at 300 K the Lorentzian collisional FWHM rises at about 5.1 GHz per atm and overtakes the 1.3 GHz Doppler Gaussian near 0.26 atm, so above roughly a quarter-atmosphere the profile is homogeneously broadened

The core equations

The single atom's radiated field is a decaying, oscillating carrier onto which each collision stamps an abrupt, random phase increment. This separates the two clocks limiting coherence: the slow radiative decay and the fast phase resets.

E(t)=E_0\,e^{-t/2\tau_{\mathrm{nat}}}\,e^{-i\omega_0 t}\,e^{i\phi(t)} \qquad (1)

where: $E(t)$ — emitted field of one atom; $\tau_{\mathrm{nat}}$ — natural (radiative) lifetime; $\omega_0$ — unperturbed transition angular frequency; $\phi(t)$ — piecewise-constant random phase, reset at each collision. The factor $\tfrac12$ in the amplitude exponent is because intensity, not amplitude, decays at rate $1/\tau_{\mathrm{nat}}$ . The measurable spectrum is built not from $E(t)$ directly but from how well the field at one instant predicts the field a time $\tau$ later — its first-order autocorrelation. By Wiener–Khinchin the spectrum is the Fourier transform of this correlation, so the whole line shape is encoded in how $g_1(\tau)$ decays.

g_1(\tau)=\langle E^*(t)\,E(t+\tau)\rangle \propto e^{-\tau/2\tau_{\mathrm{nat}}}\,e^{-\tau/\tau_{\mathrm{col}}} \qquad (2)

where: $g_1$ — first-order field correlation function; $\tau_{\mathrm{col}}$ — mean time between phase-interrupting collisions. Two independent decoherence channels multiply their survival probabilities, so their rates add in one exponential — the root of linear (not quadrature) width addition. The Fourier transform of a decaying exponential correlation is a Lorentzian, its full width set by the total decoherence rate. Collecting the radiative and collisional rates gives the homogeneous line and width in one stroke.

I(\omega)\propto \frac{\Gamma/2}{(\omega-\omega_0)^2+(\Gamma/2)^2},\qquad \Gamma=\frac{1}{\tau_{\mathrm{nat}}}+\frac{2}{\tau_{\mathrm{col}}} \qquad (3)

where: $I(\omega)$ — spectral intensity; $\Gamma$ — total FWHM in angular frequency. The two contributions enter as a plain sum; the factor of $2$ on the collision term separates the amplitude decay rate ( $1/2\tau_{\mathrm{nat}}$ ) from the phase-decoherence rate ( $1/\tau_{\mathrm{col}}$ ). Isolating the collisional part and converting to ordinary frequency gives the central working formula. The mean collision time is set by gas-kinetic quantities, where pressure enters.

\Delta\nu_{\mathrm{col}}=\frac{1}{\pi\tau_{\mathrm{col}}}=\frac{n\,\sigma\,\bar v_r}{\pi} \qquad (4)

where: $\Delta\nu_{\mathrm{col}}$ — collisional FWHM in hertz; $n$ — perturber number density; $\sigma$ — effective (optical) broadening cross section; $\bar v_r$ — mean relative speed. The width is just the collision rate divided by $\pi$ — count the phase resets per second and you have the linewidth. The mean relative speed follows from the Maxwell–Boltzmann distribution of the colliding pair. That is why the reduced mass, not either atomic mass alone, appears, and where the temperature dependence enters.

\bar v_r=\sqrt{\frac{8k_BT}{\pi\mu}},\qquad \mu=\frac{m_A m_B}{m_A+m_B} \qquad (5)

where: $k_B$ — Boltzmann constant; $T$ — gas temperature; $\mu$ — reduced mass of radiator and perturber; $m_A,m_B$ — their masses. A light perturber (small $\mu$ ) collides faster and broadens more per unit density. Eliminating the density in favour of pressure through the ideal-gas law $P=nk_BT$ exposes the experimental signature. The width comes out strictly proportional to pressure, and weakly falling with temperature.

\Delta\nu_{\mathrm{col}}=\frac{\sigma\,\bar v_r}{\pi k_BT}\,P\;\propto\;\frac{P}{\sqrt{T}} \qquad (6)

where: $P$ — gas pressure. With $\bar v_r\propto\sqrt{T}$ and $n\propto1/T$ at fixed pressure, the powers combine to $1/\sqrt{T}$ : heat the gas and the line narrows slightly. Finally, the encounters that randomize phase generally leave a nonzero mean phase per collision, displacing the line centre. The shift is a flux-weighted integral of the collision phase over impact parameter — distinct from the broadening, but from the same interaction.

\Delta_s=n\,\bar v_r\int_0^\infty 2\pi b\,\sin[\eta(b)]\,\mathrm{d}b,\qquad \frac{\Delta_s}{\Delta\nu_{\mathrm{col}}}\sim\!\text{const of the potential} \qquad (7)

where: $\Delta_s$ — shift of the line centre; $b$ — impact parameter; $\eta(b)$ — phase accumulated at impact parameter $b$ . Broadening comes from $[1-\cos\eta(b)]$ and the shift from $\sin\eta(b)$ ; their ratio is fixed by the potential — red for attraction, blue for repulsion.

Derivation AFrom Poisson-interrupted wavetrains to the Lorentzian width

The goal is to derive the collisional Lorentzian and its width from the statistics of phase-interrupting collisions alone, and thereby prove that the collisional and natural widths add linearly. We model the single-atom field as a radiatively damped oscillator whose phase is reset by collisions arriving as a Poisson process at rate $1/\tau_{\mathrm{col}}$ , compute the autocorrelation, and Fourier-transform it. The payoff is $\Delta\nu_{\mathrm{col}}=1/(\pi\tau_{\mathrm{col}})=n\sigma\bar v_r/\pi$ , plus the reason the profile is a Lorentzian and not a Gaussian.

Step 1. Start from the correlation function $g_1(\tau)=\langle E^*(t)E(t+\tau)\rangle$ , whose Fourier transform is the measured spectrum. Why this object: by the Wiener–Khinchin theorem the spectrum is the transform of the autocorrelation, so the line shape is set by how fast $g_1$ forgets its phase, not by the instantaneous field.

Step 2. From the radiative amplitude decay $e^{-t/2\tau_{\mathrm{nat}}}$ , the correlation inherits a factor $e^{-i\omega_0\tau}e^{-\tau/2\tau_{\mathrm{nat}}}$ . Why the half: intensity falls at $1/\tau_{\mathrm{nat}}$ , so the amplitude falls at half that rate.

Step 3. Average the collisional phase factor. The phase difference $\phi(t+\tau)-\phi(t)$ is zero if no collision struck and uniformly random otherwise, and a uniform phase averages to $\langle e^{i\Delta\phi}\rangle=0$ . For a Poisson process the probability of zero collisions in $\tau$ is $e^{-\tau/\tau_{\mathrm{col}}}$ , so $\langle e^{i[\phi(t+\tau)-\phi(t)]}\rangle=e^{-\tau/\tau_{\mathrm{col}}}$ . Why only collision-free intervals survive: one randomizing collision erases all memory of the earlier phase.

Step 4. Multiply the independent factors: $g_1(\tau)\propto e^{-i\omega_0\tau}\,e^{-(1/2\tau_{\mathrm{nat}}+1/\tau_{\mathrm{col}})\tau}$ . Why multiplication: independent processes multiply their survival probabilities, and exponentials multiply by adding exponents (rates).

Step 5. Transform. The Fourier transform of $e^{-i\omega_0\tau}e^{-\gamma\tau}$ (for $\tau>0$ , with $\gamma=1/2\tau_{\mathrm{nat}}+1/\tau_{\mathrm{col}}$ ) is a Lorentzian centred at $\omega_0$ with HWHM $\gamma$ , hence FWHM $\Gamma=2\gamma=1/\tau_{\mathrm{nat}}+2/\tau_{\mathrm{col}}$ . Why a Lorentzian: the transform of a one-sided exponential is $1/(\gamma+i(\omega-\omega_0))$ , whose modulus squared is the Lorentzian profile.

Step 6. Take the collisional part and convert to ordinary frequency: $\Delta\nu_{\mathrm{col}}=(1/2\pi)(2/\tau_{\mathrm{col}})=1/(\pi\tau_{\mathrm{col}})$ , with $1/\tau_{\mathrm{col}}=n\sigma\bar v_r$ . Why divide by $2\pi$ : it converts angular-frequency FWHM to hertz, and kinetic theory supplies the rate.

Go deeper · Derivation B · 1 minThe collision rate and the pressure–temperature scaling

Derivation BThe collision rate and the pressure–temperature scaling

The goal is to justify the substitution $1/\tau_{\mathrm{col}}=n\sigma\bar v_r$ from elementary kinetic theory and extract the experimental signature of collisional broadening — its linear dependence on pressure and weak $1/\sqrt{T}$ dependence on temperature. This is what lets a spectroscopist read a pressure off a linewidth, and what distinguishes a collisional Lorentzian from a Doppler Gaussian, whose width instead scales as $\sqrt{T}$ .

Step 1. In time $\mathrm{d}t$ a radiator moving at mean relative speed $\bar v_r$ through perturbers of density $n$ sweeps a collision cylinder of volume $\sigma\bar v_r\,\mathrm{d}t$ , suffering $n\sigma\bar v_r\,\mathrm{d}t$ collisions, so $1/\tau_{\mathrm{col}}=n\sigma\bar v_r$ . Why the relative speed: it is the perturber's motion past the radiator that brings them together, so the reduced mass governs $\bar v_r$ .

Step 2. Insert this rate into $\Delta\nu_{\mathrm{col}}=1/(\pi\tau_{\mathrm{col}})$ to obtain $\Delta\nu_{\mathrm{col}}=n\sigma\bar v_r/\pi$ , linear in $n$ . Why linearity is exact here: each collision is an independent phase-erasing event, so doubling the density doubles the rate — until collisions overlap in time at very high density.

Step 3. Eliminate density with $n=P/(k_BT)$ to get $\Delta\nu_{\mathrm{col}}=\sigma\bar v_r\,P/(\pi k_BT)$ . Why swap $n$ for $P$ : pressure and temperature are the measured variables, making the width a direct barometer once $\sigma$ and $T$ are known.

Step 4. Use $\bar v_r=\sqrt{8k_BT/\pi\mu}\propto\sqrt{T}$ , so the explicit $T$ in the denominator combines with $\sqrt{T}$ in the numerator to give $\Delta\nu_{\mathrm{col}}\propto P/\sqrt{T}$ . Why the line narrows on heating: raising $T$ at fixed pressure lowers the density faster ( $1/T$ ) than it raises the speed ( $\sqrt{T}$ ), so the width falls as $1/\sqrt{T}$ .

Limiting cases

The profile depends on how the collision duration $\tau_c$ (time within interaction range) compares with the mean interval $\tau_{\mathrm{col}}$ between collisions and with the inverse detuning $1/|\Delta\omega|$ .

Case	Result	Meaning
Impact (Lorentz) limit	$\tau_c\ll\tau_{\mathrm{col}}$ , $\;\|\Delta\omega\|\,\tau_c\ll1$	Collisions effectively instantaneous; the line core is a Lorentzian of FWHM proportional to the collision rate — the regime of equation (4).
Quasi-static (statistical) limit	$\|\Delta\omega\|\,\tau_c\gg1$	The perturber is frozen during emission; the far-wing shape maps the distribution of level shifts $V(R)$ , giving non-Lorentzian wings.
Low-pressure limit	$\Delta\nu_{\mathrm{col}}\ll\Delta\nu_{\mathrm{nat}}$	Collisional width negligible; the line reduces to the natural Lorentzian, a Voigt dominated by the Gaussian once Doppler is added.
High-pressure limit	$\Delta\nu_{\mathrm{col}}\gg\Delta\nu_{\mathrm{Doppler}}$	The homogeneous Lorentzian dominates; width grows linearly with density and the centre shifts with the collision-averaged potential.

Common trapAdding collisional and natural widths in quadrature

It is tempting to combine the natural and collisional widths as $\sqrt{\Delta\nu_{\mathrm{nat}}^2+\Delta\nu_{\mathrm{col}}^2}$ , by analogy with adding independent uncertainties — but that is wrong. Both processes damp the same field correlation function exponentially, so their decoherence rates add and the two Lorentzians convolve into a single Lorentzian of FWHM equal to the plain sum $\Delta\nu_{\mathrm{nat}}+\Delta\nu_{\mathrm{col}}$ . Quadrature-like addition applies only to a Lorentzian combined with the inhomogeneous Doppler Gaussian — and that convolution is a Voigt profile. The rule: Lorentzian widths add linearly; a Lorentzian and a Gaussian make a Voigt.

Worked examples

1 · Collisional width of the sodium D line in argon at 1 atm (confidence builder)

A trace of sodium radiates its D line (589 nm, natural FWHM ≈ 10 MHz) in 1 atm of argon at 300 K. With an optical broadening cross section $\sigma=1\times10^{-18}\,\mathrm{m^2}$ , estimate the collisional FWHM, as a per-Torr coefficient, and compare it with the natural width.

Step 1. Number density (the Loschmidt number): $n=P/(k_BT)=(1.013\times10^5)/[(1.381\times10^{-23})(300)]=\mathbf{2.45\times10^{25}\ m^{-3}}$ .

Step 2. Mean relative speed. The reduced mass of Na ( $22.99\,\mathrm u$ ) with Ar ( $39.95\,\mathrm u$ ) is $\mu=(22.99\times39.95)/(22.99+39.95)=14.59\,\mathrm u=2.42\times10^{-26}\ \mathrm{kg}$ , so $\bar v_r=\sqrt{8k_BT/\pi\mu}=\mathbf{6.6\times10^{2}\ m\,s^{-1}}$ .

Step 3. Collision rate: $1/\tau_{\mathrm{col}}=n\sigma\bar v_r=(2.45\times10^{25})(1\times10^{-18})(660)=1.6\times10^{10}\ \mathrm{s^{-1}}$ , so $\tau_{\mathrm{col}}\approx\mathbf{62\ ps}$ — the number quoted in the first figure.

Step 4. Collisional FWHM: $\Delta\nu_{\mathrm{col}}=1/(\pi\tau_{\mathrm{col}})=(1.6\times10^{10})/\pi=\mathbf{5.1\ GHz}$ , or $5.1\ \mathrm{GHz}/760\ \mathrm{Torr}=\mathbf{6.7\ MHz/Torr}$ — squarely in the tabulated range for alkali–noble-gas broadening.

Step 5. Compare: $\Delta\nu_{\mathrm{col}}/\Delta\nu_{\mathrm{nat}}\approx5.1\times10^{9}/1.0\times10^{7}\approx\mathbf{5\times10^{2}}$ . At one atmosphere the line is broadened roughly five hundredfold beyond its natural width — made almost entirely by collisions.

2 · At what pressure do collisions overtake Doppler broadening?

For the same sodium D line at 300 K, the Doppler FWHM is $\Delta\nu_D=(\nu_0/c)\sqrt{8k_BT\ln2/m_{\mathrm{Na}}}$ . Find $\Delta\nu_D$ and the pressure at which the collisional width equals it.

Step 1. Line frequency: $\nu_0=c/\lambda_0=(2.998\times10^8)/(589\times10^{-9})=5.09\times10^{14}\ \mathrm{Hz}$ .

Step 2. Doppler width uses the sodium mass alone, $m_{\mathrm{Na}}=22.99\,\mathrm u=3.82\times10^{-26}\ \mathrm{kg}$ : $\Delta\nu_D=(\nu_0/c)\sqrt{8k_BT\ln2/m_{\mathrm{Na}}}=(1.70\times10^{6})(776\ \mathrm{m\,s^{-1}})=\mathbf{1.3\ GHz}$ .

Step 3. Crossover. The collisional width scales linearly, $\Delta\nu_{\mathrm{col}}=5.1\ \mathrm{GHz}\times P/(1\ \mathrm{atm})$ , while $\Delta\nu_D$ is pressure-independent. Setting them equal: $P_\times=(1\ \mathrm{atm})\times(1.3/5.1)=\mathbf{0.26\ atm}\approx\mathbf{195\ Torr}$ .

Step 4. Interpret: below about a quarter-atmosphere the line is Doppler-dominated (inhomogeneous, defeatable by saturation spectroscopy); above it the homogeneous collisional Lorentzian takes over — the design knob distinguishing a low-pressure cell from a broadened high-pressure lamp. Doppler ≈ 1.3 GHz; collisions win above ≈ 0.26 atm.

Physics of the portrait

Collisional broadening is where spectroscopy first met the many-body problem: a line is no longer a property of one atom but of an atom in a moving crowd, read correctly only by averaging over the statistics of encounters.

Origins · 1 minHendrik Lorentz and how this was discovered

Hendrik Lorentz

1853–1928 · the impact theory of collision broadening

Hendrik Antoon Lorentz is remembered for the electron theory of matter, the transformations that bear his name, and the Lorentzian line shape — but among spectroscopists his signature contribution is the impact theory of pressure broadening, laid out in 1906. Building on his model of the radiating atom as a damped electron oscillator, Lorentz asked what a gas of such oscillators would radiate if each were repeatedly interrupted by collisions. His decisive idealization was to treat a collision as instantaneous compared with the interval between collisions and as completely destroying the memory of the optical phase. With that assumption the wavetrain becomes a sequence of phase-uncorrelated segments, and averaging over the Poisson statistics of collision times yields a Lorentzian whose width is the sum of the radiative and collisional rates — precisely equation (3). The picture survived the quantum revolution nearly intact, refined by Weisskopf, Anderson, and Baranger into the modern quantum theory of line broadening. Every modern code that fits a Voigt profile to a combustion or planetary spectrum still rests on Lorentz's insight that a collision is, to the radiation field, simply the end of one coherent wavetrain and the start of another.

History

1895

— Michelson attributes the broadening of spectral lines to molecular collisions

1906

— Lorentz formulates the impact theory: collisions interrupt the radiating phase, giving a Lorentzian

1932

— Weisskopf computes the collision phase from the interatomic potential, defining the optical cross section and the line shift

1949

— Anderson develops the quantitative theory of pressure broadening and shift for microwave and infrared lines

1958

— Baranger gives the general quantum-mechanical theory of impact line broadening

Going further

Texts. Foot, Atomic Physics, §8.2 (collision and pressure broadening) · Loudon, The Quantum Theory of Light, 3rd ed., §2.5 (interrupted wavetrains and the Lorentzian) · Demtröder, Laser Spectroscopy, Vol. 1, §3.3 (pressure broadening, shift, and cross sections) · Hecht, Optics, 5th ed., §7.4 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §14.3. Two key results anchor the numbers throughout this article. First, the Fourier transform of $e^{-\gamma\tau}$ is a Lorentzian of angular FWHM $2\gamma$ , which is why the natural and collisional widths add linearly rather than in quadrature. Second, the Na–Ar worked example at 1 atm and 300 K gives $n=2.45\times10^{25}\ \mathrm{m^{-3}}$ , $\bar v_r\approx660\ \mathrm{m\,s^{-1}}$ , $\tau_{\mathrm{col}}\approx62\ \mathrm{ps}$ , and $\Delta\nu_{\mathrm{col}}\approx5.1\ \mathrm{GHz}$ ( $\approx6.7\ \mathrm{MHz/Torr}$ ) — some five hundred times the natural width — while the Doppler width is $\approx1.3\ \mathrm{GHz}$ and the collisional width overtakes it near $0.26\ \mathrm{atm}$ . The optical cross section $\sigma\sim10^{-18}\ \mathrm{m^2}$ is a few times the gas-kinetic value $\sim10^{-19}\ \mathrm{m^2}$ , reflecting the long-range phase sensitivity of the optical interaction.

Prerequisites. Lorentzian Line Shape · Radiative Lifetime and Natural Linewidth · Spontaneous Emission · Einstein A and B Coefficients. Leads to. Voigt Profile · Resonant Absorption · Detailed Balance and the Planck Spectrum.

Ask the tutor: Ask me to fit a real Voigt profile — say a water line in a combustor — and pull the temperature from its Gaussian part and the pressure from its Lorentzian part, step by step.

Why does chopping the wavetrain into shorter pieces make the line broader?

It is the Fourier uncertainty principle. A signal that stays phase-coherent only for a time $\tau$ cannot be resolved in frequency to better than about $1/\tau$ : a pure tone needs an infinitely long coherent wavetrain, and anything shorter is a spread of frequencies. Collisions cut the coherent segment to $\tau_{\mathrm{col}}$ , so the spectrum acquires a width of order $1/\tau_{\mathrm{col}}$ . Precisely, the spectrum is the Fourier transform of the field autocorrelation, and an exponential correlation transforms to a Lorentzian whose width is the decay rate.

What is the difference between collisional broadening and Doppler broadening?

They are opposite in almost every respect. Doppler broadening is inhomogeneous: each atom radiates a sharp line, but different velocities Doppler-shift those lines into a Gaussian — a spread across atoms, not a broadening of any one. Collisional broadening is homogeneous: every atom's own line is genuinely broadened, giving a Lorentzian. Their scalings also differ — Doppler as $\sqrt{T}$ and pressure-free, collisional as $P/\sqrt{T}$ — and Doppler-free methods strip the Doppler width but not the collisional one.

Why do the natural and collisional widths add linearly instead of in quadrature?

Because both processes damp the identical quantity — the field autocorrelation — as decaying exponentials, and exponentials multiply by adding their exponents. The radiative factor $e^{-\tau/2\tau_{\mathrm{nat}}}$ and the collisional $e^{-\tau/\tau_{\mathrm{col}}}$ combine into a single exponential whose rate is the sum, and the transform of one exponential is one Lorentzian of that summed width. Quadrature addition is reserved for a Lorentzian combined with a Gaussian — different shapes whose convolution is the Voigt profile.

Does every collision contribute equally to the width?

No. What matters is whether a collision accumulates a phase of order one radian or more; the cross section $\sigma$ is defined by the impact parameter inside which that happens. Distant, glancing encounters barely dephase the atom; close encounters with $|\eta|\gtrsim1$ do the work. This is why the optical broadening cross section is typically a few times larger than the gas-kinetic one: the phase feels the long-range tail of the potential, registering collisions at larger separations than a billiard-ball picture would count.

What decides whether the pressure shift is to the red or the blue?

The sign of the collision phase, which follows the sign of the interatomic potential during the encounter. An attractive potential lowers the transition frequency while the perturber is near, shifting the light to lower frequency — a red shift — while a repulsive interaction gives a blue shift. Quantitatively the shift is the flux-weighted integral of $\sin\eta(b)$ and the broadening the integral of $1-\cos\eta(b)$ ; a van der Waals $-C_6/R^6$ interaction gives a red shift roughly a third of the width, so the shift-to-width ratio reads off which part of the potential dominates.

Why can't saturation spectroscopy remove the collisional width the way it removes Doppler?

Saturation (Doppler-free) spectroscopy uses a pump beam to select a single velocity class, cancelling the inhomogeneous spread that different velocities produce. But collisional broadening is homogeneous: it broadens every atom's line identically, so there is no narrower velocity subset to select — picking out one class leaves its fully collision-broadened Lorentzian. The only ways to reduce homogeneous width are physical: lower the pressure or the collision rate. This is why precision optical spectroscopy is done in low-pressure cells, atomic beams, or traps.