Linewidths and Line Broadening

Every spectral line has some width, even when recorded on a perfect instrument. Zoom into any absorption or emission feature and you find a smooth peak, not an infinitely sharp spike. This width is not experimental sloppiness. It encodes real physics: how long a state lives, how fast molecules move, and how often they collide. Linewidths matter because they set the resolution limits of spectroscopy and because their shape and breadth are themselves measurements. A Doppler width gives a gas temperature. A pressure-broadened width gives a collision rate. A natural linewidth gives an excited-state lifetime. Astronomers, atmospheric chemists, and laser spectroscopists all read linewidths as data, not noise. This article builds the three classic broadening mechanisms from first principles. You will see how each one produces a characteristic line shape. You will learn the equations that connect width to lifetime, temperature, and pressure. Two worked examples then turn the theory into numbers you can check by hand.

Key takeaways

Natural linewidth comes from the finite lifetime of excited states via the energy–time uncertainty relation, giving a Lorentzian shape.
Doppler broadening arises from the thermal motion of molecules and gives a Gaussian shape whose width scales with $\sqrt{T/M}$ .
Collisional (pressure) broadening comes from interruption of the emission/absorption process by molecular collisions, also giving a Lorentzian shape whose width scales with pressure.
At low pressure in the gas phase, Doppler broadening usually dominates; at high pressure, collisions take over.
The full line shape is a convolution of Lorentzian and Gaussian contributions, called a Voigt profile.
Key idea: a spectral line's width is not instrumental noise — it is a direct measurement of a molecular lifetime, a temperature, or a collision rate.

The core insight

Every mechanism that shortens the effective time over which a molecule oscillates coherently at its transition frequency — whether that time is cut short by spontaneous decay, by a collision, or by the Doppler-shifted spread of velocities in an ensemble — broadens the spectral line, and the energy–time uncertainty relation $\Delta E \Delta t \gtrsim \hbar$ ties every one of these timescales directly to an observable linewidth.

The physical picture

Consider a large ensemble of gas-phase molecules undergoing a transition between two energy levels, populated according to the Boltzmann distribution (see Boltzmann Populations and Spectral Intensity). If every molecule behaved identically and lived forever in the excited state, the transition would occur at one exact frequency $\nu_0$ , producing an infinitely narrow line. Real lines are broadened by three physically distinct mechanisms.

Natural broadening is intrinsic to a single, isolated molecule. An excited state has a finite lifetime $\tau$ because it eventually decays, spontaneously emitting a photon. The wavefunction describing that decaying state is not a pure sinusoid at $\nu_0$ ; it is a damped oscillation, and Fourier analysis of a damped oscillation always yields a spread of frequencies. This spread is the natural linewidth, and it exists even for a single, motionless, collision-free molecule.

Doppler broadening is a statistical effect of the ensemble. Molecules move with a thermal (Maxwell–Boltzmann) distribution of velocities. A molecule moving toward the detector sees, and emits, a slightly blue-shifted frequency; one moving away is red-shifted. Averaged over the whole ensemble, the sharp natural line is smeared into a bell-shaped Gaussian profile.

Collisional (pressure) broadening arises because molecules do not evolve in isolation at finite pressure. Collisions interrupt the phase of the emitted or absorbed radiation, effectively shortening the coherence time in the same way that spontaneous decay does. Because the physical mechanism (interruption of phase) is mathematically identical to natural decay, collisional broadening also produces a Lorentzian shape — but with a width that grows linearly with pressure (collision frequency).

Figure

Figure 1. Three broadening mechanisms and the resulting line shapes

The two shapes differ most in their wings. A Gaussian falls off very quickly, as $\exp(-x^2)$ , so far from the line center a Doppler-broadened line is essentially zero. A Lorentzian falls off only as $x^{-2}$ , so natural- and pressure-broadened lines retain long, slowly-decaying tails. This difference is diagnostic: high-resolution spectra can distinguish the two contributions by comparing the shape near the peak with the shape in the wings.

Figure

Figure 2. Lorentzian versus Gaussian profiles of equal peak height and equal FWHM

The core relations

\Delta \nu_{\text{nat}} = \frac{1}{2\pi\tau}

where $\tau$ is the excited-state lifetime and $\Delta\nu_{\text{nat}}$ is the natural (Lorentzian) FWHM in Hz.

I(\nu) \propto \frac{\Delta\nu_{1/2}/2\pi}{(\nu-\nu_0)^2 + (\Delta\nu_{1/2}/2)^2}

where the Lorentzian line shape has full width at half maximum $\Delta\nu_{1/2}$ and peak at $\nu_0$ .

\Delta\nu_{\text{Doppler}} = \nu_0 \sqrt{\frac{8 k_B T \ln 2}{M c^2}}

where $M$ is the molar mass, $T$ the temperature, $k_B$ Boltzmann's constant, $c$ the speed of light, giving the Gaussian FWHM.

I(\nu) \propto \exp\left[-\frac{4\ln 2\,(\nu-\nu_0)^2}{\Delta\nu_{\text{Doppler}}^2}\right]

where this is the Gaussian line shape normalized to the same FWHM definition.

\Delta\nu_{\text{coll}} = \frac{1}{\pi \tau_{\text{coll}}} = \frac{z}{\pi}

where $\tau_{\text{coll}}$ is the mean time between collisions and $z = 1/\tau_{\text{coll}}$ the collision frequency, giving another Lorentzian FWHM.

z = p\, \sigma \sqrt{\frac{8}{\pi \mu k_B T}}

where $p$ is pressure, $\sigma$ the collision cross-section, and $\mu$ the reduced mass of the colliding pair, so $\Delta\nu_{\text{coll}} \propto p$ .

\Delta\nu_{\text{total,L}} = \Delta\nu_{\text{nat}} + \Delta\nu_{\text{coll}}

where Lorentzian widths from independent broadening processes simply add.

Worked examples

Derivation ANatural linewidth from the energy–time uncertainty relation

Goal. Show that a state with lifetime $\tau$ produces a Lorentzian line of FWHM $\Delta\nu_{\text{nat}} = 1/(2\pi\tau)$ .

Step 1. Model the excited state's transition dipole as a classical oscillator at angular frequency $\omega_0 = 2\pi\nu_0$ that decays exponentially because the population decays as $e^{-t/\tau}$ . The amplitude of oscillation (not intensity) then decays at half that rate, giving the emitted electric field

E(t) = E_0\, e^{-t/2\tau}\, e^{-i\omega_0 t}, \qquad t \ge 0.

Step 2. The spectrum is the Fourier transform of $E(t)$ . Compute

\tilde{E}(\omega) = \int_0^\infty E_0\, e^{-t/2\tau}\, e^{-i\omega_0 t}\, e^{i\omega t}\, dt = \frac{E_0}{\dfrac{1}{2\tau} + i(\omega_0-\omega)}.

Step 3. The observed intensity is proportional to $|\tilde{E}(\omega)|^2$ :

I(\omega) \propto \frac{1}{\left(\dfrac{1}{2\tau}\right)^2 + (\omega-\omega_0)^2}.

This is exactly the Lorentzian functional form.

Step 4. Identify the half-width at half maximum (HWHM) in angular frequency: the intensity falls to half its peak value when $(\omega - \omega_0)^2 = (1/2\tau)^2$ , so the HWHM is $1/(2\tau)$ and the FWHM in $\omega$ is $1/\tau$ .

Step 5. Convert to ordinary frequency using $\omega = 2\pi\nu$ , so a FWHM of $1/\tau$ in $\omega$ corresponds to

\Delta\nu_{\text{nat}} = \frac{1}{2\pi}\cdot\frac{1}{\tau} = \frac{1}{2\pi\tau}.

Step 6. This result is a direct manifestation of the energy–time uncertainty relation. Since $\Delta E = h\Delta\nu$ and $\Delta E\,\tau \sim \hbar$ , we recover $\Delta\nu \sim 1/(2\pi\tau)$ up to the numerical factor fixed by the exact Lorentzian shape.

\boxed{\Delta\nu_{\text{nat}} = \dfrac{1}{2\pi\tau}}

Example 1 — Natural linewidth of an electronically excited state (foundational). A molecule in an excited electronic state decays with a fluorescence lifetime $\tau = 5.0\ \text{ns} = 5.0\times10^{-9}\ \text{s}$ , typical of an allowed electronic transition. Find the natural linewidth in Hz, and express it as a fraction of a visible transition at $\nu_0 = 5.0\times10^{14}\ \text{Hz}$ (600 nm). Step 1 — apply the formula:

\Delta\nu_{\text{nat}} = \frac{1}{2\pi\tau} = \frac{1}{2\pi (5.0\times10^{-9}\ \text{s})} = \frac{1}{3.1416\times10^{-8}\ \text{s}} = 3.18\times10^{7}\ \text{Hz}.

Step 2 — express as a resolving power:

\frac{\Delta\nu_{\text{nat}}}{\nu_0} = \frac{3.18\times10^{7}\ \text{Hz}}{5.0\times10^{14}\ \text{Hz}} = 6.4\times10^{-8}.

Sanity check: this corresponds to $\Delta\lambda \approx \lambda \times 6.4\times10^{-8} \approx 3.8\times10^{-5}\ \text{nm}$ , an extremely narrow line compared with typical Doppler widths (below) — confirming that natural broadening is usually negligible for electronic transitions in the gas phase at room temperature unless the lifetime is very short (as in a short-lived predissociative state).

Example 2 — Doppler versus collisional broadening in a rotational microwave line (applied, harder). Consider the $J = 0 \to 1$ rotational absorption line of carbon monoxide, $^{12}$ C $^{16}$ O, near $\nu_0 = 115{,}271\ \text{MHz}$ (this transition foreshadows Rotational Microwave Spectroscopy: Principles). Molar mass $M = 28.0\ \text{g mol}^{-1} = 0.0280\ \text{kg mol}^{-1}$ . Take $T = 300\ \text{K}$ .

Part (a): Doppler width.

\Delta\nu_{\text{Doppler}} = \nu_0\sqrt{\frac{8 k_B T \ln 2}{M c^2}}.

Insert numbers, using $k_B = 1.381\times10^{-23}\ \text{J K}^{-1}$ , $c = 2.998\times10^{8}\ \text{m s}^{-1}$ , and molar mass converted per molecule by dividing by Avogadro's number $N_A = 6.022\times10^{23}\ \text{mol}^{-1}$ so that the molecular mass is

m = \frac{0.0280\ \text{kg mol}^{-1}}{6.022\times10^{23}\ \text{mol}^{-1}} = 4.65\times10^{-26}\ \text{kg}.

Now compute the dimensionless factor under the root:

\frac{8 k_B T \ln 2}{m c^2} = \frac{8\,(1.381\times10^{-23})(300)(0.6931)}{(4.65\times10^{-26})(2.998\times10^{8})^2}.

Numerator: $8 \times 1.381\times10^{-23} \times 300 = 3.314\times10^{-20}$ ; times $0.6931$ gives $2.297\times10^{-20}\ \text{J}$ . Denominator: $(4.65\times10^{-26})(8.988\times10^{16}) = 4.180\times10^{-9}\ \text{J}$ . Ratio: $2.297\times10^{-20}/4.180\times10^{-9} = 5.495\times10^{-12}$ (dimensionless). Square root: $\sqrt{5.495\times10^{-12}} = 2.344\times10^{-6}$ . So

\Delta\nu_{\text{Doppler}} = (115{,}271\ \text{MHz})(2.344\times10^{-6}) = 0.270\ \text{MHz} = 2.70\times10^{5}\ \text{Hz}.

Part (b): collisional width at 1.0 Torr. Take a typical CO–CO collision cross-section $\sigma \approx 4.3\times10^{-19}\ \text{m}^2$ (kinetic-theory value consistent with a collision diameter near 0.37 nm) and reduced mass for CO–CO collisions $\mu = m/2 = 2.33\times10^{-26}\ \text{kg}$ . Convert pressure: $1.0\ \text{Torr} = 133.3\ \text{Pa}$ . Mean relative speed factor:

\sqrt{\frac{8}{\pi \mu k_B T}} = \sqrt{\frac{8}{\pi (2.33\times10^{-26})(1.381\times10^{-23})(300)}}.

Denominator inside root: $\pi \times 2.33\times10^{-26} \times 1.381\times10^{-23} \times 300 = 3.031\times10^{-46}$ . So the ratio is $8/3.031\times10^{-46} = 2.640\times10^{46}$ , and its square root is $1.625\times10^{23}\ \text{Pa}^{-1}\text{s}^{-1}$ (units chosen to make $z$ come out in Hz once multiplied by pressure and cross-section, verified below). Collision frequency:

z = p\,\sigma\sqrt{\frac{8}{\pi\mu k_B T}} = (133.3\ \text{Pa})(4.3\times10^{-19}\ \text{m}^2)(1.625\times10^{23}\ \text{Pa}^{-1}\text{s}^{-1}\text{m}^{-2}).

Multiply stepwise: $133.3 \times 4.3\times10^{-19} = 5.73\times10^{-17}$ ; then $5.73\times10^{-17} \times 1.625\times10^{23} = 9.32\times10^{6}\ \text{s}^{-1}$ . So $z \approx 9.3\times10^{6}\ \text{Hz}$ , and

\Delta\nu_{\text{coll}} = \frac{z}{\pi} = \frac{9.3\times10^{6}}{3.1416} = 3.0\times10^{6}\ \text{Hz} = 3.0\ \text{MHz}.

Comparison. At 1 Torr, $\Delta\nu_{\text{coll}} \approx 3.0\ \text{MHz}$ dwarfs $\Delta\nu_{\text{Doppler}} \approx 0.27\ \text{MHz}$ : collisional broadening dominates even at this modest pressure, because rotational transitions have small $\nu_0$ so the fractional Doppler width is tiny, while collision rates scale directly with pressure and are large for polar, strongly-interacting molecules like CO. This is why microwave rotational spectroscopy is typically done at very low pressures (millitorr range) to recover sharp, well-resolved lines.

Limiting cases & common traps

Regime	Dominant mechanism	Typical width scale
Very low pressure gas, visible/UV transition	Doppler	MHz to GHz
Very low pressure gas, microwave/rotational transition	Natural or Doppler (both tiny)	kHz to sub-MHz
High pressure gas (torr–atm)	Collisional	MHz to tens of GHz
Condensed phase (liquid, solid)	Collisional/dephasing analogs dominate; lines often very broad	GHz and up
Short-lived excited state (predissociation, fast IC)	Natural	can exceed Doppler even at low pressure

Natural and collisional broadening both give Lorentzian shapes and simply add in FWHM; Doppler is Gaussian and combines with a Lorentzian by convolution (Voigt profile), not simple addition.
Increasing temperature increases Doppler width but does not directly change natural linewidth (though it can shorten lifetimes indirectly via faster nonradiative decay).
Pressure broadening scales linearly with pressure; at sufficiently high density, "pressure narrowing" (Dicke narrowing) can actually occur due to velocity-changing collisions confining the emitter — an advanced correction beyond this Tier 2 treatment.

The chemist behind it

Line-broadening theory grew from atomic spectroscopy, but its clearest chemical application came through microwave and laser spectroscopists who needed lineshape analysis to extract molecular lifetimes and collision cross-sections from real spectra. Arthur Schawlow, working alongside Charles Townes, used laser spectroscopy to resolve natural linewidths that had been buried under Doppler and collisional broadening for decades, pioneering Doppler-free techniques such as saturation spectroscopy that let chemists finally see the "true" Lorentzian core of a transition.

Origins · 1 minArthur L. Schawlow and how this was discovered

Arthur L. Schawlow

1921–1999 · Co-invented the laser and developed Doppler-free laser spectroscopy to isolate natural linewidths from broadened profiles

Ask the tutor: How would you fit a real experimental line shape to extract separate Doppler and collisional widths using a Voigt profile?

Why do gas-phase spectra usually look broader than expected from natural lifetime alone?

Because natural linewidths for most allowed transitions are extremely small — often kHz to tens of MHz — while Doppler broadening from thermal motion, or collisional broadening at ordinary pressures, produces widths that are orders of magnitude larger. Unless a sample is cooled, held at very low pressure, and probed with a Doppler-free technique, the natural linewidth is completely masked by these other, larger contributions.

How can spectroscopists separate Doppler and collisional contributions in a real spectrum?

Because Doppler broadening gives a Gaussian shape and collisional (plus natural) broadening gives a Lorentzian shape, the full profile is a Voigt function, a convolution of the two. Fitting the observed line shape to a Voigt profile, using the very different wing behavior of Gaussian versus Lorentzian functions, lets researchers extract both widths separately from a single spectrum, provided signal-to-noise is high enough.

Does increasing temperature always broaden a spectral line?

Increasing temperature increases Doppler broadening directly, since $\Delta\nu_{\text{Doppler}} \propto \sqrt{T}$ . It also affects collisional broadening, but in a more complex way: raising temperature increases average molecular speed, which raises the collision frequency, but the collision cross-section itself typically decreases slightly at higher energy, so the net effect on collisional width is weaker than the direct Doppler scaling.

Why does microwave rotational spectroscopy require very low sample pressures?

Rotational transition frequencies are relatively low, in the GHz range, so the fractional Doppler width is tiny even though the absolute Doppler width in Hz is modest. Collisional broadening, however, scales with pressure regardless of transition frequency, and molecules like CO have large collision cross-sections. At ordinary pressures collisions dominate and blur the closely spaced rotational lines, so spectroscopists work at millitorr pressures to keep collisional width below the natural line spacing.