---
title: Lifetimes, Line Widths, and Line Shapes
module: Radiative Transitions and Spectral Lines
moduleNumber: 7
lessonNumber: 4
order: 704
summary: >
  A spectral line is never infinitely sharp. The finite lifetime of the excited
  state gives every line a natural Lorentzian width set by the total decay rate,
  the Fourier transform of an exponentially damped emission. Thermal motion adds a
  Gaussian Doppler width that usually dominates in a gas; collisions add a further
  Lorentzian pressure width; the observed profile is the Voigt convolution of the
  Gaussian and Lorentzian parts. Strong driving fields broaden the line further
  through saturation. Each mechanism has a distinct dependence on temperature,
  density, and intensity that lets it be identified and, where possible, removed.
topics: [Radiative Transitions and Spectral Lines]
sources:
  - book: Foot
    ref: "Ch. 7 — The Interaction of Atoms with Radiation; §7.5 Lifetimes and Line Widths; App. D"
  - book: Demtröder
    ref: "Ch. 7 — Emission and Absorption; §7.3–7.5 Natural, Doppler, Pressure Broadening, Line Profiles"
  - book: Bransden & Joachain
    ref: "Ch. 4 — Interaction of One-Electron Atoms with EM Radiation; §4.6"
draft: false
---

The [spontaneous-emission rate](/atomic-physics/radiative-transitions-and-line-shapes/dipole-approximation-einstein-coefficients)
$A_{21}$ sets how fast an excited atom decays. That finite lifetime has a spectral
consequence: a line of nonzero width. A state that lived forever would emit a pure
sinusoid at a single frequency; a state that decays emits a truncated wave train,
and a truncated wave train contains a spread of frequencies. This lesson works out
the width and shape of a spectral line from three mechanisms — the natural width
from the finite lifetime, the Doppler width from thermal motion, and the pressure
width from collisions — and combines them into the observed profile. The shapes
differ (Lorentzian, Gaussian, and their Voigt convolution), and so do their
dependences on temperature, density, and driving intensity, which is what lets an
experimenter diagnose and defeat them.

## Lifetime and the exponential decay

An atom in an excited state $\ket{2}$ that can decay to several lower states has a
total decay rate equal to the sum of the individual spontaneous rates,

$$
\Gamma = \sum_k A_{2k},
$$

and the excited-state population of an ensemble decays exponentially,

$$
N_2(t) = N_2(0)\,e^{-\Gamma t} = N_2(0)\,e^{-t/\tau},
\qquad
\tau = \frac{1}{\Gamma}.
$$

The **lifetime** $\tau$ is the reciprocal of the total decay rate, and a level with
several decay channels lives no longer than its fastest channel allows. For a
strong optical transition $\Gamma\sim 10^8\ \mathrm{s^{-1}}$ and $\tau\sim
10\ \mathrm{ns}$; a
[metastable](/atomic-physics/radiative-transitions-and-line-shapes/selection-rules-forbidden-transitions)
level with only forbidden channels can have $\tau$ of seconds or longer.

$$
% caption: The excited-state population decays exponentially at the total rate
% Gamma, with 1/e lifetime tau. The emitted wave train is correspondingly damped,
% and its finite duration is what gives the line a nonzero width.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->] (-0.3,0) -- (8.2,0) node[right, black] {time};
\draw[->] (0,-0.2) -- (0,3.4) node[above, black] {population $N_2$};
% exponential decay
\draw[acc, very thick] plot[smooth] coordinates{
 (0,3.0) (0.5,2.34) (1,1.82) (1.5,1.42) (2,1.10) (2.5,0.86)
 (3,0.67) (3.5,0.52) (4,0.41) (5,0.25) (6,0.15) (7,0.09) (7.8,0.06)};
% 1/e marker
\draw[black, dashed] (0,1.10) -- (2,1.10) -- (2,0);
\node[black, anchor=north] at (2,-0.05) {lifetime};
\node[black, anchor=west] at (0.1,1.25) {1/e level};
\end{tikzpicture}
$$

## The natural line width

Treat the emitted classical field as a damped oscillator. While the atom decays,
the radiated field amplitude follows the excited-state amplitude, which decays at
half the population rate (the population goes as amplitude squared):

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

and $E(t)=0$ for $t<0$. The spectrum of the emitted light is the squared modulus
of the Fourier transform of $E(t)$:

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

Taking the modulus squared gives the **Lorentzian** line shape,

$$
I(\omega) \propto \frac{1}{(\omega-\omega_0)^2 + (\Gamma/2)^2},
\qquad \Gamma = \frac{1}{\tau},
$$

a symmetric peak centered at $\omega_0$ with full width at half maximum in angular
frequency equal to $\Gamma$. In ordinary frequency the **natural line width** is

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

A quantum treatment (the Weisskopf-Wigner theory) reproduces the same Lorentzian
with the same width; the classical damped oscillator is not a coincidence but the
correct envelope. The natural width is the time-energy uncertainty relation made
quantitative: a state of lifetime $\tau$ has an energy uncertainty $\Delta E =
\hbar\Gamma = \hbar/\tau$, so

$$
\Delta E\,\tau \approx \hbar.
$$

> **Definition (Natural line width).** The Lorentzian full width $\Gamma = 1/\tau$
> (angular frequency) or $1/2\pi\tau$ (frequency) that a spectral line has from the
> finite lifetime of its levels alone, independent of the atom's environment. It is
> the minimum width of the transition and the ultimate limit on spectroscopic
> resolution for a given level.

For the sodium $D$ lines $\tau\approx 16\ \mathrm{ns}$, so $\Delta\nu_{\text{nat}}
\approx 10\ \mathrm{MHz}$ — a few parts in $10^{8}$ of the optical frequency. This
is narrow, but in a room-temperature vapor it is swamped by thermal motion.

## Doppler broadening

Atoms in a gas move, and a moving atom radiating at rest-frame frequency $\omega_0$
is seen shifted by the first-order Doppler effect. An atom with velocity component
$v_z$ along the line of sight is observed at

$$
\omega = \omega_0\left(1 + \frac{v_z}{c}\right).
$$

In thermal equilibrium the velocity component $v_z$ has a Maxwell-Boltzmann
distribution,

$$
P(v_z)\,\d v_z = \sqrt{\frac{m}{2\pi k_B T}}\,
e^{-m v_z^2/2 k_B T}\,\d v_z,
$$

a Gaussian of width $\sqrt{k_B T/m}$. Mapping velocities to frequencies through the
Doppler relation transfers the Gaussian to the frequency axis: the line shape is

$$
I(\omega) \propto
\exp\!\left[-\frac{m c^2 (\omega-\omega_0)^2}{2 k_B T\,\omega_0^2}\right],
$$

a **Gaussian** centered at $\omega_0$. Its full width at half maximum, converted to
frequency, is the **Doppler width**

$$
\Delta\nu_{\text{D}}
= \frac{\nu_0}{c}\sqrt{\frac{8 k_B T \ln 2}{m}}.
$$

Two dependences distinguish Doppler broadening from natural broadening: it scales
as $\sqrt{T}$ and as $1/\sqrt{m}$, so it grows with temperature and shrinks for
heavier atoms. For sodium at $500\ \mathrm{K}$, $\Delta\nu_{\text{D}}\approx
1.7\ \mathrm{GHz}$, more than a hundred times the natural width. In a vapor cell the
observed line is Doppler-dominated and Gaussian, and its narrow Lorentzian core is
hidden — the reason Doppler-free techniques exist.

$$
% caption: Doppler width grows as the square root of temperature (and as one over
% the square root of atomic mass). A hotter or lighter gas gives a broader Gaussian
% line; cooling the sample narrows it.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->] (-0.3,0) -- (8.0,0) node[right, black] {temperature};
\draw[->] (0,-0.2) -- (0,3.4) node[above, black] {Doppler width};
% sqrt curve
\draw[acc, very thick] plot[smooth] coordinates{
 (0,0) (0.5,1.0) (1,1.41) (1.5,1.73) (2,2.0) (2.5,2.24)
 (3,2.45) (4,2.83) (5,3.16) (5.4,3.28)};
\node[acc, anchor=south west] at (3.2,2.3) {width grows as $T^{\frac{1}{2}}$};
\end{tikzpicture}
$$

> **Worked example (Doppler versus natural width, hydrogen Lyman-$\alpha$).** The
> $2p\to1s$ line at $\lambda = 121.6\ \mathrm{nm}$ has $\nu_0 = 2.47\times
> 10^{15}\ \mathrm{Hz}$. At $T = 300\ \mathrm{K}$ the most probable speed of a
> hydrogen atom is a few thousand metres per second, and the Doppler width is
> $$
> \Delta\nu_{\text{D}} = \frac{\nu_0}{c}\sqrt{\frac{8k_B T\ln2}{m_{\mathrm H}}}
> \approx 3.0\times 10^{10}\ \mathrm{Hz} = 30\ \mathrm{GHz}.
> $$
> The $2p$ lifetime $\tau = 1.6\ \mathrm{ns}$ gives a natural width
> $\Delta\nu_{\text{nat}} = 1/2\pi\tau \approx 99\ \mathrm{MHz}$. The Doppler width
> exceeds the natural width by a factor of about $300$: in a room-temperature gas
> the line is a Gaussian three hundred times wider than its Lorentzian core, and
> resolving the natural line shape requires a Doppler-free method or a cold sample.

## Pressure (collisional) broadening

Collisions with other atoms interrupt the radiating wave train. Each collision
randomizes the phase of the emitted field, so instead of one long damped sinusoid
the atom emits a sequence of shorter uncorrelated segments. Shorter coherent
segments mean a broader spectrum, by the same Fourier argument as the natural
width. If the mean time between phase-interrupting collisions is $\tau_{\text{col}}$,
the resulting profile is again **Lorentzian**, with an added width

$$
\Delta\nu_{\text{col}} = \frac{1}{\pi\tau_{\text{col}}}
= \frac{n\,\bar v\,\sigma}{\pi},
$$

where $n$ is the perturber number density, $\bar v$ the mean relative speed, and
$\sigma$ the collision cross section. The key signature is the **linear dependence
on density** (hence pressure at fixed temperature): pressure broadening grows in
proportion to how many perturbers there are. Extrapolating the width to zero
pressure isolates the natural width, and the slope measures the collision cross
section.

$$
% caption: Pressure broadening grows linearly with perturber density. The intercept
% at zero density is the pressure-independent width (natural plus Doppler); the
% slope encodes the collision cross section.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->] (-0.3,0) -- (8.0,0) node[right, black] {number density};
\draw[->] (0,-0.2) -- (0,3.4) node[above, black] {line width};
% linear with intercept
\draw[acc, very thick] (0,0.7) -- (6.8,3.15);
\node[acc, anchor=south east] at (5.6,2.6) {width grows as $n$};
% intercept marker
\draw[black, dashed] (0,0.7) -- (1.6,0.7);
\node[black, anchor=west] at (1.65,0.7) {zero-pressure width};
\fill[black] (0,0.7) circle (2pt);
\end{tikzpicture}
$$

## The Voigt profile

A real line carries both kinds of broadening at once: the Lorentzian natural and
pressure widths, and the Gaussian Doppler width. Because the Doppler shift and the
lifetime broadening are statistically independent, the observed profile is their
**convolution**,

$$
I_{\text{V}}(\omega) = \int_{-\infty}^{\infty}
G(\omega')\,L(\omega-\omega')\,\d\omega',
$$

the **Voigt profile**. It has no elementary closed form, but its limits are clear:
where the Gaussian dominates (low pressure, light atom, high temperature) the core
is Gaussian; the Lorentzian always wins in the far wings, because a Gaussian falls
off as $e^{-x^2}$ while a Lorentzian falls only as $1/x^2$. The wings of any real
line are therefore Lorentzian, and the natural or pressure width can be read from
them even when the Gaussian Doppler core hides it near the center.

$$
% caption: Three line profiles of equal width and height. The Gaussian (Doppler)
% falls off fastest; the Lorentzian (natural, pressure) has heavy far wings; the
% Voigt convolution has a Gaussian-like core and Lorentzian wings.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->] (-5.6,0) -- (5.8,0) node[right, black] {frequency};
\draw[->] (0,-0.2) -- (0,3.5) node[above, black] {intensity};
% Gaussian
\draw[black!70, very thick, densely dotted] plot[smooth] coordinates{
 (-3,0.006) (-2.5,0.04) (-2,0.19) (-1.5,0.63) (-1,1.5) (-0.5,2.52)
 (0,3) (0.5,2.52) (1,1.5) (1.5,0.63) (2,0.19) (2.5,0.04) (3,0.006)};
\node[black!70, anchor=south] at (-1.2,2.0) {Gaussian};
% Lorentzian
\draw[acc, very thick] plot[smooth] coordinates{
 (-5,0.115) (-4,0.176) (-3,0.3) (-2.5,0.41) (-2,0.6) (-1.5,0.92)
 (-1,1.5) (-0.5,2.4) (0,3) (0.5,2.4) (1,1.5) (1.5,0.92)
 (2,0.6) (2.5,0.41) (3,0.3) (4,0.176) (5,0.115)};
\node[acc, anchor=west] at (2.7,0.55) {Lorentzian};
% Voigt
\draw[black, very thick, dashed] plot[smooth] coordinates{
 (-5,0.06) (-4,0.09) (-3,0.16) (-2.5,0.24) (-2,0.42) (-1.5,0.85)
 (-1,1.6) (-0.5,2.5) (0,3) (0.5,2.5) (1,1.6) (1.5,0.85)
 (2,0.42) (2.5,0.24) (3,0.16) (4,0.09) (5,0.06)};
\node[black, anchor=east] at (-2.6,0.7) {Voigt};
\end{tikzpicture}
$$

## Saturation and power broadening

The mechanisms above are properties of a weakly probed atom. A strong driving
field adds one more. When the field is intense enough to cycle population between
the levels faster than they can decay, the transition **saturates**, and the line
broadens. The saturated width is

$$
\Delta\nu = \Delta\nu_{\text{nat}}\sqrt{1 + s},
\qquad
s = \frac{I}{I_{\text{sat}}},
$$

where $s$ is the saturation parameter, $I$ the intensity, and $I_{\text{sat}}$ the
saturation intensity of the transition. Power broadening is the driven-transition
version of the width: the more strongly the atom is interrogated, the broader the
line it presents, which sets a floor on how hard a precision measurement can push
before the resolution it seeks is destroyed. The effect connects directly to the
Rabi picture of the
[first lesson](/atomic-physics/radiative-transitions-and-line-shapes/time-dependent-perturbation-golden-rule):
a large Rabi frequency $\Omega_R$ broadens the response, and $s = 2\Omega_R^2/\Gamma^2$.

## Transit-time and instrumental widths

Two further contributions limit real measurements. **Transit-time broadening**
arises when an atom crosses a finite laser beam of diameter $d$ in a time
$\tau_t \sim d/\bar v$; the interaction is a truncated wave train just as a
collision-limited one is, so it contributes a width $\Delta\nu_t \sim \bar v/d$.
Widening the beam or slowing the atoms reduces it, which is one more reason cold
samples sharpen spectra. **Instrumental broadening** is the finite resolution of
the spectrometer or the finite linewidth of the probe laser; it convolves with the
atomic profile exactly as Doppler broadening does and must be measured separately
(often with a reference line of known width) and deconvolved.

Because the Lorentzian and Gaussian contributions convolve rather than add, their
widths do not combine linearly. The Lorentzian widths (natural, pressure,
transit-time) add directly,

$$
\Gamma_L = \Gamma_{\text{nat}} + \Gamma_{\text{col}} + \Gamma_t,
$$

while independent Gaussian widths (Doppler, some instrumental) add in quadrature,

$$
\Gamma_G^2 = \Gamma_{\text{D}}^2 + \Gamma_{\text{inst}}^2.
$$

The observed Voigt profile carries both $\Gamma_L$ and $\Gamma_G$ as independent
parameters, and fitting the full shape recovers each: the Gaussian width reports
the temperature, and the Lorentzian width, extrapolated to zero pressure and zero
probe power, isolates the natural width and thereby the lifetime. Reading a line
shape is a small inverse problem, and the distinct parameter dependences in the
table below are what make it solvable.

## The mechanisms compared

The four broadening mechanisms are distinguished by how their widths depend on the
controllable parameters — temperature, density, intensity — and by their shapes.

| Mechanism | Shape | Width scales as | Controlled by |
| --- | --- | --- | --- |
| Natural | Lorentzian | $1/\tau$, fixed | the transition itself |
| Doppler | Gaussian | $\sqrt{T/m}$ | cooling, heavier isotope |
| Pressure | Lorentzian | $n\,\bar v\,\sigma$ | lowering density |
| Power | Lorentzian | $\sqrt{1+I/I_{\text{sat}}}$ | weaker probe |

The hierarchy in a typical room-temperature vapor cell runs Doppler ($\mathrm{GHz}$)
$\gg$ pressure ($\mathrm{MHz}$ to $\mathrm{GHz}$, depending on pressure) $\gtrsim$
natural ($\mathrm{MHz}$). Reaching the natural width therefore means removing the
Doppler and pressure contributions: work at low pressure to kill collisions, and
use a Doppler-free method (saturated absorption, two-photon excitation, or a cold
atomic sample) to eliminate the Gaussian. Those techniques are the subject of the
[spectroscopy lesson](/atomic-physics/lasers-and-spectroscopy/spectroscopy-techniques),
and driving the natural width to its floor is the whole design goal of an
[optical clock](/atomic-physics/modern-atomic-physics/optical-clocks-precision).

$$
% caption: The width hierarchy in a room-temperature vapor. Doppler broadening
% dominates at the gigahertz scale; pressure broadening is intermediate; the natural
% width is the megahertz-scale floor reached only after the others are removed.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black] (-0.3,0) -- (9.4,0);
% Doppler wide
\fill[acc!18, draw=acc, very thick] (0.4,0) rectangle (3.6,2.6);
\node[anchor=south] at (2.0,2.63) {Doppler};
\node[anchor=north, black] at (2.0,-0.1) {about 1 GHz};
% pressure medium
\fill[black, draw=black, very thick] (4.4,0) rectangle (6.2,1.5);
\node[anchor=south] at (5.3,1.53) {pressure};
\node[anchor=north, black] at (5.3,-0.1) {MHz to GHz};
% natural narrow
\fill[black, draw=black, very thick] (7.0,0) rectangle (7.9,0.7);
\node[anchor=south] at (7.45,0.73) {natural};
\node[anchor=north, black] at (7.45,-0.1) {about 10 MHz};
\end{tikzpicture}
$$

A spectral line is a measurement of everything that limited the coherence of the
emission that produced it. Its center gives the transition energy, corrected by
[fine](/atomic-physics/fine-structure-and-the-dirac-atom/darwin-term-fine-structure-formula),
[hyperfine](/atomic-physics/qed-corrections-and-hyperfine-structure/hyperfine-structure-21cm),
and [QED](/atomic-physics/qed-corrections-and-hyperfine-structure/lamb-shift-qed)
shifts; its width and shape give the lifetime, the temperature, the pressure, and
the probe intensity. Reading a line is reading all of them at once, and separating
the contributions — Lorentzian from Gaussian, natural from collisional, intrinsic
from instrumental — is the daily work of precision spectroscopy. This closes the
module: the golden rule gave the rate, the dipole approximation gave the coupling,
the selection rules gave which lines appear, and the line shape gives what each one
carries.