Line Shape Reference

A linewidth is not a complete physical quantity until four choices are declared: the spectral coordinate, the profile family, the width convention, and the object whose response is being measured. A value quoted as “$\Gamma=10$ MHz” could mean a Lorentzian half width, a full width, an ordinary-frequency decay scale, or an angular-frequency rate divided by $2\pi$. Those interpretations differ by factors of two or $2\pi$ before any physics is discussed.

This page is a convention-safe lookup for standard isolated-line profiles and broadening scales. The canonical derivations, failure modes, and fitting discussion live in Line Shapes and Broadening. The driven two-level derivation of saturation and power broadening lives in Optical Bloch Equations.

Canonical Scope

This page owns:

• normalized Lorentzian, Gaussian, and Voigt formulas in one explicit coordinate convention;
• HWHM, FWHM, standard-deviation, lifetime, and coherence conversions;
• compact formulas for natural, Doppler, impact-collision, and power broadening;
• rules for combining independent broadening stages;
• a profile-selection and reporting checklist; and
• warnings about instrumental resolution, unresolved structure, and beyond-Voigt physics.

It does not own:

• the Fourier derivation of profile shapes;
• density-matrix dynamics and saturation curves;
• microscopic collision theory or Stark-profile calculations;
• radiative-transfer line mixing and continua;
• laser phase-noise linewidth theory;
• many-body spectral functions; or
• a universal prescription for fitting every spectrum.

A tabulated formula is valid only under the assumptions stated beside it.

Convention Ledger

Unless a section says otherwise, use angular frequency

$$x = \delta\omega = \omega-\omega_0$$

and the following symbols:

Symbol	Meaning	Units on an angular-frequency axis
$\gamma_{\mathrm L}$	Lorentzian HWHM	$\mathrm{rad\,s^{-1}}$
$w_{\mathrm L}=2\gamma_{\mathrm L}$	Lorentzian FWHM	$\mathrm{rad\,s^{-1}}$
$\sigma_{\mathrm G}$	Gaussian standard deviation	$\mathrm{rad\,s^{-1}}$
$w_{\mathrm G}$	Gaussian FWHM	$\mathrm{rad\,s^{-1}}$
$\Gamma_1$	population-decay rate	$\mathrm{s^{-1}}$
$\Gamma_2$	coherence-decay rate	$\mathrm{s^{-1}}$
$T_1=1/\Gamma_1$	population lifetime	$\mathrm s$
$T_2=1/\Gamma_2$	coherence time	$\mathrm s$

Radians are dimensionless in SI, but retaining rad in a label makes the $2\pi$ convention visible.

Coordinate conversions

Angular frequency $\omega$, ordinary frequency $\nu$, photon energy $E$, and spectroscopic wavenumber $\widetilde\nu$ obey

$$\omega=2\pi\nu, \qquad E=h\nu=\hbar\omega, \qquad \nu=c\widetilde\nu.$$

Therefore corresponding narrow-line widths satisfy

$$\Delta\omega = 2\pi\Delta\nu,$$ $$\Delta E = \hbar\Delta\omega = h\Delta\nu,$$ $$\Delta\widetilde\nu = \frac{\Delta\nu}{c}.$$

Do not use $\widetilde\nu$ for ordinary frequency merely because both are pronounced “nu.” Spectroscopic wavenumber is commonly reported in $\mathrm{cm^{-1}}$.

Width vocabulary

Name	Definition	Symmetric-profile relation
peak center	location of maximum	equals the location parameter only for the stated symmetric model
HWHM	positive offset at half maximum	$x_{1/2}$
FWHM	distance between the two half-maximum points	$2x_{1/2}$
standard deviation	square root of second central moment	finite for a Gaussian; undefined for an ideal Lorentzian
$1/e$ half width	offset where the profile falls to $1/e$ of peak	profile dependent
equivalent width	integrated absorption relative to a continuum	not a line-profile FWHM

The symbol $\Gamma$ has no universal width meaning. Always replace it with a named HWHM, FWHM, decay rate, or energy width when comparing sources.

Normalized Profiles

Let $g_x(x)$ be a line profile normalized on coordinate $x$:

$$\int_{-\infty}^{\infty} g_x(x)\,dx = 1.$$

Its units are the reciprocal units of $x$. If an integrated line strength is $S$, a linear optically thin spectral contribution can be written

$$k_x(x) = S\,g_x(x).$$

The area is $S$; the profile redistributes that area over the spectral axis. Peak height and width are therefore not independent at fixed area.

Changing the spectral coordinate

For a monotonic change $y=y(x)$, normalization requires

$$g_y(y) = g_x\!\bigl(x(y)\bigr) \left| \frac{dx}{dy} \right|.$$

Frequency and wavelength are related nonlinearly:

$$\nu=\frac{c}{\lambda}, \qquad \left| \frac{d\nu}{d\lambda} \right| = \frac{c}{\lambda^2}.$$

A profile that is exactly symmetric in frequency is not exactly symmetric in wavelength. The familiar relation

$$\frac{|\Delta\lambda|}{\lambda_0} \simeq \frac{|\Delta\nu|}{\nu_0}$$

is a first-order narrow-line approximation, not an exact profile transformation.

Profile Comparison

Property	Lorentzian	Gaussian	Voigt
parameters	center and HWHM $\gamma_{\mathrm L}$	center and standard deviation $\sigma_{\mathrm G}$	center, $\gamma_{\mathrm L}$, and $\sigma_{\mathrm G}$
common origin	exponential coherence, isolated damped pole, impact-limit phase interruption	normal distribution of static shifts, thermal one-axis velocities, Gaussian instrument response	independent Gaussian and Lorentzian stages
wings	algebraic, proportional to $x^{-2}$	exponential in $-x^2$	Lorentzian asymptote if $\gamma_{\mathrm L}>0$
variance	does not exist on an infinite domain	$\sigma_{\mathrm G}^2$	does not exist if $\gamma_{\mathrm L}>0$
convolution closure	Lorentzian widths add	variances add	not closed under arbitrary profile changes
one-width description	HWHM or FWHM is sufficient after normalization	standard deviation or FWHM is sufficient after normalization	requires both component widths

Similar-looking cores do not imply similar wings. This matters for weak neighboring lines, optical-depth retrievals, and truncated fit windows.

Lorentzian

The area-normalized Lorentzian is

$$L(x;\gamma_{\mathrm L}) = \frac{1}{\pi} \frac{\gamma_{\mathrm L}} {x^2+\gamma_{\mathrm L}^2}.$$

Its lookup properties are:

$$L(0) = \frac{1}{\pi\gamma_{\mathrm L}},$$ $$\text{HWHM} = \gamma_{\mathrm L}, \qquad \text{FWHM} = w_{\mathrm L} = 2\gamma_{\mathrm L},$$ $$L(x) \sim \frac{\gamma_{\mathrm L}} {\pi x^2} \qquad (|x|\gg\gamma_{\mathrm L}).$$

A Lorentzian does not have a finite ordinary variance:

$$\int_{-\infty}^{\infty} x^2L(x)\,dx \quad\text{diverges}.$$

Consequently, a reported “Lorentzian standard deviation” either refers to a finite window, a different distribution, or an undocumented convention.

Time-domain dictionary

An exponentially decaying coherence

$$C(t) \propto e^{-\Gamma_2|t|} e^{-i\omega_0t}$$

produces a Lorentzian frequency dependence with

$$\gamma_{\mathrm L,\omega} = \Gamma_2,$$

and hence

$$w_{\mathrm L,\omega} = 2\Gamma_2 = \frac{2}{T_2}.$$

On an ordinary-frequency axis,

$$\gamma_{\mathrm L,\nu} = \frac{\Gamma_2}{2\pi},$$ $$w_{\mathrm L,\nu} = \frac{\Gamma_2}{\pi} = \frac{1}{\pi T_2}.$$

These relations assume exponential coherence over the times relevant to the measurement.

Gaussian

The area-normalized Gaussian is

$$G(x;\sigma_{\mathrm G}) = \frac{1} {\sigma_{\mathrm G}\sqrt{2\pi}} \exp\!\left( -\frac{x^2} {2\sigma_{\mathrm G}^2} \right).$$

Its lookup properties are:

$$G(0) = \frac{1} {\sigma_{\mathrm G}\sqrt{2\pi}},$$ $$\text{HWHM} = \sqrt{2\ln2}\, \sigma_{\mathrm G},$$ $$w_{\mathrm G} = 2\sqrt{2\ln2}\, \sigma_{\mathrm G}.$$

Numerically,

$$w_{\mathrm G} \simeq 2.35482\,\sigma_{\mathrm G}.$$

The $1/e$ half width is

$$x_{1/e} = \sqrt2\,\sigma_{\mathrm G}.$$

Do not confuse this with the half width at half maximum.

Combining Gaussian stages

For independent Gaussian shifts,

$$\sigma_{\mathrm{tot}}^2 = \sum_i\sigma_i^2.$$

Because every Gaussian FWHM contains the same multiplicative factor,

$$w_{\mathrm{G,tot}}^2 = \sum_i w_{\mathrm{G},i}^2.$$

This quadrature rule applies to Gaussian convolution parameters. It is not a generic rule for all quoted uncertainties or all spectral widths.

Voigt

The Voigt profile is the convolution of a Gaussian and a Lorentzian:

$$\begin{aligned} V(x;\sigma_{\mathrm G},\gamma_{\mathrm L}) &= \int_{-\infty}^{\infty} G(x';\sigma_{\mathrm G}) \\ &\quad\times L(x-x';\gamma_{\mathrm L}) \,dx'. \end{aligned}$$

It is area normalized if both components are normalized.

Faddeeva representation

Define

$$w(z) = e^{-z^2} \operatorname{erfc}(-iz),$$

and

$$z = \frac{x+i\gamma_{\mathrm L}} {\sigma_{\mathrm G}\sqrt2}.$$

Then

$$V(x;\sigma_{\mathrm G},\gamma_{\mathrm L}) = \frac{ \operatorname{Re}w(z) } {\sigma_{\mathrm G}\sqrt{2\pi}}.$$

Use a tested complex-error-function implementation. Direct numerical quadrature or naive evaluation of $e^{-z^2}\operatorname{erfc}(-iz)$ can lose accuracy in parts of the complex plane.

Limits

The Voigt profile has the expected limits:

$$\lim_{\gamma_{\mathrm L}\to0} V(x;\sigma_{\mathrm G},\gamma_{\mathrm L}) = G(x;\sigma_{\mathrm G}),$$ $$\lim_{\sigma_{\mathrm G}\to0} V(x;\sigma_{\mathrm G},\gamma_{\mathrm L}) = L(x;\gamma_{\mathrm L}).$$

Any nonzero $\gamma_{\mathrm L}$ controls the far-wing asymptote, even when the core looks nearly Gaussian.

Approximate Voigt FWHM

Let

$$w_{\mathrm L}=2\gamma_{\mathrm L}, \qquad w_{\mathrm G} = 2\sqrt{2\ln2}\,\sigma_{\mathrm G}.$$

A widely used approximation is

$$\begin{aligned} w_{\mathrm V} &\simeq 0.5346\,w_{\mathrm L} \\ &\quad+ \sqrt{ 0.2166\,w_{\mathrm L}^2 + w_{\mathrm G}^2 }. \end{aligned}$$

This estimates the resulting FWHM. It does not convert a fitted single $w_{\mathrm V}$ into unique Lorentzian and Gaussian components.

Voigt is not pseudo-Voigt

A pseudo-Voigt profile is a weighted sum,

$$P(x) = \eta L(x) + (1-\eta)G(x),$$

with widths arranged according to a chosen approximation. A Voigt profile is a convolution. A pseudo-Voigt can be computationally convenient, but its mixing fraction $\eta$ is not a physical Lorentzian fraction unless a particular model establishes that interpretation.

Width-Combination Rules

Independent stages	Correct combination	Resulting family
Lorentzian + Lorentzian	add HWHMs: $\gamma_{\mathrm{tot}}=\gamma_1+\gamma_2$	Lorentzian
Gaussian + Gaussian	add variances: $\sigma_{\mathrm{tot}}^2=\sigma_1^2+\sigma_2^2$	Gaussian
Lorentzian + Gaussian	convolve	Voigt
unresolved discrete components	sum shifted component profiles with weights	generally not one standard profile
saturation plus Doppler distribution	average the intensity-dependent response over velocities	not automatically a fixed Voigt
line mixing or interference	use coupled response amplitudes or a relaxation matrix	can be asymmetric or dispersive

Blindly adding all FWHMs linearly is wrong. Blindly adding all FWHMs in quadrature is also wrong. First classify the profile family and statistical relationship.

Natural and Lifetime Width

For a transition between upper and lower levels with population-loss rates $\Gamma_u$ and $\Gamma_l$, and with additional Markovian pure dephasing $\gamma_\phi$, the optical coherence decays at

$$\Gamma_2 = \frac{\Gamma_u+\Gamma_l}{2} + \gamma_\phi.$$

The Lorentzian angular-frequency widths are

$$\gamma_{\mathrm L,\omega} = \Gamma_2,$$ $$w_{\mathrm L,\omega} = 2\Gamma_2 = \Gamma_u+\Gamma_l+2\gamma_\phi.$$

On an ordinary-frequency axis,

$$w_{\mathrm L,\nu} = \frac{ \Gamma_u+\Gamma_l+2\gamma_\phi } {2\pi}.$$

Simplest natural-width limit

If the lower state is stable, the upper state decays only radiatively, and there is no pure dephasing,

$$\Gamma_l=0, \qquad \gamma_\phi=0,$$ $$\Gamma_u = \sum_l A_{ul} = \frac{1}{\tau_u}.$$

Then

$$w_{\mathrm{nat},\omega} = \Gamma_u,$$ $$\Delta\nu_{\mathrm{nat}} = \frac{\Gamma_u}{2\pi} = \frac{1}{2\pi\tau_u},$$ $$\Delta E_{\mathrm{nat}} = \hbar\Gamma_u.$$

The relation “linewidth equals inverse lifetime” is correct here only for angular-frequency FWHM. The ordinary-frequency FWHM contains $2\pi$.

What counts as lifetime loss

The total upper-state loss can include

$$\Gamma_u^{\mathrm{tot}} = \Gamma_u^{\mathrm{rad}} + \Gamma_u^{\mathrm{nr}},$$

where nonradiative loss may include quenching, predissociation, autoionization, reaction, or escape from the selected state. Such loss can broaden the transition without contributing to the detected fluorescence branch.

“Natural linewidth” should identify the electromagnetic environment. Cavities and structured reservoirs can modify radiative decay, while strong coupling can replace one broadened peak with resolved normal modes.

Doppler Width

For a thermal gas in equilibrium, one velocity component is Gaussian with variance

$$\sigma_v^2 = \frac{k_{\mathrm B}T}{m}.$$

For a wavevector of magnitude

$$k_0 = \frac{\omega_0}{c},$$

the first-order shift is

$$\delta\omega \simeq -k_0v_\parallel.$$

The angular-frequency standard deviation is therefore

$$\sigma_{\mathrm D,\omega} = \frac{\omega_0}{c} \sqrt{ \frac{k_{\mathrm B}T}{m} }.$$

The angular-frequency Doppler FWHM is

$$w_{\mathrm D,\omega} = 2\sqrt{2\ln2}\, \frac{\omega_0}{c} \sqrt{ \frac{k_{\mathrm B}T}{m} }.$$

Equivalently,

$$\Delta\nu_{\mathrm D} = 2\nu_0 \sqrt{ \frac{2k_{\mathrm B}T\ln2} {mc^2} },$$

and

$$\frac{\Delta\nu_{\mathrm D}}{\nu_0} = \sqrt{ \frac{8k_{\mathrm B}T\ln2} {mc^2} }.$$

The same fractional formula applies to spectroscopic wavenumber:

$$\frac{ \Delta\widetilde\nu_{\mathrm D} } {\widetilde\nu_0} = \sqrt{ \frac{8k_{\mathrm B}T\ln2} {mc^2} }.$$

Assumptions and diagnostics

The formula assumes:

• a Maxwell–Boltzmann distribution;
• a nonrelativistic first-order Doppler shift;
• thermal equilibrium;
• the stated particle or isotopologue mass;
• the temperature projected along the optical wavevector; and
• no unresolved structure folded into the fitted Gaussian width.

A Gaussian fit parameter is not automatically a thermodynamic temperature. Laser jitter, isotope structure, field gradients, spatially varying light shifts, and instrumental response can also produce or enlarge a Gaussian component.

Collisional Width and Shift

In the impact limit, short statistically independent collisions add a homogeneous coherence-decay rate. A useful form is

$$C(t) \propto e^{- (\Gamma_2^{(0)}+\gamma_{\mathrm{col}})t } e^{-i (\omega_0+\Delta_{\mathrm{col}})t }.$$

Thus the collision-induced Lorentzian HWHM adds:

$$\gamma_{\mathrm L,tot} = \gamma_{\mathrm L,0} + \gamma_{\mathrm{col}}.$$

At low perturber density,

$$\gamma_{\mathrm{col}} \sim n_{\mathrm p} \left\langle v_{\mathrm{rel}} \sigma_{\mathrm{broad}} \right\rangle.$$

The broadening cross section describes phase interruption. It need not equal an elastic, reactive, or momentum-transfer cross section.

Pressure-coefficient convention

Molecular line lists often tabulate pressure-broadened Lorentzian HWHM coefficients at

$$T_{\mathrm{ref}}=296\ \mathrm K, \qquad p_{\mathrm{ref}}=1\ \mathrm{atm}.$$

With coefficients expressed per atmosphere and pressures inserted in atmospheres, a common HITRAN-style form is

$$\begin{aligned} \gamma_{\mathrm L}(p,T) &= \left( \frac{T_{\mathrm{ref}}}{T} \right)^n \\ &\quad\times \left[ \gamma_{\mathrm{air}} (p-p_{\mathrm{self}}) + \gamma_{\mathrm{self}} p_{\mathrm{self}} \right]. \end{aligned}$$

The result is a spectroscopic-wavenumber HWHM when the coefficients are in $\mathrm{cm^{-1}\,atm^{-1}}$. If pressure is represented as a dimensionless ratio to $p_{\mathrm{ref}}$, the equation must display that ratio instead.

A pressure shift is separate:

$$\widetilde\nu_0(p,T) = \widetilde\nu_0(0) + \delta(p,T)\,p.$$

Never absorb a fitted center shift into the width parameter.

When the impact Lorentzian fails

Pressure broadening can require more than one Lorentzian parameter because of:

• speed-dependent broadening and shifting;
• velocity-changing collisions and Dicke narrowing;
• correlations between state- and velocity-changing collisions;
• line mixing;
• quasistatic Stark or van der Waals wings;
• state-changing collisions; and
• nonideal density or temperature dependence.

The temperature exponent $n$ is line and perturber dependent. It is not a universal kinetic-theory constant.

Power Broadening

For a Markovian two-level system driven with Rabi frequency $\Omega$, let $\Gamma_1$ be the population-decay rate and $\Gamma_2$ the coherence-decay rate. Define the on-resonance saturation parameter

$$s_0 = \frac{\Omega^2} {\Gamma_1\Gamma_2}.$$

The steady-state response has ideal angular-frequency HWHM

$$\gamma_{\mathrm{pow},\omega} = \Gamma_2\sqrt{1+s_0},$$

and FWHM

$$w_{\mathrm{pow},\omega} = 2\Gamma_2\sqrt{1+s_0}.$$

The weak-drive width is

$$w_{0,\omega} = 2\Gamma_2,$$

so the compact ratio is

$$\frac{w_{\mathrm{pow}}}{w_0} = \sqrt{1+s_0}.$$

For a stable lower state, purely radiative upper decay, and no pure dephasing,

$$\Gamma_2=\frac{\Gamma_1}{2},$$

and

$$w_{\mathrm{pow},\omega} = \Gamma_1\sqrt{1+s_0}.$$

Do not convolve power broadening blindly

Power broadening changes the driven response; it is not an independent random frequency offset applied after an intrinsic line is formed. For a thermal ensemble, the forward model is generally

$$S(\Delta) = \int dv\, p(v)\, S_{\mathrm{2L}} (\Delta-kv;\Omega,\Gamma_1,\Gamma_2),$$

possibly with spatially varying $\Omega$. Strong drive can also create AC Stark shifts, optical pumping, dark resonances, Autler–Townes splitting, and Mollow sidebands. A single broadened Lorentzian is then insufficient.

Rabi-frequency and saturation conventions differ among authors. The half-maximum width should be derived from the stated Hamiltonian and master equation when factors of two matter.

Other Common Contributions

Contribution	Typical profile or forward model	Key caution
transit time	depends on beam geometry and trajectory distribution	not necessarily a Lorentzian and not an intrinsic level lifetime
finite square interrogation	sinc-squared probability envelope	Fourier-limited width is a measurement response, not a decay width
Ramsey interrogation	central fringe plus side fringes	report pulse separation and estimator
Gaussian instrument response	convolution; variances add to other Gaussian stages	calibrate with a source narrower than the instrument
finite spectrometer slit	apparatus-specific line-spread function	may be triangular, sinc-like, or asymmetric
laser frequency noise	depends on noise spectrum and observation time	one stationary linewidth may not exist
unresolved hyperfine or isotope components	weighted sum of shifted lines	can mimic broadening, asymmetry, or line pulling
opacity and saturation in propagation	nonlinear radiative-transfer model	transmitted intensity need not preserve the microscopic profile

For laser phase noise and coherence, use Linewidth and Coherence. For finite-time estimators and systematic shifts, use Precision Spectroscopy.

Homogeneous and Inhomogeneous

The distinction is operational:

Class	Meaning	Standard examples	Can subensemble selection or refocusing reduce it?
homogeneous	every selected member has the same dynamical profile	lifetime decay, Markovian pure dephasing, impact-limit collisions	not by merely selecting a narrower static class
inhomogeneous	members have different static or slowly varying centers	Doppler shifts, site disorder, field gradients, slow detuning drift	often, if the offset is resolvable or reversible

For a static center distribution $p(\Omega)$ and a center-independent homogeneous profile,

$$L_{\mathrm{ens}}(\omega) = \int_{-\infty}^{\infty} p(\Omega) L_{\mathrm{hom}}(\omega-\Omega) \,d\Omega.$$

A Gaussian $p$ and Lorentzian $L_{\mathrm{hom}}$ produce a Voigt profile. However, the same environment can appear inhomogeneous during a short scan and homogeneous during a long coherence interval if its fluctuation time changes relative to the measurement.

The ensemble dephasing time $T_2^*$ can therefore be shorter than the homogeneous $T_2$. An echo can refocus sufficiently slow reversible offsets, but not population loss or arbitrarily fast dephasing.

Beyond Voigt

The ordinary Voigt model assumes an independent Gaussian distribution of centers and a velocity-independent Lorentzian width and shift. High-resolution gas spectra can resolve violations of those assumptions.

Escalation ladder

Residual or physical evidence	Candidate extension
width or shift changes systematically across molecular speed classes	speed-dependent Voigt
pressure-dependent narrowing of the Doppler core	velocity-changing-collision or Dicke-narrowing model
both speed dependence and velocity-changing collisions matter	Hartmann–Tran-family profile
neighboring lines exchange intensity or create dispersive residuals	line-mixing model
dense bands or continua defeat isolated-line decomposition	band or cross-section model

An IUPAC task group recommended the Hartmann–Tran profile as a beyond-Voigt isolated-line model in 2014. HITRAN2024 now recommends the modified Hartmann–Tran profile when beyond-Voigt effects are relevant, while retaining Voigt parameters as a broadly available baseline. The mHT model addresses limitations in the earlier velocity-changing-collision parameterization and supports updated temperature dependences.

This is not an instruction to fit the largest possible model. Use the simplest profile that is physically justified, identifiable over the measured pressure and temperature range, and supported by structure-free residuals.

Selection Guide

Start from mechanism, not appearance

Evidence	First model to test	Immediate alternative
exponential coherence or isolated weakly damped transition	Lorentzian	nonexponential correlation model
Maxwellian one-axis velocity spread	Gaussian Doppler	nonthermal velocity distribution
Gaussian Doppler plus impact-limit homogeneous width	Voigt	speed-dependent or narrowing profile
known discrete hyperfine or isotope components	explicit weighted sum	coupled-line model if components mix
strong resonant drive	optical Bloch forward model	multilevel or dressed-state model
calibrated Gaussian apparatus function	convolution with instrument response	measured nonparametric response
asymmetric or dispersive residuals	shifted components, interference, or line mixing	apparatus asymmetry check

Limiting-case checks

A fitted implementation should satisfy:

$$\int g(x)\,dx=1,$$ $$V(x;\sigma,0)=G(x;\sigma),$$ $$V(x;0,\gamma)=L(x;\gamma),$$ $$\gamma_{\mathrm{tot}} = \gamma_1+\gamma_2 \quad \text{for Lorentzian convolution},$$ $$\sigma_{\mathrm{tot}}^2 = \sigma_1^2+\sigma_2^2 \quad \text{for Gaussian convolution}.$$

Also test positivity, area conservation under grid refinement, parameter recovery from synthetic data, and invariance under a consistent axis-unit conversion.

Fit and Reporting Ledger

Record at least:

Field	Required information
observable	absorption coefficient, optical depth, fluorescence counts, transmitted power, heterodyne amplitude, or another quantity
spectral coordinate	$\omega$, $\nu$, $\widetilde\nu$, $\lambda$, or energy, with units and vacuum/medium convention
profile	Lorentzian, Gaussian, Voigt, mHT, explicit line sum, or measured response
width convention	HWHM, FWHM, standard deviation, or another named definition
amplitude convention	area, peak height, oscillator strength, line intensity, or column density
center model	zero-pressure center and all pressure, Stark, Zeeman, recoil, or light shifts
apparatus	instrument line-spread function, laser noise, sampling, and finite interrogation
environment	temperature, pressure, perturber composition, isotope, fields, and drive intensity
statistics	likelihood or weighting, covariance, parameter correlations, and uncertainty model
validation	residual plots, fit-window variation, pressure/temperature series, and synthetic recovery

Residual diagnostics

Residual pattern	Possible cause
symmetric excess in far wings	missing Lorentzian component, opacity, or baseline error
narrow central cusp or dip	Dicke narrowing, crossover feature, coherent resonance, or detector nonlinearity
antisymmetric residual	center error, dispersive admixture, pressure shift, or line mixing
shoulders	unresolved components or dressed-state splitting
width changes with fit window	wrong wings, baseline coupling, or neighboring lines
Gaussian width changes with pressure	beyond-Voigt collision physics or parameter covariance

Residuals are evidence, not automatic mechanism labels. Challenge apparatus and baseline models before assigning new microscopic physics.

Common Mistakes

Quoting a width without the axis

A value in $\mathrm{rad\,s^{-1}}$, Hz, $\mathrm{cm^{-1}}$, eV, or nm is not interchangeable without the appropriate $2\pi$, $h$, $c$, or Jacobian.

Calling HWHM and FWHM by one symbol

Write the word or define the symbol. This avoids the most common factor-of-two error in Lorentzian formulas.

Using inverse lifetime without a convention

For a stable lower level with lifetime $\tau$, $1/\tau$ is the angular-frequency FWHM, while $1/(2\pi\tau)$ is the ordinary-frequency FWHM.

Adding every FWHM linearly

Only Lorentzian convolution widths add linearly. Gaussian variances add; mixed stages form a Voigt or another joint model.

Calling every Gaussian width Doppler

A Gaussian component can include instrument resolution, laser jitter, unresolved structure, field gradients, or static disorder.

Calling every Lorentzian width a lifetime

Impact collisions, Markovian pure dephasing, power broadening, and apparatus responses can also generate Lorentzian-like cores.

Treating a Voigt fit as proof of two mechanisms

Voigt parameters can be highly correlated over a limited signal-to-noise ratio or fit window. Mechanism attribution needs pressure, temperature, time-domain, or independent calibration evidence.

Subtracting the instrument FWHM

Deconvolution depends on profile family. Two Gaussian FWHMs subtract in quadrature at the parameter level; Lorentzian HWHMs subtract linearly; mixed profiles require a forward convolution.

Treating power broadening as intrinsic

It depends on drive strength, polarization, spatial mode, level structure, and saturation convention.

Ignoring shifts while fitting widths

Pressure, light, Zeeman, Stark, recoil, and line-mixing shifts can correlate with widths and produce asymmetric residuals.

Replacing unresolved structure with one wide line

A weighted sum of narrow lines can move the fitted center and change shape with temperature, polarization, or state preparation.

Exercises

Exercise 1: Convert Gaussian width conventions

A Gaussian line has standard deviation $\sigma_\nu=12.0$ MHz. Find its HWHM, FWHM, and $1/e$ half width.

Solution

For a Gaussian,

$$\mathrm{HWHM} = \sqrt{2\ln2}\,\sigma_\nu,$$ $$\mathrm{FWHM} = 2\sqrt{2\ln2}\,\sigma_\nu,$$

and

$$x_{1/e} = \sqrt2\,\sigma_\nu.$$

Using

$$\sqrt{2\ln2}\simeq1.17741,$$

gives

$$\mathrm{HWHM} \simeq 14.13\ \mathrm{MHz},$$ $$\mathrm{FWHM} \simeq 28.26\ \mathrm{MHz},$$ $$x_{1/e} \simeq 16.97\ \mathrm{MHz}.$$

Exercise 2: Lifetime to natural width

An upper state has radiative lifetime $\tau=30.0$ ns. The lower state is stable and pure dephasing is negligible. Find the angular-frequency FWHM, ordinary-frequency FWHM, and Lorentzian HWHM in MHz.

Solution

The population-decay rate is

$$\Gamma_u = \frac{1}{\tau} = 3.333\times10^7\ \mathrm{s^{-1}}.$$

In the stated limit, this is the angular-frequency FWHM:

$$w_{\mathrm{nat},\omega} = 3.333\times10^7\ \mathrm{rad\,s^{-1}}.$$

The ordinary-frequency FWHM is

$$\Delta\nu_{\mathrm{nat}} = \frac{\Gamma_u}{2\pi} \simeq 5.31\ \mathrm{MHz}.$$

The ordinary-frequency HWHM is half of that:

$$\gamma_{\mathrm L,\nu} \simeq 2.65\ \mathrm{MHz}.$$

Exercise 3: Both levels are unstable

An optical transition has upper and lower population-loss rates $\Gamma_u=8.0\times10^6\ \mathrm{s^{-1}}$ and $\Gamma_l=2.0\times10^6\ \mathrm{s^{-1}}$. Additional pure dephasing is $\gamma_\phi=1.0\times10^6\ \mathrm{s^{-1}}$. Find the angular-frequency HWHM and ordinary-frequency FWHM.

Solution

The coherence-decay rate is

$$\begin{aligned} \Gamma_2 &= \frac{\Gamma_u+\Gamma_l}{2} + \gamma_\phi \\ &= \frac{8.0+2.0}{2} \times10^6 + 1.0\times10^6 \\ &= 6.0\times10^6\ \mathrm{s^{-1}}. \end{aligned}$$

Thus the angular-frequency HWHM is

$$\gamma_{\mathrm L,\omega} = 6.0\times10^6\ \mathrm{rad\,s^{-1}},$$

and the angular-frequency FWHM is

$$w_{\mathrm L,\omega} = 1.20\times10^7\ \mathrm{rad\,s^{-1}}.$$

The ordinary-frequency FWHM is

$$w_{\mathrm L,\nu} = \frac{1.20\times10^7}{2\pi} \simeq 1.91\ \mathrm{MHz}.$$

Exercise 4: Doppler width of rubidium

Estimate the ordinary-frequency Doppler FWHM of a $780$ nm transition in a $^{87}\mathrm{Rb}$ vapor at $300$ K. Use $m=87\,u$, $u=1.66054\times10^{-27}$ kg, and $k_{\mathrm B}=1.380649\times10^{-23}\ \mathrm{J\,K^{-1}}$.

Solution

The center frequency is

$$\nu_0 = \frac{c}{\lambda_0} \simeq 3.8435\times10^{14}\ \mathrm{Hz},$$

and the mass is

$$m = 87u \simeq 1.4447\times10^{-25}\ \mathrm{kg}.$$

Use

$$\Delta\nu_{\mathrm D} = 2\nu_0 \sqrt{ \frac{2k_{\mathrm B}T\ln2} {mc^2} }.$$

Substitution gives

$$\Delta\nu_{\mathrm D} \simeq 5.11\times10^8\ \mathrm{Hz},$$

or about

$$511\ \mathrm{MHz}.$$

This is much broader than the few-megahertz natural width of a typical alkali resonance, which is why room-temperature vapor spectroscopy often uses Doppler-free methods to resolve hyperfine-scale structure.

Exercise 5: Estimate a Voigt FWHM

A Voigt profile has Lorentzian FWHM $w_{\mathrm L}=20$ MHz and Gaussian FWHM $w_{\mathrm G}=80$ MHz. Estimate the total FWHM.

Solution

Use

$$w_{\mathrm V} \simeq 0.5346w_{\mathrm L} + \sqrt{ 0.2166w_{\mathrm L}^2 + w_{\mathrm G}^2 }.$$

Then

$$\begin{aligned} w_{\mathrm V} &\simeq 0.5346(20) \\ &\quad+ \sqrt{ 0.2166(20)^2 + (80)^2 } \\ &\simeq 91.2\ \mathrm{MHz}. \end{aligned}$$

The result is not $100$ MHz from linear addition and not $\sqrt{20^2+80^2}$ MHz from Gaussian quadrature.

Exercise 6: Remove a Gaussian instrument width

A measured Gaussian FWHM is $60$ MHz. A calibration gives a Gaussian instrument FWHM of $25$ MHz. Assuming independent Gaussian convolution, find the sample FWHM.

Solution

Gaussian variances, and therefore squared Gaussian FWHMs, add:

$$w_{\mathrm{meas}}^2 = w_{\mathrm{sample}}^2 + w_{\mathrm{inst}}^2.$$

Thus

$$\begin{aligned} w_{\mathrm{sample}} &= \sqrt{ (60\ \mathrm{MHz})^2 - (25\ \mathrm{MHz})^2 } \\ &\simeq 54.5\ \mathrm{MHz}. \end{aligned}$$

Direct subtraction would give $35$ MHz and would be wrong for Gaussian convolution.

Exercise 7: Pressure-broadened HWHM

At $296$ K, a molecular line has air-broadened HWHM coefficient $\gamma_{\mathrm{air}}=0.070\ \mathrm{cm^{-1}\,atm^{-1}}$ and self-broadened coefficient $\gamma_{\mathrm{self}}=0.30\ \mathrm{cm^{-1}\,atm^{-1}}$. The total pressure is $0.80$ atm and the absorber partial pressure is $0.020$ atm. Neglect temperature correction. Find the Lorentzian HWHM and FWHM.

Solution

The air partial pressure is

$$p_{\mathrm{air}} = 0.80-0.020 = 0.780\ \mathrm{atm}.$$

The HWHM is

$$\begin{aligned} \gamma_{\mathrm L} &= \gamma_{\mathrm{air}}p_{\mathrm{air}} + \gamma_{\mathrm{self}}p_{\mathrm{self}} \\ &= (0.070)(0.780) + (0.30)(0.020) \\ &= 0.0606\ \mathrm{cm^{-1}}. \end{aligned}$$

Therefore,

$$w_{\mathrm L} = 2\gamma_{\mathrm L} = 0.1212\ \mathrm{cm^{-1}}.$$

A pressure shift coefficient, if supplied, would modify the center rather than this width.

Exercise 8: Diagnose a profile failure

A series of gas spectra is fit with Voigt profiles. As pressure increases, the fitted Gaussian width also increases, the residual develops a narrow central dip with symmetric shoulders, and the inferred temperature disagrees with an independent thermometer. What should be concluded and tested next?

Solution

One should not conclude that the gas temperature rises merely because the Voigt Gaussian parameter rises. In an ordinary Voigt model, the Doppler width is fixed by temperature and mass, while collision physics enters the Lorentzian component. Pressure-dependent covariance can transfer omitted collision structure into the fitted Gaussian width.

A defensible next sequence is:

1. verify pressure, temperature, frequency-axis, detector-linearity, and instrument-response calibrations;
2. fit all pressures jointly with the Doppler width constrained by the independent temperature;
3. test speed-dependent broadening and velocity-changing-collision or Dicke narrowing models;
4. inspect neighboring-line and line-mixing contributions;
5. compare a modified Hartmann–Tran implementation if the required parameters are identifiable; and
6. report model dependence rather than assigning a unique Doppler temperature from the unstable Voigt decomposition.

The residual is evidence that the Voigt assumptions fail, not by itself proof of one specific collision model.

Key Takeaways

• A linewidth requires an axis, a profile, and a named width convention.
• Lorentzian HWHMs add linearly; Gaussian variances add; a Gaussian and a Lorentzian convolve to a Voigt profile.
• A Lorentzian has no finite ordinary variance.
• For the simplest radiative transition, $\Delta\nu_{\mathrm{nat}}=1/(2\pi\tau)$ is the ordinary-frequency FWHM.
• Thermal Doppler width is Gaussian only for the relevant Maxwellian velocity component.
• Impact-limit collision broadening adds a Lorentzian HWHM and generally also shifts the line center.
• Power broadening is part of a driven-response model, not a passive convolution kernel.
• A Voigt fit does not by itself identify natural and Doppler mechanisms.
• HITRAN2024 recommends modified Hartmann–Tran when beyond-Voigt effects are relevant, but model complexity must remain identifiable.

Cross-Links

• Line Shapes and Broadening is the canonical derivation and mechanism article.
• Spectroscopy Nomenclature fixes spectral-coordinate, attenuation, cross-section, optical-depth, and generic width language before a profile is selected.
• Optical Bloch Equations derives saturation, scattering, and power-broadened response.
• Precision Spectroscopy treats line-center estimators, finite interrogation, line pulling, and uncertainty budgets.
• Linewidth and Coherence treats laser phase diffusion, technical frequency noise, and measurement time.
• Transition Rates and Einstein Coefficients connect radiative rates to natural widths.
• Absorption and Emission connects normalized profiles to cross sections, optical depth, and radiative transfer.
• Gaussian Distributions develops variance, transformations, and normal-distribution conventions.
• Spectral Functions treats poles, continua, self-energies, and many-body broadening.

References

1. W. C. Martin, W. L. Wiese, and A. Kramida, “Spectral Line Shapes, Widths, and Shifts”, in Atomic Spectroscopy: An Introduction, National Institute of Standards and Technology, updated 2025.
2. NIST Digital Library of Mathematical Functions, “Voigt Functions”, release 1.2.7 (2026).
3. HITRAN, “Definitions and Units”, including Doppler, pressure-broadening, line-intensity, and Voigt conventions, accessed 2026-07-26.
4. I. E. Gordon et al., “The HITRAN2024 molecular spectroscopic database,” Journal of Quantitative Spectroscopy and Radiative Transfer 353, 109807 (2026), official PDF.
5. J. Tennyson et al., “Recommended isolated-line profile for representing high-resolution spectroscopic transitions,” Pure and Applied Chemistry 86, 1931–1943 (2014), doi:10.1515/pac-2014-0208.
6. P. Wcisło et al., “New beyond-Voigt line-shape profile recommended for the HITRAN database,” Journal of Quantitative Spectroscopy and Radiative Transfer 347, 109596 (2025), doi:10.1016/j.jqsrt.2025.109596.
7. J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: A brief review,” Journal of Quantitative Spectroscopy and Radiative Transfer 17, 233–236 (1977), doi:10.1016/0022-4073(77)90161-3.
8. P. F. Bernath, Spectra of Atoms and Molecules, 5th ed., Oxford University Press (2025), doi:10.1093/oso/9780197754498.001.0001.
9. W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed., Springer (2014), doi:10.1007/978-3-642-53859-9.
10. B. H. Armstrong, “Spectrum line profiles: The Voigt function,” Journal of Quantitative Spectroscopy and Radiative Transfer 7, 61–88 (1967), doi:10.1016/0022-4073(67)90057-X.