---
title: Spectral-Line Formation and Broadening
draft: false
module: Radiation and Matter
moduleNumber: 3
lessonNumber: 3
order: 303
summary: >
  A spectral line is a bound-bound transition whose strength is set by an
  oscillator strength and whose shape is set by three broadening mechanisms: the
  Lorentzian natural and collisional wings, the Gaussian thermal Doppler core, and
  their Voigt convolution. Equivalent width measures the total absorption, and the
  curve of growth relates it to the number of absorbers through a linear,
  saturated, and damping regime, turning line strengths into abundances.
topics: [Radiation and Matter]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 9 — Stellar Atmospheres; §9.5 The Profiles of Spectral Lines; Ch. 5 §5.2"
  - book: Maoz
    ref: "Ch. 3 — Stellar Physics; Ch. 4 — Stellar Remnants (line diagnostics)"
---

A spectral line is the signature of a bound-bound transition: an electron jumping
between two discrete atomic levels absorbs or emits a photon at a definite
frequency. Its strength encodes the number of absorbing atoms and the quantum
probability of the transition; its shape encodes the physical conditions in the
gas, through the several mechanisms that spread the absorption over a finite band.
Reading a line means separating these two, and the curve of growth is the tool
that converts an observed line strength into an abundance.

## Bound-bound transitions and oscillator strength

An atom in level $i$ can absorb a photon of frequency $\nu_0 = (E_j - E_i)/h$ and
jump to level $j$. The probability of this happening is carried by the
**oscillator strength** $f_{ij}$, a dimensionless number that measures the
transition relative to a classical oscillating electron. The frequency-integrated
absorption cross section of the line is

$$
\int \sigma_\nu \, \d\nu = \frac{\pi e^2}{m_e c} \, f_{ij} ,
$$

in Gaussian units, with $e$ and $m_e$ the electron charge and mass.[^bob-lines]
Strong resonance lines have $f$ of order unity; forbidden lines have $f$ many
orders of magnitude smaller. The oscillator strength connects to the Einstein
coefficients that govern spontaneous and stimulated transitions, and it is the
atomic datum a line-analysis needs.

The cross section is not a spike at $\nu_0$; it is spread over frequency by a
normalized **profile function** $\phi(\nu)$ with $\int \phi(\nu)\,\d\nu = 1$, so
that

$$
\sigma_\nu = \frac{\pi e^2}{m_e c}\, f_{ij}\, \phi(\nu) .
$$

The remainder of the line's physics is the shape of $\phi(\nu)$.

## The natural profile

An excited level has a finite lifetime $\Delta t$ against radiative decay. By the
energy-time uncertainty relation, its energy is uncertain by
$\Delta E \sim \hbar / \Delta t$, so the emitted photon frequency is uncertain by
a corresponding amount. Modelling the radiating electron as a classical damped
oscillator gives a **Lorentzian** profile,

$$
\phi(\nu) = \frac{\gamma / 4\pi^2}{(\nu - \nu_0)^2 + (\gamma / 4\pi)^2} ,
$$

where $\gamma$ is the damping constant, the sum of the decay rates of the two
levels. The profile has a sharp core and broad wings that fall only as
$(\nu - \nu_0)^{-2}$, far more slowly than a Gaussian. Its full width at half
maximum in frequency is

$$
\Delta\nu_{\text{nat}} = \frac{\gamma}{2\pi} .
$$

Natural broadening is usually the smallest of the three mechanisms, but its wings
dominate far from line center because they decay so slowly.

## Doppler broadening

Atoms in a gas move with a thermal velocity distribution. An atom moving with
line-of-sight velocity $v$ absorbs at a Doppler-shifted frequency
$\nu = \nu_0(1 + v/c)$. Averaging the Maxwell-Boltzmann distribution of $v$ over
the line produces a **Gaussian** profile,

$$
\phi(\nu) = \frac{1}{\sqrt{\pi}\, \Delta\nu_D} \, \exp\!\left[ -\frac{(\nu - \nu_0)^2}{\Delta\nu_D^2} \right] ,
$$

with **Doppler width**

$$
\Delta\nu_D = \frac{\nu_0}{c}\sqrt{\frac{2kT}{m} + \xi^2} .
$$

Here $m$ is the atomic mass and $\xi$ is a **microturbulent** velocity added in
quadrature to represent small-scale mass motions beyond pure thermal agitation.
The Gaussian core is much wider than the natural core in a typical photosphere,
because $\sqrt{2kT/m}$ far exceeds the natural width, but its wings vanish
exponentially, so the Lorentzian wings win at large detuning. Heavier atoms have
narrower Doppler cores at fixed temperature, since $\Delta\nu_D \propto m^{-1/2}$.

## The Voigt profile

Both mechanisms act at once: each atom has a Lorentzian response, and the ensemble
of atoms is Doppler-shifted with a Gaussian spread. The observed profile is the
**convolution** of the two,

$$
\phi_V(\nu) = \int_{-\infty}^{\infty} \phi_{\text{Lorentz}}(\nu - \nu') \, \phi_{\text{Gauss}}(\nu') \, \d\nu' ,
$$

the **Voigt profile**. It has a Gaussian (Doppler) core and Lorentzian (natural
and collisional) wings, the two joining in a transition region. The relative
weight of core and wings is set by the ratio of the Lorentzian to the Gaussian
width; a small ratio gives a nearly Gaussian line, a large ratio a nearly
Lorentzian one.

$$
% caption: The Gaussian core falls off fastest, the Lorentzian has the broadest
% wings, and the Voigt profile combines a Gaussian core with Lorentzian wings by
% convolving the two.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {detuning from line center};
\draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {line shape};
% Gaussian (dashed) — narrow, fast decay
\draw[black, thick, densely dashed]
  (0.4,0.05) .. controls (2.8,0.1) and (3.6,3.7) .. (4.2,3.7)
  .. controls (4.8,3.7) and (5.6,0.1) .. (8.0,0.05);
\node[black, anchor=south east] at (3.6,3.4) {Gaussian};
% Lorentzian (dotted) — broad wings
\draw[black, thick, densely dotted]
  (0.4,0.55) .. controls (2.6,0.75) and (3.4,2.5) .. (4.2,2.5)
  .. controls (5.0,2.5) and (5.8,0.75) .. (8.0,0.55);
\node[black, anchor=south west] at (5.9,0.95) {Lorentzian};
% Voigt (solid accent) — Gaussian core, Lorentzian wings
\draw[acc, very thick]
  (0.4,0.4) .. controls (2.6,0.6) and (3.5,3.9) .. (4.2,3.9)
  .. controls (4.9,3.9) and (5.8,0.6) .. (8.0,0.4);
\node[acc, anchor=south west] at (4.3,3.7) {Voigt};
\end{tikzpicture}
$$

## Pressure broadening

Collisions with neighbouring particles perturb the energy levels of a radiating
atom, interrupting the phase of its emission. The result is again a Lorentzian,
with a width set by the collision rate rather than the radiative decay rate,

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

where $\tau_{\text{col}}$ is the mean time between collisions, $n$ the perturber
density, $\sigma$ the collision cross section, and $\langle v \rangle$ the mean
relative speed. **Pressure broadening** grows with density, so its strength is a
luminosity indicator: a dwarf, with a dense compact photosphere, shows broader
collisional wings than a giant of the same temperature, whose extended
low-gravity atmosphere is rarefied. This is the physical basis of the luminosity
classification of stellar spectra. The collisional and natural widths add,
because both produce a Lorentzian, so the damping constant of the Voigt wings is
$\gamma = \gamma_{\text{nat}} + \gamma_{\text{coll}}$.

## Equivalent width

The total strength of a line is measured by its **equivalent width** $W_\lambda$,
the width of a fully black rectangular band that removes the same flux from the
continuum,

$$
W_\lambda = \int \frac{F_c - F_\lambda}{F_c} \, \d\lambda ,
$$

with $F_c$ the continuum flux and $F_\lambda$ the flux in the line. The integrand
is the fractional depression of the spectrum, so $W_\lambda$ is the area of the
line normalized to the continuum, in wavelength units. It is independent of the
instrumental resolution, since smearing the line preserves its area, which makes
$W_\lambda$ the robust observable for abundance work.

$$
% caption: The equivalent width replaces the line profile by a black rectangle of
% the same area cut from the continuum; the shaded trough and the box remove equal
% flux.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {wavelength};
\draw[->, black] (0,0) -- (0,3.6) node[above, black!70] {normalized f\/lux};
% continuum level
\draw[black, densely dashed] (0.2,3.0) -- (8.0,3.0);
\node[black, anchor=south east] at (8.0,3.0) {continuum};
% line profile with shaded trough
\draw[acc, very thick] (0.4,3.0) -- (2.8,3.0)
  .. controls (3.3,3.0) and (3.4,0.7) .. (3.9,0.7)
  .. controls (4.4,0.7) and (4.5,3.0) .. (5.0,3.0) -- (8.0,3.0);
\fill[acc!14] (2.8,3.0) .. controls (3.3,3.0) and (3.4,0.7) .. (3.9,0.7)
  .. controls (4.4,0.7) and (4.5,3.0) .. (5.0,3.0) -- (5.0,3.0) -- (2.8,3.0);
% equal-area rectangle
\draw[black, thick] (6.0,3.0) rectangle (6.7,0.0);
\fill[black] (6.0,0.0) rectangle (6.7,3.0);
\node[black, anchor=south] at (6.35,3.55) {equal area};
\draw[black, <->] (6.0,-0.35) -- (6.7,-0.35);
\node[black, anchor=north] at (6.35,-0.4) {width W};
\end{tikzpicture}
$$

## The curve of growth

How $W_\lambda$ grows as more absorbing atoms are added defines the **curve of
growth**, plotted as $\log(W_\lambda / \lambda)$ against $\log(N f)$, the column
of absorbers times the oscillator strength. It has three regimes.[^maoz-cog]

- **Linear regime** (weak lines). Few absorbers, the line is optically thin at
  every frequency, and its depth is proportional to the number of atoms. The area,
  hence $W_\lambda$, grows in direct proportion, $W_\lambda \propto N f$, a slope
  of unity on the log-log plot.
- **Saturated regime** (flat part). The line core reaches zero flux and cannot
  deepen; adding atoms only widens the black core slightly, drawing on the
  Gaussian Doppler wings. Growth slows to $W_\lambda \propto \sqrt{\ln(N f)}$, a
  nearly flat plateau.
- **Damping regime** (strong lines). The Lorentzian wings, decaying only as
  $(\nu - \nu_0)^{-2}$, take over. Their area grows as the square root of the
  column, $W_\lambda \propto \sqrt{N f}$, giving a slope of one-half.

$$
% caption: The curve of growth relates equivalent width to the column of
% absorbers. A unit slope in the optically thin linear part flattens as the core
% saturates, then rises with slope one-half when the damping wings dominate.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {log column times f};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {log equivalent width};
% linear rise slope 1
\draw[acc, very thick] (0.4,0.5) -- (2.8,2.9);
\node[acc, anchor=south east, rotate=45] at (1.6,1.85) {slope 1};
% saturated plateau
\draw[acc, very thick] (2.8,2.9) .. controls (4.0,3.2) and (5.0,3.25) .. (5.6,3.35);
\node[black, anchor=south] at (4.3,3.35) {saturated};
% damping slope 1/2
\draw[acc, very thick] (5.6,3.35) -- (8.0,4.05);
\node[acc, anchor=north west] at (6.6,3.7) {slope one-half};
% regime dividers
\draw[black, densely dotted] (2.8,0) -- (2.8,2.9);
\draw[black, densely dotted] (5.6,0) -- (5.6,3.35);
\node[black, anchor=north] at (1.4,0.0) {linear};
\node[black, anchor=north] at (4.2,0.0) {f\/lat};
\node[black, anchor=north] at (6.8,0.0) {damping};
\end{tikzpicture}
$$

Reading an abundance is now direct. Measure $W_\lambda$ for a set of lines of
known $f$, place each on the theoretical curve of growth computed for the star's
temperature and pressure, and read off the column $N$ that reproduces the observed
equivalent widths. Weak lines on the linear part give the cleanest abundances,
since there $W_\lambda \propto N$ without saturation ambiguity; strong lines on
the damping part constrain the damping constant and the pressure instead.

## Rotational and turbulent broadening

Motions of the gas as a whole broaden lines without changing their equivalent
width, since they only redistribute the same absorption over a wider band. A
rotating star presents its approaching limb blueshifted and its receding limb
redshifted; integrated over the disk this convolves the line with a rotation
profile of half-width $v \sin i$, the projected equatorial velocity. The line
becomes shallower and broader while conserving area.

$$
% caption: Stellar rotation Doppler-shifts the approaching and receding limbs in
% opposite senses; summed over the disk the line broadens and shallows while its
% area, the equivalent width, is conserved.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% star disk with rotation
\draw[black, thick] (1.5,2.0) circle (1.3);
\draw[->, black] (1.5,3.5) arc (90:120:1.6);
\node[black, anchor=south] at (1.5,3.55) {rotation};
\draw[->, black, thick] (0.2,2.0) -- (0.7,2.0);
\node[black, anchor=north] at (0.4,1.95) {toward};
\draw[->, black, thick] (2.3,2.0) -- (2.8,2.0);
\node[black, anchor=north] at (2.6,1.95) {away};
% line profiles
\draw[->, black] (4.2,0.4) -- (8.4,0.4) node[right, black!70] {wavelength};
\draw[->, black] (4.2,0.4) -- (4.2,3.8) node[above, black!70] {f\/lux};
\draw[black, densely dashed] (4.4,3.4) -- (8.2,3.4);
% narrow deep (no rotation)
\draw[black, thick, densely dashed] (4.5,3.4) -- (6.0,3.4)
  .. controls (6.15,3.4) and (6.15,0.9) .. (6.25,0.9)
  .. controls (6.35,0.9) and (6.35,3.4) .. (6.5,3.4) -- (8.0,3.4);
% broad shallow (rotation)
\draw[acc, very thick] (4.5,3.4) -- (5.5,3.4)
  .. controls (6.0,3.4) and (6.0,1.9) .. (6.25,1.9)
  .. controls (6.5,1.9) and (6.5,3.4) .. (7.0,3.4) -- (8.0,3.4);
\node[acc, anchor=north] at (6.25,1.85) {rotating};
\end{tikzpicture}
$$

Macroturbulence, large-scale convective motions comparable to or larger than the
line-forming region, broadens lines similarly and is separated from rotation by
the different shape of its broadening kernel. The distinction between velocity
fields that conserve equivalent width (rotation, macroturbulence) and those folded
into the Doppler width that do not (microturbulence) is set by whether the
motion's length scale exceeds the photon mean free path.

With the line profile and its strength in hand, the missing quantity is the
continuous opacity that fixes the continuum against which lines are measured and
sets the optical-depth scale. The final lesson of this module treats the
microphysics of that opacity and its frequency average.

[^bob-lines]: Carroll & Ostlie, §9.5 and §5.2 — oscillator strengths, the natural, Doppler, and pressure-broadening mechanisms, and the Voigt profile.
[^maoz-cog]: Maoz, Ch. 3 — equivalent width, the curve of growth, and abundance determination from line strengths.