Spectral-Line Formation and Broadening

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

in Gaussian units, with $e$ and $m_e$ the electron charge and mass.¹ 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 $\varphi (\nu )$ with $\int \varphi (\nu )\mathbf{d}\nu =1$, so that

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

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 $\Delta E\sim \hslash /\Delta t$, so the emitted photon frequency is uncertain by a corresponding amount. Modelling the radiating electron as a classical damped oscillator gives a Lorentzian profile,

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

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,

with Doppler width

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,

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.

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,

where ${\tau }_{\text{col}}$ is the mean time between collisions, $n$ the perturber density, $\sigma$ the collision cross section, and $⟨v⟩$ 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,

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.

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 $\log (Nf)$, the column of absorbers times the oscillator strength. It has three regimes.²

• 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, $W_{\lambda }\propto Nf$, 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 $W_{\lambda }\propto \sqrt{\ln (Nf)}$, 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, $W_{\lambda }\propto \sqrt{Nf}$, giving a slope of one-half.

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.

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.