Widths and Profiles of Spectral Lines

Abstract

Spectral lines in discrete absorption or emission spectra are never strictly monochromatic. Even with the very high resolution of interferometers, one observes a spectral distribution I(ν) of the absorbed or emitted intensity around the central frequency ν ₀ = (E _i  − E _k )∕h corresponding to a molecular transition with the energy difference ΔE = E _i  − E _k between upper and lower levels. The function I(ν) in the vicinity of ν ₀ is called the line profile. The frequency interval δν = |ν ₂ − ν ₁| between the two frequencies ν ₁ and ν ₂ for which I(ν ₁) = I(ν ₂) = I(ν ₀)∕2 is the full-width at half-maximum of the line (FWHM), often shortened to the linewidth or halfwidth of the spectral line.

The halfwidth is sometimes written in terms of the angular frequency ω = 2 π ν with δω = 2 π δ ν, or in terms of the wavelength λ (in units of nm or Å) with δλ = |λ ₁ − λ ₂|. From λ = c∕ν, it follows that

$$\begin{aligned}\displaystyle\updelta\lambda=-(c/\nu^{2})\updelta\nu\;.\end{aligned}$$

(3.1)

The relative halfwidths, however, are the same in all three schemes:

$$\begin{aligned}\displaystyle\left|\frac{\updelta\nu}{\nu}\right|=\left|\frac{\updelta\omega}{\omega}\right|=\left|\frac{\updelta\lambda}{\lambda}\right|\;.\end{aligned}$$

(3.2)

The spectral region within the halfwidth is called the kernel of the line, the regions outside (ν < ν ₁ and ν > ν ₂) are the line wings.

In the following sections we discuss various origins of the finite linewidth. Several examples illustrate the order of magnitude of different line-broadening effects in different spectral regions and their importance for high-resolution spectroscopy [68, 69, 70, 71]. Following the usual convention we shall often use the angular frequency ω = 2 π ν to avoid factors of 2 π in the equations.