Spectral Lines, Width of

Spectral lines in optical spectra of atoms, molecules, and other quantum systems are characterized by a range of frequencies v or a range of wavelengths λ = c/v, where c is the speed of light. Such a frequency or wavelength range is called the width of spectral lines.

A certain range Δv_ki of frequencies near the frequency v_ki of a radiative transition between discrete energy levels ℰ_k and ℰ_i corresponds to every such transition; the transition frequency vM_ki = (ℰ_k - ℰ_i)lh = (ℰ_k - ℰ_j)/2πhℏ, where h = 2πℏ is Planck’s constant. The value of Δv_ki gives the width of a spectral line, that is, the extent to which a given spectral line is nonmonochromatic. A spectral-line profile φ(v)—that is, the frequency dependence of the intensity of emission or absorption—usually has a maximum at or near the transition frequency v_ki (see Figure 1). The frequency range between points where the intensity falls to half the maximum intensity is taken as the width of a spectral line. Hence, the width of a spectral line is often referred to as the half-width of a spectral line. If the Doppler effect is not taken into account, the width Δv_ki of a spectral line is given by the sum of the widths of the energy levels ℰ_k and ℰ_i: Δv_ki = (Δℰ_k – Δℰ_i)\\h ≈ (1/τ_k + 1/τ_i)/2π. In other words, the shorter the lifetimes τ_k and τ_i at the energy levels, the greater the value of Δv_ki.

Figure 1. A symmetrical spectral-line profile. The maximum emission intensity φ(v) corresponds to the frequency v_ki; the width of the spectral line (Δv_ki) is equal to the frequency range between the points where the intensity falls to half the maximum intensity.

The natural width of a spectral line is given by the expression (Δv_ki)_nat = (A_k + A_i)/2π, where A_k and A_i are the total probabilities of spontaneous transitions from the levels ℰ_k and ℰ_i to all lower-lying levels. The natural width is very small.

For atoms and molecules, the width of spectral lines is governed mainly by the broadening of the energy levels of the atoms or molecules during interactions with surrounding particles and by the broadening of the spectral lines as a result of the Doppler effect. In a gas or a plasma, the energy levels of the particles are broadened during collisions. Depending on the type of broadening, a symmetrical or an asymmetrical spectral-line profile is produced. Figure 1 shows a symmetrical profile, which is characteristic of radiation broadening.