# Spectral Lines, Width of

## 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 Δ*v _{ki}* of frequencies near the frequency

*v*of a radiative transition between discrete energy levels ℰ

_{ki}_{k}and ℰ

_{i}corresponds to every such transition; the transition frequency

*vM*= (ℰ

_{ki}_{k}- ℰ

*)*

_{i}*lh*= (ℰ

_{k}- ℰ

_{j})/2π

*hℏ*, where

*h*= 2

*πℏ*is Planck’s constant. The value of Δ

*v*gives the width of a spectral line, that is, the extent to which a given spectral line is nonmonochromatic. A spectral-line profile φ(

_{ki}*v*)—that is, the frequency dependence of the intensity of emission or absorption—usually has a maximum at or near the transition frequency

*v*(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 Δ

_{ki}*v*of a spectral line is given by the sum of the widths of the energy levels ℰ

_{ki}_{k}and ℰ

_{i}: Δ

*v*= (Δℰ

_{ki}_{k}– Δℰ

*)\*

_{i}*h*≈ (1/τ

*+ 1/τ*

_{k}*)/2π. In other words, the shorter the lifetimes τ*

_{i}_{k}and τ

_{i}at the energy levels, the greater the value of Δ

*v*.

_{ki}The natural width of a spectral line is given by the expression (Δ*v _{ki}*)

_{nat}= (

*A*+

_{k}*A*)/2π, where

_{i}*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.

### REFERENCES

Heitler, W.*Kvantovaia teoriia izlucheniia*. Moscow, 1956. (Translated from English.)

Sobel’man, I. I.

*Vvedenie v teoriiu atomnykh spektrov*, 2nd ed. Moscow, 1977.

M. A. EL’ASHEVICH