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Spectral Line Broadening

Doppler (Gaussian) and pressure plus natural (Lorentzian) widths of a gas line, combined into the exact Voigt profile.

Model: Exact

Gaussian FWHM Δ_G

0.2265 cm⁻¹

= 6.790 GHz = 5.662 pm; v_p = 2039 m/s

Lorentzian FWHM Δ_L

0.2010 cm⁻¹

collisional 2γP = 0.2000 cm⁻¹; natural 0.001000 cm⁻¹ (τ = 5.309 ns)

Voigt FWHM (exact)

0.3525 cm⁻¹

= 10.57 GHz = 8.812 pm; mixed (Voigt) regime

Voigt FWHM approximations

0.3525 cm⁻¹

Olivero +0.0035 %; TCH 0.3517 (-0.23 %), η = 0.646; TCH profile within 1.3 % of the peak

The Voigt width is not Δ_G + Δ_L: widths of the same kind add (Lorentzians linearly, Gaussians in quadrature), but a Gaussian and a Lorentzian combine by convolution.

Line profiles

00.20.40.60.81.0-1.0-0.500.51.0Offset from line centre (cm⁻¹)Normalised intensity
  • Gaussian (Doppler), peak 1
  • Lorentzian (collisional + natural), peak 1
  • Voigt (exact), peak 1
  • TCH pseudo-Voigt, same scale

Δ_G = σ̃₀ √(8 k_B T ln 2 / (m c²)), v_p = √(2 k_B T / m)

Δ_L = 2γP + Δ_nat, Δν_nat = 1/(2πτ)

V(x) = Re w((x + iΔ_L/2) / (σ√2)) / (σ√(2π)), σ = Δ_G / (2√(2 ln 2))

Δ_V ≈ 0.5346 Δ_L + √(0.2166 Δ_L² + Δ_G²)

Thermal motion of a gas in equilibrium gives a Gaussian (inhomogeneous) line; the finite lifetime of the levels and collisions in the impact approximation give a Lorentzian (homogeneous) line. The Voigt profile is their exact convolution, computed here from the Faddeeva function w(z) (Weideman's 32-term expansion, ≈ 1e-13). The Olivero-Longbothum formula gives its width to 0.02 %; the Thompson-Cox-Hastings pseudo-Voigt mixes a Lorentzian and a Gaussian of one width f with weight η, which is cheap to fit but deviates by about 1 % of the peak. γ is the HWHM coefficient at the gas temperature (HITRAN lists it at 296 K, scaling as (296 K/T)ⁿ). Not modelled: Dicke narrowing, speed-dependent broadening, line mixing, and pressure shifts.

Model and references

Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers

Exact Voigt convolution (Faddeeva function, ≈ 1e-13) of a Maxwell-Doppler Gaussian and a Lorentzian from natural decay plus impact-approximation collisions, with γ entered at the gas temperature. Not modelled: Dicke narrowing, speed-dependent broadening, line mixing and pressure shifts.

  1. Demtröder W. (2014). Laser Spectroscopy 1: Basic Principles, 5th ed. Springer, ch. 3 (line widths and profiles).
  2. Olivero J. J., Longbothum R. L. (1977). Empirical fits to the Voigt line width: a brief review. J. Quant. Spectrosc. Radiat. Transfer 17, 233–236. doi:10.1016/0022-4073(77)90161-3
  3. Thompson P., Cox D. E., Hastings J. B. (1987). Rietveld refinement of Debye–Scherrer synchrotron X-ray data from Al₂O₃. J. Appl. Cryst. 20, 79–83 (pseudo-Voigt). doi:10.1107/S0021889887087090
  4. Weideman J. A. C. (1994). Computation of the complex error function. SIAM J. Numer. Anal. 31, 1497–1518. doi:10.1137/0731077
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