How it is calculated
Doppler broadening. Thermal motion along the line of sight shifts each emitter's frequency by \(\nu_0 v_z/c\). A Maxwell–Boltzmann velocity distribution turns this into a Gaussian line shape with full width at half maximum
Natural broadening. A state that decays with lifetime \(\tau\) has an energy uncertainty that gives a Lorentzian line with \(\Delta\nu_N = 1/(2\pi\tau)\). If the lower level also decays, the rates add: \(\Delta\nu_N = (1/\tau_u + 1/\tau_l)/2\pi\). Pressure (collisional) and Stark broadening are also mainly Lorentzian. Enter those as the additional FWHM, and the Lorentzian widths add linearly.
Voigt profile. When both mechanisms act, the line is the convolution of the Gaussian and the Lorentzian. Its FWHM is well approximated by Olivero and Longbothum's fit, which is accurate to about 0.02 %:
The calculator also computes the Voigt profile numerically and reports the FWHM it measures from that profile, as a cross-check. Widths convert between units with \(\Delta\lambda = \lambda_0^2\Delta\nu/c\) and \(\Delta\tilde\nu = \Delta\nu/c\).
Worked example
Hydrogen Balmer-α (656.28 nm) emitted from a 10 000 K gas: \(\Delta\lambda_D = 656.28\ \text{nm}\times 7.162\times10^{-7}\sqrt{10\,000/1.008} = 46.8\) pm, equal to 32.6 GHz or 1.087 cm⁻¹. A 10 ns lifetime adds only 15.9 MHz (0.023 pm) of natural width, so the line is almost purely Gaussian. Adding 10 pm of Stark broadening widens the Voigt FWHM to about 52.4 pm.
Assumptions and limits
- Thermal equilibrium. The emitters have a Maxwellian velocity distribution at a single temperature, with no bulk flow and no sight-line averaging over temperature gradients.
- Inhomogeneous vs. homogeneous. Doppler broadening is inhomogeneous; natural, pressure, and Stark broadening are treated as homogeneous Lorentzians. Real Stark profiles can be asymmetric and shifted.
- No hyperfine or isotope structure. Unresolved components can make a line appear wider than its Doppler width.
- Instrument function. A spectrometer's slit function is often closer to Gaussian or triangular. For a quick estimate, add a Gaussian instrument width in quadrature with \(\Delta\lambda_D\).
References
- W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed., Springer (2014) — widths and profiles of spectral lines.
- J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: A brief review,” J. Quant. Spectrosc. Radiat. Transfer 17, 233–236 (1977).