Line shapes: Lorentzian or Voigt?

Lorentzian lines

A single, well-defined iron environment gives absorption lines with a Lorentzian shape. Fitting with Lorentzian lines assumes that every iron atom in a site has exactly the same hyperfine parameters. This works well for well-crystalline minerals, and it keeps the number of fitted parameters small.

Each line is described by the Lorentzian function

L (v) = \frac{A}{π} \cdot \frac{Γ / 2}{{(v - v_{0})}^{2} + {(Γ / 2)}^{2}}

$v$ is the velocity (mm/s), and $v_{0}$ the position of the line
$Γ$ is the linewidth (FWHM, mm/s), so $Γ / 2$ is the half width at half maximum
$A$ is the area of the line

A doublet is two of these lines and a sextet is six, with their positions set by the hyperfine parameters.

Why real lines are often broader

In natural and many synthetic samples, no two iron atoms sit in quite the same place. Substitution (for example aluminium in place of iron), vacancies, defects and small particle sizes give each atom a slightly different quadrupole splitting or hyperfine field. The spectrum is then the sum of many Lorentzian lines spread around an average value. If that spread follows a Gaussian distribution, the combined line has a Voigt shape.

Voigt-based fitting (VBF)

Voigt-based fitting, described by Rancourt and Ping (1991), describes each site by an average value and a Gaussian spread, σ. For a doublet the spread is in the quadrupole splitting. For a sextet it is in the hyperfine field.

In simplified form, a Voigt line is a Lorentzian smeared out by a Gaussian spread of line positions. Each small shift $u$ away from the average position contributes a Lorentzian, weighted by how common that shift is.

V (v) = \int G (u) L (v - u) d u

G (u) = \frac{1}{σ \sqrt{2 π}} e^{- u^{2} / 2 σ^{2}}

$L$ is the Lorentzian above, and $G$ a Gaussian of width $σ$
When $σ$ shrinks to zero the Voigt becomes a Lorentzian, and when $Γ$ shrinks to zero it becomes a Gaussian
In VBF, $σ$ is a spread in ΔE_Q or B_hf, which moves each line by a different amount, so the lines of a site are not all broadened equally

The integral has no simple closed form, so fitting programs, including MinSight, calculate it with a fast numerical approximation.

Compare the two models below.

A spread of hyperfine fields broadens the outer lines of a sextet more than the inner ones, because the outer lines move furthest when the field changes. This is typical of poorly crystalline or substituted goethite.

Choosing a model

Start with the simplest model that describes the spectrum, and look at the residuals (the difference between the data and the fit).
Use a distribution model such as VBF when the lines are clearly broad or uneven, or when a Lorentzian fit only works by adding extra sites that have no physical meaning.
More parameters will always fit a little better. Check that each one is justified and that the fitted values make physical sense.
Report which model you used. Parameters from different models are not always directly comparable.
For soils and sediments, Thompson and others (2025) give decision trees for choosing a fitting approach.

In MinSight

The fitting app offers Lorentzian, Squared Lorentzian and Voigt models. In the database, a site is simulated with VBF when the paper gives a σ value, and with Lorentzian lines otherwise.

Line shapes: Lorentzian or Voigt?

Lorentzian lines

Why real lines are often broader

Voigt-based fitting (VBF)

Choosing a model

Further reading