Curve type

2 br. Pseudo-Voigt

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

A pseudo-Voigt peak with an independent width and Gaussian-Lorentzian mix on each side of its maximum.

Left of x0 it uses sigma and eta, right of x0 it uses sigmaright and etaright; both sides reach the height A at x0, so the curve is continuous there. A is the peak height, not its area.

Giving each branch its own shape follows strongly asymmetric peaks more freely than Asym. Pseudo-Voigt, at the price of two more parameters.

Limitations

• Four shape parameters for one peak correlate strongly; on noisy data the branches can trade width for mix without improving the fit.
• The slope is not continuous at x0 when the two branches differ, which a physical profile would not show.

Asym. Pseudo-Voigt

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

A pseudo-Voigt peak made asymmetric by giving its two sides different widths, sigma + deltasigma on the right and sigma - deltasigma on the left.

Each side is (1 - eta) Gaussian + eta Lorentzian with the width of that side, both scaled to height A at x0. Unlike Pseudo-Voigt, A here is the peak height, not its area.

Splitting the width is an empirical way to follow an asymmetric peak; this software chose that form, it is not taken from a published profile function.

Limitations

• The asymmetry is descriptive only: it does not model a physical cause of asymmetry such as axial divergence.
• deltasigma must stay smaller than sigma, or the left side has no width.
• A is a height, so its value is not comparable with the area-normalised types.

Doniach-Sunjic

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

An asymmetric photoemission line of metals: a Lorentzian of half-width sigma skewed by the singularity index alpha.

A cos(pi alpha / 2 + (1 - alpha) arctan((x - x0) / sigma)) / (sigma^2 + (x - x0)^2)^((1 - alpha) / 2). alpha = 0 gives a Lorentzian; larger alpha gives a heavier tail on one side.

It describes core-level X-ray photoemission lines of metals, where screening by conduction electrons makes the line asymmetric.

I(E) = cos(pi alpha / 2 + (1 - alpha) arctan(E / gamma)) / (E^2 + gamma^2)^((1 - alpha) / 2)

Limitations

• For alpha > 0 the integral of the line diverges, so a peak area is not defined without a cut-off.
• In practice the line is convolved with a Gaussian instrument function; that convolution is not part of this type.

Sources

• S. Doniach and M. Sunjic, Many-electron singularity in X-ray photoemission and X-ray line spectra from metals, J. Phys. C 3 (1970), pp. 285-291

Exponentially Modified Gaussian

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

A Gaussian convolved with a one-sided exponential decay, giving a peak with a tail of time constant tau.

A Gaussian of standard deviation sigma centred at x0, convolved with an exponential of time constant tau, and scaled by A.

It is the classic model of chromatographic peak tailing, where the detector or column adds an exponential lag to a Gaussian band.

f(t) = Gaussian(t; t_G, sigma) convolved with exp(-t / tau) / tau

Limitations

• The tail is on one side only; a peak fronting on the other side needs a different model.
• x0 is the centre of the underlying Gaussian, not the position of the maximum, which moves toward the tail as tau grows.

Sources

• E. Grushka, Characterization of exponentially modified Gaussian peaks in chromatography, Anal. Chem. 44 (1972), pp. 1733-1738

Gaussian

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

A symmetric bell-shaped peak whose area is A, centred at x0, with standard deviation sigma.

The normal (Gaussian) density scaled by A: A / (sigma sqrt(2 pi)) exp(-(x - x0)^2 / (2 sigma^2)). A is therefore the area under the peak, not its height; the height is A / (sigma sqrt(2 pi)).

x0 is the position of the maximum and sigma the standard deviation; the full width at half maximum is 2 sqrt(2 ln 2) sigma, about 2.355 sigma.

It is the shape of broadening made of many small independent contributions, such as instrumental resolution.

f(x) = exp(-(x - mu)^2 / (2 sigma^2)) / (sigma sqrt(2 pi))

Limitations

• Its tails fall off very quickly, so it underestimates the wings of a peak that also has lifetime or size (Lorentzian) broadening - use Voigt or Pseudo-Voigt there.
• It is symmetric; an asymmetric peak fitted with it leaves a systematic residual on one side.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1, Normal Distribution https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm

Lorentzian

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

A symmetric peak with heavy tails, whose area is A, centred at x0, with full width at half maximum sigma.

The Cauchy (Lorentzian) density scaled by A: A (2 / (pi sigma)) / (1 + (2 (x - x0) / sigma)^2). A is the area and sigma here is the full width at half maximum, not a standard deviation.

It is the natural line shape of a damped oscillator and of lifetime broadening, and in diffraction the shape small crystallite size tends to give.

f(x) = 1 / (pi s (1 + ((x - t) / s)^2))

Limitations

• Its tails decay so slowly that its variance is undefined; the fitted area depends noticeably on how much of the tail the fit interval includes.
• Real peaks rarely have pure Lorentzian wings and a Gaussian core together - use Voigt or Pseudo-Voigt for a mixture.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.3, Cauchy Distribution https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm

Moffat

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

A symmetric peak of height A with core width sigma and power-law wings set by m, used for stellar images.

A / (1 + ((x - x0) / sigma)^2)^m. m = 1 is a Lorentzian; larger m gives weaker wings.

It was introduced for the point-spread function of stars, whose wings are heavier than a Gaussian predicts.

I(r) = I0 (1 + (r / alpha)^2)^(-beta)

Limitations

• sigma is a core scale, not the full width at half maximum; the half width depends on m as well.
• A is the height, not the area.

Sources

• A. F. J. Moffat, A theoretical investigation of focal stellar images in the photographic emulsion and application to photographic photometry, Astron. Astrophys. 3 (1969), pp. 455-461

Pearson VII

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

A symmetric peak of height A and full width at half maximum sigma, whose tail weight m runs from Lorentzian to Gaussian.

A / (1 + (2^(1/m) - 1) (2 (x - x0) / sigma)^2)^m. With this scaling sigma is exactly the full width at half maximum for every m.

m = 1 gives a Lorentzian; as m grows the wings shrink and the shape tends to a Gaussian. It is widely used for X-ray diffraction peaks.

y = A / (1 + (2^(1/m) - 1) (2 (x - x0) / w)^2)^m

Limitations

• A is the height, not the area.
• Very large m is numerically indistinguishable from a Gaussian, so m is poorly determined for nearly Gaussian peaks.

Sources

• M. M. Hall, V. G. Veeraraghavan, H. Rubin and P. G. Winchell, The approximation of symmetric X-ray peaks by Pearson type VII distributions, J. Appl. Cryst. 10 (1977), pp. 66-68

Pseudo-Voigt

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

A weighted sum of a Gaussian and a Lorentzian with one shared width, used as a fast stand-in for the Voigt profile.

A ((1 - eta) G(x) + eta L(x)), where G and L are the area-normalised Gaussian and Lorentzian with the same full width at half maximum sigma. A is the area.

eta, between 0 and 1, sets the mix: 0 is a pure Gaussian and 1 a pure Lorentzian. It is the standard peak shape of powder-diffraction profile fitting.

pV = eta L + (1 - eta) G

Limitations

• It approximates the Voigt profile rather than computing it; the approximation is close but not exact, so eta has no strict physical meaning.
• Both components share one width, so it cannot represent a peak whose Gaussian and Lorentzian widths differ strongly - use Voigt for that.
• It is symmetric; see Asym. Pseudo-Voigt and 2 br. Pseudo-Voigt for asymmetric peaks.

Sources

• P. Thompson, D. E. Cox and J. B. Hastings, Rietveld refinement of Debye-Scherrer synchrotron X-ray data from Al2O3, J. Appl. Cryst. 20 (1987), pp. 79-83

Skewed Gaussian

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

A Gaussian made asymmetric by the shape parameter beta, following the skew-normal distribution.

A Gaussian density of scale sigma at x0, multiplied by (1 + erf(beta (x - x0) / (sigma sqrt 2))) and scaled by A. beta = 0 gives the ordinary Gaussian; positive beta skews the peak to the right.

It adds asymmetry with one parameter while keeping Gaussian tails.

f(x) = 2 phi(x) Phi(alpha x)

Limitations

• x0 is the location parameter, not the position of the maximum, once beta is not zero.
• The tails stay Gaussian, so it cannot represent a long exponential tail - use EMG for that.

Sources

• A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171-178

Step (erf)

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

A smooth step of height A centred at x0, whose sharpness is set by sigma.

(A / 2) (1 + erf((x - x0) / (sigma sqrt 2))): the integral of a Gaussian, rising from 0 to A. sigma is the width of the transition.

Use it for an edge or a change of level rather than a peak, for example an absorption edge or a baseline shift.

erf z = (2 / sqrt(pi)) integral from 0 to z of exp(-t^2) dt

Limitations

• It is not a peak, so automatic peak finding does not place it sensibly; place it by hand.
• It is currently shown on the diffraction angle axis, because it inherits that from the formula base class; that is a known framework coupling, not a property of a step.

Sources

• NIST Digital Library of Mathematical Functions, section 7.2, Error Functions https://dlmf.nist.gov/7.2

User Defined

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

A curve whose formula you type yourself, with parameters named in that formula.

The expression is parsed and evaluated as written; every name in it other than x becomes a parameter the fit can vary.

Use it for a model the built-in types do not provide, without writing a module.

Limitations

• Nothing checks that the formula means something physical, or that its parameters are identifiable from the data.
• A parameter that does not appear in the expression cannot vary, so the fit silently holds it at its starting value.
• The expression is interpreted rather than compiled, so it is slower than a built-in type.

Voigt

Canonical — Stated by the field's authoritative sources, which the explanation quotes and cites.

The convolution of a Gaussian of width sigma with a Lorentzian of half-width gamma, scaled by A.

The exact profile of a peak broadened by two independent mechanisms at once: a Gaussian one (instrument, strain) with standard deviation sigma and a Lorentzian one (lifetime, size) with half-width gamma.

It is what Pseudo-Voigt approximates; use it when the two widths matter separately.

U(x, t) + i V(x, t) = sqrt(pi / (4 t)) exp(z^2) erfc(z), z = (1 - i x) / (2 sqrt(t))

Limitations

• It is more expensive to evaluate than Pseudo-Voigt.
• When one mechanism dominates, sigma and gamma become strongly correlated and the smaller one is poorly determined.

Sources

• NIST Digital Library of Mathematical Functions, section 7.19, Voigt Functions https://dlmf.nist.gov/7.19