Peak-Shapes  –  Gaussian, Lorentzian, pseudo-Voigt

 

Overlay of 100% Gaussian peak-shape with 100% Lorentzian peak-shape

Overlay of Two Voigt peak-shapes with Gaussian peak-shape and Lorentzian peak-shape

Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.

 

The relevant parameters are:

  • Peak maximum (or peak height), denoted as Imax, is fairly self explanatory, however it should not be used as a measure of diffracted intensity.
  • Peak area is indicated (in yellow) as the area under the curve (and above the background should it not have been subtracted). The peak area is generally taken to be synonymous with “peak intensity” since the area represents the true sum of all the diffracted X-ray photons (or neutrons) that have been detected regardless of the peak shape.
  • Peak width is a measure of the broadness of the peak. For example at low angles a small value (e.g. 0.05°) would indicate a sharp peak whereas a large value (e.g. 0.2°) would indicate a broad peak. Several parameters can be used but the most common is the FWHM (often denoted as H) which stands for full width at half the maximum; to obtain this parameter, one draws as indicated a horizontal line at half the maximum peak height (0.5 Imax) and measures its full width residing within the peak bounds. An alternative parameter is the integral breadth, β, which is defined as the ratio of the first two parameters above (peak area/peak maximum). In some of the following discussion it is not important which definition is used and in these cases the term B will be used to denote either.
  • The tails represent the extremities at each side of the peak as it asymptotically approaches zero intensity (or the background if not subtracted); although they might not contribute very much to the total intensity (i.e. the peak area) they can provide a sensitive test of how well a peak is fitted by a mathematical function or model.
  • Peak asymmetry is related to the mirror plane (or lack of) that passes vertically through the peak maximum; if the LHS (left hand side) to this “mirror line” is an identical reflection of the RHS, the peak is said to be “symmetrical”; if the two sides are not reflections of each other, the peak is said to be “asymmetrical”. Peak asymmetry, for example, is particularly noticeable with powder patterns collected from instruments on neutron spallation sources.

Gaussian

The Gaussian function is possibly the best-known peak function in the whole of science since many physical and chemical processes are governed by Gaussian statistics. Translated into powder diffraction terms, the function for the intensity at any value of 2θ near the peak becomes:

I(2θ) = Imax exp [ − π (2θ − 2θ0)2 / β2 ]
where Imax is the peak intensity, 2θ0 is the 2θ position of the peak maximum, and the integral breadth, β, is related to the FWHM peak width, H, by β = 0.5 H (π / loge2)1/2. The most important features of the Gaussian function are:

  • that it is easy to calculate
  • it is a familiar and well-understood function
  • it is a good function to describe both neutron and energy-dispersive X-ray powder diffraction peaks (it is however not good at describing angle-dispersive X-ray diffraction peaks)
  • it has a convenient convolution property (see later)
  • it is symmetrical

Because the intensity of a peak is essentially the peak area, it is often convenient to normalise the above Gaussian function so the peak area is unity; i.e.

G = √ (4 loge2 / π) (1 / H) exp {− 4 loge2 (2θ − 2θ0 )2 / H2 }

 

Lorentzian (Cauchy)

The Lorentzian is also a well-used peak function with the form:
 

I(2θ) = w2

w2 + (2θ − 2θ0)2

where w is equal to half of the peak width (w = 0.5 H). The main features of the Lorentzian function are:

  • that it is also easy to calculate
  • that, relative to the Gaussian function, it emphasises the tails of the peak
  • its integral breadth β = π H / 2
  • it has a convenient convolution property (see later)
  • it is symmetrical
  • instrumental peak shapes are not normally Lorentzian except at high angles where wavelength dispersion is dominant

We note again that since peak intensity is identified with peak area, it is often convenient to also have a form of Lorentz function normalised so that the area is unity; i.e.

 

L = 2 / H π

1 + 4 (2θ &minus 2θ0)2 / H2

 

Combination Functions

These combine different functions in an attempt to get the “best of both worlds” as far as peak shape is concerned. The combination can be by convolution (e.g. the Voigt function) or by simple addition (e.g. pseudo-Voigt which is a close approximation to the Voigt function). For example the latter case could take the form:

I(2θ) = Ihkl [η L (2θ − 2θ0) + (1 − η) G (2θ − 2θ0) ]
where, respectively, L (2θ − 2θ0) and G (2θ − 2θ0) represent suitably normalised Lorentz and Gaussian functions, and η (the “Lorentz fraction”) and (1 − η) represent the fractions of each used. The main features of combination functions are:

  • that their precise form of combination can be tailored to a specific peak shape
  • The Voigt and pseudo-Voigt (together with the Pearson VII) are popular functions for modelling peak shapes
  • that their calculation tends to be more labour-intensive though as stated before this is not significant in computing terms

Split Functions

These are used when the peak is noticeably asymmetric. Basically a given function is doctored so that it has a parameter with one value for the left hand side and another value for the right hand side of the peak. For example with the Gaussian function, the peak width, through FWHM, could be set at one value Hleft for 2θ < 2θ0 and at another Hright for 2θ > 2θ0 . The degree of asymmetry can then be said to be measured by the magnitude of  Hleft − Hright / (Hleft + Hright ),   the sign indicating whether skewing is towards higher or lower angles.

 

 

A List of Line-Shapes  (N. Fairley)

The line-shapes offered in CasaXPS are based around the following fundamental functional forms.

The Voigt functional form has been the basis for most quantitative analysis of XPS spectra. Unfortunately, an analytic form for the convolution of a Gaussian with a Lorentzian is not available [3] and so practical systems have adopted two approximations to the true Voigt function.

Gaussian/Lorentzian Product Form

Gaussian/Lorentzian Sum Form

Exponential Asymmetric Blend Based upon Voigt-type Line-Shapes

Given either of the above Gaussian/Lorentzian symmetric line-shapes, an asymmetric profile is obtained from a blend function as follows.

Where

 

Alternative Asymmetric Line-Shapes

An asymmetric line-shape due to Ulrik Gelius (Uppsala, Sweden) offers a class of profiles by modifying the Voigt function via an ad hoc adjustment. The profile is given by

Where

and

.

The parameters a and b determine the shape of the asymmetric portion of the curve.

Doniach Sunjic: A Theoretically Based Asymmetric Line-Shape

Doniach Sunjic [1] performed an analysis for both photoemission and X-ray line-shapes, both of which result in an underlying profile given by the expression below. The formula includes an asymmetry parameter a that characterizes the asymmetry for a particular metal-like material. F is related to the FWHM and the position E is again related to but not equal to the position of the maximum intensity for the line-shape. It is therefore difficult to relate optimization parameters determined from the Doniach-Sunjic profile to similar quantities determined from Voigt-type line-shapes.

 Nevertheless, the Doniach-Sunjic profile offers an asymmetric shape that is particularly appropriate for non-monochromatic X-ray induced transitions; the profile is potentially present in both the photoemission process as well as the excitation source.

Line-Shapes Based upon Backgrounds

The above profiles are assumed to be entirely due to intrinsic electron energy variations, where a background subtraction algorithm is required before these line-shapes can be used to model the spectra. An alternative approach is to include the background shape as part of the model [4]. An analytic form for the Shirley background can be determined for each of the line-shapes and the sum of these backgrounds plus line-shapes is used to approximate the variation in the spectrum. A simple constant background is all that is required for this procedure although other forms for the background are still an option.

Castle et al [4] have developed a Shirley-type adjustment to a Voigt line-shape. The Shirley approximation is calculated from the current Gaussian/Lorentzian shape and a polynomial b0+b1(x-E)  is used to scale the background in order to provide a fit to the observed spectra. The procedure yields a “Kappa” parameter (given by b0) that characterizes the “intrinsic” step in the spectrum observed for a particular sample.