Peak Shape Functions IV. Pseudo-Voigt and Other Functions

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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:

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 H_left for 2θ < 2θ₀ and at another H_right for 2θ > 2θ₀ . The degree of asymmetry can then be said to be measured by the magnitude of   H_left − H_right / (H_left + H_right ),   the sign indicating whether skewing is towards higher or lower angles.

One-dimensional Fourier Series

Fourier series are good at reproducing a given arbitrary shape provided sufficient terms in the series are taken. Once the series is determined it can be easily manipulated and convoluted in various mathematical operations. However this can be computationally expensive and therefore this approach is less useful for applications, such as structure refinement, where the peak shape needs to be computed many times over, however it is used widely for the study of microstructure.

Non-analytical Functions

A quite different approach is to dispense with the use of mathematical functions and instead set up a list of parameters (e.g. a generic look-up table) which describes the peak shape. The main features are:

• that no assumptions, mathematical or otherwise, are made about the peak shape
• the deficiencies associated with any given mathematical function are removed
• the shape cannot be manipulated mathematically
• the shape will appear noisy when the intensity is low (e.g. a weak peak)

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