Pseudo-voigt profiles

The Voigt function is a convolution product ( ) between L and G. As the convolution is expensive from a computational point of view, the pseudo-Voigt form is more often used. The pseudo-Voigt is characterized by a mixing parameter r], representing the fraction of Lorentzian contribution, i.e. r] = 1(0) means pure Lorentzian (Gaussian) profile shape. Gaussian and Lorentzian breadths can be treated as independent parameters in some expressions. 

So far no hypotheses are required concerning the true shape of the peak profile. Flowever, in order to avoid or reduce the difficulties related to the overlapping of the peaks, the experimental noise, the resolution of the data and the separation peak-background, the approach most frequently used fits by means of a least squared method the diffraction peaks using some suitable functions that allow the analytical Fourier transform, as, for example, Voigt or pseudo-Voigt functions (4) which are the more often used. 

Another typical problem met in this kind of analysis is known as the hook effect . It is due to an overestimation of the background line to the detriment of the peak tails. As a consequence, the low order Fourier coefficients of the profile are underestimated. In the fitting procedure by pseudo-Voigt functions, this problem occurs if the Gauss content is so high that the second derivative of the Fourier coefficients is negative this is obviously physically impossible because it represents a probability density. 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

The complete powder XRD profile (either for an experimental pattern or a calculated pattern) is described in terms of the following components (1) the peak positions, (2) the background intensity distribution, (3) the peak widths, (4) the peak shapes, and (5) the peak intensities. The peak shape depends on characteristics of both the instrument and the sample, and different peak shape functions are appropriate under different circumstances. The most common peak shape for powder XRD is the pseudo-Voigt function, which represents a hybrid of Gaussian and Lorentzian character, although several other types of peak shape function may be applicable in different situations. These peak shape functions and the types of function commonly used to describe the 20-dependence of the peak width are described in detail elsewhere [22]. 

In these and in many other reports, the size analysis is based on a full profile fit determined by using predefined line profiles (Voigt, pseudo-Voigt) and an intensity fitting according to the Rietveldt method (McCusker et al., 1999). This procedure, which requires substantial effort in data analysis, is reliable, provided that no changes in sample geometry... 

In some advanced implementations of the modified pseudo-Voigt function, an asymmetric peak can be constructed as a convolution of a symmetric peak shape and a certain asymmetrization function, which can be either empirical or based on the real instrumental parameters. For example, as described in section 2.9.1, and using the Simpson s multi-term integration rule this convolution can be approximated using a sum of several (usually 3 or 5) symmetric Bragg peak profiles ... 

Figure 4.16. Observed and calculated intensity in the powder diffraction pattern of NiMn02(0H) after the completion of profile fitting employing the WinCSD program. Pseudo-Voigt function was employed and all present Bragg peaks were included in the fit. The box at the bottom shows the difference between the observed and calculated intensities using the scale identical to that on the plot of both y and...

Figure 6.38. The results of Le Bail s full pattern decomposition of the powder diffraction pattern of GdsGe. The discrepancies between the observed and calculated profiles are small and all residuals are low, indicating that the unit cell dimensions are accurately determined and that the chosen peak shape function (pseudo-Voigt) is a good choice for this experiment.

Of the functions with three parameters IL is best suited for X-ray reflections. For Rietveld analyses mostly the four-parameter pseudo-Voigt function is used, for pure profile fitting (without structure refinement) often the Pearson-Vll function is also used. 

Terms included in Equation (26) are the IP, which can be expressed by the FT of a pseudo-Voigt function (Tp ) obtained from a suitable line profile standard (Section 13.2.3), contributions from size ( / ), dislocations ( / )> faulting (Ahk + anti-phase domain boundaries (Aff f). Expressions for various... 

A parametric description of the IP can be obtained by modelling the pattern of a line profile standard with pseudo-Voigt functions (see Chapters 4 and 5 and Section 13.2.3 in this chapter) ... 

The method most commonly used consists of considering that each diffraction peak corresponds to two half-Pearson VIIs or two half pseudo-Voigts each one defined on one of the two sides of the peak s maximum. This is called a spht-Pearson VII or a split pseudo-Voigt . The parameters that characterize each of the parts of the profile are independent and the resulting overall function is dissymmetric. 

Figure 3.10. Dissymmetric fitting profile, use of a split pseudo-Voigt...

The shapes of the peaks change with the diffraction angle because, among other things, the deformation effects caused by the machine are functions of 0. Therefore, for this whole pattern refinement method it is neeessary to use a peak profile describing function with a mathematical expression that varies with 0. This condition is met, for example, by the pseudo-Voigts if parameter ti is a function of 0. 

Figure 3 Fitting of a part of the pattern of a sample of nanocrystalline ZnO, using pseudo-Voigt functions for modelling individual diffraction lines, CuKa, radiation, = 1.2%. The observed intensity data are plotted as circles and the calculated pattern is shown as a continuous line. The lower trace is the difference curve. The xlO scale expansion shows the fit in the line-profile tails.

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