Pseudo-Voigt

It is often convenient modeling the peak shape assuming some analytical functions [25]. The most commonly used functions are, at present, the Voigt and pseudo-Voigt functions, a combination of a Gaussian (G) and a Lore-ntzian (L) function centered at 20(y. An expression for Gaussian and Lorentzian contributions is ... 

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. 

In order to properly take into account the instrumental broadening, the function describing the peak shape must be considered. In the case of Lorentzian shape it is Psize = Pexp - instr while for Gaussian shape p = Pl -Pl tr- In the case of pseudo-Voigt function, Gaussian and Lorentzian contributions must be treated separately [39]. 

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. 

Lorentzians, Gaussians, and combinations of both like pseudo-Voigt functions 38Frequently the effect of instrumental broadening is tacitly considered as already eliminated. 

The presence of the central spot (the primary beam) and diffuse rings Idiff from the film support brings significant errors into estimated intensities. The shape of the primary beam feam can be approximated by one of several peak-shape functions such as pseudo-Voigt, Gaussian or Lorentzian [16], The diffuse background can be described by a polynomial function of order 12. Then equation (1) becomes... 

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]. 

Figure 1. Results of full pattern fitting (after [53]) of the PXD pattern of an acid-treated phase A sample, based on an orthorhombic unit cell (refined lattice constants a = 7.940(3)A, b = 10.338(4)A, c = 11.591(5)A). The low angle peaks are somewhat asymmetric and are hence not fitted well by the assumed symmetric pseudo-Voigt peak shape.

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... 

The most notable exception is the shape of peaks in neutron powder diffraction (apart from the time-of-flight data), which is typically close to the pure Gaussian distribution. Peak shapes in TOP experiments are usually deseribed by a convolution of exponential and pseudo-Voigt functions. 

The mixing coefficient, p for pseudo-Voigt function and the exponent, p for Pearson-VII function, generally vary for a particular powder diffraction pattern. Their behavior is typically modeled with a different empirical parabolic function of tan0 and 20, respectively, as follows from equations... 

In the modified pseudo-Voigt functions described above (Eqs. 2.54 to 2.58), both the Gaussian to Lorentzian mixing parameter (q, Eq. 2.54) and their individual contributions to the total peak width H, Eq. 2.55) are tabulated. This feature may be used to lower the number of free parameters and to obtain more realistic peak shape parameters that are due to the physical state of the specimen. Either or both may be achieved by using one of the following approaches ... 

From this point of view, some applications of the modified pseudo-Voigt function (e.g. third and fourth peak shape functions employed in GSAS) are in a way similar to the fundamental parameters approach as they use instrumental parameters to describe certain aspects of peak shape. 

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 ... 

Peak shape parameters, which include full width at half maximum H), asymmetry (a), and exponent (P) for Pearson-VII or mixing parameter (ti) for pseudo-Voigt functions. All peak shape parameters are typically refined for Kai reflections. The corresponding Ktt2 components are assumed to have H, a, P (or p) identical to Ka In some applications, peak shape parameters may be fixed at certain commonly observed values, or they may only be adjusted manually. 

It is worth noting, that when software on hand does not employ a Gauss peak shape function, it can be easily modeled by the pseudo-Voigt function using the fixed mixing parameter, t) = 1. 

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...

Peak shape function Pearson VII or split Pearson VII pseudo-Voigt... 

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