Peak shape function pseudo-Voigt

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.

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]. 

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. 

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. 

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

Pseudo-Voigt peak shape function with Howard s asymmetry, see Eq. 2.63. 

Pseudo-Voigt peak shape function. Asymmetry was changed to Finger, Cox and Jephcoat approximation, which better represents peak shapes measured on this powder diffractometer using Mo Ka radiation, see section 6.10. 

The peak shape function was a Thompson modified pseudo-Voigt. It is referred as No. 2 in GSAS (also see section 2.9.1 and relevant equations). 

Table 7.31. Coordinates of atoms and individual isotropic displacement parameters in the crystal structure of Gd5Si4 fully refined by Rietveld technique using the pseudo-Voigt peak shape function. The space group is Pnma. The unit cell dimensions are a = 7.4896(4), b = 14.7544(8), c = 7.7519(4) A. All sites are fully occupied. ...

Table 7.34. Atomic parameters in the crystal structure of Gd5Si2Ge2 fully refined by Rietveld technique using pseudo-Voigt peak shape function. The space group is PI 12i/a. The unit cell...

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 ... 

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. 

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 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. 

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 ... 

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. 

Figure 6.19. Full widths at half maximum (FWHM) and mixing parameters (p) of the pseudo-Voigt function used to approximate peak shapes in the x-ray Figure 6.16) and neutron Figure 6.18) powder diffraction patterns collected from the same CeRhQea powder.

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. 

The Thompson-Cox-Hastings function is often used to refine profiles with broad diffraction peaks because it is the more appropriate model for line-broadening analysis where the Lorentzian and Gaussian contributions for crystallite size and for microstrains are weighted. So in this case, the peak shape is simulated by the pseudo-Voigt function, which is a Unear combination of a Gaussian and a Lorentzian function (Table 8.5). 

The diffraction data were analysed using the Rietveld technique as implemented in the Fullprof Suite Program. Peak shape was described by a pseudo-Voigt function, and the background level was fitted with linear interpolation between a set of given points. 

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