Pearson VII profile

A test series e.g. a raw file) can be viewed as a superposition of signals plus noise. Since a Fourier transform is additive, the frequency spectrum of a measurement is the sum (superposition) of the signal frequency spectrum and the noise frequency spectrum. If a reflection can be fitted by a Pearson VII profile (Chap. 4.7) and if the maximum of this even function is put in x = 0,... 

Figure 4.11 A series of Pearson VII profiles with equal peak position, peak height, and half width, but with different slope shapes (given by the exponent m), and area. For m= Lorentzian (L), m=1.5 intermediate Lorentzian (IL), m = 2 modified Lorentzian (ML). Already with m = 10 a Gaussian is approximated (exact at oo). X ray peaks mostly exhibit m values between 1.5 and 2. (After Howard and Preston, 1989. )...

Figure 4.14 Fitting of a measured, asymmetrical Si(l 11) reflection (points) by a Gaussian, a Lorentzian, and a split Pearson VII profile (full lines). Besides the not considered asymmetry (above and middle) the flanks above are too narrow (Gaussian) and in the middle too wide (Lorentzian. See also figure 4.17. (After Kern, 1992. )...

Figure 4.16 Patterns of the changes of half widths (top) and of slope parameters m (bottom) for split Pearson VII profiles (SP7), which were taken with a Gninier camera for a PbN03 sample. By such curves the number of profile parameters to be refined can appreciably be reduced (After Brown and Edmonds, 1980. )...

X-ray powder diffraction (XRD) was carried out using a Phillips MPD 1880 diffractometer, equipped with a Cu Ka source (A = 0.15418 nm), at 40 kV and 40 mA. The profiles were recorded at 0.02° (26) and step recording time of 5 s. The Ni(200) peak was fitted to a Pearson-VII profile shape function, as described elsewhere [5], with a residual error always lower than 1%. The method of Bertaut-Warren-Averbach (BWA) [11] was used to obtain CSDs and surface-average crystallite sizes, from which estimates of metal dispersion were obtained considering 0.065 nmVnickel atom and spherical crystallites [1]. Transmission electron microscopy (TEM) was carried out on a Jeol 200C working at 100 kV. The... 

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 profile functions applied are Gaussians, Lorentzians, Pearson VII, and other functions. Here, the Gaussian (G) and the Lorentzian (L) functions are described... 

In Eq. 2.61 a is a free variable, i.e. the asymmetry parameter, which is refined during profile fitting and z,- is the distance fi om the maximum of the symmetric peak to the corresponding point of the peak profile, i.e. z,-= 20yfc - 20 . This modification is applied separately to every individual Bragg peak, including Kaj and Ka2 components. Since Eq. 2.61 is a simple intensity multiplier, it may be easily incorporated into any of the peak shape functions considered above. Additionally, in the case of the Pearson-VII function, asymmetry may be treated differently. It works nearly identical to Eq. 2.61 and all variables have the same meaning as in this equation but the expression itself is different ... 

Figure 4.9. Observed (thick line) and calculated (thin line) intensity profiles in a fragment of the powder diffraction pattern of NiMn02(OH). The position of the missing peak is indicated by a downward facing arrow. Symmetrical Pearson-VII function with default peak shape parameters was used in this example.

Figure 4.15. Observed and calculated intensity in the powder diffraction pattern of NiMn02(0H) after the completion of profile fitting using the DMSNT program. A symmetrical Pearson-VII 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 same scale as on the plot of both and...

Figure 6.39. The results of Le Bail s full pattern decomposition of the powder diffraction pattern of Gd5Si4- 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 (Pearson-VII) is a good choice for this experiment. The observed data are available in the data files Ch6Exll MoKa.xy and Ch6Exll MoKa.dat on the CD.

Pearson-VII all profile, then unit cell, coordinate and overall displacement parameters, preferred orientation along [001] (parameters were released sequentially) 6.16 8.28 3.16 17.3... 

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. 

OrientExpress V3.3 freeware was used to index the Laue diffraction patterns. We developed a special algorithm to measure the sample-detector distance (d) and to determine the coordinates of the projection of the sample position in the recording plane of the CCD detector. The algorithm implies the analysis of two Laue diffraction patterns detected at two different distances dpi at the same angular position of the sample. A Pearson VII two-variable function was used to approximate the profiles of the Bragg reflections. 

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