Pearson VII

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. 

Curve fitting is an important tool for obtaining band shape parameters and integrated areas. Spectroscopic bands are typically modeled as Lorenzian distributions in one extreme and Gaussian distributions in the other extreme [69]. Since many observable spectroscopic features lie in between, often due to instrument induced signal convolution, distributions such as the Voight and Pearson VII have been developed [70]. Many reviews of curve fitting procedures can be found in the literature [71]. 

The Pearson VII model contains four adjustable parameters and is particularly well suited for the curve fitting of large spectral windows containing numerous spectral features. The adjustable parameters a, p, q and v° correspond to the amplitude, line width, shape factor and band center respectively. As q —the band reduces to a Lorenzian distribution and as q approaches ca. 50, a more-or-less Gaussian distribution is obtained. If there are b bands in a data set and... 

Normalized Pearson VII Function for various m values. Note that m controls the shape and not the width of the peak. 

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

Figure 2.43. The example of using Pearson-VII function to fit experimental data (open circles) representing a single Bragg peak containing Kai and Ktt2 components.

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 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 ... 

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 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 4.14. Observed and ealculated intensity in the powder diffraction pattern of NiMn02(0H) after fitting using a Pearson-VII function. Downward facing dash-dotted arrows indicate the positions of six weak Bragg peaks not included in the fit.

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

Asymmetry Individual using split Pearson VII Common for all peaks... 

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... 

Pearson-VII all, plus individual isotropic displacement parameters and preferred orientation along [001] 6.07 8.19 3.02 16.9... 

Hence, we will continue the refinement and employ a different peak shape function. The use of the Pearson-VII function to represent peak shapes results in lower residuals (see rows 7 and 8 in Table 7.30). Nonetheless, individual isotropic parameters of Sil atoms remain unphysical and we may conclude that this is due to the low scattering power of Si and other errors present in the measured powder diffraction pattern. The errors were likely introduced during sample preparation, as it is easy to overlook... 

Figure 7.46. The observed and calculated powder diffraction patterns of Gd5Si4 after the completion of the refinement using Pearson-VII peak shape function. A total of 808 independent Bragg reflections are possible in the examined range of Bragg angles.

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,... 

The advantage of X-ray powder patterns over other spectra is the roughly common shape of the individual reflections (equal half width and equal shape of the flanks). Therefore, one can use peak search methods that presume a special peak shape. Sanchez (1991) reports a peak search algorithm for Gaussian peaks with an average half width 2D. This method can be easily adapted for Lorentzian peaks (y = A/[l + x — u)/b) ] with FWHM = 2b) or Pearson-VII peaks (y = A/[l + ((x- )/b) f with FWHM = 2b-7( 72-l)). X-ray peaks very often exhibit a peak shape with m between 1.5 and 2. 

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. )...

Pearson VII (P7) (Table 4.7a) With shape exponent m (or individual mt) ... 

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