Peak shape function parameters

I. All possible variables (positions and shapes) are refined independently for each peak or with some constraints. For example, an asymmetry parameter is usually a variable, common for all peaks full width at half maximum or even all peak shape function parameters may be common for all peaks, especially if a relatively narrow range of Bragg angles is processed. When justified by the quality of data, an independent fit of all or most parameters produces best results. A major problem in this approach (i.e. all parameters are free and unconstrained) occurs when clusters of reflections include both strong and weak Bragg peaks. Then, peak shape parameters corresponding to weak Bragg peaks may become... 

The notations used in Eq. 6.6 are identical to Eq. 2.48. Individual integrated intensities are treated as free least squares parameters. Peak shape function parameters are represented as described in section 2.9, and Bragg peak positions, which affect the values of Xk, are established by the unit cell dimensions, see section 2.8. The background, where 1 < n... 

There is a variety of freely available software, which enables one to deconvolute a powder diffraction pattern and determine either or all individual intensities, lattice and peak shape function parameters, and observed structure factors of all possible Bragg reflections. Freeware codes include EXPO, FullProf, GSAS, LHPM-Rietica, and others. In addition to free programs, nearly all manufacturers of commercial powder diffractometers offer software for sale either as a package with the sale of the equipment or as stand-alone products. ... 

The normalized peak-shape function PS introduced by equation (1) must be determined in order to figure out the dependence of PS on several crystallite parameters, such as average size of crystallites, misorientation of crystallites in the sample etc. These parameters lead to a broadening of reflections, which must be taken into account. 

The shape of Bragg peaks is usually represented by a bell-like function -the so-called peak shape function. The latter is weakly dependent on the crystal structure and is the convolution of various individual functions, established by the instrumental parameters and to some extent by the properties of the specimen, see Table 2.7. The shape of each peak can be modeled using instrumental and specimen characteristics, although in reality ab initio modeling is difficult and most often it is performed using various empirically selected peak shape functions and parameters. If the radiation is not strictly monochromatic, i.e. when both Kai and Kaa components are present in the diffracted beam, the resultant peak should include contributions from both components as shown in Figure 2.39. 

In general, three different approaches to the description of peak shapes can be used. The first employs empirical peak shape functions, which fit the profile without attempting to associate their parameters with physical quantities. The second is a semi-empirical approach that describes instrumental and wavelength dispersion functions using empirical functions, while specimen properties are modeled using realistic physical parameters. In the third, the so-called fundamental parameters approach, all three components of the peak shape function (Eq. 2.45) are modeled using rational physical quantities. 

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 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 ... 

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. 

When peak shape functions and their parameters, including Bragg reflection positions, are known precisely and the background is modeled by a polynomial function with j coefficients, the solution of Eq. 6.6 is trivial because all equations are linear with respect to the unknowns (Bj, see Eq. 4.1, and / ). It facilitates the use of a linear least squares algorithm described in section 5.13.1. In practice, it is nearly always necessary to refine both peak shape and lattice parameters in addition to Bj and h to achieve a better precision of the resultant integrated intensities. Thus, a non-linear least squares minimization technique (see next section) is usually employed during full pattern decomposition using Eq. 6.6. 

The use of Mo Ka radiation shifts all diffraction peaks to lower Bragg angles and therefore, asymmetry effects are more severe than in the previous example, where Cu Ka radiation was used. As a result, the order in which parameters were refined was changed to avoid potential least squares instability problems. When all parameters were refined in essentially the same approximation as in the previous example (see row 5 in Table 6.9), the resultant figures of merit were satisfactory, but a carefiil analysis of Figure 6.15 indicates that the selected peak shape function does not adequately describe the observed peak shapes at low Bragg angles. 

Refinement of the individual isotropic parameters of all atoms yields a small negative 5 of Si 1. It is unfeasible that Nd atoms are statistically mixed in the same sites with Si because their volumes are too different ( 27 for Nd versus 1 K for Si). Given the density of the alloy, it is also impossible that all sites except this one are partially occupied. Therefore, the negative 5sii is likely due to the fact that Si atoms have only a fraction of the scattering ability of Nd atoms, and individual displacement parameters of the former cannot be reliably determined from this experiment. Another possible reason is the non-ideality of the selected peak shape function, or other small but unaccounted systematic errors. One of these is an unknown polarization constant of the employed monochromator (see Eq. 2.69). Another possibility is a more complex preferred orientation. As a result, the isotropic displacement parameters of two independent sites occupied by Si were constrained to be identical in a way, the Si atoms were refined in an overall isotropic approximation. 

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... 

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...

IL and ML represent already hybrid functions between G and L and simulate rather well the shape of a symmetrical X-ray reflection. If a variable slope width is introduced as a 4th parameter one gets the two following, widely used peak shape functions ... 

The Mossbauer parameters are derived from the peak parameters (base line parameters, peak position, peak width, and peak area/height) via the fitting process by computer evaluation of spectra in the case of the so-called model-dependent evaluation. In this case, an exact a priori knowledge about the spectrum decomposition (peak-shape function, number, and type of subspectra corresponding to the interactions assumed for each microenvironment in the model) is inevitably necessary. (Incorrectly chosen number of peaks renders the analysis itself incorrect.)... 

This likelihood function has to be maximized for the parameters in f. The maximization is to be done under a set of constraints. An important constraint is the knowledge of the peak-shapes. We assume that f is composed of many individual... 

Once MOD parameters are obtained a spectrum can be simulated with a suitable choice of the band-shape function f. We have used Gaussian and Lorentzian functions for this purpose. In either case, a width parameter must be chosen. This parameter is generally chosen such that the widths of the peaks in the simulated spectrum are similar to those in the observed spectrum. 

Spectral Manipulation Techniques. Many sophisticated software packages are now available for the manipulation of digitized spectra with both dedicated spectrometer minicomputers, as well as larger main - frame machines. Application of various mathematical techniques to FT-IR spectra is usually driven by the large widths of many bands of interest. Fourier self - deconvolution of bands, sometimes referred to as "resolution enhancement", has been found to be a valuable aid in the determination of peak location, at the expense of exact peak shape, in FT-IR spectra. This technique involves the application of a suitable apodization weighting function to the cosine Fourier transform of an absorption spectrum, and then recomputing the "deconvolved" spectrum, in which the widths of the individual bands are now narrowed to an extent which depends on the nature of the apodization function applied. Such manipulation does not truly change the "resolution" of the spectrum, which is a consequence of instrumental parameters, but can provide improved visual presentations of the spectra for study. 

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