Peak shape functions

The calculated intensity z Xi, yj) at any point x, y, of a dififaction pattern is expressed as a function of the integrated intensity h of the reflections contained in the pattern and a normalized analytical peak shape function PS x, y) is used to model the individual profiles. It is given by... 

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 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]. 

Although approximate peak positions could be obtained from many different types of software, programs which use peak shape functions to fit the powder diffraction peak profiles are more appropriate to determine the peak position more precisely. Some of the common programs are Xfit,29 TOPAS, etc. 

As the name indicates, fitting the complete powder diffraction profile as a function of intensities, background and a peak shape function is full-profile fitting. ... 

Figure 2.38. The appearance of the powder diffraction pattern a) - only Bragg peak positions (e.g. see Eq. 2.28) are represented by the vertical bars of equal length b) - in addition to peak positions, their intensities are indicated by using the bars with variable lengths (the higher the intensity, the longer the bar) c) - peak shapes have been introduced by convoluting individual intensities with appropriate peak shape functions, and a constant background has been indicated by the dash-double dotted line d) - the resultant powder diffraction pattern is the sum of all components shown separately in (c), i.e. discrete but partially overlapped peaks and continuous background.

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. 

Figure 2.39. The two individual peak shape functions corresponding to monochromatic Ka, and Ktt2 wavelengths (left) and the resulting combined peak shape function for a Ka /Ka2 doublet as the sum of two peaks (right). Since both Kai and Kaj peaks correspond to the same d hM, their positions, 0i and 02, are related as sin0iAKai = sin02/XKa2 (see Eq. 2.21), while their areas (intensities) are related as 2 to 1 (see Figure 2.5).

Finally, the resultant powder diffraction pattern is a sum of the individual peak shape functions and a background function as illustrated in Figure 2.38d, where the background function was assumed constant for clarity. 

The observed peak shapes are best described by the so-called peak shape function (PSF), which is a convolution of three different functions instrumental broadening, Q, wavelength dispersion. A, and specimen function, E. Thus, PSF can be represented as follows ... 

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. 

The four most commonly used empirical peak shape functions (y) are as follows ... 

The two simplest peak shape functions (Eqs. 2.49 and 2.50) represent Gaussian and Lorentzian distributions, respectively, of the intensity in the Bragg peak. They are compared in Figure 2.42, from which it is easy to see that the Lorentz function is sharp near its maximum but has long tails on each side near its base. On the other hand, the Gauss function has no tails at the base but has a rounded maximum. Both functions are centrosymmetric, i.e. G(x) = G -x) and L x) = L -x). 

Figure 2.42. The illustration of Gauss (dash-dotted line) and Lorentz (solid line) peak shape functions. Both functions have been normalized to result in identical definite integrals...

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. 

Since as5mimetry cannot be completely eliminated, it should be addressed in the profile fitting procedure. Generally, there are three ways of treating the asymmetry of Bragg peaks, all achieved by various modifications of the selected peak shape function ... 

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. 

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... 

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

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... 

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. 

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 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. 

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.

---