**6. Peak-shape function**

The peak shape is one of the most important parameters in Rietveld refinement due to its dependence on crystallite/domain size, stress/strain, defects/vacancies, source/geometry, slit-size/detector resolution and 2θ/hkl indices [55]. An accurate description of the shapes of the peaks in a powder pattern is critical to the success of a Rietveld refinement. Poor description can lead to unsatisfactory refinement results, false minima and divergence. Peak shape analysis/function is the most complex parameter in Rietveld refinement, with dimensions into the space of unattainable and non-realistic. It is therefore essential for a working algorithm to make some assumptions/compromises on peak shape and sometimes neglect the otherwise essential aspect of peak shape. For x-ray and constant wavelength neutron data, the use of pseudo-Voigt approximated peak function is widely used. The pseudo-Voigt function is essentially a combination of Lorentzian and Gaussian peak function in a linear mode [30, 31, 56–61].

Voigt function is mathematically defined as

$$V(\mathbf{x}, p, \boldsymbol{\uprho}) = \int\_{-\infty}^{+\infty} G(\mathbf{x}, p) L(\mathbf{x} - \boldsymbol{\uprho}', \boldsymbol{\uprho}) d\mathbf{x}' \tag{1}$$

Where *<sup>G</sup>* <sup>¼</sup> *<sup>e</sup>*�*x*2*=*2*p*<sup>2</sup> *<sup>p</sup>* ffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> is Gaussian function and *<sup>L</sup>* <sup>¼</sup> *<sup>y</sup> <sup>π</sup> <sup>x</sup>*2þ*y*<sup>2</sup> ð Þ*<sup>m</sup>* is Lorentzian function with m = 1 for symmetry.

The pseudo-Voigt function is described as

$$W\_p(\mathfrak{x}, f) = \eta L(\mathfrak{x}, f) + (1 - \eta)G(\mathfrak{x}, f) \tag{2}$$

With

$$0 < \eta < 1$$

*η* is the full width half maximum parameter and the ratio of Gaussian and Lorentzian functions *η=*ð Þ 1 � *n* determines the mixing of these functions.

The graphical representation of the pseudo-Voigt function with variable *η* is shown in **Figure 5**.

Pearson VII peak-shape function (**Figure 6**) is used alternatively where the exponent m (Eq. 1) varies differently, but the same trends in line shape are observed. Although the Gaussian and Lorentzian components of Voigt function can be devolved into meaningful physical interpretations of stress/strain, microstructure and line broadening effects, no such interpretation can be drawn from Pearson

**Figure 5.** *Pseudo-Voigt peak function (black) and variation of peak shape (color) with η.*

**Figure 6.** *Pearson VII peak function (black) and variation of peak shape (color) with m.*

VII function. Another advantage of pseudo-Voigt peak function over other functions is the separation of sample and instrument contributions.

$$I = I\_o \frac{\mathcal{Y}^{2m}}{\left(\mathcal{Y}^2 + (\infty - \mathcal{X}')^2\right)^m} \tag{3}$$

centre and Intensity offset. The property of these peak functions can be employed more efficiently by empirical evaluation of the peak shapes. Although both the functions provide similar results when the variation of peak shapes with 2θ is accounted.

*Variation of fitting between two different shapes of peaks (triangles and circles) using (a), (b) pseudo-Voigt*

Peak 1 Peak 2 pseudo-Voigt Fit Pearson VII Fit

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

*DOI: http://dx.doi.org/10.5772/intechopen.96065*

 **(b)**

The profile parameters include every detail that a structural model packs in, except (background, peak shape and FWHM). Although FWHM is considered a part of profile in XRD, it is necessarily a variant under peak shape function. Therefore most of the available programs for Rietveld refinement list it under profile section. Practically, Clubbing of the asymmetry parameter, preferred orientation parameter and FWHM together due to their interdependence makes more sense. The structural model which is available should be complete otherwise the calculated profile will significantly deviate. The incorrect profile parameters during refinement process generally leads to refinement of FWHM, peak asymmetry, zero shift, etc. In such cases, it is more prudent to use methods that are structure independent. Le-Bail [23, 39, 62–65], Pawley [66, 67] etc. are suitable for obtaining initial values of profile parameters and extract a list of integrated intensities. The integrated intensities can then be used to calculate electron scattering densities and possible structure determination. In addition, the initial parameters can be refined to obtain more agreeable profile parameters. The information like crystallite size, defect concentration, microstrain etc. which can be extracted from XRD are derived from the profile

Voigt and Pearson VII function can be visualized in **Figure 7(a)-(d)**.

*<sup>2</sup> 0.01*) of peak and base in both pseudo-

The variation in chi-square fitting (*χ*

*(red) and (c), (d) Pearson VII function (blue).*

**(c) (d)**

**7. Profile parameters**

**(a)**

**Figure 7.**

**261**

It is also imperative to point out that pseudo-Voigt peak fitting accounts for peak base asymmetry more rigorously while Pearson VII is more inclined towards the peak *Challenges in Rietveld Refinement and Structure Visualization in Ceramics DOI: http://dx.doi.org/10.5772/intechopen.96065*

**Figure 7.**

*Variation of fitting between two different shapes of peaks (triangles and circles) using (a), (b) pseudo-Voigt (red) and (c), (d) Pearson VII function (blue).*

centre and Intensity offset. The property of these peak functions can be employed more efficiently by empirical evaluation of the peak shapes. Although both the functions provide similar results when the variation of peak shapes with 2θ is accounted.

The variation in chi-square fitting (*χ <sup>2</sup> 0.01*) of peak and base in both pseudo-Voigt and Pearson VII function can be visualized in **Figure 7(a)-(d)**.

### **7. Profile parameters**

The profile parameters include every detail that a structural model packs in, except (background, peak shape and FWHM). Although FWHM is considered a part of profile in XRD, it is necessarily a variant under peak shape function. Therefore most of the available programs for Rietveld refinement list it under profile section. Practically, Clubbing of the asymmetry parameter, preferred orientation parameter and FWHM together due to their interdependence makes more sense. The structural model which is available should be complete otherwise the calculated profile will significantly deviate. The incorrect profile parameters during refinement process generally leads to refinement of FWHM, peak asymmetry, zero shift, etc. In such cases, it is more prudent to use methods that are structure independent. Le-Bail [23, 39, 62–65], Pawley [66, 67] etc. are suitable for obtaining initial values of profile parameters and extract a list of integrated intensities. The integrated intensities can then be used to calculate electron scattering densities and possible structure determination. In addition, the initial parameters can be refined to obtain more agreeable profile parameters. The information like crystallite size, defect concentration, microstrain etc. which can be extracted from XRD are derived from the profile

VII function. Another advantage of pseudo-Voigt peak function over other

*y*<sup>2</sup>*<sup>m</sup>*

It is also imperative to point out that pseudo-Voigt peak fitting accounts for peak base asymmetry more rigorously while Pearson VII is more inclined towards the peak

*<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> ð Þ<sup>2</sup> *<sup>m</sup>* (3)

functions is the separation of sample and instrument contributions.

*Pearson VII peak function (black) and variation of peak shape (color) with m.*

*Pseudo-Voigt peak function (black) and variation of peak shape (color) with η.*

**Figure 5.**

*Advanced Ceramic Materials*

**Figure 6.**

**260**

*I* ¼ *Io*

**8. Rietveld refinement: procedure and guidelines**

*DOI: http://dx.doi.org/10.5772/intechopen.96065*

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

**Figure 9.**

**263**

*and Bragg positions (black bars).*

After getting a complete structural model, suitable unit-cell parameters, the sufficient profile parameters and agreeable background, the Rietveld refinement of structural parameters can be started [68–71]. Refinement is usually done in sets of two to five cycles at a time but for effective refinements in simpler crystal systems hundreds of cycles will be required and thousands for complex systems. While the refinement is underway, we can monitor the progress either graphically or numerically [49]. While the numerical parameters can give us a statistical idea about the refinement, the most useful information about the profile fit is best seen graphically. However the parameter shifts are much more rigorously visualized numerically. Introduction of Reliability factors or R values by Rietveld enabled us to visualize profile fit between observed and calculated patterns more effective, although the graphical inspections still retain their superiority. The difference plot is also a good indication of the quality of profile fit, however the actual difference between the observed and calculated profiles and the origination of the deviation is not always quite evident from it. **Figure 9** shows the full profile Rietveld refinement of LaMnO3 (couple more example of Rietveld refinement are given at the end of the chapter) while **Figure 10** shows the observed and calculated profile for a certain peak along with difference plots, while the corresponding R values for the whole profile are listed in the table. The increased R-values can be due to insufficient structural model, or inaccurate profile parameter. The atomic coordinates and positional parameters can cause changes in relative peak intensities at both high and low angles. The refinement of all the parameters has to be either done simultaneously or in a particular order to avoid numerical and statistical errors. The order and the explanation of the various refinable parameters and reliability factors will be done

*Typical Rietveld refinement plot (LaMnO3) with observed (black circles), calculated (red), difference (blue)*

**Figure 8.**

*The interplay of peak width, shape and its effect on FWHM with (a) symmetric profile (b) asymmetric profile without significant peak shift and (c) asymmetric profile resulting in significant peak shifting.*

parameters. Although, the independence of profile parameters and peak shape is questionable due to their correlated nature, to make physical sense from the variation of either, the profile parameter needs a separate part in X-ray diffraction. The asymmetry in peak profile is another feature that arises from convolution of closely spaced multiple peaks. The two common and prevalent reasons for such conditions are; the evolution of microstrain and the defects. However the contributions from the instrument and the sample holder cannot be undermined. The interplay between FWHM, Peak position, shape and width is best visualized graphically (**Figure 8**).
