International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk 
International Tables for Crystallography (2018). Vol. H, ch. 3.1, pp. 238241
Section 3.1.5. Dataanalysis methods^{a}National Institute of Standards and Technology, Gaithersburg, Maryland, USA 
Dataanalysis procedures can range from the entirely nonphysical, using arbitrary analytical functions that have been observed to yield reasonable fits to the observation, to those that exclusively use model functions, derived to specifically represent the effect of some physical aspect of the experiment. The nonphysical methods serve to parameterize the performance of the instrument in a descriptive manner. The origins of two of the most common measures of instrument performance are illustrated in Fig. 3.1.25. The first is the difference between the apparent position, in 2θ, of the profile maximum and the position of the Bragg reflection computed from the certified lattice parameter. These data are plotted versus 2θ to yield a Δ(2θ) curve; a typical example is shown in Fig. 3.1.26. An illustration of the halfwidthathalfmaximum (HWHM), which is defined as the width of either the right or left half of the profile at one half the value of maximum intensity after background subtraction, is also shown in Fig. 3.1.25. These values can be summed to yield the FWHM, and plotted versus 2θ to yield an indication of the profile breadth as it varies with 2θ (Fig. 3.1.27). In addition, the left and right HWHM values of Fig. 3.1.28 gauge the variation of profile asymmetry with 2θ; additional parameters of interest, such as the degree of Lorentzian and Gaussian contribution to profile shape, can be plotted versus 2θ to describe the instrument and evaluate its performance.

Simulated and actual FWHM data from SRM 660b using the two Voigt PSFs with (`with Caglioti') and without constraints. 

Left and right HWHM data from SRM 660b using the split pseudoVoigt PSF fitted with uniform weighting. 
The least computationally intensive methods for the analysis of XRPD data, which have been available since the onset of automated powder diffraction, are based on first or secondderivative algorithms. These methods report peak positions as the 2θ value at which a local maximum in diffraction intensity is detected in the raw data. Typical software provides `tuning' parameters so that the operation of these algorithms can be optimized for the noise level, step width and profile width of the raw data. These methods are highly mature and offer a quick and reliable means of analysing data in a manner suitable for qualitative analysis and latticeparameter refinement. However, they only give information about the position of the top of the peak. Calibration of the diffractometer via this method is useful only for subsequent analyses that also use such peaklocation methods.
Profile fitting with an analytical profileshape function offers the potential for greater accuracy, because the entire profile is used in the analysis. As with the derivativebased methods, profile fitting also reports the observed 2θ position of maximum intensity, in addition to parameters describing profile shape and breadth. The discussion of the IPF in Section 3.1.1, as well as a quick look at Figs. 3.1.26–3.1.28, shows the complexity in the line profile shape from a Bragg–Brentano instrument. The profiles are symmetric only in a limited region of 2θ; in other regions, the degree and direction of profile asymmetry also vary as a function of 2θ. To a first approximation, the optics of an instrument contribute to the Gaussian nature of the profiles; this Gaussian nature will be constant with respect to 2θ. The Lorentzian contribution is primarily from the emission spectrum; given the dominance of angulardispersion effects at high angle, one can expect to see an increase in the Lorentzian character of the profiles with increasing 2θ. While it can be argued that it is physically valid to model specific contributions to the IPF with Gaussian and Lorentzian PSFs, either of these two analytical functions alone cannot be expected to fit the complexities of the IPF and yield useful results. Combinations of these two functions, however, using shape parameters that vary as a function of 2θ, have given credible results for fitting of data from the Bragg–Brentano diffractometer and have been widely incorporated into Rietveld structurerefinement software. The Voigt function is a convolution of a Gaussian with a Lorentzian, while the pseudoVoigt is the sum of the two. The parameters that are refined consist of an FWHM and shape parameter that indicates the ratio of the Gaussian to Lorentzian character. The Voigt, being a true convolution, is the more desirable PSF as it is more physically realistic; the pseudoVoigt tends to be favoured as it is less computationally intensive and the differences between the two PSFs have been demonstrated to be minimal (Hastings et al., 1984), although there is not universal agreement about this.
Refining the profile shapes independently invariably leads to errors when analysing patterns with peak overlap, as correlations occur between shape parameters of neighbouring profiles. This problem can be addressed by constraining the shape parameters to follow some functional form with respect to 2θ. Caglioti et al. (1958) developed such a function specifically for constantwavelength neutron powder diffractometers; it has been incorporated in many Rietveld codes for use with XRPD data. It constrains the FWHM of the Gaussian contribution to the Voigt or pseudoVoigt PSF:where the refineable parameters are U, V and W. The term U can be seen to correspond with microstrain broadening from the sample, and broadening due to the angulardispersion component of the IPF. In GSAS an additional term, GP, in 1/cos θ, is included to account for Gaussian size broadening. The Lorentzian FWHM in GSAS can vary aswhere LX and LY are the refineable parameters. Here LX varies with size broadening while LY is the Lorentzian microstrain and angulardispersion term. Given that the emission spectrum is described with Lorentzian profiles, we would expect the LY term to model the effects of angular dispersion. Within the code HighScore Plus, the Lorentzian contribution is allowed to vary aswhere γ_{1}, γ_{2}, and γ_{3} are the refineable parameters. Alternatives to the Caglioti function have been proposed that are arguably more appropriate for describing the FWHM data from a Bragg–Brentano instrument (Louër & Langford, 1988; Cheary & Cline, 1995). However, they have not yet been incorporated into many computer codes.
The asymmetry in the observed profiles can be fitted with the use of a split profile, where the two sides of the PSF are refined with independent shape and HWHM parameters. This approach will improve the quality of the fit to the observations; however, it is empirical in nature. The more physically valid approach is the use of models to account for the origins of profile asymmetry. The Finger et al. (1994) model for axial divergence has been widely implemented in various Rietveld codes. It is formulated to model the axialdivergence effects of a synchrotron powder diffraction experiment where the incident beam is essentially parallel. The two refineable parameters, S/L and H/L, refer to the ratios of sample and receivingslit length, relative to the goniometer radius; they define the level of axial divergence in the diffracted beam. This model is not in precise correspondence with the optics of a Bragg–Brentano diffractometer where both the incident and diffracted beams exhibit divergence in the axial direction. It does, however, give quality fits to such data. The use of such a model, as opposed to the sole use of a symmetric or split PSF, will yield peak positions and/or lattice parameters that are `corrected' for the effects of the aberration in question. Therefore, results from the use of model(s) cannot be directly compared with empirical methods that simply characterize the form of the observation. In the case of the Bragg–Brentano experiment, the correction that the Finger model applies is not rigorously correct. However, the impact of axial divergence, regardless of the details of diffractometer optics, is universal; as such the use of the Finger model results in a more accurate assessment of `true' peak position and, therefore, lattice parameters.
A third PSF that is in common use is the Pearson VII, or split Pearson VII, that was proposed by Hall et al. (1977) for fitting Xray line profiles. No a priori physical justification exists for the use of this PSF. The refineable parameters are the FWHM, or HWHM, and an exponent, m. The exponent can range from 1, approximating a Lorentzian PSF, to infinity, where the function tends to a Gaussian. Owing to the lack of a clear physical justification for use of this PSF, it is not often used in Rietveld analysis software.
Convolutionbased profile fitting, as shown in Fig. 3.1.4, was proposed by Klug and Alexander in 1954 (see Klug & Alexander, 1974) and much of the formalism of the aberration functions shown in Table 3.1.1 was developed by Wilson (1963). However, limitations in computing capability largely prevented the realization of the full fundamentalparameters approach method until 1992, with the work of Cheary & Coelho. This was made available to the community through the publicdomain programs Xfit, and later KoalaRiet (Cheary & Coelho, 1996) and more recently via TOPAS. Other FPA programs are available, most notably BGMN (Bergman et al., 1998); more recently, PDXL 2 has had some FPA models incorporated. Within the FPA there are no PSFs other than the Lorentzians used to describe the emission spectrum, the shapes of which are not typically refined. All other aspects of the observation are characterized with the use of model functions that yield parameters descriptive of the experiment. Plausibility of the analysis is determined through evaluation of these parameters with respect to known or expected values. Direct comparison of the results from an FPA to those from methods using analytical PSFs is difficult because of the fundamental difference in the output from the techniques; for example, FWHM values are not obtained directly from the FPA method. However, the NIST program FPAPC can be used to determine FWHM values numerically.
The FPA models of TOPAS, BGMN and PDXL 2 were developed specifically for the analysis of data from a laboratory diffractometer of Bragg–Brentano geometry. Analyses using this method would be expected to result in the lowest possible residual error terms that characterize the difference between calculation and observation. As has been discussed, the various aberrations affecting the diffraction line shape are such that the observed profile maxima do not necessarily correspond to the dspacing of the diffracting plane (hkl), except perhaps in a limited region of 2θ, emphasizing the need for physically valid modelling of the observed line shape to realize a credible value for the lattice parameter. At NIST, we are particularly interested in the capabilities of the FPA method, as one of the primary interests of the NIST Xray metrology program is obtaining the correct values for lattice parameters. Furthermore, experience has demonstrated that the refined parameters obtained through the use of FPA models can be used in a `feedback loop' to isolate problems and anomalies with the equipment.
The instrument response, i.e. the diffracted intensity as a function of 2θ, is measured by Rietveld analysis using models for intensitysensitive parameters such as crystalstructure parameters and Lorentz–polarization factors. The extraction of plausible crystalstructure parameters from standards via a Rietveld analysis serves as an effective and independently verifiable means of calibrating instrument performance. Considering these refined values provides an effective way to detect defects that vary smoothly over the full range of 2θ. However, errors that are only observable within limited regions of 2θ may be difficult to detect with a wholepattern method; these should be investigated with secondderivative or profilefitting methods. SRM 676a (alumina) is well suited to assessing instrument response because it is nonorienting and of high purity. Alumina is of lower symmetry than either silicon or lanthanum hexaboride; it has a considerable number of diffraction lines and has well established structure parameters. A Rietveld analysis of SRM 660c, however, yields the IPF in terms of codespecific profile shape terms and verifies that peakpositionspecific aspects of the equipment and analysis are working correctly.
The instrument response may be evaluated with the more conventional dataanalysis methods with use of SRM 1976b. Measurements of peak intensities are obtained from the test instrument, typically by profile fitting, and compared with the certified values. However, the use of SRM 1976b with diffraction equipment with different optical configurations may require the application of a bias to the certified values to render them appropriate for the machine to be qualified. This bias is needed to account for differences in the polarization effects from the presence, absence and character of crystal monochromators. The polarization factor for a diffractometer that is not equipped with a monochromator is (Guinier, 1994)The polarization factor for a diffractometer equipped with only an incidentbeam monochromator is (Azároff, 1955)where 2θ_{m} is the 2θ angle of diffraction for the monochromator crystal. The polarization factor for a diffractometer equipped with only a diffractedbeam postmonochromator is (Yao & Jinno, 1982)where 2θ_{m} is the 2θ angle of the monochromator crystal. Equations (3.1.5) and (3.1.6) are appropriate when the crystal has an ideal mosaic structure, i.e. the diffracting domains are uniformly small and, therefore, the crystal is diffracting in the kinematic limit. This is in contrast to a `perfect' crystal, which would diffract in accordance with dynamical scattering theory. Note that equations (3.1.5) and (3.1.6) both have the cos^{2} 2θ_{m} multiplier operating on the cos^{2} 2θ term. Since this multiplier is less than unity, the intensity change on machines equipped with a monochromator exhibits a weaker angular dependence.
The certification data for SRM 1976b were collected with the NIST machine equipped with the Johansson IBM and a scintillation detector. The simplified IPF of this machine is advantageous for the accurate fitting of the profiles and, therefore, intensity measurement. The validity of the `ideal mosaic' assumption embodied in equation (3.1.5) was evaluated using this diffractometer; the validity of equation (3.1.6) was evaluated with data from the machine configured with the postmonochromator. With respect to equation (3.1.5), for a Ge crystal (111) reflection, 2θ_{m} was set to 27.3°; with regard to equation (3.1.6), for a pyrolytic graphite crystal (0002) basalplane reflection, 2θ_{m} was set to 26.6°. Rietveld analyses of data from SRMs 660b, 1976b and 676a included a refinement of the polarization factor, modelled according to equations (3.1.5) and (3.1.6) in TOPAS, and yielded fits of high quality, indicating that these models were appropriate for these crystals and configurations. Equations (3.1.4), (3.1.5) and (3.1.6) were used to bias the certified values to correspond to those of alternative configurations. These values are included in the SRM 1976b CoA as ancillary data.
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