International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 
International Tables for Crystallography (2018). Vol. H, ch. 3.9, pp. 358359
Section 3.9.6.2. Modelling of structural disorder^{a}CSIRO Mineral Resources, Private Bag 10, Clayton South 3169, Victoria, Australia,^{b}TU Bergakademie Freiberg, Institut für Mineralogie, Brennhausgasse 14, Freiberg, D09596, Germany, and ^{c}Bruker AXS GmbH, Oestliche Rheinbrückenstr. 49, 76187 Karlsruhe, Germany 
One major challenge for QPA is the treatment of stacking disorder. An alternative to the use of calibrated models is to develop extended structure models that more effectively represent the phases present in the sample than the simple structure models. Stacking disorder occurs in layered structures where longrange order is present within the layers but there is only partial or even no relationship from one layer to another. It is a commonly occurring type of microstructure and is of great interest in various fields including mineralogy and material science.
The most common types of stacking faults in lamellar structures are:
Mixedlayer (interstratified) systems contain different types of layers in a single stack, hence it is necessary to distinguish these from the types above. In this case, the layer types have different basal spacings and atomic coordinates (for example, illite–smectite interstratifications; Reynolds & Hower, 1970). Combinations of several of these types of disorder frequently occur in natural clay minerals. Intricate structural analysis using modelling techniques can give a reliable picture of the disorder of selected pure clay minerals, but such information is difficult to obtain from multiphase samples. Therefore, the type and degree of disorder of the components in natural rocks is one of the major unknowns when starting a quantitative analysis of such samples. The field of clay mineralogy represents a discipline where QPA has a long tradition, but has struggled with issues arising from a wide variety of disorder types. This complexity has led practitioners away from the use of crystallographic models and encouraged modification of the classical methods of quantitative analysis to incorporate empirical, calibrationbased techniques such as those described earlier in this section.
An alternative approach is the application of a robust mathematical description of the observed features in the diffraction pattern, thus minimizing their impact on the QPA. In QPA, the existence of disorder contributes to inaccuracy through line broadening and shifting, which results in difficulties in the extraction of integral intensities or scale factors. A range of tools for the modelling of diffraction patterns of disordered layer structures has existed since the middle of the last century (Hendricks & Teller, 1942; Warren, 1941); these have been summarized by Drits & Tchoubar (1990).
In clay mineralogy, highly oriented samples are used for phase identification and characterization. Onedimensional diffraction patterns are collected initially from these, commonly airdried, oriented samples and contain the information along c* that is characteristic of the type, composition and sequence of the layers comprising the clay. Based on this information, the clay minerals are classified into layer types, a classification which is a precursor to more precise identification of mineral species. Diffraction patterns are often collected again following various treatments of the oriented samples (e.g. solvation with ethylene glycol, heating to predetermined temperatures for specified times, wetting and drying cycles). Changes in peak positions, shapes and intensities between treatments are also diagnostic for identification of the clay mineral type present.
From a mathematical point of view, the onedimensional calculation of intensities is much less laborious than a threedimensional one, because only z coordinates are used and a–b translations and rotations are not considered. In 1985 Reynolds introduced the software package NEWMOD for the simulation of onedimensional diffraction patterns for the study of interstratified systems of two clay minerals (Reynolds, 1985). This simulation was based upon a suite of parameters including instrumental, chemical and structural factors, and has been widely applied to the QPA of interstratified clays via the `patternmixing' approach. An updated version (NEWMOD+; Yuan & Bish, 2010) has since been developed that incorporates improvements in claystructure modelling, an improved GUI and the calculation of various fitting parameters that improve the operator's ability to estimate the quality of the profile fit.
The principal drawback of onedimensional pattern approaches to QPA is that they are limited to the quantification of the ratio of layered structures only. Other minerals within the sample cannot be quantified at the same time. The degree of preferred orientation achieved in the oriented specimens may also differ between the mineral species present depending upon the method of sample preparation (Lippmann, 1970; Taylor & Norrish, 1966; Zevin & Viaene, 1990). This will affect the intensities of the observed peaks, which in turn affects the modelling of the relative proportions of the constituent minerals (Dohrmann et al., 2009; Reynolds, 1989). Therefore, the quantification of minerals from severely oriented samples such as these is frequently inaccurate, as existing correction models are unable to describe the intensity aberrations adequately (Reynolds, 1989).
Quantification of clay minerals within multiphase specimens requires the modelling of the threedimensional pattern of the randomly ordered clay. There are a number of approaches incorporated in various software packages for the calculation of these threedimensional diffraction patterns of disordered structures. WILDFIRE (Reynolds, 1994) calculates threedimensional diffraction patterns of randomly oriented illite and illite–smectite powders with various types and quantities of rotational disorder. This is limited, however, to specific mineral types (the procedure has provided much information about the structural disorder of illite, for example) and is computationally demanding. Another approach is the general recursive method of Treacy et al. (1991), which simulates diffraction effects from any crystal with stacking disorder. This uses the intensity calculations of Hendricks & Teller (1942) and Cowley (1976) along with Michalski's recurrence relations describing disorder (Michalski, 1988; Michalski et al., 1988). The calculation process for this method is less time consuming than that of WILDFIRE, but has the drawback of requiring the user to define the complete stacking sequence including stackingtransition probabilities and interlayer vectors. The original software for this method, DIFFAX (Treacy et al., 1991), was extended by a refinement algorithm to DIFFAX+ (Leoni et al., 2004) and FAULTS (CasasCabanas et al., 2006), but multiphase analysis is not possible within either package.
The application of Rietveldbased methods is widespread with many industrial applications, but their application to samples containing disordered materials is not yet routine. As the classical Rietveld method is based on the calculation of intensity for discrete reflections, the question of how the diffraction patterns of disordered phases may be modelled arises.
In principle, every atomic arrangement can be described in the space group P1 if the cell parameters are sufficiently large and a reflectionintensity calculation using the Rietveld method could then be performed. But the absence of symmetry in such `large cell' models makes them inflexible, and parameters describing probabilities of translational and rotational stacking faults and layertype stacking may not be directly included and refined. Nevertheless, some applications of such externally generated, largecell structures in Rietveld phase analysis have been published; for example the phase analysis of montmorillonite (Gualtieri et al., 2001).
The use of small, ideal cells in a traditional Rietveld approach for the calculation of diffraction patterns is hampered by the fact that the number of reflections generated by such models is insufficient to fit the asymmetric peak shapes of disordered layer structures. Standard anisotropic linebroadening models exist, such as ellipsoids (Le Bail & Jouanneaux, 1997), spherical harmonics (Popa, 1998) or the distribution of lattice metric parameters (Stephens, 1999), but these are typically unable to fit the patterns of disordered layered structures. They may also become unstable when physically unrealistic parameters are introduced, such as higherorder spherical harmonics. The application of such standard broadening models to clay minerals has therefore not proved successful.
Other Rietveldbased methods attempt to approximate the diffraction features of disordered layered materials by empirical enhancement of the number of reflections. The simplest method is the splitting of the reflections of a traditional cell into two or three separate reflections that can be separately broadened and shifted, following prescribed rules (Bergmann & Kleeberg, 1998). In this way, the broadening of special classes of peaks, for example reflections with k ≠ 3n, can be modelled. This method is particularly suitable for structures showing well defined stacking faults, such as b/3 translations or multiples of 120° rotations. However, when structures show more complex disorder, such as turbostratic stacking, simple geometric dependencies of broadening and shifting are not sufficient to approximate their diffraction patterns.
Turbostratically disordered structures can be depicted in reciprocal space as infinite rods perpendicular to the ab plane and parallel to ; see Fig. 3.9.12 (Ufer et al., 2004). The diffraction features from such disordered materials consist of twodimensional asymmetric bands, as can be observed typically for smectites and some other clay minerals (Brindley, 1980). One method for approximating the diffraction effects along the reciprocallattice rods within the Rietveld method is via the `singlelayer' approach (Ufer et al., 2004). Here, a single layer is placed in a cell elongated along c^{*}, which is effectively a `supercell'. In doing this, an enhanced number of discrete lattice points are generated along the rods, according to the factor of elongation of the cell. This elongation generates a continuous distribution of additional hkl positions on the reciprocal rods. The inclusion of only a single layer in the supercell destroys periodicity, which is lacking in turbostratically disordered structures. By treating the pseudopeaks of the supercell in the same manner as other structures within the Rietveld method (i.e., introducing additional broadening, scaling the intensity) and separately calculating the peaks of the 00l series, the patterns of turbostratic structures like smectites can be reliably fitted. The model generated in this fashion can be used directly in phase quantification (Ufer, Kleeberg et al., 2008; Ufer, Stanjek et al., 2008).

Section of the reciprocal lattice of a turbostratically disordered pseudohexagonal Ccentred structure. 
However, this approach is limited to the turbostratic case. Moreover, the basal 00l series points are conventionally calculated, assuming rational diffraction from constant basal spacings in the stack. So the method cannot be applied to mixedlayered structures.
In order to overcome this limitation, Ufer et al. (Ufer, Kleeberg et al., 2008; Ufer et al., 2012) combined the recursive calculation method of Treacy et al. (1991) and the supercell approach in the structuredescription code of the Rietveld software BGMN (Bergmann et al., 1998). In this method a supercell is used to generate numerous discrete hkl spots along c^{*}, but the partial structure factors are calculated by the recursive algorithm. This allows the refinement of structural parameters of mixedlayered structures and simultaneous Rietveld QPA to be performed (Ufer et al., 2012). A broader introduction of such models in Rietveld phase analysis can be expected with the development of reliable structure models and enhanced computational power (Coelho et al., 2016, 2015; Bette et al., 2015).
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