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. 356360
Section 3.9.6. Quantification of phases with partial or no known crystal structures^{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 
While the Rietveldbased methods described in Section 3.9.3.6 work for well ordered phases with known crystal structures, they are limited when published structure data do not accurately represent the phase actually present in the sample, are incomplete or do not exist. Poor agreement with published structure data is a common occurrence in mineralogical research where disorder exists and observed diffraction data deviate significantly from the ideal; this situation occurs with many of the clay minerals. The issue of incomplete or nonexistent structure data can occur in almost any area where new materials are either synthesized or discovered. The growing demand for the analysis of materials from the nanotechnology community, where phases are at the boundary of what can be considered crystalline, serves to further highlight these limitations. Recent developments in diffraction methods have sought to address these issues and have used approaches that include the development of calibrated models or, where appropriate, the extension of existing structure data to incorporate systematic disorder such as stacking faults in clay minerals.
Calibrated models are generally developed in one of two ways. The first (which uses what is referred to hereafter as an hkl_phase) is obtained via the use of partial structure information. Here the peak positions are constrained by a unit cell and space group but the relative intensities, in the absence of atom types and locations in the unit cell, are determined empirically from a pure sample or one where the phase is present in a mixture at a known concentration. The second method involves the use of a discrete set of peaks whose positions, intensities, width and shape are all determined empirically. Once determined using a standard sample, this group of peaks may then be scaled as a single unit and is referred to hereafter as a peaks_phase.
The software SIROQUANT (Taylor & Rui, 1992) employs the simultaneous use of observed and calculated standard profiles within the framework of the Rietveld method. It draws on a library of structures that are stored as lists of reflections and intensities (hkl files). These are calculated on a cyclebycycle basis for well described crystalline materials but are read directly from the hkl files for poorly defined materials such as clay minerals. This method still requires some knowledge of the crystal chemistry of all phases involved and that they be included within the programme's database. By the inclusion of reflection information in this way some aberrations such as preferred orientation may be allowed for. This approach to clay mineralogy also provides for the refinement of two sets of halfwidth parameters in order to model the coexisting sharp and broad reflections generated by such minerals.
A subsequent development of the wholepattern approach is the `partial or no known crystal structure' (PONKCS) method (Scarlett & Madsen, 2006). This method operates within the framework of the Rietveld method but replaces the traditional crystal structure of the phases in question with an empirical set of peaks (either as an hkl_phase or a peaks_phase). These can then be scaled as a single unit in the course of refinement in similar fashion to the set of structure factors derived from a crystal structure. Since the full structure information is not available, it is not possible to calculate the ZMV phase constant normally required for quantification via equation (3.9.26) (Hill & Howard, 1987); hence, an empirical value must be derived through calibration.
The generation of a suitable PONKCS model requires that:
The initial step in the generation of a PONKCS model is to describe the contribution to the diffraction pattern of the phase with a series of peaks. If the phase of interest has been indexed, the Le Bail or Pawley methods (see Chapter 3.5 ) can be used to constrain peak positions to the space group and unitcell parameters while the individual reflection intensities are allowed to vary to best match the observed peaks (i.e. an hkl_phase). If the phase has not been indexed, a series of unrelated peaks can be refined using a standard material and scaled as a group during analysis (i.e. a peaks_phase). While this approach is effective in most cases, it restricts the refinable parameters that may be used in the treatment of systematic errors such as preferred orientation.
The next step is to calibrate the hkl_phase or peaks_phase and derive a `phase constant' that is equivalent to the ZMV value in crystalstructurebased quantification. This is achieved by the preparation of a mixture in which there are known amounts W_{α} and W_{s} of the unknown and standard, respectively. Recalling equation (3.9.25), the ratio of the weight fractions is then given bywhere S_{α} and S_{s} are the refined scale factors for the unknown and standard, respectively.
Rearrangement of equation (3.9.42) then provides the means for determining an empirical value of (ZMV)_{α}, which is required for the calibration of a peaks_phase:For an hkl_phase the value of V can be determined from the refined unitcell parameters and hence can be removed from the phase constant resulting inUnlike the ZMV value derived from the unitcell contents of a crystal structure, the phase constants derived using equations (3.9.43) and (3.9.44) have no physical meaning, since they have been derived by empirical measurement. For an hkl_phase, a more physically meaningful value of ZM can be obtained by deriving the true unitcell mass from the measured phase density according toThe empirical `structure factor' values in the hkl_phase could then be scaled according to the relation ZM_{α(true)}/ZM_{α}, making them approximate `real' structure factors for the material. Note that this final step is not necessary for quantification, but may make the method more generally applicable.
The PONKCS method is applicable to any mixture in which there are one or more phases that are not fully characterized crystallographically, including essentially amorphous material, provided appropriate calibration samples can be obtained. In the mineralogical context, it may not be possible to obtain pure phase specimens typical of those found in the bulk mixtures, but it may be possible to concentrate them to a point where they can be used. Methods of achieving this may include gravity or magnetic separation, or selective chemical dissolution.
The original paper describing this method (Scarlett & Madsen, 2006) gives a detailed example based upon sample 1 from the IUCr CPD round robin on QPA (Madsen et al., 2001; Scarlett et al., 2002). There, corundum was regarded as the unknown phase, fluorite as an impurity of known crystal structure and zincite a standard material added at known weight fraction. In the same paper, there is a more realistic example regarding the poorly ordered clay mineral nontronite, which is of commercial significance but difficult to quantify via traditional structurebased Rietveld methodology. Further details regarding quantification of this mineral via the PONKCS method is given in articles detailing its importance in lowgrade nickel laterite ores (Scarlett et al., 2008; Wang et al., 2011).
A calibrationbased method such as PONKCS may also find increasing application with phases that have a known crystal structure. It has the greatest potential for accuracy, as the calibration process may obviate residual aberrations in the data such as microabsorption. Assuming that the sample suite has the same absorption characteristics as that used for calibration, such aberrations will be included in the calibration function and require no further correction during the sample analysis. This is a realistic scenario for routine analyses in industries as diverse as mineral processing, cement production and pharmaceutical production.
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).
Traditionally, most activity in diffractionbased QPA has been concerned with the assessment of the crystalline components. However, all materials possess a nondiffracting surface layer with some degree of disorder or contain some surface reaction products and adsorbed species. While such a layer can easily account for ∼1 wt% of the entire sample in a finely divided solid, the fraction of this surface layer will increase as the particle size decreases (Cline et al., 2011). In addition, some materials can contain separate phases that may be amorphous or at least poorly crystalline. The advent of nanotechnology has served to further blur the boundaries between what is defined by powder XRD as crystalline or amorphous.
During in situ studies, some phases undergo transformations via amorphous intermediate components; the presence of these phases has the potential to influence our understanding of reaction mechanisms. Given the potential for these amorphous components to influence bulkmaterial properties, the need to quantify them is an increasingly important issue for analysts using diffractionbased methods. Many of the traditional phasequantification techniques described in this chapter fail to take into account the occurrence of amorphous material in the sample and, without careful attention by the analyst, its presence may remain undetected.
Madsen et al. (2011) recently reviewed a range of techniques for the determination of amorphous content and assessed their applicability for various analytical situations. The study used both singlepeak and wholepattern methodology and applied it in two distinct ways.
In general, for the determination of amorphous material the problem will dictate the method(s) used. All methods discussed in the study of Madsen et al. (2011) are, in principle, capable of determining the concentration of amorphous material in mixtures with similar levels of accuracy and precision as is possible for crystalline phases (down to ∼1% absolute or better). The limitations are similar to those for the QPA of crystalline phases, and are dictated by sample properties and the analytical techniques used.
A summary of the recommendations resulting from the study include:

Some materials contain more than one amorphous phase and there may be a desire to quantify these separately rather than as a group. This provides a significant challenge since their broad diffraction patterns will be highly overlapped, thus leading to a high degree of correlation during analysis. However, Williams et al. (2011) have demonstrated that, with careful experimentation and data analysis, it is possible to provide QPA for two poorly crystalline components in geopolymers.
Phase abundances reported in the literature are often provided in a manner that suggests they are absolute values. Where no specific allowance for amorphous content has been made and reported, it is better to assume that the reported phase abundances are correct relative to one another, but may be overestimated in an absolute sense. Therefore, standard practice in QPA should be to use methodology which produces absolute rather than relative phase abundances. Any positive difference between unity and the sum of the absolute weight fractions will alert the analyst to the presence of nonanalysed material in the sample.
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