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. 364370
Section 3.9.10. Improving accuracy^{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 
There are many factors that influence the accuracy and precision of QPA results where (i) accuracy is defined as the agreement between the analytical result and the true value, and (ii) precision is the agreement between results if the analysis is repeated under the same conditions. Precision may further be split into (i) repeatability, which is the agreement between repeated measurement and analysis of the same specimen, and (ii) reproducibility, which additionally includes repreparation, measurement and analysis of the sample.
Determination of the actual accuracy of an analysis is not a trivial task in a standardless method. In fact, it cannot be achieved without recourse to another measure of the sample that does incorporate standards. Too often, analysts will report Rietveld errors calculated in the course of refinement as the errors in the final quantification. However, these numbers relate purely to the mathematical fit of the model and have no bearing on the accuracy of the quantification itself.
Consider, for example, a threephase mixture of corundum, magnetite and zircon. Such a sample was presented as sample 4 in the IUCr CPD round robin on QPA (Scarlett et al., 2002). Its components were chosen with the deliberate aim of creating a sample in which severe samplerelated aberrations occur. Table 3.9.4 shows the weighed amounts of each component and the results of replicate analyses of three different subsamples of this material.

It is apparent that the standard deviation of the mean abundances of the three replicates, which represents the expected precision in the analysis, is 3 to 4 times greater than the errors reported by the Rietveld software. The good level of fit achieved in conducting these analyses (evidenced by low R factors) could lead the analyst to conclude that the mean value ± the standard deviation of the mean is an adequate measure of the phase abundances and their errors.
However, both the Rietveld errors and the precision are at least an order of magnitude smaller than the bias. The large bias, in this case due to the presence of severe microabsorption, represents the true accuracy that can be achieved in this example. Unfortunately, there is nothing in the XRD data and Rietveld analysis process that indicates that there may be a problem. It is only when the QPA is compared with other estimates, in this case derived from XRF chemicalanalysis results, that the problem becomes apparent. The analyst must take further steps to identify samplepreparation and/or datacollection protocols that may improve accuracy and, importantly, seek ways to verify the results.
The fundamental measured quantities in a diffraction pattern are the integrated intensities of the observed peaks. The precision of these measurements can be improved by: (i) increasing the primary intensity of the diffractometer using optics or higherpower Xray sources; (ii) using scanning linear detectors (see Chapter 2.1 ), which have multiple detector elements to collect individual intensities many times; these are then summed to achieve higher accumulated counts; (iii) increasing the number of counts accumulated at each step, that is increasing the step counting time T; and (iv) increasing the number of points, N, measured across the peak.
Often, the temptation is to collect data with large values of N and T to maximize counting statistics. However, the resulting increased precision is only useful up to the point where counting variance becomes negligible in relation to other sources of error; thereafter datacollection time is wasted. For example, if the sample is affected by the presence of severe samplerelated aberrations, the collection of highly precise data will not improve the accuracy of the resulting analysis significantly.
Therefore, the most important approach to improving the accuracy of an analysis is to eliminate the systematic errors. Given that the largest sources of error in QPA are experimental (Chung & Smith, 2000) and relate to sampling and specimen preparation, then this is the area on which the most careful attention needs to be focused. A detailed discussion of sample preparation and datacollection procedures is beyond the scope of this chapter but further details can be found in Chapter 2.10 , and in Hill & Madsen (2002) and Buhrke et al. (1998).
Crystallite size is considered here as the length of a coherent scattering domain and should not be confused with the terms grain or particle size used frequently in powder diffraction to describe the macroscopic size of the components in the sample. The macroscopic size of the particle is somewhat irrelevant (as in ceramics or other solid pieces of samples) as long as the crystallites (or domains) that comprise the particle are (i) sufficiently small to ensure that there are enough crystallites contributing to the diffraction process (Smith, 1992) and (ii) randomly oriented, thus ensuring a true powderaverage representation of intensities. However, for large domains or crystallites this assumption is usually not fulfilled and therefore it is necessary to reduce the crystallite size by reducing the size of the particles or grains that constitute the macroscopic objects of a powder.
Most issues in sample preparation are related to crystallite size and preferred orientation of the particles in the sample holder. For QPA a representative sampling of all possible orientations of crystallites with respect to the diffraction geometry is required. Rotation of the sample improves the particle statistics, since more crystallites can satisfy the diffraction condition (Elton & Salt, 1996).
Largecrystallite issues are easily detected using twodimensional (2D) detectors, where the Debye rings show a `spotty' intensity distribution. However, most QPA measurements are performed using 0D (point) or 1D (strip) detectors. The effect of large crystallites in a 1D pattern is that a few crystallites may contribute to irregularly high intensities for selected reflections. In the diffraction pattern, this situation is usually identified by intense reflections having a sharp peak profile compared with the surrounding peaks in the pattern. Furthermore, in a Rietveld refinement this situation is manifested by large intensity differences between the observed and calculated pattern that may not be associated with a particular crystallographic direction and hence to preferred orientation. Another way of detecting inhomogeneous crystallite distributions is to measure a series of scans from the same specimen at various rotation angles and comparing the relative peak intensities. It is worth noting that the push towards ever higher resolution in both laboratory and synchrotron instruments serves to further exacerbate the crystallitesize issue. This arises from the use of beams with decreased divergence, resulting in fewer crystallites likely to satisfy the diffraction condition.
There is no simple mathematical correction for largecrystallite issues and the effect is often misinterpreted in Rietveld refinement as preferred orientation. In this case, the correction would typically involve use of several directions for March–Dollasetype functions (Dollase, 1986) or an increasing order of sphericalharmonics coefficients (Ahtee et al., 1989). In any case, this is an improper use of these corrections and the necessity to do so clearly points to deficiencies in the sample preparation and datacollection regime.
The best way to minimize the largecrystallite issue is to reduce the crystallite size through grinding of the sample. However, sizereduction methods need to be carefully assessed, since overgrinding can cause peak broadening due to (i) a decrease of longrange order and hence crystallite size and (ii) the introduction of microstrain (Hill & Madsen, 2002). The practical effect of peak broadening is increasing peak overlap, which may complicate the phase identification. For wholepatternbased QPA, overgrinding is not as serious as long as it does not yield nanometresized particles or amorphous materials. This is because the integral intensity of the peaks is preserved. It should be noted that some phases can undergo transformation to other polymorphs or decompose to other phases during grinding (Hill & Madsen, 2002).
In practice, there is no generally applicable comminution strategy. For each material, a suitable milling device and grinding strategy needs to be identified. Inhomogeneous materials such as ores, concentrates and other mineralogical materials may have very different comminution properties for their constituents, leading to size fractionation during grinding. Largecrystallite issues are frequently observed for hard minerals (e.g. quartz, feldspar) while the grain size of soft minerals (e.g. talc) is reduced more rapidly.
A practical way of finding a best compromise for the milling conditions of a mixture may be the analysis of a series of samples of the same material where, for example, the grinding time is successively increased and the quantification results are compared. Fig. 3.9.17 shows the variation of analysed wt% with grinding time for two minerals: a stable result is eventually obtained.
There is a more extensive discussion of the impact of largecrystallite size on observed diffraction data (Smith, 1992) and ways to minimize its effect (Elton & Salt, 1996) in the published literature.
In order to generate peak intensities that accurately represent the intensityweighted reciprocal lattice, the crystallites in the powder must not only be sufficient in number, but they must also be randomly oriented. In other words, each crystal orientation should have the same probability of diffracting. Preferred orientation can arise when particles align in the sample holder according to their morphology. This is most common with platy or needlelike materials and the effect on the diffraction pattern is the observation of enhanced intensity along specific crystallographic directions with a subsequent decrease of intensity along other directions.
A number of samplepresentation methods can be used to minimize preferred orientation. For flat specimens, back pressing and side drifting into the sample holder can be effective. These methods tend to produce much less preferred orientation than frontmounted samples, but tend not to be very effective for chronic preferred orientation such as that exhibited by phases like clays, feldspars and chlorite. Reducing the size of the crystallites improves the probability of achieving random alignment of the crystallites in the sample holder. Gradually milling a sample and monitoring the preferredorientation coefficients as a function of grinding time may again help to find the correct, or at least reproducible, grinding conditions (Fig. 3.9.18).
A major advantage of wholepatternbased QPA over singlepeak methods is that all classes of reflections are considered in the calculation. In this sense, the method is less prone to preferred orientation of a particular class of peaks. Furthermore, orientation effects may be corrected by applying March–Dollase (Dollase, 1986) or sphericalharmonics (Ahtee et al., 1989) corrections. A properly applied correction may be of high importance for QPA in cases where a phase is present at low concentration and only a few peaks can clearly be identified in the pattern. If those peak(s) are affected by preferred orientation, the March–Dollase coefficient correlates strongly with scale factors and leads to biased QPA results. Examples of this effect occur with layered materials that have sheetlike morphology perpendicular to the c axis, including mica and clay minerals, which typically show stronger than expected intensity for the 00l reflections.
The crucial factor seems to be to what extent the orientation parameters correlate with the Rietveld scale factor. An example where the correlation is only minor is sample 2 from the IUCr CPD round robin on QPA (Scarlett et al., 2002). In that example, brucite [Mg(OH)_{2}] shows strong preferred orientation along the 00l direction. This may be corrected by the March–Dollase model, which returns a refined value of 0.66. However, the introduction of this preferredorientation correction only changes the brucite concentration from 35 to 36 wt% (weighed = 36.36 wt%); this is surprising because the orientation is strong and the weighted residual R_{wp} changes from 30 to 15%. Close examination of the correlations reveals a strong correlation between the brucite scale factor and preferredorientation factor. However, the correlation of the brucite preferredorientation parameter to the other scale factors (zincite, corundum and fluorite) is close to zero; this explains why in this example the QPA is not highly dependent on preferred orientation. In cases of strong correlation between the orientation parameter of one phase and the scale factors of other phases, preferred orientation should probably not be refined, or at least it should be verified carefully. It is worth noting that, in all Rietveldbased analyses, users should examine the correlation matrix as a matter of general practice to establish which parameters might be affecting parameters of interest.
It should be noted that sample rotation around the scattering vector (typically employed in flatplate Bragg–Brentano geometry) during the scan does not reduce preferred orientation, since there is no change between the preferredorientation direction and the diffraction vector. Using capillaries in transmission geometry assists in the reduction of preferred orientation, but the timeconsuming nature of packing capillaries makes this technique infeasible in industrial applications where diffractionbased QPA is used for routine quality control.
The strongest ongoing impediment to accuracy in QPA using XRD data is microabsorption. The microabsorption effect occurs when a multiphase sample contains both low and highly absorbing phases. For the highly absorbing phases, the Xray beam is more likely to be absorbed in the surface layers of the grain; thus, the fraction of the grain contributing to the diffraction pattern will decrease as the size of the grain increases above the beampenetration depth. For the lowabsorbing phases, the beam penetrates further into the particle resulting in a greater likelihood of the desired `volume diffraction' occurring (Brindley, 1945). The overall effect is the observation of a disproportionate amount of observed intensity from individual grains relative to what would be expected for the average absorption of the sample; the highly absorbing phases are underrepresented relative to the lowabsorbing phases. There is extensive discussion of the microabsorption issue in Zevin & Kimmel (1995).
Brindley (1945) has described the particle absorption contrast factor τ_{α} aswhere V is the particle volume, and μ_{α} and are the linear absorption coefficients of phase α and the entire sample, respectively. While it is relatively easy to calculate the absorption coefficients, equation (3.9.48) implies knowledge of the particle size of each component; this information is only available through independent microscope or lightscattering characterization.
This correction term is commonly incorporated into QPA through a modification to equation (3.9.26) of the formBrindley has also devised criteria by which to assess whether a microabsorption problem is likely to be present or not. Calculation of μD (where μ is the linear absorption coefficient and D is the particle diameter) yields the following criteria:
It is difficult for the analyst encountering a new sample to determine whether a correction for microabsorption is required without first obtaining additional information. A minimum requirement should be to calculate μD for each phase present. However, this requires knowledge of the particle size which, in a multiphase sample, can be very difficult to obtain unambiguously. Even when the particle size is measured by, for example, dynamic light scattering or optical or SEM image analysis, the applicability of the correction can still be unclear. In addition, the correction factor embodied in equations (3.9.48) and (3.9.49) makes the assumption that the particles of the phase of interest are spherical and of uniform size. This assumption is unrealistic in almost all samples; in reality, each phase is likely to be present at a wide range of particle sizes and the particles are highly unlikely to be spherical.
Table 3.9.5 shows the calculated values of μD for Cu Kα radiation for some commonly encountered phases in mineralogical analysis. For the least absorbing phase (corundum), the upper range of applicability of the Brindley model (medium powder) is reached at about 5 µm; by 8 µm, the coarse powder criterion has been reached and the correction model is no longer applicable. For magnetite, these limits are reached an order of magnitude earlier at about 0.5 and 0.9 µm, respectively.

Fig. 3.9.19 shows an SEM image of a mixture of approximately equal amounts of corundum, magnetite and zircon. The individual components of the sample were weighed and the mixture ground in ethanol in a McCrone micronizing mill (McCrone Research Associates, London) for 10 min g^{−1}. This approach to sample preparation is generally accepted as best practice for powder XRD because it minimizes structural damage during grinding (Hill & Madsen, 2002). After decanting and drying, the sample was backpacked into a cavity sample holder for XRD data collection; the same sample was then used to obtain the SEM image in Fig. 3.9.19. Visual observation shows a wide range of particle sizes (from submicron to greater than 10 µm) and shapes that do not even approximate spheres. Even if this information is obtained, selection of a particle size that best represents each individual phase is a difficult task. In addition, in many sample suites, the component phases exhibit a range of hardness resulting in different rates of grinding and hence difference size ranges. Regrettably, what happens too often in practice is that analysts will micronize the sample and then select an arbitrary particle size in order to derive a `preferred' value for the final analysis. Therefore, caution is advised in the application of these correction models. The IUCr CPD round robin on QPA (Madsen et al., 2001; Scarlett et al., 2002) showed that many participants severely degraded their results by applying a correction when none was necessary.
Equation (3.9.48) shows that there two ways to minimize microabsorption. The first is to reduce the absorption contrast by, for example, changing the Xray wavelength. While corundum and magnetite have very different linear absorption coefficients for Cu Kα radiation (126 and 1123 cm^{−1}, respectively), the difference is reduced to 196 and 231 cm^{−1}, respectively, for Co Kα radiation. The second approach is to reduce the particle size in order to meet Brindley's fine or mediumpowder criteria.
However, even these steps may not be sufficient to eliminate the microabsorption effect. Slightly different absorption coefficients, or different particle sizes for phases with the same absorption coefficients, may still introduce a bias between expected and analysed concentrations. In this situation, it may be better to use a calibrated hkl_phase or peaks_phase (Section 3.9.6) instead of a Rietveld, structurebased phase. The calibration step involved in the generation of such a phase incorporates the microabsorption problem into the calibration constant.
Fig. 3.9.20 shows the bias between known concentrations (derived from chemical analysis) and QPAdetermined concentrations for a series of salt samples. The samples contain halite (NaCl), sylvite (KCl) and kieserite (MgSO_{4}·H_{2}O) as major phases and small amounts of anhydrite (CaSO_{4}), langbeinite [K_{2}Mg_{2}(SO_{4})_{3}] and carnallite [KMgCl_{3}·6(H_{2}O)]. The linear absorption coefficient of sylvite (254 cm^{−1}) is much higher than halite (165 cm^{−1}). Using crystalstructurebased analysis, there is a systematic deviation of up to 3% with an overestimation of the low absorber (halite) and an underestimation of the high absorber (sylvite). After replacing sylvite by a calibrated hkl_phase, the bias is reduced to about 1% and does not show systematic deviations.
It should be noted, however, that the phase constants developed using such a calibration approach will only be applicable to the sample suite and preparation conditions for which it was developed. The calibration process will need to be repeated if there are significant changes to the sample suite or samplepreparation conditions.
One of the distinct advantages of structurebased wholepattern fitting for QPA is that no standards need to be prepared because the structure for each phase provides the phase constant ZMV; the unitcell dimensions allow the calculation of the cell volume V and the unitcell contents provide the mass ZM (Bish & Howard, 1988; Hill & Howard, 1987). These values are used, along with the Rietveld scale factor S, in equation (3.9.26) to derive the phase abundance. This is especially useful for complex systems where the preparation of multiple standards would add considerably to the analytical complexity.
An additional advantage is the ability to refine the crystal structure (unitcell dimensions and siteoccupation factors, for example), when the data are of sufficiently high quality, in order to obtain the best fit between observed and calculated patterns. In addition to updating the ZMV value, the site occupancies are contained in the structurefactor calculation and, therefore, will change the relative reflection intensities and have an impact on the scale factor and QPA. Other structural parameters that have a strong effect on the scale factor and QPA are the atomic displacement parameters (ADPs). Strong correlation between the ADPs and amorphous material concentration has been shown by Gualtieri (2000) and Madsen et al. (2011).
This leads to the question: which crystal structure should be selected for QPA? Databases contain multiple entries for the same phase with the structures determined using different methods. While ADPs and siteoccupation factors determined using neutron diffraction and singlecrystal analysis should be favoured over those determined using Xray powder data, many database entries do not have refined ADPs for all (and in some cases, any) atoms. Often, arbitrarily chosen default values of 0.5 or 1.0 Å^{2} for B_{eq} are entered for all atoms, but this should be viewed or used with great caution. There is clearly a need to carefully evaluate the crystalstructure data used for QPA. This is particularly worth mentioning in view of the advent of new `userfriendly' software that automatically assigns crystal structures after having performed the phase identification.
Empirical profileshape models contribute significantly to the complexity (and correlations) of wholepowderpattern fitting for QPA because of the large number of phases and multiple parameters required to model the profile shape of each phase. The use of convolutionbased profile fitting [in, for example, BGMN (Bergmann et al., 1998, 2000) and TOPAS (Bruker AXS, 2013)] greatly reduces the number of parameters, because the instrumentresolution function (which is constant for a given setup) can be separated from samplerelated peak broadening. The instrument component can be refined using a standard and then fixed for subsequent analysis. The sample contribution to peak width and shape can then be related directly to crystallite size and microstrain using a minimal number of parameters. The reduction of the total number of parameters reduces the refinement complexity and the chance of parameter correlation.
The choice of the function used to model the pattern background may also have a strong influence on amorphous content (Gualtieri, 2000; Madsen et al., 2011). Given that the intensities of both the background and the amorphous contribution vary slowly as a function of 2θ, it is inevitable that there will be a high degree of correlation between them. Hence, any errors in determining the true background will result in errors in amorphous phase determination. A simple approach is to use a background function with a minimal number of parameters. A more exact approach requires the separation of the amorphous contribution from background components such as Compton scattering and parasitic scattering by the sample environment and air in the beam path. This is routinely done in pair distribution function (PDF) analysis; details can be found in Chapter 5.7 in this volume and in Egami & Billinge (2003).
Another parameter that correlates with the pattern background is the width of broad peaks for phases of low concentration. If allowed to refine to very large width values, the peaks are `smeared' over a broad range of the pattern with no clear distinction between peaks and background. The same issue applies when there is a high degree of peak overlap, particularly at high angles, leading to severe under or overestimation of the phase. The careful use of limits for either crystallite size or corresponding parameters in empirical peakshape modelling assists in minimizing this effect.
There can be a subtle interplay between the profileshape function and the pattern background that has an impact on wholepattern fitting (Hill, 1992). The data in Fig. 3.9.21, collected using a Cu tube and an Ni Kβ filter, exhibit lowangle truncation of the peak tails at the βfilter absorption edge. On the highangle side, the anatase peak displays a wide tail which extends to the position of the strongest rutile peak at about 27.5° 2θ. In this case, rutile is present as a minor phase and the error in the background determination using conventional peakprofile modelling (Fig. 3.9.21a) introduces about 0.5% bias in the rutile QPA. The use of a more accurate profile model that incorporates the effect of the βfilter absorption edge (Fig. 3.9.21b) serves to improve the accuracy (Bruker AXS, 2013).
XRDbased derivation of elemental abundances relies on (i) the QPA abundances, and (ii) the assumed or measured stoichiometry of the crystalline phases. The accuracy of the QPA result may then be evaluated by comparing the calculated elemental abundances with those determined by traditional chemicalanalysis techniques. However, for the best level of agreement, this method requires that the composition of the crystalline phases be well defined. A complication, in particular for minerals, is that idealized compositions may be reported but do not necessarily match the actual composition of the species present in the sample. Where possible, detailed phase analysis using microbeam techniques should be undertaken to establish the true composition for each phase. A complication that serves to decrease the agreement is that chemically based compositional analysis does not distinguish between crystalline and amorphous phase content, while the diffractionbased QPA usually measures only the crystalline phases. Generally, the composition of amorphous phases may not be known accurately and even highly crystalline material can contain amorphous components because of nondiffracting surface layers of the grains (Cline et al., 2011).
An example demonstrating the level of agreement that can be achieved is that of the ironore certified reference material SX 1114 from Dillinger Hütte (Fig. 3.9.22). The material is moderately complex and consists of nine distinct mineral species. The data were measured with Co Kα radiation and analysed using Rietveldbased QPA in TOPAS (Bruker AXS, 2013). The phase abundances are converted to elemental and oxide compositions for comparison with the certified elemental analyses (Table 3.9.6). There is excellent agreement between the XRD results and the chemical analysis with bias values better than ±1 wt%.

In contrast to elemental compositional analysis, where standard reference materials (SRMs) are widespread, there are only a very limited number of SRMs available for diffractionbased QPA. Prominent examples are SRMs for the cement industry [NIST reference material clinker 8486 (Stutzman & Leigh, 2000) and ordinary Portland cement NIST SRM 2686] or ceramics materials (silicon nitride CRM BAMS001) (Peplinski et al., 2004). Similar to elemental standards, the certified values do not necessarily represent the true composition. Rather, they are published values that are typically averaged over the results from different independent methods, instruments and laboratories. Therefore, confidence limits of concentrations are provided that may be much larger than estimated standard deviations of concentrations within a single laboratory.
Finally, a number of interlaboratory tests, or round robins, have been conducted on synthetic mixtures in order to set benchmarks for particular materials and/or the application of methods. Examples range from well ordered, highsymmetry phases discussed in earlier sections of this chapter (Madsen et al., 2001; Scarlett et al., 2002) to standard mixtures of geological material, granite and bauxites (Bish & Post, 1993), and technical products like artificial Portland cements (De la Torre & Aranda, 2003) where relative biases of 2–3% for the main phases and 5–10% for minor phases were found.
Very recently, the precision and accuracy of QPA for the analysis of Portland clinker and cement were determined for synthetic mixtures and commercial samples. The scatter of the results from the interlaboratory comparison, and the fact that individual errors are much smaller than the standard deviations of all submitted results, points to the widespread presence of userdependent systematic errors (LéonReina et al., 2009).
One of the most challenging round robins is the Reynolds Cup (Ottner et al., 2000; McCarty, 2002; Kleeberg, 2005; Omotoso et al., 2006; Raven & Self, 2017), organized biannually since the year 2000 by the Clay Minerals Society. Synthetic mixtures representing typical sedimentary rock types are analysed and require a very high level of sample preparation and analytical skills because of the presence of a variety of clay minerals.
While most round robins have dealt with inorganic materials, one for pharmaceutical materials was organised by the International Centre for Diffraction Data (ICDD) together with the Pharmaceutical Powder XRD symposium series (PPXRD) (Fawcett et al., 2010). A major outcome was the identification of operator errors in all steps of the analysis to be the largest source of error. This highlights the importance of reducing systematic errors for improving accuracy in QPA.
As a concluding remark, a variety of factors may influence the precision and accuracy of QPA. Nonetheless, better than 1 wt% agreement may be achieved for simple systems of well crystallized material. Moderately complex mixtures such as those routinely observed in cement plants and in the mining industry can be typically analysed at a 1 wt% level of accuracy provided that the analyst chooses the most appropriate samplepreparation, datacollection and analysis methodologies for the samples in question.
References
Ahtee, M., Nurmela, M., Suortti, P. & Järvinen, M. (1989). Correction for preferred orientation in Rietveld refinement. J. Appl. Cryst. 22, 261–268.Google ScholarBergmann, J., Friedel, P. & Kleeberg, R. (1998). Bgmn – a new fundamental parameters based Rietveld program for laboratory Xray sources; its use in quantitative analysis and structure investigations. IUCr Commission on Powder Diffraction Newsletter, 20, 5–8.Google Scholar
Bergmann, J., Kleeberg, R., Haase, A. & Breidenstein, B. (2000). Advanced fundamental parameters model for improved profile analysis. Mater. Sci. Forum, 347–349, 303–308.Google Scholar
Bish, D. L. & Howard, S. A. (1988). Quantitative phase analysis using the Rietveld method. J. Appl. Cryst. 21, 86–91.Google Scholar
Bish, D. L. & Post, J. E. (1993). Quantitative mineralogical analysis using the Rietveld fullpattern fitting method. Am. Mineral. 78, 932–940.Google Scholar
Brindley, G. W. (1945). The effect of grain or particle size on Xray reflections from mixed powders and alloys, considered in relation to the quantitative determination of crystalline substances by Xray methods. London Edinb. Dubl. Philos. Mag. J. Sci. 36, 347–369.Google Scholar
Bruker AXS (2013). Topas v5: General profile and structure analysis software for powder diffraction data. Version 5. https://www.bruker.com/topas.Google Scholar
Buhrke, V. E., Jenkins, R. & Smith, D. K. (1998). A Practical Guide for the Preparation of Specimens for XRay Fluorescence and Xray Diffraction Analysis. New York: WileyVCH.Google Scholar
Chung, F. H. & Smith, D. K. (2000). Industrial Applications of Xray Diffraction, edited by F. H. Chung & D. K. Smith, ch. 1, pp. 3–10, and ch. 2, pp. 13–32. New York: Marcel Dekker.Google Scholar
Cline, J. P., Von Dreele, R. B., Winburn, R., Stephens, P. W. & Filliben, J. J. (2011). Addressing the amorphous content issue in quantitative phase analysis: the certification of NIST standard reference material 676a. Acta Cryst. A67, 357–367.Google Scholar
De la Torre, A. G. & Aranda, M. A. G. (2003). Accuracy in Rietveld quantitative phase analysis of Portland cements. J. Appl. Cryst. 36, 1169–1176.Google Scholar
Dollase, W. A. (1986). Correction of intensities for preferred orientation in powder diffractometry: application of the March model. J. Appl. Cryst. 19, 267–272.Google Scholar
Egami, T. & Billinge, S. J. L. (2003). Underneath the Bragg Peaks: Structural Analysis of Complex Materials. Oxford: Elsevier.Google Scholar
Elton, N. J. & Salt, P. D. (1996). Particle statistics in quantitative Xray diffractometry. Powder Diffr. 11, 218–229.Google Scholar
Fawcett, T. G., Needham, F., Faber, J. & Crowder, C. E. (2010). International Centre for Diffraction Data round robin on quantitative Rietveld phase analysis of pharmaceuticals. Powder Diffr. 25, 60–67.Google Scholar
Gualtieri, A. F. (2000). Accuracy of XRPD QPA using the combined Rietveld–RIR method. J. Appl. Cryst. 33, 267–278.Google Scholar
Hill, R. J. (1992). The background in Xray powder diffractograms: a case study of Rietveld analysis of minor phases using Nifiltered and graphitemonochromated radiation. Powder Diffr. 7, 63–70.Google Scholar
Hill, R. J. & Howard, C. J. (1987). Quantitative phase analysis from neutron powder diffraction data using the Rietveld method. J. Appl. Cryst. 20, 467–474.Google Scholar
Hill, R. J. & Madsen, I. C. (2002). Structure Determination from Powder Diffraction Data, edited by W. David, K. Shankland, L. McCusker & C. Baerlocher, ch. 6, pp. 98–116. Oxford University Press.Google Scholar
Kleeberg, R. (2005). Results of the second Reynolds Cup contest in quantitative mineral analysis. IUCr Commission on Powder Diffraction Newsletter, 30, 22–26.Google Scholar
LeónReina, L., De la Torre, A. G., PorrasVázquez, J. M., Cruz, M., Ordonez, L. M., Alcobé, X., GispertGuirado, F., LarrañagaVarga, A., Paul, M., Fuellmann, T., Schmidt, R. & Aranda, M. A. G. (2009). Round robin on Rietveld quantitative phase analysis of Portland cements. J. Appl. Cryst. 42, 906–916.Google Scholar
McCarty, D. K. (2002). Quantitative mineral analysis of claybearing mixtures: The `Reynolds Cup' contest. IUCr Commission on Powder Diffraction Newsletter, 27 12–16.Google Scholar
Madsen, I. C., Scarlett, N. V. Y., Cranswick, L. M. D. & Lwin, T. (2001). Outcomes of the International Union of Crystallography Commission on Powder Diffraction Round Robin on Quantitative Phase Analysis: samples 1a to 1h. J. Appl. Cryst. 34, 409–426.Google Scholar
Madsen, I. C., Scarlett, N. V. Y. & Kern, A. (2011). Description and survey of methodologies for the determination of amorphous content via xray powder diffraction. Z. Kristallogr. 226, 944–955.Google Scholar
Omotoso, O., McCarty, D. K., Hillier, S. & Kleeberg, R. (2006). Some successful approaches to quantitative mineral analysis as revealed by the 3^{rd} Reynolds Cup contest. Clays Clay Miner. 54, 748–760.Google Scholar
Ottner, F., Gier, S., Kuderna, M. & Schwaighofer, B. (2000). Results of an interlaboratory comparison of methods for quantitative clay analysis. Appl. Clay Sci. 17, 223–243.Google Scholar
Peplinski, B., Kleeberg, R., Bergmann, J. & Wenzel, J. (2004). Quantitative phase analysis using the Rietveld method – estimates of possible problems based on two interlaboratory comparisons. Mater. Sci. Forum, 443444, 45–50.Google Scholar
Raven, M. D. & Self, P. G. (2017). Outcomes of 12 years of the Reynolds Cup quantitative mineral analysis round robin. Clays Clay Miner. 65, 122–134.Google Scholar
Scarlett, N. V. Y., Madsen, I. C., Cranswick, L. M. D., Lwin, T., Groleau, E., Stephenson, G., Aylmore, M. & AgronOlshina, N. (2002). Outcomes of the International Union of Crystallography Commission on Powder Diffraction Round Robin on Quantitative Phase Analysis: samples 2, 3, 4, synthetic bauxite, natural granodiorite and pharmaceuticals. J. Appl. Cryst. 35, 383–400.Google Scholar
Smith, D. K. (1992). Particle statistics and whole pattern methods in quantitative xray powder diffraction analysis. Adv. Xray Anal. 35, 1–15.Google Scholar
Stutzman, P. E. & Leigh, S. (2000). Proceedings of the TwentySecond International Conference on Cement Microscopy, 11–13 September 2000, Quebec City, Canada. https://cemmicro.org/publications/.Google Scholar
Zevin, L. S. & Kimmel, G. (1995). Quantitative Xray Diffractometry. SpringerVerlag New York, Inc.Google Scholar