Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.9, pp. 356-358

Section Use of calibrated models

I. C. Madsen,a* N. V. Y. Scarlett,a R. Kleebergb and K. Knorrc

aCSIRO Mineral Resources, Private Bag 10, Clayton South 3169, Victoria, Australia,bTU Bergakademie Freiberg, Institut für Mineralogie, Brennhausgasse 14, Freiberg, D-09596, Germany, and cBruker AXS GmbH, Oestliche Rheinbrückenstr. 49, 76187 Karlsruhe, Germany
Correspondence e-mail: Use of calibrated models

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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[link]) 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 cycle-by-cycle 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 co-existing sharp and broad reflections generated by such minerals.

A subsequent development of the whole-pattern approach is the `partial or no known crystal structure' (PONKCS) method (Scarlett & Madsen, 2006[link]). 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)[link] (Hill & Howard, 1987[link]); hence, an empirical value must be derived through calibration. Generation of calibrated PONKCS models

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The generation of a suitable PONKCS model requires that:

  • (1) The unknown phase is available as either a pure specimen or as a component of a mixture where its abundance is known (in some instances, this may be achieved by other means, such as the measurement of bulk and/or microchemical composition.)

  • (2) The unknown phase does not vary considerably from the material used to derive the relative intensities of the model. Preferred orientation and other sample-related effects may be compensated for based upon an indexed diffraction pattern.

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[link] ) can be used to constrain peak positions to the space group and unit-cell 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 crystal-structure-based quantification. This is achieved by the preparation of a mixture in which there are known amounts Wα and Ws of the unknown and standard, respectively. Recalling equation (3.9.25)[link], the ratio of the weight fractions is then given by[{{{W_\alpha }} \over {{W_s}}} = {{{S_\alpha }{{(ZMV)}_\alpha }} \over {{S_s}{{(ZMV)}_s}}} ,\eqno(3.9.42)]where Sα and Ss are the refined scale factors for the unknown and standard, respectively.

Rearrangement of equation (3.9.42)[link] then provides the means for determining an empirical value of (ZMV)α, which is required for the calibration of a peaks_phase:[{(ZMV)}_{\alpha }={{{W}_{\alpha }}\over{{W}_{s}}}{{{S}_{s}}\over{{S}_{\alpha }}} {(ZMV)}_{s}. \eqno(3.9.43)]For an hkl_phase the value of V can be determined from the refined unit-cell parameters and hence can be removed from the phase constant resulting in[{(ZM)}_{\alpha }={{{W}_{\alpha }}\over{{W}_{s}}}{{{S}_{s}}\over{{S}_{\alpha }}} {{{(ZMV)}_{s}}\over{{V}_{\alpha }}}. \eqno(3.9.44)]Unlike the ZMV value derived from the unit-cell contents of a crystal structure, the phase constants derived using equations (3.9.43)[link] and (3.9.44)[link] 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 unit-cell mass from the measured phase density according to[{(ZM)_{\alpha ({\rm true})}} = {{{\rho _\alpha }{V_\alpha }} \over {1.6604}}. \eqno(3.9.45)]The empirical `structure factor' values in the hkl_phase could then be scaled according to the relation ZMα(true)/ZMα, 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. Application of the model

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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[link]) gives a detailed example based upon sample 1 from the IUCr CPD round robin on QPA (Madsen et al., 2001[link]; Scarlett et al., 2002[link]). 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 structure-based Rietveld methodology. Further details regarding quantification of this mineral via the PONKCS method is given in articles detailing its importance in low-grade nickel laterite ores (Scarlett et al., 2008[link]; Wang et al., 2011[link]).

A calibration-based 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.


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
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
Scarlett, N. V. Y. & Madsen, I. C. (2006). Quantification of phases with partial or no known crystal structures. Powder Diffr. 21, 278–284.Google Scholar
Scarlett, N. V. Y., Madsen, I. C., Cranswick, L. M. D., Lwin, T., Groleau, E., Stephenson, G., Aylmore, M. & Agron-Olshina, 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
Scarlett, N. V. Y., Madsen, I. C. & Whittington, B. I. (2008). Time-resolved diffraction studies into the pressure acid leaching of nickel laterite ores: a comparison of laboratory and synchrotron X-ray experiments. J. Appl. Cryst. 41, 572–583.Google Scholar
Taylor, J. C. & Rui, Z. (1992). Simultaneous use of observed and calculated standard profiles in quantitative XRD analysis of minerals by the multiphase Rietveld method: the determination of pseudorutile in mineral sands products. Powder Diffr. 7, 152–161.Google Scholar
Wang, X., Li, J., Hart, R. D., van Riessen, A. & McDonald, R. (2011). Quantitative X-ray diffraction phase analysis of poorly ordered nontronite clay in nickel laterites. J. Appl. Cryst. 44, 902–910.Google Scholar

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