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. 3.9, pp. 361-362

Section 3.9.7.1. Data analysis

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:  ian.madsen@csiro.au

3.9.7.1. Data analysis

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There are usually a series of steps involved in the analysis of in situ diffraction data. Given the large number of data sets collected, it is generally not practicable to undertake detailed analysis of every pattern individually. Since any changes to the component phases are transitions generally observed in a sequence of patterns, data analysis focused on extracting QPA could be undertaken using the following steps:

  • (1) Cluster the data into a number of groups necessary to describe the major phase regions present during the reaction. This can be achieved (i) visually, using software that allows the plotting of three-dimensional data sets of the type shown in Fig. 3.9.14[link], or (ii) through the use of automatic clustering algorithms using, for example, principal-component analysis.

    [Figure 3.9.14]

    Figure 3.9.14 | top | pdf |

    Raw in situ XRD data (Co Kα radiation) collected during the synthesis of the iron-ore sinter bonding phase SFCA-I (Webster et al., 2013[link]). The data, collected as a function of heating temperature, are viewed down the intensity axis with red representing the highest intensity and blue the lowest intensity. The identified phases include gibbsite Al(OH)3, calcite CaCO3, haematite Fe2O3, lime CaO, calcium ferrites CF and CFF, calcium alumina-ferrite C2F1−xAx, magnetite Fe3O4, and SFCA-I.

  • (2) Select the `most typical' pattern of each cluster as well as the two `least typical' patterns at the extreme ends of the cluster. These patterns are often identified by clustering software based on the statistical similarity between patterns in the cluster.

  • (3) Identify the phases present in each cluster using the most typical pattern. This is not always a trivial task since (i) new phases that are not currently present in databases may have been generated; (ii) effects such as thermal expansion or variation of chemical composition may have changed the peak positions so that search/match procedures are no longer successful; or (iii) impurity elements may have stabilized phases that are not expected from related phase-diagram studies.

  • (4) For the discussion here, it will be assumed that the quantification process will be via a whole-pattern method.

    • (a) Develop appropriate (crystal structure or PONKCS) models for every phase observed within the data suite.

    • (b) Optimize the pattern and phase-analysis parameters using the most typical pattern selected from each cluster.

    • (c) Set the relevant parameter refinement limits using the least typical patterns. It is necessary to limit the range over which refined parameters can vary to avoid the return of physically unrealistic values.

  • (5) Owing to the large number of data sets, analysis for QPA will generally be approached as a batch process with limited refinement of structural parameters. This limitation on the total number of refinable parameters is necessary during batch processing in order to avoid instability in the refined values as the phases progress from major to minor concentration.

  • (6) Batch processing of data suites may be conducted in a variety of ways including:

    • (a) Sequential refinement, beginning with either the first or final pattern of the suite and including all phases present in the entire suite. This methodology must be tempered by a means to either remove or severely restrict refinement of any phases that are not present in all patterns of the suite in order to avoid the reporting of `false positives' where absent phases have been included. Some software packages allow phases to be removed from the analysis if their abundance is below a selected level or has an error that exceeds some predefined criteria (Bruker AXS, 2013[link]).

    • (b) Parametric Rietveld refinement (Stinton & Evans, 2007[link]), where the entire suite of diffraction data is analysed simultaneously. Selected parameters are constrained to the applied external variable (e.g. temperature) with a function describing their evolution throughout the data sequence. For example, the unit-cell parameters for a phase can be constrained to vary according to their thermal coefficients of expansion. This method can bring stability to refined parameters and allows the refinement of noncrystallographic parameters such as temperature and reaction rate constants directly from the diffraction data. This methodology is particularly suited to relatively simple phase systems, but is difficult to develop for complex multiphase mineralogical systems.

  • (7) In selecting a model for use in QPA, it is highly recommended that one of the approaches that generate absolute phase abundances is used. Many reactions generate intermediate amorphous phases that convert to crystalline components later in the reaction. If relative phase abundances [such as those produced by the ZMV approach embodied in equation (3.9.26)[link]] are used, the amounts of the crystalline phases will be overestimated and this will give misleading indications about the reaction mechanism and kinetics.

Whichever method is employed, it is always necessary to examine a sample of individual results as a test of veracity rather than just accepting the suite of numbers for parameter values and QPA resulting from batch processing.

The study of Webster et al. (2013[link]) demonstrates many of these points by following the formation mechanisms of the iron-ore sinter bonding phase, SFCA-I, where SFCA = silico-ferrite of calcium and aluminium (Scarlett, Madsen et al., 2004[link]; Scarlett, Pownceby et al., 2004[link]; Webster et al., 2013[link]). The starting material, comprising a synthetic mixture of gibbsite, Al(OH)3, haematite, Fe2O3, and calcite, CaCO3, was heated to about 1573 K using an Anton Paar heating stage. The laboratory-based XRD data, collected using an Inel CPS120 diffractometer, are shown in Fig. 3.9.14[link], while the QPA results are shown in Fig. 3.9.15[link]. Both figures show that there are several phase changes, including the formation of transient intermediate phases before the final production of SFCA.

[Figure 3.9.15]

Figure 3.9.15 | top | pdf |

Results of Rietveld-based QPA of the in situ data sequence shown in Fig. 3.9.14[link] (Webster et al., 2013[link]). The relative phase abundances (upper) are derived using the Hill/Howard algorithm (Hill & Howard, 1987[link]) in equation (3.9.26)[link], while the absolute phase abundances (lower) have been derived from the external-standard approach (O'Connor & Raven, 1988[link]) embodied in equation (3.9.21)[link].

In Fig. 3.9.15[link](a) the QPA results are derived using the Hill/Howard algorithm (Hill & Howard, 1987[link]) in equation (3.9.26)[link]: this is the `default' value reported by most Rietveld analysis software and normalizes the sum of the analysed components to 100 wt%. The apparent increase in haematite concentration at about 533 and 868 K results from the decomposition of gibbsite and calcite, respectively. There are no possible mechanisms in this system that could lead to an increase in haematite concentration at these temperatures; the reported increases are an artefact derived from normalizing the sum of all analysed phases to 100 wt%. Fig. 3.9.15[link](b) shows the correct result derived using the external-standard approach (O'Connor & Raven, 1988[link]) embodied in equation (3.9.21)[link], which has placed the values on an absolute scale. Fig. 3.9.15[link] demonstrates the importance of putting the derived phase abundances on an absolute scale for a realistic derivation of reaction mechanism and kinetics.

References

Bruker AXS (2013). Topas v5: General profile and structure analysis software for powder diffraction data. Version 5. https://www.bruker.com/topas.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
O'Connor, B. H. & Raven, M. D. (1988). Application of the Rietveld refinement procedure in assaying powdered mixtures. Powder Diffr. 3, 2–6.Google Scholar
Scarlett, N. V. Y., Madsen, I. C., Pownceby, M. I. & Christensen, A. N. (2004). In situ X-ray diffraction analysis of iron ore sinter phases. J. Appl. Cryst. 37, 362–368.Google Scholar
Scarlett, N. V. Y., Pownceby, M. I., Madsen, I. C. & Christensen, A. N. (2004). Reaction sequences in the formation of silico-ferrites of calcium and aluminum in iron ore sinter. Metall. Mater. Trans. B, 35, 929–936.Google Scholar
Stinton, G. W. & Evans, J. S. O. (2007). Parametric Rietveld refinement. J. Appl. Cryst. 40, 87–95.Google Scholar
Webster, N. A. S., Pownceby, M. I. & Madsen, I. C. (2013). In situ X-ray diffraction investigation of the formation mechanisms of silico-ferrite of calcium and aluminium-I-type (SFCA-I-type) complex calcium ferrites. ISIJ Int. 53, 1334–1340.Google Scholar








































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