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. 352-353

Section 3.9.4.5. Rietveld-based methods

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

3.9.4.5. Rietveld-based methods

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The strengths and weaknesses of some of the methods described in Section 3.9.3[link] are highlighted through a study of the mechanism and kinetics of nucleation and crystal growth in the context of the Bayer process for the extraction of aluminium from bauxite ores (Webster et al., 2010[link]). Specifically, the experiments utilize synthetic Bayer liquors, consisting of Al-loaded caustic solutions to which a variety of seed material is added. Several polymorphs of Al(OH)3 (gibbsite, bayerite and nordstrandite) crystallize from solution onto the seed material. The rate of crystallization and the ratio of the phases formed depend on the sample conditions used, including the Al and caustic concentrations in solution, as well as sample temperature.

The mechanism and rate of crystallization were followed by collecting XRD data at the powder-diffraction beamline of the Australian Synchrotron4 over a period of about 3 h. The diffractometer incorporates a Mythen detector (Schmitt et al., 2003[link]) which allows for the simultaneous collection of 80° 2θ of the diffraction pattern. A wavelength of 0.826 Å was used to ensure adequate penetration of the beam in the sample. The sample environment (Madsen et al., 2005[link]; Norby et al., 1998[link]) consisted of a 1-mm quartz glass capillary containing a slurry of the seed and Bayer liquor heated to temperatures between 333 and 348 K using a hot-air blower.

The data were analysed using TOPAS (Bruker AXS, 2013[link]), where a learned-profile approach to peak modelling was used with an empirical instrument width and shape contribution determined using the NIST SRM660 LaB6 profile standard. For the samples in the study, refined parameters included 2θ zero offset, a Chebychev polynomial pattern background and, for each phase, the Rietveld scale factor, crystallite size and strain, and unit-cell dimensions.

A number of different approaches were used to extract the phase abundances at each stage of the reaction. Initially, QPA was derived using equation (3.9.26)[link]; the value that many Rietveld analysis programs output as their first estimate of phase abundance. Fig. 3.9.5[link] shows the QPA output from an in situ experiment in which goethite (FeOOH) was added as the seed.

[Figure 3.9.5]

Figure 3.9.5 | top | pdf |

The results of QPA of the in situ XRD data collected during the seeding experiments of Webster et al. (2010[link]). The values were derived using the Hill/Howard (Hill & Howard, 1987[link]) relationship in equation (3.9.26)[link]. Note the decrease in apparent goethite concentration (left axis) as the polymorphs of Al(OH)3 (right axis) crystallize from solution.

At the start of the experiment, prior to the crystallization of any of the Al(OH)3 polymorphs, Fig. 3.9.5[link] shows that the reported concentration of the goethite seed is 100 wt% since it is the only phase represented in the analysis at that time. On formation of gibbsite, bayerite and nordstrandite, the goethite concentration appears to decrease progressively to about 65 wt% while the total Al(OH)3 concentration reaches about 35 wt% at the end of the experiment. However, these figures are in disagreement with (i) the fact that goethite is unlikely to dissolve or otherwise be consumed in this system (Murray et al., 2009[link]), (ii) the known addition of goethite to the sample (14.13 wt%) and (iii) the total amount of Al(OH)3 available from solution (15.92 wt%). The problem with the QPA in this case arises from the fact that only the crystalline components are considered in the analysis and that equation (3.9.26)[link] normalizes the sum of their analysed weight fractions to unity. However, aluminium, which is in solution at the start of the run, forms crystalline phases continuously throughout the reaction after an initial induction period. In order to overcome the anomalies in the QPA results, it is necessary to consider the sample as a whole; that is, the concentration of both the solid and liquid components in the X-ray beam for the duration of the experiment.

In this sample, the concentration of the goethite seed was 14.13 wt% in the slurry injected into the sample capillary. If the assumption is made that, in this environment, goethite is unreactive and its concentration will not change during the reaction, it can be used as an internal standard to put the Al(OH)3 concentrations on an absolute basis. The QPA results derived using the internal standard or `spiked' approach in equation (3.9.25)[link] are shown in Fig. 3.9.6[link].

[Figure 3.9.6]

Figure 3.9.6 | top | pdf |

The results of QPA of the in situ XRD data collected during the seeding experiments of Webster et al. (2010[link]).The values are absolute phase abundances derived using the internal standard relationship in equation (3.9.25)[link].

The goethite concentration is fixed at the known addition (14.13 wt%) at the start of the experiment. However, the concentrations of the Al(OH)3 polymorphs are now put on an absolute scale, thus allowing derivation of more meaningful reaction mechanisms.

If, however, there is residual doubt about the reactivity of the goethite, it may be necessary to use the external standard approach embodied in equation (3.9.21)[link]. In this case, the value for the instrument constant, K, can be derived using the Rietveld scale factor, ZMV and the known addition of goethite in a rearranged equation (3.9.21)[link]. For this determination, the goethite scale factor from the first few data sets, prior to the start of the reaction, was averaged to minimize any errors that may be introduced by counting statistics. The value of the sample mass absorption coefficient [\mu_m^*] was set to an arbitrary value of unity for both the determination of K and all subsequent analyses, since the overall chemical content of the capillary, and hence the attenuation of the X-ray beam, does not change during the reaction.

This experimental work was conducted at the Australian Synchrotron where the storage-ring current was boosted every 12 h. Between these times the current, and hence the incident-beam intensity, decays, resulting in what amounts to a change in the `instrument configuration'. This requires a modification of the K value and subsequent calculation of concentration to compensate for the changing incident intensity using equation (3.9.22)[link].

Fig. 3.9.7[link] now shows the results of QPA derived from equation (3.9.22)[link]. In this case the concentrations of the Al(OH)3 polymorphs are similar to those in Fig. 3.9.6[link]. However, since the phase abundances are derived using an external standard approach, any changes in the apparent goethite concentration can now be monitored. Fig. 3.9.7[link] shows that the goethite concentration did not change significantly in the early stages of the experiment (t < 10 min) before Al(OH)3 crystallization was observed but there is a small, systematic decrease in the apparent goethite concentration as the experiment progresses. At the end of the experiment, the goethite concentration appears to be lower by about 1% relative to the concentration at the start.

[Figure 3.9.7]

Figure 3.9.7 | top | pdf |

The results of QPA of the in situ XRD data collected during the seeding experiments of Webster et al. (2010[link]). The values are absolute phase abundances derived using the external standard relationship in equation (3.9.22)[link]. Note the slight decrease in the goethite concentration (left axis) during the run.

This apparent decrease could be due to a number of causes including (i) poor correction for beam-intensity changes or (ii) solid material moving about in the capillary with some movement out of the X-ray beam. Alternatively, the decrease could be attributed to the `shielding' of the goethite from the X-ray beam by the Al(OH)3 phases as they form and coat the goethite particles. This decrease could then be used to obtain an average thickness of the Al(OH)3 phases on the seed particles. This layer was calculated to be about 5.5 µm (assuming a linear absorption coefficient of 9.5 cm−1 for gibbsite at 0.826 Å) resulting in an overall particle size of about 11 µm at the end of the run (the goethite particles are about 0.2 × 2 µm and hence do not contribute significantly to the overall particle size). These values are in good agreement with independent studies (Webster et al., 2010[link]) where the gibbsite was examined using scanning electron microscopy (SEM) techniques (Fig. 3.9.8[link]) following crystallization under similar conditions to those used here.

[Figure 3.9.8]

Figure 3.9.8 | top | pdf |

SEM image of Al(OH)3 (grey hexagon) which has crystallized on goethite seed (light grey needles) (Webster et al., 2010[link]).

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
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Murray, J., Kirwan, L., Loan, M. & Hodnett, B. K. (2009). In-situ synchrotron diffraction study of the hydrothermal transformation of goethite to hematite in sodium aluminate solutions. Hydrometallurgy, 95, 239–246.Google Scholar
Norby, P., Cahill, C., Koleda, C. & Parise, J. B. (1998). A reaction cell for in situ studies of hydrothermal titration. J. Appl. Cryst. 31, 481–483.Google Scholar
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Webster, N. A. S., Madsen, I. C., Loan, M. J., Knott, R. B., Naim, F., Wallwork, K. S. & Kimpton, J. A. (2010). An investigation of goethite-seeded Al(OH)3 precipitation using in situ X-ray diffraction and Rietveld-based quantitative phase analysis. J. Appl. Cryst. 43, 466–472.Google Scholar








































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