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. 362-364

Section 3.9.9. QPA using energy-dispersive diffraction data

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:

3.9.9. QPA using energy-dispersive diffraction data

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Energy-dispersive diffraction (EDD) involves the use of high-energy white-beam radiation, often from a synchrotron source. This provides very high penetration and is, therefore, ideal as a probe to examine the internal features of relatively large objects (Barnes et al., 2000[link]; Cernik et al., 2011[link]; Hall et al., 1998[link], 2000[link]). In an experimental arrangement such as that in Fig. 3.9.16[link], diffraction data can be measured by energy-dispersive detectors producing a spectrum of diffracted intensity as a function of energy.

[Figure 3.9.16]

Figure 3.9.16 | top | pdf |

Basic experimental arrangement for energy-dispersive diffraction. The length of the active area or lozenge (dark grey region), L, is given by the function relating the incident- and diffracted-beam heights (Hi and Hd, respectively) and the angle of diffraction (2θ).

Traditional angle-dispersive diffraction (ADD) satisfies Bragg's law by using a fixed wavelength and varying 2θ to map the d-spacings. In contrast, EDD data are collected directly on an energy scale at a constant 2θ and the energy is measured to map the d-spacings. This impinges upon the use of Rietveld methodology for QPA since, in contrast to ADD, the structure factors now vary as a function of energy. Energy is related to wavelength via[E\ ({\rm keV})= {{hc} \over \lambda } \simeq {{12.395} \over \lambda }, \eqno(3.9.46)]where E is the energy of the incident radiation in keV, h is Planck's constant, c is the speed of light and λ is the wavelength associated with that energy in ångstroms. Rearrangement of equation (3.9.46)[link] and substitution for λ in Bragg's law enables the mapping of the measured energy scale to d-spacings:[E\ ({\rm keV})= {{6.197} \over {d\sin \theta }}, \eqno(3.9.47)]where 2θ is the angle between the incident beam and the detector slit.

EDD data can be analysed using structureless profile-fitting methods such as those of Le Bail et al. (see Chapter 3.5[link] ) once the energy scale has been converted to a d-spacing scale (Frost & Fei, 1999[link]; Larson & Von Dreele, 2004[link]; Zhao et al., 1997[link]). If the distribution of intensities in the incident spectrum can be measured, it is possible to normalize the EDD data, correct for absorption and convert the pattern to an ADD form using a `dummy' wavelength (Ballirano & Caminiti, 2001[link]). Access to the incident spectrum, however, is not always possible, especially at synchrotron-radiation sources where the highly intense incident beam could damage the detector.

An alternative approach is to model the pattern directly on the energy scale via equation (3.9.47)[link] (Rowles et al., 2012[link]; Scarlett et al., 2009[link]) and extract phase abundances using the methodologies described earlier in this chapter.

However, the major impediment to achieving this is the nonlinearity of the intensity distribution in the incident spectrum. This is due to (i) the nonlinear distribution of intensity as a function of energy in the incident beam, (ii) nonlinear detector responses (Bordas et al., 1977[link]) and (iii) absorption along the beam path (by the sample and air), which skews the energy distribution to the higher energies. This overall nonlinearity can be modelled empirically by functions such a lognormal curve (Bordas et al., 1977[link]; Buras et al., 1979[link]) or by an expansion of a power function (Glazer et al., 1978[link]). Alternatively, it may be determined experimentally by the use of standards measured under the same conditions as the experiment (Scarlett et al., 2009[link]). This latter approach allows some separation of the contributions from the instrument and the sample, and allows some degrees of freedom in the refinement of sample-related parameters that may be of benefit in dynamic experiments. Other contributions to the diffraction pattern that must also be accounted for include any fluorescence peaks arising from the sample or shielding or collimators, and any detector escape peaks from both diffracted and fluorescence peaks. Fluorescence peak positions and relative intensities should be constant throughout the measurement and may therefore be modelled using a fixed `peak group' whose overall intensity can be refined during analysis. Escape peaks can be accounted for by the inclusion of a second phase identical to the parent phase but with an independent scale factor and a constant energy offset determined by the nature of the detector (Rowles et al., 2012[link]).

Currently, few Rietveld software packages are capable of dealing directly with the differences between EDD and ADD, specifically (i) the variance of structure factors as a function of energy, (ii) the nonlinear distribution of intensity in the incident beam as a function of energy further modified by a nonlinear detector response, and (iii) the preferential absorption of lower-energy X-rays by the sample/air. TOPAS (Bruker AXS, 2013[link]) embodies algorithms that allow the pattern to be modelled directly on the energy scale and also the inclusion of equations to account for intensity variations arising from the experimental conditions. This allows quantification from such data to be achieved directly using Rietveld-based crystal-structure modelling incorporating the Hill and Howard algorithm in equation (3.9.26)[link] (Hill & Howard, 1987[link]). The application of TOPAS to a complex EDD experiment investigating the changes to the anode during molten-salt electrochemistry conducted in molten CaCl2 at about 1223 K has been described by Rowles et al. (2012[link]) and Styles et al. (2012[link]).


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