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. 3.6, p. 290

Section Beyond the Warren–Averbach method

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Beyond the Warren–Averbach method

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The work of Alexander (see, for example, Alexander, 1954[link]; Klug & Alexander, 1974[link]; and references therein) was definitely pioneering here. Alexander proposed a set of formulae for the synthesis of the instrumental profile, with the aim of obtaining a correction curve to subtract the instrumental contribution to the measured breadth of the profiles (thus improving the accuracy of the size determination). The true power of this idea was not fully exploited, as the Scherrer formula was still used for microstructure analysis. A few decades later, Adler, Houska and Smith (Adler & Houska, 1979[link]; Houska & Smith, 1981[link]) proposed the use of simplified analytical functions to describe the instrument, size and strain broadening and to perform the convolution of the equation numerically via Gauss–Legendre quadrature. Rao & Houska (1986[link]) improved the procedure by carrying out part of the integration analytically (for monodisperse spheres). The microstructure parameters are directly obtained from a fit of this numerical peak to the experimental data. The fit partly solves the problem of peak overlap: in traditional methods it is in fact impossible to establish the extent of overlap between the peaks and therefore to correctly extract the area or maximum intensity. The method is a major step forward, but is still related to the Warren–Averbach approach, as just two multiple-order peaks are considered.

Cheary & Coelho (1992[link], 1994[link], 1998a[link],b[link]) pushed the idea forward with the fundamental parameters approach (FPA). The FPA is based on the intuition of Alexander (1954[link]) and the general idea of the Rietveld (1969[link]) method: the calculation speed is greatly improved to facilitate widespread use. The convolution is performed directly in 2θ space, where instrumental aberrations, which were extensively explored by these researchers, occur. Very simplistic models were employed to describe the broadening due to the specimen; the whole pattern (and therefore all of the measured information) is considered in place of one or more peaks and of the extracted information. For structural analysis and for the Rietveld method, the FPA is a huge step forward, as it allows a more accurate determination of lattice parameters and integrated intensities. Moreover, it enables some line-profile analysis on low-quality patterns or on data affected by strong peak asymmetry.


Adler, T. & Houska, C. R. (1979). Simplifications in the X-ray line-shape analysis. J. Appl. Phys. 50, 3282–3287.Google Scholar
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Cheary, R. W. & Coelho, A. (1992). A fundamental parameters approach to X-ray line-profile fitting. J. Appl. Cryst. 25, 109–121.Google Scholar
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Rietveld, H. M. (1969). A profile refinement method for nuclear and magnetic structures. J. Appl. Cryst. 2, 65–71.Google Scholar

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