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. 3.8, p. 326

Section 3.8.2.4. Full-profile qualitative pattern matching

C. J. Gilmore,a G. Barra and W. Donga*

aDepartment of Chemistry, University of Glasgow, University Avenue, Glasgow, G12 8QQ, UK
Correspondence e-mail:  chris@chem.gla.ac.uk

3.8.2.4. Full-profile qualitative pattern matching

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Before performing pattern matching, some data pre-processing may be necessary. In order not to produce artefacts, this should be minimized. Typical pre-processing activities are:

  • (1) The data are normalized such that the maximum peak intensity is 1.0.

  • (2) The patterns need to be interpolated if necessary to have common increments in 2θ. High-order polynomials using Neville's algorithm can be used for this (Press et al., 2007[link]).

  • (3) If backgrounds are large they should be removed. High-throughput data are often very noisy because of low counting times and the sample itself. If this is the case, smoothing of the data can be carried out. The SURE (Stein's Unbiased Risk Estimate) thresholding procedure (Donoho & Johnstone, 1995[link]; Ogden, 1997[link]) employing wavelets is ideal for this task since it does not introduce potentially damaging artefacts, for example ringing around peaks (Barr et al., 2004a[link]; Smrčok et al., 1999[link]).

After pre-processing, which needs to be carried out in an identical way for each sample, the following steps are carried out:

  • (1) The intersecting 2θ range of the two data sets is calculated, and each of the pattern correlation coefficients is calculated using only this region.

  • (2) A minimum intensity is set, below which profile data are set to zero. This reduces the contribution of background noise to the matching process without reducing the discriminating power of the method. We usually set this to 0.1Imax as a default, where Imax is the maximum measured intensity.

  • (3) The Pearson correlation coefficient is calculated.

  • (4) The Spearman R is computed in the same way.

  • (5) An overall ρ value is calculated using (3.8.3)[link].

  • (6) A shift in 2θ values between patterns is often observed, arising from equipment settings and data-collection protocols. Three possible simple corrections are[\Delta \left({2\theta } \right) = {a_0} + {a_1}\cos \theta, \eqno(3.8.4)]which corrects for the zero-point error via the a0 term and, via the a1 cos θ term, for varying sample heights in reflection mode, or[\Delta \left({2\theta } \right) = {a_0} + {a_1}\sin \theta, \eqno(3.8.5)]which corrects for transparency errors, for example, and[\Delta \left({2\theta } \right) = {a_0} + {a_1}\sin 2\theta, \eqno(3.8.6)]which provides transparency coupled with thick specimen error corrections, where a0 and a1 are constants that can be determined by shifting patterns to maximize their overlap as measured by ρ. It is difficult to obtain suitable expressions for the derivatives [\partial {a_0}/\partial {\rho_{ij}}] and [\partial {a_1}/\partial {\rho_{ij}}] for use in the optimization, so we use the downhill simplex method (Nelder & Mead, 1965[link]), which does not require their calculation.

References

Donoho, D. L. & Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90, 1200–1224.Google Scholar
Gilmore, C. J., Barr, G. & Paisley, W. (2004). High-throughput powder diffraction. I. A new approach to qualitative and quantitative powder diffraction pattern analysis using full pattern profiles. J. Appl. Cryst. 37, 231–242.Google Scholar
Nelder, J. A. & Mead, R. (1965). A simplex method for function minimization. Comput. J. 7, 308–313.Google Scholar
Ogden, R. T. (1997). Essential Wavelets for Statistical Applications and Data Analysis, pp. 144–148. Boston: Birkhäuser.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. (2007). Numerical Recipes. 3rd ed. Cambridge University Press..Google Scholar
Smrčok, Ĺ., Ďurík, M. & Jorík, V. (1999). Wavelet denoising of powder diffraction patterns. Powder Diffr. 14, 300–304.Google Scholar








































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