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. 3.4, pp. 277-278

Section 3.4.4.3. Non-traditional indexing programs

A. Altomare,a* C. Cuocci,a A. Moliternia and R. Rizzia

aInstitute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy
Correspondence e-mail:  angela.altomare@ic.cnr.it

3.4.4.3. Non-traditional indexing programs

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The indexing programs described above are based on using, for a limited number of lines, the measured positions of peak maxima as directly obtained from the experimental powder diffraction pattern. Conograph (Oishi-Tomiyasu, 2014b[link]), which has been more recently proposed, also belongs to that group of programs. A brief description of Conograph follows

3.4.4.3.1. Conograph: indexing via the topographs method

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Conograph is based on the topographs method, and its main functions are the determination of the primitive unit cell and lattice symmetry, and refinement of lattice parameters. Among the main features we note:

  • (1) A new Bravais-lattice determination algorithm (Oishi-Tomiyasu, 2012[link]), which has been proved to be stable with respect to peak-position errors under very general conditions. The algorithm applies the Minkowski reduction to primitive cells and the Delaunay reduction (Delaunay, 1933[link]) to face-centred, body-centred, rhombohedral and base-centred cells in such a way that the computational efficiency of the process is better than the Andrews & Bernstein (1988[link]) method.

  • (2) The two figures of merit [M_n^{\rm Rev}] and [M_n^{\rm Sym}] proposed by Oishi-Tomiyasu (2013[link]) are used for for selecting the true unit cells. They are also used to estimate the zero-point shift.

  • (3) The use of many observed peaks in the default setting, which aims to make Conograph robust against dominant zones and missing or false peaks (Oishi-Tomiyasu, 2014b[link]).

  • (4) The method for exhaustively searching unit cells that involve geometrical ambiguity (Oishi-Tomiyasu, 2014a[link], 2016[link]). The geometrical ambiguities that are detected also include lattices with very similar calculated lines, because of the error tolerance in the d spacings.

Programs that use only the measured positions of peak maxima are particularly vulnerable to experimental errors in the measured peak positions and to the presence of impurity peaks. For these reasons, at the end of the 1990s new indexing strategies were developed that do not require the peak locations in the experimental pattern. These approaches are completely different from the methods described above because they use the whole diffraction profile. They try to explore the parameter space (direct space) exhaustively by applying different optimization techniques in order to find the cells in best agreement with the experimental powder diffraction pattern. Some of the most widely used indexing programs in direct space are described here.

3.4.4.3.2. GAIN: indexing via a genetic-algorithm search method

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The use of genetic algorithms (GAs) for indexing powder diffraction data by exploiting the diffraction geometry (as in the traditional indexing methods) was firstly proposed by Tam & Compton (1995[link]) and Paszkowicz (1996[link]). Subsequently, Kariuki et al. (1999[link]) applied GA techniques by using whole profile fitting with the aim of exploring the parameter space {a, b, c, α, β, γ} and finding the global minimum of the R-factor {a, b, c, α, β, γ} hypersurface, yielding the parameter set able to generate the best agreement between the observed and calculated powder diffraction patterns.

This new strategy has been implemented in the program GAIN (Harris et al., 2000[link]), whose main features are:

  • (1) Starting from a population of Np sets of lattice parameters and using the evolutionary operations of mating, mutation and natural selection, the population is allowed to evolve through several generations, with the aim of generating sets of possible trial cell parameters.

  • (2) The search procedure, using a GA, is performed in restricted, sensible cell-volume ranges consistent with the knowledge of the system under study.

  • (3) For each set of trial parameters a calculated powder diffraction pattern is constructed. The peak positions and parameters describing the shape and width of each peak are used in the Le Bail profile-fitting procedure (Chapter 3.5[link] ).

  • (4) The pattern is split into different regions (defined by the user), and the weighted profile R factor is calculated for each region; all the values are summed to obtain the overall [R_{\rm wp}^{\prime}]:[R_{\rm wp}^{\prime} = \sum\limits_{\rm regions} \left[{\textstyle\sum_i w_i(y_i - y_{ci})^2 \over \textstyle\sum_i w_iy_i^2} \right] ^{1 / 2}, ]where the summation is over the regions, i runs over the experimental points belonging to each region and [{y_i}] and [{y_{ci}}] are the observed and calculated profile at the ith experimental step, respectively. Via the [R_{\rm wp}^{\prime}] formula the residual for each region is scaled according to the total intensity in the region, so a region with only low-intensity peaks can make an important contribution to [R_{\rm wp}^{\prime}].

This approach is robust at handling the problems that may affect the experimental powder pattern: peak overlap, (hkl)-dependent effects and zero-point errors. It is time consuming (particularly in the case of low symmetry) but not very sensitive to the presence of minority impurity phases.

3.4.4.3.3. McMaille: indexing via a Monte Carlo search method

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The information in the whole powder diffraction profile is exploited by the program McMaille (Le Bail, 2004[link]), which is based on the random generation of cell parameters and uses the Monte Carlo optimization technique. Once the trial cell parameters have been generated and the Miller indices and the peak positions have been calculated, the quality of the cell is assessed by using, as figure of merit, the conventional Rietveld profile reliability factor Rp (Young, 1993[link]) or McM20 (see Section 3.4.2.1[link]). The program uses some tricks that can increase the success of the Monte Carlo algorithm:

  • (1) Only the trial cells corresponding to a value of Rp that is smaller than a user-defined value (∼50%) are retained for successive refinement.

  • (2) If all the observed peaks, except for a user-defined number of tolerated impurity peaks, are `explained' whatever the Rp value, the cell is retained for successive examination.

  • (3) If either of the conditions (1) or (2) is fulfilled, the cell parameters are randomly changed in 200 to 5000 attempts (for cubic to triclinic cases, respectively) in which small random parameter variations via the Monte Carlo algorithm are carried out. The new parameters are preserved if an improvement of Rp is verified in 85% of the attempts.

This procedure is not sensitive to impurity lines, provided that the sum of their intensities is less than 10–15% of the total intensity. A zero-point error up to 0.05° is tolerated. To reduce the long computing time required to successfully complete the procedure, a significant increase in speed has been obtained by using idealized profiles generated by applying simplified line profiles to extracted line positions. A parallelized version of McMaille has also been developed. The indexing problem can usually be solved in few minutes if: (a) no triclinic symmetry is handled (because this requires more computing time); (b) the cell volume is less than 2000 Å3; (c) no cell length is longer than 20 Å.

References

Andrews, L. C. & Bernstein, H. J. (1988). Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections. Acta Cryst. A44, 1009–1018.Google Scholar
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Kariuki, B. M., Belmonte, S. A., McMahon, M. I., Johnston, R. L., Harris, K. D. M. & Nelmes, R. J. (1999). A new approach for indexing powder diffraction data based on whole-profile fitting and global optimization using a genetic algorithm. J. Synchrotron Rad. 6, 87–92.Google Scholar
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