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, p. 275

Section 3.4.3.2.2. Global-optimization methods

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

3.4.3.2.2. Global-optimization methods

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Global-optimization methods, widely adopted for solving crystal structures from powder data, have also been successfully applied to indexing. Among them, we provide brief descriptions of genetic algorithms, and Monte Carlo and grid-search methods.

3.4.3.2.2.1. Genetic-algorithm search method

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The use of genetic algorithms (GAs) for solving the indexing problem was proposed by Tam & Compton (1995[link]) and Paszkowicz (1996[link]). Since then, Kariuki and co-workers (Kariuki et al., 1999[link]) have combined GAs with a whole-profile-fitting procedure for indexing powder diffraction patterns. This approach exploits the information of the full powder diffraction pattern. It is inspired by the Darwinian evolutionary principle based on mating, mutation and natural selection of the member of a population that survives and evolves to improve future generations. The initial population consists of a set of trial cell parameters, chosen randomly within a given volume range; a full pattern-decomposition process is performed using the Le Bail algorithm (Chapter 3.5[link] ) and the agreement between the calculated and observed profiles is derived and used for assessing the goodness of an individual member (i.e., a set of unit-cell parameters). The most plausible cell is therefore found by exploring a six-dimensional hypersurface [R_{wp}^{\prime}(a,b,c,\alpha, \beta, \gamma)] and searching for the global minimum of [R_{wp}^{\prime}] (see Section 3.4.4.3.2[link]). In contrast to the main traditional methods, whose outcomes depend on the reliability of a set of peak positions, this procedure has the advantage of being insensitive to the presence of small impurity peaks that have a negligible influence on the agreement factor between the experimental and calculated profiles: the global minimum of [R_{wp}^{\prime}] is reached if the majority phase is correctly indexed. The main disadvantage of the method is the computing time required, in particular in the case of low symmetry.

3.4.3.2.2.2. Monte Carlo search method

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The Monte Carlo approach has also been applied to indexing powder diffraction patterns (Le Bail, 2004[link]; Bergmann et al., 2004[link]; Le Bail, 2008[link]). It exploits all the information contained in the full pattern, randomly generates and selects trial cell parameters, and calculates peak positions to which it assigns the corresponding Miller indices. An idealized powder pattern consisting of peak positions d and extracted intensities I is considered to test the trial cell. The cell reliability is assessed by suitable figures of merit (e.g. Rp and McM20, see Section 3.4.2.1[link]). The main drawback of this approach is the significant computing time required, in particular for triclinic systems.

3.4.3.2.2.3. Grid-search method

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This performs an iterated `step-and-repeat search' in the parameter space. It has the advantage of being flexible, exhaustive and not particularly sensitive to impurities or errors, and the disadvantage of being slow (Shirley, 2003[link]).

References

Bergmann, J., Le Bail, A., Shirley, R. & Zlokazov, V. (2004). Renewed interest in powder diffraction data indexing. Z. Kristallogr. 219, 783–790.Google Scholar
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
Le Bail, A. (2004). Monte Carlo indexing with McMaille. Powder Diffr. 19, 249–254.Google Scholar
Le Bail, A. (2008). Structure solution. In Principles and Applications of Powder Diffraction, edited by A. Clearfield, J. H. Reibenspies & N. Bhuvanesh, pp. 261–309. Oxford: Wiley-Blackwell.Google Scholar
Paszkowicz, W. (1996). Application of the smooth genetic algorithm for indexing powder patterns – tests for the orthorhombic system. Mater. Sci. Forum, 228–231, 19–24.Google Scholar
Shirley, R. (2003). Overview of powder-indexing program algorithms (history and strengths and weaknesses). IUCr Comput. Comm. Newsl. 2, 48–54. http://www.iucr.org/resources/commissions/crystallographic-computing/newsletters/2 .Google Scholar
Tam, K. Y. & Compton, R. G. (1995). GAMATCH – a genetic algorithm-based program for indexing crystal faces. J. Appl. Cryst. 28, 640–645.Google Scholar








































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