International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.4, p. 275
Section 3.4.3.2. Non-traditional indexing methods^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy |
New indexing procedures that provide alternatives to the traditional approaches outlined in Section 3.4.3.1 have recently been proposed.
This method (Oishi et al., 2009) is based on the Ito equation (de Wolff, 1957):where Q(h) is the length of the reciprocal vector corresponding to the Miller index vector h = (hkl). It uses Conway's topograph (Conway & Fung, 1997), a connected tree obtained by associating a graph to each equation of type (3.4.9) and consisting of infinite directed edges. According to Ito's method, if quadrupoles (Q_{1}, Q_{2}, Q_{3}, Q_{4}) detected among the observed Q_{i} values satisfy the condition 2(Q_{1} + Q_{2}) = Q_{3} + Q_{4}, two Miller-index vectors h_{1} and h_{2} are expected to exist such that Q_{1} = Q(h_{1}), Q_{2} = Q(h_{2}), Q_{3} = Q(h_{1} − h_{2}) and Q_{4} = Q(h_{1} + h_{2}). If an additional value Q_{5} satisfying the condition 2(Q_{1} + Q_{4}) = Q_{2} + Q_{5} is found, the graph of the quadrupole (Q_{1}, Q_{2}, Q_{3}, Q_{4}) grows via the addition of the Q_{5} contribution; this procedure is iterated. If topographs share a Q value that corresponds to the same reciprocal-lattice vector, then a three-dimensional lattice is derived containing the two-dimensional lattices associated with the original topographs. Three-dimensional lattices are also obtained by combining topographs. The probability that topographs correspond to the correct cell increases with the number of edges of the graph structure. The method is claimed by the authors to be insensitive to the presence of impurity peaks.
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.
The use of genetic algorithms (GAs) for solving the indexing problem was proposed by Tam & Compton (1995) and Paszkowicz (1996). Since then, Kariuki and co-workers (Kariuki et al., 1999) 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 ) 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 and searching for the global minimum of (see Section 3.4.4.3.2). 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 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.
The Monte Carlo approach has also been applied to indexing powder diffraction patterns (Le Bail, 2004; Bergmann et al., 2004; Le Bail, 2008). 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. R_{p} and McM_{20}, see Section 3.4.2.1). The main drawback of this approach is the significant computing time required, in particular for triclinic systems.
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