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. 274
Section 3.4.3.1.3. Index-heuristics strategy^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy |
The index-heuristics strategy searches for the correct cell via a trial-and-error approach, assigning tentative Miller indices to a few experimental peak positions (basis lines), usually belonging to the low 2θ region of the experimental pattern. It was first proposed by Werner (1964), then successively refined (Werner et al., 1985) and made more robust and effective (Altomare et al., 2000, 2008, 2009). This approach, which works in the index space, was defined by Shirley as semi-exhaustive (Shirley, 1980). The search starts from the highest-symmetry crystal system (cubic) and, if no plausible solution is found, it is extended to lower symmetry down to triclinic. The number of selected basis lines increases as the crystal symmetry lowers. A dominant zone occurs when one cell axis is significantly shorter than the other two; in this case most of the first observed lines (in terms of increasing 2θ_{hkl} values) can be indexed with a common zero Miller index. Special short-axis tests, aimed at finding two-dimensional lattices, have been proposed for monoclinic symmetry in order to detect the presence of dominant zones (Werner et al., 1985). The index-heuristics method is based on the main indexing equation [see equation (3.4.2)] that can be rewritten (Werner et al., 1985) aswhere {x_{i}} = X is the vector of unknown parameters, which are derived by solving a system of linear equationswhere M is a matrix of Miller indices and Q is the vector of the selected Q(hkl) values corresponding to the basis lines. The dimensions of M, X and Q change according to the assumed symmetry. From the inverse matrix M^{−1} the corresponding X is obtained via X = M^{−1}Q. In the case of monoclinic and higher symmetry, {x_{i}} are calculated by Cramer's rule. Different X vectors are derived by using a different selection of basis lines. The possible solutions are checked by using the full list of peak positions (up to the first 25 experimental lines). The method is sensitive to errors on peak positions and to the presence of impurities (the presence of only one impurity peak is not critical). The correctness of the {x_{i}} strongly depends on the accuracy of the observed Q values, especially for low-2θ region lines, which are the most dominant ones for this indexing procedure. The possibility of testing different combinations of basis-line sets enables the correct cell to be found by bypassing the cases for which errors in the basis lines occur.
The method has been recently enhanced (Altomare et al., 2000, 2009) by introducing new procedures that are able to increase the probability of successful indexing (see Section 3.4.4.2.1); among them are: (1) a correction for systematic errors in the experimental 2θ values (positive and negative trial 2θ zero shifts are taken into account); this correction should, in principle, describe a real diffractometer error; in practice, it also approximates the specimen displacement error well (perhaps coupled with transparency for organic samples); (2) a more intensive search in solution space for orthorhombic and monoclinic systems; (3) an improvement of the triclinic search; (4) a new figure of merit, WRIP20, which is more powerful than M_{20} in identifying the correct solution among a set of possible ones (see Section 3.4.2.1); (5) a check for geometrical ambiguities; (6) an automatic refinement of the possible cells; and (7) a statistical study of the parity of the Miller indices, performed at the end of the cell refinement, aimed at detecting doubled axes or additional lattice points (for A-, B-, C-, I-, R- or F-centred cells) (such information is used in the successive steps).
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