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, pp. 273-275
Section 3.4.3.1. Traditional indexing methods^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy |
The traditional indexing approaches adopted over the last century are based on the following strategies and search methods: (1) zone indexing, (2) SIW heuristic, (3) index heuristics, (4) index permutation and (5) successive dichotomy. All of them exploit information about a limited number of observed peak positions.
The zone-indexing strategy was originally developed by Runge (1917), successively proposed by Ito (1949, 1950), generalized by de Wolff (1957, 1958) and enhanced by Visser (1969). This approach is based on the search for zones, i.e., crystallographic planes, in the reciprocal lattice, defined by the origin O and two lattice points. If and are two vectors in reciprocal space, i.e. the positional vectors of the lattice points A and A′, they describe a zone containing any lattice point B whose positional vector is of type where m and n are positive integers. If ω is the angle between and , the squared distance of B from O (i.e., ) can be expressed by (de Wolff, 1958; Visser, 1969)where and are the squared distances of A and A′ from O, respectively, and . R can be derived as
The method is applied as follows: and are chosen among the first experimental Q_{i} values; the {Q_{i}}, up to a reasonable resolution, are introduced in (3.4.7) in place of ; and a few positive integer values are assigned to m and n. Equation (3.4.7) provides a large number of R values; equal R values (within error limits) define a zone, for which the ω angle can easily be calculated. The search for zones is performed using different () pairs. The R values that are obtained many times identify the most important crystallographic zones. The zones are sorted according to a quality figure, enabling selection of the best ones. In order to find the lattice, all possible combinations of the best zones are tried. For every pair of zones the intersection line is found, then the angle between them is determined and the lattice is obtained.
The method has the advantage of being very efficient for indexing low-symmetry patterns. The main disadvantage is its sensitivity to errors in the peak positions, particularly in the low 2θ region.
This needs only one single well-established zone, then it arbitrarily chooses the 001 line from the first-level lines. The indexing problem is thus lowered to two dimensions and an exhaustive search is carried out.
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).
This strategy was proposed by Taupin (1973), and is based on a systematic permutation of indices associated to observed lines for obtaining candidate cells. Because this trial-and-error strategy is similar to the index-heuristics approach, we do not describe it here.
The successive-dichotomy method, first developed by Louër & Louër (1972), is based on an exhaustive strategy working in direct space (except for triclinic systems, where it operates in reciprocal space) by varying the lengths of the cell axes and the interaxial angles within finite intervals. The search for the correct cell is performed in an n-dimensional domain D (where n is the number of cell parameters to be determined). If no solution belongs to D, the domain is discarded and the ranges for the allowed values of cell parameters are increased; on the contrary, if D contains a possible solution, it is explored further by dividing the domain into 2^{n} subdomains via a successive-dichotomy procedure. Each subdomain is analyzed and discarded if it does not contain a solution. The method was originally applied to orthorhombic and higher-symmetry systems (Louër & Louër, 1972), but it has been successively extended to monoclinic (Louër & Vargas, 1982) and to triclinic systems (Boultif & Louër, 1991). The search can be performed starting from cubic then moving down to lower symmetries (except for triclinic) by partitioning the space into shells of volume ΔV = 400 A^{3}. For triclinic symmetry ΔV is related to the volume V_{est} suggested by the method proposed by Smith (1977), which is able to estimate the unit-cell volume from only one line in the pattern:where d_{N} is the value for the Nth observed line; in the case N = 20 the triclinic cell volume is .
Let us consider, as an example, the monoclinic case; in terms of direct cell parameters, Q(hkl) is given by (Boultif & Louër, 1991)where , , , and B = b. The search using the successive-dichotomy method is performed in a four-dimensional space that is covered by increasing the integer values i, l, m and n in the intervals [A_{−}, A_{+}] = [A_{−} = A_{0} + ip, A_{+} = A_{−} + p], [B_{−}, B_{+}] = [B_{−} = B_{0} + lp, B_{+} = B_{−} + p], [C_{−}, C_{+}] = [C_{−} = C_{0} + mp, C_{+} = C_{−} + p] and [β_{−}, β_{+}] = [β_{−} = 90 + nθ, β_{+} = β_{−} + θ], where the step values of p and θ are 0.4 Å and 5°, respectively, and A_{0}, B_{0} and C_{0} are the lowest values of A, B and C (based on the positions of the lowest-angle peaks), respectively. Each quartet of intervals defines a domain D and, by taking into account the current limits for the parameters A, B, C and β, a calculated pattern is generated, not in terms of discrete Q(hkl) values but of allowed intervals . D is retained only if the observed Q_{i} values belong to the range [Q_{−}(hkl) − ΔQ_{i}, Q_{+}(hkl) + ΔQ_{i}], where ΔQ_{i} is the absolute error of the observed lines (i.e., impurity lines are not tolerated). If D has been accepted, it is divided into 2^{4} subdomains by halving the original intervals [A_{−}, A_{+}], [B_{−}, B_{+}], [C_{−}, C_{+}] and [β_{−}, β_{+}] and new limits are calculated; if a possible solution is found, the dichotomy method is applied iteratively. In case of triclinic symmetry the expression for Q(hkl) in terms of direct cell parameters is too complicated to be treated via the successive-dichotomy method; therefore the basic indexing equation (3.4.2) is used. In this case, the [Q_{−}(hkl), Q_{+}(hkl)] intervals are set in reciprocal space according to the A_{ij} parameters of (3.4.2). To reduce computing time the following restrictions are put on the (hkl) Miller indices associated with the observed lines: (1) maximum h, k, l values equal to 2 in case of the first five lines; (2) h + k + l < 3 for the first two lines.
The outcome of the successive-dichotomy method is not strongly influenced by the presence of a dominant zone. New approaches have been devoted to overcome the limitations of the method with a strict dependence on data accuracy and on impurities (Boultif & Louër, 2004; Louër & Boultif, 2006, 2007), see Section 3.4.4.2).
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