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. 274-275
Section 3.4.3.1.5. Successive-dichotomy search method^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy |
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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