Tables for
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. 277

Section Conograph: indexing via the topographs method

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: Conograph: indexing via the topographs method

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Conograph is based on the topographs method, and its main functions are the determination of the primitive unit cell and lattice symmetry, and refinement of lattice parameters. Among the main features we note:

  • (1) A new Bravais-lattice determination algorithm (Oishi-Tomiyasu, 2012[link]), which has been proved to be stable with respect to peak-position errors under very general conditions. The algorithm applies the Minkowski reduction to primitive cells and the Delaunay reduction (Delaunay, 1933[link]) to face-centred, body-centred, rhombohedral and base-centred cells in such a way that the computational efficiency of the process is better than the Andrews & Bernstein (1988[link]) method.

  • (2) The two figures of merit [M_n^{\rm Rev}] and [M_n^{\rm Sym}] proposed by Oishi-Tomiyasu (2013[link]) are used for for selecting the true unit cells. They are also used to estimate the zero-point shift.

  • (3) The use of many observed peaks in the default setting, which aims to make Conograph robust against dominant zones and missing or false peaks (Oishi-Tomiyasu, 2014b[link]).

  • (4) The method for exhaustively searching unit cells that involve geometrical ambiguity (Oishi-Tomiyasu, 2014a[link], 2016[link]). The geometrical ambiguities that are detected also include lattices with very similar calculated lines, because of the error tolerance in the d spacings.

Programs that use only the measured positions of peak maxima are particularly vulnerable to experimental errors in the measured peak positions and to the presence of impurity peaks. For these reasons, at the end of the 1990s new indexing strategies were developed that do not require the peak locations in the experimental pattern. These approaches are completely different from the methods described above because they use the whole diffraction profile. They try to explore the parameter space (direct space) exhaustively by applying different optimization techniques in order to find the cells in best agreement with the experimental powder diffraction pattern. Some of the most widely used indexing programs in direct space are described here.


Andrews, L. C. & Bernstein, H. J. (1988). Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections. Acta Cryst. A44, 1009–1018.Google Scholar
Delaunay, B. (1933). Neue Darstellung der geometrischen Kristallographie. Z. Kristallogr. 84, 109–149.Google Scholar
Oishi-Tomiyasu, R. (2012). Rapid Bravais-lattice determination algorithm for lattice parameters containing large observation errors. Acta Cryst. A68, 525–535.Google Scholar
Oishi-Tomiyasu, R. (2013). Reversed de Wolff figure of merit and its application to powder indexing solutions. J. Appl. Cryst. 46, 1277–1282.Google Scholar
Oishi-Tomiyasu, R. (2014a). Method to generate all the geometrical ambiguities of powder indexing solutions. J. Appl. Cryst. 47, 2055–2059.Google Scholar
Oishi-Tomiyasu, R. (2014b). Robust powder auto-indexing using many peaks. J. Appl. Cryst. 47, 593–598.Google Scholar
Oishi-Tomiyasu, R. (2016). A table of geometrical ambiguities in powder indexing obtained by exhaustive search. Acta Cryst. A72, 73–80.Google Scholar

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