International
Tables for
Crystallography
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. 275

## Section 3.4.3.2.1. 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:  angela.altomare@ic.cnr.it

#### 3.4.3.2.1. The topographs method

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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 (Q1, Q2, Q3, Q4) detected among the observed Qi values satisfy the condition 2(Q1 + Q2) = Q3 + Q4, two Miller-index vectors h1 and h2 are expected to exist such that Q1 = Q(h1), Q2 = Q(h2), Q3 = Q(h1h2) and Q4 = Q(h1 + h2). If an additional value Q5 satisfying the condition 2(Q1 + Q4) = Q2 + Q5 is found, the graph of the quadrupole (Q1, Q2, Q3, Q4) grows via the addition of the Q5 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.

### References

Conway, J. H. & Fung, F. Y. C. (1997). The Sensual (Quadratic) Form. Washington, DC: The Mathematical Association of America.Google Scholar
Oishi, R., Yonemura, M., Hoshikawa, A., Ishigaki, T., Mori, K., Torii, S., Morishima, T. & Kamiyama, T. (2009). New approach to the indexing of powder diffraction patterns using topographs. Z. Kristallogr. Suppl. 30, 15–20.Google Scholar
Wolff, P. M. de (1957). On the determination of unit-cell dimensions from powder diffraction patterns. Acta Cryst. 10, 590–595.Google Scholar