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. 275
Section 3.4.3.2.1. The topographs method^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy |
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 (Q_{1}, Q_{2}, Q_{3}, Q_{4}) detected among the observed Q_{i} values satisfy the condition 2(Q_{1} + Q_{2}) = Q_{3} + Q_{4}, two Miller-index vectors h_{1} and h_{2} are expected to exist such that Q_{1} = Q(h_{1}), Q_{2} = Q(h_{2}), Q_{3} = Q(h_{1} − h_{2}) and Q_{4} = Q(h_{1} + h_{2}). If an additional value Q_{5} satisfying the condition 2(Q_{1} + Q_{4}) = Q_{2} + Q_{5} is found, the graph of the quadrupole (Q_{1}, Q_{2}, Q_{3}, Q_{4}) grows via the addition of the Q_{5} 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 ScholarOishi, 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