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, pp. 273-274

Section 3.4.3.1.1. Zone-indexing strategy

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

3.4.3.1.1. Zone-indexing strategy

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The zone-indexing strategy was originally developed by Runge (1917[link]), successively proposed by Ito (1949[link], 1950[link]), generalized by de Wolff (1957[link], 1958[link]) and enhanced by Visser (1969[link]). 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 [{\bf r}_{hkl}^*] and [{\bf r}_{h'k'l'}^*] 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 [m {\bf r}_{hkl}^* \pm n {\bf r}_{h'k'l'}^*,] where m and n are positive integers. If ω is the angle between [{\bf r}_{hkl}^*] and [{\bf r}_{h'k'l'}^*], the squared distance of B from O (i.e., [{Q_{m,n}}]) can be expressed by (de Wolff, 1958[link]; Visser, 1969[link])[Q_{m,n} = m^2 Q_{\rm A} + n^2 Q_{\rm A'} \pm m n R,\eqno(3.4.6)]where [Q_{\rm A} = Q(hkl)] and [Q_{\rm A'}=Q(h'k'l')] are the squared distances of A and A′ from O, respectively, and [R = ] [2(Q_{\rm A} Q_{\rm A'})^{1/2} \cos \omega ]. R can be derived as[R = |{Q_{m,n}} - {m^2} {Q_{\rm A}} - {n^2} {Q_{\rm A'}}|/m n.\eqno(3.4.7)]

The method is applied as follows: [Q_{\rm A}] and [Q_{\rm A'}] are chosen among the first experimental Qi values; the {Qi}, up to a reasonable resolution, are introduced in (3.4.7)[link] in place of [{Q_{m,n}}]; and a few positive integer values are assigned to m and n. Equation (3.4.7)[link] 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 ([Q_{\rm A},Q_{\rm A'}]) 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.

References

Ito, T. (1949). A general powder X-ray photography. Nature, 164, 755–756.Google Scholar
Ito, T. (1950). X-ray Studies on Polymorphism. Tokyo: Maruzen Company.Google Scholar
Runge, C. (1917). Die Bestimmung eines Kristallsystems durch Rontgenstrahlen. Phys. Z. 18, 509–515.Google Scholar
Visser, J. W. (1969). A fully automatic program for finding the unit cell from powder data. J. Appl. Cryst. 2, 89–95.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
Wolff, P. M. de (1958). Detection of simultaneous zone relations among powder diffraction lines. Acta Cryst. 11, 664–665.Google Scholar








































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