International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 616-618   | 1 | 2 |

## Section 4.6.3.3.3.1. Indexing

W. Steurera* and T. Haibacha

aLaboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland
Correspondence e-mail:  walter.steurer@mat.ethz.ch

#### 4.6.3.3.3.1. Indexing

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There are several indexing schemes in use. The generic one uses a set of six rationally independent reciprocal-basis vectors pointing to the corners of an icosahedron, , , , , with , the angle between two neighbouring fivefold axes (setting 1) (Fig. 4.6.3.28). In this case, the physical-space basis corresponds to a simple projection of the 6D reciprocal basis . Sometimes, the same set of six reciprocal-basis vectors is referred to a differently oriented Cartesian reference system (C basis, with basis vectors along the twofold axes) (Bancel et al., 1985). The reciprocal basis is

An alternate way of indexing is based on a 3D set of cubic reciprocal-basis vectors (setting 2) (Fig. 4.6.3.32): The Cartesian C basis is related to the V basis by a rotation around , yielding , followed by a rotation around : Thus, indexing the diffraction pattern of an icosahedral phase with integer indices, one obtains for setting 1 . These indices transform into the second setting to with the fractional cubic indices , , . The transformation matrix is

 Figure 4.6.3.32 | top | pdf |Perspective parallel-space view of the two alternative reciprocal bases of the 3D Penrose tiling: the cubic and the icosahedral setting, represented by the bases , and , respectively.

### References

Bancel, P. A., Heiney, P. A., Stephens, P. W., Goldman, A. I. & Horn, P. M. (1985). Structure of rapidly quenched Al–Mn. Phys. Rev. Lett. 54, 2422–2425.