InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 616-618
## Section 4.6.3.3.3.1. Indexing |

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

### 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.