A symmetrical indexing scheme for decagonal quasicrystals analogous to Miller-Bravais indexing of hexagonal crystals

The problems of redundancy and superfluous indices in indexing the planes and axes in a decagonal quasicrystal are considered, using a scheme of five coplanar vectors in the quasiperiodic plane and one perpendicular vector. Of all the indexing schemes in use, this scheme offers the maximum advantage. An analogy is drawn to the hexagonal system using Miller-Bravais indices. Based on this, a symmetry-based indexing system for decagonal phases is devised that follows a simplified approximate zone law analogous to the exact zone law for the hexagonal case. The indices based on this scheme will be designated as `Frank indices'. High-symmetry electron diffraction zone-axis patterns as well as powder X-ray diffraction patterns are indexed using Frank indices and compared with those of other indexing schemes.