Derivation of the proper basis of quasicrystals

Abstract

A method based on the physical space Patterson ͑pair correlation͒ function is derived which allows the determination of the correct n-D Bravais lattice of quasicrystals. ͓Hereafter, we will abbreviate n-D for n dimensional. We will also call perpendicular space the (nϪ3)-D orthogonal space that is added to physical 3D space to form the n-D embedding space.͔ The optimum unit cell can be chosen and therefore the proper indexing of the diffraction pattern. The size of the integrated maxima of the Patterson function depends on their multiplicity and on their perpendicular space component. Lifting the positions of these maxima into n-D space allows the set of ''quasilattice'' vectors to be distinguished from the set of decoration vectors. This procedure leads to a unique n-D lattice. Taking advantage of scaling symmetries, the best choice of the n-D unit cell can be found. A detailed analysis of the decoration vectors reveals all possible positions of the hyperatoms therein. This powerful technique is illustrated on simulated data of a decorated Fibonacci chain and on experimental data of

The ordering of a Xe monolayer on quasicrystalline Al–Ni–Co

“…The bulk composition for these experimental studies was Al 70 Ni 10 Co 17 , which is consistent with the basic Co-rich phase [4], although the surface composition may differ somewhat. The surface structure derived in the most recent of these studies [3] is consistent with a relaxed bulk model of Cervellino et al [5] in which quasicrystalline planes, each having fivefold rotation symmetry, are stacked in an ABAB arrangement for overall tenfold symmetry. The surface layers *Corresponding author.…”

Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals

“…Mukhopadhyay, Lord, 2002;Lord, 2003). Nevertheless an optimum basis (low indices are assigned to strong reflections) can be derived: not the metrics, as for regular periodic crystals, but the intensity distribution (related to the Patterson peaks in direct space) characterizes the best choice of indexing (see, e.g., Cervellino, Haibach, Steurer, 1998). An appropriate set of reciprocal basis vectors can be identified experimentally in the following way:…”

Quasiperiodicity in decagonal phases forced by inclined net planes?

“…4 This is related to the fact that a Penrose tiling of edge ( 3 a r has been found (Cervellino et al, 1998) to have structural relevance, in the sense that (a) on such a scale a tiling can ideally represent the DQC but not at a lower scale (particularly at the bond-length scale, with tile edge a r ) because of structural disorder; (b) accordingly, in¯ation symmetry is ideally ful®lled from this scale up. Accordingly, the named pentagon is the (…”