Modelling of Lattices of Two-Dimensional Quasi-Crystals

Abstract

We propose the method for modelling of quasi-periodic structures based on an algorithm being a geometrical interpretation of the Fibonacci-type numerical sequences. The modelling consists in a recurrent multiplication of basis groups of the sites, which possess the 10-th, 8-th or 12-th order rotational symmetry. The advantage of the proposed method consists in an ability to operate with only two-dimensional space coordinates rather than with hypothetical spaces of dimension more than three. The correspondence between the method of projection of quasi-periodic lattices and the method of recurrent multiplication of basis-site groups is shown. As established, the six-dimensional reciprocal lattice for decagonal quasi-crystals can be obtained from orthogonal six-dimensional lattice for icosahedral quasi-crystals by changing the scale along one of the basis vectors and prohibiting the projection of sites, for which the sum of five indices (corresponding to other basis vectors) is not equal to zero. It is shown the sufficiency of using only three indices for describing diffraction patterns from quasi-crystals with 10-th, 8-th and 12-th order symmetry axes. Original algorithm enables direct obtaining of information about intensity of diffraction reflexes from the quantity of self-overlaps of sites in course of construction of reciprocal lattices of quasi-crystals.

Keywords

• quasi-periodic structures
• fibonacci sequence
• projection method
• basis vectors
• rotation symmetry
• reciprocal lattice