The reference lattice concept and its application to the analysis of diffraction patterns

Abstract

The concept of the reference lattice has been thoroughly discussed in physical and in higher-dimensional spaces. Such an approach has been applied to di raction pattern analysis of di erent periodic, modulated and other aperiodic structures. It has been shown that two periodicities can be found in di raction patterns of the above structures, namely the periodicity of the envelope curves describing the di raction peak intensities, and the periodicity of peak positions ascribed to each envelope curve. The formula for the envelope curve has been given and it depends on the statistical moments of a specially de® ned real-space variable describing spatial¯uctuations in atomic positions. Full equivalence of the presented physical space approach to the higher-dimension analysis of perfect quasicrystals has been shown.

Average unit cell of a square Fibonacci tiling

“…2 and 4 in [2]) and obtained as a intersection of the physical space with rational direction [1,1]. More generally the average unit cell is an oblique projection of the atomic surface onto physical space [5,7].…”

“…Its 2D diffraction pattern is shown in Fig. 2, where different cutting-lines: [1,1], [1,s] and [1,2] have been also marked.…”

“…For incommensurate modulation this requires another wave vector q 0 -the modulation vector. To reconstruct the diffraction pattern of such structure one has to use two average unit cells for two different wave vectors [2][3][4]. Then the structure factor for k ¼ nk 0 þ mq 0 where n is an index of the main reflection and m -of the satellite, reads…”

“…The average unit cell has been constructed using the statistical approach and the reference lattice concept [2]. The reference lattice is a set of parallel lines periodically arranged in 2D and perpendicular to the corresponding scattering vector.…”

Two-dimensional quasicrystals and their unit cells (elliptical or Gaussian approximations)

“…The concept of average unit cell was first introduced in the theory of incommensurately modulated structures [1]. It has already been shown [2][3][4][5][6][7][8] that for quasicrystals one can also construct the average unit cell in physical space.…”