Six-dimensional structure model for the icosahedral quasicrystal Al6CuLi3.

Abstract

The structure of icosahedral quasicrystals can be described either by a quasiperiodic density function in three-dimensional (physical) space, or by a periodic density function in six-dimensional (6D) space. The real structure is obtained as a particular 3D section of the 6D density function. The 6D description involves 6D bodies, which on intersection by 3D space give rise to atoms in physical space. In this paper we derive all possible perpendicular-space shapes belonging to the 6D atoms, which arise for 6D structures describing any decoration of the 3D Penrose tiling. These results are applied to icosahedral ${\mathrm{Al}}_{6}$${\mathrm{CuLi}}_{3}$, for which a 6D structure model is proposed. Refinement of this model on single-crystal x-ray-diffraction data shows the structure to be close to that of a decorated Penrose tiling. Unlike ${\mathrm{Al}}_{73}$${\mathrm{Mn}}_{21}$${\mathrm{Si}}_{6}$, it is found that a perpendicular-space shape is necessary which has a lower internal symmetry than given by the icosahedral point group.