Geometry of quasicrystal-to-crystal transformations

The geometrical relationships between quasiperiodic structures and the cubic β-phase (CsCl-type) formed in a continuous phase transformation is discussed on the examples of decagonal Al-Co-Me (Me = Cu, Ni) and icosahedral Al-Mn-Pd. For these purposes, first the alternative higher-dimensional embedding of decagonal Al-Co-Ni as incommensurately modulated structure is introduced. By projection onto the physical space, a periodic average structure of the decagonal quasicrystal can be obtained. It is demonstrated that the lattice of the periodic average structure of the quasiperiodic phases matches very well the lattice of the cubic β-phase. In this way, the experimentally observed orientational order and registry of the cubic β-phase with regard to the quasiperiodic structure can be nicely explained.