The Schur rotation as a simple approach to the transition between quasiperiodic and periodic phases

The recently observed transition of CrNiSi from the octahedral to a cubic phase has a natural mathematical counterpart in terms of a one-parameter Schur rotation. The authors present the corresponding tilings and the diffraction patterns of delta -scatterers at vertex positions for a series of rotation angles. These angles show up as a length scaling in the patterns and provide a measurable order parameter. They comment on the rational reductions and give two further examples, one on a connection between octagonal and dodecagonal patterns and one on a transition between icosahedral and primitive or body-centred cubic phases.