A continuous transition from twins to quasi-crystals

The authors discuss different structures, built up using two types of Robinson triangle tiling the plane without defects and having diffraction patterns with symmetry forbidden by classical crystallography. A universal behaviour of the diffraction patterns of the different structures allows definition of a similarity parameter which distinguishes 'quasicrystals' (i.e. structures for which the diffraction peak intensities scale as N2) from all other types of structure. It is shown that the similarity parameter depends only on the concentration of small Robinson triangles. The concentration dependence of the similarity parameter exhibits singular behaviour at the Penrose concentration, indicating that a continuous transition takes place from twinned structures to 'quasi-crystals'. The critical exponent for this behaviour was determined to be 0.55+or-0.02.