Cubic approximants in quasicrystal structures

2014 The regular deviations from the exact icosahedral symmetry, usually observed at the diffraction patterns of quasicrystal alloys, are analyzed. It is shown that shifting, splitting and asymmetric broadening of reflections can be attributed to crystalline phases with the cubic symmetry very close to the icosahedral one (such pseudo-icosahedral cubic approximants may be called the Fibonacci crystals). The Fibonacci crystal is labelled as Fn+1/Fn>, if in this crystal the most intense vertex reflections have the Miller indices {0, Fn, Fn + 1} where Fi are the Fibonacci numbers (Fi = 1, 1, 2, 3, 5, 8, 13, 21, 34...). The deviations of x-ray and electron reflections from their icosahedral positions are calculated. The comparison with available experimental data shows that at least four different Fibonacci crystals have been observed in Al-Mn and Al-Mn-Si alloys : 2/1> (MnSi structure), 5/3> (03B1-Al-Mn-Si), 13/8>, and 34/21> with the lattice constants 4.6 Å, 12.6 Å, 33.1 Å, 86.6 Å respectively. It is interesting to note that there are no experimental evidences for the intermediate approximants 3/2>, 8/5> and 21/13>. The possible space groups of the Fibonacci crystals and their relationships with quasicrystallographic space groups are discussed. J. Phys. France 51 ( 1990) 2717-2732 1 er DÉCEMBRE 1990,