The crystallographic restriction shows that a periodic plane or space lattice can…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction
Invariance, Ch Point groups and crystallographic symmetry
Problem
The crystallographic restriction shows that a periodic plane or space lattice can carry only one-, two-, three-, four- or six-fold rotation axes, because a lattice rotation must be represented by an integer matrix, and hence have an integer trace 1+2cos(2π/n), in a primitive lattice basis. The Ammann-Beenker tiling is a quasiperiodic tiling of the plane by squares and 45∘ rhombi. It has long-range order: the Fourier transform of its vertex positions, the analogue of a diffraction pattern, consists of sharp spots, as in Shechtman's Al-Mn alloy. The pattern of spots shows eight-fold symmetry. Identify the specific step of the restriction's proof that fails for the Ammann-Beenker tiling, and explain why an eight-fold diffraction pattern is not a contradiction of the restriction, even though no periodic lattice has an eight-fold axis.