Study of atomic hopping in quasicrystals

It is a well-known fact that fivefold rotational symmetry is incompatible with crystalline lattice periodicity. Suppose that a Bravais lattice would contain a regular pentagon (Figure l), then it is possible to construct a smaller pentagon that also belongs to the lattice, as can be seen in Figure 1. This implies that a periodic lattice with fivefold symmetry can not satisfy the Delaunay condition. Thus, it came as a big surprise when Shechtman et al. (1) discovered quasicrystals (QC) (2) that exhibited diffraction patterns with five-fold (icosahedral) symmetry and sharp Bragg peaks characteristic of long-range order. Later, QC with eight-fold, tenfold and twelve-fold 2D symmetry were reported. The origin of this apparent paradox lies in a confusion between the concepts of long-range order and periodicity. Quasiperiodic lattices or tilings, such as illustrated in Figure 2 (for eightfold 2D symmetry), are not periodic but do give rise to diffraction patterns with sharp Bragg peaks due to their und...