Theory of Electronic Structure in Quasicrystals

Abstract

Shechtman, Blech, Gratias, and Cahn discovered a sharp diffraction pattern of icosahedral (i) symmetry in the Al-Mn alloy rapidly quenched from the melt and reported the i phase of metallic alloys (Shechtman et al. 1984). Levine and Steinhardt (1984) showed that the three-dimensional (3D) Penrose lattice gives a diffraction pattern of densely distributed $δ$ functions with i symmetry. The notion of quasicrystals (QCs) is characterized by the following facts (Steinhart and Ostlund 1987):1the scattering intensity is a sum of densely distributed $δ$ functions2$δ$ function spots can be specified with a set of integers whose number exceeds the space dimension, and3the pattern of scattering spots has a rotational symmetry forbidden in crystals.