Hierarchical nature of the Brillouin zones in icosahedral A1₅₉B₃CU_25.5Fe_12.5

Abstract

Brillouin-zone widths are deduced from fivefold electron diffraction patterns for a quasicrystalline sample of Al₅₉Cu_25.5Fe_12.5B₃. Optical measurements in the infrared were also performed on the same alloy. The results were analysed by the Kramers-Krönig method in order to obtain the absorption coefficient as a function of the energy. The optical study allows the observation of critical energies that are directly correlated to singularities in the density of states at the edge of Brillouin zones. Recurrent pseudogaps in the density of states are discussed in terms of the hierarchical nature of Brillouin zones, in agreement with theoretical predictions.