Localized modes and topological frustration in rational approximants to quasicrystals

The existence of localized vibrational states is demonstrated for rational approximants of icosahedral Al-Zn-Mg quasicrystals. It is shown that the origin of localized modes is in a local topological frustration, i.e., in a local deviation from ideal icosahedral packing. As the localized mode involves only atoms in a single tile of the three-dimensional Penrose lattice, it can be expected to exist in the infinite quasicrystal.