Lattice dynamics of 3-dimensional tilings modelling icosahedral quasicrystals

Abstract

A study of the lattice dynamics of three-dimensional tilings modelling icosahedral quasicrystals is presented. The phonon density of states is calculated, and the character of the eigenstates is determined. Three different types of commensurate approximants are considered, namely symmetrized, perfect and randomized approximants. It appears that the density of states is smoothed by randomization. The participation ratio, which measures the rate of localization of an eigenmode, is given as a function of frequency. Only the states at the very upper end of the frequency spectrum appear to be localized, whereas all other states are extended. The density of states at low frequencies is analyzed in more detail, by applying a Brillouin zone integration over the lowest branches. It is found that these lowest branches scale for successive approximants.