Scaling properties of vibrational spectra and eigenstates for tiling models of icosahedral quasicrystals

(Received 11 August 1992, accepted in final form 1 October 1992)

Abstract
A study of the lattice dynamics of 3-dimensional tilings modelling icosahedral quasicrystals is presented, both in commensurate approximations and cluster approximations. In the commensurate approximation this is done for three different types of tilings, namely: perfect, symmetrized and randomized approximants. It turns out that the density of states as a function of frequency is smoothed by randomization. A multifractal analysis of the spectrum shows that mainly at high frequencies the scaling behaviour of the spectrum is different from that for periodic structures. Also the eigenvectors are examined and it appears that only the states at the very upper end of the spectrum have a relatively small participation fraction, i.e. are more localized. The majority of the states scale as normal extended states, as is shown by a multifractal analysis of the eigenvectors for systematic approximants. Also, for most of the states localization is not enhanced by randomization. Throughout the paper the results are compared with those for a 1-dimensional quasicrystal, the Fibonacci chain.