Phonons in models for icosahedral quasicrystals : low frequency behaviour and inelastic scattering properties

Abstract

Abstract. -A detailed study of the low frequency behaviour of the phonon spectrum for 3-dimensional tiling models of icosahedral quasicrystals is presented, in commensurate approximations with up to 10 336 atoms per unit cell. The scaling behaviour of the lowest phonon branches shows that the widths of the gaps relative to the bandwidths vanish in the low frequency limit. The density of states at low frequencies is calculated by Brillouin zone integration, using either local linear or local quadratic interpolation of the branch surface. For perfect approximants it appears that there is a deviation from the normal w 2-behaviour already at relatively low frequencies, in the form of pseudogaps. Also randomized approximants are considered, and it turns out that the pseudogaps in the density of states are flattened by randomization. When approaching the quasiperiodic limit, the dispersion of the acoustic branches becomes more and more isotropic. and the two transversal sound velocities tend to the same value. The dynamical structure factor isdetermined for several approximants, and it is shown that the linearity and the isotropy of the dispersion are extended far beyond the range of the acoustic branches inside the Brillouin zone. A sharply peaked response is observed at low frequencies, and broadening at higher frequencies. To obtain these results, an efficient algorithm based on Lanczos tridiagonalisation is used.

Phonon excitations in quasicrystals

“…Introducing bending forces besides stretching forces shifts the peak to higher frequencies (Los and Janssen, 1990;Los et al, 1993bLos et al, , 1993c)-see Fig. 19.…”

“…21). First, there is a collection of very flat bands close together (Los and Janssen 1990;Los et al, 1993bLos et al, , 1993c, D ue to the scaling behavior, these topic branches also occur at lower frequencies, albeit with a much smaller contribu tion to S(q,a)). The other reason is that for higher fre quencies the modes are m ore localized.…”

“…This implies an infinity of gaps accumulat ing at zero frequency. The size of these gaps goes expo nentially to zero, so that the relevance for experiment is rapidly lost (Los and Janssen, 1990;Los et ah, 1993;1993c). The fast closing of gaps in the limit of zero fre quency allows one to find an average density of states.…”

“…In going to higher approximants this value diminishes, however, which is an indication that the states are not exponentially localized. A multifractal analysis gives an indication that the states are critical (Los and Janssen, 1990;Los et al, 1993bLos et al, , 1993c.…”

Clusters and localization of electrons in quasicrystals

“…On the theoretical side, band-structure calculations for approximants also reveal specific properties. [5][6][7][8][9] The dispersion relations are flat, corresponding to small velocities. Fine peaks in density of states are associated with the flat bands.…”