Special points in the reciprocal space of an icosahedral quasi-crystal and the quasi-dispersion relation of electrons

It is shown that there exist special points in the reciprocal space of an icosahedral quasi-lattice; they correspond to high-symmetry points in the Brillouin zone of a periodic lattice. The translationally equivalent special points are distributed quasi-periodically with different intensities in the reciprocal space. On the other hand, electronic wavefunctions of the icosahedral quasi-lattice are investigated using a numerical method based on the tight-binding model. Their Fourier spectra are mapped in the energy versus wavenumber plane along several axes in reciprocal space. Dispersion-relation-like patterns are observed. It is found that critical points (stationary points) of the quasi-dispersion relation appear at the special points. The quasi-dispersion relation recurs quasi-periodically all over reciprocal space.