The acoustic phonon spectrum of penrose tilings

The spectral weight function associated with the vibrational modes of a two-dimensional quasicrystal is studied numerically. It is found that plane waves corresponding to special symmetry points in reciprocal space are at least very nearly eigenstates of the system. Near periodicities in lineshape parameters plotted as a function of wavevector are found to occur, and the first Brillouin zone edge is identified with lattice parameters. The contribution of these modes to the specific heat is found to obey a two-dimensional Debye Law.