Phonons in One-Dimensional Quasi-Lattices

Vibrational properties in one-dimensional quasi-lattices constructed by the Fibonacci substitution rule are studied numerically within a harmonic approximation. Examining the generation dependence of the substitution, we find that the gross spectrum structure is generation-independent and self-similar, and that the ratio of the numbers of the eigenfrequencies contained in each band tends to the golden mean with the generation. The localization property of the eigenfunctions and other class of the quasi-lattice are discussed.