Gap structures in the phonon spectrum of a linear chain with varying masses

Abstract

The phonon spectrum of a one-dimensional lattice with masses varying incommensurately with the underlying lattice is studied by a discrete version of a multiple-scale perturbation theory. From the resonances in the higher order correction terms we determine the location of the gaps in the spectrum. These resonances also provide a gap labeling in terms of two integers. Approximate analytic expressions are found predicting the behavior of the integrated density of states close to the two widest gaps and the exponential growth rate of the displacements within a gap. In the continuum limit the equation of motion of the discrete model reduces to the Mathieu equation.