Optical absorption and neutron scattering by lattice vibrations of incommensurate lattices

The lattice dynamics of discrete incommensurate lattices are studied numerically by linearizing the equations of motion for the Frenkel-Kontorova model (which consists of a chain of masses connected by springs sitting in a sinusoidal potential) about the equilibrium positions of the atoms, which are determined numerically. The optical-absorption and neutron scattering cross sections are studied. It is found that despite the fact that the incommensurate system lacks translational invariance (because there does not exist a unit cell), the optical-absorption and neutron scattering spectra always appear to consist of a few sharp lines, no matter how strong the coupling of the two periodic systems. The calculations should be applicable to incommensurate charge-density waves, ionic order in potassium hollandite, a one-dimensional ionic conductor, and other systems exhibiting two incommensurate periodicities.