The harmonic lattice vibrations of a one-dimensional incommensurate lattice

Abstract

The properties of the dynamical matrix for a one-dimensional incommensurate lattice are found, as are the normal mode vibrational frequencies and eigenvectors. A theory of the equilibrium state of the incommensurate structure is developed before the dynamical matrix is obtained, and a first harmonic lattice distortion of wave vector 2Q is found, in addition to the fundamental distortion of wave vector Q. New first harmonic phase and amplitude modes are also found, in addition to the previously discovered fundamental phase and amplitude modes; also, the previously discovered (Overhauser) phase mode is found to be mixed with the long-wavelength phonon mode. Several of the parameters of interest (e.g. the amplitude of the first harmonic distortion, the phason velocity, the frequency of the amplitude mode, and the effective mass) are evaluated using the microscopic Frohlich electron–phonon model, and are found to depend strongly on the magnitude of the dimensionless Frohlich coupling constant. The implications for various experiments, including inelastic neutron scattering, are discussed.