Dynamics of lattice vibrations for one-dimensional commensurate and incommensurate composites with harmonic interaction

Abstract

Abstract. We introduce a model thr.t generalizes the Frenkel-Kontorova model and describes :i one-dimensional (ID ) composite system made of iwo subsystems which are treated on an equal footing. The gap structure o f the phonon energy spectrum and the character of lattice vibration are studied. We show that the phason mode may be pinned or not depending on the form o f the intersubsystem potential. For a weak intersubsystem interaction the structure of the phonon energy spectrum is universal and may be revealed by a gnomonic projection of l ) onto the plane. For a short-range harmonic interaction there are similarities between this model of composite and ID quasi crystals.

Sliding on a quasiperiodic substrate

“…If the modulation function is smooth this yields a zero frequency vibration mode. This has been found in the FK model [3] and in the DCM [1,4]. This same situation occurs if the substrate is quasiperiodic of rank 2.…”

“…To study the motion of a crystal on the surface of another (periodic or aperiodic) crystal, a simple model can be used that has its origin in the the study of incommensurate composites, the double chain model (DCM) [1]. However, still simpler is the model that usually is called the Frenkel-Kontorova model, or a generalization of it [2].…”

Theory of Phasons in Aperiodic Crystals

“…For incommensurate composites the situation is similar, as follows from numerical calculations in the double chain model [5][6][7]. When the modulation function is smooth, there is a zero frquency mode.…”

KAM theory for quasi-periodic equilibria in 1D quasi-periodic media: II. Long-range interactions

“…In contrast, if the KAM tori are not present, the system is pinned. There have been numerical explorations of these issues in [vE99,vEFRJ99,vEFJ01,vEF02,RJ97,RJ99]. In particular, the above references pay special attention to the boundary of the set of parameters for which there is an analytic solution (breakdown of analyticity, Aubry transition), which corresponds physically to the boundary between sliding and pinning.…”