Rigid-Unit Modes in Layers and Nanotubes

Abstract

Among the vibrational modes of a complex system, apart from the rigid-body displacements of the system as a whole, there can exist also the rigid-body displacements of the subsystems. While in the standard considerations of the layered structures, it is understood that rigid-layer modes emerge due to the relatively weaker inter-layer (than intra-layer) coupling, here we derive universal dynamical condition which provides exactly rigid modes of subsystems. It clarifies that weak inter-layer interaction itself is sufficient to induce the layer modes which are only approximately rigid. The obtained exact condition separates effective dynamical matrix, which in the layered systems corresponds to the well-known linear chain model. The analysis of quasi-one-dimensional systems is more refined, due to additional rigid-body degrees of freedom: the twisting modes and their interference with other modes (e.g., mixing of twisting and breathing modes in multiwalled nanotubes). It is also shown that condition for exactly rigid modes of subsystems can be fulfilled due to symmetry, used also to determine infrared and Raman activity of rigid-layer modes in uniform multilayers (with one repeated monolayer).