Long-wavelength vibrational modes in quasi-2D and tubular quasi-1D structures.

We propose a continuum elasticity theory approach to predict long-wavelength vibrational modes of quasi-two-dimensional and quasi-one-dimensional tubular structures, such as empty and liquid-filled tubules, which are very hard to reproduce using the force-constant-matrix based atomistic approach based on {\em ab initio} calculation. We characterize the elastic behavior of these structures by a $(3{\times}3)$ elastic matrix characterizing a 2D membrane or a tubular wall, as well as the flexural rigidity of the membrane or the wall structure. We derive simple quantitative expressions for frequencies of long-wavelength acoustic modes, which we determine using 2D elastic constants calculated by {\em ab initio} density functional theory. Our results accurately reproduce observed and calculated long-wavelength phonon spectra of 2D graphene, 2D phosphorene, 1D microtubules of tubulin and 1D carbon nanotubes.