  The radial-breathing-like phonon modes (RBLMs) of the double-walled carbon
nanotubes are studied in a simple analytical model, in which the interaction
force constants (FCs) can be obtained analytically from the continuous model.
The RBLMs frequencies are obtained by solving the dynamical matrix, and their
relationship with the tube radii can be obtained analytically, offering a
powerful experimental tool for determining precisely the radii of the
multi-walled carbon nanotubes.
