  The electronic spectra of carbon nanotubes and other nanoscale systems are
quantized because of their small radii. Similar quantization in the phonon
spectra has been difficult to observe because of the far smaller energy scale.
We probed this regime by measuring the temperature-dependent specific heat of
purified single-wall nanotubes. Thee data show direct evidence of
one-dimensional quanitized phonon subbands. Above 4 K, they are in excellent
agreement with model calculations of individual nanotubes and differ markedly
from the specific heat of two-dimensional graphene or three-dimensional
graphite. Detailed modeling yields an energy of 4.3 meV for the lowest
quantized phonon subband and a tube-tube (or "lattice") Debye energy of 1.1
meV, implying a small intertube coupling in bundles.

Mesoscale and Nanoscale Physics;Materials Science