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. The data show direct evidence of one-dimensional quantized phonon subbands. Above 4 kelvin, 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 millielectron volts for the lowest quantized phonon subband and a tube-tube (or "lattice") Debye energy of 1.1 millielectron volts, implying a small intertube coupling in bundles.

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The thermal properties of carbon nanotubes are strongly dependent on their unique structure and size, and show promise as an ideal material for thermal management on the micro- and macro-scale. The specific heat of nanotubes is similar to that of two-dimensional graphene at high temperatures, but is sensitive to the effects of rolling the the graphene sheet into a small cylinder at low temperatures. Specifically, the acoustic phonon modes are stiffened due to the cylindrical geometry, and the phonon spectrum is quantized due to the small diameter of the tube. In bundles of single-walled nanotubes, the specific heat is a sensitive probe of inter-tube mechanical coupling. Measurements of the specific heat show that inter-tube coupling is relatively weak, and show direct evidence for quantum effects. The thermal conductivity of nanotubes should reflect the on-tube phonon structure. Aligned bundles of SWNTs show a high thermal conductivity (>200 W/m-K at room temperature), and possible quantization effects at low temperature.