  We present wide-range (3 meV - 6 eV) optical studies on freestanding
transparent carbon nanotube films, made from nanotubes with different diameter
distributions. In the far-infrared region, we found a low-energy gap in all
samples investigated. By a detailed analysis we determined the average
diameters of both the semiconducting and metallic species from the near
infrared/visible features of the spectra. Having thus established the
dependence of the gap value on the mean diameter, we find that the frequency of
the low energy gap is increasing with increasing curvature. Our results
strongly support the explanation of the low-frequency feature as arising from a
curvature-induced gap instead of effective medium effects. Comparing our
results with other theoretical and experimental low-energy gap values, we find
that optical measurements yield a systematically lower gap than tunneling
spectroscopy and DFT calculations, the difference increasing with decreasing
diameter. This difference can be assigned to electron-hole interactions.
