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Low-frequency phonons in carbon nanotubes are studied using a continuum model which allows consideration of an arbitrary wall thickness for the nanotube. Phonon dispersion relations are calculated for two archetypal examples of carbon nanotubes, the ͑5,5͒ and ͑10, 10͒ tubes. The dependence of the radial breathing mode frequency at ⌫ on the inverse nanotube diameter is verified within this model; furthermore, we prove it to hold for all pure modes within the thin-shell approximation. The effect of the nanotube wall thickness on the eigenfrequencies of carbon nanotubes is also analyzed, and a criterion to fix this parameter within a continuum model is presented. We compare our results to other continuum approaches, and show that by choosing the appropriate parameters, excellent agreement with recent first-principles calculations can be achieved.

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The optical-absorption spectra associated with transitions between the n = 1 valence subband and the donor-impurity band and between the acceptor-impurity band and the n = 1 conduction subband were calculated for both the infinite and the finite GaAs-Ga& Al As quantum wells. An absorption edge associated with transitions involving impurities at the center of the well and a peak related with impurities at the edges of the well were the main features observed in the theoretical spectra. Results obtained in the case of an infinite quantum well were rather different from those previously reported in the literature.