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This Demonstration shows the dependence of the value of the optical matrix element ... for light polarized parallel to the nanotube axis over the Brillouin zone (BZ) of a single-walled carbon nanotube (SWNT). A general analytical expression for ... for any SWNT chirality ... and unit cell unrolled in 2D has been obtained in the literature (see details) from the tight-binding electronic structure of graphene. The profile of ... is plotted over the 2D graphene BZ as a function of the graphene wavevectors ... . The optical selection rules obey the azimuthal ... and axial ... quantum number conservation, hence only vertical optical transitions between electronic bands with the same quantum number ... and at the same ... are allowed. The profile of ... over the nanotube 1D BZ is obtained by applying the zone-folding method to the 2D analytical function ... and evaluating this expression along the allowed SWNT cutting lines, which are line sections of the 2D BZ and which also provide the SWNT electronic bands. The 1D plots so obtained are then superimposed to show how the matrix element changes value and sign over each cutting line of index ... . The plots shown in this Demonstration show that the properties of ... are closely related to the SWNT geometry and determine the profile of the optical absorption spectrum for a given ... chirality.
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