  We perform density functional theory calculations on a series of armchair and
zigzag nanotubes of diameters less than 1nm using the all-electron
Full-Potential(-Linearised)-Augmented-Plane-Wave (FPLAPW) method. Emphasis is
laid on the effects of curvature, the electron beam orientation and the
inclusion of the core-hole on the carbon electron energy loss K-edge. The
electron energy loss near-edge spectra of all the studied tubes show strong
curvature effects compared to that of flat graphene. The curvature induced
$\pi-\sigma$ hybridisation is shown to have a more drastic effect on the
electronic properties of zigzag tubes than on those of armchair tubes. We show
that the core-hole effect must be accounted for in order to correctly reproduce
electron energy loss measurements. We also find that, the energy loss near edge
spectra of these carbon systems are dominantly dipole selected and that they
can be expressed simply as a proportionality with the local momentum projected
density of states, thus portraying the weak energy dependence of the transition
matrix elements. Compared to graphite, the ELNES of carbon nanotubes show a
reduced anisotropy.
