  A computational method is developed whereby the reflectivity of low-energy
electrons from a surface can be obtained from a first-principles solution of
the electronic structure of the system. The method is applied to multilayer
graphene. Two bands of reflectivity minima are found, one at 0 - 8 eV and the
other at 14 - 22 eV above the vacuum level. For a free-standing slab with n
layers of graphene, each band contains n-1 zeroes in the reflectivity. Two
additional image-potential type states form at the ends of the graphene slab,
with energies just below the vacuum level, hence producing a total of 2n
states. A tight-binding model is developed, with basis functions localized in
the spaces between the graphene planes (and at the ends of the slab). The
spectrum of states produced by the tight-binding model is found to be in good
agreement with the zeros of reflectivity (i.e. transmission resonances) of the
first-principles results.
