  Conventional X-ray methods use incoming plane waves and result in discrete
diffraction patterns when scattered at crystals. Here we find, by a systematic
method, incoming waveforms which exhibit discrete diffraction patterns when
scattered at helical structures. As examples we present simulated diffraction
patterns of carbon nanotubes and tobacco mosaic virus.
  The new incoming waveforms, which we call twisted waves due to their
geometric shape, are found theoretically as closed-form solutions to Maxwell's
equations. The theory of the ensuing diffraction patterns is developed in
detail. A twisted analogue of the Von Laue condition is seen to hold, with the
peak locations encoding the symmetry and the helix parameters, and the peak
intensities indicating the electronic structure in the unit cell.
  If suitable twisted X-ray sources can in the future be realized
experimentally, it appears from our mathematical results that they will provide
a powerful tool for directly determining the detailed atomic structure of
numerous biomolecules and nanostructures with helical symmetries. This would
eliminate the need to crystallize those structures or their subunits.
