Conventional X-ray methods use incoming plane waves which 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.
Bibliographical noteFunding Information:
This author's research was supported by the AFOSR (FA9550-15-1-0207) and partially supported by the ONR (N00014-14-0714), an NSF/PIRE grant (OISE-0967140), the MURI program (FA9550-12-1-0458), and a John Von Neumann visiting professorship at Technical University of Munich.
© 2016 Society for Industrial and Applied Mathematics.
- Helical structure
- Maxwell equations
- Poisson summation
- X-ray diffraction