Reprint of: Inversion of dynamical Bragg intensities to complex structure factors by iterated projections. For Ultramic. 2020. ("Pico" Festschrift, May 2021)

Highlights: • Remove multiple scattering artifacts from many-beam Bragg intensities in transmission electron diffraction. • Iterated projection algorithm doesn't need sample thickness, covers wide beam-energy range. • Solves phase problem , includes mean absorption, unaffected by lens phases. • Allows determination of chirality (hand) for enantiomorphs. Symmetry reduction discussed. • Extension to X-ray, neutron, micro-ED diffraction and cryo-EM may be possible. A method for recovering complex structure factors from many simultaneously excited Bragg beam in- tensities is described. The method is applied to simulated transmission electron diffraction data over a wide range of crystal thickness and beam energies. The method is based on iterated projections between structure and scattering matrices, which are related by a matrix unit ary transformation, exponential, which we invert. The algorithm removes multiple-scattering perturbations from diffraction data and might be extended to other fields, including X-ray and neutron diffraction and cryo-electron microscopy. Because coherent multiple scattering involves interference between Bragg beams, the method also solves the phase problem. Unlike dynamical inversion from electron microscope images or ptychography data, the method, which starts with Bragg beam intensities, provides complex structure factors unaffected by focusing errors or resolution limitations imposed by lenses. We provide inversions from simulated data with 441 simultaneously excited Bragg beams over a range of thickness and beam energy. We discuss the retrieval of chirality information from enantiomorphs, the efficient incorporation of symmetry information using the irreducible representation of the group of structure matrices, and the effect of HOLZ lines to provide three-dimensional information.