Direct inversion of dynamical electron diffraction patterns to structure factors

An exact nonperturbative inversion method is described for transmission electron diffraction which allows crystal structure factors to be obtained directly from the intensities of multiply scattered Bragg beams. These amplitudes are required at several thicknesses (or electron beam energies) and orientations of a thin crystal. The analysis applies to centrosymmetric crystals with anomalous absorption and to noncentrosymmetric crystals if the mean absorption potential only is included. Phases of stucture factors from noncentrosymmetric crystals are correctly recovered. Dynamical coherent convergent-beam microdiffraction patterns with overlap of adjacent diffraction orders, provide the required data and may be obtained from nanometer-sized regions. The method allows the direct synthesis of charge-density maps of unknown crystal structures at high resolution from electron microdiffraction patterns, using a scanning transmission electron microscope. Whereas this microscope must be capable of resolving only the first-order lattice spacing, much higher order reflections may in principle be determined. Such a charge-density map provides fractional atomic coordinates and the identification of atomic species (as in X-ray crystallography) from microcrystalline samples and other multiphase inorganic materials for which large single crystals cannot be obtained or X-ray powder patterns obtained or analyzed. In other language, this paper solves the inversion problem of quantum mechanics for the case of electron scattering from a periodic potential, described by the Schroedinger equation, in which the scattering is given as a function of some parameter and the potential sought. The diagonalization of large matrices is required - the method does not provide a closed-form solution.