Crystal structure determination by direct inversion of dynamical microdiffraction patterns

The intensity at points where coherent convergent-beam transmission diffraction discs overlap is shown to be described by interference between elements of the same row but different columns of the dynamical scattering matrix for an axial orientation. These intensities are used as the basis for an exact, nonperturbative inversion of the multiple electron scattering problem, allowing crystal structure factors to be obtained directly from the intensities of multiply scattered Bragg beams. Eigenvectors of the structure matrix are obtained using coherent CBED patterns from many crystal orientations. Unique eigenvalues are obtained from these patterns recorded at two accelerating voltages. The inevitable variation in electron probe position at different crystal tilts is considered. The analysis applies to centrosymmetric crystals with anomalous absorption, to centrosymmetric projections of acentric crystals and to acentric crystals if the mean absorption potential only is included. The method would allow the direct synthesis of charge-density maps of unknown crystal structures at high resolution from multiple scattering data, using a scanning transmission electron microscope (STEM). The resolution of this map may be much higher than the first-order d-spacing; however, the STEM need only be capable of resolving this first-order spacing. 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 analysed. In summary, we solve the inversion problem of quantum mechanics for the case of electron scattering from a periodic potential, described by the nonrelativistic Schrodinger equation, in which the scattering is given as a function of some parameter, and the potential sought.