On the Inversion of Schroedinger's Equation for Periodic Potentials: Dynamical Ptychography

摘要

The inversion of H. Bethe's 1929 multiple-scattering theory of electron scattering is treated in the transmission geometry. We consider a 1-1000 keV electron beam traversing a thin slab of crystalline material. The problem of obtaining the Fourier coefficients (structure matrix) of the periodic scattering potential from measurements of the intensities of transmitted Bragg beams is considered when there is arbitrarily strong multiple scattering of the electron. The scattering problem, described by the solution of a relativistically corrected Schroedinger equation, is formulated as a unitary transformation with a scattering matrix of order N, where N beams are considered. Inversion of this transformation is shown to represent an ill-posed problem unless data are collected under at least two different boundary conditions, such as crystal thickness or electron beam energy. The intensity at points where coherent convergent-beam transmission diffraction (CBED) discs overlap is shown to be described by interference between elements of the same row but different columns of the scattering matrix for an axial orientation. Tilting experiments then allow all of the complex scattering matrix to be determined, and hence the eigenvectors of the scattering and structure matrices can be found. An exact, non-perturbative inversion of the multiple electron scattering problem is then possible. Unique eigenvalues are obtained from these patterns recorded at two accelerating voltages. 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 be capable only of resolving this first-order spacing. Such a charge-density map provides fractional atomic co-ordinates and the identification of atomic species from microcrystalline regions.