Direct Inversion of Dynamical Electron Diffraction Patterns to Structure Factors

Abstract

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.

Coherent nano‐area electron diffraction

“…Inversion of exit wave function to object potential is not as straightforward as an inverse Fourier transform. Theory developed by several research groups (led by Spence at ASU and Allen in Australia) shows the principle of inversion using a data set of multiple thicknesses, orientations, and overlapping coherent electron diffraction (Spence, 1998). The inversion is based on the scattering matrix that relates scattered wave to the incident wave through the Fourier transform of the object potential.…”

Electron nanodiffraction

“…Also, the restriction to the weak-phase object approximation, which implies the use of very thin crystals of light atoms, has prevented any practical application of this method to real specimens. However, recent methods proposed for the reconstruction of the potential distribution from the amplitudes and phases of the exit waves from quite thick crystals (Spence, 1997) may change this situation.…”

Characteristics of precession electron diffraction intensities from dynamical simulations

“…1) extends to large resolution compared to the tilt vector h. The eigenvalues of A are unaffected by tilting to K t =h, except for the addition of a constant factor |S h |=g 2 /2K (the excitation error of beam h referred to the case of K t =0). The only effect of this constant on the S matrix is multiplication by a constant phase factor (see Spence (Spence 1998)). The result is that the off-central columns of the structure matrix S contain Fourier coefficients for the scattered electron wave for orientations differing by reciprocal lattice vectors h relative to the central column.…”