Structure determination at the atomic level from dynamical electron diffraction data under systematic row conditions

We discuss a method to obtain structural information on crystals at the atomic level in high-resolution transmission electron microscopy from dynamical diffraction data under systematic row conditions. Working at a fixed incident energy and within an N-beam approximation, data is required at a well defined set of N incident beam orientations to determine the scattering matrix, one orientation for each column in the matrix. At each orientation the corresponding column of the scattering-matrix is obtained by Fourier transformation of the exit surface wave function. Thus, in addition to each exit surface image, we must recover the phase of the wave function for that orientation in the image plane. We show that retrieval of the phase using algorithms based on conservation of flux, which assume continuity of the phase, can yield incorrect solutions for the phase. This is because singularities can occur in the phase of the wave field at points where the intensity is zero, which can lead to edge dislocations in the phase. We demonstrate, using a model example, how these edge dislocations arise. We will show that phase retrieval from a through focal series of measurements or using the Gerchberg-Saxton algorithm (starting from measurements of an image and the corresponding diffraction pattern), correctly retrieves the phase and hence the exit surface wave function for all the orientations required to obtain the scattering-matrix. The dynamical (multiple) scattering can then be inverted to uniquely obtain the projected potential.