Inversion of Dynamical Scattering from Large-Angle Rocking-Beam Electron Diffraction Patterns

A method for ab initio structure factor retrieval from large-angle rocking-beam electron diffraction data of thin crystals is described and tested with experimental and simulated data. No additional information, such as atomicity or information about chemical composition, has been made use of. Our numerical experiments show that the inversion of dynamical scattering works best, if the beam tilt range is large and the specimen not too thick, because for moderate multiple scattering, the large tilt amplitude effectively removes local minima in this global optimization problem.

Figure 1

LARBED patterns of ${\mathrm{SrTiO}}_3$ oriented close to the (001) zone axis simulated for a 120 keV electron beam, with 45 beams selected. Within the discs of radius ${\theta }_{\max }=5$, 10, 20, and 40 mrad, 1024 different beam tilt vectors uniformly distributed within these discs have been chosen. The different rows represent specimen thicknesses of 5, 10, 20, and 40 nm.

Figure 2

LARBED patterns simulated from the structure factor fitted to the simulated data shown in Fig. 1 by minimizing expression (5).

Figure 3

Maps of the projected potential of $2\times 2$ unit cells corresponding to the recovered sets of structure factors. The reconstructed potential maps are shown on the left-hand side, the differences between the reconstructed and the theoretical maps are shown on the right-hand side. The numbers in the top right corner of each of these maps represent $E_{\text{triplet}}$, i.e., the mean of 1189 phase triplet errors computed from those ${\mathbf{U}}_g$ for which intensity measurements have been included in the optimization, i.e., which appear in the $m$th column of the structure factor matrix. The numbers in the top left corner of each map indicate the $R$ value calculated from Eq. (6).

Figure 4

(a) Experimental LARBED pattern of ${\mathrm{SrTiO}}_3$ comprising 1009 different beam tilts. (b) LARBED pattern simulated from the set of reconstructed structure factors, assuming no additional information about the specimen other than the experimental data. (c) Difference of the intensities shown in (a) and (b). (d) Theoretical map of the (001)-projected potential of ${\mathrm{SrTiO}}_3$ as represented by the first 121 reflections in this orientation. The projected potential of $2\times 2$ unit cells is shown [same in (e) and (f)]. (e) Reconstructed potential map corresponding to the first 121 beams, applying no additional (symmetry) constraints. The real part of the projected potential is shown on the left-hand side, and the imaginary part on the right-hand side [same in (f)]. (f) Reconstructed potential map with all 456 different structure factors, but constraining some $|{\mathbf{U}}_g|$’s as explained in the text.