A method for ab-initio structure factor retrieval from large-angle rocking-beam electron diffraction (LARBED)[1] data of thin crystals is described and tested.
This method determines crystal structure factors and specimen thickness from the intensities of the diffraction spots alone, solving a nonlinear least-squares problem. No additional information, such as atomicity or information about chemical composition, have been made use of.

In addition to a demonstration of the method on 120 keV experimental data of SrTiO₃, where 456 structure factors have been retrieved, we analyze the dependence of the success of the reconstruction on specimen thickness and tilt range by applying this algorithm to simulated data. Our numerical experiments show that the dynamical inversion by gradient optimization works best if the beam tilt range is large and the specimen not too thick. At specimen thicknesses which allow for moderate multiple scattering, the large tilt amplitude effectively removes local minima in this global optimization problem, making ab-initio structure factor retrieval possible.

In addition, a dynamic parallelism framework  based on the Compute Unified Device Architecture (CUDA) is introduced, reducing the runtime of the optimization routine by two orders of magnitude when compared to running on a CPU.

The authors acknowledge funding by the Carl Zeiss Foundation as well as the German Research Foundation (Grant no. KO 2911/7-1 and SFB 951).

[1] Koch, C.T., 2011. Aberration-compensated large-angle rocking-beam electron diffraction. Ultramicroscopy, 111, 828–840. doi:10.1016/j.ultramic.2010.12.014

Figures:

(1) Simulated target (001) LARBED pattern of SrTiO3 for a thickness of 20 nm and a max tilt angle of 20 mrad. (2) Pattern fitted to (1). (3) Potential map reconstructed from (1). (4) Target LARBED pattern of SrTiO3 for a thickness 20 nm and a max tilt angle of 40 mrad. (5) Pattern fitted to (4). (6) Potential map reconstructed from (4).

(a) Experimental LARBED pattern of SrTiO3 comprising 1009 different beam tilts. (b) LARBED pattern fitted to (a). (c) Difference of the intensities shown in (a) and (b). (d) Theoretical map of the (001)-projected potential of SrTiO3 as represented by the first 121 reflections in this orientation. (e) Reconstructed potential map corresponding to the first 121 beams, applying no additional constraints. (f) Reconstructed potential map with all 456 different structure factors, but enforcing Friedel's law.

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