Conventions Special aspects of the scattering of high- energetic electrons at crystals Axel Rother*, Kurt Scheerschmidt**, Hannes Lichte* *Triebenberg Laboratory, Institute of Structure Physics, Dresden University **Max Planck Institute of Microstructure Physics, Halle 1. Influence of the backscattered electrons I II III crystal vacuum incoming wave object exit wave y x z thickness d 0 Model for the scattering experiment: Equation to solve: Solving Method [1] : a.Finding complete set of eigenfunctions for each region b.Representing the wavefunction with these eigenfunctions c.Determining the excitation coefficients by applying boundary conditions between the different regions 2. Singularities in the scattering potential Scattering potential  sum of screened coulomb potentials of isolated atoms (Fig. 2) Perturbation condition not fulfilled  treatment of scattering (on heavy atoms) in the framework of Dirac equation ? Simulation of object exit waves: GaAs [001]-orientation, 300 kV Doyle-Turner atomic potentials Sampling rate of unit cell: only 11x11x11 due to limited computer power Reference simulation (blue line): numeric forward integration of Schrödinger equation  Further calculations have to be performed on a mainframe computer  low sampling rates  no high angle scattering (HOLZ not included)  no backscattering  high sampling rates  high angle scattering included  backscattering ? Simulated object exit wave of GaAs [3] [001], 17nm thickness, 300 kV, forward scattering Fig. 2: Atomic potentials [2] The correct understanding of the scattering of high energetic electrons is crucial for a thorough and quantitative interpretation of HREM images. The standard procedure incorporates a quantitative comparison between simulated and experimentally recorded images. The simulation calculates an object exit wave, which is experimentally accessible by means of electron holography. However, the simulated object exit waves do not match those of the experiment. Some of the possible reasons are presented here for discussion: 1. The influence of the backscattered electrons2. The influence of the singularities, contained in the scattering potential3. The influence of the thermal motion of the atoms. 3. Thermal motion of the atoms Mean potential approach (Debye-Waller damping) facilitates analytic solution Holographic imaging including incoherent averaging Probability space of image wave (e.g. thermal motion, lens aberrations) Probability density FOLZ 87.8% 91.1% Intensity Large scattering angles (Fig. 3) caused by singular scattering potentials can lead to backscattering (1.) and thermal diffuse scattering (3.) ? Au As C 0.1·|p|·c 200 kV 300 kV Radius in Å Potential in eV Fig. 3a: 64x64x64 Pixel Fig. 3b: 128x128x128 Pixel [1] E. Lamla (1938), Annalen der Physik, 5. Folge, Band 32 [2] E. Kirkland (1998), Advanced Computing in Electron Microscopy, Plenum Press [3] Atomic potentials: A. Weickenmeier, H. Kohl (1991), Acta Cryst. A47, Fully coherent holographic imaging Reconstructed wave coherently averaged! Fully coherent scattering Coherent averaging over time Simulated image wave No analytic solution available Region I and III (Zero potential) Region II (Periodic potential) general complex Bloch vector (only z-component not fixed) a.Finding basis of eigenfunctions for each region Excitation coefficients c.Applying boundary conditions b.Representing wave function in this basis 2. Order polynomial eigenvalue problem c.Applying boundary conditions Summary: Dirac equation How to incorporate total intensity damping of the reflections due to the intensity in the thermal diffuse background ? Experimental considerations Theoretical considerations References / Acknowledgements Abstract Fig. 1: [00]-beam amplitude vs. thickness thickness in nm normed amplitude Presented model Forward scattering Dresden University of Technology We thank all the members of the Triebenberg Laboratory for their assistance and fruitful discussions Max Planck Institute of Microstructure Physics Electron density ,<< Reziprocal lattice vector Triebenberg Laboratory