Electron crystallography based on inverse dynamic scattering

1. A1-11 A1-1-1 A000 1. object modeling Kx(i,j)/a* 2. wave simulation A-111 A-11-1 A-220 object wave amplitude r e p e t i t i o n FT set q1: Ge set q2: CdTe dVo/Vo = 0.02% dV’o/V’o = 0.8% Ge-CdTe, 300kV Sample: D. Smith Holo: H. Lichte, M.Lehmann ? 3. image process wave reconstruction Ky(i,j)/a* The inversion needs generalized matrices due to different numbers of unknowns in X and measured reflexes in y disturbed by noise Generalized Inverse (Penrose-Moore): X= X0+(JMTJM)-1JMT.[Fexp- F(X0)] ? P000 P1-11 P1-1-1 4. likelihood measure image 10nm object wave phase t(i,j)/Å P-111 P-11-1 P-220 trial-and-error image analysis direct object reconstruction parameter & potential reconstruction ... y = M(X) y0 A0 Ag1 Ag2 Ag3 Fexp ... P0 Pg1 Pg2 Pg3 Assumptions: - object: weakly distorted crystal - structure as mixed type potential V => SDqkVk - described by unknown parameter set X={t, K, qk, u} - approximation of start t0, K0 a priori known X= X0+(JMTJM)-1JMT.[Fexp- F(X0)] X X X X j j j j ... i i i i y = M(X0) y0 + JM(X0)(X-X0) y0 t(i,j) Kx(i,j) Ky(i,j) qk(i,j) Input data t, Kx, Ky (and random qi to vary the model) for retrieval test and estimating the model error Retrieved t, Kx, Ky for solely the basic potential Retrieved t, Kx, Ky, q1 assuming 1 additional mixed type potential Retrieved t, Kx, Ky, q1, q2 assuming 2 different mixed type potentials Retrieved t, Kx, Ky, q1, q2,…,q5 assuming 5 different mixed type potentials Kurt Scheerschmidt Max Planck Institute of Microstructure Physics, Halle/Saale, Germany, schee@mpi-halle.de, http://mpi-halle.de Electron crystallography: Determination of atoms in (small volumes of) solids by TEM-methods Method: Replacement of trial & error image matching by direct object (parameter) retrieval without data information loss by linearizing and regularizing the dynamical scattering theory Problems: Stabilization and including further parameter as e.g. potential and atomic displacements Extension: Second (3rd) order perturbation solution and mixed type potential approximation electron crystallographywhatatom determination in solids: position, type, …Hannes Lichte: Which atoms are where?whystrong electron interaction: small volumes, excitations, defects,…howtrial & error vs. direct analysis: inverse problem, tomography, methodic mixa priori unit cell and space group SAED, CBEDlocal lattices and defects CTEM, HRTEMexit wave phases HOLO, Defokus-Rekochemical analyses EFTEM,EELS Electron crystallography based on inverse dynamic scattering multi-slice inversion (van Dyck, Griblyuk, Lentzen, Allen, Spargo, Koch) kinematic + refinement (Hovmoeller) Pade-inversion (Spence) non-Convex sets (Spence) local linearization deviations from reference structures: displacement field (Head) algebraic discretization parameter & potential Inversion ? atomic displacements exit object wave direct interpretation by data reduction: Fourier filtering QUANTITEM Fuzzy & Neuro-Net Srain analysis image reference beam (holography) defocus series (Kirkland, van Dyck …) Gerchberg-Saxton (Jansson) tilt-series, voltage variation EXPERIMENTS Example 1: Tilted and twisted grains in Au BASIS: Linearized dynamical theory Single reflex reconstruction Generalized Inverse Open Questions: Stability increased, but confidence ? Potential variations recoverable, but in 3D ? Modeling error problem to be solved ? Analytic inverse solution JM needs analytic solutions for inversion Perturbation: eigensolution g, C for K, V yields analytic solution of y and its derivatives for D={K+DK, V+SDqkVk} the perturbed EWs are l = g + tr(D) + D{1/(gi-gj)}D + . . . . (3.rd order) M = C-1(1+D)-1 {exp(2pil(t+Dt)} (1+D)C+ . . . . (2.nd) no iteration same ambiguities additionalinstabilities Likelihood measure of wave F differences log(e) of F0+dXdF vs |dX| log(e) of F0+dXdF Regularization Extend the Penrose-Moore inverse by regularization (r regularization, C1 reflex weights, C2 pixels smoothing) X=X0+(JMTC1JM + rC2)-1JMTDF equivalent to a Maximum-Likelihood error distribution: ||Fexp-Fth||2 + r||DX||2 = Min Example 2: Grains in GeCdTe with different composition and scattering potential Retrieval error e vs regularization parameter r lg(e) 3 4 -lg(r) 5 Kx(i,j)/a* 3 Ky(i,j)/a* 4 5 t(i,j)/Å