Direct Retrieval of Object Information using Inverse Solutions of Dynamical Electron Diffraction Max Planck Institute of Microstructure Physics Halle/Saale, Germany Kurt Scheerschmidt Quantitative Analysis: Trial-&-Error or Inverse Problems Confidence: a priori Data versus Regularization

trial-and-error image analysis direct object reconstruction 1. object modeling 2. wave simulation 3. image process 4. likelihood measure repetitionrepetition parameter & potential reconstruction wave reconstruction ? image ?

Inversion ? no iteration same ambiguities additional instabilities parameter & potential atomic displacements exit object wave image direct interpretation by data reduction: Fourier filtering QUANTITEM Fuzzy & Neuro-Net Srain analysis deviations from reference structures: displacement field (Head) algebraic discretization reference beam (holography) defocus series (Kirkland, van Dyck …) Gerchberg-Saxton (Jansson) tilt-series, voltage variation multi-slice inversion (van Dyck, Griblyuk, Lentzen, Allen, Spargo, Koch) Pade-inversion (Spence) non-Convex sets (Spence) local linearization

 = M(X)  0  = M(X 0 )  0 + M(X 0 )(X-X 0 )  0 Assumptions: - object: weakly distorted crystal - described by unknown parameter set X={t, K,V g, u} - approximations of t 0, K 0 a priori known

 M needs analytic solutions for inversion Perturbation: eigensolution , C for K, V yields analytic solution of  and its derivatives for K+  K, V+  V with  tr(  ) +  {1/(  i -  j )}   = C -1 (1+  ) -1 {exp(2  i (t+  t)} (1+  )C The inversion needs generalized matrices due to different numbers of unknowns in X and measured reflexes in  disturbed by noise Generalized Inverse (Penrose-Moore): X= X 0 +( M T M) -1 M T.[  exp -  X   ]

A0A0 A g1 A g2 A g3 P0P0 P g1 P g2 P g3...  exp X= X 0 +( M T M) -1 M T.[  exp -  X   ] i ii jjj XXX... t(i,j)K x (i,j)K y (i,j)

-lg(  ) lg(  ) Regularization K x (i,j)/a* K y (i,j)/a* t(i,j)/Å

Retrieval with iterative fit of the confidence region lg(  ) step / Å relative beam incidence to zone axis [110] [-1,1,0] [002] i ii iii i ii iii (i-iii increasing smoothing)

Ge-CdTe, 300kV Sample: D. Smith Holo: H. Lichte, M.Lehmann 10nm object wave amplitude object wave phase FT A000 P000 A1-11 P1-11 A1-1-1 A-111 P-111 P1-1-1 A-11-1 P-11-1 A-220 P-220 K x (i,j)/a* K y (i,j)/a* t(i,j)/Å set 1: Ge set 2: CdTe dV o /V o = 0.02% dV’ o /V’ o = 0.8%

K y (i,j)/a* K x (i,j)/a* K(i,j)/a* t(i,j)/ Å model/reco input 7 / 7 15 / / 9 15 / 7 beams used Influence of Modeling Errors

Thanks for your attention Thanks for cooperation: H.Lichte, M.Lehmann (Uni-Dresden)

regularization physically motivated Assumption:complex amplitudes are regular Cauchy relations: a/ x = a. / y a/ y = -a. / x Linear inversion:t(x+1,y)-2t(x,y)+t(x-1,y)=0 t(x,y+1)-2t(x,y)+t(x,y-1)=0

Confidence range? K x (i,j)/a*K y (i,j)/a*K(i,j)/a*t(i,j)/ Å

Properly posed problems (J. Hadamard 1902) Existence Uniqueness Stability if at least one solution But: exists which is unique and continuous with data implies determinism (Laplacian deamon, classical physics) of integrable systems for known initial/boundary conditions suitable theory/model & a priori knowledge inverse 1.kind solution via construction but small confidence (uniqueness/stability)

Direct & Inverse: black box gedankenexperiment operator A f input g output wave image thickness local orientation structure & defects composition microscope theory, hypothesis, model of scattering and imaging direct: g=A < f experiment, measurement invers 1.kind: f=A -1 < g parameter determination invers 2.kind: A=g $ f -1 identification, interpretation a priori knowledge intuition & induction additional data if unique & stable inverse A -1 exists ill-posed & insufficient data => least square

restricted information channel (D. van Dyck) a priori information: object & additional experiments amorph coordinates FT white noise medium range order PDF, ADF FT dense but structured S(r) crystal space group with basis / displacements FT discrete convolution with defects and shape

Uniqueness (D.M. Barnett):  / z ~  gu  / z => series expansion of  u => unique coefficient relations