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, [email protected], 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