Features Direct Methods for Image Processing in HREM

Features Direct Methods for Image Processing in HREM Direct Methods of Solving Aperiodic Structures Searching Algorithm for Finding Modulation waves in 4 D Fourier Maps

Deconvolution of a Single EM Original image far from the Scherzer Image defocus of the final model Image after Averaged. Deconvoluted Fourier image recycling image

Symmetry averaging Original EM ED from. Prof. N. from Uyeda N. Uyeda Two-Step Image Processing Search for defocus Partial Complete structure model Image Deconvolution Phase extension Fourier recycling

Image Processing of Bi-2212 EM image from Dr. S. Horiuchi Space group: N [Bbmb] 1 -1 1 a = 5. 42, b = 5. 44, c = 30. 5Å; q = 0. 21 b* + c* Deconvolution Phase extension Bi Sr Cu Ca Sr Cu Bi FT-1 c b Oxygen in Cu-O layer

Features Direct Methods for Image Processing in HREM Direct Methods of Solving Aperiodic Structures Searching Algorithm for Finding Modulation waves in 4 D Fourier Maps

What’s a Modulated Structure ? TT T t T T T= ¹ 0 (mod t) or MOD (T, t) = ¹ 0 Commensurate modulation Incommensurate modulationÞ Þ superstructures incommensurate structures

Schematic diffraction pattern of an incommensurate modulated structure b* q a*

Conclusion In the reciprocal space: The diffraction pattern of an incommensurate modulated crystal is the projection of a 4 - or higher-dimensional weighted lattice In the direct space: An incommensurate modulated structure is the “hypersection” of a 4 - or higherdimensional periodic structure cut with the 3 dimensional physical space

Representation of one-dimensionally modulated incommensurate structures Lattice vectors in real- and reciprocal- space

Structure-factor formula Modulated atoms situated at their average positions

Modified Sayre Equations in multi-dimensional space

Strategy of solving incommensurate modulated structures i) Derive phases of main reflections using ii) Derive phases of satellite reflections using iii) Calculate the multi-dimensional Fourier map iv) Cut the resulting Fourier map with the 3 -D ‘hyperplane’ (3 -D physical space) v) Parameters of the modulation functions are measured directly on the multi-dimensional Fourier map

DIMS: direct methods for incommensurate structures

Modulated atoms in g - Na 2 CO 3 Na O 1, 3

Bi-2223 superconductor Incommensurate modulation revealed by the direct method

(Pb. S)1. 18 Ti. S 2 composite structure 4 -dimensional average structure solved by the direct method

Features Direct Methods for Image Processing in HREM Direct Methods of Solving Aperiodic Structures Searching Algorithm for Finding Modulation waves in 4 D Fourier Maps

MIMS: automatic search in 4 D Fourier maps

MIMS: searching in 3 -dimensional space

MIMS: searching in 4 -dimensional space

MIMS: output structure model

4 -Dimensional Structure Refinement

Multislice Method for conventional structures and aperiodic crystals

Using experimental thermal motion (B) & modulation (M) Setting parameters Setting M=0 B=0 Setting B=0 & M=0 Bi-2201 Variation of dynamicaldiffraction amplitudes with sample thickness

Potential Maps of Bi-2201 calculated with dynamical-diffraction amplitudes ~100Å ~200Å ~300Å

Fourier sections of the superconductor Bi-2212

2 D section in a 4 D Fourier map

Contour mapping

Contrast Adjustment