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

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

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

Image Processing of Bi-2212 Deconvolution FT-1 Phase extension EM image from Dr. S. Horiuchi Space group: N [Bbmb] 1 -1 1 a = 5.42, b = 5.44, c = 30.5Å; q = 0.21b* + c* Deconvolution FT-1 c b Bi Sr Cu Ca Oxygen in Cu-O layer Phase extension

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

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

Schematic diffraction pattern of an incommensurate modulated structure b* q

Conclusion In the reciprocal space: The diffraction pattern of an incommen-surate 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 higher-dimensional 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

incommensurate modulated structures 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 - Na2CO3 Na O1,3

Bi-2223 superconductor Incommensuratemodulation revealed by the direct method

(PbS)1.18TiS2 composite structure 4-dimensional average structure solved by the direct method

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

MIMS: automatic search in 4D 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

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

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

Fourier sections of the superconductor Bi-2212

2D section in a 4D Fourier map

Contour mapping

Contrast Adjustment