Features

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

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

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

6. Deconvolution Bi Sr Cu Ca Cu Sr FT-1 Bi c b Oxygen in Cu-O layer 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 = 0.21b* + c* Phase extension

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

8. T T T T T t T = 0 (mod t) or MOD (T, t) = 0 Commensurate modulation Þ superstructures T¹ 0 (mod t) or MOD (T, t) ¹ 0 Incommensurate modulation Þ incommensurate structures What’s a Modulated Structure ?

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

10. 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

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

12. situated at their average positions Modulated atoms Structure-factor formula

13. Modified Sayre Equations in multi-dimensional space

14. using using Strategy of solving incommensurate modulated structures i) Derive phases of main reflections ii) Derive phases of satellite reflections 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

15. DIMS: direct methods for incommensurate structures

16. Modulated atoms in g - Na2CO3 Na O1,3

17. Bi-2223 superconductor Incommensuratemodulation revealed by the direct method

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

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

20. MIMS:automatic search in 4D Fourier maps

21. MIMS: searching in 3-dimensional space

22. MIMS: searching in 4-dimensional space

23. MIMS: output structure model

24. 4-Dimensional Structure Refinement

25. Multislice Method for conventional structures and aperiodic crystals

26. 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

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

28. Fourier sections of the superconductor Bi-2212

29. 2D section in a 4D Fourier map

30. Contour mapping

31. Contrast Adjustment