Electron diffraction and electron microscopy in structure determination

Cowley, J. M.; Goodman, P.; Vainshtein, B. K.; Zvyagin, B. B.; Dorset, D. L.

doi:10.1107/97809553602060000558

International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 276-345 | 1 | 2 |
https://doi.org/10.1107/97809553602060000558

Chapter 2.5. Electron diffraction and electron microscopy in structure determination

2.5. Electron diffraction and electron microscopy in structure determination (pp. 276-345) | html | pdf | chapter contents |
J. M. Cowley, P. Goodman, B. K. Vainshtein, B. B. Zvyagin and D. L. Dorset
- 2.5.1. Foreword (pp. 276-277) | html | pdf |
  J. M. Cowley
- 2.5.2. Electron diffraction and electron microscopy (pp. 277-285) | html | pdf |
  J. M. Cowley
  - 2.5.2.1. Introduction (pp. 277-278) | html | pdf |
  - 2.5.2.2. The interactions of electrons with matter (pp. 278-279) | html | pdf |
  - 2.5.2.3. Recommended sign conventions (pp. 279-280) | html | pdf |
  - 2.5.2.4. Scattering of electrons by crystals; approximations (pp. 280-281) | html | pdf |
  - 2.5.2.5. Kinematical diffraction formulae (pp. 281-282) | html | pdf |
  - 2.5.2.6. Imaging with electrons (pp. 282-283) | html | pdf |
  - 2.5.2.7. Imaging of very thin and weakly scattering objects (pp. 283-284) | html | pdf |
  - 2.5.2.8. Crystal structure imaging (p. 284) | html | pdf |
  - 2.5.2.9. Image resolution (pp. 284-285) | html | pdf |
  - 2.5.2.10. Electron diffraction in electron microscopes (p. 285) | html | pdf |
- 2.5.3. Space-group determination by convergent-beam electron diffraction (pp. 285-306) | html | pdf |
  P. Goodman
  - 2.5.3.1. Introduction (pp. 285-286) | html | pdf |
    - 2.5.3.1.1. CBED (pp. 285-286) | html | pdf |
    - 2.5.3.1.2. Zone-axis patterns from CBED (p. 286) | html | pdf |
  - 2.5.3.2. Background theory and analytical approach (pp. 286-288) | html | pdf |
    - 2.5.3.2.1. Direct and reciprocity symmetries: types I and II (pp. 286-287) | html | pdf |
    - 2.5.3.2.2. Reciprocity and Friedel's law (p. 287) | html | pdf |
    - 2.5.3.2.3. In-disc symmetries (pp. 287-288) | html | pdf |
    - 2.5.3.2.4. Zero-layer absences (p. 288) | html | pdf |
  - 2.5.3.3. Pattern observation of individual symmetry elements (pp. 288-289) | html | pdf |
  - 2.5.3.4. Auxiliary tables (pp. 289-291) | html | pdf |
  - 2.5.3.5. Space-group analyses of single crystals; experimental procedure and published examples (pp. 291-292) | html | pdf |
    - 2.5.3.5.1. Stages of procedure (pp. 291-292) | html | pdf |
    - 2.5.3.5.2. Examples (p. 292) | html | pdf |
  - 2.5.3.6. Use of CBED in study of crystal defects, twins and non-classical crystallography (pp. 292-295) | html | pdf |
  - 2.5.3.7. Present limitations and general conclusions (p. 295) | html | pdf |
  - 2.5.3.8. Computer programs available (pp. 295-306) | html | pdf |
- 2.5.4. Electron-diffraction structure analysis (EDSA) (pp. 306-310) | html | pdf |
  B. K. Vainshtein and B. B. Zvyagin
  - 2.5.4.1. Introduction (p. 306) | html | pdf |
  - 2.5.4.2. The geometry of ED patterns (pp. 306-308) | html | pdf |
  - 2.5.4.3. Intensities of diffraction beams (pp. 308-309) | html | pdf |
  - 2.5.4.4. Structure analysis (pp. 309-310) | html | pdf |
- 2.5.5. Image reconstruction (pp. 310-315) | html | pdf |
  B. K. Vainshtein
  - 2.5.5.1. Introduction (pp. 310-311) | html | pdf |
  - 2.5.5.2. Thin weak phase objects at optimal defocus (pp. 311-312) | html | pdf |
  - 2.5.5.3. An account of absorption (p. 312) | html | pdf |
  - 2.5.5.4. Thick crystals (p. 312) | html | pdf |
  - 2.5.5.5. Image enhancement (pp. 313-315) | html | pdf |
- 2.5.6. Three-dimensional reconstruction (pp. 316-320) | html | pdf |
  B. K. Vainshtein
  - 2.5.6.1. The object and its projection (pp. 315-316) | html | pdf |
  - 2.5.6.2. Orthoaxial projection (pp. 316-317) | html | pdf |
  - 2.5.6.3. Discretization (p. 317) | html | pdf |
  - 2.5.6.4. Methods of direct reconstruction (p. 317) | html | pdf |
  - 2.5.6.5. The method of back-projection (p. 318) | html | pdf |
  - 2.5.6.6. The algebraic and iteration methods (p. 318) | html | pdf |
  - 2.5.6.7. Reconstruction using Fourier transformation (pp. 318-319) | html | pdf |
  - 2.5.6.8. Three-dimensional reconstruction in the general case (pp. 319-320) | html | pdf |
- 2.5.7. Direct phase determination in electron crystallography (pp. 320-326) | html | pdf |
  D. L. Dorset
  - 2.5.7.1. Problems with `traditional' phasing techniques (pp. 320-321) | html | pdf |
  - 2.5.7.2. Direct phase determination from electron micrographs (pp. 321-322) | html | pdf |
  - 2.5.7.3. Probabilistic estimate of phase invariant sums (pp. 322-323) | html | pdf |
  - 2.5.7.4. The tangent formula (pp. 323-324) | html | pdf |
  - 2.5.7.5. Density modification (p. 324) | html | pdf |
  - 2.5.7.6. Convolution techniques (p. 324) | html | pdf |
  - 2.5.7.7. Maximum entropy and likelihood (p. 325) | html | pdf |
  - 2.5.7.8. Influence of multiple scattering on direct electron crystallographic structure analysis (pp. 325-326) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 2.5.2.1. The variation with accelerating voltage of electrons of (a) the wavelength, λ and (b) the quantity $[\lambda [1 + (h^{2} / m_{0}^{2} c^{2} \lambda^{2})] = \lambda_{c} / \beta]$ (p. 279) | html | pdf |
  - Fig. 2.5.2.2. Diagram representing the critical components of a conventional transmission electron microscope (TEM) and a scanning transmission electron microscope (STEM) (p. 282) | html | pdf |
  - Fig. 2.5.2.3. The functions $[\chi (U)]$ , the phase factor for the transfer function of a lens given by equation (2.5.2.33), and $[\sin \chi (U)]$ for the Scherzer optimum defocus condition (p. 283) | html | pdf |
  - Fig. 2.5.3.1. Vector diagram in semi-reciprocal space, using Ewald-sphere constructions to show the `incident', `reciprocity' and `reciprocity × centrosymmetry' sets of vectors (p. 287) | html | pdf |
  - Fig. 2.5.3.2. Diagrammatic representation of a CBED disc with symmetry lines m, m_R (alternate labels G, S) and the central point $[{\bf K}_{g} = 0]$ (p. 287) | html | pdf |
  - Fig. 2.5.3.3. Diagrammatic representation of the influence of non-symmorphic elements (p. 289) | html | pdf |
  - Fig. 2.5.3.4. (a) Zone-axis patterns from hexagonal structures β- $[\hbox{Si}_{3}\hbox{N}_{4}]$ (left) and β-GaS (right) together with the appropriate planar figures for point symmetries 6 and 6mm, respectively (p. 293) | html | pdf |
  - Fig. 2.5.3.5. Zone-axis patterns from cubic structures gahnite (left) (Ishizuka & Taftø, 1982) and χ-phase precipitate (right) (Steeds & Evans, 1980) (p. 294) | html | pdf |
  - Fig. 2.5.3.6. (a) CBED pattern from the exact [001] (cyclic) zone-axis orientation of FeS₂ (p. 294) | html | pdf |
  - Fig. 2.5.4.1. Ewald spheres in reciprocal space (p. 306) | html | pdf |
  - Fig. 2.5.4.2. Triclinic reciprocal lattice (p. 307) | html | pdf |
  - Fig. 2.5.4.3. Formation of ellipses on an electron-diffraction pattern from an oblique texture (p. 307) | html | pdf |
  - Fig. 2.5.5.1. The χ function and two components of the Scherzer phase function sin $[\chi (U)]$ and cos $[\chi (U)]$ (p. 311) | html | pdf |
  - Fig. 2.5.5.2. (a) Diagram of an optical diffractometer (p. 314) | html | pdf |
  - Fig. 2.5.6.1. A three-dimensional object $[\varphi_{3}]$ and its two-dimensional projection $[\varphi_{2}]$ (p. 316) | html | pdf |
  - Fig. 2.5.6.2. The projection sphere and projection $[\varphi_{2}]$ of $[\varphi_{3}]$ along τ onto the plane $[{\bf x} \perp {\bf \boldtau}]$ (p. 316) | html | pdf |
  - Fig. 2.5.6.3. Orthoaxial projection (p. 317) | html | pdf |
  - Fig. 2.5.6.4. Discretization and oblique projection (p. 317) | html | pdf |
  - Fig. 2.5.6.5. (a) Formation of a projection function; (b) superposition of these functions (p. 318) | html | pdf |
  - Fig. 2.5.6.6. Relative position of the particle and planes of projection (p. 320) | html | pdf |
  - Fig. 2.5.6.7. Section of a three-dimensional Fourier transform of the density of the particles, corresponding to plane projections of this density (p. 320) | html | pdf |
  - Fig. 2.5.6.8. Plane projections of a three-dimensional body (p. 320) | html | pdf |
  - Fig. 2.5.7.1. Potential map for diketopiperazine (p. 323) | html | pdf |
- Tables
  - Table 2.5.2.1. Standard crystallographic and alternative crystallographic sign conventions for electron diffraction (p. 280) | html | pdf |
  - Table 2.5.3.1. Listing of the symmetry elements relating to CBED patterns under the classifications of `vertical' (I), `horizontal' (II) and combined or roto-inversionary axes (p. 286) | html | pdf |
  - Table 2.5.3.2. Diagrammatic illustrations of the actions of five types of symmetry elements (given in the last column in Volume A diagrammatic symbols) on an asymmetric pattern component, in relation to the centre of the pattern at $[{\bf K}_{00} = {\bf 0}]$ , shown as `⊕', or in relation to the centre of a diffraction order at $[{\bf K}_{0g} = {\bf 0}]$ , shown as `+' (p. 288) | html | pdf |
  - Table 2.5.3.3. Diffraction point-group tables, giving whole-pattern and central-beam pattern symmetries in terms of BESR diffraction-group symbols and diperiodic group symbols (p. 290) | html | pdf |
  - Table 2.5.3.4. Tabulation of principal-axis CBED pattern symmetries against relevant space groups given as numbers (pp. 296-297) | html | pdf |
  - Table 2.5.3.5. Conditions for observation of GS bands for the 137 space groups exhibiting these extinctions (pp. 298-305) | html | pdf |

Chapter 2.5. Electron diffraction and electron microscopy in structure determination

Contents