Electron diffraction

Colliex, C.; Cowley, J. M.; Dudarev, S. L.; Fink, M.; Gjønnes, J.; Hilderbrandt, R.; Howie, A.; Lynch, D. F.; Peng, L. M.; Ren, G.; Ross, A. W.; Smith, V. H. Jr; Spence, J. C. H.; Steeds, J. W.; Wang, J.; Whelan, M. J.; Zvyagin, B. B.

doi:10.1107/97809553602060000593

International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 259-429
https://doi.org/10.1107/97809553602060000593

Chapter 4.3. Electron diffraction

4.3. Electron diffraction (pp. 259-429) | html | pdf | chapter contents |
C. Colliex, J. M. Cowley, S. L. Dudarev, M. Fink, J. Gjønnes, R. Hilderbrandt, A. Howie, D. F. Lynch, L. M. Peng, G. Ren, A. W. Ross, V. H. Smith Jr, J. C. H. Spence, J. W. Steeds, J. Wang, M. J. Whelan and B. B. Zvyagin
- 4.3.1. Scattering factors for the diffraction of electrons by crystalline solids (pp. 259-262) | html | pdf |
  J. M. Cowley
  - 4.3.1.1. Elastic scattering from a perfect crystal (p. 259) | html | pdf |
  - 4.3.1.2. Atomic scattering factors (pp. 259-260) | html | pdf |
  - 4.3.1.3. Approximations of restricted validity (p. 260) | html | pdf |
  - 4.3.1.4. Relativistic effects (pp. 260-261) | html | pdf |
  - 4.3.1.5. Absorption effects (p. 261) | html | pdf |
  - 4.3.1.6. Tables of atomic scattering amplitudes for electrons (p. 261) | html | pdf |
  - 4.3.1.7. Use of Tables 4.3.1.1 and 4.3.1.2 (pp. 261-262) | html | pdf |
- 4.3.2. Parameterizations of electron atomic scattering factors (p. 262) | html | pdf |
  J. M. Cowley, L. M. Peng, G. Ren, S. L. Dudarev and M. J. Whelan
- 4.3.3. Complex scattering factors for the diffraction of electrons by gases (pp. 262-391) | html | pdf |
  A. W. Ross, M. Fink, R. Hilderbrandt, J. Wang and V. H. Smith Jr
  - 4.3.3.1. Introduction (p. 262) | html | pdf |
  - 4.3.3.2. Complex atomic scattering factors for electrons (pp. 262-390) | html | pdf |
    - 4.3.3.2.1. Elastic scattering factors for atoms (pp. 262-389) | html | pdf |
    - 4.3.3.2.2. Total inelastic scattering factors (pp. 389-390) | html | pdf |
    - 4.3.3.2.3. Corrections for defects in the theory of atomic scattering (p. 390) | html | pdf |
  - 4.3.3.3. Molecular scattering factors for electrons (pp. 390-391) | html | pdf |
- 4.3.4. Electron energy-loss spectroscopy on solids (pp. 391-412) | html | pdf |
  C. Colliex
  - 4.3.4.1. Definitions (pp. 391-394) | html | pdf |
    - 4.3.4.1.1. Use of electron beams (pp. 391-392) | html | pdf |
    - 4.3.4.1.2. Parameters involved in the description of a single inelastic scattering event (p. 392) | html | pdf |
    - 4.3.4.1.3. Problems associated with multiple scattering (pp. 392-393) | html | pdf |
    - 4.3.4.1.4. Classification of the different types of excitations contained in an electron energy-loss spectrum (pp. 393-394) | html | pdf |
  - 4.3.4.2. Instrumentation (pp. 394-397) | html | pdf |
    - 4.3.4.2.1. General instrumental considerations (pp. 394-395) | html | pdf |
    - 4.3.4.2.2. Spectrometers (pp. 395-397) | html | pdf |
    - 4.3.4.2.3. Detection systems (p. 397) | html | pdf |
  - 4.3.4.3. Excitation spectrum of valence electrons (pp. 397-404) | html | pdf |
    - 4.3.4.3.1. Volume plasmons (pp. 397-399) | html | pdf |
    - 4.3.4.3.2. Dielectric description (pp. 399-401) | html | pdf |
    - 4.3.4.3.3. Real solids (pp. 401-403) | html | pdf |
    - 4.3.4.3.4. Surface plasmons (pp. 403-404) | html | pdf |
  - 4.3.4.4. Excitation spectrum of core electrons (pp. 404-411) | html | pdf |
    - 4.3.4.4.1. Definition and classification of core edges (pp. 404-406) | html | pdf |
    - 4.3.4.4.2. Bethe theory for inelastic scattering by an isolated atom (Bethe, 1930; Inokuti, 1971; Inokuti, Itikawa & Turner, 1978, 1979) (pp. 406-408) | html | pdf |
    - 4.3.4.4.3. Solid-state effects (pp. 408-410) | html | pdf |
    - 4.3.4.4.4. Applications for core-loss spectroscopy (pp. 410-411) | html | pdf |
  - 4.3.4.5. Conclusions (pp. 411-412) | html | pdf |
- 4.3.5. Oriented texture patterns (pp. 412-414) | html | pdf |
  B. B. Zvyagin
  - 4.3.5.1. Texture patterns (p. 412) | html | pdf |
  - 4.3.5.2. Lattice plane oriented perpendicular to a direction (lamellar texture) (pp. 412-413) | html | pdf |
  - 4.3.5.3. Lattice direction oriented parallel to a direction (fibre texture) (pp. 413-414) | html | pdf |
  - 4.3.5.4. Applications to metals and organic materials (p. 414) | html | pdf |
- 4.3.6. Computation of dynamical wave amplitudes (pp. 414-416) | html | pdf |
  - 4.3.6.1. The multislice method (pp. 414-415) | html | pdf |
    D. F. Lynch
  - 4.3.6.2. The Bloch-wave method (pp. 415-416) | html | pdf |
    A. Howie
- 4.3.7. Measurement of structure factors and determination of crystal thickness by electron diffraction (pp. 416-419) | html | pdf |
  J. Gjønnes and J. W. Steeds
- 4.3.8. Crystal structure determination by high-resolution electron microscopy (pp. 419-429) | html | pdf |
  J. C. H. Spence and J. M. Cowley
  - 4.3.8.1. Introduction (pp. 419-421) | html | pdf |
  - 4.3.8.2. Lattice-fringe images (pp. 421-422) | html | pdf |
  - 4.3.8.3. Crystal structure images (pp. 422-424) | html | pdf |
  - 4.3.8.4. Parameters affecting HREM images (pp. 424-425) | html | pdf |
  - 4.3.8.5. Computing methods (pp. 425-427) | html | pdf |
  - 4.3.8.6. Resolution and hyper-resolution (p. 427) | html | pdf |
  - 4.3.8.7. Alternative methods (pp. 427-428) | html | pdf |
  - 4.3.8.8. Combined use of HREM and electron diffraction (pp. 428-429) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 4.3.4.1. Definition of the regions in (E, q) space that can be investigated with the different primary sources of particles available at present (p. 392) | html | pdf |
  - Fig. 4.3.4.2. A primary electron of energy E₀ and wavevector k is inelastically scattered into a state of energy E₀ − ΔE and wavevector k′ (p. 392) | html | pdf |
  - Fig. 4.3.4.3. Excitation spectrum of aluminium from 1 to 250 eV, investigated by EELS on 300 keV primary electrons (p. 393) | html | pdf |
  - Fig. 4.3.4.4. Complete electron energy-loss spectrum of a thin rhodizite crystal (thickness ~60 nm) (p. 393) | html | pdf |
  - Fig. 4.3.4.5. Schematic energy-level representation of the origin of a core-loss excitation (from a core level C at energy E_c to an unoccupied state U above the Fermi level E_f) and its general shape in EELS, as superimposed on a continuously decreasing background (p. 393) | html | pdf |
  - Fig. 4.3.4.6. Energy-loss spectrum, in the meV region, of an evaporated germanium film (thickness $[\simeq]$ 25 nm) (p. 394) | html | pdf |
  - Fig. 4.3.4.7. Schematic drawing of a uniform magnetic sector spectrometer with induction B normal to the plane of the figure (p. 395) | html | pdf |
  - Fig. 4.3.4.8. Different factors contributing to the energy resolution in the dispersion plane (p. 395) | html | pdf |
  - Fig. 4.3.4.9. Optical coupling of a magnetic sector spectrometer on a STEM column (p. 396) | html | pdf |
  - Fig. 4.3.4.10. Principle of the Wien filter used as an EELS spectrometer: the trajectories are shown in the two principal (dispersive and focusing) sections (p. 396) | html | pdf |
  - Fig. 4.3.4.11. Principle of the Castaing & Henry filter made from a magnetic prism and an electrostatic mirror (p. 396) | html | pdf |
  - Fig. 4.3.4.12. A commercial EELS spectrometer designed for parallel detection on a photodiode array (p. 397) | html | pdf |
  - Fig. 4.3.4.13. The dispersion curve for the excitation of a plasmon (curve 1) merges into the continuum of individual electron–hole excitations (between curves 2 and 4) for a critical wavevector q_c (p. 398) | html | pdf |
  - Fig. 4.3.4.14. Measured angular dependence of the differential cross section dσ/dΩ for the 15 eV plasmon loss in Al (dots) compared with a calculated curve by Ferrell (solid curve) and with a sharp cut-off approximation at θ_c (dashed curved) (p. 399) | html | pdf |
  - Fig. 4.3.4.15. Variation of plasmon excitation mean free path Λ_p as a function of accelerating voltage V in the case of carbon and aluminium (p. 399) | html | pdf |
  - Fig. 4.3.4.16. Dielectric and optical functions calculated in the Drude model of a free-electron gas with ħω_p = 16 eV and τ = 1.64 × 10⁻¹⁶ s (p. 400) | html | pdf |
  - Fig. 4.3.4.17. Same as previous figure, but for a Lorentz model with an oscillator of eigenfrequency ħω₀ = 10 eV and relaxation time τ₀ = 6.6 × 10⁻¹⁶ s superposed on the free-electron term (p. 401) | html | pdf |
  - Fig. 4.3.4.18. Dielectric coefficients $[\varepsilon_1]$ , $[\varepsilon_2]$ and $[{\rm Im}(-1/\varepsilon)]$ from a collection of typical real solids (p. 402) | html | pdf |
  - Fig. 4.3.4.19. Dielectric functions in graphite derived from energy losses for E ⊥ c (i.e. the electric field vector being in the layer plane) and for E||c (p. 403) | html | pdf |
  - Fig. 4.3.4.20. Geometric conditions for investigating the anisotropic energy-loss function (p. 404) | html | pdf |
  - Fig. 4.3.4.21. Definition of electron shells and transitions involved in core-loss spectroscopy (p. 404) | html | pdf |
  - Fig. 4.3.4.22. Chart of edges encountered in the 50 eV up to 3 keV energy-loss range with symbols identifying the types of shapes (p. 405) | html | pdf |
  - Fig. 4.3.4.23. A selection of typical profiles (K, L_2,3, M_4,5, and N_2,3) illustrating the most important behaviours encountered on major edges through the Periodic Table (p. 406) | html | pdf |
  - Fig. 4.3.4.24. Bethe surface for K-shell ionization, calculated using a hydrogenic model (p. 407) | html | pdf |
  - Fig. 4.3.4.25. A novel technique for simulating an energy-loss spectrum with two distinct edges as a superposition of theoretical contributions (hydrogenic saw-tooth for O K, Lorentzian white lines and delayed continuum for Fe L_2,3 calculated with the Hartree–Slater description) (p. 407) | html | pdf |
  - Fig. 4.3.4.26. Definition of the different fine structures visible on a core-loss edge (p. 408) | html | pdf |
  - Fig. 4.3.4.27. High-energy resolution spectra on the L_2,3 titanium edge from two phases (rutile and anatase) of TiO₂ (p. 408) | html | pdf |
  - Fig. 4.3.4.28. The dramatic change in near-edge fine structures on the L₃ and L₂ lines of Cu, from Cu metal to CuO (p. 409) | html | pdf |
  - Fig. 4.3.4.29. Illustration of the single and multiple scattering effects used to describe the final wavefunction on the excited site (p. 409) | html | pdf |
  - Fig. 4.3.4.30. Comparison of the experimental O K edge (solid line) with calculated profiles in the multiple scattering approach (p. 409) | html | pdf |
  - Fig. 4.3.4.31. The conventional method of background subtraction for the evaluation of the characteristic signals S_{O K} and S_{Fe L_2,3} used for quantitative elemental analysis (p. 410) | html | pdf |
  - Fig. 4.3.4.32. Values of n_eff for metallic aluminium based on composite optical data (p. 411) | html | pdf |
  - Fig. 4.3.5.1. The relative orientations of the direct and the reciprocal axes and their projections on the plane ab, with indication of the distances B_hk and D_hkl that define the positions of reflections in lamellar texture patterns (p. 412) | html | pdf |
  - Fig. 4.3.5.2. (a) Part of the OTED pattern of the clay mineral kaolinite and (b) the intensity profile of a characteristic quadruplet of reflections recorded with the electron diffractometry system (p. 413) | html | pdf |
  - Fig. 4.3.5.3. The projections of the reciprocal axes on the plane ab of the direct lattice, with indications of the distances B and D of the hk rows from the fibre-texture axes a or [hk] (p. 413) | html | pdf |
  - Fig. 4.3.7.1. (a) Dispersion-surface section for the symmetric four-beam case (0, g, g + h, m), γk is a function of k_x, referred to (b), where k_x = k_y = 0 corresponds to the exact Bragg condition for all three reflections (p. 418) | html | pdf |
  - Fig. 4.3.8.1. Atomic resolution image of a tantalum-doped tungsten trioxide crystal (pseudo-cubic structure) showing extended crystallographic shear-plane defects (C), pentagonal-column hexagonal-tunnel (PCHT) defects (T), and metallization of the surface due to oxygen desorption (JEOL 4000EX, crystal thickness less than 200 Å, 400 kV, C_s = 1 mm) (p. 419) | html | pdf |
  - Fig. 4.3.8.2. Imaging conditions for few-beam lattice images (p. 420) | html | pdf |
  - Fig. 4.3.8.3. A summary of three- (or five-) beam axial imaging conditions (p. 422) | html | pdf |
  - Fig. 4.3.8.4. The contrast of few-beam lattice images as a function of focus in the neighbourhood of the stationary-phase focus (p. 422) | html | pdf |
  - Fig. 4.3.8.5. (a) The transfer function for a 400 kV electron microscope with a point resolution of 1.7 Å at the Scherzer focus; the curve is based on equation (4.3.8.17) (p. 424) | html | pdf |
  - Fig. 4.3.8.6. Structure image of a thin lamella of the 6H polytype of SiC projected along [110] and recorded at 1.2 MeV (p. 426) | html | pdf |
- Tables
  - Table 4.3.1.1. Atomic scattering amplitudes (Å) for electrons for neutral atoms (pp. 263-271) | html | pdf |
  - Table 4.3.1.2. Atomic scattering amplitudes (Å) for electrons for ionized atoms (pp. 272-281) | html | pdf |
  - Table 4.3.2.1. Parameters useful in electron diffraction as a function of accelerating voltage (p. 281) | html | pdf |
  - Table 4.3.2.2. Elastic atomic scattering factors of electrons for neutral atoms and s up to 2.0 Å⁻¹ (pp. 282-283) | html | pdf |
  - Table 4.3.2.3. Elastic atomic scattering factors of electrons for neutral atoms and s up to 6.0 Å⁻¹ (pp. 284-285) | html | pdf |
  - Table 4.3.3.1. Partial wave elastic scattering factors for neutral atomsinteractive version (pp. 286-377) | html | pdf |
  - Table 4.3.3.2. Inelastic scattering factors (pp. 378-388) | html | pdf |
  - Table 4.3.4.1. Different possibilities for using EELS information as a function of the different accessible parameters (r, $[\boldtheta]$ , ΔE) (p. 394) | html | pdf |
  - Table 4.3.4.2. Plasmon energies measured (and calculated) for a few simple metals (p. 397) | html | pdf |
  - Table 4.3.4.3. Experimental and theoretical values for the coefficient α in the plasmon dispersion curve together with estimates of the cut-off wavevector (p. 398) | html | pdf |
  - Table 4.3.4.4. Comparison of measured and calculated values for the halfwidth ΔE_1/2(0) of the plasmon line (p. 398) | html | pdf |

Chapter 4.3. Electron diffraction

Contents