International Tables for Crystallography

Electron diffraction
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. International Tables for Crystallography (2006). Vol. C, ch. 4.3, pp. 259-429  [ doi:10.1107/97809553602060000593 ]


The first section of this chapter concerns scattering factors for the diffraction of electrons by crystalline solids. An explanation of the theory of scattering by a perfect crystal is followed by a discussion of the kinematical, two-beam and phase-grating approximations. Relativistic and absorption effects are considered. Extensive tables of atomic scattering amplitudes for electrons for neutral and ionized atoms are presented. The second section of the chapter briefly discusses the parameterization of electron atomic scattering factors. Tables of useful parameters as a function of accelerating voltage and elastic atomic scattering factors for neutral atoms are given. Complex scattering factors for the diffraction of electrons by gases are discussed in the third section of the chapter. This section includes tables of scattering factors of interest for gas-phase electron diffraction from atoms and molecules in the keV energy region. In addition to the tables and a description of their uses, a discussion of the theoretical uncertainties related to the material in the tables is also provided. The tables give scattering factors for elastic and inelastic scattering from free atoms. The theory of molecular scattering based on these atomic quantities is also discussed. Electron energy-loss spectroscopy on solids is discussed in the fourth section of the chapter. Topics covered include: the use of electron beams; single and multiple scattering; the classification of the different excitations in a spectrum; instrumentation; and the excitation spectra of valence and core electrons. The fifth section of the chapter describes oriented texture patterns. Lamellar and fibre texture patterns are discussed and applications to metals and organic materials are mentioned. The computation of dynamic wave amplitudes in then described in the sixth section of the chapter. This section deals first with the multislice method. The numerical procedure is outlined and factors that influence the choice of thickness of the slice are discussed. Two checks that can be performed during a multislice calculation are noted. The Bloch-wave method is then described. The use of Bloch waves to describe electron diffraction and electron imaging in thin crystals is outlined together with the concept of the dispersion surface. These emerge as natural solutions of the Schrödinger equation with a periodic optical potential to generate the elastic scattering and also the loss of intensity from the coherent wave field due to thermal diffuse and inelastic scattering. The Bloch-wave approach is a useful complement to the multislice method and provides a clear picture of wave propagation in perfect and imperfect crystals. In the seventh section of the chapter, the measurement of structure factors and the determination of crystal thickness by electron diffraction are described. The use of convergent-beam electron diffraction to obtain integrated intensities is discussed and the relationship between intensity features and the dispersion surface is explained. The last section of the chapter concerns crystal-structure determination by high-resolution microscopy. This technique allows the arrangement of atomic columns in thin crystals to be observed directly. The resolution of the best instruments is now slightly below 0.1 nm. The images usually show a projection through a slice of crystal about 20 nm thick, however tomographic (three-dimensional) reconstruction is now possible at nanometre resolution. The images show the host of microphases, grain boundaries, twins, line and planar defects which broad-beam methods, such as X-ray diffraction, provide the average scattering from. These defects often control the properties of crystals, engineering materials and electronic devices. Individual nanostructures, such as carbon nanotubes and catalyst particles, may be imaged at atomic resolution. Fine twinning, polytypes, intergrowth of oxide phases etc. can be identified, and increasingly the detailed atomic structure of defects (such as oxide, superconductor and semiconductor interfaces) is being determined. Substitutional dopant atoms have recently been imaged for the first time. In biology the method is limited by radiation damage; however by summing many images of identical randomly oriented macromolecules, tomographic density maps can be reconstructed at subnanometre resolution from hydrated proteins which cannot be crystallized (e.g. membrane proteins). This section reviews the theoretical principles of high-resolution electron microscopy, including few-beam and structure image formation, effects of electron-optical lens aberrations, partial coherence, resolution-limiting factors, image-simulation methods, dynamical effects, and a summary of super-resolution schemes.

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About International Tables for Crystallography

International Tables for Crystallography is the definitive resource and reference work for crystallography. The series consists of nine volumes and comprises articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of materials.