Chapter 2.5 Electron diffraction and electron microscopy in structure determination

Reciprocal space

First Online Edition (2006)

Part 2. Reciprocal space in crystal-structure determination

  1. J. M. Cowley1,
  2. P. Goodman2,
  3. B. K. Vainshtein3,
  4. B. B. Zvyagin4,
  5. D. L. Dorset5

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000558

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Cowley, J. M., Goodman, P., Vainshtein, B. K., Zvyagin, B. B. and Dorset, D. L. 2006. Electron diffraction and electron microscopy in structure determination. International Tables for Crystallography. B:2:2.5:276–345.

Author Information

  1. 1

    Arizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287-1504, USA

  2. 2

    School of Physics, University of Melbourne, Parkville, Australia 3052

  3. 3

    Institute of Crystallography, Academy of Sciences of Russia, Leninsky prospekt 59, Moscow B-117333, Russia

  4. 4

    Institute of Ore Mineralogy (IGEM), Academy of Sciences of Russia, Staromonetny 35, 109017 Moscow, Russia

  5. 5

    ExxonMobil Research and Engineering Co., 1545 Route 22 East, Clinton Township, Annandale, New Jersey 08801, USA

Publication History

  1. Published Online: 1 JAN 2006


This chapter deals with electron diffraction and imaging in the transmission geometry at kilovolt and higher beam energies for the purposes of crystal structure determination. Techniques such as low-energy electron diffraction, gas electron diffraction and reflection electron diffraction are not considered. Following a brief review of the history of the subject and its relationship to X-ray diffraction, the fundamental theory is presented by J. M. Cowley in Section 2.5.2. Unlike X-rays, which diffract from the electron density in a crystal, electrons are scattered elastically by the Coulomb potential, which is related to the density through Poisson’s equation and includes the nuclear contribution. Electrons are much more strongly scattered than X-rays and have much smaller wavelengths. (This leads to a very ‚flat’ Ewald sphere and the simultaneous excitation of many Bragg beams.) Electron sources have comparable brightness to third-generation synchrotrons fitted with an undulator, so that Bragg intensities are very high. Because of their limited penetration, samples for transmission electron microscopes (TEMs) usually have thicknesses of less than a micron, while for interpretable atomic resolution images, thicknesses of tens of nanometres are preferred. Multiple scattering complicates TEM image interpretation and diffraction for larger thicknesses. The current (2006) spatial resolution of the best aberration-corrected TEM instruments is about 0.8 nm. Microdiffraction patterns may be obtained using a beam width of sub-nanometre dimensions, while the analysis of characteristic X-rays excited by such a probe provides chemical information. For organic monolayers (such as proteins), the amount of structural information obtained per unit of radiation damage (which is a strong function of resolution) exceeds that obtainable by X-ray diffraction. Cowley’s treatment includes sections on electron scattering factors, Bethe’s 1928 multiple scattering theory, Born’s series, sign conventions, two-beam dynamical theory and single-scattering theory. This is followed by the theory of electron microscope imaging at high resolution, including the weak-phase object and other approximations, and the Scherzer focusing condition. The section ends with treatments of atomic resolution imaging in crystals and the factors which limit it, with Fourier or Talbot self-imaging, and with a brief discussion of coherent nanodiffraction. Section 2.5.3 by P. Goodman describes the convergent-beam electron diffraction (CBED) method used for space-group determination. This information is obtained in the microdiffraction mode, allowing nanocrystals to be analysed. A cone of illumination is used, which broadens the Bragg spots into discs, whose internal symmetries are analysed together with the symmetry of the whole pattern of discs. This section reviews the history of the subject, and provides tables which allow a systematic deduction of point and space groups from these data in most cases. Multiple scattering effects, in particular, can be used to distinguish between centrosymmetric and noncentrosymmetric crystals, while cancellation along symmetry-related multiple scattering paths allows identification of screw and glide elements. Examples from the literature are cited and sources of useful multiple-scattering software for simulating CBED patterns are noted. Section 2.5.4 by B. K. Vainshtein and B. B. Zvyagin is devoted to the use of transmission electron diffraction patterns to solve the structures of thin crystal structures using the kinematic theory. Texture and polycrystalline patterns are considered, as is the relationship to X-ray work. Section 2.5.5 by B. K. Vainshtein again outlines the theory of high-resolution electron imaging, extending this to include image processing, image cross-correlation and alignment, and image filtering and enhancement. Section 2.5.6 by B. K. Vainshtein discusses the problem of three-dimensional image reconstruction from projections, widely used in cryo-electron microscopy of proteins, and increasingly also now in materials science. The real-space methods of Radon, the method of back-projection, iterative methods and reciprocal-space techniques are described in full. Section 2.5.7 by D. L. Dorset summarizes solutions to the phase problem which may be applied to electron diffraction data. These include many of those currently in use for X-ray diffraction, including Patterson maps, direct methods and trial-and-error search techniques. Much of the section concerns electron diffraction data from thin organic films, analysed using the three-phase invariants of the direct methods approach to phasing. The tangent formula is reviewed and useful software is suggested. Density modification and maximum entropy methods are outlined. Because of the strong dependence of multiply scattered electron diffraction intensities on the thickness of the sample, and on local orientation changes (effects which are not accounted for in the structure analysis of the data), the thinnest possible samples must be used, and phasing measures are sought which are robust with respect to multiple scattering perturbations. Conversely, the ability to obtain data from nanometre-sized regions can greatly assist the effort to obtain high-quality perfect-crystal data free of defects, bending or thickness variation.


  • electron crystallography;
  • electron diffraction;
  • electron microscopy;
  • EDSA;
  • electron-diffraction structure analysis;
  • kinematical diffraction;
  • weak phase objects;
  • crystal structure imaging;
  • image reconstruction;
  • convergent-beam electron diffraction;
  • CBED;
  • space-group determination;
  • zone-axis patterns;
  • Friedel’s law;
  • in-disc symmetries;
  • symmetry elements;
  • crystal defects;
  • twins;
  • Ewald sphere;
  • optimal defocus;
  • non-classical crystallography;
  • image enhancement;
  • three-dimensional reconstruction;
  • back-projection method;
  • Fourier transformation;
  • discretization;
  • direct phase determination;
  • phase invariant sums;
  • tangent formula;
  • density modification;
  • convolution techniques;
  • maximum entropy;
  • maximum likelihood;
  • absorption in electron diffraction;
  • image resolution