## International Tables for Crystallography

Electron diffraction and electron microscopy in structure determination International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 297-402 [ doi:10.1107/97809553602060000767 ] Abstract 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 M. Tanaka describes how the point groups and space groups of ordinary (three-dimensional or 3D) crystals, one-dimensionally incommensurate (4D) crystals and quasicrystals (5D and 6D) can be determined by convergent-beam electron diffraction. Useful tables and examples of point- and space-group determination are provided. 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. In Section 2.5.6, B. K. Vainshtein and P. A. Penczek discuss algorithms for three-dimensional reconstruction from sets of ray projections, with emphasis on algorithms used in cryo-electron microscopy, including single-particle reconstruction and objects with icosahedral and helical symmetries. The general feasibility of the reconstruction problem as well as the limitations posed by discretization and interpolation are discussed. A detailed analysis of three classes of reconstruction methods is provided: algebraic and iterative, filtered backprojection, and direct Fourier inversion. In each case, the efficiency of the respective method is discussed and its performance for typical cryo-electron microscopy data sets is evaluated. In Section 2.5.7, P. A. Penczek describes macromolecular structure determination using cryo-electron microscopy and the single-particle approach. A general overview of the analytical steps is given with a detailed analysis of the pivotal computational methods involved and with emphasis on the evaluation of the reliability of the results. Examples of near-atomic resolution as well as intermediate resolution structure-determination projects are given, accompanied by a discussion of the methods used to present and analyse the results. Section 2.5.8 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. |

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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 eight 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.