International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 297402
https://doi.org/10.1107/97809553602060000767 Chapter 2.5. Electron diffraction and electron microscopy in structure determination
J. M. Cowley,^{a}^{‡} J. C. H. Spence,^{b} M. Tanaka,^{f} B. K. Vainshtein,^{c}^{‡} B. B. Zvyagin,^{d}^{‡} P. A. Penczek^{g} and D. L. Dorset^{e}
^{a}Arizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287–1504, USA, ^{b}Department of Physics, Arizona State University, Tempe, AZ 95287–1504, USA, ^{c}Institute of Crystallography, Academy of Sciences of Russia, Leninsky prospekt 59, Moscow B117333, Russia, ^{d}Institute of Ore Mineralogy (IGEM), Academy of Sciences of Russia, Staromonetny 35, 109017 Moscow, Russia, ^{e}ExxonMobil Research and Engineering Co., 1545 Route 22 East, Clinton Township, Annandale, New Jersey 08801, USA,^{f}Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Japan, and ^{g}The University of Texas – Houston Medical School, Department of Biochemistry and Molecular Biology, 6431 Fannin, MSB 6.218, Houston, TX 77030, USA 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 lowenergy 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 Xray diffraction, the fundamental theory is presented by J. M. Cowley in Section 2.5.2. Unlike Xrays, 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 Xrays 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 thirdgeneration 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 aberrationcorrected TEM instruments is about 0.8 nm. Microdiffraction patterns may be obtained using a beam width of subnanometre dimensions, while the analysis of characteristic Xrays 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 Xray diffraction. Cowley's treatment includes sections on electron scattering factors, Bethe's 1928 multiple scattering theory, Born's series, sign conventions, twobeam dynamical theory and singlescattering theory. This is followed by the theory of electron microscope imaging at high resolution, including the weakphase 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 selfimaging, 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 (threedimensional or 3D) crystals, onedimensionally incommensurate (4D) crystals and quasicrystals (5D and 6D) can be determined by convergentbeam electron diffraction. Useful tables and examples of point and spacegroup determination are provided. Section 2.5.4 by B. K. Vainshtein and B. B. Zvyagin is devoted to the use of transmission electrondiffraction patterns to solve the structures of thin crystal structures using the kinematic theory. Texture and polycrystalline patterns are considered, as is the relationship to Xray work. Section 2.5.5 by B. K. Vainshtein again outlines the theory of highresolution electron imaging, extending this to include image processing, image crosscorrelation and alignment, and image filtering and enhancement. In Section 2.5.6, B. K. Vainshtein and P. A. Penczek discuss algorithms for threedimensional reconstruction from sets of ray projections, with emphasis on algorithms used in cryoelectron microscopy, including singleparticle 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 cryoelectron microscopy data sets is evaluated. In Section 2.5.7, P. A. Penczek describes macromolecular structure determination using cryoelectron microscopy and the singleparticle 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 nearatomic resolution as well as intermediate resolution structuredetermination 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 electrondiffraction data. These include many of those currently in use for Xray diffraction, including Patterson maps, direct methods and trialanderror search techniques. Much of the section concerns electrondiffraction data from thin organic films, analysed using the threephase 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 electrondiffraction 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 nanometresized regions can greatly assist the effort to obtain highquality perfectcrystal data free of defects, bending or thickness variation. 
Given that electrons have wave properties and the wavelengths lie in a suitable range, the diffraction of electrons by matter is completely analogous to the diffraction of Xrays. While for Xrays the scattering function is the electrondensity distribution, for electrons it is the potential distribution which is similarly peaked at the atomic sites. Hence, in principle, electron diffraction may be used as the basis for crystal structure determination. In practice it is used much less widely than Xray diffraction for the determination of crystal structures but is receiving increasing attention as a means for obtaining structural information not readily accessible with Xray or neutrondiffraction techniques.
Electrons having wavelengths comparable with those of the Xrays commonly used in diffraction experiments have energies of the order of 100 eV. For such electrons, the interactions with matter are so strong that they can penetrate only a few layers of atoms on the surfaces of solids. They are used extensively for the study of surface structures by lowenergy electron diffraction (LEED) and associated techniques. These techniques are not covered in this series of volumes, which include the principles and practice of only those diffraction and imaging techniques making use of highenergy electrons, having energies in the range of 20 keV to 1 MeV or more, in transmission through thin specimens.
For the most commonly used energy ranges of highenergy electrons, 100 to 400 keV, the wavelengths are about 50 times smaller than for Xrays. Hence the scattering angles are much smaller, of the order of 10^{−2} rad, the recording geometry is relatively simple and the diffraction pattern represents, to a useful first approximation, a planar section of reciprocal space. Extinction distances are hundreds of ångstroms, which, when combined with typical lattice spacings, produces rockingcurve widths which are, unlike the Xray case, a significant fraction of the Bragg angle.
The elastic scattering of electrons by atoms is several orders of magnitude greater than for Xrays. This fact has profound consequences, which in some cases are highly favourable and in other cases are serious hindrances to structure analysis work. On the one hand it implies that electrondiffraction patterns can be obtained from very small singlecrystal regions having thicknesses equal to only a few layers of atoms and, with recently developed techniques, having diameters equivalent to only a few interatomic distances. Hence singlecrystal patterns can be obtained from microcrystalline phases.
However, the strong scattering of electrons implies that the simple kinematical singlescattering approximation, on which most Xray diffraction structure analysis is based, fails for electrons except for very thin crystals composed of lightatom materials. Strong dynamical diffraction effects occur for crystals which may be 100 Å thick, or less for heavyatom materials. As a consequence, the theory of dynamical diffraction for electrons has been well developed, particularly for the particular special diffracting conditions relevant to the transmission of fast electrons (see Chapter 5.2 ), and observations of dynamical diffraction effects are commonly made and quantitatively interpreted. The possibility has thus arisen of using the observation of dynamical diffraction effects as the basis for obtaining crystal structure information. The fact that dynamical diffraction is dependent on the relative phases of the diffracted waves then implies that relative phase information can be deduced from the diffraction intensities and the limitations of kinematical diffraction, such as Friedel's law, do not apply. The most immediately practicable method for making use of this possibility is convergentbeam electron diffraction (CBED) as described in Section 2.5.3.
A further important factor, determining the methods for observing electron diffraction, is that, being charged particles, electrons can be focused by electromagnetic lenses. Many of the resolutionlimiting aberrations of cylindrical magnetic lenses have now been eliminated through the use of aberrationcorrection devices, so that for weakly scattering samples the resolution is limited to about 1 Å by electronic and mechanical instabilities. This is more than sufficient to distinguish the individual rows of atoms, parallel to the incident beam, in the principal orientations of most crystalline phases. Thus `structure images' can be obtained, sometimes showing direct representation of projections of crystal structures [see IT C (2004), Section 4.3.8 ]. However, the complications of dynamical scattering and of the coherent imaging processes are such that the image intensities vary strongly with crystal thickness and tilt, and with the defocus or other parameters of the imaging system, making the interpretation of images difficult except in special circumstances. Fortunately, computer programs are readily available whereby image intensities can be calculated for model structures [see IT C (2004), Section 4.3.6 ]. Hence the means exist for deriving the projection of the structure if only by a process of trial and error and not, as would be desirable, from a direct interpretation of the observations.
The accuracy with which the projection of a structure can be deduced from an image, or series of images, improves as the resolution of the microscope improves but is not at all comparable with the accuracy attainable with Xray diffraction methods. A particular virtue of highresolution electron microscopy as a structural tool is that it may give information on individual small regions of the sample. Structures can be determined of `phases' existing over distances of only a few unit cells and the defects and local disorders can be examined, one by one.
The observation of electrondiffraction patterns forms an essential part of the technique of structure imaging in highresolution electron microscopy, because the diffraction patterns are used to align the crystals to appropriate axial orientations. More generally, for all electron microscopy of crystalline materials the image interpretation depends on knowledge of the diffraction conditions. Fortunately, the diffraction pattern and image of any specimen region can be obtained in rapid succession by a simple switching of lens currents. The ready comparison of the image and diffraction data has become an essential component of the electron microscopy of crystalline materials but has also been of fundamental importance for the development of electrondiffraction theory and techniques. The development of the nanodiffraction method in the fieldemission scanning transmission electron microscope (STEM) has allowed microdiffraction patterns to be obtained from subnanometresized regions, and so has become the ideal tool for the structural analysis of the new microcrystalline phases important to nanoscience. The direct phasing of these coherent nanodiffraction patterns is an active field of research.
The individual specimen regions giving singlecrystal electrondiffraction patterns are, with few exceptions, so small that they can be seen only by use of an electron microscope. Hence, historically, it was only after electron microscopes were commonly available that the direct correlations of diffraction intensities with crystal size and shape could be made, and a proper basis was available for the development of the adequate dynamical diffraction theory.
For the complete description of a diffraction pattern or image intensities obtained with electrons, it is necessary to include the effects of inelastic scattering as well as elastic scattering. In contrast to the Xray diffraction case, the inelastic scattering does not produce just a broad and generally negligible background. The average energy loss for an inelastic scattering event is about 20 eV, which is small compared with the energy of about 100 keV for the incident electrons. The inelastically scattered electrons have a narrow angular distribution and are diffracted in much the same way as the incident or elastically scattered electrons in a crystal. They therefore produce a highly modulated contribution to the diffraction pattern, strongly peaked about the Bragg spot positions (see Chapter 4.3 ). Also, as a result of the inelastic scattering processes, including thermal diffuse scattering, an effective absorption function must be added in the calculation of intensities for elastically scattered electrons.
The inelastic scattering processes in themselves give information about the specimen in that they provide a measure of the excitations of both the valenceshell and the innershell electrons of the solid. The innershell electron excitations are characteristic of the type of atom, so that microanalysis of small volumes of specimen material (a few hundreds or thousands of atoms) may be achieved by detecting either the energy losses of the transmitted electrons or the emission of the characteristic Xray [see IT C (2004), Section 4.3.4 ]. The development of the annular dark field (ADF) mode in STEM provides a favourable detector geometry for microanalysis, in which the forward scattered beam may be passed to an electron energyloss spectrometer (EELS) for spectral analysis, while scattering at larger angles is collected to form a simultaneous scanning image. The arrangement is particularly efficient because, using a magnetic sector dispersive spectrometer, electrons of all energy losses may be detected simultaneously (parallel detection). Fine structure on the EELS absorption edges is analysed in a manner analogous to soft Xray absorption spectroscopy, but with a spatial resolution of a few nanometres. The spectra are obtained from points in the corresponding ADF image which can be identified with subnanometre accuracy.
An adverse effect of the inelastic scattering processes, however, is that the transfer of energy to the specimen material results in radiation damage; this is a serious limitation of the application of electronscattering methods to radiationsensitive materials such as organic, biological and many inorganic compounds. The amount of radiation damage increases rapidly as the amount of information per unit volume, derived from the elastic scattering, is increased, i.e. as the microscope resolution is improved or as the specimen volume irradiated during a diffraction experiment is decreased. At the current limits of microscopic resolution, radiation damage is a significant factor even for the radiationresistant materials such as semiconductors and alloys.
In the historical development of electrondiffraction techniques the progress has depended to an important extent on the level of understanding of the dynamical diffraction processes and this understanding has followed, to a considerable degree, from the availability of electron microscopes. For the first 20 years of the development, with few exceptions, the lack of a precise knowledge of the specimen morphology meant that diffraction intensities were influenced to an unpredictable degree by dynamical scattering and the impression grew that electrondiffraction intensities could not meaningfully be interpreted.
It was the group in the Soviet Union, led initially by Dr Z. G. Pinsker and later by Dr B. K. Vainshtein and others, which showed that patterns from thin layers of a powder of microcrystals could be interpreted reliably by use of the kinematical approximation. The averaging over crystal orientation reduced the dynamical diffraction effects to the extent that practical structure analysis was feasible. The development of the techniques of using films of crystallites having strongly preferred orientations, to give patterns somewhat analogous to the Xray rotation patterns, provided the basis for the collection of threedimensional diffraction data on which many structure analyses have been based [see Section 2.5.4 and IT C (2004), Section 4.3.5 ].
In recent years improvements in the techniques of specimen preparation and in the knowledge of the conditions under which dynamical diffraction effects become significant have allowed progress to be made with the use of highenergy electrondiffraction patterns from thin single crystals for crystal structure analysis. Particularly for crystals of lightatom materials, including biological and organic compounds, the methods of structure analysis developed for Xray diffraction, including the direct methods (see Section 2.5.8), have been successfully applied in an increasing number of cases. Often it is possible to deduce some structural information from highresolution electronmicroscope images and this information may be combined with that from the diffraction intensities to assist the structure analysis process [see IT C (2004), Section 4.3.8.8 ].
The determination of crystal symmetry by use of CBED (Section 2.5.3) and the accurate determination of structure amplitudes by use of methods depending on the observation of dynamical diffraction effects [IT C (2004), Section 4.3.7 ] came later, after the information on morphologies of crystals, and the precision electron optics associated with electron microscopes, became available. This powerful convergentbeam microdiffraction method has now been widely adopted as the preferred method for spacegroup determination of microphases, quasicrystals, incommensurate, twinned and other imperfectly crystalline structures. Advantage is taken of the fact that multiple scattering preserves information on the absence of inversion symmetry, while the use of an electron probe which is smaller than a mosaic block allows extinctionfree structurefactor measurements to be made. Finally, an enhanced sensitivity to ionicity is obtained from electrondiffraction measurements of structure factors by the very large difference between electron scattering factors for atoms and those for ions at small angles. This section by M. Tanaka replaces the corresponding section by the late P. Goodman in previous editions , which researchers may also find useful.
In spite of the problem of radiation damage, a great deal of progress has been made in the study of organic and biological materials by electronscattering methods. In some respects these materials are very favourable because, with only light atoms present, the scattering from thin films can be treated using the kinematical approximation without serious error. Because of the problem of radiation damage, however, special techniques have been evolved to maximize the information on the required structural aspects with minimum irradiation of the specimen. Imageprocessing techniques have been evolved to take advantage of the redundancy of information from a periodic structure and the means have been devised for combining information from multiple images and diffraction data to reconstruct specimen structure in three dimensions. These techniques are outlined in Sections 2.5.5, 2.5.6 and 2.5.7. Section 2.5.6, written for the first and second editions by Boris Vainshtein, has been revised and extended for this third edition by Pawel Penczek. It deals with the general theory of threedimensional reconstruction from projections and compares several popular methods. Section 2.5.7 describes the application of electronmicroscope imaging to the structure analysis of proteins which cannot be crystallized, and so addresses a crucial problem in structural biology. This is done by the remarkably successful method of singleparticle image reconstruction, in which images of the same protein, lying in random orientations within a thin film of vitreous ice, are combined in the correct orientation to form a threedimensional reconstructed chargedensity map at nanometre or better resolution. The summation over many particles achieves the same radiationdamagereduction effect as does crystallographic redundancy in protein crystallography. Finally, Section 2.5.8 describes experience with the application of numerical direct methods to the phase problem in electron diffraction. Although direct imaging `solves' the phase problem, there are many practical problems in combining electronmicrodiffraction intensities with corresponding highresolution images of a structure over a large tilt range. In cases where multiple scattering can be minimized, some success has therefore been obtained using direct phasing methods, as reviewed in this section.
For most inorganic materials the complications of manybeam dynamical diffraction processes prevent the direct application of these techniques of image analysis, which depend on having a linear relationship between the image intensity and the value of the projected potential distribution of the sample. The smaller sensitivities to radiation damage can, to some extent, remove the need for the application of such methods by allowing direct visualization of structure with ultrahighresolution images and the use of microdiffraction techniques.
2.5.2. Electron diffraction and electron microscopy^{1}
The contributions of electron scattering to the study of the structures of crystalline solids are many and diverse. This section will deal only with the scattering of highenergy electrons (in the energy range of 10^{4} to 10^{6} eV) in transmission through thin samples of crystalline solids and the derivation of information on crystal structures from diffraction patterns and highresolution images. The range of wavelengths considered is from about 0.122 Å (12.2 pm) for 10 kV electrons to 0.0087 Å (0.87 pm) for 1 MeV electrons. Given that the scattering amplitudes of atoms for electrons have much the same form and variation with as for Xrays, it is apparent that the angular range for strong scattering of electrons will be of the order of 10^{−2} rad. Only under special circumstances, usually involving multiple elastic and inelastic scattering from very thick specimens, are scattering angles of more than 10^{−1} rad of importance.
The strength of the interaction of electrons with matter is greater than that of Xrays by two or three orders of magnitude. The singlescattering, first Born approximation fails significantly for scattering from single heavy atoms. Diffracted beams from single crystals may attain intensities comparable with that of the incident beam for crystal thicknesses of 10^{2} Å, rather than 10^{4} Å or more. It follows that electrons may be used for the study of very thin samples, and that dynamical scattering effects, or the coherent interaction of multiply scattered electron waves, will modify the diffracted amplitudes in a significant way for all but very thin specimens containing only light atoms.
The experimental techniques for electron scattering are largely determined by the possibility of focusing electron beams by use of strong axial magnetic fields, which act as electron lenses having focal lengths as short as 1 mm or less. Electron microscopes employing such lenses have been produced with resolutions approaching 1 Å. With such instruments, images showing individual isolated atoms of moderately high atomic number may be obtained. The resolution available is sufficient to distinguish neighbouring rows of adjacent atoms in the projected structures of thin crystals viewed in favourable orientations. It is therefore possible in many cases to obtain information on the structure of crystals and of crystal defects by direct inspection of electron micrographs.
The electromagnetic electron lenses may also be used to form electron beams of very small diameter and very high intensity. In particular, by the use of cold fieldemission electron guns, it is possible to obtain a current of 10^{−10} A in an electron beam of diameter 10 Å or less with a beam divergence of less than 10^{−2} rad, i.e. a current density of 10^{4} A cm^{−2} or more. The magnitudes of the electron scattering amplitudes then imply that detectable signals may be obtained in diffraction from assemblies of fewer than 10^{2} atoms. On the other hand, electron beams may readily be collimated to better than 10^{−6} rad.
The cross sections for inelastic scattering processes are, in general, less than for the elastic scattering of electrons, but signals may be obtained by the observation of electron energy losses, or the production of secondary radiations, which allow the analysis of chemical compositions or electronic excited states for regions of the crystal 100 Å or less in diameter.
On the other hand, the transfer to the sample of large amounts of energy through inelastic scattering processes produces radiation damage which may severely limit the applicability of the imaging and diffraction techniques, especially for biological and organic materials, unless the information is gathered from large specimen volumes with low incident electron beam densities.
Structure analysis of crystals can be performed using electron diffraction in the same way as with Xray or neutron diffraction. The mathematical expressions and the procedures are much the same. However, there are peculiarities of the electrondiffraction case which should be noted.
The material of this section is also reviewed in the text by Spence (2003).
There are two alternative sets of signs for the functions describing wave optics. Both sets have been widely used in the literature. There is, however, a requirement for internal consistency within a particular analysis, independently of which set is adopted. Unfortunately, this requirement has not always been met and, in fact, it is only too easy at the outset of an analysis to make errors in this way. This problem might have come into prominence somewhat earlier were it not for the fact that, for centrosymmetric crystals (or indeed for centrosymmetric projections in the case of planar diffraction), only the signs used in the transmission and propagation functions can affect the results. It is not until the origin is set away from a centre of symmetry that there is a need to be consistent in every sign used.
Signs for electron diffraction have been chosen from two points of view: (1) defining as positive the sign of the exponent in the structurefactor expression and (2) defining the forward propagating freespace wavefunction with a positive exponent.
The second of these alternatives is the one which has been adopted in most solidstate and quantummechanical texts.
The first, or standard crystallographic convention, is the one which could most easily be adopted by crystallographers accustomed to retaining a positive exponent in the structurefactor equation. This also represents a consistent International Tables usage. It is, however, realized that both conventions will continue to be used in crystallographic computations, and that there are by now a large number of operational programs in use.
It is therefore recommended (a) that a particular sign usage be indicated as either standard crystallographic or alternative crystallographic to accord with Table 2.5.2.1, whenever there is a need for this to be explicit in publication, and (b) that either one or other of these systems be adhered to throughout an analysis in a selfconsistent way, even in those cases where, as indicated above, some of the signs appear to have no effect on one particular conclusion.
electron interaction constant ; (relativistic) electron mass; electron wavelength; (magnitude of) electron charge; Planck's constant; ; volume of the unit cell; continuous reciprocalspace vector, components u, v; discrete reciprocalspace coordinate; crystal potential averaged along beam direction (positive); slice thickness; absorption potential [positive; typically ]; defocus (defined as negative for underfocus); spherical aberration coefficient; excitation error relative to the incidentbeam direction and defined as negative when the point h lies outside the Ewald sphere; atomic scattering factor for electrons, , related to the atomic scattering factor for Xrays, , by the Mott formula . Fourier transform of periodic slice transmission function.

The forwardscattering approximation to the manybeam dynamical diffraction theory outlined in Chapter 5.2 provides the basis for the calculation of diffraction intensities and electronmicroscope image contrast for thin crystals. [See Cowley (1995), Chapter 5.2 and IT C (2004) Sections 4.3.6 and 4.3.8 .] On the other hand, there are various approximations which provide relatively simple analytical expressions, are useful for the determination of diffraction geometry, and allow estimates to be made of the relative intensities in diffraction patterns and electron micrographs in favourable cases.


Electron optics. Electrons may be focused by use of axially symmetric magnetic fields produced by electromagnetic lenses. The focal length of such a lens used as a projector lens (focal points outside the lens field) is given bywhere is the relativistically corrected accelerating voltage and is the z component of the magnetic field. An expression in terms of experimental constants was given by Liebman (1955) as where is a constant, NI is the number of ampere turns of the lens winding, S is the length of the gap between the magnet pole pieces and D is the bore of the pole pieces.
Lenses of this type have irreducible aberrations, the most important of which for the paraxial conditions of electron microscopy is the thirdorder spherical aberration, coefficient , giving a variation of focal length of for a beam at an angle α to the axis. Chromatic aberration, coefficient , gives a spread of focal lengths for variations and of the accelerating voltage and lens currents, respectively.
The objective lens of an electron microscope is the critical lens for the determination of image resolution and contrast. The action of this lens in a conventional transmission electron microscope (TEM) is described by use of the Abbe theory for coherent incident illumination transmitted through the object to produce a wavefunction (see Fig. 2.5.2.2).

Diagram representing the critical components of a conventional transmission electron microscope (TEM) and a scanning transmission electron microscope (STEM). For the TEM, electrons from a source A illuminate the specimen and the objective lens forms an image of the transmitted electrons on the image plane, B. For the STEM, a source at B is imaged by the objective lens to form a small probe on the specimen and some part of the transmitted beam is collected by a detector at A. 
The amplitude distribution in the back focal plane of the objective lens is written where is the Fourier transform of and T(u, v) is the transfer function of the lens, consisting of an aperture function and a phase function exp where the phase perturbation due to lens defocus and aberrations is usually approximated as and u, v are the reciprocalspace variables related to the scattering angles , by
The image amplitude distribution, referred to the object coordinates, is given by Fourier transform of (2.5.2.31) as where , given by Fourier transform of , is the spread function. The image intensity is then
In practice the coherent imaging theory provides a good approximation but limitations of the coherence of the illumination have appreciable effects under highresolution imaging conditions.
The variation of focal lengths according to (2.5.2.30) is described by a function . Illumination from a finite incoherent source gives a distribution of incidentbeam angles . Then the image intensity is found by integrating incoherently over and :where denotes the Fouriertransform operation.
In the scanning transmission electron microscope (STEM), the objective lens focuses a small bright source of electrons on the object and directly transmitted or scattered electrons are detected to form an image as the incident beam is scanned over the object (see Fig. 2.5.2.2). Ideally the image amplitude can be related to that of the conventional transmission electron microscope by use of the `reciprocity relationship' which refers to point sources and detectors for scalar radiation in scalar fields with elastic scattering processes only. It may be stated: `The amplitude at a point B due to a point source at A is identical to that which would be produced at A for the identical source placed at B'.
For an axial point source, the amplitude distribution produced by the objective lens on the specimen is If this is translated by the scan to X, Y, the transmitted wave is
The amplitude on the plane of observation following the specimen is then and the image signal produced by a detector having a sensitivity function H(u, v) is If H(u, v) represents a small detector, approximated by a delta function, this becomes which is identical to the result (2.5.2.35) for a plane incident wave in the conventional transmission electron microscope.



Currently most electrondiffraction patterns are obtained in conjunction with images, in electron microscopes of one form or another, as follows.

Because the cross section for electron scattering is at least a thousand times greater than that for Xrays, and because multiple Bragg scattering preserves information on symmetry (such as the absence of inversion symmetry), electron diffraction is exquisitely sensitive to symmetry. The additional ability of modern electronoptical lenses to focus an electron probe down to nanometre dimensions, and so allow the study of nanocrystals too small for analysis by Xrays, has meant that the method of convergentbeam diffraction described here has now become the preferred method of symmetry determination for very small crystals, domains, twinned structures, quasicrystals, incommensurate structures and other imperfectly crystalline materials.
Convergentbeam electron diffraction (CBED) originated with the experiments of Kossel & Möllenstedt (1938). However, modern crystallographic investigations by CBED began with the studies performed by Goodman & Lehmpfuhl (1965) in a modified transmission electron microscope. They obtained CBED patterns by converging a conical electron beam with an angle of more than 10^{−3} rad on an ~30 nm diameter specimen area, which had uniform thickness and no bending. Instead of the usual diffraction spots, diffraction discs (in Laue or transmission geometry) were produced. The diffraction intensity within a disc shows a specific symmetry, which enables one to determine the point groups and space groups of microcrystals. Unlike Xray diffraction, the method is extremely sensitive to the presence or absence of inversion symmetry.
The method corresponding to CBED in the field of light optics is the conoscope method. Using a conoscope, we can identify whether a crystal is isotropic, uniaxial or biaxial, and determine the optic axis and the sign of birefringence of a crystal. When CBED, a conoscope method using an electron beam, is utilized, more basic properties of a crystal – the crystal point group and space group – can be determined.
Point and spacegroup determinations are routinely also carried out by Xray diffraction. This method, to which kinematical diffraction is applicable, cannot determine whether a crystal is polar or nonpolar unless anomalous absorption is utilized. As a result, the Xray diffraction method can only identify 11 Laue groups among 32 point groups. CBED, based fully upon dynamical diffraction, can distinguish polar crystals from nonpolar crystals using only a nanometresized crystal, thus allowing the unique identification of all the point groups by inspecting the symmetries appearing in CBED discs.
As pointed out above, an unambiguous experimental determination of crystal symmetry, in the case of Xray diffraction, is usually not possible because of the apparent centrosymmetry of the diffraction pattern, even for noncentrosymmetric crystals. However, methods based on structurefactor and Xray intensity statistics remain useful for the resolution of spacegroup ambiguities, and are routinely applied to structure determinations from Xray data. These methods are described in Chapter 2.1 of this volume.
In the field of materials science, correct spacegroup determination by CBED is often requested prior to Xray or neutron structure refinement, in particular in the case of Rietveld refinements based on powder diffraction data.
CBED can determine not only the point and space groups of crystals but also crystal structure parameters – lattice parameters, atom positions, Debye–Waller factors and loworder structure factors. The lattice parameters can be determined from submicron regions of thin crystals by using higherorder Laue zone (HOLZ) reflections with an accuracy of 1 × 10^{−4}. Cherns et al. (1988) were the first to perform strain analysis of artificial multilayer materials using the largeangle technique (LACBED) (Tanaka et al., 1980). Since then, many strain measurements at interfaces of various multilayer materials have been successfully conducted. In recent years, strain analysis has been conducted using automatic analysis programs, which take account of dynamical diffraction effects (Krämer et al., 2000). We refer to the book of Morniroli (2002), which carries many helpful figures, clear photographs and a comprehensive list of papers on this topic.
Vincent et al. (1984a,b) first applied the CBED method to the determination of the atom positions of AuGeAs. They analysed the intensities of HOLZ reflections by applying a quasikinematical approximation. Tanaka & Tsuda (1990, 1991) and Tsuda & Tanaka (1995) refined the structural parameters of SrTiO_{3} by applying the dynamical theory of electron diffraction. The method was extended to the refinements of CdS, LaCrO_{3} and hexagonal BaTiO_{3} (Tsuda & Tanaka, 1999; Tsuda et al., 2002; Ogata et al., 2004). Rossouw et al. (1996) measured the order parameters of TiAl through a Blochwave analysis of HOLZ reflections in a CBED pattern. Midgley et al. (1996) refined two positional parameters of AuSn_{4} from the diffraction data obtained with a small convergence angle using multislice calculations.
Loworder structure factors were first determined by Goodman & Lehmpfuhl (1967) for MgO. After much work on loworder structurefactor determination, Zuo & Spence determined the 200 and 400 structure factors of MgO in a very modern way, by fitting energyfiltered patterns and manybeam dynamical calculations using a leastsquares procedure. For the loworder structurefactor determinations, the excellent comprehensive review of Spence (1993) should be referred to. Saunders et al. (1995) succeeded in obtaining the deformation charge density of Si using the loworder crystal structure factors determined by CBED. For the reliable determination of the loworder Xray crystal structure factors or the charge density of a crystal, accurate determination of the Debye–Waller factors is indispensable. Zuo et al. (1999) determined the bondcharge distribution in cuprite. Simultaneous determination of the Debye–Waller factors and the loworder structure factors using HOLZ and zerothorder Laue zone (ZOLZ) reflections was performed to determine the deformation charge density of LaCrO_{3} accurately (Tsuda et al., 2002).
CBED can also be applied to the determination of lattice defects, dislocations (Cherns & Preston, 1986), stacking faults (Tanaka, 1986) and twins (Tanaka, 1986). Since this topic is beyond the scope of the present chapter, readers are referred to pages 156 to 205 of the book by Tanaka et al. (1994).
We also mention the book by Spence & Zuo (1992), which deals with the whole topic of CBED, including the basic theory and a wealth of literature.
When an electron beam traverses a thin slab of crystal parallel to a zone axis, one can easily imagine that symmetries parallel to the zone axis should appear in the resulting CBED pattern. It is, however, more difficult to imagine what symmetries appear due to symmetries perpendicular to the incident beam. Goodman (1975) pioneered the clarification of CBED symmetries for the twofold rotation axis and mirror plane perpendicular to the incident beam, and the symmetry of an inversion centre, with the help of the reciprocity theorem of scattering theory. Tinnappel (1975) solved many CBED symmetries at various crystal settings with respect to the incident beam using a grouptheoretical treatment. Buxton et al. (1976) also derived these results from first principles, and generalized them to produce a systematic method for the determination of the crystal point group. Tanaka, Saito & Sekii (1983) developed a method to determine the point group using simultaneously excited manybeam patterns. The pointgroupdetermination method given by Buxton et al. (1976) is described with the aid of the description by Tanaka, Saito & Sekii (1983) in the following.
Since CBED uses the Laue geometry, Buxton et al. (1976) assumed a perfectly crystalline specimen in the form of a parallelsided slab which is infinite in two dimensions. The symmetry elements of the specimen (as distinct from those of an infinite crystal) form `diffraction groups', which are isomorphic to the point groups of the diperiodic plane figures and Shubnikov groups of coloured plane figures. The diffraction groups of a specimen are determined from the symmetries of CBED patterns taken at various orientations of the specimen. The crystal pointgroup of the specimen is identified by referring to Fig. 2.5.3.4, which gives the relation between diffraction groups and crystal point groups.
A specimen that is parallelsided and is infinitely extended in the x and y directions has ten symmetry elements. The symmetry elements consist of six twodimensional symmetry elements and four threedimensional ones. The operation of the former elements transforms an arbitrary coordinate (x, y, z) into (x′, y′, z), with z remaining the same. The operation of the latter transforms a coordinate (x, y, z) into (x′, y′, z′), where . A vertical mirror plane m and one, two, three, four and sixfold rotation axes that are parallel to the surface normal z are the twodimensional symmetry elements. A horizontal mirror plane m′, an inversion centre i, a horizontal twofold rotation axis 2′ and a fourfold rotary inversion are the threedimensional symmetry elements, and are shown in Fig. 2.5.3.1. The fourfold rotary inversion was not recognized as a symmetry element until the point groups of the diperiodic plane figures were considered (Buxton et al., 1976). Table 2.5.3.1 lists these symmetry elements, where the symbols in parentheses express symmetries of CBED patterns expected from threedimensional symmetry elements.

The diffraction groups are constructed by combining these symmetry elements (Table 2.5.3.2). Twodimensional symmetry elements and their combinations are given in the top row of the table. The third symmetry m in parentheses is introduced automatically when the first two symmetry elements are combined. Threedimensional symmetry elements are given in the first column. The equations given below the table indicate that no additional threedimensional symmetry elements can appear by combination of two symmetry elements in the first column. As a result, 31 diffraction groups are produced by combining the elements in the first column with those in the top row. Diffraction groups in square brackets have already appeared ealier in the table. In the fourth row, three columns have two diffraction groups, which are produced when symmetry elements are combined at different orientations. In the last row, five columns are empty because a fourfold rotary inversion cannot coexist with threefold and sixfold rotation axes. In the last column, the number of independent diffraction groups in each row is given, the sum of the numbers being 31.

It is difficult to imagine the symmetries in CBED patterns generated by the threedimensional symmetry elements of the sample. The reason is that if a threedimensional symmetry element is applied to a specimen, it turns it upside down, which is impractical in most experiments. The reciprocity theorem of scattering theory (Pogany & Turner, 1968) enables us to clarify the symmetries of CBED patterns expected from these threedimensional symmetry elements. A graphical method for obtaining CBED symmetries due to sample symmetry elements is described in the papers of Goodman (1975), Buxton et al. (1976) and Tanaka (1989). The CBED symmetries of the threedimensional symmetries do not appear in the zoneaxis patterns, but do in a diffraction disc set at the Bragg condition, each of which we call a darkfield pattern (DP). The CBED symmetries obtained are illustrated in Fig. 2.5.3.2. A horizontal twofold rotation axis 2′, a horizontal mirror plane m′, an inversion centre i and a fourfold rotary inversion produce symmetries m_{R} (m_{2}), 1_{R}, 2_{R} and 4_{R} in DPs, respectively.

Illustration of symmetries appearing in darkfield patterns (DPs). (a) m_{R} and m_{2}; (b) 1_{R}; (c) 2_{R}; (d) 4_{R}, originating from 2′, m′, i and , respectively. 
Next we explain the symbols of the CBED symmetries. (1) Operation m_{R} is shown in the lefthand part of Fig. 2.5.3.2(a), which implies successive operations of (a) a mirror m with respect to a twofold rotation axis, transforming an open circle beam (○) in reflection G into a beam (+) in reflection G′ and (b) rotation R of this beam by π about the centre point of disc G′ (or the exact Bragg position of reflection G′), resulting in position ○ in reflection G′. The combination of the two operations is written as m_{R}. When the twofold rotation axis is parallel to the diffraction vector G, two beams (○) in the lefthand part of the figure become one reflection G, and a mirror symmetry, whose mirror line is perpendicular to vector G and passes through the centre of disc G, appears between the two beams (the righthand side figure of Fig. 2.5.3.2a). The mirror symmetry is labelled m_{2} after the twofold rotation axis. (2) Operation 1_{R} (Fig. 2.5.3.2b) for a horizontal mirror plane is a combination of a rotation by 2π of a beam (○) about a zone axis O (symbol 1), which is equivalent to no rotation, and a rotation by π of the beam about the exact Bragg position or the centre of disc G. (3) Operation 2_{R} is a rotation by π of a beam (○) in reflection G about a zone axis (symbol 2), which transforms the beam into a beam (+) in reflection −G, followed by a rotation by π of the beam (+) about the centre of disc −G, resulting in the beam (○) in disc −G (Fig. 2.5.3.2c). The symmetry is called translational symmetry after Goodman (1975) because the pattern of disc +G coincides with that of disc −G by a translation. It is emphasized that an inversion centre is identified by the test of translational symmetry about a pair of ±G darkfield patterns – if one disc can be translated into coincidence with the other, an inversion centre exists. We call the pair ±DP. (4) Operation 4_{R} (Fig. 2.5.3.2d) can be understood in a similar manner. It is noted that regular letters are symmetries about a zone axis, while subscripts R represent symmetries about the exact Bragg position. We call a pattern that contains an exact Bragg position (if possible at the disc centre) a darkfield pattern. As far as CBED symmetries are concerned, we do not use the term darkfield pattern if a disc does not contain the exact Bragg position.
The four threedimensional symmetry elements are found to produce different symmetries in the DPs. These facts imply that these symmetry elements can be identified unambiguously from the symmetries of CBED patterns.
Twodimensional symmetry elements that belong to a zone axis exhibit their symmetries in CBED patterns or zoneaxis patterns (ZAPs) directly, even if dynamical diffraction takes place. A ZAP contains a brightfield pattern (BP) and a whole pattern (WP). The BP is the pattern appearing in the brightfield disc [the central or `direct' (000) beam]. The WP is composed of the BP and the pattern formed by the surrounding diffraction discs, which are not exactly excited. The twodimensional symmetry elements m, 1, 2, 3, 4 and 6 yield symmetry m_{v} and one, two, three, four and sixfold rotation symmetries, respectively, in WPs, where the suffix v for m_{v} is assigned to distinguish it from mirror symmetry m_{2} caused by a horizontal twofold rotation axis.
It should be noted that a BP shows not only the zoneaxis symmetry but also threedimensional symmetries, indicating that the BP can have a higher symmetry than the symmetry of the corresponding WP. The symmetries of the BP due to threedimensional symmetry elements are obtained by moving the DPs to the zone axis. As a result, the threedimensional symmetry elements m′, i, 2′ and produce, respectively, symmetries 1_{R}, 1, m_{2} and 4 in the BP, instead of 1_{R}, 2_{R}, m_{2} and 4_{R} in the DPs (Fig. 2.5.3.2). We mention that the BP cannot distinguish whether a specimen crystal has an inversion centre or not, because an inversion centre forms the lowest symmetry 1 in the BP.
In conclusion, all the twodimensional symmetry elements can be identified from the WP symmetries.
All the symmetry elements of the diffraction groups can be identified from the symmetries of a WP and DPs. But it is practical and convenient to use just the four patterns WP, BP, DP and ±DP to determine the diffraction group. The symmetries appearing in these four patterns are given for the 31 diffraction groups in Table 2.5.3.3 (Tanaka, Saito & Sekii, 1983), which is a detailed version of Table 2 of Buxton et al. (1976). All the possible symmetries of the DP and ±DP appearing at different crystal orientations are given in the present table. When a BP has a higher symmetry than the corresponding WP, the symmetry elements that produce the BP are given in parentheses in column II except only for the case of 4_{R}. When two types of vertical mirror planes exist, these are distinguished by symbols m_{v} and m_{v′}. Each of the two or three symmetries given in columns IV and V for many diffraction groups appears in a DP or ±DP in different directions.

It is emphasized again that no two diffraction groups exhibit the same combination of BP, WP, DP and ±DP, which implies that the diffraction groups are uniquely determined from an inspection of these pattern symmetries. Fig. 2.5.3.3 illustrates the symmetries of the DP and ±DP appearing in Table 2.5.3.3, which greatly eases the cumbersome task of determining the symmetries. The first four patterns illustrate the symmetries appearing in a single DP and the others treat those in ±DPs. The pattern symmetries are written beneath the figures. The other symbols are the symmetries of a specimen. The crosses outside the diffraction discs designate the zone axis. The crosses inside the diffraction discs indicate the exact Bragg position.

Illustration of symmetries appearing in darkfield patterns (DPs) and a pair of darkfield patterns (±DP) for the combinations of symmetry elements. 
When the four patterns appearing in three photographs are taken and examined using Table 2.5.3.3 with the aid of Fig. 2.5.3.3, one diffraction group can be selected unambiguously. It is, however, noted that many diffraction groups are determined from a WP and BP pair without using a DP or ±DP (or from one photograph) or from a set of a WP, a BP and a DP without using a ±DP (or from two photographs).
Fig. 2.5.3.4 provides the relationship between the 31 diffraction groups for slabs and the 32 point groups for infinite crystals given by Buxton et al. (1976). When a diffraction group is determined, possible point groups are selected by consulting this figure. Each of the 11 highsymmetry diffraction groups corresponds to only one crystal point group. In this case, the point group is uniquely determined from the diffraction group. When more than one point group falls under a diffraction group, a different diffraction group has to be obtained for another zone axis. A point group is identified by finding a common point group among the point groups obtained for different zone axes. It is clear from the figure that highsymmetry zones should be used for quick determination of point groups because lowsymmetry zone axes exhibit only a small portion of crystal symmetries in the CBED patterns. Furthermore, it should be noted that CBED cannot observe crystal symmetries oblique to an incident beam or horizontal three, four or sixfold rotation axes. The diffraction groups to be expected for different zone axes are given for all the point groups in Table 2.5.3.4 (Buxton et al., 1976). The table is useful for finding a suitable zone axis to distinguish candidate point groups expected in advance.

We shall explain the pointgroup determination procedure using an Si crystal. Fig. 2.5.3.5(a) shows a [111] ZAP of the Si specimen. The BP has threefold rotation symmetry and mirror symmetry or symmetry 3m_{v}, which are caused by the threefold rotation axis along the [111] direction and a vertical mirror plane. The WP has the same symmetry. Figs. 2.5.3.5(b) and (c) are and DPs, respectively. Both show symmetry m_{2} perpendicular to the reflection vector. This symmetry is caused by a twofold rotation axis parallel to the specimen surface. One DP coincides with the other upon translation. This translational or 2_{R} symmetry indicates the existence of an inversion centre. By consulting Table 2.5.3.3, the diffraction group giving rise to these pattern symmetries is found to be 6_{R}mm_{R}. Fig. 2.5.3.4 shows that there are two point groups and causing diffraction group 6_{R}mm_{R}. Fig. 2.5.3.6 shows another ZAP, which shows symmetry 4mm in the BP and the WP. The point group which has fourfold rotation symmetry is not but . The point group of Si is thus determined to be .

CBED patterns of Si taken with the [111] incidence. (a) BP and WP show symmetry 3m_{v}. (b) and (c) DPs show symmetry m_{2} and DP symmetry 2_{R}m_{v′}. 
HOLZ reflections appear as excess HOLZ rings far outside the ZOLZ reflection discs and as deficit lines in the ZOLZ discs. By ignoring these weak diffraction effects with components along the beam direction, we may obtain information about the symmetry of the sample as projected along the beam direction. Thus when HOLZ reflections are weak and no deficit HOLZ lines are seen in the ZOLZ discs, the symmetry elements found from the CBED patterns are only those of the specimen projected along the zone axis. The projection of the specimen along the zone axis causes horizontal mirror symmetry m′, the corresponding CBED symmetry being 1_{R}. When symmetry 1_{R} is added to the 31 diffraction groups, ten projection diffraction groups having symmetry symbol 1_{R} are derived as shown in column VI of Table 2.5.3.3. If only ZOLZ reflections are observed in CBED patterns, a projection diffraction group instead of a diffraction group is obtained, where only the pattern symmetries given in the rows of the diffraction groups having symmetry symbol 1_{R} in Table 2.5.3.3 should be consulted. Two projection diffraction groups obtained from two different zone axes are the minimum needed to determine a crystal point group, because it is constructed by the threedimensional combination of symmetry elements. It should be noted that if a diffraction group is determined carelessly from CBED patterns with no HOLZ lines, the wrong crystal point group is obtained.
In the sections above, the pointgroup determination method established by Buxton et al. (1976) was described, where two and threedimensional symmetry elements were determined, respectively, from ZAPs and DPs.
The Laue circle is defined as the intersection of the Ewald sphere with the ZOLZ, and all reflections on this circle are simultaneously at the Bragg condition. If many such DPs are recorded (all simultaneously at the Bragg condition), many threedimensional symmetry elements can be identified from one photograph. Using a grouptheoretical method, Tinnappel (1975) studied the symmetries appearing in simultaneously excited DPs for various combinations of crystal symmetry elements. Based upon his treatment, Tanaka, Saito & Sekii (1983) developed a method for determining diffraction groups using simultaneously excited symmetrical hexagonal sixbeam, square fourbeam and rectangular fourbeam CBED patterns. All the CBED symmetries appearing in the symmetrical manybeam (SMB) patterns were derived by the graphical method used in the paper of Buxton et al. (1976). From an experimental viewpoint, it is advantageous that symmetry elements can be identified from one photograph. It was found that twenty diffraction groups can be identified from one SMB pattern, whereas ten diffraction groups can be determined by Buxton et al.'s method. An experimental comparison between the two methods was performed by Howe et al. (1986).
SMB patterns are easily obtained by tilting a specimen crystal or the incident beam from a zone axis into an orientation to excite loworder reflections simultaneously. Fig. 2.5.3.7 illustrates the symmetries of the SMB patterns for all the diffraction groups except for the five groups 1, 1_{R}, 2, 2_{R} and 21_{R}. For these groups, the twobeam method for exciting one reflection is satisfactory because manybeam excitation gives no more information than the twobeam case. In the sixbeam and square fourbeam cases, the CBED symmetries for the two crystal (or incidentbeam) settings which excite respectively the +G and −G reflections are drawn because the vertical rotation axes create the SMB patterns at different incidentbeam orientations. [This had already been experienced for the case of symmetry 2_{R} (Goodman, 1975; Buxton et al., 1976).] In the rectangular fourbeam case, the symmetries for four settings which excite the +G, +H, −G and −H reflections are shown. For the diffraction groups 3m, 3m_{R}, 3m1_{R} and 6_{R}mm_{R}, two different patterns are shown for the two crystal settings, which differ by π/6 rad from each other about the zone axis. Similarly, for the diffraction group 4_{R}mm_{R}, two different patterns are shown for the two crystal settings, which differ by π/4 rad. Illustrations of these different symmetries are given in Fig. 2.5.3.7. The combination of the vertical threefold axis and a horizontal mirror plane introduces a new CBED symmetry 3_{R}. Similarly, the combination of the vertical sixfold rotation axis and an inversion centre introduces a new CBED symmetry 6_{R}.

Illustration of symmetries appearing in hexagonal sixbeam, square fourbeam and rectangular fourbeam darkfield patterns expected for all the diffraction groups except for 1, 1_{R}, 2, 2_{R} and 21_{R}. 
There is an empirical and conventional technique for reproducing the symmetries of the SMB patterns which uses three operations of twodimensional rotations, a vertical mirror at the centre of disc O and a rotation of π about the centre of a disc (1_{R}) without involving the reciprocal process. For example, we may consider 3_{R} between discs F and F′ in Table 2.5.3.5 in the case of diffraction group 31_{R}. Disc F′ is rotated anticlockwise not about the zone axis but about the centre of disc O by 2π/3 rad (symbol 3) to coincide with disc F, and followed by a rotation of π rad (symbol R) about the centre of disc F′, resulting in the correct symmetry seen in Fig. 2.5.3.7. When the symmetries appearing between different SMB patterns are considered, this technique assumes that the symmetry operations are conducted after discs O and are superposed. Another assumption is that the vertical mirror plane perpendicular to the line connecting discs O and acts at the centre of disc O when the symmetries between two SMB patterns are considered. As an example, symmetry 3_{R} between discs S and appearing in the two SMB patterns is reproduced by a threefold anticlockwise rotation of disc S about the centre of disc O (or ) and followed by a rotation of π rad (R) about the centre of disc .
Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7 express the symmetries illustrated in Fig. 2.5.3.7 with the symmetry symbols for the hexagonal sixbeam case, square fourbeam case and rectangular fourbeam case, respectively. In the fourth rows of the tables the symmetries of zoneaxis patterns (BP and WP) are listed because combined use of the zoneaxis pattern and the SMB pattern is efficient for symmetry determination. In the fifth row, the symmetries of the SMB pattern are listed. In the following rows, the symmetries appearing between the two SMB patterns are listed because the SMB symmetries appear not only in an SMB pattern but also in the pairs of SMB patterns. That is, for each diffraction group, all the possible SMB symmetries appearing in a pair of symmetric sixbeam patterns, two pairs AB and AC of the square fourbeam patterns and three pairs AB, AC and AD of the rectangular fourbeam patterns are listed, though such pairs are not always needed for the determination of the diffraction groups. It is noted that the symmetries in parentheses are the symmetries which add no new symmetries, even if they are present. In the last row, the point groups which cause the diffraction groups listed in the first row are given.



By referring to Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7, the characteristic features of the SMB method are seen to be as follows. CBED symmetry m_{2} due to a horizontal twofold rotation axis can appear in every disc of an SMB pattern. Symmetry 1_{R} due to a horizontal mirror plane, however, appears only in disc G or H of an SMB pattern. In the hexagonal sixbeam case, an inversion centre i produces CBED symmetry 6_{R} between discs S and S′ due to the combination of an inversion centre and a vertical threefold rotation axis (and/or of a horizontal mirror plane and a vertical sixfold rotation axis). This indicates that one hexagonal sixbeam pattern can reveal whether a specimen has an inversion centre or not, while the method of Buxton et al. (1976) requires two photographs for the inversion test. All the diffraction groups in Table 2.5.3.5 can be identified from one sixbeam pattern except groups 3 and 6. Diffraction groups 3 and 6 cannot be distinguished from the hexagonal sixbeam pattern because it is insensitive to the vertical axis. In the square fourbeam case, fourfold rotary inversion produces CBED symmetry 4_{R} between discs F and F′ in one SMB pattern, while Buxton et al.'s method requires four photographs to identify fourfold rotary inversion. Although an inversion centre itself does not exhibit any symmetry in the square fourbeam pattern, it causes symmetry 1_{R} due to the horizontal mirror plane produced by the combination of an inversion centre and the twofold rotation axis. Thus, symmetry 1_{R} is an indication of the existence of an inversion centre in the square fourbeam case. All of the seven diffraction groups in Table 2.5.3.6 can be identified from one square fourbeam pattern. One rectangular fourbeam pattern can distinguish all the diffraction groups in Table 2.5.3.7 except the groups m and 2mm. It is emphasized again that the inversion test can be carried out using one sixbeam pattern or one square fourbeam pattern.
Fig. 2.5.3.8 shows CBED patterns taken from a [111] pyrite (FeS_{2}) plate with an accelerating voltage of 100 kV. The space group of FeS_{2} is . The diffraction group of the plate is 6_{R} due to a threefold rotation axis and an inversion centre. The zoneaxis pattern of Fig. 2.5.3.8(a) shows threefold rotation symmetry in the BP and WP. The hexagonal sixbeam pattern of Fig. 2.5.3.8(b) shows no symmetry higher than 1 in discs O, G, F and S but shows symmetry 6_{R} between discs S and S′, which proves the existence of a threefold rotation axis and an inversion centre. The same symmetries are also seen in Fig. 2.5.3.8(c), where reflections , , , , and are excited. Table 2.5.3.5 indicates that diffraction group 6_{R} can be identified from only one hexagonal sixbeam pattern, because no other diffraction groups give rise to the same symmetries in the six discs. When Buxton et al.'s method is used, three photographs or four patterns are necessary to identify diffraction group 6_{R} (see Table 2.5.3.3). In addition, if the symmetries between Figs. 2.5.3.8(b) and (c) are examined, symmetry 2_{R} between discs G and and symmetry 6_{R} between discs F and are found. All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7 and Table 2.5.3.5.

CBED patterns of FeS_{2} taken with the [111] incidence. (a) Zoneaxis pattern, (b) hexagonal sixbeam pattern with excitation of reflection +G, (c) hexagonal sixbeam pattern with excitation of reflection −G. Symmetry 6_{R} is noted between discs S and S′ and discs and . 
Fig. 2.5.3.9 shows CBED patterns taken from a [110] V_{3}Si plate with an accelerating voltage of 80 kV. The space group of V_{3}Si is Pm3n. The diffraction group of the plate is 2mm1_{R} due to two vertical mirror planes and a horizontal mirror plane, a twofold rotation axis being produced at the intersection line of two perpendicular mirror planes. The zoneaxis pattern of Fig. 2.5.3.9(a) shows symmetry 2mm in the BP and WP. The rectangular fourbeam pattern of Fig. 2.5.3.9(b) shows symmetry 1_{R} in disc H due to the horizontal mirror plane and symmetry m_{2} in both discs and F′ due to the twofold rotation axes in the [001] and [110] directions, respectively. The same symmetries are also seen in Fig. 2.5.3.9(c), where reflections , S′ and are excited. Table 2.5.3.7 implies that the diffraction group 2mm1_{R} can be identified from only one rectangular fourbeam pattern, because no other diffraction groups give rise to the same symmetries in the four discs. When Buxton et al.'s method is used, two photographs or three patterns are necessary to identify diffraction group 2mm1_{R} (see Table 2.5.3.3). One can confirm the theoretically predicted symmetries between Fig. 2.5.3.9(b) and Fig. 2.5.3.9(c). All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7 and Table 2.5.3.7.

CBED patterns of V_{3}Si taken with the [110] incidence. (a) Zoneaxis pattern, (b) rectangular fourbeam pattern with excitation of reflections H, and F, (c) rectangular fourbeam pattern with excitation of reflections , S and . 
These experiments show that the SMB method is quite effective for determining the diffraction group of slabs. Buxton et al.'s method identifies twodimensional symmetry elements in the first place using a zoneaxis pattern, and threedimensional symmetry elements using DPs. On the other hand, the SMB method primarily finds many threedimensional symmetry elements in an SMB pattern, and twodimensional symmetry elements from a pair of SMB patterns, as shown in Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7. Therefore, the use of a ZAP and SMB patterns is the most efficient way to find as many crystal symmetry elements in a specimen as possible.
When the point group of a specimen crystal is determined, the crystal axes may be found from a spot diffraction pattern recorded at a highsymmetry zone axis, using the orientations of the symmetry elements determined in the course of pointgroup determination. Integralnumber indices are assigned to the spots of the diffraction patterns. The systematic absence of reflections indicates the lattice type of the crystal. It should be noted that reflections forbidden by the lattice type are always absent, even if dynamical diffraction takes place. (This is true for all sample thicknesses and accelerating voltages.) By comparing the experimentally obtained absences and the extinction rules known for the lattice types [P, C (A, B), I, F and R], a lattice type may be identified for the crystal concerned.
There are three spacegroup symmetry elements of diperiodic plane figures: (1) a horizontal screw axis , (2) a vertical glide plane g with a horizontal glide vector and (3) a horizontal glide plane g′. These are related to the pointgroup symmetry elements 2′, m and m′ of diperiodic plane figures, respectively. (It is noted that these symmetry elements and ten pointgroup symmetry elements form 80 space groups.)
The ordinary extinction rules for screw axes and glide planes hold only in the approximation of kinematical diffraction. The kinematically forbidden reflections caused by these symmetry elements appear owing to Umweganregung of dynamical diffraction. Extinction of intensity, however, does take place in these reflections at certain crystal orientations with respect to the incident beam (i.e. in certain regions within a CBED disc). This dynamical extinction was first predicted by Cowley & Moodie (1959) and was discussed by Miyake et al. (1960) and Cowley et al. (1961). Goodman & Lehmpfuhl (1964) first observed the dynamical extinction as dark cross lines in kinematically forbidden reflection discs of CBED patterns of CdS. Gjønnes & Moodie (1965) discussed the dynamical extinction in a more general way considering not only ZOLZ reflections but also HOLZ reflections. They completely clarified the dynamical extinction rules by considering the exact cancellation which may occur along certain symmetryrelated multiplescattering paths. Based on the results of Gjønnes & Moodie (1965), Tanaka, Sekii & Nagasawa (1983) tabulated the dynamical extinctions expected at all the possible crystal orientations for all the space groups. These were later tabulated in a better form on pages 162 to 172 of the book by Tanaka & Terauchi (1985).
Fig. 2.5.3.10(a) illustrates Umweganregung paths to a kinematically forbidden reflection. The 0k0 (k = odd) reflections are kinematically forbidden because a bglide plane exists perpendicular to the a axis and/or a 2_{1} screw axis exists in the b direction. Let us consider an Umweganregung path a in the zerothorder Laue zone to the 010 forbidden reflection and path b which is symmetric to path a with respect to axis k. Owing to the translation of one half of the lattice parameter b caused by the glide plane and/or the 2_{1} screw axis, the following relations hold between the crystal structure factors:That is, the structure factors of reflections hk0 and have the same phase for even k but have opposite phases for odd k.

Illustration of the production of dynamical extinction lines in kinematically forbidden reflections due to a bglide plane and a 2_{1} screw axis. (a) Umweganregung paths a, b and c. (b) Dynamical extinction lines A are formed in forbidden reflections 0k0 (k = odd). Extinction line B perpendicular to the lines A is formed in the exactly excited 010 reflection. 
Since an Umweganregung path to the kinematically forbidden reflection 0k0 contains an odd number of reflections with odd k, the following relations hold:whereand the functions including the excitation errors are omitted because only the cases in which the functions are the same for all the paths are considered. The excitation errors for paths a and b become the same when the projection of the Laue point along the zone axis concerned, L, lies on axis k. Since the two waves passing through paths a and b have the same amplitude but opposite signs, these waves are superposed on the 0k0 discs (k = odd) and cancel out, resulting in dark lines A in the forbidden discs, as shown in Fig. 2.5.3.10(b). The line A runs parallel to axis k passing through the projection point of the zone axis.
In path c, the reflections are arranged in the reverse order to those in path b. When the 010 reflection is exactly excited, two paths a and c are symmetric with respect to the bisector m′–m′ of the 010 vector having the same excitation errors. The following equation holds:Since the waves passing through these paths have the same amplitude but opposite signs, these waves are superposed on the 010 discs and cancel out, resulting in dark line B in this disc, as shown in Fig. 2.5.3.10(b). Line B appears perpendicular to line A at the exact Bragg positions. When Umweganregung paths are present only in the zerothorder Laue zone, the glide plane and screw axis produce the same dynamical extinction lines A and B. We call these lines A_{2} and B_{2} lines, subscript 2 indicating that the Umweganregung paths lie in the zerothorder Laue zone.
The dynamical extinction effect is analogous to interference phenomena in the Michelson interferometer. That is, the incident beam is split into two beams by Bragg reflections in a crystal. These beams take different paths, in which they suffer a relative phase shift of π and are finally superposed on a kinematically forbidden reflection to cancel out.
When the paths include higherorder Laue zones, the glide plane produces only extinction lines A but the screw axis causes only extinction lines B. These facts are attributed to the different relations between structure factors for a 2_{1} screw axis and a glide plane,In the case of the glide plane, extinction lines A are still formed because two waves passing through paths a and b have opposite signs to each other according to equation (2.5.3.5), but extinction lines B are not produced because equation (2.5.3.4) holds only for the 2_{1} screw axis. In the case of the 2_{1} screw axis, only the waves passing through paths a and c have opposite signs according to equation (2.5.3.4), forming extinction lines B only. We call these lines A_{3} and B_{3} dynamical extinction lines, suffix 3 indicating the Umweganregung paths being via higherorder Laue zones.
It was predicted by Gjønnes & Moodie (1965) that a horizontal glide plane g′ gives a dark spot at the crossing point between extinction lines A and B (Fig. 2.5.3.10b) due to the cancellation between the waves passing through paths b and c. Tanaka, Terauchi & Sekii (1987) observed this dynamical extinction, though it appeared in a slightly different manner to that predicted by Gjønnes & Moodie (1965). Table 2.5.3.8 summarizes the appearance of the dynamical extinction lines for the glide planes g and g′ and the 2_{1} screw axis. The three spacegroup symmetry elements can be identified from the observed extinctions because these three symmetry elements produce different kinds of dynamical extinctions.

In principle, a horizontal screw axis and a vertical glide plane can be distinguished by observations of the extinction lines A_{3} and B_{3}. It is, however, not easy to observe the extinction lines A_{3} and B_{3} because broad extinction lines A_{2} and B_{2} appear at the same time. The presence of the extinction lines A_{3} and B_{3} can be revealed by inspecting the symmetries of fine defect HOLZ lines appearing in the forbidden reflections instead of by direct observation of the lines A_{3} and B_{3} (Tanaka, Sekii & Nagasawa, 1983). That is, if HOLZ lines form lines A_{3} and B_{3}, HOLZ lines are symmetric with respect to the extinction lines A_{2} and B_{2}. If HOLZ lines do not form lines A_{3} and B_{3}, HOLZ lines are asymmetric with respect to the extinction lines A_{2} and B_{2}. When the HOLZ lines are symmetric about the extinction lines A_{2}, the specimen crystal has a glide plane. When the HOLZ lines are symmetric with respect to lines B_{2}, a 2_{1} screw axis exists. It should be noted that a relatively thick specimen area has to be selected to observe HOLZ lines in ZOLZ reflection discs.
Fig. 2.5.3.11 shows CBED patterns taken from (a) thin and (b) thick areas of FeS_{2}, whose space group is , at the 001 Bragg setting with the [100] electronbeam incidence. In the case of the thin specimen (Fig. 2.5.3.11a), only the broad dynamical extinction lines formed by the interaction of ZOLZ reflections are seen in the oddorder discs. On the other hand, fine HOLZ lines are clearly seen in the thick specimen (Fig. 2.5.3.11b). The HOLZ lines are symmetric with respect to both A_{2} and B_{2} extinction lines. This fact proves the presence of the extinction lines A_{3} and B_{3}, or both the c glide in the (010) plane and the 2_{1} screw axis in the c direction, this fact being confirmed by consulting Table 2.5.3.9. Fig. 2.5.3.12 shows a [110] zoneaxis CBED pattern of FeS_{2}. A_{2} extinction lines are seen in both the 001 and discs. Fine HOLZ lines are symmetric with respect to the A_{2} extinction lines in the disc but asymmetric about the A_{2} extinction line in the 001 disc, indicating formation of the A_{3} extinction line only in the disc. This proves the existence of a 2_{1} screw axis in the [001] direction and an a glide in the (001) plane. The appearance of HOLZ lines is easily changed by a change of a few hundred volts in the accelerating voltage. Steeds & Evans (1980) demonstrated for spinel changes in the appearance of HOLZ lines in the ZOLZ discs at accelerating voltages around 100 kV.

CBED patterns obtained from (a) thin and (b) thick areas of (001) FeS_{2}. (a) Dynamical extinction lines A_{2} and B_{2} are seen. (b) Extinction lines A_{3} and B_{3} as well as A_{2} and B_{2} are formed because HOLZ lines are symmetric about lines A_{2} and B_{2}. 

CBED pattern of FeS_{2} taken with the [110] electronbeam incidence. In the 001 and discs, HOLZ lines are asymmetric with respect to extinction lines A_{2}, indicating the existence of a 2_{1} screw axis parallel to the c axis. In the and discs, HOLZ lines are symmetric with respect to extinction lines A_{2}, indicating existence of a glide plane perpendicular to the c axis. 
Another practical method for distinguishing between glide planes and 2_{1} screw axes is that reported by Steeds et al. (1978). The method is based on the fact that the extinction lines are observable even when a crystal is rotated with a glide plane kept parallel and with a 2_{1} screw axis perpendicular to the incident beam. With reference to Fig. 2.5.3.10(a), extinction lines A_{3} produced by a glide plane remain even when the crystal is rotated with respect to axis h but the lines are destroyed by a rotation of the crystal about axis k. Extinction lines B_{3} originating from a 2_{1} screw axis are not destroyed by a crystal rotation about axis k but the lines are destroyed by a rotation with respect to axis h.
We now describe a spacegroup determination method which uses the dynamical extinction lines caused by the horizontal screw axis and the vertical glide plane g of an infinitely extended parallelsided specimen. We do not use the extinction due to the glide plane g′ because observation of the extinction requires a laborious experiment. It should be noted that a vertical glide plane with a glide vector not parallel to the specimen surface cannot be a symmetry element of a specimen of finite thickness; however, the component of the glide vector perpendicular to the incident beam acts as a symmetry element g. (Which symmetry elements are observed by CBED is discussed in Section 2.5.3.3.5.) The 2_{1}, 4_{1}, 4_{3}, 6_{1}, 6_{3 }and 6_{5} screw axes of crystal space groups that are set perpendicular to the incident beam act as a symmetry element because two or three successive operations of 4_{1}, 4_{3}, 6_{1}, 6_{3} and 6_{5} screw axes make them equivalent to a 2_{1} screw axis: (4_{1})^{2} = (4_{3})^{2} = (6_{1})^{3} = (6_{3})^{3} = (6_{5})^{3} = 2_{1}. The 4_{2}, 3_{1}, 3_{2}, 6_{2} and 6_{4} screw axes that are set perpendicular to the incident beam do not produce dynamical extinction lines because the 4_{2} screw axis acts as a twofold rotation axis due to the relation (4_{2})^{2} = 2, the 3_{1} and 3_{2} screw axes give no specific symmetry in CBED patterns, and the 6_{2 }and 6_{4} screw axes are equivalent to 3_{1} and 3_{2} screw axes due to the relations (6_{2})^{2} = 3_{2} and (6_{4})^{2} = 3_{1}. Modifications of the dynamical extinction rules were investigated by Tanaka, Sekii & Nagasawa (1983) when more than one crystal symmetry element (that gives rise to dynamical extinction lines) coexists and when the symmetry elements are combined with various lattice types. Using these results, dynamical extinction lines A_{2}, A_{3}, B_{2} and B_{3} expected from all the possible crystal settings for all the space groups were tabulated.
Table 2.5.3.9 shows all the dynamical extinction lines appearing in the kinematically forbidden reflections for all the possible crystal settings of all the space groups. The first column gives space groups. In each of the following pairs of columns, the lefthand column of the pair gives the reflection indices and the symmetry elements causing the extinction lines and the righthand column gives the types of the extinction lines. The (second) suffixes 1, 2 and 3 of a 2_{1} screw axis in each column distinguish the first, the second and the third screw axis of the space group (as in the symbols 2_{11} and 2_{12} of space group P2_{1}2_{1}2). The glide symbols in the [001] column for space group P4/nnc have two suffixes (n_{21} and n_{22}). The first suffix 2 denotes the second glide plane of the space group. The second suffixes 1 and 2, which appear in the tetragonal and cubic systems, distinguish two equivalent glide planes which lie in the x and y planes. The equivalent planes are distinguished only for the cases of [100], [010] and [001] electronbeam incidences, for convenience. The cglide planes of space group Pcc2 are distinguished with symbols c_{1} and c_{2} (the first suffix only), because the equivalent planes do not exist. The glide symbol in the [001] column for space group P4/mbm has only one suffix 1 or 2. The suffix distinguishes the equivalent glide planes lying in the x and y planes. The first suffix to distinguish the first and the second glide planes is not necessary because the space group has only one glide symbol b. When the index of the incidentbeam direction is expressed with a symbol like [h0l] for point groups 2, m and 2/m, the index h or l can take a value of zero. That is, the extinction rules are applicable to the [100] and [001] electronbeam incidences. However, if columns for [100], [010] and [001] incidences are present, as in the case of point group mm2, [hk0], [0kl] and [h0l] incidences are only for nonzero h, k and l. The reflections in which the extinction lines appear are always perpendicular to the corresponding incidentbeam directions . The indices of the reflections in which extinction lines appear are odd if no remark is given. For cglide planes of space groups R3c and and for dglide planes, the reflections in which extinction lines appear are specified as 6n + 3 and 4n + 2 orders, respectively.
