International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 2.5, pp. 276345
https://doi.org/10.1107/97809553602060000558 Chapter 2.5. Electron diffraction and electron microscopy in structure determination
J. M. Cowley,^{a}^{‡} P. Goodman,^{b}^{‡} B. K. Vainshtein,^{c}^{‡} B. B. Zvyagin^{d}^{‡} and D. L. Dorset^{e}
^{a}Arizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 852871504, USA, ^{b}School of Physics, University of Melbourne, Parkville, Australia 3052, ^{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, and ^{e}ExxonMobil Research and Engineering Co., 1545 Route 22 East, Clinton Township, Annandale, New Jersey 08801, 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 P. Goodman describes the convergentbeam electron diffraction (CBED) method used for spacegroup 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 symmetryrelated multiple scattering paths allows identification of screw and glide elements. Examples from the literature are cited and sources of useful multiplescattering 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 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. Section 2.5.6 by B. K. Vainshtein discusses the problem of threedimensional image reconstruction from projections, widely used in cryoelectron microscopy of proteins, and increasingly also now in materials science. The realspace methods of Radon, the method of backprojection, iterative methods and reciprocalspace 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 Xray diffraction, including Patterson maps, direct methods and trialanderror search techniques. Much of the section concerns electron diffraction 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 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 nanometresized regions can greatly assist the effort to obtain highquality perfectcrystal data free of defects, bending or thickness variation. 
2.5.1. Foreword^{1}
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 rad, the recording geometry is relatively simple and the diffraction pattern represents, to a useful first approximation, a planar section of reciprocal space.
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. The irreducible aberrations of cylindrical magnetic lenses have, to date, limited the resolution of electron microscopes to the extent that the least resolvable distances (or `resolutions') are about 100 times the electron wavelength. However, with microscopes having a resolution of better than 2 Å it is possible to distinguish the individual rows of atoms, parallel to the incident electron beam, in the principal orientations of many 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 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 ].
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 electron diffraction 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 2.5.7), 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.
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 and 2.5.6. They are based essentially on the application of the kinematical approximation and have been used very effectively within that limitation.
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 to 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 rad. Only under special circumstances, usually involving multiple elastic and inelastic scattering from very thick specimens, are scattering angles of more than 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 Å, rather than 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 A in an electron beam of diameter 10 Å or less with a beam divergence of less than rad, i.e. a current density of 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 atoms. On the other hand, electron beams may readily be collimated to better than 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.
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 by where 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.

2.5.3. Spacegroup determination by convergentbeam electron diffraction^{2}
Convergentbeam electron diffraction, originating in the experiments of Kossel and Möllenstedt (Kossel & Möllenstedt, 1938) has been established over the past two decades as a powerful technique for the determination of space group in inorganic materials, with particular application when only microscopic samples are available. Relatively recently, with the introduction of the analytical electron microscope, this technique – abbreviated as CBED – has become available as a routine, so that there is now a considerable accumulation of data from a wide range of materials. A significant extension of the technique in recent times has been the introduction of LACBED (largeangle CBED) by Tanaka & Terauchi (1985). This technique allows an extensive angular range of single diffraction orders to be recorded and, although this method cannot be used for microdiffraction (since it requires an extensive singlecrystal area), new LACBED applications appear regularly, particularly in the field of semiconductor research (see Section 2.5.3.6).
The CBED method relies essentially on two basic properties of transmission electron diffraction, namely the radical departure from Friedel's law and the formation of characteristic extinction bands within spacegroupforbidden reflections. Departure from Friedel's law in electron diffraction was first noted experimentally by Miyake & Uyeda (1950). The prediction of spacegroupforbidden bands (within spacegroupforbidden reflections) by Cowley & Moodie (1959), on the other hand, was one of the first successes of Nbeam theory. A detailed explanation was later given by Gjønnes & Moodie (1965). These are known variously as `GM' bands (Tanaka et al., 1983), or more simply and definitively as `GS' (glide–screw) bands (this section). These extinctions have a close parallel with spacegroup extinctions in Xray diffraction, with the reservation that only screw axes of order two are accurately extinctive under Nbeam conditions. This arises from the property that only those operations which lead to identical projections of the asymmetric unit can have Nbeam dynamical symmetries (Cowley et al., 1961).
Additionally, CBED from perfect crystals produces highorder defect lines in the zeroorder pattern, analogous to the defect Kikuchi lines of inelastic scattering, which provide a sensitive measurement of unitcell parameters (Jones et al., 1977; Fraser et al., 1985; Tanaka & Terauchi, 1985).
The significant differences between Xray and electron diffraction, which may be exploited in analysis, arise as a consequence of a much stronger interaction in the case of electrons (Section 2.5.2). Hence, thin, approximately parallelsided crystal regions must be used in highenergy (100 kV–1 MV) electron transmission work, so that diffraction is produced from crystals effectively infinitely periodic in only two dimensions, leading to the relaxation of threedimensional diffraction conditions known as `excitation error' (Chapter 5.2 ). Also, there is the ability in CBED to obtain data from microscopic crystal regions of around 50 Å in diameter, with corresponding exposure times of several seconds, allowing a survey of a material to be carried out in a relatively short time.
In contrast, singlecrystal Xray diffraction provides much more limited symmetry information in a direct fashion [although statistical analysis of intensities (Wilson, 1949) will considerably supplement this information], but correspondingly gives much more direct threedimensional geometric data, including the determination of unitcell parameters and threedimensional extinctions.
The relative strengths and weaknesses of the two techniques make it useful where possible to collect both convergentbeam and Xray singlecrystal data in a combined study. However, all parameters can be obtained from convergentbeam and electrondiffraction data, even if in a somewhat less direct form, making possible spacegroup determination from microscopic crystals and microscopic regions of polygranular material. Several reviews of the subject are available (Tanaka, 1994; Steeds & Vincent, 1983; Steeds, 1979). In addition, an atlas of characteristic CBED patterns for direct phase identification of metal alloys has been published (Mansfield, 1984), and it is likely that this type of procedure, allowing Nbeam analysis by comparison with standard simulations, will be expanded in the near future.
Symmetry analysis is necessarily tied to examination of patterns near relevant zone axes, since the most intense Nbeam interaction occurs amongst the zerolayer zoneaxis reflections, with in addition a limited degree of upperlayer (higherorder Laue zone) interaction. There will generally be several useful zone axes accessible for a given parallelsided single crystal, with the regions between axes being of little use for symmetry analysis. Only one such zone axis can be parallel to a crystal surface normal, and a microcrystal is usually chosen at least initially to have this as the principal symmetry axis. Other zone axes from that crystal may suffer mild symmetry degradation because the Nbeam lattice component (`excitation error' extension) will not have the symmetry of the structure (Goodman, 1974; Eades et al., 1983).
Upperlayer interactions, responsible for imparting threedimensional information to the zero layer, are of two types: the first arising from `overlap' of dynamic shape transforms and causing smoothly varying modulations of the zerolayer reflections, and the second, caused by direct interactions with the upperlayer, or higherorder Laue zone lines, leading to a sharply defined fineline structure. These latter interactions are especially useful in increasing the accuracy of spacegroup determination (Tanaka et al., 1983), and may be enhanced by the use of lowtemperature specimen stages. The presence of these defect lines in convergentbeam discs, occurring especially in lowsymmetry zoneaxis patterns, allows symmetry elements to be related to the threedimensional structure (Section 2.5.3.5; Fig. 2.5.3.4c).
To the extent that such threedimensional effects can be ignored or are absent in the zerolayer pattern the projection approximation (Chapter 5.2 ) can be applied. This situation most commonly occurs in zoneaxis patterns taken from relatively thin crystals and provides a useful starting point for many analyses, by identifying the projected symmetry.
Convergentbeam diffraction symmetries are those of Schrödinger's equation, i.e. of crystal potential, plus the diffracting electron. The appropriate equation is given in Section 2.5.2 [equation (2.5.2.6)] and Chapter 5.2 [equation (5.2.2.1)] in terms of the realspace wavefunction ψ. The symmetry elements of the crystal responsible for generating pattern symmetries may be conveniently classified as of two types (I and II) as follows.

A minimal summary of basic theoretical points, otherwise found in Chapter 5.2 and numerous referenced articles, is given here.
For a specific zerolayer diffraction order the incident and diffracted vectors are and . Then the threedimensional vector has the patternspace projection, . The point gives the symmetrical Bragg condition for the associated diffraction disc, and is identifiable with the angular deviation of from the vertical z axis in threedimensional space (see Fig. 2.5.3.1). also defines the symmetry centre within the twodimensional disc diagram (Fig. 2.5.3.2); namely, the intersection of the lines S and G, given by the trace of excitation error, , and the perpendicular line directed towards the reciprocalspace origin, respectively. To be definitive it is necessary to index diffracted amplitudes relating to a fixed crystal thickness and wavelength, with both crystallographic and momentum coordinates, as , to handle the continuous variation of (for a particular diffraction order), with angles of incidence as determined by , and registered in the diffraction plane as the projection of .

Vector diagram in semireciprocal space, using Ewaldsphere constructions to show the `incident', `reciprocity' and `reciprocity × centrosymmetry' sets of vectors. Dashed lines connect the full vectors to their projections in the plane of observation. 
Reciprocity was introduced into the subject of electron diffraction in stages, the essential theoretical basis, through Schrödinger's equation, being given by Bilhorn et al. (1964), and the Nbeam diffraction applications being derived successively by von Laue (1935), Cowley (1969), Pogany & Turner (1968), Moodie (1972), Buxton et al. (1976), and Gunning & Goodman (1992).
Reciprocity represents a reverseincidence configuration reached with the reversed wavevectors and , so that the scattering vector is unchanged, but is changed in sign and hence reversed (Moodie, 1972). The reciprocity equation, is valid independently of crystal symmetry, but cannot contribute symmetry to the pattern unless a crystalinverting symmetry element is present (since belongs to a reversed wavevector). The simplest case is centrosymmetry, which permits the righthand side of (2.5.3.1) to be complexconjugated giving the useful CBED pattern equation Since K is common to both sides there is a pointbypoint identity between the related distributions, separated by 2g (the distance between g and reflections). This invites an obvious analogy with Friedel's law, , with the reservation that (2.5.3.2) holds only for centrosymmetric crystals. This condition (2.5.3.2) constitutes what has become known as the ±H symmetry and, incidentally, is the only reciprocityinduced symmetry so general as to not depend upon a disc symmetrypoint or line, nor on a particular zone axis (i.e. it is not a point symmetry but a translational symmetry of the pattern intensity).

Horizontal glides, a′, n′ (diperiodic, primed notation), generate zerolayer absent rows, or centring, rather than GS bands (see Fig. 2.5.3.3). This is an example of the projection approximation in its most universally held form, i.e. in application to absences. Other examples of this are: (a) appearance of both G and S extinction bands near their intersection irrespective of whether glide or screw axes are involved; and (b) suppression of the influence of vertical, nonprimitive translations with respect to observations in the zero layer. It is generally assumed as a working rule that the zerolayer or ZOLZ pattern will have the rotational symmetry of the pointgroup component of the vertical screw axis (so that ). Elements included in Table 2.5.3.1 on this pretext are given in parentheses. However, the presence of rather than 2 ( rather than 3 etc.) should be detectable as a departure from accurate twofold symmetry in the firstorderLauezone (FOLZ) reflection circle (depicted in Fig. 2.5.3.3). This has been observed in the cubic structure of Ba_{2}Fe_{2}O_{5}Cl_{2}, permitting the space groups I23 and to be distinguished (Schwartzman et al., 1996). A summary of all the symmetry components described in this section is given diagrammatically in Table 2.5.3.2.

Diagrammatic representation of the influence of nonsymmorphic elements: (i) Alternate rows of the zerolayer pattern are absent owing to the horizontal glide plane. The pattern is indexed as for an `a' glide; the alternative indices (in parentheses) apply for a `b' glide. (ii) GS bands are shown along the central row of the zero layer, for oddorder reflections. 
The following guidelines, the result of accumulated experience from several laboratories, are given in an experimentally based sequence, and approximately in order of value and reliability.

These results are illustrated in Table 2.5.3.2 and by actual examples in Section 2.5.3.5.
Space groups may very well be identified using CBED patterns from an understanding of the diffraction properties of realspace symmetry elements, displayed for example in Table 2.5.3.2. It is, however, of great assistance to have the symmetries tabulated in reciprocal space, to allow direct comparison with the pattern symmetries.
There are three generally useful ways in which this can be done, and these are set out in Tables 2.5.3.3 to 2.5.3.5. The simplest of these is by means of point group, following the procedures of Buxton et al. (1976). Next, the CBED pattern symmetries can be listed as diperiodic groups which are space groups in two dimensions, allowing identification with a restricted set of threedimensional space groups (Goodman, 1984b). Finally, the dynamic extinctions (GS bands and zerolayer absences) can be listed for each nonsymmorphic space group, together with the diffraction conditions for their observation (Tanaka et al., 1983; Tanaka & Terauchi, 1985). Descriptions for these tables are given below.




The list of examples given here must necessarily be regarded as unsatisfactory considering the vastness of the subject, although some attempt has been made to choose a diverse range of problems which will illustrate the principles involved. Some particular aspects, however, need further mention.
One of these concerns the problem of examining largeunitcell materials with a high diffractionpattern density. This limits the possible convergence angle, if overlap is to be avoided, and leaves numerous but featureless discs [for example Goodman (1984b)]. Technical advances which have been made to overcome this problem include the beamrocking technique (Eades, 1980) and LACBED (Tanaka et al., 1980), both of which are reviewed by Tanaka & Terauchi (1985) and Eades et al. (1983). The disadvantage of these latter methods is that they both require a significantly larger area of specimen than does the conventional technique, and it may be that more sophisticated methods of handling the crowded conventional patterns are still needed.
Next, the matter of accuracy must be considered. There are two aspects of the subject where this is of concern. Firstly, there is a very definite limit to the sensitivity with which symmetry can be detected. In a simple structure of mediumlight atoms, displacements of say 0.1 Å or less from a pseudomirror plane could easily be overlooked. An important aspect of CBED analysis, not mentioned above, is the Nbeam computation of patterns which is required when something approaching a refinement (in the context of electron diffraction) is being attempted. Although this quantitative aspect has a long history [for example see Johnson (1972)], it has only recently been incorporated into symmetry studies as a routine (Creek & Spargo, 1985; Tanaka, 1994). Multislice programs which have been developed to produce computersimulated pattern output are available (Section 2.5.3.8).
Next there is concern as to the allocation of a space group to structures which microscopically have a much lower symmetry (Goodman et al., 1984). This arises because the volume sampled by the electron probe necessarily contains a large number of unit cells. Reliable microscopic interpretation of certain nonstoichiometric materials requires that investigations be accompanied by highresolution microscopy. Frequently (especially in mineralogical samples), nonstoichiometry implies that a space group exists only on average, and that the concept of absolute symmetry elements is inapplicable.
From earlier and concluding remarks it will be clear that combined Xray/CBED and CBED/electronmicroscopy studies of inorganic materials represents the standard ideal approach to spacegroup analysis at present; given this approach, all the spacegroup problems of classical crystallography appear soluble. As has been noted earlier, it is important that HREM be considered jointly with CBED in determining space group by electron crystallography, and that only by this joint study can the socalled `phase problem' be completely overcome. The example of the spacegroup pairs and has already been cited. Using CBED, it might be expected that FOLZ lines would show a break from twofold symmetry with the incident beam aligned with a axis. However, a direct distinction should be made apparent from highresolution electron micrographs. Other less clearcut cases occur where the HREM images allow a spacegroup distinction to be made between possible space groups of the same arithmetic class, especially when only one morphology is readily obtained (e.g. , , ).
The slightly more subtle problem of distinguishing enantiomorphic spacegroup pairs can be solved by one of two approaches: either the crystal must be rotated around an axis by a known amount to obtain two projections, or the required threedimensional phase information can be deduced from specific threebeaminteraction data. This problem is part of the more general problem of solving handedness in an asymmetric structure, and is discussed in detail by Johnson & Preston (1994).

2.5.4. Electrondiffraction structure analysis (EDSA)^{4}
Electrondiffraction structure analysis (EDSA) (Vainshtein, 1964) based on electron diffraction (Pinsker, 1953) is used for the investigation of the atomic structure of matter together with Xray and neutron diffraction analysis. The peculiarities of EDSA, as compared with Xray structure analysis, are defined by a strong interaction of electrons with the substance and by a short wavelength λ. According to the Schrödinger equation (see Section 5.2.2 ) the electrons are scattered by the electrostatic field of an object. The values of the atomic scattering amplitudes, , are three orders higher than those of Xrays, , and neutrons, . Therefore, a very small quantity of a substance is sufficient to obtain a diffraction pattern. EDSA is used for the investigation of very thin singlecrystal films, of 5–50 nm polycrystalline and textured films, and of deposits of finely grained materials and surface layers of bulk specimens. The structures of many ionic crystals, crystal hydrates and hydrooxides, various inorganic, organic, semiconducting and metalloorganic compounds, of various minerals, especially layer silicates, and of biological structures have been investigated by means of EDSA; it has also been used in the study of polymers, amorphous solids and liquids.
Special areas of EDSA application are: determination of unit cells; establishing orientational and other geometrical relationships between related crystalline phases; phase analysis on the basis of and sets; analysis of the distribution of crystallite dimensions in a specimen and inner strains in crystallites as determined from line profiles; investigation of the surface structure of single crystals; structure analysis of crystals, including atomic position determination; precise determination of lattice potential distribution and chemical bonds between atoms; and investigation of crystals of biological origin in combination with electron microscopy (Vainshtein, 1964; Pinsker, 1953; Zvyagin, 1967; Pinsker et al., 1981; Dorset, 1976; Zvyagin et al., 1979).
There are different kinds of electron diffraction (ED) depending on the experimental conditions: highenergy (HEED) (above 30–200 kV), lowenergy (LEED) (10–600 V), transmission (THEED), and reflection (RHEED). In electrondiffraction studies use is made of special apparatus – electrondiffraction cameras in which the lens system located between the electron source and the specimen forms the primary electron beam, and the diffracted beams reach the detector without aberration distortions. In this case, highresolution electron diffraction (HRED) is obtained. ED patterns may also be observed in electron microscopes by a selectedarea method (SAD). Other types of electron diffraction are: MBD (microbeam), HDD (highdispersion), CBD (convergentbeam), SMBD (scanningbeam) and RMBD (rockingbeam) diffraction (see Sections 2.5.2 and 2.5.3). The recent development of electron diffractometry, based on direct intensity registration and measurement by scanning the diffraction pattern against a fixed detector (scintillator followed by photomultiplier), presents a new improved level of EDSA which provides higher precision and reliability of structural data (Avilov et al., 1999; Tsipursky & Drits, 1977; Zhukhlistov et al., 1997, 1998; Zvyagin et al., 1996).
Electrondiffraction studies of the structure of molecules in vapours and gases is a large special field of research (Vilkov et al., 1978). See also Stereochemical Applications of GasPhase Electron Diffraction (1988).
In HEED, the electron wavelength λ is about 0.05 Å or less. The Ewald sphere with radius has a very small curvature and is approximated by a plane. The ED patterns are, therefore, considered as plane cross sections of the reciprocal lattice (RL) passing normal to the incident beam through the point 000, to scale (Fig. 2.5.4.1). The basic formula is where r is the distance from the pattern centre to the reflection, h is the reciprocalspace vector, d is the appropriate interplanar distance and L is the specimentoscreen distance. The deviation of the Ewald sphere from a plane at distance h from the origin of the coordinates is . Owing to the small values of λ and to the rapid decrease of depending on , the diffracted beams are concentrated in a small angular interval ( rad).
Singlecrystal ED patterns image one plane of the RL. They can be obtained from thin ideal crystalline plates, mosaic singlecrystal films, or, in the RHEED case, from the faces of bulk single crystals. Point ED patterns can be obtained more easily owing to the following factors: the small size of the crystals (increase in the dimension of RL nodes) and mosaicity – the small spread of crystallite orientations in a specimen (tangential tension of the RL nodes). The crystal system, the parameters of the unit cell and the Laue symmetry are determined from point ED patterns; the probable space group is found from extinctions. Point ED patterns may be used for intensity measurements if the kinematic approximation holds true or if the contributions of the dynamic and secondary scattering are not too large.
The indexing of reflections and the unitcell determination are carried out according to the formulae relating the RL to the DL (direct lattice) (Vainshtein, 1964; Pinsker, 1953; Zvyagin, 1967).
Under electrondiffraction conditions crystals usually show a tendency to lie down on the substrate plane on the most developed face. Let us take this as (001). The vectors a and b are then parallel, while vector is normal to this plane, and the RL points are considered as being disposed along direct lines parallel to the axis with constant hk and variable l.
The interpretation of the point patterns as respective RL planes is quite simple in the case of orthogonal lattices. If the lattice is triclinic or monoclinic the pattern of the crystal in the position with the face (001) normal to the incident beam does not have to contain hk0 reflections with nonzero h and k because, in general, the planes ab and do not coincide. However, the intersection traces of direct lines hk with the plane normal to them (plane ab) always form a net with periods (Fig. 2.5.4.2). The points hkl along these directions hk are at distances from the ab plane.
By changing the crystal orientation it is possible to obtain an image of the plane containing hk0 reflections, or of other RL planes, with the exception of planes making a small angle with the axis .
In the general case of an arbitrary crystal orientation, the pattern is considered as a plane section of the system of directions hk which makes an angle ϕ with the plane ab, intersecting it along a direction [uv]. It is described by two periods along directions 0h, 0k; with an angle γ″ between them satisfying the relation and by a system of parallel directions The angles are formed by directions 0h, 0k in the plane of the pattern with the plane ab. The coefficients depend on the unitcell parameters, angle ϕ and direction [uv]. These relations are used for the indexing of reflections revealed near the integer positions hkl in the pattern and for unitcell calculations (Vainshtein, 1964; Zvyagin, 1967; Zvyagin et al., 1979).
In RED patterns obtained with an incident beam nearly parallel to the plane ab one can reveal all the RL planes passing through which become normal to the beam at different azimuthal orientations of the crystal.
With the increase of the thickness of crystals (see below, Chapter 5.1 ) the scattering becomes dynamical and Kikuchi lines and bands appear. Kikuchi ED patterns are used for the estimation of the degree of perfection of the structure of the surface layers of single crystals for specimen orientation in HREM (IT C, 2004, Section 4.3.8 ). Patterns obtained with a convergent beam contain Kossel lines and are used for determining the symmetry of objects under investigation (see Section 5.1.2 ).
Texture ED patterns are a widely used kind of ED pattern (Pinsker, 1953; Vainshtein, 1964; Zvyagin, 1967). Textured specimens are prepared by substance precipitation on the substrate, from solutions and suspensions, or from gas phase in vacuum. The microcrystals are found to be oriented with a common (developed) face parallel to the substrate, but they have random azimuthal orientations. Correspondingly, the RL also takes random azimuthal orientations, having as the common axis, i.e. it is a rotational body of the point RL of a single crystal. Thus, the ED patterns from textures bear a resemblance, from the viewpoint of their geometry, to Xray rotation patterns, but they are less complicated, since they represent a plane cross section of reciprocal space.
If the crystallites are oriented by the plane (hkl), then the axis is the texture axis. For the sake of simplicity, let us assume that the basic plane is the plane (001) containing the axes a and b, so that the texture axis is , i.e. the axis . The matrices of appropriate transformations will define a transition to the general case (see IT A , 2005). The RL directions , parallel to the texture axis, transform to cylindrical surfaces, the points with are in planes perpendicular to the texture axis, while any `tilted' lines transform to cones or hyperboloids of rotation. Each point hkl transforms to a ring lying on these surfaces. In practice, owing to a certain spread of axes of single crystals, the rings are blurred into small band sections of a spherical surface with the centre at the point 000; the oblique cross section of such bands produces reflections in the form of arcs. The main interference curves for texture patterns are ellipses imaging oblique plane cross sections of the cylinders hk (Fig. 2.5.4.3).
At the normal electronbeam incidence (tilting angle ) the ED pattern represents a cross section of cylinders perpendicular to the axis , i.e. a system of rings.
On tilting the specimen to an angle ϕ with respect to its normal position (usually ) the patterns image an oblique cross section of the cylindrical RL, and are called obliquetexture (OT) ED patterns. The ellipses and layer lines for orthogonal lattices are the main characteristic lines of ED patterns along which the reflections are arranged. The shortcoming of obliquetexture ED patterns is the absence of reflections lying inside the cone formed by rotation of the straight line coming from the point 000 at an angle around the axis and, in particular, of reflections 00l. However, at ϕ ≲ 60–70° the set of reflections is usually sufficient for structural determination.
For unitcell determination and reflection indexing the values d (i.e. ) are used, and the reflection positions defined by the ellipses hk to which they belong and the values η are considered. The periods are obtained directly from and values. The period , if it is normal to the plane ( being arbitrary), is calculated as For obliqueangled lattices In the general case of obliqueangled lattices the coaxial cylinders hk have radii and it is always possible to use the measured or calculated values in (2.5.4.5a) instead of , since In OT patterns the and η values are represented by the lengths of the small axes of the ellipses and the distances of the reflections hkl from the line of small axes (equatorial line of the pattern)
Analysis of the values gives a, b, γ, while p, s and q are calculated from the values. It is essential that the components of the normal projections of the axis c on the plane ab measured in the units of a and b are Obtaining one can calculate Since
The α, β values are then defined by the relations
Because of the small particle dimensions in textured specimens, the kinematic approximation is more reliable for OT patterns, enabling a more precise calculation of the structure amplitudes from the intensities of reflections.
Polycrystal ED patterns. In this case, the RL is a set of concentric spheres with radii . The ED pattern, like an Xray powder pattern, is a set of rings with radii
The intensities of scattering by a crystal are determined by the scattering amplitudes of atoms in the crystal, given by (see also Section 5.2.1 ) where is the potential of an atom and . The absolute values of have the dimensionality of length L. In EDSA it is convenient to use without K. The dimensionality of is [potential ]. With the expression of in V Å^{3} the value in (2.5.4.13) is 47.87 V Å^{2}.
The scattering atomic amplitudes differ from the respective Xray values in the following: while (electron shell charge), the atomic amplitude at is the `full potential' of the atom. On average, , but for small atomic numbers Z, owing to the peculiarities in the filling of the electron shells, exhibits within periods of the periodic table of elements `reverse motion', i.e. they decrease with Z increasing (Vainshtein, 1952, 1964). At large , . The atomic amplitudes and, consequently, the reflection intensities, are recorded, in practice, up to values of , i.e. up to .
The structure amplitude of a crystal is determined by the Fourier integral of the unitcell potential (see Chapter 1.2 ), where Ω is the unitcell volume. The potential of the unit cell can be expressed by the potentials of the atoms of which it is composed: The thermal motion of atoms in a crystal is taken into account by the convolution of the potential of an atom at rest with the probability function describing the thermal motion: Accordingly, the atomic temperature factor of the atom in a crystal is where the Debye temperature factor is written for the case of isotropic thermal vibrations. Consequently, the structure amplitude is This general expression is transformed (see IT I, 1952) according to the space group of a given crystal.
To determine the structure amplitudes in EDSA experimentally, one has to use specimens satisfying the kinematic scattering condition, i.e. those consisting of extremely thin crystallites. The limit of the applicability of the kinematic approximation (Blackman, 1939; Vainshtein, 1964) can be estimated from the formula where is the averaged absolute value of (see also Section 5.2.1 ). Since are proportional to , condition (2.5.4.20) is better fulfilled for crystals with light and medium atoms. Condition (2.5.4.20) is usually satisfied for textured and polycrystalline specimens. But for mosaic single crystals as well, the kinematic approximation limit is, in view of their real structure, substantially wider than estimated by (2.5.4.20) for ideal crystals. The fulfillment of the kinematic law for scattering can be, to a greater or lesser extent, estimated by comparing the decrease of experimental intensity averaged over definite angular intervals, and sums calculated for the same angular intervals.
For mosaic singlecrystal films the integral intensity of reflection is for textures Here is the incident electronbeam density, S is the irradiated specimen area, t is the thickness of the specimen, α is the average angular spread of mosaic blocks, R′ is the horizontal coordinate of the reflection in the diffraction pattern and p is the multiplicity factor. In the case of polycrystalline specimens the local intensity in the maximum of the ring reflection is measured, where ΔS is the measured area of the ring.
The transition from kinematic to dynamic scattering occurs at critical thicknesses of crystals when (2.5.4.20). Mosaic or polycrystalline specimens then result in an uneven contribution of various crystallites to the intensity of the reflections. It is possible to introduce corrections to the experimental structure amplitudes of the first strong reflections most influenced by dynamic scattering by applying in simple cases the twowave approximation (Blackman, 1939) or by taking into account multibeam theories (Fujimoto, 1959; Cowley, 1981; Avilov et al. 1984; see also Chapter 5.2 ).
The application of kinematic scattering formulae to specimens of thin crystals (5–20 nm) or dynamic corrections to thicker specimens (20–50 nm) permits one to obtain reliability factors between the calculated and observed structure amplitudes of , which is sufficient for structural determinations.
With the use of electron diffractometry techniques, reliability factors as small as 2–3% have been reached and more detailed data on the distribution of the innercrystalline potential field have been obtained, characterizing the state and bonds of atoms, including hydrogen (Zhukhlistov et al., 1997, 1998; Avilov et al., 1999).
The applicability of kinematics formulae becomes poorer in the case of structures with many heavy atoms for which the atomic amplitudes also contain an imaginary component (Shoemaker & Glauber, 1952). The experimental intensity measurement is made by a photo method or by direct recording (Avilov, 1979). In some cases the amplitudes can be determined from dynamic scattering patterns – the bands of equal thickness from a wedgeshaped crystal (Cowley, 1981), or from rocking curves.
The unit cell is defined on the basis of the geometric theory of electrondiffraction patterns, and the space group from extinctions. It is also possible to use the method of converging beams (Section 5.2.2 ). The structural determination is based on experimental sets of values or (Vainshtein, 1964).
The trialanderror method may be used for the simplest structures. The main method of determination is the construction of the Patterson functions and their analysis on the basis of heavyatom methods, superposition methods and so on (see Chapter 2.3 ). Direct methods are also used (Dorset et al., 1979). Thus the phases of structure factors are calculated and assigned to the observed moduli
The distribution of the potential in the unit cell, and, thereby, the arrangement in it of atoms (peaks of the potential) are revealed by the construction of threedimensional Fourier series of the potential (see also Chapter 1.3 ) or projections The general formulae (2.5.4.26a) and (2.5.4.26b) transform, according to known rules, to the expressions for each space group (see IT I, 1952). If are expressed in V and the volume Ω or the cell area S in and , respectively, then the potential ϕ is obtained directly in volts, while the projection of the potential is in V Å. The amplitudes are reduced to an absolute scale either according to a group of strong reflections or using the Parseval equality or Wilson's statistical method The term defines the mean inner potential of a crystal, and is calculated from [(2.5.4.13), (2.5.4.19)] The Fourier series of the potential in EDSA possess some peculiarities (Vainshtein, 1954, 1964) which make them different from the electrondensity Fourier series in Xray analysis. Owing to the peculiarities in the behaviour of the atomic amplitudes (2.5.4.13), which decrease more rapidly with increasing compared with , the peaks of the atomic potential are more `blurred' and exhibit a larger halfwidth than the electrondensity peaks . On average, this halfwidth corresponds to the `resolution' of an electrondiffraction pattern – about 0.5 Å or better. The potential in the maximum (`peak height') does not depend as strongly on the atomic number as in Xray analysis: while in Xray diffraction . In such a way, in EDSA the light atoms are more easily revealed in the presence of heavy atoms than in Xray diffraction, permitting, in particular, hydrogen atoms to be revealed directly without resorting to difference syntheses as in Xray diffraction. Typical values of the atomic potential (which depend on thermal motion) in organic crystals are: H ∼ 35, C ∼ 165, O 215 V; in Al crystals 330 V, in Cu crystals 750 V.
The EDSA method may be used for crystal structure determination, depending on the types of electrondiffraction patterns, for crystals containing up to several tens of atoms in the unit cell. The accuracy in determination of atomic coordinates in EDSA is about 0.01–0.005 Å on average. The precision of EDSA makes it possible to determine accurately the potential distribution, to investigate atomic ionization, to obtain values for the potential between the atoms and, thereby, to obtain data on the nature of the chemical bond.
If the positions in the cell are occupied only partly, then the measurement of gives information on population percentage.
There is a relationship between the nuclear distribution, electron density and the potential as given by the Poisson equation This makes it possible to interrelate Xray diffraction, EDSA and neutrondiffraction data. Thus for the atomic amplitudes where Z is the nuclear charge and the Xray atomic scattering amplitude, and for structure amplitudes where is the Xray structure amplitude of the electron density of a crystal and is the amplitude of scattering from charges of nuclei in the cell taking into account their thermal motion. The values can be calculated easily from neutrondiffraction data, since the charges of the nuclei are known and the experiment gives the parameters of their thermal motion.
In connection with the development of highresolution electronmicroscopy methods (HREM) it has been found possible to combine the data from direct observations with EDSA methods. However, EDSA permits one to determine the atomic positions to a greater accuracy, since practically the whole of reciprocal space with 1.0–0.4 Å resolution is used and the threedimensional arrangement of atoms is calculated. At the same time, in electron microscopy, owing to the peculiarities of electron optics and the necessity for an objective aperture, the image of the atoms in a crystal is a convolution, with the aperture function blurring the image up to 1.5–2 Å resolution. In practice, in TEM one obtains only the images of the heaviest atoms of an object. However, the possibility of obtaining a direct image of a structure with all the defects in the atomic arrangement is the undoubted merit of TEM.
2.5.5. Image reconstruction ^{5}
In many fields of physical measurements, instrumental and informative techniques, including electron microscopy and computational or analogue methods for processing and transforming signals from objects investigated, find a wide application in obtaining the most accurate structural data. The signal may be radiation from an object, or radiation transmitted through the object, or reflected by it, which is transformed and recorded by a detector.
The image is the twodimensional signal on the observation plane recorded from the whole threedimensional volume of the object, or from its surface, which provides information on its structure. In an object this information may change owing to transformation of the scattered wave inside an instrument. The real image is composed of and noise from signal disturbances:
Imagereconstruction methods are aimed at obtaining the most accurate information on the structure of the object; they are subdivided into two types (Picture Processing and Digital Filtering, 1975; Rozenfeld, 1969):

These two methods may be used separately or in combination.
The image should be represented in the form convenient for perception and analysis, e.g. in digital form, in lines of equal density, in points of different density, in halftones or colour form and using, if necessary, a change or reversal of contrast.
Reconstructed images may be used for the threedimensional reconstruction of the spatial structure of an object, e.g. of the density distribution in it (see Section 2.5.6).
This section is connected with an application of the methods of image processing in transmission electron microscopy (TEM). In TEM (see Section 2.5.2), the sourceemitted electrons are transmitted through an object and, with the aid of a system of lenses, form a twodimensional image subject to processing.
Another possibility for obtaining information on the structure of an object is structural analysis with the aid of electron diffraction – EDSA. This method makes use of information in reciprocal space – observation and measurement of electrondiffraction patterns and calculation from them of a twodimensional projection or threedimensional structure of an object using the Fourier synthesis. To do this, one has to find the relative phases of the scattered beams.
The wavefunction of an electronmicroscopic image is written as Here is the incident plane wave. When the wave is transmitted through an object, it interacts with the electrostatic potential [ is the threedimensional vector in the space of the object]; this process is described by the Schrödinger equation (Section 2.5.2.1). As a result, on the exit surface of an object the wave takes the form where q is the transmission function and x is the twodimensional vector . The diffraction of the wave is described by the twodimensional Fourier operator:
Here, we assume the initial wave amplitude to be equal to unity and the initial phase to be zero, so that , which defines, in this case, the wavefunction in the back focal plane of an objective lens with the reciprocalspace coordinates . The function Q is modified in reciprocal space by the lens transfer function . The scattered wave transformation into an image is described by the inverse Fourier operator .
The process of the diffraction , as seen from (2.5.5.1), is the same in both TEM and EDSA. Thus, in TEM under the lens actions the image formation from a diffraction pattern takes place with an account of the phases, but these phases are modified by the objectivelens transfer function. In EDSA, on the other hand, there is no distorting action of the transfer function and the `image' is obtained by computing the operation .
The computation of projections, images and Fourier transformation is made by discretization of twodimensional functions on a twodimensional network of points – pixels in real space and in reciprocal space .
The intensity distribution of an electron wave in the image plane depends not only on the coherent and inelastic scattering, but also on the instrumental functions. The electron wave transmitted through an object interacts with the electrostatic potential which is produced by the nuclei charges and the electronic shells of the atoms. The scattering and absorption of electrons depend on the structure and thickness of a specimen, and the atomic numbers of the atoms of which it is composed. If an object with the threedimensional distribution of potential is sufficiently thin, then the interaction of a plane electron wave with it can be described as the interaction with a twodimensional distribution of potential projection , where b is the specimen thickness. It should be noted that, unlike the threedimensional function of potential with dimension , the twodimensional function of potential projection has the potentiallength dimension which, formally, coincides with the charge dimension. The transmission function, in the general case, has the form (2.5.2.42), and for weak phase objects the approximation is valid.
In the back focal plane of the objective lens the wave has the form where is the Scherzer phase function (Scherzer, 1949) of an objective lens (Fig. 2.5.5.1), is the aperture function, the spherical aberration coefficient, and Δf the defocus value [(2.5.2.32) –(2.5.2.35)].
The brightfield image intensity (in object coordinates) is where . The phase function (2.5.5.7) depends on defocus, and for a weak phase object (Cowley, 1981) where , which includes only an imaginary part of function (2.5.5.6). While selecting defocus in such a way that under the Scherzer defocus conditions [(2.5.2.44), (2.5.2.45)] , one could obtain In this very simple case the image reflects directly the structure of the object – the twodimensional distribution of the projection of the potential convoluted with the spread function . In this case, no image restoration is necessary. Contrast reversal may be achieved by a change of defocus.
At high resolution, this method enables one to obtain an image of projections of the atomic structure of crystals and defects in the atomic arrangement – vacancies, replacements by foreign atoms, amorphous structures and so on; at resolution worse than atomic one obtains images of dislocations as continuous lines, inserted phases, inclusions etc. (Cowley, 1981). It is also possible to obtain images of thin biological crystals, individual molecules, biological macromolecules and their associations.
Image restoration. In the case just considered (2.5.5.10), the projection of potential , convoluted with the spread function, can be directly observed. In the general case (2.5.5.9), when the aperture becomes larger, the contribution to image formation is made by large values of spatial frequencies U, in which the function sin χ oscillates, changing its sign. Naturally, this distorts the image just in the region of appropriate high resolution. However, if one knows the form of the function sin χ (2.5.5.7), the true function can be restored.
This could be carried out experimentally if one were to place in the back focal plane of an objective lens a zone plate transmitting only onesign regions of sin χ (Hoppe, 1971). In this case, the information on is partly lost, but not distorted. To perform such a filtration in an electron microscope is a rather complicated task.
Another method is used (Erickson & Klug, 1971). It consists of a Fourier transformation of the measured intensity distribution TQ (2.5.5.6) and division of this transform, according to (2.5.5.7a,b), by the phase function sin χ. This gives Then, the new Fourier transformation yields (in the weakphaseobject approximation) the true distribution The function sin χ depending on defocus Δf should be known to perform this procedure. The transfer function can also be found from an electron micrograph (Thon, 1966). It manifests itself in a circular image intensity modulation of an amorphous substrate or, if the specimen is crystalline, in the `noise' component of the image. The analogue method (optical Fourier transformation for obtaining the image ) can be used (optical diffraction, see below); digitization and Fourier transformation can also be applied (Hoppe et al., 1973).
The thin crystalline specimen implies that in the back focal objective lens plane the discrete kinematic amplitudes are arranged and, by the above method, they are corrected and released from phase distortions introduced by the function sin χ (see below) (Unwin & Henderson, 1975).
For the threedimensional reconstruction (see Section 2.5.6) it is necessary to have the projections of potential of the specimen tilted at different angles α to the beam direction (normal beam incidence corresponds to ). In this case, the defocus Δf changes linearly with increase of the distance l of specimen points from the rotation axis . Following the above procedure for passing on to reciprocal space and correction of sin χ, one can find (Henderson & Unwin, 1975).
Elastic interaction of an incident wave with a weak phase object is defined on its exit surface by the distribution of potential projection ; however, in the general case, the electron scattering amplitude is a complex one (Glauber & Schomaker, 1953). In such a way, the image itself has the phase and amplitude contrast. This may be taken into account if one considers not only the potential projection , but also the `imaginary potential' which describes phenomenologically the absorption in thin specimens. Then, instead of (2.5.5.5), the wave on the exit surface of a specimen can be written as and in the back focal plane if and Usually, μ is small, but it can, nevertheless, make a certain contribution to an image. In a sufficiently good linear approximation, it may be assumed that the real part cos χ of the phase function (2.5.5.7a) affects , while , as we know, is under the action of the imaginary part sin χ.
Thus, instead of (2.5.5.6), one can write and as the result, instead of (2.5.5.10),
The functions and can be separated by object imaging using the throughfocus series method. In this case, using the Fourier transformation, one passes from the intensity distribution (2.5.5.15) in real space to reciprocal space. Now, at two different defocus values and [(2.5.5.6), (2.5.5.7a,b)] the values and can be found from the two linear equations (2.5.5.14). Using the inverse Fourier transformation, one can pass on again to real space which gives and (Schiske, 1968). In practice, it is possible to use several throughfocus series and to solve a set of equations by the leastsquares method.
Another method for processing takes into account the simultaneous presence of noise and transfer function zeros (Kirkland et al., 1980). In this method the space frequencies corresponding to small values of the transfer function modulus are suppressed, while the regions where such a modulus is large are found to be reinforced.
When the specimen thickness exceeds a certain critical value (50–100 Å), the kinematic approximation does not hold true and the scattering is dynamic. This means that on the exit surface of a specimen the wave is not defined as yet by the projection of potential (2.5.5.3), but one has to take into account the interaction of the incident wave and of all the secondary waves arising in the whole volume of a specimen.
The dynamic scattering calculation can be made by various methods. One is the multislice (or phasegrating) method based on a recurrent application of formulae (2.5.5.3) for n thin layers thick, and successive construction of the transmission functions (2.5.5.4), phase functions , and propagation function (Cowley & Moodie, 1957).
Another method – the scattering matrix method – is based on the solution of equations of the dynamic theory (Chapter 5.2 ). The emerging wave on the exit surface of a crystal is then found to diffract and experience the transfer function action [(2.5.5.6), (2.5.5.7a,b)].
The dynamic scattering in crystals may be interpreted using Bloch waves: It turns out that only a few (bound and valence Bloch waves) have strong excitation amplitudes. Depending on the thickness of a crystal, only one of these waves or their linear combinations (Kambe, 1982) emerges on the exit surface. An electronmicroscopic image can be interpreted, at certain thicknesses, as an image of one of these waves [with a correction for the transfer function action (2.5.5.6), (2.5.5.7a,b)]; in this case, the identical images repeat with increasing thickness, while, at a certain thickness, the contrast reversal can be observed. Only the first Bloch wave which arises at small thickness, and also repeats with increasing thickness, corresponds to the projection of potential , i.e. the atom projection distribution in a thin crystal layer.
An image of other Bloch waves is defined by the function , but their maxima or minima do not coincide, in the general case, with the atomic positions and cannot be interpreted as the projection of potential. It is difficult to reconstruct from these images, especially when the crystal is not ideal and contains imperfections. In these cases one resorts to computer modelling of images at different thicknesses and defocus values, and to comparison with an experimentally observed pattern.
The imaging can be performed directly in an electron microscope not by a photo plate, but using fastresponse detectors with digitized intensity output on line. The computer contains the necessary algorithms for Fourier transformation, image calculation, transfer function computing, averaging, and correction for the observed and calculated data. This makes possible the interpretation of the pattern observed directly in experiment (Herrmann et al., 1980).
The real electronmicroscope image is subdivided into two components: The main of these, , is a twodimensional image of the `ideal' object obtained in an electron microscope with instrumental functions inherent to it. However, in the process of object imaging and transfer of this information to the detector there are various sources of noise. In an electron microscope, these arise owing to emissioncurrent and acceleratingvoltage fluctuations, lenssupplying current (temporal fluctuations), or mechanical instabilities in a device, specimen or detector (spatial shifts). The twodimensional detector (e.g. a photographic plate) has structural inhomogeneities affecting a response to the signal. In addition, the specimen is also unstable; during preparation or imaging it may change owing to chemical or some other transformations in its structure, thermal effects and so on. Biological specimens scatter electrons very weakly and their natural state is moist, while in the electronmicroscope column they are under vacuum conditions. The methods of staining (negative or positive), e.g. of introducing into specimens substances containing heavy atoms, as well as the freezeetching method, somewhat distort the structure of a specimen. Another source of structure perturbation is radiation damage, which can be eliminated at small radiation doses or by using the cryogenic technique. The structure of stained specimens is affected by stain graininess. We assume that all the deviations of a specimen image from the `ideal' image are included in the noise term . The substrate may also be inhomogeneous. All kinds of perturbations cannot be separated and they appear on an electron microscope image as the full noise content .
The image enhancement involves maximum noise suppression and hence the most accurate separation of a useful signal from the real image (2.5.5.1). At the signal/noise ratio such a separation appears to be rather complicated. But in some cases the real image reflects the structure sufficiently well, e.g. during the atomic structure imaging of some crystals . In other cases, especially of biological specimen imaging, the noise N distorts substantially the image, . Here one should use the methods of enhancement. This problem is usually solved by the methods of statistical processing of sets of images . If one assumes that the informative signal is always the same, then the noise error may be reduced.
The image enhancement methods are subdivided into two classes:
These methods can be used separately or in combination. The enhancement can be applied to both the original and the restored images; there are also methods of simultaneous restoration and enhancement.
The image can be enhanced by analogue (mainly optical and photographic) methods or by computational methods for processing digitized functions in real and reciprocal space.
The cases where the image has translational symmetry, rotational symmetry, and where the image is asymmetric will be considered.
Periodic images. An image of the crystal structure with atomic or molecular resolution may be brought to selfalignment by a shift by a and b periods in a structure projection. This can be performed photographically by printing the shifted image on the same photographic paper or, vice versa, by shifting the paper (McLachlan, 1958).
The Fourier filtration method for a periodic image with noise N is based on the fact that in Fourier space the components and are separated. Let us carry out the Fourier transformation of the periodic signal with the periods a, b and noise N: The left part of (2.5.5.18) represents the Fourier coefficients distributed discretely with periods and in the plane . This is the twodimensional reciprocal lattice. The righthand side of (2.5.5.18) is the Fourier transform distributed continuously in the plane. Thus these parts are separated. Let us `cut out' from distribution (2.5.5.18) only values using the `window' function . The window should match each of the real peaks which, owing to the finite dimensions of the initial periodic image, are not points, as this is written in an idealized form in (2.5.5.18) with the aid of δ functions. In reality, the `windows' may be squares of about , in size, or a circle. Performing the Fourier transformation of product (2.5.5.18) without , and set of windows , we obtain: the periodic component without the background, . The zero coefficient in (2.5.5.19) should be decreased, since it is due, in part, to the noise. When the window w is sufficiently small, in (2.5.5.19) represents the periodic distribution (average over all the unit cells of the projection) included in (2.5.5.18). Nevertheless, some error from noise in an image does exist, since with we also introduced into the inverse Fourier transformation the background transform values which are within the `windows'.
This approach is realized by an analogue method [optical diffraction and filtering of electron micrographs in a laser beam (Klug & Berger, 1964)] and can also be carried out by computing.
As an example, Fig. 2.5.5.2(b) shows an electron micrograph of the periodic structure of a twodimensional protein crystal, while Fig. 2.5.5.2(c) represents optical diffraction from this layer. In order to dissect the aperiodic component in a diffraction plane, according to the scheme in Fig. 2.5.5.2(a), one places a mask with windows covering reciprocallattice points. After such a filtration, only the component makes a contribution during the image formation by means of a lens, while the component diffracted by the background is delayed. As a result, an optical pattern of the periodic structure is obtained (Fig. 2.5.5.2d).

(a) Diagram of an optical diffractometer. D is the object (an electron micrograph), is the diffraction plane and a mask that transmits only , is the plane of the (filtered) image; (b) an electron micrograph of a crystalline layer of the protein phosphorylase b; (c) its optical diffraction pattern (the circles correspond to the windows in the mask that transmits only the diffracted beams from the periodic component of the image); (d) the filtered image. Parts (b)–(d) are based on the article by Kiselev et al. (1971). 
Optical diffractometry also assists in determining the parameters of a twodimensional lattice and its symmetry.
Using the same method, one can separate the superimposed images of twodimensional structures with different periodicity and in different orientation, the images of the `near' and `far' sides of tubular periodic structures with monomolecular walls (Klug & DeRosier, 1966; Kiselev et al., 1971), and so on.
Computer filtering involves measuring the image optical density , digitization, and Fourier transformation (Crowther & Amos, 1971). The sampling distance usually corresponds to onethird of the image resolution. When periodic weak phase objects are investigated, the transformation (2.5.5.18) yields the Fourier coefficients. If necessary, we can immediately make corrections in them using the microscope transfer function according to (2.5.5.6), (2.5.5.7a,b) and (2.5.5.11a), and thereby obtain the true kinematic amplitudes . The inverse transformation (2.5.5.16) gives a projection of the structure (Unwin & Henderson, 1975; Henderson & Unwin, 1975).
Sometimes, an observed image is `noised' by the to a great extent. Then, one may combine data on real and reciprocal space to construct a sufficiently accurate image. In this case, the electrondiffraction pattern is measured and structurefactor moduli from diffraction reflection intensities are obtained: At the same time, the structure factors are calculated from the processed structure projection image by means of the Fourier transformation. However, owing to poor image quality we take from these data only the values of phases since they are less sensitive to scattering density distortions than the moduli, and construct the Fourier synthesis
Here the possibilities of combining various methods open up, e.g. for obtaining the structurefactor moduli from Xray diffraction, and phases from electron microscopy, and so on (Gurskaya et al., 1971).
Images with point symmetry. If a projection of an object (and consequently, the object itself) has a rotational Nfold axis of symmetry, the structure coincides with itself on rotation through the angle . If the image is rotated through arbitrary angles and is aligned photographically with the initial image, then the best density coincidence will take place at a rotation through which defines N. The pattern averaging over all the rotations will give the enhanced structure image with an times reduced background (Markham et al., 1963).
Rotational filtering can be performed on the basis of the Fourier expansion of an image in polar coordinates over the angles (Crowther & Amos, 1971). The integral over the radius from azimuthal components gives t