International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 297-402   | 1 | 2 |
https://doi.org/10.1107/97809553602060000767

Chapter 2.5. Electron diffraction and electron microscopy in structure determination

J. M. Cowley,a J. C. H. Spence,b M. Tanaka,f B. K. Vainshtein,c B. B. Zvyagin,d P. A. Penczekg and D. L. Dorsete

aArizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287–1504, USA, bDepartment of Physics, Arizona State University, Tempe, AZ 95287–1504, USA, cInstitute of Crystallography, Academy of Sciences of Russia, Leninsky prospekt 59, Moscow B-117333, Russia, dInstitute of Ore Mineralogy (IGEM), Academy of Sciences of Russia, Staromonetny 35, 109017 Moscow, Russia, eExxonMobil Research and Engineering Co., 1545 Route 22 East, Clinton Township, Annandale, New Jersey 08801, USA,fInstitute of Multidisciplinary Research for Advanced Materials, Tohoku University, Japan, and gThe University of Texas – Houston Medical School, Department of Biochemistry and Molecular Biology, 6431 Fannin, MSB 6.218, Houston, TX 77030, USA

This chapter deals with electron diffraction and imaging in the transmission geometry at kilovolt and higher beam energies for the purposes of crystal structure determination. Techniques such as low-energy electron diffraction, gas electron diffraction and reflection electron diffraction are not considered. Following a brief review of the history of the subject and its relationship to X-ray diffraction, the fundamental theory is presented by J. M. Cowley in Section 2.5.2[link]. Unlike X-rays, which diffract from the electron density in a crystal, electrons are scattered elastically by the Coulomb potential, which is related to the density through Poisson's equation and includes the nuclear contribution. Electrons are much more strongly scattered than X-rays and have much smaller wavelengths. (This leads to a very `flat' Ewald sphere and the simultaneous excitation of many Bragg beams.) Electron sources have comparable brightness to third-generation synchrotrons fitted with an undulator, so that Bragg intensities are very high. Because of their limited penetration, samples for transmission electron microscopes (TEMs) usually have thicknesses of less than a micron, while for interpretable atomic resolution images, thicknesses of tens of nanometres are preferred. Multiple scattering complicates TEM image interpretation and diffraction for larger thicknesses. The current (2006) spatial resolution of the best aberration-corrected TEM instruments is about 0.8 nm. Microdiffraction patterns may be obtained using a beam width of sub-nanometre dimensions, while the analysis of characteristic X-rays excited by such a probe provides chemical information. For organic monolayers (such as proteins), the amount of structural information obtained per unit of radiation damage (which is a strong function of resolution) exceeds that obtainable by X-ray diffraction. Cowley's treatment includes sections on electron scattering factors, Bethe's 1928 multiple scattering theory, Born's series, sign conventions, two-beam dynamical theory and single-scattering theory. This is followed by the theory of electron microscope imaging at high resolution, including the weak-phase object and other approximations, and the Scherzer focusing condition. The section ends with treatments of atomic resolution imaging in crystals and the factors which limit it, with Fourier or Talbot self-imaging, and with a brief discussion of coherent nanodiffraction. Section 2.5.3[link] by M. Tanaka describes how the point groups and space groups of ordinary (three-dimensional or 3D) crystals, one-dimensionally incommensurate (4D) crystals and quasicrystals (5D and 6D) can be determined by convergent-beam electron diffraction. Useful tables and examples of point- and space-group determination are provided. Section 2.5.4[link] by B. K. Vainshtein and B. B. Zvyagin is devoted to the use of transmission electron-diffraction patterns to solve the structures of thin crystal structures using the kinematic theory. Texture and polycrystalline patterns are considered, as is the relationship to X-ray work. Section 2.5.5[link] by B. K. Vainshtein again outlines the theory of high-resolution electron imaging, extending this to include image processing, image cross-correlation and alignment, and image filtering and enhancement. In Section 2.5.6[link], B. K. Vainshtein and P. A. Penczek discuss algorithms for three-dimensional reconstruction from sets of ray projections, with emphasis on algorithms used in cryo-electron microscopy, including single-particle reconstruction and objects with icosahedral and helical symmetries. The general feasibility of the reconstruction problem as well as the limitations posed by discretization and interpolation are discussed. A detailed analysis of three classes of reconstruction methods is provided: algebraic and iterative, filtered backprojection, and direct Fourier inversion. In each case, the efficiency of the respective method is discussed and its performance for typical cryo-electron microscopy data sets is evaluated. In Section 2.5.7[link], P. A. Penczek describes macromolecular structure determination using cryo-electron microscopy and the single-particle approach. A general overview of the analytical steps is given with a detailed analysis of the pivotal computational methods involved and with emphasis on the evaluation of the reliability of the results. Examples of near-atomic resolution as well as intermediate resolution structure-determination projects are given, accompanied by a discussion of the methods used to present and analyse the results. Section 2.5.8[link] by D. L. Dorset summarizes solutions to the phase problem which may be applied to electron-diffraction data. These include many of those currently in use for X-ray diffraction, including Patterson maps, direct methods and trial-and-error search techniques. Much of the section concerns electron-diffraction data from thin organic films, analysed using the three-phase invariants of the direct methods approach to phasing. The tangent formula is reviewed and useful software is suggested. Density modification and maximum entropy methods are outlined. Because of the strong dependence of multiply scattered electron-diffraction intensities on the thickness of the sample, and on local orientation changes (effects which are not accounted for in the structure analysis of the data), the thinnest possible samples must be used, and phasing measures are sought which are robust with respect to multiple scattering perturbations. Conversely, the ability to obtain data from nanometre-sized regions can greatly assist the effort to obtain high-quality perfect-crystal data free of defects, bending or thickness variation.

2.5.1. Foreword

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J. M. Cowleya and J. C. H. Spenceb

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 X-rays. While for X-rays the scattering function is the electron-density 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 X-ray diffraction for the determination of crystal structures but is receiving increasing attention as a means for obtaining structural information not readily accessible with X-ray- or neutron-diffraction techniques.

Electrons having wavelengths comparable with those of the X-rays 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 low-energy 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 high-energy 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 high-energy electrons, 100 to 400 keV, the wavelengths are about 50 times smaller than for X-rays. Hence the scattering angles are much smaller, of the order of 10−2 rad, the recording geometry is relatively simple and the diffraction pattern represents, to a useful first approximation, a planar section of reciprocal space. Extinction distances are hundreds of ångstroms, which, when combined with typical lattice spacings, produces rocking-curve widths which are, unlike the X-ray case, a significant fraction of the Bragg angle.

The elastic scattering of electrons by atoms is several orders of magnitude greater than for X-rays. 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 electron-diffraction patterns can be obtained from very small single-crystal 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 single-crystal patterns can be obtained from microcrystalline phases.

However, the strong scattering of electrons implies that the simple kinematical single-scattering approximation, on which most X-ray diffraction structure analysis is based, fails for electrons except for very thin crystals composed of light-atom materials. Strong dynamical diffraction effects occur for crystals which may be 100 Å thick, or less for heavy-atom 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[link] ), 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 convergent-beam electron diffraction (CBED) as described in Section 2.5.3[link].

A further important factor, determining the methods for observing electron diffraction, is that, being charged particles, electrons can be focused by electromagnetic lenses. Many of the resolution-limiting aberrations of cylindrical magnetic lenses have now been eliminated through the use of aberration-correction devices, so that for weakly scattering samples the resolution is limited to about 1 Å by electronic and mechanical instabilities. This is more than sufficient to distinguish the individual rows of atoms, parallel to the incident beam, in the principal orientations of most crystalline phases. Thus `structure images' can be obtained, sometimes showing direct representation of projections of crystal structures [see IT C (2004[link]), Section 4.3.8[link] ]. 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[link]), Section 4.3.6[link] ]. 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 X-ray diffraction methods. A particular virtue of high-resolution 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 electron-diffraction patterns forms an essential part of the technique of structure imaging in high-resolution 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 electron-diffraction theory and techniques. The development of the nanodiffraction method in the field-emission scanning transmission electron microscope (STEM) has allowed microdiffraction patterns to be obtained from subnanometre-sized regions, and so has become the ideal tool for the structural analysis of the new microcrystalline phases important to nanoscience. The direct phasing of these coherent nanodiffraction patterns is an active field of research.

The individual specimen regions giving single-crystal electron-diffraction 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 X-ray 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[link] ). 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 valence-shell and the inner-shell electrons of the solid. The inner-shell 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 X-ray [see IT C (2004[link]), Section 4.3.4[link] ]. The development of the annular dark field (ADF) mode in STEM provides a favourable detector geometry for microanalysis, in which the forward scattered beam may be passed to an electron energy-loss spectrometer (EELS) for spectral analysis, while scattering at larger angles is collected to form a simultaneous scanning image. The arrangement is particularly efficient because, using a magnetic sector dispersive spectrometer, electrons of all energy losses may be detected simultaneously (parallel detection). Fine structure on the EELS absorption edges is analysed in a manner analogous to soft X-ray absorption spectroscopy, but with a spatial resolution of a few nanometres. The spectra are obtained from points in the corresponding ADF image which can be identified with subnanometre accuracy.

An adverse effect of the inelastic scattering processes, however, is that the transfer of energy to the specimen material results in radiation damage; this is a serious limitation of the application of electron-scattering methods to radiation-sensitive 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 radiation-resistant materials such as semiconductors and alloys.

In the historical development of electron-diffraction 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 electron-diffraction 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 X-ray rotation patterns, provided the basis for the collection of three-dimensional diffraction data on which many structure analyses have been based [see Section 2.5.4[link] and IT C (2004[link]), Section 4.3.5[link] ].

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 high-energy electron-diffraction patterns from thin single crystals for crystal structure analysis. Particularly for crystals of light-atom materials, including biological and organic compounds, the methods of structure analysis developed for X-ray diffraction, including the direct methods (see Section 2.5.8[link]), have been successfully applied in an increasing number of cases. Often it is possible to deduce some structural information from high-resolution electron-microscope images and this information may be combined with that from the diffraction intensities to assist the structure analysis process [see IT C (2004[link]), Section 4.3.8.8[link] ].

The determination of crystal symmetry by use of CBED (Section 2.5.3[link]) and the accurate determination of structure amplitudes by use of methods depending on the observation of dynamical diffraction effects [IT C (2004[link]), Section 4.3.7[link] ] came later, after the information on morphologies of crystals, and the precision electron optics associated with electron microscopes, became available. This powerful convergent-beam microdiffraction method has now been widely adopted as the preferred method for space-group determination of microphases, quasicrystals, incommensurate, twinned and other imperfectly crystalline structures. Advantage is taken of the fact that multiple scattering preserves information on the absence of inversion symmetry, while the use of an electron probe which is smaller than a mosaic block allows extinction-free structure-factor measurements to be made. Finally, an enhanced sensitivity to ionicity is obtained from electron-diffraction measurements of structure factors by the very large difference between electron scattering factors for atoms and those for ions at small angles. This section by M. Tanaka replaces the corresponding section by the late P. Goodman in previous editions[link] , which researchers may also find useful.

In spite of the problem of radiation damage, a great deal of progress has been made in the study of organic and biological materials by electron-scattering 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. Image-processing 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[link], 2.5.6[link] and 2.5.7[link]. Section 2.5.6[link], written for the first and second editions by Boris Vainshtein, has been revised and extended for this third edition by Pawel Penczek. It deals with the general theory of three-dimensional reconstruction from projections and compares several popular methods. Section 2.5.7[link] describes the application of electron-microscope imaging to the structure analysis of proteins which cannot be crystallized, and so addresses a crucial problem in structural biology. This is done by the remarkably successful method of single-particle image reconstruction, in which images of the same protein, lying in random orientations within a thin film of vitreous ice, are combined in the correct orientation to form a three-dimensional reconstructed charge-density map at nanometre or better resolution. The summation over many particles achieves the same radiation-damage-reduction effect as does crystallographic redundancy in protein crystallography. Finally, Section 2.5.8[link] describes experience with the application of numerical direct methods to the phase problem in electron diffraction. Although direct imaging `solves' the phase problem, there are many practical problems in combining electron-microdiffraction intensities with corresponding high-resolution images of a structure over a large tilt range. In cases where multiple scattering can be minimized, some success has therefore been obtained using direct phasing methods, as reviewed in this section.

For most inorganic materials the complications of many-beam 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 ultra-high-resolution images and the use of microdiffraction techniques.

2.5.2. Electron diffraction and electron microscopy1

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J. M. Cowleya

2.5.2.1. Introduction

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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 high-energy electrons (in the energy range of 104 to 106 eV) in transmission through thin samples of crystalline solids and the derivation of information on crystal structures from diffraction patterns and high-resolution 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 [(\sin \theta)/\lambda] as for X-rays, it is apparent that the angular range for strong scattering of electrons will be of the order of 10−2 rad. Only under special circumstances, usually involving multiple elastic and inelastic scattering from very thick specimens, are scattering angles of more than 10−1 rad of importance.

The strength of the interaction of electrons with matter is greater than that of X-rays by two or three orders of magnitude. The single-scattering, 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 102 Å, rather than 104 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 field-emission electron guns, it is possible to obtain a current of 10−10 A in an electron beam of diameter 10 Å or less with a beam divergence of less than 10−2 rad, i.e. a current density of 104 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 102 atoms. On the other hand, electron beams may readily be collimated to better than 10−6 rad.

The cross sections for inelastic scattering processes are, in general, less than for the elastic scattering of electrons, but signals may be obtained by the observation of electron energy losses, or the production of secondary radiations, which allow the analysis of chemical compositions or electronic excited states for regions of the crystal 100 Å or less in diameter.

On the other hand, the transfer to the sample of large amounts of energy through inelastic scattering processes produces radiation damage which may severely limit the applicability of the imaging and diffraction techniques, especially for biological and organic materials, unless the information is gathered from large specimen volumes with low incident electron beam densities.

Structure analysis of crystals can be performed using electron diffraction in the same way as with X-ray or neutron diffraction. The mathematical expressions and the procedures are much the same. However, there are peculiarities of the electron-diffraction case which should be noted.

  • (1) Structure analysis based on electron diffraction is possible for thin specimens for which the conditions for kinematical scattering are approached, e.g. for thin mosaic single-crystal specimens, for thin polycrystalline films having a preferred orientation of very small crystallites or for very extensive, very thin single crystals of biological molecules such as membranes one or a few molecules thick.

  • (2) Dynamical diffraction effects are used explicitly in the determination of crystal symmetry (with no Friedel's law limitations) and for the measurement of structure amplitudes with high accuracy.

  • (3) For many radiation-resistant materials, the structures of crystals and of some molecules may be determined directly by imaging atom positions in projections of the crystal with a resolution of 2 Å or better. The information on atom positions is not dependent on the periodicity of the crystal and so it is equally possible to determine the structures of individual crystal defects in favourable cases.

  • (4) Techniques of microanalysis may be applied to the determination of the chemical composition of regions of diameter 100 Å or less using the same instrument as for diffraction, so that the chemical information may be correlated directly with morphological and structural information.

  • (5) Crystal-structure information may be derived from regions containing as few as 102 or 103 atoms, including very small crystals and single or multiple layers of atoms on surfaces.

The material of this section is also reviewed in the text by Spence (2003[link]).

2.5.2.2. The interactions of electrons with matter

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  • (1) The elastic scattering of electrons results from the interaction of the charged electrons with the electrostatic potential distribution, [\varphi ({\bf r})], of the atoms or crystals. An incident electron of kinetic energy eW gains energy [e\varphi ({\bf r})] in the potential field. Alternatively it may be stated that an incident electron wave of wavelength [\lambda = h / mv] is diffracted by a region of variable refractive index [n ({\bf r}) = k / K_{0} = \{[W + \varphi ({\bf r})] / W\}^{1/2} \simeq 1 + \varphi ({\bf r}) / 2 W.]

  • (2) The most important inelastic scattering processes are:

    • (a) thermal diffuse scattering, with energy losses of the order of 2 × 10−2 eV, separable from the elastic scattering only with specially devised equipment; the angular distribution of thermal diffuse scattering shows variations with [(\sin \theta) / \lambda] which are much the same as for the X-ray case in the kinematical limit;

    • (b) bulk plasmon excitation, or the excitation of collective energy states of the conduction electrons, giving energy losses of 3 to 30 eV and an angular range of scattering of 10−4 to 10−3 rad;

    • (c) surface plasmons, or the excitation of collective energy states of the conduction electrons at discontinuities of the structure, with energy losses less than those for bulk plasmons and a similar angular range of scattering;

    • (d) interband or intraband excitation of valence-shell electrons giving energy losses in the range of 1 to 102 eV and an angular range of scattering of 10−4 to 10−2 rad;

    • (e) inner-shell excitations, with energy losses of 102 eV or more and an angular range of scattering of 10−3 to 10−2 rad, depending on the energy losses involved.

    • (3) In the original treatment by Bethe (1928)[link] of the elastic scattering of electrons by crystals, the Schrödinger equation is written for electrons in the periodic potential of the crystal; i.e. [\nabla^{2} \psi ({\bf r}) + K^{2}_0 [1 + \varphi ({\bf r}) / W] \psi ({\bf r}) = 0, \eqno (2.5.2.1)]where [\eqalignno{\varphi ({\bf r}) &= \textstyle\int V ({\bf u}) \exp \{-2 \pi i{\bf u} \cdot {\bf r}\} \,\hbox{d}{\bf u}\cr &= \textstyle\sum\limits_{\bf h} V_{\bf h} \exp \{-2 \pi i{\bf h} \cdot {\bf r}\}, &(2.5.2.2)}]K0 is the wavevector in zero potential (outside the crystal) (magnitude [2\pi / \lambda]) and W is the accelerating voltage. The solutions of the equation are Bloch waves of the form [\psi ({\bf r}) = \textstyle\sum\limits_{\bf h} C_{\bf h} ({\bf k}) \exp \{-i ({\bf k}_{0} + 2 \pi {\bf h}) \cdot {\bf r}\}, \eqno (2.5.2.3)]where [{\bf k}_{0}] is the incident wavevector in the crystal and h is a reciprocal-lattice vector. Substitution of (2.5.2.2)[link] and (2.5.2.3)[link] in (2.5.2.1)[link] gives the dispersion equations [(\kappa^{2} - k_{\bf h}^{2}) C_{\bf h} + \textstyle\sum\limits_{\bf g}^{}{}^{\prime}\,V_{\bf h-g} C_{\bf g} = 0. \eqno (2.5.2.4)]Here κ is the magnitude of the wavevector in a medium of constant potential [V_{0}] (the `inner potential' of the crystal). The refractive index of the electron in the average crystal potential is then [n = \kappa / K = (1 + V_{0} / W)^{1/2} \simeq 1 + V_{0} / 2 W. \eqno (2.5.2.5)]Since [V_{0}] is positive and of the order of 10 V and W is [10^{4}] to [10^{6}] V, [n - 1] is positive and of the order of [10^{-4}].

      Solution of equation (2.5.2.4)[link] gives the Fourier coefficients [C_{\bf h}^{(i)}] of the Bloch waves [\psi^{(i)} ({\bf r})] and application of the boundary conditions gives the amplitudes of individual Bloch waves (see Chapter 5.2[link] ).

    • (4) The experimentally important case of transmission of high-energy electrons through thin specimens is treated on the assumption of a plane wave incident in a direction almost perpendicular to an infinitely extended plane-parallel lamellar crystal, making use of the small-angle scattering approximation in which the forward-scattered wave is represented in the paraboloidal approximation to the sphere. The incident-beam direction, assumed to be almost parallel to the z axis, is unique and the z component of k is factored out to give [\nabla^{2} \psi + 2k \sigma \varphi \psi = \pm i2k {\partial \psi \over \partial z}, \eqno (2.5.2.6)]where [k = 2\pi / \lambda] and [\sigma = 2\pi me\lambda / h^{2}]. [See Lynch & Moodie (1972)[link], Portier & Gratias (1981)[link], Tournarie (1962)[link], and Chapter 5.2[link] .]

      This equation is analogous to the time-dependent Schrödinger equation with z replacing t. Retention of the ± signs on the right-hand side is consistent with both ψ and [\psi^{*}] being solutions, corresponding to propagation in opposite directions with respect to the z axis. The double-valued solution is of importance in consideration of reciprocity relationships which provide the basis for the description of some dynamical diffraction symmetries. (See Section 2.5.3[link].)

    • (5) The integral form of the wave equation, commonly used for scattering problems, is written, for electron scattering, as [{\psi ({\bf r}) = \psi^{(0)} ({\bf r}) + (\sigma / \lambda) \int {\exp \{- i{\bf k}| {\bf r - r}'|\} \over |{\bf r - r}'|} \varphi ({\bf r}') \psi ({\bf r}') \,\hbox{d}{\bf r}'}. \eqno (2.5.2.7)]

      The wavefunction [\psi ({\bf r})] within the integral is approximated by using successive terms of a Born series [\psi ({\bf r}) = \psi^{(0)} ({\bf r}) + \psi^{(1)} ({\bf r}) + \psi^{(2)} ({\bf r}) + \ldots .\eqno (2.5.2.8)]

      The first Born approximation is obtained by putting [\psi ({\bf r}) = \psi^{(0)} ({\bf r})] in the integral and subsequent terms [\psi^{(n)} ({\bf r})] are generated by putting [\psi^{(n - 1)} ({\bf r})] in the integral.

      For an incident plane wave, [\psi^{(0)} ({\bf r}) = \exp \{- i{\bf k}_{0} \cdot {\bf r}\}] and for a point of observation at a large distance [{\bf R} = {\bf r} - {\bf r}'] from the scattering object [(|{\bf R}| \gg |{\bf r}'|)], the first Born approximation is generated as [\psi^{(1)} ({\bf r}) = {i\sigma \over \lambda R} \exp \{- i{\bf k} \cdot {\bf R}\} \int \varphi ({\bf r}') \exp \{i{\bf q} \cdot {\bf r}'\} \,\hbox{d}{\bf r}',]where [{\bf q} = {\bf k} - {\bf k}_{0}] or, putting [{\bf u} = {\bf q}/2\pi] and collecting the pre-integral terms into a parameter μ, [\Psi ({\bf u}) = \mu \textstyle\int \varphi ({\bf r}) \exp \{2\pi i{\bf u} \cdot {\bf r}\} \,\hbox{d}{\bf r}. \eqno (2.5.2.9)]This is the Fourier-transform expression which is the basis for the kinematical scattering approximation. It is derived on the basis that all [\psi^{(n)} ({\bf r})] terms for [n \neq 0] are very much smaller than [\psi^{(0)} ({\bf r})] and so is a weak scattering approximation.

      In this approximation, the scattered amplitude for an atom is related to the atomic structure amplitude, [f({\bf u})], by the relationship, derived from (2.5.2.8)[link], [\eqalignno{\psi ({\bf r}) &= \exp \{- i{\bf k}_{0} \cdot {\bf r}\} + i {\exp \{- i{\bf k} \cdot {\bf r}\} \over R\lambda} \sigma f({\bf u}),&\cr f({\bf u}) &= \textstyle\int \varphi ({\bf r}) \exp \{2\pi i{\bf u} \cdot {\bf r}\} \,\hbox{d}{\bf r}.&(2.5.2.10)\cr}]For centrosymmetrical atom potential distributions, the [f({\bf u})] are real, positive and monotonically decreasing with [|{\bf u}|]. A measure of the extent of the validity of the first Born approximation is given by the fact that the effect of adding the higher-order terms of the Born series may be represented by replacing [f({\bf u})] in (2.5.2.10)[link] by the complex quantities [f({\bf u}) = |{\bf f}| \exp \{i\eta ({\bf u})\}] and for single heavy atoms the phase factor η may vary from 0.2 for [|{\bf u}| = 0] to 4 or 5 for large [|{\bf u}|], as seen from the tables of IT C (2004[link], Section 4.3.3[link] ).

    • (6) Relativistic effects produce appreciable variations of the parameters used above for the range of electron energies considered. The relativistic values are [\eqalignno{m &= m_{0} (1 - v^{2} / c^{2})^{-1/2} = m_{0} (1 - \beta^{2})^{-1/2}, &(2.5.2.11)\cr \lambda &= h[2m_{0}|e| W (1 + |e| W / 2m_{0}c^{2})]^{-1/2} &(2.5.2.12)\cr &= \lambda_{\rm c} (1 - \beta^{2})^{1/2} / \beta, &(2.5.2.13)}%(2.5.2.13)]where [\lambda_{\rm c}] is the Compton wavelength, [\lambda_{\rm c} = h / m_{0}c = 0.0242\,\hbox{\AA}], and [\eqalignno{\sigma &= 2\pi me\lambda / h^{2} = (2\pi m_{0}e / h^{2})(\lambda_{\rm c} / \beta)\cr &= 2\pi / \{\lambda W[1 + (1 - \beta^{2})^{1/2}]\}. &(2.5.2.14)}]Values for these quantities are listed in IT C (2004[link], Section 4.3.2[link] ). The variations of λ and σ with accelerating voltage are illustrated in Fig. 2.5.2.1[link]. For high voltages, σ tends to a constant value, [2\pi m_{0}e\lambda_{\rm c} / h^{2} = e / \hbar c].

      [Figure 2.5.2.1]

      Figure 2.5.2.1 | top | pdf |

      The variation with accelerating voltage of electrons of (a) the wavelength, λ and (b) the quantity [\lambda [1 + (h^{2} / m_{0}^{2} c^{2} \lambda^{2})] = \lambda_{\rm c} / \beta] which is proportional to the interaction constant σ [equation (2.5.2.14)[link]]. The limit is the Compton wavelength [\lambda_{\rm c}] (after Fujiwara, 1961[link]).

2.5.2.3. Recommended sign conventions

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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 structure-factor expression and (2) defining the forward propagating free-space wavefunction with a positive exponent.

The second of these alternatives is the one which has been adopted in most solid-state and quantum-mechanical 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 structure-factor 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[link], 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 self-consistent way, even in those cases where, as indicated above, some of the signs appear to have no effect on one particular conclusion.

Table 2.5.2.1| top | pdf |
Standard crystallographic and alternative crystallographic sign conventions for electron diffraction

 StandardAlternative
Free-space wave [\exp [- i({\bf k} \cdot {\bf r} - \omega t)]] [\exp [+ i({\bf k} \cdot {\bf r} - \omega t)]]
Fourier transforming from real space to reciprocal space [\textstyle\int \psi ({\bf r}) \exp [+ 2\pi i({\bf u} \cdot {\bf r})]\,\hbox{d}{\bf r}] [\textstyle\int \psi ({\bf r}) \exp [- 2\pi i({\bf u} \cdot {\bf r})]\,\hbox{d}{\bf r}]
Fourier transforming from reciprocal space to real space [\psi ({\bf r}) = \textstyle\int \Psi ({\bf u}) \exp [- 2\pi i({\bf u} \cdot {\bf r})]\,\hbox{d}{\bf u}] [\textstyle\int \Psi ({\bf u}) \exp [+ 2\pi i({\bf u} \cdot {\bf r})]\,\hbox{d}{\bf u}]
Structure factors [V({\bf h}) = (1/\Omega) \textstyle\sum_{j} f_{j} ({\bf h}) \exp (+ 2\pi i{\bf h} \cdot {\bf r}_{j})] [(1/\Omega) \textstyle\sum_{j} f_{j} ({\bf h}) \exp (- 2\pi i{\bf h} \cdot {\bf r}_{j})]
Transmission function (real space) [\exp [- i\sigma \varphi (x, y) \Delta z]] [\exp [+ i\sigma \varphi (x, y) \Delta z]]
Phenomenological absorption [\sigma \varphi ({\bf r}) - i\mu ({\bf r})] [\sigma \varphi ({\bf r}) + i\mu ({\bf r})]
Propagation function P(h) (reciprocal space) within the crystal [\exp (- 2\pi i\zeta_{\bf h} \Delta z)] [\exp (+ 2\pi i\zeta_{\bf h} \Delta z)]
Iteration (reciprocal space) [\Psi_{n + 1} ({\bf h}) = [\Psi_{n} ({\bf h}) \cdot P({\bf h})] \ast Q({\bf h})]  
Unitarity test (for no absorption) [T({\bf h}) = Q({\bf h}) \ast Q^{*} (- {\bf h}) = \delta ({\bf h})]  
Propagation to the image plane-wave aberration function, where [\chi (U) = \pi \lambda \Delta fU^{2} + \textstyle{1 \over 2} \pi C_{\rm s} \lambda^{3} U^{4}], [U^{2} = u^{2} + v^{2}] and [\Delta f] is positive for overfocus [\exp [i\chi (U)]] [\exp [- i\chi (U)]]

[\sigma =] electron interaction constant [= 2\pi me\lambda/h^{2}]; [m =] (relativistic) electron mass; [\lambda =] electron wavelength; [e =] (magnitude of) electron charge; [h =] Planck's constant; [k = 2\pi/\lambda]; [\Omega =] volume of the unit cell; [{\bf u} =] continuous reciprocal-space vector, components u, v; [{\bf h} =] discrete reciprocal-space coordinate; [\varphi (x, y) =] crystal potential averaged along beam direction (positive); [\Delta z =] slice thickness; [\mu ({\bf r}) =] absorption potential [positive; typically [\leq 0.1 \sigma \varphi ({\bf r})]]; [\Delta f =] defocus (defined as negative for underfocus); [C_{\rm s} =] spherical aberration coefficient; [\zeta_{\bf h} =] excitation error relative to the incident-beam direction and defined as negative when the point h lies outside the Ewald sphere; [f_{j} ({\bf h}) =] atomic scattering factor for electrons, [f_{\rm e}], related to the atomic scattering factor for X-rays, [f_{\rm X}], by the Mott formula [f_{\rm e} = (e/\pi U^{2}) (Z - f_{\rm X})]. [Q({\bf h})=] Fourier transform of periodic slice transmission function.

2.5.2.4. Scattering of electrons by crystals; approximations

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The forward-scattering approximation to the many-beam dynamical diffraction theory outlined in Chapter 5.2[link] provides the basis for the calculation of diffraction intensities and electron-microscope image contrast for thin crystals. [See Cowley (1995)[link], Chapter 5.2[link] and IT C (2004[link]) Sections 4.3.6[link] and 4.3.8[link] .] 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.

  • (a) The kinematical approximation, derived in Section 2.5.2.2[link] from the first Born approximation, is analagous to the corresponding approximation of X-ray diffraction. It assumes that the scattering amplitudes are directly proportional to the three-dimensional Fourier transform of the potential distribution, [\varphi ({\bf r})]. [V({\bf u}) = \textstyle\int \varphi ({\bf r}) \exp \{2\pi i{\bf u} \cdot {\bf r}\} \,\hbox{d}{\bf r}, \eqno (2.5.2.15)]so that the potential distribution [\varphi ({\bf r})] takes the place of the charge-density distribution, [\rho ({\bf r})], relevant for X-ray scattering.

    The validity of the kinematical approximation as a basis for structure analysis is severely limited. For light-atom materials, such as organic compounds, it has been shown by Jap & Glaeser (1980)[link] that the thickness for which the approximation gives reasonable accuracy for zone-axis patterns from single crystals is of the order of 100 Å for 100 keV electrons and increases, approximately as [\sigma^{-1}], for higher energies. The thickness limits quoted for polycrystalline samples, having crystallite dimensions smaller than the sample thickness, are usually greater (Vainshtein, 1956[link]). For heavy-atom materials the approximation is more limited since it may fail significantly for single heavy atoms.

  • (b) The phase-object approximation (POA), or high-voltage limit, is derived from the general many-beam dynamical diffraction expression, equation (5.2.13.1), Chapter 5.2[link] , by assuming the Ewald sphere curvature to approach zero. Then the scattering by a thin sample can be expressed by multiplying the incoming wave amplitude by the transmission function [q(xy) = \exp \{-i\sigma \varphi (xy)\}, \eqno (2.5.2.16)]where [\varphi (xy) = \int \varphi ({\bf r})\,\hbox{d}z] is the projection of the potential distribution of the sample in the z direction, the direction of the incident beam. The diffraction-pattern amplitudes are then given by two-dimensional Fourier transform of (2.5.2.16)[link].

    This approximation is of particular value in relation to the electron microscopy of thin crystals. The thickness for its validity for 100 keV electrons is within the range 10 to 50 Å, depending on the accuracy and spatial resolution involved, and increases with accelerating voltage approximately as [\lambda^{-1/2}]. In computational work, it provides the starting point for the multislice method of dynamical diffraction calculations (IT C, 2004[link], Section 4.3.6.1[link] ).

  • (c) The two-beam approximation for dynamical diffraction of electrons assumes that only two beams, the incident beam and one diffracted beam (or two Bloch waves, each with two component amplitudes), exist in the crystal. This approximation has been adapted, notably by Hirsch et al. (1965)[link], for use in the electron microscopy of inorganic materials.

    It forms a convenient basis for the study of defects in crystals having small unit cells (metals, semiconductors etc.) and provides good preliminary estimates for the determination of crystal thicknesses and structure amplitudes for orientations well removed from principal axes, and for electron energies up to 200–500 keV, but it has decreasing validity, even for favourable cases, for higher energies. It has been used in the past as an `extinction correction' for powder-pattern intensities (Vainshtein, 1956[link]).

  • (d) The Bethe second approximation, proposed by Bethe (1928)[link] as a means for correcting the two-beam approximation for the effects of weakly excited beams, replaces the Fourier coefficients of potential by the `Bethe potentials' [U_{\bf h} = V_{\bf h} - 2 k_{0}\sigma \sum\limits_{\bf g} {V_{\bf g} \cdot V_{{\bf h} - {\bf g}}\over \kappa^{2} - k_{\bf g}^{2}}. \eqno (2.5.2.17)]Use of these potentials has been shown to account well for the deviations of powder-pattern intensities from the predictions of two-beam theory (Horstmann & Meyer, 1965[link]) and to predict accurately the extinctions of Kikuchi lines at particular accelerating voltages due to relativistic effects (Watanabe et al., 1968[link]), but they give incorrect results for the small-thickness limit.

2.5.2.5. Kinematical diffraction formulae

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  • (1) Comparison with X-ray diffraction. The relations of real-space and reciprocal-space functions are analogous to those for X-ray diffraction [see equations (2.5.2.2)[link], (2.5.2.10)[link] and (2.5.2.15)[link]]. For diffraction by crystals [\eqalignno{\varphi ({\bf r}) &= \sum\limits_{\bf h} V_{\bf h} \exp \{-2 \pi i{\bf h \cdot r}\},\cr V_{\bf h} &= \int \varphi ({\bf r}) \exp \{2 \varphi i{\bf h \cdot r}\}\,\hbox{d} {\bf r} &(2.5.2.18)\cr &= {1\over \Omega} \sum\limits_{i} f_{i} ({\bf h}) \exp \{2 \pi i{\bf h \cdot r}_{i}\}, &(2.5.2.19)}%(2.5.2.19)]where the integral of (2.5.2.18)[link] and the summation of (2.5.2.19)[link] are taken over one unit cell of volume (see Dawson et al., 1974[link]).

    Important differences from the X-ray case arise because

    • (a) the wavelength is relatively small so that the Ewald-sphere curvature is small in the reciprocal-space region of appreciable scattering amplitude;

    • (b) the dimensions of the single-crystal regions giving appreciable scattering amplitudes are small so that the `shape transform' regions of scattering power around the reciprocal-lattice points are relatively large;

    • (c) the spread of wavelengths is small (10−5 or less, with no white-radiation background) and the degree of collimation is better (10−4 to 10−6) than for conventional X-ray sources.

      As a consequence of these factors, single-crystal diffraction patterns may show many simultaneous reflections, representing almost-planar sections of reciprocal space, and may show fine structure or intensity variations reflecting the crystal dimensions and shape.

  • (2) Kinematical diffraction-pattern intensities are calculated in a manner analogous to that for X-rays except that

    • (a) no polarization factor is included because of the small-angle scattering conditions;

    • (b) integration over regions of scattering power around reciprocal-lattice points cannot be assumed unless appropriate experimental conditions are ensured.

      For a thin, flat, lamellar crystal of thickness H, the observed intensity is [I_{\bf h}/I_{{\bf 0}} = |\sigma (V_{\bf h}/\Omega) (\sin \pi \zeta_{\bf h} H)/(\pi \zeta_{\bf h})|^{2}, \eqno (2.5.2.20)]where [\zeta_{\bf h}] is the excitation error for the h reflection and Ω is the unit-cell volume.

      For a single-crystal diffraction pattern obtained by rotating a crystal or from a uniformly bent crystal or for a mosaic crystal with a uniform distribution of orientations, the intensity is [I_{\bf h} = I_{{\bf 0}} {\sigma^{2} |V_{\bf h}|^{2} V_{\rm c} d_{\bf h} \over 4\pi^{2} \Omega^{2}}, \eqno (2.5.2.21)]where [V_{\rm c}] is the crystal volume and [d_{\bf h}] is the lattice-plane spacing. For a polycrystalline sample of randomly oriented small crystals, the intensity per unit length of the diffraction ring is [I_{\bf h} = I_{{\bf 0}} {\sigma^{2} |V_{\bf h}|^{2} V_{\rm c} d_{\bf h}^{2} M_{\bf h} \over 8\pi^{2} \Omega^{2} L\lambda}, \eqno (2.5.2.22)]where [M_{\bf h}] is the multiplicity factor for the h reflection and L is the camera length, or the distance from the specimen to the detector plane. The special cases of `oblique texture' patterns from powder patterns having preferred orientations are treated in IT C (2004[link], Section 4.3.5[link] ).

    • (3) Two-beam dynamical diffraction formulae: complex potentials including absorption. In the two-beam dynamical diffraction approximation, the intensities of the directly transmitted and diffracted beams for transmission through a crystal of thickness H, in the absence of absorption, are [\eqalignno{I_{\bf0} &= (1 + w^{2})^{-1} \left[w^{2} + \cos^{2} \left\{{\pi H (1 + w^{2})^{1/2} \over \xi_{\bf h}}\right\}\right] &(2.5.2.23)\cr I_{\bf h} &= (1 + w^{2})^{-1} \sin^{2} \left\{{\pi H (1 + w^{2})^{1/2} \over \xi_{\bf h}}\right\}, &(2.5.2.24)}%(2.5.2.24)]where [\xi_{\bf h}] is the extinction distance, [\xi_{\bf h} = (2\sigma |V_{\bf h}|)^{-1}], and [w = \xi_{\bf h} \zeta_{\bf h} = \Delta \theta / (2\sigma |V_{\bf h}| d_{\bf h}), \eqno (2.5.2.25)]where [\Delta \theta] is the deviation from the Bragg angle.

      For the case that [\zeta_{\bf h} = 0], with the incident beam at the Bragg angle, this reduces to the simple Pendellösung expression [I_{\bf h} = 1 - I_{\bf 0} = \sin^{2} \{2 \pi \sigma |V_{\bf h}|H\}. \eqno (2.5.2.26)]

      The effects on the elastic Bragg scattering amplitudes of the inelastic or diffuse scattering may be introduced by adding an out-of-phase component to the structure amplitudes, so that for a centrosymmetric crystal, [V_{\bf h}] becomes complex by addition of an imaginary component. Alternatively, an absorption function [\mu({\bf r})], having Fourier coefficients [\mu_{\bf h}], may be postulated so that [\sigma V_{\bf h}] is replaced by [\sigma V_{\bf h} + i \mu_{\bf h}]. The [\mu_{\bf h}] are known as phenomenological absorption coefficients and their validity in many-beam diffraction has been demonstrated by, for example, Rez (1978)[link].

      The magnitudes [\mu_{\bf h}] depend on the nature of the experiment and the extent to which the various inelastically or diffusely scattered electrons are included in the measurements being made. If measurements are made of purely elastic scattering intensities for Bragg reflections or of image intensity variations due to the interaction of the sharp Bragg reflections only, the main contributions to the absorption coefficients are as follows (Radi, 1970[link]):

      • (a) from plasmon and single-electron excitations, [\mu_{0}] is of the order of 0.1 V0 and [\mu_{\bf h}], for [{\bf h} \neq 0], is negligibly small;

      • (b) from thermal diffuse scattering; [\mu_{\bf h}] is of the order of 0.1 Vh and decreasing more slowly than [V_{\bf h}] with scattering angle.

        Including absorption effects in (2.5.2.26)[link] for the case [\zeta_{\bf h} = 0] gives [\eqalign{I_{\bf 0} &= {\textstyle{1 \over 2}} \exp \{ - \mu_{0}H\} [\cosh \mu_{\bf h}H + \cos (2 \pi \sigma V_{\bf h}H)],\cr I_{\bf h}&= {\textstyle{1 \over 2}} \exp \{ - \mu_{0}H\} [\cosh \mu_{\bf h}H - \cos (2 \pi \sigma V_{\bf h}H)].\cr} \eqno (2.5.2.27)]The Borrmann effect is not very pronounced for electrons because [\mu_{\bf h} \ll \mu_{\bf 0}], but can be important for the imaging of defects in thick crystals (Hirsch et al., 1965[link]; Hashimoto et al., 1961[link]).

        Attempts to obtain analytical solutions for the dynamical diffraction equations for more than two beams have met with few successes. There are some situations of high symmetry, with incident beams in exact zone-axis orientations, for which the many-beam solution can closely approach equivalent two- or three-beam behaviour (Fukuhara, 1966[link]). Explicit solutions for the three-beam case, which displays some aspects of many-beam character, have been obtained (Gjønnes & Høier, 1971[link]; Hurley & Moodie, 1980[link]).

2.5.2.6. Imaging with electrons

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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[f_{\rm p}^{-1} = {e \over 8mW_{\rm r}} \int\limits_{-\infty}^{\infty} H_{z}^{2}(z) \,\hbox{d}z, \eqno (2.5.2.28)]where [W_{\rm r}] is the relativistically corrected accelerating voltage and [H_{z}] is the z component of the magnetic field. An expression in terms of experimental constants was given by Liebman (1955)[link] as [{1 \over f} = {A_{0}(NI)^{2} \over W_{\rm r}(S + D)}, \eqno (2.5.2.29)]where [A_{0}] 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 third-order spherical aberration, coefficient [C_{\rm s}], giving a variation of focal length of [C_{\rm s}\alpha^{2}] for a beam at an angle α to the axis. Chromatic aberration, coefficient [C_{\rm c}], gives a spread of focal lengths [\Delta f = C_{\rm c} \left({\Delta W_{0} \over W_{0}} + 2{\Delta I \over I}\right) \eqno (2.5.2.30)]for variations [\Delta W_{0}] and [\Delta I] 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 [\psi_{0} (xy)] (see Fig. 2.5.2.2[link]).

[Figure 2.5.2.2]

Figure 2.5.2.2 | top | pdf |

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 [\Psi_{0} (u, v) \cdot T (u, v), \eqno (2.5.2.31)]where [\Psi_{0} (u, v)] is the Fourier transform of [\psi_{0} (x, y)] and T(u, v) is the transfer function of the lens, consisting of an aperture function [A (u, v) = \cases{1 &${\rm for }\,\,(u^{2} + v^{2})^{1/2} \leq A$\cr 0 &elsewhere\cr} \eqno (2.5.2.32)]and a phase function exp [\{i\chi (u, v)\}] where the phase perturbation [\chi (uv)] due to lens defocus [\Delta f] and aberrations is usually approximated as [\chi (uv) = \pi \cdot \Delta f \cdot \lambda (u^{2} + v^{2}) + {\pi\over 2}C_{\rm s} \lambda^{3} (u^{2} + v^{2})^{2}, \eqno (2.5.2.33)]and u, v are the reciprocal-space variables related to the scattering angles [\varphi_{x}], [\varphi_{y}] by [\eqalign{u &= (\sin \varphi_{x})/\lambda,\cr v &= (\sin \varphi_{y})/\lambda.\cr}]

The image amplitude distribution, referred to the object coordinates, is given by Fourier transform of (2.5.2.31)[link] as [\psi (xy) = \psi_{0} (xy) \ast t (xy), \eqno (2.5.2.34)]where [t (xy)], given by Fourier transform of [{T (u, v)}], is the spread function. The image intensity is then [I (xy) = |\psi (xy)|^{2} = |\psi_{0} (xy) \ast t (xy)|^{2}. \eqno (2.5.2.35)]

In practice the coherent imaging theory provides a good approximation but limitations of the coherence of the illumination have appreciable effects under high-resolution imaging conditions.

The variation of focal lengths according to (2.5.2.30)[link] is described by a function [G (\Delta f)]. Illumination from a finite incoherent source gives a distribution of incident-beam angles [H (u_{1}, v_{1})]. Then the image intensity is found by integrating incoherently over [\Delta f] and [u_{1}, v_{1}]:[ \eqalignno{I (xy) &= \textstyle\int \textstyle\int G (\Delta f) \cdot H (u_{1} v_{1})\cr &\quad\times |{\scr F} \{\Psi_{0} (u - u_{1}, v - v_{1}) \cdot T_{\Delta f} (u, v)\}|^{2} \,\hbox{d} (\Delta f) \cdot \,\hbox{d} u_{1} \,\hbox{d}v_{1},\cr& &(2.5.2.36)}]where [ {\scr F}] denotes the Fourier-transform 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[link]). 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 [ {\scr F} [T (u, v)] = t (xy). \eqno (2.5.2.37)]If this is translated by the scan to X, Y, the transmitted wave is [\psi_{0} (xy) = q (xy) \cdot t (x - X, y - Y). \eqno (2.5.2.38)]

The amplitude on the plane of observation following the specimen is then [\Psi (uv) = Q (u, v) \ast \{T (uv) \exp [2 \pi i(uX + vY)]\}, \eqno (2.5.2.39)]and the image signal produced by a detector having a sensitivity function H(u, v) is [\eqalignno{I (X, Y) &= \textstyle\int H (u, v) |Q (u, v) \ast T (u, v)&\cr &\quad\times \exp \{2 \pi i (uX + vY)\}|^{2} \,\hbox{d}u \,\hbox{d}v. &(2.5.2.40)}]If H(u, v) represents a small detector, approximated by a delta function, this becomes [I (x, y) = |q (xy) \ast t (xy)|^{2}, \eqno (2.5.2.41)]which is identical to the result (2.5.2.35)[link] for a plane incident wave in the conventional transmission electron microscope.

2.5.2.7. Imaging of very thin and weakly scattering objects

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  • (a) The weak-phase-object approximation. For sufficiently thin objects, the effect of the object on the incident-beam amplitude may be represented by the transmission function (2.5.2.16)[link] given by the phase-object approximation. If the fluctuations, [\varphi (xy) - \bar{\varphi}], about the mean value of the projected potential are sufficiently small so that [\sigma [\varphi (xy) - \bar{\varphi}] \ll 1], it is possible to use the weak-phase-object approximation (WPOA) [q (xy) = \exp \{- i\sigma \varphi (xy)\} = 1 - i\sigma \varphi (xy), \eqno (2.5.2.42)]where [\varphi (xy)] is referred to the average value, [\bar{\varphi}]. The assumption that only first-order terms in [\sigma \varphi (xy)] need be considered is the equivalent of a single-scattering, or kinematical, approximation applied to the two-dimensional function, the projected potential of (2.5.2.16)[link]. From (2.5.2.42)[link], the image intensity (2.5.2.35)[link] becomes [I (xy) = 1 + 2\sigma \varphi (xy) \ast s (xy), \eqno (2.5.2.43)]where the spread function s(xy) is the Fourier transform of the imaginary part of T(uv), namely [A(uv)\sin\chi (uv)].

    The optimum imaging condition is then found, following Scherzer (1949)[link], by specifying that the defocus should be such that [|\sin \chi|] is close to unity for as large a range of [U = (u^{2} + v^{2})^{1/2}] as possible. This is so for a negative defocus such that [\chi (uv)] decreases to a minimum of about [-2 \pi /3] before increasing to zero and higher as a result of the fourth-order term of (2.5.2.33)[link] (see Fig. 2.5.2.3[link]). This optimum, `Scherzer defocus' value is given by [{\hbox{d}\chi \over \hbox{d}u} = 0 \quad \hbox{for} \quad \chi = -2 \pi /3]or [\Delta f = -\left(\textstyle{4 \over 3} C_{\rm s} \lambda \right)^{1/2}. \eqno (2.5.2.44)]

    [Figure 2.5.2.3]

    Figure 2.5.2.3 | top | pdf |

    The functions [\chi (U)], the phase factor for the transfer function of a lens given by equation (2.5.2.33)[link], and [\sin \chi (U)] for the Scherzer optimum defocus condition, relevant for weak phase objects, for which the minimum value of [\chi (U)] is [-2\pi /3].

    The resolution limit is then taken as corresponding to the value of [U = 1.51 C_{\rm s}^{-1/4} \lambda^{-3/4}] when [\sin\chi] becomes zero, before it begins to oscillate rapidly with U. The resolution limit is then [\Delta x = 0.66 C_{\rm s}^{1/4} \lambda^{3/4}. \eqno (2.5.2.45)]For example, for [C_{\rm s} = 1] mm and [\lambda = 2.51 \times 10^{-2}] Å (200 keV), [\Delta x = 2.34] Å.

    Within the limits of the WPOA, the image intensity can be written simply for a number of other imaging modes in terms of the Fourier transforms [c ({\bf r})] and [s ({\bf r})] of the real and imaginary parts of the objective-lens transfer function [T ({\bf u}) =] [ A ({\bf u}) \exp \{i \chi ({\bf u})\}], where r and u are two-dimensional vectors in real and reciprocal space, respectively.

    For dark-field TEM images, obtained by introducing a central stop to block out the central beam in the diffraction pattern in the back-focal plane of the objective lens, [I ({\bf r}) = [\sigma \varphi ({\bf r}) \ast c ({\bf r})]^{2} + [\sigma \varphi ({\bf r}) \ast s ({\bf r})]^{2}. \eqno (2.5.2.46)]Here, as in (2.5.2.42)[link], [\varphi ({\bf r})] should be taken to imply the difference from the mean potential value, [\varphi ({\bf r}) - \bar{\varphi}].

    For bright-field STEM imaging with a very small detector placed axially in the central beam of the diffraction pattern (2.5.2.39)[link] on the detector plane, the intensity, from (2.5.2.41)[link], is given by (2.5.2.43)[link].

    For a finite axially symmetric detector, described by [D ({\bf u})], the image intensity is [{I ({\bf r}) = 1 + 2 \sigma \varphi ({\bf r}) \ast \{s ({\bf r}) [d ({\bf r}) * c ({\bf r})] - c ({\bf r}) [d ({\bf r}) \ast s ({\bf r})]\}}, \eqno (2.5.2.47)]where [d ({\bf r})] is the Fourier transform of [D ({\bf u})] (Cowley & Au, 1978[link]).

    For STEM with an annular dark-field detector which collects all electrons scattered outside the central spot of the diffraction pattern in the detector plane, it can be shown that, to a good approximation (valid except near the resolution limit) [I ({\bf r}) = \sigma^{2} \varphi^{2} ({\bf r}) \ast [c^{2} ({\bf r}) + s^{2} ({\bf r})]. \eqno (2.5.2.48)]Since [c^{2} ({\bf r}) + s^{2} ({\bf r}) = |t ({\bf r})|^{2}] is the intensity distribution of the electron probe incident on the specimen, (2.5.2.48)[link] is equivalent to the incoherent imaging of the function [\sigma^{2} \varphi^{2} ({\bf r})].

    Within the range of validity of the WPOA or, in general, whenever the zero beam of the diffraction pattern is very much stronger than any diffracted beam, the general expression (2.5.2.36)[link] for the modifications of image intensities due to limited coherence may be conveniently approximated. The effect of integrating over the variables [\Delta f, u_{1}, v_{1}], may be represented by multiplying the transfer function T(u, v) by so-called `envelope functions' which involve the Fourier transforms of the functions [G (\Delta f)] and [H (u_{1}, v_{1})].

    For example, if [G(\Delta f)] is approximated by a Gaussian of width [epsilon] (at [e^{-1}] of the maximum) centred at [\Delta f_{0}] and [H(u_{1} v_{1})] is a circular aperture function [H(u_{1} v_{1}) = \cases{1 &${\rm if}\,\,u_{1}, v_{1}\,\lt\, b$\cr 0 &otherwise,}]the transfer function [T_{0}(uv)] for coherent radiation is multiplied by [\exp \{-\pi^{2} \lambda^{2} \varepsilon^{2} (u^{2} + v^{2})^{2}/4\} \cdot J_{1} (\pi B \eta)/(\pi B \eta)]where [\eqalignno{\eta &= f_{0}\lambda (u + v) +C_{\rm s}\lambda^{3} (u^{3} + v^{3})&\cr &\quad + \pi i \varepsilon^{2} \lambda^{2} (u^{3} + u^{2} v + uv^{2} + v^{3})/2. &(2.5.2.49)}]

  • (b) The projected charge-density approximation. For very thin specimens composed of moderately heavy atoms, the WPOA is inadequate. Within the region of validity of the phase-object approximation (POA), more complicated relations analagous to (2.5.2.43)[link] to (2.5.2.47)[link] may be written. A simpler expression may be obtained by use of the two-dimensional form of Poisson's equation, relating the projected potential distribution [\varphi (xy)] to the projected charge-density distribution [\rho (xy)]. This is the PCDA (projected charge-density approximation) (Cowley & Moodie, 1960[link]), [I (xy) = 1 + 2\Delta f \cdot \lambda \sigma \rho (xy). \eqno (2.5.2.50)]

    This is valid for sufficiently small values of the defocus [\Delta f\!], provided that the effects of the spherical aberration may be neglected, i.e. for image resolutions not too close to the Scherzer resolution limit (Lynch et al., 1975[link]). The function [\rho (xy)] includes contributions from both the positive atomic nuclei and the negative electron clouds. For underfocus ([\Delta f] negative), single atoms give dark spots in the image. The contrast reverses with defocus.

2.5.2.8. Crystal structure imaging

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  • (a) Introduction. It follows from (2.5.2.43)[link] and (2.5.2.42)[link] that, within the severe limitations of validity of the WPOA or the PCDA, images of very thin crystals, viewed with the incident beam parallel to a principal axis, will show dark spots at the positions of rows of atoms parallel to the incident beam. Provided that the resolution limit is less than the projected distances between atom rows (1–3 Å), the projection of the crystal structure may be seen directly.

    In practice, theoretical and experimental results suggest that images may give a direct, although nonlinear, representation of the projected potential or charge-density distribution for thicknesses much greater than the thicknesses for validity of these approximations, e.g. for thicknesses which may be 50 to 100 Å for 100 keV electrons for 3 Å resolutions and which increase for comparable resolutions at higher voltage but decrease with improved resolutions.

    The use of high-resolution imaging as a means for determining the structures of crystals and for investigating the form of the defects in crystals in terms of the arrangement of the atoms has become a widely used and important branch of crystallography with applications in many areas of solid-state science. It must be emphasized, however, that image intensities are strongly dependent on the crystal thickness and orientation and also on the instrumental parameters (defocus, aberrations, alignment etc.). It is only when all of these parameters are correctly adjusted to lie within strictly defined limits that interpretation of images in terms of atom positions can be attempted [see IT C (2004[link], Section 4.3.8[link] )].

    Reliable interpretations of high-resolution images of crystals (`crystal structure images') may be made, under even the most favourable circumstances, only by the comparison of observed image intensities with intensities calculated by use of an adequate approximation to many-beam dynamical diffraction theory [see IT C (2004[link], Section 4.3.6[link] )]. Most calculations for moderate or large unit cells are currently made by the multislice method based on formulation of the dynamical diffraction theory due to Cowley & Moodie (1957)[link]. For smaller unit cells, the matrix method based on the Bethe (1928)[link] formulation is also frequently used (Hirsch et al., 1965[link]).

  • (b) Fourier images. For periodic objects in general, and crystals in particular, the amplitudes of the diffracted waves in the back focal plane are given from (2.5.2.31)[link] by [\Psi_{0} ({\bf h}) \cdot T ({\bf h}). \eqno (2.5.2.51)]For rectangular unit cells of the projected unit cell, the vector h has components [h/a] and [k/b]. Then the set of amplitudes (2.5.2.34)[link], and hence the image intensities, will be identical for two different sets of defocus and spherical aberration values [\Delta f_{1}, C_{\rm s1}] and [\Delta f_{2}, C_{\rm s2}] if, for an integer N, [\chi_{1} (h) = \chi_{2} (h) = 2 N\pi\semi]i.e. [{\pi \lambda \left({h^{2} \over a^{2}} + {k^{2} \over b^{2}}\right) (\Delta f_{1} - \Delta f_{2}) + {1 \over 2} \pi \lambda^{3} \left({h^{2} \over a^{2}} + {k^{2} \over b^{2}}\right)^{2} (C_{\rm s1} - C_{\rm s2}) = 2\pi N.}]

    This relationship is satisfied for all h, k if [a^{2}/b^{2}] is an integer and [\Delta f_{1} - \Delta f_{2} = 2na^{2}/\lambda]and [C_{\rm s1} - C_{\rm s2} = 4ma^{4}/\lambda^{3}, \eqno (2.5.2.52)]where m, n are integers (Kuwabara, 1978[link]). The relationship for [\Delta f] is an expression of the Fourier image phenomenon, namely that for a plane-wave incidence, the intensity distribution for the image of a periodic object repeats periodically with defocus (Cowley & Moodie, 1960[link]). Hence it is often necessary to define the defocus value by observation of a nonperiodic component of the specimen such as a crystal edge (Spence et al., 1977[link]).

    For the special case [a^{2} = b^{2}], the image intensity is also reproduced exactly for [\Delta f_{1} - \Delta f_{2} = (2n + 1) a^{2}/\lambda, \eqno (2.5.2.53)]except that in this case the image is translated by a distance [a/2] parallel to each of the axes.

2.5.2.9. Image resolution

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  • (1) Definition and measurement. The `resolution' of an electron microscope or, more correctly, the `least resolvable distance', is usually defined by reference to the transfer function for the coherent imaging of a weak phase object under the Scherzer optimum defocus condition (2.5.2.44)[link]. The resolution figure is taken as the inverse of the U value for which [\sin\chi (U)] first crosses the axis and is given, as in (2.5.2.45)[link], by [\Delta x = 0.66C_{\rm s}^{1/4} \lambda^{3/4}. \eqno (2.5.2.45)]

    It is assumed that an objective aperture is used to eliminate the contribution to the image for U values greater than the first zero crossing, since for these contributions the relative phases are distorted by the rapid oscillations of [\sin \chi (U)] and the corresponding detail of the image is not directly interpretable in terms of the projection of the potential distribution of the object.

    The resolution of the microscope in practice may be limited by the incoherent factors which have the effect of multiplying the WPOA transfer function by envelope functions as in (2.5.2.49)[link].

    The resolution, as defined above, and the effects of the envelope functions may be determined by Fourier transform of the image of a suitable thin, weakly scattering amorphous specimen. The Fourier-transform operation may be carried out by use of an optical diffractometer. A more satisfactory practice is to digitize the image directly by use of a two-dimensional detector system in the microscope or from a photographic recording, and perform the Fourier transform numerically.

    For the optical diffractometer method, the intensity distribution obtained is given from (2.5.2.43)[link] as a radially symmetric function of U, [ \eqalignno{I (U) &= |{\scr F} I (xy)|^{2}&\cr &= \delta (U) + 4\sigma^{2} |\Phi (u)|^{2} \cdot \sin^{2} \chi (U) \cdot E^{2} (U), &(2.5.2.54)}]where [E(U)] is the product of the envelope functions.

    In deriving (2.5.2.54)[link] it has been assumed that:

    • (a) the WPOA applies;

    • (b) the optical transmission function of the photographic record is linearly related to the image intensity, [I(xy)];

    • (c) the diffraction intensity [|\Phi (U)|^{2}] is a radially symmetric, smoothly varying function such as is normally produced by a sufficiently large area of the image of an amorphous material;

    • (d) there is no astigmatism present and no drift of the specimen; either of these factors would remove the radial symmetry.

      From the form of (2.5.2.54)[link] and a preknowledge of [|\Phi (U)|^{2}], the zero crossings of [\sin\chi] and the form of [E(U)] may be deduced. Analysis of a through-focus series of images provides more complete and reliable information.

  • (2) Detail on a scale much smaller than the resolution of the electron microscope, as defined above, is commonly seen in electron micrographs, especially for crystalline samples. For example, lattice fringes, having the periodicity of the crystal lattice planes, with spacings as small as 0.6 Å in one direction, have been observed using a microscope having a resolution of about 2.5 Å (Matsuda et al., 1978[link]), and two-dimensionally periodic images showing detail on the scale of 0.5 to 1 Å have been observed with a similar microscope (Hashimoto et al., 1977[link]).

    Such observations are possible because

    • (a) for periodic objects the diffraction amplitude [\Psi_{0} (uv)] in (2.5.2.31)[link] is a set of delta functions which may be multiplied by the corresponding values of the transfer function that will allow strong interference effects between the diffracted beams and the zero beam, or between different diffracted beams;

    • (b) the envelope functions for the WPOA, arising from incoherent imaging effects, do not apply for strongly scattering crystals; the more general expression (2.5.2.36)[link] provides that the incoherent imaging factors will have much less effect on the interference of some sets of diffracted beams.

      The observation of finely spaced lattice fringes provides a measure of some important factors affecting the microscope performance, such as the presence of mechanical vibrations, electrical interference or thermal drift of the specimen. A measure of the fineness of the detail observable in this type of image may therefore be taken as a measure of `instrumental resolution'.

2.5.2.10. Electron diffraction in electron microscopes

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

  • (a) Selected-area electron-diffraction (SAED) patterns are obtained by using intermediate and projector lenses to form an image of the diffraction pattern in the back-focal plane of the objective lens (Fig. 2.5.2.2).[link] The area of the specimen from which the diffraction pattern is obtained is defined by inserting an aperture in the image plane of the objective lens. For parallel illumination of the specimen, sharp diffraction spots are produced by perfect crystals.

    A limitation to the area of the specimen from which the diffraction pattern can be obtained is imposed by the spherical aberration of the objective lens. For a diffracted beam scattered through an angle α, the spread of positions in the object for which the diffracted beam passes through a small axial aperture in the image plane is [C_{\rm s} \alpha^{3}], e.g. for [C_{\rm s} = 1] mm, [\alpha = 5 \times 10^{-2}] rad (10,0,0 reflection from gold for 100 keV electrons), [C_{\rm s} \alpha^{3} = 1250] Å, so that a selected-area diameter of less than about 2000 Å is not feasible. For higher voltages, the minimum selected-area diameter decreases with [\lambda^{2}] if the usual assumption is made that [C_{\rm s}] increases for higher-voltage microscopes so that [C_{\rm s}\lambda] is a constant.

  • (b) Convergent-beam electron-diffraction (CBED) patterns are obtained when an incident convergent beam is focused on the specimen, as in an STEM instrument or an STEM attachment for a conventional TEM instrument.

    For a large, effectively incoherent source, such as a conventional hot-filament electron gun, the intensities are added for each incident-beam direction. The resulting CBED pattern has an intensity distribution[I (uv) = \textstyle\int |\Psi_{u_{1} v_{1}} (uv)|^{2} \,\hbox{d}u_{1} \,\hbox{d}v_{1}, \eqno (2.5.2.55)]where [\Psi_{u_{1} v_{1}} (uv)] is the Fourier transform of the exit wave at the specimen for an incident-beam direction [u_{1}, v_{1}].

  • (c) Coherent illumination from a small bright source such as a field emission gun may be focused on the specimen to give an electron probe having an intensity distribution [|t(xy)|^{2}] and a diameter equal to the STEM dark-field image resolution [equation (2.5.2.47)[link]] of a few Å. The intensity distribution of the resulting microdiffraction pattern is then [|\Psi (uv)|^{2} = |\Psi_{0}(uv) \ast T (uv)|^{2}, \eqno (2.5.2.56)]where [\Psi_{0}(uv)] is the Fourier transform of the exit wave at the specimen. Interference occurs between waves scattered from the various incident-beam directions. The diffraction pattern is thus an in-line hologram as envisaged by Gabor (1949)[link].

  • (d) Diffraction patterns may be obtained by using an optical diffractometer (or computer) to produce the Fourier transform squared of a small selected region of a recorded image. The optical diffraction-pattern intensity obtained under the ideal conditions specified under equation (2.5.2.54)[link] is given, in the case of weak phase objects, by [I (uv) = \delta (uv) + 4\sigma^{2}| \Phi (uv)|^{2} \cdot \sin^{2} \chi (uv) \cdot E^{2} (uv) \eqno (2.5.2.57)]or, more generally, by [I(uv) = c\delta (uv) + |\Psi (uv) \cdot T(uv) \ast \Psi^{*}(uv) \cdot T^{*}(uv)|^{2},]where [\Psi (uv)] is the Fourier transform of the wavefunction at the exit face of the specimen and c is a constant depending on the characteristics of the photographic recording medium.

2.5.3. Point-group and space-group determination by convergent-beam electron diffraction

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M. Tanakaf

2.5.3.1. Introduction

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Because the cross section for electron scattering is at least a thousand times greater than that for X-rays, and because multiple Bragg scattering preserves information on symmetry (such as the absence of inversion symmetry), electron diffraction is exquisitely sensitive to symmetry. The additional ability of modern electron-optical lenses to focus an electron probe down to nanometre dimensions, and so allow the study of nanocrystals too small for analysis by X-rays, has meant that the method of convergent-beam diffraction described here has now become the preferred method of symmetry determination for very small crystals, domains, twinned structures, quasicrystals, incommensurate structures and other imperfectly crystalline materials.

Convergent-beam electron diffraction (CBED) originated with the experiments of Kossel & Möllenstedt (1938[link]). However, modern crystallographic investigations by CBED began with the studies performed by Goodman & Lehmpfuhl (1965[link]) in a modified transmission electron microscope. They obtained CBED patterns by converging a conical electron beam with an angle of more than 10−3 rad on an ~30 nm diameter specimen area, which had uniform thickness and no bending. Instead of the usual diffraction spots, diffraction discs (in Laue or transmission geometry) were produced. The diffraction intensity within a disc shows a specific symmetry, which enables one to determine the point groups and space groups of microcrystals. Unlike X-ray diffraction, the method is extremely sensitive to the presence or absence of inversion symmetry.

The method corresponding to CBED in the field of light optics is the conoscope method. Using a conoscope, we can identify whether a crystal is isotropic, uniaxial or biaxial, and determine the optic axis and the sign of birefringence of a crystal. When CBED, a conoscope method using an electron beam, is utilized, more basic properties of a crystal – the crystal point group and space group – can be determined.

Point- and space-group determinations are routinely also carried out by X-ray diffraction. This method, to which kinematical diffraction is applicable, cannot determine whether a crystal is polar or nonpolar unless anomalous absorption is utilized. As a result, the X-ray diffraction method can only identify 11 Laue groups among 32 point groups. CBED, based fully upon dynamical diffraction, can distinguish polar crystals from nonpolar crystals using only a nanometre-sized crystal, thus allowing the unique identification of all the point groups by inspecting the symmetries appearing in CBED discs.

As pointed out above, an unambiguous experimental determination of crystal symmetry, in the case of X-ray diffraction, is usually not possible because of the apparent centrosymmetry of the diffraction pattern, even for noncentrosymmetric crystals. However, methods based on structure-factor and X-ray intensity statistics remain useful for the resolution of space-group ambiguities, and are routinely applied to structure determinations from X-ray data. These methods are described in Chapter 2.1[link] of this volume.

In the field of materials science, correct space-group determination by CBED is often requested prior to X-ray or neutron structure refinement, in particular in the case of Rietveld refinements based on powder diffraction data.

CBED can determine not only the point and space groups of crystals but also crystal structure parameters – lattice parameters, atom positions, Debye–Waller factors and low-order structure factors. The lattice parameters can be determined from sub-micron regions of thin crystals by using higher-order Laue zone (HOLZ) reflections with an accuracy of 1 × 10−4. Cherns et al. (1988[link]) were the first to perform strain analysis of artificial multilayer materials using the large-angle technique (LACBED) (Tanaka et al., 1980[link]). Since then, many strain measurements at interfaces of various multilayer materials have been successfully conducted. In recent years, strain analysis has been conducted using automatic analysis programs, which take account of dynamical diffraction effects (Krämer et al., 2000[link]). We refer to the book of Morniroli (2002[link]), which carries many helpful figures, clear photographs and a comprehensive list of papers on this topic.

Vincent et al. (1984a[link],b[link]) first applied the CBED method to the determination of the atom positions of AuGeAs. They analysed the intensities of HOLZ reflections by applying a quasi-kinematical approximation. Tanaka & Tsuda (1990[link], 1991[link]) and Tsuda & Tanaka (1995[link]) refined the structural parameters of SrTiO3 by applying the dynamical theory of electron diffraction. The method was extended to the refinements of CdS, LaCrO3 and hexagonal BaTiO3 (Tsuda & Tanaka, 1999[link]; Tsuda et al., 2002[link]; Ogata et al., 2004[link]). Rossouw et al. (1996[link]) measured the order parameters of TiAl through a Bloch-wave analysis of HOLZ reflections in a CBED pattern. Midgley et al. (1996[link]) refined two positional parameters of AuSn4 from the diffraction data obtained with a small convergence angle using multislice calculations.

Low-order structure factors were first determined by Goodman & Lehmpfuhl (1967[link]) for MgO. After much work on low-order structure-factor determination, Zuo & Spence determined the 200 and 400 structure factors of MgO in a very modern way, by fitting energy-filtered patterns and many-beam dynamical calculations using a least-squares procedure. For the low-order structure-factor determinations, the excellent com­pre­hensive review of Spence (1993[link]) should be referred to. Saunders et al. (1995[link]) succeeded in obtaining the deformation charge density of Si using the low-order crystal structure factors determined by CBED. For the reliable determination of the low-order X-ray crystal structure factors or the charge density of a crystal, accurate determination of the Debye–Waller factors is indispensable. Zuo et al. (1999[link]) determined the bond-charge distribution in cuprite. Simultaneous determination of the Debye–Waller factors and the low-order structure factors using HOLZ and zeroth-order Laue zone (ZOLZ) reflections was performed to determine the deformation charge density of LaCrO3 accurately (Tsuda et al., 2002[link]).

CBED can also be applied to the determination of lattice defects, dislocations (Cherns & Preston, 1986[link]), stacking faults (Tanaka, 1986[link]) and twins (Tanaka, 1986[link]). Since this topic is beyond the scope of the present chapter, readers are referred to pages 156 to 205 of the book by Tanaka et al. (1994[link]).

We also mention the book by Spence & Zuo (1992[link]), which deals with the whole topic of CBED, including the basic theory and a wealth of literature.

2.5.3.2. Point-group determination

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When an electron beam traverses a thin slab of crystal parallel to a zone axis, one can easily imagine that symmetries parallel to the zone axis should appear in the resulting CBED pattern. It is, however, more difficult to imagine what symmetries appear due to symmetries perpendicular to the incident beam. Goodman (1975[link]) pioneered the clarification of CBED symmetries for the twofold rotation axis and mirror plane perpendicular to the incident beam, and the symmetry of an inversion centre, with the help of the reciprocity theorem of scattering theory. Tinnappel (1975[link]) solved many CBED symmetries at various crystal settings with respect to the incident beam using a group-theoretical treatment. Buxton et al. (1976[link]) also derived these results from first principles, and generalized them to produce a systematic method for the determination of the crystal point group. Tanaka, Saito & Sekii (1983[link]) developed a method to determine the point group using simultaneously excited many-beam patterns. The point-group-determination method given by Buxton et al. (1976[link]) is described with the aid of the description by Tanaka, Saito & Sekii (1983[link]) in the following.

2.5.3.2.1. Symmetry elements of a specimen and diffraction groups

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Since CBED uses the Laue geometry, Buxton et al. (1976[link]) assumed a perfectly crystalline specimen in the form of a parallel-sided slab which is infinite in two dimensions. The symmetry elements of the specimen (as distinct from those of an infinite crystal) form `diffraction groups', which are isomorphic to the point groups of the diperiodic plane figures and Shubnikov groups of coloured plane figures. The diffraction groups of a specimen are determined from the symmetries of CBED patterns taken at various orientations of the specimen. The crystal point-group of the specimen is identified by referring to Fig. 2.5.3.4[link], which gives the relation between diffraction groups and crystal point groups.

A specimen that is parallel-sided and is infinitely extended in the x and y directions has ten symmetry elements. The symmetry elements consist of six two-dimensional symmetry elements and four three-dimensional ones. The operation of the former elements transforms an arbitrary coordinate (x, y, z) into (x′, y′, z), with z remaining the same. The operation of the latter transforms a coordinate (x, y, z) into (x′, y′, z′), where [z'\ne z]. A vertical mirror plane m and one-, two-, three-, four- and sixfold rotation axes that are parallel to the surface normal z are the two-dimensional symmetry elements. A horizontal mirror plane m′, an inversion centre i, a horizontal twofold rotation axis 2′ and a fourfold rotary inversion [\bar 4] are the three-dimensional symmetry elements, and are shown in Fig. 2.5.3.1[link]. The fourfold rotary inversion was not recognized as a symmetry element until the point groups of the diperiodic plane figures were considered (Buxton et al., 1976[link]). Table 2.5.3.1[link] lists these symmetry elements, where the symbols in parentheses express symmetries of CBED patterns expected from three-dimensional symmetry elements.

Table 2.5.3.1| top | pdf |
Two- and three-dimensional symmetry elements of an infinitely extended parallel-sided specimen

Symbols in parentheses show CBED symmetries appearing in dark-field patterns.

Two-dimensional symmetry elementsThree-dimensional symmetry elements
1 m′ (1R)
2 i (2R)
3 2′ (m2, mR)
4 [\bar 4] (4R)
5  
6  
m  
[Figure 2.5.3.1]

Figure 2.5.3.1 | top | pdf |

Four symmetry elements m′, i, 2′ and [\bar 4] of an infinitely extended parallel-sided specimen.

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

Table 2.5.3.2| top | pdf |
Symmetry elements of an infinitely extended parallel-sided specimen and diffraction groups

 12346m2m(m)3m4m(m)6m(m) 
1 1 2 3 4 6 m 2m(m) 3m 4m(m) 6m(m) 10
(m′) 1R 1R 21R 31R 41R 61R m1R 2m(m)1R 3m1R 4m(m)1R 6m(m)1R 10
(i) 2R 2R [21R] 6R [41R] [61R] 2Rm(mR) [2m(m)1R] 6Rm(mR) [4m(m)1R] [6m(m)1R] 4
            [2Rm(mR)] [2m(m)1R] [3m1R]      
(2′) mR mR 2mR(mR) 3mR 4mR(mR) 6mR(mR) [m1R] [4R(m)mR] [6Rm(mR)] [4m(m)1R] [6Rm(mR)] 5
[(\bar 4)] 4R   4R   [41R]   4Rm(mR) [4Rm(mR)]   [4m(m)1R]   2

1R × 2R = 2, 2R × 2R = 1, mR × 2R = m, 4R × 2R = 4, 1R × mR = m × mR, 1R × 4R = 4 × 1R, mR × 4R = m × 4R.

2.5.3.2.2. Identification of three-dimensional symmetry elements

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It is difficult to imagine the symmetries in CBED patterns generated by the three-dimensional symmetry elements of the sample. The reason is that if a three-dimensional symmetry element is applied to a specimen, it turns it upside down, which is impractical in most experiments. The reciprocity theorem of scattering theory (Pogany & Turner, 1968[link]) enables us to clarify the symmetries of CBED patterns expected from these three-dimensional symmetry elements. A graphical method for obtaining CBED symmetries due to sample symmetry elements is described in the papers of Goodman (1975[link]), Buxton et al. (1976[link]) and Tanaka (1989[link]). The CBED symmetries of the three-dimensional symmetries do not appear in the zone-axis patterns, but do in a diffraction disc set at the Bragg condition, each of which we call a dark-field pattern (DP). The CBED symmetries obtained are illustrated in Fig. 2.5.3.2[link]. A horizontal twofold rotation axis 2′, a horizontal mirror plane m′, an inversion centre i and a fourfold rotary inversion [\bar 4] produce symmetries mR (m2), 1R, 2R and 4R in DPs, respectively.

[Figure 2.5.3.2]

Figure 2.5.3.2 | top | pdf |

Illustration of symmetries appearing in dark-field patterns (DPs). (a) mR and m2; (b) 1R; (c) 2R; (d) 4R, originating from 2′, m′, i and [\bar 4], respectively.

Next we explain the symbols of the CBED symmetries. (1) Operation mR is shown in the left-hand part of Fig. 2.5.3.2[link](a), which implies successive operations of (a) a mirror m with respect to a twofold rotation axis, transforming an open circle beam (○) in reflection G into a beam (+) in reflection G′ and (b) rotation R of this beam by π about the centre point of disc G′ (or the exact Bragg position of reflection G′), resulting in position ○ in reflection G′. The combination of the two operations is written as mR. When the twofold rotation axis is parallel to the diffraction vector G, two beams (○) in the left-hand part of the figure become one reflection G, and a mirror symmetry, whose mirror line is perpendicular to vector G and passes through the centre of disc G, appears between the two beams (the right-hand side figure of Fig. 2.5.3.2[link]a). The mirror symmetry is labelled m2 after the twofold rotation axis. (2) Operation 1R (Fig. 2.5.3.2[link]b) for a horizontal mirror plane is a combination of a rotation by 2π of a beam (○) about a zone axis O (symbol 1), which is equivalent to no rotation, and a rotation by π of the beam about the exact Bragg position or the centre of disc G. (3) Operation 2R is a rotation by π of a beam (○) in reflection G about a zone axis (symbol 2), which transforms the beam into a beam (+) in reflection −G, followed by a rotation by π of the beam (+) about the centre of disc −G, resulting in the beam (○) in disc −G (Fig. 2.5.3.2[link]c). The symmetry is called translational symmetry after Goodman (1975[link]) because the pattern of disc +G coincides with that of disc −G by a translation. It is emphasized that an inversion centre is identified by the test of translational symmetry about a pair of ±G dark-field patterns – if one disc can be translated into coincidence with the other, an inversion centre exists. We call the pair ±DP. (4) Operation 4R (Fig. 2.5.3.2[link]d) can be understood in a similar manner. It is noted that regular letters are symmetries about a zone axis, while subscripts R represent symmetries about the exact Bragg position. We call a pattern that contains an exact Bragg position (if possible at the disc centre) a dark-field pattern. As far as CBED symmetries are concerned, we do not use the term dark-field pattern if a disc does not contain the exact Bragg position.

The four three-dimensional symmetry elements are found to produce different symmetries in the DPs. These facts imply that these symmetry elements can be identified unambiguously from the symmetries of CBED patterns.

2.5.3.2.3. Identification of two-dimensional symmetry elements

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Two-dimensional symmetry elements that belong to a zone axis exhibit their symmetries in CBED patterns or zone-axis patterns (ZAPs) directly, even if dynamical diffraction takes place. A ZAP contains a bright-field pattern (BP) and a whole pattern (WP). The BP is the pattern appearing in the bright-field disc [the central or `direct' (000) beam]. The WP is composed of the BP and the pattern formed by the surrounding diffraction discs, which are not exactly excited. The two-dimensional symmetry elements m, 1, 2, 3, 4 and 6 yield symmetry mv and one-, two-, three-, four- and sixfold rotation symmetries, respectively, in WPs, where the suffix v for mv is assigned to distinguish it from mirror symmetry m2 caused by a horizontal twofold rotation axis.

It should be noted that a BP shows not only the zone-axis symmetry but also three-dimensional symmetries, indicating that the BP can have a higher symmetry than the symmetry of the corresponding WP. The symmetries of the BP due to three-dimensional symmetry elements are obtained by moving the DPs to the zone axis. As a result, the three-dimensional symmetry elements m′, i, 2′ and [\bar 4] produce, respectively, symmetries 1R, 1, m2 and 4 in the BP, instead of 1R, 2R, m2 and 4R in the DPs (Fig. 2.5.3.2[link]). We mention that the BP cannot distinguish whether a specimen crystal has an inversion centre or not, because an inversion centre forms the lowest symmetry 1 in the BP.

In conclusion, all the two-dimensional symmetry elements can be identified from the WP symmetries.

2.5.3.2.4. Diffraction-group determination

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All the symmetry elements of the diffraction groups can be identified from the symmetries of a WP and DPs. But it is practical and convenient to use just the four patterns WP, BP, DP and ±DP to determine the diffraction group. The symmetries appearing in these four patterns are given for the 31 diffraction groups in Table 2.5.3.3[link] (Tanaka, Saito & Sekii, 1983[link]), which is a detailed version of Table 2 of Buxton et al. (1976[link]). All the possible symmetries of the DP and ±DP appearing at different crystal orientations are given in the present table. When a BP has a higher symmetry than the corresponding WP, the symmetry elements that produce the BP are given in parentheses in column II except only for the case of 4R. When two types of vertical mirror planes exist, these are distinguished by symbols mv and mv. Each of the two or three symmetries given in columns IV and V for many diffraction groups appears in a DP or ±DP in different directions.

Table 2.5.3.3| top | pdf |
Symmetries of different patterns for diffraction and projection diffraction groups

(II) Bright-field patterns (BPs); (III) whole patterns (WPs); (IV) dark-field patterns (DPs); and (V) ±dark-field patterns (±DPs) for diffraction groups (I) and projection diffraction groups (VI).

IIIIIIIVVVI
1 1 1 1 1 1R
1R 2 1 2 = 1R 1
(1R)
2 2 2 1 2 21R
2R 1 1 1 2R
21R 2 2 2 21R
mR m 1 1 1 m1R
(m2) mR
m2 1
m mv mv 1 1
mv
mv 1
m1R 2mm mv 2 1
[mv + m2 + (1R)] mv1R
2mvm2 1
2mRmR 2mm 2 1 2 2mm1R
(2 + m2) m2 2mR(m2)
2mm 2mvmv 2mvmv 1 2
mv 2mv(mv)
2RmmR mv mv 1 2R
m2 2Rmv(m2)
mv 2RmR(mv)
2mm1R 2mvmv 2mvmv 2 21R
2mvm2 21Rmv(mv)
4 4 4 1 2 41R
4R 4 2 1 2
41R 4 4 2 21R
4mRmR 4mm 4 1 2 4mm1R
(4 + m2) m2 2mR(m2)
4mm 4mvmv 4mvmv 1 2
mv 2mv(mv)
4RmmR 4mm 2mvmv 1 2
(2mvmv + m2) m2 2mR(m2)
mv 2mv(mv)
4mm1R 4mvmv 4mvmv 2 21R
2mvm2 21Rmv(mv)
3 3 3 1 1 31R
31R 6 3 2 1
(3 + 1R)
3mR 3m 3 1 1 3m1R
(3 + m2) mR
m2 1
3m 3mv 3mv 1 1
mv
mv 1
3m1R 6mm 3mv 2 1
[3mv + m2 + (1R)] mv1R
2mvm2 1
6 6 6 1 2 61R
6R 3 3 1 2R
61R 6 6 2 21R
6mRmR 6mm 6 1 2 6mm1R
(6 + m2) m2 2mR(m2)
6mm 6mvmv 6mvmv 1 2
mv 2mv(mv)
6RmmR 3mv 3mv 1 2R
m2 2Rmv(m2)
mv 2RmR(mv)
6mm1R 6mvmv 6mvmv 2 21R
2mvm2 21Rmv(mv)

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

[Figure 2.5.3.3]

Figure 2.5.3.3 | top | pdf |

Illustration of symmetries appearing in dark-field patterns (DPs) and a pair of dark-field patterns (±DP) for the combinations of symmetry elements.

When the four patterns appearing in three photographs are taken and examined using Table 2.5.3.3[link] with the aid of Fig. 2.5.3.3[link], one diffraction group can be selected unambiguously. It is, however, noted that many diffraction groups are determined from a WP and BP pair without using a DP or ±DP (or from one photograph) or from a set of a WP, a BP and a DP without using a ±DP (or from two photographs).

2.5.3.2.5. Point-group determination

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Fig. 2.5.3.4[link] provides the relationship between the 31 diffraction groups for slabs and the 32 point groups for infinite crystals given by Buxton et al. (1976[link]). When a diffraction group is determined, possible point groups are selected by consulting this figure. Each of the 11 high-symmetry diffraction groups corresponds to only one crystal point group. In this case, the point group is uniquely determined from the diffraction group. When more than one point group falls under a diffraction group, a different diffraction group has to be obtained for another zone axis. A point group is identified by finding a common point group among the point groups obtained for different zone axes. It is clear from the figure that high-symmetry zones should be used for quick determination of point groups because low-symmetry zone axes exhibit only a small portion of crystal symmetries in the CBED patterns. Furthermore, it should be noted that CBED cannot observe crystal symmetries oblique to an incident beam or horizontal three-, four- or sixfold rotation axes. The diffraction groups to be expected for different zone axes are given for all the point groups in Table 2.5.3.4[link] (Buxton et al., 1976[link]). The table is useful for finding a suitable zone axis to distinguish candidate point groups expected in advance.

Table 2.5.3.4| top | pdf |
Diffraction groups expected at various crystal orientations for 32 point groups

This table is adapted from Buxton et al. (1976[link]).

Point groupZone-axis symmetries
<111><100><110><uv0><uuw>[uvw]
m3m 6RmmR 4mm1R 2mm1R 2RmmR 2RmmR 2R
[\bar 4 3 m] 3m 4RmmR m1R mR m 1
432 3mR 4mRmR 2mRmR mR mR 1

Point groupZone-axis symmetries
<111><100><uv0>[uvw]
m3 6R 2mm1R 2RmmR 2R
23 3 2mRmR mR 1

Point groupZone-axis symmetries
[0001][\langle 11\bar 2 0\rangle][\langle 1\bar 1 00\rangle][uv.0][uu.w][[u\bar u.w]][uv.w]
6/mmm 6mm1R 2mm1R 2mm1R 2RmmR 2Rmm 2RmmR 2R
[\bar 6 m 2] 3m1R m1R 2mm m mR m 1
6mm 6mm m1R m1R mR m m 1
622 6mRmR 2mRmR 2mRmR mR mR mR 1

Point groupZone-axis symmetries
[0001][uv.0][uv.w]
6/m 61R 2RmmR 2R
[\bar 6] 31R m 1
6 6 mR 1

Point groupZone-axis symmetries
[0001][\langle 11\bar 2 0\rangle][[u\bar u.w]][uv.w]
[\bar 3 m] 6RmmR 21R 2RmmR 2R
3m 3m 1R m 1
32 3mR 2 mR 1

Point groupZone-axis symmetries
[0001][uv.w]
[\bar 3] 6R 2R
3 3 1

Point groupZone-axis symmetries
[001]<100><110>[u0w][uv0][uuw][uvw]
4/mmm 4mm1R 2mm1R 2mm1R 2RmmR 2RmmR 2RmmR 2R
[\bar 4 2 m] 4RmmR 2mRmR m1R mR mR m 1
4mm 4mm m1R m1R m mR m 1
422 4mRmR 2mRmR 2mRmR mR mR mR 1

Point groupZone-axis symmetries
[001][uv0][uvw]
4/m 41R 2RmmR 2R
[\bar 4] 4R mR 1
4 4 mR 1

Point groupZone-axis symmetries
[001]<100>[u0w][uv0][uvw]
mmm 2mm1R 2mm1R 2RmmR 2RmmR 2R
mm2 2mm m1R m mR 1
222 2mRmR 2mRmR mR mR 1

Point groupZone-axis symmetries
[010][u0w][uvw]
2/m 21R 2RmmR 2R
m 1R m 1
2 2 mR 1

Point groupZone-axis symmetry
[uvw]
[\bar 1] 2R
1 1
[Figure 2.5.3.4]

Figure 2.5.3.4 | top | pdf |

Relation between diffraction groups and crystal point groups (after Buxton et al., 1976[link]).

We shall explain the point-group determination procedure using an Si crystal. Fig. 2.5.3.5[link](a) shows a [111] ZAP of the Si specimen. The BP has threefold rotation symmetry and mirror symmetry or symmetry 3mv, which are caused by the threefold rotation axis along the [111] direction and a vertical mirror plane. The WP has the same symmetry. Figs. 2.5.3.5[link](b) and (c) are [2\bar{2}0] and [\bar{2}20] DPs, respectively. Both show symmetry m2 perpendicular to the reflection vector. This symmetry is caused by a twofold rotation axis parallel to the specimen surface. One DP coincides with the other upon translation. This translational or 2R symmetry indicates the existence of an inversion centre. By consulting Table 2.5.3.3[link], the diffraction group giving rise to these pattern symmetries is found to be 6RmmR. Fig. 2.5.3.4[link] shows that there are two point groups [\bar{3}m] and [m\bar{3}m] causing diffraction group 6RmmR. Fig. 2.5.3.6[link] shows another ZAP, which shows symmetry 4mm in the BP and the WP. The point group which has fourfold rotation symmetry is not [\bar{3}m] but [m\bar{3}m]. The point group of Si is thus determined to be [m\bar{3}m].

[Figure 2.5.3.5]

Figure 2.5.3.5 | top | pdf |

CBED patterns of Si taken with the [111] incidence. (a) BP and WP show symmetry 3mv. (b) and (c) DPs show symmetry m2 and DP symmetry 2Rmv.

[Figure 2.5.3.6]

Figure 2.5.3.6 | top | pdf |

CBED pattern of Si taken with the [100] incidence. The BP and WP show symmetry 4mm.

2.5.3.2.6. Projection diffraction groups

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HOLZ reflections appear as excess HOLZ rings far outside the ZOLZ reflection discs and as deficit lines in the ZOLZ discs. By ignoring these weak diffraction effects with components along the beam direction, we may obtain information about the symmetry of the sample as projected along the beam direction. Thus when HOLZ reflections are weak and no deficit HOLZ lines are seen in the ZOLZ discs, the symmetry elements found from the CBED patterns are only those of the specimen projected along the zone axis. The projection of the specimen along the zone axis causes horizontal mirror symmetry m′, the corresponding CBED symmetry being 1R. When symmetry 1R is added to the 31 diffraction groups, ten projection diffraction groups having symmetry symbol 1R are derived as shown in column VI of Table 2.5.3.3[link]. If only ZOLZ reflections are observed in CBED patterns, a projection diffraction group instead of a diffraction group is obtained, where only the pattern symmetries given in the rows of the diffraction groups having symmetry symbol 1R in Table 2.5.3.3[link] should be consulted. Two projection diffraction groups obtained from two different zone axes are the minimum needed to determine a crystal point group, because it is constructed by the three-dimensional combination of symmetry elements. It should be noted that if a diffraction group is determined carelessly from CBED patterns with no HOLZ lines, the wrong crystal point group is obtained.

2.5.3.2.7. Symmetrical many-beam method

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In the sections above, the point-group determination method established by Buxton et al. (1976[link]) was described, where two- and three-dimensional symmetry elements were determined, respectively, from ZAPs and DPs.

The Laue circle is defined as the intersection of the Ewald sphere with the ZOLZ, and all reflections on this circle are simultaneously at the Bragg condition. If many such DPs are recorded (all simultaneously at the Bragg condition), many three-dimensional symmetry elements can be identified from one photograph. Using a group-theoretical method, Tinnappel (1975[link]) studied the symmetries appearing in simultaneously excited DPs for various combinations of crystal symmetry elements. Based upon his treatment, Tanaka, Saito & Sekii (1983[link]) developed a method for determining diffraction groups using simultaneously excited symmetrical hexagonal six-beam, square four-beam and rectangular four-beam CBED patterns. All the CBED symmetries appearing in the symmetrical many-beam (SMB) patterns were derived by the graphical method used in the paper of Buxton et al. (1976[link]). From an experimental viewpoint, it is advantageous that symmetry elements can be identified from one photograph. It was found that twenty diffraction groups can be identified from one SMB pattern, whereas ten diffraction groups can be determined by Buxton et al.'s method. An experimental comparison between the two methods was performed by Howe et al. (1986[link]).

SMB patterns are easily obtained by tilting a specimen crystal or the incident beam from a zone axis into an orientation to excite low-order reflections simultaneously. Fig. 2.5.3.7[link] illustrates the symmetries of the SMB patterns for all the diffraction groups except for the five groups 1, 1R, 2, 2R and 21R. For these groups, the two-beam method for exciting one reflection is satisfactory because many-beam excitation gives no more information than the two-beam case. In the six-beam and square four-beam cases, the CBED symmetries for the two crystal (or incident-beam) settings which excite respectively the +G and −G reflections are drawn because the vertical rotation axes create the SMB patterns at different incident-beam orientations. [This had already been experienced for the case of symmetry 2R (Goodman, 1975[link]; Buxton et al., 1976[link]).] In the rectangular four-beam case, the symmetries for four settings which excite the +G, +H, −G and −H reflections are shown. For the diffraction groups 3m, 3mR, 3m1R and 6RmmR, two different patterns are shown for the two crystal settings, which differ by π/6 rad from each other about the zone axis. Similarly, for the diffraction group 4RmmR, two different patterns are shown for the two crystal settings, which differ by π/4 rad. Illustrations of these different symmetries are given in Fig. 2.5.3.7[link]. The combination of the vertical threefold axis and a horizontal mirror plane introduces a new CBED symmetry 3R. Similarly, the combination of the vertical sixfold rotation axis and an inversion centre introduces a new CBED symmetry 6R.

[Figure 2.5.3.7]
[Figure 2.5.3.7]
[Figure 2.5.3.7]

Figure 2.5.3.7 | top | pdf |

Illustration of symmetries appearing in hexagonal six-beam, square four-beam and rectangular four-beam dark-field patterns expected for all the diffraction groups except for 1, 1R, 2, 2R and 21R.

There is an empirical and conventional technique for reproducing the symmetries of the SMB patterns which uses three operations of two-dimensional rotations, a vertical mirror at the centre of disc O and a rotation of π about the centre of a disc (1R) without involving the reciprocal process. For example, we may consider 3R between discs F and F′ in Table 2.5.3.5[link] in the case of diffraction group 31R. Disc F′ is rotated anticlockwise not about the zone axis but about the centre of disc O by 2π/3 rad (symbol 3) to coincide with disc F, and followed by a rotation of π rad (symbol R) about the centre of disc F′, resulting in the correct symmetry seen in Fig. 2.5.3.7[link]. When the symmetries appearing between different SMB patterns are considered, this technique assumes that the symmetry operations are conducted after discs O and [\bar{O}] are superposed. Another assumption is that the vertical mirror plane perpendicular to the line connecting discs O and [\bar{O}] acts at the centre of disc O when the symmetries between two SMB patterns are considered. As an example, symmetry 3R between discs S and [\bar{S}] appearing in the two SMB patterns is reproduced by a threefold anticlockwise rotation of disc S about the centre of disc O (or [\bar{O}]) and followed by a rotation of π rad (R) about the centre of disc [\bar{S}].

Tables 2.5.3.5[link], 2.5.3.6[link] and 2.5.3.7[link] express the symmetries illustrated in Fig. 2.5.3.7[link] with the symmetry symbols for the hexagonal six-beam case, square four-beam case and rectangular four-beam case, respectively. In the fourth rows of the tables the symmetries of zone-axis patterns (BP and WP) are listed because combined use of the zone-axis pattern and the SMB pattern is efficient for symmetry determination. In the fifth row, the symmetries of the SMB pattern are listed. In the following rows, the symmetries appearing between the two SMB patterns are listed because the SMB symmetries appear not only in an SMB pattern but also in the pairs of SMB patterns. That is, for each diffraction group, all the possible SMB symmetries appearing in a pair of symmetric six-beam patterns, two pairs AB and AC of the square four-beam patterns and three pairs AB, AC and AD of the rectangular four-beam patterns are listed, though such pairs are not always needed for the determination of the diffraction groups. It is noted that the symmetries in parentheses are the symmetries which add no new symmetries, even if they are present. In the last row, the point groups which cause the diffraction groups listed in the first row are given.

Table 2.5.3.5| top | pdf |
Symmetries of hexagonal six-beam CBED patterns for diffraction groups

 Projection diffraction group
31R3m1R61R
Diffraction group 3 31R 3mR 3m 3m1R 6 6R 61R
Two-dimensional symmetry 3 3 3 3m 3m 6 3 6
Three-dimensional symmetry   m 2′   m′, (2′)   i m′, (i)
Zone-axis pattern Bright-field pattern 3 6 3m 3m 6mm 6 3 6
Whole-field pattern 3 3 3 3m 3m 6 3 6
Hexagonal six-beam pattern O 1 1 1 m2 1 mv m2 mv 1 1 1
G 1 1R m2 1 1 mv 1R 1Rmv(m2) 1 1 1R
F 1 1 m2 1 1 1 1 m2 1 1 1
S 1 1 1 m2 1 1 m2 1 1 1 1
FF 1 3R 1 1 1 mv 3R 3Rmv 1 1 3R
SS 1 1 1 1 1 mv 1 mv 1 6R 6R
A pair of symmetrical six-beam patterns [Scheme scheme13] ±O 1 1R m2 1 mv 1 mv1R 1Rm2 2 1 2(1R)
±G 1 1 1 mR mv 1 mvmR 1 2 2R 21R
±F 1 1 1 1 mv 1 mv 1 1 6R 6R
±S 1 3R 1 1 mv 1 3Rmv 3R 1 1 3R
[F'\bar F] 1 1 1 mR 1 1 mR 1 2 1 2
[S'\bar S] 1 1 mR 1 1 1 1 mR 2 1 2
Point group 23, 3 [\bar 6] 432, 32 [\bar 43m], 3m [\bar 6m2] 6 m3, 3 6/m

 Projection diffraction group
6mm1R
Diffraction group 6mRmR 6mm 6RmmR 6mm1R
Two-dimensional symmetry 6 6mm 3m 6mm
Three-dimensional symmetry 2′   i, (2′) m′, (i, 2′)
Zone-axis pattern Bright-field pattern 6mm 6mm 3m 6mm
Whole-field pattern 6 6mm 3m 6mm
Hexagonal six-beam pattern O m2 mv 1 mv(m2) mv(m2)
G m2 mv m2 mv 1Rmv(m2)
F m2 1 m2 1 m2
S m2 1 1 m2 m2
FF 1 mv 1 mv 3Rmv
SS 1 mv 6R 6Rmv 6Rmv
A pair of symmetrical six-beam patterns [Scheme scheme113] ±O 2m2 2mv′ mv(m2) 1 2(1R)mv(m2)
±G 2mR 2mv′ 2Rmv 2RmR 21Rmv′(mR)
±F 1 mv 6Rmv 6R 6Rmv′
±S 1 mv′ mv 1 3Rmv′
[F'\bar F] 2mR 2 1 mR 2mR
[S'\bar S] 2mR 2 mR 1 2mR
Point group 622 6mm m3m, [\bar 3 m] 6/mmm

Table 2.5.3.6| top | pdf |
Symmetries of square four-beam CBED patterns for diffraction groups

 Projection diffraction group
41R4mm1R
Diffraction group 4 4R 41R 4mRmR 4mm 4RmmR 4mm1R
Two-dimensional symmetry 4 (2) 4 4 4mm (2mm) 4mm
Three-dimensional symmetry   [\bar 4] m′, (i, [\bar 4]) 2′   [\bar 4], 2′ m′, (i, 2′, [\bar 4])
Zone-axis pattern Bright-field pattern 4 4 4 4mm 4mm 4mm 4mm
Whole-field pattern 4 2 4 4 4mm 2mm 4mm
Square four-beam pattern   O 1 1 1 m2 mv m2 mv mv(m2)
G 1 1 1R m2 mv m2 mv 1Rmv(m2)
F 1 1 1 m2 1 1 m2 m2
FF 1 4R 4R 1 mv 4R 4Rmv 4Rmv
Two pairs of square four-beam patterns [Scheme scheme14] AB ±O 2 2 2(1R) 2m2 2mv′ 2m2 2mv′ 2(1R)mv(m2)
±G 2 2 21R 2mR 2mv′ 2mR 2mv′ 21Rmv(mR)
FF 2 2 2 2mR 2 2 2mR 2mR
±F 1 4R 4R 1 mv′ 4R 4Rmv′ 4Rmv
AC OO 4 4 4 4m2 4mv′′ 4mv 4m2 4mv′′(m2)
GG 4 4R 41R 4mR 4mv′′ 4Rmv 4RmR 41Rmv′′(mR)
FS 4 1 4 4mR 4 mR 1 41Rmv′′(mR)
FS 1 1 1R 1 mv′′ mv 1 1Rmv′′
Point group 4 [\bar 4] 4/m 432, 422 4mm [\bar 4 3 m], [\bar 4 2 m] m3m, 4/mmm

Table 2.5.3.7| top | pdf |
Symmetries of rectangular four-beam CBED patterns for diffraction groups

 Projection diffraction group
m1R2mm1R
Diffraction group mR m m1R 2mRmR 2mm 2RmmR 2mm1R
Two-dimensional symmetry   m m 2 2mm m 2mm
Three-dimensional symmetry 2′   m′, 2′ 2′   2′, i m′, 2′, i
Zone-axis pattern Bright-field pattern m m 2mm 2mm 2mm m 2mm
Whole-field pattern 1 m m 2 2mm m 2mm
Rectangular four-beam pattern   O 1 1 1 1 1 1 1
G 1 1 1R 1 1 1 1R
F m2 1 m2 m2 1 m2 m2
S 1 1 1 m2 1 1 m2
Three pairs of rectangular four-beam patterns [Scheme scheme15] AB [O_GO_{\bar H}] m2 1 m2 m2 mv mv(m2) mv(m2)
[G\bar H] 1 1 1 mR mv mv mvmR
[F \bar F] 1 1 1 1 mv 2Rmv 2Rmv
SS 1 1 1R 1 mv mv mv1R
AC OGOH 1 mv mv m2 mv′ 1 mv(m2)
GH mR mv mvmR mR mv′ mR mv′mR
FF 1 mv mv1R 1 mv′ 1 mv′1R
[S\bar S] 1 mv mv 1 mv′ 2R 2Rmv′
AD [O_GO_{\bar G}] 1 1 1R 2 2 1 2(1R)
GG 1 1 1 2 2 2R 21R
[F\bar F'] 1 1 1 2mR 2 1 2mR
[S\bar S'] mR 1 mR 2mR 2 mR 2mR
Point group 2, 222, mm2, 4, [\bar 4], 422, 4mm, [\bar 4 2 m], 32, 6, 622, 6mm, [\bar 6m2], 23, 432, [\bar 43m] m, mm2, 4mm, [\bar 42m], 3m, [\bar 6], 6mm, [\bar 6 m 2], [\bar 4 3 m] mm2, 4mm, 42m, 6mm, [\bar 6 m 2], [\bar 4 3 m] 222, 422, [\bar 42m], 622, 23, 432 mm2, [\bar 6 m2] 2/m, mmm, 4/m, 4/mmm, [\bar 3 m], [\bar 6/m], 6/mmm, m3, m3m mmm, 4/mmm, m3, m3m, 6/mmm

By referring to Tables 2.5.3.5[link], 2.5.3.6[link] and 2.5.3.7[link], the characteristic features of the SMB method are seen to be as follows. CBED symmetry m2 due to a horizontal twofold rotation axis can appear in every disc of an SMB pattern. Symmetry 1R due to a horizontal mirror plane, however, appears only in disc G or H of an SMB pattern. In the hexagonal six-beam case, an inversion centre i produces CBED symmetry 6R between discs S and S′ due to the combination of an inversion centre and a vertical threefold rotation axis (and/or of a horizontal mirror plane and a vertical sixfold rotation axis). This indicates that one hexagonal six-beam pattern can reveal whether a specimen has an inversion centre or not, while the method of Buxton et al. (1976[link]) requires two photographs for the inversion test. All the diffraction groups in Table 2.5.3.5[link] can be identified from one six-beam pattern except groups 3 and 6. Diffraction groups 3 and 6 cannot be distinguished from the hexagonal six-beam pattern because it is insensitive to the vertical axis. In the square four-beam case, fourfold rotary inversion [\bar{4}] produces CBED symmetry 4R between discs F and F′ in one SMB pattern, while Buxton et al.'s method requires four photographs to identify fourfold rotary inversion. Although an inversion centre itself does not exhibit any symmetry in the square four-beam pattern, it causes symmetry 1R due to the horizontal mirror plane produced by the combination of an inversion centre and the twofold rotation axis. Thus, symmetry 1R is an indication of the existence of an inversion centre in the square four-beam case. All of the seven diffraction groups in Table 2.5.3.6[link] can be identified from one square four-beam pattern. One rectangular four-beam pattern can distinguish all the diffraction groups in Table 2.5.3.7[link] except the groups m and 2mm. It is emphasized again that the inversion test can be carried out using one six-beam pattern or one square four-beam pattern.

Fig. 2.5.3.8[link] shows CBED patterns taken from a [111] pyrite (FeS2) plate with an accelerating voltage of 100 kV. The space group of FeS2 is [P2_1/a\bar{3}]. The diffraction group of the plate is 6R due to a threefold rotation axis and an inversion centre. The zone-axis pattern of Fig. 2.5.3.8[link](a) shows threefold rotation symmetry in the BP and WP. The hexagonal six-beam pattern of Fig. 2.5.3.8[link](b) shows no symmetry higher than 1 in discs O, G, F and S but shows symmetry 6R between discs S and S′, which proves the existence of a threefold rotation axis and an inversion centre. The same symmetries are also seen in Fig. 2.5.3.8[link](c), where reflections [\bar{O}], [\bar{G}], [\bar{F}], [\bar{S}], [\bar{F}'] and [\bar{S}'] are excited. Table 2.5.3.5[link] indicates that diffraction group 6R can be identified from only one hexagonal six-beam pattern, because no other diffraction groups give rise to the same symmetries in the six discs. When Buxton et al.'s method is used, three photographs or four patterns are necessary to identify diffraction group 6R (see Table 2.5.3.3[link]). In addition, if the symmetries between Figs. 2.5.3.8[link](b) and (c) are examined, symmetry 2R between discs G and [\bar{G}] and symmetry 6R between discs F and [\bar{F}] are found. All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7[link] and Table 2.5.3.5[link].

[Figure 2.5.3.8]

Figure 2.5.3.8 | top | pdf |

CBED patterns of FeS2 taken with the [111] incidence. (a) Zone-axis pattern, (b) hexagonal six-beam pattern with excitation of reflection +G, (c) hexagonal six-beam pattern with excitation of reflection −G. Symmetry 6R is noted between discs S and S′ and discs [\bar F] and [\bar F'].

Fig. 2.5.3.9[link] shows CBED patterns taken from a [110] V3Si plate with an accelerating voltage of 80 kV. The space group of V3Si is Pm3n. The diffraction group of the plate is 2mm1R due to two vertical mirror planes and a horizontal mirror plane, a twofold rotation axis being produced at the intersection line of two perpendicular mirror planes. The zone-axis pattern of Fig. 2.5.3.9[link](a) shows symmetry 2mm in the BP and WP. The rectangular four-beam pattern of Fig. 2.5.3.9[link](b) shows symmetry 1R in disc H due to the horizontal mirror plane and symmetry m2 in both discs [\bar{S}] and F′ due to the twofold rotation axes in the [001] and [110] directions, respectively. The same symmetries are also seen in Fig. 2.5.3.9[link](c), where reflections [\bar{H}], S′ and [\bar{F}] are excited. Table 2.5.3.7[link] implies that the diffraction group 2mm1R can be identified from only one rectangular four-beam pattern, because no other diffraction groups give rise to the same symmetries in the four discs. When Buxton et al.'s method is used, two photographs or three patterns are necessary to identify diffraction group 2mm1R (see Table 2.5.3.3[link]). One can confirm the theoretically predicted symmetries between Fig. 2.5.3.9[link](b) and Fig. 2.5.3.9[link](c). All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7[link] and Table 2.5.3.7[link].

[Figure 2.5.3.9]

Figure 2.5.3.9 | top | pdf |

CBED patterns of V3Si taken with the [110] incidence. (a) Zone-axis pattern, (b) rectangular four-beam pattern with excitation of reflections H, [\bar S] and F, (c) rectangular four-beam pattern with excitation of reflections [\bar H], S and [\bar F].

These experiments show that the SMB method is quite effective for determining the diffraction group of slabs. Buxton et al.'s method identifies two-dimensional symmetry elements in the first place using a zone-axis pattern, and three-dimensional symmetry elements using DPs. On the other hand, the SMB method primarily finds many three-dimensional symmetry elements in an SMB pattern, and two-dimensional symmetry elements from a pair of SMB patterns, as shown in Tables 2.5.3.5[link], 2.5.3.6[link] and 2.5.3.7[link]. Therefore, the use of a ZAP and SMB patterns is the most efficient way to find as many crystal symmetry elements in a specimen as possible.

2.5.3.3. Space-group determination

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2.5.3.3.1. Lattice-type determination

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When the point group of a specimen crystal is determined, the crystal axes may be found from a spot diffraction pattern recorded at a high-symmetry zone axis, using the orientations of the symmetry elements determined in the course of point-group determination. Integral-number indices are assigned to the spots of the diffraction patterns. The systematic absence of reflections indicates the lattice type of the crystal. It should be noted that reflections forbidden by the lattice type are always absent, even if dynamical diffraction takes place. (This is true for all sample thicknesses and accelerating voltages.) By comparing the experimentally obtained absences and the extinction rules known for the lattice types [P, C (A, B), I, F and R], a lattice type may be identified for the crystal concerned.

2.5.3.3.2. Identification of screw axes and glide planes

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There are three space-group symmetry elements of diperiodic plane figures: (1) a horizontal screw axis [2_1'], (2) a vertical glide plane g with a horizontal glide vector and (3) a horizontal glide plane g′. These are related to the point-group symmetry elements 2′, m and m′ of diperiodic plane figures, respectively. (It is noted that these symmetry elements and ten point-group symmetry elements form 80 space groups.)

The ordinary extinction rules for screw axes and glide planes hold only in the approximation of kinematical diffraction. The kinematically forbidden reflections caused by these symmetry elements appear owing to Umweganregung of dynamical diffraction. Extinction of intensity, however, does take place in these reflections at certain crystal orientations with respect to the incident beam (i.e. in certain regions within a CBED disc). This dynamical extinction was first predicted by Cowley & Moodie (1959[link]) and was discussed by Miyake et al. (1960[link]) and Cowley et al. (1961[link]). Goodman & Lehmpfuhl (1964[link]) first observed the dynamical extinction as dark cross lines in kinematically forbidden reflection discs of CBED patterns of CdS. Gjønnes & Moodie (1965[link]) discussed the dynamical extinction in a more general way considering not only ZOLZ reflections but also HOLZ reflections. They completely clarified the dynamical extinction rules by considering the exact cancellation which may occur along certain symmetry-related multiple-scattering paths. Based on the results of Gjønnes & Moodie (1965[link]), Tanaka, Sekii & Nagasawa (1983[link]) tabulated the dynamical extinctions expected at all the possible crystal orientations for all the space groups. These were later tabulated in a better form on pages 162 to 172 of the book by Tanaka & Terauchi (1985[link]).

Fig. 2.5.3.10[link](a) illustrates Umweganregung paths to a kinematically forbidden reflection. The 0k0 (k = odd) reflections are kinematically forbidden because a b-glide plane exists perpendicular to the a axis and/or a 21 screw axis exists in the b direction. Let us consider an Umweganregung path a in the zeroth-order Laue zone to the 010 forbidden reflection and path b which is symmetric to path a with respect to axis k. Owing to the translation of one half of the lattice parameter b caused by the glide plane and/or the 21 screw axis, the following relations hold between the crystal structure factors:[\eqalignno{F(h,k) &= F(\bar{h},k)\quad \hbox{for }k=2n,&\cr F(h,k) &= -F(\bar{h},k)\quad\hbox{for }k=2n+1. &(2.5.3.1)}]That is, the structure factors of reflections hk0 and [\bar{h}k0] have the same phase for even k but have opposite phases for odd k.

[Figure 2.5.3.10]

Figure 2.5.3.10 | top | pdf |

Illustration of the production of dynamical extinction lines in kinematically forbidden reflections due to a b-glide plane and a 21 screw axis. (a) Umweganregung paths a, b and c. (b) Dynamical extinction lines A are formed in forbidden reflections 0k0 (k = odd). Extinction line B perpendicular to the lines A is formed in the exactly excited 010 reflection.

Since an Umweganregung path to the kinematically forbidden reflection 0k0 contains an odd number of reflections with odd k, the following relations hold:[\eqalignno{&F(h_1,k_1)F(h_2,k_2)\ldots F(h_n,k_n) \quad\hbox{for path }a&\cr &\quad = -F(\bar{h}_1,k_1)F(\bar{h}_2,k_2)\ldots F(\bar{h}_n,k_n)\quad\hbox{for path }b,&\cr&&(2.5.3.2)}]where[\textstyle\sum\limits_{i=1}^n h_i=0,\quad \textstyle\sum\limits_{i=1}^n k_i = k \,\,(k={\rm odd})]and the functions including the excitation errors are omitted because only the cases in which the functions are the same for all the paths are considered. The excitation errors for paths a and b become the same when the projection of the Laue point along the zone axis concerned, L, lies on axis k. Since the two waves passing through paths a and b have the same amplitude but opposite signs, these waves are superposed on the 0k0 discs (k = odd) and cancel out, resulting in dark lines A in the forbidden discs, as shown in Fig. 2.5.3.10[link](b). The line A runs parallel to axis k passing through the projection point of the zone axis.

In path c, the reflections are arranged in the reverse order to those in path b. When the 010 reflection is exactly excited, two paths a and c are symmetric with respect to the bisector m′–m′ of the 010 vector having the same excitation errors. The following equation holds:[\eqalignno{&F(h_1,k_1)F(h_2,k_2)\ldots F(h_n,k_n) \quad\hbox{for path }a&\cr &\quad = -F(\bar{h}_n,k_n)F(\bar{h}_{n-1},k_{n-1})\ldots F(\bar{h}_1,k_1)\quad\hbox{for path }c.&\cr&&(2.5.3.3)}]Since the waves passing through these paths have the same amplitude but opposite signs, these waves are superposed on the 010 discs and cancel out, resulting in dark line B in this disc, as shown in Fig. 2.5.3.10[link](b). Line B appears perpendicular to line A at the exact Bragg positions. When Umweganregung paths are present only in the zeroth-order Laue zone, the glide plane and screw axis produce the same dynamical extinction lines A and B. We call these lines A2 and B2 lines, subscript 2 indicating that the Umweganregung paths lie in the zeroth-order Laue zone.

The dynamical extinction effect is analogous to interference phenomena in the Michelson interferometer. That is, the incident beam is split into two beams by Bragg reflections in a crystal. These beams take different paths, in which they suffer a relative phase shift of π and are finally superposed on a kinematically forbidden reflection to cancel out.

When the paths include higher-order Laue zones, the glide plane produces only extinction lines A but the screw axis causes only extinction lines B. These facts are attributed to the different relations between structure factors for a 21 screw axis and a glide plane,[\eqalignno{F(hkl)&=(-1)^kF(\bar{h}k\bar{l})\,\,\,\hbox{for a }2_1\hbox{ screw axis in the [010] direction,}&\cr &&(2.5.3.4)\cr F(hkl)&=(-1)^kF(\bar{h}kl)\,\,\,\hbox{for a }b\hbox{ glide in the (100) plane.}&\cr&&(2.5.3.5)}%fd2o5o3o5]In the case of the glide plane, extinction lines A are still formed because two waves passing through paths a and b have opposite signs to each other according to equation (2.5.3.5)[link], but extinction lines B are not produced because equation (2.5.3.4)[link] holds only for the 21 screw axis. In the case of the 21 screw axis, only the waves passing through paths a and c have opposite signs according to equation (2.5.3.4)[link], forming extinction lines B only. We call these lines A3 and B3 dynamical extinction lines, suffix 3 indicating the Umweganregung paths being via higher-order Laue zones.

It was predicted by Gjønnes & Moodie (1965[link]) that a horizontal glide plane g′ gives a dark spot at the crossing point between extinction lines A and B (Fig. 2.5.3.10[link]b) due to the cancellation between the waves passing through paths b and c. Tanaka, Terauchi & Sekii (1987[link]) observed this dynamical extinction, though it appeared in a slightly different manner to that predicted by Gjønnes & Moodie (1965[link]). Table 2.5.3.8[link] summarizes the appearance of the dynamical extinction lines for the glide planes g and g′ and the 21 screw axis. The three space-group symmetry elements can be identified from the observed extinctions because these three symmetry elements produce different kinds of dynamical extinctions.

Table 2.5.3.8| top | pdf |
Dynamical extinction rules for an infinitely extended parallel-sided specimen

Symmetry elements of plane-parallel specimenOrientation to specimen surfaceDynamical extinction lines
Two-dimensional (zeroth Laue zone) interactionThree-dimensional (HOLZ) interaction
Glide planes perpendicular: g A2 and B2 A3
parallel: g intersection of A3 and B3
Twofold screw axis perpendicular: 21
parallel: [2_1^\prime] A2 and B2 B3

In principle, a horizontal screw axis and a vertical glide plane can be distinguished by observations of the extinction lines A3 and B3. It is, however, not easy to observe the extinction lines A3 and B3 because broad extinction lines A2 and B2 appear at the same time. The presence of the extinction lines A3 and B3 can be revealed by inspecting the symmetries of fine defect HOLZ lines appearing in the forbidden reflections instead of by direct observation of the lines A3 and B3 (Tanaka, Sekii & Nagasawa, 1983[link]). That is, if HOLZ lines form lines A3 and B3, HOLZ lines are symmetric with respect to the extinction lines A2 and B2. If HOLZ lines do not form lines A3 and B3, HOLZ lines are asymmetric with respect to the extinction lines A2 and B2. When the HOLZ lines are symmetric about the extinction lines A2, the specimen crystal has a glide plane. When the HOLZ lines are symmetric with respect to lines B2, a 21 screw axis exists. It should be noted that a relatively thick specimen area has to be selected to observe HOLZ lines in ZOLZ reflection discs.

Fig. 2.5.3.11[link] shows CBED patterns taken from (a) thin and (b) thick areas of FeS2, whose space group is [P2_1/a\bar{3}], at the 001 Bragg setting with the [100] electron-beam incidence. In the case of the thin specimen (Fig. 2.5.3.11[link]a), only the broad dynamical extinction lines formed by the interaction of ZOLZ reflections are seen in the odd-order discs. On the other hand, fine HOLZ lines are clearly seen in the thick specimen (Fig. 2.5.3.11[link]b). The HOLZ lines are symmetric with respect to both A2 and B2 extinction lines. This fact proves the presence of the extinction lines A3 and B3, or both the c glide in the (010) plane and the 21 screw axis in the c direction, this fact being confirmed by consulting Table 2.5.3.9[link]. Fig. 2.5.3.12[link] shows a [110] zone-axis CBED pattern of FeS2. A2 extinction lines are seen in both the 001 and [\bar{1}10] discs. Fine HOLZ lines are symmetric with respect to the A2 extinction lines in the [\bar{1}10] disc but asymmetric about the A2 extinction line in the 001 disc, indicating formation of the A3 extinction line only in the [\bar{1}10] disc. This proves the existence of a 21 screw axis in the [001] direction and an a glide in the (001) plane. The appearance of HOLZ lines is easily changed by a change of a few hundred volts in the accelerating voltage. Steeds & Evans (1980[link]) demonstrated for spinel changes in the appearance of HOLZ lines in the ZOLZ discs at accelerating voltages around 100 kV.

[Figure 2.5.3.11]

Figure 2.5.3.11 | top | pdf |

CBED patterns obtained from (a) thin and (b) thick areas of (001) FeS2. (a) Dynamical extinction lines A2 and B2 are seen. (b) Extinction lines A3 and B3 as well as A2 and B2 are formed because HOLZ lines are symmetric about lines A2 and B2.

[Figure 2.5.3.12]

Figure 2.5.3.12 | top | pdf |

CBED pattern of FeS2 taken with the [110] electron-beam incidence. In the 001 and [00\bar 1] discs, HOLZ lines are asymmetric with respect to extinction lines A2, indicating the existence of a 21 screw axis parallel to the c axis. In the [\bar 1 10] and [1\bar10] discs, HOLZ lines are symmetric with respect to extinction lines A2, indicating existence of a glide plane perpendicular to the c axis.

Another practical method for distinguishing between glide planes and 21 screw axes is that reported by Steeds et al. (1978[link]). The method is based on the fact that the extinction lines are observable even when a crystal is rotated with a glide plane kept parallel and with a 21 screw axis perpendicular to the incident beam. With reference to Fig. 2.5.3.10[link](a), extinction lines A3 produced by a glide plane remain even when the crystal is rotated with respect to axis h but the lines are destroyed by a rotation of the crystal about axis k. Extinction lines B3 originating from a 21 screw axis are not destroyed by a crystal rotation about axis k but the lines are destroyed by a rotation with respect to axis h.

2.5.3.3.3. Space-group determination

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We now describe a space-group determination method which uses the dynamical extinction lines caused by the horizontal screw axis [2_1'] and the vertical glide plane g of an infinitely extended parallel-sided specimen. We do not use the extinction due to the glide plane g′ because observation of the extinction requires a laborious experiment. It should be noted that a vertical glide plane with a glide vector not parallel to the specimen surface cannot be a symmetry element of a specimen of finite thickness; however, the component of the glide vector perpendicular to the incident beam acts as a symmetry element g. (Which symmetry elements are observed by CBED is discussed in Section 2.5.3.3.5[link].) The 21, 41, 43, 61, 63 and 65 screw axes of crystal space groups that are set perpendicular to the incident beam act as a symmetry element [2_1'] because two or three successive operations of 41, 43, 61, 63 and 65 screw axes make them equivalent to a 21 screw axis: (41)2 = (43)2 = (61)3 = (63)3 = (65)3 = 21. The 42, 31, 32, 62 and 64 screw axes that are set perpendicular to the incident beam do not produce dynamical extinction lines because the 42 screw axis acts as a twofold rotation axis due to the relation (42)2 = 2, the 31 and 32 screw axes give no specific symmetry in CBED patterns, and the 62 and 64 screw axes are equivalent to 31 and 32 screw axes due to the relations (62)2 = 32 and (64)2 = 31. Modifications of the dynamical extinction rules were investigated by Tanaka, Sekii & Nagasawa (1983[link]) when more than one crystal symmetry element (that gives rise to dynamical extinction lines) coexists and when the symmetry elements are combined with various lattice types. Using these results, dynamical extinction lines A2, A3, B2 and B3 expected from all the possible crystal settings for all the space groups were tabulated.

Table 2.5.3.9[link] shows all the dynamical extinction lines appearing in the kinematically forbidden reflections for all the possible crystal settings of all the space groups. The first column gives space groups. In each of the following pairs of columns, the left-hand column of the pair gives the reflection indices and the symmetry elements causing the extinction lines and the right-hand column gives the types of the extinction lines. The (second) suffixes 1, 2 and 3 of a 21 screw axis in each column distinguish the first, the second and the third screw axis of the space group (as in the symbols 211 and 212 of space group P21212). The glide symbols in the [001] column for space group P4/nnc have two suffixes (n21 and n22). The first suffix 2 denotes the second glide plane of the space group. The second suffixes 1 and 2, which appear in the tetragonal and cubic systems, distinguish two equivalent glide planes which lie in the x and y planes. The equivalent planes are distinguished only for the cases of [100], [010] and [001] electron-beam incidences, for convenience. The c-glide planes of space group Pcc2 are distinguished with symbols c1 and c2 (the first suffix only), because the equivalent planes do not exist. The glide symbol in the [001] column for space group P4/mbm has only one suffix 1 or 2. The suffix distinguishes the equivalent glide planes lying in the x and y planes. The first suffix to distinguish the first and the second glide planes is not necessary because the space group has only one glide symbol b. When the index of the incident-beam direction is expressed with a symbol like [h0l] for point groups 2, m and 2/m, the index h or l can take a value of zero. That is, the extinction rules are applicable to the [100] and [001] electron-beam incidences. However, if columns for [100], [010] and [001] incidences are present, as in the case of point group mm2, [hk0], [0kl] and [h0l] incidences are only for nonzero h, k and l. The reflections in which the extinction lines appear are always perpendicular to the corresponding incident-beam directions [(0k'l'\perp [0kl],\, h'k'0 \perp [hk0],\ldots)]. The indices of the reflections in which extinction lines appear are odd if no remark is given. For c-glide planes of space groups R3c and [R\bar{3}c] and for d-glide planes, the reflections in which extinction lines appear are specified as 6n + 3 and 4n + 2 orders, respectively.

Table 2.5.3.9| top | pdf |
Dynamical extinction lines appearing in ZOLZ reflections for all crystal space groups except Nos. 1 and 2

Point groups 2, m, 2/m (second setting, unique axis b)

Space groupIncident-beam direction
[h0l]
3 P2      
4 P21 0k0 A2 B2
21   B3
5 C2      
6 Pm      
7 Pc h0lo A2 B2
c A3  
8 Cm      
9 Cc he0lo A2 B2
c A3  
10 P2/m      
11 P21/m 0k0 A2 B2
21   B3
12 C2/m      
13 P2/c h0lo A2 B2
c A3  
14 P21/c 0k0 A2 B2
21   B3
h0lo A2 B2
c A3  
15 C2/c he0lo A2 B2
c A3  

Point group 222

Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
16 P222                                    
17 P2221 00l A2 B2 00l A2 B2       00l A2 B2            
21   B3 21   B3       21   B3            
18 P21212 0k0 A2 B2 h00 A2 B2 h00 A2 B2       h00 A2 B2 0k0 A2 B2
212   B3 211   B3 211   B3       211   B3 212   B3
            0k0                      
            212                      
19 P212121 0k0 A2 B2 h00 A2 B2 h00 A2 B2 00l A2 B2 h00 A2 B2 0k0 A2 B2
212   B3 211   B3 211   B3 213   B3 211   B3 212   B3
00l     00l     0k0                      
213     213     212                      
20 C2221 00l A2 B2 00l A2 B2       00l A2 B2            
21   B3 21   B3       21   B3            
21 C222                                    
22 F222                                    
23 I222                                    
24 I212121                                    

Point group mm2

Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
25 Pmm2                                    
26 Pmc21 00l A2 B2 00l           00l A2 B2       h0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3  
27 Pcc2 00l     00l                 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
28 Pma2             h00 A2 B2             ho0l A2 B2
            a A3               a A3  
29 Pca21 00l     00l A2 B2 h00 A2 B2 00l A2 B2 0klo A2 B2 ho0l A2 B2
21   B3 c, 21 A3 B3 a A3   21   B3 c A3   a A3  
30 Pnc2 00l
c

A3
  00l
n

A3
  0k0
n
A2
A3
B2       0kl: k + l = 2n + 1
n
A2
A3
B2 h0lo
c
A2
A3
B2
31 Pmn21 00l
n, 21
A2
A3
B2
B3
00l
21
 
B3
h00
n
A2
A3
B2 00l
21
A2 B2
B3
      h0l: h + l = 2n + 1
n
A2
A3
B2
32 Pba2             h00 A2 B2       0kol A2 B2 ho0l A2 B2
            a A3         b A3   a A3  
            0k0                      
            b                      
33 Pna21 00l
21
 
B3
00l
n, 21
A2
A3
B2
B3
h00
a
0k0
n
A2
A3
B2 00l
21
A2 B2
B3
0kl: k + l = 2n + 1
n
A2
A3
B2 ho0l
a
A2
A3
B2
34 Pnn2 00l
n2

A3
  00l
n1

A3
  h00
n2
0k0
n1
A2
A3
B2       0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
35 [\matrix{\hfill Cmm2\cr \hfill ba2}]                                  
36 [\matrix{\hfill Cmc2_1\cr\hfill bn2_1}] 00l A2 B2 00l           00l A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3  
37 [\matrix{\hfill Ccc2\cr\hfill nn2}] 00l     00l                 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
38 [\matrix{\hfill Amm2\phantom{_1}\cr\hfill nc2_1}]                                    
39 [\matrix{\hfill Abm2\phantom{_1}\cr\hfill cc2_1}]                         0kolo A2 B2      
                        b A3        
40 [\matrix{\hfill Ama2\phantom{_1}\cr\hfill nn2_1}]             h00 A2 B2             ho0le A2 B2
            a A3               a A3  
41 [\matrix{\hfill Aba2\phantom{_1}\cr\hfill cn2_1}]             h00 A2 B2       0kolo A2 B2 ho0le A2 B2
            a A3         b A3   a A3  
42 Fmm2                                    
43 [\matrix{\hfill Fdd2\phantom{_1}\cr\hfill dd2_1}] 00l: l = 4n + 2
d2

A3
  00l: l = 4n + 2
d1

A3
  h00: h = 4n + 2
d2
0k0: k = 4n + 2
d1
A2
A3
B2       0kele: ke + le = 4n + 2
d1
A2
A3
B2 he0le: he + le = 4n + 2
d2
A2
A3
B2
44 [\matrix{\hfill Imm2\phantom{_1}\cr\hfill nn2_1}]                                    
45 [\matrix{\hfill Iba2\phantom{_1}\cr\hfill cc2_1}]                         0kolo A2 B2 ho0lo A2 B2
                        b A3   a A3  
46 [\matrix{\hfill Ima2\phantom{_1}\cr\hfill nc2_1}]                               ho0lo A2 B2
                              a A3  

Point group mmm

Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
47 P2/m2/m2/m                                    
48 P2/n2/n2/n 00l
n2
0k0
n3

A3
  00l
n1
h00
n3

A3
  0k0
n1
h00
n2

A3
  hk0: h + k = 2n + 1
n3
A2
A3
B2 0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
49 P2/c2/c2/m 00l     00l                 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
50 P2/b2/a2/n 0k0
n

A3
  h00
n

A3
  0k0
b
h00
a

A3
  hk0: h + k = 2n + 1
n
A2
A3
B2 0kol
b
A2
A3
B2 ho0l
a
A2
A3
B2
51 P21/m2/m2/a       h00 A2 B2 h00     hok0 A2 B2 h00 A2 B2      
      21, a A3 B3 21   B3 a A3   21   B3      
52 P2/n21/n2/a 00l
n2

A3
  00l
n1
h00
a

A3
  0k0
n1, 21
A2
A3
B2
B3
hok0
a
A2
A3
B2 0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
0k0           h00                 0k0 A2 B2
21   B3       n2 A3               21   B3
53 P2/m2/n21/a 00l
n, 21
A2
A3
B2
B3
h00
a

A3
  h00
n

A3
  hok0
a
A2
A3
B2       h0l: h + l = 2n + 1
n
A2
A3
B2
      00l           00l A2 B2            
      21   B3       21   B3            
54 P21/c2/c2/a 00l     00l     h00     hok0 A2 B2 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3   21   B3 a A3   c1 A3   c2 A3  
      h00 A2 B2             h00 A2 B2      
      a, 21 A3 B3             21   B3      
55 P21/b21/a2/m 0k0     h00     0k0 A2 B2       0kol A2 B2 ho0l A2 B2
212   B3 211   B3 b, 212 A3 B3       b A3   a A3  
            h00           h00 A2 B2 0k0 A2 B2
            a, 211           211   B3 212   B3
56 P21/c21/c2/n 00l
c2

A3
  00l
c1

A3
  0k0
212
h00
211
 
B3
hk0: h + k = 2n + 1
n
A2
A3
B2 0klo
c1
A2
A3
B2 h0lo
c2
A2
A3
B2
0k0 A2 B2 h00 A2 B2             h00 A2 B2 0k0 A2 B2
212, n A3 B3 211, n A3 B3             211   B3 212   B3
57 P2/b21/c21/m 00l A2 B2 00l     0k0 A2 B2 00l A2 B2 0kol A2 B2 h0lo A2 B2
c, 212 A3 B3 212   B3 b, 211 A3 B3 212   B3 b A3   c A3  
0k0                             0k0 A2 B2
211   B3                         211   B3
58 P21/n21/n2/m 00l
n2

A3
  00l
n1

A3
  0k0
n1, 212
h00
n2, 211
A2
A3
B2
B3
      0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
0k0     h00                 h00 A2 B2 0k0 A2 B2
212   B3 211   B3             211   B3 212   B3
59 P21/m21/m2/n 0k0
n, 212
A2
A3
B2
B3
h00
n, 211
A2
A3
B2
B3
0k0
212
h00
211
 
B3
hk0: h + k = 2n + 1
n
A2
A3
B2 h00
211
A2 B2
B3
0k0
212
A2 B2
A3
60 P21/b2/c21/n 00l
c, 212
A2
A3
B2
B3
h00
n, 211
A2
A3
B2
B3
0k0
b

A3
  hk0: h + k = 2n + 1
n
A2
A3
B2 0kol
b
A2
A3
B2 h0lo
c
A2
A3
B2
0k0     00l     h00     00l A2 B2 h00 A2 B2      
n A3   212   B3 211   B3 212   B3 211   B3      
61 P21/b21/c21/a 00l A2 B2 00l     0k0 A2 B2 hok0 A2 B2 0kol A2 B2 h0lo A2 B2
c, 213 A3 B3 213   B3 b, 212 A3 B3 a A3   b A3   c A3  
0k0     h00 A2 B2 h00     00l A2 B2 h00 A2 B2 0k0 A2 B2
212   B3 a, 211 A3 B3 211   B3 213   B3 211   B3 212   B3
62 P21/n21/m21/a 00l
213
0k0
212
 
B3
00l
n, 213
h00
a, 211
A2
A3
B2
B3
0k0
n, 212
A2
A3
B2
B3
hok0
a
A2
A3
B2 0kl: k + l = 2n + 1
n
A2
A3
B2 0k0
212
A2 B2
B3
            h00     00l A2 B2 h00 A2 B2      
            211   B3 213   B3 211   B3      
63 C2/m2/c21/m 00l A2 B2 00l           00l A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3  
64 C2/m2/c21/a 00l A2 B2 00l           hoko0 A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       a A3         c A3  
                  00l A2 B2            
                  21   B3            
65 C2/m2/m2/m                                    
66 C2/c2/c2/m 00l     00l                 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
67 C2/m2/m2/a                   hoko0 A2 B2            
                  a A3              
68 C2/c2/c2/a 00l     00l           hoko0 A2 B2 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3         a A3   c1 A3   c2 A3  
69 F2/m2/m2/m                                    
70 F2/d2/d2/d 00l: l =
4n + 2
d2
0k0: k =
4n + 2
d3

A3
  h00: h =
4n + 2
d3
00l: l =
4n + 2
d1

A3
  0k0: k =
4n + 2
d1
h00: h =
4n + 2
d2

A3
  heke0: he + ke = 4n + 2
d3
A2
A3
B2 0kele: ke + le = 4n + 2
d1
A2
A3
B2 he0le: he + le = 4n + 2
d2
A2
A3
B2
71 I2/m2/m2/m                                    
72 I2/b2/a2/m                         0kolo A2 B2 ho0lo A2 B2
                        b A3   a A3  
73 I21/b21/c21/a                   hoko0 A2 B2 0kolo A2 B2 ho0lo A2 B2
                  a A3   b A3   c A3  
74 I21/m21/m21/a                   hoko0 A2 B2            
                  a A3              

Point groups [4, \bar 4, 4/m]

Space groupIncident-beam direction
[hk0]
75 P4      
76 P41 00l A2 B2
41   B3
77 P42      
78 P43 00l A2 B2
43   B3
79 I4      
80 I41      
81 [P\bar 4]      
82 [I\bar 4]      
83 P4/m      
84 P42/m      
85 P4/n hk0: h + k = 2n + 1 A2 B2
n A3  
86 P42/n hk0: h + k = 2n + 1 A2 B2
n A3  
87 I4/m      
88 I41/a hoko0 A2 B2
a A3  

Point group 422

Space groupIncident-beam direction
[hk0][0kl]
89 P422            
90 P4212       h00 A2 B2
      21   B3
91 P4122 00l A2 B2      
41   B3      
92 P41212 00l A2 B2 h00 A2 B2
41   B3 21   B3
93 P4222            
94 P42212       h00 A2 B2
      21   B3
95 P4322 00l A2 B2      
43   B3      
96 P43212 00l A2 B2 h00 A2 B2
43   B3 21   B3
97 I422            
98 I4122            

Point group 4mm. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column.

Space groupIncident-beam direction
[100][001][110][h0l][hhl]
99 P4mm                              
100 P4bm