International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 2.4, pp. 8082

The electron wavelengths normally used to obtain powder patterns from thin films of polycrystalline materials lie in the range 8 × 10^{−2} to 2 × 10^{−2} Å (20 to 200 kV accelerating voltages). The maximum scattering angles () observed are usually less than 10^{−1} rad.
Patterns are usually recorded on flat photographic plates or films and a smallangle approximation is applied. For a camera length L, the distance from the specimen to the photographic plate in the absence of any intervening electron lenses, the approximation is made that, for a diffraction ring of radius r, or the interplanar spacing, d, is given by For a scattering angle of 10^{−1} rad, the error in this expression is 0.5%. A better approximation, valid to better than 0.1% at 10^{−1} rad, is The `camera constant' may be obtained by direct measurement of L and the accelerating voltage if there are no electron lenses following the specimen.
Direct electronic recording of intensities has great advantages over photographic recording (Tsypursky & Drits, 1977).
In recent years, electron diffraction patterns have been obtained most commonly in electron microscopes with three or more postspecimen lenses. The cameraconstant values are then best obtained by calibration using samples of known structure.
With electronoptical instruments, it is possible to attain collimations of 10^{−6} rad so that for scattering angles of 10^{−1} rad an accuracy of 10^{−5} in d spacings should be possible in principle but is not normally achievable. In practice, accuracies of about 1% are expected. Some factors limiting the accuracy of measurement are mentioned in the following sections. The smallanglescattering geometry precludes application of any of the special camera geometries used for highaccuracy measurements with Xrays (Chapter 2.3 ).
The specimens used in electron microscopes may be selfsupporting thin films or fine powders supported on thin films, usually made of amorphous carbon. Specimen thicknesses must be less than about 10^{3} Å in order to avoid perturbations of the diffraction patterns by strong multiplescattering effects. The selectedarea electrondiffraction (SAED) technique [see Section 2.5.2 in IT B (2001)] allows sharply focused diffraction patterns to be obtained from regions 10^{3} to 10^{5} Å in diameter. For the smaller ranges of selectedarea regions, specimens may give singlecrystal patterns or very spotty ring patterns, rather than continuous ring patterns, because the number of crystals present in the field of view is small unless the crystallite size is of the order of 100 Å or less. By use of the convergentbeam electrondiffraction (CBED) technique, diffraction patterns can be obtained from regions of diameter 100 Å [see Section 2.5.3 in IT B (2001)] or, in the case of some specialized instruments, regions less than 10 Å in diameter. For these reasons, the methods for phase identification from electron diffraction patterns and the corresponding databases (see Subsection 2.4.1.6) are increasingly concerned with singlecrystal spot patterns in addition to powder patterns.
Instrument manufacturers usually provide values of camera lengths, L, or camera constants, Lλ, for a wide range of designated lenscurrent settings. It is advisable to check these calibrations with samples of known structure and to determine calibrations for nonstandard lens settings.
The effective camera length, L, is dependent on the specimen height within the objectivelens polepiece. If a specimenheight adjustment (a zlift) is provided, it should be adjusted to give a predetermined lens current, and hence focal length, of the objective lens.
In some microscopes, at particular lens settings the projector lenses may introduce a radial distortion of the diffraction pattern. This may be measured with a suitable standard specimen.
The techniques of specimen preparation may result in a strong preferred orientation of the crystallites, resulting in strong arcing of powderpattern rings, the absence of some rings, and perturbations of relative intensities.
For example, small crystals of flaky habit deposited on a flat supporting film may be oriented with one reciprocallattice axis preferentially perpendicular to the plane of the support. A ring pattern obtained with the incident beam perpendicular to the support then shows only those rings for planes in the zone parallel to the preferred axis. Such orientation is detected by the appearance of arcing and additional reflections when the supporting film is tilted. Tilted specimens give the socalled oblique texture patterns which provide a rich source of threedimensional diffraction information, used as a basis for crystal structure analysis.
A full discussion of the texture patterns resulting from preferred orientations is given in Section 2.5.4 of IT B (2001).
In the kinematical approximation, the expression for intensities of electron diffraction follows that for Xray diffraction with the exception that, because only small angles of diffraction are involved, no polarization factor is involved. Following Vainshtein (1964), the intensity per unit length of a powder line is where is the incidentbeam intensity, is the structure factor, is the unitcell volume, V is the sample volume, and M is the multiplicity factor.
The kinematical approximation has limited validity. The deviations from this approximation are given to a first approximation by the twobeam approximation to the dynamicalscattering theory. Because an averaging over all orientations is involved, the manybeam dynamicaldiffraction effects are less evident than for singlecrystal patterns.
By integrating the twobeam intensity expression over excitation error, Blackman (1939) obtained the expression for the ratio of dynamical to kinematical intensities: where is the zeroorder Bessel function, with the interaction constant , and H is the crystal thickness. Careful measurements on ring patterns from thin aluminium films by Horstmann & Meyer (1962) showed agreement with the `Blackman curve' [from equation (2.4.1.4)] to within about 5% with some notable exceptions. Deviations of up to 40 to 50% from the Blackman curve occurred for several reflections, such as 222 and 400, which are secondorder reflections from strong inner reflections. A practical algorithm for implementing Blackman corrections has been published by Dvoryankina & Pinsker (1958).
Such deviations result from pluralbeam systematic interactions, the coherent multiple scattering between different orders of a strong inner reflection. When the Bragg condition is satisfied for one order, the excitation errors for the other orders are the same for all possible crystal orientations and these other orders contribute systematically to the ringpattern intensities. A correction for the effects of systematic interactions may be made by use of the Bethe second approximation (Bethe, 1928) (see Chapter 8.8 ).
For nonsystematic reflections, corresponding to reciprocallattice points not collinear with the origin and the reciprocallattice point of interest, the averaging over all crystal orientations ensures that the powderpattern intensity calculated from the twobeam formula will not be appreciably affected. Appreciable effects from nonsystematic interactions may, however, occur when the averaging is over a limited range of crystal orientations, as in the case of strong preferred orientations. It was shown theoretically by Turner & Cowley (1969) and experimentally by Imamov, Pannhorst, Avilov & Pinsker (1976) that appreciable modifications of intensities of obliquetexture patterns may result from nonsystematic interactions for particular tilt angles, especially for heavyatom materials [see also Avilov, Parmon, Semiletov & Sirota (1984)].
The techniques for the measurement of electron diffraction intensities are described in Chapter 7.2 . Most commonly electron diffraction powder patterns are recorded by photographic methods and a microdensitometer is used for quantitative intensity measurement. The Grigson scanning method, using a scintillator and photomultiplier to record intensities as the pattern is scanned over a fine slit, has considerable advantages in terms of linearity and range of the intensity scale (Grigson, 1962). This method also has the advantage that it may readily be combined with an energy filter so that only elastically scattered electrons (or electrons inelastically scattered with a particular energy loss) may be recorded.
Smallangle electron diffraction may give useful information in some cases, but must be interpreted carefully because the features may result from multiple scattering or other artefacts. It may give additional details of periodicity (superperiods) and deviations of the real symmetry from the ideal symmetry suggested by other data. Care must be taken with the interpretation of additional reflections, as they may relate to the structure of small regions that are not typical of the bulk specimens such as are examined by Xray diffraction.
The techniques for interpretation of electron diffraction powderpattern intensities follow those for Xray patterns when the kinematical approximation is valid. For very small crystals, giving very broad rings, it is possible to use the method, commonly applied for diffraction by gases, of performing a Fourier transform to obtain a radial distribution function (Goodman, 1963).
The methods used in Xray diffraction for the determination of average crystal size or size distributions may be applied to electron diffraction powder patterns. Except in the case of very small crystal dimensions, several factors peculiar to electrons should be taken into consideration.
For crystallites of regular habit, such as the small cubic crystals of MgO smoke, the ring broadening from this source is strongly dependent on the crystallographic planes involved (Sturkey & Frevel, 1945; Cowley & Rees, 1947; Honjo & Mihama, 1954). For more isometric crystal shapes, this dependence is less marked and the broadening has been estimated (Cowley & Rees, 1947) as equivalent to that due to a particle size of about 200 Å.
To a limited extent, the compilations of data for Xray diffraction, such as the ICDD Powder Diffraction File, may be used for the identification of phases from electron diffraction data. The nature of the electron diffraction data and the circumstances of its collection have prompted the compilation of databases specifically for use with electron diffraction. Factors taken into consideration include the following.
With these points in mind, databases specially designed for use with electron diffraction have been developed. The NIST/Sandia/ICDD Electron Diffraction Database follows the design principles of Carr, Chambers, Melgaard, Himes, Stalick & Mighell (1987). The 1993 version contains crystallographic and chemical information on over 81 500 crystalline materials with, in most cases, calculated patterns to ensure that diagnostic highdspacing reflections can be matched. It is available on magnetic tape or floppy disks. The MAXd index (Anderson & Johnson, 1979) has been expanded to 51 580 NSIbased entries (Mighell, Himes, Anderson & Carr, 1988) in book form for manual searching.
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