International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 2.4, pp. 80-82

Section 2.4.1. Electron techniques

J. M. Cowleya

2.4.1. Electron techniques

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2.4.1.1. Powder-pattern geometry

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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 ([2\theta_B]) observed are usually less than 10−1 rad.

Patterns are usually recorded on flat photographic plates or films and a small-angle 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, [\lambda/d = 2\sin\theta\simeq\tan 2\theta=r/L,]or the interplanar spacing, d, is given by [d=L\lambda/r.\eqno (2.4.1.1)]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 [d=(L\lambda/r)(1+3r^2/8L^2).\eqno (2.4.1.2)]The `camera constant' [L\lambda] 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[link]).

In recent years, electron diffraction patterns have been obtained most commonly in electron microscopes with three or more post-specimen lenses. The camera-constant values are then best obtained by calibration using samples of known structure.

With electron-optical 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 small-angle-scattering geometry precludes application of any of the special camera geometries used for high-accuracy measurements with X-rays (Chapter 2.3[link] ).

2.4.1.2. Diffraction patterns in electron microscopes

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The specimens used in electron microscopes may be self-supporting thin films or fine powders supported on thin films, usually made of amorphous carbon. Specimen thicknesses must be less than about 103 Å in order to avoid perturbations of the diffraction patterns by strong multiple-scattering effects. The selected-area electron-diffraction (SAED) technique [see Section 2.5.2[link] in IT B (2001[link])] allows sharply focused diffraction patterns to be obtained from regions 103 to 105 Å in diameter. For the smaller ranges of selected-area regions, specimens may give single-crystal 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 convergent-beam electron-diffraction (CBED) technique, diffraction patterns can be obtained from regions of diameter 100 Å [see Section 2.5.3[link] in IT B (2001[link])] 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[link]) are increasingly concerned with single-crystal 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 lens-current settings. It is advisable to check these calibrations with samples of known structure and to determine calibrations for non-standard lens settings.

The effective camera length, L, is dependent on the specimen height within the objective-lens pole-piece. If a specimen-height adjustment (a z-lift) 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.

2.4.1.3. Preferred orientations

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The techniques of specimen preparation may result in a strong preferred orientation of the crystallites, resulting in strong arcing of powder-pattern 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 reciprocal-lattice 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 so-called oblique texture patterns which provide a rich source of three-dimensional 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[link] of IT B (2001[link]).

2.4.1.4. Powder-pattern intensities

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In the kinematical approximation, the expression for intensities of electron diffraction follows that for X-ray diffraction with the exception that, because only small angles of diffraction are involved, no polarization factor is involved. Following Vainshtein (1964[link]), the intensity per unit length of a powder line is [I(h)=J_0\lambda^2\bigg |{\Phi_h\over \Omega}\bigg | ^2 V{d^2_hM\over 4\pi L\lambda},\eqno (2.4.1.3)]where [J_0] is the incident-beam intensity, [\Phi_h] is the structure factor, [\Omega] is the unit-cell 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 two-beam approximation to the dynamical-scattering theory. Because an averaging over all orientations is involved, the many-beam dynamical-diffraction effects are less evident than for single-crystal patterns.

By integrating the two-beam intensity expression over excitation error, Blackman (1939[link]) obtained the expression for the ratio of dynamical to kinematical intensities: [I_{\rm dyn}/I_{\rm kin}=A^{-1}_{h} \textstyle\int\limits_{0}^{A_{h}} J_0(2x)\,{\rm d}x, \eqno (2.4.1.4)]where [J_o(x)] is the zero-order Bessel function, [A_h=\sigma H\Phi _h] with the interaction constant [\sigma = 2\pi me\lambda /h^2], and H is the crystal thickness. Careful measurements on ring patterns from thin aluminium films by Horstmann & Meyer (1962[link]) showed agreement with the `Blackman curve' [from equation (2.4.1.4)[link]] 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 second-order reflections from strong inner reflections. A practical algorithm for implementing Blackman corrections has been published by Dvoryankina & Pinsker (1958[link]).

Such deviations result from plural-beam 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 ring-pattern intensities. A correction for the effects of systematic interactions may be made by use of the Bethe second approximation (Bethe, 1928[link]) (see Chapter 8.8[link] ).

For non-systematic reflections, corresponding to reciprocal-lattice points not collinear with the origin and the reciprocal-lattice point of interest, the averaging over all crystal orientations ensures that the powder-pattern intensity calculated from the two-beam formula will not be appreciably affected. Appreciable effects from non-systematic 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[link]) and experimentally by Imamov, Pannhorst, Avilov & Pinsker (1976[link]) that appreciable modifications of intensities of oblique-texture patterns may result from non-systematic interactions for particular tilt angles, especially for heavy-atom materials [see also Avilov, Parmon, Semiletov & Sirota (1984[link])].

The techniques for the measurement of electron diffraction intensities are described in Chapter 7.2[link] . 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[link]). 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.

Small-angle 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 (super-periods) 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 X-ray diffraction.

The techniques for interpretation of electron diffraction powder-pattern intensities follow those for X-ray 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[link]).

2.4.1.5. Crystal-size analysis

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The methods used in X-ray 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.

  • (a) Unless energy filtering is used to remove inelastically scattered electrons, a component is added to the rings broadened by the effects of inelastic scattering involving electronic excitations. Since the mean free paths for such processes are of the order of 103 Å and the angular spread of the scattered electrons is 10−3 to 10−4 rad, the ring broadening for thick samples may be equivalent to the broadening for a crystal size of the order of 100 Å.

  • (b) When a powder sample consists of separated crystallites having faces not predominantly parallel or perpendicular to the incident beam, the diffraction rings may be appreciably broadened by refraction effects. The refractive index for electrons is given, to a first approximation, by [n=1+\Phi _0/2E,]where [\Phi _0] is the mean inner potential of the crystal, typically 10 to 20 V, and E is the accelerating voltage of the incident electron beam. For the beam passing through the two faces of a 90° wedge, each at an angle α = 45° to the beam, for example, the beam is deflected by an amount [\Delta= \Phi_0/E=1.5\times10^{-4}\, {\rm rad}]for an inner potential of 15 V and E = 100 kV. The broadening of the ring by such deflections can correspond to the broadening due to a particle size of λ/2Δ [\simeq] 120 Å.

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[link]; Cowley & Rees, 1947[link]; Honjo & Mihama, 1954[link]). For more isometric crystal shapes, this dependence is less marked and the broadening has been estimated (Cowley & Rees, 1947[link]) as equivalent to that due to a particle size of about 200 Å.

2.4.1.6. Unknown-phase identification: databases

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To a limited extent, the compilations of data for X-ray 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.

  • (a) Because of the increasing use of single-crystal patterns obtained in the SAED mode in an electron microscope, the use of single-crystal spot patterns, in addition to powder patterns, must be considered for purposes of identification. Methods for the analysis of single-crystal patterns are summarized in Section 5.4.1[link] .

  • (b) The deviations from kinematical scattering conditions may be large, especially for single-crystal patterns, so that little reliance can be placed on relative intensities, and reflections kinematically forbidden may be present.

  • (c) Compositional information may be obtained by use of X-ray microanalysis (or electron-energy-loss spectroscopy) performed in the electron microscope and this provides an effective additional guide to identification.

  • (d) Electron diffraction data often extend to smaller d spacings than X-ray data because there is no wavelength limitation.

  • (e) The electron diffraction d-spacing information is rarely more precise than 1% and the uncertainty may be 5% for large d spacings.

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[link]). The 1993 version contains crystallographic and chemical information on over 81 500 crystalline materials with, in most cases, calculated patterns to ensure that diagnostic high-d-spacing reflections can be matched. It is available on magnetic tape or floppy disks. The MAX-d index (Anderson & Johnson, 1979[link]) has been expanded to 51 580 NSI-based entries (Mighell, Himes, Anderson & Carr, 1988[link]) in book form for manual searching.

References

Anderson, R. & Johnson, G. G. Jr (1979). The MAX-d alphabetical index to the JCPDS data base: a new tool for electron diffraction analysis. 37th Annu. Proc. Electron Microsc. Soc. Am., edited by G. W. Bailey, pp. 444–445. Baton Rouge: Claitors.
Avilov, A. S., Parmon, V. S., Semiletov, S. A. & Sirota, M. I. (1984). Intensity calculations for many-wave diffraction of fast electrons in polycrystal specimens. Kristallografiya, 29, 11–15. [In Russian.]
Bethe, H. A. (1928). Theorie der Beugung von Elektronen an Kristallen. Ann. Phys. (Leipzig), 87, 55–129.
Blackman, M. (1939). On the intensities of electron diffraction rings. Proc. R. Soc. London, 173, 68–82.
Carr, M. J., Chambers, W. F., Melgaard, D. K., Himes, V. L., Stalick, J. K. & Mighell, A. D. (1987). NBS/Sandia/ICDD Electron Diffraction Data Base. Report SAND87-1992-UC-13. Sandia National Laboratories, Albuquerque, NM 87185, USA.
Cowley, J. M. & Rees, A. L. G. (1947). Refraction effects in electron diffraction. Proc. Phys. Soc. 59, 287–302.
Dvoryankina, G. G. & Pinsker, Z. G. (1958). The structural study of Fe4N. Kristallografiya, 3, 438–445. [In Russian.]
Goodman, P. (1963). Investigation of arsenic trisulphide by the electron diffraction radial distribution method. Acta Cryst. 16, A130.
Grigson, C. W. B. (1962). On scanning electron diffraction. J. Electron. Control, 12, 209–232.
Honjo, G. & Mihama, K. (1954). Fine structure due to refraction effect in electron diffraction pattern of powder sample. J. Phys. Soc. Jpn, 9, 184–198.
Horstmann, M. & Meyer, G. (1962). Messung der elastischen Electronenbeugungsintensitaten polykristalliner Aluminium-Schichten. Acta Cryst. 15, 271–281.
Imamov, R. M., Pannhorst, V., Avilov, A. S. & Pinsker, Z. G. (1976). Experimental study of dynamic effects associated with electron diffraction in partly oriented films. Kristallografiya, 21, 364–369.
International Tables for Crystallography (2001). Vol. B, Reciprocal space, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.
Mighell, A. D., Himes, V. L., Anderson, R. & Carr, M. J. (1988). d-spacing and formula index for compound identification using electron diffraction. 46th Annu. Proc. Electron Microsc. Soc. Am., edited by G. W. Bailey, pp. 912–913. San Francisco Press.
Sturkey, L. & Frevel, L. K. (1945). Refraction effects in electron diffraction. Phys. Rev. 68, 56–57.
Tsypursky, S. I. & Drits, V. A. (1977). The efficiency of the electronometric measurement of intensities in electron diffraction structural studies. Izv. Akad. Nauk SSSR Ser. Phys. 41, 2263–2271. [In Russian.]
Turner, P. S. & Cowley, J. M. (1969). The effect of n-beam dynamical diffraction in electron diffraction intensities from polycrystalline materials. Acta Cryst. A25, 475–481.
Vainshtein, B. K. (1964). Structure analysis by electron diffraction. Oxford: Pergamon Press. [Translated from the Russian: Strukturnaya Electronografiya.]








































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