Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 5.2, p. 652   | 1 | 2 |

Section 5.2.14. Approximations

A. F. Moodie,a J. M. Cowleyb and P. Goodmanc

aDepartment of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia,bArizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287–1504, USA, and cSchool of Physics, University of Melbourne, Parkville, Australia

5.2.14. Approximations

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So far, only the familiar first Born and two-beam approximations and the projection approximation have been mentioned. Several others, however, have a considerable utility.

A high-voltage limit can be calculated in standard fashion to give [U_{\rm HVL} (h, k) = \hbox{\scr F} \exp \left\{ -i \sigma_{c} \textstyle\int\limits_{0}^{T} \varphi (x,y,z)\, \hbox{d}z\right\}, \eqno(]where [\hbox{\scr F}] is the Fourier transform operator, and [\sigma_{c} = 2\pi m_{0}e\lambda_{c} / h^{2}] with [\lambda_{c} = (h / m_{0}c)], the Compton wavelength. The phase-grating approximation, which finds application in electron microscopy, involves the assumption that equation ([link] has some range of validity when [\sigma_{c}] is replaced by [\sigma]. This is equivalent to ignoring the curvature of the Ewald sphere and can therefore apply to thin crystals [see Section 2.5.2[link] and IT C (2004[link], Section 4.3.8[link] )].

Approximations that involve curtailing the number of beams evidently have a range of validity that depends on the size of the unit cell. The most explored case is that of three-beam inter­actions. Kambe (1957[link]) has demonstrated that phase information can be obtained from the diffraction data; Gjønnes & Høier (1971[link]) analysed the confluent case, and Hurley & Moodie (1980[link]) have given an explicit inversion for the centrosymmetric case. Analyses of the symmetry of the defining differential equation, and of the geometry of the noncentrosymmetric case, have been given by Moodie et al. (1996[link], 1998[link]).

Niehrs and his co-workers (e.g. Blume, 1966[link]) have shown that, at or near zones, effective two-beam conditions can sometimes obtain, in that, for instance, the central beam and six equidistant beams of equal structure amplitude can exhibit two-beam behaviour when the excitation errors are equal. Group-theoretical treatments have been given by Fukuhara (1966[link]) and by Kogiso & Takahashi (1977[link]). Explicit reductions for all admissible noncentrosymmetric space groups have been obtained by Moodie & Whitfield (1994[link]). Extensions of such results have application in the interpretation of lattice images and convergent-beam patterns.

The approximations near the classical limit have been extensively explored [for instance, see Berry (1971[link])] but channelling has effectively become a separate subject and cannot be discussed here.


International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.
Berry, M. V. (1971). Diffraction in crystals at high voltages. J. Phys. C, 4, 697–722.
Blume, J. (1966). Die Kantenstreifung im Elektronen-Mikroskopischen Bild Wurfelformiger MgO Kristalle bei Durchstrahlung im Richtung der Raumdiagonal. Z. Phys. 191, 248–272.
Fukuhara, A. (1966). Many-ray approximations in the dynamical theory of electron diffraction. J. Phys. Soc. Jpn, 21, 2645–2662.
Gjønnes, J. & Høier, R. (1971). The application of non-systematic many-beam dynamic effects to structure-factor determination. Acta Cryst. A27, 313–316.
Hurley, A. C. & Moodie, A. F. (1980). The inversion of the three-beam intensities for scalar scattering by a general centrosymmetric crystal. Acta Cryst. A36, 737–738.
Kambe, K. (1957). Study of simultaneous reflection in electron diffraction by crystal. J. Phys. Soc. Jpn, 12, 13–31.
Kogiso, M. & Takahashi, H. (1977). Group-theoretical method in the many-beam theory of electron diffraction. J. Phys. Soc. Jpn, 42, 223–229.
Moodie, A. F., Etheridge, J. & Humphreys, C. J. (1996). The symmetry of three-beam scattering equations: inversion of three-beam diffraction patterns from centrosymmetric crystals. Acta Cryst. A52, 596–605.
Moodie, A. F., Etheridge, J. & Humphreys, C. J. (1998). The Coulomb interaction and the direct measurement of structural phase. In The Electron, edited by A. Kirkland & P. Brown, pp. 235–246. IOM Communications Ltd. London: The Institute of Materials.
Moodie, A. F. & Whitfield, H. J. (1994). The reduction of N-beam scattering from noncentrosymmetric crystals to two-beam form. Acta Cryst. A50, 730–736.

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