Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 306   | 1 | 2 |

Section Crystal structure imaging

J. M. Cowleya Crystal structure imaging

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  • (a) Introduction. It follows from ([link] and ([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 ([link] by [\Psi_{0} ({\bf h}) \cdot T ({\bf h}). \eqno (]For rectangular unit cells of the projected unit cell, the vector h has components [h/a] and [k/b]. Then the set of amplitudes ([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 (]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 (]except that in this case the image is translated by a distance [a/2] parallel to each of the axes.


International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited E. Prince, 3rd ed. Dordrecht: Kluwer Academic Publishers.
Bethe, H. A. (1928). Theorie der Beugung von Elektronen an Kristallen. Ann. Phys. (Leipzig), 87, 55–129.
Cowley, J. M. & Moodie, A. F. (1957). The scattering of electrons by atoms and crystals. I. A new theoretical approach. Acta Cryst. 10, 609–619.
Cowley, J. M. & Moodie, A. F. (1960). Fourier images. IV. The phase grating. Proc. Phys. Soc. London, 76, 378–384.
Hirsch, P. B., Howie, A., Nicholson, R. B., Pashley, D. W. & Whelan, M. J. (1965). Electron Microscopy of Thin Crystals. London: Butterworths.
Kuwabara, S. (1978). Nearly aberration-free crystal images in high voltage electron microscopy. J. Electron Microsc. 27, 161–169.
Spence, J. C. H., O'Keefe, M. A. & Kolar, H. (1977). Image interpretation in crystalline germanium. Optik (Stuttgart), 49, 307–323.

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