Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 307-308   | 1 | 2 |

Section Introduction

M. Tanakaf Introduction

| top | pdf |

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.


Cherns, D., Kiely, C. J. & Preston, A. R. (1988). Electron diffraction studies of strain in epitaxial bicrystals and multilayers. Ultramicroscopy, 24, 355–370.
Cherns, D. & Preston, A. R. (1986). Convergent beam diffraction studies of crystal defects. Proc. XI Int. Congr. Electron Microsc., Kyoto, Japan, p. 721.
Goodman, P. & Lehmpfuhl, G. (1965). Elektronenbeugungsunter­suchungen im konvergenten bundel mit dem Siemens Elmiskop I. Z. Naturforsch. Teil A, 20, 110–114.
Goodman, P. & Lehmpfuhl, G. (1967). Electron diffraction study of MgO h00-systematic interactions. Acta Cryst. 22, 14–24.
Kossel, W. & Möllenstedt, G. (1938). Electron interference in a convergent beam. Nature (London), 26, 660.
Krämer, S., Mayer, J., Witt, C., Weikenmeier, A. & Rühle, M. (2000). Ultramicroscopy, 81, 245–262.
Midgley, P. A., Sleight, M. E. & Vincent, R. (1996). The structure of a metastable Au–Sn phase determined by convergent beam electron diffraction. J. Solid State Chem. 124, 132–142.
Morniroli, J. P. (2002). Large-Angle Convergent Beam Electron Diffraction. Paris: French Society of Microscopy.
Ogata, Y., Tsuda, K., Akishige, Y. & Tanaka, M. (2004). Refinement of the crystal structural parameters of the intermediate phase of h-BaTiO3 using convergent-beam electron diffraction. Acta Cryst. A60, 525–531.
Rossouw, C. J., Gibson, M. A. & Forwood, C. T. (1996). Dynamical electron diffraction analysis of lattice parameters, Debye–Waller factors and order in Ti–Al and Ti–Ga alloys. Ultramicroscopy, 66, 193–209.
Saunders, M., Bird, D. M., Zaluzee, N. J., Burgess, W. G., Preston, A. R. & Humphreys, C. J. (1995). Measurement of the low-order structure factors for silicon from zone-axis CBED pattern. Ultramicroscopy, 60, 311–323.
Spence, J. C. H. (1993). On the accurate measurement of structure-factor amplitudes and phases by electron diffraction. Acta Cryst. A49, 231–260.
Spence, J. C. H. & Zuo, J. M. (1992). Electron Microdiffraction. New York: Plenum Press.
Tanaka, M. (1986). Conventional transmission-electron-microscopy techniques in convergent-beam electron diffraction. J. Electron Microsc. 35, 314–323.
Tanaka, M., Saito, P., Ueno, K. & Harada, Y. (1980). Large angle convergent-beam electron diffraction. J. Electron. Microsc. 29, 408–412.
Tanaka, M., Terauchi, M. & Tsuda, K. (1994). Convergent-Beam Electron Diffraction III. Tokyo: JEOL–Maruzen.
Tanaka, M. & Tsuda, K. (1990). Determination of positional parameters by convergent-beam electron diffraction. Proc. XIIth Int. Congr. Electron Microsc., Seattle, edited by L. D. Peachy & D. B. Williams, Vol. 2, pp. 518–519. San Francisco: San Francisco Press.
Tanaka, M. & Tsuda, K. (1991). Microbeam Analysis, edited by D. G. Howitt, pp. 145–146. San Francisco: San Francisco Press.
Tsuda, K., Ogata, Y., Takagi, K., Hashimoto, T. & Tanaka, M. (2002). Refinement of crystal structural parameters and charge density using convergent-beam electron diffraction – the rhombohedral phase of LaCrO3. Acta Cryst. A58, 514–525.
Tsuda, K., Saito, M., Terauchi, M., Tanaka, M., Tsai, A. P., Inoue, A. & Masumoto, K. (1993). Electron microscope study of decagonal quasicrystals of Al70Ni15Fe15. Jpn. J. Appl. Phys. 32, 129–134.
Tsuda, K. & Tanaka, M. (1995). Refinement of crystal structure parameters using convergent-beam electron diffraction: the low-temperature phase of SrTiO3. Acta Cryst. A51, 7–19.
Tsuda, K. & Tanaka, M. (1999). Refinement of crystal structural parameters using two-dimensional energy-filtered CBED patterns. Acta Cryst. A55, 939–954.
Vincent, R., Bird, D. M. & Steeds, J. W. (1984a). Structure of AuGeAs determined by convergent-beam electron-diffraction. 1. Derivation of basic structure. Philos. Mag. A, 50, 745–763.
Vincent, R., Bird, D. M. & Steeds, J. W. (1984b). Structure of AuGeAs determined by convergent-beam electron-diffraction. 2. Refinement of structural parameters. Philos. Mag. A, 50, 765–786.
Zuo, J. M., Kim, M., O'Keeffe, M. & Spence, J. C. H. (1999). Observation of d holes and Cu–Cu bonding in cuprite. Nature (London), 401, 49–52.

to end of page
to top of page