Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 5.2, p. 552   | 1 | 2 |

Section 5.2.2. The defining equations

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 3052

5.2.2. The defining equations

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No many-body effects have yet been detected in the diffraction of fast electrons, but the velocities lie well within the relativistic region. The one-body Dirac equation would therefore appear to be the appropriate starting point. Fujiwara (1962[link]), using the scattering matrix, carried through the analysis for forward scattering, and found that, to a very good approximation, the effects of spin are negligible, and that the solution is the same as that obtained from the Schrödinger equation provided that the relativistic values for wavelength and mass are used. In effect a Klein–Gordon equation (Messiah, 1965[link]) can be used in electron diffraction (Buxton, 1978[link]) in the form [\nabla^{2} \psi_{b} + {8\pi^{2} m|e|\varphi\over h^{2}} \psi_{b} + {8 \pi^{2} m_{0}|e|W\over h^{2}} \left(1 + {|e|W\over 2m_{0}c^{2}}\right) \psi_{b} = 0.] Here, W is the accelerating voltage and [\varphi], the potential in the crystal, is defined as being positive. The relativistic values for mass and wavelength are given by [m = m_{0} (1 - v^{2} / c^{2})^{-1/2}], and taking `e' now to represent the modulus of the electronic charge, [|e|], [\lambda = h[2m_{0}eW (1 + eW / 2m_{0}c^{2})]^{-1/2},] and the wavefunction is labelled with the subscript b in order to indicate that it still includes back scattering, of central importance to LEED (low-energy electron diffraction).

In more compact notation, [[\nabla^{2} + k^{2} (1 + \varphi / W)] \psi_{b} = (\nabla^{2} + k^{2} + 2 k\sigma \varphi) \psi_{b} = 0. \eqno(] Here [k = |{\bf k}|] is the scalar wavenumber of magnitude [2\pi / \lambda], and the interaction constant [\sigma = 2\pi me\lambda / h^{2}]. This constant is approximately [10^{-3}] for 100 kV electrons.

For fast electrons, [\varphi / W] is a slowly varying function on a scale of wavelength, and is small compared with unity. The scattering will therefore be peaked about the direction defined by the incident beam, and further simplification is possible, leading to a forward-scattering solution appropriate to HEED (high-energy electron diffraction).


Buxton, B. (1978). Graduate lecture-course notes: dynamical diffraction theory. Cambridge University, England.
Fujiwara, K. (1962). Relativistic dynamical theory of electron diffraction. J. Phys. Soc. Jpn, 17, Suppl. B11, 118–123.
Messiah, A. (1965). Quantum mechanics, Vol. II, pp. 884–888. Amsterdam: North-Holland.

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