Bloch-wave formulations

Moodie, A. F.; Cowley, J. M.; Goodman, P.

doi:10.1107/97809553602060000570

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 5.2, p. 555 | 1 | 2 |

Section 5.2.10. Bloch-wave formulations

A. F. Moodie,^a J. M. Cowley^b ^‡ and P. Goodman^c ^‡

^a Department 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

5.2.10. Bloch-wave formulations

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In developing the theory from the beginning by eigenvalue techniques, it is usual to invoke the periodicity of the crystal in order to show that the solutions to the wave equation for a given wavevector k are Bloch waves of the form $[\psi = C({\bf r}) \exp \{i {\bf k} \cdot {\bf r}\},]$ where $[C({\bf r})]$ has the periodicity of the lattice, and hence may be expanded in a Fourier series to give $[\psi = {\textstyle\sum\limits_{\bf h}} C_{\bf h} ({\bf k}) \exp \{i ({\bf k} + 2\pi {\bf h}) \cdot {\bf r}\}. \eqno(5.2.10.1)]$ The $[C_{\bf h} ({\bf k})]$ are determined by equations of consistency obtained by substitution of equation (5.2.10.1) into the wave equation.

If N terms are selected in equation (5.2.10.1) there will be N Bloch waves where wavevectors differ only in their components normal to the crystal surface, and the total wavefunction will consist of a linear combination of these Bloch waves. The problem is now reduced to the problem of equation (5.2.8.2).

The development of solutions for particular geometries follows that for the X-ray case, Chapter 5.1 , with the notable differences that:

(1) The two-beam solution is not adequate except as a first approximation for particular orientations of crystals having small unit cells and for accelerating voltages not greater than about 100 keV. In general, many-beam solutions must be sought.
(2) For transmission HEED, the scattering angles are sufficiently small to allow the use of a small-angle forward-scattering approximation.
(3) Polarization effects are negligible except for very low energy electrons.

Humphreys (1979) compares the action of the crystal, in the Bloch-wave formalism, with that of an interferometer, the incident beam being partitioned into a set of Bloch waves of different wavevectors. `As each Bloch wave propagates it becomes out of phase with its neighbours (due to its different wavevector). Hence interference occurs. For example, if the crystal thickness varies, interference fringes known as thickness fringes are formed.' For the two-beam case, these are the fringes of the pendulum solution referred to previously.

References

Humphreys, C. J. (1979). The scattering of fast electrons by crystals. Rep. Prog. Phys. 42, 1825–1887.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 5.2, p. 555