International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 5.2, p. 650
Section 5.2.10. Bloch-wave formulations^{a}Department of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia,^{b}Arizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287–1504, USA, and ^{c}School of Physics, University of Melbourne, Parkville, Australia |
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 where has the periodicity of the lattice, and hence may be expanded in a Fourier series to give The 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:
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.