J. Phys. I France
Volume 1, Numéro 8, August 1991
Page(s) 1187 - 1193
DOI: 10.1051/jp1:1991199
J. Phys. I France 1 (1991) 1187-1193

Diffraction in crystals and topology

V. E. Dmitrienko

Institute of Crystallography, Leninsky pr. 59, Moscow, 117333 U.S.S.R.

(Received 20 March 1991, revised 10 April 1991, accepted 25 April 1991)

It is shown that some simple ideas of topology can be applied to the diffraction theory. As an example, the phase of the wave transmitted through the crystal is considered for the Laue diffraction geometry. When the angle of incidence varies across the diffraction region, the phase of the transmitted wave can be changed on $2\pi N$, where N is an integer. The number N is a topological invariant (topological charge) of the diffraction region: in the general case, this number does not change its value when any parameters of experiment are slightly varied, but it changes by unity for some fixed values of those parameters. Therefore, from the topological point of view, the regions of diffraction are analogous to the domain walls in ferroelectrics. Then, within this approach, the singular points are revealed inside the diffraction region. It is shown that the topological ideas are especially fruitful for those cases where the solution of diffraction problem is absent or formidable (for instance, the multiple-beam diffraction or diffraction in imperfect crystals). The topological charge is also important for the dispersion relations connecting the phase and the amplitude of the transmitted wave in the diffraction region.

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