International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B, ch. 5.1, pp. 548-549
Section 5.1.8.2. Borrmann triangle^{a}Laboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France |
The first property of the Borrmann triangle is that the angular density of the wavefield paths is inversely proportional to the curvature of the dispersion surface around their tie points. Let us consider an incident wavepacket of angular width . It will generate a packet of wavefields propagating within the Borrmann triangle. The angular width (Fig. 5.1.8.2) between the paths of the corresponding wavefields is related to the radius of curvature of the dispersion surface by where α is the angle between the wavefield path and the lattice planes [equation (5.1.2.26)] and is called the amplification ratio. In the middle of the reflecting domain, the radius of curvature of the dispersion surface is very much shorter than its value, k, far from it (about 10^{4} times shorter) and the amplification ratio is therefore very large. As a consequence, the energy of a wavepacket of width in reciprocal space is spread in direct space over an angle given by (5.1.8.1). The intensity distribution on the exit surface BC (Fig. 5.1.8.1a) is therefore proportional to . It is represented in Fig. 5.1.8.3 for several values of the absorption coefficient:
References
Borrmann, G. (1959). Röntgenwellenfelder. Beit. Phys. Chem. 20 Jahrhunderts, pp. 262–282. Braunschweig: Vieweg und Sohn.Kato, N. (1960). The energy flow of X-rays in an ideally perfect crystal: comparison between theory and experiments. Acta Cryst. 13, 349–356.
Kato, N. & Lang, A. R. (1959). A study of Pendellösung fringes in X-ray diffraction. Acta Cryst. 12, 787–794.
Laue, M. von (1960). Röntgenstrahl-Interferenzen. Frankfurt am Main: Akademische Verlagsgesellschaft.