Departure from Bragg's law of the incident wave

Authier, A.

doi:10.1107/97809553602060000569

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

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International Tables for Crystallography (2006). Vol. B. ch. 5.1, p. 538 | 1 | 2 |

Section 5.1.3.1. Departure from Bragg's law of the incident wave

A. Authier^a ^*

^a Laboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France
Correspondence e-mail: authier@lmcp.jussieu.fr

5.1.3.1. Departure from Bragg's law of the incident wave

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The wavefields excited in the crystal by the incident wave are determined by applying the boundary condition mentioned above for the continuity of the tangential component of the wavevectors (Section 5.1.2.3). Waves propagating in a vacuum have wavenumber $[k = 1/\lambda]$ . Depending on whether they propagate in the incident or in the reflected direction, the common extremity, M, of their wavevectors $[{\bf OM} = {\bf K}_{{\bf o}}^{(a)} \hbox{ and } {\bf HM} = {\bf K}_{{\bf h}}^{(a)}]$ lies on spheres of radius k and centred at O and H, respectively. The intersections of these spheres with the plane of incidence are two circles which can be approximated by their tangents $[T'_{o}]$ and $[T'_{h}]$ at their intersection point, $[L_{a}]$ , or Laue point (Fig. 5.1.3.1).

Figure 5.1.3.1| top | pdf |

Departure from Bragg's law of an incident wave.

Bragg's condition is exactly satisfied according to the geometrical theory of diffraction when M lies at $[L_{a}]$ . The departure Δθ from Bragg's incidence of an incident wave is defined as the angle between the corresponding wavevectors OM and $[{\bf OL}_{a}]$ . As Δθ is very small compared to the Bragg angle in the general case of X-rays or neutrons, one may write $[\eqalign{{\bf K}_{{\bf o}}^{(a)} &= {\bf OM} = {\bf OL}_{a} + {\bf L}_{a}{\bf M},\cr \Delta \theta &= \overline{L_{a}M} / k.} \eqno(5.1.3.1)]$

The tangent $[T'_{o}]$ is oriented in such a way that Δθ is negative when the angle of incidence is smaller than the Bragg angle.

References

International Tables for Crystallography (2006). Vol. B. ch. 5.1, p. 538