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

International Tables for Crystallography (2006). Vol. B, ch. 5.1, pp. 540-541   | 1 | 2 |

Section 5.1.3.7. Geometrical interpretation of the solution in the zero-absorption case

A. Authiera*

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

5.1.3.7. Geometrical interpretation of the solution in the zero-absorption case

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5.1.3.7.1. Transmission geometry

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In this case (Fig. 5.1.3.4[link]) [\hbox{S}(\gamma_{h})] is +1 and (5.1.3.10)[link] may be written [\xi_{j} = -\hbox{S}(C) \left[\eta \pm (\eta^{2} + 1)^{1/2}\right] / \gamma^{1/2}. \eqno(5.1.3.11)]

Let [A_{1}] and [A_{2}] be the intersections of the normal to the crystal surface drawn from the Lorentz point [L_{o}] with the two branches of the dispersion surface (Fig. 5.1.3.4[link]). From Sections 5.1.3.3[link] and 5.1.3.4[link], they are the tie points excited for [\eta = 0] and correspond to the middle of the reflection domain. Let us further consider the tangents to the dispersion surface at [A_{1}] and [A_{2}] and let [I_{o1}], [I_{o2}] and [I_{h1}], [I_{h2}] be their intersections with [T_{o}] and [T_{h}], respectively. It can be shown that [\overline{I_{o1}I_{h2}}] and [\overline{I_{o2}I_{h1}}] intersect the dispersion surface at the tie points excited for [\eta = -1] and [\eta = +1], respectively, and that the Pendellösung distance [\Lambda_{L} = 1/\overline{A_{2}A_{1}}], the width of the rocking curve [2\delta = \overline{I_{o1}I_{o2}}/k] and the deviation parameter [\eta = \overline{M_{o}M_{h}}/\overline{A_{2}A_{1}}], where [M_{o}] and [M_{h}] are the intersections of the normal to the crystal surface drawn from the extremity of any incident wavevector OM with [T_{o}] and [T_{h}], respectively.

5.1.3.7.2. Reflection geometry

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In this case (Fig. 5.1.3.5[link]) [\hbox{S}(\gamma_{h})] is now −1 and (5.1.3.10)[link] may be written [\xi_{j} = \hbox{S}(C) \left[\eta \pm (\eta^{2} - 1)^{1/2}\right] / |\gamma|^{1/2}. \eqno(5.1.3.12)]

Now, let [I_{1}] and [I_{2}] be the points of the dispersion surface where the tangent is parallel to the normal to the crystal surface, and further let [I_{o1}, I_{h1}, I'] and [I_{o2}, I_{h2}, I''] be the intersections of these two tangents with [T_{o}, T_{h}] and [T'_{o}], respectively. For an incident wave of wavevector OM where M lies between I′ and I″, the normal to the crystal surface drawn from M has no real intersection with the dispersion surface and [\overline{I'I''}] defines the total-reflection domain. The tie points [I_{1}] and [I_{2}] correspond to [\eta = -1] and [\eta = +1], respectively, the extinction distance [\Lambda_{L} = 1/\overline{I_{o1}I_{h1}}], the width of the total-reflection domain [2\delta = \overline{I_{o1}I_{o2}}/k = \overline{I'I''}/k] and the deviation parameter [\eta = - \overline{M_{o}M_{h}} / \overline{I_{o1}I_{h1}}], where [M_{o}] and [M_{h}] are the intersections with [T_{o}] and [T_{h}] of the normal to the crystal surface drawn from the extremity of any incident wavevector OM.








































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