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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, p. 543
Section 5.1.6.3.2. Amplitudes – Pendellösung
a
Laboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France |
We shall assume from now on that the crystal is plane parallel. Two wavefields arrive at any point of the exit surface. Their constituent waves interfere and generate emerging waves in the refracted and reflected directions (Fig. 5.1.6.4![[link]](/graphics/greenarr.gif)
). Their respective amplitudes are given by the boundary conditions ![[\eqalign{D_{o}^{(d)} \exp (-2\pi i {\bf K}_{{\bf o}}^{(d)} \cdot {\bf r}) &= D_{o1} \exp (-2\pi i {\bf K}_{{{\bf o}1}} \cdot {\bf r})\cr &\quad + D_{o2} \exp (-2\pi i {\bf K}_{{{\bf o}2}} \cdot {\bf r})\cr D_{h}^{(d)} \exp (-2\pi i {\bf K}_{{\bf h}}^{(d)} \cdot {\bf r}) &= D_{h1} \exp (-2\pi i {\bf K}_{{{\bf h}1}} \cdot {\bf r})\cr &\quad + D_{h2} \exp (-2\pi i {\bf K}_{{{\bf h}2}} \cdot {\bf r}),} \eqno(5.1.6.4)]](/teximages/bach5o1/bach5o1fd52.svg)
where r is the position vector of a point on the exit surface, the origin of phases being taken at the entrance surface.
![[Figure 5.1.6.4]](/figures/Bafig5o1o6o4thm.gif) | Decomposition of a wavefield into its two components when it reaches the exit surface. |
![[{\bf S}_{1}]](/teximages/bach5o1/bach5o1fi302.svg)
and ![[{\bf S}_{2}]](/teximages/bach5o1/bach5o1fi303.svg)
are the Poynting vectors of the two wavefields propagating in the crystal belonging to branches 1 and 2 of the dispersion surface, respectively, and interfering at the exit surface.In a plane-parallel crystal, (5.1.6.4) reduces to ![[\eqalign{D_{o}^{(d)} &= D_{o1} \exp \left(-2\pi i \overline{MP_{1}} \cdot t\right) + D_{o2} \exp \left(-2\pi i \overline{MP_{2}} \cdot t\right)\cr D_{h}^{(d)} &= D_{h1} \exp \left(-2\pi i \overline{MP_{1}} \cdot t\right) + D_{h2} \exp \left(-2\pi i \overline{MP_{2}} \cdot t\right),}]](/teximages/bach5o1/bach5o1fd53.svg)
where t is the crystal thickness.
In a non-absorbing crystal, the amplitudes squared are of the form ![[\left|D_{o}^{(d)}\right|^{2} = |D_{o1}|^{2} + |D_{o2}|^{2} + 2 D_{o1} D_{o2} \cos 2\pi \overline{P_{2}P_{1}} t.]](/teximages/bach5o1/bach5o1fd54.svg)
This expression shows that the intensities of the refracted and reflected beams are oscillating functions of crystal thickness. The period of the oscillations is called the Pendellösung distance and is ![[\Lambda = 1/\overline{P_{2}P_{1}} = \Lambda_{L}/(1 + \eta_{r}^{2})^{1/2}.]](/teximages/bach5o1/bach5o1fd55.svg)
