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. 542-543
Section 5.1.6.3. Boundary conditions at the exit surface^{a}Laboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France |
When a wavefield reaches the exit surface, it breaks up into its two constituent waves. Their wavevectors are obtained by applying again the condition of the continuity of their tangential components along the crystal surface. The extremities, and , of these wavevectors lie at the intersections of the spheres of radius k centred at O and H, respectively, with the normal n′ to the crystal exit surface drawn from (j = 1 and 2) (Fig. 5.1.6.3).
If the crystal is wedge-shaped and the normals n and n′ to the entrance and exit surfaces are not parallel, the wavevectors of the waves generated by the two wavefields are not parallel. This effect is due to the refraction properties associated with the dispersion surface.
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). Their respective amplitudes are given by the boundary conditions where r is the position vector of a point on the exit surface, the origin of phases being taken at the entrance surface.
In a plane-parallel crystal, (5.1.6.4) reduces to where t is the crystal thickness.
In a non-absorbing crystal, the amplitudes squared are of the form 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