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

International Tables for Crystallography (2006). Vol. B, ch. 5.1, p. 536   | 1 | 2 |

Section 5.1.2.3. Boundary conditions at the entrance surface

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

5.1.2.3. Boundary conditions at the entrance surface

| top | pdf |

The choice of the `o' component of expansion (5.1.2.15)[link] is arbitrary in an infinite medium. In a semi-infinite medium where the waves are created at the interface with a vacuum or air by an incident plane wave with wavevector [{\bf K}_{{\bf o}}^{(a)}] (using von Laue's notation), the choice of [{\bf K}_{{\bf o}}] is determined by the boundary conditions.

This condition for wavevectors at an interface demands that their tangential components should be continuous across the boundary, in agreement with Descartes–Snell's law. This condition is satisfied when the difference between the wavevectors on each side of the interface is parallel to the normal to the interface. This is shown geometrically in Fig. 5.1.2.2[link] and formally in (5.1.2.18)[link]: [{\bf K}_{{\bf o}} - {\bf K}_{{\bf o}}^{(a)} = {\bf OP} - {\bf OM} = \overline{MP} \cdot {\bf n}, \eqno(5.1.2.18)] where n is a unit vector normal to the crystal surface, oriented towards the inside of the crystal.

[Figure 5.1.2.2]

Figure 5.1.2.2 | top | pdf |

Boundary condition for wavevectors at the entrance surface of the crystal.

There is no absorption in a vacuum and the incident wavevector [{\bf K}_{{\bf o}}^{(a)}] is real. Equation (5.1.2.18)[link] shows that it is the component normal to the interface of wavevector [{\bf K}_{{\bf o}}] which has an imaginary part, [{\bf K}_{{\bf o}i} = {\cal I} (\overline{MP}) \cdot {\bf n} = - \mu {\bf n} / (4\pi \gamma_{o}), \eqno(5.1.2.19)] where [{\cal I}(\;f)] is the imaginary part of f, [\gamma_{o} = \cos ({\bf n} \cdot {\bf s}_{{\bf o}})] and [{\bf s}_{{\bf o}}] is a unit vector in the incident direction. When there is more than one wave in the wavefield, the effective absorption coefficient μ can differ significantly from the normal value, [\mu_{o}], given by (5.1.2.13)[link] – see Section 5.1.5[link].








































to end of page
to top of page