Boundary conditions for the amplitudes at the entrance surface – intensities of the reflected and refracted waves

Authier, A.

doi:10.1107/97809553602060000569

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. B. ch. 5.1, p. 542 | 1 | 2 |

Section 5.1.6.2. Boundary conditions for the amplitudes at the entrance surface – intensities of the reflected and refracted waves

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

5.1.6.2. Boundary conditions for the amplitudes at the entrance surface – intensities of the reflected and refracted waves

| top | pdf |

Let us consider an infinite plane wave incident on a crystal plane surface of infinite lateral extension. As has been shown in Section 5.1.3, two wavefields are excited in the crystal, with tie points $[P_{1}]$ and $[P_{2}]$ , and amplitudes $[D_{o1}, D_{h1}]$ and $[D_{o2}, D_{h2}]$ , respectively. Maxwell's boundary conditions (see Section A5.1.1.2 of the Appendix) imply continuity of the tangential component of the electric field and of the normal component of the electric displacement across the boundary. Because the index of refraction is so close to unity, one can assume to a very good approximation that there is continuity of the three components of both the electric field and the electric displacement. As a consequence, it can easily be shown that, along the entrance surface, for all components of the electric displacement $[\eqalign{D_{o}^{(a)} &= D_{o1} + D_{o2}\cr 0 &= D_{h1} + D_{h2},} \eqno(5.1.6.2)]$ where $[D_{o}^{(a)}]$ is the amplitude of the incident wave.

Using (5.1.3.11), (5.1.5.2) and (5.1.6.2), it can be shown that the intensities of the four waves are $[\eqalign{|D_{oj}|^{2} &= |D_{o}^{(a)}|^{2} \exp (- \mu_{j}z/\gamma_{o}) \left[(1 + \eta_{r}^{2})^{1/2} \mp \eta_{r}\right]^{2}\cr &\quad \times [4 (1 + \eta_{r}^{2})]^{-1},\cr |D_{hj}|^{2} &= |D_{o}^{(a)}|^{2} \exp (- \mu_{j}z/\gamma_{o}) |F_{h}/F_{\bar{h}}| [4\gamma (1 + \eta_{r}^{2})]^{-1}\hbox{;}}\eqno(5.1.6.3)]$ top sign: [j = 1] ; bottom sign: [j = 2] .

Fig. 5.1.6.2 represents the variations of these four intensities with the deviation parameter. Far from the reflection domain, $[|D_{h1}|^{2}]$ and $[|D_{h2}|^{2}]$ tend toward zero, as is normal, while $[\eqalign{|D_{o1}|^{2} &\gg |D_{o2}|^{2} \hbox{ for } \eta_{r} \Rightarrow - \infty,\cr |D_{o1}|^{2} &\ll |D_{o2}|^{2} \hbox{ for } \eta_{r} \Rightarrow + \infty.}]$

Figure 5.1.6.2| top | pdf |

Variation of the intensities of the reflected and refracted waves in an absorbing crystal for the 220 reflection of Si using Mo Kα radiation, t = 1 mm (μt = 1.42). Solid curve: branch 1; dashed curve: branch 2.

This result shows that the wavefield of highest intensity `jumps' from one branch of the dispersion surface to the other across the reflection domain. This is an important property of dynamical theory which also holds in the Bragg case and when a wavefield crosses a highly distorted region in a deformed crystal [the so-called interbranch scattering: see, for instance, Authier & Balibar (1970) and Authier & Malgrange (1998)].

References

Authier, A. & Balibar, F. (1970). Création de nouveaux champs d'onde généralisés dus à la présence d'un objet diffractant. II. Cas d'un défaut isolé. Acta Cryst. A26, 647–654.Google Scholar

Authier, A. & Malgrange, C. (1998). Diffraction physics. Acta Cryst. A54, 806–819.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 5.1, p. 542