Diffraction Physics

The main theories of diffraction are brie ̄y described and their more important results compared. The limitations of the geometrical theory are discussed and the concept of extinction introduced. The main features of the diffraction by a perfect crystal are brie ̄y reviewed: total re ̄ection and Darwin width associated with the Bragg gap, standing waves, anomalous absorption, ray tracing, plane-wave and spherical-wave PendelloÈsung, polarization properties. Real crystals are seldom perfect. They may be nearly perfect with small strains and/or individual lattice defects faults or they may be highly deformed with large strains and a high density of defects. The diffraction by the former is handled using extensions of the dynamical theory of diffraction by perfect crystals using ray tracing. The results are analytical in the case of a constant strain gradient and are otherwise described by simulations which can be compared to the experimental results. The latter case is more dif®cult but can be approached by more sophisticated theories such as that of Takagi and Taupin. 1. Introduction Diffraction of waves by crystals has permitted the development of crystallography in the 20th century. It all started with Ewald's thesis and his theory of re ̄ection and refraction, which relates the macroscopic properties of dispersion and refraction in a crystal to the interaction of the propagating waves with a microscopic distribution of resonators, that is with its atomic structure. The derivation does not depend on the wavelength and it is this remark by him in January 1912 in answer to a question by Laue that started off Laue's reasoning and led to Friedrich & Knipping's decisive experiment. It was promptly followed by Laue's geometrical theory and Darwin's geometrical and dynamical theories (Darwin, 1914a,b). Ewald's extension of his theory to the case of X-rays (Ewald, 1916, 1917) shows that refraction and re ̄ection of light waves and X-ray diffraction are essentially the same physical phenomenon. The scope of diffraction physics is very wide, ranging from the interaction of waves with matter to diffraction theory for perfect and imperfect crystals, powders, modulated structures, paracrystals etc., extinction theory, X-ray optics, interferometry, imaging of defects, . and only limited aspects will be broached upon in this paper. 2. The theories of diffraction 2.1. Geometrical theory The basis of Laue's `geometrical theory' of X-ray diffraction is given in the very ®rst of the two papers that gave the account of the discovery of X-ray diffraction (Friedrich et al., 1912): the amplitude diffracted by a three-dimensional periodic assembly of atoms is derived by adding the amplitudes of the waves diffracted by each atom, simply taking into account the optical path differences between them, but neglecting the interaction of the propagating waves and matter. This can be expressed simply using Fourier transforms. The expression of the distribution of electronic density (or more generally of diffracting centres) of a triply periodic in®nite medium, 1 r, can be written as the convolution of the electron density in one cell, 0 r, by a triply AndreÂ Authier studied at the University of Paris and at the Ecole Normale SupeÂrieure. He obtained a DSci from the University of Paris in 1961 and was Professor at the University of Paris (now UniversiteÂ P. et M. Curie) from 1965 to 1996. He became Professor Emeritus in 1997. He was the ®rst President of the European Crystallographic Committee (now European Crystallographic Association) from 1972 to 1975. He was President of the International Union of Crystallography from 1990 to 1993. He is Editor of Section A of Acta Crystallographica and of Volume D of International Tables for Crystal-