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This chapter provides a historical introduction to X-ray dynamical diffraction. It starts with an account of Ewald's thesis on the dispersion of light and of the famous experiment of the diffraction of X-rays by crystals by M. Laue, W. Friedrich, and P. Knipping. The successive steps in the development of the theory of X-ray diffraction are then summarized: Laue's and Darwin's geometrical theories; Darwin's, Ewald's, and Laue's dynamical theories; early experimental proofs, the notion of extinction and the mosaic crystal model, observation in the fifties and sixties of the fundamental properties of the X-ray wavefields in crystals (anomalous absorption and the Borrmann effect, double refraction, Pendellösung, bent trajectories in deformed crystals), extension of the dynamical theory to the case of deformed crystals, modern applications for the characterization of crystal defects and X-ray optics for synchrotron radiation.

This chapter is the first of the next few chapters devoted to plane-wave advanced dynamical theory. The fundamental equations of dynamical diffraction are derived for vector waves and the expression of the dispersion equation is given in the two-beam case and for absorbing crystals, the following discussion being limited to geometrical situations where neither the incidence nor the emergence angle is grazing. The notion of wavefields and the dispersion surface are introduced, and it is shown that the Poynting vector, which gives the direction of propagation of the energy, is normal to it. The boundary conditions at the entrance surface are then introduced. Transmission and reflection geometries are treated separately. For each case, the deviation parameter is introduced geometrically and the coordinates of the tiepoints determined, the Pendellösung distance (extinction distance in the reflection geometry), Darwin width, the anomalous absorption coefficient, index of refraction, the phase and amplitude ratios of the reflected and refracted waves are calculated. Borrmann's standing wave interpretation of the anomalous absorption effect is given. The last section is to the case where Bragg's angle is close to π/2.

This chapter presents the basic properties of dynamical diffraction in an elementary way. The relationship with the band theory of solids is explained. The fundamental equations of dynamical theory are given for scalar waves as a simplification; the solutions of the propagation equation are then derived for an incident plane wave in the 2-beam case; and the amplitude ratio between reflected and refracted waves deduced. The notions of wavefields, dispersion surface, and tie points are introduced. Two experimental set-ups are considered: transmission and reflection geometries. The boundary conditions at the entrance surface of the crystal are expressed in each case and the intensities of the refracted and reflected waves calculated as well as the anomalous absorption coefficient, due to the Borrmann effect, the Pendellösung interference fringe pattern and the integrated intensity. It is shown that the geometrical diffraction constitutes a limit of dynamical diffraction by small crystals. At the end of the chapter dynamic diffraction by quasicrystals is considered.

This chapter is concerned with the cases where several reciprocal lattice points are close to the Ewald sphere and several waves simultaneously excited (multiple-beam or n-beam diffraction). The principle of Renninger-scans is given and it is shown how the solutions of the fundamental equations of the dynamical theory are obtained in the general case. The particular case of the three-beam coplanar case is then considered. One section in this chapter is devoted to the determination of absolute phases using n-beam absorption and its application for structure determinations. The last section explains the enhancement of the anomalous absorption effect (super-Borrmann effect) in specific three-beam cases.