Dynamical Theory of X-Ray Diffraction

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1 Dynamical Theory of X-Ray Diffraction ANDRE AUTHIER Universite P. et M. Curie, Paris OXFORD UNIVERSITY PRESS

2 Contents I Background and basic results 1 1 Historical developments Prologue The discovery of X-ray diffraction The geometrical theory of diffraction Darwin's dynamical theory of diffraction Extinction theories Ewald's dynamical theory Early confirmations of the dynamical theory Laue's dynamical theory Umweganregung and Aufhellung The properties of wavefields Anomalous absorption (the Borrmann effect) Wavefield trajectories Pendellosung Diffraction by deformed crystals Modern times 26 2 Properties of the electromagnetic field propagation and scattering Maxwell's equations The electrodynamic potentials in vacuum The vector and scalar potentials The retarded potentials The electrodynamic potentials in polarized media Hertz vectors (polarization potentials) Propagation of an electromagnetic wave in vacuum Scattering of X-rays by an electron Polarizability of matter for X-rays Elementary dispersion theory Fourier expansion of the polarizability Index of refraction Absorption Ewald's dispersion theory Propagation equation of an electromagnetic wave in materials in Laue's dynamical theory Laue's basic assumption 49

3 2.9.2 Propagation equation Specular reflection Fresnel relations 50 Geometrical theory of X-ray diffraction Classical scattering by an electron polarization Amplitude diffracted by a periodic electron distribution Intensity diffracted by a small crystal Reflectivity Integrated intensity Mosaic crystals 67 Elementary dynamical theory Limitations of the geometrical theory Introduction of the dispersion surface Analogy with the band theory of solids Propagation equation Fundamental equations of dynamical theory Amplitude ratio of the refracted and reflected waves Solutions of plane-wave dynamical theory Boundary conditions Departure from Bragg's angle of the incident wave Transmission and reflection geometries Deviation parameter Determination of the tiepoints Effective absorption coefficient The diffracted waves in the transmission geometry Double refraction Boundary conditions for the amplitudes at the entrance surface Intensities of the reflected and refracted waves Anomalous absorption Boundary conditions at the exit surface Reflectivity Pendellosung Integrated intensity The diffracted waves in the reflection geometry Tiepoints Thick crystals total reflection Thin crystals Influence of the asymmetry on the position and width of the rocking curve" and of the angular distribution of the reflected beam Comparison with geometrical theory Dynamical diffraction by quasicrystals 110

4 xi II Advanced dynamical theory Properties of wavefields Relations between the field vectors Fundamental equations of the dynamical theory The dispersion equation in the two-beam case Poynting vector of the wavefields Determination of the tiepoints geometrical interpretation of the deviation parameter Boundary condition for the wavevectors Deviation from Bragg's angle of the middle of the reflection domain Coordinates of the tiepoint Deviation parameter, Pendellosung distance and Darwin width in the transmission geometry Deviation parameter, extinction distance, penetration depth and Darwin width in the reflection geometry Index of refraction for dynamical diffraction The deviation parameter in absorbing crystals Amplitude ratio of the refracted and reflected waves Phase of the amplitude ratio in the transmission geometry Phase of the amplitude ratio in the reflection geometry Anomalous absorption Effective absorption coefficient in the transmission geometry Absorption coefficient in the propagation direction Discussion of anomalous absorption properties of the standing wavefield Anomalous absorption in the reflection geometry penetration depth Dispersion surface when the Bragg angle is close to n/ Deviation from Bragg's angle and Darwin width Dispersion surface Penetration depth Applications Intensities of plane waves in the transmission geometry Boundary conditions for the amplitudes at the entrance surface Amplitudes of the refracted and reflected waves 157

5 6.3 Boundary conditions for the wavevectors at the exit surface Condition for the existence of two outgoing waves Wavevectors of the outgoing waves (Laue-Laue geometry) Laue-Bragg geometry Rocking curves of the reflected and refracted beams Boundary conditions for the amplitudes at the exit surface Reflectivity Properties of the rocking curves Integrated intensity 170 Intensities of plane waves in the reflection geometry Thick absorbing crystals Reflectivity Shape of the rocking curves Standing waves Thin crystals Boundary conditions for the amplitudes Reflectivity 186 Dynamical diffraction in highly asymmetric coplanar and non-coplanar geometries Introduction Diffraction at grazing incidence or grazing emergence Deviation from Bragg's incidence of the middle of the reflection domain Grazing incidence and Bragg geometry Grazing incidence, Laue geometry Grazing emergence Variation of the Darwin width for a grazing incidence Variation of the width of the diffracted beam for a grazing emergence Equation of the dispersion surface Relation with the traditional dynamical theory Specularly and Bragg-reflected intensities Boundary conditions for the amplitudes at the entrance surface Specularly and Bragg-reflected intensities for a grazing incidence and the Bragg geometry (semi-infinite crystal) Specularly and Bragg-reflected intensities for a grazing incidence and the Laue geometry Grazing incidence diffraction (non-coplanar geometry) 213

6 xiii Introduction Three-dimensional representation of the dispersion surface Tiepoints excited by the incident wave Equation of the dispersion surface Amplitudes of the waves n-beam dynamical diffraction Introduction The general three-beam case Renninger-scans Fundamental equations of the dynamical theory Solution in the general case Energy flow The three-beam coplanar case Determination of phases using n-beam diffraction The super-borrmann effect Experimental evidence Solution of the 111,111 case Anomalous absorption coefficient Spherical-wave dynamical theory: I. Kato's theory Extension of the dynamical theory to any kind of incident wave Fourier expansion of a spherical wave in plane waves Principle of Kato's spherical-wave theory The incident wave is a scalar wave The incident wave is a vector wave Direct integration in the transmission geometry The geometrical conditions Stationary phase method Amplitude distribution on the exit surface reflected wave Amplitude distribution on the exit surface refracted wave Intensity distribution on the exit surface Equal-intensity (Pendellosung) fringes Integration by the stationary phase method Integrated intensity Influence of polarization Bragg geometry 269 Appendix: Geometrical interpretation of rj/^s(yh) + rj 2 in the transmission geometry 274

7 xiv CONTENTS 11 Spherical-wave dynamical theory: II. Takagi's theory Introduction Generalized fundamental equations Modulated waves Takagi's equations Boundary conditions for the amplitudes at the entrance surface Reduction of Takagi's equations in the plane-wave case Absorbing crystals Analytical resolution of Takagi's equations for perfect crystals Analytical solution for a point source using the method of integral equations Transmission geometry Reflection geometry Analytical resolution of Takagi's equations using the Riemann function Hyperbolic nature of Takagi's equations General expression of the reflected and refracted waves Determination of the Riemann function General solution of Takagi's equations Analytical solution for an incident spherical wave using the method of Riemann functions The incident wave is a point source located on the entrance surface The incident wave is a point source located away from the entrance surface Conservation of energy 298 Appendix: Hyperbolic partial differential equations 299 Characteristics 299 Adjoint differential expression Ray tracing in perfect crystals Ray tracing The structure of real waves Wavepackets made of the superposition of separate plane waves Wavepackets made of a continuous distribution of wavevectors Group velocity and Poynting vector Angular amplification Intensity distribution along the base of the Borrmann triangle (transmission geometry) 317

8 xv 12.8 Geometrical properties of wavefield trajectories within the Borrmann triangle Wavefields propagating along the median, AE, of the Borrmann triangle Properties of the trajectories of the two wavefields excited by a plane wave Experimental proof of double refraction Experimental observation of the separation of the wavefield paths Experimental setup Focalization of the various wavelengths Separation of wavefield paths in the transmission case Plane-wave Pendellosung Application to the measurement of the index of refraction Fresnel diffraction near the Bragg incidence Ray tracing in finite crystals Introduction Bragg-Laue geometry pseudo-plane waves Bragg-Bragg geometry; multiple reflections of a pseudo-plane wave in thin crystals Laue-Bragg geometry Borrmann-Lehmann fringes Coherence of extended, non-strictly monochromatic sources 349 III Extension of the dynamical theory to deformed crystals Ray tracing in slightly deformed crystals X-ray propagation in deformed materials The different degrees of deformation Principle of ray theories for weak deformations Effective misorientation Local reciprocal lattice vector Effective misorientation in direct space Effective misorientation in reciprocal space Strain gradient Polarizability of a deformed crystal The Eikonal approximation Justification of the concept of local dispersion surface Fermat's principle Ray trajectories Local dispersion surface Local wavevectors Differential equation of the wavefield trajectories 369

9 13.6 The case of a constant strain gradient Equation of the ray trajectory with respect to the lattice planes Ray trajectories in the transmission geometry Pure bending Temperature gradient Ray trajectories in the reflection geometry Diffracted intensities plane-wave case Zero absorption Absorbing crystals (transmission geometry) Expression of the diffracted intensities for a constant strain gradient Discussion of the intensity distribution for a constant strain gradient Diffracted intensities spherical-wave case Pendellosung in slightly deformed crystals Phase of the refracted wave in a deformed crystal Expression of the phase in terms of the coordinates in direct space Shape of the Pendellosung fringes in a deformed crystal Propagation of X-rays in highly deformed crystals Introduction Takagi's equations in a deformed crystal Resolution of Takagi's equations in the deformed crystal case Small deformations, limit of the validity of the Eikonal approximation Analytical resolution of Takagi's equations Numerical integration Applications Ray concept applied to highly distorted crystals Generalization of the notion of wavefields, interbranch scattering Example: X-ray propagation in a crystal with a concentration gradient (Keitel et al. 1999) Statistical dynamical theories Introduction Principle of Kato's statistical dynamical theory Experimental tests of the statistical dynamical theory 431 Appendix: Resolution of Takagi's equations in the case of a constant strain gradient using Laplace transforms (Katagawa and Kato 1974) 432

10 xvii IV Applications X-ray optics X-ray sources X-ray tubes Synchrotron radiation Flat monochromators Introduction Monochromator crystals Multiple-reflection monochromators Applications of multiple-crystal arrangements to beam conditioning Suppression of tails Wavelength scanner Production of beams with a very narrow angular spread Harmonic suppression Production of polarized radiation Focusing optics Introduction Mirrors Multilayers Curved crystals Fresnel zone plates Bragg-Fresnel lenses Refractive lenses X-ray wave-guides X-ray interferometers Principle Applications Imaging with X-rays Introduction Phase contrast imaging 16 Location of atoms at surfaces and interfaces using X-ray standing waves Principle Theory Fluorescence yield Influence of thermal vibrations 16.3 Bulk crystals Extinction effect Determination of the polarity of heteropolar crystals 16.4 Solution to the surface resistration oroblem

11 Thin films and buried interfaces Simple model Calculation of the standing pattern in an overlayer with the dynamical theory Standing waves in deformed crystals Standing waves due to specular reflection X-ray diffraction topography Introduction Single-crystal reflection topography (Berg-Barrett technique) Principle Image formation Penetration depth Stereographic views Applications Single-crystal transmission topography Early history Principle of section topographs Projection topographs Dislocation images Images of planar defects Applications Double- or multiple-crystal topography Principle of double-crystal topography Plane-wave topography Synchrotron double-crystal topography Mapping of distortions and of lattice parameter variations Equal-strain or equal-lattice parameter contours Double-crystal setting for high spatial resolution topography Appendices Appendix 1 Useful formulae 571 Appendix 2 The early days of dynamical theory 576 References 583 List of symbols 637 Index 641

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