Multiple Diffraction of X-Rays in Crystals

The three-dimensional arrangement of atoms and molecules in crystals and the comparable magnitude of x-ray wavelengths and interatomic distances make it possible for crystals to have more than one set of atomic planes that satisfy Bragg's law and simultaneously diffract an incident x-ray beam - this is the so-called multiple diffraction. This type of diffraction should, in prin­ ciple, reflect three-dimensional information about the structure of the dif­ fracting material. Recent progress in understanding this diffraction phenome­ non and in utilizing this diffraction technique in solid-state and materials sciences reveals the diversity as well as the importance of multiple diffraction of x-rays in application. Unfortunately, there has been no single book written that gives a sys­ tematic review of this type of diffraction, encompasses its diverse applica­ tions, and foresees future trends gf development. It is for this purpose that this book is designed. It is hoped that its appearance may possibly turn more attention of condensed-matter physicists, chemists and material scientists toward this particular phenomenon, and that new methods of non-destructive analysis of matter using this diffraction technique may be developed in the future.

1. Introduction.- 2. Geometry, Peak Indexing, and Experimental Techniques.- 2.1 Geometry of Multiple Diffraction.- 2.1.1 Real-Space and Reciprocal-Space Representations.- 2.1.2 Persistent and Coincidental Multiple Diffractions.- 2.1.3 Lattice-Symmetry Dependence―Intrinsic Multiple Diffraction.- 2.1.4 Multiple-Diffraction Possibilities.- 2.1.5 Decomposition of Multiple Diffraction.- 2.2 Experimental Techniques for Obtaining Multiple Diffraction.- 2.2.1 Collimated-Beam Technique.- 2.2.2 Divergent-Beam Techniques.- 2.3 Indexing Multiple Diffraction Patterns.- 2.3.1 Reference Vector Method.- 2.3.2 Orientation Matrix Method.- 2.4 Indexing Kossel Patterns.- 2.4.1 Stereographic Projection Method.- 2.4.2 Combined Gnomonic-Stereographic Projection Method.- 3. Kinematical Theory of Diffraction.- 3.1 Equation of Power Transfer for Multi-Beam Cases.- 3.2 Approximate Solutions to the Equation of Power Transfer.- 3.3 Integrated Intensity and the Lorentz-Polarization Factors.- 3.4 Path Lengths of X-Ray Beams in Crystals.- 3.5 Exact Solution to the Power-Transfer Equation.- 3.6 Iterative Calculation for Reflection Power.- 3.7 Dynamical Treatment for Kinematical Reflections.- 3.8 Diffraction in Multi-Layered Crystals.- 3.9 Peak Width, Beam Divergence, and Mosaic Spread.- 4. Dynamical Theory of X-Ray Diffraction.- 4.1 Fundamental Equation of Wavefields.- 4.2 Polarization of Wavefields.- 4.3 Dispersion Surface.- 4.3.1 Geometry of the Dispersion Surface.- 4.3.2 Excitation of the Dispersion Surface.- 4.4 Energy Flow.- 4.4.1 Poynting Vectors and the Dispersion Surface.- 4.4.2 Group Velocity and Energy Flow.- 4.5 Modes of Wave Propagation.- 4.5.1 Wavefields of Modes.- 4.5.2 Number of Permitted Modes.- 4.6 Absorption.- 4.7 Boundary Conditions.- 4.8 Excitation of Mode and Excitation of Beam.- 4.9 Intensity of Wavefield (Standing-Wave) in Crystal.- 4.10 Consideration of the Spherical-Wave Nature of the Incident X-Rays.- 5. Approximations, Numerical Computing, and Other Approaches.- 5.1 Two-Beam Approximation for Three-Beam Diffraction.- 5.1.1 General Considerations.- 5.1.2 Three-Beam Transmission Case (Borrmann Diffraction).- 5.1.3 Diffracted Intensity of Three-Beam Bragg Reflection.- 5.1.4 Integrated Reflection for Non-Absorbing Crystals.- 5.1.5 Diffraction in Absorbing Crystals.- 5.2 Procedures for Numerical Computing.- 5.3 Quantum Mechanical Approach.- 5.4 N-Beam Diffraction in Other Types of Interaction.- 5.4.1 Two-Beam Diffraction with Specular Reflection.- 5.4.2 Interaction of X-Rays with Phonons.- 6. Case Studies.- 6.1 Bragg-Type Multiple Diffraction from Gallium Arsenide, Indium Arsenide and Indium Phosphide―Kinematical Interpretation.- 6.2 Three-Beam Borrmann Diffraction―Dynamical Calculation.- 6.3 Simultaneous Four-Beam Borrmann Diffraction.- 6.4 Three-Beam Bragg-Laue and Bragg-Bragg Diffraction.- 6.5 Four-Beam Bragg-Laue Diffraction.- 7. Applications.- 7.1 Experimental Determination of X-Ray Reflection Phases; Application to Crystal Structure Determination.- 7.1.1 General Consideration of the X-Ray Phase Problem.- 7.1.2 Reflection Phases and Multiple Diffraction.- 7.1.3 Experimental Methods for Phase Determination Using Multiple Diffraction.- 7.1.4 Determination of Centrosymmetric Crystal Structures in Practice.- 7.1.5 Phase Determination for Non-Centrosymmetric Crystals.- 7.2 Determination of Lattice Constants of Single Crystals.- 7.2.1 Divergent-Beam Photographic Methods.- 7.2.2 Collimated-Beam Method.- 7.3 Determination of Lattice Mismatch in Thin Layered Materials.- 7.4 Multi-Beam X-Ray Topography.- 7.5 Multi-Beam X-Ray Interferometer.- 7.6 Monochromatization of X-Ray Beams.- 7.7 Plasma Diagnosis.- 7.8 Determination of Mosaic Spread of Crystals.- 7.9 Multi-Beam X-Ray Standing-Wave Excited Fluorescence Technique for Surface Studies―A Proposed Method.- 7.10 Possible Future Trend of Development.- References.