Dynamical Scattering of X-Rays in Crystals

(Historical Survey) The discovery of X-ray diffraction in crystals by LAUE, FRIDRICH and KNIPPING in 1912 [1.1] served as the starting pOint for the development of scientific research along a number of important lines. We shall discuss just a few of them. The above discovery convincingly demonstrated the wave properties of X-rays. This, together with the previously established electromagnetic nature of radiation, confirmed the hypothesis that X-rays form the short-wave part of the electromagnetic spectrum. Further, this discovery was the first and decisive experimental proof of the periodic structure of crystals. In fact, theoretical crystallography had already arrived at this conclusion, mainly as an outcome of the theory of the space groups of symmetry elaborated by FEDOROV [1.2] and SCHOENFLIES [1.3]. From the optics of visible light we know that the radiation of a wave length of the same order as, and preferably less than, the period of a grat ing suffers diffraction on periodic objects of the type of optical grating. Thus, the discovery proved that the wavelength of an X-ray must be of the order of interatomic distances. It became clear why the visible light of wavelengths exceeding the crystal lattice periods by about 500 to 1000 times failed to reveal the periodic structure of crystals in diffraction experi ments.

Contenu

1. Introduction.- 2. Wave Equation and Its Solution for Transparent Infinite Crystal.- 2.1 Wave Equation and Its Solution.- 2.2 Two-Wave Approximation. Dispersion Surface.- 3. Transmission of X-Rays Through a Transparent Crystal Plate. Laue Reflection.- 3.1 Wave Fields Inside a Crystal.- 3.1.1 Semi-infinite Crystal. Connection with Experimental Conditions. Refraction Effect.- 3.1.2 Wave Amplitudes; Pendulum Solution. Extinction. Quasi-standing Waves.- 3.2 Transmission and Reflection Coefficients. Analysis of Pendulum Solution in the Case of Plane-Parallel Plate.- 3.3 Transmission Through a Wedge-Shaped Plate.- 4. X-Ray Scattering in Absorbing Crystal. Laue Reflection.- 4.1 Atomic Scattering and Absorption.- 4.2 Complex Form of Dynamical Scattering Parameters.- 4.3 Derivation of Exact Formulae for Transmission (T) and Reflection (R) Coefficient in the Case of an Absorbing Crystal.- 4.4 Derivation of Approximate Equations for Transmission Coefficient T and Reflection Coefficient R.- 4.5 Analysis of Approximate Equations for the Transmission Coefficient T and the Reflection Coefficient R.- 4.5.1 Symmetrical Reflection.- 4.5.2 Asymmetrical Reflection.- 4.6 Integrated Values of Reflection Ri and Transmission Ti in the Case of Absorbing Crystal.- 4.7 Analysis of Expressions for Integrated Values Ri and Ti as Applied to Important Particular Cases.- 5. Poynting's Vectors and the Propagation of X-Ray Wave Energy.- 5.1 Averaged Poynting's Vector in the General Case.- 5.2 The Triply Averaged Poynting's Vector in Transparent Crystal.- 5.3 Triply Averaged Poynting's Vector in Absorbing Centro-symmetrical Crystal.- 5.4 Energy Propagation in Absorbing Crystal Without a Centre of Symmetry, Taking into Account the Periodic Component of Poynting's Vector. Additional Remarks.-6. Dynamical Theory in Incident-Spherical-Wave Approximation.- 6.1 Dynamical Theory in a Two-Wave Approximation with Spherical Wave Incident on Crystal. Application to Scattering in Transparent Plane-Parallel and Wedge-Shaped Crystals.- 6.2 Application of the Theory Described to Scattering in Absorbing Crystal.- 7. Bragg Reflection of X-Rays. I. Basic Definitions. Coefficients of Absorption; Diffraction in Finite Crystal.- 7.1 Reflection from Transparent Crystal.- 7.2 True Absorption in Bragg Reflection. Investigation of Coefficient of Absorption ? from Plane-Parallel Plate.- 7.3 Diffraction in Finite Crystal in Incident-Spherical Wave or Incident Wave Packet Approximation.- 8. Bragg Reflection of X-Rays. II. Reflection and Transmission Coefficients and Their Integrated Values.- 8.1 Deriving General Expressions for Reflection and Transmission Coefficients.- 8.2 Bragg Reflection from Transparent Crystal.- 8.3 Bragg Reflection from Thick Absorbing Crystal.- 8.4 Integrated Reflection from Absorbing Crystal in Bragg Case.- 9. X-Ray Spectrometers Used in Dynamical Scattering Investigations. Some Results of Experimental Verification of the Theory.- 9.1 Estimating Wavelength Spread and Angular Divergence of X-Ray Tube Radiation.- 9.2 Two-Crystal Spectrometer, Using Bragg Reflections in Both Crystals (Bragg-Bragg Scheme).- 9.3 Three-Crystal Spectrometer.- 9.4 Other Types of Diffractometers.- 9.4.1 Double-Crystal Spectrometers of the Bragg-Laue and Laue-Laue Type.- 9.4.2 Multi-Crystal Diffractometers with MCC, with Symmetrical and Asymmetrical Bragg Reflections.- 9.4.3 Rigorous Theory of X-Ray Diffractometers.- 9.4.4 Investigation and Utilization of Bragg-Reflection Curves.- 9.4.5 Investigations into the Interference Effects of the Pendulum Solution.- Determining the AbsoluteValues of Atomic Amplitudes.- Some Other Pendulum Solution Investigations.- 10. X-Ray Interferometry. Moiré Patterns in X-Ray Diffraction.- 10.1 Three-Crystal Interferometers.- 10.2 Two-Crystal Interferometer.- 10.3 Formation and Utilization of X-Ray Moiré Patterns.- 10.4 Experimental Investigations. Three-Crystal Interferometer.- 10.4.1 Double-Crystal Interferometer.- 11. Generalized Dynamical Theory of X-Ray Scattering in Perfect and Deformed Crystals.- 11.1 Deriving Fundamental Equations in the General Case of Deformed Crystal.- 11.2 X-Ray Diffraction in Perfect Crystal Under Conditions of Space-Inhomogeneous Dynamical Problem. Influence Functions of Point Source.- 11.3 Laue Reflection in Perfect Crystal.- 11.4 Bragg Reflection in Perfect Crystal.- 11.5 Application of Generalized Theory to Deformed Crystal. Relationship Between Angular Variable ?n and Deformation Field.- 11.6 Fundamental Equations of Geometrical Optics of X-Rays.- 11.7 Approaches Based Upon Wave Theory.- 11.7.1 Rigorous Theory of Laue Diffraction of X-Rays in Crystal with Uniform Strain Gradient.- 11.7.2 Integral Formulation of Huygens-Fresnel Principle.- 11.7.3 Quasi-classical Wave Field Asymptotes.- 11.7.4 Ray Trajectories.- 11.7.5 Integrated Intensity of Diffracted Wave.- 11.7.6 Conclusion.- 12. Dynamical Scattering in the Case of Three Strong Waves and More.- 12.1 Scattering in Nonabsorbing Crystal. Reference Coordinate Systems.- 12.2 System of Fundamental Equations in the Case of Three Strong Waves, and the Dispersion Surface Equation.- 12.3 Another Method of Deriving the Dispersion Surface Equation in a Three-Wave Case.- 12.4 Analysis of the Dispersion Surface Equation in the Case of Nonabsorbing Crystal.- 12.5 Deriving the Dispersion Surface Equation in the Case of Four Strong Waves.- 12.6Coefficients of Transmission and Reflection for a Plane-Parallel Plate. Laue Reflection.- 12.7 Scattering in Absorbing Crystal. Introducing Complex Parameters of Scattering and the Coefficient of Absorption.- 12.8 The Relationship Between the Coefficient of Absorption and the Shape of the Dispersion Surface. EWALD's Criterion.- 12.9 Asymptotic Properties of the Dispersion Surface. Transition from the Multiwave to the Two-Wave Region.- 12.10 Symmetrical Cases of Multiwave Diffraction. Nonlinear Borrmann Effect.- 12.11 Scattering in Germanium and Silicon Crystals.- 12.12 Bragg Reflection of X-Rays in Multiwave Diffraction.- 12.13 Methods for Numerical Determination of Dispersion Surface and Electric Displacement Vectors.- Appendix A.- Appendix B.- Appendix C.- References.