Diffraction Optics of Complex Structured Periodic Media

Probing matter with beams of photons, neutrons and electrons provides the main source of information about both the microscopic and macroscopic structure of materials. This is particularly true of media, such as crystals and liquid crystals, that have a periodic structure. This book discusses the interaction of waves (which may represent x-rays, gamma rays, electrons, or neutrons) with various kinds of ordered media. After two chapters dealing with exact and approximate solutions to the scattering problem in periodic media in general, the author discusses: the diffraction of Mobbauer radiation in magnetically ordered crystals; the optics of chiral liquid crystals; the radiation of fast particles in regular media (Cherenkov radiation); nonlinear optics of periodic media; neutron scattering in magnetically ordered media; polarization phenomena in x-ray optics; magnetic x-ray scattering; and Mobbauer filtration of synchrotron radiation.

1 Waves in Media with One-Dimensional Periodicity (Exact Solution).- 1.1 Layered Medium.- 1.1.1 Harmonic Modulation of Media.- 1.1.2 Properties of Eigenwaves.- 1.2 Optics of Cholesterics (Exact Solution).- 1.2.1 Dielectric Properties of Cholesterics.- 1.2.2 Eigenwaves.- 1.2.3 Properties of Eigensolutions.- 1.3 Solution of Boundary Problem.- 1.3.1 Planar Cholesteric Texture.- 1.3.2 Reflection from Thick Layers.- 1.3.3 The Case Where Medium and Cholesteric Have Equal Dielectric Constants.- 1.3.4 Dependence of Reflection on Polarization.- 1.3.5 Effect of Dielectric Boundaries.- 1.3.6 Method of Characteristic Matrices.- References.- 2 Approximate Description of Interaction of Radiation with Regular Media.- 2.1 Kinematical Approximation.- 2.1.1 Scattering Cross Section.- 2.1.2 Polarization Characteristics.- 2.1.3 Scattering of Light in Cholesterics.- 2.1.4 Kinematical Approximation Limitations.- 2.2 Dynamical Theory.- 2.2.1 Set of Dynamical Equations.- 2.2.2 Dispersion Surfaces.- 2.2.3 Solution of the Boundary Problem.- References.- 3 Diffraction of Mössbauer Radiation in Magnetically Ordered Crystals.- 3.1 Diffraction of Mössbauer Radiation.- 3.2 Amplitude of Coherent Mössbauer Scattering.- 3.2.1 Amplitude of Elastic Scattering.- 3.2.2 Isotope and Spin Incoherence.- 3.2.3 Coherent Amplitude in Limiting Cases of Completely Split and Unsplit Lines.- 3.3 Kinematical Theory of Mössbauer Diffraction.- 3.3.1 Structure Amplitude.- 3.3.2 Phase Determination.- 3.3.3 Diffraction at Magnetically Ordered Crystals.- 3.3.4 Resolved Zeeman Splitting of a Mössbauer Line.- 3.3.5 Crystals with Mössbauer Nuclei at Sites with Inhomogeneous Electric Field (EFG).- 3.4 Dynamical Theory of Mössbauer Diffraction.- 3.4.1 Unsplit Mössbauer Line.- 3.4.2 Suppression of Inelastic Channels of Nuclear Reactions.- 3.4.3 Bragg Reflection from Crystals.- 3.4.4 Solution of the Dynamical Equations for Hyperfine Splitting of Mössbauer Line.- 3.4.5 Polarization-Independent Amplitude of Scattering at Zero Angle.- 3.4.6 Completely Resolved Zeeman Splitting of Mössbauer Line.- References.- 4 Optics of Chiral Liquid Crystals.- 4.1 Optics of Cholesteric Liquid Crystals (CLC).- 4.1.1 The Fundamental Equations.- 4.1.2 Eigensolutions.- 4.1.3 The Solution of the Boundary Problem.- 4.1.4 High-Order Reflection for Oblique Incidence.- 4.1.5 A Planar Layer.- 4.1.6 Optical Rotation.- 4.2 Absorbing Cholesteric Liquid Crystals.- 4.2.1 The Dielectric Tensor of Absorbing CLCs.- 4.2.2 Suppression of Absorption.- 4.2.3 Optical Rotation.- 4.2.4 Borrmann Effect for Oblique Incidence.- 4.3 Chiral Smectic Liquid Crystals.- 4.3.1 Dielectric Permittivity Tensor of a C*.- 4.3.2 Second-Order Diffraction Reflection.- 4.3.3 First-Order Reflection.- 4.3.4 Boundary Problem.- 4.4 Blue Phase of Liquid Crystals.- 4.4.1 Observed Properties of Blue Phases.- 4.4.2 Symmetry Restrictions for the Dielectric Permittivity Tensor.- 4.4.3 Fourier Harmonics of $$ \hat{\epsilon }\left( r \right) $$.- 4.4.4 The Explicit Form of $$ \hat{\epsilon }\left( r \right) $$ (r).- 4.4.5 Optical Properties.- 4.4.6 Structure Studies of the Blue Phase.- References.- 5 Radiation of Fast Charged Particles in Regular Media.- 5.1 Structure Cherenkov Radiation (Kinematical Approximation).- 5.1.1 Formulation of the Problem.- 5.1.2 Radiation of a Particle in a Crystal.- 5.1.3 Photon Emission Cross Section.- 5.1.4 The Structure Factor.- 5.1.5 Temperature Dependence.- 5.1.6 Dimension of the Emitting Region.- 5.1.7 Estimations of the Structure Radiation Intensity.- 5.1.8 On the Phenomenological Approach.- 5.2 Coherent Coulomb Excitation of a Mössbauer Transition in a Single Crystal.- 5.2.1 Coherent Coulomb Excitation in a “Frozen” Lattice.- 5.2.2 Influence of Thermal Motion of Atoms.- 5.2.3 Radiation Intensity.- 5.3 Structure Cherenkov Radiation in a Cholesteric Liquid Crystal.- 5.3.1 Basic Equations.- 5.3.2 Intensity and Polarization Properties of Radiation.- 5.4 Dynamic Theory of Coherent Radiation of Fast Charged Particles in Regular Media.- 5.4.1 Dynamic Theory of the Structure Cherenkov Radiation.- 5.4.2 Coherent Coulomb Excitation of a Mössbauer Transition.- 5.4.3 Cherenkov Radiation in Chiral Liquid Crystals.- 5.4.4 The Shift of Cherenkov Radiation Threshold.- References.- 6 Nonlinear Optics of Periodic Media.- 6.1 Enhancement of Efficiency of the Nonlinear Transformation of Optical Frequencies in Periodic Medium.- 6.2 Third Harmonic Generation in a Cholesteric Liquid Crystal.- 6.2.1 Conditions for Phase Matching.- 6.2.2 Solution of the Boundary Problem.- 6.2.3 Third Harmonic Generation in the Band of Selective Reflection.- 6.2.4 Extreme Efficiency of Transformation.- 6.2.5 The Conditions for the Enhancement Effect.- 6.3 Phase-Matching Conditions Independent of Frequency Dispersion.- 6.4 Nonlinear Generation of Harmonics in Blue Phase of Liquid Crystals.- 6.4.1 Phase-Matching Condition.- 6.4.2 Symmetry Restrictions for the Tensor of Nonlinear Susceptibility.- 6.4.3 Fourier Components of the $$\hat X\left( r \right)$$ (r) Tensor.- References.- 7 Dynamic Scattering of Thermal Neutrons in Magnetically Ordered Crystals.- 7.1 Scattering Amplitude of Thermal Neutrons.- 7.2 System of Dynamic Equations.- 7.3 Structures with Compensated Magnetic Moment of a Unit Cell.- 7.3.1 Matrix of the Dynamic System.- 7.3.2 Eigenpolarizations.- 7.3.3 Solution to the Boundary Problem.- 7.3.4 Averaged Scattering Parameters.- 7.3.5 Observed Dynamic Effects.- 7.4 Dynamic Scattering in Magnetically Ordered Structures of the General Type.- 7.4.1 Eigensolutions to the Dynamic System.- 7.4.2 Solution to the Boundary Problem.- 7.4.3 The Cases of Analytical Solution of the Dynamic Problem.- 7.4.4 Bragg Reflection from Thick Crystals.- 7.4.5 Reflection in the Case of Strong Birefringence.- 7.4.6 Concluding Remarks.- References.- 8 Polarization Phenomena in X-Ray Optics.- 8.1 Polarization Phenomena in Diffraction.- 8.1.1 Diffraction in Perfect Crystals.- 8.1.2 Polarization Tensor and Integrated Polarization Parameters.- 8.1.3 Diffractive Birefringence and Dichroism.- 8.1.4 Diffraction in Imperfect Crystals.- 8.2 Anisotropy of X-Ray Susceptibility.- 8.2.1 Birefringence and Dichroism near Absorption Edges.- 8.2.2 Symmetry Restriction on the X-Ray Susceptibility Tensor of a Crystal.- 8.2.3 Forbidden Reflections.- 8.3 Polarization Effects in X-Ray Optical Components.- 8.3.1 Polarization of Radiation Produced by X-Ray Sources.- 8.3.2 Production and Analysis of Linear Polarization.- 8.3.3 The X-Ray Quarter-Wave Plate.- 8.304 Pendelosung.- 804 Conclusions.- References.- 9 Magnetic X-Ray Scattering.- 9.1 Magnetic Scattering.- 9.2 Polarization Properties of Magnetic Scattering.- 9.3 X-Ray Resonance Magnetic Scattering.- 904 X-Ray Magnetic Birefringence and Dichroism.- References.- 10 Mössbauer Filtration of Synchrotron Radiation.- 10.1 On Synchrotron Radiation Sources and Mössbauer Filtration.- 10.2 The First Observations of Mössbauer Filtration.- 10.3 Speed-Up of Mössbauer Scattering in Crystals.- 10.4 Temporal Quantum Beats in Mössbauer Filtration of SR.- 10.5 Future Mössbauer Experiments with Synchrotron Radiation.- 10.6 Coherent Inelastic Mössbauer Scattering.- 10.7 Recent Experiments on Mössbauer Filtration.- References.