DIFFRACTION PHYSICS THIRD REVISED EDITION JOHN M. COWLEY. Regents' Professor enzeritus Arizona State University

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1 DIFFRACTION PHYSICS THIRD REVISED EDITION JOHN M. COWLEY Regents' Professor enzeritus Arizona State University 1995 ELSEVIER Amsterdam Lausanne New York Oxford Shannon Tokyo

2 CONTENTS Preface to the first edition Preface to the second edition Preface to the third edition V VI VI Section I - PHYSICAL OPTICS Chapter 1. Fresnel and Fraunhofer diffraction Introduction Wave equations and waves Wave functions Electromagnetic waves Particle waves Superposition and coherence Superposition Independent point sources Huygen's principle Kirchhoff's formulation Application of the Kirchhoff formula Scattering theory approach Integral form of wave equation Born series Reciprocity Fresnel diffraction Small angle approximation Fresnel integrals Periodic objects - "Fourier images" Fraunhofer diffraction 21 Problems 24 Chapter 2. Fourier transform and convolutions Preliminaries Delta-functions and discontinuities Convolutions Examples of convolutions Fourier transforms: general Definitions 30

3 viii CONTENTS Properties of Fourier transforms Multiplication and convolution Space and time Fourier transforms and diffraction: examples Point source or point aperture A plane wave: the inverse of Translation of an object Slit function Slit function-alternative Straight edge Rectangular aperture Circular aperture Two very narrow slits Two slits of appreciable width Finite wave train Periodic array of narrow slits Arbitrary periodic function Diffraction grating: thin slits Diffraction grating: general Gaussian function Row of circular holes Complementary objects-babinet's principle Total intensities: Parseval's theorem 48 Problems 48 Chapter 3. Imaging and diffraction Wave theory of imaging Coherent wave optics Incoherent wave imaging Abbe theory Small angle approximation Phase contrast Phase and amplitude objects Out-of-focus contrast Aperture limitation Zernike phase contrast Holography Multi-component systems Partial coherence 69 Problems 72 Section II - KINEMATICAL DIFFRACTION Chapter 4. Radiations and their scattering by matter X-rays X-ray sources Scattering by electrons 78

4 CONTENTS ix Scattering by atoms Dispersion corrections Electrons Sources of electrons Atom scattering amplitudes Phase object approximation Failure of first Born approximation "Absorption" effects Neutrons Atomic scattering factors Nuclear spin scattering Isotopic disorder Thermal and magnetic scattering 91 Problems 92 Chapter 5. Scattering from assemblies of atoms The kinematical approximation Real and reciprocal space Reciprocal space distribution The reciprocal lattice Friedel's law and the phase problem The generalized Patterson function Examples of correlation functions Finite volume limitations Finite crystals Correlation in space and time Four-dimensional Patterson Special cases Ideal monatomic gas or liquid Real monatomic gases and liquids The hydrogen atom Diffraction geometry and intensities Practical considerations Finite sources and detectors Wavelength spread Integrated intensities Sections and projections 119 Problems 121 Chapter 6. Diffraction from crystals Ideal crystals Diffraction geometry Laue and Bragg diffraction conditions Shape transforms Special cases for electron diffraction 128

5 x CONTENTS 6.3. Crystal structure analysis The phase problem Supplementary information Structure analysis methods Trial and error Patterson function Heavy-atom & isomorphous replacement methods Direct methods Neutron diffraction structure analysis Nuclear scattering Magnetic scattering Electron diffraction structure analysis 141 Problems 142 Chapter 7. Diffraction from imperfect crystals Formulation of the diffraction problem Types of defects General diffraction formulation Patterson function approach Patterson with average periodic structure Patterson with no average structure Deviations from an average lattice Random vacancies: no relaxation Clustered vacancies Lattice relaxation Thermal vibrations - Einstein model Imperfect crystals with no average lattice Uneven separation of lattice planes Disordered orientations 160 Problems 163 Section III - DYNAMICAL SCATTERING Chapter 8. Diffraction by perfect crystals Multiple coherent scattering Theoretical approaches Bethe theory The dispersion equations Solutions of the equations Boundary conditions Two-beam approximation Bloch waves and dispersion surfaces Conduction electrons-energy representation X-ray diffraction; polarization 178

6 CONTENTS xi 8.5. The Laue (transmission) case Electron diffraction for a thin crystal Small angle approximation Laue case with absorption Bethe potentials The Bragg case 184 Chapter 9. Dynamical diffraction effects Thickness fringes, rocking curves-electron diffraction Intensity formulas Real space picture Rocking curves Extinction contours Convergent beam diffraction Diffraction and imaging of crystal wedges Absorption effects for wedges Dynamical effects of X-ray and neutron diffraction Techniques for X-ray diffraction Energy flow Dispersion surface picture Neutron diffraction Borrmann effect 205 Problems 207 Chapter 10. Extension to many beams Dynamical n-beam diffraction Extension of Bethe theory - transmission Matrix formulation Small angle approximation Bloch waves and boundary conditions The scattering matrix Derivation of the two-beam approximation The Darwin-type approach Special cases - beam reduction Computing methods Column approximation 227 Problems 229 Chapter 11. Multi-slice approaches Propagation of electrons in crystals Transmission through thin slices Three-dimensional objects Diffraction by a crystal General expression; excitation errors 236

7 xii CONTENTS Multiple-scattering series Zero-order scattering Single scattering-kinematical approximation Multiple scattering General double-summation solution General series solution Phase grating approximation Computing methods "Slice method u calculations Steps in a computation Possible errors Consistency tests Computing times Intensities from non-periodic objects Real-space formulations High-energy approximation Useful approximations A real-space basis for computing 253 Problem 254 Section IV - APPLICATIONS TO SELECTED TOPICS Chapter 12. Diffuse scattering and absorption effects Thermal diffuse scattering Phonons and vibrational waves Scattering for a longitudinal wave Diffuse scattering component Dispersion curves Three-dimensional generalizations Static atom displacements Relaxation around point defects Diffraction intensities for displaced atoms The Bragg peaks The diffuse scattering Electron excitations Inelastic X-ray scattering Electron excitation by electrons-plasmons Single-electron excitations Dynamical effects in diffuse scattering Scattering and re-scattering Coherent and incoherent scattering Analysis of diffuse scattering Absorption effects The nature of absorption parameters Absorption of X-rays and neutrons "Absorption" for electrons Absorption due to thermal vibrations Absorption from electron excitations Values of absorption coefficients 285

8 CONTENTS xiii Chapter 13. Electron microscope imaging Electron microscopes Conventional transmission e. m Scanning transmission electron microscopes Image formation Contrast for thin specimens Phase-object approximation Weak-phase object approximation Failure of weak-phase object approximation Dark-field images The imaging of crystals Imaging of thin crystals; structure images Calculation of images of crystals: envelope Imaging of crystals - inelastic scattering Lattice fringe imaging Crystal imaging without lattice resolution Image contrast in STEM STEM imaging of thin crystals STEM imaging of thicker crystals Electron holography Combining high-resolution imaging with diffraction 326 Problems 328 Chapter 14. K-line patterns and channelling Kossel lines Geometry of Kossel lines Dynamical theory of Kossel intensities Kossel lines with limited resolution Kikuchi lines External sources of divergent radiation Information from K-line patterns Channelling Secondary radiations 346 Chapter 15. Application of dynamical effects in single crystals Dependence of dynamical effects an crystal parameters X-ray interferometry n-beam and 2-beam dynamical diffraction Accurate determinations of structure amplitudes Measurements of thickness fringes Structure amplitudes from rocking curves Convergent beam electron diffraction method The use of critical voltages Intersecting K-lines The determination of crystal symmetries Coherent convergent-beam electron diffraction 367

9 xiv CONTENTS Chapter 16. Mosaic crystals and polycrystalline materials General Mosaic crystals The mosaic crystal model Kinematical integrated intensities Extinction effects Dynamical electron diffraction intensities Polycrystalline material Idealized models Kinematical diffraction intensities Line profile analysis Rietveld refinements Dynamical diffraction intensities fl-beam diffraction effects 383 Chapter 17. Ordering of atoms in crystals The nature and description of disordered states Order parameters Short-range order Long-range order Patterson function Size effects Kinematical diffraction Diffraction with ordering only Diffraction with ordering and size effects Relationship with ordering energies Dynamical scattering from disordered crystals Dynamical effects in diffuse scattering Calculations of diffuse scattering Strong scattering, multi-atom correlations High resolution imaging disordered crystals Out-of-phase domains Ordered out-of-phase superlattices Out-of-phase domains in disordered alloys Modulated structures 408 Problems 410 Chapter 18. Extended defects introduction Stacking faults-statistical, kinematical theory Patterson method for a simple case A general treatment ?aults in close-packed structures Dynamical diffraction by stacking faults Dislocations Diffraction effects 424

10 CONTENTS xv The imaging of dislocations Averaging over angles of incidence n-beam diffraction effects 427 Chapter 19. Diffraction from surfaces Introduction Surface imaging and diffraction with electrons Phase-contrast imaging Crystal terminations and superlattices Structure analysis of surface superlattices Crystal profile imaging Reflection from surfaces: grazing incidence Kinematical approximation: x-rays, neutrons Standing wave techniques RHEED and REM Reflection at normal incidence: LEED Diffraction of emitted electrons 453 References 457 Index 477

Elastic and Inelastic Scattering in Electron Diffraction and Imaging

$Elastic and Inelastic Scattering in Electron Diffraction and Imaging$ Elastic and Inelastic Scattering in Electron Diffraction and Imaging Contents Introduction Symbols and definitions Part A Diffraction and imaging of elastically scattered electrons Chapter 1. Basic kinematical