Contents 1. of a higher level of mathematics, physics, or crystallography.

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1 Contents 1 1 Diffraction and the X-Ray Powder Diffractometer Diffraction Introduction to Diffraction Bragg s Law StrainEffects SizeEffects ASymmetryConsideration MomentumandEnergy Experimental Methods TheCreationofX-Rays Bremsstrahlung CharacteristicRadiation Synchrotron Radiation TheX-RayPowderDiffractometer Practice of X-Ray Generation GoniometerforPowderDiffraction Monochromators, Filters, Mirrors X-RayDetectorsforXRDandTEM DetectorPrinciples SolidStateDetectors Position-Sensitive Detectors Charge Sensitive Preamplifier OtherElectronics Experimental X-Ray Powder Diffraction Data * Intensities of Powder Diffraction Peaks Normals of Diffracting Planes SlitWidth LorentzFactor In section titles, the asterisk, *, denotes a more specialized topic. The double dagger,, warns of a higher level of mathematics, physics, or crystallography. xi

2 xii Contents Absorption Polarization Multiplicity and Density Measured Intensities Phase Fraction Measurement Peak Ratio Method AbsorptionFactors Example:RetainedAusteniteinSteels Lattice Parameter Measurement * Refinement Methods for Powder Diffraction Data Further Reading Problems The TEM and Its Optics Introduction to the Transmission Electron Microscope WorkingwithLensesandRayDiagrams SingleLenses Multi-Lens Systems Modes of Operation of a TEM Dark-FieldandBright-FieldImaging SelectedAreaDiffraction Convergent-Beam Electron Diffraction Nanobeam Diffraction High-ResolutionImaging PracticalTEMOptics ElectronGuns IlluminationLensSystems ImagingLensSystems GlassLenses Interfaces LensesandRays Lenses and Phase Shifts Magnetic Lenses Focusing ImageRotation PolePieceGap LensAberrationsandOtherDefects Spherical Aberration ChromaticAberration Diffraction Astigmatism GunBrightness Resolution Further Reading Problems...109

3 Contents xiii 3 Neutron Scattering NeutronsandNeutronScattering NeutronScattering Properties of Neutrons Neutron Sources FissionandSpallation Moderation NeutronPowderDiffractometers Reactor-Based Powder Diffractometer Pulsed-Source-Based Powder Diffractometer Waves of Phase Phase in Elastic Scattering * Phase in Inelastic Scattering InstrumentsforMeasuringLargerStructures Small-AngleScattering * Neutron Reflectivity *InelasticScattering * Triple-Axis Spectrometer * Fermi Chopper Spectrometer *OtherInelasticInstruments *QuasielasticScattering * Magnetic Scattering NuclearScattering Further Reading Problems Scattering WavesandScattering Wavefunctions Coherent and Incoherent Scattering ElasticandInelasticScattering Wave Amplitudes and Cross-Sections X-RayScattering Electrodynamics of X-Ray Scattering *InelasticComptonScattering X-Ray Mass Attenuation Coefficients Coherent Elastic Scattering BornApproximationforElectrons AtomicFormFactors PhysicalPicture Scattering of Electrons by Model Potentials * Atomic Form Factors General Formulation Further Reading Problems Inelastic Electron Scattering and Spectroscopy InelasticElectronScattering Electron Energy-Loss Spectrometry (EELS)

4 xiv Contents Instrumentation General Features of EELS Spectra *FineStructure PlasmonExcitations PlasmonPrinciples * Plasmons and Specimen Thickness CoreExcitations Scattering Angles and Energies Qualitative InelasticFormFactor * Double-Differential Cross-Section, d 2 σ in /dφ de * Scattering Angles and Energies Quantitative * Differential Cross-Section, dσ in /de Partial and Total Cross-Sections, σ in Quantification of EELS Core Edges Energy-Filtered TEM Imaging (EFTEM) Spectrum Imaging Energy Filters Chemical Mapping with Energy-Filtered Images ChemicalAnalysiswithHighSpatialResolution Energy Dispersive X-Ray Spectrometry (EDS) Electron Trajectories Through Materials Fluorescence Yield EDSInstrumentationConsiderations ArtifactsinEDSMeasurements Quantitative EDS Thin-FilmApproximation *ZAFCorrection * Limits of Microanalysis Further Reading Problems Diffraction from Crystals SumsofWaveletsfromAtoms ElectronDiffractionfromaMaterial WaveDiffractionfromaMaterial The Reciprocal Lattice and the Laue Condition Diffraction from a Simple Lattice Reciprocal Lattice Laue Condition Equivalence of the Laue Condition and Bragg s Law Reciprocal Lattices of Cubic Crystals Diffraction from a Lattice with a Basis Structure Factor and Shape Factor StructureFactorRules Symmetry Operations and Forbidden Diffractions...256

5 Contents xv 6.4 Chemically-OrderedStructures Superlattice Diffractions OrderParameters Crystal Shape Factor Shape Factor of Rectangular Prism Other Shape Factors SmallParticlesinaLargeMatrix DeviationVector(DeviationParameter) Ewald Sphere Ewald Sphere Construction Ewald Sphere and Bragg s Law Tilting Specimens and Tilting Electron Beams Laue Zones * Effects of Curvature of the Ewald Sphere Further Reading Problems Electron Diffraction and Crystallography IndexingDiffractionPatterns IssuesinIndexing Method1 StartwithZoneAxis Method 2 Start with Diffraction Spots Stereographic Projections and Their Manipulation Construction of a Stereographic Projection Relationship Between Stereographic Projections andelectrondiffractionpatterns Manipulations of Stereographic Projections Kikuchi Lines and Specimen Orientation Origin of Kikuchi Lines Indexing Kikuchi Lines Specimen Orientation and Deviation Parameter The Sign of s Kikuchi Maps Double Diffraction Occurrence of Forbidden Diffractions Interactions Between Crystallites * Convergent-Beam Electron Diffraction Convergence Angle of Incident Electron Beam Determination of Sample Thickness MeasurementsofUnitCellParameters Determination of Point Groups Determination of Space Groups Further Reading Problems...343

6 xvi Contents 8 Diffraction Contrast in TEM Images Contrast in TEM Images DiffractionfromCrystalswithDefects Review of the Deviation Parameter, s Atom Displacements, δr Shape Factor and t Diffraction Contrast and {s,δr,t} Extinction Distance The Phase-Amplitude Diagram Fringes from Sample Thickness Variations Thickness and Phase-Amplitude Diagrams Thickness Fringes in TEM Images Bend Contours in TEM Images DiffractionContrastfromStrainFields DislocationsandBurgersVectorDetermination DiffractionContrastfromDislocationStrainFields The g b RuleforNullContrast Image Position and Dislocation Pairs or Loops Semi-Quantitative Diffraction Contrast from Dislocations Weak-Beam Dark-Field (WBDF) Imaging of Dislocations Procedure to Make a WBDF Image Diffraction Condition for a WBDF Image Analysis of WBDF Images Fringes at Interfaces Phase Shifts of Electron Wavelets Across Interfaces Moiré Fringes Diffraction Contrast from Stacking Faults KinematicalTreatment Results from Dynamical Theory Determination of the Intrinsic or Extrinsic Nature of Stacking Faults Partial Dislocations Bounding the Fault An Example of a Stacking Fault Analysis Sets of Stacking Faults in TEM Images RelatedFringeContrast Antiphase (π) Boundaries and δ Boundaries Antiphase Boundaries δ Boundaries ContrastfromPrecipitatesandOtherDefects Vacancies Coherent Precipitates Semicoherent and Incoherent Particles Further Reading Problems...420

7 Contents xvii 9 Diffraction Lineshapes Diffraction Line Broadening and Convolution Crystallite Size Broadening Strain Broadening Instrumental Broadening Convolution Fourier Transform Deconvolutions Mathematical Features * Effects of Noise on Fourier Transform Deconvolutions Simultaneous Strain and Size Broadening Diffraction Lineshapes from Columns of Crystals WaveletsfromPairsofUnitCellsinOneColumn A Column Length Distribution Intensity from Column Length Distribution Comments on Diffraction Lineshapes Further Reading Problems Patterson Functions and Diffuse Scattering The Patterson Function Overview Atom Centers at Points in Space Definition of the Patterson Function Properties of Patterson Functions PerfectCrystals DeviationsfromPeriodicityandDiffuseScattering Diffuse Scattering from Atomic Displacements Uncorrelated Displacements Homogeneous Disorder Temperature * Correlated Displacements Atomic Size Effects DiffuseScatteringfromChemicalDisorder Uncorrelated Chemical Disorder Random Alloys *SROParameters * Patterson Function for Chemical SRO SRODiffuseIntensity *IsotropicMaterials * Polycrystalline Average and Single Crystal SRO * Amorphous Materials One-Dimensional Model Radial Distribution Function Partial Pair Correlation Functions SmallAngleScattering Concept of Small Angle Scattering * Guinier Approximation (Small Δk) * Porod Law (Large Δk) * Density-Density Correlations (All Δk)...514

8 xviii Contents 10.6 Further Reading Problems High-Resolution TEM Imaging Huygens Principle Wavelets from Points in a Continuum Huygens Principle for a Spherical Wavefront Fresnel Zones FresnelDiffractionNearanEdge PhysicalOpticsofHigh-ResolutionImaging Wavefronts and Fresnel Propagator Lenses Materials Experimental High-Resolution Imaging Defocus and Spherical Aberration Lenses and Specimens LensCharacteristics * Simulations of High-Resolution TEM Images PrinciplesofSimulations PracticeofSimulations IssuesandExamplesinHigh-ResolutionTEMImaging Images of Nanostructures Examples of Interfaces * Specimen and Microscope Parameters *SomePracticalIssuesforHRTEM * Geometric Phase Analysis Further Reading Problems High-Resolution STEM and Related Imaging Techniques Characteristics of High-Angle Annular Dark-Field Imaging Electron Channeling Along Atomic Columns OpticalFiberAnalogy CriticalAngle * Tunneling Between Columns Scattering of Channeled Electrons Elastic Scattering of Channeled Electrons * Inelastic Scattering of Channeled Electrons * Comparison of HAADF and HRTEM Imaging HAADF Imaging with Atomic Resolution * Effect of Defocus Experimental Examples *LensAberrationsandTheirCorrections C s Correction with Magnetic Hexapoles Higher-Order Aberrations and Instabilities Examples of C s -Corrected Images...607

9 Contents xix Three-Dimensional Imaging High Resolution EELS ElectronTomography Further Reading Problems Dynamical Theory Chapter Overview * Mathematical Features of High-Energy Electrons inaperiodicpotential * The Schrödinger Equation Kinematical and Dynamical Theory * The Crystal as a Phase Grating First Approach to Dynamical Theory Beam Propagation Second Approach to Dynamical Theory Bloch Waves and Dispersion Surfaces Diffracted Beams, {Φ g }, are Beats of Bloch Waves, {Ψ (j) } Crystal Periodicity and Dispersion Surfaces EnergiesofBlochWavesinaPeriodicPotential General Two-Beam Dynamical Theory Essential Difference Between Kinematical and Dynamical Theories Diffraction Error, s g, in Two-Beam Dynamical Theory Bloch Wave Amplitudes and Diffraction Error DispersionSurfaceConstruction Dynamical Diffraction Contrast from Crystal Defects Dynamical Diffraction Contrast Without Absorption * Two-Beam Dynamical Theory of Stacking Fault Contrast Dynamical Diffraction Contrast with Absorption * Multi-Beam Dynamical Theories of Electron Diffraction Further Reading Problems Appendix A.1 IndexedPowderDiffractionPatterns A.2 Mass Attenuation Coefficients for Characteristic Kα X-Rays A.3 AtomicFormFactorsforX-Rays A.4 X-RayDispersionCorrectionsforAnomalousScattering A.5 Atomic Form Factors for 200 kev Electrons and Procedure for Conversion to Other Voltages A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp A.7 Stereographic Projections A.8 ExamplesofFourierTransforms A.9 Debye Waller Factor from Wave Amplitude A.10Time-VaryingPotentialsandInelasticNeutronScattering...712

10 xx Contents A.11ReviewofDislocations A.12 TEM Laboratory Exercises A.12.1 Laboratory 1 Microscope Procedures and Calibration with Au and MoO A.12.2 Laboratory 2 Diffraction Analysis of θ Precipitates A.12.3 Laboratory 3 Chemical Analysis of θ Precipitates A.12.4 Laboratory 4 Contrast Analysis of Defects A.13 Fundamental and Derived Constants Bibliography Further Reading References and Figures Index...747