1. Diffraction and the X-Ray Powder Difiractometer ............. 1
1.1. Diffraction ........................................... 1
1.1.1. Introduction to Diffraction ................... 1
1.1.2. Bragg's Law ................................... 3
1.1.3. Strain Effects ................................ 6
1.1.4. Size Effects .................................. 7
1.1.5. A Symmetry Consideration ...................... 8
1.1.6. Experimental Methods .......................... 9
1.2. The Creation of X-Rays ............................... 13
1.2.1. Bremsstrahhmg ................................ 13
1.2.2. Characteristic Radiation ..................... 16
1.2.3. Synchrotron Radiation ........................ 20
1.3. The X-Ray Powder Diffractometer ...................... 23
1.3.1. Practice of X-Ray Generation ................. 23
1.3.2. Goniometer for Powder Diffraction ............ 25
1.3.3. Monochromators and Filters ................... 27
1.4. X-Ray Detectors for XRD and ТЕМ ...................... 29
1.4.1. Detector Principles .......................... 29
1.4.2. Position-Sensitive Detectors ................. 34
1.4.3. Charge Sensitive Preamplifier ................ 35
1.4.4. Other Electronics ............................ 35
1.5. Experimental X-Ray Powder Diffraction Data ........... 37
1.5.1. * Intensities of Powder Diffraction Peaks .... 37
1.5.2. Phase Fraction Measurement ................... 45
1.5.3. Lattice Parameter Measurement ................ 50
1.5.4. * Refinement Methods for Powder Diffraction
Data ......................................... 52
1.5.5. * Pair Distribution Function Analysis ........ 55
Further Reading ...................................... 56
Problems ............................................. 57
2. The ТЕМ and its Optics ..................................... 63
2.1. Introduction to the Transmission Electron
Microscope ........................................... 63
2.2. Working with Lenses and Ray Diagrams ................. 67
2.2.1 Single Lenses ................................. 67
2.2.2 Multi-Lens Systems ............................ 70
2.3. Modes of Operation of а ТЕМ .......................... 73
2.3.1. Conventional Modes ........................... 73
2.3.2. Convergent-Beam Electron Diffraction ......... 83
2.3.3. High-Resolution Imaging ...................... 84
2.4. Real Lens Systems .................................... 89
2.4.1. Illumination Lens Systems .................... 89
2.4.2. Imaging Lens Systems ......................... 93
2.5. Glass Lenses ......................................... 94
2.5.1. Interfaces ................................... 94
2.5.2. Lenses and Rays .............................. 95
2.5.3. Lenses and Phase Shifts ...................... 98
2.6. Magnetic Lenses ..................................... 100
2.7. Lens Aberrations and Other Defects .................. 105
2.7.1. Spherical Aberration ........................ 105
2.7.2. Chromatic Aberration ........................ 106
2.7.3. Diffraction ................................. 107
2.7.4. Astigmatism ................................. 107
2.7.5. Gun Brightness .............................. 111
2.8. Resolution .......................................... 113
Further Reading ..................................... 115
Problems ............................................ 116
3. Scattering ............................................... 123
3.1. Coherence and Incoherence ........................... 123
3.1.1. Phase and Energy ............................ 123
3.1.2. Wave Amplitudes and Cross-Sections .......... 126
3.2. X-Ray Scattering .................................... 130
3.2.1. Electrodynamics of X-Ray Scattering ......... 130
3.2.2. * Inelastic Compton Scattering .............. 135
3.2.3. X-Ray Mass Attenuation Coefficients ......... 136
3.3. Coherent Elastic Scattering ......................... 138
3.3.1.  Born Approximation for Electrons .......... 138
3.3.2. Atomic Form Factors Physical Picture ....... 143
3.3.3. Scattering of Electrons by Model
Potentials .................................. 146
3.3.4. * Atomic Form Factors - General
Formulation ................................. 150
3.4. * Nuclear Scattering ................................ 155
3.4.1. Properties of Neutrons ...................... 156
3.4.2. * Inelastic Neutron Scattering .............. 158
3.4.3. * Mössbauer Scattering ...................... 161
Further Reading ..................................... 164
Problems ............................................ 164
4. Inelastic Electron Scattering and Spectroscopy ............ 167
4.1. Inelastic Electron Scattering ....................... 167
4.2. Electron Energy-Loss Spectrometry (EELS) ............ 169
4.2.1. Instrumentation ............................. 169
4.2.2. General Features of EELS Spectra ............ 170
4.2.3. * Fine Structure ............................ 172
4.3. Plasmon Excitations ................................. 177
4.3.1. Plasmon Principles .......................... 177
4.3.2. * Plasmons and Specimen Thickness ........... 179
4.4. Core Excitations .................................... 181
4.4.1. Scattering Angles and Energies -
Qualitative ................................. 182
4.4.2. Inelastic Form Factor ..................... 184
4.4.3. * Double-Differential Cross-Section,
А2аm/афАЕ ................................... 188
4.4.4. * Scattering Angles and Energies -
Quantitative ................................ 190
4.4.5. * Differential Cross-Section, dam/dE ...... 192
4.4.6. Partial and Total Cross-Sections, <rm ..... 193
4.4.7. Quantification of EELS Core Edges ........... 196
4.5. * Energy-Filtered ТЕМ Imaging (EFTEM) ............... 198
4.5.1. * Energy Filters ............................ 199
4.5.2. * Chemical Mapping with Energy-Filtered
Images ...................................... 200
4.5.3. Chemical Analysis with High Spatial
Resolution .................................. 202
4.6. Energy Dispersive X-Ray Spectrometry (EDS) .......... 203
4.6.1. Electron Trajectories Through Materials ..... 203
4.6.2. Fluorescence Yield .......................... 208
4.6.3. EDS Instrumentation Considerations .......... 210
4.6.4. Thin-Film Approximation ..................... 213
4.6.5. * ZAF Correction ............................ 216
4.6.6. Limits of Microanalysis ..................... 218
Further Reading ..................................... 220
Problems ............................................ 220
5. Diffraction from Crystals ................................. 225
5.1. Sums of Wavelets from Atoms ......................... 225
5.1.1. Electron Diffraction from a Material ........ 226
5.1.2. Wave Diffraction from a Material ............ 228
5.2. The Reciprocal Lattice and the Laue Condition ....... 232
5.2.1. Diffraction from a Simple Lattice ........... 232
5.2.2. Reciprocal Lattice .......................... 233
5.2.3. Laue Condition .............................. 235
5.2.4. Equivalence of the Laue Condition and
Bragg's Law ................................. 235
5.2.5. Reciprocal Lattices of Cubic Crystals ....... 236
5.3. Diffraction from a Lattice with a Basis ............. 237
5.3.1. Structure Factor and Shape Factor ........... 237
5.3.2. Structure Factor Rules ...................... 239
5.3.3. Symmetry Operations and Forbidden
Diffractions ................................ 244
5.3.4 Superlattice Diffractions .................... 245
5.4. Crystal Shape Factor ................................ 249
5.4.1. Shape Factor of Rectangular Prism ........... 249
5.4.2. Other Shape Factors ......................... 253
5.4.3. Small Particles in a Large Matrix ........... 254
5.5. Deviation Vector (Deviation Parameter) .............. 258
5.6. Ewald Sphere ........................................ 259
5.6.1. Ewald Sphere Construction ................... 259
5.6.2. Ewald Sphere and Bragg's Law ................ 261
5.6.3. Tilting Specimens and Tilting Electron
Beams ....................................... 261
5.7. Laue Zones .......................................... 263
5.8. * Effects of Curvature of the Ewald Sphere .......... 266
Further Reading ..................................... 267
Problems ............................................ 268
6. Electron Diffraction and Crystallography .................. 275
6.1. Indexing Diffraction Patterns ....................... 275
6.1.1. Issues in Indexing .......................... 276
6.1.2. Method 1 - Start with Zone Axis ............. 278
6.1.3. Method 2 - Start with Diffraction Spots ..... 281
6.2. Stereographic Projections and Their Manipulation 284
6.2.1. Construction of a Stereographic
Projection .................................. 284
6.2.2. Relationship Between Stereographic
Projections and Electron Diffraction
Patterns .................................... 286
6.2.3. Manipulations of Stereographic
Projections ................................. 286
6.3. Kikuchi Lines and Specimen Orientation .............. 292
6.3.1. Origin of Kikuchi Lines ..................... 292
6.3.2. Indexing Kikuchi Lines ...................... 296
6.3.3. Specimen Orientation and Deviation
Parameter ................................... 298
6.3.4. The Sign of s ............................... 301
6.3.5. Kikuchi Maps ................................ 301
6.4. Double Diffraction .................................. 304
6.4.1. Occurrence of Forbidden Diffractions ........ 304
6.4.2. Interactions Between Crystallites ........... 305
6.5. * Convergent-Beam Electron Diffraction .............. 306
6.5.1. Convergence Angle of Incident Electron
Beam ........................................ 308
6.5.2. Determination of Sample Thickness ........... 309
6.5.3. Measurements of Unit Cell Parameters ........ 311
6.5.4. * Determination of Point Groups ........... 316
6.5.5. * Determination of Space Groups ........... 327
6.6. Further Reading ..................................... 332
Problems ............................................ 332
7. Diffraction Contrast in ТЕМ Images ........................ 339
7.1. Contrast in ТЕМ Images .............................. 339
7.2. A Review of Structure and Shape Factors ............. 341
7.3. Extinction Distance ................................. 343
7.4. The Phase-Amplitude Diagram ......................... 346
7.5. Fringes from Sample Thickness Variations ............ 348
7.5.1. Thickness and Phase-Amplitude Diagrams ...... 348
7.5.2. Thickness Contours in ТЕМ Images ............ 349
7.6. Bend Contours in ТЕМ Images ......................... 354
7.7. Diffraction Contrast from Strain Fields ............. 357
7.8. Dislocations and Burgers Vector Determination ....... 360
7.8.1. Diffraction Contrast from Dislocation
Strain Fields ............................... 360
7.8.2. The g·b for Null Contrast ................... 363
7.8.3. Image Position and Dislocation Pairs or
Loops ....................................... 368
7.9. Semi-Quantitative Diffraction Contrast from
Dislocations ........................................ 370
7.10. Weak-Beam Dark-Field (WBDF) Imaging of
Dislocations ........................................ 378
7.10.1. Procedure to Make a WBDF Image .............. 379
7.10.2. Diffraction Condition for a WBDF Image ...... 380
7.10.3. Analysis of WBDF Images ..................... 381
7.11. Fringes at Interfaces ............................... 385
7.11.1. Phase Shifts of Electron Wavelets Across
Interfaces .................................. 385
7.11.2. Moire Fringes ............................... 388
7.12. Diffraction Contrast from Stacking Faults ........... 392
7.12.1. Kinematical Treatment ....................... 392
7.12.2. Results from Dynamical Theory ............... 398
7.12.3. Determination of the Intrinsic or
Extrinsic Nature of Stacking Faults ......... 400
7.12.4. Partial Dislocations Bounding the Fault ..... 400
7.12.5. An Example of a Stacking Fault Analysis ..... 401
7.12.6. Sets of Stacking Faults in ТЕМ Images ....... 403
7.12.7. Related Fringe Contrast ..................... 404
7.13. Antiphase (π) Boundaries and δ Boundaries ........... 405
7.13.1. Antiphase Boundaries ........................ 405
7.13.2. δ Boundaries ................................ 406
7.14. Contrast from Precipitates and Other Defects ........ 408
7.14.1. Vacancies ................................... 408
7.14.2. Coherent Precipitates ....................... 409
7.14.3. Semicoherent and Incoherent Particles ....... 414
Further Reading ..................................... 414
Problems ............................................ 415
8. Diffraction Lineshapes .................................... 423
8.1. Diffraction Line Broadening and Convolution ......... 423
8.1.1. Crystallite Size Broadening ................. 424
8.1.2. Strain Broadening ........................... 427
8.1.3. Instrumental Broadening Convolution ......... 430
8.2. Fourier Transform Deconvolutions .................... 434
8.2.1. Mathematical Features ....................... 434
8.2.2. Effects of Noise on Fourier Transform
Deconvolutions .............................. 437
8.3. Simultaneous Strain and Size Broadening ............. 441
8.4. * Fourier Methods with Multiple Orders .............. 447
8.4.1. * Formulation ............................. 447
8.4.2. * Strain Heterogeneity and Peak Asymmetry ... 452
8.4.3. * Column Lengths ............................ 455
8.4.4. * Size Coefficients ....................... 456
8.4.5. * Practical Issues in Warren - Averbach
Analysis .................................... 458
8.5. Comments on Diffraction Lineshapes .................. 459
Further Reading ..................................... 462
Problems ............................................ 462
9. Patterson Functions and Diffuse Scattering ................ 465
9.1. The Patterson Function .............................. 465
9.1.1. Overview .................................... 465
9.1.2. Atom Centers at Points in Space ............. 466
9.1.3. Definition of the Patterson Function ........ 467
9.1.4. Properties of Patterson Functions ........... 469
9.1.5. Perfect Crystals .......................... 471
9.2. Patterson Functions for Homogeneous Disorder
and Atomic Displacement Diffuse Scattering .......... 474
9.2.1. Deviations from Periodicity ................. 474
9.2.2. Uncorrelated Displacements .................. 475
9.2.3. * Correlated Displacements: Atomic Size
Effects ..................................... 478
9.2.4. Temperature ............................... 480
9.3. Diffuse Scattering from Chemical Disorder ........... 485
9.3.1. Randomness - Uncorrelated Chemical
Disorder .................................... 485
9.3.2. * SRO Parameters .......................... 489
9.3.3. * Patterson Function for Chemical SRO ..... 491
9.3.4. Short-Range Order Diffuse Intensity ......... 492
9.3.5. * Isotropic Materials ..................... 493
9.3.6. * Polycrystalline Average and Single
Crystal SRO ................................. 494
9.4. * Amorphous Materials ............................... 495
9.4.1. * One-Dimensional Model ................... 495
9.4.2. * Radial Distribution Function ............ 500
9.4.3. * Partial Pair Correlation Functions ...... 504
9.5. Small Angle Scattering .............................. 506
9.5.1 Concept of Small Angle Scattering ............ 506
9.5.2. * Guinier Approximation (small Δk) .......... 509
9.5.3. * Porod Law (large Δk) ...................... 511
9.5.4. * Density-Density Correlations (all Δk) ... 514
Further Reading ..................................... 516
Problems ............................................ 517
10. High-Resolution ТЕМ Imaging ............................... 521
10.1. Huygens Principle ................................... 522
10.1.1. Wavelets from Points in a Continuum ......... 522
10.1.2. Huygens Principle for a Spherical
Wavefront - Fresnel Zones ................... 527
10.1.3. Fresnel Diffraction Near an Edge .......... 531
10.2. Physical Optics of High-Resolution Imaging .......... 536
10.2.1. Wavefronts and Fresnel Propagator ......... 536
10.2.2. Lenses .................................... 538
10.2.3. Materials ................................. 540
10.3. Experimental High-Resolution Imaging ................ 542
10.3.1. Defocus and Spherical Aberration ............ 542
10.3.2. Lenses and Specimens ...................... 547
10.3.3. Lens Characteristics ........................ 550
10.4. * Simulations of High-Resolution ТЕМ Images ......... 559
10.4.1. Principles of Simulations ................... 559
10.4.2. * Practice of Simulations ................... 565
10.5. Issues and Examples in High-Resolution ТЕМ
Imaging ............................................. 566
10.5.1. Images of Nanostructures .................... 566
10.5.2. Examples of Interfaces ...................... 569
10.5.3. * Effects of Solute Misfit and Scattering
Factor Differences on Spot Intensities ...... 572
10.5.4. * Specimen and Microscope Parameters ........ 576
10.5.5. * Hints and Tricks for HRTEM ................ 583
10.6. Z-Contrast Imaging .................................. 586
10.6.1. Characteristics of Z-Contrast Imaging ....... 586
10.6.2. Comparison of Z-Contrast Imaging
with HRTEM Imaging .......................... 590
10.6.3. Z-Contrast Imaging with Atomic Resolution ... 592
10.6.4. Developments in Atomic-Resolution Imaging ... 594
Further Reading ..................................... 594
Problems ............................................ 595
11. Dynamical Theory .......................................... 597
11.1. Chapter Overview .................................... 597
11.2. * Mathematical Features of High-Energy
Electrons in a Periodic Potential ................... 599
11.2.1. * The Schrцdinger Equation ................ 599
11.2.2. Kinematical and Dynamical Theory .......... 605
11.2.3. The Crystal as a Phase Grating .............. 607
11.3. First Approach to Dynamical Theory - Beam
Propagation ......................................... 609
11.4. Second Approach to Dynamical Theory - Bloch
Waves and Dispersion Surfaces ....................... 613
11.4.1. Diffracted Beams, {Φg}, are Beats of
Bloch Waves, {Ψ(j)} ......................... 613
11.4.2. Crystal Periodicity and Dispersion
Surfaces .................................... 619
11.4.3. Energies of Bloch Waves in a Periodic
Potential ................................... 623
11.4.4. General Two-Beam Dynamical Theory ......... 626
11.5. Essential Difference Between Kinematical and
Dynamical Theories .................................. 632
11.6. X Diffraction Error, sg, in Two-Beam Dynamical
Theory .............................................. 637
11.6.1. Bloch Wave Amplitudes and Diffraction
Error ....................................... 637
11.6.2. Dispersion Surface Construction ............. 639
11.7. Dynamical Diffraction Contrast from Crystal
Defects ............................................. 641
11.7.1. Dynamical Diffraction Contrast Without
Absorption .................................. 641
11.7.2. * Two-Beam Dynamical Theory of Stacking
Fault Contrast .............................. 646
11.7.3. Dynamical Diffraction Contrast with
Absorption .................................. 650
11.8. X * Multi-Beam Dynamical Theories of Electron
Diffraction ......................................... 655
Further Reading ..................................... 658
Problems ............................................ 658
Bibliography .................................................. 663
Further Reading ........................................... 663
References and Figures .................................... 667
A. Appendix .................................................. 677
A.1. Indexed Powder Diffraction Patterns ................. 677
A.2. Mass Attenuation Coefficients for Characteristic
Kα X-Rays ........................................... 678
A.3. Atomic Form Factors for X-Rays ...................... 679
A.4. X-Ray Dispersion Corrections for Anomalous
Scattering .......................................... 683
A.5. Atomic Form Factors for 200 keV Electrons
and Procedure for Conversion to Other Voltages ...... 684
A.6. Indexed Single Crystal Diffraction Patterns: fcc,
bcc, dc, hcp ........................................ 689
A.7. Stereographic Projections ........................... 699
A.8. Examples of Fourier Transforms ...................... 703
A.9. Kα1, Kα2 Splitting and the Rachinger Correction ..... 706
A.10. Numerical Approximation for the Voigt Function ...... 707
A.11. Debye-Waller Factor from Wave Amplitude ............. 708
A.12. Review of Dislocations .............................. 709
A.13. ТЕМ Laboratory Exercises ............................ 716
A.13.1. Preliminary - JEOL 2000FX Daily Operation ... 716
A.13.2. Preliminary - Philips 400T Daily Operation. 720
A.13.3. Laboratory 1 - Microscope Procedures and
Calibration with Au and MoO3 ................ 722
A.13.4. Laboratory 2 - Diffraction Analysis of θ'
Precipitates ................................ 726
A.13.5. Laboratory 3 - Chemical Analysis of θ'
Precipitates ................................ 729
A.13.6. Laboratory 4 - Contrast Analysis of
Defects ..................................... 730
A.14. Fundamental and Derived Constants ................... 732
Index ......................................................... 735
In section titles, the asterisk, "*," denotes a more
specialized topic. The double dagger, "" warns of a higher level
of mathematics, physics, or crystallography.
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