Preface ....................................................... xiv
Acknowledgements .............................................. xix
Figure reproductions .......................................... xxi
1 Basic crystallography ...................................... 1
1.1 Introduction ............................................... 1
1.2 Direct space and lattice geometry .......................... 2
1.2.1 Basis vectors and unit cells ........................ 2
1.2.2 The dot product and the direct metric tensor ........ 5
1.3 Definition of reciprocal space ............................. 9
1.3.1 Planes and Miller indices ........................... 9
1.3.2 The reciprocal basis vectors ....................... 10
1.3.3 Lattice geometry in reciprocal space ............... 14
1.3.4 Relations between direct space and reciprocal
space .............................................. 16
1.3.5 The non-Cartesian vector cross product ............. 18
1.4 The hexagonal system ...................................... 23
1.4.1 Directions in the hexagonal system ................. 24
1.4.2 The reciprocal hexagonal lattice ................... 26
1.5 The stereographic projection .............................. 29
1.5.1 Drawing a point .................................... 32
1.5.2 Constructing a great circle through two poles ...... 32
1.5.3 Constructing a small circle around a pole .......... 33
1.5.4 Finding the pole of a great circle ................. 34
1.5.5 Measuring the angle between two poles .............. 34
1.5.6 Measuring the angle between two great circles ...... 34
1.6 Crystal symmetry .......................................... 34
1.6.1 Symmetry operators ................................. 35
1.6.2 Mathematical representation of symmetry operators .. 39
1.6.3 Point groups ....................................... 42
1.6.1 Families of planes and directions .................. 44
1.6.5 Space groups ....................................... 46
1.7 Coordinate transformations ................................ 47
1.7.1 Transformation rules ............................... 48
1.7.2 Examples of coordinate transformations ............. 50
1.7.3 Rhombohedral and hexagonal settings of the
trigonal system .................................... 53
1.8 Converting vector components into Cartesian coordinates ... 55
1.9 Crystallographic calculations on the computer ............. 59
1.9.1 Preliminary remarks ................................ 59
1.9.2 Implementing the metric tensor formalism ........... 62
1.9.3 Using space groups on the computer ................. 64
1.9.4 Graphical representation of direct and reciprocal
space .............................................. 68
1.9.5 Stereographic projections on the computer .......... 70
1.10 Recommended additional reading ............................ 77
Exercises ................................................. 77
2 Basic quantum mechanics, Bragg's Law and other tools ...... 79
2.1 Introduction .............................................. 79
2.2 Basic quantum mechanics ................................... 80
2.2.1 Scalar product between functions ................... 81
2.2.2 Operators and physical observables ................. 82
2.2.3 The Schrödinger equation ........................... 84
2.2.4 The de Broglie relation ............................ 85
2.2.5 The electron wavelength (non-relativistic) ......... 86
2.2.6 Wave interference phenomena ........................ 87
2.3 Elements of the special theory of relativity .............. 89
2.3.1 Introduction ....................................... 89
2.3.2 The electron wavelength (relativistic) ............. 91
2.3.3 Relativistic correction to the governing equation .. 94
2.4 The Bragg equation in direct and reciprocal space ......... 96
2.4.1 The Bragg equation in direct space ................. 96
2.4.2 The Bragg equation in reciprocal space ............. 98
2.4.3 The geometry of electron diffraction .............. 100
2.5 Fourier transforms and convolutions ...................... 103
2.5.1 Definition ........................................ 103
2.5.2 The Dirac delta-function .......................... 105
2.5.3 The convolution product ........................... 106
2.5.4 Numerical computation of Fourier transforms and
convolutions ...................................... 108
2.6 The electrostatic lattice potential ...................... 111
2.6.1 Elastic scattering of electrons by an individual
atom .............................................. 111
2.6.2 Elastic scattering by an infinite crystal ......... 116
2.6.3 Finite crystal size effects ....................... 119
2.6.4 The excitation error or deviation parameter sg .... 121
2.6.5 Phenomenological treatment of absorption .......... 122
2.6.6 Atomic vibrations and the electrostatic lattice
potential ......................................... 126
2.6.7 Numerical computation of the Fourier coefficients
of the lattice potential .......................... 128
2.7 Recommended additional reading ........................... 133
Exercises ................................................ 134
3 The transmission electron microscope ..................... 136
3.1 Introduction ............................................. 136
3.2 A brief historical overview .............................. 137
3.3 Overview of the instrument ............................... 138
3.4 Basic electron optics: round magnetic lenses ............. 142
3.4.1 Cross-section of a round magnetic lens ............ 142
3.4.2 Magnetic field components for a round lens ........ 144
3.4.3 The equation of motion for a charged particle in
a magnetic field .................................. 146
3.4.4 The paraxial approximation ........................ 148
3.4.5 Numerical trajectory computation .................. 150
3.4.6 General properties of round magnetic lenses ....... 156
3.4.7 Lenses and Fourier transforms ..................... 161
3.5 Basic electron optics: lens aberrations .................. 166
3.5.1 Introduction ...................................... 166
3.5.2 Aberration coefficients for a round magnetic
lens .............................................. 166
3.6 Basic electron optics: magnetic multipole lenses ......... 174
3.6.1 Beam deflection ................................... 176
3.6.2 Quadrupole elements ............................... 178
3.7 Basic electron optics: electron guns ..................... 179
3.7.1 Introduction ...................................... 179
3.7.2 Electron emission ................................. 179
3.7.3 Electron guns ..................................... 187
3.7.4 Beam energy spread and chromatic aberration ....... 190
3.7.5 Beam coherence .................................... 193
3.7.6 How many electrons are there in the microscope
column? ........................................... 195
3.8 The illumination stage: prespecimen lenses ............... 195
3.9 The specimen stage ....................................... 199
3.9.1 Types of objective lenses ........................ 199
3.9.2 Side-entry, top-entry and special purpose stages .. 201
3.9.3 The objective lens and electron diffraction
geometry .......................................... 204
3.9.4 Numerical computation of electron diffraction
patterns .......................................... 208
3.9.5 Higher-order Laue zones ........................... 211
3.10 The magnification stage: post-specimen lenses ............ 216
3.11 Electron detectors ....................................... 221
3.11.1 General detector characteristics .................. 221
3.11.2 Viewing screen .................................... 227
3.11.3 Photographic emulsions ............................ 228
3.11.4 Digital detectors ................................. 230
Exercises ................................................ 233
4 Getting started .......................................... 235
4.1 Introduction ............................................. 235
4.2 The xtalinfo.f90 program ................................. 237
4.3 The study materials ...................................... 238
4.3.1 Material I: Cu-15 at % Al ......................... 238
4.3.2 Material II: Ti ................................... 241
4.3.3 Material III: GaAs ................................ 243
4.3.4 Material IV: BaTiO3 ............................... 250
4.4 A typical microscope session ............................. 252
4.4.1 Startup and alignment ............................. 252
4.4.2 Basic observation modes ........................... 257
4.5 Microscope calibration ................................... 272
4.5.1 Magnification and camera length calibration ....... 273
4.5.2 Image rotation .................................... 276
4.6 Basic CTEM observations .................................. 277
4.6.1 Bend contours ..................................... 279
4.6.2 Tilting towards a zone axis pattern ............... 282
4.6.3 Sample orientation determination .................. 284
4.6.4 Convergent beam electron diffraction patterns ..... 288
4.7 Lorentz microscopy: observations on magnetic thin foils .. 291
4.7.1 Basic Lorentz microscopy (classical approach) ..... 291
4.7.2 Experimental methods .............................. 293
4.8 Recommended additional reading ...................... 300
Exercises ................................................ 301
5 Dynamical electron scattering in perfect crystals ........ 303
5.1 Introduction ............................................. 303
5.2 The Schrödinger equation for dynamical electron
scattering ............................................... 304
5.3 General derivation of the Darwin-Howie-Whelan equations .. 306
5.4 Formal solution of the DHW multibeam equations ........... 311
5.5 Slice methods ............................................ 313
5.6 The direct space multi-beam equations .................... 315
5.6.1 The phase grating equation ........................ 316
5.6.2 The propagator equation ........................... 317
5.6.3 Solving the full direct-space equation ............ 319
5.7 Bloch wave description ................................... 320
5.7.1 General solution method ........................... 322
5.7.2 Determination of the Bloch wave excitation
coefficients ...................................... 326
5.7.3 Absorption in the Bloch wave formalism ............ 327
5.8 Important diffraction geometries and diffraction
symmetry ................................................. 328
5.8.1 Diffraction geometries ............................ 328
5.8.2 Thin-foil symmetry ................................ 330
5.8.3 The reciprocity theorem ........................... 331
5.9 Concluding remarks and recommended reading .......... 340
Exercises ................................................ 343
6 Two-beam theory in defect-free crystals .................. 345
6.1 Introduction ............................................. 345
6.2 The column approximation ................................. 346
6.3 The two-beam case: DHW formalism ......................... 348
6.3.1 The basic two-beam equations ...................... 348
6.3.2 The two-beam kinematical theory ................... 348
6.3.3 The two-beam dynamical theory ..................... 352
6.4 The two-beam case: Bloch wave formalism .................. 361
6.4.1 Mathematical solution ............................. 362
6.4.2 Graphical solution ................................ 367
6.5 Numerical two-beam image simulations ..................... 371
6.5.1 Numerical computation of extinction distances
and absorption lengths ............................ 371
6.5.2 The two-beam scattering matrix .................... 377
6.5.3 Numerical (two-beam) Bloch wave calculations ...... 382
6.5.4 Example two-beam image simulations ................ 384
6.5.5 Two-beam convergent beam electron diffraction ..... 388
Exercises ................................................ 394
7 Systematic row and zone axis orientations ................ 395
7.1 Introduction ............................................. 395
7.2 The systematic row case .................................. 396
7.2.1 The geometry of a bend contour .................... 396
7.2.2 Theory and simulations for the systematic row
orientation ....................................... 398
7.2.3 Thickness integrated intensities .................. 412
7.3 The zone axis case ....................................... 419
7.3.1 The geometry of the zone axis orientation ......... 420
7.3.2 Example simulations for the zone axis case ........ 422
7.3.1 Bethe potentials .................................. 435
7.3.4 Application of symmetry in multi-beam
simulations ....................................... 437
7.4 Computation of the exit plane wave function .............. 439
7.4.1 The multi-slice and real-space approaches ......... 439
7.4.2 The Bloch wave approach ........................... 446
7.4.3 Example exit wave simulations ..................... 446
7.5 Electron exit wave for a magnetic thin foil .............. 450
7.5.1 The Aharonov-Bohm phase shift ..................... 450
7.5.2 Direct observation of quantum mechanical effects .. 453
7.5.3 Numerical computation of the magnetic phase
shift ............................................. 454
7.6 Recommended reading ...................................... 458
Exercises ................................................ 458
8 Defects in crystals ...................................... 460
8.1 Introduction ............................................. 460
8.2 Crystal defects and displacement fields .................. 460
8.3 Numerical simulation of defect contrast images ........... 465
8.3.1 Geometry of a thin foil containing a defect ....... 466
8.3.2 Example of the use of the various reference
frames ............................................ 470
8.3.3 Dynamical multi-beam computations for a column
containing a displacement field ................... 474
8.4 Image contrast for selected defects ...................... 478
8.4.1 Coherent precipitates and voids ................... 479
8.4.2 Line defects ...................................... 481
8.4.3 Planar defects .................................... 492
8.4.4 Planar defects and the systematic row ............. 511
8.4.5 Other displacement fields ......................... 514
8.5 Concluding remarks and recommended reading ............... 515
Exercises ................................................ 516
9 Electron diffraction patterns ............................ 518
9.1 Introduction ............................................. 518
9.2 Spot patterns ............................................ 518
9.2.1 Indexing of simple spot patterns .................. 518
9.2.2 Zone axis patterns and orientation relations ...... 523
9.2.3 Double diffraction ................................ 525
9.2.4 Overlapping crystals and Moire patterns ........... 530
9.3 Ring patterns ............................................ 535
9.4 Linear features in electron diffraction patterns ......... 537
9.4.1 Streaks ........................................... 537
9.4.2 HOLZ lines ........................................ 540
9.4.3 Kikuchi lines ..................................... 544
9.5 Convergent beam electron diffraction ..................... 550
9.5.1 Point group determination ......................... 551
9.5.2 Space group determination ......................... 554
9.6 Diffraction effects in modulated crystals ................ 556
9.6.1 Modulation types .................................. 556
9.6.2 Commensurate modulations .......................... 558
9.6.3 Incommensurate modulations and quasicrystals ...... 569
9.7 Diffuse intensity due to short range ordering ............ 571
9.8 Diffraction effects from polyhedral particles ............ 575
Exercises ................................................ 584
10 Phase contrast microscopy ................................ 585
10.1 Introduction ............................................. 585
10.1.1 A simple experimental example ..................... 586
10.2 The microscope as an information channel ................. 587
10.2.1 The microscope point spread and transfer
functions ......................................... 589
10.2.2 The influence of beam coherence ................... 606
10.2.3 Plug in the numbers ............................... 621
10.2.4 Image formation for an amorphous thin foil ........ 626
10.2.5 Alignment and measurement of various imaging
parameters ........................................ 629
10.3 High-resolution image simulations ........................ 638
10.4 Lorentz image simulations ................................ 641
10.4.1 Example Lorentz image simulations for periodic
magnetization patterns ............................ 642
10.4.2 Fresnel fringes for non-magnetic objects .......... 648
10.5 Exit wave reconstruction ................................. 649
10.5.1 What are we looking for? .......................... 649
10.5.2 Exit wave reconstruction for Lorentz microscopy ... 652
Exercises ................................................ 658
10.6 Final remarks ............................................ 659
Appendix Al Explicit crystallographic equations .............. 661
Appendix A2 Physical constants ............................... 665
Appendix A3 Space group encoding and other software .......... 666
Appendix A4 Point groups and space groups .................... 667
List of symbols ............................................... 677
Bibliography .................................................. 685
Index ......................................................... 705
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