Preface ......................................................... v
1. Introduction ................................................ 1
1.1 Where are the Atoms? .................................... 1
1.2 Brief Historical Overview ............................... 5
1.3 Scope of the Book ....................................... 8
1.4 General Outline ......................................... 8
Part I — Fundamentals .......................................... 11
2 High-Resolution Transmission Electron Microscopy ............ 13
2.1 Overview ............................................... 14
2.2 The Exit-Plane Wave .................................... 16
2.2.1 The weak phase object approximation ............. 16
2.2.2 Dynamic scattering theories ..................... 18
2.3 Image Formation ........................................ 19
2.4 The Phase Contrast Transfer Function ................... 21
2.4.1 The objective aperture .......................... 22
2.4.2 Lens effects .................................... 23
2.4.3 Partial spatial and partial temporal coherence .. 28
2.4.4 Point resolution and information limit .......... 34
2.4.5 Other factors ................................... 36
2.5 Non-Linear Imaging ..................................... 40
2.6 Summary ................................................ 43
3 Scanning Transmission Electron Microscopy ................... 45
3.1 Overview ............................................... 45
3.2 Geometrical Considerations ............................. 49
3.2.1 The diffraction limit ........................... 49
3.2.2 Lens effects — spherical aberration ............. 53
3.2.3 Partial temporal coherence — chromatic
aberration ...................................... 54
3.2.4 Partial spatial coherence — the effective
source size ..................................... 56
3.2.5 Stability ....................................... 57
3.2.6 Small electron probes ........................... 58
3.3 Wave Optical Description of an Electron Probe .......... 61
3.3.1 Diffraction ..................................... 62
3.3.2 Defocus and spherical aberration ................ 63
3.3.3 Geometrical source size ......................... 66
3.3.4 Energy spread of the electron beam .............. 67
3.3.5 Concluding remarks .............................. 68
3.4 Summary ................................................ 72
4 Limits of Conventional Atomic-Resolution Electron
Microscopy .................................................. 73
4.1 The Case of Transmission Electron Microscopy ........... 73
4.2 The Case of Scanning Transmission Electron Microscopy .. 80
4.3 Summary ................................................ 87
Part II — Electron Optics ...................................... 89
5 Basic Principles of Electron Optics ......................... 91
5.1 The Electron Microscope ................................ 92
5.2 Newton's Second Law of Motion and Electron Optics ...... 93
5.3 The Hamiltonian Analogy ................................ 95
5.4 Geometrical Electron Optics ............................ 96
5.5 Electrons in the Electrostatic Field ................... 97
5.5.1 Snell's law of refraction ....................... 98
5.5.2 Snell's law of refraction for electrons ......... 98
5.5.3 Fermat's principle: The shortest light optical
path ........................................... 103
5.5.4 The reduced principle of least action .......... 107
5.5.5 The point eikonal .............................. 109
5.6 Electrons in the Stationary Electromagnetic Field ..... 114
5.6.1 Principle of Maupertius ........................ 114
5.6.2 The refractive index of electrons .............. 117
5.6.3 Geometrical wave surfaces ...................... 119
5.7 Summary ............................................... 121
6 Gaussian Dioptrics ......................................... 123
6.1 Geometry and Coordinate Systems ....................... 123
6.2 Fields and Lenses ..................................... 127
6.3 The Paraxial Approximation ............................ 129
6.4 Path Equation of an Electrostatic Field ............... 130
6.5 Path Equation of a Stationary Magnetic Field .......... 134
6.5.1 The homogeneous rotationally symmetric
magnetic field ................................. 134
6.5.2 The rotationally symmetric magnetic field ...... 144
6.5.3 The rotationally symmetric electromagnetic
field .......................................... 152
6.6 Series Expansion of the Fields ........................ 159
6.7 Imaging Within the Paraxial Approximation ............. 161
6.7.1 Theorem of optical imaging ..................... 161
6.7.2 Generalized theorem of Lippich ................. 166
6.7.3 Real image and virtual image ................... 168
6.7.4 Asymptotic cardinal elements ................... 171
6.8 Summary ............................................... 185
Part III — Aberration Correction .............................. 187
7 Aberrations ................................................ 189
7.1 Overview .............................................. 190
7.2 Image Aberrations ..................................... 192
7.3 Types of Aberrations .................................. 193
7.3.1 Axial aberrations vs. off-axial aberrations .... 193
7.3.2 Chromatic aberrations vs. geometrical
aberrations .................................... 195
7.4 Geometrical Aberration of a Round Electron Lens ....... 201
7.5 Wave Surface, Aberration Function and Image
Aberrations ........................................... 214
7.5.1 HRTEM and STEM ................................. 216
7.5.2 Non-ideal geometrical wave surfaces ............ 218
7.5.3 The aberration function ........................ 218
7.5.4 Other notations for axial aperture
aberrations .................................... 223
7.5.5 Aberration function and image aberrations ...... 227
7.6 Summary ............................................... 228
8 Spherical Aberration Correctors ............................ 229
8.1 Overview .............................................. 229
8.1.1 Strategies to deal with the spherical
aberration ..................................... 230
8.1.2 Strategies to correct the spherical
aberration ..................................... 231
8.2 Multi-Pole Lenses ..................................... 234
8.2.1 Quadrupoles .................................... 235
8.2.2 Hexapoles ...................................... 236
8.2.3 Octupoles ...................................... 240
8.3 The Basic Principles of Spherical Aberration
Correctors ............................................ 240
8.3.1 Hexapole spherical aberration corrector ........ 242
8.3.2 Quadrupole-octupole spherical aberration
corrector ...................................... 245
8.4 Beyond the Basic Principles — Parasitic Aberrations
and Other Complications ............................... 252
8.4.1 The multitude of axial aberrations ............. 252
8.4.2 Residual intrinsic axial aberrations ........... 253
8.4.3 Balancing the effect of higher-order
aberrations .................................... 253
8.4.4 Off-axial aberrations .......................... 254
8.4.5 The chromatic aberration ....................... 255
8.4.6 Dealing with the fifth-order spherical
aberration ..................................... 256
8.5 Improved Correctors ................................... 257
8.5.1 Fifth-order correction with hexapole fields .... 258
8.5.2 Fifth-order correction with quadrupole-
octupole fields ................................ 259
8.5.3 Correction of geometrical and chromatic
aberrations .................................... 260
8.6 Summary ............................................... 264
9 Aberration-Corrected Imaging ............................... 265
9.1 Aberration Diagnosis .................................. 265
9.1.1 Geometrical resolution limit ................... 266
9.1.2 Zemlin-tableau method .......................... 267
9.1.3 Tilt tableaux for probe-forming microscopes .... 273
9.1.4 Ronchigram methods ............................. 276
9.1.5 Other methods .................................. 278
9.2 Aberration-Corrected HRTEM ............................ 279
9.2.1 The coherent phase contrast transfer function .. 280
9.2.2 Incoherent contributions ....................... 286
9.2.3 Summary ........................................ 291
9.3 Aberration-Corrected STEM ............................. 291
9.3.1 Illumination aperture .......................... 291
9.3.2 Geometrical aberrations ........................ 293
9.3.3 Partial coherence .............................. 297
9.3.4 Considerations about small electron probes ..... 300
9.4 New Possibilities and New Limits ...................... 306
Appendix A Physical Constants, Abbreviations, Acronyms and
Symbols .................................................... 313
A.l Physical Constants .................................... 313
A.2 Abbreviations and Acronyms ............................ 314
Bibliography .................................................. 315
Index ......................................................... 329
About the Author .............................................. 335
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