 | Di Bartolo B. Crystal symmetry, lattice vibrations, and optical spectroscopy of solids: a group theoretical approach / B.Di Bartolo, R.C.Powell. - Singapore: World scientific, 2014. - xviii, 515 p.: ill. - Incl. bibl. ref. - Sub. ind.: p.511-515. - ISBN 978-981-4579-20-9
|
Preface ....................................................... vii
Part I. Symmetry of Crystals .................................... 1
Chapter 1. Introduction ......................................... 3
1.1 The Hamiltonian of a CrystaUine Solid ...................... 3
1.2 The Adiabatic Approximation ................................ 6
1.3 The Role of Symmetry ....................................... 7
1.4 The Symmetries of the Hamiltonian .......................... 9
Reference ...................................................... 12
Chapter 2. Concepts of Group Theory ............................ 13
2.1 Properties of a Group ..................................... 13
2.2 Subgroups, Cosets, and Classes ............................ 15
2.3 Theory of Representations ................................. 17
2.4 Orthogonality Relations ................................... 20
2.5 Characters of a Matrix Representation ..................... 23
2.6 Reduction of a Reducible Representation ................... 25
2.7 Basis Functions for Irreducible Representations ........... 27
2.8 Direct Product Representations ............................ 29
2.9 The Fundamental Theorem for Functions Transforming
Irreducibly ............................................... 31
2.10 Product Groups and Their Representations .................. 34
2.11 Connection of Quantum Mechanics with Group Theory ......... 35
Chapter 3. Crystal Symmetries .................................. 41
3.1 Unit Cells and Space Lattices ............................. 41
3.2 Miller Indices ............................................ 44
3.3 The Crystal Systems ....................................... 47
3.3.1 The Four Two-Dimensional Crystal Systems ........... 48
3.3.2 The Seven Three-Dimensional Crystal Systems ........ 48
3.4 The Bravais Lattices ...................................... 50
3.4.1 The Five Bravais Lattices in Two Dimensions ........ 50
3.4.2 The Fourteen Bravais Lattices in Three Dimensions .. 53
Chapter 4. Group Theoretical Treatment of Crystal Symmetries ... 57
4.1 Space Groups .............................................. 57
4.2 The Crystallographic Point Groups ......................... 60
4.2.1 Two-Dimensional Crystallographic Point Groups ...... 60
4.2.2 Three-Dimensional Crystallographic Point Groups .... 61
4.2.3 Site Groups ........................................ 65
4.3 The Invariant Subgroup of Primitive Translations:
Bravais Lattices .......................................... 68
4.4 The Compatibility of Rotational and Translational
Symmetries and Its Relevance to Space Groups .............. 69
4.5 The Irreducible Representations of a Group of Primitive
Translations Brillouin Zones .............................. 72
4.6 The Irreducible Representations of Space Groups ........... 77
4.6.1 Effects of lYanslational Symmetry .................. 78
4.6.2 Effects of Rotational Symmetry ..................... 79
4.6.3 General Properties of the Irreducible
Representations .................................... 80
4.6.4 Small Representations for Different
Points of the Brillouin Zone ....................... 83
4.7 Example I. Symmorphic Group C14v .......................... 86
4.8 Example II. Nonsymmorphic Group C24v ..................... 106
References ............................................... 127
Chapter 5. Scattering of X-Rays by Crystals ................... 129
5.1 Introduction ............................................. 129
5.2 Scattering from a Single Electron ........................ 130
5.3 Scattering from a Single Atom ............................ 133
5.4 Scattering from the Atoms in the Unit Cell of a Crystal .. 136
5.5 Scattering from a Crystal ................................ 137
5.6 Interpretation of Laue Equations in Reciprocal Space ..... 141
5.7 Methods of X-Ray Diffraction ............................. 143
5.7.1 The Laue Method (see Fig. 5.8a) ................... 143
5.7.2 The Bragg Method (see Fig. 5.8b) .................. 144
5.7.3 The Debye-Scherrer Method (see Fig. 5.8c) ......... 145
References .................................................... 145
Part II. Lattice Vibrations of Crystals ....................... 147
Chapter 6. Lattice Vibrations of Crystals ..................... 149
6.1 The Infinite Linear Crystal .............................. 149
6.2 The Finite Linear Crystal ................................ 154
6.3 Normal Modes of Vibration of a Linear Crystal ............ 157
6.4 Linear Crystal with a Basis .............................. 167
6.5 Lattice Vibrations in Three Dimensions ................... 173
6.5.1 The Equations of Motion ........................... 173
6.5.2 Allowed Values of k. Density of Phonon Modes ...... 176
6.5.3 Normal Modes of Vibration ......................... 179
6.5.4 Energy Levels ..................................... 184
6.5.5 Particular Modes of Vibration ..................... 188
6.5.6 Spectrum of Lattice Vibrations .................... 190
6.6 Group Theory and Lattice Vibrations ...................... 191
6.6.1 Properties of the Normal Coordinates .............. 191
6.6.2 The Frequency Eigenvalues and the Polarization
Vectors ........................................... 193
6.6.3 Additional Degeneracies Not Due to Spacelike
Symmetries ........................................ 197
6.6.4 Time-Reversal Degeneracy .......................... 198
6.7 Group-Theoretical Analysis of the Lattice Vibrations of
a Linear Crystal ......................................... 202
6.7.1 Case of One Atom Per Unit Cell .................... 202
6.7.2 Case of Two Atoms Per Unit Cell ................... 206
6.8 Group-Theoretical Analysis of the Lattice Vibrations of
a Three-Dimensional Crystal .............................. 209
6.9 Example I. Lattice Vibrations of a Two-Dimensional
Crystal with Symmetry C14v ............................... 212
6.10 Example II. Lattice Vibrations of a Two-Dimensional
Crystal with Symmetry C24v ............................... 221
References .................................................... 232
Chapter 7. Thermodynamics of Lattice Vibrations ............... 233
7.1 Thermodynamics of Specific Heats ......................... 233
7.2 The Classical Theory of the Specific Heats
of Solids ................................................ 235
7.3 The Einstein Theory of Specific Heat ..................... 236
7.4 The Debye Theory of Specific Heat ........................ 240
7.4.1 The Specific Heat of a Linear Crystal ............. 240
7.4.2 The Debye Theory Applied to a Linear Crystal ...... 243
7.4.3 The Debye Theory Applied to a Three-Dimensional
Crystal ........................................... 244
7.5 Temperature Dependence of the Amplitude of Vibrations
Solids. The Lindemann Law of Melting ..................... 252
References .................................................... 255
Chapter 8. Effect of Lattice Vibrations on X-ray
Scattering and Neutron Scattering ............................. 257
8.1 Effect of Lattice Vibrations on the Intensity
of the Scattered Radiation ............................... 257
8.1.1 The Intensity of the Scattered Radiation .......... 257
8.1.2 The Effect of Lattice Vibrations: Einstein
Model ............................................. 258
8.1.3 The Effect of Lattice Vibrations: Normal Mode
Treatment ......................................... 261
8.2 Theory of Neutron Scattering ............................. 267
8.3 Elastic Neutron Scattering ............................... 274
8.4 Inelastic Neutron Scattering ............................. 278
8.5 Application of Neutron Scattering to the Study of
Lattice Vibrations ....................................... 283
References .................................................... 286
Part III. Optical Spectroscopy of Crystals ................... 289
Chapter 9. Interaction of Radiation with Matter ............... 291
9.1 The Classical Radiative Field ............................ 292
9.2 The Quantum Theory of the Radiative Field ................ 301
9.3 The Hamiltonian of a Charged Particle in an
Electromagnetic Field .................................... 303
9.4 The Interaction Between a Charged Particle and
a Radiative Field ........................................ 305
9.5 First-Order Processes. Absorption and Emission of
Radiation ................................................ 308
9.6 Second-Order Processes ................................... 315
9.6.1 Matrix Element Due to H1 .......................... 317
9.6.2 Matrix Elements Due to H2 ......................... 318
9.6.3 Effective Matrix Element .......................... 319
9.6.4 Transition Rates of Scattering Processes .......... 323
References .................................................... 325
Chapter 10. Optical Spectra of Impurities in Solids ........... 327
10.1 Impurities in Crystals ................................... 328
10.2 Review of the Theory of Small Vibrations (Classical) ..... 328
10.3 Harmonic and Anharmonic Relaxation ....................... 337
10.4 Review of the Theory of Small Vibrations (Quantum
Mechanical) .............................................. 341
10.5 The Effect of Impurities on Lattice Vibrations ........... 346
10.6 The Franck-Condon Principle .............................. 352
10.7 Absorption and Emission in Crystals ...................... 359
10.8 Purely Electronic (Zero-Phonon) Transitions .............. 362
10.9 Characteristics of the Zero-Phonon Lines ................. 370
10.10 Phonon-Assisted Transitions ............................. 372
10.11 Radiative Transitions in the Presence of Localized
Vibrations ............................................... 380
10.12 Classification of Vibronic Spectra ...................... 390
References .................................................... 391
Chapter 11. Optical Spectra of Impurities in Solids II ........ 393
11.1 Summary of Previous Results .............................. 393
11.2 Deviations from the Franck-Condon Approximation .......... 397
11.3 Deviations from the Adiabatic Approximation.
Radiationless Transitions ................................ 410
11.4 A Simple Model for Laser Crystals: An Effective
Hamiltonian .............................................. 413
11.5 Radiative, Vibronic, and Radiationless Transitions of
Magnetic Impurities ...................................... 416
11.6 Selection Rules for Vibronic Transitions ................. 426
11.7 Effect of Temperature on the Position and Shape of
a Purely Electronic Line ................................. 427
11.7.1 Thermal Line Shift ................................ 428
11.7.2 Thermal Broadening of Sharp Lines ................. 429
References .................................................... 433
Chapter 12. Interaction of Light with Lattice Vibrations:
Infrared Absorption and Inelastic Light Scattering ............ 435
12.1 General Characteristics of Infrared Absorption by
Crystals ................................................. 436
12.2 Infrared Transitions in a Molecular System ............... 436
12.3 Momentum and Energy Conservation in Infrared Absorption .. 438
12.4 Quantum Theory of Infrared Absorption .................... 441
12.5 Reststrahl (One-Phonon) Absorption ....................... 451
12.6 Two-Phonon Absorption .................................... 453
12.7 Selection Rules for Infrared Absorption .................. 457
12.8 The Effect of Impurities on Infrared Absorption Spectra .. 459
12.9 Infrared Absorption in Homopolar Crystals ................ 460
12.10 General Characteristics of Raman Scattering from
Crystals ................................................ 468
12.11 Theory of Raman Scattering .............................. 471
12.12 Transition Polarizability ............................... 475
12.13 Energy Scattered in Raman Scattering Experiments ........ 480
12.14 Selection Rules for Raman Scattering .................... 483
12.15 The Effect of Impurities on Raman Scattering ............ 485
12.16 Brillouin Scattering .................................... 486
References .................................................... 490
Chapter 13 Lattice Vibrations and Lasers ...................... 491
13.1 Nonradiative Transitions ................................. 494
13.2 Single Wavelength Lasers ................................. 498
13.2.1 Optical Transitions in Rare Earth Ion Lasers ..... 499
13.2.2 Radiationless Decay Processes in Rare Earth Ion
Lasers ........................................... 502
13.3 Multiple Wavelength Lasers ............................... 505
References .................................................... 509
Subject Index ................................................. 511
|
|
