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Summary
Summary
Every process in physics is governed by selection rules that are the consequence of symmetry requirements. The beauty and strength of group theory resides in the transformation of many complex symmetry operations into a very simple linear algebra. This concise and class-tested book has been pedagogically tailored over 30 years MIT and 2 years at the University Federal of Minas Gerais (UFMG) in Brazil. The approach centers on the conviction that teaching group theory in close connection with applications helps students to learn, understand and use it for their own needs. For this reason, the theoretical background is confined to the first 4 introductory chapters (6-8 classroom hours). From there, each chapter develops new theory while introducing applications so that the students can best retain new concepts, build on concepts learned the previous week, and see interrelations between topics as presented. Essential problem sets between the chapters also aid the retention of the new material and for the consolidation of material learned in previous chapters. The text and problem sets have proved a useful springboard for the application of the basic material presented here to topics in semiconductor physics, and the physics of carbon-based nanostructures.
Author Notes
Mildred Dresselhaus was born Mildred Spiewak in Brooklyn, New York on November 11, 1930. She received a bachelor's degree from Hunter College, a master's degree from Radcliffe College, and a Ph.D. from the University of Chicago. In 1960, she ended up at the Massachusetts Institute of Technology. She worked at Lincoln Laboratory, a defense research center, where she was one of two women on a scientific staff of 1,000. Her research into the fundamental properties of carbon helped transform it into the superstar of modern materials science and the nanotechnology industry. In 1968, she was the first woman to secure a full professorship at M.I.T. and worked to promote the cause of women in science. She published more than 1,700 scientific papers and co-wrote eight books. She received the National Medal of Science, the Presidential Medal of Freedom, the Kavli Prize in Nanoscience, and the Enrico Fermi Prize. She died on February 20, 2017 at the age of 86.
(Bowker Author Biography)
Table of Contents
Part I Basic Mathematics | |
1 Basic Mathematical Background: Introduction | p. 3 |
1.1 Definition of a Group | p. 3 |
1.2 Simple Example of a Group | p. 3 |
1.3 Basic Definitions | p. 6 |
1.4 Rearrangement Theorem | p. 7 |
1.5 Cosets | p. 7 |
1.6 Conjugation and Class | p. 9 |
1.7 Factor Groups | p. 11 |
1.8 Group Theory and Quantum Mechanics | p. 11 |
2 Representation Theory and Basic Theorems | p. 15 |
2.1 Important Definitions | p. 15 |
2.2 Matrices | p. 16 |
2.3 Irreducible Representations | p. 17 |
2.4 The Unitarity of Representations | p. 19 |
2.5 Schur's Lemma (Part 1) | p. 21 |
2.6 Schur's Lemma (Part 2) | p. 23 |
2.7 Wonderful Orthogonality Theorem | p. 25 |
2.8 Representations and Vector Spaces | p. 28 |
3 Character of a Representation | p. 29 |
3.1 Definition of Character | p. 29 |
3.2 Characters and Class | p. 30 |
3.3 Wonderful Orthogonality Theorem for Character | p. 31 |
3.4 Reducible Representations | p. 33 |
3.5 The Number of Irreducible Representations | p. 35 |
3.6 Second Orthogonality Relation for Characters | p. 36 |
3.7 Regular Representation | p. 37 |
3.8 Setting up Character Tables | p. 40 |
3.9 Schoenflies Symmetry Notation | p. 44 |
3.10 The Hermann-Mauguin Symmetry Notation | p. 46 |
3.11 Symmetry Relations and Point Group Classifications | p. 48 |
4 Basis Functions | p. 57 |
4.1 Symmetry Operations and Basis Functions | p. 57 |
4.2 Basis Functions for Irreducible Representations | p. 58 |
4.3 Projection Operators P[subscript kl superscript (Gamma subscript n)] | p. 64 |
4.4 Derivation of an Explicit Expression for P[subscript kl superscript (Gamma subscript n)] | p. 64 |
4.5 The Effect of Projection Operations on an Arbitrary Function | p. 65 |
4.6 Linear Combinations of Atomic Orbitals for Three Equivalent Atoms at the Corners of an Equilateral Triangle | p. 67 |
4.7 The Application of Group Theory to Quantum Mechanics | p. 70 |
Part II Introductory Application to Quantum Systems | |
5 Splitting of Atomic Orbitals in a Crystal Potential | p. 79 |
5.1 Introduction | p. 79 |
5.2 Characters for the Full Rotation Group | p. 81 |
5.3 Cubic Crystal Field Environment for a Paramagnetic Transition Metal Ion | p. 85 |
5.4 Comments on Basis Functions | p. 90 |
5.5 Comments on the Form of Crystal Fields | p. 92 |
6 Application to Selection Rules and Direct Products | p. 97 |
6.1 The Electromagnetic Interaction as a Perturbation | p. 97 |
6.2 Orthogonality of Basis Functions | p. 99 |
6.3 Direct Product of Two Groups | p. 100 |
6.4 Direct Product of Two Irreducible Representations | p. 101 |
6.5 Characters for the Direct Product | p. 103 |
6.6 Selection Rule Concept in Group Theoretical Terms | p. 105 |
6.7 Example of Selection Rules | p. 106 |
Part III Molecular Systems | |
7 Electronic States of Molecules and Directed Valence | p. 113 |
7.1 Introduction | p. 113 |
7.2 General Concept of Equivalence | p. 115 |
7.3 Directed Valence Bonding | p. 117 |
7.4 Diatomic Molecules | p. 118 |
7.4.1 Homonuclear Diatomic Molecules | p. 118 |
7.4.2 Heterogeneous Diatomic Molecules | p. 120 |
7.5 Electronic Orbitals for Multiatomic Molecules | p. 124 |
7.5.1 The NH[subscript 3] Molecule | p. 124 |
7.5.2 The CH[subscript 4] Molecule | p. 125 |
7.5.3 The Hypothetical SH[subscript 6] Molecule | p. 129 |
7.5.4 The Octahedral SF[subscript 6] Molecule | p. 133 |
7.6 [sigma]- and [pi]-Bonds | p. 134 |
7.7 Jahn-Teller Effect | p. 141 |
8 Molecular Vibrations, Infrared, and Raman Activity | p. 147 |
8.1 Molecular Vibrations: Background | p. 147 |
8.2 Application of Group Theory to Molecular Vibrations | p. 149 |
8.3 Finding the Vibrational Normal Modes | p. 152 |
8.4 Molecular Vibrations in H[subscript 2]O | p. 154 |
8.5 Overtones and Combination Modes | p. 156 |
8.6 Infrared Activity | p. 157 |
8.7 Raman Effect | p. 159 |
8.8 Vibrations for Specific Molecules | p. 161 |
8.8.1 The Linear Molecules | p. 161 |
8.8.2 Vibrations of the NH[subscript 3] Molecule | p. 166 |
8.8.3 Vibrations of the CH[subscript 4] Molecule | p. 168 |
8.9 Rotational Energy Levels | p. 170 |
8.9.1 The Rigid Rotator | p. 170 |
8.9.2 Wigner-Eckart Theorem | p. 172 |
8.9.3 Vibrational-Rotational Interaction | p. 174 |
Part IV Application to Periodic Lattices | |
9 Space Groups in Real Space | p. 183 |
9.1 Mathematical Background for Space Groups | p. 184 |
9.1.1 Space Groups Symmetry Operations | p. 184 |
9.1.2 Compound Space Group Operations | p. 186 |
9.1.3 Translation Subgroup | p. 188 |
9.1.4 Symmorphic and Nonsymmorphic Space Groups | p. 189 |
9.2 Bravais Lattices and Space Groups | p. 190 |
9.2.1 Examples of Symmorphic Space Groups | p. 192 |
9.2.2 Cubic Space Groups and the Equivalence Transformation | p. 194 |
9.2.3 Examples of Nonsymmorphic Space Groups | p. 196 |
9.3 Two-Dimensional Space Groups | p. 198 |
9.3.1 2D Oblique Space Groups | p. 200 |
9.3.2 2D Rectangular Space Groups | p. 201 |
9.3.3 2D Square Space Group | p. 203 |
9.3.4 2D Hexagonal Space Groups | p. 203 |
9.4 Line Groups | p. 204 |
9.5 The Determination of Crystal Structure and Space Group | p. 205 |
9.5.1 Determination of the Crystal Structure | p. 206 |
9.5.2 Determination of the Space Group | p. 206 |
10 Space Groups in Reciprocal Space and Representations | p. 209 |
10.1 Reciprocal Space | p. 210 |
10.2 Translation Subgroup | p. 211 |
10.2.1 Representations for the Translation Group | p. 211 |
10.2.2 Bloch's Theorem and the Basis of the Translational Group | p. 212 |
10.3 Symmetry of k Vectors and the Group of the Wave Vector | p. 214 |
10.3.1 Point Group Operation in r-space and k-space | p. 214 |
10.3.2 The Group of the Wave Vector G[subscript k] and the Star of k | p. 215 |
10.3.3 Effect of Translations and Point Group Operations on Bloch Functions | p. 215 |
10.4 Space Group Representations | p. 219 |
10.4.1 Symmorphic Group Representations | p. 219 |
10.4.2 Nonsymmorphic Group Representations and the Multiplier Algebra | p. 220 |
10.5 Characters for the Equivalence Representation | p. 221 |
10.6 Common Cubic Lattices: Symmorphic Space Groups | p. 222 |
10.6.1 The [Gamma] Point | p. 223 |
10.6.2 Points with k [not equal] 0 | p. 224 |
10.7 Compatibility Relations | p. 227 |
10.8 The Diamond Structure: Nonsymmorphic Space Group | p. 230 |
10.8.1 Factor Group and the [Gamma] Point | p. 231 |
10.8.2 Points with k [not equal] 0 | p. 232 |
10.9 Finding Character Tables for all Groups of the Wave Vector | p. 235 |
Part V Electron and Phonon Dispersion Relation | |
11 Applications to Lattice Vibrations | p. 241 |
11.1 Introduction | p. 241 |
11.2 Lattice Modes and Molecular Vibrations | p. 244 |
11.3 Zone Center Phonon Modes | p. 246 |
11.3.1 The NaCl Structure | p. 246 |
11.3.2 The Perovskite Structure | p. 247 |
11.3.3 Phonons in the Nonsymmorphic Diamond Lattice | p. 250 |
11.3.4 Phonons in the Zinc Blende Structure | p. 252 |
11.4 Lattice Modes Away from k = 0 | p. 253 |
11.4.1 Phonons in NaCl at the X Point k = ([pi]/a)(100) | p. 254 |
11.4.2 Phonons in BaTiO[subscript 3] at the X Point | p. 256 |
11.4.3 Phonons at the K Point in Two-Dimensional Graphite | p. 258 |
11.5 Phonons in Te and [alpha]-Quartz Nonsymmorphic Structures | p. 262 |
11.5.1 Phonons in Tellurium | p. 262 |
11.5.2 Phonons in the [alpha]-Quartz Structure | p. 268 |
11.6 Effect of Axial Stress on Phonons | p. 272 |
12 Electronic Energy Levels in a Cubic Crystals | p. 279 |
12.1 Introduction | p. 279 |
12.2 Plane Wave Solutions at k = 0 | p. 282 |
12.3 Symmetrized Plane Solution Waves along the [Delta]-Axis | p. 286 |
12.4 Plane Wave Solutions at the X Point | p. 288 |
12.5 Effect of Glide Planes and Screw Axes | p. 294 |
13 Energy Band Models Based on Symmetry | p. 305 |
13.1 Introduction | p. 305 |
13.2 k [middle dot] p Perturbation Theory | p. 307 |
13.3 Example of k [middle dot] p Perturbation Theory for a Nondegenerate [characters not reproducible] Band | p. 308 |
13.4 Two Band Model: Degenerate First-Order Perturbation Theory | p. 311 |
13.5 Degenerate second-order k [middle dot] p Perturbation Theory | p. 316 |
13.6 Nondegenerate k [middle dot] p Perturbation Theory at a [Delta] Point | p. 324 |
13.7 Use of k [middle dot] p Perturbation Theory to Interpret Optical Experiments | p. 326 |
13.8 Application of Group Theory to Valley-Orbit Interactions in Semiconductors | p. 327 |
13.8.1 Background | p. 328 |
13.8.2 Impurities in Multivalley Semiconductors | p. 330 |
13.8.3 The Valley-Orbit Interaction | p. 331 |
14 Spin-Orbit Interaction in Solids and Double Groups | p. 337 |
14.1 Introduction | p. 337 |
14.2 Crystal Double Groups | p. 341 |
14.3 Double Group Properties | p. 343 |
14.4 Crystal Field Splitting Including Spin-Orbit Coupling | p. 349 |
14.5 Basis Functions for Double Group Representations | p. 353 |
14.6 Some Explicit Basis Functions | p. 355 |
14.7 Basis Functions for Other [Gamma subscript 8 superscript +] States | p. 358 |
14.8 Comments on Double Group Character Tables | p. 359 |
14.9 Plane Wave Basis Functions for Double Group Representations | p. 360 |
14.10 Group of the Wave Vector for Nonsymmorphic Double Groups | p. 362 |
15 Application of Double Groups to Energy Bands with Spin | p. 367 |
15.1 Introduction | p. 367 |
15.2 E(k) for the Empty Lattice Including Spin-Orbit Interaction | p. 368 |
15.3 The k [middle dot] p Perturbation with Spin-Orbit Interaction | p. 369 |
15.4 E(k) for a Nondegenerate Band Including Spin-Orbit Interaction | p. 372 |
15.5 E(k) for Degenerate Bands Including Spin-Orbit Interaction | p. 374 |
15.6 Effective g-Factor | p. 378 |
15.7 Fourier Expansion of Energy Bands: Slater-Koster Method | p. 389 |
15.7.1 Contributions at d = 0 | p. 396 |
15.7.2 Contributions at d = 1 | p. 396 |
15.7.3 Contributions at d = 2 | p. 397 |
15.7.4 Summing Contributions through d = 2 | p. 397 |
15.7.5 Other Degenerate Levels | p. 397 |
Part VI Other Symmetries | |
16 Time Reversal Symmetry | p. 403 |
16.1 The Time Reversal Operator | p. 403 |
16.2 Properties of the Time Reversal Operator | p. 404 |
16.3 The Effect of T on E(k), Neglecting Spin | p. 407 |
16.4 The Effect of T on E(k), Including the Spin-Orbit Interaction | p. 411 |
16.5 Magnetic Groups | p. 416 |
16.5.1 Introduction | p. 418 |
16.5.2 Types of Elements | p. 418 |
16.5.3 Types of Magnetic Point Groups | p. 419 |
16.5.4 Properties of the 58 Magnetic Point Groups {{A[subscript i], M[subscript k]}} | p. 419 |
16.5.5 Examples of Magnetic Structures | p. 423 |
16.5.6 Effect of Symmetry on the Spin Hamiltonian for the 32 Ordinary Point Groups | p. 426 |
17 Permutation Groups and Many-Electron States | p. 431 |
17.1 Introduction | p. 432 |
17.2 Classes and Irreducible Representations of Permutation Groups | p. 434 |
17.3 Basis Functions of Permutation Groups | p. 437 |
17.4 Pauli Principle in Atomic Spectra | p. 440 |
17.4.1 Two-Electron States | p. 440 |
17.4.2 Three-Electron States | p. 443 |
17.4.3 Four-Electron States | p. 448 |
17.4.4 Five-Electron States | p. 451 |
17.4.5 General Comments on Many-Electron States | p. 451 |
18 Symmetry Properties of Tensors | p. 455 |
18.1 Introduction | p. 455 |
18.2 Independent Components of Tensors Under Permutation Group Symmetry | p. 458 |
18.3 Independent Components of Tensors: Point Symmetry Groups | p. 462 |
18.4 Independent Components of Tensors Under Full Rotational Symmetry | p. 463 |
18.5 Tensors in Nonlinear Optics | p. 463 |
18.5.1 Cubic Symmetry: O[subscript h] | p. 464 |
18.5.2 Tetrahedral Symmetry: T[subscript d] | p. 466 |
18.5.3 Hexagonal Symmetry: D[subscript 6h] | p. 466 |
18.6 Elastic Modulus Tensor | p. 467 |
18.6.1 Full Rotational Symmetry: 3D Isotropy | p. 469 |
18.6.2 Icosahedral Symmetry | p. 472 |
18.6.3 Cubic Symmetry | p. 472 |
18.6.4 Other Symmetry Groups | p. 474 |
A Point Group Character Tables | p. 479 |
B Two-Dimensional Space Groups | p. 489 |
C Tables for 3D Space Groups | p. 499 |
C.1 Real Space | p. 499 |
C.2 Reciprocal Space | p. 503 |
D Tables for Double Groups | p. 521 |
E Group Theory Aspects of Carbon Nanotubes | p. 533 |
E.1 Nanotube Geometry and the (n, m) Indices | p. 534 |
E.2 Lattice Vectors in Real Space | p. 534 |
E.3 Lattice Vectors in Reciprocal Space | p. 535 |
E.4 Compound Operations and Tube Helicity | p. 536 |
E.5 Character Tables for Carbon Nanotubes | p. 538 |
F Permutation Group Character Tables | p. 543 |
References | p. 549 |
Index | p. 553 |