Fundamentals of the physics of solids : structure and dynamics

Title:

Personal Author:

Solyom, J. (Jeno)

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Berlin ; New York : Springer, 2007-

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v. 1 : ill. ; 24 cm.

ISBN:

9783540725992

9783540726005

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Solid state physics -- Textbooks

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Volume I, Structure and Dynamics

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The reader is holding the ?rst volume of a three-volume textbook on sol- state physics. This book is the outgrowth of the courses I have taught for many years at Eötvös University, Budapest, for undergraduate and graduate students under the titles Solid-State Physics and Modern Solid-State Physics. The main motivation for the publication of my lecture notes as a book was that none of the truly numerous textbooks covered all those areas that I felt should be included in a multi-semester course. Especially, if the course strives to present solid-state physics in a uni?ed structure, and aims at d- cussing not only classic chapters of the subject matter but also (in more or less detail) problems that are of great interest for today's researcher as well. Besides, the book presents a much larger material than what can be covered in a two- or three-semester course. In the ?rst part of the ?rst volume the analysis of crystal symmetries and structure goes into details that certainly cannot be included in a usual course on solid-state physics. The same applies, among others, to the discussion of the methods used in the determination of band structure,the properties of Fermi liquids and non-Fermiliquids, andthe theory of unconventional superconductors in the second and third volumes. These parts canbe assignedas supplementary reading for interested students, or can be discussed in advanced courses.

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This first volume of a projected three-volume set focuses on crystal structure and lattice dynamics. Solyom (Eotvos Lorand Univ., Budapest) emphasizes the observed physics of real materials over theoretical interpretation, providing extensive discussion of both experimental results and techniques. He also provides a substantial amount of theoretical material at an advanced graduate level. Topics include bonding in solids, crystal symmetries, structure determination, defects and dislocations, lattice dynamics and phonons, and magnetic order and magnons. A chapter devoted to amorphous materials and quasicrystals is particularly notable. Extensive appendixes provide reference material in both mathematics and physics, but do not include problems for students. There is more material here than can be covered in a single semester, and this book is more detailed than the well-known books by Charles Kittel (Quantum Theory of Solids, 2nd ed., 1987) or Neil Ashcroft and N. David Mermin (Solid State Physics, 1976). This volume would be most attractive for a one- or even two-semester advanced graduate course, and is a desirable reference work anywhere condensed matter/solid state/materials research is conducted. Summing Up: Recommended. Graduate students through professionals. M. C. Ogilvie Washington University

1 Introduction	p. 1
2 The Structure of Condensed Matter	p. 13
2.1 Characterization of the Structure	p. 14
2.1.1 Short- and Long-Range Order	p. 14
2.1.2 Order in the Center-of-Mass Positions, Orientation, and Chemical Composition	p. 19
2.2 Classification of Condensed Matter According to Structure	p. 20
2.2.1 Solid Phase	p. 20
2.2.2 Liquid Phase	p. 22
2.2.3 Mesomorphic Phases	p. 23
3 The Building Blocks of Solids	p. 31
3.1 Solids as Many-Particle Systems	p. 31
3.1.1 The Hamiltonian of Many-Particle Systems	p. 32
3.1.2 Effects of Applied Fields	p. 34
3.1.3 Relativistic Effects	p. 36
3.2 The State of Ion Cores	p. 38
3.2.1 Hund's Rules	p. 41
3.2.2 Angular Momentum and Magnetic Moment	p. 44
3.2.3 The Magnetic Hamiltonian of Atomic Electrons	p. 46
3.2.4 Magnetization and Susceptibility	p. 47
3.2.5 Langevin or Larmor Diamagnetism	p. 49
3.2.6 Atomic Paramagnetism	p. 51
3.2.7 Van Vleck Paramagnetism	p. 60
3.2.8 Electron Spin Resonance	p. 61
3.3 The Role of Nuclei	p. 68
3.3.1 Interaction with Nuclear Magnetic Moments	p. 68
3.3.2 Nuclear Magnetic Resonance	p. 71
3.3.3 The Mossbauer effect	p. 72
4 Bonding in Solids	p. 75
4.1 Types of Bonds and Cohesive Energy	p. 75
4.1.1 Classification of Solids According to the Type of the Bond	p. 76
4.1.2 Cohesive Energy	p. 76
4.2 Molecular crystals	p. 78
4.2.1 Van der Waals Bonds in Quantum Mechanics	p. 79
4.2.2 Cohesive Energy of Molecular Crystals	p. 81
4.3 Ionic Bond	p. 83
4.4 Covalent Bond	p. 89
4.4.1 The Valence-Bond Method	p. 90
4.4.2 Polar Covalent Bond	p. 94
4.4.3 The Molecular-Orbital Method	p. 96
4.4.4 The LCAO Method	p. 97
4.4.5 Molecular Orbitals Between Different Atoms	p. 100
4.4.6 Slater Determinant Form of the Wavefunction	p. 101
4.4.7 Hybridized Orbitals	p. 103
4.4.8 Covalent Bonds in Solids	p. 105
4.5 Metallic Bond	p. 106
4.6 The Hydrogen Bond	p. 106
5 Symmetries of Crystals	p. 109
5.1 Translational Symmetry in Crystals	p. 110
5.1.1 Translational Symmetry in Finite Crystals	p. 110
5.1.2 The Choice of Primitive Vectors	p. 111
5.1.3 Bravais Lattice and Basis	p. 113
5.1.4 Primitive Cells, Wigner-Seitz Cells, and Bravais Cells	p. 114
5.1.5 Crystallographic Positions, Directions, and Planes	p. 118
5.2 The Reciprocal Lattice	p. 120
5.2.1 Definition of the Reciprocal Lattice	p. 120
5.2.2 Properties of the Reciprocal Lattice	p. 122
5.3 Rotations and Reflections	p. 124
5.3.1 Symmetry Operations and Symmetry Elements	p. 124
5.3.2 Point Groups	p. 127
5.4 Rotation and Reflection Symmetries in Crystals	p. 135
5.4.1 Rotation Symmetries of Bravais Lattices	p. 135
5.4.2 Crystallographic Point Groups	p. 137
5.4.3 Crystal Systems and Bravais Groups	p. 138
5.4.4 Two-Dimensional Bravais-Lattice Types	p. 142
5.4.5 Three-Dimensional Bravais-Lattice Types	p. 146
5.4.6 The Hierarchy of Crystal Systems	p. 154
5.5 Full Symmetry of Crystals	p. 157
5.5.1 Screw Axes and Glide Planes	p. 157
5.5.2 Point Groups of Crystals and Crystal Classes	p. 160
5.5.3 Space Groups	p. 162
5.5.4 Symmetries of Magnetic Crystals	p. 166
6 Consequences of Symmetries	p. 171
6.1 Quantum Mechanical Eigenvalues and Symmetries	p. 172
6.1.1 Wigner's Theorem	p. 172
6.1.2 Splitting of Atomic Levels in Crystals	p. 173
6.1.3 Spin Contributions to Splitting	p. 179
6.1.4 Kramers' Theorem	p. 182
6.1.5 Selection Rules	p. 184
6.2 Consequences of Translational Symmetry	p. 184
6.2.1 The Born-von Karman Boundary Condition	p. 185
6.2.2 Bloch's Theorem	p. 186
6.2.3 Equivalent Wave Vectors	p. 189
6.2.4 Conservation of Crystal Momentum	p. 191
6.2.5 Symmetry Properties of Energy Eigenstates	p. 194
6.3 Symmetry Breaking and Its Consequences	p. 199
6.3.1 Symmetry Breaking in Phase Transitions	p. 199
6.3.2 Goldstone's Theorem	p. 200
7 The Structure of Crystals	p. 203
7.1 Types of Crystal Structures	p. 203
7.2 Cubic Crystal Structures	p. 205
7.2.1 Simple Cubic Structures	p. 205
7.2.2 Body-Centered Cubic Structures	p. 210
7.2.3 Face-Centered Cubic Structures	p. 214
7.2.4 Diamond and Sphalerite Structures	p. 221
7.3 Hexagonal Crystal Structures	p. 224
7.4 Typical Sizes of Primitive Cells	p. 229
7.5 Layered and Chain-Like Structures	p. 229
7.6 Relationship Between Structure and Bonding	p. 233
7.6.1 The Structure of Covalently Bonded Solids	p. 233
7.6.2 Structures with Nondirectional Bonds	p. 235
8 Methods of Structure Determination	p. 241
8.1 The Theory of Diffraction	p. 242
8.1.1 The Bragg and Laue Conditions of Diffraction	p. 242
8.1.2 Structure Amplitude and Atomic Form Factor	p. 246
8.1.3 Diffraction Cross Section	p. 249
8.1.4 The Shape and Intensity of Diffraction Peaks	p. 252
8.1.5 Cancellation in Structures with a Polyatomic Basis	p. 255
8.1.6 The Dynamical Theory of Diffraction	p. 258
8.2 Experimental Study of Diffraction	p. 261
8.2.1 Characteristic Properties of Different Types of Radiation	p. 261
8.2.2 The Ewald Construction	p. 264
8.2.3 Diffraction Methods	p. 265
8.3 Other Methods of Structure Determination	p. 269
9 The Structure of Real Crystals	p. 273
9.1 Point Defects	p. 275
9.1.1 Vacancies	p. 275
9.1.2 Interstitials	p. 278
9.1.3 Pairs of Point Defects	p. 280
9.2 Line Defects, Dislocations	p. 283
9.2.1 Edge and Screw Dislocations	p. 284
9.2.2 The Burgers Vector	p. 286
9.2.3 Dislocations as Topological Defects	p. 288
9.2.4 Disclinations	p. 290
9.2.5 Dislocations in Hexagonal Lattices	p. 292
9.3 Planar Defects	p. 293
9.3.1 Stacking Faults	p. 293
9.3.2 Partial Dislocations	p. 294
9.3.3 Low-Angle Grain Boundaries	p. 298
9.3.4 Coincident-Site-Lattice and Twin Boundaries	p. 299
9.3.5 Antiphase Boundaries	p. 300
9.4 Volume Defects	p. 302
10 The Structure of Noncrystalline Solids	p. 303
10.1 The Structure of Amorphous Materials	p. 303
10.1.1 Models of Topological Disorder	p. 303
10.1.2 Analysis of the Short-Range Order	p. 305
10.2 Quasiperiodic Structures	p. 309
10.2.1 Periodic and Quasiperiodic Functions	p. 310
10.2.2 Incommensurate Structures	p. 312
10.2.3 Experimental Observation of Quasicrystals	p. 314
10.2.4 The Fibonacci Chain	p. 317
10.2.5 Penrose Tiling of the Plane	p. 323
10.2.6 Three-Dimensional Quasicrystals	p. 327
11 Dynamics of Crystal Lattices	p. 331
11.1 The Harmonic Approximation	p. 331
11.1.1 Second-Order Expansion of the Potential	p. 332
11.1.2 Expansion of the Energy for Pair Potentials	p. 334
11.1.3 Equations Governing Lattice Vibrations	p. 335
11.2 Vibrational Spectra of Simple Lattices	p. 337
11.2.1 Vibrations of a Monatomic Linear Chain	p. 337
11.2.2 Vibrations of a Diatomic Chain	p. 341
11.2.3 Vibrations of a Dimerized Chain	p. 345
11.2.4 Vibrations of a Simple Cubic Lattice	p. 349
11.3 The General Description of Lattice Vibrations	p. 354
11.3.1 The Dynamical Matrix and its Eigenvalues	p. 355
11.3.2 Normal Coordinates and Normal Modes	p. 357
11.3.3 Acoustic and Optical Vibrations	p. 360
11.4 Lattice Vibrations in the Long-Wavelength Limit	p. 363
11.4.1 Acoustic Vibrations as Elastic Waves	p. 363
11.4.2 Elastic Constants of Crystalline Materials	p. 367
11.4.3 Elastic Waves in Cubic Crystals	p. 371
11.4.4 Optical Vibrations in Ionic Crystals	p. 373
11.5 Localized Lattice Vibrations	p. 377
11.5.1 Vibrations in a Chain with an Impurity	p. 377
11.5.2 Impurities in a Three-Dimensional Lattice	p. 381
11.6 The Specific Heat of Classical Lattices	p. 383
12 The Quantum Theory of Lattice Vibrations	p. 387
12.1 Quantization of Lattice Vibrations	p. 387
12.1.1 The Einstein Model	p. 387
12.1.2 The Debye Model	p. 389
12.1.3 Quantization of the Hamiltonian	p. 390
12.1.4 The Quantum Mechanics of Harmonic Oscillators	p. 392
12.1.5 Creation and Annihilation Operators of Vibrational Modes	p. 394
12.1.6 Phonons as Elementary Excitations	p. 395
12.1.7 Acoustic Phonons as Goldstone Bosons	p. 397
12.1.8 Symmetries of the Vibrational Spectrum	p. 397
12.2 Density of Phonon States	p. 398
12.2.1 Definition of the Density of States	p. 399
12.2.2 The Density of States in One- and Two-Dimensional Systems	p. 402
12.2.3 Van Hove Singularities	p. 405
12.3 The Thermodynamics of Vibrating Lattices	p. 409
12.3.1 The Ground State of the Lattice and Melting	p. 410
12.3.2 The Specific Heat of the Phonon Gas	p. 413
12.3.3 The Equation of State of the Crystal	p. 418
12.4 Anharmonicity	p. 421
12.4.1 Higher-Order Expansion of the Potential	p. 421
12.4.2 Interaction Among the Phonons	p. 423
12.4.3 Thermal Expansion and Thermal Conductivity in Crystals	p. 425
13 The Experimental Study of Phonons	p. 429
13.1 General Considerations	p. 429
13.2 Optical Methods in the Study of Phonons	p. 431
13.2.1 Infrared Absorption	p. 431
13.2.2 Raman Scattering	p. 433
13.2.3 Brillouin Scattering	p. 436
13.3 Neutron Scattering on a Thermally Vibrating Crystal	p. 438
13.3.1 Coherent Scattering Cross Section	p. 439
13.3.2 Temperature Dependence of the Intensity of Bragg Peaks	p. 443
13.3.3 Inelastic Phonon Peaks	p. 444
13.3.4 The Finite Width of Phonon Peaks	p. 446
14 Magnetically Ordered Systems	p. 449
14.1 Magnetic Materials	p. 450
14.1.1 Ferromagnetic Materials	p. 450
14.1.2 Antiferromagnetic Materials	p. 453
14.1.3 Spiral Magnetic Structures	p. 459
14.1.4 Ferrimagnetic Materials	p. 461
14.2 Exchange Interactions	p. 463
14.2.1 Direct Exchange	p. 463
14.2.2 Indirect Exchange in Metals	p. 464
14.2.3 Superexchange	p. 466
14.2.4 Double Exchange	p. 468
14.3 Simple Models of Magnetism	p. 469
14.3.1 The Isotropic Heisenberg Model	p. 469
14.3.2 Anisotropic Models	p. 471
14.4 The Mean-Field Approximation	p. 473
14.4.1 The Mean-Field Theory of Ferromagnetism	p. 474
14.4.2 The Mean-Field Theory of Antiferromagnetism	p. 478
14.4.3 The General Description of Two-Sublattice Antiferromagnets	p. 485
14.4.4 The Mean-Field Theory of Ferrimagnetism	p. 487
14.5 The General Description of Magnetic Phase Transitions	p. 488
14.5.1 The Landau Theory of Second-Order Phase Transitions	p. 489
14.5.2 Determination of Possible Magnetic Structures	p. 492
14.5.3 Spatial Inhomogeneities and the Correlation Length	p. 494
14.5.4 Scaling Laws	p. 496
14.5.5 Elimination of Fluctuations and the Renormalization Group	p. 500
14.6 High-Temperature Expansion	p. 503
14.7 Magnetic Anisotropy, Domains	p. 504
14.7.1 A Continuum Model of Magnetic Systems	p. 505
14.7.2 Magnetic Domains	p. 508
15 Elementary Excitations in Magnetic Systems	p. 515
15.1 Classical Spin Waves	p. 516
15.1.1 Ferromagnetic Spin Waves	p. 516
15.1.2 Spin Waves in Antiferromagnets	p. 518
15.2 Quantum Mechanical Treatment of Spin Waves	p. 521
15.2.1 The Quantum Mechanics of Ferromagnetic Spin Waves	p. 521
15.2.2 Magnons as Elementary Excitations	p. 524
15.2.3 Thermodynamics of the Gas of Magnons	p. 527
15.2.4 Rigorous Representations of Spin Operators	p. 530
15.2.5 Interactions Between Magnons	p. 533
15.2.6 Two-Magnon Bound States	p. 536
15.3 Antiferromagnetic Magnons	p. 540
15.3.1 Diagonalization of the Hamiltonian	p. 541
15.3.2 The Antiferromagnetic Ground State	p. 543
15.3.3 Antiferromagnetic Magnons at Finite Temperature	p. 544
15.3.4 Excitations in Anisotropic Antiferromagnets	p. 545
15.3.5 Magnons in Ferrimagnets	p. 546
15.4 Experimental Study of Magnetic Excitations	p. 547
15.5 Low-Dimensional Magnetic Systems	p. 548
15.5.1 Destruction of Magnetic Order by Thermal and Quantum Fluctuations	p. 549
15.5.2 Vortices in the Two-Dimensional Planar Model	p. 551
15.5.3 The Spin-1/2 Anisotropic Ferromagnetic Heisenberg Chain	p. 560
15.5.4 The Ground State of the Antiferromagnetic Chain	p. 566
15.5.5 Spinon Excitations in the Antiferromagnetic Chain	p. 569
15.5.6 The One-Dimensional XY Model	p. 572
15.5.7 The Role of Next-Nearest-Neighbor Interactions	p. 575
15.5.8 Excitations in the Spin-One Heisenberg Chain	p. 578
15.5.9 Spin Ladders	p. 581
15.5.10 Physical Realizations of Spin Chains and Spin Ladders	p. 583
15.6 Spin Liquids	p. 584
A Physical Constants and Units	p. 587
A.1 Physical Constants	p. 587
A.2 Relationships Among Units	p. 588
B The Periodic Table of Elements	p. 593
B.1 The Electron and Crystal Structures of Elements	p. 593
B.2 Characteristic Temperatures of the Elements	p. 596
C Mathematical Formulas	p. 601
C.1 Fourier Transforms	p. 601
C.1.1 Fourier Transform of Continuous Functions	p. 601
C.1.2 Fourier Transform of Functions Defined at Lattice Points	p. 606
C.1.3 Fourier Transform of Some Simple Functions	p. 608
C.2 Some Useful Integrals	p. 610
C.2.1 Integrals Containing Exponential Functions	p. 610
C.2.2 Integrals Containing the Bose Function	p. 611
C.2.3 Integrals Containing the Fermi Function	p. 612
C.2.4 Integrals over the Fermi Sphere	p. 613
C.2.5 d-Dimensional Integrals	p. 615
C.3 Special Functions	p. 615
C.3.1 The Dirac Delta Function	p. 615
C.3.2 Zeta and Gamma Functions	p. 617
C.3.3 Bessel Functions	p. 620
C.4 Orthogonal Polynomials	p. 623
C.4.1 Hermite Polynomials	p. 623
C.4.2 Laguerre Polynomials	p. 624
C.4.3 Legendre Polynomials	p. 625
C.4.4 Spherical Harmonics	p. 627
C.4.5 Expansion in Spherical Harmonics	p. 629
D Fundamentals of Group Theory	p. 633
D.1 Basic Notions of Group Theory	p. 633
D.1.1 Definition of Groups	p. 633
D.1.2 Conjugate Elements and Conjugacy Classes	p. 635
D.1.3 Representations and Characters	p. 635
D.1.4 Reducible and Irreducible Representations	p. 637
D.1.5 The Reduction of Reducible Representations	p. 638
D.1.6 Compatibility Condition	p. 639
D.1.7 Basis Functions of the Representations	p. 639
D.1.8 The Double Group	p. 641
D.1.9 Continuous Groups	p. 642
D.2 Applications of Group Theory	p. 646
D.2.1 Irreducible Representations of the Group O[subscript h]	p. 646
D.2.2 Group Theory and Quantum Mechanics	p. 648
E Scattering of Particles by Solids	p. 653
E.1 The Scattering Cross Section	p. 653
E.2 The Van Hove Formula for Cross Section	p. 656
E.2.1 Potential Scattering	p. 656
E.2.2 Magnetic Scattering	p. 660
F The Algebra of Angular-Momentum and Spin Operators	p. 665
F.1 Angular Momentum	p. 665
F.1.1 Angular Momentum and the Rotation Group	p. 665
F.1.2 The Irreducible Representations of the Rotation Group	p. 667
F.1.3 Orbital Angular Momentum and Spin	p. 669
F.1.4 Addition Theorem for Angular Momenta	p. 670
F.2 Orbital Angular Momentum	p. 672
F.3 Spin Operators	p. 673
F.3.1 Two-Dimensional Representations of the Rotation Group	p. 673
F.3.2 Spin Algebra	p. 674
F.3.3 Projection Operators	p. 676
Figure Credits	p. 677
Name Index	p. 679
Subject Index	p. 683

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