Condensed matter physics

A modern, unified treatment of condensed matter physics

This new work presents for the first time in decades a sweeping review of the whole field of condensed matter physics. It consolidates new and classic topics from disparate sources, teaching "not only about the effective masses of electrons in semiconductor crystals and band theory, but also about quasicrystals, dynamics of phase separation, why rubber is more floppy than steel, electron interference in nanometer-sized channels, and the quantum Hall effect."

Six major areas are covered---atomic structure, electronic structure, mechanical properties, electron transport, optical properties, and magnetism. But rather than defining the field in terms of particular materials, the author focuses on the way condensed matter physicists approach physical problems, combining phenomenology and microscopic arguments with information from experiments. For graduate students and professionals, researchers and engineers, applied mathematicians and materials scientists, Condensed Matter Physics provides:
* An exciting collection of new topics from the past two decades.
* A thorough treatment of classic topics, including band theory, transport theory, and semiconductor physics.
* Over 300 figures, incorporating many images from experiments.
* Frequent comparison of theory and experiment, both when they agree and when problems are still unsolved.
* More than 50 tables of data and a detailed index.
* Ample end-of-chapter problems, including computational exercises.
* Over 1000 references, both recent and historically significant.

An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.

Author Notes

Michael P. Marder, Ph.D., is Associate Professor of Physics at the University of Texas at Austin and a member of the internationally known Center for Nonlinear Dynamics.

Preface	p. xix
References	p. xxii
I Atomic Structure	p. 1
1 The Idea of Crystals	p. 3
1.1 Introduction	p. 3
1.1.1 Why are Solids Crystalline?	p. 4
1.2 Two-Dimensional Lattices	p. 6
1.2.1 Bravais Lattices	p. 6
1.2.2 Enumeration of Two-Dimensional Bravais Lattices	p. 7
1.2.3 Lattices with Bases	p. 7
1.2.4 Primitive Cells	p. 9
1.2.5 Wigner-Seitz Cells	p. 10
1.3 Symmetries	p. 11
1.3.1 The Space Group	p. 11
1.3.2 Translation and Point Groups	p. 11
Problems	p. 13
References	p. 15
2 Three-Dimensional Lattices	p. 17
2.1 Introduction	p. 17
2.1.1 Distribution Among Elements	p. 17
2.2 Monatomic Lattices	p. 20
2.2.1 The Simple Cubic Lattice	p. 20
2.2.2 The Face-Centered Cubic Lattice	p. 20
2.2.3 The Body-Centered Cubic Lattice	p. 21
2.2.4 The Hexagonal Lattice	p. 22
2.2.5 The Hexagonal Close-Packed Lattice	p. 23
2.2.6 The Diamond Lattice	p. 24
2.3 Compounds	p. 24
2.3.1 Rocksalt--Sodium Chloride	p. 25
2.3.2 Cesium Chloride	p. 26
2.3.3 Fluorite--Calcium Fluoride	p. 26
2.3.4 Zincblende--Zinc Sulfide	p. 26
2.3.5 Wurtzite--Zinc Oxide	p. 28
2.3.6 Perovskite--Calcium Titanate	p. 28
2.4 Classification of Lattices by Symmetry	p. 28
2.4.1 Fourteen Bravais Lattices and Seven Crystal Systems	p. 30
2.5 Symmetries of Lattices with Bases	p. 32
2.5.1 Thirty-Two Crystallographic Point Groups	p. 32
2.5.2 Two Hundred Thirty Distinct Lattices	p. 36
2.6 Some Macroscopic Implications of Microscopic Symmetries	p. 37
2.6.1 Pyroelectricity	p. 37
2.6.2 Piezoelectricity	p. 37
2.6.3 Optical Activity	p. 38
Problems	p. 38
References	p. 41
3 Experimental Determination of Crystal Structures	p. 43
3.1 Introduction	p. 43
3.2 Theory of Scattering from Crystals	p. 44
3.2.1 Lattice Sums	p. 47
3.2.2 Reciprocal Lattice	p. 48
3.2.3 Miller Indices	p. 51
3.2.4 Scattering from a Lattice with a Basis	p. 52
3.3 Experimental Methods	p. 54
3.3.1 Laue Method	p. 55
3.3.2 Rotating Crystal Method	p. 56
3.3.3 Powder Method	p. 58
3.4 Further Features of Scattering Experiments	p. 59
3.4.1 Interaction of X-Rays with Matter	p. 60
3.4.2 Production of X-Rays	p. 60
3.4.3 Neutrons	p. 61
3.4.4 Electrons	p. 61
3.4.5 Deciphering Complex Structures	p. 63
3.4.6 Accuracy of Structure Determinations	p. 64
Problems	p. 65
References	p. 67
4 Surfaces and Interfaces	p. 69
4.1 Introduction	p. 69
4.2 Geometry of Interfaces	p. 69
4.2.1 Coherent and Commensurate Interfaces	p. 70
4.2.2 Stacking Period and Interplanar Spacing	p. 71
4.2.3 Other Topics in Surface Structure	p. 73
4.3 Experimental Observation and Creation of Surfaces	p. 73
4.3.1 Low-Energy Electron Diffraction (LEED)	p. 74
4.3.2 Reflection High-Energy Electron Diffraction (RHEED)	p. 75
4.3.3 Molecular Beam Epitaxy (MBE)	p. 76
4.3.4 Field Ion Microscopy (FIM)	p. 77
4.3.5 Scanning Tunneling Microscopy (STM)	p. 77
4.3.6 Atomic Force Microscopy (AFM)	p. 82
4.3.7 High Resolution Electron Microscopy (HREM)	p. 82
Problems	p. 82
References	p. 85
5 Complex Structures	p. 87
5.1 Introduction	p. 87
5.2 Alloys	p. 87
5.2.1 Equilibrium Structures	p. 87
5.2.2 Phase Diagrams	p. 89
5.2.3 Superlattices	p. 90
5.2.4 Phase Separation	p. 91
5.2.5 Nonequilibrium Structures in Alloys	p. 94
5.2.6 Dynamics of Phase Separation	p. 95
5.3 Simulations	p. 97
5.3.1 Monte Carlo	p. 97
5.3.2 Molecular Dynamics	p. 98
5.4 Liquids	p. 99
5.4.1 Correlation Functions	p. 99
5.4.2 Extended X-Ray Absorption Fine Structure (EXAFS)	p. 101
5.4.3 Calculating Correlation Functions	p. 103
5.5 Glasses	p. 103
5.6 Liquid Crystals	p. 107
5.6.1 Nematics, Cholesterics, and Smectics	p. 108
5.6.2 Liquid Crystal Order Parameter	p. 109
5.7 Polymers	p. 110
5.7.1 Ideal Radius of Gyration	p. 111
5.8 Quasicrystals	p. 115
5.8.1 One-Dimensional Quasicrystal	p. 116
5.8.2 Two-Dimensional Quasicrystals--Penrose Tiles	p. 121
5.8.3 Experimental Observations	p. 124
5.8.4 Fullerenes	p. 124
Problems	p. 125
References	p. 129
II Electronic Structure	p. 133
6 The Single-Electron Model	p. 135
6.1 Introduction	p. 135
6.2 The Basic Hamiltonian	p. 137
6.3 Densities of States	p. 139
6.3.1 Definition of Density of States D	p. 140
6.3.2 Results for Free Electrons	p. 141
6.4 Statistical Mechanics of Noninteracting Electrons	p. 143
6.5 Sommerfeld Expansion	p. 146
6.5.1 Specific Heat of Noninteracting Electrons at Low Temperatures	p. 149
Problems	p. 150
References	p. 153
7 The Schrodinger Equation and Symmetry	p. 155
7.1 Introduction	p. 155
7.2 Translational Symmetry--Bloch's Theorem	p. 155
7.2.1 Van Hove Singularities	p. 160
7.2.2 Fourier Analysis of Bloch's Theorem	p. 163
7.2.3 Kronig-Penney Model	p. 166
7.3 Rotational Symmetry--Group Representations	p. 169
7.3.1 Classes and Characters	p. 175
7.3.2 Consequences of point group symmetries for Schrodinger's equation	p. 178
Problems	p. 181
References	p. 184
8 Nearly Free and Tightly Bound Electrons	p. 185
8.1 Introduction	p. 185
8.2 Nearly Free Electrons	p. 185
8.2.1 Degenerate Perturbation Theory	p. 187
8.3 Brillouin Zones	p. 189
8.3.1 Nearly Free Electron Fermi Surfaces	p. 191
8.4 Tightly Bound Electrons	p. 194
8.4.1 Wannier Functions	p. 194
8.4.2 Tight Binding Model	p. 197
Problems	p. 199
References	p. 202
9 Electron-Electron Interaction	p. 203
9.1 Introduction	p. 203
9.2 Hartree and Hartree-Fock Equations	p. 204
9.2.1 Variational Principle	p. 205
9.2.2 Hartree-Fock Equations	p. 205
9.2.3 Numerical Implementation	p. 209
9.2.4 Hartree-Fock Equations for Jellium	p. 212
9.3 Density Functional Theory	p. 214
9.3.1 Thomas-Fermi Theory	p. 216
9.3.2 Kohn-Sham Equations	p. 218
9.4 Stability of Matter	p. 220
Problems	p. 223
References	p. 226
10 Calculation of Band Structures	p. 229
10.1 Introduction	p. 229
10.2 Numerical Methods	p. 230
10.2.1 Pseudopotentials and Orthogonalized Planes Waves (OPW)	p. 230
10.2.2 Linear Combination of Atomic Orbitals (LCAO)	p. 235
10.2.3 Plane Waves	p. 237
10.2.4 Linear Augmented Plane Waves (LAPW)	p. 240
10.2.5 Linearized Muffin Tin Orbitals (LMTO)	p. 243
10.3 Definition of Metals, Insulators, and Semiconductors	p. 246
10.4 Brief Survey of the Periodic Table	p. 248
10.4.1 Noble Gases	p. 248
10.4.2 Nearly Free Electron Metals	p. 250
10.4.3 Semiconductors	p. 252
10.4.4 Transition Metals	p. 252
10.4.5 Rare Earths	p. 252
Problems	p. 254
References	p. 258
III Mechanical Properties	p. 261
11 Cohesion of Solids	p. 263
11.1 Introduction	p. 263
11.1.1 Radii of Atoms	p. 263
11.2 Noble Gases	p. 265
11.3 Ionic Crystals	p. 269
11.3.1 Ewald Sums	p. 270
11.4 Metals	p. 272
11.4.1 Use of Pseudopotentials	p. 275
11.5 Band Structure Energy	p. 276
11.5.1 Peierls Distortion	p. 277
11.5.2 Structural Phase Transitions	p. 279
11.6 Hydrogen-Bonded Solids	p. 280
11.7 Cohesive Energy from Band Calculations	p. 280
11.8 Classical Potentials	p. 282
Problems	p. 283
References	p. 285
12 Elasticity	p. 287
12.1 Introduction	p. 287
12.2 General Theory of Linear Elasticity	p. 287
12.2.1 Solids of Cubic Symmetry	p. 289
12.2.2 Isotropic Solids	p. 290
12.3 Other Constitutive Laws	p. 295
12.3.1 Liquid Crystals	p. 295
12.3.2 Rubber	p. 298
12.3.3 Composite and Granular Materials	p. 301
Problems	p. 301
References	p. 303
13 Phonons	p. 305
13.1 Introduction	p. 305
13.2 Vibrations of a Classical Lattice	p. 305
13.2.1 Normal Modes	p. 307
13.2.2 Lattice with a Basis	p. 309
13.3 Vibrations of a Quantum-Mechanical Lattice	p. 313
13.3.1 Phonon Specific Heat	p. 317
13.3.2 Einstein and Debye Models	p. 321
13.3.3 Thermal Expansion	p. 324
13.4 Inelastic Scattering from Phonons	p. 326
13.4.1 Neutron Scattering	p. 327
13.4.2 Formal Theory of Neutron Scattering	p. 329
13.4.3 Averaging Exponentials	p. 333
13.4.4 Evaluation of Structure Factor	p. 335
13.4.5 Kohn Anomalies	p. 336
13.5 The Mossbauer Effect	p. 336
Problems	p. 339
References	p. 340
14 Dislocations and Cracks	p. 343
14.1 Introduction	p. 343
14.2 Dislocations	p. 345
14.2.1 Experimental Observations of Dislocations	p. 347
14.2.2 Force to Move a Dislocation	p. 350
14.2.3 One-Dimensional Dislocations: Frenkel-Kontorova Model	p. 350
14.3 Two-Dimensional Dislocations and Hexatic Phases	p. 353
14.3.1 Impossibility of Crystalline Order in Two Dimensions	p. 353
14.3.2 Orientational Order	p. 355
14.3.3 Kosterlitz-Thouless-Berezinskii Transition	p. 356
14.4 Cracks	p. 363
14.4.1 Fracture of a Strip	p. 363
14.4.2 Stresses Around an Elliptical Hole	p. 366
14.4.3 Stress Intensity Factor	p. 368
14.4.4 Atomic Aspects of Fracture	p. 368
Problems	p. 370
References	p. 373
15 Fluid Mechanics	p. 375
15.1 Introduction	p. 375
15.2 Newtonian Fluids	p. 375
15.2.1 Euler's Equation	p. 375
15.2.2 Navier-Stokes Equation	p. 377
15.3 Polymeric Solutions	p. 378
15.4 Plasticity	p. 385
15.5 Superfluid [superscript 4]He	p. 389
15.5.1 Two-Fluid Hydrodynamics	p. 392
15.5.2 Second Sound	p. 393
15.5.3 Origin of Superfluidity	p. 395
15.5.4 Lagrangian Theory of Wave Function	p. 400
15.5.5 Superfluid [superscript 3]He	p. 403
Problems	p. 404
References	p. 408
IV Electron Transport	p. 411
16 Dynamics of Bloch Electrons	p. 413
16.1 Introduction	p. 413
16.1.1 Drude Model	p. 413
16.2 Semiclassical Electron Dynamics	p. 415
16.2.1 Bloch Oscillations	p. 416
16.2.2 k.P Method	p. 417
16.2.3 Effective Mass	p. 419
16.3 Noninteracting Electrons in an Electric Field	p. 419
16.3.1 Zener Tunneling	p. 422
16.4 Semiclassical Equations from Wave Packets	p. 425
16.4.1 Formal Dynamics of Wave Packets	p. 425
16.5 Quantizing Semiclassical Dynamics	p. 430
16.5.1 Wannier-Stark Ladders	p. 432
16.5.2 de Haas-van Alphen Effect	p. 432
16.5.3 Experimental Measurements of Fermi Surfaces	p. 434
Problems	p. 437
References	p. 440
17 Transport Phenomena and Fermi Liquid Theory	p. 443
17.1 Introduction	p. 443
17.2 Boltzmann Equation	p. 443
17.2.1 Boltzmann Equation	p. 445
17.2.2 Relaxation Time Approximation	p. 446
17.2.3 Relation to Rate of Production of Entropy	p. 448
17.3 Transport Symmetries	p. 449
17.3.1 Onsager Relations	p. 450
17.4 Thermoelectric Phenomena	p. 451
17.4.1 Electrical Current	p. 451
17.4.2 Effective Mass and Holes	p. 453
17.4.3 Mixed Thermal and Electrical Gradients	p. 454
17.4.4 Wiedemann-Franz Law	p. 455
17.4.5 Thermopower--Seebeck Effect	p. 456
17.4.6 Peltier Effect	p. 457
17.4.7 Thomson Effect	p. 457
17.4.8 Hall Effect	p. 459
17.4.9 Magnetoresistance	p. 461
17.4.10 Giant Magnetoresistance	p. 462
17.5 Fermi Liquid Theory	p. 462
17.5.1 Basic Ideas	p. 462
17.5.2 Statistical Mechanics of Quasi-Particles	p. 464
17.5.3 Effective Mass	p. 466
17.5.4 Specific Heat	p. 468
17.5.5 Fermi Liquid Parameters	p. 469
17.5.6 Traveling Waves	p. 470
17.5.7 Comparison with Experiment in [superscript 3]He	p. 473
Problems	p. 474
References	p. 478
18 Microscopic Theories of Conduction	p. 481
18.1 Introduction	p. 481
18.2 Weak Scattering Theory of Conductivity	p. 481
18.2.1 General Formula for Relaxation Time	p. 481
18.2.2 Matthiessen's Rule	p. 486
18.2.3 Fluctuations	p. 487
18.3 Metal-Insulator Transitions	p. 488
18.3.1 Types of Impurities	p. 488
18.3.2 Impurity Scattering and Green's Functions	p. 492
18.3.3 Green's Functions	p. 493
18.3.4 Single Impurity	p. 497
18.4 Coherent Potential Approximation	p. 499
18.5 Localization	p. 500
18.5.1 Exact Results in One Dimension	p. 501
18.5.2 Scaling Theory of Localization	p. 505
18.5.3 Comparison with Experiment	p. 509
Problems	p. 510
References	p. 514
19 Electronics	p. 517
19.1 Introduction	p. 517
19.2 Metal Interfaces	p. 518
19.2.1 Work Functions	p. 519
19.2.2 Schottky Barrier	p. 520
19.2.3 Contact Potentials	p. 522
19.3 Semiconductors	p. 524
19.3.1 Pure Semiconductors	p. 525
19.3.2 Semiconductor in Equilibrium	p. 528
19.3.3 Intrinsic Semiconductor	p. 530
19.3.4 Extrinsic Semiconductor	p. 531
19.4 Diodes and Transistors	p. 533
19.4.1 Surface States	p. 536
19.4.2 Semiconductor Junctions	p. 537
19.4.3 Boltzmann Equation for Semiconductors	p. 540
19.4.4 Detailed Theory of Rectification	p. 542
19.4.5 Transistor	p. 545
19.5 Inversion Layers	p. 548
19.5.1 Heterostructures	p. 548
19.5.2 Quantum Point Contact	p. 550
19.5.3 Quantum Dot	p. 553
Problems	p. 556
References	p. 557
V Optical Properties	p. 559
20 Phenomenological Theory	p. 561
20.1 Introduction	p. 561
20.2 Maxwell's Equations	p. 563
20.2.1 Traveling Waves	p. 565
20.2.2 Mechanical Oscillators as Dielectric Function	p. 566
20.3 Kramers-Kronig Relations	p. 568
20.3.1 Application to Optical Experiments	p. 570
20.4 The Kubo-Greenwood Formula	p. 573
20.4.1 Born Approximation	p. 573
20.4.2 Susceptibility	p. 577
20.4.3 Many-Body Green Functions	p. 578
Problems	p. 578
References	p. 581
21 Optical Properties of Semiconductors	p. 583
21.1 Introduction	p. 583
21.2 Cyclotron Resonance	p. 583
21.2.1 Electron Energy Surfaces	p. 586
21.3 Semiconductor Band Gaps	p. 588
21.3.1 Direct Transitions	p. 588
21.3.2 Indirect Transitions	p. 589
21.4 Excitons	p. 591
21.4.1 Mott-Wannier Excitons	p. 591
21.4.2 Frenkel Excitons	p. 594
21.4.3 Electron-Hole Liquid	p. 595
21.5 Optoelectronics	p. 595
21.5.1 Solar Cells	p. 595
21.5.2 Lasers	p. 596
Problems	p. 602
References	p. 606
22 Optical Properties of Insulators	p. 609
22.1 Introduction	p. 609
22.2 Polarization	p. 609
22.2.1 Ferroelectrics	p. 609
22.2.2 Clausius-Mossotti Relation	p. 611
22.3 Optical Modes in Ionic Crystals	p. 613
22.3.1 Polaritons	p. 616
22.3.2 Polarons	p. 618
22.3.3 Experimental Observations of Polarons	p. 623
22.4 Point Defects and Color Centers	p. 623
22.4.1 Vacancies	p. 624
22.4.2 F Centers	p. 625
22.4.3 Electron Spin Resonance and Electron Nuclear Double Resonance	p. 626
22.4.4 Other Centers	p. 628
22.4.5 Franck-Condon Effect	p. 628
22.4.6 Urbach Tails	p. 632
Problems	p. 633
References	p. 635
23 Optical Properties of Metals and Inelastic Scattering	p. 637
23.1 Introduction	p. 637
23.1.1 Plasma Frequency	p. 637
23.2 Metals at Low Frequencies	p. 640
23.2.1 Anomalous Skin Effect	p. 642
23.3 Plasmons	p. 643
23.3.1 Experimental Observation of Plasmons	p. 644
23.4 Interband Transitions	p. 646
23.5 Brillouin and Raman Scattering	p. 649
23.5.1 Brillouin Scattering	p. 650
23.5.2 Raman Scattering	p. 651
23.5.3 Inelastic X-Ray Scattering	p. 651
23.6 Photoemission	p. 651
23.6.1 Measurement of Work Functions	p. 651
23.6.2 Angle-Resolved Photoemission	p. 654
23.6.3 Core-Level Photoemission and Charge-Transfer Insulators	p. 658
Problems	p. 664
References	p. 667
VI Magnetism	p. 669
24 Classical Theories of Magnetism and Ordering	p. 671
24.1 Introduction	p. 671
24.2 Three Views of Magnetism	p. 671
24.2.1 From Magnetic Moments	p. 671
24.2.2 From Conductivity	p. 672
24.2.3 From a Free Energy	p. 673
24.3 Magnetic Dipole Moments	p. 675
24.3.1 Spontaneous Magnetization of Ferromagnets	p. 678
24.3.2 Ferrimagnets	p. 679
24.3.3 Antiferromagnets	p. 681
24.4 Mean Field Theory and the Ising Model	p. 682
24.4.1 Domains	p. 684
24.4.2 Hysteresis	p. 687
24.5 Other Order-Disorder Transitions	p. 688
24.5.1 Alloy Superlattices	p. 688
24.5.2 Spin Glasses	p. 691
24.6 Critical Phenomena	p. 691
24.6.1 Landau Free Energy	p. 692
24.6.2 Scaling Theory	p. 698
Problems	p. 702
References	p. 705
25 Magnetism of Ions and Electrons	p. 707
25.1 Introduction	p. 707
25.2 Atomic Magnetism	p. 709
25.2.1 Hund's Rules	p. 710
25.2.2 Curie's Law	p. 714
25.3 Magnetism of the Free-Electron Gas	p. 717
25.3.1 Pauli Paramagnetism	p. 718
25.3.2 Landau Diamagnetism	p. 719
25.3.3 Aharonov-Bohm Effect	p. 722
25.4 Tightly Bound Electrons in Magnetic Fields	p. 724
25.5 Quantum Hall Effect	p. 728
25.5.1 Integer Quantum Hall Effect	p. 728
25.5.2 Fractional Quantum Hall Effect	p. 733
Problems	p. 739
References	p. 742
26 Quantum Mechanics of Interacting Magnetic Moments	p. 745
26.1 Introduction	p. 745
26.2 Origin of Ferromagnetism	p. 745
26.2.1 Heitler-London Calculation	p. 745
26.2.2 Spin Hamiltonian	p. 750
26.3 Heisenberg Model	p. 750
26.3.1 Indirect Exchange and Superexchange	p. 752
26.3.2 Ground State	p. 753
26.3.3 Spin Waves	p. 753
26.3.4 Spin Waves in Antiferromagnets	p. 756
26.3.5 Comparison with Experiment	p. 759
26.4 Ferromagnetism in Transition Metals	p. 759
26.4.1 Stoner Model	p. 759
26.4.2 Calculations Within Band Theory	p. 761
26.5 Kondo Effect	p. 763
26.5.1 Scaling Theory	p. 768
26.6 Hubbard Model	p. 772
26.6.1 Mean-Field Solution	p. 773
Problems	p. 776
References	p. 779
27 Superconductivity	p. 783
27.1 Introduction	p. 783
27.2 Phenomenology of Superconductivity	p. 784
27.2.1 Phenomenological Free Energy	p. 785
27.2.2 Thermodynamics of Superconductors	p. 787
27.2.3 Landau-Ginzburg Free Energy	p. 788
27.2.4 Type I and Type II Superconductors	p. 789
27.2.5 Flux Quantization	p. 794
27.2.6 The Josephson Effect	p. 796
27.2.7 Circuits with Josephson Junction Elements	p. 798
27.2.8 SQUIDS	p. 799
27.2.9 Origin of Josephson's Equations	p. 800
27.3 Microscopic Theory of Superconductivity	p. 802
27.3.1 Electron-Ion Interaction	p. 803
27.3.2 Formal Derivation	p. 806
27.3.3 Instability of the Normal State: Cooper Problem	p. 808
27.3.4 Self-Consistent Ground State	p. 812
27.3.5 Thermodynamics of Superconductors	p. 817
27.3.6 Superconductor in External Magnetic Field	p. 820
27.3.7 Derivation of Meissner Effect	p. 824
27.3.8 Comparison with Experiment	p. 827
27.3.9 High-Temperature Superconductors	p. 828
Problems	p. 833
References	p. 837
Appendices	p. 841
A Lattice Sums and Fourier Transforms	p. 843
A.1 One-Dimensional Sum	p. 843
A.2 Area Under Peaks	p. 843
A.3 Three-Dimensional Sum	p. 844
A.4 Discrete Case	p. 845
A.5 Convolution	p. 846
A.6 Using the Fast Fourier Transform	p. 846
References	p. 848
B Variational Techniques	p. 849
B.1 Functionals and Functional Derivatives	p. 849
B.2 Time-Independent Schrodinger Equation	p. 850
B.3 Time-Dependent Schrodinger Equation	p. 851
B.4 Method of Steepest Descent	p. 852
References	p. 852
C Second Quantization	p. 853
C.1 Rules	p. 853
C.1.1 States	p. 853
C.1.2 Operators	p. 853
C.1.3 Hamiltonians	p. 854
C.2 Derivations	p. 855
C.2.1 Bosons	p. 855
C.2.2 Fermions	p. 856
Index	p. 859

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