Lecture notes on field theory in Condensed matter Physics

lecture notes on field theory in condensed matter physics and condensed matter physics crystals liquids liquid crystals and polymers pdf
RyanCanon Profile Pic
RyanCanon,United Arab Emirates,Teacher
Published Date:21-07-2017
Your Website URL(Optional)
Comment
Condensed Matter Physics I Peter S. Riseborough February 5, 2015 Contents 1 Introduction 10 1.1 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . 10 2 Crystallography 18 2.0.1 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Structures 20 3.1 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Crystalline Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 The Direct Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 Primitive Unit Cells . . . . . . . . . . . . . . . . . . . . . 30 3.3.2 The Wigner-Seitz Unit Cell . . . . . . . . . . . . . . . . . 31 3.4 Symmetry of Crystals . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.1 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . 34 3.4.2 Group Multiplication Tables . . . . . . . . . . . . . . . . 35 3.4.3 Point Group Operations . . . . . . . . . . . . . . . . . . . 37 3.4.4 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.5 Limitations Imposed by Translational Symmetry . . . . . 41 3.4.6 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.7 Point Group Nomenclature . . . . . . . . . . . . . . . . . 44 3.5 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5.1 Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5.2 Cubic Bravais Lattices. . . . . . . . . . . . . . . . . . . . 53 3.5.3 Tetragonal Bravais Lattices. . . . . . . . . . . . . . . . . . 58 3.5.4 Orthorhombic Bravais Lattices. . . . . . . . . . . . . . . . 59 3.5.5 Monoclinic Bravais Lattice. . . . . . . . . . . . . . . . . . 60 3.5.6 Triclinic Bravais Lattice. . . . . . . . . . . . . . . . . . . . 61 3.5.7 Trigonal Bravais Lattice.. . . . . . . . . . . . . . . . . . . 63 3.5.8 Hexagonal Bravais Lattice. . . . . . . . . . . . . . . . . . 66 3.5.9 Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6 Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.6.1 Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.7 Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.7.1 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.8 Crystal Structures with Bases. . . . . . . . . . . . . . . . . . . . 78 3.8.1 Diamond Structure . . . . . . . . . . . . . . . . . . . . . . 78 3.8.2 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.8.3 Graphite Structure . . . . . . . . . . . . . . . . . . . . . . 79 3.8.4 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.8.5 Hexagonal Close-Packed Structure . . . . . . . . . . . . . 83 3.8.6 Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.8.7 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.8.8 Other Close-Packed Structures . . . . . . . . . . . . . . . 86 3.8.9 Sodium Chloride Structure . . . . . . . . . . . . . . . . . 88 3.8.10 Cesium Chloride Structure . . . . . . . . . . . . . . . . . 90 3.8.11 Fluorite Structure . . . . . . . . . . . . . . . . . . . . . . 92 3.8.12 The Copper Three Gold Structure . . . . . . . . . . . . . 93 3.8.13 Rutile Structure . . . . . . . . . . . . . . . . . . . . . . . 94 3.8.14 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.8.15 Zinc Blende Structure . . . . . . . . . . . . . . . . . . . . 95 3.8.16 Zincite Structure . . . . . . . . . . . . . . . . . . . . . . . 96 3.8.17 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.8.18 The Perovskite Structure . . . . . . . . . . . . . . . . . . 98 3.8.19 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.9 Lattice Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.9.1 Exercise 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.9.2 Exercise 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.9.3 Exercise 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.10 Quasi-Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4 Structure Determination 112 4.1 X Ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.1.1 The Bragg condition . . . . . . . . . . . . . . . . . . . . . 112 4.1.2 The Laue conditions . . . . . . . . . . . . . . . . . . . . . 114 4.1.3 Equivalence of the Bragg and Laue conditions . . . . . . . 118 4.1.4 The Ewald Construction . . . . . . . . . . . . . . . . . . . 119 4.1.5 X-ray Techniques . . . . . . . . . . . . . . . . . . . . . . . 120 4.1.6 Exercise 18 . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.1.7 The Structure and Form Factors . . . . . . . . . . . . . . 125 4.1.8 Exercise 19 . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.1.9 Exercise 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.1.10 Exercise 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.1.11 Exercise 22 . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.1.12 Exercise 23 . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.1.13 Exercise 24 . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.1.14 Exercise 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.2 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.2.1 Exercise 26 . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.3 Theory of the Differential Scattering Cross-section . . . . . . . . 145 24.3.1 Time Dependent Perturbation Theory . . . . . . . . . . . 146 4.3.2 The Fermi Golden Rule . . . . . . . . . . . . . . . . . . . 148 4.3.3 The Elastic Scattering Cross-Section . . . . . . . . . . . . 150 4.3.4 The Condition for Coherent Scattering . . . . . . . . . . . 152 4.3.5 Exercise 27 . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.3.6 Exercise 28 . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.3.7 Exercise 29 . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.3.8 Anti-Domain Phase Boundaries . . . . . . . . . . . . . . . 156 4.3.9 Exercise 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.4 Elastic Scattering from Quasi-Crystals . . . . . . . . . . . . . . . 158 4.5 Elastic Scattering from a Fluid . . . . . . . . . . . . . . . . . . . 162 5 The Reciprocal Lattice 164 5.0.1 Exercise 31 . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.1 The Reciprocal Lattice as a Dual Lattice. . . . . . . . . . . . . . 165 5.1.1 Exercise 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.2 Examples of Reciprocal Lattices. . . . . . . . . . . . . . . . . . . 169 5.2.1 The Simple Cubic Reciprocal Lattice . . . . . . . . . . . . 169 5.2.2 The Body Centered Cubic Reciprocal Lattice . . . . . . . 169 5.2.3 The Face Centered Cubic Reciprocal Lattice . . . . . . . 170 5.2.4 The Hexagonal Reciprocal Lattice . . . . . . . . . . . . . 171 5.2.5 Exercise 33 . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.3 The Brillouin Zones . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.3.1 The Simple Cubic Brillouin Zone . . . . . . . . . . . . . . 173 5.3.2 The Body Centered Cubic Brillouin Zone . . . . . . . . . 175 5.3.3 The Face Centered Cubic Brillouin Zone . . . . . . . . . . 177 5.3.4 The Hexagonal Brillouin Zone. . . . . . . . . . . . . . . . 178 5.3.5 The Trigonal Brillouin Zone . . . . . . . . . . . . . . . . . 180 6 Electrons 183 7 Electronic States 183 7.1 Many-Electron Wave Functions . . . . . . . . . . . . . . . . . . . 184 7.1.1 Exercise 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.2 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.4 Plane Wave Expansion of Bloch Functions . . . . . . . . . . . . . 198 7.5 The Bloch Wave Vector . . . . . . . . . . . . . . . . . . . . . . . 202 7.6 The Density of States . . . . . . . . . . . . . . . . . . . . . . . . 203 7.6.1 Exercise 35 . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.7 The Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 207 38 Approximate Models 211 8.1 The Nearly-Free Electron Model . . . . . . . . . . . . . . . . . . 211 8.1.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 212 8.1.2 Non-Degenerate Perturbation Theory . . . . . . . . . . . 213 8.1.3 Degenerate Perturbation Theory . . . . . . . . . . . . . . 215 8.1.4 Empty Lattice Approximation Band Structure . . . . . . 222 8.1.5 Exercise 36 . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.1.6 Degeneracies of the Bloch States . . . . . . . . . . . . . . 231 8.1.7 Exercise 37 . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.1.8 Exercise 38 . . . . . . . . . . . . . . . . . . . . . . . . . . 242 8.1.9 Brillouin Zone Boundaries and Fermi Surfaces. . . . . . . 242 8.1.10 The Geometric Structure Factor . . . . . . . . . . . . . . 248 8.1.11 Exercise 39 . . . . . . . . . . . . . . . . . . . . . . . . . . 258 8.1.12 Exercise 40 . . . . . . . . . . . . . . . . . . . . . . . . . . 260 8.1.13 Exercise 41 . . . . . . . . . . . . . . . . . . . . . . . . . . 261 8.1.14 Exercise 42 . . . . . . . . . . . . . . . . . . . . . . . . . . 262 8.1.15 Exercise 43 . . . . . . . . . . . . . . . . . . . . . . . . . . 263 8.2 The Pseudo-Potential Method . . . . . . . . . . . . . . . . . . . . 264 8.2.1 The Pseudo-Potential Theorem . . . . . . . . . . . . . . . 267 8.2.2 The Cancellation Theorem . . . . . . . . . . . . . . . . . 269 8.2.3 The Scattering Approach . . . . . . . . . . . . . . . . . . 272 8.2.4 The Ziman-Lloyd Pseudo-potential . . . . . . . . . . . . . 274 8.2.5 Exercise 44 . . . . . . . . . . . . . . . . . . . . . . . . . . 276 8.2.6 Exercise 45 . . . . . . . . . . . . . . . . . . . . . . . . . . 276 8.2.7 Exercise 46 . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.3 The Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . 278 8.3.1 Tight-Binding s Band Metal. . . . . . . . . . . . . . . . . 286 8.3.2 Tight-Binding Bands of Diamond Structured Semicon- ductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.3.3 Exercise 47 . . . . . . . . . . . . . . . . . . . . . . . . . . 293 8.3.4 Exercise 48 . . . . . . . . . . . . . . . . . . . . . . . . . . 294 8.3.5 Exercise 49 . . . . . . . . . . . . . . . . . . . . . . . . . . 296 8.3.6 Exercise 50 . . . . . . . . . . . . . . . . . . . . . . . . . . 296 8.3.7 Exercise 51 . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8.3.8 Exercise 52 . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8.3.9 Wannier Functions . . . . . . . . . . . . . . . . . . . . . . 297 8.3.10 Exercise 53 . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.3.11 Exercise 54 . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.3.12 Example of Tight-Binding: Graphene . . . . . . . . . . . 301 9 Electron-Electron Interactions 307 9.1 The Landau Fermi Liquid . . . . . . . . . . . . . . . . . . . . . . 307 9.1.1 The Scattering Rate . . . . . . . . . . . . . . . . . . . . . 309 9.1.2 The Quasi-Particle Energy . . . . . . . . . . . . . . . . . 316 9.1.3 Exercise 55 . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.2 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . 320 49.2.1 The Free Electron Gas. . . . . . . . . . . . . . . . . . . . 324 9.2.2 Exercise 56 . . . . . . . . . . . . . . . . . . . . . . . . . . 342 9.3 The Density Functional Method. . . . . . . . . . . . . . . . . . . 344 9.3.1 Hohenberg-Kohn Theorem. . . . . . . . . . . . . . . . . . 345 9.3.2 Functionals and Functional Derivatives . . . . . . . . . . 346 9.3.3 The Variational Principle . . . . . . . . . . . . . . . . . . 350 9.3.4 The Electrostatic Terms . . . . . . . . . . . . . . . . . . . 352 9.3.5 The Kohn-Sham Equations . . . . . . . . . . . . . . . . . 353 9.3.6 The Local Density Approximation . . . . . . . . . . . . . 355 9.4 Static Screening. . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 9.4.1 The Thomas-Fermi Approximation . . . . . . . . . . . . . 361 9.4.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . 364 9.4.3 Density Functional Response Function . . . . . . . . . . . 368 9.4.4 Exercise 57 . . . . . . . . . . . . . . . . . . . . . . . . . . 371 9.4.5 Exercise 58 . . . . . . . . . . . . . . . . . . . . . . . . . . 372 10 Stability of Structures 377 10.1 Momentum Space Representation . . . . . . . . . . . . . . . . . . 377 10.2 Real Space Representation . . . . . . . . . . . . . . . . . . . . . . 384 11 Metals 392 11.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 11.1.1 The Sommerfeld Expansion . . . . . . . . . . . . . . . . . 393 11.1.2 The Specific Heat Capacity . . . . . . . . . . . . . . . . . 395 11.1.3 Exercise 59 . . . . . . . . . . . . . . . . . . . . . . . . . . 398 11.1.4 Exercise 60 . . . . . . . . . . . . . . . . . . . . . . . . . . 398 11.1.5 Pauli Paramagnetism . . . . . . . . . . . . . . . . . . . . 398 11.1.6 Exercise 61 . . . . . . . . . . . . . . . . . . . . . . . . . . 402 11.1.7 Exercise 62 . . . . . . . . . . . . . . . . . . . . . . . . . . 402 11.1.8 Landau Diamagnetism . . . . . . . . . . . . . . . . . . . . 402 11.1.9 Landau Level Quantization . . . . . . . . . . . . . . . . . 403 11.1.10The Diamagnetic Susceptibility . . . . . . . . . . . . . . . 405 11.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . 408 11.2.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . 408 11.2.2 Scattering by Static Defects . . . . . . . . . . . . . . . . . 408 11.2.3 Exercise 63 . . . . . . . . . . . . . . . . . . . . . . . . . . 415 11.2.4 The Hall Effect and Magneto-resistance. . . . . . . . . . . 416 11.2.5 Multi-band Models . . . . . . . . . . . . . . . . . . . . . . 424 11.3 Electromagnetic Properties of Metals . . . . . . . . . . . . . . . . 427 11.3.1 The Longitudinal Response . . . . . . . . . . . . . . . . . 430 11.3.2 Electron Scattering Experiments . . . . . . . . . . . . . . 440 11.3.3 Exercise 64 . . . . . . . . . . . . . . . . . . . . . . . . . . 445 11.3.4 Exercise 65 . . . . . . . . . . . . . . . . . . . . . . . . . . 447 11.3.5 The Transverse Response . . . . . . . . . . . . . . . . . . 452 11.3.6 Optical Experiments . . . . . . . . . . . . . . . . . . . . . 457 11.3.7 Kramers-Kronig Relation . . . . . . . . . . . . . . . . . . 459 511.3.8 Exercise 66 . . . . . . . . . . . . . . . . . . . . . . . . . . 460 11.3.9 Exercise 67 . . . . . . . . . . . . . . . . . . . . . . . . . . 461 11.3.10The Drude Conductivity . . . . . . . . . . . . . . . . . . . 462 11.3.11Exercise 68 . . . . . . . . . . . . . . . . . . . . . . . . . . 466 11.3.12Exercise 69 . . . . . . . . . . . . . . . . . . . . . . . . . . 468 11.3.13The Anomalous Skin Effect . . . . . . . . . . . . . . . . . 468 11.3.14Inter-Band Transitions . . . . . . . . . . . . . . . . . . . . 472 11.4 The Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 474 11.4.1 Semi-Classical Orbits . . . . . . . . . . . . . . . . . . . . 474 11.4.2 de Haas - van Alphen Oscillations . . . . . . . . . . . . . 478 11.4.3 Exercise 70 . . . . . . . . . . . . . . . . . . . . . . . . . . 481 11.4.4 The Lifshitz-Kosevich Formulae . . . . . . . . . . . . . . . 481 11.4.5 Geometric Resonances . . . . . . . . . . . . . . . . . . . . 487 11.4.6 Cyclotron Resonances . . . . . . . . . . . . . . . . . . . . 489 11.5 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . 494 11.5.1 The Integer Quantum Hall Effect . . . . . . . . . . . . . . 495 11.5.2 Exercise 71 . . . . . . . . . . . . . . . . . . . . . . . . . . 504 11.5.3 Exercise 72 . . . . . . . . . . . . . . . . . . . . . . . . . . 505 11.5.4 The Fractional Quantum Hall Effect . . . . . . . . . . . . 506 11.5.5 Quasi-Particle Excitations . . . . . . . . . . . . . . . . . . 508 11.5.6 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 11.5.7 Composite Fermions . . . . . . . . . . . . . . . . . . . . . 520 12 Insulators and Semiconductors 523 12.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 12.1.1 Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 12.1.2 Intrinsic Semiconductors . . . . . . . . . . . . . . . . . . . 531 12.1.3 Extrinsic Semiconductors . . . . . . . . . . . . . . . . . . 533 12.1.4 Exercise 73 . . . . . . . . . . . . . . . . . . . . . . . . . . 536 12.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . 536 12.3 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 537 13 Phonons 538 14 Harmonic Phonons 538 14.1 Lattice with a Basis . . . . . . . . . . . . . . . . . . . . . . . . . 548 14.2 A Sum Rule for the Dispersion Relations. . . . . . . . . . . . . . 549 14.2.1 Exercise 74 . . . . . . . . . . . . . . . . . . . . . . . . . . 551 14.3 The Nature of the Phonon Modes . . . . . . . . . . . . . . . . . . 552 14.3.1 Exercise 75 . . . . . . . . . . . . . . . . . . . . . . . . . . 554 14.3.2 Exercise 76 . . . . . . . . . . . . . . . . . . . . . . . . . . 555 14.3.3 Exercise 77 . . . . . . . . . . . . . . . . . . . . . . . . . . 556 14.3.4 Exercise 78 . . . . . . . . . . . . . . . . . . . . . . . . . . 556 14.3.5 Exercise 79 . . . . . . . . . . . . . . . . . . . . . . . . . . 556 14.3.6 Exercise 80 . . . . . . . . . . . . . . . . . . . . . . . . . . 557 14.3.7 Exercise 81 . . . . . . . . . . . . . . . . . . . . . . . . . . 558 614.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 14.4.1 The Specific Heat . . . . . . . . . . . . . . . . . . . . . . 562 14.4.2 The Einstein Model of a Solid . . . . . . . . . . . . . . . . 563 14.4.3 The Debye Model of a Solid . . . . . . . . . . . . . . . . . 564 14.4.4 Exercise 82 . . . . . . . . . . . . . . . . . . . . . . . . . . 566 14.4.5 Exercise 83 . . . . . . . . . . . . . . . . . . . . . . . . . . 567 14.4.6 Exercise 84 . . . . . . . . . . . . . . . . . . . . . . . . . . 567 14.4.7 Exercise 85 . . . . . . . . . . . . . . . . . . . . . . . . . . 568 14.4.8 Lindemann Theory of Melting. . . . . . . . . . . . . . . . 568 14.4.9 Thermal Expansion . . . . . . . . . . . . . . . . . . . . . 572 14.4.10Thermal Expansion of Metals . . . . . . . . . . . . . . . . 573 14.5 Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 14.5.1 Exercise 86 . . . . . . . . . . . . . . . . . . . . . . . . . . 575 15 Phonon Measurements 577 15.1 Inelastic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . 577 15.2 The Scattering Cross-Section . . . . . . . . . . . . . . . . . . . . 578 15.2.1 The Zero-Phonon Scattering Process . . . . . . . . . . . . 582 15.3 The Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . 583 15.3.1 The One-Phonon Scattering Processes. . . . . . . . . . . . 585 15.3.2 Multi-Phonon Scattering. . . . . . . . . . . . . . . . . . . 590 15.3.3 Exercise 87 . . . . . . . . . . . . . . . . . . . . . . . . . . 592 15.3.4 Exercise 88 . . . . . . . . . . . . . . . . . . . . . . . . . . 593 15.3.5 Exercise 89 . . . . . . . . . . . . . . . . . . . . . . . . . . 593 15.4 Raman and Brillouin Scattering of Light . . . . . . . . . . . . . . 593 16 Phonons in Metals 599 16.1 Screened Ionic Plasmons . . . . . . . . . . . . . . . . . . . . . . . 600 16.1.1 Kohn Anomalies . . . . . . . . . . . . . . . . . . . . . . . 601 16.2 Dielectric Constant of a Metal. . . . . . . . . . . . . . . . . . . . 602 16.3 The Retarded Electron-Electron Interaction . . . . . . . . . . . . 604 16.4 Phonon Renormalization of Quasi-Particles . . . . . . . . . . . . 605 16.5 Electron-Phonon Interactions . . . . . . . . . . . . . . . . . . . . 608 16.6 Electrical Resistivity due to Phonon Scattering . . . . . . . . . . 609 16.6.1 Umklapp Scattering . . . . . . . . . . . . . . . . . . . . . 614 16.6.2 Phonon Drag . . . . . . . . . . . . . . . . . . . . . . . . . 615 17 Phonons in Semiconductors 616 17.1 Resistivity due to Phonon Scattering . . . . . . . . . . . . . . . . 616 17.2 Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 17.3 Indirect Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . 617 718 Impurities and Disorder 619 18.1 Scattering by Impurities . . . . . . . . . . . . . . . . . . . . . . . 625 18.1.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 18.2 Virtual Bound States . . . . . . . . . . . . . . . . . . . . . . . . . 633 18.2.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 18.3 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 18.4 Coherent Potential Approximation . . . . . . . . . . . . . . . . . 638 18.4.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 18.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 18.5.1 Anderson Model of Localization. . . . . . . . . . . . . . . 642 18.5.2 Scaling Theories of Localization. . . . . . . . . . . . . . . 644 19 Magnetic Impurities 647 19.1 Localized Magnetic Impurities in Metals . . . . . . . . . . . . . . 647 19.2 Mean-Field Approximation . . . . . . . . . . . . . . . . . . . . . 647 19.3 The Atomic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 652 19.4 The Schrieffer-Wolf Transformation . . . . . . . . . . . . . . . . . 656 19.4.1 The Kondo Hamiltonian . . . . . . . . . . . . . . . . . . . 659 19.5 The Resistance Minimum . . . . . . . . . . . . . . . . . . . . . . 660 20 Collective Phenomenon 668 21 Itinerant Magnetism 668 21.1 Stoner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668 21.1.1 Exercise 90 . . . . . . . . . . . . . . . . . . . . . . . . . . 671 21.1.2 Exercise 91 . . . . . . . . . . . . . . . . . . . . . . . . . . 671 21.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . 671 21.3 Magnetic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 673 21.4 Spin Fluctuations near Ferromagnetic Instabilities . . . . . . . . 677 21.4.1 Ferromagnetic Spin Waves . . . . . . . . . . . . . . . . . . 680 21.5 The Slater-Pauling Curves . . . . . . . . . . . . . . . . . . . . . . 685 21.6 The Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . 687 22 Localized Magnetism 688 22.1 Holstein-Primakoff Transformation . . . . . . . . . . . . . . . . . 690 22.2 Spin Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . 694 22.2.1 Exercise 92 . . . . . . . . . . . . . . . . . . . . . . . . . . 697 22.3 Anti-ferromagnetic Spinwaves . . . . . . . . . . . . . . . . . . . . 697 22.3.1 Exercise 93 . . . . . . . . . . . . . . . . . . . . . . . . . . 700 23 Spin Glasses 702 23.1 Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 706 23.2 The Sherrington-Kirkpatrick Solution. . . . . . . . . . . . . . . . 708 824 Magnetic Neutron Scattering 711 24.1 The Inelastic Scattering Cross-Section . . . . . . . . . . . . . . . 711 24.1.1 The Dipole-Dipole Interaction. . . . . . . . . . . . . . . . 711 24.1.2 The Inelastic Scattering Cross-Section . . . . . . . . . . . 711 24.2 Time-Dependent Spin Correlation Functions . . . . . . . . . . . . 715 24.3 The Fluctuation - Dissipation Theorem . . . . . . . . . . . . . . 718 24.4 Magnetic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 720 24.4.1 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . 720 24.4.2 Exercise 94 . . . . . . . . . . . . . . . . . . . . . . . . . . 722 24.4.3 Exercise 95 . . . . . . . . . . . . . . . . . . . . . . . . . . 722 24.4.4 Spin Wave Scattering . . . . . . . . . . . . . . . . . . . . 723 24.4.5 Exercise 96 . . . . . . . . . . . . . . . . . . . . . . . . . . 724 24.4.6 Critical Scattering . . . . . . . . . . . . . . . . . . . . . . 724 25 Superconductivity 726 25.1 Experimental Manifestation . . . . . . . . . . . . . . . . . . . . . 727 25.1.1 The London Equations . . . . . . . . . . . . . . . . . . . . 729 25.1.2 Thermodynamics of the Superconducting State . . . . . . 730 25.2 The Cooper Problem . . . . . . . . . . . . . . . . . . . . . . . . . 733 25.3 Pairing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 25.3.1 The Pairing Interaction . . . . . . . . . . . . . . . . . . . 738 25.3.2 The B.C.S. Variational State . . . . . . . . . . . . . . . . 740 25.3.3 The Gap Equation . . . . . . . . . . . . . . . . . . . . . . 742 25.3.4 The Ground State Energy . . . . . . . . . . . . . . . . . . 745 25.4 Quasi-Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 25.4.1 Exercise 97 . . . . . . . . . . . . . . . . . . . . . . . . . . 752 25.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 25.6 Perfect Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 756 25.7 The Meissner Effect . . . . . . . . . . . . . . . . . . . . . . . . . 758 25.8 Landau-Ginzburg Theory . . . . . . . . . . . . . . . . . . . . . . 760 25.8.1 Extremal Configurations . . . . . . . . . . . . . . . . . . . 764 25.8.2 Characteristic Length Scales . . . . . . . . . . . . . . . . 765 25.8.3 The Surface Energy . . . . . . . . . . . . . . . . . . . . . 768 25.8.4 The Little-Parks Experiment . . . . . . . . . . . . . . . . 770 25.8.5 The Critical Current . . . . . . . . . . . . . . . . . . . . . 772 91 Introduction Condensed Matter Physics is the study of materials in Solid and Liquid Phases. Itencompassesthestudyoforderedcrystallinephasesofsolids, aswellasdisor- dered phases such as the amorphous and glassy phases of solids. Furthermore, it also includes materials with short-ranged order such as conventional liquids andliquidcrystalswhichshowunconventionalorderintermediatebetweenthose of a crystalline solid and a liquid. Condensed matter has the quite remarkable property that, due to the large number of particles involved, the behavior of the materials may be qualitatively distinct from those of the individual con- stituents. The behavior of the incredibly large number of particles is governed by (quantum) statistics which, through the chaotically complicated motion of theparticles, producesnewtypesoforder. Theseemergentphenomenaarebest exemplified in phenomenon such as magnetism or superconductivity where the collective behavior results in transitions to new phases. In surveying the properties of materials, it is convenient to separate the properties according to two (usually) disparate time scales. One time scale is a slow time scale which governs the structural dynamics and the other is a faster time scale that governs the electronic motion. The large difference be- tween the time scales is due to the large ratio of the nuclear masses to the 3 electronic mass, M /m ∼ 10 . The long-ranged electromagnetic force binds N e these two constituents of different mass into electrically neutral material. The slow moving nuclear masses can be considered to be quasi-static, and are re- sponsible for defining the structure of matter. In this approximation, the fast movingelectronsequilibrateinthequasi-staticpotentialproducedbythenuclei. 1.1 The Born-Oppenheimer Approximation The difference in the relevant time scales for electronic and nuclear motion 1 allows one to make the Born-Oppenheimer Approximation . In this approxi- mation, the electronic states are treated as if the nuclei were at rest at fixed positions. However, when treating the slow motions of the nuclei, the electrons are considered as adapting instantaneously to the potential of the charged nu- clei, thereby minimizing the electronic energies. Thus, the nuclei charges are dressed by a cloud of electrons forming ionic or atomic-like aggregates. A qualitative estimate of the relative energies of nuclear versus electronic motion can be obtained by considering metallic hydrogen. The electronic en- ergies are calculated using only the Bohr model of the hydrogen atom. The equation of motion for an electron of mass m has the form e 2 2 Z e m v e − = − (1) 2 a a 1 M. Born and R. Oppenheimer, Ann. Phys. (Leipzig), 84, 457 (1927). 10whereZ isthenuclearchargeandaistheradiusoftheatomicorbital. Thestan- dardsemi-classicalquantizationconditionduetoBohrandSommerfeldrestricts the angular momentum to integral values of ¯h m v a = n h ¯ (2) e These equations can be combined to find the Bohr radius as 2 ¯h 2 a = n (3) 2 m Z e e and also the quantized total electronic energy of the hydrogen atom 2 2 Z e m v e E = − + e a 2 2 Z e = − 2 a 2 4 m Z e e = − (4) 2 2 2 n h ¯ which are standard results from atomic physics. Note that the kinetic energy term and the electrostatic potential term have similar magnitudes. Now consider the motion of the nuclei. The forces consist of Coulomb forces betweenthenucleiandelectronsandthequantummechanicalPauliforces. The electrostatic repulsions and attractions have similar magnitudes since the inter- nuclear separations are of the same order as the Bohr radius. In equilibrium, the sum of the forces vanish identically. Furthermore, if an atom is displaced from the equilibrium position by a small distance equal tor, the restoring force is approximately given by the dipole force 2 2 Z e − α r (5) 3 a where α is a dimensionless constant. Hence, the equation of motion for the displacement of a nuclei of mass M is N 2 2 2 Z e d r − α r = M (6) N 3 2 a dt which shows that the nuclei undergo harmonic oscillations with frequency 2 2 Z e 2 ω = α (7) 3 M a N The semi-classical quantization condition I M dr . v = 2 π n ¯h (8) N 11yields the energy for nuclear motion as E = n h ¯ ω N  1 2 4 2 m Z e 1 Z m e e 2 = n α (9) 2 M ¯h N where we have substituted the expression for the Bohr radius for a in the ex- pression for ω. Thus, the ratio of the energies of nuclear motion to electronic motion are given by the factor 1   2 E m N e ∼ (10) E M e N 1 Since the ratio of the mass of electron to the proton mass is , the nuclear 2000 kinetic energy is negligible when compared to the electronic kinetic energy. A morerigorousproofofthevalidityoftheBorn-Oppenheimerapproximationwas 2 given by Migdal . —————————————————————————————————- Example: Beyond the Born-Oppenheimer Approximation AnexampleofacorrectiontotheBorn-Oppenheimerapproximationisgiven by (incoherent) inelastic neutron-atom scattering. The scattering occurs only via the nuclear force between the neutrons and the nucleus. The electrons are bound to the nucleus and are only excited via an indirect process. We intend to show that the probability that an electronic transition occurs is governed by he ratio of the mass of the electron m to the mass of the nucleus M . First, we e N shall consider the elastic-scattering process in the Born-Oppenheimer approxi- mation and then we shall consider inelastic-scattering. In the consideration of the elastic-scattering process, the electronic states maybeignoredsincetheelectronsarenotexcited. Furthermore,consistentwith the Born-Oppenheimer approximation, the mass of the electron is neglected in comparison with the mass of the nuclei when we consider the kinetics of the scattering. Therefore, in an incoherent scattering process, a neutron of mass m is scattered from an individual nucleus of mass M . The elastic-scattering n N process satisfies conservation of energy and momentum m M m M n N 2 n N 2 2 2 v + V = v + V i i f f 2 2 2 2 m v + M V = m v + M V (11) n N n N i i f f The scattering results in a transfer of energy and momentum between the two particles. The energy and momentum transferred between the neutron and the 2 A. B. Migdal, Sov. Phys. J.E.T.P. 7, 996 (1958). 12Nuclear Compton Scattering p f θ p m i n P i M P N f Figure 1: The scattering of a neutron of massm by an atom with a nucleus of n mass M . N nucleus is given by h ¯ q = m ( v − v ) n i f m n 2 2 ¯h ω = ( v − v ) i f 2 2 ¯h 2 = h ¯ q . v − q (12) i 2 m n The neutron’s energy and momentum loss can be determined by experiment. If the nucleus is initially at rest, the magnitude of the final momentum of the neutron is related to the initial momentum via the scattering angle θ, via q M 2 N 2 cosθ + ( ) − sin θ m n v = v (13) f i M N 1 + ( ) m n M N It is seen that, if  1, the scattering is elastic. The positioning of the m n detectorselectstheneutronswhicharescatteredthroughtheangleθ. Thescat- tering of the neutrons by the nuclei is characterized by the correlation function S(q,ω) which embodies the above conservation laws in a delta function factor   2 ¯h h ¯ 2 S(q,ω) ∝ δ hω ¯ − q . p + q (14) i m 2m n n 13The correlation function can be directly expressed in terms of the initial prop- erties of the nuclei via Z   2 ¯h h ¯ 3 2 S(q,ω) = d P n(P ) δ hω ¯ − q . P − q (15) i i i M 2M N N where n(P) is the initial momentum distribution of the nuclei, which is given by 2 n(P) = φ(P) (16) and where φ(P) is the nuclear wave function in the momentum representation   Z 1 i 3 φ(P) = d R exp − P . R χ(R) (17) 3 2 ¯h ( 2 π ¯h ) in which χ(R) is the nuclear wave function in the real space representation. SincethescatteringpotentialisrepresentedbyaFermipoint-scatteringpseudo- potential, 2 2 π h ¯ V(r −R) = b δ(r −R) (18) n n m n where b is the scattering length, its Fourier transform is given by   2 2 π h ¯ V(q) = b exp i q . R (19) m n Therefore, the neutron scattering cross-section can be entirely expressed in terms of the nuclear density-density correlation function     2 2 d σ k m f n 2 = V(q) S(q,ω) (20) 2 dΩdω k i 2 π h ¯ From this, we conclude that measurements of the scattering cross-section yields information about the kinetics of the transition and the matrix-elements of the interaction potential. We shall now consider the effect of the neutron scattering from a neutral atom. Since the interaction between the neutron and the atom is identically zero in the asymptotic initial and final states, the wave function of the atom Ψ(r,R) can be analyzed in terms of its center of mass and relative coordinates. If the atomic Hamiltonian for the asymptotic in and out states is expressed as 2 2 2 ˆ P pˆ Z e ˆ H = + − (21) 2 M 2 m r−R N e then it decouples in the center of mass and relative coordinates M R + m r N e R = CM M + m N e r = r − R (22) rel 14Whenexpressedintermsofthesecoordinates,theatomicHamiltoniandecouples as 2 2 2 h ¯ h ¯ ( M + m ) Z e N e 2 2 ˆ H = − ∇ − ∇ − (23) CM rel 2 ( M + m ) 2 M m r N e N e rel Hence, the asymptotic atomic wave functions have the form of products Ψ(r,R) = φ(r ) χ(R ) (24) rel CM If we neglect the mass of the electron compared with the mass of the nucleus, m /M  1, we expect to recover the Born-Oppenheimer approximation. In e N this approximation, the center of mass coordinate reduces to the nuclear coor- dinateR →R. In the (incoherent) inelastic scattering process, the electrons CM in the atom are excited, hence the initial and final states are represented as product states involving the electron and the atomic nucleus Ψ (r,R) = φ (r−R) χ (R) i,n i n Ψ (r,R) = φ (r−R) χ (R) (25) j,m j m In the impulse approximation, the nuclear wave functions are represented by plane waves   1 χ (R) = √ exp i k . R (26) n n V The inelastic scattering cross-section for the neutron beam is represented as       2 2 X d σ k m f n 2 = Ψ V(q)Ψ δ ¯hω+E −E j,m i,n i,n j,m 2 dΩdω k i 2 π h ¯ i,n;j,m (27) since h ¯ ω and ¯h q are the energy and momentum gained by the atom. The integration over the vector R yields the condition for the conservation of mo- mentum. Hence, the scattering cross-section is proportional to the overlap of the initial and final electronic states Z 3 ∗ d r φ (r−R) φ (r−R) (28) i j expressed in terms of the relative coordinates. On using the orthonormality condition for the electronic energy eigenstates, one obtains a factor of δ , so i,j that the initial and final electronic states are identical. Therefore, in the Born- Oppenheimer approximation, the scattering is purely elastic. IngoingbeyondtheBorn-Oppenheimerapproximation,oneneedstoinclude themassoftheelectrons,m . Theconservationofmomentumandenergyrefers e to the center of mass motion of the atom, where the atomic mass is the total mass M + m . The center of mass-coordinate of the atom is defined by N e M R + m r N e R = (29) CM M + m N e 15The center of mass part of the initial and final state atomic wave function, respectively, contains the factor of   M R + m r N e exp i k . (30) n M + m N e and   M R + m r N e exp i k . (31) m M + m N e wherek andk aretheinitialandfinalmomentaoftheatom. Weshallexpress n m 0 the electronic positionr relative to the nuclear position asr = r − R. With thisnotation,theintegrationoverthenuclearcoordinatestillyieldsconservation of momentum. However, the electronic part of the scattering matrix elements now involves the factor which is a function of the momentum q gained by the nucleus Z   0 m r e 3 0 ∗ 0 0 d r φ (r ) exp − i q . φ (r ) (32) i j M + m N e The exponential factor allows inelastic transitions to take place. The scattering amplitude for inelastic transitions is proportional to   m e m e M N q = q (33) m e M + m 1 + N e M N and also to the “dipole” matrix element Z 3 0 ∗ 0 0 0 d r φ (r ) r φ (r ) (34) i j Hence, we have shown that the probability amplitude that the electrons are excited by the nuclear motion is controlled by the ratio m e (35) M N m e The Born-Oppenheimer approximation is valid whenever  1. M N —————————————————————————————————- In the first part of the course it is assumed that the Born-Oppenheimer ap- proximation is valid. First,thesubjectofCrystallographyshallbediscussed,andthecharacters of the equilibrium structures of the dressed nuclei in matter will be described. An important class of such materials are those which posses long-ranged peri- odictranslationalorderandothersymmetries. Itshallbeshownhowtheselong range ordered and amorphous structures can be effectively probed by various elasticscatteringexperimentsinwhichthewavelengthofthescatteredparticles 16is comparable to the distance between the nuclei. In the second part, the properties of the Electrons shall be discussed. On assuming the validity of the Born-Oppenheimer approximation, the nature of the electronic states that occur in the presence of the potential produced by the static nuclei shall be discussed. One surprising result of this approach is that, even though the strength of the ionic potential is quite large (of the or- der of Rydbergs), in some metals the highest occupied electronic states bear a close resemblance to the states expected if the ionic potential was very weak or negligible. In other materials, the potential due to the ionic charges can pro- duce gaps in the electronic energy spectrum. Using Bloch’s theorem, it shall be shown how periodic long-ranged order can produce gaps in the electronic spec- trum. Another surprising result is that in most metals, it appears as though the electron-electron interactions can be neglected or, more precisely, that the excitations of the interacting electron system are similar to those of a non- interacting electron gas, albeit with renormalized masses or magnetic moments. The thermodynamic properties of electrons in these Bloch states shall be treated using Fermi-Dirac statistics. Furthermore, the concepts of the Fermi energy and Fermi surface of metals will be introduced. It shall be shown how the electronic transport properties of metals are dominated by states with en- ergies close to the Fermi surface, and how the Fermi surface can be probed. The third part concerns the motion of the ions or nuclei. In particular, it will be considered how the fast motion of the electrons dresses or screens the inter-nuclear potentials. The low-energy excitations of the dressed nuclear or ionic structure of matter give rise to harmonic-like vibrations. The elementary excitationofthequantizedvibrationsareknownasPhonons. Itshallbeshown how these phonon excitations manifest themselves in experiments, in thermo- dynamic properties and, how they participate in limiting electrical transport. The final part of the course concerns some of the more striking examples of the Collective Phenomenon such as Magnetism and Superconductivity. These phenomena involve the interactions between the elementary excitations of the solid which through collective action, spontaneously break the symmetry of the Hamiltonian. In many cases, the spontaneously broken symmetry is ac- companied by the formation of a new branch of low-energy excitations. 172 Crystallography Crystallography is the study of the structure of ordered solids, disordered solids and also liquids. In this section, it shall be assumed that the nuclei are static, frozen into their average positions. Due to the large nuclear masses and strong interactions between the nuclei (dressed by their accompanying clouds of elec- trons), one may assume that the nuclear or ionic motion can be treated classi- cally. The most notable failure of this assumption occurs with the very lightest of nuclei, such as He. In the anomalous case of He, where the separation be- tween ions, d, is of the order of angstroms, the uncertainty of the momentum h ¯ is given by and so the kinetic energyE for this quantum zero point motion K d can be estimated as 2 h ¯ E ≈ (36) K 2 2 M d The kinetic energy is large since the mass M of the He atom is small. The magnitude of the kinetic energy of the zero point fluctuations is larger than the weak van der Waals or London force between theHe ions. Thus, the inter-ionic forcesareinsufficienttobindtheHeionintoasolidandthematerialremainsin a liquid-like state until the lowest attainable temperatures. For these reasons, He behaves like a quantum fluid. However, for the heavier nuclei, the quantum nature of the particles only manifest themselves in more subtle ways. First, the various types of structures and the symmetries that can be found inCondensedMatterwillbedescribedandthenthevariousexperimentalmeth- ods used to observe these structures will be discussed. —————————————————————————————————— 2.0.1 Exercise 1 Consider the interaction potential between two electrically neutral atoms po- sitioned at R and R . The positions of the electrons located on atom 1 are 1 2 denoted by the set of r and the positions of the electrons associated with the i second atom are denoted by the set r . If the atoms are sufficiently far apart, j the sets of electrons belonging to each atom may be thought of as being distin- guishable. In this case, the interaction between the two atoms can be expressed as 2 2 2 2 2 X X X Z e e Ze Ze ˆ H = + − − int R − R r − r r − R R − r 1 2 i j i 2 1 j i,j i j (37) This interaction can be expanded in inverse powers of R − R . 1 2 If the eigenstates with energy eigenvalue E of the individual atoms are n (1) (2) denoted by Ψ and Ψ , then second-order perturbation theory n n 18yields the shift of the ground state energy due to the interaction between the pair of atoms as (1) (2) (1) (2) 2 X ˆ Ψ Ψ H Ψ Ψ (1) (2) (1) (2) int n m 0 0 ˆ ΔE = Ψ Ψ H Ψ Ψ + int 0 0 0 0 2 E − E − E 0 n m n,m (38) Show that the first term is just the classical electrostatic interaction due to the charge density distributions around each atom. Using the hydrogenic-like 1s one-electron wave functions r   3 κ φ (r) = exp − κ r (39) 1s π for the ground state and the 2s and 2p wave functions r     3 κ φ (r) = 1 − κ r exp − κ r 2s π r   3 κ φ (r) = cosθ κ r exp − κ r 2p,0 π r     3 κ φ (r) = sinθ exp ± i ϕ κ r exp − κ r 2p,±1 2 π (40) etc. for the excited states, estimate the sign and magnitude of the energy shift 3 for atoms with completely filled shells . 4 Estimate the Z dependence of the strengths of the London interaction for the inert gases and compare your results with the values obtained from a part of the semi-empirical Lennard-Jones potential      12 6 a a V (R) = V − (41) int 0 R R wherethevaluesofV anda,respectively,aregiveninunitsofeVandAngstroms 0 by element Ne Ar Kr Xe V 0.0124 0.0416 0.056 0.080 0 a 2.74 3.40 3.65 3.98 3 R. Eisenschitz and F. London, Z. fur ¨ Physik, 60, 491 (1930), J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931). 4 F. London, Z. fur ¨ Physik, 63, 245 (1930). 19—————————————————————————————————— 3 Structures The structure of condensed materials is usually thought about in terms of den- sity of either electrons or nuclear matter. To the extent that the regions of non-zero density of the nuclear matter are highly localized in space, with lin- −15 ear dimensions of 10 meters, the nuclei can be discussed in terms of point objects. The electron density is more extended and varies over length scales of −10 10 meters. The length scale for the electronic density in solids and fluids is verysimilartothelengthscaleoverwhichtheelectrondensityvariesinisolated atoms. The similarity of scales occurs as electrons are partially responsible for the bonding of atoms into a solid. That is, the characteristic atomic length scale is almost equal to the characteristic separation of the nuclei in condensed matter. Due to the near equality of these two length scales, the electron den- sity in solids definitely cannot be represented in terms of a superposition of the densityofwelldefinedatoms. However, theelectrondensitydoesshowasignifi- cantvariationthatcanbeinterpretedintermsoftheelectrondensityofisolated 5 atoms,subjecttosignificantmodificationswhenbroughttogether . Astheelec- tron density for isolated atoms is usually spherically symmetric, the structure in the electronic density may, for convenience of discussion, be approximately represented in terms of a set of spheres of finite radius. 3.1 Fluids Both liquids and gases are fluids. The microscopic structure of a fluid varies locally from position to position and in time. The macroscopic characteristics offluids are thattheyarespatiallyuniform andisotropic, whichmeans thatthe average environment of any atom is identical to the average environment of any other atom. The density is defined by the function X 3 ρ(r) = δ ( r − r ) (42) i i in which r is the instantaneous position of the i-th atom. A measurement of i the density usually results in the time average of the density which corresponds 5 An example of the change in the charge density of the valence electrons of Si, caused by solid formation, can be found in the contour plot in the article authored by D. R. Hamann, Phys. Rev. Lett. 42, 622 (1979). 20

Advise: Why You Wasting Money in Costly SEO Tools, Use World's Best Free SEO Tool Ubersuggest.