Contents -- Preface -- 1. Lattice Dynamics -- 1.1 Born-Oppenheimer Approximation -- 1.2 Lattice Potential Energy and Harmonic Approximation -- 1.3 Normal Modes of a Three-Dimensional Crystal with a Multi-Atom Basis -- 1.3.1 Equations of motion of atoms -- 1.3.2 Allowed values of wave vector k -- 1.3.3 Allowed values of frequency ω -- 1.3.4 Polarization vectors -- 1.3.5 Displacements of atoms -- 1.3.6 Hamiltonian of a crystal with a multi-atom basis -- 1.4 Classical Theory of the Lattice Specific Heat -- 1.5 Quantization of Lattice Vibrations -- 1.5.1 Statistics for phonons -- 1.6 Phonon Density of States -- 1.7 Lattice Specific Heat of Solids -- 1.7.1 General expression of the lattice specific heat -- 1.7.2 High-temperature limit -- 1.7.3 Low-temperature limit -- 1.8 Debye Model -- 1.8.1 High-temperature limit -- 1.8.2 Low-temperature limit -- 1.8.3 Debye temperature -- 1.9 Einstein Model -- 1.10 Effect of Thermal Expansion on Phonon Frequencies -- 1.11 Specific Heat of a Metal -- Problems -- 2. Determination of Phonon Dispersion Relations -- 2.1 Experimental Techniques -- 2.1.1 Triple-axis spectrometer -- 2.1.2 Time-of-flight-spectrometer -- 2.2 Description of Neutron Scattering -- 2.2.1 System of the neutron and crystal -- 2.2.2 Interaction between the neutron and crystal -- 2.2.3 Scattering amplitude and differential cross-section -- 2.3 Double Differential Cross-Section -- 2.4 Elastic Scattering -- 2.5 Inelastic Scattering -- Problems -- 3. Elementary Theory of Energy Bands -- 3.1 Development of Computational Methods for Band Structures -- 3.2 Fundamental Problem in an Energy Band Theory -- 3.2.1 Independent-electron approach -- 3.2.2 Correlated-electron approach -- 3.3 Hartree-Fock Method -- 3.3.1 Hartree method -- 3.3.2 Hartree-Fock method -- 3.3.2.1 Slater determinant trial wave function -- 3.3.2.2 Energy average.

3.3.2.3 Direct and exchange terms -- 3.3.2.4 Hartree-Fock equations -- 3.3.3 Application of the Hartree-Fock method to the electron gas -- 3.3.4 Variants of the Hartree-Fock method -- 3.3.4.1 Restricted Hartree-Fock method -- 3.3.4.2 Unrestricted Hartree-Fock method -- 3.4 Plane-Wave Method -- 3.4.1 Numerical issues -- 3.4.1.1 Energy cutoff and basis set of plane waves -- 3.4.1.2 Diagonalization of the Hamiltonian matrix -- 3.4.1.3 Special wave vectors in the first Brillouin zone -- 3.4.2 Slow convergence of the plane-wave method -- 3.5 k · p Method -- 3.5.1 Effective mass tensor -- 3.5.2 Effective mass theory -- 3.6 Augmented-Plane-Wave Method -- 3.6.1 Muffin-tin spheres -- 3.6.2 Muffin-tin potential -- 3.6.3 Augmented plane waves -- 3.6.4 APW secular equation -- 3.7 Linearized-Augmented-Plane-Wave Method -- 3.7.1 New augmented plane wave basis functions -- 3.7.1.1 Energy derivatives of radial wave functions -- 3.7.1.2 New augmented plane waves -- 3.7.2 LAPW secular equation -- 3.8 Linear-Muffin-Tin-Orbitals Method -- 3.9 KKR Method -- 3.10 Orthogonalized-Plane-Wave Method -- 3.10.1 Linear combination of core orbitals -- 3.10.2 Orthogonalized plane waves -- 3.10.3 OPW secular equation -- 3.10.4 OPW pseudopotential -- 3.11 Tight-Binding Method -- 3.11.1 Basis set -- 3.11.2 Secular equation in TBA -- 3.11.3 Application of the tight-binding method to s band -- 3.11.4 Tight-binding band structure in a two-dimensional square lattice -- 3.11.4.1 Dispersion of the tight-binding band -- 3.11.4.2 Density of states of the tight-binding band -- 3.11.4.3 Tight-binding Fermi surface and its variation with the electron filling factor -- 3.11.5 Tight-binding band structures in the cubic crystal system -- 3.11.5.1 Tight-binding band structure in a simple cubic crystal -- 3.11.5.2 Tight-binding band structure in a body-centered cubic crystal.

3.11.5.3 Tight-binding band structure in a face-centered cubic crystal -- Problems -- 4. Determination of Electronic Band Structures -- 4.1 Interaction of Electrons with Electromagnetic Fields -- 4.1.1 Classical Hamiltonian -- 4.1.2 Semi-classical Hamiltonian -- 4.1.3 Quantization of electromagnetic fields -- 4.1.4 Second quantization of electrons -- 4.1.4.1 Operators nk and †nk -- 4.1.4.2 Anticommutation relations between operators nk and †nk -- 4.1.4.3 Physical meanings of operators nk and †nk -- 4.1.4.4 General anticommutation relations between nk and †nk -- 4.1.4.5 States of electrons in a solid -- 4.1.4.6 Quantum field operator of electrons -- 4.1.5 Quantum Hamiltonian -- 4.2 De Haas-van Alphen Effect -- 4.2.1 De Haas-van Alphen effect in a three-dimensional electron gas -- 4.2.1.1 Single-electron levels in a magnetic field -- 4.2.1.2 Density of states of the electron gas in a magnetic field -- 4.2.1.3 Ground-state energy -- 4.2.1.4 Magnetization -- 4.2.1.5 Magnetic susceptibility -- 4.2.1.6 Oscillation period and frequency -- 4.2.2 Lifshits-Kosevich theory of the de Haas-van Alphen effect -- 4.2.3 Techniques for the measurement of the dHvA effect -- 4.2.3.1 Torque method -- 4.2.3.2 Field-modulation method -- 4.2.4 De Haas-van Alphen frequency and amplitude -- 4.2.5 De Haas-van Alphen effect in copper -- 4.2.5.1 Fermi surface of copper -- 4.2.5.2 Experimental results on the features of the Fermi surface of copper -- 4.3 Photoemission Spectroscopy -- 4.3.1 Elementary concepts -- 4.3.1.1 Emission process of photoelectrons -- 4.3.1.2 Kinematics in photoemission process -- 4.3.1.3 Quantum yield -- 4.3.1.4 X-ray and ultraviolet photoemission -- 4.3.1.5 Energy distribution curve -- 4.3.1.6 Modes of photoemission spectroscopy -- 4.3.1.7 Angle-integrated and angle-resolved photoemission spectroscopy.

4.3.2 Methods for the determination of band structures -- 4.3.2.1 Free-electron-like final states -- 4.3.2.2 Triangulation method -- 4.3.2.3 Symmetry method -- 4.3.2.4 Appearance angle method -- 4.3.3 Three-step model -- 4.3.3.1 Photo-excitation of electrons -- 4.3.3.2 Travel of electrons to the surface -- 4.3.3.3 Escape of electrons into the vacuum -- 4.3.3.4 Photocurrent -- 4.3.4 Response theory of photoemission -- 4.3.4.1 Evaluation of matrix elements -- 4.3.5 Correlated electrons -- 4.3.6 Electronic Band Structure of Copper from ARPES -- Problems -- 5. Electron-Phonon Interaction -- 5.1 Electron-Phonon Interaction Hamiltonian -- 5.1.1 Electron-phonon interaction Hamiltonian in metals -- 5.1.1.1 Jellium model -- 5.1.2 Electron-phonon interaction Hamiltonian for ionic crystals -- 5.1.3 Electron-phonon interaction Hamiltonian in insulators -- 5.2 Electron-Phonon Interaction in Metals -- 5.2.1 Electron self-energy -- 5.2.2 Electron-phonon coupling function α2F -- 5.2.2.1 Effective electron-phonon coupling constant -- 5.2.2.2 Moments of the frequency -- 5.2.2.3 Electron self-energy -- 5.2.3 Effective electron-electron interaction -- 5.2.3.1 Effective electron-electron interaction Hamiltonian -Derivation using the formal scattering theory -- 5.2.3.2 Effective electron-electron interaction Hamiltonian -Derivation using the canonical transformation -- 5.3 Polarons -- 5.3.1 Weak-coupling Frohlich polaron -- 5.3.1.1 Effective mass of a Frohlich polaron -- 5.3.1.2 Number of virtual phonons in a Frohlich polaron -- 5.3.1.3 Lattice charge density in a Frohlich polaron -- 5.3.2 Bipolarons -- 5.4 Green's Functions at Zero Temperature -- 5.4.1 Definition of Green's functions in real time -- 5.4.2 Perturbation series of Green's functions -- 5.4.2.1 Perturbation series for the electron Green's function -- 5.4.2.2 Perturbation series of the phonon Green's function.

5.4.3 Zeroth-order Green's functions -- 5.4.3.1 Zeroth-order electron Green's function in an empty band -- 5.4.3.2 Zeroth-order electron Green's function in a degenerate electron gas -- 5.4.3.3 Zeroth-order phonon Green's function -- 5.4.4 Wick's theorem -- 5.4.4.1 Normal product, time-ordered product, and contraction -- 5.4.4.2 Statement of Wick's theorem -- 5.4.4.3 Proof of Wick's theorem -- 5.4.5 Application of Wick's theorem to phonon operators -- 5.4.6 Application of Wick's theorem to electron operators -- 5.4.7 Second-order self-energies -- 5.4.7.1 Second-order electron self-energy -- 5.4.7.2 Second-order phonon self-energy -- 5.4.8 Feynman rules -- 5.4.9 Dyson equation -- 5.4.9.1 Fourth-order electron self-energy -- 5.4.9.2 Dyson equation for electron Green's function -- 5.4.9.3 Dyson equation for phonon Green's function -- 5.4.10 Migdal's theorem -- 5.4.11 Analytic properties of Green's functions -- 5.4.11.1 Spectral functions -- 5.4.12 Retarded and advanced Green's functions -- 5.4.12.1 Spectral representations -- 5.4.12.2 Kramers-Kronig relations -- 5.4.12.3 Relations with the time-ordered Green's functions -- 5.4.12.4 Energy, lifetime, and effective mass of an electronic excitation -- 5.4.12.5 Retarded and advanced Green's functions for phonons -- 5.5 Green's Functions at Finite Temperatures -- 5.5.1 Dynamics in imaginary time -- 5.5.2 S-matrix -- 5.5.3 Definitions of Matsubara Green's functions -- 5.5.4 Matsubara frequencies -- 5.5.4.1 Matsubara frequencies for electrons -- 5.5.4.2 Matsubara frequencies for phonons -- 5.5.5 Perturbation series for Matsubara Green's functions -- 5.5.5.1 Perturbation series for the electron Matsubara Green's function -- 5.5.5.2 Perturbation series for the phonon Matsubara Green's function -- 5.5.6 Zeroth-order Matsubara Green's functions -- 5.5.7 Zeroth-order Matsubara Green's function for electrons.

5.5.8 Zeroth-order Matsubara Green's function for phonons.