Problems in Solid State Physics with Solutions

The following sections are included:

• Preface

• Contents

The following sections are included:

• Electron speed distribution

• Average and standard deviation of the collision time interval

• Two successive collisions

• Conductivity of a superconductor

• Relative dielectric function of a metal

• Propagation of electromagnetic radiation in a metal

• Thermal conduction of a one-dimensional metal

• Metal in a uniform static electric field

The following sections are included:

• Fermi-Dirac distribution function at low temperatures

• Effects of hydrostatic pressure

• Approximate expression for the Fermi-Dirac distribution function

• Uncertainty in the electron kinetic energy

• Lindhard function

• Boltzmann equation

• Two-dimensional electron gas

• Thermodynamics of an electron gas

The following sections are included:

• Primitive cells of five two-dimensional Bravais lattices

• Wigner-Seitz cells of five two-dimensional Bravais lattices

• Triangular and centered rectangular Bravais lattices

• Packing fractions in two dimensions

• Body-centered cubic crystal

• Interstices in a face-centered cubic structure

• Crystal structures and densities

• Bond lengths and angles in BCC and FCC structures

• Neighbors in cubic crystals

• Volumes of primitive cells

• Coulomb interaction energy in an SC structure

The following sections are included:

• Identification of groups

• Group of Pauli matrices

• Statements about groups

• Identification of point groups

• Expressions of rotations and reflections

• Matrix representation of a point group

• Invariant subgroup

• Subgroups of point group C_3v

• Abelian groups

• Equivalence classes

• Point group C_4v

The following sections are included:

• Centerings in the hexagonal crystal system

• Centerings in the cubic crystal system

• Relations between symmetries of crystal systems

• Lattice planes and directions in the cubic crystal system

The following sections are included:

• Identification of crystals with symmorphic space groups

• Identification of crystals with nonsymmorphic space groups

• Identification of crystals with symmorphic or nonsymmorphic space groups

• Translation vectors in the diamond structure

• Conventional and primitive unit cells of a monoclinic lattice

• BCC and FCC structures of iron

• Nearest and second-nearest neighbors in an HCP crystal

• Diamond and body-centered tetragonal structures of gray tin

• Wurtzite and zincblende structures of GaN

• Crystal structure of diamond

• Crystal structure of CaF₂

• Monatomic BCC and FCC crystals

• Hypothetical ceramic material of AX type

• Crystal structure of CsCl

The following sections are included:

• Evaluation of the differential scattering cross-section

• Wave vector transfer in X-ray diffraction

• Charged particle in a static magnetic field

• Decay of a charged particle through radiation

• Scattering by an atom

• Incident and scattered X-ray beams

• Motion of a bound electron in an atom under the influence of X-rays

• Interaction of a bound electron with a plane electromagnetic wave

The following sections are included:

• Reciprocal lattice of the reciprocal lattice

• Symmetry of the reciprocal lattice

• Reciprocal lattice of a two-dimensional Bravais lattice

• Another two-dimensional Bravais lattice

• First three Brillouin zones of a two-dimensional triangular lattice

• Lengths of first eight reciprocal lattice vectors in SC, BCC, and FCC

• Reciprocal lattice vectors and lattice planes

• Structure factors of BCC and FCC crystals

• First Brillouin zones and interplanar distances

• Simple hexagonal lattice and its primitive cell and first Brillouin zone

• Atom density in a lattice plane

• Fourier series of a function with the periodicity of the Bravais lattice

• Interplanar distances

• Sizes of first Brillouin zones

• First Brillouin zone of a simple orthorhombic Bravais lattice

• Monatomic monovalent metal with an FCC structure

The following sections are included:

• Powder diffraction on silicon

• Powder diffraction on cubic CaF₂

• Debye-Scherrer experiments on two cubic samples

• Crystal structure of BaTiO₃

• Temperature dependence of the Bragg angle for Al

• Room-temperature superconductor

• Powder diffraction on YBa₂Cu₃O_6.9

• Three phases of iron

• Powder diffraction on La_1.8Ba_0.2CuO_4−y

• X-ray diffraction on Al

The following sections are included:

• Probe wavelengths proper for given structures

• Reciprocal lattice vectors and families of lattice planes

• Geometric structure factors

• Structure factor of the HCP lattice

• Neutron diffraction on the NaCl structure

• Be as a neutron attenuator

• Diffraction on Al of neutrons of energies below 15 meV

• Neutron diffraction peak

The following sections are included:

• Molecular orbitals for a hydrogen molecule

• Energy of a hydrogen molecule in the bonding molecular orbital

• Energy of a hydrogen molecule in the antibonding molecular orbital

• Permanent dipole-permanent dipole interaction

• Van der Waals bond in a diatomic molecule

• Bond in a hypothetical diatomic molecule

The following sections are included:

• Morse potential

• One-dimensional crystal of a chain of alternating ions

• Bonding in a three-dimensional ionic crystal

• Born-Meyer theory of bonding in ionic crystals

• Bonding in NaCl

• Madelung constant of the CsCl crystal

• Alkali metals

• Exchange energy of the electron gas in an alkali metal

The following sections are included:

• Normal modes of a linear chain of ions

• Simple one-dimensional crystal with a two-atom basis

• One-dimensional crystal with next-nearest-neighbor interactions

• Linear chain of atoms with damping

• Polarizable molecules with internal degrees of freedom

• Triatomic linear chain

• Two-dimensional crystal with a square Bravais lattice

• Simple cubic crystal

• Three-dimensional monatomic crystal

• Three-dimensional crystal with a two-atom basis

The following sections are included:

• Quantum field operator of atomic momenta

• Hamiltonian for a 3D crystal with a multi-atom basis

• Thermodynamics of a gas of phonons

• Lattice specific heat of a 1D crystal of inert gas atoms

• Debye model for a 1D crystal of inert gas atoms

• Electronic and lattice contributions to the specific heat of a metal

• Specific heat of potassium at low temperatures

• Phonon density of states for an optical branch

• Phonon density of states for an acoustical branch

• Number of phonons and its variance

• Grüneisen parameter of a 1D crystal of inert gas atoms

The following sections are included:

• Debye-Waller factors in spaces of different dimensionality

• Q-ω region accessible for inelastic neutron scattering

• Cumulant expansion

• Property of the dynamical structure factor

The following sections are included:

• Quasi-momentum operator of Bloch electrons

• Free-electron model

• Infinite chain of atoms with a Peierls distortion

• Densities of states in 1D and 2D tight-binding energy bands

• Electron energies in a 2D metal with a square lattice

• Allowed wave vectors in a simple cubic crystal

• Energy bands at the center of the first Brillouin zone for aluminum

• Monovalent metal with an ideal HCP structure

• Energy bands for an FCC lattice in the [111] direction

• Band structure of a divalent FCC Sr metal

The following sections are included:

• Electrons in a 2D square lattice

• Energy bands in a 1D crystal with a two-atom basis

• Energy gap at the M point in a 2D square lattice

• Tight-binding band from localized orbitals

• Energy band structure of aluminum

The following sections are included:

• Plane-wave method for a 1D crystal

• Special k-points for a simple cubic Bravais lattice

• Special k-points for a body-centered cubic crystal

• Special k-points for a face-centered cubic crystal

• Evanescent core potential

• Green's function in the KKR method

• k · p method for a semiconductor

• Variational derivation of the tight-binding secular equation

• Tight-binding approximation for a 1D crystal

• Two-dimensional graphite sheet

The following sections are included:

• Electrons in an ellipsoidal energy band

• Electrons in the conduction band of a 1D metal

• Electrons in a 1D tight-binding conductor

• Electrons in a semiconductor superlattice

• Effective mass approximation

• Semi-classical equations of motion with damping

The following sections are included:

• Conduction and valence bands of a hypothetical semiconductor

• Effective mass, energy, momentum, and velocity of a hole

• Effective mass of electrons at the conduction band edge

• Density of states for a single k-space ellipsoid in Si

• Density of states in a nonparabolic conduction band

• Electron-hole pair excitations

• Chemical potential of an intrinsic semiconductor

• As-doped silicon crystal

• Donors in indium antimonide

• Band gap of InSb

• Fermi levels in InP

• Sb-doped silicon crystal

• As-doped germanium

• Concentrations of electrons and holes in a p-type semiconductor

• Maximum concentration of donor atoms for intrinsic conduction

• Conductivity and Fermi level of GaAs

The following sections are included:

• Electron densities in the Hartree and Fock-Hartree models

• Pair densities and correlation functions in the Hartree-Fock model

• Jellium model

• Computation of one functional derivative

• Computation of six functional derivatives

• Action functional and Euler-Lagrange equation

• One-electron system

• Thomas-Fermi screening

• N-representable density

• Electron-ion interaction functional

• Janak's Theorem

The following sections are included:

• Smooth and oscillatory functions

• Atom with a harmonic radial potential

• Numerical solution of the radial Schrödinger equation

• Kerker scheme for pseudopotentials

• Spherical averages

• Hedin-Lundqvist interpolation scheme for exchange and correlation

• Simplified OPW pseudopotential

• Equivalence of the norm-conservation condition

• Perdew-Zunger parametrization of the correlation energy

The following sections are included:

• Kohn-Sham equations for auxiliary wave functions

• General formula for local operators

The following sections are included:

• Quantization of electromagnetic fields

• Quantum field operator of electrons

• Poisson summation formula

• Application of the Lifshits-Kosevich theory

• Amplitude of dHvA oscillations in a free electron gas

The following sections are included:

• Deduction of the energy of vacancy formation from resistivity data

• Energy of vacancy formation in gold

• Number of vacant sites

• Number of occupied lattice sites for every vacancy in Ni

• Energy of vacancy formation in Al

• Energy of vacancy formation in Mo

• Schottky defects in copper

• Schottky defects in an oxide ceramic

• Burgers vectors of dislocations in FCC, BCC, and SC crystals

• Two- and three-dimensional defects

• Two perpendicular long edge dislocations

The following sections are included:

• Perturbation computations for the electron-phonon system

• Zeroth-order Green's functions from equations of motion

• Field operators and single-electron Green's function

• Feynman rules for the effective electron-electron interaction

• Fourth-order corrections to the phonon Green's function

• Spectral function, renormalization constant, and effective mass

• Real and imaginary parts of the electron retarded self-energy

• Periodic Anderson model

• Fourth-order corrections to phonon Green's function at finite temperatures

• Time-ordered product of three operators

• Evaluation of Matsubara sums

• Generalized spin susceptibility of a free electron gas

• Pairing susceptibility

• Localized electrons

• Single impurity

• Discontinuity of the momentum distribution

• Multiparticle expectation value

• Friedel oscillations

• Plasma waves

• Fermi gas in a spin-dependent external field

The following sections are included:

• Equilibrium distribution function

• Vector product of mechanical momentum with itself

• DC conductivity of Bloch electrons in the presence of damping

• Matthiessen's rule and its violation

• Linear response theory and DC conductivity

• Thermopower

• Conductivity of a metal

• Conductivity in terms of the effective mass

The following sections are included:

• Localized magnetic moments in a weak magnetic field

• Holstein-Primakoff transformation

• Heisenberg-Weiss model for a ferromagnet

• Two-site Hubbard model

• Spin waves in a two-dimensional triangular ferromagnet

• Double exchange model on two lattice sites

• Specific heat of N noninteracting ½-spins in a magnetic field

• Linear chain of three ½-spins

• Landau's theory for the ferromagnetic phase transition

• Spin operator in terms of electron annihilation and creation operators

• Dot product of spin operators in terms of electron operators

• One-dimensional spin-S Heisenberg quantum ferromagnet

• Pauli susceptibility of a non-interacting Fermi gas

• Instability of electron gas to ferromagnetism

• Schwinger boson representation

• Jordan-Wigner transformation

• Matsubara susceptibility and Curie's law

• One-dimensional anisotropic XY-model

• Phonon-ferromagnon interaction

• Weakly interacting dilute Bose gas

• XY model for a one-dimensional quantum magnet

• Schwinger-boson mean-field theory for a one-dimensional ferrimagnet

The following sections are included:

• Properties of the electric susceptibility tensor

• Reflectivity of a simple metal

• Dielectric function of a free electron gas

• Static dielectric function of a metal

• Low-frequency response of electrons to an AC field

• Dielectric function of an ideal metal

• An isotropic solid modeled by Lorentz oscillators

• Electric susceptibility in the Lorentz oscillator model

• Optical properties of NaCl

• Lyddane-Sachs-Teller relation for a crystal with a three-atom basis

• Absorption coefficient of a nanowire

• Model dielectric function for Ge

• Analysis of reflectance data

The following sections are included:

• Isotope effect in tin

• Conductivity of a superconductor in the two-fluid model

• Magnetic field inside an infinite superconducting plate

• Critical field of a type-I superconductor in the Ginzburg-Landau theory

• Ginzburg-Landau equation and superconducting current density

• Proximity effect between two planar superconductors

• Macroscopic quantum wave function of Cooper pairs

• Phonon-mediated effective electron-electron attraction

• Effect of the Fermi sea

• Size of a Cooper pair

• BCS superconducting ground state

• Finite-momentum BCS state

• Variational approach of the BCS theory

• Average internal flux density in a triangular vortex lattice

• Superconducting instability

The following sections are included:

• References

• Index