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E-grāmata: Quantum Theory of the Solid State: An Introduction

  • Formāts: PDF+DRM
  • Sērija : Fundamental Theories of Physics 136
  • Izdošanas datums: 31-May-2004
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781402021541
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  • Formāts: PDF+DRM
  • Sērija : Fundamental Theories of Physics 136
  • Izdošanas datums: 31-May-2004
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781402021541
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The book targets a broad readership. First of all, it targets young researchers (postgraduate students) in solid state physics (both physicists and theoretical chemists) as it contains a wide and comprehensive coverage of all important branches of the subject including an up-to-date survey of recent revolutionary advances in quantum mechanics which have made it possible not only to calculate many properties of molecules and solids in close agreement with experiment, but to make reliable predictions in cases when a direct experiment is not possible (e.g. the Earth core). Secondly, it should be a valuable asset to established researchers in the areas of materials science, solid-state physics and chemistry due to very detailed explanations of a wide range of phenomena ranging from symmetry, lattice vibrations, electronic structure and superconductivity to magnetic and dielectric properties. Rigour and detail in explaining complicated mathematical techniques and in providing derivations when talking of various physical concepts are essential for those who would like to really understand things they have never had a chance to. Because of that and of the fact that the book contains a lot of material from different areas of solid-state physics retold from a single viewpoint, it should be indispensable for lecturers. Not only a number of courses, both general and specialised, should be possible to set up, but these courses may also be of a different level of difficulty ranging from undergraduate, postgraduate and then to highly advanced ones. This is because of a clear marking system adopted in the book. Hence, it should also be useful for advanced third- and fourth-year undergraduate students.
Acknowledgement v
Foreword vii
Introduction ix
Structures
1(44)
Crystals: periodic arrays of atoms
1(3)
Mathematical description of crystal structures
4(22)
Definition of a group
4(1)
Translation groups
5(1)
Operators of translation
5(1)
Construction of subgroups
6(1)
Point groups
7(1)
Elementary point groups
8(2)
Symmetry groups of a tetrahedron and a cube
10(2)
Space groups
12(1)
Symmetry operations
12(2)
Types of Bravais lattices
14(5)
Crystallographic (conventional) unit cell
19(1)
Crystal classes
20(2)
Symmorphic and nonsymmorphic crystal lattices
22(1)
Close packing structures
22(1)
2D (planar) groups
23(1)
*Matrix and operator representations of a group
23(1)
Operator representation of a group
23(1)
Matrix representation of a group
24(1)
Indexing of planes and directions: definitions
25(1)
*International Tables for X-Ray Crystallography
26(4)
Examples of crystal structures
30(7)
Cubic face-centred structures
31(1)
Space group O5h (Fm3m, No. 225)
31(1)
Space group O7h (Fd3m, No. 227)
32(1)
Space group T2d (F43m, No. 216)
32(1)
Cubic body-centred structures
32(1)
Space group O9h (Im3m, No. 229)
32(1)
Space group T7h (Ia3, No. 206)
33(1)
Structures with simple cubic lattice
33(1)
Space group O1h (Pm3m, No. 221)
33(1)
Space group T6h (Pa3, No. 205)
34(1)
Tetragonal lattice
34(1)
Space group D144h (P42/mnm, No. 136)
34(1)
Structures with trigonal lattice
35(1)
Space group D63d (R3c, No. 167)
35(1)
Structures with hexagonal lattice
36(1)
Space group D46h (P63/mmc, No. 194)
36(1)
Space group C46v (P63mc, No. 186)
36(1)
Space group D43 (P3121, No. 152)
37(1)
Nonperiodic solids
37(8)
Definition of order and quasicrystals
38(2)
A road to disorder
40(1)
Point defects
40(1)
Cellular disorder
40(3)
Topological disorder
43(2)
The reciprocal lattice and X-ray diffraction
45(24)
The reciprocal lattice
45(2)
*Once again about crystal planes and Miller indices
47(3)
Brillouin zones
50(1)
Periodic functions: Fourier analysis
51(4)
Introduction to X-ray diffraction
55(14)
Diffraction intensity
55(4)
Bragg law
59(1)
Structure factor
60(3)
Interpretation of diffraction experiments
63(1)
*X-ray diffraction of nonperiodic solids
63(1)
Density-density correlation function
64(1)
Periodic systems revisited
65(1)
Application to a binary alloy
66(2)
Glass
68(1)
Binding in Crystals
69(32)
Adiabatic approximation
69(6)
Molecules: types of chemical bonding
75(13)
Simple example: a molecule with two atoms
75(2)
Ionic bond
77(1)
Covalent bond
77(1)
Hydrogen molecular ion H+2
77(1)
Hydrogen molecule: MO method
78(4)
Hydrogen molecule: VB method
82(1)
Covalent bonds for elements having the (np)3 shells
83(1)
Covalent bonds for elements having the (ns)2(np)2 shells
83(1)
Some other examples of hybrid orbitals
84(1)
Ion-Covalent bond
85(1)
Van der Waals interaction
85(3)
Hydrogen bond
88(1)
Binding in crystals
88(13)
Cohesive and lattice energies
88(1)
Electrostatic energy
89(1)
Conditional convergence
89(1)
*Ewald method: electrostatic potential
90(2)
*Ewald constant
92(1)
*Ewald method: electrostatic energy
93(1)
The Madelung energy of a finite large crystal sample
94(1)
Van der Waals crystals
94(1)
Ionic crystals
95(2)
Covalent crystals
97(1)
Hydrogen bond systems
98(1)
Metals
98(1)
Real crystals
99(2)
Atomic vibrations
101(102)
Lagrangian and Hamiltonian method
101(2)
One dimensional lattice
103(19)
Monoatomic basis
103(1)
Lagrangian and equation of motion
104(1)
General solution
105(1)
First Brillouin zone
106(1)
Elastic Waves
106(1)
Long wavelength limit
107(1)
Two atoms in the basis
107(1)
Lagrangian and equations of motion
108(1)
Analysing the solution
109(1)
Limiting case of identical atoms
110(2)
Why optical and acoustic?
112(1)
Boundary conditions
112(2)
*Normal coordinates
114(1)
Discrete Fourier transform
115(1)
Matrix form for the eigenvalue problem and the dynamical matrix of the chain
116(2)
Diagonal representation for the kinetic and potential energies of the chain
118(2)
Normal coordinates of the chain
120(1)
Density of states for the ID chain
121(1)
Three dimensional lattice: classical
122(23)
Harmonic approximation
123(2)
Phonons in a 3D crystal
125(1)
Hamiltonian and equations of motion
125(1)
Trial solution
125(1)
Dynamical matrix
126(1)
Eigenvalue and eigenvector problem for lattice vibrations
126(1)
Symmetry properties
127(1)
The wavevector
128(1)
General solution
129(1)
Limiting case of long waves
129(1)
Acoustic branches
130(1)
Optical branches
130(1)
Example: a crystal with central forces
131(1)
Crystal with central forces
131(2)
Oscillations of a binary fcc crystal
133(2)
Phonon density of states (DOS)
135(1)
Contribution from long acoustic waves: Debye model
136(1)
*Van Hove singularities
137(2)
*Normal coordinates
139(1)
3D discrete Fourier transform
139(2)
Formal introduction of normal coordinates
141(1)
Diagonalisation of the kinetic and potential energies using complex coordinates
142(2)
Introduction of real coordinates
144(1)
Lattice stability at zero temperature
145(1)
Three dimensional lattice dynamics: quantum
145(10)
A single harmonic oscillator
145(1)
Introduction of creation and destruction (annihilation) operators
146(1)
Introduction to algebra of operators a and a†
147(2)
*Some useful operator identities
149(3)
Crystal vibrations in the harmonic approximation
152(1)
Second quantisation
153(2)
Thermal properties of crystals
155(24)
Equilibrium statistical mechanics
155(1)
Classical statistical mechanics
155(1)
Quantum statistical mechanics
156(1)
Phonon statistics
156(2)
Phonons as quasiparticles
158(1)
*Some useful statistical averages
158(3)
*Displacement-displacement correlation function
161(1)
Internal energy and specific heat
162(1)
Internal energy
162(1)
Specific heat
163(1)
Debye model for acoustic branches
163(3)
Einstein model for optical branches
166(1)
Equation of states
166(1)
*Quasiharmonic approximation
166(2)
*Equation of state
168(1)
Thermal expansion
169(1)
Melting
170(2)
Thermal conductivity and anharmonicity
172(1)
Elementary kinetic theory of thermal conductivity
172(1)
*Anharmonicity
173(3)
*Debye-Waller factor
176(2)
Elastic and inelastic phonon processes
178(1)
*Elementary theory of elasticity and stability
179(24)
Main ideas of the classical theory of elasticity
179(1)
External and Lagrangian strain
180(2)
Stress
182(2)
Isotropic pressure
184(1)
Shear and normal strain and stress
184(1)
Thermodynamics
185(1)
Elastic constants
186(1)
Elastic properties of crystals
186(1)
Hooke's law
187(1)
Crystal symmetry
187(2)
Noncrystalline solids
189(2)
Stability
191(2)
Elastic waves
193(2)
Waves in a cubic crystal
195(1)
Method of homogeneous deformation
196(1)
General description of a homogeneous deformation in a crystal
196(1)
General expressions for the isothermal elastic constants
197(3)
Example: a crystal with pairwise central interactions
200(3)
Electrons in a periodic potential
203(96)
Model of a free electron gas
203(15)
Why electron gas?
203(1)
Energies and wavefunctions
204(1)
Periodic boundary conditions
204(1)
Orthogonality and completeness of plane waves
205(2)
Distribution of electrons on energy levels. Fermi sphere
207(1)
Density of states
208(1)
Quantum statistics: Fermi-Dirac distribution
208(3)
*Heat capacity and chemical potential of the electron gas
211(1)
One useful integral
211(1)
Chemical potential
212(1)
Heat capacity
213(1)
Comparison with experiment
214(1)
Transport processes
214(1)
Electrical conductivity
215(1)
Matthiessen's rule
215(1)
Motion in magnetic field. Hall effect
216(2)
Thermal conductivity
218(1)
Wiedemann-Franz law
218(1)
Energy bands
218(24)
Bloch theorem
219(1)
The meaning of vector q and periodic boundary conditions
220(1)
*Wannier functions
221(1)
Electronic band structure via plane waves
222(1)
*Density-functional theory and Kohn-Sham potential
223(2)
*Calculation of the Hartree potential
225(2)
Approximation of a nearly free electron gas
227(1)
Empty lattice approximation: reduced zone scheme
227(2)
Model of a nearly free electron gas
229(2)
Tight binding method
231(2)
An example: s bands
233(1)
Assembling a crystal from atoms
234(1)
*Kronig-Penney model
234(2)
Density of states (DOS)
236(3)
Metals and insulators
239(3)
Transport properties: electrical and thermal conductivity revisited
242(24)
Fermi surfaces
242(1)
Examples
243(1)
Quasiparticles
244(1)
Particles and holes
244(1)
Wave packets
245(3)
Effective electron mass
248(2)
Current in bands
250(2)
*Kinetic equation
252(2)
Collision term and the detailed balance
254(1)
Relaxation time approximation
255(1)
*Electrical conductivity
256(2)
*Heat transport
258(1)
*Quantum description of transport processes
259(1)
Nonequilibrium quantum statistical mechanics
259(2)
Kubo's linear response theory
261(2)
Generalised susceptibilities
263(1)
General expression for electrical conductivity
264(1)
Relaxation time approximation
265(1)
Electron-electron interaction
266(33)
Qualitative consideration
266(2)
*Elementary theory of ``plasma'' oscillations
268(2)
Excitations of plasmons by fast electrons
270(1)
Interaction with electromagnetic waves
271(2)
Interaction with longitudinal electrostatic field
273(1)
*Theory of plasma oscillations based on density fluctuations
274(1)
Electron Hamiltonian in the jellium model
274(2)
Classical treatment of plasma oscillations
276(2)
Quantum treatment of plasma oscillations
278(1)
*Screening in the electron gas
279(1)
Screening Coulomb potential of a point charge
280(3)
*Dielectric function of the electron gas
283(1)
Maxwell equations for zero magnetic field
284(1)
Tensor of the microscopic dielectric function
285(3)
General expression for electronic susceptibility
288(2)
Self-consistent consideration of the electronic response
290(1)
Susceptibility in the independent particles approximation
291(3)
Application to a free electron gas
294(5)
Magnetism
299(68)
Magnetic moment in classical electrodynamics
299(8)
Magnetic field of a system of moving charges far away from them
299(2)
Relation between the magnetic moment and angular momentum
301(1)
Movement of a charged particle in a magnetic field
301(2)
Magnetic field in matter and magnetic permeability
303(4)
Magnetic moment in quantum mechanics
307(14)
*Relativistic description of an electron
307(1)
Dirac equation
307(2)
Solution of the Dirac equation for a free relativistic electron
309(1)
Spin
310(2)
*An electron in electro-magnetic field
312(2)
Magnetic moment of an electron
314(1)
Quasi-relativistic approach
314(1)
An electron in a magnetic field
315(1)
*One electron atom in a homogeneous magnetic field
316(1)
Magnetic moment of an atom
317(1)
One electron atom (ion)
317(2)
Many-electron atom (ion)
319(1)
Hund rules and physical reasons for permanent localised magnetic moments
320(1)
Thermodynamics of magnetic materials
321(1)
Para- and diamagnetism of localised electrons
322(7)
Classical paradox
322(1)
Almost classical theory of diamagnetism
323(1)
Quantum theory of diamagnetism
324(1)
Almost classical theory of paramagnetism (Langevin)
325(2)
Quantum theory of paramagnetism
327(2)
Para- and diamagnetism of the electron gas
329(7)
Pauli paramagnetism
329(2)
*Magnetism of electrons in metals: Landau diamagnetism and the de Haas-van Alphen effect
331(1)
General expression for the grand potential
331(4)
Pauli paramagnetism versus Landau diamagnetism
335(1)
The de Haas-van Alphen effect
336(1)
Magnetic ordering
336(21)
Interaction between localised magnetic moments
336(1)
Weiss molecular field
336(1)
Curie-Weiss law
337(1)
Ferromagnetism
338(1)
Antiferromagnetism
339(3)
Ferrimagnetism
342(1)
Hysteresis and domain structure
343(1)
Hysteresis curve
343(1)
Anisotropy
344(1)
Domains
344(2)
Wall motion and rotation versus reversibility and irreversibility
346(1)
Domain energetics
346(1)
Exchange interaction and the phenomenological theory of ferromagnetism
347(1)
*Hydrogen molecule revisited
348(1)
*Spin Hamiltonians
349(1)
*Indirect exchange
350(1)
*Mean field method
351(1)
*Band theory of ferromagnetism
352(1)
Exchange interaction in metals: exchange hole
352(1)
Stoner model: general equations
353(2)
Stoner model: paramagnetism
355(1)
Stoner model: ferromagnetism
356(1)
Stoner model: specific heat
357(1)
Symmetry breaking and order parameters
357(10)
Symmetry breaking
358(2)
The Landau theory of second order phase transition
360(2)
*Bragg-Williams theory
362(5)
Superconductivity
367(54)
General properties
367(4)
Critical magnetic field and critical current
368(1)
Meissner-Ochsenfeld effect
368(2)
Superconducting phase transition
370(1)
Heat capacity
370(1)
Isotope effect
371(1)
Phenomenological theory of superconductivity
371(6)
Thermodynamics of superconductors
371(2)
London equations
373(3)
Experimental evidence
376(1)
Main ideas of the microscopic theory of superconductivity
377(32)
Attraction between electrons
377(1)
Cooper pairs
377(5)
*Ground state of the metal in the superconducting state
382(1)
Creation and annihilation operators for electrons in a normal metal
382(2)
Variational wavefunction for a superconductor
384(2)
Calculation of the ground state using a variational method
386(5)
Isotope effect
391(1)
Correlation and coherence lengths
391(2)
*Excitation energies in the superconducting state
393(4)
Energy gap
397(2)
Temperature dependence
399(2)
Supercurrents
401(2)
Existence of the critical magnetic field
403(1)
*The Meissner-Ochsenfeld effect
404(1)
Current density operator
404(1)
Coordinate representation of the wave function
405(1)
Derivation of the second London equation
406(1)
Quantisation of magnetic flux
407(2)
*Ginzburg-Landau theory of superconductivity
409(6)
Order parameter and the free energy
410(2)
Ginzburg-Landau equations
412(2)
Examples of applications
414(1)
Type II superconductors
415(1)
High Tc superconductors
416(5)
Cuprates
416(3)
Fullerenes
419(2)
Dielectric materials
421(50)
Microscopic polarisation
422(4)
Phonon contribution to the dielectric function
426(21)
The local field
426(3)
Optical vibrations of a binary ionic crystal
429(1)
Huang equations
430(4)
Dispersion formula for the dielectric function
434(2)
Long optical phonons
436(2)
*General consideration: classical
438(5)
*General consideration: quantum
443(4)
Thermodynamics of dielectrics
447(8)
Contribution of the field to thermodynamic potentials
447(1)
Isotropic dielectrics
448(1)
Crystals
448(1)
*Pyroelectrics and crystal symmetry
449(1)
Dielectric tensor and crystal symmetry
449(1)
*Effect of the elastic deformation
450(1)
Piezoelectric tensor
451(1)
Crystal symmetry allowing piezoelectricity
452(1)
Statics and dynamics of a piezoelectric crystal
453(1)
Elastic waves in piezoelectrics
454(1)
Ferroelectric transition
455(16)
General description of ferroelectrics
455(5)
Landau theory of the ferroelectric phase transition
460(1)
Second order transition
461(1)
First order transitions
462(3)
*Microscopic consideration: Effective field model of Lines
465(6)
*Modern methods of electronic structure calculations
471(124)
Many-electron wavefunction
472(21)
Antisymmetry
472(2)
Slater determinants
474(3)
Antisymmetriser
477(1)
Creation and annihilation operators
478(2)
Slater rules: matrix elements between determinants
480(4)
Non-orthogonal spin-orbitals
484(1)
Operators in second quantisation
484(1)
One-particle operator
485(2)
Two-particle operator
487(1)
Total energy
488(1)
Reduced density matrices
488(1)
Electron densities
488(2)
Reduced density matrices
490(2)
Natural orbitals and occupation numbers
492(1)
Quantum chemistry methods
493(25)
Configuration Interaction (CI) method
494(2)
Variational calculus
496(3)
Hartree-Fock theory
499(2)
Hartree-Fock energy and electronic density
501(1)
Variational method: Hartree-Fock equations
502(4)
Koopman's theorem and physical significance of εi
506(1)
Closed shell electron system
507(1)
Hartree-Fock-Roothaan method
508(2)
HF theory of the homogeneous electron gas
510(4)
Electronic correlation
514(4)
Density Functional Theory
518(21)
Hohenberg-Kohn theorems
518(3)
The Levy constrained search method
521(2)
The Kohn-Sham method
523(1)
Relation to a fictitious noninteracting electron gas
523(1)
Derivation of the KS equations from the variational principle
524(2)
Matrix form of the KS equations
526(1)
Local density approximation (LDA) and beyond
527(3)
More about the KS equations
530(1)
Meaning of the KS eigenvalues
531(2)
Spin polarised DFT
533(3)
Other extensions of the DFT
536(1)
Excited states
536(1)
Nonzero temperatures
537(1)
Time dependent DFT (TDDFT)
538(1)
Some technical details
539(14)
Basis set
539(3)
Periodic boundary conditions and k-point sampling
542(4)
Pseudopotentials: ``hard'' and ``soft''
546(5)
``Exact'' pseudopotentials: PAW method
551(1)
Order-N methods
552(1)
Ab initio simulations
553(42)
Static properties: energies and forces
554(1)
Hellmann-Feynman theorem
555(2)
Pulay forces
557(2)
Stress
559(1)
Electronic DOS
560(1)
Adsorption of atomic and molecular oxygen on the MgO (001) surface
561(7)
Ab initio molecular dynamics simulations
568(3)
Hydrolysis at stepped MgO surfaces
571(2)
Calculation of free energies
573(3)
Ab initio lattice dynamics: direct method
576(4)
Density functional perturbation theory
580(5)
Quantum polarisation
585(10)
Bibliography 595(14)
Index 609