Thermodynamics, Gibbs Method and Statistical Physics of Electron Gases (eBook)

After graduating in Physics from Baku State University, Azerbaijan, Bahram Askerov received his Ph.D. from Institute of Semiconductors, St.Petersburg, Russia, in 1962 (principal supervisor Professor A. I. Anselm). Since 1971 he is Chair of Solid State Physics, Department of Physics, Baku State University, Azerbaijan. S.Figarova received her Ph.D. in 1981 (her principal supervisor was professor B.M.Askerov), and her DSc in 2008. She is the author of a manual and numerous research articles in the field of condensed matter physics. At present her research efforts center around studies of transport properties in low-dimensional systems. She is an associate professor of the Chair of Solid State Physics, Department of Physics, Baku State Univers

Preface 6
Contents 9
1 Basic Concepts of Thermodynamicsand Statistical Physics 13
1.1 Macroscopic Description of State of Systems: Postulates of Thermodynamics 13
1.2 Mechanical Description of Systems: Microscopic State:Phase Space: Quantum States 18
1.3 Statistical Description of Classical Systems: Distribution Function: Liouville Theorem 25
1.4 Microcanonical Distribution: Basic Postulate of Statistical Physics 31
1.5 Statistical Description of Quantum Systems: Statistical Matrix: Liouville Equation 34
1.6 Entropy and Statistical Weight 39
1.7 Law of Increasing Entropy:Reversible and Irreversible Processes 43
1.8 Absolute Temperature and Pressure: Basic Thermodynamic Relationship 47
2 Law of Thermodynamics: Thermodynamic Functions 54
2.1 First Law of Thermodynamics:Work and Amount of Heat: Heat Capacity 54
2.2 Second Law of Thermodynamics: Carnot Cycle 61
2.3 Thermodynamic Functions of Closed Systems: Method of Thermodynamic Potentials 67
2.4 Thermodynamic Coefficients and General Relationships Between Them 74
2.5 Thermodynamic Inequalities: Stability of Equilibrium State of Homogeneous Systems 80
2.6 Third Law of Thermodynamics: Nernst Principle 85
2.7 Thermodynamic Relationships for Dielectrics and Magnetics 90
2.8 Magnetocaloric Effect:Production of Ultra-Low Temperatures 94
2.9 Thermodynamics of Systems with Variable Number of Particles: Chemical Potential 97
2.10 Conditions of Equilibrium of Open Systems 101
3 Canonical Distribution: Gibbs Method 104
3.1 Gibbs Canonical Distribution for Closed Systems 104
3.2 Free Energy: Statistical Sum and Statistical Integral 110
3.3 Gibbs Method and Basic Objects of its Application 113
3.4 Grand Canonical Distribution for Open Systems 114
4 Ideal Gas 120
4.1 Free Energy, Entropy and Equationof the State of an Ideal Gas 120
4.2 Mixture of Ideal Gases: Gibbs Paradox 123
4.3 Law About Equal Distribution of Energy Over Degrees of Freedom: Classical Theory of Heat Capacityof an Ideal Gas 126
4.3.1 Classical Theory of Heat Capacity of an Ideal Gas 129
4.4 Quantum Theory of Heat Capacity of an Ideal Gas: Quantization of Rotational and Vibrational Motions 131
4.4.1 Translational Motion 133
4.4.2 Rotational Motion 136
4.4.3 Vibrational Motion 139
4.4.4 Total Heat Capacity 142
4.5 Ideal Gas Consisting of Polar Molecules in an External Electric Field 144
4.5.1 Orientational Polarization 144
4.5.2 Entropy: Electrocaloric Effect 148
4.5.3 Mean Value of Energy: Caloric Equation of State 149
4.5.4 Heat Capacity: Determination of Electric Dipole Moment of Molecule 150
4.6 Paramagnetic Ideal Gas in External Magnetic Field 152
4.6.1 Classical Case 152
4.6.2 Quantum Case 154
Magnetization 156
Entropy, Mean Energy and Heat Capacity 158
4.7 Systems with Negative Absolute Temperature 161
5 Non-Ideals Gases 167
5.1 Equation of State of Rarefied Real Gases 167
5.2 Second Virial Coefficient and Thermodynamics of Van Der Waals Gas 174
5.3 Neutral Gas Consisting of Charged Particles: Plasma 179
6 Solids 185
6.1 Vibration and Waves in a Simple Crystalline Lattice 185
6.1.1 One-Dimensional Simple Lattice 188
6.1.2 Three-Dimensional Simple Crystalline Lattice 192
6.2 Hamilton Function of Vibrating Crystalline Lattice: Normal Coordinates 194
6.3 Classical Theory of Thermodynamic Properties of Solids 197
6.4 Quantum Theory of Heat Capacity of Solids: Einstein and Debye Models 204
6.4.1 Einstein's Theory 206
6.4.2 Debye's Theory 207
6.5 Quantum Theory of Thermodynamic Properties of Solids 214
7 Quantum Statistics: Equilibrium Electron Gas 223
7.1 Boltzmann Distribution: Difficulties of Classical Statistics 224
7.2 Principle of Indistinguishability of Particles: Fermions and Bosons 232
7.3 Distribution Functions of Quantum Statistics 239
7.4 Equations of States of Fermi and Bose Gases 244
7.5 Thermodynamic Properties of Weakly Degenerate Fermi and Bose Gases 247
7.6 Completely Degenerate Fermi Gas: Electron Gas: Temperature of Degeneracy 250
7.7 Thermodynamic Properties of Strongly Degenerate Fermi Gas: Electron Gas 254
7.8 General Case: Criteria of Classicity and Degeneracy of Fermi Gas: Electron Gas 259
7.8.1 Low Temperatures 260
7.8.2 High Temperatures 261
7.8.3 Moderate Temperatures: TT0 261
7.9 Heat Capacity of Metals:First Difficulty of Classical Statistics 264
7.9.1 Low Temperatures 266
7.9.2 Region of Temperatures 266
7.10 Pauli Paramagnetism: Second Difficulty of Classical Statistics 268
7.11 ``Ultra-Relativistic'' Electron Gas in Semiconductors 272
7.12 Statistics of Charge Carriers in Semiconductors 275
7.13 Degenerate Bose Gas: Bose–Einstein Condensation 287
7.14 Photon Gas: Third Difficulty of Classical Statistics 292
7.15 Phonon Gas 299
8 Electron Gas in Quantizing Magnetic Field 307
8.1 Motion of Electron in External Uniform Magnetic Field: Quantization of Energy Spectrum 307
8.2 Density of Quantum States in Strong Magnetic Field 312
8.3 Grand Thermodynamic Potential and Statistics of Electron Gas in Quantizing Magnetic Field 314
8.4 Thermodynamic Properties of Electron Gas in Quantizing Magnetic Field 320
8.5 Landau Diamagnetism 324
9 Non-Equilibrium Electron Gas in Solids 330
9.1 Boltzmann Equation and Its Applicability Conditions 330
9.1.1 Nonequilibrium Distribution Function 330
9.1.2 Boltzmann Equation 332
9.1.3 Applicability Conditions of the Boltzmann Equation 334
9.2 Solution of Boltzmann Equation in Relaxation Time Approximation 337
9.2.1 Relaxation Time 337
9.2.2 Solution of the Boltzmann Equation in the Absence of Magnetic Field 339
9.2.3 Solution of Boltzmann Equation with an Arbitrary Nonquantizing Magnetic Field 345
9.3 General Expressions of Main Kinetic Coefficients 349
9.3.1 Current Density and General Formof Conductivity Tensors 349
9.3.2 General Expressions of Main Kinetic Coefficients 351
Galvanomagnetic Effects 351
Thermomagnetic Effects 351
9.4 Main Relaxation Mechanisms 353
9.4.1 Charge Carrier Scattering by Ionized Impurity Atoms 354
9.4.2 Charge Carrier Scattering by Phonons in Conductorswith Arbitrary Isotropic Band 357
Scattering by Acoustic Phonons, Deformation Potential Method 357
Scattering by Nonpolar Optical Phonons, Deformation Potential Method 360
Scattering by Polar Optical Phonons 363
9.4.3 Generalized Formula for Relaxation Time 366
9.5 Boltzmann Equation Solution for Anisotropic Band in Relaxation Time Tensor Approximation 368
9.5.1 Current Density 368
9.5.2 The Boltzmann Equation Solution 369
9.5.3 Current Density 371
Definite Integrals Frequently Met in Statistical Physics 372
A.1 Gamma-Function or Euler Integral of Second Kind 372
A.2 Integral of Type 373
A.3 Integral of Type 374
A.4 Integral of Type 375
A.5 Integral of Type 376
Jacobian and Its Properties 378
Bibliograpy 379
Index 381