[Han,_Fuxiang]_Problems_in_solid_state_physics_wit(z-lib.org) 2.pdf

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Unformatted text preview: PROBLEMS . IN SOLIO STATE WITH SOLUTI"ONS Fuxiang Han I PROBLEMS I N SOLIO STATE ~. PHYSICS WITH SOLUTIO NS .r T hiS book provides a practical approach to consolidate one 's acquired knowledge or to learn new concepts in solid state physics through solving problems. It contains 300 problems on various subjects of solid state physics. The problems in this book can be used as homework assignments in an introductory or advanced course on solid state physics for undergraduate or graduate students. It can also serve as a desirable reference book to solve typical problems and grasp mathematical techniques in solid state physics. In practice, it is more fascinating and rewarding to learn a new idea or technique through solving challenging problems rather than through reading only. In this aspect, this book is not a plain collect ion of problems but it presents a large number of problem-solving ideas and procedures, some of which are valuable to practitioners in condensed matter physics. PROBLEMS IN SOLIO STATE PHYSICS WITH SOLUTIONS ff'QAZCAPOTZALCO l.E.Yxiang Han Dalian University of Technology, China ,� World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishjng Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ojfice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for thjs book is available from the British Library. PROBLEMS IN SOLID STATE PHYSICS WITH SOLUTIONS Copyright© 2012 by World Scientific Publishing Co. Pte. Ltd. Ali rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording ar any information storage and retrie val system now known or to be invented, without written permission from the Publisher. For photocopying of material in thjs volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Orive, Danvers, MA O 1923, USA. In this case permjssion to photocopy is not required from the publisher. ISBN-13 ISBN-10 ISBN-! 3 ISBN-1O 978-981-4365-02-4 981-4365-02-5 978-981-4366-87-8 (pbk) 981-4366-87-0 (pbk) Printed in Singapore by B & Jo Enterprise Pte Ltd To my wife Pan Yanmei Without her constant support and encouragement, 1 could not have finished writing this book. Preface " This book provides a practical approach to consolidate one's already gained knowledge or to learn new knowledge in solid state physics through solving problems. It contains 300 problems. For the convenience of those using the book, the problems are partitioned into 30 chapters that cover t he commonly taught subjects in introductory and advanced solid state physics courses. There are both simple problems that are routine exercises in solid state physics and challenging problems that call for more time and efforts for their solutions. The author has carefully worked out every problem and provided detailed solutions to all 300 problems with special attention paid to the clear presentation of physical ideas and mathematical techniques that a re also relevant to research work at the frontiers of condensed matter physics. Many commonly used mathematical methods in condensed matter physics can be used in obtaining solutions to sorne of the problems. Alt hough this book alone is not suitable for beginners in solid state physics, it can be used in combination with with any book on solid state physics [a list of reference books is given at t he end of this book]. The problems presented can b e used in homework assignments in an introductory or advanced course on solid state physics for undergraduate or graduate students. Even though being a ble to solve problems, homework problems in a course in particular , is not the ult imate goal of studying solid state physics (as a ma tter of fact , it is not the ultimate goal of studying any branch of science), it is exceedingly important for a b etter understanding of basic concepts and for a quick grasp of methodology in solid state physics. It also helps the reader to get acquainted with t he problem-solving paradigm in solid state physics. This book can also b e used as a reference book for typical problems and mathematical techniques in solid state physics. Many problems in t his book v ii viii Problems in Salid State Physics with Solutions are not just plain exercises; they are actual targets to which problem-solving ideas and mathematical tools are applied, not just for solving the problems, but also for elucidating the relevant physical ideas and mathematical techniques. In practice, it is perhaps more fascinating and rewarding to learn a new idea or technique through solving a real challenging problem than through reading only. In this aspect, this book is not a plain collection of problems but it presents a large number of problem-solving ideas and procedures, sorne of which are useful to practitioners in condensed matter physics. For the convenience of quickly reviewing the relevant materials , a succinct recapitulation of important concepts and formulas is provided at the beginning of each chapter. With the formulas not derived and the notations in them not explained in most cases, the recapitulations are not aimed at complete expositions of relevant subjects. A recapitulation in a chapter can be used only as a place for formulas or as a brief summary of major subjects covered in the chapter. If a desired piece of information cannot be found in the recapitulation, a reference book should be consulted. To avoid turning pages back and forth, the solution to a problem follows immediately the statement of the problem, with a short rule to indicate the end of the statement of the problem and the beginning of the solution. This way, when the solution to a problem is consulted, the statement of the problem can be conveniently looked up. By just reading the statement of a problem without peeking at the provided solution, the reader can work out a solution to the problem on his/her own. The writing of this book has been supported by grants for teaching reform from the Teaching Affairs Division and the Graduate School at the Dalian University of Technology. Dalian, March 2011 Fuxiang Han Contents Preface l. 2. Drude Theory of Metals 1 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 2 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 . . . . . 2 3 4 5 7 8 9 Sommerfeld Theory of Metals 11 2-1 2-2 2-3 12 14 2-4 2-5 2-6 2- 7 2-8 3. vii Fermi-Dirac distribution function at low temperatures Effects of hydrostatic pressure. . . . . . . . . . . . . . Approximate express ion 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 . 15 16 18 21 24 25 Bravais Lattice 29 3-1 3-2 30 31 Primitive cells of five two-dimensional Bravais lattices Wigner-Seitz cells of five two-dimensional Bravais lattices ix Problems in Salid State Physics with Solutions x 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 4. 5. 6. 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 31 32 33 34 36 36 37 38 39 Point Groups 43 4-1 4-2 4-3 4-4 4-5 4-6 4- 7 4-8 4-9 4-10 4-11 44 44 45 47 48 51 52 54 56 56 57 Identification of groups . Group of Pauli matrices Statements about groups Identification of point groups Expressions of rotations and refiections . Matrix representation of a point group Invariant subgroup .... Subgroups of point group C3v Abelian groups . . Equivalence classes Point group C4v Classification of Bravais Lattices 63 5-1 5-2 5-3 5-4 64 68 69 70 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 Space Groups of Crystal Structures 71 6-1 6-2 6-3 73 73 6-4 6-5 6-6 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 ...... 73 74 75 76 Contents 6-7 6-8 6-9 6-10 6-11 6-12 6-13 6-14 7. 77 78 78 79 81 82 85 85 Scattering of X-Rays by a Crystal 89 7-1 7-2 7-3 7-4 7-5 7-6 89 90 90 92 93 96 7-7 7-8 8. 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 ofAX type Crystal structure of CsCI . xi 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 ato m under the influence of X-rays . . . . . . . . . .. . . . . . . . . . . . . . . . . Interaction of a bound electron with aplane electromagnetic wave . 98 99 Reciprocal Lattice 103 8-1 8-2 8-3 8-4 8-5 104 105 105 107 8-6 8-7 8-8 8-9 8- 10 8-11 Reciprocal lattice of the reciprocallattice Symmetry of the reciprocallattice 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 reciprocallattice 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 108 109 109 110 111 113 115 Problems in Salid State Physics with Salutians xii 8-12 8-13 8-14 8-15 8-16 9. 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 . X-Ray Diffraction on Crystals 9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8 9-9 9-10 Powder diffraction on silicon. . . . . . . . . . . . . Powder diffraction on cubic CaF2 . . . . . . . . . . Debye-Scherrer experiments on two cubic samples . Crystal structure of BaTi0 3 . . . . . . . . . . . . . Temperature dependence of the Bragg angle for Al Room-temperature superconductor . Powder diffraction on YBa2Cu306.9 . . . Three phases of iron . . . . . . . . . . . . Powder diffraction on Lal.sBao.2CU04-y . X-ray diffraction on Al . . . . . . . 10. Crystal Structure by Neutron Diffraction 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 Probe wavelengths proper for given structures . Reciprocallattice vectors and families of lattice planes Geometric structure factors . . . . . . . . Structure factor of the HCP lattice . . . . Neutron diffraction on the NaCl structure Be as a neutro n attenuator . . . . . . . . Diffraction on Al of neutrons of energies below 15 me V . Neutron diffraction peak. . . . . . . . . . . . . . . . .. 11. Bonding in Solids Molecular orbitals for a hydrogen molecule Energy of a hydrogen molecule in the bonding molecular orbital . . . . . . . . . . . . . . . . . . . . . . . . . 11-3 Energy of a hydrogen molecule in the antibonding molecular orbital . . . . . . . . . . . . . . . . . 11-4 Permanent dipole-permanent dipole interaction 11-5 Van der Waals bond in a diatomic molecule 11-6 Bond in a hypothetical diatomic molecule . . . 11-1 11-2 116 117 119 120 121 123 124 125 126 128 129 130 130 134 136 138 141 142 143 144 145 147 148 149 150 153 154 155 158 159 160 161 Contents 12. Cohesion of Solids 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 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 esCl crystal . . . . . Alkali metals . . . . . . . . . . . . . . . . . . . Exchange energy of the electron gas in an alkali metal 13. Normal Modes of Lattice Vibrations 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-4 Linear chain of atoms with damping . . . . . . . . . . 13-5 Polarizable molecules with internal degrees of freedom 13-6 Triatomic linear chain . . . . . . . . . . . . . . . . . . 13-7 Two-dimensional crystal with a square Bravais lattice 13-8 Simple cubic crystal . . . . . . . . . . . . . . . . 13-9 Three-dimensional monatomic crystal . . . . . . 13-10 Three-dimensional crystal with a two-atom basis 13-1 13-2 13-3 14. Quantum Theory of Lattice Vibrations 14-1 14-2 14-3 14-4 14-5 14-6 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 ID crystal of inert gas atoms Debye model for a ID crystal of inert gas atoms Electronic and lattice contributions to the specific heat of a metal . . . . . . . . . . . . .. . . . . . . . . 14-7 Specific heat of potassium at low temperatures . 14-8 Phonon density of states for an optical branch .. 14-9 Phonon density of states for an acoustical branch 14-10 Number of phonons and its variance . . . . . . . 14-11 Grüneisen parameter of a ID crystal of inert gas atoms xiii 163 164 165 167 167 169 170 171 173 177 178 181 183 185 186 188 190 195 199 206 209 211 213 215 217 219 221 222 224 224 225 229 l Problems in Salid State Physics with Solutions xiv 15. Inelastic Neutron Scattering by Phonons 15-1 15-2 15-3 15-4 231 Debye-Waller factors in spaces of different dimensionality Q-w region accessible for inelastic neutron scattering Cumulant expansion . . . . . . . . . . . . . Property of the dynamical structure factor. 241 16. Origin of Electronic Energy Bands 16-1 16-2 16-3 16-4 Quasi-momentum operator of Bloch electrons Free-electron model. . . . . . . . . . . . . . . Infinite chain of atoms with a Peierls distortion Densities of states in ID and 2D tight-binding energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . 16-5 Electron energies in a 2D metal with a square lattice . 16-6 Allowed wave vectors in a simple cubic crystal 16-7 Energy bands at the center of the first Brillouin zone for aluminum . . . . . . . . . . . . . . . . . . . . . . . . . 16-8 Monovalent metal with an ideal HCP structure . . . . 16-9 Energy bands for an FCC lattice in the [111] direction 16-10 Band structure of a divalent FCC Sr metal . . . . . . Electrons in a 2D square lattice . Energy bands in a ID 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 Plane-wave method for a ID 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 Tight-binding approximation for a ID crystal Two-dimensional graphite sheet . . . . . . . . . . 247 249 250 251 251 255 256 259 262 264 265 268 273 18. Methods for Band Structure Computations 18-1 18-2 18-3 18-4 18-5 18-6 18-7 18-8 18-9 18-10 242 244 246 259 17. Electrons in a Weak Periodic Potential 17-1 17-2 17-3 17-4 17-5 232 236 238 240 274 279 282 285 286 287 290 equation 292 293 . . . .. 294 . . r Problems in Salid State Physics with Solutions xvi 21-8 21-9 21-10 21-11 Thomas-Fermi screening . . . . . . N-representable density . . . . . . Electron-ion interaction functional Janak's Theorem . . . . . . .. . . 22. Pseudopotentials 22-1 22-2 22-3 22-4 22-5 22-6 22-7 22-8 22-9 Smooth and oscillatory functions . . . . . . . . . . . . Atom with a harmonic radial potential . . . . . . . . . Numerical solution of the radial Schrodinger equation Kerker scheme for pseudopotentials . . . . . . . . . . . Spherical averages . . . . . . . . . . . . . . . . . . . . Hedin-Lundqvist interpolation scheme for exchange and correlation. . . . . . . . . . . . . . . . . . . . . Simplified OPW pseudopotential . . . . . . . . . . . . .. Equivalence of the norrn-conservation condition . . . . .. Perdew-Zunger parametrization of the correlation energy. 23 . Projector-Augmented Plane-Wave Method 23-1 23-2 Kohn-Sham equations for auxiliary wave functions General formula for local operators . . . . . . . . . 24. Determination of Electronic Band Structures 24-1 24-2 24-3 24-4 24-5 Quantization of electromagnetic fields Quantum field operator of electrons Poisson surnmation formula . . . . . . Application of the Lifshits-Kosevich theory Amplitude of dHvA oscillations in a free electron gas . 25. Crystal Defects Deduction of the energy of vacancy formation from resistivity data . . . . . . . . . . . . 25-2 Energy of vacancy forrnation in gold . . . . . . . . 25-3 Number of vacant sites . . . . . . . . . . . . . . . . 25-4 Number of occupied lattice sites for every vacancy in Ni 25-5 Energy of vacancy forrnation in Al . 25-6 Energy of vacancy forrnation in Mo . 25-7 Schottky defects in copper . . . . . . 25-8 Schottky defects in an oxide ceramic 354 357 359 360 365 366 369 371 375 379 381 383 387 388 391 392 395 397 399 404 406 408 409 415 25-1 417 418 419 419 420 420 420 421 Contents Burgers vectors of dislocations in FCC , BCC, and SC crystals . . . . . . . . . . . . . . . . . 25-10 Two- and three-dimensional defects .. .. 25-11 Two perpendicular long edge dislocations xvii 25-9 26. Electron-Phonon Interaction 26-1 26-2 26-3 26-4 26-5 26-6 26-7 26-8 26-9 26-10 26-11 26-12 26-13 26-14 26-15 26-16 26-17 26-18 26-19 26-20 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...
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