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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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