Quantum theory of solids

240 Pages·1955·12.504 MB·English

by  Rudolf E Peierls

OXFORD CLASSIC TEXTS IN
THE PHYSICAL SCIENCES
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QUANTUM THEORY
OF SOLIDS
BY
R. E. PEIERLS
CLARENDON PRESS • OXFORD
OXPORD
UNIVERSITY PRESS
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First published 1955
Reprinted 1956,1964,1965,1974
Published in the Oxford Classics Series 2001
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ISBNO 19 850781 X
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Redwood Books, Trowbridge, Wiltshire
PKEFACE
THE quantum theory of solids has sometimes the reputation of being
rather less respectable than other branches of modern theoretical
physics. The reason for this view is that the dynamics of many-body
systems, with which it is concerned, cannot be handled without the use
of simplifications, or without approximations which often omit essential
features of the problem. Nevertheless, the theory contains a good deal
of work of intrinsic interest in which one can either develop a solution
convincingly from first principles, or at least give a clear picture of the
features which have been omitted and therefore discuss qualitatively the
changes which they are likely to cause.
In this introduction to the subject I have concentrated on the funda-
mental problems, on the known methods for solving them in suitably
idealized situations, and on the important basic problems still awaiting
solution. Recent literature has tended to concentrate on the practical
aspects, including the discussion of the properties of specific substances,
and it seemed therefore useful to stress the fundamental aspects.
The book is, in the first place, addressed to theoretical physicists, but
it should also be useful to experimentalists with a working knowledge of
quantum mechanics who want, to understand the basis of the models
customarily used in dealing with the properties of solids. The book
developed from notes which were prepared for a course given in 1953
at the Les Houches summer school in theoretical physics (under the
auspices of the University of Grenoble).
The order in which the material is presented was determined by
convenience of exposition rather than by the history of the subject.
No attempt has been made to give credit to the original authors. Their
names can be found in the more detailed review articles and textbooks
which are quoted in the bibliography.
Some branches of the theory of solids have developed to such an extent
that it was impossible to do them justice within the scope of this small
volume. In particular, crystal structures and binding energies, the
strength and plasticity of solids, and the electric breakdown of dielectrics
are amongst the important topics which have been omitted completely.
I would like to acknowledge my indebtedness to the organizers of the
summer school at Les Houches, and in particular to its director, Mrs.
Cecile De Witt, for the invitation which provided the occasion to tidy
vi PEEFACE
up my ideas about the subject. Discussions with many members of the
summer school helped in getting some points stated more clearly. I also
received a good deal of constructive criticism from several research
students at Birmingham, particularly J. B. Taylor, W. Marshall, and
D. A. Greenwood. The latter also very kindly assisted with the proofs.
Finally, I would like to express my thanks to the publishers for help in
many ways, and to their anonymous proof-reader for many useful
suggestions about wording.
R. E. P.
BIRMINGHAM
July 1954
NOTE ON SECOND IMPRESSION
ON reprinting this book no major changes have been made, except to
correct a few errors, of which the only substantial ones are in equations
(7.30), (7.32), (7.34), and (7.40). I am grateful to Professor W. Pauli
for drawing my attention to these errors.
Notes containing further explanation, or references to recent work
have been added at the end of Chapters I, II, VII, and VIII.
CONTENTS
I. CRYSTAL LATTICES. GENERAL THEORY
1.1 Introduction. Examples of structures 1
1.2 Dynamical problem. Adiabatic approximation 4
1.3 Equilibrium 7
1.4 Types of binding 9
1.5 Atomic vibrations. Classical mechanics. Linear chain 11
1.6 Atomic vibrations. Classical mechanics. General solution 14
1.7 Properties of normal vibrations 17
1.8 Remark about elastic constants 22
1.9 Quantum theory 24
II. CRYSTAL LATTICES. APPLICATIONS
2.1 Specific heat 27
2.2 Anharmonic terms. Thermal expansion 31
2.3 Linear term in specific heat 34
2.4 Thermal conductivity 40
2.5 Boltzmann equation 45
2.6 High temperature 50
2.7 Impurities and size effect 52
III. INTERACTION OF LIGHT WITH NON-CONDUCTING
CRYSTALS
3.1 Statement of problem. Infra-red absorption 54
3.2 Diffraction of X-rays 58
3.3 Effect of the atomic vibrations 63
3.4 Scattering of light 68
3.5 Scattering of neutrons 70
IV. ELECTRONS IN A PERFECT LATTICE
4.1 Bloch theorem 75
4.2 Strong binding 79
4.3 Nearly free electrons 83
4.4 Velocity and acceleration 87
4.5 Many electrons. Statistics 89
4.6 Specific heat 93
4.7 Surface problems 95
V. COHESIVE FORCES IN METALS
5.1 General discussion 101
5.2 The Wigner-Seitz approximation 104
5.3 Distorted structures. Linear chain 108
5.4 Distorted structures. Three dimensions 112
viii CONTENTS
VI. TRANSPORT PHENOMENA
6.1 General considerations. Collision time 115
6.2 Thermal conductivity 120
6.3 Static obstacles. Impurities and imperfections 121
6.4 Effect of lattice vibrations. General 125
6.5 Collisions between electrons 131
6.6 Collisions at high temperatures 133
6.7 Low temperatures 135
6.8 Validity of assumptions 139
VII. MAGNETIC PROPERTIES OF METALS
7.1 Paramagnetiam 143
7.2 Diamagnetisra of free electrons 144
7.3 Effect' of aperiodic field 160
7.4 Hall effect and magneto-resistance 155
VIII. FERROMAGNETISM
8.1 The Weiss model 161
8.2 The spin-wave theory. One dimension 164
8.3 Spin-wave model and ferromagnetism 170
8.4 The collective electron model 173
8.6 Neutron scattering 177
8.6 Remark on magnetization curves 179
8.7 Anti-ferromagnetism 181
IX. INTERACTION OF LIGHT WITH ELECTRONS IN SOLIDS
9.1 General outline. Classical theory 184
9.2 Transitic!:1; between bands 188
9.3 Photoelectric effect 190
9.4 Non-conduoting crystals 192
X. SEMI-CONDUCTORS AND LUMINESCENCE
10.1 Semi-conductors 197
10.2 Number of carriers 198
10.3 Electrical properties 200
10.4 Density gradients and space charge 202
10.6 Rectifying contacts 205
10.6 Electrons not in thermal equilibrium 207
XI. SUPERCONDUCTIVITY
11.1 Summary of properties 211
11.2 Outline of Frohlich-Bardeen theory 215
11.3 Effect of a magnetic field 218
11.4 Objections and difficulties 220
BIBLIOGRAPHY 222
REFERENCES 223
LIST OF '•"• 'BOLS 225
INDEX 227
I
CRYSTAL LATTICES. GENERAL THEORY
1.1 Introduction. Examples of structures
By a solid we usually mean a substance which shows some stiffness
under shear. Normally such substances have a crystalline structure,
and, for the purposes of this book, we shall limit ourselves to crystalline
solids. This excludes glasses, and I shall not discuss the question
whether glasses should properly be regarded as solids, and which of our
results, if any, may be applicable to them.
A crystal lattice may be constructed by repeating a 'unit cell', which
may consist of one or more atoms, periodically. The vector leading from
a point in one unit cell to the corresponding point in another is called a
'lattice vector' and can be represented as a linear combination with
integral coefficients of a small number of basic lattice vectors. The set
of lattice vectors determines the 'translation group' of the lattice. All
lattices'with the same translation group differ by having different unit
cells; the simplest of them has just one atom in the unit cell. The transla-
tion group is therefore often specified by the name of this simplest
lattice belonging to it.
I shall make no attempt to give a complete list of even the more
important types of lattices, but I shall give a few examples, which will
be used later as illustrations.
We start with cubic lattices, which are defined as remaining un-
changed if rotated by 90° about any one of three mutually perpendicular
axes. It is then clear that the basic lattice vectors can be stated most
simply in relation to these cubic axes.
(a) Simple cubic lattice. The unit cell has a single atom, the basic
lattice vectors are three vectors of equal length a in the directions of the
three cubic axes. In other words, taking Cartesian coordinates along
these axes, the basic lattice vectors are
the general lattice vector, and hence the position of any lattice point
relative to a given one, being
with arbitrary integers (positive, negative, or zero) n n , n .
v 2 3