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Solid state physics

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SOLID STATE
* PHYSICS
SECOND EDITION

J. S. Blakemore
Department of Physics and Astronomy
Western Washington University

CAMBRIDGE
UNIVERSITY PRESS

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PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge CB2 1RP, United Kingdom
CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, UK

40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
© W. B. Saunders Company 1969,1974
This edition © Cambridge University Press 1985
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published by W. B. Saunders Company 1969
second edition first published by W. B. Saunders Company 1974
This updated second edition first published by Cambridge University Press 1985
Reprinted 1986,1988,1989,1991,1993,1995,1998
Typeset in Bodoni Book 10/12 pt
A catalogue record for this book is available from the British Library
Library of Congress Cataloguing in Publication data
Blakemore, J. S. (John Sydney), 1927Solid state physics
Includes bibliographies and indexes
1. Solid state physics I. Title
QC176.B63 1985 530.41 85-47879
ISBN 0 52130932 8 hardback
ISBN 0 521 313910 paperback
Transferred to digital printing 2004

UP

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CONTENTS


Preface
General references

vii
ix

Chapter One
CRYSTALLINITY AND THE FORM OF SOLIDS
1.1
1.2
1.3
1.4
1.5
1.6

Forms of Interatomic Binding
Symmetry Operations
Actual Crystal Structures
Crystal Diffraction
Reciprocal Space
Crystalline Defects
Problems
Bibliography

1
5
25
39
51
67

74
81
84

Chapter Two
LATTICE DYNAMICS

87

2.1

Elastic Waves, Atomic Displacements, and
Phonons
2.2 Vibrational Modes of a Monatomic Lattice
2.3 Vibrational Spectrum for a Structure with
a Basis
2.4 Phonon Statistics and Lattice Specific Heats
2.5 Thermal Conduction
Problems
Bibliography

88
92
105
120
132
144
147

Chapter Three

ELECTRONS IN METALS
3.1
3.2

Some Features of the Metallic State
Classical Free Electron Theory

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149
151
157


CONTENTS
3.3 The Quantized Free Electron Theory
3.4 The Band Theory of Solids
3.5 Dynamics of Electron Motion
3.6 Superconductivity
Problems
Bibliography

170
202
235
266
285
291

Chapter Four

SEMICONDUCTORS
4.1
4.2
4.3
4.4

293

Equilibrium Electron Statistics
Electronic Transport in a Semiconductor
Band Shapes in Real Semiconductors
Excess Carrier Phenomena
Problems
Bibliography

295
330
362
378
396
403

Chapter Five
DIELECTRIC AND MAGNETIC PROPERTIES
OF SOLIDS
5.1
5.2
5.3

Dielectric Properties

Magnetic Properties of Solids
Magnetic Resonance
Problems
Bibliography

405
407
431
455
484
488

TABLE OF SOME USEFUL NUMERICAL CONSTANTS

490

AUTHOR INDEX

491

SUBJECT INDEX

497

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PREFACE

This book was written as the text for a one quarter, or one semester,

introductory course on the physics of solids. For an undergraduate
majoring in physics, the associated course will usually be taken during the
last two undergraduate years. However, the book is designed also to meet
needs of those with other degree majors: in chemistry, electrical
engineering, materials science, etc., who may not encounter this
requirement in their education until graduate school. Some topics discussed
(band theory, for example) require familiarity with the language and
concepts of quantum physics; and an assumed level of preparedness is one
semester of "modern physics". A reader who has taken a formal quantum
mechanics course will be well prepared, but it is recognized that this is often
not possible. Thus Schrodinger's equation is seen from time to time, but
formal quantum mechanical proofs are side-stepped.
The aim is thus a reasonably rigorous - but not obscure - first exposition of solid state physics. The emphasis is on crystalline solids,
proceeding from lattice symmetries to the ideas of reciprocal space and
Brillouin zones. These ideas are then developed: for lattice vibrations, the
theory of metals, and crystalline semiconductors, in Chapters 2, 3, and 4
respectively. Aspects of the consequences of atomic periodicity comprise
some 75 % of the book's 500 pages. In order to keep the total exposition
within reasonable bounds for a first solid state course, a number of other
aspects of condensed matter physics have been included but at a relatively
brief survey level. Those topics include lattice defects, amorphous solids,
superconductivity, dielectric and magnetic phenomena, and magnetic
resonance.
The text now offered is on many pages unchanged from that of the
1974 second edition published by Saunders. However, the present
opportunity to offer this book through the auspices of Cambridge
University Press has permitted me to correct some errors, add some needed
lines of explanation (such as at the end of Section 1.5), revise some figures,
and update the bibliographies following this preface and at the end of each
chapter. The SI system of units, adopted for the second edition, is of course

retained here. Two exceptions to the SI system should be noted: retention
of the Angstrom unit in describing interatomic distances, and use of the
electron volt for discussions of energy per electron of per atom. There seems
no sign that crystallographers are ready to quote lattice spacings in
nanometers, and the 10~10 conversion factor from A to meters is an easy

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VIII

PREFACE

one. Use of the eV rather than 1.6x10 19 J also simplifies many descriptions
of energy transformation events. Questions of units are of course important
for the numerical aspects of homework problems.
These problems are grouped at the end of each chapter, and there are
125 of them altogether. Many do include a numerical part, intended to
draw the student's attention to the relative magnitudes of quantities and
influences more than to the importance of decimal place accuracy. The
problems vary (intentionally) greatly in length and difficulty; and I have
been told several times that some of these problems are too difficult for
the level of the text. These can certainly provide a worthwhile challenge
for one who has "graduated" from the present book to one of the
advanced solid state texts cited in the General Reference list which follows
this preface.
As in previous editions of this book, many more literature citation
footnotes are given than are typical in an undergraduate text. These
augment the bibliography at the end of each chapter in citing specific
sources for optional additional reading. A paper so cited in a footnote may

serve as the beginning of a literature search undertaken years after the
owner's first exposure to this book, and the footnotes have been provided
with this in mind.
The present book was written to be an account of ideas about the
physics of solids rather than a compilation of facts and numbers. Accordingly, tables of numerically determined properties are relatively few - in
contrast, for example, to nearly 60 tables of data in the fifth edition of
Kittel's well-known textbook. The reader needing quantitative physical
data on solids has a variety of places to turn to, with extensive data in the
American Institute of Physics Handbook (last revised in 1972) and in the
Handbook of Chemistry and Physics (updated annually). As noted in the list
of General References on page ix, new volumes have recently been
appearing in the Landolt—Bornstein Tables series, including data compilation for some semiconductor materials. The work of consolidating
numerical information concerning solids is indeed a continuous one.
Over the years of writing and rewriting material for successive
editions of this book, I have been helped by many people who have made
suggestions concerning the text, worked problems, and provided
illustration material. To all of those individually acknowledged in the
prefaces of the first and second editions, I am still grateful. In preparing
this updated second edition for Cambridge University Press, my principal
acknowledgement should go to L. E. Murr of the Oregon Graduate Center
for the photographs that provide a number of attractive and informative
new figures in Chapter 1, and to H. K. Henisch of Pennsylvania State
University for the print used as Figure 1.2.
Beaver ton, Oregon

j . s.

March 1985

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BLAKEMORE


GENERAL REFERENCES

Solid State Physics (Introductory/Intermediate Level)
R. H. Bube, Electrons in Solids: An Introductory Survey (Academic Press,
3rd ed., 1992).
R. H. Bube, Electronic Properties of Crystalline Solids (Academic Press, 1973).
A. J. Dekker, Solid State Physics (Prentice-Hall, 1957). [Was never revised and is
now out of print, but includes interesting discussion of several topics others
omit.]
H. J. Goldsmid (ed.), Problems in Solid State Physics (Pion, 1968).
W. A. Harrison, Electronic Structure and the Properties of Solids (Freeman, 1980).
C. Kittel, Introduction to Solid State Physics (Wiley, 6th ed., 1986).
J. P. McKelvey, Solid-State and Semiconductor Physics (Krieger, 1982).
H. M. Rosenberg, The Solid State (Oxford Univ. Press, 3rd ed., 1988).

Solid State Physics (Advanced Level)
N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, 1976).
J. Callaway, Quantum Theory of the Solid State (Academic Press, 2nd ed., 1991).
D. L. Goodstein, States of Matter (Prentice-Hall, 1975).
W. A. Harrison, Solid State Theory (Dover, 1980).
A. Haug, Theoretical Solid State Physics (Pergamon Press, 1972), 2 vols.
W. Jones and N. H. March, Theoretical Solid State Physics (Wiley, 1973), 2 vols.
C. Kittel, Quantum Theory of Solids (Wiley, 1963).
R. Kubo and T. Nagamiya (eds.), Solid State Physics (McGraw-Hill, 1969).
P. T. Landsberg (ed.), Solid State Theory, Methods & Applications (Wiley, 1969).
R. E. Peierls, Quantum Theory of Solids (Oxford, 1965). [Out of print, but a classic]

F. Seitz, Modern Theory of Solids (Dover, 1987). [Reprint of a 1940
McGraw-Hill classic]
J. Ziman, Principles of the Theory of Solids (Cambridge Univ. Press, 2nd ed., 1972).

Solid State Electronics
A. Bar-Lev, Semiconductors and Electronic Devices (Prentice-Hall, 1979).
N. G. Einspruch (ed.), VLSI Electronics: Microstructure Science (Academic Press,
Vol. 1, 1981, through vol. 8, 1984, and continuing).
R. J. Elliott and A. F. Gibson, An Introduction to Solid State Physics and its
Applications (Barnes and Noble, 1974).
A. S. Grove, Physics and Technology of Semiconductor Devices (Wiley, 1967).
A. G. Milnes, Semiconductor Devices and Integrated Circuits (Van Nostrand, 1980).
T. S. Moss, G. J. Barrett and B. Ellis, Semiconductor Optoelectronics (Butterworths,
1973).
B. G. Streetman, Solid State Electronic Devices (Prentice-Hall, 2nd ed., 1980).
S. M. Sze, Physics of Semiconductor Devices (Wiley, 2nd ed., 1981).
F. F. Y. Wang, Introduction to Solid State Electronics (North-Holland, 1980).

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GENERAL REFERENCES
Quantum Phenomena
F. J. Bockhoff, Elements of Quantum Theory (Addison-Wesley, 2nd ed., 1976).
A. P. French and E. F. Taylor, An Introduction to Quantum Physics (Norton, 1978).
B. K. Ridley, Quantum Processes in Semiconductors (Oxford Univ. Press, 1981).
D. ter Haar (ed.), Problems in Quantum Mechanics (Pion, 3rd ed., 1975).

Statistical Physics
S. Fujitta, Statistical and Thermal Physics (Krieger, 1984).

C. Kittel, Elementary Statistical Physics (Wiley, 1958).
C. Kittel and H. Kroemer, Thermal Physics (Freeman, 2nd ed., 1980).
F. Mohling, Statistical Mechanics (Wiley-Halsted, 1982).
L. E. Reichl, A Modern Course in Statistical Physics (Univ. Texas, 1980).
R. C. Tolman, The Principles of Statistical Mechanics (Dover, 1979).

Wave Phenomena
L. Brillouin, Wave Properties and Group Velocity (Academic Press, 1960).
L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1972).
I. G. Main, Vibrations and Waves in Physics (Cambridge Univ. Press, 1978).
C. F. Squire, Waves in Physical Systems (Prentice-Hall, 1971).

Numerical Data
American Institute of Physics Handbook (McGraw-Hill, 3rd ed., 1972).
Handbook of Chemistry and Physics (CRC Press, 66th ed., 1985).
Handbuch der Physik (S. Fliigge, general editor for 54 volume series)
(Springer-Verlag, 1956 through 1974).
Landolt-Bornstein Tables (Springer-Verlag). [Volumes date from the 1950s and
earlier, but new ones are now appearing on solid state topics.]

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chapter one

CRYSTALLINITY
AND THE
FORM OF SOLIDS

Solid materials can be classified according to a variety of criteria.

Among the more significant of these is the description of a solid as
being either crystalline or amorphous. The solid state physics community has tended during the period from the mid-1940's to the late 1960's
to concentrate a much larger effort on crystalline solids than on the less
tractable amorphous ones.
An amorphous solid exhibits a considerable degree of short range
order in its nearest-neighbor bonds, but not the long range order of
a periodic atomic lattice; examples include randomly polymerized
plastics, carbon blacks, allotropic forms of elements such as selenium
and antimony, and glasses. A glass may alternatively be thought of as a
supercooled liquid in which the viscosity is too large to permit atomic
rearrangement towards a more ordered form. Since the degree of ordering of an amorphous solid depends so much on the conditions of its
preparation, it is perhaps not inappropriate to suggest that the preparation and study of amorphous solids has owed rather less to science and
rather more to art than the study of crystalline materials. Intense study
since the 1960s on glassy solids such as amorphous silicon (of interest for
its electronic properties) is likely to create a more nearly quantitative basis
for interpreting both electronic and structural features of noncrystalline
materials.
In the basic theory of the solid state, it is a common practice to start
with models of single crystals of complete perfection and infinite size.
The effects of impurities, defects, surfaces, and grain boundaries are
then added as perturbations. Such a procedure often works quite well
even when the solid under study has grains of microscopic or submicroscopic size, provided that long range order extends over distances
which are very large compared with the interatomic spacing. However,

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CRYSTALLINITY AND THE FORM OF SOLIDS

it is particularly convenient to carry out experimental measurements on

large single crystals when they are available, whether they are of natural origin or synthetically prepared.1 Figures 1-1 and 1-2 show examples of microscopic and macroscopic synthetic crystals.
Large natural crystals of a variety of solids have been known to
man for thousands of years. Typical examples are quartz (SiO2), rocksalt
(NaCl), the sulphides of metals such as lead and zinc, and of course
gemstones such as ruby (A12O3) and diamond (C). Some of these natural
crystals exhibit a surprising degree of purity and crystalline perfection,
which has been matched in the laboratory only during the past few
years.2 For many centuries the word "crystal" was applied specifically
to quartz; it is based on the Greek word implying a form similar to that
of ice. In current usage, a crystalline solid is one in which the atomic
arrangement is regularly repeated, and which is likely to exhibit an external morphology of planes making characteristic angles with each
other if the sample being studied happens to be a single crystal.
When two single crystals of the same solid are compared, it will
usually be found that the sizes of the characteristic plane "faces" are
1
For discussions of single growth techniques, see the bibliography at the end of
Chapter 1.
2
Indeed, synthetically created diamonds still do not compare in quality with the
finest natural diamonds. For most other gemstones, man seems to have been able to do at
least as well as nature.

Figure 1-1 Scanning electron microscope view of small NiO crystal, with well
developed facets. (Photo courtesy of L. E. Murr, Oregon Graduate Center.) At room
temperature, antiferromagnetic ordering provides for NiO a trigonal distortion of the
(basically rocksalt) atomic arrangement.

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CRYSTALLINITY AND THE FORM OF SOLIDS

Figure 1 - 2 The growing surface of a calcium tartrate crystal, during growth in
a tartrate gel infused with calcium chloride solution. From Crystal Growth in Gels by
H. K. Henisch (Penn. State Univ. Press, 1970).

not in the same proportion (the "habit" varies from crystal to crystal).
On the other hand the interfacial angles are always the same for crystals
of a given material; this was noted in the sixteenth century and formed
the basis of the crystallography of the next three centuries. These observations had to await the development of the atomic concept for an
explanation, and it was not until Friedrich, Knipping, and Laue demonstrated in 1912 that crystals could act as three-dimensional diffraction
gratings for X-rays that the concept of a regular and periodic atomic
arrangement received a sound experimental foundation. More recently,
the periodic arrangement of atoms has been made directly visible by
field-emission microscopy.3
Whether we wish to study mechanical, thermal, optical, electronic,
or magnetic properties of crystals —be they natural ones, synthetic
single crystals (such as Ge, Si, A12O3, KBr, Cu, Al), or polycrystalline
aggregates —most of the results obtained will be strongly influenced by
the periodic arrangement of atomic cores or by the accompanying periodic electrostatic potential. The consequences of periodicity take up a
major fraction of this book, for a periodic potential has many consequences, and exact or approximate solutions are possible in many situations.
In this first chapter we shall consider how atoms are bonded
together and how symmetry requirements result in the existence of a
limited variety of crystal classes. There is no optimum order for consideration of the two topics of bonding and crystal symmetry, since each
depends on the other for illumination; it is recommended that the
3
See, for example, Figure l-56(a) on page 79, for an ion-microscope view of atoms
at the surface of an iridium crystal.

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3


CRYSTALLINITY AND THE FORM OF SOLIDS

reader skim through the next two sections completely before embarking upon a detailed study of either.
The chapter continues (in Section 1.3) with an account of some of
the simpler lattices in which real solids crystallize. The emphasis of the
section is on the structures of elements and of the more familiar inorganic binary compounds.
Sections 1.4 (Crystal Diffraction) and 1.5 (Reciprocal Space) are
closely connected, and once again it is recommended that both sections
be read through before a detailed study of either is undertaken. An understanding of the reciprocal lattice helps one to see what diffraction of
a wave in a crystal is all about, and vice versa.
Section 1.6 does little more than mention the principal types of
point and line imperfection in a crystal. Bibliographic sources are cited
for the reader who wishes to know more about dislocations, or about
the chemical thermodynamics of defect interactions in solids.

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Forms of
Interatomic Binding

All of the mechanisms which cause bonding between atoms derive
from electrical attraction and repulsion. The differing strengths and differing types of bond are determined by the particular electronic structures of the atoms involved. The weak van der Waals (or residual) bond
provides a universal weak attraction between closely spaced atoms and
its influence is overridden when the conditions necessary for ionic,
covalent, or metallic bonding are also present.

The existence of a stable bonding arrangement (whether between a
pair of otherwise isolated atoms, or throughout a large, three-dimensional crystalline array) implies that the spatial configuration of positive
ion cores and outer electrons has less total energy than any other configuration (including infinite separation of the respective atoms). The
energy deficit of the configuration compared with isolated atoms is
known as the cohesive energy, and ranges in value from 0.1 eV/atom
for solids which can muster only the weak van der Waals bond to
7 eV/atom or more in some covalent and ionic compounds and some
metals.4 The cohesive energy constitutes the reduction in potential
energy of the bonded system (compared with separate atoms) minus the
additional kinetic energy which the Heisenberg uncertainty principle
tells us must result from localization of the nuclei and outer shell electrons.
In covalent bonding the angular placement of bonds is very important, while in some other types of bonding a premium is placed upon
securing the largest possible coordination number (number of nearest
neighbors). Such factors are clearly important in controlling the most
favorable three-dimensional structure. For some solids, two or more
quite different structures would result in nearly the same energy, and a
change in temperature or hydrostatic pressure can then provoke a
change from one allotropic form of the solid to another, as envisaged in
Figure 1-3. As discussed further under the heading of the Covalent
Bond, an allotropic transition to an energetically more favorable structure can sometimes be postponed, depending on the rate of conditions
of cooling or warming.
4
The joule is a rather large energy unit for discussion of events involving a single
atom. Thus energies in this book will often be quoted in terms of electron volts per particle or per microscopic system. (It is hoped that the context will leave no doubt as to
whether an energy change in eV refers to a molecule, an atom, or a single electron.) One
elementary charge moved through a potential difference of one volt involves a potential
energy change of 1.6022 X 10~19 joule (see the table of useful constants inside the cover).
Chemists tend to cite bond energies and cohesive energies in calories per mole.
1 eV/molecule is equivalent to 23,000 calories per mole, or 9.65 X 104 joule/mole.


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CRYSTALLINITY AND THE FORM OF SOLIDS

Allotropic transition

/
/

—y
/

Structure 2

Figure 1-3 Cohesive energy
versus temperature or pressure for a
solid in which two different atomic
arrangements are possible. An allotropic transition may occur at the
pressure or temperature at which one
structure replaces the other as having
minimum energy.

^-Structure 1

Pressure or Temperature

THE VAN DER WAALS BOND
As previously noted, van der Waals bonding occurs universally
between closely spaced atoms, but is important only when the conditions for stronger bonding mechanisms fail. It is a weak bond, with a

typical strength of 0.2 eV/atom, and occurs between neutral atoms and
between molecules. The name van der Waals is associated with this
form of bond since it was he who suggested that weak attractive forces
between molecules in a gas lead to an equation of state which represents the properties of real gases rather better than the ideal gas law
does. However, an explanation of this general attractive force had to
await the theoretical attentions of London (1930).
London noted that a neutral atom has zero permanent electric
dipole moment, as do many molecules; yet such atoms and molecules
are attracted to others by electrical forces. He pointed out that the zeropoint motion, which is a consequence of the Heisenberg uncertainty
principle, gives any neutral atom a fluctuating dipole moment whose
amplitude and orientation vary rapidly. The field induced by a dipole
falls off as the cube of the distance. Thus if the nuclei of two atoms are
separated by a distance r, the instantaneous dipole of each atom creates
an instantaneous field proportional to (1/r)3 at the other. The potential
energy of the coupling between the dipoles (which is attractive) is then
(1-1)
A quantum-mechanical calculation of the strength of this dipole-dipole
attraction suggests that E attr would reach 10 eV if r could be as small as
1A. However, a spacing this small is impossible because of overlap
repulsion.
As the interatomic distance decreases, the attractive tendency
begins to be offset by a repulsive mechanism when the electron clouds
of the atoms begin to overlap. This can be understood in terms of the
Pauli exclusion principle, that two or more electrons may not occupy

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1.1 FORMS OF INTERATOMIC BINDING


the same quantum state. Thus overlap of electron clouds from two
atoms with quasi-closed-shell configurations is possible only by promotion of some of the electrons to higher quantum states, which requires
more energy.
The variation of repulsive energy with interatomic spacing can be
simulated either by a power law expression (a dependence as strong as
r~n or r~12 being necessary) or in terms of a characteristic length. The
latter form is usually found to be the most satisfactory, and the total
energy can then be written as
(1-2)
which is drawn as the solid curve in Figure 1-4. The strength of the
bond formed and the equilibrium distance r0 between the atoms so
bonded depend on the magnitudes of the parameters A, B, and p. Since
the characteristic length p is small compared with the interatomic
spacing, the equilibrium arrangement of minimum E occurs with the
repulsive term making a rather small reduction in the binding energy.5
We have spoken of van der Waals bonding so far as occurring
between a pair of otherwise isolated atoms. Within a three-dimensional
solid, the dipole-dipole attractive and overlap repulsive effects with
respect to the various neighbor atoms add to give an overall cohesive
energy still in accord with Equation 1-2. There are no restrictions on
bond angles, and solids bound by van der Waals forces tend to form in
the (close-packed) crystal structures for which an atom has the largest
possible number of nearest neighbors. (This is the case, for example, in
the crystals of the inert gases Ne, Ar, Kr, and Xe, all face-centered-cubic
structures, in which each atom has twelve nearest neighbors.) The
rapid decrease of van der Waals attraction with distance makes atoms
beyond the nearest neighbors of very little importance.
' See Problem 1.1 for an exercise of this principle.

Figure 1 - 4 Total potential energy in a van der Waals

bond (solid curve), showing the
attractive and repulsive terms
which combine to give a stable
bond at an internuclear distance r0.
Van der Waals
attraction

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8

CRYSTALLINITY AND THE FORM OF SOLIDS

The solid inert gases6 are fine examples of solids which are bound
solely by van der Waals forces, because the closed-shell configurations
of the atoms eliminate the possibility of other, stronger bonding mechanisms. Far more typically do we find solids in which van der Waals
forces bind saturated molecules together, molecules within which
much stronger mechanisms are at work. This is the case with crystals of
many saturated organic compounds and also for solid H2, N2, O2, F2, Cl2,
Br2, and I2. The example of Cl2, with a sublimation energy of 0.2
eV/molecule but a dissociation energy of 2.5 eV/molecule, shows how
the van der Waals bond between diatomic molecules can be broken
much more readily than the covalent Cl-Cl bond.

THE COVALENT BOND

The covalent bond, sometimes referred to as a valence or homopolar bond, is an electron-pair bond in which two atoms share two electrons. The result of this sharing is that the electron charge density7 is
high in the region between the two atoms. An atom is limited in the
number of covalent bonds it can make (depending on how much the

number of outer electrons differs from a closed-shell configuration), and
there is a marked directionality in the bonding. Thus carbon can be involved in four bonds at tetrahedral angles (109.5°), and the characteristic tetrahedral arrangement is seen in crystalline diamond and in
innumerable organic compounds. Other examples of characteristic
angles between adjacent covalent bonds are 105° in plastic sulphur and
102.6° in tellurium.
The hydrogen molecule, H2, serves as a simple example of the
covalent bond. Two isolated hydrogen atoms have separate Is states for
their respective electrons. When they are brought together, the interaction between the atoms splits the Is state into two states of differing
energy, as sketched in Figure 1-5. When the two nuclei are very close
together, the total energy is increased for both kinds of states by internuclear electrostatic repulsion; but for the Is state marked8 crg, which
has an even (symmetric) orbital wave-function, the energy is lowered
(i.e., there is an attractive tendency) for a moderate spacing.9
6

For helium, the zero-point motion is so violent that solidification even at absolute
zero can be accomplished only by applying an external pressure of 30 atmospheres.
7
Remember that in quantum mechanics we cannot describe a specific orbit for a
bound electron but only a wave-function \JJ whose square is proportional to the probability of finding an electron at a location on a time-averaged basis. Then if i// is a normalized
wave-function (such that \jj2 integrated over all space is unity), the average charge density
at any location is the value of —ei//2.
8
The designation of the two orbital wave-functions as crg and German terms "gerade" and "ungerade" for even and odd.
9
A principal feature of the bonding attraction is the resonance energy corresponding
to the exchange of the two electrons between the two atomic orbitals, as first discussed by
W. Heitler and F. London, Z. Physik 44, 455 (1927). For a recent account of this in
English, see E. E. Anderson, Modern Physics and Quantum Mechanics (W. B. Saunders,
1971), p. 390.

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1.1 FORMS OF INTERATOMIC BINDING

Figure 1 - 5 Variation of energy with internuclear spacing for the neutral hydrogen
molecule, after Heitler and London (1927). The figure shows the as (bonding) and au (antibonding) states. crg accommodates two electrons with anti-parallel spins.

This symmetric crg Is solution requires that the electron charge
density —ei//2 be concentrated in the region between the two nuclei.
The requirement of the Pauli principle that total wave functions combine in an anti-symmetric manner is satisfied if the crg Is state is occupied by two electrons with antiparallel spins.
The alternative cru Is state would have to be occupied by two electrons with parallel spins in order to conform with the Pauli principle,
but as Figure 1-5 demonstrates, this state is an anti-bonding (repulsive)
one at all distances. This is unimportant for H2, since the crg state can
accommodate the only two electrons in the system and a strong bond
results.
Note that this could not happen for a double bond between two
helium atoms, since the total energy would be increased by populating
both of the cru states as well as the crg states. Interestingly, the molecule-ion He 2 + is stable.
The wave-mechanical problem becomes much more formidable
when covalent bonds are considered between multi-electron atoms, but
qualitatively the picture is that sketched for the H-H bond. In all cases
the closeness of approach is limited by the Coulomb repulsion of the
nuclei, assisted in the heavier atoms by overlap repulsion of inner
closed-shell electrons.
Some of the classes of covalently bonded materials are:
1. Most bonds within organic compounds.
2. Bonds between pairs of halogen atoms (and between pairs of
atoms of hydrogen, nitrogen, or oxygen) in the solid and fluid forms of

these media.
3. Elements of Group VI (such as the spiral chains of tellurium),

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CRYSTALLINITY AND THE FORM OF SOLIDS

Group V (such as in the crinkled hexagons of arsenic), and Group IV
(such as diamond, Si, Ge, a-Sn).
4. Compounds obeying the (8-N) rule (such as InSb) when the horizontal separation in the Periodic Table is not too large.
It is often found that valence-bonded solids can crystallize in several different structures for almost the same cohesive energy. The
energetically most favored structure can be displaced from its prime
position by a change of temperature or pressure (Figure 1-3), resulting
in the situation known as allotropy or polymorphism. Thus ZnS can
exist either in a cubic form (zinc-blende) or as a hexagonal structure
(wurtzite). The coordination of nearest neighbors is the same for zincblende and wurtzite; it is the arrangement of second-nearest neighbors which creates a very slight energy difference between the two
structures. Similarly, silicon carbide has an entire range of "polytypes," from the purely cubic to the purely hexagonal, which show
subtle differences in their electronic properties.
In the cases of ZnS and SiC, the various crystalline forms can all be
maintained at room temperature without apparent risk of spontaneous
conversion to the energetically most favored form (the conditions of
crystallization accounting for the various forms capable of being studied at low temperatures). With other materials, spontaneous conversion
occurs quite readily.
Thus selenium cooled rapidly from its melting point (218°C) to
room temperature is amorphous, but crystallization begins if the solid is

warmed to 60-70°C, and the material remains crystalline on cooling
back to room temperature. Another good example of allotropic conversion is provided by tin, which is stable as a gray semimetal (a-Sn)
below 17°C, crystallizing in the diamond lattice with four tetrahedrallylocated bonds. Temperatures above 17°C, or application of pressure
even below that temperature, cause a conversion to a much more dense
white metallic form (/3-Sn) with a tetragonal structure in which each
atom has six nearest neighbors.

COVALENT-VAN DER WAALS STRUCTURES
As previously noted, this combination of bonding mechanisms is
found in materials such as solid hydrogen, in which each pair of atoms
is internally covalently bonded and van der Waals bonds create a
"molecular crystal." The same principles apply to most organic solids.
An example of another kind of covalent-residual bonding is provided by tellurium (Figure 1-6), in which successive atoms in each
spiral chain are covalently bonded. The forces between chains are
much weaker and are probably little more than van der Waals attraction. Consequently, tellurium has a low structural strength and is anisotropic in all its mechanical, thermal, and electronic properties.
Similarly, in graphite (Figure 1-7) carbon atoms are arranged in
hexagons in each layer, so that three of the four outer shell electrons
from each atom are used in valence bonds within the layer. (The fourth
electron is free.) The interlayer spacing is large, with essentially only

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1.1 FORMS OF INTERATOMIC BINDING

Figure 1 - 6 The atomic arrangement in tellurium. From Blakemore et al., Progress
in Semiconductors, Vol. 6. (Wiley, 1962). Each atom makes covalent bonds with its nearest
neighbors up and down the spiral chain. Inter-chain forces are weak. One allotropic form of
selenium adopts the same structure.

Figure 1 - 7 Atomic arrangement in the graphite form of carbon. Within a layer, each
atom makes three strong covalent bonds (r0 = 1.42 A) in order to preserve the hexagonal
array. The bonding between layers (spacing of 3.4 A) is weak, so that the layers can slide
over each other with ease.

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CRYSTALLINITY AND THE FORM OF SOLIDS
van der Waals attraction. Thus the planes can slide over each other very
easily, the property which makes graphite useful as a "solid lubricant."
The same considerations apply in MoS2.
THE IONIC BOND
An ionic crystal is made up of positive and negative ions arranged
so that the Coulomb repulsion between ions of the same sign is more
than compensated for by the Coulomb attraction of ions of opposite
sign. The alkali halides such as NaCl are typical members of the class
of ionic solids; NaCl crystallizes (almost) as Na+Cl~. Electron transfer
from Na to Cl occurs to such a major extent because the ionization potential Ie of the alkali metal is small (work el e must be done to convert
Na into the cation Na4" with a closed electronic shell configuration),
whereas the electron affinity E a of the halogen is large. (Energy E a is
provided when Cl receives an electron and becomes the anion Cl~, also
with a closed shell configuration.) Problem 1.2 looks at the energetics
of the ionic bond in a single alkali halide molecule.
When a Na+ ion and a Cl" ion approach each other in the absence
of any other atoms, as envisaged in Figure 1-8, the attractive Coulomb

energy at internuclear separation r relative to zero energy at infinite
separation is
ECoui = -e2/477-e0r

(1-3)

since the (closed-shell) electronic charge distributions are spherically
symmetrical. The approach distance is limited by repulsion when the
closed-shell electron clouds of anion and cation overlap, in consequence of the Pauli principle. The energy associated with repulsion
varies rapidly with separation, as noted in connection with van der
Waals bonding; two approximate ways of describing it are
E rep = A/rn

(n - 12)

(1-4)

or
E rep = Bexp(-r/p)

(1-5)

Figure 1 - 8 A Cl~ anion and Na+ cation
in contact, drawn in the observed ratio of sizes.
In an ionic solid the size ratio plays an important
part in determining the most favorable structure.

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1.1 FORMS OF INTERATOMIC BINDING
neither of which really does justice to the complicated quantummechanical process which constitutes repulsion.
If Equation 1-5 is adopted as at least giving some idea of how the
repulsive energy varies with internuclear separation, it becomes apparent that the stable bond length between Na+ and Cl" will be the quantity for which
i = (Ecoul + E rep ) = -e2/47re0r + B exp(-r/p)

(1-6)

is a minimum. This minimum is shown in Figure 1-9. Because the
repulsive term is much more sensitive to changes in r than is the Coulomb term, the bond energy is only slightly smaller than (e2/47re0r0),
while the restoring forces whenever r departs from r0 are dictated by
the values of B and p.
The principles noted above as being operative for a single Na + Cl"
bond hold equally well10 for solid NaCl, together with some additional
geometric considerations. We shall be talking again about the sodium
chloride structure in Section 1.3 from the viewpoint of geometry and
symmetry [using Figure l-33(a) at that time], but we need to examine
all four parts of Figure 1-10 to appreciate how the particular bonding
arrangement arises. In solid NaCl, each cation (i.e., each sodium ion)
has six anions as its nearest neighbors [and vice versa, as we can see
from Figure l-10(b)] and the interaction with nearest neighbors involves both Coulomb attraction and overlap repulsion. As can be seen
10

In the next subsection, we shall have to note that the electron transfer is not 100
per cent complete even for the most strongly "ionic" compounds, though ionic considerations are certainly the most important ones for the alkali halides.

\
X^-Repulsive energy
\

-2

-4

/ Coulomb attractive
'*~~ energy

-6

i

0

1

2
3
4
Internuclear Spacing
r(A)
Figure 1-9 Energy of a Na+Cl~ molecule compared with that of separate ions, according to Equation 1-6. The characteristic length of the repulsive energy is here assumed
to be p = 0.345 A and the magnitude of repulsion produces a minimum energy at
r0 = 2.82 A.

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CRYSTALLINITY AND THE FORM OF SOLIDS

(

c

(b)

(a)

F i g u r e 1 - 1 0 Four views of the NaCl crystal lattice. Solid circles are used for cations and open circles for anions. (a) The conventional picture, showing the arrangement of
crystallographic planes, (b) An octahedral configuration, in which each ion has six nearest
neighbors, (c) The unit cube with ions drawn to correct sizes so that cation-anion contact
occurs, (d) A section through a cube face, showing that anion-anion and cation-cation contacts do not occur. After R. C. Evans, Introduction to Crystal Chemistry (Cambridge University Press, 1964).

from parts (c) and (d) of Figure 1-10, in which cations and anions are
drawn to proper size, there is no cation-cation contact, nor do the large
anions even manage to touch. Thus the interaction of an ion with anything but its nearest neighbors involves only Coulomb terms. The
overall sum of Coulomb terms (which are both positive and negative)
must more than compensate for the overlap repulsion with the six
nearest neighbors if the solid is to have a positive cohesive energy.
(Possession of a positive cohesive energy means that Ej is negative with
respect to separated ions.) As can be seen from the simple calculation
in Problem 1.2, the energy necessary to separate an ionic solid into separate ions is larger than the amount necessary to separate the solid into
isolated neutral atoms.
The most advantageous crystal structure for an ionic solid depends
on the ratio of anion to cation radii. (Remember that an anion such as

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