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CONCEPTS OF
MODERN PHYSICS
Second Edition
Arthur Beiser
CONCEPTS OF MODERN PHYSICS
McGRAWHILL SERIES IN FUNDAMENTALS OF PHYSICS:
AN UNDERGRADUATE TEXTBOOK PROGRAM
E. U. CONDON, EDITOR, UNIVERSITY OF COLORADO
MEMBERS OF THE ADVISORY BOARD
D. Allan Bromley, Ytilc Oiiirp.sili/
Arthur F. Kip, ! nittmity of California, Berkeley
Hugh D. Young, f?rmwfftf ftftiBrwn t 'niversity
INTRODUCTORY TEXTS
Beiscr ■ Concepts of Modem Physics
Kip • luiHilmmnliils of I'.teitriciti/ and Magnetism
Young ■ Fundamentals of Mechanics and Seal
Young • Fuiuliiuieiiliilfi of Optics tint! Modem I'liysics
UPPERDIVISION TEXTS
Burger :mil ObsOfl . Ctas&oal Mechanics: A Modern Perspective
Beiser . Pnsperiirr.s of Modem I'lit/xin
Cohen ■ CoRetpti of \tolror Physics
Elmore and llculd . Physics of Waves
kraut . frmilitiw nliit.i t>! Mathenuitical Physics
Longn . fundamentals of tllementtiry Particle Physics
Meyerhof  i'.tcments (f Nuclear Physics
Rcif ■ Fundamentals of Statistical ami Thermal Physics
Trail! and Potuilla ■ Atomic Theory: An Introduction to Wave Mec.hnoii.i
CONCEPTS OF MODERN PHYSICS
Second Edition
Arthur Beiser
INTERNATIONAL STUDENT EDITION
McGRAWHILL KOGAKUSHA, LTD.
Tokyo IXisseldorf Johannesburg London Mexico
New Delhi Panama Rio de Janeiro Singapore Sydney
Library of Congress Cataloging in Publication Data
Beiser, Arthur,
Concepts of modern physics.
(McGrawHill series in fundamentals of physics)
I. MatterConstitution, 2. Quantum theory,
1. Title.
QC173.B413 1973 530.1 727089
ISBN 0070043634
CONCEPTS OF MODERN PHYSICS
INTERNATIONAL STUDENT EDITION
Exclusive rights by McGrawHill Kogakusha, Ltd., for manufacture and export. This
book cannot be reexported from the country to which it is consigned by McGrawHill.
Copyright © 1963, 1967, 1973 by McGrawHill, Inc. All rights reserved. No part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher.
PRINTED IN JAPAN
Preface
List of Abbreviations
ix
PART ONE BASIC CONCEPTS
Chapter I Special Relativity 3
1.1 The MichelsonMorlcy Experi
ment 3
1.2 The Special Theory of
Relativity g
1.3 Time Dilation 12
1.4 The Twin Paradox 16
1.5 length Contraction 17
1.6 Meson Decay 20
•1.7 The Lorentz Transformation 22
"1.8 The Inverse Lorentj; Trans
formation 27
•1.9 Velocity Addition 28
I.JO The Relativity of Mass 30
1.11 Mass and Energy 35
1.12 Mass and Energy: Alternative
Derivation 37
Problems 39
Chapter 2 Particle Properties of
Waves 43
2.1 The Photoelectric Effect 43
2.2 The Quantum TTiaory of light 47
2.3 X Rays 51
2.4 XRay Diffraction 56
2.5 The Compton Effect 60
2.6 Pair Production 63
•2.7 Gravitational Red Shift 68
Problems 70
CONTENTS
Chapter 3 Wave Properties of
Particles
3.1 De Broglie Waves
3.2 Wave Function
3.3 De Broglie Wave Velocity
3.4 Phase and Croup Velocities
3.5 The Diffraction of Particles
3.6 The Uncertainty Principle
3.7 Applications of the Uncertainty
Principle
3.8 The Waveparticle Duality
Problems
PART TWO THE ATOM
Chapter 4 Atomic Structure
4.1 Atomic Models
•4.2 Alphaparticle Scattering
•4.3 The Rutherford Scattering
Formula
4.4 Nuclear Dimensions
4.5 Electron Orbits
4.6 Atomic Spectra
4.7 The Bohr Atom
4.8 Energy levels and Spectra
4.9 Nuclear Motion
4.10 Atomic Excitation
4.11 The Correspondence Principle
Problems
Chapter 5 Quantum Mechanics
5.1 Introduction to Qiuui turn
Mechanics
5.2 The Wave Equation
73
73
74
76
79
82
91
93
96
101
101
105
wa
rn
113
in
121
125
129
131
133
135
139
139
140
5.3
1.4
5.5
5.8
Schrodingrr", Equation: Time
dfH'tnlcnt Fortn
Expect sit ion Values
Sehrodinger'i Equation: Steady
statc Form
Tin 1 Particle in » Box; Energy
Quantization
5.7 The Particle in u Bus: Wave
Functions
5.8 The Particle in a Nan rigid
Box
5.0 '['In' 1 1. milium ()s t ill, ilni
•5.10 The Harmonic Oscillator: Solu
tion of Schrddioger's Equation
Problems
Chapter 6 Quantum Theory of the
Hydrogen Atom
6.1 Schrodinger's Equation for the
Hydrogen Atom
°(j.2 Separation of Variables
"6.3 Quantum Numbers
6.4 Principal Quantum Number
6.5 Orbital Quantum Number
I'd Magnetic Quantuiii Number
6.7 1 !n Normal Zeeman Effect
fl.8 Electron Probability Density
6.9 Radiative Transitions
6. t() Selection Rules
Problems
Chapter 7 Manyelectron Atoms
7.1 Electron Spin
7.2 Spin orbit Coupling
7.3 The Exclusion Principle
7.1 El eel n )i i ( '. in i lii> 1 1 rat i i >i is
7.5 The Periodic Table
7.6 Hund's Rule
°7.7 Total Angular Momentum
•7.8 1.S Coupling
•7.9 ft Coupling
*7.10 Oneelectron Spectra
*7.1 1 Twoelectron Spectra
7.12 Xray Spectra
Problems
143
145
146
140
153
156
158
163
169
173
173
176
178
\m
ISO
184
187
169
im
198
200
203
203
207
210
213
215
222
222
226
228
229
232
235
237
8.2
•ft3
"9.4
PART THREE
PROPERTIES OF MATTER
Chapter 8 The Physics of Mole
cules £43
S. I Molecular Fun nation 243
8.2 Fleet ron Sharing 245
8.3 The 6V Molecular Ion 247
8.4 The llj Molecule 252
8.5 Molecular Orliitals 254
Mi Hybrid Orliitals 261
8.7 Carboncarbon Bonds 265
8.8 Rotational Energy Levels 269
8.9 Vibrational Energy Levels 272
8.10 Electronic Spectra of Mole
cules £81
Chapter 9 Statistical Mechanics 287
9,1 Statistical DfetrflnitfOQ Uws 287
Phase Space 288
M ax wellBol t nim n I >isl ri 1 1 1 1 
lion 289
Evaluation of Constant^ 293
9.5 Molecular Energies in an Ideal
Gas 295
9.6 Rotutional Spectra 298
"9.7 RoseEinstein Distribution 300
9.8 Rluckhody RadUlimi 3tM
•9.9 FcnniDirac Distribution M)
9.10 Comparison of Results 310
9.11 The Laser 311
Problems 314
< ' I i.i 1 1 1. 1 10 The Solid State 317
10.1 Crystalline and Amorphous
Solids 317
10.2 Ionic Crystals 318
10.3 Covalent Crystals 325
111.4 Van Dcr Waals Forces 327
10.5 The Metallic Braid 331
10.6 The Rand Theory of Solids 333
•10.7 The Fermi Bong 1 339
•10.8 Electronenergy Distribution 342
•10.9 Rrtllouin Zones 344
"10.11) Origin y( Forhiddcn Bands 346
10.11 Effect ive Mass 355
Problems 355
PART FOUR THE NUCLEUS
Chapter 1 1 The Atomic Nucleus 361
1 1.1 Atomic Masses 361
11.2 The Neutron 384
11.3 Stable Nuclei 386
11.4 Nuclear Sizes and Shapes 370
11.5 Binding Energy 372
"11.8 The Dcuteron 374
•1 1.7 Ground Stale of the Dculcrun 377
11.8 Triplet and Singlet States 379
1 1.8 The Liquiddrop Model 380
11.10 The Shell Model 383
Prohlems 397
Chapter 12 Nuclear Transformations 389
12.1 Radioactive Decay 389
12.2 Radioactive Scries 393
12.3 Alpha Decay 396
"12.4 Barrier Penetration 399
*12.5 Theory of Alpha Decay 404
12.8 Reta Decay 408
12.7 Inverse Reta Decay 411
12.8 Camma Decay 412
12.9 Cross Section
12.10 The Compound Nucleus
12.1 1 \lk [ear Fivsion
12.12 Trausuranic Elements
12.13 Thermonuclear Energy
Prohlems
Chapter 13 Elementary Particles
13.1 Antiparticlcs
13.2 Meson Theory of Nuclear
Forces
13.3 Piraiv and Muons
13.4 Kaons and Ilyperons
13.5 Systematic* of Elementary
Particles
13.6 Strangeness Number
13.7 Isotoptc Spin
13.8 Syn in let lies 10 id Conservation
Principles
13.9 Theories of Elementary Particles
Problems
413
■117
420
423
424
127
431
431
433
436
438
439
+14
448
447
•151
Answers to Oddnumliercd Problems 457
481
VI
CONTENTS
CONTENTS
VII
PREFACE
This book is intended for use with onesemester courses in modern physics
that have elementary classical physics and calculus as prerequisites. Kelativity
and quantum theory are considered first to provide a framework for under
standing the physics of atoms and nuclei. The theory of the atom is then de
veloped with emphasis on quantummechanical notions, and is followed by a
discussion of the properties of aggregates of atoms. Finally atomic nuclei and
elementary particles are examined.
The balance here deliberately leans more toward ideas than toward experi
mental methods and practical applicatioas. because I believe that the beginning
student is better served in his introduction to modern physics by a conceptual
framework than by a mass of individual details. However, all physical theories
live or die by the sword of experiment, and a nmnlier of extended derivations
are included in order lo demonstrate exactly how an abstract concept can lie
related to actual measurements. Many instructors will prefer not to hold their
students responsible for the more complicated (though not necessarily mathe
matically difficult) discussions, and I have indicated with asterisks sections
that can be passed over lightly without loss of continuity; problems based
on the contents of these sections are also marked with asterisks. Other omis
sions are also passible, of course. Relativity, For example, may well have teen
covered earlier, and Part 3 in its entirety may be skipped when its contents
will lie the subject of later work. Thus there is scope for an instructor to fash
ion the type of course he wishes, whether a general survey or a deeper inquiry
into selected subjects, and to choose the level of treatment appropriate to his
audience.
An expanded version of this book requiring no higher degree of mathematical
preparation is my Perspectives of Modern Physics, an Upper Division Text in
this series; other Upper Division Texts carry forward specific aspects of modem
physics in detail.
In preparing this edition of Concepts of Modern Physics much of the original
text has been reorganized and rewritten, the coverage of a number of topics
has been broadened, and some material of peripheral interest has been dropped.
I am grateful to Y. Beers and T. Satoh for their helpful suggestions j n this
regard.
Arthur Beiser
LIST OF ABBREVIATIONS
k
angstrom
amp
ampere
atm
atmosphere
b
barn
C
coulomb
Ci
cmie
d
day
eV
electron volt
F
farad
fm
fermi
g
gram
h
hour
Hz
hertz
]
joule
K
degree Kelvin
m
meter
mi
mile
miri
minute
mol
mole
N
newton
s
seeond
T
tesla
u
atomic mass unit
V
volt
W
watt
yr
year
PREFACE
xi
SPECIAL RELATIVITY
i
Our study of modern physics begins with a consideration of the special theory
of relativity. This is a logical starting point, since all physics is ultimately
concerned with measmeinent and relativity involves an analysis of how meas
urements depend upon the observer as weD as upon what is observed. From
relativity emerges a new mechanics in which there are intimate relationships
Iretween space and time, mass and energy. Without these relationships it would
be impossible to understand the microscopic world within the atom whose
elucidation is the central problem of modem physics.
LI THE fJIICHELSONMORLEY EXPERIMENT
The wave theory of light was devised and perfected several decades before the
electromagnetic nature of the waves became known. It was reasonable for the
pioneers in optics to regard light waves as undulations in an allpervading elastic
medium called the ether, and their successful explanation of diffraction and
interference phenomena in terms Gf ether waves made the notion of the ether
so familial that its existence was accepted without question. Maxwell's develop
ment of the electromagnetic theory of light in 1864 and Hertz's experimental
confirmation of it in 1 887 deprived the ether of most of its properties, but nobody
at the lime seemed willing to discard the hmdamentai idea represented by the
ether: thai light propagates relative to some sort of universal frame of reference.
Let us consider an example of what this idea implies with die help of a simple
analogy.
Figure 11 is a sketch of a river of width D which Hows with the speed v.
Two boats start out from one bank of the river with the same speed V, Boal A
crosses the river to a point on the other bank directly opposite the starting point
Bud then returns, white boal B heads downstream for the distance D and then
returns to the starling point. Let us calculate the time required for each round
trip.
=3
land
river
4
r
B
CD
CD
FIGURE 11 Boat A goes directly across the river and returns to its starting point, while boat B heads
downstream for an identical distance and then returns.
We begin by considering Ixmt A. If A heads perpendicularly across Lhe river,
the current will carry it downstream from its goal on the opposite bank
(Fig. 12). It must therefore head somewhat upstream in order to compensate
for the current. In order to accomplish this, its upstream component of velocity
should be exactly — v in order to cancel out the river current v, leaving the
FIGURE 12 Boat A must head upstream in order to compensate for the river current.
Jl
% —
4
BASIC CONCEPTS
component V as its net speed across the river. From Fig. 12 we see that these
speeds are related by the formula
V 2 = V' 2 + v 2
so that the actual speed with which boat A crosses the river is
V'= VV 2  v 2
Hence the time for the initial crossing is Lhe distance D divided by the
speed V, Since the reverse crossing involves exactly die same amount of time,
the total roundtrip time t A is twice D/V\ or
l.i
2.D/V
\/l  v 2 /V 2
The case of boat B is somewhat different. As it heads downstream, its speed
relative to the shore is its own speed V plus the speed t; of the river (Fig. 13),
and it travels the distance D downstream in the time D/(V + c). On its return
trip, however, B's speed relative to the shoTe is its own speed V minus the
speed u of the river. It therefore requires the longer time D/{V — c) to travel
upstream the distance D to its starting point. The total roundtrip time t H is
the sum of these times, namely,
.« =
D D
V + v Vv
Using the common denominator (V + l)(V — v) for both terms,
_ D(V ~ v) + D(V + v)
' B ~ (V + v)(V  o)
2DV
1.2
V* v"
•2D/V
1 _ oVV a
which is greater than f^, the corresponding roundtrip time for the other boat.
The ratio between the times f., and t B is
1.3
= VT  i;7 V 2
If we know the common speed Vof the two boats and measure the ratio t A /t 8 ,
we can determine the speed v of the river.
L
SPECIAL RELATIVITY
V . B
V
— i
B X v
B v v
FIGURE 13 The speed of boat H downstream relative to the shore is increased by the speed of the
river current while its speed upstream is reduced by the same amount.
The reasoning used in this problem may be transferred to the analogous
problem of the passage of light waves through the ether. If there is an ether
pervading space, we move through it with at least the 3 x 0* m/s (18.5 mi/s)
speed of the earth's orbital motion about the sun; if the sun is also in motion,
our speed through the ether is even greater (Fig. 14). From the point of view
of an observer on the earth, the ether is moving past the earth. To detect this
motion, we can use the pair of light lieams formed by a halfsilvered mirror
instead of a pair of boats (Fig. 15). One of these light beams is directed to a
FIGURE 14 Motions of the earth through a hypothetical ether
mirror A
parallel light
from
single source
mirror B
half silvered mirror
FIGURE 15 The MichetsonMortey experiment.
hypothetical
ether current
viewing screen
BASIC CONCEPTS
mirror along a path perpendicular to the ether current, while the other goes
to a mirror along a path parallel to the ether current. The optical arrangement
is such that both beams return to the same viewing screen. The purpose of the
clear glass plate is to ensure that both beams pass through the same thicknesses
of air and glass.
If the path lengths of the two beams are exactly the same, they will arrive
at the screen hi phase and will interfere constructively to yield a bright Held
of view. The presence of an ether current in the direction shown, however, would
cause the beams to have different transit limes in going from the halfsilvered
mirror to the screen, so that they would no longer arrive at the screen in phase
but would interfere destructively. In essence this is the famous experiment
performed in 1887 by the American physicists Miehelson and Morley.
In the actual experiment the two mirrors are not perfectly perpendicular, with
the result that the viewing screen appears crossed with a series of bright and
dark interference fringes due to differences in path length between adjacent light
waves (Fig. 16). If either of the optical paths in the apparatus is varied in length,
the fringes appear to move across the screen as reinforcement and cancellation
of the waves succeed one another at each point. The stationary apparatus, then,
ran tell us nothing about any time difference between die two paths. When
die apparatus is rotated by 90", however, the two padis change their orientations
relative to the hypothetical ether stream, so that the beam formerly requiring
the time t A for the round trip now requires t B and vice versa. If these times
are different, the fringes will move across the screen during the rotation.
SPECIAL RELATIVITY
FIGIWE 16 Fringe pat
tern observed in Michel
sonMorley experiment.
Let us calculate the fringe shift expected on the basis of the ether theory.
From Eqs. 1.1 and 1.2 the time difference between the two paths owing to the
ether drift is
M = t B  t A
2D/V 2D/V
1  v 2 /V 2 y/1  v 2 /V 2
Here t> is the ether speed, which we shall take as the earth's orbital speed of
3 X 10 4 rn/s, and V is the speed of light c, where c = 3x lf* w m/s. Hence
V 2
io fi
which is much smaller than 1. According to the binomial theorem, when x is
extremely small compared with 1,
(l±af « 1 ± nx
We may therefore express Af to a good approximation as
Here D is the distance between the halfsilvered mirror and each of the other
mirrors. The path difference d corresponding to a time difference At is
el = cA(.
BASIC CONCEPTS
If d corresponds to the shifting of a fringes,
d — nX
where X is the wavelength of the light used. Equating these two formulas for
d t we find that
cAt
_ Dv 2
Ac 2
In the actual experiment Michelson and Morley were able to make D about 10 m
in effective length through the use of multiple reflections, and the wavelength
of the light they used was about 5,000 A (1 A = 10~ 10 m). The expected fringe
shift in each path when the apparatus is rotated by 90° is therefore
10 m X (3 X L0 J m/s} 2
"5x Mr T m X (3 X 10 s m/s) 2
= 0.2 fringe
Since both paths experience this fringe shift, the total shift should amount to
2n or 0.4 fringe. A shift of this magnitude is readily observable, and therefore
Michelson and Morley looked forward to establishing directly the existence of
the ether.
To everybody's surprise, no fringe shift whatever was found. When the exper
iment was performed at different seasons of the year and in different locations,
and when experiments of other kinds were tried for (he same purpose, the
conshisions were always identical: no motion through the ether was detected.
The negative result of the MichelsonMorley experiment had two conse
quences. First, it rendered untenable the hypothesis of the ether by demon
strating that the ether has no measurable properties — an ignominious end for
what had once been a respected idea. Second, it suggested a new physical
principle: the speed of light in free space is the same everywhere, regardless
of any motion of source or ohserver.
1.2 THE SPECIAL THEORY OF RELATIVITY
We mentioned earlier the role of the ether as a universal frame of reference
with respect to which light waves were supposed to propagate. Whenever we
speak of "motion," of course, we really mean "motion relative to a frame of
SPECIAL RELATIVITY
10
reference." The frame of reference may be a road, the earth's surface, the sun,
the center of our galaxy; but in every case we must specify it. Stones dropped
in Bermuda and in Perth, Australia, both fall "down," and yet the two move
in exactly opposite directions relative to the earth's center. Which is the correct
location of the frame of reference in this situation, the earth's surface or its
center?
The answer is that all frames of reference are equally correct, aldiough one
may be more convenient to use in a specific case. // there were an ether
pervading all space, we could refer all motion to it, and the inhabitants of
Bermuda and Perth would escape from their quandary. The absence of an ether,
then, implies that there is no universal frame of reference, since light (or, in
general, electromagnetic waves) is the only means whereby information can Ik;
transmitted through empty space. All motion exists solely relative to the person
or instrument observing it. If we are in a free halloon above a uniform cloud
bank and see another free balloon change its position relative to us, we have
no way of knowing which balloon is "really" moving. Should we be isolated
in the universe, there would be no way in which we could determine whether
we were in motion or not, because without a frame of reference the concept
of motion has no meaning.
The theory of relativity resulted from an analysis of the physical coii.se<nenees
implied by the absence oi a universal frame of reference. The special theurv
of relativity, develox:d by Albert Einstein in 1905, treats problems involving
inertial frames of reference, which are frames of reference moving at constant
velocity with respect to one another. The general Uieory of relativity, proposed
by Einstein a decade later, treats problems involving frames of reference accel
erated with respect to one another. An observer in an isolated laboratory can
detect accelerations. Anybody who has been in an elevator or on a merrygo
round can verify this statement from his own experience. The special theory
lias had a profound influence on all physics, and we shall concentrate on it with
only a brief glance at the general theorv.
The special theory of relativity is based upon two postulates. The first states
that the laws of physics may be expressed in equations having the same form
in all frames of reference moving at constant velocity with respect to one
another. This postulate expresses the ahsence of a universal frame of reference,
if the laws of physics had different forms for different observers in relative
motion, it could be determined from these differences which objects are "sta
tionary" in space and which are "moving." But because there is no universal
frame of reference, this distinction does not exist in nature; hence the above
postulate.
The second postulate of special relativity states that the speed of light in free
space has the same value for all observers, regardless of their state of motion.
BASIC CONCEPTS
This postulate follows directly from the results of the MfcbelsonMorley experi
ment and many others.
At first sight these postulates hardly seem radical. Actually they subvert almost
all the intuitive concepts of lime and space we form on the basis of our daily
experience. A simple example will illustrate this statement. In Fig. 17 we have
the two boats A and B once more, with boat A at rest in the water while boat B
drifts at the constant velocity v. There is a lowlying fog present, and so on
neither boat does the observer have any idea which is the moving one. At the
instant that H is abreast of A, a flare is fired. The light from the flare travels
uniformly in all directions, according to the second postulate of special relativity.
An observer on either boat must find a sphere of light expanding with hlmsefy
at its center, according to the first postulate of special relativity, even though
one of them is changing his position with respect to the point where the flare
went off. The observers cannot detect which of them is undergoing such a change
in position since the fog eliminates any frame of reference other than each boat
itself, and so, since the speed of light is the same for both of (hem, they must
hoth see the identical phenomenon.
Why is the situation of Fig. 17 unusual? Let us consider a more familiar
analog. The boats are at sea on a clear day and somel>ody on one of them drops
a stone into the water when they arc abreast of each other. A circular pattern
FIGURE 17 Relativist* phenomena differ from everyday experience.
<jDJ
B
»i<3D>
A
each person sees sphere
of light expand h)* about
himself
A
A
<3D
light emitted by flare
"CD
li
A
<3Dj
<3CP
B
each person sees pattern
in different place relative
to himself
pattern of ripples from
stone dropped in water
SPECIAL RELATIVITY
11
12
of ripples spreads out, as at the bottom of Fig, ] 7, which appears different to
observers on each boat. Merely by observing whether or not he is at the center
of tiie pattern of ripples, each observer can tell whether he is moving relative
to the water or not. Water is in itself a frame of reference, and an oliserver
on a boat moving through it measures ripple speeds with respect to himself thai
are different in different directions, in contrast to the uniform ripple speed
measured by an observer on a stationary boat. It is important to recognize that
motion and waves in water are entirely different from motion and waves in space;
water is in itself a frame of reference while space is not, and wave speeds in
water vary with the observer's motion while wave speeds of light in space do
not
The only way of interpreting the fact that observers in the two boats in our
example perceive identical expanding spheres of light is to regard the coordinate
system of each observer, from the point of view of die other, as being affected
by their relative motion. When this idea is developed, using only accepted laws
of physics and Einstein's postulates, we shall see that many peculiar effects are
predicted. One of the triumphs of modem physics is the experimental confirma
tion of these effects.
1.3 TIME DILATION
We shall first use the postulates of special relativity to investigate how relative
motion affects measurements of time intervals and lengths.
A clock moving with respect to an observer appears to tick less rapidly than
it does when at rest with respect to him. That is, if someone in a spacecraft
finds that the time interval between two events in the spacecraft is („, wc on
the ground would find that the same interval has the longer duration f. The
quantity r,„ which is determined by events dial occur at the soma place in an
observer's frame of reference, is called the proper time of the interval between
the events. When witnessed from the ground, the events Uut mark the beginning
and end of the time interval occur at different places, and in consequence the
duration of the interval appears longer than the proper time. This effect is called
time dilation.
To see how rime dilation comes about, let us examine the operation of the
particularly simple clock shown in Fig. IS and inquire how relative motion
affects what we measure. This clock consists of a stick / J(1 long with a mirror
at each end. A pulse of light is reflected up and down between the mirrors,
and an appropriate device is attached to one of the mirrors to give a "tick"
of some kind each time the pulse of light strikes it. (Such a device might he
BASIC CONCEPTS
recording device
light pulse
A
photosensitive
surface
FIGURE lS A simple clock. Each "tick" corresponds to a round trip of the light
pulse from the lower mirror to the upper out and back.
a photosensitive surface on the mirror which can be arranged to give an electric
signal when the light pulse arrives,) The proper time t lt between ticks is
1.4
2L
If the stick is 1 m long,
2 m
3 X 10 s m/s
= 0.67 X 10" H s
and there are 1.5 X 10 s ticks/s. Two identical clocks of this kind are built, and
one is attached to a spaceship mounted perpendicular to the direction of morion
while the other remains at rest on the earth's surface.
Now we ask how much time t elapses between ticks in the moving clock as
measured by an observer on the ground with an identical clock that is stationary
with respect to him. Each tick involves the passage of a pulse of light at speed
SPECIAL RELATIVITY
13
14
c from the lower mirror to the upper one mid hack. During this roundtrip
passage the entire clock in the spaceship is in motion, which means that the
pulse of light, as seen from the ground, actually follows a zigzag path (Fig, 19).
On its way from the lower mirror to the upper one in the time r/2, the pulse
of light travels a horizontal distance of c(/2 and a total distance of ct/2. Since
L,, is the vertical distance between the mirrors.
(D^v + (ir
4( C *  w) = v
and
1.5
t =
t 2 =
2/..A
W
4V
c 2  v 2 c 2 {l  v l /c)
VI  v 2 /c 2
But 2Lq/c is the time interval * n l>etween ticks on the clock on the ground, as
in Eq. 1.4, and so
1.6
t =
Vl  bVc 3
Time dilation
FIGURE 19 A light clock In a spacecraft es seen by an observer at rest on the ground. The mirrors are
parallel to the direction of motion of the spacecraft.
<^>
BASIC CONCEPTS
The moving clock in the spaceship appears to tick at a slower rate than the
stationary one on the ground, as seen by an observer on the ground.
Ebtaetfy the same analysis holds for measurements of the clock on the ground
by the pilot of the spaceship. To him, the light pulse of the ground clock follows
a zigzag path which requires a total time t per round trip, while his own clock,
at rest in the spaceship, ticks at intervals of („. He too finds that
Vi  f
so the effect is reciprocal: every observer finds that clocks in motion relative
to him tick more slowly than when they are at rest.
Our discussion has been based on a somewhat unusual clock that employs a
light pulse bouncing back and forth between two mirrors. Do the same con
clusions apply to more conventional clocks that use machinery— springcontrolled
escapements, timing forks, or whatever— to produce ticks at constant time
intervals? The answer must be yes, since if a mirror clock and a conventional
clock in the spaceship agree with each other on the ground hut not when in
flight, the disagreement between them could lie used to determine the speed
of the spaceship without reference to any other object— which contradicts the
principle that all motion is relative. Detailed calculations of what happens to
conventional clocks in motion— as seen from the ground— confirm this answer.
For example, as we shall learn in Sec. 1.10, the mass of an object is greater when
it is in motion, so that the period of an oscillating object must be greater in the
moving spaceship. Therefore till clocks at rest relative to one another behave
the same to all observers, regardless of any motion at constant velocity of either
the group of clocks or the observers.
The relative character of lime has many implications. For example, events
that seem to take place sitnullaneously to one observer may not be simultaneous
to another observer in relative motion, and vice versa. Who is right? The
question is, of course, meaningless: both observers are "right," since each simply
measures what he sees.
Because simultaneity is a relative notion and not an absolute one, physical
theories which require simultaneity in events at different locations must be
discarded. The principle of conservation of energy in its elementary form states
that the total energy contenl of the universe is constant, but it does not rule
out a process in which a certain amount of energy AE vanishes at one point
while an equal amount of energy AE spontaneously conies into being somewhere
else with no actual transport of energy from one place to the other. Because
simultaneity is relative, some observers of the process will find energy not being
conserved To rescue coaservation of energy in the light of special relativity,
then, it is necessary to say that, when energy disappears somewhere and appears
SPECIAL RELATIVITY
15
16
elsewhere, it has actually flotced from the first location to the second (There
are many ways in which a flow of energy can occur, of course.) Thus energy
is conserved locally in any arbitrary region of space at any time, not merely
when the universe as a whole is coasidered— a much stronger statement of this
principle.
Although time is a relative quantity, not ah" the notions of time formed by
everyday experience are incorrect. Time does not run backward to any olxserver,
for instance: a sequence of events that occur somewhere at (,, t 2 , t 3 , . . . will
appear in the same order to all observers everywhere, though not necessarily
with the same time intervals U  <„ t 3  t 2 , . . . between each pair of events.
Similarly, no distant olwerver, regardless of his state of motion, can see an event
before it happens— more precisely, before a nearby observer sees it— since the
speed of light is finite and signals reqnire the minimum period of time L/c to
travel a distance L. There is no way to peer into the future, although temporal
(and, as we shall see, spatial) perspectives of past events may appear different
to different observers.
1.4 THE TWIN PARADOX
We are now in a position to .understand the famous relativist* phenomenon
known as the twin paradox. This paradox involves two identical clocks, one of
which remains on the earth while the other goes on a voyage into space at the
speed v and returns a time t later. It is customary to replace actual clocks with
a pair of identical male twins named A and B; this substitution is perfectly
acceptable, because the processes of lifeheartbeats, respiration, and so on
eonstitute biological clocks of reasonable regularity.
Twin A takes off when he is 20 yr old and travels at a speed of 0.99c. To
his brother B on the earth, A seems to 1% living more slowly, in fact at a rate
only
VI  cVc 2 = Vl  (0.99c)Vc 2 = 0.14 = 14 percent
as fast as B goes. For every breath that A takes, B takes 7; for every meal that
A eats, B eats 7; for every tiiought that A thinks, B thinks 7. Finally, after 70
yr have elapsed by B's reckoning, A returns home, a man of only 30 while B
is then 90 yr old.
Where is the paradox? If we examine the situation from the point of view
of twin A in the spaceship, B on the earth is in motion at 0.99c. Therefore we
might expect B to be 30 yr old upon the return of the spaceship while A is
90 at this time— the precise opposite of what was concluded in the preceding
paragraph.
BASIC CONCEPTS
The resolution of the paradox depends upon the fact that the spaceship is
accelerated at various limes in its journey: when it takes off, when it turns around,
and when it finally comes to a stop. During each of these accelerations A was
not in an inertial frame of reference, and the inertia! frames corresponding to
the outward and return trips were different. The earthbound twin B, on the
other hand, was not accelerated and stayed in the same inertial frame all the
time. What B measured may therefore lie interpreted on the basis of special
relativity, and his conclusion — that A is younger when he comes back — is correct.
Of course, A's lifespan has not been extended to A hiimclf, since however long
his 10 yr on the spacecraft may have seemed to his brother B, it has been only
10 yr as far as he is concerned. What has happened is that A's accelerations
affected his life processes, and by applying the conclusions of general relativity
for accelerated clocks we find that A is younger than B on his return by the
exact amount expected on the basis of fl's analysis using the formula for time
dilation.
1.5 LENGTH CONTRACTION
Measurements of lengths as well as of time intervals are affected by relative
motion. The length L of an object in motion with respect to an observer always
ap>ears to the observer to l>e shorter than its length L„ when it is at rest with
respect to him, a phenomenon known as the LorentzVitzCeraUl contraction. This
contraction occurs only in the direction of the relative motion. The length L,,
of an object in its rest frame is called its projjer length.
We can use the light clock of the previous section to investigate the Lorentz
contraction. For this purpose we imagine the clock oriented so that the light
pulse travels back and forth parallel to the direction in which the clock is moving
relative to the observer (Fig. 110). At ( = the light pulse starts from the rear
mirror, and it arrives at the front mirror at /,. The pulse has traveled the distance
rf, to reach the front mirror, where from the diagram
Hence
1.7
C*, = L + Ulj
where L is the distance l>etween the mirrors as measured by the observer at
rest.
The pulse is then reflected by the front mirror and returns to the rear mirror
at t after traveling the distance c(<  (,), where
c(t  f,) = L  v(t  r,)
SPECIAL RELATIVITY
17
rear
mirror
(hint
u
^>
~<*H
L + t>f,
/w
I— I.  DitIJ— *
FIGURE 110 A light clock In a spacecraft as seen by an observe* on the ground.
The mirrors are perpendicular to the direction of motion of the spacecraft.
Hence the entire lime interval (, as determined from the ground, is
1.8
( =
C + V
+ 'l
We eliminate /, with the help of Eq. 1.7 to find that
19
/ =
c + v c
L
— V
2Lc
(c + v)(c 
■V)
=
2Lc
C*  v z
2L/c
I  tVc*
Equation 1.9 gives the time interval t between ticb of the moving clock as
measured by an observer on the ground
We earlier fonnd another expression for t,
18
BASIC CONCEPTS
(IS)
t =
21^,/c
VI  oVe*
which is in terms of Lq, the proper distance l>etwecn the mirrors, instead of in
terms of /, the distance as measured by an observer in relative motion. The
two formulas must he equivalent, and hence we liave
ILJc
2L/c
1  cVc ~ VI  v 2 /c 2
1.10
L = 1^ Vl  v?/c 2
Lorentz contraction
Because the relative velocity appears only as o 8 in Eq. 1.10, the Lorentz
contraction is a reciprocal effect. To a man in a spacecraft, objects on the earth
appear shorter than tbey did when he was on the groimd by the same factor,
Vl — v s /c*, that the spacecraft appears shorter to somebody at rest. The proper
length of an object is the maximum length any observer will find.
The relativistk' length contraction is negligible for ordinary speeds, but it is
an important effect at speeds close to the speed of light. A speed of 1,000 mi/s
seems enormous to us. and yet it results in a shortening in the direction of motion
to only
/
= ./!
(],(HK)mi/s) 2
(186,000 mi/s) 2
= 0.9«W985
= 99,9985 percent
of the length at rest. On the other band, a body traveling at 0.9 the speed of
light is shortened to
r = A
(0.9tf
= 0.436
= 43.6 percent
"I I lie length at rest, a significant change.
The ratio Iwtwccu / and L„ in Eq. 1.10 is the same as that in Eq, 1.3 when
il is applied to the times of travel of the two light beams, so that we might
be kmpted to consider the MiehelsonMorley result solely as evidence for the
contraction of the length of their apparatus in the direction of the earth's motion.
SPECIAL RELATIVITY
19
20
This interpretation was tested by Kennedy and Thorndike in a similar experiment
using an interferometer with arms of unequal length. They also found no fringe
shift, which means that these experiments must lie considered evidence for the
absence of an ether with all this implies and not only for contractions of the
apparatus.
An actual photograph of an object in very rapid relative motion would reveal
a somewhat different distortion, depending upon the direction from which the
object is viewed and the ratio o/e. The reason for this effect is that light reaching
the camera (or eye, for that matter) from the more distant parts of the object
was emitted earlier than that corning from the nearer parts; the camera "sees"
a picture that is actually a composite, since the object was at different locations
when the various elements of the single image that reaches the film left it. This
effect supplements the Lorentz contraction by extending the apparent length
of a moving object in the direction of motion. As a result, a threedimensional
body, such as a cube, may be seen as rotated in orientation as well as changed
in shape, again depending upon the position of the observer and the value of
e/c. This result must be distinguished from the Lorentz contraction itself, which
is a physical phenomenon. If there were no Lorentz contraction, the appearance
of a moving body would lie also different from what it is at rest, but in another
way.
It is interesting to note that the above approach to the visual appearance of
rapidly moving objects was not made until 1959, 54 years after the publication
of the special theory of relativity.
1.6 MESON DECAY
A striking illustration of both time dilation and length contraction occurs in the
decay of unstable particles called a mesons, whose properties we shall discuss
in greater detail later. For the moment our interest lies in the fact that a ft
meson decays into an electron an average of 2 X 10 8 s after it comes into being.
Now n mesons are created high in the atmosphere by fast cosmicray particles
arriving at the earth from space, and reach sea level in profusion. Such mesons
have a typical speed of 2.994 X 10 s m/s, which is 0.998 of the velocity of tight
c. But in t n , = 2 X 10 8 s, the meson's mean lifetime, they can travel a distance
of only
y = ««o
= 2.994 x 10" m/s x 2 X I0" 6 s
= 600 in
while they are actually created at altitudes more than JO times greater than this.
BASIC CONCEPTS
We can resolve the meson paradox by using the results of the special theory
of relativity. I<et us examine the problem from the frame of reference of the
meson, in which its lifetime is 2 x 10~ fi s. In this case the distance from the
meson to the ground appears shortened by the factor
£« VI  v s /c*
That is, while we, on the ground, measure the altitude at which the meson is
produced as y n , the meson "sees" it as y. If we let y be 600 m, the maximum
distance the meson can go in its own frame of reference at the speed 0.998c
before decaying, we find that the corresponding distance y„ in our reference
frame is
in
\/T=
cVe"
600
f
(0.998c) 2
c*
600
\/T
 0.996
Will
0.063
m
a 9,500
m
Hence, despite their brief lifespans, it is possible for the mesons to reach the
ground from the considerable altitudes at which they are actually formed.
Now let OS examine the problem from the frame of reference of an observer
on the ground. From the ground the altitude at which the meson is produced
is (/,„ but its lifetime in our reference frame has l>een extended, owing to the
relative motion, to the value
/ 
\/l  Dt/c*
2X 10 8
1 
(0.998c) 2
2 X IP" 6
0.063 S
= 31.7X 10* s
SPECIAL RELATIVITY
21
almost 16 times greater than when it is at rest with respect to as. In 31.7 X I0" 6 s
a meson whose speed is 0.998e can travel a distance
Vn = i"
= 2.994 X 10 H m/s X 31.7 X 10" 6 s
= 9,500 m
the same distance obtained before. The two points of view give identical results.
*1.7 THE IORENTZ TRANSFORMATION
Let us suppose that we are in a frame of reference S and find that the coordinates
of some event that occurs at the time t are ,t\ y, z. An observer located in a
different frame of reference S' which is moving with respect to S at the constant
velocity o will find that the same event occurs at the time f and has the coordi
nates x\ y', z'. (In order to simplify our work, we shall assume that v is in the
+x direction, as in Fig. 111.) How are the measurements *, y, z, t related to
x', f. s'. W
If we are unaware of special relativity, the answer seems obvious enough.
If time in both systems is measured from the instant when the origins of S and
i
FIGURE I 11 Frame v moves In the i r direc
tion with the speed a relative to frame S.
V
n
»'
22
BASIC CONCEPTS
S ' coincided, measurements in the x direction made in S will exceed those made
iu S' bv the amount vt, which represents the distance that S' has moved in the
x direction. That is
1.11 x' = X — vt
There is no relative motion in the ;/ and z directions, and so
1.12 if = tj
1.13 z' = z
In the absence of any indication to the contrary in our everyday experience,
we further assume that
1,14
(' = *
The set of Eqs. 1.11 to 1.14 is known as the Galilean transformation.
To convert velocity components measured in the S frame to their equivalents
in the S' frame according to the Galilean transformation, we simply differentiate
.r' : ij\ and z' with respect to time:
i. is
1.16
1.17
o; =
dt'
dy'
dt'
dz'
dt'
»». — 6
= t>„
While the Galilean transformation and the velocity transformation il leads to
are both in accord with our intuitive expectations, they violate Irath of the
postulates of special relativity. The first postulate calls for identical equations
of physics in both the S and S' frames of reference, but the fundamental equations
of electricity and magnetism assume very different forms when the Galilean
transformation is used to convert quantities measured in one frame into their
equivalents in the other. The second postulate calls for the same value of the
speed of light e whether determined in S or S'. If we measure the speed of
light in the x direction in the S system to l>e a, however, in the S' system it
will be
c' = c — v
according to Eq. 1.15. Clearly a different transformation is required if the
postulates of special relativity are to be satisfied. We would expect Iwth time
dilation and length contraction to follow naturally from this new transformation.
SPECIAL RELATIVITY
23
24
A reasonable guess as to the nature of the correct relationship between x and
x' is
1.18
x' = k(x  vt)
wham k is a factor of proportionality that does not depend upon either x or
t but may be a function of o. The choice of Eq. [18 follows from several
considerations:
1 It is linear in ,v and .v', so that a single event in frame S corresponds to
a single event in frame S', as it must.
2. It is simple, and a simple solution to a problem should always be explored
first. '
3. U has the possibility of reducing to Eq. 1. 11, which we know to be correct
in ordinary mechanics.
Because the equations of physics must have the same form in both S and S',
we need only change the sign of o (in order to take into account the difference
in the direction of relative motionl to write the corresponding equation for x
in terms of .v' and /':
1.19 x = k(x' + vt')
ITie factor k must be the same in both frames of reference since there is n u
difference between S and ,S" other than in the sign of i .
As in the case of the Galilean transformation, there i.s nothing to indicate that
there might be differences Ixstween the corresponding coordinates y, tf and z,
z' which are normal to the direction of r. Hence we again take
1.20
1.21
>j' = y
The time coordinates t and f, however, are not equal. We can see this by
sulMtituting the value of x' given by Eq. LIS into Eq. 1.19. We obtain
x = k 2 (x  vt) + kvt'
from which we find that
1.22
<' = " + (^)>
Equations 1.18 and 1.20 to 1.22 constitute a coordinate transformation that
satisfies the first postulate of special relativity.
BASIC CONCEPTS
The second postulate of relativity enables us to evaluate k. At the instant ( = 0,
ihe origins of the two frames of reference S and S" are in the same place,
according to our initial conditions, and /' — then also. Suppose that a flare
is set off at the common origin of S and S' at / = I' = (), and the oKservers in
each system proceed to measure the speed with which the light from it spreads
out. Both observers must find the same speed c (Fig. 17), which means that
in the S frame
1.23 X = Ct
while in the S' frame
1.24 x' — ct'
Substituting For .v' and /' in Eq. 1.24 with the help of Eqs. I.IK and 1.22,
Jt(jr  tr) = ckt + (  \~ ki ) cx
ami solving for x,
ckt + vkt
(^)<
= c(
= ct
k + k
c
*<4*W
1 + .2,
ttW
II i is expression for x will be the same as that given by Eq. 1.23, namely .v = ct,
provided that the quantity in the brackets equals 1. Therefore
c
fe')f
= 1
and
1.25
k =
VI  »Vc 2
SPECIAL RELATIVITY
25
26
Inserting the above value of k in Eqs. 1.18 and 1.22, we have, for the complete
transformation of measurements of an event made in S to the corresponding
measurements made in S\ the equations
1.26
1.27
1.28
1.29
X" =
x — vt
Vl  v 2 /c*
y = y
z' = z
r =
Lorentz transformation
Vl  dVc !
These equations comprise the lorentz transformation. They were first obtained
by the Dutch physicist H. A. lorentz, who showed that the basic formulas of
etectromagneh'sm are the same in all frames of reference in uniform relative
motion only when these transformation equations are used. It was not until a
number of years later that Einstein discovered their full significance. It is obvious
diat the Lorentz transformation reduces to the Galilean transformation when
the relative velocity c is small compared with the velocity of light ft
The relativists length contraction follows directly from the Lorentz trans
formation. Let us consider a rod lying along the x' axis in the moving frame
S'. An observer in this frame determines the coordinates of its ends to be x\
and x' 3 , and so the proper length of the rod is
Ml = x 2 ~ x t
In order to find L = x 2 — x v the length of the rod as measured in the stationary
frame S at the time t, we make use of Eq. 1.26. We have
x; =
VI  v n /c*
*2 ~ vt
Vl  v*/c 2
and so
L = x? — *j
= (4~*.)Vit>7c*
= L Vl oVc 2
which is the same as Eq. 1.10.
BASIC CONCEPTS
•1.8 THE INVERSE LORENTZ TRANSFORMATION
In the previous section the coordinates of the ends of the moving rod were
measured in the stationary frame S at the same time r, and it was easy to use
Eq. 1.26 to find L in terms of Lq and v. If we want to examine time dilation,
though, Eq. 1.29 is not convenient, because (, and f 2 , the start and finish of the
chosen time interval, must be measured when the moving clock is at the respec
tive different positioas x l and x 2 . In situations of this kind it is easier to use
the inverse Lorentz transformation, which converts measurements made in the
moving frame S' to their equivalents in S. To obtain the inverse transformation,
primed and unprimed quantities in Eqs. 1.26 to 1.29 are exchanged, and v is
replaced by — <■:
1.30
1.31
1.32
1.33
r =
x' + vt'
Vi  v 2 /!?
y = y
Inverse Lorentz transformation
t =
V\  vVc 2
I«t us consider a clock at the point x' in the moving frame S'. When an
observer in S' finds that the time is l' Y , an observer in S will find it to be *,,
where, from Eq. 1.33,
t vx
1 ~ Vl  eVc 2
After a time interval of f (to him), the observer in the moving system finds that
the time is now t' 2 according to his clock. That is,
t = t 2 — f j
The observer in S, however, measures the end of the same time interval to be
vx J
l > + c*
Vl  dVc 2
SPECIAL RELATIVITY
27
28
so to him the duration of the interval / is
1 = 1,1,
t'a ~ 'i
or
/ =
VI  i'7r 2
6.
v'l  H<*
as we found earlier with the help of a lightpulse clock.
•1.9 VELOCITY ADDITION
One of the postulates of special relativity stales thai the speed of light c in free
space has the same value for all observers, regardless of their relative motion.
Bill "common sense" tells us that, if we throw a hall forward at 5ft ft/s from
a car moving at 80 ft/s, the hall's speed relative to the ground is 130 ft/s, the
sum of the two speeds. Hence we would expect that a ray of light emitted in
a frame of reference H' in the direction of its motion at velocity c relative to
another frame S will have a speed of r + c as measured in S, contradicting the
above postulate, "Common sense" is no more reliable as a guide in science than
it is elsewhere, and we must turn to the Lorentz transformation equations for
the correct scheme of velocity addition.
Let us consider something moving relative to both S and S'. An observer in
S measures its three velocity components to l»e
• dt
» dt
dz
V — _zl
• ~ (It
while to an observer in S' they are
v: = —
dY_
df
K =
df
v: =
dz'
df
By differentiating the inverse Lorentz transformation equations for x, y,
t, we obtain
and
dx =
dx' 4 odf
dy = ihj'
dz = dz'
BASIC CONCEPTS
dt =
df + sM.
tr
Vl  v 2 /c 2
and so
*~ dt
dx' + v (If
(If +
dx'
dx 1
dt'
+ v
1+^r
o dx 1
dt'
1.34
v; + i>
i +
vV'
RelatMstic velocity transformation
Similarly
1.35
1.36
v„ =
v'iVt
 v 2 /c 2
i +
ov;
v r =
v;VT
 v 2 /c 2
1 +
vV' t
c 2
If V^ = c, that is, if a ray of light is emitted in the moving reference frame
S' in its direction of motion relative to S, an observer in frame S will measure
the velocity
V.ss
V' + D
•♦9
c + v
l + Jf
tr
c{c 4 v)
C 4
= c
Both observers determine the same value for the speed of light, as they must.
SPECIAL RELATIVITY
29
The relativistic velocity transformation has other peculiar consequences. For
iastance, we might imagine wishing to pass a space ship whose speed with respect
to the earth in 0.9c at a relative speed of 0.5c. According to conventional
mechanics our required speed relative to the earth would have to be 1.4c, more
than the velocity of light. According to Eq. 1 .34, however, with V^ = 0.5c and
v = 0.9c, the necessary speed is only
V. =
v; + »
1 +
vV>
0.5c + 0.9c
1 +
(0.9c){0.5e)
= 0.9655c
which is less than c. We need go less than 1 percent faster than a space ship
traveling at 0.9c in order to pass it at a relative speed of 0.5c.
1.10 THE RELATIVITY OF MASS
Until now we have been considering only the purely kinematical aspects of
special relativity. The dynamical consequences of relativity are at least as
remarkable, including as they do the variation of mass with velocity and the
equivalence of mass and energy.
We begin by considering an elastic collision (that is, a collision in which kinetic
energy is conserved) between two particles A and B, as witnessed by observers
in the reference frames S and S' which are in uniform relative motion. The
properties of A and B are identical when determined in reference frames in which
they are at rest. The frames S and $' are oriented as in Fig. 112, with S' moving
in the + 1 direction with respect to S at the velocity v.
Before the collision, particle A had been at rest in frame S and particle B
in frame S', Then, at the same instant, A was thrown in the + y direction at
the speed V A while B was thrown in the — if direction at the speed V^, where
1.37
v A = v B
30
Hence the behavior of A as seen from S is exactly the same as die behavior
of B as seen from §', When the two particles collide, A rebounds in the — y
direction at the speed V A , while B rebounds in the +y' direction at the speed
V' B . If the particles are thrown from positions Y apart, an observer in S finds
that the collision occurs at y = l / 2 Y and one in S' finds that it occurs at y' = %Y.
BASIC CONCEPTS
y
kit
s
ff
k
collision as seen from frame S
collision as seen from frame S':
FIGURE 112 An elastic cplllslon is observed In two different frames of reference,
SPECIAL RELATIVITY
31
The roundtrip time T n for A as measured in frame S is therefore
V
1,38
*"*
and it is the same for H in S',
'it v ,
If momentum is conserved in the S frame, it must be true that
1.39 m A V A = m B V B
where m A and m B are the tnasses of A and B, and V^ and V B their velocities
a* measured in the S /ra»i«. In S the speed V' w is found from
1.40
V. = i
where T is the time required for H to make its round trip as measured in S.
In S', however, B's trip requires the time '/;,, where
1.41
r =
/I  d»/c
according to our previous results. Although observers in both frames see the
same event, they disagree as to the length of time the particle thrown from the
other frame requires to make the collision and return.
Replacing T in Eq. 1.40 with its equivalent in terms of T , we have
= ryi  dVc 2
B T
Prom Eq. 1.38
A r Q
Inserting these expressions for V A and V B in Eq. 1.39, we see that momentum
is conserved provided that
1.42
= m K Vl  » 2 /c
Our original hypothesis was that A and B are identical when at rest with respect
to an observer; the difference between m A and m B therefore means thai measure
ments of mass, like tho.se of space and time, depend upon the relative speed
l>etween an observer and whatever he is observing.
32
BASIC CONCEPTS
In the above example both A and 8 are moving in S. In order to obtain a
formula giving the mass m of a body measured while in motion in terms of its
mass m„ when measured at rest, we need only consider a similar example in
which V", and V^ are very small. In this case an observer in S will see B approach
A with Hie velocity v. make a glancing collision (since V' B < v), and then continue
on. In S
and
and so
1.43
m A = irtf,
m n = m
m =
Relattvistic mass
Vl  v 2 /c*
The mass of a body moving at the speed v relative to an observ er is larger
than its mass when at rest relative to the observer by the factor 1/ Vl  o'/c*.
This mass increase is reciprocal; to an observer in S'
and
m, = m
m„ = m„
Measured from the earth, h rocket ship in night is shorter than its twin still OB
the ground and its mass is greater. To somebody on the rocket ship in Hight
the ship on the ground also appears shorter and to have a greater mass. (The
effect is, of course, unobservably small for actual rocket speeds.) Equation 1.43
is plotted in Fig. 113.
Provided that momentum is defined as
] 14
\/l  v 2 /c*
conservation of momentum is valid in special relativity just as in classical physics.
However, Newton's second law of motion is correct only in the form
us
_ tt T m a v 1
~ rfiL \/l  dVc 2 !
This is not equivalent to saying that
F = ina
dt
SPECIAL RELATIVITY
33
34
FIGURE 113 The relativity of mm.
even with m given by Eq. 1.43, because
d \ dv dm
and dm/elt does not vanish if the speed of the body varies with lime. The
resultant force on a lx>dy is always equal to the time rate of change of its
momentum.
Relativistic mass increases are significant only at speeds approaching that of
light. At a speed onetenth that of light the mass increase amounts to only (1.5
percent, but this increase is over 100 percent at a speed ninetenths that of light.
Only atomic particles such as electrons, protons, mesons, and so on have suffi
ciently high speeds for relativistic effects to be measurable, and in dealing with
these particles the "ordinary" laws of physics cannot be used. Historically, the
first confirmation of Eq. 1.43 was the discovery by Bucherer in 1908 that the
ratio e/m of the electron's charge to its mass is smaller for fast electrons than
for slow ones; this equation, like the others of special relativity, has been verified
by so many experiments that it is now recognized as one of the basic formulas
of physics.
BASIC CONCEPTS
l.U MASS AND ENERGY
The most famous relationship Einstein obtained from the postulates of special
relativity concerns mass and energy. This relationship can be derived directly
from the definition of the kinetic energy T of a moving body as the work done
in bringing it from rest to its state of motion. That is,
■£**
where F is the component of the applied force in the direction of the displace
ment ds and s is the distance over which the force acts. Using the relativistic
form of the second law of motion
F =
rf(fflli)
~dT
the expression for kinetic energy becomes
[
d(mv)
da
dt
J hib
i; d{mc)
{ \y/l v 2 /c 2 I
Integrating by parts (/* dy = xy  fy dx),
T =
m„v £
VI  v 2 /c 2
vdv
Vl  He 2
+ in^Vl  i> 2 /c 2
 "h c
1.46
t/1  oVc 2
= mc 2  m^c 2
Equation 1.46 states that the kinetic energy of a body is equal to the increase
in its mass consequent upon its relative motion multiplied by the square of the
speed of light.
Eq nation 1.46 may be rewritten
1A7
mc 1 = T+ m„c 2
SPECIAL RELATIVITY 35
If we interpret mc 2 as the total energy E of the body, it follows that, when the
body is at rest and T = 0. it nevertheless possesses the energy jn (1 c a . Accordingly
m n c 2 is called the rest energy E of a l»dy whose mass at rest is m„. Equation
1.17 therefore Ixscomes
where
1.48
E = m n a 2
Rest energy
In addition to its kinetic, potential, electromagnetic, thermal, and other
familial guises, then, energy can manifest itself as mass, The conversion factor
between the unit of mass (kg) and the unit of energy (J) is c\ so 1 kg of matter
has an energy content of 9 X 10«J. Even a minute hit of matter represents
a vast amount of energy, and, in fact, the conversion of matter into energy is
the source of the power liberated in all the exothermic reactions of physics and
chemistry.
Since mass and energy are not independent entities, the separate conservation
principles of energy and mass are properly a single one, the principle of con
servation of mass energy. Mass can lie created or destroyed but when this
happens an equivalent amount of energy simultaneously vanishes or comes into
being, and vice versa. Mass and energy are different aspects of die same thing.
When the relative speed v is small compared with c, the formula for kinetic
energy must reduce to the familiar Y,m () v 2 , which has been verified by experiment
at low speeds. Let us see whether this is true. The binomial theorem of algebra
tells us that if some quantity x is much smaller than 1,
(1 ± *)» =; 1 ± nx
The relativistic formula for kinetic energy is
T = mc 2  m^c 2
m,,r"~
\/l  b=/c»
— nup
36
Expanding the first term of this formula with the help of the binomial theorem,
with d*/«* < 1 since t: is much less than c,
T = (1 + %^/c 2 )m v c 2  m a c 2
= %m v 2
1 terice at low speeds the relativistic expression for the kinetic energy of a moving
particle reduces to the classical one. The total energy of such a particle is
E = m c 2 + %m v v 2
BASIC CONCEPTS
In the foregoing calculation relativity has once again met an important test;
it has yielded exactly the same results as those of ordinary mechanics at low
speeds, where we know by experience that the latter are perfectly valid. It is
nevertheless important to keep in mind that, so far as is known, the correct
iormulation of mechanics has its basis in relativity, with classical mechanics no
more than an approximation correct only under certain circumstances.
It is often convenient to express several of the relativistic formulas obtained
above in forms somewhat different from their original ones. The new equations
:ire so easy to derive thai we shall simply state them without proof:
1.49
1.50
1.51
1.52
1.53
1.54
£ = Vm^V' + p 2 c
(■'■
 1
T = m n c 2 ( . 1 =r  l)
i A I
c V [1 + (T/nw*)] 2
= i +
The symbol ;/ is used for the magnitude of the linear momentum mv.
These formulas are particularly useful in nuclear and elementaryparticle
physics, where the kinetic energies of moving particles are customarily specified,
rather than their velocities. Equation 1.52, for Instance, permits us to find r, i:
directly from T/m lt c 2 , the ratio between the kinetic and rest energies of a particle.
1.12 MASS AND ENERGY: ALTERNATIVE DERIVATION
Tl*c equivalence of mass and energy can be demonstrated in a numlicr of
different ways. An interesting derivation that is somewhat different from the
■*■ given above, but also suggested by Einstein, makes use of the basic notion
owl the center of mass of an isolated system (one that does not Interact with
'Is surroundings) cannot be changed by any process occurring within the system,
hi this derivation we imagine a closed box from one end of which a burst of
"ferromagnetic radiation is emitted, as in Fig. 114. This radiation carries energy
aiM l momentum, and when the emission occurs, the lx>x recoils in order that the
SPECIAL RELATIVITY
37
38
L
J
initial center of mass
7 s —
1 burst of radiation is
emitted
 new center of mass
■•— radiation is absorbed and box stops
FIGURE 114 Radiant energy possesses inertial mass.
total momentum of the system remain constant. When the radiation is absorbed
at the opposite end of the box, its momentum cancels the momentum of the
box, which then comes to rest. During the time in which the radiation was in
transit, the box has moved a distance s. If the center of mass of the system is
still to be in the same location in space, the radiation must have transferred mass
from the end at which it was emitted to the end at which it was absorbed. We
shall compute the amount of mass that must be transferred if the center of mass
of the system is to remain unchanged
For simplicity we shall consider the sides of the box to lie inassless and its
ends to have the mass l / 3 M each. The center of mass is therefore at the center
of the ixjx. a distance %L from each end. A burst of electromagnetic radiation
that has the energy E carries the momentum E/c according to electromagnetic
theory, and, by hypothesis, has associated with it an amount of mass m. When
the radiation is emitted, the box, whose mass is now M — m, recoils with the
velocity v. From the principle of conservation of momentum,
Pbox = Pr«tl»Uon
(M  m)v = ^
c
and so the recoil velocity of the box is
t; =
E „ E
BASIC CONCEPTS
since m is much smaller than M. The time / during which the Ixjx moves is
,'((iial to the time required by the radiation to reach the opposite end of the
lwx, a distance L away; this means that t = L/c (assuming that v < c, which
is true when in < M). During the lime t the box is displaced to the left In
s = vt = EL/Mc 2 .
After the box has stopped, the mass of its lefthand end is '/ 2 :V/ — m and the
mass of its righthand end is '/ 2 M + m owing to the transfer of the mass m
issDciated with the energy i, of the radiation. II the renter of BOBS! li to be
in the same place it was originally,
or
(%M  m)(%I. + s) = ( l /M + m){%L  a)
Ms
m =
L
Inserting the value of the displacement s.
The mass associated with an amount of energy £ is equal to E/c 2 .
In the above derivation we assumed Uiat the box is a perfectly rigid body:
that the entire box starts to move when the radiation is emitted and the entire
box comes to a stop when the radiation is absorbed. Actually, of course, diere
is no such thing as a rigid body that meets this specification; for example, the
radiation, which travels witii the speed of light, will arrive at the righthand
end of the box before that end begins to move! When the finite speed of elastic
waves in the box is taken into account in a more elaborate calculation, however,
the same result that in = E/c 2 is obtained.
Problems
'■ A certain particle has a lifetime of 10 7 s when measured at rest. How far
does it go Ixrfore decaying if its speed is 0.99c when it is created?
2 An airplane is Hying at 300 m/s {672 mi/h). How much time must elapse
before a clock in the airplane and one on the ground differ by 1 s?
*! How fast must a spacecraft travel relative to the earth for each day on the
spacecraft to correspond to 2 d on the earth?
■ A rocket ship leaves the earth at a speed of 0.98c. How much time does
" take for the minute hand of a clock in the ship to make a complete revolution
35 measured by an observer on the earth?
SPECIAL RELATIVITY
39
40
5. An astronaut whose height on the earth is exactly 6 ft is lying parallel to
the axis of a spacecraft moving at 0.9c relative to the earth. What is his height
as measured by an observer in the same spacecraft? By an observer on the earth?
6. A meter slick is projected into space at so great a speed that its length appears
contracted to only 50 cm. I low fits! is it going in miles per second?
7. A rocket ship is 1(H) m long on the ground. When it is in flight, its leogtfe
is 99 m to an observer on the ground. What is its speed?
' 8. An observer moving in the + x direction at a speed (in the lalwratory system)
of 2.9 x 10 s m s finds (he speed of an object moving in the — x direction to
lie 2,998 x 10 H m/s. What is the speed of the object in the laboratory system?
' 9, A man on the moon sees two spacecraft, A and H, coming toward him from
opposite directions at the respective speeds of 0.8c and 0,9c. (a) What does a
man on A measure for the speed with which he is approaching the moon? Pea
the speed with which he is approaching B? (b) What does a man on II measure
for the speed with which he is approaching the moon? For the speed with which
he is approaching A?
10. It is possible for the electron beam in a television picture tube to move
across the screen at a speed faster than the speed of light. Why does this not
contradict special relativity?
1 1 . A man has a mass of KM) kg on the ground. When he is in a rocket ship
in Bight, his mass is 101 kg as determined by an observer on the ground. Win
is the speed of the rocket ship?
12. How fast must an electron move in order that its mass equal the rest mass
of the proton?
13. Find the speed of a 0. 1MeV electron according to classical and relativist ic
mechanics.
14. How much mass docs a proton gain when it is accelerated to a kinetic
energy of 500 MeV?
15. How much mass does an electron gain when it is accelerated to a kinetic
energy of 500 MeV?
16. The total energy of a particle is exactly twice its rest energy. Find its spee<
17. How much work must lie done in order to increase the speed of an electron
from 1.2 X 10 s m/s to 2.4 X 10 s m/s?
18. (a) The density of a substance is p in the S frame in which it is at rest.
BASIC CONCEPTS
Find the density p' that an observer in the $' frame moving at a speed relative
to S of c would determine. (/») Cold has a density oF 19.3 g/cnr 1 when the sample
is at rest relative to the observer. What is its density when the relative velocity
is 0.9r.
19. A certain quantity of ice at 0°C melts into water at 0°C and in so doing
gains 1 kg of mass. What was its initial mass?
20. Dynamite liberates about 5.4 X 10" J/kg when it explodes. What fraction
of its total energy content is this?
2). Solar energy reaches the earth at the rate of about 1 .100 \V"/m 2 of surface
perpendicular to the direction of the sun. By how much does the mass of the
sun decrease in each second? (The mean radius of the earth's orbit is
1.5 X I0 11 m.)
22. Prove diat '/ 2 »ii(3 z , where m. = m<,/ vl— vr/i", does not equal the kinetic
energy of a particle moving at relativistic speeds.
23. Express the relativistic form of the second law of motion, F = d(mv)/dt,
in terms of m„, v, v, and tit/ tit.
24. A man leaves the earth in a rocket ship that makes a round trip to (he
nearest star, 4 lightyears distant, at a speed of 0.9c. How much younger is he
upon Ins return than his twin brother who remained behind? (A lightyear is
the distance light travels in a year. It is equal to 9.46 X 10 15 m.)
25. Light of frequency P is emitted by a source. An observer moving away
from the source at the speed D measures a frequency of v'. By considering the
source as a clock that ticks p times per second and gives off a pulse of light
with each tick, show that
V 1 + t/
r
c/c
This constitutes the longitudinal do/ipler effect in light. (If the observer is moving
ti'utiul the source at the relative speed o, the + and — signs in the radical
™ the above formula are interchanged.) Why does this result differ from the
"""■spending one for sound waves in air?
2 " The transverse doppler effect, which has no nonrelativislic counterpart,
a pplies to measurements of light waves made by an observer in relative motion
f^rpendit'ular to the direction of propagation of the waves. (In the preceding
problem the observer moves parallel to the direction of propagation.) Show that
'" the transverse doppler effect
p' = c\/l  v 2 /c 2
SPECIAL RELATIVITY
41
27. Twin A makes a round Irip at a speed of 0,8c to a star 4 lightyears away,
while twin B stays behind on the earth. Each twin sends the other a signal once
a year by his own reckoning, {a) How many signals does A send during the trip?
How many does B send? (h) Use the doppler effect formula of Prob. 25 to
analyze this situation. How many signals does A receive during the trip? How
many does 8 receive? Are these results consistent with those of part (a)?
PARTICLE PROPERTIES OF WAVES
2
In our everyday experience there is nothing mysterious or ambiguous about the
concepts of particle and wave. A stone dropped into a lake and the ripples that
spread out from its point of impact apparently have in common only the ability
to carry energy and momentum from one place to another. Classical physics,
which mirrors the "physical reality" of our sense impressions, treats particles
and waves as separate components of that reality. The mechanics of particles
and the optics of waves are traditionally independent disciplines, each with its
Own chain of experiments and hypotheses.
The physical reality we perceive arises from phenomena that occur in the
microscopic world of atoms and molecules, electrons and nuclei, but in this world
there are neither particles nor waves in our sense of these terms. We regard
electrons as particles because they possess charge and mass and behave according
to the laws of particle mechanics in such familiar devices as television picture
tubes. We shall see, however, that there is as much evidence in favor of inter
preting a moving electron as a wave manifestation as there is in favor of inter
preting it as a particle manifestation. We regard electromagnetic waves as waves
because under suitable circumstances they exhibit diffraction, interference, and
polarization. Similarly, we shall see that under other circumstances electro
magnetic waves behave as though they consist of streams of particles. Together
witli special relativity, the waveparttcle duality is central to an understanding
of modern physics, and in this liook there are few arguments that do not draw
upon one or the other of these fundamental principles.
2.1 THE PHOTOELECTRIC EFFECT
42
BASIC CONCEPTS
Late in the nineteenth century a series of experiments revealed that electrons
are emitted from a metal surface when light of sufficiently high frequency
(ultraviolet light is required for all but the alkali metals) falls upon it. This
phenomenon is known as the photoelectric effect. Figure 21 illustrates the type
43
44
light
^ ^
electrons
evacuated tjuartz tube
— o—
rAVWW
FIGURE 21 Experimental observation of the photoelectric effect.
ni apparatus that was employed in the more precise of these experiments. An
evacuated Lube contains two electrodes connected to an external circuit like that
Shown schematically, with the metal plate whose surface is to he irradiated as
the anode. Some of the photoelcetrons that emerge from the irradiated surface
have sufficient energy to reach the cathode despite its negative polarity, and
they constitute the current that is measured by the ammeter in the circuit. As
the retarding potential V is increased, fewer and fewer electrons get to the
cathode and the currcul drops. Ultimately, when V equals or exceeds a certain
value V„, of the order of a few volts, no further electrons strike the cathode
and the current ceases.
The existence of the photoelectric effect ought not to l>e surprising; after all,
light waves carry energy, and some of the energy absorbed by the metal may
somehow concentrate on individual electrons and reappear as kinetic energy.
When we look more closely at the data, however, we find that the photoelectric
effect can hardly he interpreted so simply.
One of the features of the photoelectric effect that particularly puzzled its
discoverers ts that the energy distribution in the emitted electrons (called phato
etectmns) is independent of the intensity of the light, A strong light Ixmrn yields
more photoelcetrons than a weak one of the same frequency, but the average
electron energy is the same (Fig, 22). Also, within the limits of experimental
accuracy (about 10" s), there is no time lag between the arrival of light at a
metal surface and the emission of photoelcetrons. These observations cannot be
understood from the electromagnetic theory of light.
BASIC CONCEPTS
Let us consider violet light falling on a sodium surface, in an apparatus like
that of Fig. 21. There will be a detectable photoelectric current when
10~ n W/m* of electromagnetic energy is absorbed by the surface (a more intense
beam than this is required) of course, since sodium is a good reflector of light i.
Now there are about lu 1 ' atoms in a layer of sodium 1 atom thick and 1 in~
in area, so that, if we assume that the incident light is absorbed in the 10
uppermost layers of sodium atoms, the 10 '' \\ in'' is distributed among 10""
:itoms. Hence each atom receives energy at the average rate of 10"*" W, which
is less than 10 " eV/s. It should therefore lake more than 10" s. or almost a year,
for any single electron to accumulate the 1 eV or so of energy that the photo
electrons are found to have! In the maximum possible time of I0" !i s, an average
electron, according to electromagnetic theory, will have gained only J0"" ; eV.
Even if we call upon some kind of resonance process to explain why some
FIGURE 22 Photoelectron current is proportional to light Intensify lor all retarding voltages The eitinc
lion voltage V .. Is the same for all intensities of light of i given frequency ft
frequency = V = constant
5
I
RETARDING POTENTIAL
PARTICLE PROPERTIES OF WAVES
45
electrons acquire mare energy than others, the fortunate electrons could hardly
have more than 10" l " of the observed energy.
Equally odd from the point of view of the wave theory is the fact that the
photoelectron energy depends upon the frequency of the light employed
(Fig. 23). At frequencies below a certain critical frequency characteristic of
aacfe pint ict ilar metal, no electrons whatever are emitted. Above this threshold
frequency the photoelectrons have a range of energies from to a certain
maximum value, and this maximum energy increases linearly with increasing
frequency. High frequencies result in highmaximum photoelectron energies, low
frequencies in lowmaximum photoelectron energies. Thus a faint blue light
produces electrons with more energy than those produced by a bright red light,
although the latter yields a greater number of them.
Figure 24 is a plot of maximum photoelectron energy T mKX versus the fre
quency v of the incident light in a particular experiment that employed a sodium
surface. It is clear that the relationship between T mtx and the frequency c
involves a proportionality, which we can express in the Form
2.1
= hv — h>
FIGURE 2a Maximum photoelectron energy as a function ot the frequency of the incident light lor a
sodium surface.
where •>„ is the threshold frequency below which no photocmission occurs and
k is a constant. Significantly, the value of li
h = 6.626 X 10 34 Js
\aahwttj$ the same, although v a varies with the particular metal being illuminated.
FIGURE 23 The extinction! voltage V'„ depends upon the frequency 9 of the light. When the retarding
potential is V a II, the photoelectric current is the same for light of a given Intensity regardless of its
frequency.
light intensity = constant
3
V„ (3)
V (2) V„ (J)
RETAROING POTENTIAL
2.2 THE QUANTUM THEORY OF LIGHT
'Hit 1 electromagnetic theory of light accounts so well for such a variety of
phenomena that it must contain some measure of truth. Yet this wellfounded
theory is completely at odds with the photoelectric effect. In JH05 Albert
Einstein found that the paradox presented by the photoelectric effect could l>e
understood only by taking seriously a notion proposed five years earlier by the
German theoretical physicist Max Planck, Planck was seeking to explain the
characteristics of the radiation emitted by bodies hot enough to be luminous,
t problem notorious at the time for its resistance to solution. Planck was able
to derive a formula for the spectrum of this radiation (that is, the relative
brightness of the various colors present) as a function of the temperature of the
"°&y that was in agreement %vith experiment provided he assumed that the
radiation is emitted disconttrwouahj as little bursts of energy. These bursts of
^wgy are called quanta. Planck found that the quanta associated with a
particular frequency v of light all have the same energy and that this energy
£ is directly proportional to v. That is.
46
BASIC CONCEPTS
PARTICLE PROPERTIES OF WAVES
47
2.2
E = /if
Quantum energy
where K today known as Planck's constant, has the value
h m 6.626 X 10 •" Js
Wa shall examine BOOM uf the details of this L u tew s Ung problem and its solution
in Chap, 9.
While he had to assume that tlie electromagnetic energy radiated by a hot
object W U OtgOg intermittently. Planck did nol doubt that ii propagates continu
ously through space as electromagnetic waves. Einstein proposed that light not
only is emitted a quantum at a time, Imt also propagates as individual quanta.
In terms of this hypothesis the photoelectric effect can be readily explained.
The empirical formula Eq. 2,1 may be rewritten
2.3
'"' " I'm™ + ft, 'ti
Photoelectric effect
Einstein's proposal means that the three terms of Eq. 2.3 are to be interpreted
as follows:
hv a the energy content of each quantum of the incident light
T ual = the maximum photoelectron energy
ht>„ = the minimum energy needed to dislodge an electron from the
metal surface being ilhiiiiiuatcd
There must he a mininmm energy required by an electron in order to escape
I rum a metal surface, or else elect runs would pour out even in the absence
light. The energy /w„ characteristic oi a particular surface is called its u.oitl
[unction. Hence Eq. 2.3 states that
Quantum maximum electron work function
e a +
energy energy of surface
It is easy to sec why not all photoelectrons have the same energy, hut emerg
with all energies up to T max : hi>„ is the work that must he done to take an electron
through the metal surface from just beneath it, and mora work is required when I
the electron originates deeper in the metal.
The validity of this interpretation of the photoelectric effect is confirmed by I
studies of thermionic emission. Long ago it became known that the presents
of a very hot object increases the electrical conductivity of the surrounding air,
and late in the nineteenth century the reason for this phenomenon was found!
to be the emission of electrons from such an object. Thermionic emission makes
possible the operation of such devices as television picture lubes, in which met;)]
filaments or specially coated cathodes at high temperature supply dense streams
of electrons. The emitted electrons evidently obtain their energy from the
thermal agitation of the particles constituting tin metal, and we should expect
thai the electrons must acquire a certain minimum energy in order to escape.
This minimum energy can be determined for many surfaces, and it is always
close to the photoelectric work function for the same surfaces. In photoelectric
emission, photons of light provide the energy required by an electron to escape,
while in thermionic emission heat does so: in both cases the physical processes
involved in the emergence of an electron from a metal surface arc the same.
Let us apply Eq. 2.3 to a specific situation. The work function of potassium
is 2.2 eV. When ultraviolet light of wavelength 3,500 A ( 1 A = t angstrom
unit = 10~ ,0 m) falls on a potassium surface, what is the maximum energy in
electron volts of the photoelectrons? From Eq. 2.3,
r m™ = *» ~ '»■()
Since /ii' u is already expressed in electron volts, we need only compute the
quantum energy /»■ of 3,500 A light. This is
h» = ?£
A
_ 6.63 X 10 M Js X 3 X 10" m/s X 10 10 A/m
3.500 A
= .5.7 x HI W J
To convert this energy from joules to electron volts, we recall that
1 eV= 1.6 X 10 < 3 J
and
hv =
5.7 x IP" '"J
1.6 X 10 1B J
= 3.6 eV
Hence the maximum photoelectron energy is
'''„,« = ""  '»'„
a 3.6 eV  2.2 eV
= 1.4 eV
The view that light propagates as a series of little packets of energy (usually
called photons) is directly opposed to the wave theory of light. The latter, which
provides the sole means of explaining a host of optical effects — notably diffi action
48
BASIC CONCEPTS
PARTICLE PROPERTIES OF WAVES
49
and interference — is one of the most securely established of physical theories.
Planck's suggestion that a hot object emits light in separate quanta was nol
incompatible with the propagation of light as a wave, Einstein's suggestion in
1905 that Ught travels through space in the form of distinct photons, however,
elicited incredulity from his contemporaries. According to the wave theory, light
waves spread out from a source in the way ripples spread out on the surface
of a lake when a stone falls into it. The energy carried by the light, in Ibis
analogv. is di s tr ibu t ed continWHBry tkrou^jhoat she wave pattern. Accowbng to
the quantum theory, on the other hand, light spreads out from a source as a
scries of localized concentrations of energy, each sufficiently small to be capable
Of absorption by a single electron. Curiously, the quantum theory of light, which
treats it strictly as a particle phenomenon, explicitly involves the light frequency
j», strictly a wave concept.
The quantum theory of light is strikingly successful in explaining the photo
electric effect. It predicts correctly that the maximum photoclectron energy
should depend upon the frequency of the incident light and not upon its intensity,
contrary to what the wave theory suggests, and it is able to explain why even
the feeblest light can lead to the immediate emission of photoeleetrons, again
contrary to the wave theory. The wave theory can give no reason why there
should be a threshold frequency such that, when light of tower frequency is
employed, no photoeleetrons are observed no matter how strong the light beam,
Something that follows naturally from the quantum theory.
Which theory are we to believe? A great many physical hypotheses have had
to be altered or discarded when they were found to disagree with experiment,
but never before have we had to devise two totally different theories to account
for a single physical phenomenon. The situation here is fundamentally different
from what it is, say, in the ease of relativistic versus Newtonian mechanics, where
the tatter turns out to lie an approximation of the former. There is no way of
deriving the quantum theory of light from the wave dicory of light or vice versa.
In a specific event light exhibits either a wave or a particle nature, never both
simultaneously. The same light ljeam that is diffracted by a grating can cause
the emission of photoeleetrons from a suitable surface, but these processes occur
independently. The wave theory of light and the quantum theory of light
complement each other. Electromagnetic waves account for the observed man
ner in which light propagates, while photons account for the observed manner
in which energy is transferred between light and matter. We have iio alternative
to regarding light as something that manifests itself as a stream of discrete photons
on occasion and as a wave train the rest of the time. The "true nature'' of light
is no longer something that can be visualized in terms of everyday experience,
and we must accept both wave and quantum theories, contradictions and all,
as the closest we can get to a complete description of light.
50
BASIC CONCEPTS
2.3 X RAYS
The photoelectric effect provides convincing evidence that photons of light can
transfer energy to electrons. Is the inverse process also possible? That is, can
part or all of the kinetic energy of a moving electron be converted into a photon?
As it happens, the inverse photoelectric effect not only does occur, but had been
discovered (though not at all understood) prior to the theoretical work of Planck
and Einstein.
In 1 895 Wilhelm Roentgen made the classic observation that a highly pene
trating radiation of unknown nature is produced when fast electrons impinge
on matter. These A' rays were soon fomid to travel in straight lines, even through
electric and magnetic fields, to pass readily through opaque materials, to cause
phosphorescent substances to glow, and to expose photographic plates. The faster
the original electrons, the more penetrating the resulting X rays, and the greater
the number of electrons, the greater the intensity of die Xray beam.
Not long after this discovery it began to lie suspected that X rays are electro
magnetic wav&s. After all, electromagnetic theory predicts that an accelerated
electric charge will radiate electromagnetic waves, and a rapidly moving electron
suddenly brought to rest is certainly accelerated. Radiation produced under these
circumstances is given the German name bremsntrahhing ("braking radiation").
The absence of any perceptible Xray refraction in the early work could be
attributed to very short wavelengdis, below those in the ultraviolet range, since
the refractive index of a substance decreases to unity (corresponding to straight
line propagation) with decreasing wavelength.
The wave nature of X rays was first established in 1906 by Bark la, who was
able to exhibit their polarization. Barkk's experimental arrangement is sketched
in Fig. 25. We shall analyze this classic experiment under the assumption that
X rays are electromagnetic waves. At the left a beam of unpolarized X rays
heading in the — z direction impinges on a small block of carbon. These X rays
are scattered by the carbon; this means that electrons in the carbon atoms are
set in vibration by the electric vectors of the X rays and then reradiate. Because
the electric vector in an electromagnetic wave is perpendicular to its direction
of propagation, the initial beam of X rays contains electric vectors that lie in
the xy plane only. The target electrons dierefore are induced to vibrate in die
xy plane. A scattered X ray that proceeds in the + x direction can have an
electric vector in the y direction only, and so it is planepolarized. To demon
strate this polarization, another carbon block is placed in the path of die ray,
as at the right. The electrons in diis block are restricted to vibrate in the t/
direction and therefore reradiate X rays that propagate in the xz plane exclu
sively, and not at all in the y direction. The observed absence of scattered
X rays outside the xz plane confirms die wave character of X rays.
PARTICLE PROPERTIES OF WAVES
51
52
zero
intensity
*
first polarized
scattcrer scattered ray
maximum
intensity
second
scattcrer
uupolarizcd
Xray
FIGURE 25 Barilla's Biperfnwnt to demonstrate Xray polarization.
In 1912 a method was devised for measuring the wavelengths of X rays. A
diffraction experiment had been recognized as ideal, but. as we recall from
physical optics, the spacing between adjacent lines on a diffraction grating must
be of the same order of magnitude as the wavelength of the light tor satisfactory
results, and gratings cannot be ruled with the minute spacing required by
\ rays. In 1912. however. Max von Laue recognized that the wavelengths
hypothesized for X rays were about the same order of magnitude as the spacing
between adjacent atoms in crystals, which is alwut I A. He therefore proposed
lhat crystals be used to diffract X rays, with their regular lattices acting as a
kind of threedimensional grating. Suitable experiments were performed in the
next year, and the wave nature of X rays was successfully demonstrated. In these
experiments wavelengths from 1.3 X 10" 11 to4.8 X 10" n m were found, 0.13
to 0.48 A, 101 of those in visible light and hence having quanta 10' times as
energetic. We shall consider Xray diffraction in Sec. 2.4.
For purposes of classification, electromagnetic radiations with wavelengths in
fa approximate interval from 1(1 " to 10 * m (0.1 to 1«H> A) are today con
sidered as X rays.
Figure 26 is a diagram of an Xray lul>e. A cathode, healed by an adjacent
filament through which an electric current is passed, supplies electrons copiously
by thermionic emission. The high potential difference V maintained between
fa cathode and a metallic target accelerates the electrons toward the latter.
The face of the target is at an angle relative to the electron lieain, and the
X rays that emerge from the target pass through the side of the tulie. The tube
is evacuated to permit the electrons to get to the target unimpeded
BASIC CONCEPTS
evacuated v
tube \
r^
larKtl
FIGURE 2 6 An Xray tube
As was said earlier, classical electromagnetic theory predicts the production
of bremsstrahlung when electrons are accelerated, thereby apparently accounting
for the X rays emitted when fast electrons are stopped by the target of an Xray
tiilw. However, the agreement between the classical theory and the experimental
data is not satisfactory in certain important respects. Figures 27 and 28 show
the Xray spectra that result when tungsten and molybdenum targets are bom
barded by electrons at several different accelerating potentials. The curves
exhibit two distinctive features not explainable in terms of electromagnetic
theory:
1. In the case of molybdenum there are pronounced intensity peaks at certain
wavelengths, indicating the enhanced production of X rays. These peaks occur
at various specific wavelengths for each target material and originate in re
arrangements of the electron structures of the target atoms after having been
disturbed by the bombarding electrons. The important thing to note at this point
is the production of X rays of specific wavelengths, a decidedly nonclassical effect,
in addition to the production of a continuous Xray spectrum.
2. The X rays produced at a given accelerating potential V vary in wavelength,
bul none has a wavelength shorter than a certain value A llUn , Increasing V
decreases X m)[1 . At a particular V, A mln is the same for both the tungsten and
molybdenum targets. Duane and Hunt have found that A m[n is inversely propor
tional to V; their precise relationship is
2.4
\ _
rt ni!n —
1.24 X 10 6 Vm
Xray production
PARTICLE PROPERTIES OF WAVES
53
FIGURE 27 Xray spectra of tung
sten at various accelerating poten
tials
0.2
O.i) 0,6 0.8
WAVELENGTH. A
1.0
FIGURE 2e Xray spectra of tung
sten and molybdenum at 35 kV accel
erating potential.
54
0.2
BASIC CONCEPTS
0.4 0.6 0.8
WAVELENGTH. 8
1.0
The second observation is readily understood in terms of the quantum theory
of radiation. Most of the electrons incident upon the target lose their kinetic
energy gradually in niurreroiLS collisions, their energy going simply into heat.
(This is the reason that the targets in Xray tubes are normally of highmelting
point metals, and an efficient means of cooling the target is often employed.)
A few electrons, though, lose most or all of their energy in single collisions with
target atoms; this is the energy that is evolved as X rays. Xray production, then,
except for the peaks mentioned in observation 1 above, represents an inverse
photoelectric effect. Instead of photon energy being transformed into electron
kinetic energy, electron kinetic energy is being transformed into photon energy.
A short wavelength means a high frequency, and a high frequency means a high
photon energy hr. It is therefore logical to interpret the short wavelength limit
\ min of Eq. 2.4 as corresponding to a maximum photon energy hv mux , where
2.S
''"max =
he
Since work functions are only several electron volts while the accelerating
potentials in Xray tubes are tens or hundreds of thousands of volts, we may
assume that the kinetic energy 7* of the bombarding electrons is
2.6
7"=eV
When the entire kinetic energy of an electron is lost to create a single photon,
then
2.7
hv„,„ = T
Substituting Eqs. 2.5 and 2.6 into 2.7, we see that
Kmix = T
iw
= eV
rt mln
X — he
6.63 X IP 34 Js X 3 X H) a m/s
1.6 X I0 ,9 C X V
1.24 X lO 8 „
= Vm
V
which is just the experimental relationship of Eq. 2.4. It is therefore correct
to regard Xray production as the inverse of the photoelectric effect.
A conventional Xray machine might have an accelerating potential of
50,000 V. To find the shortest wavelength present in its radiation, we use
PARTICLE PROPERTIES OF WAVES
55
Eq. 2.4, with the result that
1.24 X 10 6 Vm
A ,,,,,, —
5 X 10* V
= 2.5x 10 "m
= 0.25 A
This wavelength corresponds to the frequency
p~„ =
56
A n.ln
_ 3 X 10" m/s
~ 2.5 X 10 »m
= 1.2 X 10 1B Hz
2.4 XRAY DIFFRACTION
Lei us now return to the question of how X rays may he demonstrated to consist
of electromagnetic waves. A crystal consists of a regular array of atoms, each
of which is able to scatter any electromagnetic waves that happen to strike it.
The mechanism of scattering is straightforward. An atom in a constant eleetric
field becomes polarized since its negatively charged electrons and positively
charged nucleus experience forces in opposite directions; these forces arc small
compared with the forces holding the atom together, and so the result is a
distorted charge distribution equivalent to an electric dipole. In the presence
ol the alternating electric field of an electromagnetic wave of frequency e, the
polarization changes back and forth with the same frequency v. An oscillating
electric dipole is dius created at the expense of some of the energy of the
incoming wave, whose amplitude is accordingly decreased. The oscillating (Spate
in turn radiates electromagnetic waves of frequency », and these secondary waves
proceed in all directions except along the dipole axis. In an assembly of atoms
exposed to unpolarized radiation, the secondary radiation is isotropic since the
contributions of the individual atoms are random. In wave terminology, the
secondary waves have spherical wavefronts in place of the plane wavefronts of
the incoming waves (Fig. 29). The scattering process, then, involves an atom
absorbing incident plane waves and reemitting spherical waves of the same
frequency.
A monochromatic team of X rays that falls upon a crystal will !>e scattered
in all directions within it, but, owing to the regular arrangement of the atoms,
in certain directions die scattered waves will constructively interfere with one
BASIC CONCEPTS
incident waves
=>
scattered
waves
unscattcred waves
^>
FIGURE 29 The scattering of electromagnetic radiation by > group ol atoms. Incident plane waves are
re emitted as spherical waves.
another while in others they will destructively interfere. The atoms in a crystal
may lie thought of a* defining families of parallel planes, as in Fig. 210, with
each family having a characteristic separation between its component planes.
This analysis was suggested in 1913 by W. L. Bragg, in honor of whom the above
planes are called Bragg planes. Tin conditions that must he fulfilled for radiation
scattered by crystal atoms to undergo constructive interference may be obtained
FIGURE 2 10 Two sett ol Bragg planes In a NaCI crystal.
PARTICLE PROPERTIES OF WAVES 57
from a diagram like that in Fig. 211. A beam containing X rays of wavelength
A is incident upon a crystal at an angle with a family of Bragg planes whose
spacing is rf. The beam goes past atom A in the first plane and atom 8 in the
next, and each of them scatters part of the beam in random directions. Con
structive interference takes place only between those scattered rays diat are
parallel and whose paths differ by exactly A, 2\, 3A, and so on. That is, the
path difference mast be nA, where n is an integer. The only rays scattered by
A and B for which this is true are those labeled I and II in Fig. 211. The first
condition upon I and II is that their common scattering angle be equal to the
angle of incidence & of the original beam. The second condition is that
2.8
2dsin0 = nA n = 1, 2, 3, .
since ray II must travel the distance 2d sin 6 farther than ray 1. The integer
n is the order of the scattered l>eam.
The schematic design of an Xray spectrometer based upon Bragg's analysis
is shown in Fig. 212. A collimated beam of X rays falls upon a crystal at an
angle &, and a detector is placed so that in records those rays whose scattering
angle is also 0. Any X rays reaching the detector therefore obey the first Bragg
condition. As is varied, the detector will record intensity peaks corresponding
to the orders predicted by Eq. 2,8. If the spacing d between adjacent Bragg
planes in the crystal is known, the Xray wavelength A may he calculated.
How can we determine the value of d? This is a simple task in the case of
crystals whose atoms are arranged in cubic lattices similar to that of rock salt
(NaCl), illustrated in Fig. 210, As an example, let us compute the separation
of adjacent atoms in a crystal of NaCl. The molecular weight of NaCl is 58.5,
which means there are 58.5 kg of NaCl per kilomole (kmol). Since there are
FIGURE 211 Xray scattering front cubic crystal.
58
BASIC CONCEPTS
path difference
= 2d sin 8
detector
collimators
FIGURE 2 12 X ray spectrometer.
,V„ = 6.02 X 10 2a molecules in a kmol of any substance (N is Avogadro's num
ber), the mass of each NaCl "molecule" — that is, of each Na + CI pair of
atoms — is given by
m KtC]  58.5
kg
kmol 6,02 X 10 2(i molecules/kmol
= 9.72 X 10" 2a kg/molecule
Crystalline NaCl has a density of 2.16 X 10 3 kg/m 3 , and so. taking into account
the presence of two atoms in each NaCl "molecule," the number of atoms in
I m 3 of NaCl is
atoms „.„,„, kg 1
„ = 2 ■ ■ .  X2.I6X 10 3 %X
molecule
= 4.45 x 10™ atoms/m 3
m 3 9.72 X 10 6 kg/ molecule
If d is the distance between adjacent atoms in a crystal, there are d l atoms/m
along any of the crystal axes and d~ 3 atoms/m;' in the entire crystal. Hence
and
d 3 = n
d = n' 3 = {4.45 x lO 28 )" 3 m
= 2.82 X l(r 10 m
= 2.82 A
PARTICLE PROPERTIES OF WAVES
59
60
2.5 THE COMPTON EFFECT
The quantum theory of light postulates that photons behave like particles except
for the absence of any rest mass. If this is tnie, then it should lie possible for
its to treat collisions between photons and, say, electrons in the same manner
as billiardball collisions are treated in elementary mechanics.
Figure 213 shows how such a collision might !>e represented, with an Xray
photon striking an electron {assumed to be initially at rest in (he laboratory
coordinate system) and being scattered away from its original direction of motion
while the electron receives an impulse and begins to move. In the collision the
photon may l>e regarded as having lost an amount of energy that is the same
as the kinetic energy T gained by the electron, though actually separate photons
are involved. If the initial photon has the frequency » associated with it, the
scattered photon has the lower frequency v\ where
Loss in photon energy = gain in electron energy
29 hv  hv' =T
From the previous chapter we recall that
£= Vm„V + p 2 c 2
so that, since the photon has no rest mass, its total energy is
£ = pc
E — hv
Since
FIGURE Z13 The Compton .Heel.
incident photon
^r^
 »*oc i, = hv'/c
p =
scattered
electron
BASIC CONCEPTS
for a photon, its momentum is
2.10
c
Momentum, unlike energy, is a vector quantity, incorporating direction as well
as magnitude, and in the collision momentum must be conserved in each of two
mutually perpendicular directions. {When more than two Inxlics participate in
a collision, of course, momentum must be conserved in each of three mutually
perpendicular directions.) The directions we choose here are that of the original
photon and one perpendicular to it in the plane containing the electron and
the scattered photon (Fig. 213), The initial photon momentum is hv/c, the
scattered photon momentum is hv'/c, and the initial and final electron momenta
are respectively and p. In the original photon direction
Initial momentum = final momentum
2.11
— + = — cos + n costf
c c
and perpendicular to this direction
Initial momentum = final momentum
2.12
(! = sin $ — p sin (i
The angle § is that between the directions of the initial and scattered photons,
andff is that between the directions of the initial photon and the recoil electron.
From Ekjs. 2.9, 2.11, and 2.12 we shall now obtain a formula relating the wave
length difference between initial and scattered photons with the angle 4> hetween
their directions, Iwth of which are readily measurable quantities.
The first step is to multiply Eqs. 2.1 1 and 2.12 by c and rewrite them as
pc cos = hv — hv' cos $
pc sin = hv' sin $
By squaring each of these equations and adding the new ones together, the angle
is eliminated, leaving
2.13 pV = {hv?  2(hv)(lu>')cos<!> + (hv') 2
Next we equate the two expressions for the total energy of a particle
K = T f m^c*
E= Vm^V + p V
PARTICLE PROPERTIES OF WAVES
61
from the previous chapter to give
Since
we have
{T + m c 2 ) 2 = wi V + p 2 c 2
p 2 c 2 = T 2 + Sm^T
T = hv  fw'
2W p V = faf _ 2{lw)(h»') + {hv'f + 2m c 2 (/i*  hr')
Substituting this value of p 2 c 2 in Eq. 2.13, we finally obtain
2.1S 2m a c(hv  hv') = 2(h>>)(hr'){l  cos <j>)
This relationship is simpler when expressed in terms of wavelength rather than
frequency. Dividing Eq. 2,15 by 2h 2 c 2 ,
r( ) = (1  cos*
and so, since v/c = 1/A and v'/c = 1/A',
™ c / 1 1\ _
h U X'l
(1  cos $)
AV
2.16
\' \=2(\ COS*)
Com p ton effect
62
Equation 2.16 was derived by Arthur H. Cotnpton in the early 1920s, and the
phenomenon it describes, which he was the first to observe, is known as the
Compton effect. It constitutes very strong evidence in support of the quantum
theory of radiation.
Equation 2.16 gives the change in wavelength expected for a photon that is
scattered through the angle 4> by a particle of rest mass m ; it is independent
of the wavelength A of the incident photon. The quantity h/m c is called the
Cwnpton wavelength of the scattering particle, which for an electron is 0.024 A
(2.4 X 10 _lz m). From Eq. 2.16 we note that the greatest wavelength change
that can occur will take place for <f> = 180°, when the wavelength change will
lie twice die Compton wavelength h/niff. Because the Compton wavelength
of an electron is 0.024 A, while it is considerably less for other particles owing
to their larger rest masses, the maximum wavelength change in die Compton
effect is 0.048 A. Changes of this magnitude or less are readily observable only
in X rays, since the shift in wavelength for visible light is less than 0.0 1 percent
of the initial wavelength, while for X rays of A = 1 A it is several percent.
The experimental demonstration of the Compton effect is straightforward.
As in Fig. 214, a beam of X rays of a single, known wavelength is directed
BASIC CONCEPTS
FIGURE 214 Experimenlal demonstration
of tho Compton effect
Xray spectrometer''
unscattered
Xray /
source of
monochrnmntii
X ravs
collimators
path of
patn oi %
spectrometer \ ,/
/
/
i
at a target, and the wavelengths of the scattered X rays are determined at various
angles 0. The results, shown in Fig. 215. exhibit the wavelength shift predicted
by Eq. 2.16, but at each angle the scattered X rays also include a substantial
proportion having the initial wavelength. This is not hard to understand. In
deriving Eq. 2.16, we assumerl that the scattering particle is able to move freely,
a reasonable assumption since many of the electroas in matter are only loosely
lx»und to their parent atoms. Other electrons, however, are very tightly Iwund
and, when struck by a photon, the entire atom recoils instead of the single
electron. In this event the value of »» to use in Eq. 2,16 is that of the entire
atom, which is tens of thousands of times greater than that of an electron, and
the resulting Compton shift is accordingly so minute as to be undetec table.
2.6 PAIR PRODUCTION
As we have seen, a photon can give up all or part of its energy hv to an electron.
It is also possible for a photon to materialize into an electron and a positron
(positive electron), a process in which electromagnetic energy is converted into
rest energy. No conservation principles are violated when an electronposit run
pair is created near an atomic nucleus (Fig. 216). The sum of the charges of
the electron (o = e) and of the positron (<7 = +e) is zero, as is the charge
of the photon; the total energy, including mass energy, of the electron and
positron equals the photon energy; and linear momentum is conserved with the
help of the nucleus, which carries away enough photon momentum for the
PARTICLE PROPERTIES OF WAVES
63
<t> = 0°
WAVELENGTH
WAVELENGTH
= 90"
WAVELENGTH
*
I
LU
z
UJ
>
P
5
UJ
a:
t /
AX
= 135°
WAVELENGTH
FIGURE 2 15 Compton scattering.
64 BASIC CONCEPTS
photon
FIGURE 216 Pair production.
/
—jf electron
X
positron
process to occur but, because of its relatively enormous mass, only a negligible
fraction of the photon energy. (Energy and linear momentum could not Ixith
!>e conserved if pair production were to occur in empty space, so it does not
occur there.)
The rest energy «j u c 2 of an electron or positron is 0.51 MeV, and so pair
production requires a photon energy of at least 1 .02 MeV. Any additional photon
energy hecomes kinetic energy of the electron and positron. The corresponding
maximum photon wavelength is 0.012 A. Electromagnetic waves with such
wavelengths are called gamma rays, and are found in nature as one of the
emissions from radioactive nuclei and in cosmic rays.
The inverse of pair production occurs when an electron and positron come
together and are annihilated to create a pair of photons. The directions of the
photons are such as to conserve both energy and linear momentum, and so no
nucleus or other particle is required for annihilation to take place.
Three processes in all are therefore responsible for the absorption of X and
gamma rays in matter. At low photon energies Compton scattering is the sole
mechanism, since there are definite thresholds for both the photoelectric effect
(several cV) and electron pair production ( 1 .02 MeV). Both Compton scattering
and the photoelectric effect decrease in importance with increasing energy, as
shown in Fig. 217 for the case of a lead absorber, while the likelihood of pair
production increases. At high photon energies the dominant mechanism of
energy loss is pair production. The curve representing the total absorption in
lead has its minimum at about 2 MeV. The ordinate of the graph is the linear
absorption coefficient /i, which is equal to the ratio between the fractional
decrease in radiation intensity —dl/I and the absorber thickness tlx. That is,
d±
i
= —fidx
PARTICLE PROPERTIES OF WAVES
65
2>^^Z. , _ pl^ojoclectric effect
4 e
PHOTON ENERGY, MeV
10
FIGURE 217 Linear absorption coefficients for photons in lead. These curves refer to
lion, not to the likelihood of Interactions in the medium.
energy absorp
whose solution is
/ = v
The intensity of the radiation decreases exponentially with the thickness of the
absorber.
2.7 GRAVITATIONAL RED SHIFT
Although a photon has no rest mass, it nevertheless behaves as though it possesses
the inertia! mass
2.17
he
Photon "mass"
66 BASIC CONCEPTS
Does a photon possess gravitational mass as well? Since the inertial and gravita
tional masses of all material bodies are found experimentally to be equal (this
principle of equivalence is one of the starting points of Einstein's general theory
of relativity), it would seem worth looking into the question of whether photons
have the same gravitational behavior as other particles.
Let us consider a photon of frequency v emitted from the surface of a star
of mass M and radius R (Fig. 218). The potential energy of a mass m on the
star's surface is
2.18
v= 
CMm
The potential energy of the photon is accordingly
GMhp
V= 
c*R
and its total energy E, the sum of V and the quantum energy hv, is
E = hv
GMhf
2.19
4 SB)
At a large distance from the star, for instance at the earth, the photon is beyond
the star's gravitational field but its total energy remains the same. The photon's
energy is now entirely electromagnetic, and
2.20
£ = hv 1
FIGURE 218 The frequency ot a photon emitted from the surf»ee of a iter decreases as It moves away
from the star.
PARTICLE PROPERTIES OF WAVES
67
where ,' Is the frequency of the arriving photon. (The potential energy of the
photon m the earth's gravitational field is negligible compared with that in the
stars field.) Hence
»»(i£)
2.21
I he photon has a lower frequency at the earth, corresponding to its loss in energy
as .t leaves the field of the star. A photon in the visible region of the spectrum
>s thus shifted toward the red end, and this phenomenon is accordingly known
as the gravitational red shift. It must 1* distinguished from the doppler'red afaifl
observed in the spectra of distant galaxies due to their apparent recession from
the earth, a recession attributed to a general expansion of the universe
As we shall learn in Chap. 4, the atoms of every element, when suitablv excited
cunt photons of certain specific frequencies only. The validity of Eq2 21 can
therefore Ik checked by comparing the frequencies found in stellar spectra with
those in spectra obtained in the lalwratory. .Since C/r is only
G _ 6.67 X 10" NmVkg 2
c 1 ~ (3xl0«m/f = 7Al X 10 " 28 m/k S
the gravitational red shift can be observed only in radiation from very dense
stars. In the ease of the sun, a more or less ordinary star, R = 6 9f> y Iff" m
and Mm LSQ X Hr 3n kg, and
— = t~ = 7.41 x 102" HL x '•'■ J x 1U k g
' C ' R kg em x m 8 m
= 2.12 X 10°
S r Q ^ r^"' ^ S raviEationaI red *» ^ solar radiation amounts to onlv
about 0.01 A for green light of wavelength 5,000 A and is undetectable due to
the doppler broadening of the spectral lines.
Ho«,v,.,, lhore is a class of stars in the final stages of their evolution called
while dwarfs that are composed of atoms whose electron structures have "col
lapsed, and such stars have quite enonnous densitieslvpicallv ~5 tons/in *
A white dwarf might have a radte of 9 X 10" m, about' 0.01 that of the sun
and a mass of 1.2 X 10* kg, about 0.6 that of the sun. so that
68
BASIC CONCEPTS
CM
— = 177T = "41 X 10
A,> _
~ c 2 R
Zz 10"^
m 1.2 X lO^kf
kg
9 X 10 B m
I lere the gravitational ret! shift would 1« :=0.5 A for light of wavelength 5,000 A ,
which is measurable under favorable circumstances. In the case of the white
dwarf Sirius B (the "companion of Sirius"), the predicted red shift is Ac/k
~ 5.9 x 10~ 5 and the observed shift is 6.C X !0~ 5 ; in view of the uncertainty
in the M/R ratio for Sirius B, these figures would seem to confirm the attribution
of gravitational mass to photons.
If there is a star for which CM/cR > 1, we see from Eq. 2.10 that no photon
can ever leave it. Such a star could not radiate and would be invisible— a "Made
hole" in space. There seems to be no fundamental reason why black holes should
not actually exist, and it should be possible to detect them by virtue of the
combination of their lightalworhin;4 ability and their gravitational effects on
astronomical objects in their vicinity. Curiously, the known universe may be
a black hole in itself: the mass and radius of the universe are believed to be
about 10 53 kg and 10 2a in respectively and, since G/c* Si 10 i " m/kg,
(CM/ C *fl) unlver8e =l.
Recently a gravitational frequency shift has been detected in a laboratory
experiment by measuring the change in frequency of gamma rays after they had
"fallen" through a height h near the earth's surface. A body of mass m that
falls a height It gains mgh of energy. If a falling photon of original frequency
v is taken to have the ccaslaut mass hv/c (the frequency shift is so small that
the change in mass may be neglected), its final energy hv' is given by
hi"'h
ht>' = hv + ingh = hv H 1—
2.22
4+£)
For h = 20 m,
A* _ gn _ 9.8 in/s 2 X 20 m
~~1?~ (3 X 10 8 m/sf
= 2.2 x 10 I5
A shift of Uiis magnitude is just detectable, and the results confirm Eq. 2.22.
PARTICLE PROPERTIES OF WAVES
69
Problems
1. The threshold wavelength for photoelectric emission in tungsten is 2,300 A.
What wavelength of light must be used in order for electrons with a maximum
energy of 1.5 eV to be ejected?
2. The threshold frequency for photoelectric emission in copper is
1.1 x 10 15 Hz. Find the maximum energy of the photoelectrons (in jotiles and
electron volts) when light of frequency 1 .5 x 10 15 Hz is directed on a topper
surface.
3. The work function of sodium is 2.3 eV. What is the maximum wavelength
of light that will cause photoelectrons to be emitted from sodium? What will
the maximum kinetic energy of the photoelectrons be if 2,000 A light falls on
a sodium surface?
4. Find the wavelength and frequency of a 100MeV photon.
5. Find the energy of a 7,000A photon.
6. Under favorable circumstances the human eye can detect 10" IH J of electro
magnetic energy. How many 6,000A photons does this represent?
7. A 1,000W radio transmitter operates at a frequency of 880 kHz. How many
photons per second does it emit?
8. How many photons per second are emitted by a 10W yellow lamp? (Assume
the light is monochromatic with a wavelength of 6,(KX) A.)
9. Light from the sun arrives at the earth at the rate of about 1 ,400 W/m 2
of area perpendicular to the direction of the light, (a) Find the maximum pressure
(in lb/in. 2 ) this light can exert on the earth s surface. (6) Assume that sunlight
consists exclusively of 6,000A photons. How many photons per m arrive at
that part of the earth directly facing the sun in each second? (c) The average
radius of the earth's orbit is 1.5 X 10" in. What is the power output of the
sun in watts, and how many photons per second does it emit? (d) How many
photons per m 3 are there near the earth?
10. What is the wavelength of the X rays emitted when 100keV electrons strike
a target? What is their frequency?
11. An Xray machine produces 0.1A X rays. What accelerating voltage does
it employ?
12. The distance between adjacent atomic planes in calcite is 3 X 10 8 cm.
What is the smallest angle between these planes and an incident beam of 0.3 A
X rays at which scattered X rays can be detected?
13. A potassium chloride crystal has a density of 1.98 X 10 3 kg/m 3 . The
molecular weight of KC1 is 74.55. Find the distance between adjacent atoms.
14. How much energy must a photon have if it is lo have the momentum of
a 10MeV proton?
15. What is the frequency of an Xray photon whose momentum is
1.1 X lO^kgm/s?
16. Prove that it is impossible for a photon to give up all its energy and
momentum to a free electron, so that the photoelectric effect can take place
only when photons strike bound electrons.
17. A beam of X rays is scattered by free electrons. At 45° from the beam
direction the scattered X rays have a wavelength of 0.022 A. What is the
wavelength of the X rays in the direct Ijcam?
18. An Xray photon whose initial frequency was 1.5 X H) in Hz emerges from
a collision with an electron with a frequency of 1.2 X 10 19 Hz. How much
kinetic energy was imparted to the electron?
19. An Xray photon of initial frequency 3 x 1 0''' 1 \v cull ides with an electron
and is scattered through 90°. Find its new frequency.
20. Find the energy of an Xray photon which can impart a maximum energy
of .50 keV to an electron.
21. A monochromatic Xray beam whose wavelength is 0.558 A is scattered
through 46". Find the wavelength of the scattered beam.
22. In Sec. 2.4 the X rays scattered by a crystal were assumed to undergo no
change in wavelength. Show that this assumption is reasonable by calculating
the Compton wavelength of a Na atom and comparing it with the typical Xray
wavelength of 1 A,
23. As discussed in Chap. 1 2, certain atomic nuclei emit photons in undergoing
transitions from "excited" energy states to their "ground" or normal states. These
photons constitute gamma rays. When a nucleus emits a photon, it recoils in
the opposite direction, (a) The Co nucleus decays by K capture to jSgFe, which
then emits a photon in losing 14.4 keV to reach its ground state. The mass of
a 5gFe atom is 9.5 X I0" 26 kg. By how much is the photon energy reduced from
70
BASIC CONCEPTS
PARTICLE PROPERTIES OF WAVES
71
the full 14.4 keV available as a result of having to share energy with the recoiling
atom? (ft) In certain crystals the atoms are so tightly Itound that the entire crystal
recoils when a gamma ray photon is emitted, instead of the individual atom.
This phenomenon is known as the Mfcshauer effect. By how much is the photon
energy reduced in this situation if the excited iJFe nucleus is part of a Ig crystal?
'(c) The essentially recoilfree emission of gamma rays in situations like that of
(ft) means (hat it is possible to construct a source of essentially monoencrgetic
and hence monochromatic photons. Such a source was used in the experiment
described in the last paragraph of Sec. 2.7. What is the original frequency and
the change in frequency of a 14.4keV gammaray photon after it has fallen 20 m
near the earth's surface?
24. A positron collides head on with an electron and both are annihilated. Each
particle had a kinetic energy of I MeV. Find the wavelength of the resulting
photons.
WAVE PROPERTIES OF PARTICLES
In retrospect it may seem odd that two decades passed between the discovery
in 1905 of the particle properties of waves and the speculation in 1924 that
particles might exhibit wave behavior. It is one thing, however, to suggest a
revolutionary hypothesis to explain Otherwise mysterious data and quite another
to advance an equally revolutionary hypothesis in the absence of a strong
experimental mandate. The latter is just what Louts de Broglie did in H)24 when
he proposed that matter possesses wave as well as particle characteristics. So
different was the intellectual climate at the lime from that prevailing at the turn
of the century that de Broglie's notion received immediate and respectful atten
tion, whereas the earlier quantum theory of light of Planck and Einstein created
hardly any stir despite its striking empirical support. The existence of
de Broglie waves was demonstrated by 1927, and the duality principle they
represent provided the starting point for Schrodinger's successful development
of quantum mechanics in the previous year.
3.1 OE BROGLIE WAVES
A photon of light of frequency v has the momentum
which can be expressed in terms of wavelength A as
P= \
since A v = c. The wavelength of a photon is therefore specified by its momentum
according to the relation
3
3.1
A = A
72
BASIC CONCEPTS
73
74
Drawing upon an intuitive expectation that nature is symmetric, de Brogiie
asserted that Eq. 3.1 is a completely general formula that applies to material
particles as well as to photons. The momentum of a particle of mass m and
velocity v is
p = mis
and consequently its de Brogiie wavelength is
h
3.2
A =
tin:
De Brogiie waves
The greater the particle's momentum, the shorter its wavelength. In Eq. 3.2
in is the relativistic mass
m =
VI  oVc*
Equation 3.2 has been amply verified by experiments involving the diffraction
of fast electrons by crystals, experiments analogous to those that showed X rays
to be electromagnetic waves. Before we consider these experiments, it is appro
priate to look into the question of what kind of wave phenomenon Is involved
in the matter waves of de Brogiie. In a light wave the electromagnetic field
varies in space and lime, in a sound wave pressure varies in space and time;
what is it whose variations constitute de Brogiie waves?
3.2 WAVE FUNCTION
The variable quantity characterizing de Brogiie waves is called the wave func
tion, denoted by the symbol + (die Greek letter psl). The value of the wave
function associated with a moving body at the particular point x, y. z in space
at the time ( is related to the likelihood of finding the body there at the time.
+ itself, however, has no direct physical significance. There is a simple reason
why + cannot he interpreted in terms of an experiment. The probability P that
something be somewhere at a given time can have any value between two limits:
0, corresponding to the certainty of its absence, and J, corresponding to the
certainty of its presence. (A probability of 0.2, for instance, signifies a 20 percent
chance of finding the body.) But the amplitude of any wave may be negative
as well as positive, and a negative probability is meaningless. Hence * itself
cannot be an observable quantity.
This objection does not apply to *J* the square of the absolute value of the
wave function. For this and other reasons +j 2 is known as probability density.
The probability of experimentally finding the laxly described by the wave
BASIC CONCEPTS
function + at the point x, y, z at the time / is proportional to the value of l+l*
there at (. A large value of l+l 2 means the strong possibility of the body's
presence, while a small value of t+l 2 means the slight possibility of its presence.
As long as + 2 is not actually somewhere, however, there is a definite chance,
however small, of delecting it there. This interpretation was first made by Max
Born in 1926.
There is a big difference between the probability of an event and the event
itself, Although we shall speak of the wave function * that describes a particle
as being spread out in space, this does not mean that the particle itself is thus
spread out. When an experiment is performed to detect electrons, for instance,
a whole electron is either foimd at a certain time and place or it is not; there
is no such thing as 20 percent of an electron. However, it is entirely possible
for there to lie a 20 percent chance that the electron be found at that time and
place, and it is this likelihood that is specified by 4M 2 .
Alternatively, if an experiment involves a great many identical bodies all
described by the same wave function +, the actual density of bodies at x, y, z
at the time t is proportional to the corresponding value of ^ 2 .
While the wavelength of the de Brogiie waves associated with a moving body
is given by the simple formula
me
determining their amplitude + as a function of position and time usually presents
a formidable problem. We shall discuss the calculation of + in Chap. 5 and
then go on to apply the ideas developed there to the structure of the atom in
Chap. 6. Until then we shall assume that we have whatever knowledge of *
is required by the situation at hand.
In the event that a wave function + is complex, with both real and i in aginary
parts, the probability density is given by the product ^° "V of + and its complex
conjugate V. The complex conjugate of any function is obtained by replacing
i { = \/— 1) by — i wherever it appears in the function. Every complex function
* can be written in the form
+ =A + iB
where A and B are real functions. The complex conjugate S* * of ^ is
** = A  iB
and so
*•* = A 2  i 2 B* = A 2 f B
since i 2 = — 1. Hence "fr* 4* is always a positive real quantity.
WAVE PROPERTIES OF PARTICLES
75
3.3 DE BROGUE WAVE VELOCITY
With what velocity do de Broglie waves travel? Since we associate a tie Broglie
wave with a moving tody, it is reasonable to expect that this wave travels
at the same velocity D as [hat of the tody. If we call the de Broglie wave ve
locity w, we may apply the usual formula
w = p\
to determine the value of .«. The wavelength A is just the de Broglie wavelength
III!
We shall take the frequency , to be that specified by the quantum equation
E = lw
Hence
or, since
we have
h
E = me 2
v —
mc
The de Broglie wave velocity is therefore
3.3 i, _ ,.,\
_ mc 2 h
h me
Since the particle velocity v must be less than the velocity of light c the
de Broglie wave velocity w is always greater than d Clearly v and w are never
equal for a moving body. In order to understand this unexpected result, we shall
digress briefly to consider the notions of plutse velocity zndgnmp velocity. (Phase
velocity is sometimes called wave velocity.)
Let us begin by reviewing how waves are described mathematically For
elanty we shall consider a string stretched along the x axis whose vibrations are
m the y direction, as in Fig. 31, and are simple harmonic in character. If we
choose  = when the displacement „ of the string at x = is a maximum its
displacement at any future time t at the same place is given by the formula
3.4 y = A cos 2xt't
76 BASIC CONCEPTS
( =
vibrating string 
FIGURE 31 Wave motion.
1 =
y = A cos 2w vt
where A is the amplitude of the vibrations {that is, iheir maximum displacement
on either side of the x axis) and c their frequency.
Equation 3.4 tells us what the displacement of a single point on the string
is as a function of time r. A complete description of wave motion in a stretched
string, however, should tell us what y is at any point on the string at any time.
What we want is a formula giving y as a function of both x and t. To obtain
such a formula, let us imagine that we shake the string at x = when t = 0,
so that a wave starts to travel down the string in the + ,% direction (Fig. 32).
This wave has some speed w that depends upon the properties of the string.
The wave travels the distance x = wt in the time f; hence the time interval
between the formation of the wave at x = (I and its arrival at the point x is
x/w. Accordingly the displacement y of the string at v at any time t is exactly
I he same as the value of y at x = at the earlier time t — x/w. By simply
replacing t in Eq. 3.4 with t — x/w, then, we have the desired formula giving
t/ in terms of both ,v and /:
3.5
y = A cos2wclr  —1
As a check, we note that Eq. 3.5 reduces to Eq, 3.4 at x = 0.
Equation 3.5 may be rewritten
y = A cas2fflcf — —J
Since
w = v \
WAVE PROPERTIES OF PARTICLES
77
*
t =
FIGURE 32 Wave propagation
we have
3.6 <j = A COs27r(j't  ij
Equation 3.6 is often more convenient to apply then Eq. 3.5.
Perhaps the most widely used description of a wave, however, is still another
form of Eq. 3.5. We define the quantities angular frequency u and wove number
k by the Formulas
3.7
3.S
3.9
to = 2trv
Angular frequency
Wave number
The unit of w is the rad/s and that of k is the rad/m. Angular frequency gets
its name from uniform circular motion, where a particle thai moves around a
circle v times per second sweeps out 2.<nv rad/s. The wave number is equal to
the number of radians corresponding to a wave train I m long, since there are
2ir rad in one complete wave. In terms of a and k, Eq. 3,5 becomes
3.10 y = A cos (at — kx)
In three dimensions k becomes a vector k normal to the wave fronts and x is
replaced by the radius vector r. The scalar product k • r is then used instead
of far in Eq. 3.10.
3.4 PHASE AND GROUP VELOCITIES
The amplitude of the de Broglie waves that correspond to a moving body reflects
the probability that it be found at a particular place at a particular time. It
is clear that de Broglie waves cannot lw represented simply by a formula re
sembling Eq. 3.10, which describes an indefinite series of waves all with the same
amplitude A. Instead, we expect the wave representation of a moving body to
correspond to a wave packet, or wn\:e group, like that shown in Fig. 33, whose
constituent waves have amplitudes upon which the likelihood of detecting the
Iwdy depends,
A familiar example of how wave groups come into being is the case of beats.
When two sound waves of the same amplitude, but of slightly different frequen
cies, are produced simultaneously, the sound we hear has a frequency equal to
the average of the two original frequencies and its amplitude rises and falls
periodically. The amplitude fluctuations occur a number of times per second
equal to the difference Iwtween the two original frequencies. If the original
sounds have frequencies of, say, 440 and 442 Hz, we will hear a fluctuating sound
of frequency 441 Hz with two loudness peaks, called heats, per second. The
production of l>eats is illustrated in Fig. 34.
A way of mathematically describing a wave group, then, is in terms of a
superposition of individual wave patterns, each of different wavelength, whose
interference with one another results in the variation in amplitude that defines
the group shape. If the speeds of the waves are the same, the speed with which
the wave group travels is identical with the common wave speed. However,
if the wave speed varies with wavelength, the different individual waves do not
proceed together, and the wave group has a speed different from that of the
waves that compose it.
It is not difficult to compute the speed tt with which a wave group travels,
I«t us suppose that a wave group arises from the combination of two waves
FIGURE 33 A wave group.
wave group
78
BASIC CONCEPTS
WAVE PROPERTIES OF PARTICLES
79
MVWWVWV
+
vwwwwwv
FIGURE 34 The production of beaiv.
with the same amplitude A but differing by an amount <iw in angular frequency
and aii amount dk in wave number. We may represent the original waves by
the formulas
;/! = A eos (oil — kx)
j/jj = A cos [(« + (/w)(  (/c f dk)x]
The resultant displacement ij at any time ( and any position * is the sum of y l
and ij t . With the help of the identity
cos a + cos ft = 2 cos %{a + ft) cos %(a — ft)
and die relation
cos{ — 9) = eostf
we find that
¥  9i + A
= 2A cos &(&> + f /«)f  (2fc + (tk)x\ cos </ 2 (rf w f  ilk x)
Since «*« and tik are small compared with t> and fe respectively,
2w + rfu ~ 2u
2* + d* sr 2k
and
3.11
y = 2A cos (w( — A*) cos
(ff')
Equation 3.11 represents a wave of angular frequency o: and wave number k
that lias superimposed upon it a modulation of angular frequency '/U/w and of
wave number l / 2 dk. The effect of the modulation is to produce successive wave
groups, as in I'ig. 34. The phase velocity w is
3.12
•I
while die velocity u of the wave groups is
do
3.13
dk
Phase velocity
Group velocity
In general, depending upon the manner in which phase velocity varies with wave
number in a particular medium, the group velocity may be greater than or less
than the phase velocity. If the phase velocity w is the same for all wavelengths,
the group and phase velocities are the same.
The angular frequency and wave number of the de Broglie waves associated
with a body of rest mass m u moving with the velocity v are
3.14
a = 2,xt>
h
1 1 vl  V 2 /r
and
3.15
k 2 *
Qmnv
'Zirm^ti
Both « and k are functions of the velocity o. The phase velocity w is, as we
found earlier,
w = —
k
80
BASIC CONCEPTS
WAVE PROPERTIES OF PARTICLES
81
82
which exceeds Ixrth the velocity of the lxidy v and the velocity of light c, since
l! < C.
The group velocity u of the de Broglie waves associated with the body is
tha
elk
\V,\\
dw/dt
dk/dv
dw
2itm v>
dv
' ft(l
 &f<?f n
dk
2irm u
and
dv ft(l  oVc 2 ) 3 ' 2
and so the group velocity is
3.16 U = V
The de Broglie wave group associated with a moving body travels with the same
velocity as the body. The phase velocity tt> of the de Broglie waves evidently
has no simple physical significance in itself.
3.5 THE DIFFRACTION OF PARTICLES
A wave manifestation having no analog in the behavior of Newtonian particles
is diffraction. In 1927 Davisson and Germer in the United States and G. P.
Thomson in England independently confirmed de Broglie's hypothesis by dem
onstrating that electrons exhibit diffraction when they are scattered from crystals
whose atoms are spaced appropriately. We shall consider the experiment of
Davisson and Germer because its interpretation is more direct.
Davisson and Germer were studying the scattering of electrons from a solid,
using an apparatus like that sketched in Fig. .35. The energy of the electrons
in the primary beam, the angle at which they are incident upon the target, and
the position of the detector can all be varied. Classical physics predicts that
the scattered electrons will emerge in all directions with only a moderate
dependence of their intensity upon scattering angle and even less upon the energy
of the primary electrons. Using a block of nickel as the target, Davisson and
Germer verified these predictions.
I r> the midst of their work there occurred an accident that allowed air to enter
their apparatus and oxidize the metal surface. To reduce the oxide to pure nickel,
the target was baked in a hightemperature oven. After this treatment, the target
BASIC CONCEPTS
electron gun
electron
detector
FIGURE 35 Tha Damson Germer experiment.
scattered
beam
was returned to the apparatus and the measurements resumed. Now the results
were very different from what had been found before the accident: instead of
a continuous variation of scattered electron intensity with angle, distinct maxima
and minima were observed whose positions depended upon the electron energy!
Typical polar graphs of electron intensity after the accident are shown in Fig.
36; the method of plotting is such that the intensity at any angle is proportional
to the distance of the curve at that angle from the point of scattering.
Two questions come to mind immediately: what is the reason for this new
effect, and why did it not appear until after the nickel target was baked?
De Broglie's hypothesis suggested the interpretation that electron waves were
being diffracted by the target, much as X rays are diffracted by planes of atoms
in a crystal. This interpretation received support when it was realized that the
effect of heating a block of iiieke! at high temperature is to cause the many
FIGURC 36 R*»ultt ol the Damson Germer experiment.
■10 V 44 V 48 V 54 V 60 V 64 V 68 V
WAVE PROPERTIES OF PARTICLES
83
smalt individual crystals of which it is normally composed to form into a single
targe crystal, all of whose atoms are arranged in a regular lattice.
Let ns sec whether we can verify that de Broglie waves arc responsible for
the findings of Davisson and Cermer. In a particular determination, a beam of
54eV elections was directed perpendicularly at the nickel target, and a sharp
maximum in die electron distribution occurred at any angle of 50° with the
original beam. Tile angles of incidence and scattering relative to the family of
Bragg planes shown in Fig. 37 will both be 65°. The spacing of the planes
in this family, which can 1m; measured by Xray diffraction, is 0.91 A. The Bragg
equation for maxima in the diffraction pattern is
n\ = '2d sin i)
Here d = 0,9 1 A and 6 = 85°; assuming that n = 1, the de Broglie wavelength
X of die diffracted electrons is
n\ = 2d sin (i
= 2 X 0.9 1 A X 65"
= 1.65 A
Now we use de Broglie's formula
to calculate the expected wavelength of the electrons. The electron kinetic
energy of 54 eV is small compared with its rest energy m n c 2 of 5.1 X 10* eV,
and so we can ignore relalivistic considerations. Since
FIGURE 37 The diffraction of de Broglie waves by the target
is responsible for the results of Davisson and Germer.
the electron momentum mr is
me = ^2mT
= \/2 X 9.1 X lO" 31 kg X 54 eV X 16 X 10~ li, J/eV
= 40 X lO" 2 ' 1 kgm/s
The electron wavelength is therefore
me
6.63 x lO"* 1 Js
single crystal
ol nickel
4.0 X 10 ' kgm/s
= 1.66 X 10 u 'm
= 1.66 A
in excellent agreement with the observed wavelength. The DavissonCenner
experiment dins provides direct verification of de Broglie's li\ poliosis ol the wave
nature of moving bodies.
The analysis of the Davisson Germer experiment is actually less straightforward
than indicated above, since the energy of an electron increases when it enters
a crystal by an amount equal to the work function of the surface. Hence the
electron speeds in the experiment were greater inside the crystal and the corre
sponding de Broglie wavelength shorter than the corresponding values outside.
\n additional complication arises from interference between waves diffracted
by different families of Bragg planes, which restricts the occurrence of maxima
to certain combinations of electron energy and angle of incidence rather than
merely to any combination that obeys the Bragg equation.
Electrons arc not the only bodies whoso wave behavior can be demonstrated.
The di (fraction of neutrons and of whole atoms when scattered by suitable crystals
has liccn observed, and in fact neutron diffvaclion, like Xray and electron
diffraction, is today a widely used tool for Investigating crystal structures,
As in the case of electromagnetic waves, the wave and particle aspects of
moving bodies can never be simultaneously observed, so that we cannot deter
mine which is die "correct" description. All we can say is thai in some respects
a moving body exhibits wave properties and in other respects it exhibits particle
properties. Which set of properties is most conspicuous depends upon how the
de Broglie wavelength compares with the dimensions of the bodies involved;
the 1.66 A wavelength of a 54eV electron is of the same order of magnitude
as the lattice spacing in a nickel crystal, but the wavelength of an automobile
moving at 60 mi h is about 5 X 10 ;is ft, far loo small to manifest itself.
84
BASIC CONCEPTS
WAVE PROPERTIES OF PARTICLES
85
3.6 THE UNCERTAINTY PRINCIPLE
The fact that a moving body must Ik; regarded as a de Broglie wave group rather
than as a localized entity suggests that there is a fundamental limit to iho
accuracy with which we can measure its particle properties. Figure 3Ha show s
a de Broglie wave group: the particle may Imj anywhere within the wave group.
If the group is very narrow, as in Fig. 38/j, the position of the particle is readily
found, hut the wavelength is impossible to establish. At the other extreme, a
wide group, as in Fig. 3Kc. permits a saHsfac lory wavelength estimate, but where
is the particle located?
A straightforward argument based upon the nature of wave groups permits
us to relate the inherent uncertainty A* in a measurement of particle position
with the inherent uncertainty Ap in a simultaneous measurement of its momen
tum.
The simplest example of the formation of wave groups is that given in
Sec. 3.4, where two wave trains slightly different in areolar frequency w and
propagation constant k were superposed to yield the series of groups shown in
Fig. 34. Here let us consider the wave groups that arise when the de Brogfie
waves
*, = A cos (wf  far)
* 2 = A [cos \*s + lu)t (k + &k)x\
FIGURE 38 The width of a wave group is a measure of the uncertainty in the location of the particle
It represents. The narrower the wave group, (he greater the uncertainty In the wavelength.
FIGURE 39 Wave groups that result
but different frequencies.
from the interference of wave trains hiving the same amplitudes
are combined. From a calculation identical with the one used in obtaining
Eq. 3.11, we find that
3.17 + = *, + %
zz 2A cos (wt  kx) cos ('/ 2 Aw /  %\k x)
which is plotted in Fig. 39. The width of each group is evidently equal to half
the wavelength X m of the modulation. It is reasonable to suppose that this width
is of the same order of magnitude as the inherent uncertainty A.t in the position
of the group, that is,
348 Ax = '/jA,,
The modulation wavelength is related to its propagation constant k m by
m
From Eq, 3.17 we see that the propagation constant of the modulation is
k m = %\k
with the result that
2*f
A,„ —
%M
and
3.19
lx ~Tk
A moving body corresponds to a single wave group, not a succession of them,
but a single wave group can also be thought of in terms of the superposition
of trains of harmonic waves. However, an infinite number of wave trains of
86
BASIC CONCEPTS
WAVE PROPERTIES OF PARTICLES
87
different frequencies, wave numbers, and amplitudes is required for an isolated
group of arbitrary shape.
At a certain time I, the wave group *(*) can be represented by the Fourier
integral
3.20
*(*) aa f g(k) cos kx (Ik
where the function g(fc) describes how the amplitudes of the waves that contribute
to *{.r) vary with wave number k. This fimetion is called the Fourier transform
of +(x), and it specifies the wave group just as completely as ♦(*) does.
Figure 310 contains graphs of the Fourier transforms of a pulse and of a wave
group. For comparison, the Fourier transform of an infinite train of harmonic
waves is also included: only a single wave uumlier is present in this ease, of course.
Strictly speaking, the wave numbers needed to represent a wave group extend
Irani A = to k m oo, but for a group whose length A.* is finite, the waves whose
amplitudes g(ftj are appreciable have wave numbers Skat lie within a finite
interval Afc. As Fig. 3 10 indicates, the shorter die group, the broader the range
of wave numbers needed to describe it, and vice versa. The relationship between
Uie distance A.r anil the wavenumber spread \k depends upon the shape of the
wave group and upon how Aa and AA are defined. The minimum value of die
product Ax A* occurs when the group has the form of a gaussian fimetion. in
which ease its Fourier transform happens to be a gaussian function also. If A*
and Afc are taken as the standard deviations of the respective functions ffcc) and
gikl [hen A.v Afc = >/,. In general, A.v Afc has I lie order of magnitude of I:
3.21
IxSk^ I
FIGURE 310 The wave functions and Fourier transforms for (a) a puis*, (b) a wa¥e gfmpi and {c , 3n
infinite wave tram, A briel disturbance requires a broader range of frequencies to describe it than a dis
turbance of greater duration.
«
A
•n
WWWh
e
"
JL_ "I
88
(a)
BASIC CONCEPTS
(b)
(c)
The de Broglie wavelength of a particle of momentum p is
A = *
The wave number corresponding to diis wavelength is
2rrp
= ~h~
Hence an uncertainty At in the wave number of the de Broglie waves associated
with the particle results in an uncertainly Ap in the particle's momentum ac
cording to the formula
, I, Afc
' 2iT
Since A.vAfc a 1. AA :« I /A.v and
3.22
A.v Ap > —
Uncertainty principle
The sign > is used because \x and Ap are irreducible minima that are conse
quences of the wave natures of mooing luufiea; any instrumental or statistical
uncertainties dial arise in the actual conduct of the measurement only augment
the product A.v Ap.
Equation 3.22 is one form of the uncirtointij principle first obtained by Werner
ITeisenberg in 1927. ft states that the product of the uncertainty A.v in the
position of a body at some instant and the uncertainty Ap in its momentum at
the same instant is equal to or greater dian ft/2w. We cannot measure simulta
neously both position anil momentum with perfect accuracy. If we arrange
matters so dial A.v is small, corresponding to the narrow wave group of
Fig. 3SrJ, Ap will lw large. If we reduce Ap in some way, corresponding to
the wide wave group of Fig. 3fic, A.t will be large. These uncertainties are due
not to inadequate apparatus but to the imprecise character in nature of the
quantities involved.
The quantity /i/2w appears quite often in modern physics because, besides its
connection with the uncertainty principle, Ii/2t also turns out to be the basic
unit of angular momentum. It is dierefore customary lo abbreviate h/2z by the
symbol ft:
ft = ^ = J. 054 X 10 :il Js
2m
In the remainder of this book we shall use ft in place of ft/2ir.
WAVE PROPERTIES OF PARTtCLES
89
"The uncertainty principle can be arrived at in a variety of ways. Let us obtain
it by liasing our argument upon the particle nature of waves instead of upon
the wave nature of particles as we did above.
Suppose that we wish to measure the position and momentum of something
at a certain moment. To accomplish this, we must prod it with something else
that is to carry the desired information back to us; that is. we have to touch
it with our finger, illuminate it with light, or interact with it in some other way.
We might be examining an electron with the help of light of wavelength A, as
in Fig. 311. In this process photons of light strike the electron and bounce off
it. Each photon possesses the momentum h/\, and when it collides with die
electron, the electron's original momentum p is ehanged. The precise change
cannot be predicted, but it Ls likely to be of the same order of magnitude as
the photon momentum h/X. Hence the act of measurement introduces an
uncertainty of
3.23
in the momentum of the electron. The longer the wavelength of the light wc
employ in "seeing" the electron, the smaller the consequent uncertainty in its
momentum.
Because light has wave properties, we cannot expect to determine the elec
tron's position with infinite accuracy mider any circumstances, but we might
reasonably hope to keep the irreducible uncertainty Ax in its position to
FIGURE 311 An electron cannol be observed without changing rts momentum.
tj incidc
X
f
viewer
nt
photon
original
momentum
of electron
viewer
reflected
photon
nal \
lenrum \
momentum
of electron
90
BASIC CONCEPTS
L wavelength of die light being used. That is,
3.Z4 A.r ~ X
The shorter the wavelength, the smaller the uncertainty in the position of the
electron.
From Ivqs. 3.23 and 3.24 it is clear that, if we employ light of short wavelength
to improve the accuracy of the position determination, there will be a corre
sponding reduction tn the accuracy of the momentum determination, while light
of long wavelength will yield an accurate momentum value but an inaccurate
position value. Substituting X = A.r into Eq, 3.23 yields
3.25 A.v Ap > h
This result is consistent with Eq. 3.22, since both Ax and Ap here were defined
rather pessimistically.
Arguments like the preceding one, though superficially attractive, must as a
rule be approached with caution. The above argument implies that the electron
can possess a definite position and momentum at any instant, and that it is the
measurement process that introduces the indeterminacy in AxAp. On the con
trary, this indeterminacy is inherent in the nature of a moving body. The
justification for the many "derivations" of this kind is, first, that diey show it
is impossible to imagine a way around the uncertainty principle, and second,
that they present a view of the principle that can lie appreciated in a more
familiar context than tiiat of wave packets.
3.7 APPLICATIONS OF THE UNCERTAINTY PRINCIPLE
Planck's constant h is so minute— only 6.63 X 10~ :H Js— that the limitations
imposed by the uncertainty principle are significant only in die realm of the
atom. On this microscopic scale, however, there are many phenomena that can
be understood in terms of this principle; we shall consider several of them here.
One interesting question is whether electrons are present in atomic nuclei.
As wc shall learn later, typical nuclei are less than If)  ' 4 m in radius. For an
electron to be confined within such a nucleus, the uncertainty in its position
may not exceed I0 1 ' 1 m. The corresponding uncertainly in the electron's mo
mentum is
Ap >
As
1.0,54 X 10' M Js
10" tn
> 1.1 x lO 20 kgm/s
WAVE PROPERTIES OF PARTICLES
91
92
If this is the uncertainty in the electron's momentum, the momentum itself must
ba at least comparable in magnitude. An electron whose momentum is
1.1 X 10" 80 kgiu s has it kinetic energy 7 many limes greater than its rest energy
m c 2 , and we may accordingly ate the extreme relativists formula
7* = )>c
to find 7". Substituting for p and c, we obtain
T= 1.1 X 10» kgm/s X 3 X IIV in/s
= 3.3 X 10~ 12 J
Since I C V = 1.6 x Wr* J. the kinetic energy of the electron must be well over
20 MeV if it k to be a unclear constituent. Experiments indicate that the electrons
associated even with unstable atoms never have more than a fraction of this
energy, and we conclude that electrons cannot be present within nuclei.
1*1 us now ask how much energy an electron needs to be confined to an atom.
The hvdmgcn atom is about 5 X 10 " in in radius, and therefore the uncertainty
in the position of its electron may not exceed this figure. The corresponding
momentum uncertainty is
Ao = 2.1 X 10 'kgm/s
An electron whose momentum is of this order of magnitude is nnnretativistie
in behavior, and its kinetic energy is
•2m
_ (2.1 x HF" kgin/s)3
2 X 9.1 X I0" kg
= 2.1 X 10 s j
or about 15 eV. This is a wholly plausible figure.
Another form oj the uncertaint) principle is sometimes useful, We might wish
to measure the energy /■; emitted sometime during the time interval A< ir, an
atomic process. If the energy is in the form of electromagnetic waves, M». limited
time available restricts the accuracy with which we can determine the frequent )
f of the waves. Let us assume that the uncertainty in the number of waves we
 OUBI n a wave group is one wave. Since the frequency of the waves under
study is equal to the number of them we count divided by the time interval,
DM uncertainty Ac in our frequency measurement is
A,=L
Af
BASIC CONCEPTS
The corresponding energy uncertainty is
AE = ft li>
uj III *>
AE = A
M
AEAl > ft
A mure realistic calculation changes this to
3.26 M. It > h
Equation 3.26 states that the product of the uncertainty AE in an energy meas
urement and the uncertainty M in the time at which the measurement was made
is equal to or greater than H.
As an example of the significance of Eq. 3,26 we can consider the radiation
of light from an "excited" atom. Such an atom divests itself of its excess energy
by emitting one or more photons of characteristic frequency. The average period
that elapses between the excitation of an atom and the time it radiates is 10~ s s.
Thus the photon energy is uncertain by an amount
AE = f
A(
1.054 X I» :i4 Js
10 51 s
= 1.1 X UH»J
and the frequent) ol die light is uncertain by
» AE
an = —r
ft
= 1.6 x 10 7 Hz
II lis is the irreducible limit to the accuracy with which we can determine the
frequency of the radiation emitted by an atom.
3.8 THE WAVEPARTICLE DUALITY
Despite the abundance of experimental confirmation, many of us find it hard
lo appreciate how what we normally think of EtS a wave can also be a particle
in nl how what we normally think of as a particle can also be a wave. The
uncertainty principle provides a valuable perspective on this question which
WAVE PROPERTIES OF PARTICLES
93
makes it possible lo put such statements as those at the end of Sec. 2.2 on a
more concrete oasis.
Figure :3!2 show., an experimental arrangement in which light that has hern
diffracted by a doable ditto detected on a "screen" that consists of manv adjacent
photoelectric cells. The photoelectric cells respond to photons, which have all
the properties we associate with particles. However, when we plot the rnanber
of photons each cell counts in a certain period of time agdnal the location of
the cell, we find the characteristic pattern produced by the interference of a
pair of coherent wave trains. This pattern even occurs when the li«ht intensity
is SO low that, on the average, aoiy one photon at a time is in the apparatus.
The problem is, how can a photon that passes through one of the slits be affected
by the presence of the other rift? In other words, how can a photon interfere
mil, itself? This problem does not arise in the ease of waves, which arc spread
out in spare, but it would seem to have meaning in the case of photons, whose
behavior suggests that thev are localized in very small regions of sp
To have meaning, every question or statement in science must ultimately be
reducible to an experiment. Here the relevant experiment is one that would
FIGURE 31 2 Hypothetical e.periment to determine which slit each photon contributing to a double si it
interference pattern has passed through
photoelectric cells ^
^path of
photon before
collision
particle after
i collision with photon
'
path of
photon after
collision is
uncertain
number of
photons
counted
94
BASIC CONCEPTS
detect which of the slits a particular photon passes through on its way to the
screen, l^et us imagine that we introduce a cloud of small particles l>etween
the sbts and the screen. A photon that passes through one of the slits strikes
a particle and gives it a certain impulse which enables us to detect it (Fig. 312).
Provided that we can establish the position of the particle with an uncertainty
lij that is less than half the space d between the slits, we can determine which
slit the photon passed through. Therefore
But if we are able to limit the uncertainty in the u coordinate of the struck
particle to Aw, the uncertainty Ap v in the w component of its momentum is
3.27
*P»
Aw d
Since the collision introduces a change of Ap v in the particle's momentum,
the same change must have occurred in the photon's momentum. A change of
Ap u in the photon's momentum means a shift of
in the location on the screen which the photon strikes; because p„ < p (the width
of the diffraction pattern is small compared with the distance L), p, ~ p. and
we can write
§.4Bu
p
The photon momentum is related to the wavelength A of the light by Eq. 3.1,
p = x
and so
_ ^PiM*
From Eq. 3.27 we have Ap„ > 2H/d, which means that the shift in die photon's
screen position is
3.28 S = — t
•ad
WAVE PROPERTIES OF PARTICLES
95
The distance t^, between a maximum (that is, a "bright line") in the interference
pattern and an adjacent minimum ("dark line"! is known from elementary optics
to he
3.29
" " 2d
96
This distance is almost the same as the minimum shift involved in establishing
which slit each photon passes through. What would otherwise lw a pattern of
alternating bright and dark lines becomes blurred owing to the Interaction!
between the photons and the particles used to trace their paths. Thus no inter
ference can lie obsenetk the price of determining the exact path of each photon
is the destruction of the interference pattern. If our interest is in the wave aspects
of a phenomenon, they can lie demonstrated; if our interest is in the particle
aspects of the same phenomenon, they too can be demonstrated; bul it is impos
sible to demonstrate Imth aspects in a simultaneous experiment. (Using photo
electric cells to detect an interference pattern is not a simultaneous experiment
in this sense, because there is no way in which a photoelectric cell can determine
through which slit a particular photon striking it has passed.)
The original question of how a photon can interfere with itself therefore turns
out to be meaningless. It is important to lie aware of the distinction between
a legitimate question that cannot be answered because our existing knowledge
is not sufficiently detailed or advanced to cope with it and a question whose
very statement is in contradiction with experiment. Questions that seek to prv
apart the elements of the waveparticle duality fall into the latter class in view
oi the uncertainly principle, whose own empirical validity has been thoroughly
established.
Problems
1. Find the dc Broglie wavelength of an electron whose speed is If)" m %
2. Find the de Broglie wavelength of a 1McV proton.
3. Nuclear dimensions are of the order of in • ' m . (a) Find the energy in eV
of an electron whose de Broglie wavelength is 10" l5 m and which is thus capal.lv
of revealing details of nuclear structure, (h) Make the same calculation for a
neutron.
1. Neutrons in equilibrium with matter at room temperature (300 Kl have
average energies of about %g eV. (Such neutroas arc often called "thermal
neutrons."} Find their tie Broglie wavelength.
BASIC CONCEPTS
5. Derive a formula expressing the de Broglie wavelength (in A) of an electron
in terms of the potential difference V (in volts) through which it is accelerated.
fi. Derive a formula for the de Broglie wavelength of a particle in terms of
its kinetic energy 7" and its rest energy nttf?. If T > "'„'"■ bow tloes tne particle s
wavelength compare with the wavelength of a photon of the same energy?
T. Assume that electromagnetic waves are a special case of de Broglie waves.
Show that photons mast travel with the velocity c and that the rest mass of the
photon must be 0.
h. Obtain the de Broglie wavelength of a moving particle in the following way.
which parallels de Broglie's original treatment. Consider a particle of rest mass
iii ( , as having a characteristic frequency of vibration of i\ v specified by the
relationship lit;, = m,f 2 . The particle travels with the speed i? relative to an
observer. With the help of special relativity, show that the observer sees a
progressive wave whose phase velocity is iv = c 2 /t: and whose wavelength is
h/mv, where in = m„/vl — v/c 2 .
9. The velocity of ocean waves is Vg\/2w, where g is the acceleration of
gravity. Find the group velocity of these waves.
1 0. The velocity of ripples on a liquid surface is \/2w/Xp, where S is the surface
tension and p the density of the liquid. Find the group velocity of these waves.
11. The position and momentum of a 1keV electron are simultaneously de
termined. If its position is located to within 1 A, what is the percentage of
uncertainly in its momentum?
12. An electron microscope uses 10keV electrons. Find its ultimate resolving
power on the assumption thai this is equal to the wavelength of the eleclrous.
13. Compare the uncertainties in the velocities of an electron and a proton
confined in a 10A box.
14. Wavelengths can l>e determined with accuracies of one part in 10 s . What
is the uncertainty in the position of a lA Xray photon when its wavelength
is simultaneously measured?
15. At a certain lime t a measurement establishes the position of an electron
with an accuracy of ±10 ' ' m. Find the uncertainty in the electron's momentum
at t atul. from ibis the uncertainty in its position 1 slater. If the latter uncertainly
is not ±10" m, account for the difference in terms of the concept of a moving
particle as a wave packet.
16. (a) How much time is needed to measure the kinetic energy of an electron
whose speed is 10 m/s with an un certainty of no more than 0.1 percent? How
WAVE PROPERTIES OF PARTICLES
97
far will the electron have traveled in this period of rime? (b) Make the same
calculations for a Ig insect whose speed is the same. What do these sets of
figures indicate?
1 . . The atoms in a solid possess a certain minimum zeropoint energy even at
K, while no such restriction holds for the molecules in an ideal gas. Use the
uncertainty principle to explain these statements.
18. Verify that the uncertainty principle can be expressed in the form
M,itf > h, where 1L is the uncertainty in the angular momentum of a Iwdy
and A0 is the uncertainty in its angular position. [Hint: Consider a particle
moving in a circle.)
98
BASIC CONCEPTS
ATOMIC STRUCTURE
4
Far in the past people began to suspect that matter, despite its appearance of
being continuous, possesses a definite structure on a microscopic level beyond
the direct reach of our senses. This suspicion did not take on a more concrete
form until a little over a century and a half ago; since then the existence of
atoms and molecules, the ultimate particles of matter in its common forms, has
been amply demonstrated, and their own ultimate particles, electrons, protons,
and neutrons, have been identified and studied as well. In this chapter and in
others to come our chief concern will Ik the structure of the atom, since it is
ihis structure that is responsible for nearly all the properties of matter that have
shaped the world around us.
Every atom consists of a small nucleus of protons and neutrons with a mnntier
of electrons some distance away. It is templing to think of the electrons as
circling the nucleus as planets do the sun, but classical electromagnetic theory
denies the possibility of stable electron orbits. In an effort to resolve this paradox,
Niels Bohr applied quantum ideas to atomic structure in 1913 to obtain a model
which, despite its serious inadequacies and subsequent replacement In a
quantummechanical description of greater accuracy and usefulness, nevertheless
remains a convenient mental picture of the atom. While it is not the general
policy of this book to go deepb into hypotheses that have had to In discarded,
we shall discuss Bohr's theory of tin: hydrogen atom hecause it provides a valuable
transition to the more abstract quantum theory of the atom. For this reason
our account of the Bohr theory differs somewhat from the original one given
by Uiilir, though all the results are identical.
4.1 ATOMIC MODELS
W bile the scientists of the nineteenth century accepted the idea that the chemical
''leiucnts consist ol atoms, thev knew virtually nothing about the atoms them
selves. The discovery of the electron and the realization that all atoms contain
101
electrons provided the first important insight into atomic structure. Electrons
contain negative electrical charges, while atoms themselves are electrically
neutral: every atom must therefore contain enough positively charged matter
to halancc the negative charge of its electrons. Furthermore, electrons are
thousands of times lighter than whole atoms; tins suggests that the positively
charged constituent of atoms is what provide! them with nearly all their mass.
When J, J. Thomson proposed in 1898 that atoms are uniform spheres of posi
tively charged matter in which electrons are embedded, his hypothesis then
seemed perfectly reasonable. Thomson's plumpudding model of the atom— so
called from its resemblance to Uiat raisinstudded delicacy is sketched in Fig.
41. Despite the importance of the problem, ].') years p^ved before a definite
experimental test of the plumpudding model was made. This experiment, as
we shall see, compelled the abandonment of this apparently plausible model,
leaving in its place a concept of atomic structure incomprehensible in the light
of classical physics.
The most direct way to find out what is inside a plain padding is to plunge
a linger into it. a technique not very different from that used by Ceiger and
Marsdeu to find out wbaJ is inside an atom. In their classic experiment, performed
in 191] at the suggestion of Finest I'.titherford, the] employed as prolics the
fast ulfilm fuirtuUs spontaneously emitted by certain radioaetivc elements. Alpha
particles arc helium atoms that have lost two electrons, leaving them with a
charge of + 2r; we shall examine their origin and properties in more detail later
Ceiger and Marsden placed a sample of an alphaparticleemitling substance
behind a lead screen that had a small hole in it. us in Fig. 12. SO that a narrow
Ixsam of alpha particles was produced. This beam WU directed at a thin eold
FIGURE 41 Th » Thomson model of lh« atom.
"
fleet run
 positively charged matin
radioactive
substance that
emits alpha
particles
lead
collimator
zinc sulfide
screen
FIGURE 42 The Rutherford ««Hering experiment.
foil. A moveable zinc sulfide screen, which gives off a visible flash of light when
struck by an alpha particle, was placed on the other side of the foil. It was
anticipated that most of the alpha particles would go right through the foil, while
the remainder would at most suffer only slight deflections. This behavior follows
from the Thomson atomic model, in which the charges within an atom are
assumed to be uniformly distributed throughout its volume. If the Thomson
model is correct, only weak electric forces are exerted on alpha particles passing
through a thin metal foil, and their initial momenta should be enough to carry
thern through with only minor departures from their original paths.
What Geiger and Marsden actually found was that, while most of the alpha
particles indeed emerged without deviation, some were scattered through very
large angles. A few were even scattered in the backward direction. Since alpha
particles are relatively heavy (over 7,(KX1 limes more massive than electrons) and
those used in this experiment traveled at high speed, it was clear that strong
forces had to be exerted upon them to cause such marked deflections. To explain
the results, Rutherford was forced to picture an atom as being composed of a
tiny nucleus, in which its positive charge and nearly all of its mass are concen
trated, with its electrons some distance away (Fig. 43). Considering an atom
as largely empty space, it is easy to see why most alpha particles go right through
a thin foil. When an alpha particle approaches a nucleus, however, it encounters
»" intense electric field and is likely to be scattered through a considerable angle.
The atomic electrons, being so light, do not appreciably affect the motion of
incident alpha particles.
102
THE ATOM
ATOMIC STRUCTURE
103
/ e
G
e X
\
\
e \
! _ +
, L^CJ positive nucleus
electron T
FIGURE 43 The Rutherford
model oi the atom.
\
\
Q
e /
e /
104
Numerical estimates of electricfield intensities within the Thomson and
Rutherford models emphasize the difference between them. If we assume with
Thomson that the positive charge within a gold atom u spread evenly throughout
its volume, and if we neglect the electrons completely, the dec trie field Intensity
at the atom's surface (where it is a maximum; is about I0 ,:1 V m. On the other
hand, if we assume with Rutherford that the positive charge within a gold atom
is mi Kent rated in a small nucleus at its center, the elect ric(ield intensity at the
surface of the nucleus exceeds 10 21 V/m— a factor of 10 s greater. Such a
strong field can deflect or even reverse the direction of an energetic alpha
particle that comes near the nucleus, while the feebler field of the Thomson
atom cannot.
The experiments of Ceiger and Marsden and later work of a similar kind also
supplied information about the nuclei of the atoms thai composed the various
target foils. The delleclion an alpha paiiidr experiences when it passes near
a nucleus depends upon the magnitude of the nuclear charge, and so comparing
the relative scattering of alpha particles by different foils provides a way of
estimating the nuclear charges of the atoms involved. All of the atoms of any
one clement wen found to have the same unique nuclear charge, and this charge
increased regularly from element to element in the periodic table. The nuclear
charges always turned out to be multiples of + e; the number of unit positive
charges in the nuclei of an element is today called the (domic number of the
element. We know now that protons, each with a charge +e, are responsible
for the charge on a nucleus, and so the atomic number of an element is the
same as the number of protons in the nuclei of its atoms.
THE ATOM
«4.2 ALPHAPARTICLE SCATTERING
Bnlberford arrived at a formula, describing the scattering of alpha particles by
thin foils on the basis of his atomic model, that agreed with the experimental
results. The derivation of this formula both illustrates the application of funda
mental physical laws tu a novel setting and introduces certain notions, such as
that of the crass section for an interaction, that are important in many other
aspects of modern physics.
Kntherford began by assuming that the alpha particle atal the nucleus it
interacts with arc both small enough to be considered as point masses and charges:
that the electrostatic repulsive force between alpha particle and nucleus (which
are both positively charged) is the only one acting; and that the nucleus is so
massive compared with the alpha particle that it does not move during their
interaction. Owing to the variation of the electrostatic force with I/r, where
i is the instantaneous separation between alpha particle and nucleus, the alpha
particle"* path is a hyperbola with the nucleus at the outer focus (Fig. 11".
The impact parameter b is the minimum distance to which the alpha particle
would approach the nucleus if there were no force between them, and the
scattering angle, is the angle between the asymptotic direction of approach
of the alpha particle and the asymptotic direction in which it recedes. Our first
task is to find a relationship between /; and (J.
FIGURE 44 Rutherford scattering.
^ alpha particle
target nucleus O v SL
" — scattering angle
™ = impact parameter
ATOMIC STRUCTURE
105
As a result of ihe impulse f F dt given il by the nucleus, the momentum of
the alpha particle changes by Ap from the initial value p, to the final value p 2 .
That is,
4.1
Ap = p 2  Pl
= JFdl
Because the nucleus remains stationary during the passage of the alpha particle,
by hypothesis, the alphaparticle kinetic energy remains constant; hence the
magnitude of its momentum also remains constant, and
Pi = ft = mo
Here v is the alphaparticle velocity far from the nucleus. From Fig. 45 we
see that, according to the law of sines,
Ap
sin 8
mo
sin — 
Since
and
sin— (tt  8) = cos —
sin 8 = 2 sin — cos —
2 2
FIGURE 45 Geometrical relallonthlps in Rutherford scattering.
M(rff)
W(ir
/ i** 5 ^— ___ S_ alpha
path of alpha particle / ^i LAj 7^ — particle
\
target nucleus
we have for the momentum change
4.2 Ap ae 2mc sin —
Because the impulse J F" df is in the same direction as the momentum change
Ap, its magnitude U
4.3 /F*sa/FOM*«B
where £ is the instantaneous angle between V and Ap along tin pall, d the alpha
particle. Inserting Eqs. 4.2 and J.J in Eij. 1.1.
f*
2m<;sin— =1 fcosCKif
2 *,,
In change the variable on the righthand side from f to </>, we note that the
limits of integration will change to  J^w  8) and + )&(*  0), corresponding
to £ at f = and I = co respectively, and so
4.4
2tnm sin
2 hr«/2
i=/.:
f " i
/ eos«f>— d$
The quantity drf>/d< is just the angular velocity U of the alpha particle about
the nucleus (this is evident from Kiu MS). The elecAfbstatiC force exerted l>\
(he nucleus on the alpha particle acts along the radius vector joining them, and
so there is no torque on the alpha particle and its angular momentum row* 8 is
constant Hence
mu'r
= constant
= mr —r
= mi h
from
whJch we
see that
fid, "
r 1
' vb
Substituting this expression for <*/<*$> in Eq, 4.4.
8 r +fc
4.s 2»jr/jsiii  I R a co«^d^
2 , '<w«/2
106
THE ATOM
ATOMIC STRUCTURE
107
As we recall, F is the electrostatic force exerted by the nucleus on the alpha
particle. The charge on the nucleus is Ze, corresponding to the atomic man
lier Z, and that on the alpha particle is le. Therefore
F =
1 2Ze 2
4«? n i"
and
AmMuPfa
■ sin
Ze a
f
I = J
te9)/2
feSl/2
= 2 cos —
cos <j> d$
The scattering angle $ is related to the impact parameter b by the equation
, 2m it tiic ,
cot T = — ?3 '>
It is more convenient to specify the alphaparticle energy T instead of its mass
and velocity .separately; with ilus substitution,
4m a T
rat 2=^M
Figure 46 is a schematic representation of Eq. i.fi; the rapid decrease in as
h increases is evident. A very near miss is required for a substantial deflection.
FIGURE 46 The scattering angle decreases with imreating impact parameter.
L
b
r
m ^ — target nucleus
urea = iri» s
*4.3 THE RUTHERFORD SCATTERING FORMULA
Equation 1.6 cannot he directly confronted with experiment since there is no
way of measuring the impact parameter corresponding to a particular o b ser v ed
scattering angle. An indirect strategy is required. Our first step is to note that
.ill alpha particles approaching a target nucleus with an impact parameter from
to h will be scattered through an angle of or inure, where is given in terms
of b by Eq. 4.6. This means that an alpha particle that is initially directed
,ui> where within the area »fc* around a nucleus will be scattered through // or
more (Fig. 46); the area <nb'~ is accordingly ralleil the era?.? xectitnt for the
interaction. The general symbol for cross section is <r, and so here
47
a = «r& a
We must keep in mind that the incident alpha particle is actually scattered Ivcfore
it reaches the immediate vicinity of the nucleus and hence does not necessarily
pass within a distance b of it.
Now we consider a foil of thickness I that contains n atoms per unit volume.
The number of target nuclei per unit area is nt. and BO alphaparticle beam
incident upon an area A therefore encounters nlA nuclei. The aggregate cross
section for scatterings of tl or more is the number of target nuclei nf.\ multiplied
liy the cross section o for such scattering per nucleus, or u/Ao. Hence the fraction
/of incident alpha particles scattered by (/ or more is the ratio between ihe
•'g^egalc cross section ittAo lor such scattering and the total target area A.
That is.
/ =
alpha particles scattered by or more
incident alpha particles
aggregate cross section
target area
_ 1 1 (An
A
Substituting for b from Eq. 4.fi.
m f S ^Lz<y*cot*f
\ lw„T7 2
bi Ihe above calculation it was assumed thai the foil is sufficiently thin so that
•lie cross sections of adjacent nuclei du not overlap and that a scattered alpha
particle receives its entire deflection I ruin an encounter with a single nucleus.
103
THE ATOM
ATOMIC STRUCTURE
109
110
Let us use Eq. 4 .8 to determine what fraction of a beam of 7.7MeV alpha
particles is scattered through angles of more than 45° when incident upon a
gold foil 3 X 10" ~ m thick. (These values are typical of the alphaparticle
energies and foil thicknesses used by Gciger and Marsdcn; for comparison, a
human hair is about Kl ' m in diameter.) We begin by finding n, the number
of gold atoms per unit volume in the foil, from the relationship
Atoms (atoms/kmol) X (mass/volume)
Volume
massAmol
\,t>
it;
where N n is Avogadro's number, p the density of gold, and us its atomic weight.
Since N u = 8.03 X 10* atoms/kmol, p = 1.93 X 10* kg/m 3 , and w = 197, we
have
6.03 X 10 2(J atoms/kmol X 1.93 X 10* kg/m 3
n —
197 kg/kmol
ss 5.91 x 10 28 atoms/m 3
The atomic number 7, of gold is 79, a kinetic energy of 7.7 MeV is equal to
1.23 x 10 vl J, and tf = 45°; from these figures we find that
/ = 7 X 10" 5
of the incident alpha particles are scattered through 45° or more — only 0.007
percent! A foil this thin is quite transparent to alpha particles.
In an actual experiment, a detector measures alpha particles scattered between
and + dO, as in Fig, 47. The fraction of incident alpha particles so scattered
is found by differentiating Eq. 4.8 with respect to 0, an operation that yields
4.9
\4wr o r/ 2 2
do
(The minus sign expresses the fact that / decreases with increasing 0.) tn the
experiment, a fluorescent screen was placed a distance r from the foil, and the
scattered alpha particles were detected by means of the scintillations they caused.
Those alpha particles scattered between and (I + dO reach a zone of a sphere
of radius r whose width is nlS. The zone radius itself is rsintf, and so the area
dS of the screen struck by these particles is
f/S s= (2vr sin ff)(rd0)
= Tnr 1 sin 6 dO
= 4irr a sin — cos — dO
2 2
THE ATOM
FIGURE 47 tn the Rutherford experiment, particles
are detected that have been scattered between S and
2.9 1 i d
area as 4 irr sin ; cos r "0 '
If a total of N t alpha particles strike the foil during the course of the experiment,
the number scattered into dB at is A/, df. The number N(0) per unit area striking
the screen at 0. which is the quantity actually measured, is
N(S) =
dS
NwB ,(^eLV* crt  cse8 . d »
* \4m Q Tf 2 2
4.10
N(0) =
4jtt 2 sin — cos — d6
2 2
(oVejVT 2 sin* (0/2)
Rutherford scattering formula
Equation 4.10 is the Rutherford scattering formula.
According to Eq. 4.10, the number of alpha particles per unit area arriving
at the fluorescent screen a distance r from the scattering foil should be directly
proportional to the thickness ( of the foil, the nuuilier of foil atoms per unit
volume n, and the square of the atomic number Z of the foil atoms, and it should
"6 inversely proportional to the square of the kinetic energy T of the alpha
particles and to sin 4 (0/2), where 8 is the scattering angle. These predictions
agreed with the measurements of Geiger and Marsden mentioned earlier, which
ATOMIC STRUCTURE
111
led Rutherford lo conclude that his assumptions, chief among them (he hypothesis
of the nuclear atom, were correct. Rutherford is therefore credited with the
"discovery" of the nucleus. Figure 48 shows how ,V(ff) varies wi(h 6.
4.4 NUCLEAR DIMENSIONS
When we say that the experimental data on the scattering of alpha particles by
thin foils verifies our assumption Uiat atomic nuclei are point particles, what
is really meant is that their dimensions are insignificant compared with the
FIGURE 48 Rutherford nattering.
.Y<0)
:\'<l8Cn
0° 20° 40° 60° 80" 100' 120" 140 s 160° 180°
minimum distance to which the incident alpha particles approach the nuclei.
Rutherford scattering therefore permits us to determine an upper limit to nuclear
dimensions. Let us compute the distance of closest approach r„ of the most
energetic alpha particles employed in the early experiments. An alpha part it 1b
will liave its smallest r when its impact parameter is b = 0, corresponding to
a headon approach followed by a 180° scattering. Al the instant of closest
approach the initial kinetic energy T of the particle is entirely converted to
fltctioKiatir potential energy, and so at thai instant
4.11
r =
1 2Ze 2
\~s ,,
since the charge of the alpha particle is 2<? and that of the nucleus 7,e. I lence
IZe 1
4*e«T
The maxiitniui /' found in alpha particles of natural origin is 7.7 MeV, which is
7.7 X 10" eV X 1.6 X 10 ,B J/eV = 1.2 X 10 > 2 J
Since l/4w„ = 9 x 10" Nm 2 /C 2
2 X ?> X 10 9 NmVC* X (1.6 X 10 "' Q 2 Z
1.2 X I0'*J
= 3.8x 10'«Zm
The atomic number uf gold, a typical foil material, is Z = 79, so that
r„ ( Au) = 3.0 X 10 M m
The radius of the gold nucleus is therefore less than 3.0 X 10 " m, well under
' n,  the radius of the atom as a whole.
hi more recent years particles of much higher energies than 7.7 MeV have
been artificially accelerated, and it haslieen found that the Rutherford scattering
formula does indeed eventually fail to agree with experiment. We shall discuss
experiments and the information (hey provide on actual nuclear dimensions
hi Chap. II.
45 ELECTRON ORBITS
> hu Rutherford model of the atom, so convincingly confirmed by experiment,
postulates a tiny, massive, positively charged nucleus surrounded at a relatively
greai distance In enough electrons to render the atom, a> a whole, electrically
112
THE ATOM
ATOMIC STRUCTURE
113
114
neutral. Thomson visualized the electrons in his model atom as cml:>cddcd in
the positively charged matter that fills it, and thus as being unable to move.
The electrons in Rutherford's model atom, however, cannot be stationary, because
there is nothing that can keep them in place against the electrostatic force
attracting them to the nucleus. If the electrons are in motion around the nucleus,
however, dynamically stable orbits (comparable with those of the planets about
the sun) are possible (Fig. 49).
Ijet us examine the classical dynamics of the hydrogen atom, whose single
electron makes it the simplest of all atoms. We shall assume a circular electron
orbit for convenience, though it might as reasonably be assumed elliptical in
shape. The centripetal force
F c~ —
holding the electron in an orbit r from the nucleus is provided by the electrostatic
force
F.=
1 e 2
4wc r 2
between them, and the condition for orbit stability is
F = F
4.12
ID 2
1 e 2
r 4ire r 2
FIGURE 49 Fore* balance In
the hydrogen atom .
THE ATOM
The electron velocity c is therefore related to its orbit radius r by the formula
e
4.13
I' =
N/4l
The total energy /.' of the electron in a hydrogen atom is the sum of its kinetic
energy
r = l / 2 mt; 2
and its potential energy
e 2
V =
4we r
: The minus sign signifies that the force on the electron is in the — r direction.)
1 Eettoe
E=T+ V
— nic2 _
2
e 2
l^f„r
Substituting for «; from J£q. 4.12,
£ =
8we r
4TO r
4.14
8OT(/
The total energy of an atomic electron is negative; this is necessary if it is to
be bound to the nucleus. If E were greater than zero, the electron would have
too much energy to remain in a closed orbit about the nucleus.
Experiments indicate that 13.6 eV is required to separate a hydrogen atom
Into a proton and an electron; that is, its binding energy E is — 13.6 eV, Since
13.6 eV = 2,2 X 10 1B J, we can find the orbital radius of the electron In a
hydrogen atom from Eq. 4.14;
Swe^E
(1.6 X 10 ,!, C)
8w X 8.85 X U> I2 F/m X (2.2 X Id ,s j)
= 5.3 X 10
rn
An atomic radius of this order of magnitude agrees with estimates made in other
Ways.
ATOMIC STRUCTURE
115
The above analysis is a straightforward application of Newton's laws of motion
and Coulomb's law of electric force — both pillars of classical physics — and is
in accord with the experimental observation that atoms are stable. However,
il is no/ in accord with electromagnetic theory — another pillar of classical
physics — which predicts that accelerated electric charges radiate energy in [he
form of electromagnetic waves. An electron pursuing a curved path is acceler
ated and therefore should continuously lose energy, rapidly spiraliug into the
nucleus (Fig. 410), Whenever they have Iwen directly tested, the predictions
of electromagnetic theory have always agreed with experiment, yet atoms do
not collapse. This contradiction can mean only one thing: The laws of phvsics
thai are valid in the macroscopic world do not hold true in the Microscopic
world of the atom.
The reason for the failure of classical physics to yield a meaningful analysis
of atomic structure is that it approaches nature exclusively in terms of the abstract
concepts of "pure" particles and "pure" waves. As we learned in the two
preceding chapters, particles ami waves have many properties in common, though
thesnialhiess of Hanek'5 constant renders the waveparticle duality imperceptible
in the macroscopic world. The validity of classical physics decreases as the scale
of the phenomena under study decreases, and full allowance must be made for
the particle behavior of waves and the wave behavior of particles if the atom
is to lie understood. In the remainder of this chapter we shall see how ihe Bohr
FIGURE 410 An atomic electron
should, classically, spiral rapidly Into the
nucleus as it radiates energy due to its
acceleration.
atomic model, which combines classical and modern notions, accomplishes part
of the latter task. Not until we consider the atom from the point of view of
quantum mechanics, which makes no compromise with intuitive notions acquired
iti our daily lives, will we find a really successful theory of the atom.
An interesting question arises at this point. In our derivation of the Rutherford
scattering formula we made use of the same laws of physics that proved such
dismal failures when applied to atomic stability. Is it not therefore possible, even
likelv, thai the formula is not correct, and that the atom in reality does not
resemble the Rutherford model of a small central nucleus surrounded by distant
electrons? This question is not a trivial one, and it is, in a way, a curious
coincidence that the quantummechanical analysis of alphaparticle scattering
From thin foils results in precisely the same formula that Rutherford obtained.
To vcrifv that a classical calculation ought to be at least approximately correct,
we note that the de Broglie wavelength of an alpha particle whose speed is
2 X R' 7 m/s is
h 6.63 X R) M Js
X ~ h77 ~ 6\6 X Kr*« kg X 2 X 10 r m/s
= 5 X 10 15 m
As we saw in Sec. 4.4, the closest an alpha particle with this wavelength ever
gets to a gold nucleus is ■'! X W~ u m, which is 6 de Broglie wavelengths, and
so it is reasonable to regard the alpha particle as a classical particle in the
interaction. We are therefore correct in thinking of the atom in terms of Ruther
ford's model, though the dynamics of the atomic electrons— which is another
matter entirely — requires a nonclassical approach.
4.6 ATOMIC SPECTRA
The ability of the Bohr theon ol the atom to explain the origin of spectral lines
is among its most spectacular accomplishments, and so it is appropriate to preface
"in exposition of the theory itself with a look at atomic spectra.
We have already incut ioued that healed solids emit radiation in which all
wavelengths are present, though with different intensities. We shall learn in
< 'hap. 9 thai the observed features of this radiation can be explained on the basis
"i the quantum theory of light independent of the details of the radiation process
itself or of the nature of the solid. From this fact it follows that, when a solid
is heated to incandescence, we are witnessing the collective behavior of a great
n m) interacting atoms rather than the characteristic behavior of the individual
atoms of a particular element.
116
THE ATOM
ATOMIC STRUCTURE
117
118
7,000 A
red
hydrogen
Itelii
mercury
6,000 A
orange yellow
5,000 A
green
FIGURE 4.11 PDrtions of the Bmjskln ^^ rf nydr()gen hcIlum
blue
and mercury.
4,000 A
violet
At the other extreme, the atoms or molecules in a rarefied gas are so far apart
on the average that their only mutual interactions occur during occasional
collisions. Under these circumstances we would expect any emitted radiation
to be characteristic of the individual atoms or molecules present, an expectation
hat IS realized experimentally. When an atomic gas or vapor at somewhat Z
than atmospheric pressure is suitably "excited," usually by the passage of an
dectnc current through it, the emitted radiation has a spectrum which contains
<*** to wavelengths only. Figure 41 J shows the atomic spectra of
•several elements; they are called emission line ^ectra. Every dement display,
a uiuque Hne spectrum when a sample of it in the vapor phase is exdted;
FIGURE 412 A portion of the band tpectrum ol PN.
Ill
THE ATOM
IHIWlll
absorption spectrum
of sodium vapor
emission spectrum
of sodium vapor
FIGURE 413 The dark lines in the absorption spectrum of an element correspond to bright lines in its
emission spactrum.
spectroscopy is therefore a useful tool for analyzing the composition of an
unknown substance.
The spectrum of an excited molecular gas or vapor contains (Hinds which
consist of many separate lines very close together (Fig. 412), Bands owe their
origin to rotations and vibrations of the atoms in an electronically excited
molecule, and we shall consider their interpretation in a later chapter.
When white light is passed through a gas, it is found to absorb light of certain
of the wavelengths present in its emission spectrum. The resulting absolution
line spectrum consists of a bright background crossed by dark lines corresponding
to the missing wavelengths (Fig. 413); emission spectra consist of bright lines
on a dark background. The dark Fraunhofer lines in the solar spectrum occur
hecause the luminous part of the sun, which radiates almost exactly according
to theoretical predictions for any object heated to 5800 K, is surrounded by an
envelope of cooler gas which absorbs light of certain wavelengths only.
In the latter part of the nineteenth century it was discovered that the wave
lengths present in atomic spectra fall into definite sets called spectral series. The
wavelengths in each series can be specified by a simple empirical formula, with
remarkable similarity among the formulas for the various series that comprise
the complete spectrum of an element. The first such spectral series was found
by J. J. Balmer in 1885 in the course of a study of the visible part of the hydrogen
spectrum. Figure 414 shows the Balmer aeries. The tine with the longest
FIGURE 414 The Balmer series of hydrogen. *
H,
I
ATOMIC STRUCTURE
119
wavelength, 6,563 A, is designated H n , the next, whose wavelength is 4.86.1 A,
is designated H fl , and so on. As the wavelength decreases, the lines are found
closer together and weaker in intensity until the series HttiU at 3,646 A is readied,
beyond which there arc no further separate lines but only a faint continuous
spectrum. Maimer's formula for the wavelengths of this series is
4.1S
x fi (i  ^)
n = 3, 4. 5,
Balmer
The quantity R, known as the Rydberg constant, has the value
R = 1.097 x lO T nr J
= 1.097 X tO" 3 A" 1
'ITie II„ line corresponds to n — 3, the H /} Hue to n = 4, and so on. The series
limit corresponds to n = oo, so that it occurs at a wavelength of 4/8, in agree
ment with experiment.
The Bahner series contains only those wavelengths in the visible portion of
the hydrogen spectrum. The spectral lines of hydrogen in the ultraviolet and
infrared regions fall inlo several other series. In the ultraviolet the Lyman .w rirs
contains the vvavelenglhs specified by the formula
4.16
A 1 1 2 n7
n = 2, 3, 4,
Lyman
In the infrared, three spectral series have been found whose component lines
have the wavelengths specified by the formulas
17 i = H (^i) —***•■•
Paschen
Brackett
Pfund
The above spectral series of hydrogen are plotted in terms of wavelength in
Fig 415; the Brackett series evidently overlaps the Paschen and Pfund series.
The value of R is the same in Eqs. 4.15 to 4.19.
'ITie existence ul such remarkable regularities hi the hydrogen spectrum,
together with similar regularities in the spectra of more complex elements, poses
a definitive test for any theory of atomic structure.
riGURE 415 The spectral series of hydrogen.
X
50,000 k
20,000 A
10,000 A
5,000 A 
2,500 A 
2,000 A
1,500 A 
1,250 A 
,000 A
j J Pfund series
J Brackett series
r Paschen series
Bal
iner series
Lvman series
4.7 THE BOHR ATOM
We saw in Sec. 45 that die principles of classical physics are incompatible with
the observed stability of the hydrogen atom. The electron in this atom is obliged
to whir] around the nucleus to keep from being pulled into it and yet must radiate
electromagnetic energy continuously. Because other apparently paradoxical
phenomena, like the photoelectric effect and the diffraction of electrons, find
Uptanarkm in terms of quantum concepts, it is appropriate to inquire whether
this might not also be true lor the atom.
120
THE ATOM
ATOMIC STRUCTURE
121
Let us start by examining the wave behavior of an electron in orbit around
a hydrogen nucleus. The dc Broglie wavelength of this electron is
X =
nit
where the electron speed v is that given by Eq. 1.13:
e
V5
w,,»ir
Hence
4.20
' m
By substituting 5.3 X lO" 11 m for the radius r of the electron orbit, we find the
electron wavelength to be
_ 6.63 X lfl 3 " Js M*
1.6 X 10 ,9 C V —
* X 8.85 X 10 12 F/m X 5.3 X 10 u
9J x 10~ 3 ' kg
= 33 X JO" 1
m
This wavelength is exactly the same as the circumference of the electron orbit,
2wr = 33 X 10" m
The orbit of the electron in a hydrogen atom corresponds to one complete
electron wave joined on itself (Fig. 416).
The fact that the electron orbit in a hydrogen atom is one electron wavelength
in circumference provides the clue we need to construct a theory of the atom.
If we consider the vibrations of a wire loop (Fig, 417), we find that their
wavelengths always fit an integral number of times into the loop's circumference
so that each wave joins smoothly with the next. If the wire were perfectly rigid,
these vibrations would continue indefinitely. Why are these the only vibrations
possible in a wire loop? If a fractional number of wavelengths is placed around
the loop, as in Fig. 418, destructive interference will occur as the waves travel
around the bop, and the vibrations will die out rapidly. By considering the
behavior of electron waves in the hydrogen atom as analogous to the vibrations
of a wire loop, then, we may postulate that an electron can circle a nucleus
indefinitely without radiating energy provided that its orbit contains an integral
number of de Broglie wavelengths.
This postulate is the clue to understanding the atom. It combines !x>th the
particle and wave characters of the electron into a single statement, since the
electron wavelength is computed from the orbital speed required to balance the
electron path
de Broglie electron wave
FIGURE 416 The orbit of the electron in 1 hydrogen atom corresponds to i complete electron de
Broglie wave joined en itself.
electrostatic attraction of the nucleus. While we can never observe these anti
thetical characters simultaneously, they are inseparable in nature.
It is a simple matter to express the condition that an electron orbit contain
AD integral number of de Broglie wavelengths. The circumference of a circiilar
orbit of radius r is 27rr, and so we may write the condition for orbit stability as
122
THE ATOM
ATOMIC STRUCTURE
123
4.21
nX t= 2T7T n n  1, 2. 3, . . .
where r n designates the radius of the orbit that contains n wavelengths. Tlie
integer n is called the ifuuntum number of the orbit. Substituting lot A, the
electron wavelength given by Kq. 4.20. yields
nh
e
x / 4 **!?* _
2.7TI
and so the stable electron orbits are those whose radii are given In
n' 2 h*E t>
4.22
n = !, 2, 3, . .
FIGURE 417 The vibrations of a wire loop.
circumference = 2 wavelengths
circumference — 4 wavelengths
circumference = S wavelengths
FIGURE 418 A fractional number of wavelength t cannot persist
because destructive interference will occur.
The radius of the innermost orbit is customarily called the Bohr rmliux of the
hydrogen atom and is denoted by the symbol a,,:
a ( ,  \\  5.3 X 10" " in
= 0.53 A
The other radii are given in terms of a„ by the formula
% ■ »"«„
so that the spacing between adjacent orbits increases progressively.
4.8 ENERGY LEVELS AND SPECTRA
The various permitted orbits involve different electron energies. The electron
energy £ n is given in terms of the orbit radius r n by Eq. 4. 14 as
K.  ■
8*V«
Substituting for r n from Eq. 4.22, we see that
124
THE ATOM
Energy levels
ATOMIC STRUCTURE
125
The energies specified by Eq. 4,23 are called the energy level's of the hydrogen
atom and are plotted in Fig. 419. These levels are all negative, signifying thai
the electron does not have enough energy to escape from the atom. The lowest
energy level £, is called the ground state of the atom, and the higher levels
E 2> &). £4* • • ■ are called excited states. As the quantum number n increases,
the corresponding energy E n approaches closer and closer to 0; in the limit of
n a oe, E x  and the electron is no longer bound to the nucleus to form an
atom. {A positive energy for a nucleuselectron combination means that the
electron is not bound to the nucleus and has no quantum conditions to fulfill;
such a combination does not constitute an atom, of course.)
It is now necessary for us to confront directly the equations we have developed
with experiment. An especially striking experimental result is that atoms exhibit
line spectra in both emission and absorption; do these spectra follow from our
atomic model?
free electron
r
excited states ■
n = as —
n = 5
n = 4
n = 3
v.
r> = 2
energy, J
0.87 x lO" 1 * 1
1.36 x 10 ,B
2.42 X 10 1! >
5.43 x 10
energy, eV
0.54
0.85
1.51
3.40
ground state n = 1
126 THE ATOM
FIGURE 4. 19 Energy tenets of the
hydrogen atom.
21.76 X 10'
 Ki.fi
The presence of definite, discrete energy levels in the hydrogen atom suggests
a connection with line spectra. Let as tentatively assert that, when an electron
in an excited state drops to a lower state, the lost energy is emitted as a single
photon of light. According to "ur model, electrons cannot exist in an atom except
in certain specific energy levels. The jump of an electron from one level to
another, with the difference in energy between the levels being given off all
at once in a photon rather than in some more gradual manner, fits in well with
this model. If the quantum number of the initial (higher energy) state is n t and
the quantum number of the final (lower energy) state is n f , we are asserting that
Initial energy — final energy = photon energy
where » is the frequency of the emitted photon.
The initial and final states of a hydrogen atom that correspond to the quantum
numbers n ( and n f have, from Eq. 4.23, the energies
Hence the energy difference I>etween these states is
\nj~~nj)
8*o
me 4
& 2 h s '
The frequency v of the photon released in this transition is
K E,
4.35
8e 2 /i 3
\n/' n { 2 /
In terms of photon wavelength X, since
we have
4.26
me 1
fk^ch 3 \nf
n _ _l
n t 2 )
Hydrogen spectrum
ATOMIC STRUCTURE
127
Equation 4.26 states that the radiation emitted by excited hydrogen atoms
should contain certain wavelengths only. These wavelengths, furthermore, fall
into definite sequences that depend upon the quantum number 8. of the final
energy level of the electron. Since the initial quantum number n 7 must always
be greater than the final quantum number n f in each case, in order that there
be an excess of energy to be given off as a photon, the calculated formulas for
the first five series are
1 ' X StjWVl 1 n 2 /
n, = 2:
n f = 3:
», = 4:
1  me * ( V J_\
I = >»^ ( 1 J_\
A 8k W\3 2 n 2 I
1 _ me A ( I J_\
A 8efdfi\$ n 2 /
1 _ me' 1 ( 1 J \
n = 2, 3, 4,
n  3, 4. 5,
n = 4, 5, 6,
ti = 5, 6, 7,
n = 6, 7, 8,
Lyman
B aimer
Paschen
Brackett
Pfyrtd
These sequences are identical in form with the empirical spectral series discussed
earlier. The Lyman series corresponds to n f = 1; the Balmer series corresponds
to n r = 2; the Paschen series corresponds to n f = 3; the Brackett series corre
sponds to n {  4; and the Pfund series corresponds to n,  5.
We still cannot consider our assertion that the line spectrum of hydrogen
originates in electron transitions from high to low energy states as proved,
however. The final step is to compare the value of the constant term in the
above equations with that of the Rydberg constant R of the empirical equations
4.15 to 4.19. The value of this constant term is
me 4 9.1 x 1Q 31 kg X (1.6 X 1(1" IB C) 1
&j,W
128
8 X (8.85 X JO" 12 F/m) 2 X 3 X 10 s m/s X (6.63 x H)~» Jsf
= 1.097 X 10*0! '
which is indeed the same as fi! This theory of the hydrogen atom, which is
essentially that developed by Bohr in 1913, therefore agrees both qualitatively
and quantitatively with experiment. Figure 42(1 shows schematically how
spectral lines are related to atomic energy levels.
THE ATOM
= E =
energy
FIGURE 420 Spectral lines originate in transitions between energy levels.
4.9 NUCLEAR MOTION
In the preceding analysis, we assumed that the hydrogen nucleus (a single proton)
remains stationary while the orbital electron revolves around it. What must
actually happen is that both nucleus and electron revolve around their common
center of mass, which is very close to the nucleus liecause the nuclear mass is
much greater than that of the electron (Fig, 421). Because the nucleus and the
electron are always on opposite sides of the center of mass, their linear momenta
are in opposite directions, and linear momentum is conserved by the atom,
A system of this kind is equivalent tu a single particle of mass m' that revolves
around the position of the heavier particle. (This equivalence is demonstrated
in most mechanics textbooks; see Sec. 8,8.) If m is the electron mass and M
the nuclear mass, the m' is given by
4.27
m + iW
The quantity m' is called the reduced mass of the electron because its value
is less than in. To correct for the motion of the nucleus in the hydrogen atom.
ATOMIC STRUCTURE
129
FIGURE 421 Both the electron and nucleus ot a hydrogen atom revolve around a common center ol
mait.
then, all we need to do is to imagine that the electron is replaced by a particle
of mass m' and charge — e. The energy levels of the atom therefore become
4.28
m'e* / 1 \
Owing to motion of the nucleus, all the energy levels of hydrogen are changed
by the fraction
m
M
M + m
1,836
~ 1,837
= 0.99945
an increase of 0.055 percent since the energies E n , being smaller in absolute value,
are therefore less negative. The use of Eq. 4.28 in place of 4,23 removes a small
but definite discrepancy between the predicted wavelengths of the various
spectral lines of hydrogen and the actual experimentally determined wave
lengths. The value of the Rydberg constant H to eight significant figures without
correcting for nuclear motion is 1.0973731 X 10 T m 1 ; the correction lowers it
to 1.0967758 X lO'm" 1 .
The notion of reduced mass played an important part in the discovery of
deuterium, an isotope of hydrogen whose atomic mass is almost exactly double
that of ordinary hydrogen owing to the presence of a neutron as well as a proton
in the nucleus. Because of the greater nuclear mass, the spectral lines of deute
rium are all shifted slightly to wavelengths shorter than those of ordinary hydro
gen. The H a line of deuterium, for example, has a wavelength of 6,561 A, while
that of hydrogen is 6.563 A: a small but definite difference, sufficient for the
identification of deuterium.
4.10 ATOMIC EXCITATION
There are two principal mechanisms that can excite an atom to an energy level
above its ground state, thereby enabling it to radiate. One mechanism is a
collision with another particle during which part of their joint kinetic energy
is absorbed by the atom. An atom excited in this way will return to its ground
state in an average of 10 8 s by emitting one or more photons. To produce an
electric discharge in a rarefied gas, an electric field is established which acceler
ates electrons and atomic ions until their kinetic energies are sufficient to excite
atoms they happen to collide with. Neon signs and mercuryvapor lamps are
familiar examples of how a strong electric field applied between electrodes in
a gasfilled tube leads to the emission of the characteristic spectral radiation of
that gas, which happens to be reddish light in the case of neon and bluish light
in the case of mercury vapor.
A different excitation mechanism is involved when an atom absorbs a photon
of light whose energy is jnsl the right amount to raise the atom to a higher energy
level. For example, a photon of wavelength 1 ,217 A is emitted when a hydrogen
atom in the n = 2 state drops to the n = 1 state; the absorption of a photon
of wavelength 1,217 A by a hydrogen atom initially in the n = 1 state will
therefore bring it up to the n = 2 state. This process explains the origin of
absorption spectra. When white light, which contains all wavelengths, is passed
through hydrogen gas, photons of those wavelengths that correspond to transitions
lietween energy levels are absorbed The resulting excited hydrogen atoms
rcradiate their excitation energy almost at once, but these photons come off in
random directions with only a few in the same direction as the original beam
of while light. The dark lines in an absorption spectrum are therefore never
completely black, but only appear so by contrast with the bright background.
We expect the absorption spectrum of any element to be identical with its
emission spectrum, then, which agrees with observation.
Atomic spectra are not the only means of investigating the presence of discrete
energy levels within' atoms. A series of experiments based on the first of the
excitation mechanisms of the previous section was performed by Franck and
Hertz starting in 1914, These experiments provided a direct demonstration thai
atomic energy levels do indeed exist and, furthermore, that these levels are the
same as those suggested by observations of line spectra.
130
THE ATOM
ATOMIC STRUCTURE
131
132
Franck and Hertz bombarded the vapors of various elements with electrons
of known energy, using an apparatus like that shown in Fig. 422. A small
potential difference V„ is maintained between the grid and collecting plate, so
that only elections having energies greater than a certain minimum contribute
to the current i through the ammeter. As the accelerating potential V is in
creased, more and more electrons arrive at the plate and i rises (Fig. 422).
If kinetic energy is conserved in a collision lwtween an electron and one of the
atoms In the vapor, the electron merely lxmiices off in a direction different from
its original one. Because an atom is so much heavier than an electron, the latter
loses almost no kinetic energy in the process. After a certain critical electron
energy is reached, however, die plate current drops abruptly. The interpretation
of this effect is that an electron colliding with one of the atoms gives up some
or all of its kinetic energy in exciting the atom to an energy level above its ground
state. Such a collision is called trtdosik, in contrast to an elastic collision in
which kinetic energy is conserved. The critical electron energy corresponds to
the excitation energy of the atom.
Then, as die accelerating potential Vis raised further, the plate current again
increases, since the electrons now have sufficient energy left after experiencing
an inelastic collision to reach the plate. Eventually another sharp drop in plate
current i occurs, which is int erpr eted as arising from the excitation of the same
higher energy level in another atom. As Fig. 423 indicates, a .series of critical
potentials for a particular atomic species is obtained in this way. Thus the
highest potentials result from several inelastic collisions and are multiples of
the lower ones.
To check the interpretation of critical potentials as being due to discrete atomic
energy levels, Frauds and Hertz observed the emission spectra of vapors (luring
electron bombardment, hi die case of mercury vapor, for example, they found
that a minimum electron energy of 4.H eV was required to excite the 2,536A
FIGURE A22 Apparatus lor the Frant*. Hertz experiment.
filament grid plate
THE ATOM
FIGURE 423 Results of the
FranckHertz experiment, showing
critical potentials,
A /
A /
ACCELERATING POTENTIAL, V
spectral line of mercury— and a photon of 2,536A light has an energy of just
1.9 eV. The FranckHertz experiments were performed shortly after llohr an
nounced his theory of the hydrogen atom, and they provided independent
confirmation of his basic ideas.
4.11 THE CORRESPONDENCE PRINCIPLE
The principles of quantum physics, so different from those of classical physics
in i he microscopic world that lies beyond the reach of our senses, must never
theless yield results identical with those of classical physics in I lie domain where
experiment indicates the latter to be valid. We have alreadv seen that this
fundamental requirement is satisfied by the theory of relativity, the quantum
theory of radiation, and l lie wave theory of matter; we shall now show Hint it
Is satisfied also by Bohr's theory of the atom.
According to electromagnetic theory, an electron moving in a circular orbit
radiates electromagnetic waves whose frequencies are equal to its frequency of
(evolution and to harmonics (that is, integral multiples) of that frequencv. In
a hydrogen atom die electron's; speed is
V4iTC mf
ATOMIC STRUCTURE
133
according to Eq. 4.13, where r is the radius of its orbit. Hence ihe frequency
of revolution /of the electron is
_ electron speed
orbit circumference
2wr
27T V^wfQmr 3
The radius r n of a stable orbit is given in terms of its quantum Bomber n by
Eq. 4,22 as
n 2 hh n
r. =
and so the frequency of revolution is
Under what circumstances should the Bohr atom behave classically? If the
electron orbit is so large that we might expect to be able to measure it directly,
quantum effects should be entirely inconspicuous. An orbit 1 cm across, for
example, meets this specification; its quantum number is very close to n = 10,000,
and, while hydrogen atoms so grotesquely large do not actually occur because
their energies would be only infinitesimal ly below the ionization energy, they
are not prohibited in theory. What does the Bohr theory predict that such an
atom will radiate? According to Eq. 4.25, a hydrogen atom dropping from the
Hjth energy level to the n f th energy level emits a photon whose frequency is
^ me* ( 1 1 \
Let us write n for the initial quantum number n { and n — p (where p = 1, 2,
3, . . .) for the final quantum number n r With this substitution,
me 4 J I
8e u 2 n 3 >p) 2
me 1
ite^h*
inp — p i
n 2 (n — pf
Now, when a, and n, are both very large, n is much greater than p, and
2np — p' 2 21 inp
(n — pf ~ n 2
so that
4.30
Se^h 3
fc 3 W 3 /
When p — 1, the frequency v of the radiation is exactly the same as the frequent)'
of rotation/of the orbital electron given in Eq. 4.29. Harmonics of this frequency
are radiated when p = 2, 3, 4, , . , . Hence both quantum and classical pictures
of the hydrogen atom make identical predictions in the limit of very large
quantum numbers. When n = 2, Eq. 4.29 predicts a radiation frequency that
differs from that given by Eq. 4.25 by almost 300 percent, while when
n = 10,000, the discrepancy is only about 0.01 percent.
The requirement that quantum physics give the same results as classical physics
in the limit of large quantum numbers was called by Bohr the correspondence
principle. It has played an important role in the development of the quantum
theory of matter.
Problems
'1. A 5MeV alpha particle approaches a gold nucleus with an impact parameter
of 2.6 X 10" 13 m. Through what angle will it be scattered?
* 2. What is the impact parameter of a 5MeV alpha particle scattered by 10"
when it approaches a gold nucleus?
* 3. What fraction of a beam of 7.7MeV alpha particles incident upon a gold
foil 3 x I0" T m thick is scattered by less than 1"?
*4. What fraction of a l>eam of 7,7MeV alpha particles incident upon a gold
foil 3 X I0" T m thick is scattered by 90° or more?
* 5. Show that twice as many alpha particles are scattered by a foil through angles
hetween 60 and 90° as are scattered through angles of 90° or more.
6. A beam of 8.3MeV alpha particles is directed at an aluminum foil. It is
found that the Rutherford scattering formula ceases to be obeyed at scattering
angles exceeding about 60°. If the alpha particle is assumed to have a radius
of 2 x 10~ 15 m, find the radius of the aluminum nucleus.
Determine the distance of closest approach of 1MeV protons incident upon
gold nuclei.
8 Find the distance of closest approach of 8MeV protons incident upon gold
nuclei.
134
THE ATOM
ATOMIC STRUCTURE
135
9. The derivation of the Rutherford scattering formula was made nonrelativisti
eally. Justih this approximation by computing the mass ratio between an 8MeV
alpha particle and an alpha particle at rest.
10. Kind iht? frequency of rotation of the electron in the classical model of the
hydrogen atom. In what region of the spectrum are electromagnetic waves of
this frequency?
11. The electricfield intensity at a distance r from the center of a uniformly
charged sphere of radius Ii and total charge (J is ()r/45re () H 3 when r < /{. Such
a sphere corresponds to the Thomson model of the atom. Show that an electron
in this sphere executes simple harmonic motion about its center and derive a
formula for the frequency of this motion. Evaluate the frequency of the electron
<ist illations for the case of the hydrogen atom and compare it with the frequencies
of the spectral lines of hydrogen.
12. Find the wavelength of the spectral line corresponding to the transition
in hydrogen from the n = fl state to the n = 3 State.
13. Find the wavelength of the photon emitted when a hydrogen atom goes
from the n = HI state to its ground state.
11. How much energy is required to remove an electron in the n = 2 slate
I ron i a hydrogen atom?
15. A beam of electrons bombards a sample of hydrogen. Through what
potential difference must the electrons have been accelerated if the first line
of the Balnier series is to l>e emitted?
HI Find the recoil speed of a hydrogen atom after it emits a photon in going
from the n = 4 stale to the n = 1 state.
17. How many revolutions does an electron in the n = 2 state of a hydrogen
alum make Ixjfore dropping to the n = I state? (The average lifetime of an
excited state is about H)~ M s.)
)H, The average lifetime of an excited atomic state is 10 N s. If the wavelength
of the spectral line associated with the decay of this state is 5.000 A, find the
width of the line.
19. At what temperature will the average molecular kinetic energy in gaseous
hydrogen eijiial the binding energ) ol ;i bytfe o gep atom?
20. Calculate the angular momentum about the nucleus of an electron in the
nth orbit of a hydrogen atom, and show from this that an alternate expression
of Bohr's first postulate is that the angular momentum of such an atom must
be nti. (In fact, the quantization of angular momentum in units of ft" was the
starting point of Bohr's original work, since the hypothesis of de Broglie waves
had not been proposed as yet. We shall see in (hap. 8 that this quantization
rule holds only for the component of the angular momentum of a system in one
particular direction, while the magnitude of the total angular momentum is
quantized in a somewhat different way.)
21. A mixture of ordinary hydrogen and tritium, a hydrogen isotope whose
nucleus is approximately three times more massive than ordinary hydrogen, b
excited and its spectrum observed. How far apart in wavelength will the H n
lines of the two kinds of hydrogen be?
22. A ji' meson (in = 207 m r ) can be captured by a proton In form a "mesne
atom." Find the radius of the first Bohr orbit of such an atom.
23. A (i~ meson is in the n — 2 state of a titanium atom. Find the energy
radiated when the mesic atom drops to its ground state.
24. A positronium atom is a system consisting of a positron (positive electron)
and an electron, (a) Compare the wavelength of the photon emitted in the
ii — 3 — » n — 2 transition in positronium with that of the H a line, (b) Compare
the ionization energy in positronium with that in h y dr ogen .
25. {a) Derive a formula for ihe energy levels of a litjtlrogenic atom, which is
an ion such as He 1 " or Li* + whose nuclear charge is + Ze and which contains
a single electron, (b) Sketch the energy levels of the He* ion and compare them
with the energy levels of the H atom, (c) An election joins a bare helium nucleus
to form a He 4 ion. Find the wavelength of the photon emitted in this process
ii die electron is assumed to have had no kinetic energy when it combined with
the nucleus.
26'. Use the uncertainty principle to determine ihe groundstate radius r l of
the hydrogen atom in Ihe following way. First find a formula for the electron
kinetic energy in terms of the momentum an electron must have if confined to
a region of linear dimension r,. Add this kinetic energy to the electrostatic
potential energy of an electron the distance r, from a proton, and different late
wiih respect to r, the resulting expression for the total electron energy I: to liiul
the value of r, for which E is a minimum. Compare the result with that given
by Eq. J.22 with it = 1.
136
THE ATOM
ATOMIC STRUCTURE
137
QUANTUM MECHANICS
5
The Bohr theory of the atom, discussed in the previous chapter, is able to account
for certain experimental data in a convincing manner, but it has a number of
severe limitations. While the Bohr theory correctly predicts the spectral series
of hydrogen, hydrogen isotopes, and hydrogenic atoms, it is incapable of being
extended to treat the spectra of complex atoms having two or more electrons
each; it can give no explanation of why certain spectral lines are more intense
than others (that is, why certain transitions between energy levels have greater
probabilities of occurrence); and it cannot account for the observation that many
spectral lines actually consist of several separate lines whose wavelengths differ
slightly. And, perhaps most important, it does not permit us to obtain what a
really successful theory of the atom should make possible: an understanding of
hew individual atoms interact with one another to endow macroscopic aggregates
of matter with the physical and chemical properties we observe.
These objections to the Bohr theory are not put forward in an unfriendly way,
for the theory was one of those seminal achievements that transform scientific
thought, but rather to emphasize that an approach to atomic phenomena of
greater generality is required. Such an approach was developed in 19251926
by Erwin Schrodinger, Werner Heisenberg, and others under the apt name of
quantum meclianics. By the early 1930s the application of quantum mechanics
to problems involving nuclei, atoms, molecules, and matter in the solid state made
it possible to understand a vast body of otherwisepuzzling data and — a vital
attribute of any theory — led to predictions of remarkable accuracy.
5.1 INTRODUCTION TO QUANTUM MECHANICS
Hie fundamental difference between Newtonian mechanics and quantum me
chanics lies in what it is that they describe. Newtonian mechanics is concerned
w »lh the motion of a particle under the influence of applied forces, and it takes
for granted that such quantities as die particle's position, mass, velocity, acceler
139
ation, etc.. vim be measured. This assumption is, of course, completely valid
in our everyday experience, and Newtonian met banks provides the "correct"
explanation far llic behavior of moving bodies in the sense that the values il
predicts for observable magnitudes agree with the measured values of those
maunitudrs.
Quantum mechanics, loo, consists of relationships between olwervable magni
tudes, but tile uncertainty principle radically alters the definition of ■'observable
magnitude" in the atomic realm. According to the uncertainly principle, the
position and momentum ol a particle cannot be accurately measured at the same
time, while in Newtonian mechanics Imth are assumed to have definite, ascer
tainable values at every instant. The quantities whose relationships quantum
mechanics explores are probabilities. Instead of asserting, for example, that the
radius of the electron's orbit in a groundstate hydrogen atom is aiwavs exactly
5,3 X 10 ■> m, quantum mechan ics states that this is the most pmbiibh radios;
if we conduct a suitable experiment, most trials will yield a different value, eitl.er
larger or smaller, but the value most likely to lie found will be 5.3 X HI" 11 m.
At first glance quantum mechanics seems a poor substitute for Newtonian
mechanics, hut closer inspection reveals a striking fact: Xnutonuin nieehimif.
It nothing but an approximate tersiott of tpiantaia nirriinnies. The certainties
proclaimed In Newtonian mechanics are illusory, and their agreement with
experiment is a consequence of the fact thai macroscopic bodies consist of so
many individual atoms that departures from average behavior are uniiotieeable.
Instead of two sets of physical principles, one for the macroscopic universe and
one lor the microscopic universe, there is only a single sel, and quantum me
chanics represents our best effort to date in formulating it.
5.2 THE WAVE EQUATION
As mentioned in Chap. 3, the quantity with which quantum mechanics is con
cerned is the warn function * of a body. While + itself has no physical
interpretation, the square of its absolute magnitude j+[ 2 (or ++* if «i' is complex)
evaluated at a particular point at a particular time is proportional to the proba
bility of experimentally finding the body there at that lime. The problem <>l
quantum mechanics is to determine 'I' for a body when its freedom of motion
is limited by the action of external forces.
liven liefore we consider the actual calculation of +, we can establish certain
requirements it must always fulfill, for one thing, since * a is proportional to
the probability /'of finding the Iwdy deserifyed by +, the integral of +  over
all space must be finite — the body is somewhere, alter all. If
S.1
J l+ldV
is 0, the particle does not exist, and the integral obviously cannot be oc awl
still mean anything: j^l cannot be negative or complex because of the way it
i, defined, and so the only possibility left is that its integral be a finite quantity
if >k is to descrilx; properly a real body.
II is usually convenient to have 4' be equal to the probability P of finding
the particle described by +, rather than merely be proportional to P. If l^l"'
is In equal /' then it must l>e true thai
5.2 j X \^\ i (IV=l
f I'dV = 1
Normalization
since
is the mathematical statement that the particle exists somewhere at all times
A wave function that obeys Eq. 5.2 is said to Ix: normalized. Kverv acceptable
wave function can be normalized by multiplying by an appropriate const. ml,
we shall shortly sec exactly how this is done.
Besides being normalizable, + must be singlevalued, since /' can have only
one value at a particular place and lime. A further condition that *!' must obey
is that it and its partial derivatives H" • t, • +  i/. ■ "\'/<)z l>e continuous every
where.
v/mWtJigrr s equation, which is the fundamental equation of quantum me
chanics in the same sense that the second law of motion is the fundamental
equation of Newtonian mechanics, is a wave equation in the variable +. Before
we tackle .Sehrodinger's equation, let ils review the general wave equation
5.3
fcr* ~ e» i fi
Wave equation
wlueli governs a wave whose variable quantity is ij thai propagates in the x
direction with the speed o. In the case of a wave in a stretched string, r/ is the
I^plaeeinenl of ibe string from the .v axis; in the ease ol a sound wave. 1/ is
me pressure d iffe r en ce; in the case of a light wave, ij is either the electric or
die magnetic field magnitude. The wave equation is derived in textbooks of
mechanics for mechanical waves and in textbooks of electricity and magnetism
•or electromagnetic waves.
Solutions to the wave equation may be of many kinds, reflecting the variety
of waves thai can occur — a single traveling pulse, a train of waves of constant
140
THE ATOM
QUANTUM MECHANICS 141
amplitude and wavelength, a train of superposed waves of the same amplitudes
and wavelengths, a train of superposed waves of different amplitudes and wave
lengths, a standing wave in a string fastened at both ends, and so on. AH solutions
must be of the form
5.4
'I
= ''(' *i)
where F is any function that can lie differentiated. The solutions F(t — x/v)
represent waves traveling in the + x direction, and the solutions F(t + x/v)
represent waves traveling in the — x direction.
Our interest here is in the wave equivalent of a "free particle," namely a
particle that is not under the influence of any forces and therefore pursues a
straight path at constant speed. This equivalent corresponds to the general
solution of Eq. 5.3 for undamped (that is, constant amplitude A), monochromatic
(constant angular frequency w) harmonic waves in the +■* direction,
5.5
— J»MI~i/»)
y = Ae
In this formula y is a complex quantity, with both real and imaginary parts.
Because
5.6 e~'* = cos — i sin
Eq. 5.5 can be written in the form
5.7 y = A cos aft  — I  iA sin ult  — 1
Only the real part of Eq. 5.7 has significance in the case of waves in a stretched
string, where ij represents the displacement of the string from its normal position
(Fig. 51); in this case the imaginary part is discarded as irrelevant.
FIGURE 51 Wxis In the xy plane traveling (n the + 1 direction along a stretched string lying on the i
5.3 SCHRODINGER'S EQUATION: TIMEDEPENDENT FORM
In quantum mechanics the wave function * corresjionds to the wave variable tj
of wave motion in general. However, +, unlike t/, is not itself a measurable
quantity and may therefore be complex. For this reason we shall assume that
4> is specified in the .t direction by
5.8 * = Ae M> * /pi
When we replace w in the above formula by 2t,v and v by Xr, we obtain
5.9 * = Ae 3 '"""'"
which is convenient since we already know what f and A are in terms of the
total energy E and momentum p of the particle being described by *. Since
and
E = he = 2irHp
\ = h. = ?lEE
we have
5,10 + = Ae <"*><£<«"'
Equation 5.10 is a mathematical description of the wave equivalent of an
unrestricted particle of tola] energy /■. and momentum p moving in the +x
direction, just as Eq. 5.5 is a mathematical description of, for example, a harmonic
displacement wave moving freely along a stretched string.
The expression for the wave function <fr given by Eq. 5.10 is correct only for
freely moving particles, while wc are most interested in situations where the
motion of a particle is subject to various restrictions. An important concern,
for example, is an electron Ixmnd to an atom by the electric field of its nucleus.
What we must now do is obtain the fundamental differential equation for 4',
which we can then solve in a specific situation.
We !>egin by differentiating Eq. 5.10 twice with respect to x, yielding
5.11
= £ +
ft*
ff = Atoswll i/o)
and once with respect to (. yielding
3l * ft
A I speeds small compared with that of light, the total energy E of a particle
142
THE ATOM
QUANTUM MECHANICS 143
is the sum of its kinetic energy p/2m and its potential energy V, where V is
in general a function of position x and time <:
5.13
/■; =
+ v
Multiplying both sides of this equation by the wave function 'I'.
5.14 FA' = *— + \'*l'
2m
From Kqs. 5.11 and 5.12 we see that
r 3/
and
s.i6 p»* = ft 2 —
< \
Substftttting these expressions for !'A' and p1' into Eq. 5.14, we obtain
3< 2m dx 2
Timedependent
Schrodinger's equation
in one dimension
Equation 5.17 is the timedependent form of Schrddktg/sr's equation. In three
dimensions the timedependent form of Schrodinger's equation is
9f 2m I &* + dtf + p J + V
where the particle's potential energy V is some function of x, ;/, ~, and /. Am
< lions that may be present on the particle's motion will ailed Ihe
polentialenergy tun. linn V. Once Vis known, Schrodinger's equation may be
solved for the wave finiction + of the particle, from which its probability density
'I' ma) be determined lor a specified v, ;/, :, /.
The manner in which Schrodinger's equation was obtained starling from llie
wave Function of a freely moving particle deserves attention, The extension of
Ncliriidingcr's equation from the special case of an unrestricted particle potential
energy V = constant) to the general case of a particle subject to arbitrary forces
that vary in time and space [V = V(.v,r/ .».()] is entirely plausible, but there is
no a priori way to print that ibis extension is correct. All we can do is to
postulate Sehriklinger's equation, solve it for a variety of physical Situations, and
compare the results of the calculations with the results of experiments. If they
agree, the postulate embodied in Schrodinger's equation is valid: if llie) disagree,
the postulate must be discarded and some other approach would have to he
t qilorcd. In other words, Schrodinger's equation cannot be derived from "first
principles," but represents a first principle itself.
In practice, Schrodinger's equation has turned out to Iw completely accurate
in predicting the results of experiments. To be sure, we must keep in mind that
Eq. 5.18 can be used only for nonrelativistic problems, and a more elaborate
formulation is required when particle speeds comparable with that of light are
involved. Because it is in accord with experiment within its range of applica
bility, we are entitled to regard Schrodinger's equation as representing a success
ful postulate concerning certain aspects of the physical world. But for all its
success, this equation remains a postulate in the same sense as the postulates
of special relativity or statistical mechanics: None of these can be derived from
some other principle, and each is a fundamental generalization neither more nor
less valid than the empirical data it is based upon. It is worth noting in this
connection that .Schrodinger's equation does not represent an increase in the
number of postulates required to descril>e the workings of the physical world,
because Newton's second law of motion, regarded in classical mechanic] as a
postulate, can be derived from Schrodinger's equation provided that the quanti
ties it relates are understood to be averages rather than definite values,
5.4 EXPECTATION VALUES
Once Schrodinger's equation has been solved for a particle in a given physical
situation, the result Mm wave function +(x,i/ r 3,r) contains all the information about
the particle that is permitted by the uncertainty principle. Except for those
variables that happen to be quantized in certain cases, this information is in the
form of probabilities and not specific numbers.
Ys an example, let us calculate the expectation value (x> of the position of
a particle confined to the X axis that is described by the wave function ^(x.i).
this is the value of ,v we would obtain if we determined experimental l\ ihe
positions of a great many particles described by the same wave function at some
idsiiiin t and then averaged the results,
1" make the procedure clear, we shall first answer a slight!) different question:
What is the average position x of a number of particles distributed along the
( axis in such a way that there are ;V, particles at x,, N 2 particles at x.„ and
J The average position in this case is the same as the center of mass of
Hie distribution, and SO
__ ,V,x, + X 2 x 2 + A ' :l .v a + •••
.Y, + V 2 + N a +
S Y r
2M
144
THE ATOM
QUANTUM MECHANICS 145
When we are dealing wilh a single particle, we mnsl replace the numlier N t
of particles at i, by the probability P, that the particle Ije found in an interval
dx at x,. This probability is
/> = i*,, 2 dx
where +, Ls the particle wave function evaluated at .r = .*,. Making this substi
tution and changing the summations to integrals, we see that the expectation
value of the position of the single particle is
5,19
<*> =
f x\*] 2 dx
— » m
j ** dx
If + is a normalized wave function, the denominator of Eq. 5.19 is equal to
the probability that the particle exists somewhere between x =  oo and x = oo,
and therefore has the value 1. Hence
5.20
<*> = fxWfdx
This formula states that <x> Ls located at the center of mass (so to speak) of
*  2 ; if +] a is plotted versus x on a graph and the area enclosed by the curve
and the x axis is cut out, the balance point will he at (*>.
The same procedure as that followed aliove can be used to obtain the ex
pectation value <C(x)> of any quantity [for instance, potential energy V(x}\ that
is a function of the position x of a particle described by a wave function +,
The result is
5.21
(C{x)> = J"c(*))**rfx
Expectation value
This formula holds even if G(x) varies with time, because <C(*}> in any event
must be evaluated at a particular time ( since + is itself a function of f.
146
5.5 SCHRODINGER'S EQUATION: STEADYSTATE FORM
In a great many situations the potential energy of a particle does not depend
upon time explicitly; the forces that act upon it, and hence V, vary with the
position of the particle only. When this is true, Schrodinger's equation may be
simplified by removing all reference to /. We note that the onedimensional wave
function + of an unrestricted particle may l>e written
THE ATOM
5,22
= A e HB/KH
That is, + is the product of a timedependent function e  "*'* 1 ' and a position
dependent function $. As it happens, the time variations of all functions of
particles acted upon by stationary forces have the same form as that of an
unrestricted particle. Substituting the * of Eq. 5.22 into the timedependent
form of Schrodinger's equation, we find that
2m dx 2 V
and so, dividing through by the common exponential factor,
dH 2m
a? + TF< E " ^ = °
Steady state
Schrodinger's equation
in one dimension
Equation 5.23 is the steadystate form of Schrodinger's etiuation. In three di
mensions it is
5.24
Bfy 3ty 3ty 2m
Steadystate
Schrodinger's equation
in three dimensions
In general, Schrodinger's steadystate equation can lie solved only for certain
values of the energy E. What is meant by this statement has nothing to do with
any mathematical difficulties Uwt may lie present, but is something much more
fundamental, To "solve" Schrodinger's equation for a given system means to
obtain a wave function ^ that not only obeys the equation and whatever boundary
conditions there are, but also fulfills the requirements for an acceptable wave
function— namely, that it and its derivatives be continuous, finite, and single
valued. If there is no such wave function, the system cannot exist in a steady
state. Thus energy quantization appears in wave mechanics as a natural element
of the theory, and energy quantization in the physical world ts revealed as a
universal phenomenon characteristic of all stable systems.
A familiar and quite close analogy to the manner in which energy quantization
occurs in solutions of Schrodinger's equal ion is with standing waves in a stretched
string of length /. that is fixed at both ends. Here, instead of a single wave
propagating indefinitely in one direction, waves are traveling in both the +x
and —x directions simultaneously subject to the condition that the displacement
V always be zero at both ends of the string. An acceptable function y(x,t) for
the displacement must, with its derivatives, obey the same requirements of
QUANTUM MECHANICS 147
continuity, finiteness, ami singlevaluedness as 4 and, in addition, must lie real
Since 1/ represents a directly measurable quantity. The only solutions of the wave
equation
dx 2
1 3^y
that are in accord with these various limitations are those in which the wave
lengths arc given by
2L
<V. =
n + 1
rr = (), I, 2, 3, . . .
as shown in Fig. 52. It is the combination of the wave equation ami I he
restrictions placed on the nature of its solution that leads ps to conclude thai
y(x,t) can exist only for certain wavelengths A„.
The values of energy E n for which Schrodinger's steadystale equation can
be solved are called eigenvalues and the corresponding wave functions ii F1 are
FIGURE 52 Standing waves In a stretched siring fastened at both ends.
\=2L
ri + I
148
THE ATOM
called eigenfuiwthiiif;. (These terms come from the German Eigtmwm, meaning
"proper or characteristic value," and Eigtinfunktion, or "proper or characteristic
function.") The discrete energy levels of the hydrogen atom
E„ =
32tt% 2 H 2
(*)
n = 1, 2, 3,
are an example of a set of eigenvalues; we shall see in Chap, 6 why these
particular values of E are the only ones that yield acceptable wave functions
for the electron in the hydrogen atom.
An important example of a dynamical variable other than total energy that
is found to be quantized in stable systems is angular momentum L. In the case
of the hydrogen atom, we shall find that die eigenvalues of the magnitude of
die total angular momentum are specified by
E t = Vt(t+ l)fi / = <>, 1, 2 (n 1)
Of course, a dynamical variable G may not be quantized. In this case meas
urements of G made on a numlier of identical systems will not yield a unique
result but instead a spread of values whose average is the expectation value
<G> = f'Gh!,\?dx
In the hydrogen atom, the electron's position is not quantized, for instance, so
that we must think of the electron as being present in the vicinity of the nucleus
with a certain probability ^; 2 per unit volume but with no predictable position
or even orbit in the classical sense. This probabilistic statement does not conflict
with the fact that experiments performed on hydrogen atoms always show that
it contains one whole electron, not 27 percent of an electron in a certain region
and 73 percent elsewhere; the probability is one of finding the electron, and
although this probability is smeared nut in space, the election itself is not.
5,6 THE PARTICLE IN A BOX: ENERGY QUANTIZATION
To solve Schrodi tiger's equation, even in its simpler steadystate form, usually
requires sophisticated mathematical techniques. For this reason the studv of
quantum mechanics has traditionally been reserved for advanced students who
have the required proficiency in mathematics. However, since quantum me
chanics is the theoretical structure whose results are closest to experimental
reality, we must explore its mediods and applications if we are to achieve any
understanding of modem physics. As we shall see, even a relatively limited
mathematical background is sufficient for us to follow the trains of thought that
have led quantum mechanics to its greatest achievements.
QUANTUM MECHANICS 149
150
Our firsl problem using Schrodinger's equation is that of a particle bouncing
back and forth l>etween the walls of a box (Fig, .53}. Our interest in this problem
is threefold: to see how Schrodinger's equation is solved when the motion of
a particle is subject to restrictions; to learn the characteristic properties of
solutions of this equation, such as the limitation of particle energy to certain
specific values only; and to compare the predictions of quantum mechanics with
those of Newtonian mechanics.
We may specify the particle's motion by saying that it is restricted to traveling
along the x axis between x = and x = L by infinitely hard walls. A particle
does not lose energy when it collides with such walls, so that its total energy
stays constant. From the formal point of view of quantum mechanics, the
potential energy V of the particle is infinite on both sides of the Iwx, while V
is a constant— say for convenience— on the inside. Since the particle cannot
have an infinite amount of energy, it cannot exist outside the box, and so its
wave function is for x < and x > /., Our task is to find what + is within
the box, namely, Ixjtween x = and x = L.
Within the box Schrodinger's equation becomes
d 2 ^ 2m
FIGURE 53 A particle confined to a
box of width I .
THE ATOM
since V = there. (The total derivative tl'^/dx' 2 is the same as the partial
derivative d^/Sx 2 because <p is a function of x only in this problem.) Equation
5.25 has the two possible solutions
5.26
5.27
tfr = A sin
f2m~E
V ft" 2
y = B cos
ft" 2
■2,i,l
f2m~E
which we can verify by substitution back into Eq. 5.25; their sum is also a
solution. A and ii are coast ants to be evaluated.
These solutions are subject to the important boundary condition that ^ =
for x = and for x = L. Since cost) = 1, the second solution cannot describe
the particle because it does not vanish at x = 0. Hence we conclude that li = (1.
Since sin = 0, the first solution always yields y = (1 at x = 0, as required, but
^ will be at jc = /.. only when
llm'E
V ft 2
5.28
2f "£ , , „
^— L = v, 2ff, .Jw, . . .
= nir it = 1, 2, 3,
This result comes about because the sines of the angles », 2v, 3^, ... are all 0,
From Eq. 5.28 it is clear that the energy of the particle can have only certain
values, which are the eigenvalues mentioned in the previous .section. These
eigenvalues, constituting the energy levels of the system, are
5.29
uMPs 2
E =
2mL'
n = I, 2, 3,
Particle in a box
The integer n corresponding to the energy level K„ ts called its quantum number.
A particle confined to a Iwx cannot have an arbitrary energy: the fact of its
confinement leads to restrictioas on its wave function that permit it to have only
those energies specified by Eq. 5.29.
It is significant that the particle cannot have zero energy; if it did, the wave
function ij, would have to be zero everywhere in the box. and tlm means that
the particle cannot be present there. The exclusion of E — as a possible value
for the energy of a trapped particle, like the limitation of E to a discrete set
of definite values, is a quantummechanical result that has no counterpart in
classical mechanics, where all energies, including zero, are presumed possible.
The uncertainty principle provides confirmation that E = is not admissible.
Because the particle is trapped in the box, the uncertainty in its position is
QUANTUM MECHANICS 151
152
Ax = L, the width of the box. The uncertainty in its momentum must therefore
be
which is not compatible with E = 0. We note that the momentum corresponding
to /■;, is, since the particle energy here is entirely kinetic,
■nh
which it in accord with the uncertainty principle.
Why are we not aware of energy quantization in our own experience? Surely
a marble rolling hack and forth between the sides of a level box with a smooth
door can have any speed, and therefore any energy, we choose to give it,
including zero. In order to assure ourselves that Eq. 5.29 does not conflict with
our direct observations while providing unique insights on a microscopic scale,
we shall compote the permitted energy levels of (1) an electron in a Ijox 1 A
widu and 2: ;i 10g marble in a !>m lOaa W 'kk.
In case I we have m =9.1 X W"* 1 kg and L = I A = 10" w m, so that the
permitted electron energies are
n 2 X iff 2 X (1054 X lQ^Js) 2
E " ~ 2 X 9.1 X 10 :,1 kgX (10 ,o m) 2
= 6.0 x 10 18 n z J
= 3H« Z eV
The minimum energy the electron can have is 38 eV, corresponding to n = f .
The sequence of energy levels continues with E^ ~ 152 eV, E, = 342 eV, E A =
60S eV, and SO on (Fig. 54). These energy levels are sufficiently far apart to
make the quantization of electron energy in such a box conspicuous if such a
box act n ally did exist.
In case 2 we have m = 10 g = I0~ s kg and L = 10 cm = 10 1 m, so that the
permitted marble energies are
IL =
n 2 %<x' 1 X (1.054 X 10 34 Js)'
2 X 10" a kg X (lO^ni) 2
= 5.5 X lO'Mfi 2 J
The minimum energy the marble can have is 5.5 X lO 84 J, corresponding to
n = 1. A marble with this kinetic energy has a speed of only 3.3 X 10~ 31 m/s
and is therefore experimentally indistinguishable from a stationary marble. A
reasonable speed a marble might have is, say, l / 3 m/s — which corresponds to the
energy level of quantum number n = 10 3u ! The permissible energy levels are
THE ATOM
700
suo ~
^00 —
FIGURE 54 Energy levels of an electron
confined to a box 1 A wide.
w 3C0 —
so very close together, then, that there is no way to determine whether the marble
can take on only those energies predicted by Eq, 5,29 or any energy whatever.
Hence in the domain of everyday experience quantum effects are imperceptible;
this accounts for the success in this domain of Newtonian mechanics.
5.7 THE PARTICLE IN A BOX: WAVE FUNCTIONS
In the previous section we found that the wave function of a particle in a b
whose energy is E is
V =b A sin / 
V ft 2
QUANTUM MECHANICS 153
Since the possible energies arc
*; =
2mL 3
substituting E„ for E yields
5.30
#„ a* A sin jr
for the cigenfunclions corresponding to the energy eigenvalues E n .
It is easy to verify that these eigenfunctions meet all the requirements we
have discussed: for each quantum number n, & n is a singlevalued function of
x, and $ n and t>4> n /Bx are continuous. Furthermore, the integral of ^J over
all space is finite, as we can see by integrating \4> n r dx from x = to x = L
(since the particle, by hypothesis, is confined within these limits!:
f\^?dx = f\^dx
■ L _ . farx\
■<m*
5.31
»A»f
To normalize y we must assign a value to A such that ^„i 2 is eoutif to the
probability Pdx of finding the particle between x and at + dk, rather than merely
proportional to /'. If ^„i 2 is to equal P, then it must be true that
5.32 f$J*dx=l
since
j Pdx = 1
is the mathematical way of stating that the particle exists somewhere at all limes.
Comparing Eqs. 5.31 and 5.32, wc see that the wave functions of a particle in
a box are normalized if
5.33
A
The normalized wave functions of the particle are therefore
154
/2 nxx
THE ATOM
The normalized wave functions i^, if.,, and ^ together with the probability
densities li^l 2 , \$ 2 \ 2 , and ^ 3 j 2 are plotted in Fig. 55. While ^„ may l>e negative
as well as positive, \$J is always positive and, since ^„ is normalized, its value
at a given x is equal to the probability P of finding the particle there. In every
case ji/J a = at x = and x m L, the boundaries of the box. At a particular
point in the box the probability of the particle being present may be verv
different for different quantum numbers. For instance, ^,p has its maximum
value of %L in the middle of the box, while \^\ 2 = there: a particle in the
lowest energy level of n = 1 is most likely to be in the middle of the box, while
a particle in the next higher state of n a 2 is never there! Classical physics,
of course, predicts the same probability for the particle Ixn'ng anywhere in the
box.
The wave functions shown in Fig. 55 resemble the possible vibrations of a
string fixed at both ends, such as those of the stretched string of Fig. 52. This
is a consequence of the fact that waves in a stretched string and the wave
representing a moving particle are descriljed by equations of the same form,
so that, when identical restrictions are placed upon each kind of wave, the formal
results are identical.
FIGURE 55 Wave functions and probability densities of a particle confined to a box with rigid walls.
* =
x = L
*=0 *=L
QUANTUM MECHANICS 155
5.8 THE PARTICLE IN A NONRIGID BOX
It is interesting to solve the problem of the particle in a box when the walls
of the box are no longer assumed to be infinitely rigid, in this case the potential
energy V outside the box is a finite quantity; the corresponding situation in the
ease of a vibrating string would involve an imperfect attachment of the string
at each end, so that the ends can move slightly. This problem is more difficult
to treat, and we shall simply present the result here. (We shall take another
look at a particle in a nonrigid box wheu we examine the theory of the deulemn
in Chap. 11.)
The first few wave hmctions for a particle in such a box are shown in
Fig. 56. The wave functions ^ n now do not equal zero outside the box. Even
though the particle's energy is smaller than the value of V outside the box, there
ts still a definite probability Chat it be found outside itl In other words, even
though the particle does not have enough energy to break through the walls
of the box according to "common sense," it may nevertheless somehow penetrate
them. This peculiar situation is readily understandable in terms of the uncertainty
principle. Because the uncertainty Sp in a particle's mo men turn is related to
the uncertainty Iv in its position hy the formula
an infinite uncertainty in particle momentiuu outside the box is the price of
definitely establishing that the particle is never there, A particle requires an
infinite amount of energy if its momentum is to have an infinite uncertainty,
implying that V = sc outside the box. If V instead has a finite value outside
the box, then, there is some pmlrability — not necessarily great, but not zero
FIGURE 56 Wave functions and probability densities of a particle confined to a box with nonrigid walls.
#,
J
A Av
z
^^^^
156
x —
THE ATOM
x = L
* =
x = L
either — that the particle will "leak" out. As we shall see in Chap, 12, the
quantummechanical prediction that particles always have some chance of
escaping from confinement (since potential energies are never infinite in the real
world, our original rigidwalled box has no physical counterpart) exactly fits the
observed behavior of those radioactive nuclei that emit alpha particles.
When the confining box has nonrigid walls, the particle wave function ^ n does
not equal zero at the walls. The particle wavelengths that can fit into the box
are therefore somewhat longer than in the case of the box with rigid walls,
corresponding to lower particle momenta and hence to lower energy levels.
The condition that the potential energy V outside the box be finite has another
consequence: it is now possible for a particle to have an energy E that exceeds
V. Such a particle is not trapped inside the box, since it always has enough energy
to penetrate its walls, and its energy is not quantized but may have any value
above V. However, the particle's kinetic energy outside the box, E — V, is always
less than its kinetic energy inside, which is just £ since V = in the box according
to our original specification. Less energy means longer wavelength, and so 4
has a longer wavelength outside the box than inside.
In the optics of light waves, it is readily observed that when a light wave
reaches a region where its wavelength changes (that is, a Tegion of different index
of refraction), reflection as well as transmission occurs. This is the reason we
see our reflections in shop windows. The effect is common to all types of waves,
and it may be shown mathematically to follow from the requirement that the
wave variable (electricfield intensity £ in the case of electromagnetic waves,
pressure p in the case of sound waves, wave height h in the case of water waves,
etc.) and its first derivative t>e continuous at the boundary where the wavelength
change takes place.
Exactly the same considerations apply to the wave function \p representing
a moving particle. The wave function of a particle encountering a region in
which it has a different potential energy, as we saw above, decreases in wave
length if V decreases and increases in wavelength if V increases. In either
situation some reflection occurs at the boundaries between the regions. What
does "some" reflection mean when we are discussing the motion of a single
particle? Since \p is related to the probability of finding the particle in a particular
place, the partial reflection of \p means that there is a chance that the particle
will be reflected. That is, if we shoot many particles at a box with nonrigid
wails, most will get through but some will be scattered.
What we have been saying, then, is that particles with enough energy to
penetrate a wall nevertheless stand some chance of bouncing off instead. This
prediction complements the "leaking" out of particles trapped in the box despite
the fact that they have insufficient energy to penetrate its walls. Both of these
predictions are unique with quantum mechanics and do not correspond to any
QUANTUM MECHANICS 157
158
l>ehavior expected in classical physics. Their confirmation in numerous atomic
and nuclear experiments supports the validity of the quantummechanical ap
proach.
5.9 THE HARMONIC OSCILLATOR
Harmonic motion occurs when a system of some kind vibrates alxmt an equilib
rium configuration. The system may lie an object supported by a spring or
floating in a liquid, a diatomic molecule, an atom in a crystal lattice — there are
countless examples in both the macroscopic and the microscopic realms. The
condition for harmonic motion to occur is the presence of a restoring force that
acts to return the system to its equilibrium configuration when it is disturbed;
the inertia of the masses involved causes them to overshoot equilibrium, and
the system oscillates indefiiutely if no dissipativc processes are also present.
In the special case of simple harmonic motion, the restoring force F on a
particle of mass m is linear; that is, /•' is proportional to the particle's displacement
x from its equilibrium position, so that
5.35
F = far
This relationship is customarily called Hooke's law. According to the second
law of motion, F = ma, and so here
— kx =
rf»x
dl 2
5.36
# + ** = <>
(It* m
There are various ways to write the solution to Eq. 5.36, a convenient one being
5,37
where
5.38
X — A COS (llivt + <M
2w V m
is the frequency of the oscillations, A is their amplitude, and <#>, the phase constant,
is a constant that depends upon the value of x at the time t = 0.
The importance of the simple harmonic oscillator in both classical and modem
physics lies not in the strict adherence of actual restoring forces to I looke's law,
which is seldom true, but in the fact that these restoring forces reduce to Hooke's
law for small displacements x. To appreciate this point we note that any force
which is a function of X can be expressed in a Maclaurin's series about the
equilibrium position x = as
THE ATOM
Fix) = F^, +
UL/ + 2W), = / + eU'L,, 1
Since x = W is the equilibrium position, F r  Q = 8, and since for small x the values
of x, x 3 , . . , are very small compared with x, the third and higher terms of
the series can be neglected. 'Ihe only term of significance when x is small is
therefore the second one. Hence
*(£L
which is Hooke's law when {dF/dx) r=0 is negative, as of course it is for any
restoring force. The conclusion, then, is that all oscillations are simple harmonic
in character when their amplitudes are sufficiently small.
The potential energy function V(x) that corresponds to a Hooke's law force
may be found by calculating the work needed to bring a particle from x =
to x = x against such a force. The result is
5.39
V(x) =  f F(x) dx = kj xdx= '/aitx 2
and is plotted in Fig. 57. If the energy of the oscillator is E, the particle vibrates
back and forth between x = — A and x == +A, where E and A arc related by
E m V 2 kA.
FIGURE 57 The potential enerpy of a harmonic oscillator is proportional to r, where t Is the displace
ment from the equilibrium position. The amplitude A of the motion Is determined by the total energy f.
of the oscillator, which classically can have any value.
V = % he'
QUANTUM MECHANICS 159
Even before we make a detailed calculation we can anticipate three quan
tummechanical modifications to this classical picture. First, there will not he
a continuous spectrum of allowed energies hut a discrete spectrum consisting
of certain specific values only. Second, the lowest allowed energy will not he
E = hut will [>e some definite minimum £ = E l) . Third, there will be a certain
probability that the particle can "penetrate" the potential well it is in and go
beyond the limits of —A and +A.
The actual results agree with these expectations. The energy levels of a
harmonic oscillator whose classical frequency of oscillation is s> (given by
Eq. 5.38) turn out to be given by the formula
5.40
e »  ( + D»
n = 0, 1, 2,
Energy levels of
harmonic oscillator
The energy of a harmonic oscillator is thus quantized in steps of ftp. The energy
levels here are evenly spaced (Fig. 58), unlike the energy levels of a particle
in a box whose spacing diverges. We note that, when n = 0,
5.41
£o = x fJ»
Zeropoint energy
which is the lowest value the energy of the oscillator can have. Tins value is
called the serojwint energy because a harmonic oscillator in equilibrium with
its surroundings would approach an energy of E — E tt and not E = as the
temperature approaches K.
The wave functions corresponding to the first six energy levels of a harmonic
oscillator are shown in Fig. 59. In each case the range to which a particle
FIGURE 58 Energy levels of a harmonic oscillator
according to quantum mechanics.
160
THE ATOM
*=— A x=+A
x=A x=+A
^
x= — A x=+A
x=A *=+A
i=— A x=+A
x=A s=+A
FIGURE 59 The first sii harmonicoscillator wave functions. The vertical lines show the limits —A and
T.l between which a classical oscillator with the same energy would vibrate.
oscillating classically with the same total energy E„ would be confined is indi
cated; evidently the particle is able to penetrate into classically forbidden
regions — in other words, to exceed the amplitude A determined by the energy —
with an exponentially decreasing probability, just as tn the situation of a particle
in a box with rtonrigid walls.
It is interesting and instructive to compare the probability densities of a
classical harmonic oscillator and a quantummechanical harmonic oscillator of
the same energy. The upper graph of Fig. 510 shows this density for the classical
QUANTUM MECHANICS 161
*=— A *=+A
r
x = A
x = +A
162
FIGURE 5 10 Probability densities lor the n = and n = 10 stales at a quantum mechanical harmonic
oscillator. The probability densities for classical harmonic oscillators with the same energies are shown
In white.
THE ATOM
oscillator: The probability P of finding the particle at a given position is greatest
at the endpoints of its motion, where it moves slowly, and least near the
equilibrium position (x = 0), where it moves rapidly. Exactly the opposite
behavior is manifested by a quantum mechanical oscillator in its lowest energy
state of n = 0. As shown, the probability density i/> 2 has its maximum value
at x = and drops off on either side of this position. However, this disagreement
becomes less and less marked with increasing n: The lower graph of Fig. 5JO
coricsponds to n = 10, and it is clear that <^ ln 2 when averaged over z has
approximately the general character of the classical probability P. This is another
example of the correspondence principle mentioned in Sec. 4.11: In the limit
of large quantum numbers, quantum physics yields the same results as classical
physics.
It might be objected that, although l^jnl 2 does indeed approach P when
smoothed out, nevertheless i^ l0  a fluctuates rapidly with r whereas P does not.
However, this objection has meaning only if the fluctuations are observable, and
the smaller the spacing of the peaks and hollows, the more strongly the un
certainty principle prevents their detection without altering the physical state
of the oscillator. The exponential "tails" of I^kJ 2 heyond * = —A also decrease
in magnitude with increasing n. Thus the classical and quantum pictures begin
to resemble each other more and more the larger the value of n, in agreement
with the correspondence principle, although they are radically different for
small n.
*5.10 THE HARMONIC OSCILLATOR: SOLUTION OF
SCHRODINGER'S EQUATION
In this section we shall see how the preceding conclusions are obtained,
Schrodinger's equation for the harmonic oscillator is, with V = y z fcr 2 .
It is convenient to simplify Eq. 5.42 by introducing the dimensionless quantities
» = (»'
11/2
5.43
I2mnv
QUANTUM MECHANICS 163
164
and
5.44
2£ ,
ft i
M
hv
where v is the classical frequency of the oscillation given by Eq. &38. In making
these substitutions, what we have essentially clone is change the units in which
X and E are expressed from meters and joules, respectively, to appropriate
tlimcnsionless units. Jn terms of y and « Sehrodinger's equation becomes
5.45
dy 2
7J + ("^ = "
We begin the solution of Eq. 5.45 by finding the asymptotic form that £ must
have as y — * ±tx>. If any wave function if is to represent an actual particle
localized in space, its value must approach zero as y approaches infinity in order
that J )^ 2 <ii/ be a finite, no n vanishing quantity. Let us rewrite Eq. 5.45 as
follows:
dy
d^j,
dtj
dH/dtf
3(^«»=fl
i£ = (9 t «K>
= 1
As y — * oo, y~ ^ « and we have
.. d^/dy 2 ,
SM hm — \/ = I
>i wo yty
A I unction t£ v that satisfies Eq. 5.46 is
5.47
since
^ = *****
lii 1 1
rfV.
= lim (if  \)e"" n = f/V :
(/•* dy 2 tjns
Equation 5.47 is the required asymptotic form of if.
THE ATOM
■;>
We are now able to write
5.48
= f(y)e* 1 <*
where /((/) is a function of y that remains to be found. By inserting the ^ of
Eq. 5.48 in Eq. 5.45 we obtain
549 ^ 2 4 + (a  1,/=n
which is the differential equation that / ol>eys.
The standard procedure for solving differential equations like Eq. 5.49 is to
assume that /( y) can t>e expanded in a power series in y, namely
f(y) = A + A t y + A 2 y 2 + A 3 y 3 + ■ ■ ■
5.50
=» S A„y n
and then to determine the values of the coefficients A,. Differentiating /yields
 = A, + 2A. z y + 3A a ,f + ■ . .
SB
= 2 n Ky'" 1
By multiplying this equation by y vve obtain
5.51
= ^ nA a y"
n0
The second derivative of / with respect to y is
x
= 2 "(*>  VAnl}"
QUANTUM MECHANICS 165
166
which is equal to
d 2 f "
J n=H
(That the latter two series are indeed equal can lie verified by working out the
first terms of each.) We now substitute Eqs. 5.50 to 5.52 in Eq. 5.49 to obtain
5.53 2 K n + 2 )(" + l K*S " ( 2 » + 1 " a ) A n]tf" = °
In order for this equation to hold for all values of y, the quantity in brackets
must be zero for all values of n. Hence we have the condition that
and so
5.54
(n + 2}(n + l)A n+2 = (2,i + 1  «)A„
^nt2 —
2n + 1  a
{n + 2)(„ + ])
Tin's recursion formula enables us to find the coefficients A 2 , A.,, A t , ... in terms
of A and A,. (Since Eq. 5,49 is a secondorder differential equation, its solution
has two arbitrary constants, winch are A n and Aj here.) Starting from A„ we
obtain the sequence of coefficients A 2 , A 4 , A e , .... and starting from A y we
obtain the other sequence A 3 , A 5 , A 7
It is necessary for us to inquire into the behavior of
as y — * oo; only if if/ — > as y — * oo can ifbti physically acceptable wave
function. Because f(tj) is multiplied by e~ v/2 , f will meet this requirement
provided that
lim /{y)<e*' /2
(As we shall see, it is unnecessary for us to specify just how much smaller / must
be in the limit than e v ' n .)
A suitable way to compare the asymptotic behaviors of f(u) and e v '~ /2 is to
express the latter in a power series (/ is already in the form of a power series)
and to examine the ratio between successive coefficients of each series as n — * oo.
From the recursion formula of Eq. 5.54 we can tell by inspection that
lim V 2
it— x A n
2
n
THE ATOM
Since
e ' = 1+ ^ + lT + S +
we can express e y ~ /2 in a power series as
e ^n = ] + ML + _JL_ + 1—
= S
2^2! 2 3 3!
I
it=n,2, ■(.... 2 n/2 [ — 1 1
G)'
= X W
n=Q,2,4....
The ratio between successive coefficients of y" here is
2(f)l 2(f),
BL
»■ "^rtwft+£\, 2 2"«(£+l)(£)!
S ( + 1 ) ■+■
In the limit of n > oo this ratio becomes
lim
1+* _ '
Thus successive coefficients in the power series for / decrease tess rapidly than
those in the power series for e*'"' /2 instead of more rapidly, which means that
f{ij)e v ' f  docs not vanish as 1/ * oo.
There is a simple way out of this dilemma. If the series representing / termi
nates at a certain value of n, so that all the coefficients A„ are zero for values
of n higher than this one, $ will go to zero as y — * oo because of the e~ v * n
factor. In other words, if/ is a polynomial with a finite number of terms Instead
of an infinite series, it is acceptable. From the recursion formula
2n + 1  n
A " +2 ~ (n + 2)(n + 1) "
QUANTUM MECHANICS 167
it is clear that if
5.55
«  2n + 1
for any value of n, then A K+S = A n+Jl = A nl(i = ■   =0, which is what we want.
(Equation 5.55 lakes care of only one sequence of coefficients, either the
sequence of even n starting with A„ or the sequence of odd n starting with A v
If n is even, it must be true that A, = and only even powers of ij appear
in the polynomial, while if « is odd, it must be true that A„  and only odd
powers of ;/ apj>ear. We shall see the result later in this section, where the
polynomial is tabulated for various values of n.)
The condition that a = 2ri + 1 is a necessary and sufficient condition for the
wave equation 5.45 lo have solutions that meet the various requirements that
$ must fulfill. From Eq. 5.44, the definition of a, we have
«. = ^ = 2n + I
or
5.56
*(•+!)*
«t = 0, 1. 2,
This is the formula that was given as Eq. 5,40 in the preceding section.
For each choice of the parameter «„ there is a different wave function ^„.
Each function consists of a polynomial ll n (ij) {called a iiennite polynomial) in
either odd or even powers of y, the exponential factor e _,, ' /2 , and a numerical
coefficient which is needed for ^„ to meet the normalization condition
ftyjdy=\ n =0,1,2,...
 »
'Hie general formula for the nth wave function is
5.57 + n = (2f.y* (2"n\)^lUy)e"^
Table 5.1.
SOME HERMITE POLYNOMIALS.
"M
K,
1
1
&>
1
%
3
y»
2
V  2
5
y>
3
Sjc 1  12y
7
w»
4
lBy*  4%> + 12
9
w»
S
32y s  leOy 3 + 120y
a
"&h*
The first six Ilermite polynomials H n (y) are listed in Table 5.1, and the
corresponding wave functions ^„ are those that were plotted in Figs. 59 and
510 of the preceding section.
Problems
1. Verify that all solutions of the wave equation
dx 2 v 2 di?
must be of the form y = F(( ± x/v) as asserted in Sec. 5.2.
2. If +,(x,0 and %{x,t) are lx>th solutions of Sehrodinger's equation for a given
potential V(x), show that the linear combination
En which a, and « a are arbitrary coastants is also a solution. (This result is in
accord with the empirical observation of the interference of dc Broglie waves,
for instance in the DavissonGermer experiment discussed in Chap. 3.)
3. Find the lowest energy of a neutron confined to a Ik>x 10" '"' in across. (The
size of a nucleus is of this order of magnitude.)
4. According to the correspondence principle, quantum theory should give the
same results as classical physics in the limit of large quantum numlsers. Show
that, as n — * oo, the probability of finding a particle trapped in a irox Iwtween
X and x + dx is independent of x, which is the classical expectation.
5. Find the zeropoint in electron volts of a pendulum whose period is I s.
6. An important property of the eigenfunctions of a system is that they are
ttrthogpnal to one another, which means that
f W m dV = Q n*m
Verify this relationship for the eigenfunelions of a particle in a oncdimcnsional
box with the help of the relationship sin = {e iB — e"'*)/2t.
*7. Show that the expectation values <T> and < V) of the kinetic and potential
energies of a harmonic oscillator are given by <T) = (V) = l£ t) /2 when it is
in the n = state. (This is true for all states of a harmonic oscillator, in fact.)
Mow docs this result compare with the classical values of 1* and V?
168
THE ATOM
QUANTUM MECHANICS 169
'8. Use the fact that « > (since E > (!) to show that the coefficients ,\„ of
Ecj. 5.50 arc all zero for negative values of n.
' 9, Show that the first three harmonicoscillator wave functions are normalized
solutions of Sehrodinger's equation.
10. According to elementary classical physics, the total energy of a harmonic
oscillator of mass in, frequency v, and amplitude A is 2^A' J f'm. Use the un
certainty principle to verify that the lowest possible energy of the oscillator is
lw/2 by assuming that Ax = A.
1 1 . Which of the wave functions shown in Fig. 5 1 1 might conceivably have
physical significance?
FIGURE 511 (Continued)
170 THE ATOM
QUANTUM MECHANICS 171
QUANTUM THEORY OF THE HYDROGEN ATOM
6
The quantummechanical theory of the atom, which was developed shortly after
the formulation of quantum mechanics itself, represents an epochal contribution
to our knowledge of the physical universe. Besides revolutionizing our approach
to atomic phenomena, this theory has made it possible for us to understand such
related matters as how atoms interact with one another to form stable molecules,
the origin of the periodic table of the elements, and why solids arc endowed
with their characteristic electrical, magnetic, and mechanical properties, all
topics we shall explore in later chapters. For the moment we shall concentrate
on the quantum theory of the hydrogen atom and how its lormal mathematical
results may be interpreted in terms of familiar concepts.
6.1 SCHRODINGER'S EQUATION FOR THE HYDROGEN ATOM
V hydrogen atom consists of a proton, a particle of electric charge + e. and an
electron, a particle of charge —e which Is 1,836 limes lighter than the proton.
For the sake of convenience vve shall consider the proton to lie stationary, with
the electron moving about in its vicinity hut prevented from escaping by the
proton's electric field. (As in the Bohr theory, the correction for proton motion
is simply a matte* of replacing the electron mass m by the reduced mass in'.}
Schrodinger's equation for the electron in three liii net is ions, which is what we
must use for the hydrogen atom, is
r^ d^d. d% 2»i
ox* r)j/ dz* n l
The potential energy V here is the electrostatic potential energy
6.2
V= 
4*V
of a charge — e when it is the distance r from another charge +e.
173
Since V is a function of r rather than of r, y, z, we cannot substitute Eq. 6,2
directly into Eq. 6.1. There are two alternatives: we can express V in terms
of the cartesian coordinates x, y, z by replacing r by y/x 2 + y 2 + z 2 , or we can
express Schrodinger's equation in terms of the spherical polar coordinates r. S,
<> defined in Fig, 61. As it happens, owing to the symmetry of the physical
situation, doing the latter makes the problem considerably easier to solve.
The spherical polar coordinates r. 0, ^ of the point !' shown in Fig. 61 have
the following interpretation:
r = length of radius vector from origin O to point P
= Vx 2 + if + z 2
(i = angle between radius vector and +z axis
= zenith angle
= cos" l
V^ + y 2 + z 2
c> = angle I>etween the projection of the radius vec
tor in the xy plane and the + x axis, measured
in the direction shown
= azimuth angle
= tan 1 2
Spherical
polar
coordinates
RGUHE 61 Spherical polar coordinate!.
*y
i — r s'm$cos<t>
U — r sin & sin <f>
~ — TCOS$
174
THE ATOM
On the surface of a sphere whose center is at O, lines of constant zenith angle
are like parallels of latitude on a globe (but we note that the value of of
a point is not the same as its latitude; = 90° at the equator, for instance, hut
the latitude of the equator is 6°), and lines of constant azimuth angle <> arc like
meridians of longitude (here the definitions coincide if the axis of the globe is
taken as the +z axis and the +.v axis is at <p = 0°).
In spherical polar coordinates Schrodinger's equation becomes
6.3
r 2 dr\ drf

1
r'sinW
i s,n — r I
I
r 2 sin 2 /y <V ft
dU 2m
+ zr( E  vw = o
6.4
sin
Substituting Eq. 6.2 for the potential energy V and multiplying the entire
equation by r 2 sin 2 tf, we obtain
2mr 2 S m 2 & f_e*_ + A, ^ Q Hydroger> atom
H \4<ne t) r /
Equation 6.4 is the partial differential equation for the wave function tf< of the
electron in a hydrogen atom. TogeUier with the various conditions ^ must obey,
as discussed in Chap. 5 (for instance, that ^ have just one value at each point
r, 0, tj>), this equation completely specifies the behavior of the electron. In order
to see just what this behavior is, we must solve Eq. 6.4 for ^.
When Eq. 6.4 is solved, it I urns out that three quantum numbers are required
to describe the electron in a hydrogen atom, in place of the single quantum
number of the Bohr theory. (In the next chapter we shall find that a fourth
quantum numlwr is needed to describe the spin of the electron.) In the Bohr
model, the electron's motion is basically onedimensional, since the only quantity
that varies as it moves is its position in a definite orbit. One quantum number
is enough to specify the state of such an electron, just as one quantum number
is enough to specify the state of a particle in a onedimensional l>ox.
A particle in a threedimensional box needs three quantum numbers for its
description, since now there are three sets of boundary conditions that the
particle's wave function i£ must obey: ^ mast to at the walls of the box in
the x, y, and z directions independently. In a hydrogen atom the electron's
motion is restricted by the inversesquare electric field of the nucleus instead
of by the walls of a box, but nevertheless the electron is free to move in three
dimensions, and it is accordingly not surprising that three quantum numbers
govern its wave function also.
QUANTUM THEORY OF THE HYDROGEN ATOM 175
176
The three quantum numbers revealed by the solution of Eq. 6.4, together with
their possible values, are as follows:
Principal quantum number — r» = 1, 2, 3, . . .
Orbital quantum number =s f = 0, 1, 2 n — 1
Magnetic quantum niimlier = m, = 0, ±1, ±2, . . . , ±(
The principal quantum number n governs the total energy of the electron, and
corresponds to the quantum number n of the Bohr theory. The orbital quantum
number J governs the magnitude of the electron's angular momentum about the
nucleus, and the magnetic quantum number m, governs die direction of the
angular momentum.
*6.2 SEPARATION OF VARIABLES
The virtue of writing Schrodinger's equation in spherical polar coordinates for
the problem of the hydrogen atom is thai in this form it may be readily separated
into three independent equations, each involving only a single coordinate. The
procedure is to look for solutions in which the wave function ^(r, 6, £) has the
form of a product of three different functions: R(r), which depends upon r alone;
8(0), which depends upon alone; and <1>(0), which depends upon $ alone. That
is, we assume that
6.5
flr, 8, +) = /l(r)8(tf )*(<*.)
Hydrogen atom wave function
The function R(r) describes how the wave function $ of the electron varies along
a radius vector from the nucleus, with and constant. The function H(fl)
dcscriljes how i£ varies with zenith angle along a meridian on a sphere centered
at the nucleus, with r and tj> constant. The function 4>(<£} describes how i/> varies
with azimudi angle <J> along a parallel on a sphere centered at the nucleus, with
r and constant.
From Eq. 6.5, which we may write more simply as
$ = R8#
we see that
3r dr
7)0 80
a* 2 dt*
THE ATOM
ilence, when we substitute flQG for ^ in Schrodinger's equation for the hydrogen
atom and divide the entire equation by H8<!>, we find thai
R 3rV dr/ + 8 W\ BB }
J_ a 2 * 2mr z sin z g / e 2 , ,,\ _ n
+ * 3^ + ft 2 We t f
The third term of Eq. 6.6 is a function of azimuth angle <£ only, while the other
terms are functions of r and 6 only. l>cl us rearrange Eq. 6.6 to read
R dr\ dr) + 8 30 l 30 }
2mr 2 sin* / e 2 V  9^
+ W 2 \4frf. r + h ) ~ ~*"3^
Thix equation can Ik correct only if both sides of it am equal to the same constant,
since they are functions of different variables. As we shall see, it is convenient
to call this constant m, 2 . The differential equation for the function 4> is therefore
6.8
1 <***  m 2
* * 2 " m '
When we sulwtitute m, 2 for the right hand side of Eq. 6.7, divide die entire
equation by sin 2 0, and rearrange the various terms, wc find dmt
6.9
±±( r2 dR\, W /_e 2 _ A
R dr \ dr / ft 2 Um n r 1
1
sin a ff 8sin0 90
H£)
Again we have an equation in which different variables appear on each side,
reqiu'ring that Ixith sides Ix: equal to the same constant. This constant we shall
call 1(1 + 1), once more for reasoas that will Ix: apparent later. The equal ions
for the functions 8 and R are therefore
6.10
6.11
sin 2 8siuffd»
6,1?
Filiations 6.8, 6.10, and 6.11 arc usually wrillen
d 2 *
d4> 2
2 + m, 2 ^ a
QUANTUM THEORY OF THE HYDROGEN ATOM 177
6.13
6.14
Mil'/ ,lil
Each of these is an ordinary differential equation for a single function of a single
variable. We have therefore accomplished our task of simplifying Schrodinger's
equation for the hydrogen atom, which began as a partial differential equation
for a function i£ of three variables.
'6.3 QUANTUM NUMBERS
The first of the alwve equations, Eq. ft 12. is readily solved, with the result
6.15 <t>($) = Ae*'"'" 1
where A is the constant of integration. We have already stated that one of the
conditions a wave function— and hence *, which is a component of the complete
wave function iffmusl olie\ is that it have a single value at a given point in
space. From Fig. 62 it is evident that $ and tj> + 2tt lx>th identify the same
FIGURE 52 The angles ?. and $. + 2v. both identify the
same meridian plane.
*y
178 THE ATOM
meridian plane. Hence it must l>e true that *(£) = <l>(£ + 2a), or
Ae im t* = Ae ,m ' ( ' lW " aT,
which can only happen when wi ( is or a positive or negative integer (±1, ±2,
±3, . . .). The constant in, is known as the magnetic quantum number of the
hydrogen atom.
The differential equation 6.13 for 0(ff) has a rather complicated solution in
terms of polynomials called the associated Ixgmdre functions. For our present
purpose, the important thing about these functions is that they exist only when
the constant / is an integer equal to or greater than m,\, the absolute value of
m,. This requirement can be expressed as a condition on m ; in the form
Nil = 0, ±1, ±2, .... ±t
The constant I is known as the orbital quantum number.
The solution of the final equation, Eq. 6.14, for the radial part fi(r) of the
hydrogenatom wave function ^ is also complicated, l>eing in terms of poly
nomials called the associated Laguerre functioiu. Equation 6.14 can be solved
only when £ is positive or has one of the negative values E n (signifying that
the electron is bound to the atom) specified by
6.1S
,__ me* (±\
where n is an integer. We recognize that this is precisely the same formula for
the energy levels of the hydrogen atom that Bohr obtained
Another condition that must be obeyed in order to solve Eq, 6,14 is that n,
known as the principal quantum number, must l>e equal to or greater than I + 1 .
This requirement may be expressed as a condition on / in the form
i=0, 1.2 ,(n 1}
Hence we may tabulate the three quantum numbers n, /, and m together with
their permissible values as follows:
n = 1, 2, 3, . . .
6.17 / = 0, 1, 2 (n  1}
m. =0, ±1, ±2,
:J
Principal quantum number
Orbital quantum number
Magnetic quantum number
It is worth noting again how inevitably quantum numbers appear in quantum
mechanical theories of particles trapped in a particular region of space.
QUANTUM THEORY OF THE HYDROGEN ATOM 179
To exhibit ihc dependence of H, B, and <l> upon the quantum numbers >i, I,
m, we may write for the electron wave Function
6.18
* = WlmPm,
The wave functions H, W, and <J> together with \l are given in Table 6. 1 for n = 1 ,
2. and 3.
6.4 PRINCIPAL QUANTUM NUMBER
It is interesting to consider the interpretation of the hydrogenatom quantum
numbers in terms of the classical model of the atom. This model, as we saw
in Chap. 4, corresponds exactly to planetary motion in the solar svstetn except
that the inversesquare force holding the electron to the nucleus is electrical
rather than gravitational. Two quantities :ire conserved— \h&t is, maintain a
constant value at all rimes— in planetary motion, as Newton was able to slum
Iron i Kepler's three empirical laws. These are the scalar total energy and the
vector angular momentum of each planet.
< SasstcaBy the total energy can have any value whatever, but it must, of course,
be negative if the planet is to be trapped permanently in the solar system. In
the quantummechanical theory of the hydrogen atom the electron ettetg) is
also a constant, but while it may have any positive value whatever, the only
negative vaJues it can have are .specified by the formula
(6.16)
*■„ =
 »»<■•' (J_\
The theory of planetary motion can also be worked out from Schrocliu^i's
equation, and it yields an energy restriction identical in form. However, the
total quantum number » fur any of the planets turns out to be so immense that
the separation of permitted energy levels is far too small to be observable. For
this reason classical physics provides an adequate description of planetary motion
hut fails within the atom.
The quantization of electron energy in the hydrogen atom is therefore de
scribed by the principal quantum number n,
6.5 ORBITAL QUANTUM NUMBER
The Interpretation of the orbital qua n tum nmnber / is a bit less obvious. Let
\is examine the differential equation for the radial part R{r) of the wave function
(6.14)
H(*%hm&+*hm*<
180 THE ATOM
I
I
S
3
f
II
i
a
mi
o
o
i
UJ
B
I
a
2
3
p
o
z
2
to
+ L
I*
I
■
S
4
n
8
<*>
■I
a
i T
s
M
<
1
s *^^* 
° k
==
i
M
5.
M
"c
.'
M
n
a
=
ft
a" —
A
i
™0
1$ •$
oe
L* 
5
CO
4
j
t
I*
c
§ &
w
# :
;
^
i
> 1
a ^
1
? „  _ a * * =i 
M
VI
H
*
H
■a
j H
k
l»
a
s
IS *
1 3
o*
i«
M
«
ee
3
BS
ce

^
a
•I
I 1 ? i^ ? ^I« \% "?i« ? ^h ?r ^1'
I
i*„L«i«i i _b _i^ ^ii _i„
i t
'i
o o
MoJiMetnpjKesrt
This equation is solely concerned with the radial aspect of the electron's motion,
(hat is, its motion toward or away from the nucleus; yet we notice the presence
of E, the total electron energy, in it. The total energy /; includes the electron's
kinetic energy of orbital motion, which should have nothing to do with its radial
motion.
This contradiction may he removed by the following argument. The kinetic
energy '/' of the electron has two parts, T raliiHi due to its motion toward or away
from the nucleus, and T nrMM due to its motion around the nucleus. The potential
energy V of the electron is the electrostatic energy
V= —
**W
Hence the total energy of the electron is
B» T
radii nl
+ T
titiilial
+ v
= Z
radial
+ T„
nrliifjil
Inserting this expression for E in Eq. 6,14 we obtain, after a slight rearrangement.
1 d / ,,dfi\ 2m r H 2 lll + 1)1
If the last two terms in the square brackets of this equation cancel each other
ont, we shall have what we want: a differential equation for H(r) that involves
functions of the radius vector r exclusively. We therefore require that
6.20
H*K1 + I)
2mr
The orbital kinetic energy of the electron is
'orbital — /a wll; "orbllal
Since the angular momentum /, of the electron is
L = « B »art*w'
we may write for the orbital kinetic energy
L
t?
2mn
Hence, from Eq. 6.20,
l 2 m(i + i)
2nir
or
6.21
/. = V/(/ + 1) ft
Electron angular momentum
Our interpretation of this result is that, since the orbital quantum number /
is restricted to the values
I = 0. 1,2 (n  1)
the electron can have only those particular angular momenta / specified by
Kq. 6.21. Like total energy K, angular momentum is hoth conserved and quoit
tizetl. The quantity
ft = /i/2w = 1.054 X 10 ;i 'Js
is thus the natural unit of angular momentum.
In macroscopic planetary motion, once again, the quantum number describing
angular momentum is so large that the separation into discrete angular
tiiouieiiliiiu slates cannot lie experimentally observed. For example, an electron
or, lot that matter, any other body) whose orbital quantum number is 2 has
(he angular momentum
L = \/2(2+ l)ft"
s \/6ft
= 2,6 x 10 M Js
B) contrast the orbital angular momentum of the earth is 2.7 x l() ,0 Jsl
ll is customary to specify aiigularinomeutum states by a letter, with s corre
sponding to I — 0, p to / = 1, and so on according to the following scheme:
I = (I 1 2 3 I 5 6,
s p d f g h i.
Angularmomentum states
I his peculiar code originated in the empirical classification of spectra into series
called sharp, principal, diffuse, and fundamental which occurred before the
theory of the atom was developed. Thus an n state is one with no angular
momentum, a p slate has the angular momentum \/2«~, etc.
The combination of the total quantum number with the letter that represents
orbital angular momentum provides a convenient and widely used notation for
atomic slates. In this notation a state in winch n = 2, I = is a 2s slate, for
example, and one in which n = 4, / = 2 is a 4d state. Table b'.2 gives the
designations of atomic states in hydrogen through n = 6, / = 5.
182
THE ATOM
QUANTUM THEORY OF THE HYDROGEN ATOM 183
184
Table 6.2.
THE SYMBOLIC DESIGNATION OF ATOMIC STATES IN HYDROGEN.
i
p
ii
/
I
/i
( =
1 = 1
J = 2
U3
; 4
(" = 9
n = 1
I*
n = 2
2j
%,
Fl = 3
3*
*
.V
ii = 4
U
*
«d
■•/
n = S
Si
3p
Hd
5/
%
n = 8
Si
6p
ad
6/
(i
eh
6.6 MAGNETIC QUANTUM NUMBER
The orbital qu a ntum number / determines the magnitude of the electron's angular
momentum. Angular momentum, however, like linear momentum, is a vector
quantity, and so to describe it completely requires that ils direction be specified
as well as its magnitude. (The vector L, we recall, is perpendicular to the plane
in which the rotational motion lakes place, and its sense is given bv the righthand
rule: when the fingers of the right hand point in the direction or the motion,
die thumb is in the direction of L. This rule is illustrated in Fig. 63.)
What possible significance can a direction in space have for a hydrogen atom?
The answer becomes clear when we reflect that an electron revolving about a
nucleus is a minute current loop and lias a magnetic field like that of a magnetic
dipole. Hence an atomic electron that possesses angular momentum interacts
with an external magnetic field B, The magnetic quantum number m t specifies
fl.
FIGURE 63 The righthand lule
lor angular momentum.
thumb in direction of angular*
momentum vector
fingers of right hand in direction
of rotational motion
THE ATOM
the direction of L by determining the component of L in the field direction.
This phenomenon is often referred to as space quantization.
If we let the magneticfield direction be parallel to the ; axis, the component
of L in this direction is
6,22
L, = m,fi
Space quantization
The possible values of m, for a given value of / range from +/ through to
— /, so that the number of possible orientations of the angularmomentum vector
L in a magnetic field is 2/ + I. When / = 0, L. can have only the single value
of (1; when / = I, /„. may be ii, 0, or — ft; when / = 2, L t may lie 2ft, ft, 0, —ft,
or — 2ft; and so on. We note that L can never be aligned exactly parallel or
autiparallel to B, since /. ; is always smaller than the magnitude \/i{l + 1 ) ft of
the total angular momentum.
The space quantization of the orbital angular momentum of the hydrogen atom
is shown in Fig. 64. We must regard an atom characterized by a certain value
FIGURE 64 Space quantization of orbital angular momentum.
f =2
= Veft
QUANTUM THEORY OF THE HYDROGEN ATOM 185
of jii, as standing ready to assume a certain orientation of its angular momentum
L relative to an external magnetic field in the event it finds itself in such a field.
In the absence of an external magnetic field, the direction of the t axis is
entirely arbitrary. Hence it must be true that the component of L in any
direction we choose is m,ti; the significance of an external magnetic field is that
it provides an experimentally meaningful reference direction. A magnetic field
is not the only such reference direction possible. For example, the line between
the two IT atoms in the hydrogen molecule H, is just as experimentally mean
ingful as the direction of a magnetic field, and along this line the components
of the angular momenta of the IT atoms are determined by their m, values.
U by is only one component of L quantized? The answer is closely related
to the fact that L can never point in any specific direction z but instead traces
out a cone in space such that its projection /. ; is m,H. The reason for the latter
phenomenon is the uncertainty principle: if L were fixed in space, so that L t
and L„ j»s well as L t had definite values, (he electron would lie confined to a
definite plane. For instance, if L were in the z direction, the electron would
have to t>e in the xy plane at all times (Fig, 65«). This can occur only if (he
electron \s momentum component p. in the r direction is infinitely uncertain,
which of coarse is impossible if it is to l>e part of a hydrogen atom. However,
since in reality only one component L t of L together with its magnitude L have
definite values and £ > L_, the electron is not limited to a single plane Fig.
FIGURE 65 The uncertainty principal prohibit* the angularmomentum vector I. from having a definite
direction in space.
L = VW + l)fi
FIGURE 6 6 The angularmomentum vector L
processes constantly about the a axis.
r^y
1=2
65/)), and there is a builtin uncertainty, as it were, in the electron's z coordinate.
The direction of L is constantly changing, as in Fig. 6fl, and so the average
values of L t and L y are 0, although L s always has the specific value mfl.
6.7 THE NORMAL ZEEMAN EFFECT
In an external magnetic field B, a magnetic dipole has an amount of potential
energy V m that depends upon lioth the magnitude /i of its magnetic moment
and the orientation of this moment with respect to the field (Fig. 67).
The torque t on a magnetic dipole in a magnetic field of flux density B is
t = fiB sin
FIGURE 6 7 A magnetic dipole of
moment r. at the angle relative to
a magnetic held li
QUANTUM THEORY OF THE HYDROGEN ATOM 187
188
where is the angle between p and B. The torque is a maximum when the
dipole is perpendicular to the field, and zero when it is parallel or antiparallcl
to it. To calculate the potential energy V m , we must first establish a reference
configuration at which V m is zero by definition. (Since only clianges in potential
energy are ever experimentally observed, the choice of a reference configuration
is arbitrary.) It is convenient to set V m = when — 90°, that is, when u, is
perpendicular to B. The potential energy at any other orientation of ji is equal
to lire externa! work that must be done to rotate the dipole from ll, y  90° to
ihe angle 6 characterizing that orientation. Hence
sin 8 dll
6.23
= — //Bcosff
When fi points in the same direction as B, then, V,,, = fiB, its minimum value.
This is a natural consequence of the fact that a magnetic dipole tends to align
itself with an external magnetic field.
Since the magnetic moment of the orbital electron in a hydrogen atom depends
upon its angular momentum L, both the magnitude of I and its orientation with
respect to the field determine the extent of the magnetic contribution to the
total energy of the atom when it is in a magnetic field. The magnetic moment
of a current loop is
/t = iA
where i is the current and A the area it encloses. An electron which makes )■
icv ■, in .1 circular nihil of radius r Is equivalent to a correal of m since me
electronic charge is ~e), and its magnetic moment is therefore
fi = — evzr*
The linear speed v of the electron is 2WIT, and so its angular momentum is
L = mcr
= 2w in t'f 2
Comparing the formulas for magnetic moment (i and angular momentum L
shows that
6.24
— te>
Electron magnetic moment
for an orbital electron. The quantity {— e/2m), which involves the charge and
mass of ihe electron only, is called its gymmagnetiv ratio. The minus sign means
THE ATOM
that ,i is in the opposite direction to L While the alwve expression for the
magnetic moment of an orbital electron has been obtained by a classical calcula
tion, quantum mechanics yields the same result. The magnetic potential energy
of an atom in a magnetic field is therefore
6 25
■fe)'
B cos
which is a function of l>oth B and 9.
From Fig. 65 we see that the angle 9 Iietwcen L and the z direction can
have only the values specified by the relation
cosfl =
while the permitted values of £i are specified by
To find the magnetic energy that an atom of magnetic quantum number m, has
when it is in a magnetic field B, we insert the above expressions for cos0 and
L in Eq. 6.25, which yields
6.26
v  —<£)>
Magnetic energy
The quantity cti/2m is called the Bohr magneton; its value is 9 27 x 10 M I/tesla
(T).
fa a Magnetic field, then, the energy of a particular atomic state depends upon
the value of m, as well as upon that of 0. A state of total quantum number
u breaks up into several subslates when the atom is in a magnetic field, and
their energies are slightly more or slightly less than the energy of the state in
tbe absence of the field. This phenomenon leads to a "splitting" of individual
J**ral Hw» ''"to separate lines when atoms radiate in a magnetic field, with
I he spacing of the lines dependent upon the magnitude of the field. The splitting
of spectral lines by a magnetic field is called the Zm««„ rffhi alter the Dutch
physicist Zeeman, who first observed it in 1896. The Zeeman effect is a vivid
confirmation of space quantization; it is further discussed in Chap. 7.
68 ELECTRON PROBABILITY DENSITY
In Bohr's model of the hydrogen atom the electron is visualized as revolving
around the nucleus in a circular path. This model is pictured in a spherical polar
"■ordinate system in Fig, 68. We see it implies that, if a suitable experiment
QUANTUM THEORY OF THE HYDROGEN ATOM 189
190
Bohr electron orbit
FIGURE 58 The Bohr model of the hydrogen
atom in a spherical polar coordinate system.
were performed, the electron would always be found a distance of r = n'a n
(where n is the quantum number of the orbit and a,, = 0.53 A is the radius of
the innermost orbit) from the nucleus and in the equatorial plane = 90°, while
its azimuth angle £ changes with time.
The quantum toeory of the hydrogen atom modifies the straightforward
prediction of the Bohr model in two ways. First, no definite values for r, 0, or
* °"> ,le S'ven, but only the relative probabilities for finding the electron at
various locations. This imprecision is, of course, a consequence of the wave
nature of the electron. Second, we cannot even think of the electron as moving
around die nucleus in any conventional sense since the probability density W?
is independent of lime and may vary considerably from place to place.
The electron wave function \p in a hydrogen atom is given by
where
R = R ol (r)
describes how $ varies with r when the orbital and total quantum numbers have
the values n and I;
describes how tf. varies with $ when the magnetic and orbital quantum numbers
have the values / and m,; and
* = 'KW
THE ATOM
describes how ^ varies with <J> when the magnetic quantum number is »»,. The
probability density i£> 2 may therefore lie written
6.27
l+l a = \Rf\&f\<
where it is understood that the square of any function is to l>e replaced by the
product of it and its complex conjugate if the function is a complex quantity.
The a/imulhal probability density 'l', which is a measure of the likelihood
i)l lading lln' electron at a particular azimuth angle <£, is a constant that does
not depend upon (J> at all. Hence the electron's probability density is symmetrical
about the z axis regardless of the quantum state it is in, so that the electron has
the same chance of being found at one angle ci as at another.
The radial part R of the wave function, in contrast to '!>, not only varies with
r but does so in a different way for each combination of quantum numliers n
and /. Figure 69 contains graphs of R versus r for Is, 2,s\ 2w, 3«, 3p, and 3d
slaues of the hydrogen atom. Evidently R is a maximum at r = 0— that is, at
the nucleus itself — for all * states, while it is zero at r = for states that possess
angular momentum.
The pwlmhiiity density of the electron at the point r, 0, £ is proportional to
l^l 2 , but the actual probability of finding it in the infinitesimal volume element
dV there is Ivfj^jV. Now in spherical polar coordinates
</V = r'^mOdrdOd^
so that, since and <l» are normalized functions, the actual numerical probability
F[r)dr of finding the electron in a hydrogen atom somewhere between r and r + dr
from the nucleus is
2:7
IWdr = r 2  R \*dr f 0 2 sin d0 f <l> »d$
6.28
= r 2 R\dr
Equation 6.28 is plotted in Fig. 610 for the same states whose radial functions
R were shown in Fig. 69; the curves are quite different as a rule. We note
immediately that P is not a maximum at the nucleus for s states, as R itself is,
but has its maximum at a finite distance from it. Interestingly enough, the most
probable value of r for a la electron is exactly «„, the orbital radios of a ground
state electron in the Bohr model. However, die average value of r for a Is 
electron is I.5« , which seems puzzling at first sight because the energy levels
are the same in both the quanlummechanical and Bohr atomic models. This
apparent discrepancy is removed when we recall that the electron energy
depends upon 1/r rather than upon r directly, and, as it happens, the average
value of 1/r for a l.s electron is exactly l/a^.
QUANTUM THEORY OF THE HYDROGEN ATOM 191
5"n
10«o
15h„
192
FIGURE 69 The variation with distance from trie nucleus or ihe radial part of the
electron wave function In hydrogen tor various quantum states. The quantity
« r . = IP fme' = 0.S3 A is the radius of the first Bohr orbit.
The function varies with zenith angle for all quantum numbers . and m,
except I = m, = (), which are a stales. The probability density 9j* for an s state
is a constant <% in fact), which means that, since <t> 2 is also a constant, the
electron probability density \ip\ 2 has the same value at a given r in all directions.
Electrons in other states, however, do have angular preferences, sometimes quite
complicated ones. This can be seen in Fig. 611, in which electron probability
THE ATOM
densities as functions of r and B are shown for several atomic states. (The quantity
plotted is lufrj*, not l^f 2 dV.) Since \$>, 2 is independent of <;>, a threedimensional
picture of l^l 2 is obtained by rotating a particular representation about a vertical
axis. When this is done, the probability densities for s states are evidently
spherically symmetric, while others are not. The pronounced lobe patterns
characteristic of many of the stales turn out to be significant in chemistry since
these patterns help determine the manner in which adjacent atoms in a molecule
interact; we shall refer to this notion once more in Chap. H.
\ study of Fig, 6 1 1 also reveals qaairttanmecliardcai stales wftt a remaikable
resemblance to those of the Bohr model. The electron probabilitydensity dis
tribution for a 2p state with in, = ±1, for instance, is like a doughnut in the
equatorial plane centered at the nucleus, and calculation shows the most probable
distance of the electron from the nucleus to lie 4%— precisely the radius of the
Bohr orbit for the same total quantum number. Similar correspondences exist
for 3d states with m, a ±2, if states with m, = ±3, and so on: in every case
the highest angular momentum possible for that energy level, and in every case
the angular momentum vector as near the z axis as possible so that the probability
density lx: as close as possible lo the equatorial plain 1'hus [he Hohi model
predicts the most probable location of the electron in one of the several possible
states in each energy level.
FIGURE 610 The probability of finding the electron in a hydrogen atom at a distance be
tween r and r ■ •>• from the nucleus for the quantum states of Fig. 69.
5u n
10u n
15d
20.i ( , 25,.,,
QUANTUM THEORY OF THE HYDROGEN ATOM 193
FIGURE 611 Photographic represent a lion of the electron probability density distribution \i  for several
energy stares These man be regarded as sectional views of the distributions in a plane containing the
polar aiis. which is vertical and in the plane of the paper. The scale varies from figure to figure.
M t » .. ♦
• t
m = ±3
6.9 RADIATIVE TRANSITIONS
bfcnpahtfag his theory of the hydrogen atom, Bohr was obliged to postulate
 n^a, b constant in time doe, not radiate, whil, if it ve " "
relative to the nucleus oscillates, electromagnetic wave* „, cmiL w
recjuency is the j. a, that of the oscillation" For ^^StjX
the component of electron motion i„ the , direction only
The tunedependent wave fiction ^ ()f an e]eclr0]1 , n
U) „b er n and energy ^ is the product of a timeindependent wav Z
*„ and a hmevarying function whose frequency is
/:.
Hence
6.29
% = l^C^.'*"
and
6.30
+; = ,£«.+(«,,/»»
The average position of such an electron is the
v, namely.
expectation value (Sec. 5.4) of
Inserting the wave functions of Eqs. 6.29 and 6.30,
<*> = J x^^eiHVitlHE./AHi dx
winch * comta* in time since *„ and « are, by definition, functions of position
only. The dec ron does not oscillate, and no radiation occur, This q!,,,,,,,
mechanics predicts that „ atom in a specific quantum state docs „ t "2
tins agrees wdh observation, though not with classical physics
196 THE ATOM
We art' now n. a position to coruddM an rlcrlnin nhflflghtg Inuii dih enefgy
state to another. Let ns formulate a definite problem: an atom is in its ground
state when, at ( — 0, an excitation process of some kind (a beam of radiation,
say, or collisions with other particles) begins to act upon it. Subsequently we
find that the atom emits radiation corresponding to a transition from an excited
state of energy li. m to the ground state, and we conclude that at some time during
the intervening period the atom existed in the state hi. What is the frequency
of the radiation?
The wave function + of an electron capable of existing in states n or m may be
written
633
+ = «+„ + ft+ m
wham a*a is the probability that the electron is in state n and b*h the
probability that it is in state m. Of course, it must always be true that a*a +
b*b = 1. At I = 0, o — 1 and b — by hypothesis; when the electron is in the
excited state, a = and b = 1; and ultimately a = 1 and // = it once more.
While the electron is in eitiier state, there is no radiation, but when it is in the
midst of the tratisiliun from rn to n (that is, when both a and b have uonvanishinu
values), electromagnetic waves are produced. Substituting the composite wave
function of Eq. 6.33 into Eq. 6.31, we obtain for the average electron position
6.34
i Here, as before, we let a* a = a' 1 and b*b = b 2 ,) The first and last integrals are
constants, according to Eq. 6.32, and so the second and third integrals are the only
ones capable of contributing to a time variation in (x).
Willi the help of Eqs. 6.29 to 6.31 we may expand Eq. 6.34 to obtain
6.35 <*> = « a j * x^„ dx + b*af" x^;e +t »V' ,1 V J1 <r (,fi " / ' 11 ' dx
+ a'bf x4%****</ mi 4f i jr**J**tk + b z f* x^ m dx
In ibe case of a finite bound system of two states, which is what we have here,
and so we can combine the timevarying terms of Eq. 6.35 into the single term
— »
QUANTUM THEORY OF THE HYDROGEN ATOM
197
Now
*"+£'*= 2 OOSl9
Hence Eq. 6.36 simplifies to
which contains the time varying factor
= cos 2irvt
The electron's position therefore oscillates sinu.snidally at the frequency
E — F
6.37 V a iS ±»
h
and the hill expression for <x), the average position of the electron, is
6.38 <*>=^J xW»dx+b*fxr m t m dx
+ 2«*fc cos %m [ x^„, dx
When die electron is in state n or state m. the probabilities ¥ or o» respec
tively are zero, and the expectation value of the electron's position is constant
When the electron is undergoing a transition between these states, its position
oscillates with the frequency r. This frequency is identical with that postulated
by Bohr and verified by experiment, and. as we have seen. Eq H.37 car, be derived
using quantum mechanics without making any special assumptions.
II is interesting to note that the frequency of the radiation is the same frecmency
as the beats we might imagine to l>e produced if the electron simultaneously
existed in both the n and m slates, whose characteristic frequencies are respec
tively EJh and EJh.
6.10 SELECTION RULES
It was not necessary for us to know the values of the probabilities a and b as
factions of time, nor the electron wave functions fc, and * in order to deter
imne v. We must know these quantities, however, if we wish to compute the
likelihood for a given transition to occur. The general condition necessary for
an atom in an excited state to radiate is that the integral
' —x
not be zero, since the intensity of the radiation is proportional to it. Transitions
for which this integral is finite are called allowed Iramilions, while those for
which it is zero are called forfridden &mwi*fon#,
In the case of the hydrogen atom, three quantum numbers are needed to specify
the initial and final stales involved in a radiative transition. If the total, orbital,
and magnetic quantum numbers of die initial state are n', V, m' t respectively
and those of the final slate are n, /, m„ and the coordinate u represents either
the x, y, and z coordinate, the condition for an allowed transition is
6.39
J* «sS,,,.„, ( ^V, m > * °
When u is taken as .t, for example, the radiation referred to is that which would
he produced by an ordinary dipole antenna lying along the x axis. Since the
wave functions'^ i mi for the hydrogen atom are known, Eq. 6.39 can be evalu
ated for ii = x, a = tj, and u = z for all pairs of states differing in one or more
quantum numbers. When this is done, it is found that the only transitions that
can occur are those in which the orbital quantum number i changes by + I or
 1 and the magnetic quantum number m, does not change or changes by + 1
or 1; in other words, the condition for an allowed transition is that
Selection rules
198 THE ATOM
6.40 A/ = ±1
6.4i Am, — (1, ±1
The change in total quantiun number n is not restricted. Equations 6.40 and
6.4 1 are known as the selection rute.s for allowed transitions.
In order to get an intuitive idea of the physical basis for these selection rules,
let us refer again to Fig. 611. There we sec thai, for instance, a transition from
a 2p state to a U state involves a change from one probabilitydensity distribution
to another such that the oscillating charge during the transitions behaves like
an electric dipole antenna. On the other hand, a transition from a 2s state to
a l.v state involves a change from a spherically symmetric probabilitydensity
distribution to another spherically symmetric distribution, which means thai the
oscillations that take place are like those of a charged sphere thai alternately
expands and contracts. Oscillations of this kind do not lead to the radiation of
rk'ctroniagnetic waves.
The selection rule requiring that / change by ±1 if an atom is to radiate means
lhat an emitted photon carries off angular momentum equal to the difference
between the angular momenta of the atom's initial and final states. The classical
QUANTUM THEORY OF THE HYDROGEN ATOM 199
analog of a photon with angular momentum is a circularly polarized elect ro
inagneric wave, so that this notion is not unique with <iianlum theory.
The preceding analysis of radiative transitions in an atom is based on a mixture
of classical and quantum concepts. As we have seen, the expectation value of
the position of an atomic electron oscillates at the frequency << of Eq. 6.37 while
passing from an initial eigenstate to another one of lower energy. Classically
such an oscillating charge gives rise to electromagnetic waves of the same
frequency v, and indeed the observed radiation has this frequency. However,
classical concepts are not always reliable guides to atomic processes, and a deeper
treatment is required. Such a treatment, called quantum rkctrodtjnamics. modi
fies the preceding picture by showing that a single photon of energy ht> is emitted
during the transition from state m to state n instead of eleetricdipole radiation
in all directions except that of the electron's motion, which would be the classical
prediction.
In addition, quantum electrodynamics provides an explanation for the mecha
nism that causes the "'spontaneous" transition of an atom from one energy state
to a lower one. All electric and magnetic fields turn out to fluctuate constantly
about the E and B that would be expected on purely classical grounds. Such
fluctuations occur even when electromagnetic waves are absent and when,
classically, E = B = 0. It is diese fluctuations (often called "vacuum fluctua
tions" and analogous in a sense to the zeropoint vibrations of a harmonic
oscillator) that induce the "spontaneous" emission of photons by atoms in exeited
states.
Problems
1. Verify that Eq.t. 6.1 and 6.3 are equivalent
"2. Show that
\/io
&sd0}^~r€iaj^9  I)
4
is a solution of Eq. 6.13 and that it is normalized.
°3. Show that
«iofo =
3/2'
r/a„
is a solution of Im. 6.14 and that it is normalized.
4, In Sec. 6.8 it is stated that the probability P of finding the electron
200 THE ATOM
!>) .1
hydrogen atom between rand r + drhom the nucleus is P dr s r'^RJ" dr. Verify
that P dr has its maximum value for a Is electron at r = a a , the Bohr radius.
5. According to Fig. 610, Pdr has two maxima for a 2s electron. Find the
values of r at which these maxima occur.
6. The wave function for a hydrogen atom in a 2p state varies with direction
as well as with distance from the nucleus. In the case of a 2u electron for which
m , = 0, where does P have its maximum in the % direction? In the xy plane?
7. The probability of finding an atomic electron whose radial wave function
is Il(r) outside a sphere of radius r„ centered on the nucleus is
f fl(r)V(/r
The wave function R 1(J (r) of Prob, 3 corresponds to the groiuid state of a hydrogen
atom, and «„ there is the radius of the Bohr orbit corresponding to that state,
(a) Calculate the probability of finding a groundstate electron in a hydrogen
atom at a distance greater than «„ from the nucleus, (b) When the electron in
a groundstate hydrogen atom is 2«„ from the nucleus, all its energy is potential
niergy. According to classical physics, the electron therefore cannot ever exceed
the distance 2a from the nucleus. Find the probability dial r > 2a for the
electron in a groundstate hydrogen atom.
8. Unsold's theorem stales that, for any value of the orbital quantum number
L the probability densities summed over all possible states from m, — — / to
m, = +/ yield a constant independent of angles or tf>; that is,
V ]8p«> 2 = constant
Ibis theorem means that every closed subshell atom or ion (Sec. 7.5) has a
spherically symmetric distribution of electric charge. Verify Unsold's theorem
for / = (),/=], and I = 2 with the help of Table 6.1.
9. Find the percentage difference between L and the maximum value of L t
fur an atomic electron in p, d, and / states.
° 10. The selection rule for transitions l>elween states in a harmonic oscillator
is in =s ± 1 . (a) Justify this rule on classical grounds, (b) Verify from the relevant
wave functions that the transitions n = 1 * n = and n = 1 » n = 2 are
possible for a harmonic oscillator while n = 1 » n = 3 is prohibited.
1 1. With the help of the wave functions listed in Table 6.1 verify that 11 — ±1
for n = 2— » n = 1 transitions in the hydrogen atom.
QUANTUM THEORY OF THE HYDROGEN ATOM 201
MANYELECTRON ATOMS
7
Despite the accuracy with which the quantum theory accounts for certain of
the properties of the hydrogen atom, and despite the elegance and essential
simplicity of this theory, it cannot approach a complete description of this atom
or of other atoms without the further hypothesis of electron spin and the exclusion
principle associated with it. In this chapter wc shall be introduced to the role
of electron spin in atomic phenomena and to the reason why the exclusion
principle is the key to understanding the structures of complex atomic systems.
7.1 ELECTRON SPIN
Let us !>egin by citing two of the most conspicuous shortcomings of the theory
developed in the preceding chapter. The first is the experi mental fact that many
spectral lines actually consist of two separate lines that are very close together.
An example of this /me structure is the first line of the Maimer series of hydrogen,
which arises from transitions between the n = 3 and n = 2 levels in hydrogen
alums. Here the theoretical prediction is for a single line of wavelength fi.563 A,
while in reality there are two lines 1.4 A apart — a small effect, but a conspicuous
failure for the theory.
The second major discrepancy between the simple quantum theory of the atom
and the experimental data occurs in the Zceman effect. In Sec. 6.7 we saw that
;i hydrogen atom of magnetic quantum number m, has the magnetic energy
7.1
V m = m.—B
when it is located in a magnetic field of Hux density B. Now m, can have the
21 f 1 values of +1 through to — f, so a state of given orbital quantum number
I is split into 2/ + 1 substates differing in energy by {cH/2m)B when the atom
is in a magnetic field. However, because changes in m, are restricted to im, = 0,
— 1, a given spectral line that arises from a transition between two states of
203
different / is split into only three components, as shown in Fig. 71. The normal
Y.tnnun efftvt, then, consists of the splitting of a spectral line of frequency i>
into three components whose frequencies are
eh B e
1 ° 2m h ° 4wm
7.2
Normal Zeeman effect
eH B
'■* = "" + I^T = '°
4ffwi
While the normal Zeeman effect is indeed observed in the spectra of a few
elements under certain circumstances, more often it is not: four, six, or even
more components may appear, and even when three components are present
their spacing may not agree with Eq. 7.2. Several anomalous Zeeman patterns
are shown in Fig. 72 together with the predictions of Eq. 7.2,
FIGURE 7>1 The normal leeman effect.
No magnetic field
Magnetic field present
1 = 2
hv a
1= 1
K#>
?r\ hv * + 2n\
ntf —
a
m,=
l
m,=
1
m,=
i
pii/ —
2
(*i+tf)
■m, = 1
■to, =
' m t = ~ I
Am, = —1
Ami = 1
Spectrum without
magnetic field
Am t —
("°~4JFW v ("° + 4W
Spectrum with magnetic
field present
no magnetic field
magnetic field present
V
expected splitting
■■HUH
s
no magnetic
Reld
magnetic field
present
\
expected splitting
FIGURE 72 The normal and anomalous Zeeman effects in various spectral lines.
In an effort to account for Ixith fine structure in spectra! lines and the anoma
lous Zeeman effect, S. A. Coudsmit and G. E. Uhlenbeck proposed in 1925 that
the electron possesses an intrinsic angular momentum independent of any orbital
angular momentum it might have and, associated with this angular momentum,
a certain magnetic moment. What GoudsmH and Uhlenbeck had in mind was
a classical picture of an electron as a charged sphere spinning on its axis. The
rotation involves angular momentum, and because I he electron is negatively
charged, it has a magnetic moment u, opposite in direction to its angular
momentum vector L,. The notion of electron spin proved to be successful in
explaining not only line structure and the anomalous Zeeman effect but a wide
variety of other atomic effects as well.
(If course, the idea that electrons are spinning charged spheres is hardlv faj
accord with quantum mechanics, but in 1928 Dirac was able to show on the
basis of a relalivistie quantumtheoretical treatment that particles having ibe
charge and mass of the electron must have just the intrinsic angular momentum
and magnetic moment attributed to them by Coudsmit and Uhlenbeck.
The quantum number S is used to describe the spin angular momentum of
the electron. The only value a can have is s = %; this restriction follows from
Dirac *s theory and, as we shall see below, may also lie obtained empirically from
spectral data. The magnitude S of the angular momentum due to electron spin
Is gfoen in terms of the spin quantum number .v by the formula
S = \A(* + \)H
7.3 = *—«
204
THE ATOM
MANYELECTRON ATOMS
205
which is the same formula as that giving the magnitude L of the orbital angular
momentum in terms of the orbital quantum number h
L= Vl(l + 1) ft
The space quantization of electron spin is described by the spin magnetic
quantum number m„. Just as the orbital angularmomentum vector can have
the 21 + 1 orientations in a magnetic field frotn + 1 to  /, the spin angular
momentum vector can have the 2s + I = 2 orientations specified by m, = + %
and m„ =  \/ 2 (Fig, 71). The component S t of the spin angular momentum
of an electron along a magnetic field in the z direction is determined by the
m = Vi
FIGURE 73 The two possibl* orien
tations of the spin angularmomentum
vector.
Virt
m = — V4
206 THE ATOM
M)iu magnetic quantum numl)er, so that
7 4 % = 'n„fr
The gyromagnetic ratio characteristic of electron spin is almost exactly twice
that characteristic of electron orbital motion. Thus, taking this ratio as equal
to 2, the spin magnetic moment p, t of an electron is related to its spin angular
momentum S by
7.5
m
The passible components of ft along any axis, say the a axis, are therefore limited
to
7.6
M„=^
2m
We recognize the quantity (eft/2m) as the Bohr magneton.
Space quantization was first explicitly demonstrated by O. Stern and
W. Gertach in 1921. They directed a Iream of neutral silver atoms from an oven
through a set of collimating slits into an inhomogeneous magnetic field, as shown
in Fig, 74. A photographic plate recorded the configuration of the beam after
its passage through the field. In its normal state, the entire magnetic moment
of a silver atom is due to the spin of one of its electrons. In a uniform magnetic
field, such a dipole would merely experience a torque tending to align it with
dm field. In an inhomogeneous field, however, each * , pole" of the dipole is subject
to a force of different magnitude, and therefore there is a resultant force on
the dipole that varies with its orientation relative to the field. Classically, all
orientations should be present in a beam of atoms, which would result tmivly
in a broad trace on the photographic plate instead of the thin line formed in
the absence of any magnetic field. Stem and Cerlaeh found, however, that the
initial beam split into two distinct parts, corresponding to the two opposite spin
orientations in the magnetic field that are permitted by space quantization.
7.2 SPINORBIT COUPLING
The finestructure doubling of spectral lines may be explained on the basis of
I magnetic interaction between the spin and orbital angular momenta of atomic
electrons. This spinorbit coupling can be understood in terms of a straight
forward classical model. An electron revolving about a proton finds itself in a mag
netic field because, in its own frame of reference, the proton is circling alxnit it.
MANYELECTRON ATOMS 207
magnet pole
** inhomogeii MBS
magnetic field
field off
photographic
plate
classical
pattern
actual
pattern
FIGURE 74 The SternGerlKh eiperimenl.
This magnetic field then acts upon the electron's own spin magnetic moment
to produce a kind of internal Zeeman effect. The energy V m of a magnetic
dipole of moment u in a magnetic field of flux density B is, in general.
7.7
V m = —pBcos$
where is the angle between u and B. The quantity /icostf is the component
of p parallel to B, which in the case of the spin magnetic moment of the election
is ti si . 1 lence, letting
u cos = a,. =
we find that
7.8
2m
ci
V = ± B
2m
Depending upon the orientation of its spin vector, the energy of an election
in a given atomic quantum state will he higher or lower by (eh/2m)ll than its
energy in the absence of spinorbit coupling. The result is the splitting of every
208
THE ATOM
quantum state (except 8 states) into two separate substates and, consequently,
the splitting of every spectral line into two component lines.
The assignment of s m % is the only one that conforms to the observed
finestructure doubling. The fact that what should lie single states are in fact
twin states imposes the condition that the St + 1 possible orientations of the
spin angularmomentum vector S must total 2. Hence
2s + 1 = 2
To check whether the observed fine structure in spectral lines corresponds to
the energy shifts predicted by Eq. 7.8, we must compute the magnitude B of
the magnetic field experienced by an atomic electron. An estimate is easy to
obtain. A circular wire loop of radius r that carries the current i has a magnetic
field of (lux density
fttrg
2r
at its center. An orbital electron, say in a hydrogen atom, "sees" itself circled
/"times each second by a proton of charge +e, for a resulting flux density of
B =
Hfe
In the groundstate Bohr atom /= 6.8 x 10 15 and r = 5.3 X 10  " m, so that
B=: 1ST
which is a very strong magnetic field. The value of the Bohr magneton is
^ = 9.27 X 1« u J/T
2m
Hence the magnetic energy V m of such an electron is, from Eq. 7.8,
V =^B
" 2m
S 9.27 X 10 '' J/T X 13 T
7S 1.2 X 10"*"J
Hie wavelength shift corresponding to such a change in energy is about 2 A
for a spectral line of unperturbed wavelength 6.5B"} A , somewhat more than the
observed splitting of the line originating in the n = 3 — * n — 2 transition.
However, the ilux density of the magnetic field at orbits of higher order is less
than for groundslate orbits, which accounts for the discrepancy.
MANY ELECTRON ATOMS
209
210
7.3 THE EXCLUSION PRINCIPLE
In the normal configuration of a hydrogen atom, the electron is in its lowest
quantum state. What are the normal configurations of more complex atoms?
Are all 92 electrons of a uranium atom in the same quantum slate, to Im: envi
sioned perhaps as circling the nucleus crowded together in a single Bohr orbit?
Many lines of evidence make this hypothesis unlikely. One example is the great
(infe ren ce in chemical behavior exhibited by certain elements whose atomic
structures differ by just one electron: for instance, the elements having atomic
numbers 9, 10, and 1 1 are respectively the halogen gas fluorine, the inert gas
neon, and the alkali metal sodium. Sinn the electronic Structure of an atom
controls its interactions with other atoms, it is hard to understand why the
chemical properties of the elements should change so abruptly with a small
change in atomic number if all the electrons in an atom exist together in the
same quantum state.
In 1925 Wolfgang Pauli discovered the fundamental principle that governs
the electronic configurations of atoms having more than one electron. His
exclusion principle states that no two electrons in an atom can exist in the same
quantum state. Each electron in an atom must have a different set of quantum
numbers n, /, mi,, m,.
Pauli w» led to the exclusion principle by a study of atomic spectra. It is
possible to determine the various states of an atom from its spectrum, and the
quantum numbers of these states can be inferred. In the spectra of every element
but hydrogen a number of lines are miwmg that correspond to transitions to
and from states having certain combinations of quantum numbers. Thus no
transitions are observed in helium to or from the groundstate configuration in
which the spins of both electrons arc in the same direction to give a total spin
of I, although transitions are observed to and from the other groundstate
configuration, in which the spins are in opposite directions to give a total spin
of 0. In the absent state the quantum numbers of hath electrons would be it — I .
/ = 0, m t = 0, wi j = '/_,, while in the slate known to exist one of the electrons
has hi, = '/j and the other m, = — '/,. Pauli showed that every unobserved
atomic state involves two or mora electrons with identical quantum numbers,
and the exclusion principle is a statement ol this empirical finding.
Before we explore the role of the exclusion principle i" determining atomic
structures, let us look into its quantummechanical implications. \\V saw in the
ue\ ions chapter that the complete wave function * of the electron in a hydrogen
atom can l>e expressed as the product of three separate wave functions, each
describing that part of * which is a function of one of the three coordinates
r, '/, o. It is possible to show in an analogous waj lliat fcMBmplete wave function
THE ATOM
y(I, 2, 3, . , , , n) of a system of n particles can l*j approximately expressed
as the product of the wave functions *( 1 ), ^(2), t£{3), • ■ ■ . >£(.'0 of the individual
particles. That is,
7.9
«l,2,3,...,n) = +<l)iKS)iK3)...Mii)
We shall use this result to investigate the kinds of wave functions that can be
used to describe a system of two identical particles.
l^et us suppose that one of the particles is in quantum state a and the other
in state h. Because the particles are identical, it should make no difference in
the probability density «^ 2 of the system if the particles are exchanged, with
the one in state a replacing the one in state b and vice versa. Symbolically,
we require that
7.10
C, 1,2) = 1*2(2,1)
Hence the wave function *(2,1), representing the exchanged particles, can be
given by either
7.11
or
7.12
*(2,1) = *(1,2)
*<2 t l) = W2)
Symmetric
Antisymmetric
and still fulfill F.q. 7. 10. The wave function of the system is not itself a measurable
quantity, and so it can l>e altered in sign by the exchange of the particles. Wave
functions that are unaffected by an exchange of particles arc said to l>e sym
metric, while those that reverse sign upon such an exchange are said to be
antisymmetric.
If particle I is in state a and particle 2 is in state b, the wave function of
Hie system is, according to Eq. 7.9,
7.13, *,=WI)W2)
while if particle 2 is in stale a and particle 1 is in state h, the wave function is
7.13b
*n = *.(W)
Because the two particles are in fact indistinguishable, we have no way of
knowing at any moment whether C, or * u describes ihe system. The likelihood
that C i is correct al any moment is the same as the likelihood that ii lr is correcl.
''■'(uivalently, we can say lhat the system spends half the time in the configuration
whose wave function is y, and the other half in the configuration whose wave
function is * n . Therefore a linear combination of *, and \fi u is the proper
description of the system. There are two such combinations possible —
MANYELECTRON ATOMS
21]
212
the symmetric one
™ * s = ~k [U]) U2) + * a{2) wl)1
and the antisymmetric one
\/2
The factor 1/ y2 is required to normalize <fr B and ty A . Exchanging particles 1
and 2 leaves ^ s unaffected, while it reverses the sign of $ A , Doth i£ s and $ 4
uhiv l",t. 7.10.
There are a number of important distinctions between the behavior of particles
in systems whose wave functions are symmetric and that of particles in systems
whose wave functions arc antisymmetric. The most obvious is that, in the former
case, both particles 1 and 2 can simultaneously exist in the same state, with a = b,
while in the latter case, if we set a = b, we find that s> i = 0: the two particles
cannot be in the same quantum slate. Comparing this conclusion with Pauli's
empirical exclusion principle, which states that no two electrons in an atom can
be in the same quantum state, we conclude that systems of electrons are described
by wave Functions that reverse sign upon the exchange of any pair of them.
The results of various experiments show thai all particles which have a spin
of Y 2 have wave functions that are antisymmetric to an exchange of any pair
of them. Such particles, which include protons and neutrons as well as electrons,
obey the exclusion principle when they are in the saime system; that is, when
they move in a common force Held, each member of the system must be in a
different quantum state. Particles of spio '/j, are often referred to as Fermi
particles or fermiom because, as we shall learn in Chap. 9, the behavior of
aggregates of them is governed by a statistical distribution law discovered by
Fermi and Dirac.
Particles whose spins are or an integer have wave functions that arc sym
metric to an exchange of any pair of them. These particles do not obey the
exclusion principle. Particles of or integral spin arc often referred to as Bote
pitrtich's or hawm because the statistical distribution law that describes aggre
gates ol them was discovered by Hose and Einstein. Photons, alpha particles,
and helium atoms are Base particles,
'['here are other important consequences of the symmetry or antisymmetry
nl particle wave functions lwsides that expressed in the exclusion principle. It
is these consequences that make it useful to classify particles according to the
nature of their wave functions rather than simply according to whether or not
they oliey the exclusion principle.
THE ATOM
7.4 ELECTRON CONFIGURATIONS
Two basic rules determine the electron structures of manyelectron atoms:
1. A system of particles is stable when its total energy is a minimum.
2. Only one electron can exist in any particular quantum state in an atom.
Before we apply these rules to actual atoms, let us examine the variation of
electron energy with quantum state.
While the various electrons in a complex atom certainly interact directly with
one another, much about atomic structure can be understood by simply con
sidering each electron as though it exists in a constant mean force field. For
a given electron this field is approximately the electric field of the nuclear charge
Ze decreased by the partial shielding of those other electrons that are closer
to the nucleus. All the electrons that have the same total quantum number n
are, on the average, roughly the same distance from the nucleus. These electrons
therefore interact with virtually the same electric field and have similar energies.
It is conventional to speak of such electrons as occupying the same atomic shell.
Shells are denoted by capital letters according to the following scheme:
n at J 2 3 4 5 . . .
K I U V O . . .
Atomic shells
The energy of an electron in a particular shell also depends to a certain extent
upon its orbital quantum number /, though this dependence is not so great as
that upon n. In a complex atom the degree to which the full nuclear charge
is shielded from a given electron by intervening shells of other electrons varies
with its probabilitydensity distribution. An electron of small / is more likely
to lie found near the nucleus (where it is poorly shielded by the other electrons)
than one of higher I (see Fig. 61 1), which results in a lower total energy (that
is, higher binding energy) for it. The electrons in each shell accordingly increase
in energy with increasing /. This effect is illustrated in Fig. 75, which is a plot
of the binding energies of various atomic electrons as a function of atomic
number.
Electrons that share a certain value of / in a shell are said to occupy the same
mbshelt. All the electrons in a subshell have almost identical energies, since
the dependence of electron energy upon m, and m t is comparatively minor.
The occupancy of the various subshelts in an atom is usually expressed with
Ihe help of the notation introduced in the previous chapter for the various
quantum states of the hydrogen atom. As indicated in Table 6.2, each subshell
is identified by its total quantum number n followed by the letter corresponding
'" its orbital quantum number /. A superscript after the letter indicates the
MANYELECTRON ATOMS
213
r
H ; Li  BNFNe
He Be C O
ATOMIC NUMBER
214
FIGURE 75 The binding energies ol atomtc electroni tn Ry, (1 Rjj =; 1 Rydberg =
13.6 sV ■ groundslate energy of H atom.)
number of electrons in that subshell. For example, the electron configuration
of sodium is written
ls a 2s 2 2p a V
which means that the Is (n = 1, I = 0) and 2s (n = 2, I = 0) subshells contain
two electroas each, tlie 2p (n = 2,1= 1) subshell contains six electrons, and the
3s (n = 3, f = 0) subshell contains one electron.
THE ATOM
7.5 THE PERIODIC TABLE
When the elements arc listed in order of atomic number, elements with similar
chemical and physical properties recur at regular intervals. This empirical
observation, known as the periodic law, was first formulated by Dmitri Mendeleev
about a century ago. A tabular arrangement of the elements exhibiting this
recurrence of properties is called a periodic table, Table 7.1 is perhaps the
simplest form of periodic table; though more elaborate periodic tables have been
devised to exhibit the periodic law in finer detail. Table 7, 1 is adequate for our
purposes.
Kleinents with similar properties form the groups shown as vertical columns
in Table 7.1. Thus group I consists of hydrogen plus the alkali metals, all of
which are extremely active chemically and all of which have valences of +1.
Group VII consists of the halogens, volatile, active nnmnetals that have valences
of — 1 and form diatomic molecules in the gaseous state. Croup VIII consists
of the inert gases, elements so inactive that not only do they almost never form
compounds with other elements, but their atoms do not join together into
molecules like the atoms of oilier gases.
The horizontal rows in Table 7.1 are called periods. Across each period is
a more or less steady transition from an active metal through less active metals
and weakly active nonmetaJs to highly active nonmetals and finally to an inert
gas. Within each column there are also regular changes in properties, but they
are far less conspicuous than those in each period. For example, increasing atomic
number in the alkali metals is accompanied by greater chemical activity, while
the reverse is true in the halogens.
A series of transition elements appears in each period after the third between
the group II and group III elements. The transition elements are metals with
a considerable chemical resemblance to one another but no pronounced resem
blance to the elements in the major groups. Fifteen of the transition elements
in period 6 are virtually indistinguishable in their properties, and arc known
as the kmthantde elements (or rare earths). A similar group of closely related
metals, the actinide elements, is found in period 7.
The notion of electron shells and subshells fits perfectly into the pattern of
the periodic table, which is just a mirror of the atomic structures of the elements.
Let us see how this pattern arises.
The exclusion principle places definite limits on the number of electrons that
can occupy a given subshell. A subshell is characterized by a certain total
quantum number n and orbital quantum number /, where
t = (\l,2,...,(nl)
There are 21 + 1 different values of the magnetic quantum number m, for any
MANY ELECTRON ATOMS
215
E
■ „
lj
1 i
I!
i =
a B
* e
X 1
... ° ?
a .g —
Sig
«8
£■£§
a.
n —
e >
= U.
"1
sa
, 8
3g
£<§
lJU
a
g >
*o
<8 M =
S sn e4
8
8*
S£0
§j
a
i»
(9
»k3
« a. o>
 8
1^
= 8
8p"
sag
EL
h
»u
,8
8
$5*
S*
gg
a.
3 _
E =
(9
««l
go
2 <i
IN
1T
B»3
c *
s ^i
*s
_ «8
? = S
8
2
So
R 3 "5
c
gas
»«1
t J« s
*"3
a *l
*2
8
8
*(;
tflj
Sfl
M
fc i «i
t " a
S[S
Ml
M U. jjj
  —
'Be
i
5fi
"5 4= ^
8  a
?£f
sdS
Rji
ao
<m 5 ?
**8
•if
3
a>3
8
*2gj
' P Z
B*
s£§
8*!!
as
sag
*
—  *
a,
8
B5
■*8
a.
s =
o
**I
x t. 8
if? at fc
Sdg
1 i
a.
3
s ™
(3
 = !
n>3 5
_.8
"*1
l~ pfi f
&£
i .1
■ *
1 
cn n
/, since
= 0, ±1.±2,...,±(
and two possible values of the spin magnetic quantum number hi, ( + % and  %)
for any »t ( . Hence each snbshell caj) contain a maximum of 2(2/ + 1) electrons
and each shell a maximum of
^ 2(2/ + 1) = 2[1 + 3 + 5 + ■ ■ • + 2(n  1) + lj
(0
= 2[1 + 3 + 5 + • ■  + 2n  1]
"ITie quantity in the brackets contains n terms whose average value is %[1 +
(2n  I)], so thai the maximum number of electrons in the nth shell is
2 x [1+ {2»  1)] = 2n*
An atomic shell or subshell that contains its full quota of electrons is said to
he closed. A closed s subshell (1 = 0) holds two electrons, a closed p subshell
(/ = 1) six electrons, a closed </ snbshell (I = 2) ten electrons, and so on.
The total orbital and spin angular momenta of the electrons in a closed subshell
are zero, and their effective charge distributions are perfectly symmetrical (see
Prob. 8 of Chap. 6). The electrons in a closed shell are all very tightly bound,
since tlie positive nuclear charge is large relative to the negative charge of the
inner shielding electrons (Fig. 76). Since an atom containing only closed shells
has no dipole moment, it does not attract other electrons, and its electrons cannot
lie readily detached. Such atoms we expect to be passive chemically, like the
inert gases— and the inert gases all turn out to have closedshell electron con
figurations or their equivalents.
Those atoms with bul a single electron in their outermost shells tend to lose
lliis electron, which is relatively far from the nucleus and is shielded by the inner
electrons from all but an effective nuclear charge of +e. Hydrogen and the
alkali metals are in this category and accordingly have valences of + 1 . Atoms
whose outer shells lack a single electron of being closed tend to acquire such
an electron through the attraction of the imperfectly shielded strong nuclear
charge, which accounts for the chemical behavior of the halogens. In this manner
the similarities of the members of the various groups of the periodic table may
be accounted for.
Table 7.2 shows the electron configurations of the elements. The origin of
the transition elements evidently lies in the tighter binding of s electrons than
" or / electrons in complex atoms, discussed in the previous section. The first
element to exhibit this effect is potassium, whose outermost electron is in a 4s
MANYELECTRON ATOMS 217
Na
218
FIGURE 76 Electron shielding in sodium and »rgon. Each outer electron in an Ar atom is acted upon
by an effective nuclear charge 8 times greater than that acting upon the outer electron in a Na atom,
even though the outer electrons In both cases are In the M(» ■ 3) shell.
instead of a 3d substate. The difference in binding energy between 3d and 4s
electrons is not very great, as can be seen in the configurations of chromium
and copper. In both of these elements an additional 3d electron is present at
the expense of a vacancy in the 4s subshell. In tiiis connection another glance
at Fig. 7.5 will be instructive.
The order in which electron subshells are filled in atoms is
Is. 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4/, 5d, fip, 7s, 6d
as we can see from Table 7.2 and Fig. 77. The remarkable similarities in
chemical behavior among the lanthanidcs and actinides are easy to understand
on the basis of this sequence. All of the lanlhanides have the same 5s 2 5p 1 W i
configurations but have incomplete 4/ subshells. The addition of 4/ electrons
has virtually no effect on the chemical properties of the lanthanide elements,
THE ATOM
Table 7.2.
ELECTRON CONFIGURATIONS OF THE ELEMENTS,
L
M
Z> 2> 3t 3p 3,1 4i 4(1 4rf 4/
5j> 5d 5/ 6, 6p 6il 7 1
1 li
1
2 lie
2
3 1 J
2
1
1 Be
2
2
., K
2
2
1
6C
2
2
2
7N
2
2
3
SO
2
2
4
9P
2
2
5
10 Ne
2
2
11 Na
2
o
e
i
12 Mg
2
2
o
2
13 Al
2
2
6
2
1
14 Si
2
2
6
2
1
ISP
2
2
2
3
10 S
2
2
2
4
it a
2
2
6
2
5
ISA
2
2
6
2
8
19 K
2
2
e
2
6
1
20 C»
2
2
e
2
2
21 St
2
2
2
e
I
2
22 Ti
2
2
b
2
2
2
23V
2
2
1
2
6
3
2
24 Cr
I
2
6
2
6
5

23 \ln
2
2
6
2
6
5
2
26 Fe
2
2
6
2
6
e
2
27 Co
2
2
6
2
e
7
2
28 Ni
2
2
2
e
8
2
29 Cu
2
2
6
2
e
10
I
30 Zn
2
2
6
2
8
10
2
31 Ca
2
2
2
6
10
2
1
32 Ce
2
2
6
2
e
10
2
2
33 As
2
2
2
e
10
2
3
34 Se
2
2
6
2
a
10
2
4
35 Br
2
2
6
2
a
10
2
5
36 Kr
2
2
6
2
e
10
2
6
37 Hl>
2
2
6
2
e
10
2
e
1
38 Sr
2
2
e
2
e
10
2
e
2
39 Y
2
2
6
2
e
10
2
1
2
40 Zr
2
2
6
2
n
10
2
2
2
II Nb
2
2
(i
2
6
10
2
4
1
■12 Mo
2
2
6
2
6
10
2
6
5
1
43 Tc
2
2
6
2
6
10
2
6
5
2
! 1 fill
2
2
G
2
6
10
2
B
7
1
45 Rh
2
2
6
2
6
10
2
6
8
1
46 Pd
2
2
6
2
6
10
2
6
in
47 Ag
2
2
a
2
6
10
2
e
10
1
IS CI
2
2
6
2
6
10
2
6
10
2
49 In
2
2
6
2
6
10
2
10
2 1
50 Sri
2
2
e
2
6
10
2
6
10
2 2
51 Sb
2
2
a
2
e
10
2
10
2 3
Table 7.2 {Continued)
K
1.
U
,v
4
)
P
Q
li
2i
2P
3.
3;
3./
4,
4<
4,1
«/
b
3p
U
5/
6.
6,, 6,1
7.
52 Te
2
2
6
2
6
10
2
6
III
2
4
53!
2
2
6
2
6
10
2
6
10
2
5
34 Xc
2
2
6
2
6
10
2
li
10
2
6
55 Cs
2
2
6
2
6
10
2
6
10
2
6
1
58 ll..
2
2
6
2
e
10
2
6
10
2
6
2
57 La
2
2
6
2
6
10
2
6
10
2
6
1
2
58 Cc
2
2
6
2
6
10
2
6
10
2
2
8
2
59 Pi
2
2
6
2
6
10
2
6
10
3
2
8
2
60 V.I
2
2
6
2
6
10
s
6
10
4
2
6
2
fil Pm
2
2
6
2
6
10
2
6
10
5
2
6
2
62 Km
2
2
6
2
6
10
2
e
10
8
2
8
2
63 l.,i
2
2
6
2
6
10
2
6
10
7
2
2
64 Cd
2
2
6
2
6
10
2
6
10
7
2
6
1
2
6,5 Tb
2
2
6
2
6
10
2
e
10
9
2
6
2
66 Dy
2
2
6
2
6
[0
2
6
10
10
2
6
2
67 l[o
2
2
6
2
e
10
2
8
10
11
2
6
2
68 Er
2
2
6
2
e
10
2
6
10
12
2
6
2
69 Tin
2
2
6
2
6
10
2
8
10
13
2
6
2
Til Vh
2
2
6
2
6
10
2
6
10
14
2
6
2
71 Lu
2
2
6
2
e
10
2
8
10
14
2
6
1
2
72 llf
2
2
6
2
6
10
2
10
14
2
6
2
2
73 Ta
2
2
6
2
e
10
2
8
10
11
2
3
2
74 VV
2
2
6
2
e
10
2
8
10
14
2
6
4
2
75 Re
2
2
6
2
6
10
2
6
10
14
2
a
5
2
76 Os
2
2
6
2
6
10
2
6
10
14
2
6
6
2
77 Ir
2
2
6
2
6
10
2
8
10
14
2
6
7
2
7fi PI
2
2
a
2
6
10
2
6
10
14
2
6
a
I
79 An
2
2
6
2
e
10
2
ll
10
14
2
6
10
t
SO lig
2
2
e
2
e
10
2
8
10
14
2
6
10
2
81 Tl
2
2
6
2
6
10
2
8
10
14
2
8
10
2
t
82 Fh
2
2
6
2
6
10
2
6
10
14
2
6
10
2
2
S3 Bi
2
2
6
2
e
10
2
e
10
14
2
6
10
2
3
84 1'. .
2
2
6
2
6
10
2
8
10
14
2
6
10
2
4
85 At
2
2
6
2
6
10
2
6
10
14
2
10
2
5
Mi Ki,
2
2
6
2
6
10
2
ll
to
14
2
6
10
2
6
87 Ft
2
2
6
2
6
10
2
li
10
14
1
6
10
2
1
88 Eta
2
2
g
2
6
10
2
e
to
14
2
6
to
2
6
2
89 Ac
2
2
6
2
6
10
2
6
to
14
2
6
10
2
6 I
2
90TTi
2
2
6
2
e
10
2
8
10
14
2
(1
10
2
8 2
2
91 Pa
2
2
6
2
6
10
2
8
10
14
2
6
to
2
2
6 I
2
92
2
2
6
2
6
10
2
e
10
14
2
8
10
3
2
8 1
2
!Ki Np
2
2
6
2
6
10
2
ii
10
14
2
8
10
4
2
6 1
2
94 Pu
2
2
B
2
6
10
2
8
10
14
2
6
to
5
2
6 t
2.
95 Am
2
2
6
2
6
10
2
6
10
14
2
6
10
6
2
6 1
2
!M> I in
2
2
B
2
6
10
2
6
10
14
2
6
10
7
2
6 1
2
97 Ilk
2
2
6
2
e
10
2
1
10
14
2
8
10
8
2
6 1
2
98 Cf
2
2
6
2
6
10
2
li
10
14
2
6
10
10
2
6
2
99E
I
2
6
o
6
10
2
li
10
14
ft
6
10
11
2
6
2
100 Fm
2
2
6
2
6
10
2
6
10
14
2
6
10
12
2
6
2
1111 Md
2
2
6
2
6
10
2
6
to
14
2
10
13
2
6
2
1(12 \„
2
2
6
2
6
10
2
6
10
14
2
6
10
14
2
6
2
103 Lw
2
2
6
2
i;
to
2
6
10
14
2
6
10
14
2
6 1
2
J_
)'
f
)'
»
n = 1 2 3 4 S fl 7
FIGURE 77 The sequence of quantum states In an atom. Not to scale.
which arc determined by the outer electrons. Similarly, all of the actinides have
6s 2 6p fi 7s 2 configurations, and differ only in the numbers of their 5/ and Bel
electrons.
These irregularities in the binding energies of atomic electrons are also re
sponsible for the lack of completely full outer shells in the heavier inert gases.
Helium (Z = 2) and neon (Z = 10) contain closed K and L shells respectively,
I "it argon (Z = 18) has only 8 electrons in its M shell, corresponding to closed
3s and 3p subshells. The reason the 3d suhshell is not filled next is simply that
4s electrons have higher binding energies than 3d electrons, as we have said,
and so the 4s subshell is filled first in potassium and calcium. As the 3d subshell
is filled in successively heavier transition elements, there are still one or two
outer 4s electrons that make possible chemical activity. Not until krypton
(Z = 36) is another inert gas reached, and here a similarly incomplete outer shell
occurs with only the 4s and 4p subshells filled. Following krypton is rubidium
I Z = 37), which skips both the 4d and 4/ subshells to have a 5s electron. The
next inert gas is xenon (Z = 54), which has filled 4d, 5s, and 5p subshells, but
now even the inner 4/ subshell is empty as well as the 5d and 5/ subshells. The
snrne pattern recurs with the remainder of the inert gases.
While we have sketched the origins of only a few of the chemical and physical
properties of the elements in terms of their electron configurations, many more
can be quantitatively understood by similar reasoning.
MANYELECTRON ATOMS 221
222
7.6 HUND'S RULE
In general, the electrons in an atom remain unpaired— that is, have parallel
spins— whenever possible. This principle is called Hund's rule. The ferro
magnetism of iron, cobalt, and nickel is a consequence of Hund's rule; their 3cf
subshells are only partially occupied, and the electrons in these subshells do not
pair off to permit their spin magnetic moments to cancel out. In iron, for
instance, five of the six 3d electrons have parallel spins, so that each iron atom
has a large resultant magnetic moment. We shall examine other consequences
of Hund's rule in Chap, i) in connection with molecular (wilding.
The origin of Hund's rule lies in the mutual repulsion of atomic electrons.
Because of this repulsion, the farther apart the electrons in an atom are, the
lower the energy of the atom. Electrons in the same subshcll with the same
spin must have different m, values and accordingly are described by wave
functions whose spatial distributions are different. Electroas with parallel spins
are therefore more separated in space than if they paired off. and this arrange
ment, having less energy, is the more stable one.
*7.7 TOTAL ANGULAR MOMENTUM
Each electron in an atom has a certain orbital angular momentum L and a certain
spin angular momentum S, both of which contribute to the total angular mo
mentum J of the atom. Like all angular momenta, J is quantized, with a magni
tude given by
Total atomic angular momentum
7.16
/= VJ(J + 1)«
and a component J t in the z direction given by
7.17 /, = Mj n
z component of total atomic angular momentum
where J and M j are the quantum numbers governing / and } t . Our task in the
remainder of this chapter is to look into the properties oF J and their effect on
atomic phenomena. We shall do this in terms of the semiclassical vector model
of the atom, which provides a more intuitively accessible framework for under
standing angularmomentum considerations than does a purely quantum
mechanical approach.
Let us first consider an atom whose total angular momentum is provided by
a single electron. Atoms of the elements in group I of the periodic table
hydrogen, lithium, sodium, and so on— are of this kind since they have single
electrons outside closed inner shells (except for hydrogen, which has no inner
electrons) and the exclusion principle assures that the total angular momentum
THE ATOM
and magnetic moment of a closed shell are zero. Also in this category are the
ions He 4 , Be + , Mg + , B + \ Al ++ , and so on.
The magnitude L of the orbital angular momentum L of an atomic electron
is determined by its orbital quant' im number / according to the formula
7.18
l= Vi(iTT)H
while (he component L t of L along the z axis is determined by the magnetic
quantum number m, according to the formula
7.19
L t = m ( fi
Similarly the magnitude S of the spin angular momentum S is determined by
the spin quantum number s (which has the sole value + 1 / 2 ) according to the
formula
7.M S = V«(s + l)ft
while the component S t of S along the z axis is determined by the magnetic
spin quantum number m, according to the formula
7.21
S, = m.ft
Because L and S are vectors, they must be added vectorially to yield the total
angular momentum J:
7.22
J = L + S
It is customary to use the symbols /and rtij for the quantum numbers that describe
J and } t for a single electron, so that
7.23 /= \//{/ + 1) ft
7.24 / ( = mfi
To obtain the relationships among the various angularmomentum quantum
numbers, it is simplest to start with the Z components of the vectors J, L, and
S. Since } t , L^, and S^ are scalar quantities,
sad
7,25
ffijfi = nijft ± m,h~
in, = m.±m.
The possible values of iij, range from + / through to /, and tho.se of m, are
~s. The quantum number J is always an integer or while s = l / 2 , and as a
r, " i 'ili m, must l>e halfintegral. The possible values of m f also range from +/
through to /in integral steps, and so, for any value of i.
MANYELECTRON ATOMS 223
7.26
j = l±S
Like trip j is always half integral.
Because of the simultaneous quantization of J, L, and S they can have only
certain specific relative orientations. This is a general conclusion; in the ease
of a oneelectron atom, there are only two relative orientations possible. One
of these corresponds to j = / + s, so that / > «., and the other to /  / — s, so
that J < L, Figure 78 shows the two ways in which L and S can combine to
form J when / = 1 . Evidently the orbital and spin angularmomentum vectors
can never be exactly parallel or antiparallel to each other or to the total angular
momentum vector.
The angular momenta L and S interact magnetically, as we saw in Sec. 7.2,
and as a result exert torques on each other. If there is no external magnetic
field, the total angular momentum J is conserved in magnitude and direction,
and I he effect of the internal torques can only be the precession of L and S
around the direction of their resultant J (Fig. 79). However, if there is an
external magnetic field B present, then J precesses about the direction of B while
L and S continue precessing about J, as in Fig. 71 (J. The precession of J about
B is what gives rise to the anomalous Zeeman effect, since different orientations
of J involve slightly different energies in the presence of B.
Atomic nuclei also have intrinsic angular momenta and magnetic moments,
as we shall see in Chap. II, and these contribute to the total atomic angular
momenta and magnetic moments. These contributions are small because nuclear
FIGURE 78 The tvro ways In which I. and S can be added to form J when I = 1, t = !*.
,=! + ,= 3/2
224 THE ATOM
,=/»= 1/2
FIGURE 79 The orbital and
spin angular momentum vec
tors I. and S precess about J
according to the vector model
of the atom.
cone traced out by L
cone traced out by S
the atom
magnetic moments are —lO 3 the magnitude of electronic moments, and they
lead to the hi/perfine .■structure of spectra! lines with typical spacings between
components of — I0~ 2 A as compared with typical finestructure spacings of
several angstroms.
FIGURE 710 In the
presence ol an enter
"ill magnetic tie Id B,
the total angular
momentum vector J
Precesses about fl ac
cording to the vector
model of the atom.
cone traced
out by J
i the atom
MANYELECTRON ATOMS
225
*7.8 LS COUPLING
When more than one electron contributes orbital and spin angular momenta
to the total angular momentum J of an atom, J is still the vector sum of these
individual momenta. Because the electrons involved interact with one another,
the manner in which their individual momenta L, and S, add together to form
J follows certain definite patterns depending upon the eireum stances. The usual
pattern for all but the heaviest atoms is that the orbital angular momenta L (
of the various electrons are coupled together electrostatically into a single
resultant L and the spin angular momenta 8, are coupled together independently
into another single resultant S; we shall examine the reasons for this behavior
later in this section. The momenta L and S then interact magnetically via the
spinorbit effect to form a total angular momentum J. This scheme, called LS
coupling, may be summarized as follows:
LS coupling
7.27 S e 2 S 4
J = L + S
As usual, L, S, j, L t , S,, and } c are quantized, with the respective quantum
numbers being L, S, J, M L , M s , and Mj. Hence
7.28
7.29
7.30
7.31
7.32
7.33
l= \A.(L + i) n
L, = M L fi
S = \/S(S + 1) ft
S, = MsR
/= \/J(J + 1) ft
! : = Mj fi
Both L and M L are always integers or 0, while the other quantum numbers are
halfintegral if an odd number of electrons is involved and integral or 11 if an
even munl>er of electrons is involved.
As an example, let us consider two electrons, one with /, = 1 and the other
with / 2 = 2. There are three ways in which L, and L*, can be combined into
a single vector L that is quantized according to Eq. 7.28, as shown in Fig. 711,
These correspond to L = 1,2, and 3, since all values of L are possible from I , + l^
to f 1 — l 2 . The spin quantum number s is always + l / 2 , so that there are two
possibilities for the sum Sj + S a , as in Fig. 711, that correspond to S = and
S = 1. We note that L, and Lj can never be exactly parallel to L, nor S t and
S 2 to S, except when the vector sum is 1). The quantum number J can have
all values between L + S and L — S, so here J can be 11, 1, 2, 3, or 4.
226
THE ATOM
L=3
L=2
L=l
S=l
s 'li 8
s=o
FIGURE 711 When I, = 1, i, = W, and 1. = 2, t, = W, there are three ways in which L, and I., can
combine to form I. and two ways in which S, and Bj can combine to form S.
The LS scheme owes its existence to the relative strengths of the electrostatic
forces that couple the individual orbital angular momenta trito a resultant 1 and
the individual spin angular momenta into a resultant S. The origins of these
forces are interesting. The coupling Iwtwccn orbital angular momenta can be
understood by reference to Fig. 612, which shows how the electron probability
density ^ 2 varies in space for different quantum states in hydrogen. The corre
sponding patterns for electrons in more complex atoms will be somewhat differ
ent, of course, but it remaias true in general that ht 2 is not spherically symmetric
except for s states. (In the latter case / = and the electron has no orbital angular
momentum to contribute anyway.) Because of the asymmetrical distributions
of their charge densities, the electrostatic forces between the electrons in an atom
vary with the relative orientations of their angularmomentum vectors, and only
certain relative orientations are stable. These stable configurations correspond
to a total orbital angular momentum that is quantized according to the formula
L = \/L(L + 1) ft.
The coupling between the various Lj is usually such that the configuration
of lowest energy is the one for which L is a maximum. This effect is easy to
understand if we imagine two electrons in the same Bohr orbit. Because the
electrons repel each other electrostatically, they tend to revolve around the
nucleus in the same direction, which maximizes L If they revolved in opposite
directions to minimize /,, the electrons would pass each other more frequently,
leading to a higher energy for the system.
The origin of the strong coupling between electron spins is harder to visualize
because it is a purely quantummechanical effect with no classical analog. (The
direct interaction between the intrinsic electron magnetic moments, it should
he noted, is insignificant and not responsible for the coupling between electron
MANYELECTRON ATOMS 227
spin angular momenta.) The basic idea is that the complete wave function
^{1,2, ...,n) of a system of n electrons is the product of a wave function
h{1, 2, .... n) that describes the coordinates of the electrons and a spin function
$(1,2 n) that describes the orientations of their spins. As we saw in Sec.
7,3, the complete function ^(1, 2, . . , , n) must l>e antisymmetric, which means
that u(l, 2 n) is not independent of s{l, 2, . . . , n). A change in the relative
orientations of the electron spin angularmomentum vectors must therefore tie
accompanied by a change in the spatial electronic configuration of the atom,
which means a change in the atom s electrostatic potential energy. To go from
one total spin angular momentum S to a different one involves altering the
structure of the atom, and therefore a strong electrostatic force, besides altering
the directions of the spin angular momenta S„ S 2 , . . . , S B , which requires only
a weak magnetic force. This situation is what is described when it is said (hat
the spin momenta S, are strongly coupled together electrostatically.
The S i always combine into a groundstate configuration in which S is a
maximum. This is an example of Hund's rule; as mentioned earlier, electrons
with parallel spins have different m t values and are described by different wave
functions, which means that there is a greater average separation in space of
the eleclroas and accordingly a lower total energy.
Although wc shall not try to justify this conclusion, the combination of L and
S that makes / a minimum results in the lowest energy.
•7.9 jj COUPLING
The electrostatic forces that couple the Lj into a single vector L and the S t into
another vector S are stronger than the magnetic spinorbit forces that couple
L and S to form J in light atoms, and dominate the situation even when a
moderate external magnetic field is applied. (In the latter case the precession
of J around B is accordingly slower than the precession of L and S around J.)
However, in heavy atoms the nuclear charge becomes great enough to produce
spinorbit interactions comparable in magnitude to the electrostatic ones between
the L, and between the S f , and the IS coupling scheme l>egins to break down.
\ similar breakdown occurs in strong external magnetic fields (typically — 10 T),
which produces the PaschenBack effect in atomic spectra. In the limit of the
failure of LS coupling, I lie total angular momenta J, of the individual electrons
add together directly to form the angular momentum J of the entire atom, a
situation referred to as jj coupling since each J, is described by a quantum number
/ in the manner described earlier. Hence
7.34
h = U + s (
jj coupling
238
THE ATOM
In Sec, 6.4 we saw that individual orbital angularmomentum states are
customarily described by a lowercase letter, with s corresponding to I = 0, p
to I = 1, d to / = 2, and so on. A similar scheme using capital letters is used
to designate the entire electronic state of an atom according to its total orbital
angidarm omen turn quantum munl>er L as follows;
L = 1 2 3 4 5 6...
S P D I G H /...
A superscript number Iwfore the letter { 2 P for instance) is used to indicate the
multiplicity of the state, which is the number of different possible orientations
of L and S and hence the numtwr of different possible values of J. The multi
plicity is equal to 2S + 1 in the usual situation where L > S, since J ranges
from L + S through to L  S. Thus when S = 0, the multiplicity is 1 (a singlet
state) and J = L; when S = y 2> the multiplicity is 2 (a clotiblet state) and J = L
± %; when S = 1, the multiplicity is 3 (a triplet state) and J = L 4 1, L, or
L — 1; and so on. (In a configuration in which S > L, the multiplicity is given
by 2L + 1.) The total angularmomentum quantum number J is used as a sub
script after the letter, so that a a P 3/2 state {read as "doublet P threehalves")
refers to an electronic configuration in which S = l / 2 < L = *> a "d J ** %■ l*° r
historical reasons, these designations are called term symbols.
In the even! that the angular momentum of the atom arises from a single outer
electron, the principal quantum numltcr n of this electron is used as a prefix:
thus the ground state of the sodium atom is described by 3 2 S t/2 , since its elec
tronic configuration has an electron with n = 3, / = 0, and * = '/ 2 (and hence
/ = V?) outside closed n = 1 and n = 2 shells. For consistency it is conventional
to denote the above state by 3 2 S, /;! with the superscript 2 indicating a doublet,
even though there is only a single possibility for J since L = 0.
'7,10 ONEELECTRON SPECTRA
We are now in a position to understand the chief features of the spectra of the
various elements. Before we examine some representative examples, it should
be mentioned thai further complications exist which have not been considered
here, for instance those that originate in relativistic effects and in the coupling
lictween electrons and vacuum fluctuations in the electromagnetic field (see Sec.
6.10). These additional factors split certain energy states into closely spaced
*ul .states and therefore represent other sources of fine structure in spectral lines.
Figure 712 shows the various states of the hydrogen atom classified by their
lotal quantum number n and orbital angularmomentum quantum number /. The
selection rule for allowed transitions here is if = ±1, which is illustrated by
MANYELECTRON ATOMS
229
230
excitation
energy, eV
13.0 ■ n = o»S
D
10 Un= S
0" n= 1
FIGURE 712 Energylevel diagram lor hydrogen showing the origins of soma of the
mora prominent spectral lines. The detailed structures of the n = 2 and »  3 levels and
the transitions that lead to the various components of the H . Una are pictured In the
the transitions shown. To indicate some of the detail that is omitted in a simple
diagram of this kind, the detailed structures of the ti = 2 and n = 3 levels are
pictured; not only arc all substates of the same n and different j separated in
energy, but the same is true of states of the same n and / but with different
I. The latter effect is most marked for states of small n and /, and was first
established in 1947 in the "Lamb shift" of the % 2 S l/2 state relative to the 2 z P l/2
state. The various separations eoaspire to split the H B spectral line (ti =
3 — > n = 2) into seven closely spaced components.
The sodium atom has a single 3s electron outside closed inner shells, and so,
if we assume that the 10 electrons in its inner core completely shield + \i)e of
nuclear charge, the outer electron is acted upon by an effective nuclear charge
of +c just as in the hydrogen atom. Hence we expect, as a first approximation,
that the energy levels of sodium will be the same as those of hydrogen except
THE ATOM
that the lowest one will correspond to n =3 instead of n = 1 because of the
exclusion principle. Figure 713 is the energylevel diagram for sodium and, by
comparison with the hydrogen levels also shown, there is indeed agreement for
the states of highest /, that is, for the states of highest angular momentum.
To understand the reason for the discrepancies at lower values of /, we need
only refer to Fig. fi11 to see how the probability for finding the electron in
a hydrogen atom varies with distance from the nucleus. The smaller the value
of / for a given n, the closer the electron gets to the nucleus on occasion.
Although the sodium wave functions are not identical with those of hydrogen,
their general behavior is similar, and accordingly we expect the outer electron
excitation
energy, eV
5.13 m
FIGURE 713 Energy,
level diagram for so.
dium. The energy
levels of hydrogen are
included for com
parison.
3
1
D F hydrogen
7 S 7p 6d 6/ n = 6
6p 5d 5/ " = 5
Ss / /
Sp
4d — 4,
»=4
'*'M .=>
— 43
rt = 2
MANY ELECTRON ATOMS 231
in a sodium atom to penetrate the core oF inner electrons most often when it
is in an s state, lass often when it is in a p state, still less often when it is in
a d state, and so on. The less shielded an outer electron is from the full nuclear
charge, the greater the average force acting on it, and ilu smaller (that is. the
more negative) its total energy. For this reason the states of small / in sodium
are displaced downward from their equivalents in hydrogen, as in Fig. 713, and
there are pronounced differences in energy Iretween states of the same n but
different I.
*7.11 TWOELECTRON SPECTRA
A single electron is responsible for the energy levels of both hydrogen and
sodium. However, there are two 1$ electrons in the ground slate of helium, and
it is interesting to consider the effect of LS coupling on the properties and
behavior of the helium atom. To do this wc first note the selection rules for
allowed transitions under LS coupling:
735
7.36
7.37
AL = 0, ±1
AJ =0, ±1
AS a
LS selection rules
232
When only a single electron is involved, AL = is prohibited and AL = Af = :£l
is the only possibility. Furthermore, J must change when the initial state has
J = 0, so that J = » J = is prohibited.
The helium energylevel diagram is shown in Fig. 714. The various levels
represent configurations in which one electron is in its ground state and the other
is in an excited state, but because the angular momenta of the two electrons
are coupled, it is proper to consider the levels as characteristic of the entire
atom. Three differences between this diagram and the corresponding ones for
hydrogen and sodium are conspicuous.
First, there is the division into singlet and triplet states, which are, respectively,
states in which the spins of the two electrons arc an ti para I lei (to give S = 0)
and parallel (to give S = 1). Becan.se of the selection rule AS = 0, no allowed
transitions can occur between singlet states and triplet states, and the helium
spectrum arises from transitions in one set or the other. Helium atoms in singlet
states .in il iparallel spins) constitute pamhelium and those in triplet states (parallel
spins) constitute ortltohelium. An orthohelium atom can lose excitation energy
in a collision and become one of parallel ium, while a parallel ium atom can gain
excitation energy in a collision and become one of orthohelium; ordinary liquid
or gaseous helium is therefore a mixture of both. The lowest triplet states are
THE ATOM
excitation
energy, cV
24.6
singlet states
(parahelium)
20
15
to
5
triplet states
(orthohelium)
~
1
FIGURE 714 Energylevel diagram for helium showing the division Into singlet (paraheilum) and triplet
(orthohelium) states. There is no 1 'S state.
called metastabk liecause, in the absence of collisions, an atom in one of them
can retain its excitation energy for a relatively long time (a second or more)
before radiating.
The second obvious peculiarity in Fig, 714 is the absence of the PS state.
Hie lowest triplet state is 2\S, although the lowest singlet state is PS". The PS
state is missing as a consequence of the exclusion principle, since in this state
the two electrons would have parallel spins and therefore identical sets of
MANYELECTRON ATOMS 233
quantum numlicrs. Third, the energy difference between the ground state and
the lowest excited state is relatively large, which reflects the tight binding of
closedshell electrons discussed earfier in this chapter. The ionization energy
of helium — the work that must be done to remove an electron from a helium
atom — is 24,6 eV, the highest of any element.
The last energylevel diagram we shall consider is that of mercury, which has
two electrons outside an inner core of 78 electrons in closed shells or suljshells
(Table 7.2}, We expect a division into singlet and triplet states as in helium,
but because the atom is so heavy W8 might also expect signs of a breakdown
in the LS coupling of angular momenta. As Fig. 715 reveals, both of these
expectations are realized, and several prominent lines in the mercury spectrum
arise from transitions that violate the AS = selection rule. The transition
FIGURE 715 EnorgyJevel diagram for mercury. In each excited level one outer electron Is In the
ground state, and the designation of the levels in the diagram corresponds to (he state of the other elec
tron.
•6/
tj> ( _» is is an example, and is responsible for the strong 2,537A line in the
ultraviolet. To be sure, this does not mean that the transition probability is
necessarily very high, since the three a P, states are the lowest of the triplet set
and therefore tend to be highly populated in excited mercury vapor. The
3p () _> ic^ ant j 3p a _► is u transitions, respectively, violate the rules that forbid
transitions from J = to J = and that limit AJ to or ±1, as well as violating
AS = 0, and hence are considerably less likely to occur than the 3 P, » 'S
transition. The 3 P and 3 P 2 states are therefore metastable and, in the absence
of collisions, an atom can persist in either of them for a relatively long time.
The strong spinorbit interaction in mercury that leads to the partial failure of
LS coupling is also respoasible for the wide spacing of the elements of the 3 P
triplets.
7,12 XRAY SPECTRA
In Chap. 2 we learned that the Xray spectra of targets l>onibarded by fast
electrons exhibit narrow spikes at wavelengths characteristic of the target mate
ria] in addition to a continuous distribution of wavelengths down to a minimum
wavelength inversely proportional to the electron energy. The continuous Xray
spectrum is the result of the inverse photoelectric effect, with electron kinetic
energy being transformed into photon energy hf. Hie discrete spectrum, on the
other hand, has its origin in electronic transitions within atoms that have been
disturbed by the incident electrons.
Transitions involving the outer electrons of an atom usually involve only a
few electron volts of energy, and even removing an outer electron requires at
most 24.6 eV (for heliiun). These transitions accordingly are associated with
photons whose wavelengths lie in or near the visible part of the electromagnetic
spectrum, as is evident from the diagram in the back endpapers of this book.
The inner electrons of heavier elements arc a quite different matter, because
these electrons experience all or much of the full nuclear charge without nearly
complete shielding by intervening electron shells and in consequence are very
tightly bound. In sodium, for example, only 5.13 eV is needed to remove the
outermost 3s electron, while the corresponding figures for the inner ones are
3] eV for each 2p electron, 63 eV for each 2a electron, and 1,041 eV for each
Is electron. Transitions that involve the inner electrons in an atom are what
give rise to discrete Xray spectra because of the high photon energies involved.
Figure 716 shows the energy levels (not to scale) of a heavy atom classed
by total quantum number n; energy differences between angular momentum
states within a shell are minor compared with the energy differences between
shells. Let us consider what happens when an energetic electron strikes the atom
MANYELECTRON ATOMS
235
N
U
,
— !
Afcty
M a
St S
My
t«
h
Ly
h
r '
K a
%
Ky
Kj
K<
1
11=4
n = 3
n = 2
I
FIGURE 716 The origin of Xray spectra.
n = l
236
and knocks oul one of the Kshell electrons. (The K electron could also be
elevated to one of the unfilled upper quantum states of the atom, but ihe
difference between the energy needed to do this and that needed to remove the
electron completely is insignificant, only 11.2 percent in sodium and still less in
heavier atoms.) An atom with a missing K electron gives up most of its consid
erable excitation energy in the form of an Xray photon when an electron from
an outer shell drops into the "hole" in the K shell. As indicated in Fig, 736,
the K series of lines in the Xray spectrum of an element consists of wavelengths
THE ATOM
arising in transitions from the L, M , N, , . . levels to the K level. Similarly the
longerwavelength L series originates when an L electron is knocked out of the
atom, the M series when an M electron is knocked out, and so on. The two
spikes in the Xray spectrum of molybdenum in Fig, 28 are the K a and Kp lines
of its K series.
An atom with a missing inner electron can also lose excitation energy by the
Auger effect without emitting an Xray photon. In the Auger effect an outershell
electron is ejected from the atom at the same time that another outershell
electron drops to the incomplete inner shell; the ejected electron carries off the
atom's excitation energy instead of a photon doing this. In a sense the Auger
effect represents an internal photoelectric effect, although the photon never
actually comes into being within the atom. The Auger process is competitive
with Xray emission in most atoms, but the resulting electrons are usually
absorbed in the target material while the X rays readily emerge to be detected.
Problems
1. If atoms could contain electrons with principal quantum numbers up to and
including n = 6, how many elements would there he?
2. The ionization energies of the elements of atomic numbers 2(1 through 29
are very nearly equal. Why should this be so when considerable variations exist
in the ionization energies of other consecutive sequences of elements?
3. The atomic radius of an element can be determined from measurements made
on crystals of which it is a constituent. The results are shown in Fig. H)'3,
Account for the general trend of the variation of radius with atomic number.
4. Many years ago it was pointed out that the atomic numbers of the rare gases
are given by the following scheme:
Z(He) = 2(1*) = 2
ZTNe) = 2(I 2 + 2*} = 10
Z{Ar) = 2(l a + 2 2 + 2 Z ) = 18
Z(Kr) = 2(1* + 2* + 2* + 3 s ) = 36
Z(Xe) = 2(1* + 2* + 2* + 3* + 3 2 ) = 54
Z(Rn) = 2(1* + 2* + 2* + 3* + 3* + 4 2 ) = 86
Kxplain the origin of this scheme in terms of atomic theory.
5. A beam of electrons enters a uniform magnetic field of flux density 1.2 T.
Find the energy difference between electrons whose spins are parallel and
anliparallel to the field.
MANYELECTRON ATOMS
237
238
fi. Mow does the agreement lierween observations of the hid ma I Zeeman effect
and the theory of (his effect tend to confirm the existence of electrons as inde
pendent entities within atoms?
7. A sample of a certain element is placed in a magnetic field of flux density
0.3 T. How far apart are the Zeeman components of a spectral line of wavelength
4,500 A?
8, Why does the normal Zeeman effect occur only in atoms with an even
number of electrons?
*9. Find the S, L, and J values that correspond to each of the following slates:
S () , P z , 'D 3ri , f' 5 , // a/2 .
* 10. The carbon atom has two 2s electrons and two 2p electrons outside a filled
inner shell. Its ground state is 3 P , What are the tenn symbols of the other
allowed states, if any? Why would you think the :i P state is the ground state?
*I1. The lithium atom has one 2s electron outside a filled inner shell. Its ground
state is 2 S t/:! . What are the term symbols of the other allowed states, if any?
Why would you think the a Si /2 state is the ground state?
*12. The magnesium atom has two 3s electrons outside filled inner shells. Find
the term symbol of its ground state,
* 13. The aluminum atom has two 3s electrons and one 3p electron outside filled
inner shells. Find the term symbol of its ground state.
*14. The magnetic moment fij of an atom in which IS coupling holds has the
magnitude
ixj = VJ(J + l)firfi fl
where n B = eft"/2m is the Bohr magneton and
, =] J(J + 1)  L(L + 1) + S(S + 1)
^ 2J(J + 1)
is the iMtule g factor, (a) Derive this result with the help of the law of cosines
starting from the fact that, averaged over time, only the components of p L and
u s parallel to J contribute to /z,. (b) Consider an atom that obeys IS coupling
that is in a weak magnetic field B in which the coupling is preserved. How
many substates are there for a given value of J? What is the energy difference
tic t ween different substates?
* 15. The ground state of chlorine is z l% ri . Find its magnetic moment (see previous
problem). Into how many substates will the ground state split in a weak magnetic
field?
THE ATOM
Show that, if the angle between the directions of L and S in Fig. 78 is 8,
CDS tl —
i(j +1)  1(1 + 1)  S (S +1)
2Vf(( + lWs + 1)
17. The spinorbit effect splits the 3P ► 3S transition in sodium (which gives
rise to the yellow light of sodiumvapor highway lamps) into two lines, 5,890 A
corresponding to 3? 3/2 — * 3S 1/Z and 5,896 A corresponding to 3P 1/2 — » 3S 1/2 .
Use these wavelengths to calculate the effective magnetic induction experienced
by the outer electron in the sodium atom as a result of its orbital motion.
18, Show that the frequency of the K„ Xray line of an clement of atomic
number Z is given by
Zct\{Z  if
" = 4
where R is the Rydberg constant. Assume that each L electron in an atom may
be regarded as the single electron in a hydrogenic atom whose effective nuclear
charge is reduced by the presence of whatever K electrons are present, [The
proportionality between * and (Z — I) 2 was used by Moseley in 1913 to establish
the atomic numbers of the elements from their Xray spectra. This propor
tionality is referred to as Moseley "s law.]
19. What element has a K a Xray line of wavelength 1.785 A? Of wavelength
0.712 A?
20, Explain why the Xray spectra of elements of nearby atomic numbers are
qualitatively very similar, while the optical spectra of these elements may differ
considerably.
MANY ELECTRON ATOMS 239
THE PHYSICS OF MOLECULES
8
What is ihe nature of the forces that bond atoms together to form molecules?
This question, of fundamental importance to the chemist, is hardly less important
to the physicist, whose theory of the atom cannot lie correct unless it provides
a satisfactory answer. The ability of the quantum theory of the atom not only
to explain chemical bonding but to do so partly in tonus of an effect thai has
no classical analog is further testimony to the power of this approach.
8.1 MOLECULAR FORMATION
A molecule is a stable arrangement of two or more atoms. By "stable" is meant
thai a molecule must he given energy from an outside source in order to break
up into its constituent atoms, in other words, a molecule exists because the
energy of the joint system is less than that of the system of separate noninteracting
atoms. If the interactions among a certain group of atoms reduce their total
energy, a molecule can be formed; if the interactions increase their total energy,
the atoms repel one another.
Let us coasider what happens when two atoms are brought closer and closer
together. Three extreme situations may occur:
1. A covalent Ixmd is formed. One or more pairs of electrons are shared by
the two atoms. As these electrons circulate lw;tween the atoms, they spend more
time lietween the atoms than elsewhere, which produces an attractive force.
An example is H 2 , the hydrogen molecule, whose electrons belong jointly to the
two protons (Fig. tilu).
2. An ionic Iwid is formed. One or more electrons from one atom may transfer
to the other, and the resulting positive and negative ions attract each other.
An example is NaCl, where the liond exists Iwtween Na* and CI ions and not
between Na and CI atoms (Fig. olfc).
3. No bond is formed. When the electron structures of two atoms overlap,
they constitute a single system, and according to the exclusion principle no two
243
electrons in such a system can exist in the same quantum state. If some of the
interacting electrons are thereby forced into higher energy states than they
occupied in the separate atoms, the system may have much more energy than
before and be unstable. To visualize this effect, we may regard the electrons
as fleeing as far away from one another as possible to avoid forming a single
system, which leads to a repulsive force between the nuclei, (Even when the
exclusion principle can lie olieyed with no increase in energy, there will be an
electrostatic repulsive force between the various electrons; this is a much less
significant factor than the exclusion principle in influencing bond formation,
however.)
Ionic tends usually do not result in the formation of molecules. A molecule
is an electrically neutral aggregate of atoms that is held together strongly enough
to be experimentally observable as a particle. Thus the individual units that
constitute gaseous hydrogen each coasist of two hydrogen atoms, and we are
entitled to regard them as molecules. On the other hand, the crystals of rock
salt (N'aCl) are aggregates of sodium and chlorine ions which, although invariably
arranged in a certain definite structure (Fig. 82), do not pair off into discrete
molecules consisting of one Na* ion and one CI ion; rock salt crystals may in
Fact be of almost any size. There are always equal numbers of Na + and CI "
ions in rock salt, so that the formula NaCI correctly represents its composition.
However, these ions form molecules rather than crystals only in the gaseous state.
FIGURE 81 (a) Covalent bonding. The shared electrons spend more time on the average between their
parent nuclei and therefore lead to an attractive force, (b) Ionic bonding. Sodium and chlorine combine
chemically by the transfer of electrons from sodium atoms to chlorine atoms; the resulting Ions attract
electrostatically.
+ 17 A \
. o o
(«)
cr... O H
Na"
I'M
m\
244 PROPERTIES OF MATTER
FIGURE 8 2 Scale model of NaCI crystal.
In Hj the bond is purely covalent and in NaCI it is purely ionic, hut in many
other molecules an intermediate type of bond occurs in which the atoms share
electrons to an unequal extent. An example is the MCI molecule, where the CI
atom attracts the shared electrons more strongly than the H atom. A strong
argument can be made for thinking of the ionic Iwnd as no more than an extreme
case of the covalent bond.
8.2 ELECTRON SHARING
The simplest possible molecular system is l\ 2 ' , the hydrogen molecular ion, in
which a single electron bonds two protons. Before we consider the bond in H.,~
in detail, let us look in a general way into how it is possible for two protons
to share an electron and why such sharing should lead to a lower total energy
and hence to a stable system.
In Chap. 5 we discussed the phenomenon of quantummechanical barrier
penetration: a particle can "leak" out of a box even though it does not have
enough energy to break through the wall because the particle's wave function
extends l>eyond it. Only if the wall is infinitely strong is the wave function wholly
inside the box. The electric held around a proton is in effect a Ijox for an electron,
and two nearby protons correspond to a pair of boxes with a wall between them
(Fig. 83). There is no mechanism in classical physics by which the electron
in a hydrogen atom can transfer spontaneously to a neighboring proton more
distant than its parent proton. In quantum physics, however, such a mechanism
does exist. There is a certain proI>ability that an electron trapped in one box
will tunnel through the wall and get into the other box, and once there it hie*
the same probability for tunneling back. This situation can be described bv saving
that the electron is shared by the protons.
To lie sure, the likelihood that an electron will pass through the region of
high potential energy— the "wall"— between two protons depends strongly upon
how far apart the protons are. If the pro ton proton distance is 1 A, the electron
THE PHYSICS OF MOLECULES
245
246
FIGURE 83 (i) Potential energy of an electron in the electric field o! two nearby
protoni. The lotal enafgy of a groundslate electron In the hydrogen atom Is indi
cated, (o) Two nearby protoni correspond quantummethanfeally to a pair of
boxes separated by a wall.
may lie regarded as going from one proton to the other about everv 10 !3 s,
which means that we can legitimately consider the electron as being shared by
both. If the protonproton distance is 10 A, however, the electron shifts across
an average of only about once per second, which is practically an infinite time
on an atomic scale. Since the effective radius of the Is wave function in hydrogen
is 0.53 A, we conclude that electron sharing can take place only between atoms
whose wave functions overlap appreciably.
Granting that two protons can share an electron, there is a simple argument
that shows why the energy of such a system could !>e less than that of a separate
hydrogen atom and proton. According to the uncertainty principle, the smaller
the region to which we restrict a particle, die greater must be its momentum
PROPERTIES OF MATTER
and hence kinetic energy. An electron shared by two protons is less confined
than one belonging to a single proton, which means that it has less kinetic energy.
The total energy' of the electron in 11./ is therefore less than that of the electron
in 11 4 II + , and provided the magnitude of the protonproton repulsion in lh'
is not too great, H 2 * ought to l)e stable.
The preceding arguments are quantummechanical ones, while we normally
tend to consider the interactions between charged particles in terms of elec
trostatic forces. There is a very important theorem, independently proved
by Feynman and by Hellmann and named after them, which states in essence
that both types of approach always yield identical results. According to the
FeynmanIIelhnann theorem, if the electron probability distribution in a mole
cule is known, the calculation of the system energy can proceed classically and
will lead to the same conclusions as a purely quantummechanical calculation.
The FcynmatiHelhnann theorem is not an obvious one, because treating a
molecule in terms of electrostatic forces does not explicitly take into account
electron kinetic energy, while a quantum treatment involves the total electron
energy; nevertheless, once die electron wave function i£ has lieen determined,
either way of proceeding may be used.
8.3 THE IV MOLECULAR ION
What we would like to know is the wave function $ of the electron in H 2 + ,
since from i£ we can calculate the energy of the system as a function of the
separation ft of the protons. If E(R) has a minimum, we will know thai a bond
can exist, and we can also determine the bond energy and the equilibrium spacing
of the protons.
Instead of solving Schrodinger's equation fort£, which is a lengthy and compli
cated procedure, we shall use an intuitive approach. Let as begin bv trying to
predict what i£ Ls when R, the distance between the protons, is large compared
with «„, the radiiLs of the smallest Bohr orbit in the hydrogen atom. In this event
4> near each proton must closely rasemble the l.v wave function of the hydrogen
atom, as pictured in Fig. 84 where the ls wave function around proton a is
called 4>„ and that around proton h is called tf 6 .
We also know what ^ looks like when li is 0, that is, when the protons are
imagined to l>e fused together. Here the situation is that of the lie 1 ion, since
the electron Ls now in the presence of a single nucleus whose charge is + 2e .
The U wave function of He* has ihe same form as that of II but with a greater
amplitude at the origin, as in Fig. 84e. Evidently ^ is going to be something
like the wave function sketched in Fig. 84rf when R is comparable with a ir
There is an enhanced likelihood of finding the electron in the region between
the protons, which we have spoken of in terms of sharing of the electron by
THE PHYSICS OF MOLECULES
247
n
"0
A*.
contours of
electron probability
(«)
+ '
\J/u
/&
(6)
* +
U— «.
(c)
1^1
* s
+•
• +
r*j
^,i
<rf)
A
I
R=0
• 2 +
(»)
FIGURE 84 The combination of two hydrogenatom 1, wave (unctions to
wave (unction ^
form the symmetric H 2 +
the protons. Thus there is on the average an excess of negative charge between
the protons, and this attracts them together. We have still to establish whether
the magnitude of this attraction is enough to overcome the mutual repulsion
of the protons.
The combination of ^ a and ^ 6 in Fig. 84 is symmetric, since exchanging a
and /; does not affect ^ (sec Sec. 7.3). However, it is also conceivable that we
could have an antisymmetric combination of t^ n and ^ 6 , as in Fig. 85, Here
there is a node between a and /; where t^ = 0, which implies a diminished
likelihood of finding the electron between the protons. Now there is on the
average a deficiency of negative charge between the protons, and in consequence
a repulsive force. With only repulsive forces acting, bonding cannot occur.
An interesting question concerns the behavior of the antisymmetric ll 2 * wave
function t^ H as R — ► 0. Obviously ty A does not become the Is wave function of
He + when R = 0. However, ij> A dm® approach the 2;j wave function of I ie^
(Fig. 85e), which has a node at the origin. Since the 2p state of He' is an excited
state while the Is state is the ground state, H a + in the antisymmetric state ought
to have more energy than when it is in the symmetric state, which agrees with
our inference from the shapes of the wave functions $ A and C K that in the former
case there is a repulsive force and in the latter an attractive one.
A line of reasoning similar to the preceding one enables us to estimate how
the total energy of the H;,* system varies with R. We first consider the syitnuct
rieal state. When R is large, the electron energy E s mast lie the — 13.6eV energy
of the hydrogen atom, while the electrostatic potential energy V of the protons.
8.1
V„ =
4™„R
falls to (J as R — ► oo. (V p is a positive quantity, corresponding to a repulsive
force.) When R = 0, the electron energy must equal that of the He + ion, which
is Z a or 4 limes that of the II atom. (See Prob. 25 of Chap. 4; the same
result is obtained from the quantum theory of oneelectron atoms.) Hence
E s = 54,4 eV when R = 0, Also, when R * 0, V,, » oo as l/R. Both E s
and Vj, are sketched in Fig. 86 as functions of /{; the shape of ihc curve for
E s can only be approximated without a detailed calculation, but we do have
its value for both R = t) and R = oo and, of course, V fi <»l«ys Eq. 8.1.
The total energy /.j,""" 1 of the system is the sum of the electron energy E 8
and the potential energy V p of the protons. Evidently E s iotai has a minimum,
which corresponds to a stable molecular state. This result is confirmed by the
experimental data on H 2 + which indicate a bond energy of 2.(55 eV and an
equilibrium separation R of 1,06 A. By "bond energy" is meant the energy
needed to break 11./ into H + H + ; the total energy of H 2 + is the 13.6 eV
of the hydrogen atom plus the — 2.65eV bond energy, or — 16.3 eV in all.
THE PHYSICS OF MOLECULES
249
iffn
contours of
electron probability
_^
(■)
4><\
b
— — *■
(6)
FIGURE 95 Tha combination at two hydrogenatom 1. wave functions to form tha antisymmetric H +
wave function * A . *
FIGURE 86 Electronic, proton repulsion, and total energy In H,' as a function of nuclear separation N
for the symmetric and antisymmetric states , The antisymmetric state has no minimum in its total en
ergy.
1 11 the case of the antisymmetric state, the analysis proceeds in the same way
except that the electron energy E A when R — is that of the 2p state of He + .
This energy is proportional lo Z/n 2 ; hence with Z = 2 and n = 2 it is just equal
to the — 13.6 eV of the groundstate hydrogen atom. Since E A — * — 13.6 eV
also as ft — * oo, we might think that the electron energy is constant, hut actually
there is a small dip at intermediate distances. However, the dip is not nearly
enough to yield a minimum in the total energy curve for the antisymmetric state,
as indicated in Fig. 86, and so in tins state no bond is formed.
THE PHYSICS OF MOLECULES
251
8,4 THE H, MOLECULE
The H 2 molecule contains two electrons instead of the single electron of 11.,".
According to the exclusion principle, both electrons can share the same orbital
{that is, lie described by the same wave function f ntmi ) provided their spins are
antiparallel. With two electrons to contribute to the bond, H a ought to be more
stable than 11./— at first glance, twice as stable, with a bond energy of 53 cV
compared with 2.65 eV for H»+ However, the H, orhitals arc not quite the
same as those of H, + because of the electrostatic repulsion Ix;tween the two
electrons in H,. a factor absent in the case of H 2 ' . The latter repulsion Wakens
the bond in H 2 , so that die actual bond energy is 4,5 eV instead of 5.3 cV.
For the same reason, the bond length in B, is 0.74 A, which is somewhat larger
than the use of unmodified H./ wave Functions would indicate. The general
conclusion in the case of 11/ that the symmetric wave function + s leads to a
hound state and the antisymmetric wave function £ to an unbound one remains
valid for H 2 ,
In See. 7.3 the exclusion principle was formulated in terms of the symmetry
and antisymmetry of wave functions, and it was concluded that systems of
electrons arc always described by antisymmetric wave functions (that is, by wave
functions that reverse sign upon the exchange of any pair of electrons). I lowever,
we have just said that the bound state in H, corresponds to Ixjth electrons being
descriliod by a symmetrical wave function ^ which seems to contradict the
above conclusion.
A closer look shows that there is really no contradiction here. The complete
wave Function +(1,2) of a system of two electrons is the product of a spatial
wave function #1,2) which describes the coordinates of the electrons and a spin
I unction 412} which describes the orientations of their spins. The exclusion
principle requires diat the complete wave function
+(1,2) = iHUHU)
be antisymmetric to an exchange of Iwth coordinates and spins, not tijl 2) by
itself, and what we have teen calling a molecular orbital is the same as # 2)
An antisymmetric complete wave function +„ can result from the combination
of a symmetric coordinate wave function + fl and an antisymmetric spin function
s A or from the combination of an antisymmetric coordinate wave function *
and a symmetric spin function fc That is, only
and
+ = ****
252 PROPERTIES OF MATTER
arc acceptable. If the spins of the two electrons are parallel, their spin function
is symmetric since it does not change sign when the electrons are exchanged.
Hence the coordinate wave function ^ for two electrons whose spins are parallel
must lie antisymmetric: we may express this by writing
*TT = f.
On the other hand, if the spins of the two electrons are antiparallel, their spin
function is antisymmetric since it reverses sign when the electrons are exchanged.
Hence the coordinate wave function 4 for two electrons whose spins are anti
parallel must be symmetric, and we may express this by writing
m = <h
Schrodinger's equation for the H z molecule has no exact solution. In fact, only
for H./ is an exact solution possible, and all other molecular systems must be
treated approximately. The results of a detailed analysis of the II Z molecule are
shown in Fig. 87 for the case when the electrons have their spins parallel and
the case when their spins arc antiparallel. The difference between the two on v es
is due to the exclusion principle, which prevents two electrons in the same
quantum state in a system from having the same spin and therefore leads to a
dominating repulsion when the spins are parallel.
FIGURE 87 The variation of the eneigy of the system H •+ H with their distances apart when
the electron spins are parallel and antiparallel.
H + II, spins antiparallel
6'
B 1
2 3
NUCLEAR SEPARATION H, A
THE PHYSICS OF MOLECULES
253
8.5 MOLECULAR ORBITALS
Covalcnt bonding in molecules other than H.,, diatomic as well as polyatomic,
is usually a more complicated story. It would be yet more complicated but for
the fact that any alteration in the electronic structure of an atom clue to ihc
proximity of another atom is confined to its outermost (or valence) electron shell.
There are two reasoas for this. First, the inner electrons are much more tightlv
bound and hence less responsive to external influences, partly because they are
closer to their parent nucleus and partly because they are shielded from the
nuclear charge by fewer intervening electrons. Second, the repulsive interatomic
forces in a molecule become predominant while the inner shells of its atoms
are still relatively far apart. Direct evidence in support of the idea that only
the valence electrons are involved in chemical bonding is available from the
Xray spectra that arise from transitions to innershell electron states; it is found
that these spectra are virtually independent of how the atoms are combined in
molecules or solids.
In discussing chemical bonding il is helpful to be able to visualize the dig
tributions in space of the various atomic orbitals. which qualitatively resemble
those of hydrogen. The pictures in Fig. 611 are limited to two dimensioas
and hence are not suitable for this purpose. It is more appropriate here to
draw Ixmndary surfaces of constant V 2 in each case that outline the regions
within which the total prol>ability of finding the electron has some definite value,
say 90 or 95 percent. Further, the sign of the wave function ^ can l>e indicated
in each lobe of such a drawing, even though what is being represented is W*.
Figure 88 contains boundarysurface diagrams for s, p, and d orbitals. These
diagrams show 0«j 2 in each case; for the corresponding radial probability
densities )R\ 1 , Fig. 610 can lie consulted. The total probability density tyf is,
of coarse, equal to the product of 0«l> a and fi 2 .
In a munber of cases the orbitals shown in Fig. 88 are derived from linear
combinations of two atomic wave functions representing states of the .same
energy; such combinations are also solutions of Schrodinger's equation. For
example, a p t orbital is the result of adding together the I = 1 wave function
for iw, = +1 and m, = — 1:
(The factor 1/ V§ is required to normalize ty p ,) Similarly the p„ orbital is given
by
orbital n I
in I
* 1,2,3
P s 2,3.4,.,. I ±1
P y 2,3,4,... 1 ±1
P« 2,3,4..,, 1 o
254
PROPERTIES OF MATTER
FIGURE 8 8 Boundary surface diagrams for ( p. and d atomic orbitals. The + and  signs refer to the
sl(n of the wave function In each region.
orbital n I „i,
dty .1,4,5, ... 2 i2
<*= 3.4A... 2 ±1
<k 3,4,5,... 2 ±1
d,i 3,4,5. .
d s %_^ 3,4,5,... 2 :£S
FIGURE 88 (Continued.)
The p. orbital, however, is identical with the I = I,m, = wave function. The
wave functions which are combined to form the (/„, d vt , d lv , and d,t_*« orbitais
are indicated in Fig. 88. The interactions between two atoms that yield a
covalent bond between them have, as a consequence, different probabilitydensity
distributions for the electrons that participate in the bond than those which are
characteristic of the atoms when alone in space, and these new distributions are
easiest to understand in terms of the orbitais shown in Fig. 88.
When two atoms come together, their orbitais overlap and the result will he
either an increased electron probability density between them that signifies a
Inmding molecular orbital or a decreased concentration that signifies an tmtt
Ixmding molecular orbital. In the previous section we saw how the Is orbitais
of two hydrogen atoms could combine to form either the bonding orbital i£ s
or the antibonding orbital $ A . In the terminology of molecular physics, $ s is
referred to as a Is o orbital and $ A as a lstr° orbital. The "Is" identifies the
atomic orbitais that are imagined to combine to form the molecular orbital.
The Creek letter o signifies that the molecular state has no angular momentum
about the bond axis (which is taken to be the ; axis). This component of the
angular momentum of a molecule is quantized, and is restricted to the values
\ti where X = 0, 1,2 Molecular states for which A = are denoted by u,
those for which A m 1 by w, those for which A = 2 by 6, and so on in alphabetical
order. Finally, an antibonding orbital is labeled with an asterisk, as in lso° for
the antibonding II, orbital y_ r
Figure 89 contains boundarysurface diagrams that show the formation of o
and 77 molecular orbitais from s and p atomic orbitais in homonuclear diatomic
molecules. Evidently a orbitais show rotational symmetry about the bond axis,
while ■n orbitais change sign upon a 180° rotation aliout the bond axis. Since
the lobes of p s orbitais arc on the Iwnd axis, these atomic orbitais form o
molecular orbitais. The p x and p v orbitais both form v molecular orbitais.
A heteronuclear diatomic molecule consists of two unlike atoms. In general,
the atomic orbitais are not the same in such molecules, so that the bonding elec
trons in them are not equally shared by both atoms. 1J1I is the simplest hetero
nuclear molecule and is a good example of this effect. The normal configuration
of the II atom is Is and that of the Li atom is l.v J 2\. which means in each case
that there is a single valence electron. The Is orbital of II and the 2s orbital
of l.i form a a 1 wilding orbital in Lill that is occupied by the two valence
electrons (Fig. 810). In both atoms the effective nuclear charge acting on a
valence electron is te (in Li the core of two Is electrons shields + 2p of the
total nuclear charge of + 3e), but in Li a valence electron is on the average
several times farther from the nucleus than in H. (The respective ionization
energies reflect this difference, with the ionization energy of H being 13.6 eV
while it is 5,4 eV in Li.) Hence the electrons in the a bonding orbital of LiH
THE PHYSICS OF MOLECULES
257
o
GO jO
bonding
;iiilil)ini(liim
*tr
:iiiiiliiin<liiib:
P,
P :
r>a
bond mil;
bonding
".
p.ir
.in! iliunit iiij!
''x p x p.* '
FIGURE 89 Boundary surface diagrams showing the formation of molecular orbitals from » and ,,
atomic orbitals in homonuclear diatomic molecules. The . aiis is along the internuelear axis of the i
cute in each case, and the plane of the paper Is the nj plane. The ;V r and ,i,r orhHals are the same as
the p,!: and p,v* orbital* eicept that they are rotated through 90'.
258 PROPERTIES OF MATTER
FIGURE 8 10 The bonding
electrons In Lift occupy a ■■•■
molecular orbital formed
from (he U orbital of the
H atom and the 2t orbital
of the LI atom.
Li H LiH
La
SJK7
favor the H nucleus, and there is a partial separation of charge in the
molecule.
If there were a complete separation of charge in LiH, as there is in NaCl,
the molecule would consist of an Li* ion and a H~ ion, and the hond would
be purely ionic. Instead the bond is only partially ionic, with the two bonding
electrons spending perhaps 80 percent of the time in the neighborhood of the
H nucleus and 20 percent in the neighlwrhood of the Li nucleus. In contrast,
the bonding electrons in a homonuclear molecule such as H 2 or O a spend .50
percent of the time near each nucleus. Molecules whose bonds are neither purely
covalent nor purely ionic are sometimes called polar covalent, since they possess
electric dipole moments. The relative tendency of an atom to attract shared
electrons when it is part of an atom is called its electronegativity. In the LiH
molecule, for instance, H is more electronegative than Li.
In a hctcroiuiclear molecule the atomic orbitals that are imagined to combine
to form a molecular orbital may be of different character in each atom. An
example is IIF, where the b atomic orbital of II joins with the 2p. orbital of
F, There are two possibilities, as in Fig. h II , a bonding spo molecular orbital
and an antibonding spn° orbital. Since the Is orbital of 11 and the 2p. orbital
of F each contain one electron (Table 8.1), the spa orbital in IIF is occupied
by two electrons, and we may regard HF as t>eing held together by a single
covalent bond. 'Hie electron structure of the HF molecule is shown in Fig. 812,
FIGURE 811 Bonding and antibonding molecular orbitals In HF.
F H HF
2P,
Q
U
II
*fw
HF
OO Q  OO
2P,
U
spa
THE PHYSICS OF MOLECULES 259
Table 8. 1 ,
ATOMIC STRUCTURES OF FIRST AND SECOND PED(00 ELEMENTS *„«.
tion, ol electrons. According to Hund's rule (See 76 .n.«. » 1 """ el " B ipln *«■
Element
Hydrogen 11
Helium He
Lithium l.i
Beryllium Iti
Boron B
Cordon C
Nitrogen \
Oxygen O
Fluorine F
Neon \,
number
Atomic
structure
Occupancy of orbital;
S
3
4
S
e
7
8
9
to
u
w
1 '2, : 2»; !
Ij2*'2rj' 1
T
Ti
n
n
u
u
u
u
n
u
r
Ti
U
n
n
u
u
ti
i
t
r
ti
u
T
T
t
U
n
T
I
?
Ti
to jom with . orlntals to form lading molecular orbitals make it poarfHe to
H*Ofe an example. Offhand we might expect a linear molecule, II OH s
oxygen is more electronegative than hydrogen and each H atom in I L accord!
2 y t ; « : sma ' positive charge  ■"« "»*** «*»**» be^LTH
2TS t° P I 11 '"" US far Rpart " P^ 1 " 6  —'>' «" "PPte sides rf
the O atom, In reality, however, the water molecule h„ . *E ckS to
O II will, an angle of 104.5° between the two OH bonds.
II
FIGURE 812 Valence atomic orbital* In
HF. the atomic orbitals shown as over
lapping form a a bonding molecular
orbrtal.
The l»ent shape of the water molecule is easy to explain. From Table 8.1 we
fine! that the 2p v and 2n. orbitals in O are only singly occupied, so that each
can join the U orbital of an II atom to form a spa landing orbital (Fig. 813),
The y and z axes are 90° apart, and the larger 104.5° angle that is actually found
may plausibly lit' attributed to the mutual repulsion of the II atoms. In support
of the latter idea is the fact that the bond angles in the otherwise similar
molecules II,S and H a Se are 92° and 90°, respectively, which we may ascribe
to the greater separation of the II atoms around the larger atoms S (atomic
Dumber Z = 16) and Se (Z = 34).
A similar argument explains the pyramidal shape of the ammonia molecule
NIL,. From Table 8,1 we find that the Zp f , 2p v , and 2p„ atomic orbitals in N
are singly occupied, which means that each of them can form a spa (winding
orbital with the Is orbital of an H atom. The bonding molecular orbitals in NIL,
should therefore be centered along the x, (/, and t axes (Fig. 814) with N— II
bonds 90° apart. As in I J 2 0. the actual bond angles in Nil, are somewhat greater
than 90°, in this case 107.5°, owing to repulsions among the H atoms. The similar
hydrides of the larger atoms P (Z = 15) and As (Z = 33) exhibit the smaller bond
angles of 94° and 90°, respectively, again in consequence of the reduced mutual
repulsions among the more distant H atoms.
8.6 HYBRID ORBITALS
The straightforward way in which the stapes of the IL.0 and NTL, molecules
are explained is a conspicuous failure in the case of methane, CH,. A carbon
atom has two electrons in its 2a orbital and one each in its 2p t and 2p u orbitals.
FIGURE 813 Valence atomic orbitals in
H 0. The bond angle is actually 104.5'.
260 PROPERTIES OF MATTER
THE PHYSICS OF MOLECULES
261
262
FIGURE 814 Valence atom!; orbilals In
NH ,. The bond angle* are actually
107.5*,
Thus we would expect the hydride of carton to he CI I,, with two spo bonding
orhitals and a hond angle of 90" or a little more. Yet CH, exists, and, furthermore,
it is perfectly symmetrical in structure with tetrahedral molecules whose C— H
bonds are exactly equivalent to one another.
We cannot consider carbon as an isolated exception, a sort of freak atom in
which a fortuitous combination of circumstances leads to CH, instead of CH,,
because the same phenomenon occurs in other atoms as well, A boron atom,
for example, has the configuration \.<t*2s*2p, and it forms BF, and BC1 3 instead
of BFand BC1.
Clearly, what is happening in carbon and toron is that the 2s orbitals, despite
toing occupied by electron pairs and having less energy— more stability —  than
the 2p orbilals, somehow enter into the formation of molecular orbitals and
thereby permit the 2s electrons to contribute to the bonds formed by C and
B with other atoms. A C atom has four n = 2 electrons in all and forms CH,,,
while a B atom has three n = 2 electrons in all and forms BF 3 , respectively
sharing four and three electrons with their partners.
The easiest way to explain the existence of CH,, is to assume that one of the
two 2* electrons in C is "promoted" to the vacant 2p. orbital, so that there is
now one electron each in the 2s, 2p r , %p v , and 2u r orbitals and four bonds can
be formed. To raise an electron from a 2s to a 2j state means increasing the
energy of the C atom, but it is reasonable to suppose that the formation of four
bonds (to yield CH 4 ) in place of two (to yield CH 2 ) lowers the energy of I he
resulting molecule more than enough to compensate for this. The foregoing
picture suggests that three of the bonds in CI I,, are spa bonds and one of them
is a s.w bond involving the Is orbital of II and now singly occupied 2s orbital
of Cj experimentally, however, all four bonds are found to be identical.
PROPERTIES OF MATTER
The correct explanation for CH 4 is based on a phenomenon called hybridization
which can occur when the iv and 2j> states of an atom in a molecule are close
together in energy. In this case the atom can contribute a linear combination
of Imth its 2s and 2l> atomic orbitals to each molecular orbital if in this way
the resulting bonds are more stable than otherwise. Thai such composite atomic
orbitals can occur follows from the nature of Schrodiiiger's equation, which i;>
a partial differential equation. The 2s and 2p wave functions of an atom are
Iwth solutions of the same equation if the corresponding energies are the same,
and a linear combination of solutions of a partial differential equation is always
itself a solution. In an isolated atom, an electron in a 2s orbital has less energy
(is more lightly bound) than an electron in a 2p orbital, and there is accordingly
no tendency for hybrid atomic orbitals to occur. On the other hand, when an
atom in a certain molecule contributes superposed s and p orbitals to the molec
ular orbitals, the resulting bonds may be stronger than the bonds that the s and
p orbitals by themselves would lead lo, even though the p parts of the hybrids
had higher energies in the isolated atom. Hybrid orbitals thus occur when the
bonding energy they give rise to is greater than that which pure orbilals would
produce, which happens in practice when the s and p levels of an atom are close
together.
In CH.,, then, carbon has four equivalent hybrid orbitals which participate
in bonding. These four orbitals are hybrids of one 2s and three 2p orbitals, and
we may consider each one as a combination of '/,s and %p. This particular
combination is therefore called a .v/) :l hybrid. Its configuration can be visualized
in terms of boundarysurface diagrams as shown in Fig. 815, Evidently a sp 3
hybrid orbital is strongly concentrated in a single direction, which accounts for
its ability to produce an exceptionally strong bond — strong enough to compensate
for the need to promote a 2* electron lo a 2p slate.
FIGURE 815 In tji 1 hybridization, an < orbital and three ,. orbitals In (he same atom combine to form
tour jjj 3 hybrid orbHais.
Q
«p
THE PHYSICS OF MOLECULES
263
It inusl l>e kept in mind that hybrid orbitals do not exist in an isolated atom,
even when it is in an excited state, but arise while that atom is interacting with
others to form a molecule.
Figure fi16 is a representation of the CB4 molecule. Also shown is a model
of this molecule that consists of a C atom in the center of a unit cube which
has B atoms at alternate comers. A triangle with the C atom at one vertex and
Bay tm II atoms at the other vertexes has sides V5/2, 1J3/2, and \/2 in length.
If the angle between the C— H bonds is 0, from the law of cosines {tr = & +
er — 2hccas0) we have
COS0 = 
a*
iV
c 2
2bc
2
%
%.
2X%
= 109.5°
which is what is determined experimentally.
The bond angles of 104.5° in H a O and 107.5° in NH 3 are evidently closer
to lire tetrahedral angle of 109.5° that occurs in sp 1 hybrid landing than to the
90° expected if only p orbitals in the O and N atoms are involved. This fact
provides a way to explain how the repulsions among the H atoms in these
molecules that were spoken of earlier can l>e incorporated into the molecular
orbital description of bonding. In NII 3 there arc three doubly occupied bonding
orbitals, leaving one unshared pair of electrons that we earlier supposed to be
FIGURE 816 The tetrahedraf methane (CHJ molecuto. The overlapping ,,' hybrid orbitals of the C
atom and 1* orbitals of the lout H atoms form bonding molecular orbital*.
FIGURE 817 Valence atomic orbitals in th* am
monia (NH .) molecule based on the assumption of
v/j 1 hybridization in the N orbitals. One of the sp J
orbitals is occupied by two N electrons and does
not contribute to bonding.
264 PROPERTIES OF MATTER
in the 2s orbital of N. If there were spP hybrid orbitals furnished by H instead
of p orbitals, the more separated bonds would mean a lower energy for the
system. Opposing this gain in molecular stability is the need to promote the
pair of nonbonding 2$ electrons to the higherenergy sp 3 hybrid state without
any contribution by them to the bonding process (unlike the case of Cll, where
all four sp 3 orbitals participate in bonds). Hence we ma)' regard the 107.5° bond
angle in NH 3 as (he result of a compromise lietween the two extremes of four
.vp 3 hybrid orbitals tn N with one of them nonbondin» and three 2p bonding
orbitals and one 2s nonbonding (but lowenergy) orbital. Figure 817 is a repre
sentation of die NIL, molecule on the basis of up 3 hybridization, which may lie
compared with Fig. 814 which was drawn on the basis of p orbitals.
In H./), since there are two Ixmding orbitals and two nonbonding orbitals
in O, the tendency to form hybrid sp 3 orbitals is less than in NH a where there
are three bonding orbitals and only one nonbonding orbital. The smaller bond
angle in H 2 is in agreement with this conclusion. The structure of the H a O
molecule is further discussed in Sec. 10.4.
8.7 CARBONCARBON BONDS
Two other types of hybrid orbital in addition to .sp 3 can occur in carl>on atoms.
In sp 2 hybridization, one valence electron is in a pure p orbital and the other
three are in hybrid orbitals that are %v and %p in character. In .vp hybridization,
two valence electrons are in pure p orbitals and the other two are in hybrid
orbitals that are l / 2 .v and %p in character.
Fthyleue, CgH*, is an example of sp 2 hybridization in which the two carbon
atoms are joined by two bonds. Figure 818 contains a boundary surface diagram
THE PHYSICS OF MOLECULES
265
266
(»)
m
(O
FIGURE 818 (a) The ethylene (C ; H.) molecule. All the atoms Ha In a plan* perpendicular to
tho plane of the paper. (6) Top view, showing the 5J > ; hybrid orbital* (hat form n bonds be
tween the C atoms and between each C atom and two H atoms, (c) Side view, showing th*
pure i>, orbitals that form a  bond between the C atoms.
showing ihe three sp' 2 hybrid orbitals, which are 120° apart in the plane of the
paper, and the pure p T orbital in each C atom. Two of the y/j orbitals in each
C atom overlap s orbitals in II atoms to form o bonding orbitals, and the third
s}} 2 orbital in each C atom forms a o bonding orbital with the same orbital in
the other C atom. The p t orbitals of the C atoms form a w bond with each
other, so that one of the bonds between the carbon atoms is a a bond and the
other is a it bond. The conventional structural formula of ethylene is accordingly
PROPERTIES OF MATTER
>^ :
Ethylene
Acetylene, C 2 U.„ is an example of sp hybridization in which the two carbon
atoms are joined by three bonds. One sp hybrid orbital in each C atom forms
a o bond with an H atom, and the second forms a o bond with the other atom.
The 2p I and 2j)„ orbitals in each C atom form v bonds, so that one of the three
Itonds i>etween the carbon atoms is a o. bond and the others are tt, and x^ bonds
{Fig. 8 19). The conventional struetnral formula of acetylene is
=C— H
Acetylene
In both ethylene and acetylene the clcctroas in the w orbitals are "exposed"
on the ontsidc of the molecules. These compounds arc much more reactive
chemically than molecules with only single <r bonds between carbon atoms, such
as ethane,
II II
H— C— C— H
Ethane
[
H II
in which all the l»onds are formed from ip 8 hybrid orbitals in th. carbon atoms.
Carbon compounds with double and triple bonds are said to be unsaturated
because they can add other atoms to their molecules in such reactions as
>°
\
H
II 11
+ HCI
H
H— CCll
H CI
FIGURE 819 The acetylene (C.H..) molecule. There are three bonds between the C
between ijj hybrid orbitals and two ■ bonds between pure p, and p, orbitals.
one i bond
THE PHYSICS OF MOLECULES 267
0>
<©
w
9
H £
'\Q Oh H
^ — x
x
w
HGURE B20 The benzene molecule. (a) The overlaps between the .,,,• hybrid orbitals in .he C
atoms with each other and with the . orbitals of the H atoms lead to „ bonds, <b) Each C atom
has a pure f orbital occupied by one electron. (e> The bonding  molecular orbitals formed by
the sl« ,,, atomic orbitals constitute a continuous electron probability distribution around the
molecule that contains si* detocalized electrons.
In a saturated compound such as methane or ethane only single bonds are present,
1,1 l "'" / "'= <  ( ,;Hv ii"' sw I atoms an arranged In ■ f ;1 i hexagonal ring.
Becau.se the carboncarbon bonds in the benzene ring are I20« apart, we conclude
that die basic structure of the molecule is ihe result ei bonding by v/r' hybrid
orbitals. Of the three sp» orhitab per C atom, one forms a a bonding orbital
with the I.v orbital oj aa H atom and the other two form a bondim orbilals
with the corresponding ap* orbitals of the C atoms on either side (Fig 820)
Hits leaves one 2p, orbital per C atom, which has lobes above and below the
plane of the ring. The total of six 2», orbitals i„ the molecule combine to produce
bonding * orbitals which take the form of a continuous electron prohabihlv
distnhi.tm,. .move and below the plane of the ring. The six electrons belong
to the molecule as a whole and not to any particular pair ofatoms: these electrons
are debvafizetl.
268 PROPERTIES OF MATTER
8.8 ROTATIONAL ENERGY LEVELS
Molecular energy stales arise from the rotation of a molecule as a whole and
from the vibrations of its constituent atoms relative to one another as well as
from changes in its electronic configuration. Rotational states are separated by
quite small energy intervals ( 10 3 eV is typical), and the spectra that arise from
transitions between these states are in the microwave region with wavelengths
of 0.1 mm to 1 cm. Vibrational slates are separated by somewhat larger energy
intervals (0.1 eV is typical), and vibrational spectra are in the infrared region
with wavelengths of 10,000 A to 0.1 mm. Molecular electronic states have higher
energies, with typical separations between the energy levels of valence electrons
of several electron volts and spectra in the visible and ultraviolet regions. A
detailed picture of a particular molecule can often be obtained from its spectra,
including bond lengths, force constants, and Iwind angles. For simplicity the
treatment here will l>e restricted to diatomic molecules, bill tin: main ideas apply
(u more complicated ones as well.
The lowest energy levels of a diatomic molecule arise from rotation about
its center of mass. We may picture such a molecule as consisting of atoms of
masses in, and in., a distance H apart, as in Fig. S 21 . The moment of inertia
of this molecule alxiut an axis passing through its center of mass and perpen
dicular lo a line joining the atoms is
8.2
/ = n^r, 2 + "*j*j a
where r, and t 2 are the distances of atoms 1 and 2 respectively from the center
of mass. Since
8.3 m x r { = in.,r.,
by definition, the moment of inertia may be written
( m v »u \
\«ii + m.,1
8.4
= llt'tt 5
center
of moss
FIGURE 8 2] A diatomic molecule can ro
tate about Its center of mass.
THE PHYSICS OF MOLECULES
269
270
where
S.5
in = —
Ht] + m.
Reduced mass
is the reduced mass of the molecule as mentioned in Sec, 4.9, Equation HA
states that the rotation of a diatomic molecule is equivalent to the rotation of
a single particle of mass m' about an axis located a distance of R away.
The angular momentum L of the molecule has the magnitude
S.G
L = !u
where to is its angular velocity. Angular momentum is always quantized in nature,
as we know. If we denote the rotational quantum number by /, we have here
8,7 /. = y/jtj + 1) ft / = 0, 1, 2,3, . . .
The energy of a rotating molecule is %/w 2 , and so its energy levels are specified
by
Ej = y 2 tu s
' 21
J(J + I)* 2
8.S
11
Rotational energy levels
Let us see what sorts of energies and angular velocities are involved in molec
ular rotation. The carbon monoxide (CO) molecule has a bond length R of 1.13 A
and the masses of the 12 C and lfl O atoms are respectively 1.99 X 10" 2S kg and
2.66 X 10 26 kg. The reduced mass m' of the CO molecule is therefore
X 10" kg
Hi, + m„
_ 1.99 X 2.66
" 1.99 + 2.66
= 1.11 X 10 M kg
and its moment of inertia / is
/ = m'R 2
= 1.14 X I0» kg X (1.13 X 10'°m) 2
= 1.46 X 10 » kgm 2
The lowest rotational energy level corresponds to J = 1 , and for this level in
CO
PROPERTIES OF MATTER
^,=
JU+ Qft
2f
ft*
T
_ (1.054 X lO" 3 ' Js) 2
~ 1.46 X 10 4 «kgm 2
= 7.61 X 10" 23 J
= 5.07 X 10 * eV
This is not a great deal of energy, and at room temperature, when AT C 2,6 x
10" 2 eV, nearly all the molecules in a sample of CO are in excited rotational
states. The angular velocity of the CO molecule when / = 1 is
2E
I
2X7.61 X IP' 23 J
1.46 X I0 ,o kgm 2
= 3.23 X 10" rad/s
Thus far we have been considering only rotation alxntt an axis perpendicular
to the ads of symmetry of a diatomic molecule, as in Fig, H21 — eudoverend
rotations. What about rotations about the axis of symmetry itself? The reason
the latter can be neglected is that the mass of an atom is almost entirely concen
trated in its nucleus, whose radius is only — 10 "' of the radius of the atom itself.
The principal contribution to the moment of inertia of a diatomic molecule about
its symmetry axis theretore comes from its electrons, which are concentrated
in a region whose radius about the axis is roughly half the bond length R but
whose total mass is only about % / Ajaoa of the total molecular mass. Since the
allowed rotational energy levels are proportional to 1 '/. rotation a! unit the
symmetry a\is must involve energies — M)' 1 times the Ej values for endoverend
rotations. Hence energies of at least several eV would be involved in any rotation
about the symmetry axis of a diatomic molecule. Since bond energies are of
this order of magnitude too, the molecule would be likely to dissociate in any
environment in which such a rotation could be excited.
Rotational spectra arise from transit ions between rotational energy stales. Onlv
molecules that have electric dipole moments can absorb or emit electromagnetic
photons in such transitions, which means that nonpolar diatomic molecules such
as H, and symmetric polyatomic molecules such as OOj (0=C=0) and CI 1 ,
(Fig. 816) do not exhibit rotational spectra, [Trausi lions between rotational states
in molecules like H a , CO,, and CH, can take place during collisions, however.)
Furthermore, even in molecules that possess permanent dipole moments, not all
transitions between rotational states involve radiation. As in the case of atomic
THE PHYSICS OF MOLECULES
271
spectra (Sec. 6.10), certain selection rules summarize the conditions for a radiative
transition between rotational stales to tx* possible. For a rigid diatomic molecule
the selection rule for rotational transitions is
s.9
&J=±l
In practice, rotational spectra are always obtained in absorption, so that each
transition that is found involves a change from some initial state of quantum
number / in the next higher state of tjuantum number 7+1. In the case of a
rigid molecule, the frequency of die photon absorbed is
''jj+l —
8.10
IE
h
*Wi
%
h
=
n
U+ 1)
Rotational spectra
272
where I is the moment of inertia for endoverend rotations. The spectrum of
a rigid molecule therefore consists of equally spaced lines, as in Fig. H22, The
frequency of each line can be measured, and the transition it corresponds to
can often Ix: ascertained from the sequence of lines; from these data the moment
of inertia of the molecule can be readily calculated, (Alternatively, the fre
quencies of any two successive lines may be used to determine I if the spec
trometer used does not record the lowest frequency lines in a particular, spectral
sequence.) In CO, for instance, the / = » / = 1 absorption line occurs at a
frequency of 1.153 X lit" 11/. Hence
'r„ = ^(/+ I)
2ffi>
1.054 X HI" 3 ' Js
~ 2ir X 1.153 X 10" s'
 1 .46 X 10" 10 kgm
Since the reduced mass of the CO molecule is 1.14 X l»"" 2,i kg, the bead length
fi co is \/7/m' = 1.13 A. This is the way in which the bond length for CO quoted
earlier in this section was determined
8.9 VIBRATIONAL ENERGY LEVELS
When soilicicntly excited, a molecule can vibrate as well as rotate. As licfnre.
we shall only consider diatomic molecules. Figure 823 shows how the potential
energy of a molecule varies with the mternuclear distance R. In the neighlwrhood
PROPERTIES OF MATTER
FIGURE 822 Energy levels and spectrum of
molecular rotation.
%"
rotational
energy
levels
1 = 4
/ = 3
J = 2
l = \
1 =
I I I I
rotational
spectrum
of the minimum of this curve, which corresponds to the normal configuration
of the molecule, the shape of the curve is very nearly a parabola. In this region,
then,
8.ii V = V + V 2 k{R  R a f
where R { , is the equilibrium separation of the atoms. The interatomic force that
gives rise to this potential energy may be found by differentiating V:
*~~lR
8.12 = k(R  R )
The force is just the restoring force that a stretched or compressed spring
exerts— a Hooke's law force— and, as with a spring, a molecule suitably excited
can undergo simple harmonic oscillations.
THE PHYSICS OF MOLECULES 273
parabolic approximation
FIGURE 823 The potential energy of a diatomic molecule si a function of intern uclear distance.
Classically, the frequency of a vibrating body of mass m connected to a spring
of force constant k is
1
8.13
: k
III
What we have in the case of a diatomic molecule is the somewhat different
situation of two bodies of masses m, and in., joined by a spring, as in Fig. 8,24.
In the absence of external forces the linear momentum of the system remains
constant, and the oscillations of the bodies therefore cannot affect the mot ion
of their center of mass. For this reason m, and nt, vibrate hack and forth relative
to their center of mass in opposite directions, and both reach the extremes of
I heir respective motions at the same times. The frequency of oscillation of such
a twobody oscillator is given by Eq. S.I. '3 with the reduced mass in' of Eq. 8.5
force constant k
"'o^MimmmKT 1
FIGURE 824 A Iwobotfy oscillator.
274 PROPERTIES OF MATTER
substituted for m:
8.14
2s
in
Two body oscillator
When the harmonicoscillator problem is solved quantum mechanically, as was
done in Chap. 5, the energy of the oscillator is found to lie restricted to the
values
8. is /.'„ = ((.•■+ V 2 ) /»<„
where u, the vibrational quantum number, may have the values
f = (>, 1 , 2, 3, . . .
The lowest vibrational state (e = 0) has the finite energy ^''"o" ,mt tne classical
value of 0; as discussed in ("hap. 5, this result is in accord with the uncertainty
principle, because if the oscillating particle were stationary, the uncertainty in
its position would l>c it = and its momentum uncertainty would then have
to be infinite— and a particle with E = cannot have an infinitely uncertain
momentum. In view of Eq. 8.14 the vibrational energy levels of a diatomic
molecule are specified by
E v = (v + V 2 )HJ—
m
Vibrational energy levels
1 et us calculate the frequency of vibration of the CO molecule and the spacing
between its vibrational energy levels. The force constant k of the bond in CO
is 187 N/m (which is 1(1 lb/in. — not an exceptional figure for an ordinary spring)
and, as we found in Sec. 8.8, the reduced mass of the CO molecule is m' =
1.14 X 10 2a kg. The frequency of vibration is therefore
2*
m'
2v V 1.
187 N/m
.14 X lO^kg
= 2.04 X K> 13 Hz
The separation AE lietween the vibrational energy levels in CO is
M: m /■;,,,,  E F = hv n
= 6M X 10 M Js X 2.04 X 10 ,:, s '
= 8.44 X IlH eV
which is considerably more than the spacing between its rotational energy levels.
Because AE > kf for vibrational states in a sample at room temperature, Most
THE PHYSICS OF MOLECULES
275
of the molecules in such a sample exist in the = state with (inly their
zeropoint energies. This situation is very different from that diaraeterislie of
rotational states, where the much smaller energies mean (hat the majority of
the molecules in a roomtemperature sample are excited to higher states.
The higher vibrational stales of a molecule do not obey Kc. 8.15 l>ceause the
parabolic approximation to its potentialenergy curve becomes less and less valid
with increasing energy. As a result, the spacing between adjacent energy levels
of high B is less than the spacing Iwtween adjacent levels of low c, which is
shown in Fig. .S25. This diagram also shows the finer structure in the vibrational
levels caused by the simultaneous excitation of rotational levels.
The selection rule for transitions between vibrational states is At = ±1 in
the harmonic oscillator approximation. This rule is easy to understand. An
oscillating dipole whose frequency isi' y can only absorb or emit electromagnetic
radiation of the same frequency, and all quanta of frequency *„ have the energy
FIGURE 825 The potential energy of a diatomic molecule as a function of interatomic distance, show
ing vibrational and rotational energy levels.
vibrational energy levels
rotational energy levels
276 PROPERTIES OF MATTER
/w> . The oscillating dipole accordingly can only alvsorb AE = hi\, at a time, in
which case its energy increases from {t + l / 2 )rW u to [v + % + l)ht> , and it can
only emit AE = hi> u at a time, in which case its energy decreases from (u + y»)hv„
to (c + Y 2 — l)hv . Hence the selection rule At) = ±1,
Pure vibrational spectra arc observed only in liquids where interactions
between adjacent molecules inhibit rotation. Because the excitation energies
involved in molecular rotation are considerably smaller than those involved in
vibration, the freely moving molecules in a gas or vapor nearly always are
rotating, regardless of their vibrational state. The spectra of such molecules do
not show isolated lines corresponding to each vibrational transition, but instead
a large number of closely spaced lines due to transitions between the various
rotational states of one vibrational level and the rotational states of the other.
In spectra obtained using a spectrometer with inadequate resolution, the lines
appear as a broad streak called a vibrationrotation band.
To a first approximation, the vibrations and rotations of a molecule take place
independently of each other, and we can also ignore the effects of centrifugal
distortion and anharnionicity. Under these circumstances the energy levels of
a diatomic molecule arc specified by
S.17
+ /(/+l»J
Figure 8.26 shows the / = 0, 1, 2, 3, and 4 levels of a diatomic molecule for
the e = and o = I vibrational states, together with the spectra! lines in
absorption that are consistent with the selection rules Ac = + 1 and A/ = ±1.
lite D = — • r;  1 transitions fall into two categories, the P branch in which
A/ =  I (that is, / > }  1 ) and the R branch in which A/ = + 1 (/ » J + I).
From Eq, 8.17 the frequencies of the spectral lines in each branch are given
by
''p = h u i  f(i,j
ft
■Iwl
8. IB
and
= ^/5 + W 11 '*' +
ft
*^>fcr /= 1.2,3..
— ^IJt I '"11.,/
I' branch
8.19
= i'o + (/ + !)■
2<nl
J = 0, 1, 2,
H branch
THE PHYSICS OF MOLECULES 277
1
I
1
., J
1
1
1
1
J
&} = l
AJ = + I
P branch
K I ir :m cl>
;=4
J = 3
J = 2
J=l
v = l
1 = 4
/ = 3
J = 2
J = l
} =
>v = [
FIGURE 8 26 The rotational structure of the ■  r 1 transitions In • diatomic molecule.
There is no Una at r a >„{1he t,) branch} because of the selection rule li = ^L
278 PROPERTIES OF MATTER
There is no line at v = i>„ because transitions for which SJ = f) are forbidden
in diatomic molecules. The spacing between the lines in both the P and the
R branch is &» = fi/2ir/; hence the moment of inertia of a molecule can be
ascertained from its infrared vibrationrotation spectrum as well as from its
microwave pure rotation spectrum. Figure 827 shows the v = — > t = 1
vibrationrotation absorption band in CO.
A molecule that consists of many atoms may have a large number of different
normal modes of vibration. Some of these modes involve the entire molecule,
but others involve only groups of atoms whose vibrations occur more or less
independently of the rest of the molecule. Thus the —OH group has a charac
teristic vibrational frequency of 1.1 X 10 1 ' 1 Hz and the — NH Z group has a
frequency of 1 .0 X 10 M Hz. The characteristic vibrational frequency of a
carlx>nearl>on group depends upon the number of bonds between the C atoms:
the — £ — C— group vibrates at about 3.3 X 10 13 Hz, the U=C^ group
vibrates at about 5.0 X I0 13 Hz, and the — C^C — group vibrates at about
6.7 X lO 13 Hz (Figs. 828 and 829). {As we would expect, the greater the
number of carbon carbon bonds, the larger the value of the force constant k
and the higher the frequency.) In each case the frequency docs not depend on
the particular molecule or the location in the molecule of the group. This in
dependence makes vibrational spectra a valuable tool in determining molecular
FIGURE 827 The r = — c = > vibratkjnrotalion absorption band fn CO undw high resolution. The
lines are identified by the value of 1 in the Initial rotational state.
6.1 6.2
6.3 6.4 6.5
FREQUENCY
6.6
6.7 x 10" Hz
THE PHYSICS OF MOLECULES 279
E*
' *
"10.1915 eV
E*
0.4527 eV
D°
.4658 «fV
symmetric bending symmetric stretching asymmetric stretching
FIGURE S28 The normal mode* of vibration of the HX molecule and the energy levels ot each mode.
structures. An example is thioaeetie acid, whose structure might conceivably
be either CH 3 CO— SIX or CH 3 CS — OH. The infrared absorption spectrum of
thioacetic acid contains lines at frequencies equal to the vibrational frequencies
of the jC=0 and — SH groups, but no lines corresponding to the /C=S
or — OH groups, so the former alternative is the correct one.
FIGURE 8 29 The norma! modes of vibration of the CO. molecule and the energy levels of each mode.
The symmetric bending mode can occur In two perpendicular planes.
:> 0.827 cV
.0.1649 eV
11.2912 eV
o£e>— r^0« O
T
c
o
symmetric bending
symmetric stretching
o c o
O* *~o — O*
asymmetric stretching
8.10 ELECTRONIC SPECTRA OF MOLECULES
The energies of rotation and vibration in a molecule are clue to the motion of
its atomic nuclei, since the nuclei contain essentially all of the molecule's mass.
The molecular electrons also can be excited to higher energy levels than those
corresponding to I In ground state of the molecule, (hough the spacing of these
levels is much greater than the spacing of rotational or vibrational levels.
Electronic transitions involve radiation in the visible or ultraviolet parts of the
spec t nun, with each transition appearing as a scries or closely spaced lines, called
a band, due to the presence of different rotational and vibrational states in each
electronic state (see Fig, 412), All molecules exhibit electronic spectra, since
a dipole moment change always accompanies a change in the electronic con
figuration of a molecule. Therefore hoi noun clear molecules, such as 1 1, and N 2 ,
which have neither rotational nor vibrational spectra becawBB they lack perma
nent dipole moments, nevertheless have electronic spectra which possess rota
lional and vibrational line structures that permit their moments of inertia and
Ixind force constants to l>e ascertained.
Electronic excitation in a polyatomic molecule often leads to a change in its
shape, which can be determined from the rotational fine structure in its band
spectrum. The origin of such changes lies in the different characters of the wave
I'uiiclioiis ol electrons in different states, which lead in UBTespandfogjIy different
types of Ixmd. For example, a possible electronic transition in a molecule whose
hinds involve sp hybrid orbitals is to a higherenergy state in which the Ixmds
involve pure p orbitals. From the sketches earlier in this chapter we can see
that, in a molecule such as Bell 2 , the Ixmd angle in the ease of sp hybridization
is 180° and the molecule is linear (II — Be — H), while the Ixmd angle in the case
of pure p orbitals is 90° and the molecule is bent (H — Be).
11
There arc various ways in which a molecule in an excited electronic state can
lose energy and return to its ground state. The molecule may, of course, simply
emit a phi>ton of the same frequency as that of the photon it absorl>eei. thereby
returning to tile ground slate in a single step. Another possibility is fluorescence;
the molecule may give up some of its vibrational energy in collisions with oilier
molecules, so that the downward radiative transition originates from a lower
vibrational level in the upper electronic State (Fig. 830). Fluorescent radiation
is therefore of lower frequency than that of the absorbed radiation.
In molecular spectra, as in atomic spectra, radiative transitions !>elween
electronic states of different total spin are prohibited (see Sec. 711). Figure 83 1
280 PROPERTIES OF MATTER
THE PHYSICS OF MOLECULES
281
C3
vibrational transition
excited state
ground state
282
REPRESENTATIVE COORDINATE
FIGURE 830 The origin of fluorescence,
shows a situation in which the molecule in its singlet (S = 0) ground state absorbs
a photon and is elevated to a singlet excited state. In collisions the molecule
can undergo radiationless tratisitions to a lower vibrational level that may happen
to have about the same energy as one of the levels in the triplet {S — 1 ) excited
state, and there is then a certain probability for a shift to the triplet state to
occur. Further collisions in the triplet state bring the molecule's energy below
that of the crossover point, so that it is now trapped in the triplet state and
PROPERTIES OF MATTER
ultimately reaches the v = level. A radiative transition from a triplet to a
Mnglet state is "forbidden" by the selection rules, which realty means not that
it is impossible to occur but that it has only a minute likelihood of doing so.
.Such transitions accordingly have very long half lives, and the resulting phos
phorescmt radiation may l>e emitted minutes or even hours after the initial
absorption.
FIGURE 831 The origin ol phosphorescence. The final transition Is delayed because ft violates the s*.
lection rules lor electronic transitions.
singlet excited state
vibrational transition
triplet excited
state
1^^ — forbidden transition
singlet ground state
REPRESENTATIVE COORDINATE
THE PHYSICS OF MOLECULES 283
Problems
1. At what temperature would the average kinetic energy of the molecule*
in a hydrogen sample be equal to their binding energy?
2. Although the molecule He 2 is unstable and docs not occur, the molecular
ion He, 1 is stable and has a bond energy about equal to that of H, + . Explain
this observation.
3. Which would you expect to have the highest ixmd energy, F,, ¥./, or K, ?
The lowest bond
energy:"
4. The ionization energy of II, is l.>.7 eV and that of H is 13.6. Why arc they
so different?
5. The/ = — c y = 1 rotational absorption line occurs at 1,153 X 10" Hz in
I2 C 1B and al 1.1(12 X 16° Hz in ? C ia O. Find the mass number of the unknown
carbon isotope.
6. Calculate the energies of the four lowest rotational energy states of the H 2
and D 2 molecules, where I) represents the deuterium atom fll.
7. The rotational spectrum of HCI contaias the following wavelengths:
12.(13 X 10 5 m
9.60 X I0 S m
8.04 X HH m
6.89 x 10~ n m
6.04 x 10 s m
If the isotopes involved are }H and ffCt, find the distance between the hvdrogen
and chlorine nuclei in an HCI molecule. (The mass of :i3 Cl is 5.81 X W~ M kg.
8. Calculate the classical frequency of rotation of a rigid body whose energy
is given by Eq. 8.8 for states of } = / and / = / + 1, and show that the frequency
of the spectral line associated with a transition between these states is interme
diate between the rotational frequencies of the states.
9. A ""'Hg^'CI molecule emits a 4.4cm photon when it undergoes a rotational
transition from / = I to / = 0. Find the interatomic distance in this molecule.
(The masses of " ,H, Hg and as Cl are, respectively, 3,32 X 10 25 kg and 5.81 x
10 ^kg.'
10. Assume that the H 2 molecule behaves exactly like a harmonic oscillator
with a force constant of 573 N'/ni and find (he vibrational quantum number
corresponding to its 4.5eV dissociation energy.
284 PROPERTIES OF MATTER
11. The bond between the hydrogen and chlorine atoms in a 'H 3S CI molecule
has a force constant of 516 N/m. Is it likely that a HCI molecule will be vibrating
in its first excited vibrational state at room temperature?
1 2. The hydrogen isotope deuterium has an atomic mass approximately twice
that of ordinary hydrogen. Does H 2 or HD have the greater zeropoint energy?
How does this affect the binding energies of the two molecules?
13. The force constant of the 'H 1!, F molecule is 966 N/m, Find the frequency
of vibration of the molecule.
14. The observed molar specific heat of hydrogen gas at constant volume is
plotted in Fig. 832 versus absolute temperature. (The temperature scale is
logarithmic.) Since each degree of freedom (that is, each mode of energy posses
sion) in a gas molecule contributes — 1 kcal/kmol K to the specific heal of the
gas, this curve is interpreted as indicating thai only translalional motion, with
three degrees of freedom, is possible for hydrogen molecules at very low temper
atures. At higher temperatures the specific heat rises to ~5 kcal/kmol K,
indicating that two more degrees of freedom are available, and at still higher
temperatures the specific heat is —7 kcal/kmol K, indicating two further degrees
of freedom. The additional pairs of degrees of freedom represent, respectively,
rotation, which can take place about two independent axes perpendicular to the
axis of symmetry of the U 2 molecule, and vibration, in which the two degrees
of freedom correspond to the kinetic and potential modes of energy possession
by the molecule, (a) Verify this interpretation of Fig. 832 by calculating the
temperatures at which kT is equal to the minimum rotational energy and to the
FIGURE 832 Molar specific heat of hydrogen at constant volume.
c„
,1
7
6
4 *
I 3
2
1 
100
200
500 1000
TEMPERATURE, K
2000
5000
*T
THE PHYSICS OF MOLECULES 285
minimum vibrational energy a H„ molecule can have. Assume that the force
constant of the bond in H.. is 573 N/m and that the H atoms are 7.42 X 10~" m
apart. (At these temperatures, approximately half the molecules are rotating or
vibrating, respectively, though in each case some are in higher states than J = 1
or v = I.) (b) To justify coasidering only two degrees of rotational freedom in
the H 2 molecule, calculate the temperature at which AT is equal to the minimum
rotational energy a H 2 molecule can have for rotation al>out its axis of symmetry,
(c) How many rotations does a li 2 molecule with } = 1 and t = 1 make per
vibration?
286
PROPERTIES OF MATTER
STATISTICAL MECHANICS
9
The branch of physics known as statistical mechanics attempts to relate the
macroscopic properties of an assembly of particles to the microscopic properties
of the particles themselves. Statistical mechanics, as its name implies, is DOt
concerned with the actual motions or interactions of individual particles, but
investigates instead their most probable behavior. While statistical mechanics
cannot help us determine the life history of a particular particle, it is able to
inform us of the likelihood that a particle (exactly which one we cannot know
in advance) has a certain position and momentum at a certain instant. Because
so many phenomena in the physical world involve assemblies of particles, the
value oJ a Statistical rather than deterministic approach is clear. Owing to the
generality of its arguments, statistical mechanics can be applied with equal
facility to classical problems (such as that of molecules in a gas) and qiiantum
incchanical problems (such as those of free electrons in a metal or photons in
a lx>x), and it is one of the most powerful tools of the theoretical physicist
9.1 STATISTICAL DISTRIBUTION LAWS
We shall use statistical mechanics to determine the most probable way in which
a fixed total amount of energy is distributed among the various members of an
assembly of identical particles; that is, how many particles are likely to have
the energy e v how many to have the energy t 2 , and so on. The particles are
assumed to interact with one another (or with the walls of their container) to
an extent sufficient to establish thermal equilibrium in the assembly but not
sufficient to result in any correlation between the motions of individual particles.
We shall consider assemblies of three kinds of particles:
1. Identical particles of any spin that are sufficiently widely separated to be
distinguished. The molecules of a gas are particles of this kind, and the Max
u eQBohmatm distribution law holds for them.
287
2. Identical particles of or integral spin that cannot be distinguished one
from another. Such particles do not obey the exclusion principle, and the
BaseEinstein distribution law holds for them. Photons are Rose particles, or
bosom; and we shall use the Bosc Einstein distribution law to explain the spec
trum of radiation from a black body,
3. Identical particles of spin )' 2 that cannot be distinguished one from another.
Such particles obey the exclusion principle, and Ihe FermiDime distribution low
holds for them. Electrons are Fermi particles, or fennions, and we shall use the
FenniDirae distribution law to explain the behavior of the free electrons in a
metal.
9.2 PHASE SPACE
The state of a system of particles is completely specified classically at a particular
instant if the position and momentum of each of its constituent particles are
known. Since position and momentum are vectors with three components apiece,
we must know six quantities,
x, tj, z, p r . Pl) , &
for each particle.
The position of a particle is a point having the coordinates x, y, z in ordinary
threedimensional space. It is convenient to generalize this conception bv
imagining a sixdimensional space in which a point has the six coordinates x.
tj, z, p z , p v , p s . This combined position and momentum space is called phase
space. The notion of phase space is introduced to enable us to develop statistical
mechanics in a geometrical framework, thereby permitting a simpler and niore
straightforward method of analysis than an equivalent one wholly abstract in
character. A point in phase space corresponds to a particular position and
momentum, while a point in ordinary space corresponds to a particular position
only. Thus every particle is completely specified by a point in phase space, and
the state of a system of particles corresponds to a certain distribution of points
in phase space.
'Hie uncertainty principle compels us to elaborate what we mean by a "point"
in phase space. Let us divide phase space into tiny sLxditncnsional cells whose
sides are dx, dy, dz, dp x , dp y , dp z . As we reduce the size of die colls, we approach
more and more closely to die limit of a point in phase space. However, the
volume of each of these cells is
t = dx dy dz dp x dp y dp,.
and, according to the uncertainty principle,
dx dp t > ft
dy dp u > ft
cfo d Pl > ft
I lenee we see that
t > ft 1
A "point" in phase space is actually a cell whose minimum volume is of the
order of ft 3 . We must think of a particle in phase space as being located some
where in such a cell centered at some location x, y, z, p x , p tf , p ? instead of being
precisely at the point itself.
A more detailed analysis shows that each cell in phase space actually has the
volume lr\ which does not contradict the uncertaintyprinciple argument since
h 3 > ft 3 . In general, each cell in a phase space consisting of k coordinates and
fc momenta occupies a volume of h k . It is the task of statistical mechanics to
determine the state of a system by investigating how the particles constituting
the system distribute themselves among the cells in phase space.
While the notion of a point of infinitesimal size in phase space can have no
physical significance, since it violates the uncertainty principle, the notion of
a point of infinitesimal size in either position space or momentum space alone
is perfectly acceptable: we can in principle determine the position of a particle
with as much precision as we like merely by accepting an unlimited uncertainty
in our knowledge of its momentum, and vice versa.
*9.3 MAXWELLBOITZMANN DISTRIBUTION
l^t us consider an assembly of ;V molecules whose energies are limited to
e., 6o, .... tj, ... . These energies may represent either discrete quantum stales
or average energies within a sequence of energy intervals, and more than one
cell in phase space may conespond to a given energy. What we would like
to know is the most probable distribution of molecules among the various possible
energies.
A fundamental premise of statistical mechanics is that the greater the number
W of different ways in which die molecules can be arranged among the cells
in phase space to yield a particular distribution of molecules among the different
energy levels, the more probable is the distribution. The most probable distri
bution is therefore the one for which W is a maximum. Our first step, then,
is to find a general expression for W. We assume dial each cell in phase space
is equally likely to l>e occupied; this assumption is plausible, but the ultimate
justification for it (as in the case of Schrodinger's equation) is that the conclusions
arrived at with its help agree with experimental results.
288
PROPERTIES OF MATTER
STATISTICAL MECHANICS
289
290
If there are g ( cells with the energy f„ the number of ways in which one
molecule can have the energy t, is g,. The total number of ways in which two
molecules can have the energy e, is g, 2 , and the total mimlicr of ways that n,
umlcc nlis can have the energy e { is (&)"'■ Hence the number of ways in which
all \ molecules ran In distributed among the various micrgies is [Ik product
of factors of the fonn (g,)"', namely,
M {gxTAfkTAgs)" 3 ■ • ■
subject to the condition thai
9.2
in. = n. + Ha + n, +
= A'
Equation 9.1 does not equal W, however, since we must take into account
the possible permutations of the molecules among the different energy levels.
The total number of permutations possible for A' molecules is >V!; in other words,
.V molecules can he arranged in A'! different sequences. As an example, we might
have four molecules, a, b, c, and d. Hie value of 4! is
4! = 4 X 3 X 2 x 1 = 24
and there are indeed 24 ways of arranging them:
abctt
hurt!
cnbd
dahc
abdc
bade
cadb
dacb
acbd
bead
chad
dhac
tin lb
beda
rbdii
dlmi
adbc
bdac
cdtib
dcab
udch
l/tlrtt
alba
del hi
When more than one molecule is in an energy level, however, permuting than
among themselves has no significance in this situation. For instance, if molecules
a, b, and c happen to be in level /, it does not matter here whether we enumerate
them as abc, acb, bat, bat; cab, or cbtr, these six distributions are equivalent,
since all we care about is the fact that n } = 3. Thus the n, molecules in the
ith level contribute n,! irrelevant permutations. If there are ri L molecules in level
1, n, molecules in level 2, and so on, there are n,!n 2 !n 3 ! . . . irrelevant permuta
tions. What we want is the total number of possible permutations A r ! divided
by the total number of irrelevant ones, or
9.3
V!
«,».,!
The total number of ways in which the .V molecules can Ik distributed among
the possible energy levels is the product of Eqs. 9.1 and 9.3:
PROPERTIES OF MATTER
9.4
w =
A'!
n^btgln,! .
(^"'(feNgy)"*
What we now mast do is determine just which distribution of the molecules
is most probable, that is, which distribution yields the largest value of W. Our
first step is to obtain a suitable analytic approximation for the factorial of a large
number. We note that, since
ti! = n(n  l)(n  2) . . . (4) (3) (2)
the natural logarithm of n! is
Inn! = In 2 + In 3 + In 4 + ■■■ + ln{n  1) + tan
Figure 91 is a plot of In n versus n. The area under the stepped curve is In n!
When n is very large, the stepped curve and the smooth curve of In n become
indistinguishable, and we can find Inn! by merely integrating In n from n — I
to n = n:
In ni = I In n dn
= n In r» — n + 1
Because we are assuming that « > 1, we may neglect the 1 in the above result,
and so we obtain
9.5
In n! = n In r* — n n > 1
Stirling's formula
Equation 9.5 is known as Stirling's formula.
The natural logarithm of Eq. 9.4 is
In W = In A'!  X In n ( ! + X n, In g,
FIGURE 9J The area under
the stepped curve is In pit
Whan ■ ts very large, the
smooth curve Is a good approx
imation of the stepped curve,
and In n! can be found by inte
grating In n from n = 1 to
a = n.
STATISTICAL MECHANICS
291
292
Stirling's formula enables us to write this expression as
In W = JVln N  N  2 n, In n, + 2 n, + 2 n, ln 6
Since Sn,= .V,
96 liiH'=iVlnW^n l ln.i l + 2n, Ing,
While we have an equation for In W rather than for W itself, this Is no handicap
since
(In W) m „ = In W .
\ 'mai "■ "max
The condition for a distribution to be the mast probable one is that small
changes dn, in any of the B,*s not affect the value of W. (If the n/s were
continuous variables instead of being restricted to integral values, we could
express this condition in the usual way as BW/Sn, = 0.) If the change in In W
corresponding to a change in n, of 6n ( is S In W, from Eq. 9.6 we see that
97 8 In W mla =  v n ,S In n f  2 In n,&i, + 2 In gjSn, =
since ,V In N is constant. Now
and so
8 In n, = — 8n,
2 n t 8 In n ( = 2 Sn,
Because tlie total mimlwr of molecules is coastant, die sum 2 5Hj of all the
changes in the number of molecules in each energy level must lie (1, which means
that
2 nfi In n, =
Hence Eq. 9.7 liecomes
9.8 2 In RfSn, + 2 In g,6*n, =
While Eq. 9.8 must be fulfilled by the most probable distribution of the
molecules among the energy levels, it does not by itself completely specify this
distribution. We must also take into account the conservation of particles
(9.2) 2n, = ri[ + n 2 + n 3 + ... = N
and the conservation of energy
9.9 2n, ej = iijfj + n 2 t 2 + n,E, + •■=£
where E is the total energy of the assembly of molecules. In consequence the
PROPERTIES OF MATTER
variations Sn,, Sn.,, ... in the number of molecules in each energy level are not
independent of one another but must obey the relationships
9.10
9.11
2 Sn, = Sn, + «n 2 + Sn 3 + ■ • ■ =
2 rjfin, = tiSrif + e 2 8n, 2 + £ 3 fin 3 + ■ ■  =
To incorporate the above conditions on the various 8n t into Eq, 9.8 we make
use of ^grange's method of undetermined multipliers, which is simply a conve
nient mathematical device. What we do is multiply Eq. 9.10 by « and
Eq. 9.11 by — fi, where a and (i are quantities independent of the n/s, and add
these expressions to Eq. 9.8. We obtain
9.12
2(ln n, + lng j  a  /SfJSn, = 8
In each of the separate equations added together to give Eq. 9.12, the variation
fin, is effectively an independent variable. In order for Eq. 9.12 to hold, then,
the quantity in parentheses must lie for each value of t. Hence
— In n, + In g, — o — (it t =
from which we obtain the MaxwellBolt/.mann distribution law:
9.13
"i = & e ~° e
fit,
MaxwellBoltzmann
distribution law
This formula gives the number of molecules n, that have the energy e, in terms
of the number of cells in phase space g s that have the energy e ( and the constants
a and p. We must now evaluate &, «, and /S.
*9.4 EVALUATION OF CONSTANTS
Energy quantization is inconspicuous in the Iranslational motion of the molecules
in a gas, and the total number of molecules in a sample is usually very large.
1 1 is therefore more convenient to consider a continuous distribution of molecular
energies rather than the discrete set e,, <? z , c 3 If n{()tk is the number of
molecules whose energies lie between t and e + dr, Eq. 9. 13 becomes
9.14 n(t) de = g(e)e~"e~&' de
In terms of molecular momentum, since
f =
2>u
we have
9.13 n{p) dp = g(p)e •**"** dp
STATISTICAL MECHANICS 293
294
The quantity g(p) is equal to the number of cells in phase space in which
a molecule has a momentum between p and u + dp. Since each cell has the
volume ft 1 ,
gp) dp = ffSS)'<lxdydzdp z <lp ll dp l
where the numerator is the phasespace volume occupied by particles with the
specified momenta. Here
fffdxdydz= V
where V is the volume occupied by the gas in ordinary position space, and
ff tl P! d l\<'P: = 4*p 2 <Ip
where 4wp 2 dp is the volume of a spherical shell of radius p and thickness dp
in momentum space. Hence
9,16
and
9.17
^dp^^pl
4wV»V D e'" ,:/2m
n tP) d P = j£ < { P
We are now able to find e~". Since
f n {p)dp = N
We find by integrating teq, 9,17 that
47re°V r"
4r»«u «
N = — — J p *ea9*/2m Jj,
It" *n
where we have made use of the definite integral
A 4 V n J
I lence
and
9.18
n(p) dp = 4ttA' /^V ,/ ' ! p2«AP , /2. dp
PROPERTIES OF MATTER
To find ft, we compute the total energy E of the assembly of molecules. Since
m ffr
p 2 = 2mi and dp —
we can write Eq. 9.18 in the form
2AyJ 3/a
/2lHF
9.19
»1{e) ffr =
Ve^ e  '*' df
The total energy is
E = f ?n(f)df
2 f" e^V**
9.30
3 JV
2 /J
where we have made use of the definite integral
3 fn
I
x 3/2 e ** dx =
4a a V a
According to the kinetic theory of gases, the total energy E of A* molecules ul
an ideal gas (which is what we have been considering) at the absolute temperature
7/ is
9.21
E = £ NkT
2
where J; is Bollzmann's constant
Jt = 1.380 X 10" 23 J/moleculedegree
Equations 9.20 and 9.21 agree if
9.22
li = w
9.5 MOLECULAR ENERGIES IN AN IDEAL GAS
Now that the parameters « and ft have been evaluated, we can write the
lioll/.mann distribution law in its final form,
9.23
n(t) df =
2wA'
■VeV''""*
Boltzmann distribution
of energies
STATISTICAL MECHANICS
295
This equation gives the number of molecules with energies between e anil t + tie
in a sample of an ideal gas that contains a total of N molecules and whose absolute
temperature is T. The Boltzmann energy distribution is plotted in Tig. 92 in
terms of kT. The curve is not symmetrica! because the lower limit to r'is e =
while there is, in principle, no upper limit (although the likelihood of energies
many times greater than kT is small).
According to Eq. 9.20, the total energy £ of an assembly of V molecules is
e 3 JV
£= 2/f
The average energy t per molecule Ls E/N, so that
2ft
9.24
.{*
Average molecular energy
At 300 K, which is approximately room temperature,
e = 6.21 X 10" Z1 J/molecule
= 54s eV/inolecule
FIGURE 92 MaiwellBottimann enargy distribution.
296 PROPERTIES OF MATTER
This average energy is the same for all molecules at .100 K, regardless of their
mass. The Boltzmann distributions of molecular momenta and speeds can be
obtained From Ec. 9.23 by noting dial
P l 2
2m 2
tie = —dp = me dv
m
We find that
9.25 n{p) dp = ^jfi 2 e pl/2nif dp
Boltzmann distribution
of momenta
is the number of molecules having momenta between p and p + dp, and
\/2wjYm' ,/a ., __,,.„.„ , Boltzmann distribution
9.26
n(v)tlr =
■v^e mr ' l '' 2kT tk
of speeds
[vkTff*
is the number of molecules having speeds between p and v + dv. The last
Formula, which was first obtained by Maxwell in IMS9, is plotted in Fig. ft3.
The speed of a molecule with the average energy of %AT is
3*f
in
9. 27
= yjv* =
Rms speed
since %mv = %kT. This speed is denoted r nils lweause it is the square root
of the average of the squared molecular speeds— the rtnitiiieanxquare speed —
and is not the same as the simple arithmetical average speed F. The relationship
between r and v ral> depends upon the distributinn law that governs the molecular
speeds being considered. For a Bolt/.tnann distributinn,
rms \i si
m> fad the rms speed is about percent greater than the arithmetical average
speed.
Because the Boll/inann distribution oF speeds Ls not symmetrical, the most
probable speed v p is smaller than cidier F or v Tta3 . To find c p , \vc set equal to
zero the derivative of n(c) with respect to o and solve the resulting equation
for v. We obtain
9.28
■ ! 2kT
Most probable speed
Molecular speeds in a gas vary considerably on either side of D p , Figure 94
shows the distribution of molecular speeds in oxygen at 73 K (200°C), in oxygen
STATISTICAL MECHANICS
297
s_
298
k » *
o = rootmeansquare speed = \f3kTfm
L « = average speed = ifikT/wm
Op = most probable speed = ~\j2kTfm
FIGURE 93 Maxwell Boltimonn velocity distribution
at 273 K (0°C). and in hydrogen at 273 K. The most probable molecular speed
increases with temperature and decreases with molecular mass. Accordingly
molecular speeds in oxygen at 73 K are on the whole less than at 273 K, and
at 273 K molecular speeds in hydrogen are on the whole greater than in oxygen
at the same temperature. (The average molecular energy is the same in lx>lh
oxygen and hydrogen at 273 K, of course.)
9.6 ROTATIONAL SPECTRA
A continuous distribution of energies occurs only in the translational motions
of molecules. As we saw in Chap. 8, molecular rotations and vibrations are
quantized, with only certain specific energies £, l>eing possible. The Bolt/.mann
distribution law for modes of energy possession of the latter sort may !>e written
9.29 n ( = n„ g, eV* r
PROPERTIES OF MATTER
800 1.200 1.600
MOLECULAR SPEED, m/s
FIGURE 94 The distribution* of molecular speeds In oxygen st 73 K. In oxygen at 273 K. and In
hydrogen at 273 K.
which is simply Eq. 9.13 with % replacing e  " and E^kT replacing 0w ( . The
factor e~ K,/kT , often called the liollzmonn factor, expresses the relative proba
bility that a quantum state of energy E t be occupied at the temperature T. The
factor g,, the multiplicity (or statistical weight) of the level, is the number of
quantum states that have the same energy £,.
Let us apply Eq, 9.29 to the rotational energy levels of a molecule. (The
relative populations of atomic energy levels can be treated in the same way.)
As we know, more than one rotational state may correspond to a particular
rotational quantum niimlier /. The degeneracy arises because the component
L, in any specified direction of the angular momentum L may have any value
in multiples of ti from }h through to —Jli, for a total of 2/ + 1 possible values.
That is, there are 2/ + I possible orientations of L relative to the specified (z)
direction, with each of these orientations constituting a separate quantum state.
I fence an energy level whose rotational quantum number is / has a statistical
weight of
gr = 2/+l
For a rigid diatomic molecule,
4 = /(/ + %
STATISTICAL MECHANICS
299
and so the Boltzmann factor corresponding to the quantum numlicr J is
e JU+ U» V2rtr
The Boltzmann distribution formula for the probabilities of occupant')' of the
rotational energy levels of a rigid diatomic molecule is therefore
9.30 Uj = (2/ + I) HpflHW+liM/MW
Here the quantity » is the number of molecules in the / = rotational state.
In Sec. tt.8 we found that the moment of inertia of the CO molecule is
1.46 x H) 10 kgm 2 . For a sample of carbon monoxide gas at room temperature
(293 K, which is 20° C)
(1.054 X 10" s,i
. ,■
and so
fr 2
2f*T ~ 2 X 1.46 X 10"« kgm* X 1.38 X 10" 23 J/K X 293 K
= 0.00941
>ij = (21+ J)„ 0<r o.<w!MiJtf+i>
Figure 95 contains graphs of the statistical weight 2J + I, the Boltzmann factor
e o.omm ju* B md lhc reUuivc p0p(ll . uj01 tlj/H() for CQ a( 2{r(; a , n fmMam
off. The/ = 7 rotational energy level is evidently the most highly populated,
and about as many molecules in a sample of CO at room temperature are in
the /= 29 level as are in the / = () level.
The intensities of the rotational lines in a molecular spectrum are proportional
to the relative populations of the various rotational energy levels. Figure 827
shows the vibrationrotation band of CO for the o = n> = 1 vibrational
transition under high resolution; lines are identified according to the / value of
the initial rotational level. The P and ft branches both have their maxima at
/ = 7, as expected.
*§.7 BOSEEINSTEIN DISTRIBUTION
The basic distinction between MaxwellBoltzmann statistics and Bosc Einstein
statistics is that the former governs identical particles which can be distinguished
from one another in some way, while the latter governs identical particles which
cannot be distinguished, though they can be counted. In BoseKimlein statistics,
as before, all quantum states are assumed to have equal probabilities of occu
pancy, so that g, represents the numlier of states that have the same energy e ( .
Kuch quantum state corresponds to a cell in phase spaa*, and our first step is
to determine the number of ways in which n, indistinguishable particles can lie
distributed in g f cells.
300 PROPERTIES OF MATTER
£ L0
4 6 8 10 12 14 16
ROTATIONAL QUANTUM NUMBER, /
W
18 20
6 8 10 12 14 16
ROTATIONAL QUANTUM NUMBER,/
(h)
20
4 6 8 10 12 14 16
ROTATIONAL QUANTUM NUMBER, /
(c)
18 20
FIGURE 95 The multiplicities (a), Boltzmann factors (bl. and relative population* (c) of the
rotational energy levels of the CO molecule at 20' C.
To carry out the required enumeration, we consider a series of n, + g,  I
objects placed in a line (Fig. 96). We note thai ft  1 of the objects can l>e
regarded as partitions separating a total of g, intervals, with the entire series
therefore representing n t particles arranged in g, cells. In the picture g f = 12
and n, = 20; J 1 partitions separate the 20 particles into 12 colls. The first cell
contains two particles, the second none, the third one particle, the fourth three
particles, and so on. There are (it, + g,  1)1 possible permutations among
". + ft  1 objects, but of these the n,! permutations of the n, particles among
themselves and the (g,  1)! permutations of theft  I partitions among them
selves do not affect the distribution and are irrelevant. Hence there are
(n, + ft  1)!
"t«g. " W
possible distinguishably different arrangements of the n, indistinguishable parti
cles among the ft cells.
The number of ways W in which the W particles can be distributed is the
product
3.31
W= II
("i + ft  I)'
»l(ft *" 1)'
of the numbers of distinct arrangements of particles among the states having
each energy. We now assume that
(«i + ft) > I
so that (it, + ft  1) can be replaced by (n, + ft), and take the natural logarithm
of both sides of Eq, 9.31 to give
In W = 2 [In (n, + & )! _ ra „,! _ fa ( & _ i)i]
FIGURE 96 A series of n, Indistinguishable particles separated by tt  1 partitions Into g, cells.
I
fc •
2 1
2 1
• particle
partition
number of indistinguishable particles — » ( . = 20
number of partitions = g^ — 1 == 11
number of cells = g t = 12
Stirling's formula
In nl = n In n — r»
permits us to rewrite In W as
9.32 In W = 2 I(n, + ft) \n(n { +&)«, In n,  In (ft  1)!  ft]
As before, the condition that this distribution be the most probable one is that
small changes Sn, in any of the individual n f 's not affect the value of W. If a
change in In YV of 8 In W occurs when ii, changes by on,, the above condition
may be written
Hence, if the Wof Eq. 9.32 represents a maximum,
9.33 ft In W nm = V [In (n, + ft)  In nj Sn t =
where we have made use of the fact that
S In n =  Sn
n
As in Sec. 9.3 we incorporate the conservation of particles, expressed in the
form
2 Sn, =
and the conservation of energy, expressed in the form
S e, S», =
by multiplying the former equation by —a and the latter by — /? and adding
to Eq. 9.33. The result is
2 [hi (n, + ft)  In n,  a  fc,] Sn, =
Since the 8n,'s are indepetident. the quantity in brackets must vanish for each
value of /. Hence
In "' + fe  a  fr, = ()
1 + ii = <?«$•
and
9.34
s*"  1
302
PROPERTIES OF MATTER
STATISTICAL MECHANICS
303
Substituting for fi from Kq. 9.22,
(9.22»
""IF
we arrive Hi the liaseKifintein tlistrifwtimi law:
ft
935
n. =
e n e" nT — 1
BoseEinstein distribution taw
9.8 BLACKBODY RADIATION
Every substance emits elect roinagnette radiation, the character «r which depends
upon the nature and temperature of the substanee. We have already discussed
the discrete spectra of excited gases which arise from electronic transitions within
isolated atoms At the other extreme, dense lx>dics such as solids radiate cent inn
ous spectra in whieh all frequencies are present; the atoms in a solid are so close
together that their mutual interactions result in ■ multitude of adjacent quantum
states indistinguishable from a continuous band of pennilted energies.
The ability of a Ixjdy to radiate is closely related to its ability to absorb
radiation. This is to be expected, since a body at a constant temperature is in
thermal equilibrium with its surroundings and must absorb energy from llieni
at the same rate as it emits energy. It b convenient to consider as an ideal body
one that absorbs all radiation incident upon it, regardless of frequency. Such
a body is called a black htuhj.
It is easy to show experimentally that a black body is a twlter emitter of
radiation than anything else. The experiment, illustrated in Fig 97, involves
two identical pairs of dissimilar surfaces. No temperature difference is observed
between surfaces 1' and I!'. At a given temperature the surfaces 1 and 1' radiate
at the rate of g, W/ni 2 , while II and IF radiate at tin different rate r z . The
surfaces I and F absorb some fraction a, or the radiation falling on them, while
II and IF absorb some other fraction «... Hence V absorbs energy from II at
a rate proportion] to rt,e a , and IF absorbs energy from L at a rate proportional
to 0,6,. Because F and 11' remain at the same temperature, it must l>e true that
and
a, «,,
The ability of a Ixxly to emit radiation is proportions] to its ability to absorb
radiation. I>et us suppose that I and F are black lx>dies, so that a, = I, while
A
n
FIGURE 97 Surfaces I and I' are
identical to each other and are dif
ferent from the identical pair of sur
faces 1! and II'.
/
■v
<d
II and II' arc not, so that <%, < 1. Hence
e, = —
and, since a., < 1, e, > e%. A black body at a given temperature radiates energy
al a faster rate than any other Ijody.
The point of introducing the idealized black lx>dy in a discussion of thermal
radiation is that we can now disregard the precise nature of whatever is radiating,
since all black ixxiics behave identically. In the lalwratory a black body can
1ms approximated by a hollow object with a very small hole leading to its Interior
(Fig. 98). Any radiation striking the hole enters the cavity, where it is trapped
FIGURE 9 8 A hole In
the wan of a hollow
object is an excellent
approximation of a black
body.
304
PROPERTIES OF MATTER
STATISTICAL MECHANICS
305
by reflection Ixick and forth until it is absorbed. The cavity walls are constantly
emitting and absorbing radiation, and it is in the properties of this radiation
(blackbody mdittliott) that we are interested. Experimentally we can sample
blackbody radiation simply by inspecting what emerges from the hole. The
results agree with our everyday experience; a black Ixxly radiates move when
it is hot than when it is cold, and the spectrum of a hot black body has its peak
at a higher frequency than the peak in the spectrum of a cooler one. We recall
the familiar behavior of an iron bar as it is heated to progressively higher
temperatures: at first it glows dull red, then bright orangered, and eventually
lxjenmes "white hot." The spectrum of blackbody radiation is shown in Fig.
99 for two temperatures.
The principles of classical physics are unable to account for the observed
blackbody spectrum. In fact, it was this particular failure of classical phvsics
that led Max 1'Ianck in 1900 to suggest that light emission is a quantum phe
nomenon. We shall use quantumstatistical mechanics to derive the Planck radia
tion formula, which predicts the same spectrum as that found by experiment.
T=lSWt
FIGURE 9 9 Black. body spec
tra. The spectral distribution of
energy In the radiation depends
only upon the temperature at the
body.
2 x 10 M 4 x 10 M 6 x 10" Hz
visible light
Our theoretical model of a black Ixxly will lx* the same as the laboratory
version, namely, a cavity in some opaque material. This cavity has some volume
V, and it contains a large nmnlx*r of indistinguishable photons of various fre
quencies. Photons do not obey the exclusion principle, and so they are Bose
particles that follow the BoseEinstein distribution law. The number of states
g(p) in which a photon can have a momentum between p and p + dp is equal
to twice the number of cells in phase space within which such a photon may
exist. The reason for the possible double occupancy of each cell is that photons
of the same frequency can have two different directions of polarization (circularly
clockwise and circularly counterclockwise). Hence, using the argument that led
to Eq. 9.16,
g(p) dp = ^
Since the momentum of a photon is p = hv/c,
p 2 dp = '"
and
... 8wV
9.36 g(t') dv = — t— f 4 (/f
c J
We must now evaluate the Iigrangian multiplier a in Eq, 9.35, To do this,
we note that the numlier of photons in the cavity need not be conserved. Unlike
gas molecules or electrons, photons ma\ lie ere.ifci! .mil destrmei!. ;iitil si. while
the total radiant energy within the cavity must remain constant, the number
of photons that incorporate this energy can change. For instance, two photons
of energy hi' can be emitted simultaneously with the absorption of a single photon
of energy 2hi>. Hence
2 on, /
which we can express by letting « = since it multiplies 2 fin, as 0.
Substituting Eq. 9.36 for g, and hi> for t t , and letting o = in the BoseEinstein
distribution law (Eq. 9.35), we find that the number of photons with frequencies
between v and p> + dp in the radiation within a cavity of volume V whose walls
are at the absolute temperature T is
9.37
n(i') di' =
&VV pa fo
 1
FREQUENCY, v
The corresponding spectral energy density viv) dv, which is the energy per unit
volume in radiation between v aixl v + dv in frequency, is given by
306 PROPERTIES OF MATTER
STATISTICAL MECHANICS
307
308
f(c) dv =
/ic»j(c) dv
9.38
fcrft p dv
c 3 e hr/kT _ 
Planck radiation formula
Equation 9.38 is the Planck radiation fannida, which agrees with experiment.
Two interesting results can be obtained from the Planck radiation formula.
In find the wavelength whose energy density is greatest, we express Kq. 9.3fS
in terms of wavelength and set
dX
= ()
and then solve for A = \ max , We obtain
he
= ■1.965
which is more conveniently expressed as
v T he
n„„l —
9.39
496Sfc
= £wS98 X 10 3 n>K
Equation 9.39 is known as Wien's displacement tan:. It quantitatively expresses
the empirical fact that the peak in the blackbody spectrum shifts to progressively
shorter wavelengths (higher frequencies! as the temperature is increased.
Another result we can obtain from Eq. 9.3S is the total energy density i w i thin
the cavity. This is the integral of the: cnerg) tieusih over all frequencies,
t ■ = I f(i») dv
_ far 8 ** *
' I5c 3 /r<
= al*
where n is a universal constant. The total energy density is proportional to the
fourth power of the absolute temperature of the cavity walk We therefore
expect that the energy a radiated by a black body per second per unit area is
also proportional to T\ a conclusion embodied in the StefunHultzniann late:
9.40
e=oT'
The value of Stefan's constant a is
a = 5.67 X W * VV/m 2 K'
PROPERTIES OF MATTER
Both Wien's displacement law and the StefanBoll/.mann law are evident in
qualitative fashion in Fig. 99; the maxima in the various curves shift to higher
frequencies and the total areas underneath them increase rapidly with rising
temperature.
"9.9 FERMIDIRAC DISTRIBUTION
FermiDirac statistics apply to indistinguishable particles which are governed
by the exclusion principle. Our derivation of the FenniDirae distribution law
will therefore parallel that of the Base Einstein distribution law except that now
each cell (that is, quantum state) can lie occupied by at most one particle.
If there are g, cells having the same energy r, and n, particles, ii, cells are
filled and (g, — n,) are vacant. The & cells can be rearranged in g t i different
ways, but the n ( ! permutations of the filled cells among themselves are irrelevant
since the particles are indistinguishable and the (g, — n,)! permutations of the
vacant cells among themselves are irrelevant since the cells are not occupied.
The number of distinguishable arrangements of the particles among the cells
is therefore
ft!
9.41
n l'(gi  "() !
The probability W of the entire distribution of particles is the product
ft]
U' =11
Taking the natural logarithm of both sides.
In W = v [In g ( !  In n,!  In (ft  n,)t]
which Stirling's formula
In rtl = n In n — n
permits us to rewrite as
9.4 Z In W = 2 [ft Lift  n, In n,  (ft  n,) In (g,  n,)]
For this distribution to represent maximum probability, small changes fin, in any
of the individual n/s must not alter W. Hence
9.43
« 1" W BM = 2 [In n, + In (g,  n,)] 5n, =
As before, we take into account the conservation of particles and of energy by
adding
STATISTICAL MECHANICS
309
— «2 6n ( =
9.47
and
to Eq. 9.43, with the result that
9.44 S [ — In ri( + In (g ( — n t ) — « — fit,] 5n, =
Since the Bn,'s are independent, the quantity in brackets must vanish Tor each
value of i, and so
In & ~ "'  «  fc = !)
n,
■i*  1 = eV"
9.45
n, =
ft
1 eV" + 1
Substituting
P kT
yields the FermiDirac distribution law,
., ft
9.46
1 e a e" /kT + 1
FermiDirac distribution (aw
The most important application of the FermiDirac distribution law is in the
freeelectron theory of metals, which we shall examine in the next chapter.
9.10 COMPARISON OF RESULTS
The three statistical distribution laws are as follows:
n ( =
«, =
e a e'> /kT
ft
e « e </kT _ 1
ft
ff e 'i' kr + 1
Max welt Boltzmann
Bose Einstein
FermiDirac
310
In these formulas n, is the number of particles whose energy is f i and g, is the
number of states that have the same energy e,. The quantity
PROPERTIES OF MATTER
jwj
Occupation index
called the occupation index of a state of energy f,, is therefore the average number
of particles in each of the stales of that energy. The occupation index docs not
depend upon how the energy levels of a system of particles are distributed, and
for this reason it provides a convenient way of comparing the essential natures
of the three distribution laws.
The MaxwellBoltzmann occupation index is a pure exponential, dropping by
the factor \/e for each increase in f ( of kT. While J\t t ) depends upon the
parameter «, the ratio between the occupation indices/^) and /?<',) of the two
energy levels t , and r^ does not;
9,48
fa)
_ e U r r,\/kT
Bolkmann factor
This formula is useful because when/(r)< 1, the BoseEinstein and FermiDirac
distributions resemble the MaxwellBoltzmann distribution, and it then permits
us to determine the relative degrees of occupancy of two quantum states in a
simple way.
In the case of a photon gas, a m 0, and the BoscEinstein occupation index
approaches the MaxwellBoltzmann one when e t > kT, whereas when e, < kT
the — 1 term in the denominator of the formula for the former occupation index
causes it to exceed the latter. The FermiDirac occupation index never goes
above 1, signifying one particle per state at most, which is a consequence of
the obedience of Fermi particles to the exclusion principle. At low temperatures
virtually all the lower energy states are filled, with the occupation index dropping
rapidly near a certain critical energy known as the Fermi energy. At high
temperatures the occupation index is sufficiently small at all energies for the
effects of the exclusion principle to be unimportant, and the FermiDirac dis
tribution liecomes similar to the MaxwellBoltzmann one.
9.11 THE LASER
Three kinds of transition involving electromagnetic radiation can occur between
two energy levels in an atom, a lower one i and an upper one / (Fig. 910),
If the atom is initially in state i, it can be raised to state / by absorbing a photon
of light whose energy is /ic = E, — £,. This process is called induced absorption.
If the atom is initially in the upper state /, it con drop to state i by emitting
a photon of energy hi>; this is spontaneous emission.
There is also a third possibility, induced emission, in which an incident photon
of energy hi' causes a transition from the upper state to the lower one. Induced
STATISTICAL MECHANICS
311
312
h»
AW*
hi'
hi
WW*
t,r \AA/»
hf
induced
absorption
spontaneous
emission
induced
emission
FIGURE 910 Transitions between Iwo energy levels In an atom can occur by Induced absorption, soon
(annus emission, and induced emission.
emission involves DO novel concepts. An analogy is a harmonic oscillator, for
instance a pendulum, which has a sinusoidal force applied to it whose period
is the same as its natural period of vibration. If the applied force is exactly in
phase with the pendulum swings, the amplitude of the latter increases; this
corresponds lo induced absorption of energy. However, if the applied force is
180° out of phase with the pendulum swings, the amplitude of the latter de
crctiM v this corresponds to induced emission of energy.
Since hi is normally much greater than kT for atomic and molecular radiations,
at thermal equilibrium the population of upper energy stales in an atomic system
is considerably smaller than that of the lowest stale. Suppose we shim; light
of frequency t> upon a system in which the energy difference between the ground
state and an excited state is hi>. With the upper State largely unoccupied, there
will lie little stimulated emission, and the chief events dial occur will he absorp
tion ot incident photons by atoms in the ground state and the subsequent sponta
neous random rcradiation of photons of the same frequency, (A certain propor
tion of excited atoms will give up their energies in cotlisioas.)
Certain atomic systems can sustain inverted energy populations, with an upper
slale occupied to a greater extenl men the ground state. Figure 911 shows a
threelevel system in which the intermediate slale I is met as table, which means
that the transition from it lo the ground slate is forbidden In selection rules.
The system can be "pumped" lo the upper state 2 by radiation of frequency
i'' = (E s — E[,)//i. (Electron impacts are another wa\ to raise the system to the
upper state.) Atoms in slale 2 have lifetimes of about 10 " s against spontaneous
emission via an allowed transition, so they fall to the metastable slate 1 (or to
the ground state) almost at once. Metastable states may have lifetimes of well
over I s against spontaneous emission, and it is therefore possible to continue
pmnping until there is a higher population in state I than there is in slate 0.
If now we direct radiation of frequency v = (£, — £,,)/& on the system, Ihe
induced emission of pholous of this frequency will exceed their absorption since
PROPERTIES OF MATTER
a \. iV it iS i.
w
(0
J *b*
I ill
3 W « g
■S5.9
I*?J1
ggsgs
e 2 "3 2«
.so. &•%!
w
U
.3 >
c —
&'£
I «
U —
>■ a
M
w
—
% B
E
W
tu>
w
03
1 
.2 2 gH.1
S3 2 c II
— E S ** CJ
a "4
~0 U)
&"§ s '
o a O c
more atoms are in the higher state, and the net result will be an output of
radiation of frequency f that exceeds the input. This is the principle of the mascr
(microwave amplification by stimulated emission of radiation) and the laser (fight
amplification by stimulated emission of radiation).
The radiated waves from spontaneous emission are, as might be expected,
incoherent, with random phase relationships in space and time since there is
no coordination among the atoms involved. The radiated waves from induced
emission, however, are in phase with the inducing waves, which makes it possible
for a mascr or laser to produce a completely coherent beam. A typical laser
is a gasfilled tulic or a transparent solid that has mirrors at both ends, one of
them partially transmitting to allow some of the light produced to emerge. The
pumping light of frequency v is directed at the active medium from the sides
of the tube, while the baekand forth traversal of the trapped light stimulate
emissions of frequency P thai maintain the emerging beam collimated. A wide
variety of outsets and lasers have been devised: usually, the required inverted
energy distribution is obtained less direct 1> than by the straightforward meeha
msni described above.
Problems
1. Verify that the most probable speed of a molecule of an ideal gas is equal
to V2JtT/m.
2. Verify that the average speed of a molecule of an ideal gas is equal to
\/SkT/irm.
3. Find the average value of l/v in a gas allying MaxwellBoltzmann statistics.
4. What proportion of the molecules of an ideal gas have components of
velocity in any particular direction greater than twice the most probable speed?
5. A flux of 10 l  neutrons/ m 2 emerges each second from a port in a nuclear
reactor. If these neutrons have a MaxwellBoltzmann energy distribution corre
sponding to T = 300 K, calculate the density of neutrons in the beam.
6. The frequency of vibration of the H 2 molecule is 1.32 X I0 N Hz. («) Find
the relative populations of the v = 0. 1, 2, 3, and 4 vibrational states at 5000 K.
(b) Can the populations of the v — 2 and v = 3 states ever be equal? If so, at
what temperature does this occur?
7. The moment of inertia of the H, molecule is 4,64 X 10 48 kgnf. (a) Find
the relative populations of the / = 0, 1, 2, 3, and 4 rotational states at 300 K.
(b) Can the populations of the J = 2 and / = 3 states ever be equal? If so, at
what temperature does this occur?
8. The N 2 molecule is linear with an N — N bond length of 1.126' A and an
N— O bond length of 1.191 A. The mass of the 18 atom is 2.66 X 1CT 2G kg
and that of the 14 N atom is 2,32 X 10~ 26 kg. (a) What is the quantum number
of the most populated rotational energy level at 3(X* K? (b) Plot rij/n,, versus
J at 300 K.
9. The temperature of the sun's chromosphere is approximately 5000 K. Find
the relative numbers of hydrogen atoms in the chromosphere in the n = 1, 2,
3, and 4 energy levels. Be sure to take into account the multiplicity of each
level.
10. If the tungsten filament of a light bulb is equivalent to a black body at
2900 K, find the percentage of the emitted radiant energy in the form of visible
light with frequencies between 4 X 10 14 and 7 X 10 14 Hz.
1 1. Sunlight arrives at the earth at the rate of about 1,400 W/m 2 when the
sun is directly overhead. The sun's radias is 6.96 X 10 8 m and the mean radius
of the earth's orbit is 1.49 X 10" m. From these data find the surface tempera
ture of the sun on the assumption that it radiates like a black body. (The actual
surface temperature of the sun is slightly less than this value.)
1 2. The problem of the blackbody spectrum was examined at die end of the
nineteenth century by Rayleigh and Jeans, using classical physics, since the notion
of electromagnetic quanta was as yet unknown. They obtained the formula
e(y)dv =
Hirv'^kT eb>
(a) Why is it impossible for a formula with this dependence on frequency to
be correct? (fo) Show that, in the limit of i> — * 0, the Planck radiation law reduces
to the B ay leigh Jeans formula,
13. At the same temperature, will a gas of classical molecules, a gas of bosom
( particles that obey BoseEinstein statistics), or a gas of fennions (particles that
oliey FermiDirac statistics) exert the greatest pressure? The least pressure?
Why?
14. Derive the StefanBoltzmann law in the following way. Consider a Camot
engine that consists of a cylinder and piston whose inside surfaces are perfect
reflectors and which uses electromagnetic radiation as its working substance.
Hie operating cycle of this engine has four steps: an isothermal expansion at
the temperature T during which the pressure remains constant at p; an adiahatic
314
PROPERTIES OF MATTER
STATISTICAt MECHANICS
315
expansion during which the temperature drops by (IT and the pressure drops
by dp, an isothermal compression al the temperature T — dT and pressure
p — dp; and an adiabatic compression to the original temperature, pressure, and
volume. The pressure exerted by radiation of energy density u in a container
with reflecting walls is n/3, and the efficiency of all Caraol engines is dW/Q =
1 _ (7" — dT)/T, where Q is the heat input during the isothermal expansion and
dW is the work done by the engine during the entire cycle. Calculate the
efficiency of this particular engine in tenns of u and T with the help of a pV
diagram and show that u = aT\ where a is a constant.
15. In a continuous heliumneon laser. He and Ne atoms are pumped to meta
stable states respectively 2tl.fi 1 and 20.88 eV above their ground states by electron
impact. Some of the excited I Ifl atoms transfer energy to .Ne atoms in collisions,
with the 0.05 eV additional energy provided by the kinetic energy of the atoms.
An excited Ne atom emits a 6328 A photon in the forbidden transition that leads
to laser action. Then a fifi80A photon is emitted in an allowed transition to
another me tas table state, and the remaining excitation energy is lost in collisions
with the tube walls. Find the excitation energies of the two intermediate states
in Ne. Wby are He atoms needed?
316 PROPERTIES OF MATTER
THE SOLID STATE
10
A solid consists of atoms, ioas, or molecules packed closely together, and their
proximity is responsible for the characteristic properties of this state of matter.
The covalent bonds involved in the formation of a molecule are also present
in certain solids. In addition, ionic, run der Winds, and metallic bauds provide
the cohesive forces in solids whose structural elements are, respectively, ions,
molecules, and metal atoms. All these Iwnds involve electric forces, so that the
chief distinctions among them lie in the distribution of electrons around the
various particles whose regular arrangement forms a solid.
10.1 CRYSTALLINE AND AMORPHOUS SOLIDS
The majority of solids are crystalline, with the atoms, ions, or molecules of which
they are composed falling into regular, repeated threedimensional patterns. The
presence of longrange order is thus the defining property of a crystal. Other
solids lack longrange order in the arrangements of their constituent particles
and may properly be regarded as supercooled liquids whose stiffness is due to
an exceptionally high viscosity. Class, pitch, and many plastics are examples
of such amorphous ("without form") solids.
Amorphous solids do exhibit shortrange order in their structures, however.
The distinction between the two kinds of order is nicely exhibited in boron
(rioxide (B 2 3 ), which can occur in both crystalline and amorphous forms. In
each case every boron atom is surrounded by three oxygen atoms, which repre
sent a shortrange order. In a B,,0. { crystal the oxygen atoms are present in
hexagonal arrays, as in Fig. 101 . which is a longrange ordering, while amorphous
BjOg, a vitreous or "glassy" substance, lacks this additional regularity. A con
spicuous example of shortrange order in a liquid occurs in water just above the
melting [Mint, where the result is a lower density than at higher temperatures
because I UC) molecules are less tightly packed when linked in crystals than when
free to move.
Ihe analogy between an amorphous solid and a liquid is worth pursuing as
a means of better understanding lioth states of matter. Liquids are usually
317
• boron a lorn
oxygen atom
(a)
w
FIGURE 101 Twodimensional representation of B ; 0,. <a> Amorphous BjO., eithlbrti only shortrange
order, (0) Crystalline GO. exhibits kingrange order as well.
regarded as resembling gases more closely than solids; after all, liquids and gases
are both fluids, and at temperatures above the critical point ihe two become
indistinguishable. However, from a microscopic point of view, liquids and solids
also have much in common. The density of a given liquid is usually close to
that of the corresponding solid for instance, which suggests that the degree of
packing is similar, an inference supported by the compressibilities of these stales.
Furthermore, Xray diffraction indicates that many liquids have definite short
range structures at any instant, quite simitar to those of amorphous solids except
that the groupings of liquid molecules are continually shifting.
Since amorphous solids are essentially liquids, they have no sharp melting
points. We can interpret this behavior on a microscopic basis by noting that,
since an amorphous solid lacks longrange order, the bonds between its molecules
vary in strength. When the solid is heated, the weakest bonds rupture at lower
temperatures than the others, so that it softens gradually. In a crystalline solid
the transition between longrange and shortrange order (or no order at all)
involves the breaking of bonds whose strengths are more or less identical, and
melting occurs at a precisely defined temperature.
10.2 IONIC CRYSTALS
Covalent bonds come into being when atoms share pairs of electrons in such
a way that attractive forces are produced. Ionic bonds come into being when
atoms that have low ionization energies, and hence lose electrons readily, interact
with other atoms that tend to acquire excess electrons. The former atoms give
up electrons to the latter, and they thereupon l>ecome positive and negative ions
318 PROPERTIES OF MATTER
respectively. In an ionic crystal these ions come together in an equilibrium
configuration in which the attractive forces between positive and negative ions
predominate over the repulsive forces lictween similar ions. As in the case of
molecules, crystals of all types are prevented from collapsing under the influence
of the cohesive forces present by the action of the exclusion principle, which
requires the occupancy of higher energy states when electron shells of different
atoms overlap and mesh together.
Table 10. 1 contains the ionization energies of the elements, and Fig. 102 shows
how these energies vary with atomic number. It is not hard to see why the
ionization energies of the elements vary as they do. For instance, an atom of
any of the alkali metals of group I has a single .s electron outside a closed subshell.
The electrons in the inner shells partially shield the outer electron from the
nuclear charge + Ze, so that the effective charge holding the outer electron to
the atom is just +e rather than +Ze. Relatively little work must be done to
detach an electron from such an atom, and the alkali metals form positive ions
readily. The larger the atom, the farther the outer electron is from the nucleus
(Fig, 103) and the weaker is the electrostatic force on it; this is why the ionization
energy generally decreases as we go down any group. The increase in ionization
energy from left to right across any period is accounted for by the increase in
nuclear charge while the number of inner shielding electrons stays constant.
There are two electrons in the common inner shell of period 2 elements, and
the effective nuclear charge acting on the outer electrons of these atoms is
FIGURE 102 The variation of Ionization energy with atomic number,
30,
30 40 50 60
ATOMIC NUMBER
70 80 90
THE SOLID STATE
319
Table 10.1.
IONIZATION ENERGIES OF THE ELEMENTS. In
i.'i.n
2
II.
84.6
S 6 7 8 * M
li f: N F Mb
N.3 113 I4.S 1.3.6 171 2l.fi
1.1 14 15 16 IT IS
Al Si P S CI! Ar
rut s.i n.o io. i i W 15.8
19 20 21 22 23 24 25 26 27 28 28 :»0 31 32 33 31 35 30
K Gi & Ti V Cr Vln ft Co \i Cu Zn Ca O As So Br Kr
4.3 8.1 6.6 6.8 67 6.8 T.l 7M 7<) 7 6 7.7 ').) fill 7.!) B.« 'is lis lio
37 38 39 40 41 42 43 44 45 40 17 48 49 50 51 52 53 54
Kb Sr V Zt \l> Mo TV Ru Itli M tg CI l.> Si, si, IV I Xe
1.2 "7 fi.i 7.0 ff.S 7.1 7..! 7.4 7.5 8,3 7.6 9.0 5.t> 7,3 H.fi !>.!! 10,5 12.1
3
4
Li
Be
5.4
0.3
II
12
N»
Mg
5.1
7.0
55
50 '
Cs
l)»
3,9
3.2
87
88 1
Fr
Ri
—
5.3
72 73 74 75 70
111 H, W Ha Os
5,5 T.'i Mi 7.8 8.7 9.2 8.0 \>.2 III. I 0.1 7.4 7..1 S.4  10.7
77 78 79 80 61 82 83 84 85 88
It Pi An lit: Tl I'h Hi I'o Al Rn
57 58 50 00 01 02 03 04 65 60 67 68 68 70 71
La Co Pr Nd Pin Sm Kit Ccl Til l>v 11,. Ec Tin VI. I.u
">.(> (>.') is [>..) 5.0 5.7 62 6.7 B.8 — 6,1 5.8 6.2 5.0
68 SO 81 82 93 94 US 90 97 98 99 100 ID! 102 103
\c Hi Pa I! Np Pii Am Cm HL U K* I'm Mil N'n l.w
_ 7.11 — 6.1 — 5.1 6.0 — — — ____
320
therefore +(z — 2)e. The outer electron in a lithium atom is held to the atom
by an effective charge of +e, while each outer electron in beryllium, boron,
carbon, etc., atoms is held to its parent atom by effective charges of + 2e, f 3e,
+4e, etc.
At the other extreme from alkali metal atoms, which tend to lose their outer
most electrons, are halogen atoms, which tend to complete their outer p suhshells
by picking up an additional electron each. The electron affinity of an element
is defined as the energy released when an electron is added to an atom of each
element. The greater the electron affinity, the more tightly bound is the added
PROPERTIES OF MATTER
10
20
30
40 50 60 70
ATOMIC NUMBER
FIGURE 103 Atomic radii of (he elements. Several have two radii, corresponding to different crystal
i tinctures.
electron. Table 10.2 shows the electron affinities of the halogens. In general.
electron affinities decrease going down any group of the periodic table and
increase going from left lo right across any period. The experimental determi
nation of electron affinities is quite difficult, and those for only a few elements
are accurately known.
\u ionic bond between two atoms can occur when one of them has a low
ionization energy, and hence a tendency to become a positive ion, while the
other one has a high electron affinity, and hence a tendency to become a negative
Table 10.2.
ELECTRON AFFINITIES OF THE HALOGENS.
in electron volts.
!■') limine
3,45
Clilurmc
3.61
Bniiiiiiii"
3.36
Iodine
3.06
THE SOLID STATE
321
ion. Sodium, with an ionization energy of 5.14 eV, is an example of the former
and chlorine, with an electron affinity of 1.61 eV, is an example of the fatter.
When a Na + ion and a Cl~ ion are in the same vicinity and are free to move,
the attractive electrostatic force between them brings them together. The
i nniliiion that a stable molecule of NaCI result is simply that the total energy
of the system of the two ions be less than the total energy of a system of two
atoms of the same elements; otherwise the surplus electron on the Cl~ ion would
transfer to the Na* ion, and the neutral Na and CI atoms would no longer be
bound together. Let us sec how this criterion is met by NaCI.
In general, in an ionic crystal each ion is surrounded by as many ions of the
opposite sign as can fit closely, which leads to maximum stability. The relative
sizes of the ions involved therefore govern the type of structure that occurs.
Two common types of structure found in ionic crystals are shown in Figs. 104
and 105. In a sodium chloride crystal, the ions of either kind may be thought
of as being located at the comers and at the centers of the faces of an assembly
of culies, with the Na + and CI assemblies interleaved. Each ion thus has six
nearest neighbors of the other kind, a consequence of the considerable difference
in the sizes of the Na + and CI" ions. Such a structure is called focfrcentered
cubic. A different arrangement is found in cesium chloride crystals, where each
ion is located at the center of a cube at whose corners are ions of the other
kind. Each ion has eight nearest neighbors of the other land in such a Imdij
cenlered vulrir structure, which results when the participating ions are compara
ble in size.
The cohesive energy of an ionic crystal is the energy that would be tilierated
by the formation of the crystal from individual neutral atoms. Cohesive energy
FIGURE 104 (a) The Facecentared cubic structure of 3 NaCI crystal. Tha coordination number (num
ber of nearest neighbors about each Ion) is 6. (6) Scale model of NaCI crystal.
(b)
FIGURE 105 (a) The bodycentered cubic structure of a CsCI crystal. The coordination
number is 8 (b) Scale model of CsCI crystal.
is usually expressed in eV/atom, in eV/molecule, or in kcal/mol, where '"mole
cule" and "inol" here refer to sets of atoms specified by the formula of the
compound involved (for instance NaCI in the case of a sodium chloride crystal)
even though molecules as such do not exist in the crystal
The principal contribution to the cohesive energy of an ionic crystal is the
electrostatic potential energy v* KOUlomb of the ions. l.et us consider an Na + ion
in NaCI. Its nearest neighbors are six CI" ioas, each one the distance r away.
The potential energy of the Na + ion due to these six CI ions is therefore
4 7 rf l) r
The next nearest neighbors are 12 Na* ions, each one the distance \/2 r away
since the diagonal of a square r long on a side is v2 r. The potential energy
of the Na* ion due to the 12 Na* ions is
V, = +
12e 2
4KF a \/2r
When the summation is continued over all the + and — ions in a crystal of
infinite size, the result is
Veouio ™ b "w( 6 "^i + '")
=  1.748
2
•k7f„r
10.1
=: — «
»"<„ r
Coulomb energy of ionic crystal
THE SOLID STATE
323
322
PROPERTIES OF MATTER
(This result holds for the potential energy of a Cl~ ion as well, of course.) The
quantity a is called the Mtithtun« ronslant of the crystal, and it has the same
value for all crystals of the same structure. Similar calculations for other crystal
varieties yield different Made) ting constants: crystals whose structures are like
that of cesium chloride (Fig. 105), for instance, have a = 1.763, and those with
structures like that of zinc blende (one form of 7.nS) have a m 1.638. Simple
crystal Structures have Madelung constants that lie between 1.0 and 1.8.
The potential energy contribution of the repulsive forces due to the action
of the exclusion principle can l)e expressed to a fair degree of approximation
in the form
10.2
niniKiv.'
r"
The sign of V repihlve is positive, which corresponds to a repulsive interaction,
and the dependence on r~ n {where n is a large number) corresponds to a short
range force that increases rapidly with decreasing intemuclear distance r. The
total potential energy V of each ion due to its interactions with all the other
ions is therefore
V= V.
CMiHiHiib
+ K,
ijmtiiH'i*
10.3
ate'
4ro r r
B
Ai the equilibrium separation r of the ions, V is a minimum by definition, and
so (dV/dr) = when r = r„. Hence
10.4
'''r= r „
*
r n 1
ae
=
nB
r Bl
'0
<*Ve 8
B

•1tti (j ii
r ■"'
1
=
and the total potential energy is
io.s v= ^L(il)
It is possible to evaluate the exponent n from the observed compressibilities
of ionic crystals. The average result isnsi), which means that the repulsive
force varies quite sharply with r: the ions are "hard" rather than "soft" and
strongly resist (wing packed too tightly. At the equilibrium ion spacing, the
mutual repulsion due to the exclusion principle (as distinct from the electrostatic
324 PROPERTIES OF MATTER
repulsion lietweeu like ions) decreases the potential energy by about 11 percent.
A really precise knowledge of n is evidently not essential; if n = 10 instead of
n = 9, V would change by only 1 percent.
In an NaCI crystal, the equilibrium distance r u between ions is 2.8J A, Since
a ■= 1.748 and n = 9, the potential energy of an ion of either sign is
4ire r \ nf
9 x It*" NmVC a X 1.748 X (1.60 X 10 lfl C) 2 / _ IV
2.S1 x K) 1 " m
= 1.27 X 10'" J
= 7.97 eV
Because we may not count each ion more than once, only half <>1 tins potential
energy, or —3.99 eV, represents the contribution per ion to the cohesive energy
of the crystal.
We must also take into account the energy needed to transfer an electron from
a Na atom to a CI atom to yield a Na + — CI" ion pair. This electron transfer
energy is the difference between the +5.14eV ionization energy of Na and Aw
— 3,61eV electron affinity of CI, or + 1.53 eV. Each atom therefore contributes
+ 0.77 eV to the cohesive energy from this source. The total cohesive energy
per atom is thus
I:
eobetfvs
= (3.99 + 0.77) eV/atom = 3.22 eV/atom
An empirical ligure for the cohesive energy of an ionic crystal can be obtained
from measurements of its heat of vaporization, dissociation energy, and electron
exchange energy. The result for Nat Tl is :?.2S eV. in close agreement with the
calculated value.
Most ionic solids are hard, owing to the strength of the I wauls between their
constituent ions, and have high melting points. They are usually brittle as well,
since the slipping of atoms past one another that accounts for the ductility of
metals is prevented l>v the ordering of positive and negative ions imposed by
the nature of the bonds. Polar liquids such as water are able to dissolve many
ionic crystals, hut covalent liquids such as gasoline generally cannot.
10.3 COVALENT CRYSTALS
The cohesive forces in covalent crystals arise from the presence of electrons
between adjacent atoms. Each atom participating in a covalent bond contributes
an electron to the bond and these electrons are shared by imth atoms rather
THE SOLID STATE
325
FIGURE 106 (a) The tetrahedral structure of diamond. The coordination number is 4. (b) Scale model
of diamond crystal
lhan being the virtually exclusive property of one of theni as in an ionic bond.
Diamond is an example of a crystal whase atoms are linked by covalent bonds.
Figure 106 shows the structure of a diamond crystal; the tetrahedral arrangement
is a consequence of the ability of each carbon atom to form covalent bonds with
four other atoms (sec Fig. 816).
Purely covalent crystals are relatively few in number. In addition to diamond,
some examples are silicon, germanium, and silicon carbide; in SiC each atom
is surrounded by four atoms of the other kind in the same tetrahedral structure
as that of diamond. All covalent crystals arc hard (diamond is the hardest
substance known, and SiC is the industrial abrasive carborundum), have high
melting points, and are insoluble in all ordinary liquids, behavior which reflects
the strength of the covalent bonds. Cohesive energies of 3 to 5 eV/atom arc
typical of covalent crystals, which is the same order of magnitude as the cohesive
energies in tonic crystals.
There are several ways to ascertain whether the bonds in a given nonmctallic,
iioiiiiiolecular crystal are predominantly ionic or covalent. In general, a com
pound of an element from group I or 11 of the periodic table with one from
group VI or VII exhibits ionic bonding in the solid state. Another guide is the
coordination number of the crystal, which is the mimtwr of nearest neighlmrs
about each constituent particle. A high coordination numlier suggests an tonic
crystal, since it is hard to see how an atom can form purely covalent Irauds with
six other atoms (as in a facecentered cubic structure like that of NaCl) or with
eight other atoms (as in a bodycentered cubic structure like that of CsCl). A
coordination number of 4, however, as in the diamond structure, is compatible
with an exclusively covalent character. To Ix; sure, as with molecules, it is tiot
always possible to classify a particular crystal as Ixjing wholly ionic or covalent:
AgCl, whose structure is the same as that of NaCl, and CuCI, whose structure
resembles that of diamond, both have bonds of intermediate character, as do
a great many other solids.
10.4 VAN DER WAALS FORCES
AM atoms and molecules — even inertgas atoms such as those of helium and
argon — exhibit weak, shortrange attractions for one another due to van der
Wauls forces. These forces arc responsible for the condensation of gases into
liquids and the freezing of liquids into solids despite the absence of ionic,
covalent. or metallic landing mechanisms. Such familiar aspects of the liehavior
of matter in bulk as friction, surface tension, viscosity, adhesion, cohesion, and
so on also arise from van der Waals forces. The van der Waals attraction between
two molecules r apart is proportional to r~ 7 , so that it is significant only for
molecules very close together.
We begin by noting that many molecules (called polar molecules) possess
permanent electric dtpole moments. An example is the H 2 molecule, in which
the concentration of electrons around the oxygen atom makes that end of the
molecule more negative than the end where the hydrogen atoms arc. Such
molecules tend to align themselves so diat ends of opposite sign are adjacent,
as in Fig. 107, and in this orientation the molecules strongly attract each other.
A polar molecule is also able to attract molecules which do not normally have
a permanent dipole moment. The process is illustrated in Fig. 10fS: the electric
field of the polar molecule causes a separation of charge in the other molecule,
with the induced moment the same in direction as that of the polar molecule.
FIGURE 107 Polar molecules attract each other.
326
PROPERTIES OF MATTER
THE SOLID STATE
327
FIGURE 10 8 Polar molecules attract potarizable molecules.
The result is an attractive force. (The effect is the same as that involved in the
attraction of an innnagneli/.ed piece of iron by a magnet.)
More remarkably, two nonpolar molecules can attract each other by the above
mechanism. Even though the electron distribution in a nonpolar molecule is
symmetric on tlte average, the electrons themselves are in constant motion and
at am/ giiYn instant one part or another of the molecule has an excess of them.
Instead of the fixed charge asymmetry of a polar molecule, a nonpolar molecule
has a constantly shifting asymmetry. When two nonpolar molecules are close
enough, their fluctuating charge distributions tend to shift together, adjacent ends
always having opposite sign (Fig. 109) and so always causing an attractive force.
This kind of force is named after the Dutch physicist van der Waals, who
suggested its existence nearly a century ago to explain observed departures from
the idealgas law; the explanation of the actual mechanism of the force, of course,
is more recent.
Van der Waals forces are much weaker than those found in ionic and covalent
bonds, and as a result molecular crystals generally have low melting and Ixnling
points and little mechanical strength. Cohesive energies are low, only 0.08
eV/atom in solid argon (melting point — 1W)*C), (M)l eV/molecule in solid
hydrogen (mp — 25S)°C), and 0.1 eV/molecule in solid methane, C1I, imp
 183 8 C).
An especially strong type of van der Waals l>ond called a hydrogen bond occurs
between certain molecules containing hydrogen atoms. The electron distribution
in such an atom is so distorted by the affinity of the "parent" atom for electrons
that each hydrogen atom in essence has donated most of its negative charge to
the parent atom, leaving behind a poorly shielded proton. The result is a
molecule with a localized positive charge which can link up with the concen
tration of negative charge elsewhere in another molecule of the same kind. The
key factor here is the small effective size of the poorly shielded proton, since
electric forces vary as r .
Water molecules are exceptionally prone to form hydrogen bonds because the
four pairs of electrons around the O atom occupy &•{)'* hybrid orhitals that project
outward as though toward the vertexes of a tetrahedron (Fig. 1010). Hydrogen
atoms are at two of these vertexes, which accordingly exhibit localized positive
charges, while the other two vertexes exhibit somewhat more diffuse negative
charges. Kach ll 2 molecule can therefore form hydrogen bonds with four other
H a O molecules; in two of these bonds the central molecule provides the bridging
protons, and in the other two the attached molecules provide them. In the liquid
state, the hydrogen bonds between adjacent H.,0 molecules are continually being
broken and reformed owing to thermal agitation, but even so at any instant the
molecules are combined in definite clusters. In the solid state, these clusters are
large and stable and constitute ice crystals.
The characteristic hexagonal pattern (Fig. 101 1) of an ice crystal arises from
the tetrahedral arrangement of the four hydrogen bonds each H 2 molecule can
participate in. With only four nearest neighbors around each molecule, ice
crystals have extremely open structures, which is the reason for the exceptionally
low density of ice. Because the molecular clusters are smaller and less stable
in the liquid state, water molecules on the average are packed more closely
rj o
FIGURE 109 On the average, nonpolar molecules
have symmetric charge distributions, but at any In
stant the distributions are asymmetric. The fluctua
tions in the charge distributions of nearby molecules
are coordinated as shown, which leads to an atir.it
tive force between them whose magnitude is propor
tional to 1/r.
+ + + +

o o
328
PROPERTIES OF MATTER
THE SOUD STATE
329
hybrid orbital^
3.
3
FIGURE 10 10 In an HO molecule, the four pairs of valence electrons around
the oxygen atom (sin contributed by the atom and one each by the H atoms) oc
cupy lour *j/ L hybrid orbital* that form a tetrahedral pattern. Each H.0 molecule
can form hydrogen bonds with four other HO molecules.
FIGURE 1011 Top view of an ice crystal, showing the open hexagonal arrangement of H.O molecules.
Each molecule has four nearest neighbors to which It is attached by hydrogen bonds.
330 PROPERTIES OF MATTER
together than are ice molecules, and water has the higher density: hence ice
floats. The density of water increases from 0°C to a maximum at 4 "C as large
clusters of H z O molecules are broken up into smaller ones that occupy less space
in the aggregate; only past 4°C does the normal thermal expansion of a liquid
manifest itself in a decreasing density with increasing temperature.
10.5 THE METALLIC BONO
The underlying theme of the modern theory of metals is that the valence electrons
of the atoms comprising a metal are common to the entire aggregate, so that
a kind of "gas" of free electrons pervades it. The interaction between this
electron gas and the positive metal ions leads to a strong cohesive force. The
presence of such free electrons accounts nicely for the high electrical and thermal
conductivities, opacity, surface luster, and other unique properties of metals.
To be sure, no electrons in any solid, even a metal, are able to move about its
interior with total freedom. All of them are influenced to some extent by the
other particles present, and when the theory of metals is refined to include these
complications, there emerges a comprehensive picture that is in excellent accord
with experiment.
Some insight into the ability of metal atoms to lx>nd together to form crystals
of unlimited size can be gained by viewing the metallic bond as an unsaturated
covalent Irond. Let us compare the bonding processes in hydrogen and in lithium,
Iwlh members of group 1 of the periodic table. A H a molecule contains two
\.i electrons with opposite spins, the maximum number of K electrons that can
be present. The 1 1 2 molecule is therefore saturated, since the exclusion principle
requires that any additional electrons be in states of higher energy and the stable
attachment of further H atoms cannot occur unless their electrons are in Is states.
Superficially lithium might seem obliged to behave in a similar way, having the
electron configuration ls 2 2s. There are, however, six unfilled 2j> states in every
Li atom whose energies are only very slightly greater than those of the 2s slates.
When a Li atom comes near a Li 2 molecule, it readily becomes attached with
a covalent bond without violating the exclusion principle, and the resulting Li 3
molecule is stable since all its valence elections remain in L shells. There is
no limit to the number of Li atoms that can join together in this way, since
lithium forms bodycentered cubic crystals (Fig, 105) in which each atom has
eighl nearest iitiuhlmrs. With only one electron per atom available to enter
into bonds, each l»nd involves onefourth of an electron on the average instead
of two electrons as in ordinary covalent bonds. Hence the bonds are far Iroin
being saturated; this is true of the bonds in other metals as well.
THE SOLID STATE
331
332
One consequence of the unsaturated nature of the metallic bond is the fact
that the properties of a mixture of different metal atoms do not depend critically
on the proportion of each kind of atom, provided their sizes are similar. Thus
the characteristics of an alloy often vary smoothly with changes in its composi
tion, in contrast to the specific atomic proportions found in ionic solids and in
covalent solids such as SiC
The most striking consequence of the unsaturated bonds in a metal is the ability
of the valence electrons to wander freely from atom to atom. To understand
this phenomenon intuitively, we can think of each valence electron av constantly
moving from bund to bond. In solid Li. each electron participates in eight bonds,
so that it only spends a short time between am pair fit Li* ions. The electron
cannot remember (so to Speak) which of the two ions it really belongs to, and
it is just as likely to move on to a liond that does not involve its parent ion
at all. The valence electrons in a metal therefore liehave in a manner quite
similar to that of molecules in a gas.
As in the case of any other solid, metal atoms cohere liecause their collective
energy is lower when they are bound together than when they exist as separate
atoms. To understand why this reduction in energy occurs in a metallic crvstal,
we note that, because of the proximity of the ions, each valence electron is an
the average closer to one nucleus or another than it would be if it Iwlonged
to an isolated atom. Hence the potential energy of the electron is less in the
crystal than in the atom, and it is this decrease in potential energy that is
responsible for the metallic Imud.
There is another factor to be considered, however. Whereas the electron
potential energy is reduced in a metallic crystal, the electron kinetic energy is
increased. The free electrons in a metal constitute a single system of electrons,
and the exclusion principle prohibits more than two of them one with each spin
from occupying each energy level, It would seem at first glance that, again using
lithium as an example, only eight valence electrons in an entire Li crystal could
occupy n = 2 quantum states, with the rest being forced into higher and higher
states of such great energy as to disrupt the entire structure. What actually
happens is less dramatic. The valence energy levels of the various metal atoms
are all slightly altered by their interactions, and an energy hand comes into l>cing
that consists of as many closely spaced energy levels as the total muiilicr of
valence energy levels in all the atoms in the crystal. The free electrons accord
ingly range in kinetic energy from to some maximum n F , called the I'cnni
energy; the Fermi energy in lithium, for example, is 4,72 eV, and the average
kinetic energy of the free electrons in metallic lithium is 2.6 eV. Since electron
kinetic energy is a positive quantity, its increase in the metal over what it was
in separate atoms leads to a repulsion.
Metallic bonding occurs when the attraction between the positive metal ions
PROPERTIES OF MATTER
. S3
(V, J
ci ►.
 i
5
1
13
1}
= t
*1
1 a
°& u
' _c £
 1
= £ 3
>
1
—
5 .'i
11
JB E
S 5
E ■=
« 4
fi
I 11
2
5 
I 1
C u ■<
J: "5
J £
o s i
I B
IS
« a
= ■
£ a
S .=
ill
a
l?3
i = i
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U
is
f 1
2 "3
a a
i I
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and the electron gas exceeds the mutual repulsion of the electrons in that gas;
that Ls, when the reduction in electron potential energy exceeds in magnitude
the concomitant increase in electron kinetic energy. The greater the number
of valence electrons per atom, the higher the average kinetic energy will be
in a metallic crystal, but without a commensurate drop in the potential energy.
For this reason the metallic elements are nearly all Found in the first three groups
of the periodic table. Some elements are right on the tjordcr line and may form
both metallic and covalent crystals. Tin is a notable example. Above 13.2°C
the metal "white tin" exists whose atoms each have six nearest neighbors. Below
13.2°C the covalent solid "gray tin" exists whose structure is the same as that
of diamond. Gray tin and white tin are quite different substances; they have
the respective densities of 5.8 and 7.3 g/cm 3 , for instance, and gray tin is a
semiconductor whereas white tin has the typically high electric conductivity of
a metal.
10.6 THE BAND THEORY OF SOLIDS
The atoms in almost every crystalline solid, whether a metal or not, are so close
together that their valence electrons constitute a single system of electrons
common to the entire crystal. The exclusion principle is obeyed by such an
electron system because the energy slates of the outer electron shells of the atoms
are all altered somewhat by their mutual interactions. In place of each precisely
defined characteristic energy level of an individual atom, the entire crystal
possesses an energy liand composed of myriad separate levels very close together.
Since there are as many of these separate levels as there are atoms in the crystal,
the band cannot be distinguished from a continuous spread of permitted energies.
The presence of energy bands, the gaps that may occur between them, and the
extent to which the)' are filled by electrons not only determine the electrical
behavior of a solid but also have important bearing on other of its properties.
There are two ways to consider the origin of energy bands. The simplest is
to look into what happens to the energy levels of isolated atoms as they are
brought closer and closer together to fonn a solid. We shall introduce the subject
in this way, and then examine some of the consequences of the notion of energy
bands. taler in the chapter we shall analyze energy bands in terms of the
restrict ions imposed by the periodicity of a crystal lattice on the motion of
electrons, a more powerful approach that provides the basis of much of the
modem theory of solids.
Figure 1012 shows the energy levels in sodium plotted versus intcrnuclear
distance. The 3.1 level is the firs! occupied level in the sodium atom to broaden
into a band; the 2p level does not begin to spread out until a quite small
334 PROPERTIES OF MATTER
10
20
30
3.67 5 10
INTERMUCLEAR DISTANCE. A
15
FIGURE 1012 The energy levels of sodium atoms become bands as their intemuclesr
distance decreases. The observed internuclear distance In solid sodium is 3.67 A.
THE SOLID STATE
335
I overlapping
energy bands
)
FIGURE 1013 The energy bands In a solid may overlap,
inlemuclear separation. This behavior reflects the order in which the electron
subshells of sodium atoms interact as the atoms are brought together. The
average energies in the 3/j and 3s bands drop at first, implying attractive forces.
The actual intemudear distance in solid sodium is indicated, and it corresponds,
as it should, to a situation of minimum average energy.
The energy bands in a solid correspond to the energy levels in an atom, and
an electron in a solid can possess only those energies that fall within these energy
bunds. The various energy bands in a sobd may overlap, as in Fig. HI 13, in
which case its electrons have a continuous distribution of permitted energies.
In other solids the bands may not overlap (Fig. 1014), and the intervals between
Ihem represent energies which their elections can not po.ssess. Such intervals
arc called forbiiitlen bonds. The electrical behavior of a crystalline solid is
determined lx>th by its energyband structure and by how these bands arc
normally filled by electrons.
Figure 1(115 is a simplified diagram of the energy levels of a sodium atom
and the energy bands of solid sodium. A sodium atom has a single 3s electron
in its outer shell. This means that the 3s band in a sodium crystal is only half
occupied, since each level in the band, like each level in the atom, is able to
contain two electrons. When an electric Held is set up across a piece of solid
forbidden
band
FIGURE 1014 A forbidden band separates non overlap ping
energy bands.
solid sodium
sodium atom
FIGURE 10 15 Energy levels in the sodium atom and
the corresponding situation in solid sodium (not to
scale). See Fig. 1012.
3a
2p
2s
Is 
sodium, electrons easily acquire additional energy while remaining in their
original energy band. The additional energy is in the form of kinetic energy,
and the moving electrons constitute an electric current. Sodium is therefore a
good conductor of electricity, as are odier crystalline solids with energy bands
that are onty partially filled.
Figure 1016 is a simplified diagram of the energy bands of diamond There
is an energy band completely filled with electrons separated by a gap of 6 eV
from an empty band above it. This means that at least 6 eV of additional energy
must be provided to an electron in a diamond crystal if it is to have any kinetic
energy, since it cannot have an energy lying in the forbidden band. An energy
increment of this magnitude cannot readily be given to an electron in a crystal
by an electric field. An electron moving through a crystal undergoes a collision
with an imperfection in the crystal lattice an average of everv — I0~"m, and
it loses much of the energy it gained from any electric field in the collision.
An electricfield intensity of (i X 10 s V/m is necessary if an electron is to gain
6 eV in a path length of I0" 8 m, well over 10 ln times greater than the electric
licit! intensity needed to cause a current to flow in sodium. Diamond is therefore
a very poor conductor of electricity and is accordingly classed as an insulator.
Silicon has a crystal structure resembbng that of diamond, and, as in diamond,
a gap separates the top of a filled energy band from a vacant higher band. The
lorbidden band in silicon, however is only Ll'eV wide. At low temperatures
if
li
n
fleV
FIGURE 1016 Energy bands In diamond (not to scale).
28
Is
336
PROPERTIES OF MATTER
THE SOLID STATE
337
338
silicon is little better than diamond as a conductor, but at room temperature
a small proportion of its electrons have sufficient kinetic energy of thermal origin
to jump the forbidden hand and enter the energy band above it. These electrons
are sufficient to permit a limited amount of current to flow when an electric
field is applied. Thus silicon has an electrical resistivity intermediate between
those of conductors and those of insulators, and it is termed a semicoiuhtctar.
The resistivity of semiconductors can lie altered considerably by small amounts
of impurity. Let us incorporate a few arsenic atoms in a silicon crystal. Arsenic
atoms have five electrons in their outermost shells, while silicon atoms have four,
{These shells have the configurations 4s 2 4p 3 and Zs'^p' 1 respectively.) When an
arsenic atom replaces a silicon atom in a silicon crystal, four of its electrons
are incorporated in eovalent bonds with its nearest neighbors. The fifth electron
requires little energy to be detached and move about in the crystal (see Probs.
23 and 24). As shown in Fig. 1017, die presence of arsenic as an impurity
provides ener^v levels just below the band which electrons must occupy for
conduction to take place. Such levels are termed donw keels, and the substance
is called an ntype semiconductor because electric current in it is carried by
negative charges.
If we alternatively incorporate gallium atoms in a silicon crystal, a different
effect occurs. Gallium atoms have only three electrons in their outer shells, whose
configuration is 4.v 2 4p, and their presence leaves vacancies called holes in the
electron structure of the crystal. An electron needs relatively little energy to
enter a hole, hut as it does so, it leaves a new hole in its former locution. When
an electric field is applied across a silicon crystal containing a trace of gallium,
electrons move toward the anode by successively filling holes. The flow of current
here is conveniently described with reference to the holes, whose behavior is
like that of positive charges since they move toward the negative electrode.
A substance of this kind is called a ptype semiconductor. (Certain metals, such
as zinc, conduct current primarily by the motion of holes.) In the energyband
diagram of Fig. 1018 we see that the presence of gallium provides energy levels,
termed acceptor letels, just above the highest filled baud. Any electrons thai
occupy these levels leave behind them, in the formerly filled baud, vacancies
winch permit electric current to flow.
donor « r _ ,
impurity ^
levels
1 empty
J band
forbidden
band
] filled
f bund
FIGURE 1017 A trace of arsenic in a
silicon crystal provides donor levels In
the normally forbidden band, producing
an Ji.fype semiconductor.
PROPERTIES OF MATTER
FIGURE 1018 A trace of gallium In a
silicon crystal provides acceptor levels in
the normally forbidden band, producing a
jjtype semiconductor.
acceptor
impurity y
levels ^^
1 empty
J band
I forbidden
band
1
filled
band
* 10.7 THE FERMI ENERGY
We shall now look more closely into the properties of the free electrons in a
DMtal, Electrons are Fermi particles since they obey the exclusion principle,
and hence the electron gas in a metal has a Fermi Dirac distribution of energies
(Sec. 9.9). The FermiDirac distribution law for the number of electrons n t with
the energy r.j is
10.6
", 
a
eV
JkT
+ 1
ft is more convenient to consider a continuous distribution of electron energies
than the discrete distribution of Eq. 10.6, so that the distribution law becomes
10.7
n(e) dt is
g(ej(fe
eV /kr + 1
To find g(r) de, the number of quantum states available to electrons with
energies between f and e + de, we use the same reasoning as for the photon
gas involved in blackbody radiation. The correspondence is exact because there
are two possible spin states, tn, = +% and m s = — %, for electrons, thus dou
bling the number of available phasespace cells just as the existence of two
possible directions of polarization for otherwise identical photons doubles the
number of cells for a photon gas. In terms of momentum we found in Sec. 9,8
that
gC») d P =
SirVp 2 dp
For nonrelativistic electrons
p" dp = (2m%) l/2 de
with the result that
,vj 8\/2 W Vm 3/2 ,_ ,
io.S g e) de =  ,, e 1/2 de
h J
THE SOLID STATE
339
340
The next step is to evaluate the parameter «. In order to do this, we consider
the condition of the electron gas at low temperatures. As observed in Sec. 9.10.
the occupation index when 7" is small is 1 from r  until near the Fermi energy
t>, where it drops rapidly to 0. This situation reflects the effect of the exclusion
principle: no states tan contain more than one electron, and so the minimum
energy configuration of an electron gas is one in which the lowest states are
filled and the remaining ones are empty. If we set
10.9
kl
the occupation index becomes
10.10
n t "^
g(t) e { ' •&** + I
The formula is in accord with the exclusion principle. At T = K,
f{t) = 1 when t < f F
= when f > e F
As the temperature increases, the occupation index changes from 1 to more
and more gradually, as in Fig. 10 19. At all temperatures
f(e) z= '/ 2 when e = i F
If a particular metal sample contains N free electrons, we can calculate its
Fermi energy r f . by filling up its energy states with these electrons in order of
increasing energy starting from * = 0. The highest energy state to be filled will
then have the energy f  *> by definition. The number of electrons that can
have the same energy < is equal to the number of states g(t? ) that have this energy,
since each state is limited to one electron. Hence
10.11
C'r
g(e) dt = «
Substituting Eq. 10.8 for g(e)de yields
N =
and
10.12
h 3
16\/2^Vw' /2
3/i 3
_ M. ( 3  v \ 2/
fF 2m \HvVt
i
/<*•
3/2
Fermi energy
PROPERTIES OF MATTER
r = ok
10
o.s
T>0K
FIGURE 1019 The occupation indei lor a Fermi Durac distribution at absolute zero and at a higher
temperature.
The quantity A'/V is the density of free electrons; hence t F is independent of
the dimensions of whatever metal sample is being considered.
Let us use Eq, 10.12 to calculate the Fermi energy in copper. The electron
configuration of the ground state of the copper atom is la t 2s Sl 2p*3a i 3p e 3d i ' > 4s;
that is. each atom has a single 4s electron outside closed inner shells. It is
therefore reasonable to assume that each copper atom contributes one free
electron to the electron gas. The electron density tj = A'/V is accordingly equal
to the number of copper atoms per unit volume, which is given by
Atoms (atoms/ kmol) X (mass/ volume)
Volume
mass/ kmol
10 13
I iere
A>
tr
so that
X„ = Avogadro's number = 6.02 X 10" atoms/kmol
p = density of copper = 8.94 X I0 ; * kg/m a
it = atomic mass of copper = 8.'3.5 kg/kmol
_ 6.02 X IP 2 " atoms/kmol X S04 X I0 a kg/m 3
'' ~ 63.5 kg/kmol
= 8.5 X 10* atoms m ;
= 8.5 X I0 S cleetrons/m 3
The corresponding Fermi energy is, from Eq. 10.12,
_ (6.63 X 10 ^ lsf / 3 X 8.5 X 10** cleetrons/nv' V ^ a
''' " 2 X 9.11 X 10 31 kg/electron I 8n }
= 1.13 X 10 S J
= 7.04 eV
THE SOLID STATE
341
At absolute zero, T = K, there would be electrons with energies of up to 7.04
eV in copper. By contrast, all the molecules in an ideal gas al absolute zero
would have zero energy. Because of its decidedly nnnelassical behavior, the
electron gas in a metal is said to l>e degenerate. Table 10.4 gives l he Fermi
energies of several common metals.
*10.8 ELECTRONENERGY DISTRIBUTION
We may now substitute for a and g(c) cfe in Eq. 10.7 to obtain a formula for
the number of electrons in an electron gas having energies between some value
e and e + de. This formula is
10.14 n{e)de =
(SySr Vm A/ yh 3 y /s de
If we express the numerator of Eq. 10.14 in terms of the Fermi energy <^., we
obtain
. . . f&ffifc,**** de
10.15 n[e) d, =  — ——f :
Equation 10.15 is plotted in Fig. 1020 for the temperatures T = 0, 300, and
1200 K.
It is interacting to determine the average electron energy at absolute zero.
To do this, we first obtain the total energy V at K, which is
r'r
I ■'„ = I fn(e) de
Table 10.4.
SOME FERMI ENERGIES.
Metal
Fermi energy, eV
Lithium
Li
4.72
Sodium
Na
3.12
Aluminum A]
11.8
I'cita^itim
K
2.14
Cesium
Ci
1.53
Copper
Cu
7.04
Ziue
Zti
11.0
Silver
Ag
5.51
Cold
Au
5.54
342
PROPERTIES OF MATTER
FIGURE 10.20 Distribution of electron energies in 3 metal at various temperatures.
Since at T = K all the electrons have energies less than or equal to the Fermi
energy t F , we may let
and
10.16
 fjfc.
The average electron energy *,, is this total energv dh ided by ihe number of
electrons present .V, which yields
10.17 ~<> = ^ f F
Since Fermi energies for metals are usually several electron volts, the average
electron energy in them at K will also be of this order of magnitude. The
temperature of an ideal gas whose molecules have an average kinetic energy
of 1 eV is 11,600 K; this means that, if free electrons Ijehaved classically, a sample
of copper would have to l>e at a temperature of about 50,000 K for lis electrons
to have the same average energy they actually have at K.
THE SOLID STATE
343
344
The considerable amount of kinetic energy possessed by the valence electrons
in the electron gas of a metal represents a repulsive influence, as noted in Sec.
10.5. The act of assembling a group of metal atoms into a solid requires that
additional energy be given to the valence electrons in order to elevate them
to the higher energy states required by the exclusion principle. The atoms in
a metallic solid, however, are closer together because of their bonds than they
would be otherwise. As a result the valence electrons are, on the average, closer
to an atomic nucleus in a metallic solid than they arc in an isolated metal atom.
These electrons accordingly have lower potential energies in the former case
than in the latter, sufficiently lower to lead to a net cohesive force even when
the added electron kinetic energy is taken into account.
•10.9 BRILLOU1N ZONES
We now turn to a more detailed examination of how allowed and forbidden bands
in a solid originate. The fundamental idea is that an electron in a crystal moves
in a region of periodically varying potential (Fig. 1021) rather than one of
constant potential, and as a result diffraction effects occur that limit the electron
to certain ranges of momenta that correspond to the allowed energy bauds. In
this way of thinking the interactions among the atoms influence valenceelectron
behavior indirectly through the lattice of the crystal these interactions bring
about, instead of directly as in the approach described in Set:. 10.fi An intuitive
approach will be used here, rather than a formal treatment based on Sehriidi tiger's
equation.
The de Broglic wavelength of a free electron of momentum p is
10.18
x = A
Free electron
FIGURE 10 21 The potential energy of in electron in a periodic array of positive ions.
positive km
^nnnnr
PROPERTIES OF MATTER
Unbound lowenergv electrons can travel freely through a crystal since their
wavelengths are long relative to the lattice spacing «. More energetic electrons,
such as those with the Kermi energy in a metal, have wavelengths comparable
utlli u, and such electrons are diffracted in precisely the same way as X ravs
(Sec. 2.4) or electrons in a l>eani (Sec. 3.5) directed at the crystal from the outside.
{When A is near a, 2a, 3m, ... in value, Eq. 10.18 no longer holds, as discussed
later.) An electron of wavelength A undergoes Bragg "reflection" from one of
the atomic planes in a crystal when it approaches the plane at the angle 0, where
from Eq. 2,8
10.19
»iA = 2« sin 8
it m 1,2,3,
It is customary to treat the situation of electron waves in a crystal hv replacing
A by the wave number k introduced in Sec. 3,3, where
i 25T
10.20 k = —
A
Wave number
The wave number is equal to the numlier of radians per meter in the wave train
it describes. Since the wave train moves in the same direction as the particle,
we can describe the wave train by means of a vector k. Bragg's formula in terms
of k is
10,21
k =
a sin 6
Figure 1 1122 shows Bragg reflection in a twodimensional square lattice.
Evidently we can express the Bragg condition by saying that reflection from the
vertical rows of ions occurs when the component of k in the .r direction, IL,
is equal to nn/u. Similarly, reflection from the horizontal rows occurs when
A„ = nx/a.
I jet us consider first those electrons whose wave numbers are sufficiently small
for them to avoid diffraction. U k is less than tr/o, the electron can move freelv
through the lattice in any direction. When k = <n/a, they are prevented from
ruining in the x or ;/ directions by diffraction. The more k exceeds ■a/a, the
more limited the possible directions of motion, until when k = ir/«sin45* =
V2n/u the electrons are diffracted even when the)' move diagonally through
the lattice.
The region in fcspacc that lowfc electrons can occupy without teing diffracted
is called the first lirillauin ^i»u?and is shown in Fig. 1023. The second Brillouin
/one is also shown; it contains electrons with k > t/u that do not fit into the
first zone yet which have sufficiently small propagation constants to avoid
diffraction by the diagonal sets of atomic planes in Fig. 1022. The second zone
contains electrons with k values from <n/a to 2w/« for electrons moving in the
THE SOLID STATE
345
positive ions
r
a
I
k
_ Mr
a sin 9
K
= ksinB
_ H7T
a
10.22
E =
2»i
It) the case of an electron in a crystal for which k < ir/a, there is practically
no interaction with the lattice, and I£ci. 10.22 is valid. Since the energy of such
an electron depends upon A, the contour lines of constant energy in a two
dimensional itspace are simply circles of constant A\ as in Fig. 1025. With
increasing k the constantenergy contour lines become progressively closer
together and also more anil more distorted. The reason for the first effect is
merely that E varies with k 2 . The reason for the second is almost equally
straightforward. The closer an electron is to die boundary of a Brillouiu zone
in Jtspaee, the closer it is to being diffracted by the actual crystal lattice. But
FIGURE 10 23 The first and second Brillouin tones of a twodimensional square lattice.
♦ *..
U^ + °
FIGURE 1022 Bragg reflection from the vertical raws of ions occurs when A, = iir/n.
" \ and "ij directions, with ili< possible range ol k values narrowing as the
diagonal directions arc approached. Further Brillouin /ones can he constructed
in the same manner. The extension of this analysis to actual threedimensional
structures leads to the Brilbuin zones shown in Fig. 1024.
"10.10 ORIGIN OF FORBIDDEN BANDS
The significance of the Brillouin zones becomes apparent when we examine the
energies of the electrons in each zone. The energy of a free electron is related
to its momentum p by
E =
2m
346
and hence to its wave numher k by
PROPERTIES OF MATTER
k =I
* a
^
I.
first Brillouin zone
t
second Brillouin
zone
*+f
k — X
V N
THE SOLID STATE
347
(a)
(b)
FIGURE 1024 First and second Brillouin zones in (a) lacecentered cubic structure and
(o) body ctniered cubic structure (tee Figs. 104 and 105)
in particle terms the diffraction occurs by virtue of the interaction of the electron
with the periodic arniv ol positive ions that occupy the lattice points, and the
stronger the interaction, the more the election's energy is affected.
Figure 1026 shows how E varies with k in the x direction. As I; approaches
17/a, E increases more slowly than ti' 2 k~/2m, the freeparticle figure. At k = w/a.
E has two values, the lower lielougiug to the first Brillouin zone and the higher
to the second zone. There is a definite gap between the possible energies in
the first and second brillouin zones which corresponds to the forbidden band
spoken of earlier. The same pattern continues as successively higher Brillonii,
•/ones are reached.
The energy discontinuity at the boundary of a Brillouin /one follows from
the fact that the limiting values of k correspond to standing waves rather than
traveling waves. For clarity we shall consider electrons moving in the X direction;
the extension of the argument to any other direction is straightforward. When
348 PROPERTIES OF MATTER
second Brillouin zone
First Brillouin /xmc
FIGURE 1025 Energy contours in electron volts In the first and second Brillouin rones of a
hypothetical square lattice.
FIGURE 1026 Electron energy E versus wave number k In the k, direction. The dashed line shows how
B varies wtth It lor a free electron.
allowed energies
47T 37T 27T IT
a a a a
u a a a
THE SOLID STATE 349
it = ±ir/fl, as we have seen, the waves are Braggreflected back and forth, and
so the only solutions of Sehrodinger's equation consist of standing waves whose
wavelength is equal to the periodicity of the lattice. There are two possibilities
for these standing waves for n a* 1, namely
10.3.1
jf>, = A sin —
a
irx
10.24 ip., = A cos —
a
The probability densities ^, a and if< 2  2 are plotted in Fig. 1027. Evidently
tf, z has its minima at the lattice points occupied by the positive ions, while
1^1' has its maxima at the lattice points. Since the charge density corresponding
to an electron wave function i£> is e#\ 2 , the charge density in the case of ^, is
concentrated between the positive ions while in the case of ^ it is concentrated
FIGURE 1027 Distributions ot the probability densities V', ' and y, '.
350 PROPERTIES OF MATTER
nCEJ
liirbiddcn
band
10
15
E,eV
FIGURE 1028 The distributions o! electron energies in the Brlllouin lones of Fig. 1025. The dashed
line Is the distribution predicted by the freeelectron theory.
at the positive ions. The potential energy of an electron in a lattice of positive
ions is greatest midway Itctwccn each pair of ions and least at the ions themselves,
so the electron energies E { and E, associated with the standing waves ^ and
v., are different. No other solutions are possible when k = ±^/a, and accordingly
no electron can have an energy between £, and E^.
Figure 1028 shows the distribution of electron energies that corresponds to
die Rrillouin /.ones pictured in Fig. 1025. At low energies (in this hypolhetical
situation for E < —2 eV) the curve is almost exactly the same as thai of Fig.
1020 based on the freeelectron theory, which is not surprising since at tow
energies k is small and the electrons in a periodic lattice then do behave like
free electrons. With increasing energy, however, the number of available energy
states goes beyond that of the freeelectron theory owing to the distortion of
the energy contours by the lattice: there are more different k values for each
energy. Then, when k = ^.v/a. I he energy contours reach the boundaries of
the first zone, and energies higher than about 4 eV (in this particular model)
are forbidden for electrons in the k r and k v directions although permitted in
other directions. As the energy goes farther and farther lieyond 4 eV, the
available energy states liccome restricted more and more to the corners of the
/one, and n{E) falls. Finally, at approximately 6% eV, there are no more states
and n(E) m 0. The lowest possible energy in the second /.one is somewhat less
than 10 eV, and another curve similar in shape to the first begins, (fere the
gap between the possible energies in the two zones is about 3 eV, and so die
forbidden Italic! is about 5 eV wide.
THE SOLID STATE
351
Although there must be an energy gap between successive Brillouin zones in
any given direction, the various gaps may overlap pe: nitted energies in other
directions so that there is no forbidden hand in the crystal as a whole. Figure
1029 contains graphs of E versus k for three directions in a crystal that has
a forbidden band and in a crystal whose allowed bands overlap sufficiently to
avoid having a forbidden band.
The electrical Ijehavior of a solid depends upon the degree of occupancy of
its energy bands as well as upon the nature of the band structure, as noted earlier.
There are two available energy states (one for each spin) in each band for each
Structura l unit in the crystal. (By "structural unit" in this context is meant an
atom in a metal or covalent elemental solid Such as diamond, a molecule in a
molecular solid, and an ion pair in an ionic solid.) A solid will he an insulator
if two conditions are met: (1) It must have an even number of valence electrons
per structural unit, and (2) the band that contains the highestenergy electrons
must be separated from the allowed band above it by an energy gap large
compared with kT. The reason for condition (1) is that it ensures that the
highestenergy band lie completely filled, and the reason for (2) is that none of
the electrons lie able to cross the gap to reach unfilled slates. Thus diamond,
with four valence electrons per atom, solid hydrogen, with two valence electrons
per II,, molecule, and NaCI, with eight valence electrons per Ha 4 — CI" ion pair,
all have wide forbidden bands in addition and are insulators. Figure 1030o shows
the energy contours of a hypothetical insulator.
A conductor is characterized by its violation of either (or both) of the above
conditions. Thus the alkali metals, with an odd number of valence electrons per
structural unit (namely one per atom), are conductors, as are such divalent metals
as magnesium and zinc which have overlapping energy bands. Figure lU'Mb
and c respectively show the energy contours of these two types of metal. When
the forbidden band in an insulator is narrow or the amount of overlap in a metal
is small, the electrical conductivity falls in the semiconductor region, and it is
not reallv correct to speak of the substance as either a metal or a nonmetal.
The boundary between filled and empty electron energy states in threedimen
sional fcspace is called the Fenni surface.
Experiments indicate that the conductivity of the divalent metals beryllium,
/inc. and cadmium is largely due to positive charge carriers, not to electrons.
This unexpected finding is readily accounted for on the basis of the band picture
by assuming that the overlap of the Fenni surface into the highest band is small,
leaving vacant states — which are holes — in the band below it. The holes in the
lower band carry the bulk of the current while the electrons in the upper band
play a minor role.
352
PROPERTIES OF MATTER
FIGURE 1029 £ versus k curves for three directions in two crystals. In (a) there is a forbid
den band, while In (b) the allowed energy bands overlap and there Is no forbidden band.
THE SOLID STATE
353
7.0nC IllHllillai icv
vacant iiuthv levels
Fermi level
L occupied
energy levels
FIGURE 1030 Electron energy contours and Fermi levels in three types ot solid: (a) Insulator; (b) mono
valent metal; (c) divalent metal. Energies are in electron votts.
10.11 EFFECTIVE MASS
An electron in a crystal interacts with the crystal lattice, and because of this
interaction its response to external forces is not, in general, the same as that
of a free electron. There is nothing unusual alxnit this phenomenon — no particle
subject to constraints behaves like a free particle. What is unusual is that the
deviations of an electron in a crystal from freeelectron behavior under the
influence of external forces can all he incorporated into the simple statement
that the effective mass of such an electron is not the same as its actual mass.
The most important results of the freeelectron theory of metals discussed in
Sections 10,7 and 10.8 can l>e incorporated in the more realistic band theory
merely by replacing the electron mass m by the average effective mass m" at
the Fermi surface. Thus the Fermi energy in a metal is given by
io. a
M„ =
ft a / 3JV V'
2m' WW
Fermi energy
where N/V is the density of valence electrons. Table 10.5 is a list of effective
mass ratios m°/m in metals.
Table 10.5.
EFFECTIVE MASS RATIOS »• u, AT THE
FERMI SURFACE IN SOME METALS.
Metal
mVm
Lithium l.i
J lt'T\ I 111 I Ml Ht
Sodium Va
Aluminum Al
Cnlwlt C«
Nickel Ni
Copper Cu
/in? Zn
Silver Ag
Platinum ft
1.2
1.6
1.2
0.97
N
28
1.01
0.85
0,99
13
Problems
) . What is the effect on the cohesive energy of ionic and eovalent crystals of
i«) van der Waals forces and (b) zeropoint oscillations of the ions and atoms
about their equilibrium positions?
THE SOLID STATE
355
356
2. The van dor Waals attraction between two lie atoms leads to it binding
energy of about 6 X Id ' cV at an equilibrium separation of about 3 A, Use
the uncertainty principle to show that, at ordinary pressures (<25 atm), solid
He cannot exist.
3. The Joulr Thomson effect refers to the drop in temperature a gas undergoes
when it passes slowly from a full container to an empty one through a porous
plug. Since the expansion is into a rigid container, no mechanical work is done.
F.xplain the JouleThomson effect in terms of the van der Waals attraction
between molecules.
4. The inu spaefngs and melting points of the sodium halides are as follows:
N»F NaCI NaBr Mat
Imi v)LH'iji!;, \
Mult ins I" 1 "' 1 ' *C
2.3
2.8
801
2.0
740
3.2
060
Explain the regular variation in these quantities with halogen atomic number.
5. I'se the notion of energy bands to explain the lullowing optical properties
of solids: (o) All metals are opaque to light of all wavelengths, (b) Semiconductors
arc transparent to infrared light although opaque to visible light, (c) Most
insulators are transparent lo visible tight.
6. The energy gap in silicon is ].] eV and in diamond it is 6 eV. Discuss the
transparency of these substances to visible light.
7. A small proportion of indium is incorporated in a germanium crystal. Is
the crystal an rttype or a jjtype semiconductor?
8. A small proportion of antimony is ineoq>orated in a germanium crystal. Is
the crystal an ntype or a jjtype semiconductor?
9. What is the connection between the fact that the free electrons in a metal
obey Fermi statistics and the fact that the photoelectric effect is virtually tem
po rat ii re  i nde pe nden t ?
10. (a) How much energy is required to form a K' and I ion pair from a pair
of these atoms? (b) What must the separation be lietwcen a K' and an 1" ion
if their total energy is to lie zero?
1 1. {a) How much energy is required lo form a Li' and Br" ion pair from a
pair of these atoms? (b) What must the separation be between a l.i' and a Hi
ion if their total energy is to be zero?
12. Show that the first five terms in the series for the Madclung constant of
PROPERTIES OF MATTER
Nat '1 are
«««%+ S
6 j 24
V5 ' V5 2 vS
13. (a) The ionization energy of potassium is 4.34 eV and the electron affinity
of chlorine is 3.61 cV. The Madehmg constant for the KG structure is 1.74S
and the distance between ions of opposite sign is 3.14 A. On the basis of these
data only, compute the cohesive energy of KC1. (b) The observed cohesive energy
of KG is 6.42 eV per ion pair. On the assumption that the difference l>etvveen
this figure and that obtained in (a) is due to the exclusion principle repulsion,
find the exponent n in the formula Br~" for the potential energy arising from
this source,
14. Repeat Proh. 13 for LiCl, in which the Madclung constant is 1.748. the
inn spacing is 2.57 A, and the observed cohesive energy is 6.8 eV per ion pair.
The ionization energy of Li is 5.4 eV.
15. The potential energy V{x) of a pair of atoms in a solid that arc displaced
by x from their equilibrium separation at K may be written V(*) = as 8 
tor 3 — ex*, where the anharmonic terms —far 8 and — ex' represent, respectively,
the asymmetry introduced by the repulsive forces between the atoms and the
leveling off of the attractive forces at large displacements. At a temperature
T the likelihood that a displacement x will occur relative to the likelihood of
no displacement is e~ v/kT , so that the average displacement .vat this temperature
is
x =
f xe v ' kT <lx
 X ___
J* e VJkT dx
Show that, for small displacements, x =; 3hfcT/4fl a . (This is the reason that the
change in length of a solid when its temperature changes is proportional to AT.)
° 16. The Fermi energy iu copper is 7.04 eV. (a) Approximately what percentage
of the free electrons in copper are in excited states at room temperature? {!>)
At the melting point of copper, 1083"C?
' 17. The Fermi energy in silver is 5.51 eV. (a) What is the average energy of
the free electrons in silver at K? (b) What temperature is necessary for the
average molecular energy in an ideal gas to have this value? (c) What is the
speed of an election with this energy?
* 1 8. The density of aluminum is 2.70 g/em 3 and its atomic weight is 26,97, The
THE SOLID STATE
357
358
electronic structure of aluminum is given in Table 7,2 (note: the energy difference
lie! ween 3* and 3p electrons is very small), and the effective mass of an electron
in aluminum is 0.97 m r . Calculate the Fermi energy in aluminum.
*19. The density of zinc is 7.13 g/cm a and its atomic weight is 65.4. The
electronic structure of zinc is given in Table 7,2, and the effective mass of an
electron in zinc is 0.85 m r . Calculate the Fermi energy in /.inc.
20. Kxplain why the free electrons in a metal make only a minor contribution
to its specific heat,
"21. Find the ratio between the kinetic energies of an electron in a two
dimensional square lattice which has Jt, = k^ = it la and an electron which has
k x = 7T la, Jr„ s 0.
°22. Draw the third Brillouin zone of the twodimensional square lattice whose
first two Brillouin /onus arc shown in Fig. 1023.
23. Phosphorus is present in a germanium sample. Assume that one of its live
valence electrons revolves in a Bohr orbit around each P" ion in the germanium
lattice, in'* if tile effective mass >! the electron i< H.ITm, and the iliefelnc
constant of germanium is 1 ft, find the radius of the first Bohr orbit of the electron.
''■ The energy gap l)Ctweeu the valence and conduction bands in germanium
is 0.65 eV. How does the ionization energy of the above electron compare with
this energy' and with kT at room temperature?
24. Repeal Prob. 23 for a silicon sample that contains arsenic. The effective
mass ol an electron in silicon is about 0.31 m r , the dielectric constant of silicon
is 12, and the energy gap in silicon is 1.1 eV.
"25. The calculation of the Fenni energy in copper made in Sec, 10.7 did not
take into account the difference between m r and m°, and yet the n F value
obtained was approximately correct. Why?
26. The effective mass m° of a current carrier in a semiconductor can be
directly determined by means of a cyclotron resonance experiment in which the
carriers (whether electrons or holes) move in spiral orbits about the direction
of an externally applied magnetic field B. An alternating electric field is applied
perpendicular to B, and resonant absorption of energy from this field occurs when
i Is I requeue) V is equal to the frequency of tv\ olulion v r of the carrier, (a) Derive
an equation for r r in terms of »i°, e, and B. (ft) In a certain experiment, B = 0.1 T
and maximum absorption is found to occur at v = 1.1 X H) 1 " Hz. Find m'.
(c) Find the maximum orbital radius of a charge carrier in this experiment whose
speed is 3 X H) 4 m/s.
PROPERTIES OF MATTER
THE ATOMIC NUCLEUS
11
Thus far we have regarded the atomic nucleus solely as a point mass that possesses
positive charge. The behavior of atomic electrons is responsible for the chief
properties (except mass) of atoms, molecules, and solids, not the behavior of
atomic nuclei. But the nucleus itself is far from insignificance in the grand scheme
of things. For instance, the elements exist because of the ability of nuclei to
possess multiple electric charges, and explaining this ability is the central problem
of nuclear physics. Furthermore, the energy that powers the continuing e volut ion
of the universe apparently can all be traced to nuclear reactions and trans
formations. And, of course, the mundane applications of nuclear energy are
familiar enough.
11.1 ATOMIC MASSES
The nucleus of an atom contains nearly all its mass, and a good deal of information
on nuclear properties can lie inferred from a knowledge of atomic masses. An
instrument used to measure atomic masses is called a nuiits spectrometer; modern
spectrometers and techniques are capable of precisions of better than 1 part
in 108.
Atomic masses refer to the masses of neutral atoms, not of stripped nuclei.
ThaH the masses of the orbital electrons and the mass equivalent of their binding
energies are incorporated in the figures, Atomic masses are conventionally
expressed in mass units (u) such that the mass of the most abundant type of
carbon atom is, by definition, exactly 12.(HX) . . . u. The value of a mass unit is,
to five significant figures,
I u = l.fifiOi X 10 "kg
and its energy equivalent is 931.48 MeV.
Not long after the development of methods for determining atomic masses
early in this century, it was discovered that not all of the atoms of a particular
361
element have the same mass. The different varieties of the same element sire
called its isotopes. Another widely used term, nuclide, refers to a particular
species of nucleus; thus each isotope of ;m clement is a nuclide.
The atomic masses listed in Table 7. 1 refer to the murage atomic mass of each
stamen I, which is the quantity of primary interest to chemists. Table 11.1
contains the atomic masses and relative abundances of the five stable isotopes
of zinc. The individual masses range from 63.92914 u to 69.92535 ii, and the
relative abundances range from 0.fi2 percent lo 48.89 percent The average mass
is 65.38 u, which is accordingly the atomic mass of /inc. Twenty elements are
composed of only a single nuclide each; beryllium, Huorine, sodium, and alumi
num are examples.
Even hydrogen is found to have isotopes, though the two heavier ones make
up only about 0.015 percent of natural hydrogen. Their atomic masses are
1.007825, 2.1H4102, and 3.01605 u; Ihe heavier isotopes are known as t Ututatjut n
and Mliutit respectively. (Tritium nuclei, called Iritom, are unstable and decay
i adioaetively into an isotope of helium.) The nucleus of the lightest isotope is
the proton, whose mass of
m„= 1.0072766 u
= 1.6725 X 10' 27 kg
is, within experimental error, equal to the mass of the entire atom minus the
mass of the electron it also contains. The proton, like the electron, is an elemen
tary particle rather than a composite ol other particles. (The notion of elemen
tary particle is considered in some detail in Ghap. 13.)
An interesting regularity is apparent in listings of nuclide masses: the values
are always very close to being integral multiples of the mass of the hydrogen
atom. 1 .007825 u. For example, the deuterium atom is approximately twice as
anSHfra .is the hydrogen atom, and the tritium atom approximately three times
as massive. The masses of the zinc isotopes listed in Table 11.1 further illustrate
this pattern, lieing quite near to 64, 66, 67, 68, and 70 times the hydrogenatom
Table 11.1.
PROPERTIES OF THE STABLE ISOTOPES OF ZINC, / = 30.
Mass number
of Isotope
Atomic mass, U
Relative abundance. %
64
SO
87
68
70
B.1.929I4
05,92005
00.02715
07.92436
09.92535
48,89
27.81
4.11
tN.se
0.62
mass. It is therefore tempting to regard all nuclei as consisting of protons —
hydrogen nuclei— somehow bound together. However, a closer look rotes out
this notion, since a nuclide mass is invariably greater than the mass of a number
of hydrogen atoms equal to its atomic number Z— and the nuclear charge of
an atom is fZe. The atomic number of /.inc is 30, but its isotopes all have
masses more than double that of 30 hydrogen atoms.
Another possibility comes to mind. Perhaps electrons may be present in nuclei
which neutralize the positive charge of some of the protons. Thus the helium
nucleus, whose mass is four times that of the proton though its charge is only
+ 2e, would be regarded as being composed of four protons and two electrons.
This explanation is buttressed by the fact that certain radioactive nuclei sponta
neouslv emit electrons, a phenomenon called Ixtta decay, whose occurence is
easy to account for if electrons are present in nuclei.
Despite the superficial attraction of the hypothesis of nuclear electrons, how
ever, there are a number of strong arguments against it:
1. Nuclear size. Nuclei are only —10 M m across. To confine an electron
to so small a region requires, by the uncertainly principle, an uncertainty in
its momentum of \p > 1.1 X 10~*° kgm/s, as was calculated in Sec. 3.7. The
electron momentum must l>e at least as large as the minimum value of Ap. The
electron kinetic energy that corresponds to a momentum of 1.1 X 10" kgm/s
is 21 MeV. (This figure may also lw obtained by calculating the lowest energy
level of an electron in a box of nuclear dimensions; since T > m t) c 2 , the latter
calculation must be made relativistically.) However, the electrons emitted during
beta decay have energies of only 2 or 3 MeV— an order of magnitude smaller
than the energies they must have had within the nucleus if they were to have
existed there.
We might remark that the uncertainly principle yields a very different result
when applied lo protons wilhin a nucleus. For a proton with a momentum of
l.l X 10" ' i0 kgm/s. T < «i[,<? 2 , and its kinetic energy can be calculated
classically. We have
T= Pi
2m
(1.1 X 10 2 " kgm/s)"
2 X 1.67 X Hr* 7 kg
= 3.6 X 10 " j
= 0.23 MeV
The presence of protons with such kinetic energies in a nucleus is entirely
plausible.
362
THE NUCLEUS
THE ATOMIC NUCLEUS 363
364
2. Suclear spin. Protons and electrons are Fermi particles with spins of %,
that is, angular momenta of ] ,Ji. Thus nuclei with an even iiumlrer of protons
plus electrons should have integral spins, while those with an odd munlier of
protons plus electrons should have halfintegral spias. This prediction is not
obeyed The fact that the denteron, which is the nucleus of an isotope of
hydrogen, has an atomic number of 1 and a mass number of 2, would he inter
preted as implying the presence of two protons and one electron. Depending
upon the orientations of the particles, the nuclear spin of '\\\ should therefore
be %> Yf —%. <"■ —%■ However, the observed spin of the denteron is J,
something that cannot be reconciled with the hypothesis of unclear electrons.
3. Mttgae U e moment. The proton has a magnetic moment only about
1.5 x 10"' that of the electron, so that nuclear magnetic moments ought to Iw
of the same order of magnitude as that of the electron if electrons are present
in nuclei. However, the observed magnetic moments of nuclei arc comparable
with that of the proton, not with that of the electron, a discrepancy that cannot
he understood if electrons arc nuclear constituents,
4. Elect mn nuclear interaction. It is observed that the forces that act between
nuclear particles lead to binding energies of the order of 8 MeV per particle.
It is therefore hard to see why, if electrons can interact strongly enough with
protons to form nuclei, the orbital electrons in an atom interact onlv elec
trostatically with its nucleus. That is, how can half the electrons in an atom
escape the strong binding of the other half? Furthermore, when fast electrons
are scattered by nuclei, they hehave as though acted upon solely by electrostatic
forces, while the nuclear scattering of fast protons reveals departures from
electrostatic influences that can Ik ascribed only to a specifically nuclear force.
The difficulties of the nuclear electron hypothesis were known for some time
Iwfore the correct explanation for nuclear masses came to light, but there seemed
to l>e no serious alternative. The problem of the mysterious ingredient Iwsides
the proton in atomic nuclei was not solved until 1932.
11.2 THE NEUTRON
The composition of atomic nuclei was finally understood in 1932. Two years
earlier the German physicists W. Bothe and II. Becker had liombarded beryllium
widi alpha particles from a sample of polonium and found that radiation was
emitted which was able to penetrate matter readily, Bothe and Becker ascer
tained that the radiation did not coasist of charged particles and assumed, quite
naturally, that it consisted of gamma rays, (Camilla rays are electromagnetic
waves of extremely short wavelength.) The ability of the radiation to pass
THE NUCLEUS
through as much as several centimeters of lead without being absorbed suggested
gamma rays of unpreccdentedly short wavelength. Other physicists became
interested in this radiation, and a number of experiments were performed to
determine its piopertics in detail. In one such experiment Irene Curie and
P. Joliot observed that when the radiation struck a slab of paraffin, a hydrogen
rich substance, protons were knocked out. At first glance this is not very sur
prising: X rays can give energy to electrons in Compton collisions, and there
is no reason why shorterwavelength gamma rays cannot give energy to protoas
in similar processes.
Curie and Joliot found proton recoil energies of up to about 5.3 MeV. From
Fq. 2.15 for the Compton effect the minimum gammaray photon energy £ = hr
needed to transfer the kinetic energy T lo a proton can be calculated. The result
is a minimum initial gammaray photon energy of 53 MeV, This seemed peculiar
because no nuclear radiation known at the time had more than a small fraction
of this considerable energy. The peculiarity liecame even more striking when
it was calculated that the presumed reaction of an alpha particle and a beryllium
nucleus to yield a carbon nucleus would result in a mass decrease of ().(H 144 u,
which is equivalent to only 10.7 MeV — onefifth the energy needed by a
gammafay photon if it is to knock 5.3 MeV protons out of paraffin.
In 1932 James Chadwick, an associate of Hutherford, proposed an alternative
hypothesis for the nowmysterious radiation emitted by beryllium when bon
barded by alpha particles. lie assumed that the radiation consisted of neutral
particles whose mass is approximately the same as that of the proton. The
electrical neutrality of these particles, which were called iwulmws, accounted
Tor their ability to penetrate matter readily. Their mass accounted nicely for
die observed proton recoil cnci»ics: ,i moving particle colliding headon with
one at rest whose mass is the same can transfer all of its kinetic energy to the
latter. A maximum proton energy of 5.3 MeV thus implies a neutron energy
of 5,3 MeV, not the 53 MeV required by a gamma ray to cause the same effect.
Other experiments had shown that such light nuclei as those ul helium, carbon,
and nitrogen could also be knocked mil of appropriate absorbers In the bcrvllium
radiation, and the measurements tnade of the energies of these nuclei fit in well
with the neutron hypothesis. In fact, Chadwick arrived at the neutronmass figure
of m n iiii p from an analysis of observed proton and nitrogen nuclei recoil
energies; no other mass gave as good agreement with the experimental data.
Before we consider the role of the neutron in nuclear structure, we should
note that it is not a stable particle outside nuclei. The free neutron decays
radioactively into a proton, an electron, and an antiueutrino: the half life of the
free neutron is I0A min.
Immediately after its discovery the neutron was recognized as the missing
ingredient in atomic nuclei. Its mass of
THE ATOMIC NUCLEUS 365
366
m n  1.0086654(1
= 1.6748 X 10 2T kg
which is slightly more than thai of the proton, its electrical neutrality, and its
spin of % all fit in perfectly with the observed properties of nuclei when it is
assumed that nuclei are composed solely of neutrons and protons.
The Following teniis and symlrols arc widely used to describe a nucleus:
Z = atomic number = number of protons
.V = neutron number = number of neutrons
A = Z + .V = mass number = total number of neutrons and protons
The term nucleon refers to both protons and neutrons, so that the mass number
A is the number of nucleoos in a particular nucleus. Nuclides are identified
according to the scheme
where X is the chemical symlxj] of the species. Thus the arsenic isotope of mass
Dumber 75 is denoted by
Mm i' the atomic number of arsenic is 33. Similarly a nucleus of ordinary hydro
gen, which is a proton, is denoted by
JH
Here the atomic and mass numbers are the same because no neutrons are present.
lire fact that nuclei are composed of neutrons as well as protons immediately
explains die existence of isotopes: the isotopes of an element all contain the same
numbers of protons hut have different numl>ers of neutrons. Since its nuclear
charge is what is ultimately responsible for the characteristic properties of an
iitom. the isotopes of an element all have identical chemical behavior and differ
conspicuously only in mass.
11.3 STABLE NUCLEI
Not all combinations of neutrons and protons form stable nuclei. In general,
light nuclei (A < 20) contain approximately equal numbers of neutrons and
protons, while in heavier nuclei the proportion of neutrons becomes progressively
greater. Tin's is evident from Fig. 111, which is a plot of N versus Z for stable
nuclei. The tendency for .V to equal Z follows from the existence of nuclear
energy levels, whose origin and properties we shall examine shortly. Nucleons,
THE NUCLEUS
10 20 30 40 50 60
PROTON NUMBER (Z)
70 80 90
FIGURE 111 Neutron proton diagram for stable nuclides. There are no stable nuclides with
/ 43 or 61. with \ 19. 35, 39, 15. 61. 89, 115, or 126, or with I m%  .\ 5 or 8. All
nuclides with / > 83, \ 126, and A • 209 are unstable.
THE ATOMIC NUCLEUS
367
i
368
which have spins of V,. oliev the exclusion prim iplc. As a result, each nuclear
energy level can contain two neutrons of opposite spins and two protons of
opposite spins. Energy levels in nuclei are filled in sequence, just as energy levels
in atoms are, to achieve configurations of minimum energy and therefore maxi
mum stability. A nucleus with, say, three neutrons anil one proton outside tiled
inner levels will have more energy than one with two neutrons and two protons
in the same situation, since in the former case one of the neutrons must go inlo
a higher level while in the latter case all four nucleons fit into the lowest available
level. Figure 1 12 shows how this notion accounts for the absence of a stable
'i(B isotope while permitting 'jjC to exist.
The preceding argument is only part of the story. Protons are positively
charged and repel one another electrostatically . This repulsion becomes SO greai
in nuclei with more than K) protons or so that an excess of neutrons, which
produce only attractive forces, is required for stability; thus the curve of Fig.
111 departs more and more from the .V — 7, line as 7. increases. Even in light
nuclei .V may exceed A. hut is never smaller; 'jB is stable, for instance, but not
Iff;
Nuclear forces are limited in range, and as a result nucleons interact strongly
only with Iheir nearest neighbors. This effect is referred to as the saturation
of nuclear forces. Because the coulomb repulsion of the protons is appreciable
throughout the entire nucleus, Ihere is a limit to the ability of neutrons to prevent
the disruption of a large nucleus. This limit is represented by the bismuth isotope
"gBi, which is the heaviest stable nut lick*. All nuclei with Z > 83 and A > 2(19
spontaneously transform themselves into lighter ones through the emission of
one or more alpha particles, which are II le nuclei. Since an alpha particle
FIGURE 1 12 Simplified energy 'evel diagrams of stable boron and carbon Isotopes. The exclusion princi
ple limits the occupancy of each level to two neutrons of opposite spin and two protons of apposite spin.
Q neutron
m proton
?»
THE NUCLEUS
>>B
12,
13,.
UJ
CD
Z
o
er.
stability curve
negative beta decay
iV decreases
7 &
Z decreases by 2
positive beta decay
Z decreases by 1
/ Z decreases by 1
* N decreases by t
PROTON NUMBER (7.)
FIGURE 113 Alpha and beta decays permit en unstable nucleus to (each a stable configuration.
consists of two protons and two neutrons, an alpha decay reduces the Z and
the .V of the original nucleus by two each. If the resulting daughter nucleus
has either too smalt or loo large a neutron/proton ratio for stability, it mav beta
decay to a more appropriate configuration. In negative beta decay, a neutron
is transformed into a proton and an electron:
u — * p + e~
The electron leaves the nucleus and is observed as a "l>eta particle." In positive
beta decay, a proton becomes a neutron and a positron is emitted:
Tims negative beta decay decreases the proportion of neutrons and positive beta
decay increases it. Figure 1 13 shows how alpha and beta decays enable stability
to be achieved. Kadioactivity is considered in more detail in Chap. 12.
THE ATOMIC NUCLEUS 369
370
11.4 NUCLEAR SIZES AND SHAPES
The Rutherford scattering experiment provided the firsl evidence that nuclei an:
of finite size. In that experiment, as we saw in Chap. 4, an incident alpha particle
is deflected by a target nucleus in a manner consistent with Coulomb's law
provided die distance between them exceeds about 10 " in. For smaller separa
tions the predictions of Coulomb's law are not obeyed because the nucleus no
longer appears as a point charge to the alpha particle.
Since Rutherford's time a variety of experiments have been performed to
determine nuclear dimensions, with particle scattering still a Favored technique.
Fast electrons and neutrons are ideal for this purpose, since an electron interacts
with a nucleus only through electrical forces while a neutron interacts only
through specifically nuclear forces. Thus election scattering provides information
on the distribution of charge in a nucleus and neutron scattering provides
information on the distribution of nuclear matter. In both cases the de liroglie
wavelength of the particle must lie smaller than the radius of the nucleus wider
study {see Frob. 3 of Chap. 3),
Actual experiments on the sizes of nuclei have employed electrons of several
hundred MeV to over 1 CeV (1 CeV = J r 0<K) MeV  111" eV) and neutrons of
20 MeV and up. In every case it is found that the volume of a nucleus is directly
proportional to the number of nucleons it contains, which is its mass number
A. If a nuclear radius is R. the corresponding volume is %srH 3 and so R :l is
proportional lo \. T]u, relationship is iisualls expressed in inVGtSe form as
n.i
R tx fl A ,/;s
Nuclear radii
The value of R u is
/{„= 1.2 x 10 ls m
The indefiniteness in fl„ Ls a consequence, not just of experimental error, but
of die characters oi the various experiments: electrons and neutrons interact
differently with nuclei. The value of R„ is slightly smaller when it is deduced
from electron scattering, which implies that nuclear matter and nuclear charge
are not identically distributed throughout a nucleus.
As we saw in the earlier part of this book, the angstrom unit (1 A = 10 "' m)
is a convenient unit of length for expressing distances in the atomic realm. For
example, the radius of the hydrogen atom is 0.53 A, the C and O atoms in a
CO molecule are 1.13 A apart, and die Na + and CI'" ions in crystalline XaCI
are 2.81 A apart. Nuclei are so small diat the fermi (fin), only 10 9 the size
of the angstrom, is an appropriate unit of length:
THE NUCLEUS
1 fermi = 1 fin = I() ls m
Hence we can write
11.2 fl= l,2A 1/a fin
for nuclear radii. From this formula we find that the radius of the *C nucleus
is
fi = 1.2 X(12)" 3 fm=:2.7fm
Similarly, the radius of the 'JAg nucleus is 5,7 rrn and that of the *§§U nucleus
is 7.4 fin.
Xow that we know nuclear sizes as well as masses, we can compute the density
of nuclear matter. In the case of ^C, whose atomic mass is 12.0 u, we have
for the nuclear density (the masses and binding energies of the six electrons may
be neglected here)
P =
_ 12.nuXl.66x 10"kg/u
%* X(2.7x lO^m) 3
 2 X 10" kg/m :i
This figure— equivalent to 3 billion tons per cubic inch!— is essentially the same
for all nuclei. Certain stars, known as "white dwarfs," are composed of atoms
whose electron shells have collapsed owing to enormous pressure, and the
densities of such stars approach that of pure nuclear matter.
We have been assuming that nuclei are spherical. How can nuclear shapes
be determined? If the distribution of charge in a nucleus is not spherically
symmetric, the nucleus will have an electric quadrupole moment. A nuclear
quadrupole moment will interact with the orbital electrons of the atom, and
the consequent shifts in atomic energy levels will lead to hyperfine splitting of
the spectral lines. Of course, this source of hyperfine structure must be distin
guished from that due to the magnetic moment of the nucleus, but when this
is done it is found that deviations from sphericity actually do occur in nuclei
whose spin quantum numbers are ! or more. Such nuclei may be prolate or
oblate spheroids, but the difference between major and minor axes never exceeds
20 percent and is usually much less. For almost all purposes it is sufficient
to regard nuclei as being spherical; nevertheless the departures from sphericity,
small as they are, furnish valuable information on nuclear structure.
THE ATOMIC NUCLEUS
371
372
11.5 BINDING ENERGY
\ Stable iitnni Invariably has a smaller mass than the ABU of the moans of its
constituent purlitks. Tlic deuterium atom JH, For example, has a mass of
2.014l()2ii T while the mass of a hydrogen atom (}ll) plus thai of a neutron is
"Vd^™ + '». = 10OT825U + LOOSOflSti
= 2.016490 V
which is O.0023N8 n greater. Since a detiteritiin nucleus — called a deuteron—K
composed of a proton and a neutron, and Iwth \U and j'H have single orbital
electrons, it is evident that the mass tiefcrt of 0.002388 n is related to the binding
of a proton and a neutron to form a dci ileum. V mass of 0.002388 11 is equal
to
0.002388 u X 931 MeV/u = 2.2.3 MeV
When a deuteron is formed from a free proton and neutron, then, 2.23 MeV
of energy is liberated. Conversely, 2.23 MeV must be supplied from an external
source to break a deuteron up into a proton and a neutron. This inference is
supported by experiments on the photodisiutegrulion of the deuteron, which show
that u gammaray photon must have an energy of at least 2.23 MeV to disrupt
a deuteron (Fig. 114).
The energy equivalent of the mass defect in a nucleus is called its binding
energy and is a measure of the stability of the nucleus. Binding energies arise
from the action of the forces that hold uucleons together to form nuclei, jusl
as ionization energies of atoms, which must be provided to remove electrons
from them, arise from the action of electrostatic forces. Binding energies range
from 2.23 MeV for the deuteron, which is the smallest compound nucleus, up
to 1,640 MeV for '$Bi, the heaviest stable nucleus,
FIGURE 114 The binding energy at the deuteron is 2.23 MeV. which li confirmed by experiments that
show that a gammaray p hoi on with a minimum energy of 2.23 MaV can split a deuteron into a tree neu
tron and a free proton.
2.2.1 MeV
/WW
photon
tic nil' mi i
®
proton
o
neutron
THE NUCLEUS
100 150
MASS NUMBER, A
200
250
FIGURE 1 15 Binding energy per nucleon as a function of mass number. The peak st V = A corresponds
to the !He nucleus.
The binding energy per nucleoli, arrived at by dividing the total binding energy
of a nucleus by the number of nucleons it contains, is a most interesting miaiitilv .
The binding energy per nucleon is plotted as a function of mass number A in
Fig. 1 15. The curve rises steeply at first and then more gradually until it reaches
a maximum of 8.79 MeV at A = 56, corresponding lo the iron nucleus fgFc,
and then drops slowly to about 7.6 MeV at the highest mass numbers. Evidently
nuclei of intermediate mass are the most stable, since the greatest amount of
energy must be supplied to liberate each of their nucleons. This fact suggests
that energy will be evolved if heavy nuclei can somehow lie split into lighter
ones or if light nuclei can somehow be joined to form heavier ones. The former
process is known as nuclear fission and the latter as nuclear fusion, and both
indeed occur under proper circumstances and do evolve energy as predicted.
Nuclear binding energies are strikingly large. To appreciate their magnitude,
it is helpful to convert the figures from MeV/ nucleon to more familiar units,
say kcal/kg. Since 1 eV = 1.60 X 10 ,n J and 1 J = 2.39 X 10" kcal, we find
that 1 MeV = 3.83 X 10 IT kcal. One mass unit is equal to 1.66 X 10 ^ kg,
and each nucleoli in a nucleus has a mass of very nearly 1 u. Hence
1
MeV _ 3.83 X IP" 17 kcal
nucleon
3.83 X 10~" kcal  i v U \u> kcal
1.66 X 102 kg ~ Ail x kg
A binding energy of 8 MeV/nucleon, a typical value, is therefore equivalent
to 1.85 x 10" kcal/kg. By contrast, the heat of vaporization of water is a mere
540 kcal/kg, and even the heat of combustion of gasoline, 1.13 x KH kcal/kg,
is 10 million times smaller.
THE ATOMIC NUCLEUS
373
374
•11.6 THE DEUTERON
The unique shortrange forces that hind nucleonsso securely into nuclei constitute
by far the strongest class of forces known. Unfortunately nuclear forces are
nowhere near as well understood as electrical forces, and in consequence the
theory of nuclear structure is still primitive as compared with the theory oi
atomic structure. However, even without a satisfactory understanding of nuclear
forces, considerable progress has been made in recent years in interpreting the
properties and behavior of nuclei in terms of detailed models, and we shall
examine some of the concepts embodied in these models in this chapter. Before
looking intq any of these theories, though, it is instructive to see what can l>c
revealed by even a very general approach. The simplest nucleus containing more
than one uucleoo is the deuteron, which consists of a proton and a neutron.
The deuteron binding energy is 2.23 MeV, a figure that can be obtained either
From the discrepancy in mass between m dcula . on and »i p + m. or from photo
disintegration experiments which show that only gamma rays with hv > 2.23
MeV can disrupt dcuterons into their constituent nucleons. In Chap. 6 we
analyzed another twobody system, the hydrogen atom, with the help of quantum
mechanics, but in that case the precise nature of the force between the proton
and the electron was known. If a force law is known for an interaction, the
corresponding potential energy function V can be found and substituted into
Schrodinger's equation. Our understanding of nuclear forces is less complete than
our understanding of coulomb forces, however, and so it is not possible to discuss
the deuteron in as much quantitative detail as the hydrogen atom.
The actual potential energy V of the deuteron, that is. the potential energy
of either nueleon with respect to the other, depends upon the distance r between
the centers of the neutron and proton more or less as shown by the solid line
in Fig. 116. (The repulsive "core" perhaps 0,4 X W~ a m in radius expresses
the inability of nucleons to mesh together more than a certain amount.) We
shall approximate this V(r) by the "square well" shown as a dashed line in the
figure. This approximation means that we consider the nuclear force between
neutron and proton to be zero when they .are more than r„ apart, and to have
a constant magnitude, leading to the constant potential energy  V , when they
are closer together than r„. Thus the parameters V n and r are representative
of the strength and range, respectively, of the interaction holding the deuteron
together, and the squarewell potential itself is representative of the shortrange
character of the interaction.
A squarewell potential means that V is a function of r alone, and therefore,
as in the case of other centralforce potentials, it is easiest to examine the problem
in a spherical polarcoordinate system ;si:e Rg. fill In spherical polar coordi
nates Sehrodinger's equation for a particle of mass m is, with ft = fc/2w,
THE NUCLEUS
11,3
&H9
i
r sin a0
(2)
Let us choose the particle in question tn be the neutron, so that we imagine
it moving in the force field of the proton. (The opposite choice would, of course,
yield identical results.) We note from Fig. 1 1 6 that E, the total energy of the
neutron, is negative and is the same as the binding energy of the deuteron.
In analyzing the hydrogen atom, where one particle is much heavier than the
other, it is still necessary to consider the effects of nuclear motion, and we did
this in Chaps. 4 and ft by replacing the electron mass m f by its reduced mass
in'. In this way tin; problem of a proton and an electron moving about a common
center of mass is replaced by the problem of a single particle of mass in' moving
about a fixed point. A similar procedure is even more appropriate here, since
neutron and proton masses are almost the same. According to Kq. 4,27, the
reduced mass of a neutronproton system is
11.4
hi in
m'= " "
in. + in.
and so we replace the m of Eq. 11.3 with in' as given alnive.
FIGURE 1 16 The actual potential energy of either proton or neutron In a deuteron and the squarewell
approximation to this potential energy at functions of the distance between proton and neutron.
total energy E
2.23 MeV
kinetic energy T
potential energy V
actual potential
energy of nueleon
squarewell approximation
THE ATOMIC NUCLEUS 375
376
We now assume that the solution of Eq. 11.3 can be written as the prod net
of radial and angular functions,
115
+</,*,♦) = fl(l)©(ff)*fc)
As before, the function R(r) describes how the wave function ^ varies along a
radius vector from the nucleus, with $ and <* constant; the function 8(0 ) describes
how if, varies with zenith angle along a meridian on a sphere centered at r = 0,
with r and <f> coaslant; and the function *{$>) describes how ^ varies with azimuth
angle <ji along a parallel on a sphere centered at r = 0, with r and 9 constant.
Although angular mutton can occur in a squarewell potential, uur interest
lor the moment is in radial motion, that is, in oscillations of the neutron and
proton about their center of mass. If there is no angular motion, <> and <t> are
both constant and their derivatives are zero. With dip/riS = d^ift/d^ 2 = 0, Eq.
11.3 becomes
11.6
W^f)*****"
A further simplification can be made by letting
11.7 u(r) = tR{t)
In terms of the new function u the wave equation becomes
dr fi 2
Because V has two different values, V =  V u inside the well and V = outside
it, there are two different solutions to Eq. 1 1.8, ti t for r ^ r a and i/ ([ for r > r .
Inside the well the wave equation is
2m'
^? + p<£ + v ) Ul = o
or, if we let
11.9
2m'
fl 2 = ^(E + VJ
(P
it becomes simply
rf 2 w,
11.10
ir* + a2 "> * "
(We note from Fig. 116 that, since V  > £, the quantities E + V and hence
a 2 are positive.) Equation 1 1 . 10 is the same as the wave equation for a particle
in a box of Chap. 5, and again the solution is
THE NUCLEUS
u.ii
ii 1 = A cos ar + B sin ar
We recall that the radial wave function fi is given by fi = »/r, which means
that the cosine solution must lie discarded if fi is not to be infinite at r = 0.
Hence A = and «j inside the well is just
11.12 II  = B '.ill ar
Outside the well V* = and so
11.13
<i 2 «„ 2m' _,
The total energy E of the neutron is a negative quantity since it is bound to
the proton. Therefore
11.14
fi* = ~(E)
is a positive quantity, and we can write
d 2 u,
11.15
dr
P.  ft* = o
The solution of Eq. 1 1.15 is
n.16 «„ = Ce~ bT + De 1 "
Because it must be true that u — * as r — ► oo, we conclude that D = 0. Hence
outside the well
ill? u n = Ce' bf
*11.7 GROUND STATE OF THE DEUTERON
We now have expressions for u (and hence for >£■) both inside and outside the
well, and it remains to match these expressions and their first derivatives at the
well boundary since it is necessary that Ixith u and du/dr !>e continuous every
where in the region for which they are defined. At r = r n , then.
11.18
fi sin ar a » Ce  "*
and
du, du u
dr dr
11.19
«Bcosflr„ = —bCe~ br >
THE ATOMIC NUCLEUS
377
378
By dh iding Eq. ! 1.18 by Eq. 11.19 we eliminate the coefficients B and C and
obtain the transcendental equation
11.20
tun rir = — —
Equation 11.20 cannot lie solved analytically . but it can be solved either
graphically or numerically to any desired degree of accuracy. We note that
11.21
h
\Z2m\E + V„)/ft
VW(E)/fi
=m
where — £ is the binding energy of the deuteron and V;, is the depth of the
potential well. Since j V  > i E, in order to obtain a first crude approximation
we might assume that a/b is so large that
tan ur n a oo
Since tan becomes infinite at = 77/2, w, .377/2 htt/2, in this approxi
mation the ground state of the deuteron, for which » = 1, correspond 1 ; to
(In fact, this Is the only hound state of the deuteron.) Hence
V2m'{£ + vy vSS% *
ft r °~ n r "~2
since we are assuming that E is negligible compared with V , and
11.22
v„ ss
Hm'r r 
The above approximation is equivalent to assuming that the function ii, inside
the well is at its maximum (corresponding to or = 90*) at the boundary ol the
well. Actually, h, must l>e somewhat past its maximum there in order to join
smoothly with the function n u outside the well, as shown in Rg. 117; a more
detailed calculation shows that ar zz 1 16° at r = r„. The difference between the
two results is due to our neglect of the binding energy — E relative to V ( , in
obtaining Eq. 1 1.22, and when this neglect is remedied, the l>etter approximation
11.23
V'„ =
is obtained.
Equation 1 1 .23 is a relationship among r tl , the radius of the potential well
THE NUCLEUS
HI,
first approximation
FIGURE 117 The wave function «(r) ol either proton or neutron In a deuteron.
and therefore representative of the range of nuclear forces, V n , the depth of the
well and therefore representative of the interaction potential energy fwtween
two nucleons due to nuclear forces, and — E, the binding energy of the deuteron.
What we now ask is whether this relationship corresponds to reality in the sense
that a reasonable choice of r„ leads to a reasonable value for V . (There is nothing
in our simple model that can point to a unique value for either r () or V .) Such
a choice for r„ might l>e 2 fm. Substituting for the known quantities in Eq. 1 1.23
and expressing energies in MeV yields
, . 1.0 X 10 ™ MeVm* , 1.9 X HI' 14 MeVm
and so, for r = 2 fm = 2 x 10 u m,
V„ = 35 MeV
This is a plausible figure for V () , from which we conclude that the basic features
of our model — nucleons that maintain their identities in the nucleus instead of
(using together, strong nuclear forces with a short range and relatively minor
angular dependence — are valid.
11.8 TRIPLET AND SINGLET STATES
Equation 11.23 contains all the information about the neutron pro ton force that
can be obtained from the observation that the deuteron is a stable system with
a binding energy of — E. To go further we require additional experimental data.
Among the most significant such data is that concerning angular momentum,
THE ATOMIC NUCLEUS 379
380
which was ignored in the preceding analysis of the deuteron when it was assumed
that the neutronproton potential is a function of r alone. The latter is not
correct; in general, angular momentum dot's play a significant role in nuclear
structure, although certain aspects of this structure are virtually independent
of it. In the case of the deuteron, for example, the proton and neutron interact
in such a manner that binding occurs only when their spins are parallel to produce
a triplet state, not when their spins arc ant [parallel to produce a ving/*/ State.
Evidently the neutronproton lone ilepemls on the relative orientation of the
spins and is weaker when the spins are anliparallel.
The difference between the triplet and singlet potentials, together with the
Pauli exclusion principle, makes it possible to see why diprntons and di neutrons
do not exist despite the observed stability of the deuteron antl the charge
independence of nuclear forces. The exclusion principle prevents a diproton or
a rli neutron from occurring in a triplet stale, since with parallel spins both
nucleoli's in each system would be in identical quantum states. No such restriction
applies to the deuteron, since the neutron and proton of which it consists are
distinguishable particles even with parallel spins. However, while a diproton
or dineutron could occur in principle in a singlet state, the singlet nuclear force
is not sufficiently strong to produce binding — and, indeed, diprotons and dineu
trons have never been found.
11.9 THE LIQUIDDROP MODEL
While the attractive forces that nucleons exert upon one another are very strong,
their range is so small that each particle in a nucleus interacts solely with Its
nearest neighbors. This situation is the same as that of atoms in a solid, which
ideally vibrate about fixed positions in a crystal lattice, or that of molecules in
a liquid, which ideally are free to move about while maintaining a fixed inter
molecular distance. The analog)' with a solid cannot be pursued because a
calculation shows that the vibrations of the nucleons about their average positions
would be too great for the nucleus to be stable. The analogy with a liquid, on
the other hand, turns out to be extremely useful in understanding certain aspects
of nuclear behavior.
Let us see how the picture of a nucleus as a drop of liquid accounts for the
observed variation of binding energy per nucleoli with mass number. We shall
start by assuming that the energy associated with each nucleoli micleon bond
has some value I'; this energy is really negative, since attractive forces are
involved, but is usually written as positive [>ecause binding energy is considered
a positive quantity for convenience. Because each liond energy V is shared by
two nucleons, each has a binding energy of %['. When an assembly of spheres
THE NUCLEUS
of the same size is packed together into the smallest volume, as we suppose is
the case of nucleons within a nucleus, each interior sphere has 12 other spheres
in contact with it (Fig. 1 1S). Hence each interior nucleoli in a nucleus has a
binding energy of 12 X 14L7 or 6L'. If all A nucleons in a nucleus were in its
interior, the total binding energy of the nucleus would he
H.Z4 E V = 6AU
Equation 1 1 .'24 is often written simply as
11.25 E B = a t A
The energy £',, is called the volume energy of a nucleus and is directly propor
tional to A.
Actually, of course, some nucleons are on the surface of every nucleus and
therefore have fewer than 12 neighbors. The number of such nucleons depends
upon the surface area of the nucleus in question. A nucleus of radius R has an
area of
Hence the number of nucleons with fewer than the maximum number of bonds
is proportional to A' i/3 , reducing the total binding energy by
11.26 E, = «jA 2/;!
The negative energy E t is called the surface energy of a nucleus; it is most
significant for the lighter nuclei since a greater fraction of their nucleons are
on the surface. Because natural systems always tend to evolve toward configura
tions of minimum potential energy, nuclei lend toward configurations of maxi
FIGUREll3 In a lightly packed assem
bly ot Identical spheres, each Interior
sphere li In contact with twelve others.
( 9>
THE ATOMIC NUCLEUS
381
382
mum binding energy, {We recall that binding energy is the massenergy differ
ence between a nucleus and the same numbers of free neutrons and protons.)
Hence a nucleus should exhibit the same surfacetension effects as a liquid drop,
and in the absence of external forces it should be spherical since a sphere has
the least surface area for a given volume.
The electrostatic repulsion between each pair of protons in a nucleus also
contributes toward decreasing its binding energy. The coulomb energy 2*. of a
nucleus is the work that must be done to bring together Z protons from infinity
into a volume equal to that of the nucleus. Hence E c is proportional to
Z(Z  l)/2, the number of proton pairs in a nucleus containing Z protons, and
inversely proportional to the nuclear radius li = /i,,A 1/3 :
11,27
£, = a
Z{Z  1)
1 1/3
The coulomb energy is negative because it arises from a force that opposes
nuclear stability.
The total binding energy E b of a nucleus is the sum of its volume, surface,
and coulomb energies:
11.26
Efc = £ v + E, + E,
= a,A — a 2 A 2/3 — a
Z(Z  1)
1 A 1/3
The binding energy per nucleon is therefore
Z{Z  1)
11.29
E,
= a i " ^TTa " "3
A V3
Each of the terms of Eq. 1 1.29 is plotted in Fig. 119 versus A, together with
their sum, E^/A. The latter is reasonably close to the empirical curve ot /.. \
shown in Fig. 115. Hence the analogy of a nucleus with a liquid drop has some
validity at least, and we may be encouraged to see what further aspects of nuclear
liehavior it can illuminate. We shall do this in Chap. 12 in connection with
nuclear reactions.
Before leaving the subject of nuclear bindiog energy, it should be noted that
effects other than those we have here considered also are involved. For instance,
nuclei with equal numbers of protons and neutrons are especially stable, as are
nuclei with even numbers of protons and neutrons. Thus such nuclei as He.
'gC, and l gO appear as peaks on the empirical binding energy per nucleon curve.
These peaks imply that the energy states of neutrons and protons in a nucleus
are almast identical and that each state can be occupied by two particles of
opposite spin, as discussed in Sec. 11.3.
THE NUCLEUS
FIGURE 119 The binding energy per nu
cleon is ttic sum of the volume, surface,
and co u tomb energies.
i
10
11.10 THE SHELL MODEL
The basic assumption of the liquiddrop model is that the constituents of a nucleus
interact only with their nearest neighbors, like the molecules of a liquid. There
is a good deal of empirical support for Ibis assumption. There is also, however,
extensive experimental evidence for the contrary hypothesis that the nucleoli.,
in a nucleus interact primarily with a general force field raider than directly
with one another. This latter situation resembles that of electrons in an atom,
where only certain quantum slates are permitted and no more than two electrons,
which are Fermi particles, can occupy each state, N'ucleons are also Fermi
particles, and several nuclear properties vary periodically with Z and S in a
manner reminiscent of the periodic variation of atomic properties with Z.
The electrons in an atom may be thought of as occupying positions in "shells"
designated by the various principal quantum numlters, and the degree of occu
pancy of the outermost shell is what determines certain important aspects of
an atom's liehavior. For instance, atoms with 2. 10, IS, 36, 54, and M6 electrons
have all their electron shells completely filled. Such electron si met ores are stable,
thereby accounting for the chemical inertness of the rare gases. The same kind
of situation is observed with respect to nuclei; nuclei having 2, H, 20, 28. 50, B2,
and 120 neutrons or protons are more abondant than other nuclei of similar mass
numliers, suggesting that their structures are more stable. Since complex nuclei
arose from reactions among lighter ones, the evolution of heavier and heavier
nuclei became retarded when each relatively inert nucleus was formed; this
accounts for their abundance.
THE ATOMIC NUCLEUS
383
384
Other evidence also points up the significance of the numbers 2, 8, 20, 28,
50, 82, and 126, which have become known as magjic numbers, in nuclear
structure. An example is the observed pattern of nuclear electric qnadrupole
moments, which are measures of the departures oF nuclear charge distributions
from sphericity. A spherical nucleus has no quadrupote moment, while one
shaped like a football has a positive moment and one shaped like a pumpkin
has a negative moment. Nuclei of magic N and Z are found to have zero
quadrupole moments and hence are spherical, while other nuclei arc distorted
in shape.
The shell model of the nucleus is an attempt to account for the existence of
magic numbers and certain other nuclear properties in terms of interactions
Iwtween an individual nuctcon and a force field produced by all the other
nudeotLs. A potential energy function is used that corresponds to a square well
about 50 MeV deep with rounded corners, so that there is a more realistic gradual
change from V = V to V = than the sudden change of the pure squarewell
potential we used in treating the deuteron, Schrodinger's equation for a particle
in a potential well of this kind is then solved, and it is found that stationary
states of the system occur characterized by quantum numbers «, /, and m, whose
significance is the same as in the analogous case of stationary states of atomic
electrons. Neutrons and protons occupy separate sets of states in a nucleus since
the latter interact electrically as well as through the specifically nuclear charge.
In order to obtain a series of energy levels that leads to the observed magic
numbers, it is merely necessary to assume a spinorbit interaction whose magni
tude is such that the consequent splitting of energy levels into subtevels is large
for large I, that is, for large orbital angular momenta. It is assumed that L$
coupling holds only for the very lightest nuclei, in which the / values are neces
sarily small in their normal configurations. In this scheme, as we saw in Chap. 7,
the intrinsic spin angular momenta S ( of the particles concerned (the neutrons
form one group and the protons another) are coupled together into a total spin
momentum S, and the orbital angular momenta L f are separately coupled
together into a total orbital momentum L; S and L are then coupled to form
a total angular momentum J of magnitude \/J(J + 1) ft. After a transition region
in which an intermediate coupling scheme holds, the heavier nuclei exhibit fj
coupling. In this case the S, and L t of each particle are first coupled to font
a J, for that particle of magnitude \/j(f + 1) ft, and the various J, then couple
together to form the total angular momentum J. The fj coupling scheme holds
for the great majority of nuclei.
When an appropriate strength is assumed for the spinorbit interaction, the
energy levels of either class of nucleoli fall into the sequence shown in Fig. 1 110.
The levels are designated by a prefix equal to the total quantum number n, a
letter that indicates I for each particle in that level according to the usual pattern
THE NUCLEUS
4s
5d
6g"
7i
spinorbit coupling
villi spinorbit coupling
■*^
«£9/2
nutieons nucleons
per level per total
2j + 1 shell nucleons
16
4
2
8
12
6
10
* 7i
13/2
4p
5i"
6ft
3s
4d
3p
4/
4PI/2
""* 6h
11/2
__ Ad.
AJt,
3/2
.,'■:
„.*<•«■■ " %7/s
Sge/2
■*Pl/2
4 'S/2
4/
7/!
14
2
4
6
8
10
12
2
4
6
8
10
6
4
14 12C
.12 82
22 50
8 28
Xk
2s.
3d'
2p
Is.
_„__ 3d
3d
3/2
1/2
5/2
2 J>l/2
2p, V2
1/2
FIGURE 1110 Sequence of nuclecn energy levels according to the shell model (not to scale).
IS
ill
THE ATOMIC NUCLEUS
385
386
(s, p, d,f, g, . . . correspond respectively to / = 0, J, 2. 3, 4, . . ,). ami a subscript
equal to j. The spinorbit interaction splits each state of given ; into 2/ + I
substates, since there are 2/ 4 1 allowed orientations of J ( . Larftic energy gaps
appear in the spacing of the levels at intervals that are consistent with the notion
of separate shells. The number of available nuclear states in each nuclear shell
is. in ascending order of energy, 2, (i, 12, 8, 22, 32, and 44; hence shells arc
filled when there arc 2, 8, 20, 28, 50, 82, or I2K neutrons or protons in a nucleus.
The shell model is able to account for several nuclear phenomena in addition
to magic numbers. Since each energy sublcvel can contain two particles (spin
up and spin down), only filled sublevels are present when there are even numl>ers
of neutrons and protons in a nucleus ("eveneven" nucleus). At the other extreme.
a nucleus with odd numbers of neutrons and protons ("oddodd" nucleus) contains
unfilled sublevels for lx)lh kinds of particle. The stability we expect to he
conferred by filled sublevels is liorne out by the fact that 160 stable eveneven
nuclides arc known, as against only four stable oddodd nuclides: ]\, !*Li, 'JJB,
and 'IN.
Further evidence in favor of the shell model is its ability to predict total nuclear
angidar momenta. In eveneven nuclei, all the protons and neutrons should pair
off so as to cancel out one another's spin and orbital angular momenta. Thus
eveneven nuclei ought to have zero nuclear angular momenta, as observed. In
evenodd (even Z, odd N) and oddeven (odd Z, even jV) nuclei, the halfintegral
spin of the single "extra" nucieon should be combined with the integral angular
momentum of the rest of the nucleus for a halfintegral total angular momentum,
and oddodd nuclei each have an extra neutron and an extra proton whose
halfintegral spins should yield integral total angular momenta. Both of these
prediction are experimental!) confirmed
It the nuclenns in a nucleus are so close together and interact so strongly thai
the nucleus can be considered as analogous to a liquid drop, how can these same
nucleons be regarded as moving independently of each other in a common force
field as required by die shell model? It would seem that the points of view are
mutually exclusive, since a nucleoli moving alxiut in a liquiddrop nucleus must
surely undergo frequent collisions with other nucleons.
A closer look shows that there is no contradiction. In the ground state of a
nucleus, the neutrons and protons fill the energy levels available to them in order
ol increasing energy in such a way as to ol>ey the exclusion principle (see Fig,
1 12). In a collision, energy is transferred from one nucieon to another, leaving
the former in a state of reduced energy and the latter in one of increased energy.
But all the available levels of lower energy are already filled, so such an energy
transfer can take place only if the exclusion principle is violated. Of course,
it is possible for two nucleons of the same kind to merely exchange their respec
tive energies, but such a collision is hardly significant since the system remains
THE NUCLEUS
in exactly the same state it was in initially. In essence, then, the exclusion
principle prevents nucleonuucleon collisions even in a tightly packed nucleus,
and thereby justifies the independentparticle approach to nuclear structure.
Both the liquiddrop and shell models of the nucleus are, in their very different
ways, able to account for much that is known of nuclear liehavior, Recently
attempts have been made to devise theories that combine tin best features of
each of these models in a consistent scheme, and some success has been achieved
in the endeavor. The resulting model includes the possibility of a nucleus
vibrating and rotating as a whole. The situation is complicated by the non
spherical shape of all but eveneven nuclei and the centrifugal distortion experi
enced by a rotating nucleus; the detailed theory is consistent with the spacing
of excited nuclear levels inferred from the gammaray spectra of nuclei and in
other ways.
Problems
1 . A lieam of singly charged ions of § I A with energies of 4(X) eV enters a uniform
magnetic field of flux density 0.08 T. The ions move perpendicular to the field
direction. Find the radius of their path in the magnetic field. (The §Li atomic
mass is 6.01513 u.)
2. A beam of singly charged lx>ron ions with energies of 1,000 eV enters a
uniform magnetic field of dux density 0.2 T. The ions move perpendicular to
the field direction. Find the radii of the path of the '°B (10.013 u) and ",H
(1 1.009 u) isotopes in the magnetic field.
3. Ordinary boron is a mixture of the l JB and 'JB isotopes and has a composite
atomic weight of 10.82 u. What percentage of each isotope is present in ordinary
boron?
4. Show that the nuclear density of [H is 10 M times greater than its atomic
density. (Assume the atom to have the radius of the first Bohr orbit.)
5. The binding energy of ffCl is 298 MeV. Find its mass in u.
6. The mass of jj{]Ne is 19.9924 u. Find its binding energy in MeV.
7. Find the average binding energy per nucieon in '!jO (the mass of the neutral
'Oatom is 15.9H49u).
8. How much energy is required to remove one proton from 'fO? (The mass
of the neutral '?N atom is 15.0001 u; that of the neutral 'jjO atom is 15.0030 u.)
9. How much energy is required to remove one neutron from ! gO?
THE ATOMIC NUCLEUS
387
388
tO, Compare the minimum energy a gammaray photon must possess if it is
to disintegrate an alpha particle into a triton and a proton with that it mast
posses; if it is to disintegrate an alpha particle into a file nucleus and a neutron.
(The atomic masses of fU and pie are respectively 3.01605 u and 3.01603 u.i
1 1. Show that the electrostatic potential energy of two protons 1.7 X 10" l5 m
apart is of the correct order of magnitude to account for the difference in binding
energy between jU and He. How does this result l>ear upon the problem of
the charge independence of nuclear forces? (The masses of the neutral alums
are, respectively, 3.(116049 and 3.016029 u.)
12. Protons, neutrons, and electrons all have spins of l / 2 . Why do iHe atoms
oliey BoseEinstein statistics while He atoms ol>ey FermiDirac statistics?
13. Show that the results of Sec. 1 1.7— a potential well for the deuteron that
is alx>ul 35 MeV deep and 2 fin in radius— are consistent with the uncertainty
principle.
14. Calculate the approximate value of ^ in Eq. 1 1.27 using whatever assump
tions seem appropriate.
15. According to the Fermi gas model of the nucleus, its protons and neutrons
exist in a box of nuclear dimensions and fill the lowest available quantum states
to the extent permitted by the exclusion principle. Since both neutrons and
protons have spins of % they are Fermi particles and must obey FermiDirac
statistics, (a) Starting from Eq. 10.12, derive an equation for the Fermi energy
in a nucleus under the assumption that equal numbers of neutrons and pro
tons are present, (h) Find the Fermi energy for such a nucleus in which R, =
1.2 fm.
THE NUCLEUS
NUCLEAR TRANSFORMATIONS
Despite the strength of the forces that hold their constituent nucleons together,
atomic nuclei are not immutable. Many nuclei are unstable and spontaneously
alter their compositions through radioactive decay. And all nuclei can be trans
formed by reactions with nucleons or other nuclei that collide with them. In
fact, complex nuclei came into being in the first place through successive nuclear
reactions, probably in stellar interiors. The principal aspects of radioactivity and
nuclear reactions are discussed in this chapter.
12.1 RADIOACTIVE DECAY
Perhaps no single phenomenon has played so significant a role in the development
of lx>th atomic and nuclear physics as radioactivity. A nucleus undergoing
radioactive decay spontaneously emits a ^He nucleus (alpha particle), an electron
(beta particle), or a photon {gamma ray), thereby ridding itself of nuclear excita
tion energy or achieving a configuration that is or will lead to one of greater
stability.
Tile activity of a sample of any radioactive material is the rate at which the
nuclei of its constituent atoms decay. If X Ls the number of nuclei present at
a certain time in the sample, its activity R is given by
12.1
R=M
dt
The minus sit>n i.s inserted to make R a positive quantity, since di\/di is, of course,
intrinsically negative. While the natural units for activity are disintegrations
per second, it is customary to express ft in terms of the curie (Ci) anil its
submultiples, the millicurie (mCi) and miaocurie (fiCi). By definition.
1 Ci = 3.70 X 10 10 dismtegrations/s
1 mc = H> 3 Ci = 3.70 X 1<) 7 dismtegrations/s
1 fie = 10 6 Ci = 3.70 X lO" disintegrations/!!
12
Experimental measurements on the activities of radioactive samples Indicate
that, in every ease, they fall off exponentially with time. Figure 121 is a graph
of fi versus t for a typical radioisotope. We note that in every 5h period,
ic g& l dle ss of when the period starts, the activity drops to half of what it was
at the start of the period. Accordingly the half life T m of the isotope is 5 li.
Every radioisotope has a characteristic half life; some have half lives of ■
millionth of a second, others have half lives that range up to billions of years.
When the observations plotted in Fig. 121 began, the activity of the sample
was R„. Five h later it decreased to 0.5fl (( . After another 3 h, fl again decreased
by a factor of 2 to 0.25fl„. That is, the activity of the sample WW only 0.25
its initial value after an interval of 2T lri . With the lapse of another half life
of 5 h. corresponding to a total interval of 3T, fl became l ,j;{0.25R u ), or 0.I25R,,.
The behavior illustrated in Fig. 12 1 indicates that we can express our empirical
information alxiut the time variation of activity in the form
12.2
fl = R e ""
where X, called the tlvanj constant, has a different value for each radioisotope.
The connection between decay constant A and half life T h . 2 is easy to establish.
After a half life lias elapsed, that is, when t = T 1/2 , the activity fl drops to %R (I
by definition. Hence
R = fl,,**'
'/.fl,, = /v "* ••
e^ T \/t = 2
Taking natural logarithms of lioth sides of this etuation,
12.3
XT, /2 = ln2
_ _ In 2 0.693
,1/2_ X " X
Half life
The decay constant of the radioisotope whose half life is 5 h is therefore
0.693
X =
'1/2
0.693
5 h X 3,600 s/ta
= 3.85 X 10 5 s"'
The fact that radioactive decay follows the exponential law of Eq. 12.2 is strong
evidence that this phenomenon is statistical in nature: every nucleus in a sample
of radioactive material has a certain probability of decaying, but there is no way
of knowing in advance which nuclei will actually decay in a particular time span.
390 THE NUCLEUS
FIGURE 12. 1 The activity of a radioisotope decreases exponentially with time
If the .sample is large enough — that is, if many nuclei are present — the actual
fraction of it that decays in a certain time span will be very close to the proba
bility for any individual nucleus to decay. The statement that a certain radioi
sotope has a half life of 5 h, then, signifies that every nucleus of this isotope
has a 50 percent chance of decaying in any 5h period. This does not mean
a 100 percent probability of decaying in 10 h; a nucleus does not have a memory,
and its decay probability per unit time is constant until it actually does decay.
A half life of 5 h implies a 75 percent probability of decay in 10 h, which increases
to 87.3 percent in 15 h, to 93.75 percent in 20 h, and so on, liecause in every
5h interval the probability is 30 percent.
The empirical activity law of Eq. 12.2 follows directly from the assumption
of a ((instant probability X per unit time for the decay of each nucleus of a given
isotope. Since X is the probability per unit time, A (It is the probability that any
nucleus will undergo decay in a time interval dt. If a sample contains V uu
decayed nuclei, the number dX that decay in a time dt is the product of the
number of nuclei X and the probability A dt that each will decay in dt. That is.
VIA
dN= XXdt
where the minus sign is required because \" decreases with increasing f. Equation
12.4 can lie rewritten
dN
N
= \dt
NUCLEAR TRANSFORMATIONS
391
392
and each side can now be integrated:
r* dK
12.5
f rf ' V v f'
lnJV— lnN = At
JV=,V e*'
A
Equation 12.5 is a formula that gives the number N of undccayed nuclei at the
time J in terms of the decay probability per unit time X of the isotope involved
and the number N n of undecayed nuclei at t = 0.
Since the activity of a radioactive sample is defined as
fl =
dN
dt
we see that, from Eq. 12.5,
R = kN e~ M
This agrees with the empirical activity law if
R„ = XN U
or, in general, if
12.6 R = KN
Evidently the decay constant X of a radioisotope is the same as the probability
per unit lime for the decay of a nucleus of that isotope.
Equation 12.6 permits its to calculate the activity of a radioisotope sample
if we know its mass, atomic mass, and decay constant. As an example, let
us determine the activity of a 1gm sample of jgSr, whose half life against beta
decay is 28 yr. The decay constant of ^Sr is
A =
(1.693
"1/2
0.693
28yr X3.I6 X 10 T s/yr
= 7.83 X 10 1 "* 1
A kmol of an isotope has a mass very nearly equal to the mass number of that
isotope expressed in kilograms. Hence 1 gm of jgSr contains
ltl a kg
90 kg/kmol
= 1.1 1 X H)" a kmol
THE NUCLEUS
One kmol of any isotope contains Avogadro's numlier of atoms, and so 1 gm
of $Sr contains
1.1 1 X 10 5 kmol X 6.025 x 10 M aloms/kmol
= 6.69 x 10 21 atoms
Thus the activity of the sample is
R =\N
= 7.83 X TO" 10 X 6.69 X 10 21 s" 1
= 5.23 X 10 12 s'
= 141 Ci
It is worth keeping in mind thai the half life of a radioisotope is not the same
as its mean lifetime T. The mean lifetime of an isotope is the reciprocal of its
decay probability per unit time:
12.7
Hence
12.8
'*
f  i — Tu2  I 447'
1 " A  0.693 " ■ ' u2
Mean lifetime
T is nearly half again more than 7' l/2 . The mean lifetime of an isotope whose
half life is 5 h is 7.2 h.
12.2 RADIOACTIVE SERIES
Most of the radioactive elements found in nature are members of four radioactive
series* with each series consisting of a succession of daughter products all ul
timately derived from a single parent nuclide. The reason that there are exactly
four such series follows from the fact that alpha decay reduces the mass number
of a nucleus by 4. Thus the nuclides whose mass numl>ers are all given by
12.9
A = 4rj
where n is an integer, can decay into one another in descending order of mass
number. Radioactive nuclides whose mass numbers otiey Eq. 12.9 are said to
he tnemljers of the 4n scries. The members of the 4n + I series have mass
numbers specified by
12.10
A = 4n + I
NUCLEAR TRANSFORMATIONS
393
394
Table 12.1.
FOUR RADIOACTIVE SERIES.
Mass numbers
Half life, v
)M
Thorium
VjTli
1.39 X 10'"
>M'I.
4» 4 1
Neptunium
:N>
2.S5 X 10"
H
*n + 1
I'rrmium
,,,.
4.51 X 10"
>gft
tn + 3
Actinium
•go
7.07 x 10*
=!gpb
and mcmticrs of the 4n + 2 and 4n + .1 series have mass numbers specified
respectively In
12.11
12.12
A = 4n + 2
,4 =ln + 3
The meml>ers of eaeli of these series, too, can decay into one another in descend
ing order of mass number.
Table 12.1 is a list of the names of four important radioactive scries, their
parent nuclides and the half lives of these parents, and the stable daughters which
are end products of the series, The half life of neptunium is so short compared
with the estimated age (— 10"' yr) of the universe that the members of this
series are not found in nature today. They have, however, f>een produced in
the laboratory by the neutron bombardment of other heavy nuclei; a brief dis
cussion is given in Sec. 12.12. The sequences of alpha and lieta decays that lead
FIGURE 122 The thorium decay
■arias (i = 4»). The decay at Br
may proceed either by alpha emis
sion and then beta emission or In
lha reverse order.
fc— a decay
— • decay
THE NUCLEUS
FIGURE 123 The neptunium
decay series i\  4n r 1). The
decay of '!,',B\ may proceed either
by alpha emission and then beta
emission or In the reverse order.
>
80
84
SB
92
from parent to stable end product in each series are shown in Figs. 122 to 125.
Some nuclides may decay either by beta or alpha emission, so that the decav
chain branches at them. Tims '^jjBi, a member of the thorium scries, has a n'li.'S
percent chance of beta decaying into "iifPo and a 33.7 percent chance of alpha
FIGURE 124 The uranium decay ,
series {A ■ 4n  2). The decay of ^
'i'Bi may proceed either by alpha >1
emission and then beta emission or
In the reverse order. 130
NUCLEAR TRANSFORMATIONS
395
I
FIGURE 125 The actinium decay
series (\ = 4n * 3). The decays
of ■; Ac and ^/Bi may proceed ei
fher by alpha emission and then
beta emission or in the reverse
order.
decaying into 2 J$T1. The beta decay is followed by a subsequent alpha decay
and ihc alpha decay is followed by a subsequent beta decay, so that both branches
lead to 2 ^1 J I».
396
Several alpharadioactive nuclides whose atomic numbers are less than 82 are
found in nature, though they are not very abundant.
12.3 ALPHA DECAY
Because the attractive forces l)etwecu nudeons arc of short range, the total
binding energy in a nucleus is approximately proportional to its mass number
A, the number of nudeons it contains. The repulsive electrostatic forces lietween
protons, however, are of unlimited range, and the total disruptive energy in a
nucleus is approximately proportional to '/.". Nuclei which contain 210 or more
uuclcons are so large that the shortrange nuclear forces that hold them together
are barely able to counterbalance the mutual repulsion of their protons. Alpha
decay occurs in such nuclei as a means of increasing their stability by reducing
their size.
Why are alpha particles almost invariably emitted rather than, say, individual
protons or r]He nuclei? The answer follows from the high binding energy of
the alpha particle. To escape from a nucleus, a particle must have kinetic energy,
and the alphaparticle mass is sufficiently smaller than that of its constituent
THE NUCLEUS
nudeons for such energy to be available. To illustrate this point, we can compute,
from the known masses of each particle and the parent and daughter nuclei,
the kinetic energy Q released when various particles are emitted bv a heaw
nucleus. This is given by
Q  (m,  m,  m
where m, is the mass of the initial nucleus. m f the mass of the final nucleus.
and m n the alphaparticle mass. We find that only the emission of an alpha
particle is energetically possible: other decay modes would require energy to
be supplied from outside the nucleus. Thus alpha decaj la ?gU is accompanied
by the release of 5.4 MeV, while 6.1 MeV would somehow have to be furnished
if a proton is to be emitted and 9.6 MeV if a jjHc nucleus Ls to be emitted.
The observed disintegration energies in alpha decay agree with the corresponding
predicted values based upon the nuclear masses involved.
The kinetic energy T a of the emitted alpha particle is never quite equal to
the disintegration energy Q because, since momentum must be conserved, the
nucleus recoils with a small amount of kinetic energy when the alpha particle
emerges. It is easy to show that, as a consequence of momentum and energy
conservation, T a is related to Q and the mass number A of the original nucleus
by
A v
The mass numbers of nearly all alpha emitters exceed 210, and so most of the
disintegration energy appears as the kinetic energy of the alpha particle. In the
decay of 2 ^Rn. Q = 5.587 MeV while 7;, = 5.486 MeV.
While a heavy nucleus can, in principle, spontaneously reduce its bulk by alpha
decay, there remains the problem of how an alpha particle can actually escape
from the nucleus. Figure 126 is a plot of the potential energy V of an alpha
particle as a function of its distance r from the center of a certain heaw nucleus.
The height of the potential barrier is about 25 MeV, which is equal to the work
thai must be done against the repulsive electrostatic force to bring an alpha
particle from infinity to a position adjacent to the nucleus but just outside the
range of its attractive forces. We may therefore regard an alpha particle in such
a nucleus as being inside a box whose walls require an energy of 25 MeV to
be mm mounted However, decay alpha particles have energies that range from
4 to 9 MeV, depending upon the particular nuclide involved— 16 to 21 MeV
short of die energy needed for escape.
Although alpha decay is inexplicable on the basis of classical arguments,
quantum mechanics provides a straightforward explanation. In fact, the theory
of alpha decay developed independently in 192K by Carnow and by Curney and
NUCLEAR TRANSFORMATIONS 397
398
Condon WW greeted as an especially striking confirmation of quantum me
chanics. In the following two sections we shall show how even a simplified
treatment of the problem of the escape of an alpha particle from a nucleus gives
results in agreement with experiment.
The basic notions of this theory are:
1. An alpha particle may exist as an entity within a heavy nucleus;
2. Such a particle Es in constant motion and is contained in the nucleus by
the surrounding potential harrier;
■'3. There is a small — but definite — likelihood that the particle nay pass through
the harrier (despite its height) each lime a collision with it occurs.
11ms the decay probability per unit lime \ can be expressed as
A = vf
where i' is the number of times per second an alpha parlit le within a nucleus
strikes the potential barrier aroimd it and F is the probability that the particle
will he transmitted through the harrier. If we suppose that at any moment only
one alpha particle exists as such in a nucleus and that it moves hack and forth
along a nuclear diameter,
c
2H
FIGURE 126 The potential energy of in alpha particle n a function of its distance from the center of a
nucleus.
potential energy of alpha particle
4ire i
alpha particle
cannot escape "'""*■
(classically)
alpha particle cannot enter (classical!*)
"T"
T — kinetic
energy of
alpha particle
r .
THE NUCLEUS
Thick mirror
thin
FIGURE 127 An incident wave pene
trates the surface of even a perfect re
flecting surface 'or a short distance and
may pass through it If the surface is suffi
cienlly thin.
Tola) reflection
Partial reflection
when i is the alphaparticle velocity when it eventually leaves the nucleus anil
R is the nuclear radius. Typical values for r ,mt.\ B might l«2x Id 7 m/s and
K)"'" 1 in respectively, so that
■ = Id*' S 1
The alpha particle knocks at its confining wall 10' times per second and yet
may have to wail an average of as much as 10 1 " yr to escape from some nuclei!
Since V > /;, classical physics predicts a transnu'ssioii probability /' of zero.
In ijuantuiu mechanics a moving alpha paiticlc is regarded ils a wave, anil the
result is a m null I mi definite value lor /'. The optical analog of this effect is well
known: a light wave ondergoing reHcclion from even a perfect mirror never
theless penetrates it with an exponential!) decreasing amplilude before reversing
direction iFig. 127'.
•12.4 BARRIER PENETRATION
Let us consider the case ol a beam of particles of kinetic energy '/'incident from
the left on a potential harrier of height V and width /., as in Fig 128. On
both sides ol the barrier V = 0, which means that no forces act upon the particles
there in these regions Sehrodingcr's equation for the particles is
12.13
and
12.14
• V . 2 m
f '^ni . 2 »<
^%a =
Let us assume that
K.15 ^, s Ae** + Be ""
12.16 trV tll = fie"" + Fe ,ax
NUCLEAR TRANSFORMATIONS
399
40Q
II
[]]
*
FIGURE 128 A beam of
particles can "leak" through
a finite barrier.
, exponential
■ sinusoidal
are .solutions to Eqs. 12.13 and 12.14 respectively. The various terms in these
solutions are not hard to interpret. As shown schematically in Fig. 128, Ae tar
is a wave of amplitude A incident from the left on the harrier. That is,
ia.17 u/ l+ = Ae ial
This wave corresponds to the incident beam of particles in the sense that \4>t+\ 2
is their probability density. If v is the group velocity of the wave, which equals
the particle velocity,
12.18
I^i+I'p
is the (lux of particles that arrive at the barrier. At x = the incident wave
strikes the barrier and is partially reflected, with
12.19 ^t = Be'""
representing the reflected wave (Fig. 129). Hence
12.20 ^ = uV, + + V
On the far side of the barrier (x > L) there can be only a wave
traveling in the +x direction, since, by hypothesis, there is nothing in region
III that could reflect the wave. Hence
F =
THE NUCLEUS
FIGURE 129 Schematic
representation of barrier
penetration.
<h+ — 
*n+ 
tl/
Vm+
— *,_
 — ^u
* =
x=L
and
ia.21
^111 = "rSll+
= Ee iat
By substituting ^, and ^ lu hack into their respective differential equations, we
find that
12.22
; 2n,r
It is evident that the transmission probability P for a particle to pass through
the barrier is the ratio
12.23 p  IT '"' =
W,! 2 AA'
between its probability density in region III and its probability density in re
gion I. Classically /' = () because the particle cannot exist inside the barrier; let
ns see what the quantummechanical result is.
In region 11 Schrodinger's equation for the particles is
12.24
B^n 2m
ta« + fi* (T  V ^=°
Its solution is
12.Z5 <£ H = Ce ib * + De""
where
/2„,(TV)
ft 2
NUCLEAR TRANSFORMATIONS 401
402
Since V > T. h Is imaginary and we may define a new wave number V by
IS = ih
•>*
2tn V  T)
12.27
I lei ice
iz.28 C„  Gs •* + De*<
The U'iui
12.29 ^„ + = Ce
is an exponentially decreasing wave function thai corresponds to a nonoscillatory
disturbance moving to the right through the barrier. Within the liarricr part
of I lie disturbance is reflected, and
12.30
*n
 f>'
is an exponentially decreasing wave function that corresponds to the reflected
disturbance moving to the left.
Even though £„ does not oscillate, and therefore does not represent a moving
particle of positive kinetic energy, the probability density i£ n  2 is not zero. There
is a Suite probability of finding a particle within the barrier. A panicle at the
far end of the barrier that is not reflected there will emerge into region 111 with
the same kinetic energy T it originally had. and its wave function will be vni
as it continues muvitig unimpeded in the +.v direction. In the limit of an
iiifiuileb thick harrier, $ nl = 0, which implies that ail the incident particles
are reflected. The reflection process takes place within the barrier, however,
not at Its lefthand wall, and a harrier of finite width therefore permits a frafi
tion P of the initial lieain to pass through it.
In order to calculate P, we must apply certain Ixmndary conditions to i^,, \}> lv
and >£,„. Figure 128 is a schematic representation of the wave functions in
redans I. 11, and 111 which may help in visualizing the boundary conditions.
\$ discussed earlier, both $ and its derivative ty/dx must be continuous every
where. With reference to Fig. 12H, these conditions mean thai, at each wall
of the harrier, the wave functions inside and outside must not only have the
same value but also the same slope, so that they match up perfectly, lien...
at the lefthand wall of the barrier
12.31a
12.31b
*1
= ^u
tyl
_ rty u
dx
dx
x =
and at the righthand wall
THE NUCLEUS
12.32*
12.32 b
^11 = ^iri
3* dx
x = L
Substituting £,, ^ n , and ^„, from Eqs. 12.15, 12.2S, and 12.21 into the aliove
equations yields
1233 A + H = C + D
"*» iaA  iaB = b'C + b'D
1235 Ce bL + De iL = Ee*' L
12.36 aCe b ' L + aDe hL a iaEe' aI '
Equations 12.33 to 12.36 may be readily solved to yield
The complex conjugate of A/E, which we require to compute the transmission
probability P, is found by replacing i by  i wherever it occurs in A/E:
Let us assume that the potential harrier is high relative to (he kinetic energy
of an incident particle; this means that // "> tt and
12.39
("£) = 
V a b I a
Let us also assume that the barrier is wide enough for t£„ to l>e severely attenu
ated (>etween .v = and x =■ L; this means that b'L > 1 and
12,40 e"' L > e VL
Hence Eqs. 12.37 and 12.38 may lie approximated by
12.11
and
12.42
(*)*(*D*~
NUCLEAR TRANSFORMATIONS 403
\iultiplviiig (A/E) and (A/E)* yields
(i)ft)'(i**)
and so tlic transmission probability /" is
i2.4 3 = [ , if,, J «» 7 
L 4 + (ft'/a) 2 J
Since from the definitions of a (Eq. 12.22) and ft' (Eq. 12.27)
the variation in the coefficient of the exponential of Eq. 12.43 with T and V
is negligible compared with the variation in ihc exponential itself. The co
efficient, furthermore, is never far From unity, and so
12.44
p~ e 2bl.
is a good approximation for the transmission probability. We shall find it conven
ient to write Eq. 12.44 as
In/' 2ft7.
12.45
♦12.5 THEORY OF ALPHA DECAY
Equation 12.45 is derived for a rectangular potential barrier, while an alpha
particle inside a nucleus is faced with a barrier of varying height, as in Fig. 126.
We tnust therefore replace In I' = — 2ft'/., by
12.46
In P = 2 I b'[x) t/.r = 2 I ft'(.t) (lx
where R„ is the radius of the nucleus and R the distance from its center at which
V = 7'. Beyond R the kinetic energy of the alpha particle is positive, and it
is able to move freely (Fig. 1210). Now
404
V(x) =
THE NUCLEUS
27,t?
tef r t
FIGURE 12 10 Alpha decay from the point at view of wave mechanic!.
is the electrostatic potential energy of an alpha particle at a distance x from
the center of a nucleus of charge 7.e (that is, Ze is the nuclear charge minus
the alphaparticle charge of 2e). We therefore have
y = / 2m ( v  r >
= L\ /Zei _ T \
\ ft 2 / \4Tre„x /
1/2
and, since T = V when x = R,
H¥nf')'
NUCLEAR TRANSFORMATIONS 405
406
I lence
1S.47
biP = 2 J* b'(x)dx
= fen; ("')"**
=(^)'""h'(T)" z ®" 2 ('^)'i
Because the potential barrier is relatively wide, R > fi () , and
cos
with the result that
,nP=  2 (ir) fi U 2 lT) J
Replacing /I by
R =
2Ze a
47r Eo r
we obtain
1218
^=f{%Y'*"'«°"'ik(fY' ZT "'
The result of evaluating the various constants in Eq. 12.48 is
InP = 2.97Z> /2 fi ,/a  3.95ZT 1 '*
where T (the alphaparticle kinetic energy) is expressed in MeV, R (the nuclear
radius) is expressed in fm(l fin = 10 13 m), and Z is the atomic number of the
nucleus minus the alpha particle. The decay constant A, given by
A = vP
= ?P
w
may therefore be written
hi A = In(^) + 2.97Z I/2 R ,/S  3.95Z7" 2
12.49
Alpha decay
THE NUCLEUS
To express Eq. 12,49 in terms of common logarithms, we note that
In A =
logi,, e " 0.4343
and so
l°gu>A = ,0 g!«(2R") + ° 4343 ( 2  97Z1/i! V" ~ 3H5Zr"2)
= lo Kio (^~) + ! 29ZU2 V /2 " 1.72ZT'"
Figure 121 1 is a plot of log [0 A versus ZT~ ,n for a number of alpharadio
active nuclides. The straight line fitted to the experimental data has the —1.72
slope predicted throughout the entire range of decay constants. We can use the
position of the line to determine R„, the nuclear radius. The result is just about
what is obtained from nuclear scattering experiments like that of Rutherford,
namely, —10 fm in such very heavy nuclei. This approach constitutes an
independent means for determining nuclear sizes.
FIGURE 1211 Experimental verifica
tion of the theory of alpha decay.
10
NUCLEAR TRANSFORMATIONS 407
408
The quantummechanical analysis of alpha particle emission, which is in
complete accord with the observed data, is significant on two grounds. First,
1 1 makes understandable the enormous variation in half life with disintegration
energy. The slowest decay is that of 2 g^Th, whose half life is 1.3 X 10 10 years,
and the fastest decay is that of ^Po, whose half life is 3.0 X H)" T sec. While
its half life is 10 24 greater, the disintegration energy of a HoTh (4.05 MeV) is only
about half that of 2 £fPo (8.95 MeV)— behavior predicted by Eq. 12.19.
The second significant feature of the theory of alpha decay is its explanation
of this phenomenon in terms of the penetration of a potential barrier by a particle
which does not have enough energy to surmount the barrier. In classical physics
such penetration cannot occur: a baseball thrown against the Great Wall of China
has, classically, a zero probability of getting through. In quantum mechanics
the probability is not much more than zero, but it is not identically equal to
zero.
12.6 BETA DECAY
Beta decay, like alpha decay, is a means whereby a nucleus can alter its Z/iV
ratio to achieve greater stability. Beta decay, however, presents a rather different
problem to the physicist who seeks to understand natural phenomena. The most
obvious difficulty is that in beta decay a nucleus emits an electron, while, as
we have seen in the previous chapter, there are strong arguments against the
presence of electrons in nuclei. Since beta decay is essentially the spontaneous
conversion of a nuclear neutron into a proton and electron, this difficulty is
disposed of if we simply assume that the electron leaves the nucleus immediately
after its creation. A more serious problem is that observations of l>eta decay
reveal that three conservation principles, those of energy, momentum, and
angular momentum, are apparently being violated.
The electron energies observed in the beta decay of a particular nuclide are
found to vary continuously from to a maximum value r mM characteristic of
the nuclide. Figure 1212 shows the energy spectrum of the electrons emitted
in the beta decay of 2 " Bi; here T max =1.17 MeV. In every case the maximum
energy
^miut = m t» c + 'nu
carried off by the decay electron is equal to the energy equivalent of the mass
difference l>etween the parent and daughter nuclei. Only seldom, however, is
an emitted electron found with an energy of T mtx .
It was suspected at one time that the "missing" energy was lost during collisions
between the emitted electron and the atomic electrons surrounding the nucleus.
THE NUCLEUS
FIGURE 1212 Energy spectrum of
electron s (mm the beta deciy of *ijBt.
energy equivalent
of mass lost by
decaying nucleus
0.4 0.6 o.s
ELECTRON ENERGY. MeV
An experiment first performed in 1927 showed that this hypothesis is not correct.
In the experiment a sample of a fueta radioactive nuclide is placed in a calo
rimeter, and the heat evolved after a given number of decays is measured. The
evolved heat divided by the number of decays gives the average energy per
decay. In the case of 2 ^Bi the average evolved energy was found to be 0.35
MeV, which is very close to the 0,39MeV average of the spectrum in Fig. 1212
but far away indeed from the T max value of 1.17 MeV. The conclusion is that
the observed continuous spectra represent the actual energy distributions of the
electrons emitted by [>etaradioaetive nuclei.
Linear and angular momenta are also found not to be conserved in beta decay.
In the beta decay of certain nuclides the directions of the emitted electrons and
of the recoiling nuclei can be observed; they arc almost never exactly opposite
as retmired for momentum conservation. The nonconservation of angular mo
mentum follows from the known spins of '/ 2 of the electron, proton, and neutron.
Beta decay involves the conversion of a nuclear neutron into a proton:
n—*p + e~
Since the spin of each of the particles involved is %. this reaction cannot take
place if spin (and hence angular momentum) is to be conserved.
In 1930 Pauli proposed that if an uncharged particle of small or zero mass
and spin % is emitted in beta decay together with the electron, the energy,
momentum, and angularmomentum discrepancies discussed above would be
removed. It was supposed that this particle, later christened the neutrino, carries
off an energy equal to the difference between T mix and the actual electron kinetic
energy (the recoil nucleus carries away negligible kinetic energy) and, in so doing,
has a momentum exactly balancing those of the electron and the recoiling
daughter nucleus. Subsequently it was found that there are two kinds of neutrino
NUCLEAR TRANSFORMATIONS 409
involved in beta decay, the neutrino itself (symbol w) and the antineutrino (symlxil
T). We shall discuss the distinction lietween them in Chap. 13. In ordinary beta
decay it is an antineutrino that is emitted:
12.50
Beta decay
n — * p + e + v
Hie neutrino hypothesis has turned out to be completely successful. The
neutrino mass was not expected to be more than a small fraction of the electron
mass because T m!a is observed to be equal (within experimental error) to the
value calculated from the parentdaughter mass difference; the neutrino mass
is now believed to be zero. The reason neutrinos were not experimentally
detected until recently is that their interaction with matter is extremely feeble.
Lacking charge and mass, and not electromagnetic in nature as is the photon,
the neutrino can pass unimpeded through vast amounts of matter. A neutrino
would have to pass through over l(K) lighttjeurx of solid iron on the average
before interacting! The only interaction with matter a neutrino can experience
is through a process called inverse Ijeta decay, which we shall consider shortly.
Positive electrons, usually called positrons, were discovered in 1932 and two
years later were found to lie spontaneously emitted by certain nuclei. The
properties of the positron are identical with those of the electron except that
it carries a charge of +e instead of e. Positron emission corresponds to the
conversion of a nuclear proton into a neutron, a positron, and a neutrino:
12.51
n + e + + v
Positron emission
While a neutron outside a nucleus can undergo negative beta decay into a proton
because its mass is greater than that of the proton, the lighter proton cannot
be transformed into a neutron except within a nucleus. Positron emission leads
to a daughter nucleus of lower atomic number Z while leaving the mass num
ber A unchanged.
Closely connected with positron emission is the phenomenon of electron
capture. In electron capture a nucleus absorbs one of its inner orbital electrons,
with the result that a nuclear proton becomes a neutron and a neutrino is
emitted. Thus the hindamental reaction in electron capture is
12.52
p + er
n + v
Electron capture
Electron capture is competitive with positron emission since both processes lead
to the same nuclear transformation. Electron capture occurs more often than
positron emission in heavy elements Ijccause the electron orbits in such elements
have smaller radii; the closer proximity of the electrons promotes their interaction
with the nucleus. Since almost the only unstable nuclei found in nature are of
high Z, positron emission was not discovered until several decades after electron
emission had been established.
12.7 INVERSE BETA DECAY
The beta decay of a proton within a nucleus follows the scheme
12.53 p — » n + e + + c
Because the absorption of an electron by a nucleus is equivalent to its emission
of a positron, the electron capture reaction
12.54
p + e~
n + c
is essentially the same as the beta decay of Eq. 12.53. Similarly, the absorption
of an antineutrino is equivalent to the emission of a neutrino, so that the reaction
12.55
p +
n + e"
also involves the same physical process as that of Eq, 12.53, This latter reaction,
called inverse beta tiecaij, is interesting because it provides a method for estab
lishing the actual existence of neutrinos.
Starting in 1953, a series of experiments were begun by F. Reines, C, L. Cowan,
and others to detect the immense flux of neutrinos from the beta decays that
occur in a nuclear reactor. A tank of water containing a cadmium compound
in solution supplied the protons which were to interact with the incident neu
trinos. Surrounding the tank were gammaray detectors. Immediately after a
proton absorbed a neutrino to yield a positron and a neutron, the positron
encountered an electron and both were annihilated. The gammaray detectors
responded to the resulting pair of 0.5IMeV photons. Meanwhile the newly
formed neutron migrated through the solution until, after a few microseconds,
it was captured by a cadmium nucleus. The new, heavier cadmium nucleus then
released about 8 MeV of excitation energy divided among three or four photons,
which were picked up by the detectors several microseconds after those from
the positronelectron annihilation. In principle, then, the arrival of the above
sequence of photons at the detector is a sure sign that the reaction of Eq. 12.55
has occurred. To avoid any uncertainty, the experiment was performed with
the reactor alternately on and off, and the expected variation in the frequency
of neutrinocapture events was observed. Thus the existence of the neutrino may
l)c regarded as experimentally established.
Inverse beta decay is the sole known means whereby neutrinos and antineu
trinos interact with matter:
p + v
n + c
n + e +
p + e~
The probability for these reactions is almost vanishingly small; this is why
410
THE NUCLEUS
NUCLEAR TRANSFORMATIONS
411
412
neutrinos are able to traverse freely such vast amounts of matter. Once liberated,
neutrinos travel freely through space and matter indefinitely, constituting a kind
of independent universe within the universe of other particles.
12.8 GAMMA DECAY
Nuclei can exist in states of definite energies, just as atoms can. An excited
nucleus is denoted by an asterisk after its usual symbol; thus {JgSr* refers to Sr
in an excited state. Excited nuclei return to their ground states by emitting
photons whose energies correspond to the energy differences between the various
initial and final states in the transitions involved. The photons emitted by nuclei
range in energy up to several MeV, and are traditionally called gamma rni/s.
A simple example of the relationship between energy levels and decay schemes
is shown in Fig. 1213, which pictures the beta decay of ^Mg to gAl. The half
life of the decay is 9.5 min, and it may take place to either of the two excited
states of fAl. The resulting ^Al" nucleus then tmdergoes one or two gamma
decays to reach the ground state.
As an alternative to gamma decay, an excited nucleus in some cases may return
to its ground state by giving up its excitation energy to one of the orbital electrons
aroiuid it. While we can think of this process, which is known as internal
conversion, as a kind of photoelectric effect in which a nuclear photon is absorbed
by an atomic electron, it is in better accord with experiment to regard internal
conversion as representing a direct transfer of excitation energy from a nucleus
to an electron. The emitted electron has a kinetic energy equal to the lost nuclear
excitation energy minus the binding energy of the electron in the atom.
Most excited nuclei have very short half lives against gamma decay, but a
few remain excited for as long as several hours. A longlived excited nucleus
is called an isomer of the same nucleus in its ground state. The excited nucleus
fJSr* has a half life of 2.8 h and is accordingly an isomer of f£Sr.
1.015 MeV
0.834 MeV
FIGURE 1213 Successive beta anal
gamma em is lion fn the decay of J'Mg I
;ai.
y\ \y
s*
THE NUCLEUS
incident
particles
target
nucleus
only these
particles will
interact
□
□
geometrical
cross section
interaction
cross section
FIGURE 1214 The concept of cross faction. The interaction cross section may be smaller than, equal
to, or larger than the geometrical cross section.
12.9 CROSS SECTION
Nuclear reactions, like chemical reactions, provide both information and a means
of utilizing this information in a practical way. Most of what we know about
atomic nuclei has come from experiments in which energetic bombarding parti
cles collide with stationary target nuclei. A very convenient way to express the
probability that a bombarding particle will interact in a certain way with a target
particle employs the idea of cross section that was introduced in Chap. 4 in
connection with the Rutherford scattering experiment. What we do is visualize
each target particle as presenting a certain area, called its cross section, to the
incident particles, as in Fig. 1214. Any incident particle that is directed at tlii*
area interacts with the target particle. Hence the greater the cross section, the
greater the likelihood of an interaction. 'I*he interaction cross section of a target
particle varies with the nature of the process involved and with the energy of
the incident particle; it may be greater or less than the geometrical cross section
of the particle.
NUCLEAR TRANSFORMATIONS
413
414
Suppose we have a slab of some material whose area is A and whose thickness
is dx (Fig. 1215), If the material contains n atoms per unit volume, there are
a total of nAdx nuclei in the slab, since its volume is Adx. Each nucleus has
a cross section of a for some particular interaction, so that the aggregate cross
section of all the nuclei in the slab is nAa dx. If there are N incident particles
in a bombarding beam, the number tl.X that interact with nuclei in the slab is
therefore specified by
Interacting particles
incident particles
,l\
N
12.56
aggregate cross section
target area
_ nAadx
A
= na dx
Cross section
Equation 12.56 is valid only for a slab of infinitesimal thickness. To find the
proportion of incident particles that interact with nuclei in a stab of finite
thickness, we must integrate tlN/N. If we assume that each incident particle
is capable of only a single interaction, d\' particles may be thought of as being
removed from the beam in passing through the first dx of the slab. Hence we
must introduce a minus sign in Eq. 12.56, which becomes
Denoting the initial number of incident particles by <V U , we have
* dN
1257
r" (IN C
In ,V — In JV„ = — nax
N = N e""
dx
The number of surviving particles N decreases exponentially with increasing stab
thickness jr.
While cross sections, which are areas, should lie expressed in irr, it is conve
nient and customary to express them in hams (b), where 1 b = 10~ 38 m 2 . The
bam is of the order of magnitude of the geometrical cross section of a nucleus.
The cross sections for most nuclear reactioas depend upon the energy of the
incident particle. Figure 1216 shows how the neutronabsorption cross section
of '.JgCd varies with neutron energy; the narrow peak at 0.176 eV is associated
with a specific energy level in the resulting "JCd nucleus.
The mean free path I of a particle in a material is the average distance it
can travel in the material before interacting with a target nucleus. The proba
bility / that an incident particle will undergo an interaction in a slab ir thick
THE NUCLEUS
n atoms/m
N incident
particles
tr= cross section/atom
area = A
NdN
particles
emerge
from slab
dN/N = no dx
FIGURE 12 IS The relationship between cross section and beam intensity.
FIGURE 1316 The cross section lor neutron absorption ;:Cd varies with neutron energy.
10.000 r
 l.ooo
100
0.001 0.01
0.1 1.0 10
NEUTRON ENERGY, eV
100
1,000 10,000
NUCLEAR TRANSFORMATIONS
415
416
of the material is
iz.58 / = no A.\
The number of times II it must traverse the slab before interacting is therefore
12.59 // =
!
MO Aa
on the average. Accordingly the average distance the particle travels before
interacting is
HAx = —
no
which is, by definition, the mean free path. Hence
/ = !
no
Mean free path
The cross section for the interaction of a neutrino with matter has been found
to be approximately 10 47 m a . Let us use Eq. 12.60 to find the mean free path
of neutrinos in solid iron. The atomic mass of iron is 55.9, so that the mass of
an iron atom is, on the average,
m Ft = 55.9 u/atom X 1.66 X HI' 27 kg/u
= 9.3 X 10" M kg/atom
Since the density of iron is 7.8 X 10 3 kg/m 3 , the number of atoms per m :l in
iron is
7.8 X 10 3 kg/m 3
n = —
9.3 X 10~ 2fi kg/atom
= 8.4 X 10 as atoms/m 3
The mean free path for neutrinos in iron is therefore
1
no 8.4 x 10* 8 atoms/m 3 x lO"" m 2
= 1.2 X K) 18 m
A lightyear (the distance light travels in free space in a year) is equal to
9.46 X If) 1 '' m, and so the mean free path turns out to be
1 =
1.2 x H) ]a m
9.46 X 10 15 m/lightyear
= 130 lightyears
THE NUCLEUS
in solid iron' An immense flux of neutrinos is produced in the sun and other
stars in the course of the nuclear reactions that occur within them, and this flux
moves practically unimpeded through the universe. There are already far more
neutrinos than atoms in the universe, and their number continues to increase.
The energy these neutrinos carry is— apparently — lost forever in the sense of
being unavailable for conversion into other forms.
12.10 THE COMPOUND NUCLEUS
Many nuclear reactions actually involve two separate stages. In the first, an
incident particle strikes a target nucleus and the two combine to form a new
nucleus, called a comjxwrul nucleus, whose atomic and mass numbers are re
spectively the sum of the atomic numbers of the original particles and the sum
of their mass numbers. The compound nucleus has no "memory" of how it was
formed, since its uuclcons are mixed together regardless of origin and the energy
brought into it by the incident particle is shared among all of them. A given
compound nucleus may therefore be formed in a variety of ways. To illustrate
this, Table 12.2 shows six reactions whose product is the compound nucleus 'fN'.
(The asterisk signifies an excited state; compound nuclei are invariably excited
by amounts equal to at least the binding energies of the incident particles in
them.) While J N and '^C arc beta radioactive with such short half lives as to
preclude the detailed study of their reactions to form l \*, there Ls no doubt
that these reactions can occur.
Compound nuclei have lifetimes of the order of 10~ 16 s or so, which, while
so short as to prevent actually observing such nuclei directly, are nevertheless
long relative to the 10~ 21 s or so required for a nuclear particle with an energy
of several MeV to pass through a nucleus. A given compound nucleus may decay
Table 12.2.
NUCLEAR REACTIONS WHOSE PRODUCT IS THE COMPOUND
N U C L E U 5 ' ; X * . The exc itation e nergies gi ve n a re c alcu lat ed I rom
the masses of the particles involved; the kinetic energy of an
incident particle will add to the excitation energy of its reaction
by an amount depending upon the dynamics of the reaction.
NUCLEAR TRANSFORMATIONS 417
in one or more different ways, depending upon its excitation energy 1 . Thus 'jN*
with an excitation energy of, say, 12 MeV can decay via the reactions
■i\*
'C + ;n
'IC + f H
or simply einit one or more gamma rays whose energies total 12 MeV, but it
cannot decay hy the emission of a triton (^H) or a lielium3 (He) particle since
it does not have enough energy to liberate them. Usually a particular decay
mode is favored by a compound nucleus in a specific excited state.
The formation and decay of a compound nucleus has an interesting inter
pretation on the basis of the liquiddrop nuclear model described in Chap. 11.
In terms of this model, an excited nucleus is analogous to a drop of hot liquid,
with the binding energy of the emitted particles corresponding to the heat of
vaporization of the liquid molecules. Such a drop of liquid will eventually
evaporate one or more molecules, thereby cooling down. The evaporation
process occurs when statistical fluctuations in the energy distribution within the
drop cause a particular molecule to have enough energy For escape. Similarly,
a compound nucleus persists in its excited state until a particular nucleoli or
group of nucleons momentarily happeas to have a sufficiently large fraction of
the excitation energy to leave the nucleus. The time interval between the
formation and decay of a compound nucleus fits in nicely with this picture.
The analysis of the reaction that occurs when a moving nucleon or nucleus
strikes another one at rest is greatly simplified by the use of a coordinate system
moving with the center of mass of the colliding particles. To an observer located
at the center of mass, the particles have equal and opposite momenta (Fig. 1 2 1 7).
Hence if a particle of mass m, and speed t; is incident upon a stationary particle
of mass i»2 as viewed by an observer in the laboratory, the speed V of the center
of mass is defined by the condition
m,(« — V) = m^V
\fii, + m s f
418
In most nuclear reactions, c < c, and so a nonrelalivistic treatment is satisfactory.
In the laboratory system, the total kinetic energy is that of the incident particle
only:
T ub = l /2 m i» 2
In the centerofmass system, lx>th particles arc moving and contribute to the
total kinetic energy:
THE NUCLEUS
= T lBh  (.«, + m 2 )V 2
12.61
mi.
The total kinetic energy of the particles relative to the center of mass is their
total kinetic energy in the laboratory system minus the kinetic energy '/^{m, +
rH,)V 2 of the moving center of mass. Thus we can regard T tm as the kinetic
energy of the relative motion of the particles. When the particles collide, the
maximum amount of kinetic energy that can be converted to excitation energy
of the resulting compound nucleus while still conserving momentum is T em , which
is always less than 7j, h .
Information about the excited states of nuclei can be gained from nuclear
reactions as well as from radioactive decay. The presence of an excited state
may l>e detected by a peak in the cross section versus energy curve of a particular
reaction, as in the neutroncapture reaction of Fig. 1216. Such a peak is called
FIGURE 1217 Laboratory and center of mi si coordinate syitemi.
(a) Motion in the laboratory coordinate system before collision.
center of moss
m.
\
V= r m 2
m l tm ! /\
(h) Motion in the centerofmass coordinate system before collision.
center of mass
i«i c — V
\
— V ^?
o
(c) A completely inelastic collision as seen In laboratory and centerofmass coordinate systems.
before
collision
laboratory
coordinate system
o
Ceaterofmats
coordinate system
o
after
collision
o
o
NUCLEAR TRANSFORMATIONS 419
a resonance by analogy with ordinary acoustic or ac circuit resonances: a com
pound nucleus is more likely to be formed when the excitation energy provided
exactly matches one of its energy levels than if the excitation energy has some
other value.
The reaction of Fig. 1216 has a resonance at 0.176 eV whose width (at
halfmaximum) is r = 0.115 eV. The uncertainty principle in the form
\E&t > ft enables us to relate the level width I" of an excited state with the
mean lifetime t of the state. The width I" evidently correspond"; to die uncer
tainty A£ in the energy of the state, and the mean lifetime t corresponds to
the uncertainty At in the time when the state will decay, in the present example
by the emission of a gamma ray. The mean lifetime of an excited state is defined
in general as
12 62
ft
Mean lifetime of excited state
In the case of the above reaction, the level width of 0,1 15 eV implies a mean
lifetime for the compound nucleus of
UKH X UH« JS
O.llSeV X 1.60 X 10 ia J/eV
= 5.73 X 10 * s
12.11 NUCLEAR FISSION
Another type of nuclearreaction phenomenon that can lie analyzed with the
help of the liquiddrop model is fission, in which a heavy nucleus (A > — 230}
splits into two lighter ones. When a liquid drop is suitably excited, it may
oscillate in a variety of ways. A simple one is shown in Fig. 1218: the drop
successively becomes a prolate spheroid a sphere, an oblate spheroid, a sphere,
a prolate spheroid again, and so on. The restoring force of its surface tension
always returns the drop to spherical shape, but the inertia of the moving liquid
molecules causes the drop to overshoot sphericity and go to the opposite extreme
of distortion.
W hile nuclei may be regarded as exhibiting surface tension, and so can vibrate
like a liquid drop when in an excited state, they also are subject lo disruptive
forces due to the mutual electrostatic repulsion of their protons. When a nucleus
is distorted from a spherical shape, the shortrange restoring force of surface
tension must cope with the latter longrange repulsive force as well as with the
inertia of the nuclear matter. If the degree of distortion is small, the surface
420
THE NUCLEUS
oCooo
time
FIGURE 1218 The oscillations of a liquid drop.
tension is adequate to do both, and the nucleus vibrates back and forth until
it eventually loses its excitation energy by gamma decay. If the degree of
distortion is sufficiently great, however, the surface tension is not adequate to
bring back together the now widely separated groups of protoas, and the nucleus
splits into two parts. This picture of fission is illustrated in Fig. 1219,
The new nuclei that result from fission are called fission fragments. Usually
fission fragments are of unequal size (Fig. 3220), and, because heavy nuclei have
a greater neutron/proton ratio than lighter ones, they contain an excess of
neutrons. To reduce this excess, two or three neutrons are emitted by the
fragments as soon as they are formed, and subsequent beta decays bring their
neutron/ proton ratios to stable values.
A heavy nucleus undergoes fission when it acquires enough excitation energy
(5 MeV or so) to oscillate violently. Certain nuclei, notably 2 (gU, are sufficiently
excited by the mere absorption of an additional neutron to split in two. Other
nuclei, notably ^U (which composes 99.3 percent of natural uranium, with a gU
composing the remainder), require more excitation energy for fission than the
binding energy released when another neutron is absorbed, and undergo fission
only by reaction with fast neutrons whose kinetic energies exceed about 1 MeV.
Fission can occur after excitation by other means besides neutron capture, for
instance, by gammaray or proton bombardment. Some nuclides are so unstable
as to be capable of spontaneous fission, but they are more likely to undergo alpha
decay before this takes place.
The most striking aspect of nuclear fission is the magnitude of the energy
evolved. This energy is readily computed. The heavy fissionable nuclides, whose
mass numbers are about 240, have binding energies of —7.6 MeV/nucleon, while
fission fragments, whose mass numbers are about 120, have binding energies of
—8.5 MeV/nucleon. Hence 0,9 MeV/nucleon is released during fission— over
FIGURE 12 19 Nuclear fist ion according to the liquid drop modal.
NUCLEAR TRANSFORMATIONS 421
0.001
150 160 170
100 110 120 130
MASS NUMBER
FIGURE 1220 The distribution of mass numbers in the fragments from the fission of =£U.
200 MeV for the 240 or so nucleons involved! Ordinary chemical reactioas, such
as those that participate in the combustion of coal and oil, liberate only a few
electron volts per individual reaction, and even nuclear reactions (other than
fission) liberate no more than several million electron volts. Most of the energy
that is released during fission goes into the kinetic energy of the fission fragments:
the emitted neutrons, beta and gamma rays, and neutrinos carry off perhaps 20
percent of the total energy.
Almost immediately after the discovery of nuclear fission in 1939 it was
rrcn^ni/ed ihai, because a neutron can induce fission in ;i suitable nucleus WH4
the consequent evolution of additional neutrons, a selfsustaining sequence of
fissions is, in principle, possible. The condition for such a chain reaction to occur
in an assembly of fissionable material is simple: at least one neutron produced
during each fission must, on the average, initiate another fission. If too few
neutrons initiate fissions, the reaction will slow down and stop; if precisely one
neutron per fission causes another fission, energy will be released at a constant
rate (which is the case in a nuclear reactor); and if the frequency of fissioas
increases, the energy release will be so rapid that an explosion will occur (which
is the case in an atomic bomb). These situations are respectively called sub
critical, critical, and supercritical.
12.12 TRANSURANIC ELEMENTS
Elements of atomic number greater than 98, which is that of uranium, have such
short half lives that, had they been formed when the vuiiverse came into being,
they would have disappeared long ago. Such transuranic elements may lie
produced in the laboratory by the bombardment of certain heavy nuclides with
neutrons. Thus 2 $)U may absorb a neutron to become 2 jjgU, which betadecays
(T 1/2 ss 23 min) into "IJjNp, an isotope of the transuranic element neptunium:
*gfU + Jn
«gu
+ e"
This neptunium isotope is itself radioactive, undergoing beta decay with a half
life of 2.3 d into an isotope of the transuranic element plutoniwn:
*IN P ^ l 8Pu + «r
Pliitonium alphadecays into «gU with a half life of 24,000 yr:
«gP«
>SU + IHe
a
It is interesting to note that "gPu, like 2 jgU, is fissionable and can be used in
nuclear reactors and weapons. Plutonium is chemically different from uranium;
its separation from the remaining 2 jgU after neutron irradiation is more easily
accomplished than the separation of "jgU from the much more abundant ^U
in natural uranium.
Transuranic elements past einsteinum (Z = 99) have halflives too short for
their isolation in weighable quantities, though they can l>e identified by chemical
means. The transuranic element of the highest atomic number yet discovered
has Z = 105.
422
THE NUCLEUS
NUCLEAR TRANSFORMATIONS 423
12.13 THERMONUCLEAR ENERGY
The basic exothermic reaction in stars— and hence the source of nearly all the
energy in the universe— is the fusion of hydrogen nuclei into helium nuclei. This
can take place under stellar conditions in two different series of processes. In
one of them, the proton proton cycle, direct collisions of protons result in the
formation of heavier nuclei whose collisions in rum yield helium nuclei. The
other, the carbon cycle, is a series of steps in which carlxm nuclei absorh a
succession of protons until they ultimately disgorge alpha particles to become
carbon nuclei once more.
The initial reaction in the protonproton cycle is
}H + JH * «H + e + + 9
the formation of deuterons by the direct combination of two protons accom
panied by the emission of a positron. A cleuteron may then join with a proton
to form a ;]He nucleus:
ill + *H » PBe + y
Finally two He nuclei react to produce a jjHe nucleus plus two protons:
£He I He • ^He + }H + JH
The total evolved energy is {&m)c\ where Am is the difference between the mass
of four protons and the mass of an alpha particle plus two positrons; it buns
out to be 24.7 MeV. Figure 1221 shows the entire sequence.
The carbon cycle proceeds in the following way:
'§C
>?N
1C + e + + v
ill i
fN
fS + »C » »*N + y
iH + »N *■ l p + y
'JO* '»N + «» + p
JH + i N » >JC + JHe
Carbon cycle
The net result again is the formation of an alpha particle and two positrons from
four protons, with the evolution of 24.7 MeV; the initial "gC acts as a kind of
catalyst for the process, since it reappears at its end (Fig. 1222).
Selfsustaining fusion reactions can occur only under conditions of extreme
temperature and pressure, to ensure that the participating nuclei have enough
energy to react despite their mutual electrostatic repulsion and that reactions
occur frequently enough to counterbalance losses of energy to the surroundings.
Stellar interiors meet these specifications, in the sun, whose interior temperature
424 THE NUCLEUS
FIGURE 1221 The protonproton cycle. This is one of the two nuclear reaction sequences that take
place in the sun and that involve the combination of tour hydrogen nuclei to form a helium nucleus with
the evolution of energy.
is estimated to be 2 X It) 8 K, the protonproton cycle has the greater probability
for occurrence. In general, the carbon cycle is more efficient at high tempera
tures, while the protonproton cycle is more efficient at low temperatures. Hence
shirs hotter than the sun obtain their energy largely from the former cycle, while
those cooler than the sun obtain the greater part of their energy from the latter
cycle. The neutrinos carry away about 10 percent of the energy produced by
a typical star.
The energy liberated in the fusion of light nuclei into heavier ones is often
called fliermonuclear energy, particularly when the fusion takes place under man's
control. On the earth neither the protonproton nor carbon cycle offers any hope
of practical application, since their several steps require a great deal of time.
Two fusion reactions that seem promising as terrestrial energy sources are the
NUCLEAR TRANSFORMATIONS
425
426
FIGURE 1222 The carbon cycle also Involves the combination of (out hydrogen nuclei to form a helium
nucleus with the evolution of energy. The ;'C nucleus is unchanged by the series of reactions.
direct combination of Ivvo deuterons in either of the following ways:
fll + fH > pie + },n + 3.3 MeV
?H + ?H * JH + {H + 4.0 MeV
THE NUCLEUS
Another is the direct combination of a deuteron and a Iriton to form an alpha
particle,
\H + JB » He 4 in + 17.8 MeV
Capitalizing upon the above reactions requires an abundant, cheap source of
deuterium. Such a source is the oceans and seas of the world, which contain
about 0.015 percent deuterium— a total of perhaps 10 15 tons! In addition, a more
efficient means of promoting fusion reactions than merely bombarding a target
with fast particles from an accelerator is required, since the operation of an
accelerator consumes far more power than can l>e evolved by the relatively few
reactions that occur in the target. Current approaches to this problem all involve
very hot plasmas (fully ionized gases) of deuterium or deuteriumtritium mixtures
which are contained by strong magnetic fields. The purpose of high temperature
is to ensure that the individual *II and ?H nuclei have enough energy to come
together and react despite their electrostatic repulsion. A magnetic field is used
as a container to keep the reactive gas from contacting any other material which
might cool it down or contaminate it; there is little likelihood that the wall will
melt since the gas. though at a temperature of several million degrees K, actually
does iidI have a high energy density. While nuclear fusion reactors present more
severe practical difficulties than fission reactors, there is little doubt that they
will eventually become a reality.
Problems
(The masses in u of neutral atoms of nuclides mentioned below are: {H, 1.007825;
m 3.016050; ilUe, 3.01fi030; ^le, 4.002603; JLI, 7.0160; jBe, 7.0169; $B,
10.0129; '? P B, 12.0)44; i§C, 12.0000; »gC. 13.0034; "X, I 1.0031; 'gO, 15.9949;
'£0, 16.9994. The neutron mass is 1.008665 u.
Atomic masses of the elements
are listed in Table 7.1.)
1. Tritium gfij has a half life of 12.5 yr against beta decay. What fraction
of a sample of pure tritium "ill remain undeeayed after 25 yr?
2. The half life of ffXa is 15 h. How long does it take for 93.75 percent of
a sample of this isotope to decay?
3. One g of radium has an activity of 1 Ci. From this fact determine the half
life of radium.
4. The mass of a millicurie of *JJPb is 3 X 10" " kg. From this fact find the
decay constant of z JPb. (Assume atomic mass in u equal to mass number in
Probs. 4 to 6.)
NUCLEAR TRANSFORMATIONS 427
428
5. The half life of ^U against alpha decay is 4.5 X 10 9 yr. How many disinte
grations per second occur in 1 g of 23 JU?
6. The potassium isotope JgK undergoes beta decay with a half life of 1 .83 X 10"
yr. Find the number of licta decays that occur per second in I g of pure ^K.
7. A 5.78MeV alpha particle is emitted in the decay of radium. If the diameter
of the radium nucleus is 2 X 10 M m, how many alphaparticle de Broglie
wavelengths fit inside the nucleus?
8. Calculate the maximum energy of the electrons emitted in the beta decay
of ] fB.
9. Why does .{Be invariably decay by electron capture tastead of by positron
emission? Note that ]Be contains one more atomic electron than does jLi.
10. Positron emission resembles electron emission in all respects except that
the shapes of their respective energy spectra are different: there are many
lowenergy electrons emitted, but few towenergy positrons. Thus the average
electron energy in beta decay is about 0.3r max> whereas the average positron
energy is about 0AT mhs . Can you suggest a simple reason for this difference?
11. Determine the ground and lowest excited states of the 39th proton in ";V
with the help of Fig. 1110. Use this information to explain the isomerism of
!I Y together with the fact, noted in Sec. 6.10, that radiative transitions between
states with very different angular momenta are extremely improbable.
12. Find the minimum energy in the laboratory system that a neutron must
have in order to initiate the reaction
£n + x $) + 2,20 MeV » lC + pie
13. Find the minimum energy in the laboratory system that a proton must have
in order to initiate the reaction
p + (I + 2.22 MeV » p + p + n
14. Find the minimum energy in the laboratory system that an alpha particle
must have in order to initiate the reaction
JHc + '{N + 1.18 MeV ► >0 + jH
15. The cross sections for comparable neutron and protoninduced nuclear
reactions vary with energy in approximately the manner shown in Fig. 1223.
Why does the neutron cross section decrease with increasing energy whereas
the proton cross section increases?
THE NUCLEUS
Neutron capture
FIGURE 1223 Neutron and proton
capture cross section! wary differently
with particle energy.
ENERGY
Proton capture
ENERGY
16. When a neutron is absorbed by a target nucleus, the resulting compound
nucleus Ls usually more likely to emit a gamma ray than a proton, deuteron,
or alpha particle. Why?
17. There are approximately 6 X 10 28 atoms/m 3 in solid aluminum. A beam
of 0.5MeV neutrons is directed at an aluminum foil 0. 1 mm thick. If the capture
cross section for neutrons of this energy in aluminum is 2 X 10~ 31 m 2 , find the
fraction of incident neutrons that are captured.
18. The density of l "B is 2.5 X I0 3 kg/m 3 . The capture cross section of l ,'B
is about 4,<XX) b for "thermal" neutrons, that is, neutrons in thermal equilibrium
with matter at room temperature. How thick a layer of 'jB is required to absorb
99 percent of an incident beam of thermal neutrons?
19. The density of iron is about 8 X H> 3 kg/m 3 . The neutroncapture cross
section of iron is about 2.5 b. What fraction of an incident beam of neutrons
is absorbed by a sheet of iron 1 cm thick?
NUCLEAR TRANSFORMATIONS 429
20. The cross section of iron for neutron capture is 2.5 b. What is the mean
free path of neutrons in iron?
21. The fission of $U releases approximately 200 MeV. What percentage of
the original mass of 2 jj§U + n disappears?
22. Certain stars obtain part of their energy by the fusion of three alpha
particles to form a 'jfC nucleus. How much energy does each such reaction
evolve?
430
THE NUCLEUS
ELEMENTARY PARTICLES
13
While nuclei are apparently composed solely of protons and neutrons, several
score other elementary particles have heen observed to be emitted by nuclei
under appropriate circumstances. These particles, christened "straiiijc particles"
soon after their discovery about two decades ago, bring the total number merely
of relatively stable elementary particles to over .30. To discern order in this
multiplicity of particles has not proved to be an easy task. While certain
regularities in elementaryparticle properties have been established, and while
such particles as the electron, the neutrino, and the * meson arc relatively well
understood, no comprehensive theory of elemental? particles lias yet found wide
acceptance. It is fitting to conclude our survey of modern physics with this topic,
then, as a reminder that there remains much to Ix.' learned about the natural
world.
13.1 ANTI PARTICLES
The electron is the only elementary particle for which a satisfactory theory is
known. This theory was developed in l l )2S by P. A. M. Dirac, who obtained
a wave equation for a charged particle in an electromagnetic field that incorpo
rated the results of special relativity. When the observed mass and charge of
the electron are inserted in the appropriate solutions of this equation, the intrinsic
angular momentum of the electron Is found to be %H (that is, spin H,< and its
magnetic moment is found to lie efi/2in. one Bohr magneton. These predictions
agree with experiment, and the agreement is strong evidence for the correctness
of the Dirac theory.
An unexpected result of the Dirac theory was its prediction that positive as
well as negative electrons should exist. At first it was thought that the proton
was the positive counterpart of the electron despite the difference in their masses,
but in 1932 a positive electron was unambiguously detected in the mix of com n it
radiation at the earth's surface. Positive electrons, as mentioned earlier, are
431
usually called pttsitrom. The materialization of an electronpositron pair from
a photon of sufficient energy (>L02 MeV) and the annihilation of an electron
and a positron that come together were described in Sec. 2.6.
The positron is often spoken of as the antipartide of the electron, since it
is able to undergo mutual annihilation with an electron. All other known
elementary particles except for the photon and the «r* and V mesons also have
autiparttele counterparts: the latter constitute their own ant i part teles. The
antipartide of a particle has the same mass, spin, and lifetime if unstable, but
its charge (if any) has the opposite sign and the alignment or antialignment
between its spin and magnetic moment is also opposite to that of (he particle.
The distinction lietween the neutrino and the antineutrino is a particularly
interesting one. The spin of the neutrino is opposite in direction to the direction
of its motion; viewed from behind, as in Fig. 131, the neutrino spins counter
clockwise. The spin of the antineutrino, on the other hand, is in the same
direction as its direction of motion; viewed from behind, it spins clockwise. Thus
the neutrino moves through space in the manner of a lefthanded screw, while
the antineutrino does so in the maimer of a righthanded screw.
Prior to 1956 it had been universally assumed that neutrinos could be either
lefthanded or righthanded, implying that, since no difference was possible
l>etwcen them except one of spin direction, the neutrino and antineutrino are
identical. This assumption had roots going all the way back to Leibniz, Newton's
contemporary and an independent inventor of calculus. The argument may be
stated as follows: if we observe an object or a physical process of some kind
^^^^•^ neutrino
o:
/
antineutrino
FIGURE 131 Neutrinos and antineutrinos have apposite elec
trons af spin.
432 THE NUCLEUS
both directly and in a mirror, we cannot ideally distinguish which object or
process is being viewed directly and which by reflection. By definition, distinc
tions in physical reality mast lie capable of discernment or they are meaningless.
Now the only difference between something seen directly and the same thing seen
in a mirror is the interchange of right and left, and so all objects and processes
must occur with equal probability with right and left interchanged. This plausible
doctrine is indeed experimentally valid for nuclear and electromagnetic inter
actions, but until 1U56 its applicability to neutrinos had never been actually
tested. In that year T. D. l^e and C. N. Yang suggested that several serious
theoretical discrepancies would l>e removed if neutrinos and anti neutrinos have
different handedness, even though it meant that neither particle could therefore
be reflected in a mirror. Experiments performed soon after their proposal showed
unequivocally that neutrinos and antineutrinos are distinguishable, having left
handed and righthanded spins respectively. We might note that the absence
of rightleft symmetry in neutrinos can occur only if the neutrino mass is exact ly
zero, thereby resolving what had liecn the very difficult experimental problem
of measuring the neutrino mass.
13.2 MESON THEORY OF NUCLEAR FORCES
If nuclear forces were exclusively attractive, a nucleus would be stable only if
its size were so small (about 2 fm in radius) that each nucleon interacted with
all the others. The binding energy per nucleon would then be proportional to
A, the uumt)cr of nuclcons present. In fact, nuclear volumes are found to be
proportional to A and the binding energy per nucleoli is roughly the same for
all nuclei; each nucleon interacts only with a small number of its nearest neigh
bors. Thus there must lie a repulsive component in nuclear forces that keeps
nuclei from collapsing, as indicated in Fig. 116, which means that these forces
are not analogous to the "ordinary" gravitational and electrical forces.
We encountered a somewhat similar situation in Sec. 8.3, where the forces
present in the H./ molecular ion can he thought of as including an exchange
force which arises because of the possibility that the electron can shift from one
of the protons to the other. Depending on whether the wave amotion of the
system is symmetric or antisymmetric for the particle exchange, the exchange
force is either attractive or repulsive. It is tempting to consider the interaction
between nuclcons to be, at least in part, a consequence of some kind of exchange
force as well. For instance, exchange forces provide an explanation for the
stability of the triplet state of the deuteron, which is descril>ed by a symmetric
wave function since the spins are parallel, and the instability of the singlet state,
which is described by an antisymmetric wave function. Since the nudeons in
ELEMENTARY PARTICLES 433
434
,i nucleus are all in different quantum states (by the exclusion principle), both
attractive and repulsive exchange forces would occur, and a mixture of an
"ordinary" attractive nuclear force and such exchange forces is able to account
in a general way for a great many nuclear properties.
The next question is, what kinds of particles are exchanged between nearby
nueleons? In 1932, lleisenherg suggested that electrons and positrons shift back
and forth between micleons: for instance, a neutron might emit an electron and
become a proton, while a proton absorbing the electron would then become
a neutron. However, calculations based on betadecay data showed that the
forces resulting from electron and positron exchange by micleons are too small
by the huge factor of 10" to Ik significant in nuclear structure. Then, in 1935,
the Japanese physicist llideki Yukawa proposed that particles called masons,
heavier than electrons, arc involved in nuclear forces, and he was able to show
that the interactions they produce between micleons arc of the correct order
ol magnitude.
According to the meson theory of nuclear forces, all nueleons consist of
identical cores surrounded by a "cloud" of one or more mesons. Mesons may
lie neutral or curry cither charge, and the sole difference between neutrons and
protons is supposed to lie in the composition of their respective meson clouds.
The forces that act between one neutron And another and between one proton
and another are the result of the exchange of neutral mesons (designated rr")
between them. The force lietween a neutron and a proton is the result of the
exchange of charged mesons ^* and 57" > between them. Thus a neutron emits
a tt~ meson and is converted into a proton;
ii * p + it
while the absorption of the tt
converts it into a neutron;
by the proton the neutron was interacting with
p + it  > n
In the reverse process, a proton emits a ** meson whose absorption by a neutron
converts it into a proton:
p — * n + it*
n + 7T — » p
While there is no simple mathematical way of demons! rating how the exchange
of particles between two bodies can lead to attractive and repulsive forces, a
rough analogy may make the process intuitively meaningful. Let us imagine
two boys exchanging basketballs (Fig. b'32). If they throw the lralls at each other,
they each move backward, and when they catch the balls thrown at them, their
backward momentum increases. Thus this method of exchanging the basketballs
THE NUCLEUS
FIGURE 132 Attractive and repulsive
forces can bath ariic From particle ex
change.
repulsive force due to particle exchange
attractive force due to particle exchange
yields the same effect as a repulsive force between the boys. If the boys snatch
the basketballs from each other's hands, however, the result will l>e equivalent
to an attractive force acting between them.
A fundamental problem presents itself at this point. If nueleons constantly
emit and absorb mesons, why are neutrons or protons never found with other
than their usual masses? The answer is based upon the uncertainty principle.
The laws of physics refer exclusively to experimentally measurable quantities,
and the uncertainty principle limits the accuracy with which certain combina
tions of measurements can be made. The emission of a meson by a nuclcon which
does not change in mass — a clear violation of the law of conservation of
energy— can occur provided that the nucleon absorbs a meson emitted by the
neighl>oring nucleon it is interacting with so soon afterward that even in principle
it is impossible to determine whether or not any mass change actually has been
involved. Since the uncertainty principle may be written
13.1
IE \t > ft
an event in which an amount of energy IE is not conserved is not prohibited
so long as the duration of the event does not exceed approximately fi/A£.
ELEMENTARY PARTICLES 435
We know that nuclear forces have a maximum range R of about 1 .7 fm, so
that if we assume a meson travels lietween nuclei at approximately the speed
of light ft the time interval At during which it is in flight is
11.2
*«£
Tire emission of a meson of mass »i T represents the nonconservalion of
13.3 AE = m^c 2
of energy. According to Eq. 13.1 this can occur if AE At > ft; that is, if
Hence the minimum meson mass is specified by
13.4
436
> 1.9 X H> 2S kg
which is about 200 m g , that is, 200 electron masses.
13.3 PIONS AND MUONS
Twelve years after the meson theory was formulated, particles with the predicted
properties were actually found outside nuclei. Today tt mesons arc usually called
pious.
Two factors contributed to the belated discovery of the free pion. First, enough
energy must be supplied to a nucleoli so that its emission of a pion conserves
energy. Thus at least m T c 2 of energy, about 140 MeV, is required. To furnish
a stationary nucleoli with this much energy in a collision, the incident particle
must have considerably more kinetic energy than m v c 2 in order that momentum
as well as energy be conserved. Particles with kinetic energies of several hundred
MeV are therefore required to produce free pions, and such particles are found
in nature only in the diffuse stream of cosmic radiation that bombards the earth.
I lence the discovery of the pion had to await the development of sufficiently
sensitive and precise methods of investigating cosmicray interactions. More
recently highenergy accelerators were placed in operation; they yielded the
necessary particle energies, and the profusion of pions thai were created with
their help could be studied readily.
The second reason for the lag between the prediction and experimental
discovery of the pion is its instability: the half life of the charged pion is only
THE NUCLEUS
1.8 X 10" s, and that of the neutral pion is 7 X 10" 1T s. The lifetime of the
tt" is so short, in fact, that its existence was not established until 1050,
Charged pions almost invariably decay into lighter mesons called ii mesons
(or m i jo m) and MllUlUMa
9
These neutrinos are not the same as those involved in beta decay, which is why
their symbols are ^ and Fy The existence of two classes of neutrino was estab
lished in 1962. A metal target was bombarded with highenergy protons, and
pions were created in profusion. Inverse reactions traceable to the neutrinos
from the decay of these pious produced unions only, and no electrons. Hence
these neutrinos must lie somehow different from those associated v. h lieta decay.
The neutral pion decays into a pair of gamma rays:
w° > 7 + y
The it* and tr" have rest masses of 273 m,, while that of the «* is slightly
less, 264 in,. The tt" is the antiparticle of the w + , and the tf" is its own anti
particle, a distinction it shares only with the photon and the ij° meson.
Whereas the existence oF pions is so readily understandable that they were
predicted many years licfore their actual discovery, unions even today represent
something of a puzzle. Their physical properties are known quite accurately.
Positive and negative unions have the same rest mass, 207 m„, and the same spin,
l / a . Both decay with a half life of 1.5 X 10 u s into electrons and neutrino
iintiiieutrino pairs:
/i + — * e +
+
r.
+
',
ii" — » e~
+
h
+
''.
As with electrons, the positive charge state of the union represents the anti
particle. There is no neutral muon.
Unlike the case of pions which, as we would expect, interact strongly with
nuclei, the only interaction between muons and matter is an electric one.
Accordingly muons readily penetrate considerable amounts of matter before
being absorbed. The majority of cosmicray particles at sea level are muons from
the decay of pions created in nuclear collisions caused by fast primary cosmicray
atomic nuclei, siuce nearly all the other particles in the cosmicray stream either
decay or lose energy rapidly and are absorbed far above the earth's surface.
The mysterious aspect of the muon is its function— or, rather, its apparent
lack of any function. Only in its mass and instability does the muon differ
significantly from the electron, leading to the hypothesis that the muon is poet)
a kind of "heavy electron" rather than a unique entity. Other evidence, which
ELEMENTARY PARTICLES 437
we shall examine later in this chapter, is less unflattering to the mnon, although
it is still not wholly clear why, for instance, pious should pre ferenti ally decay
Into unions rather than directly into electrons; only about (1.0] percent of pious
decay directly into electrons and neutrinos.
13.4 KAONS AND HYPERONS
Pions and inuons do not exhaust the list of known particles with masses interme
diate between those of the electron and proton. A third class of mesons, called
K mesons (or toons), has Iwen discovered whose members may decay in a variety
of ways. Charged kaons have rest masses of 96ftn r , spins of Q, and half lives
of 8 X 10" 9 s. The following decay schemes are possible for K + mesons, in order
of relative probability:
' J»* + »V
» w~ + it"
» JT + + 5T + + V~
* ifl + e* 4 v t
* w° + fi + + *„
+ V+ + V° + 57°
There are apparently two distinct varieties of neutral K mesons, the A", and
K 3 °. Both have rest masses of 974 m, and spins of 0, but the former has a half
life of about 7 X 10"" s, while that of the latter is about 4 X I0 R s. The
following decay modes are known for neutral K mesons, again in order of relative
probability:
K<
,
17*
+ W
»
n*
+ w°
K z
*
w"
+ e T
+
•',
— »
IT*
+ e
+
';
— »
7T
+ M +
+
r.
*
ff+
+r
+
1
»
**
+ ir
+
"
»
ir+
+ v~
+
■*
In addition to their electromagnetic interaction with matter through which
they pass, K mesons exhibit varying degrees of specifically nuclear interactions.
The K + and K° mesons interact only weakly with nuclei, while their antiparticle
counterparts are readily scattered and absorbed by nuclei in their paths.
Elementary particles heavier than protons are called hijpemm. The known
hyperons fall into four classes, A, 2, 2, and ft hyperons, in order of increasing
438
THE NUCLEUS
mass. (A, 2, E, and ft are, respectively, the Creek capital letters lambda, sigpui.
xi, and omega.) All are unstable with extremely brief mean lifetimes. The spin
of all hyperons is % except that of the ft hyperon, which is %. The masses,
half lives, and decay schemes of various hyperons are given in Table 13.1,
Like pions and kaons (but unlike inuons), hyperons exhibit definite interactions
with nuclei. The A" hyperon is even able to act as a nuclear constituent. A
nucleus containing a bound A" hyperon is called a hyperfragpient; eventually
the A decays, of course, with the resulting nucleoli and w meson either reacting
with the parent nucleus or emerging from it entirely.
13.5 SYSTEMATICS OF ELEMENTARY PARTICLES
Despite the multiplicity of elementary particles and the diversity of their prop
erties, it is possible to discern an underlying order in their behavior. The fact
of this order does not constitute a theory of elementary particles, however, any
more than the order found in atomic spectra constitutes a theory of the atom,
but it does provide hope that there may indeed be a single theoretical picture
that can encompass elementaryparticle phenomena in the manner that the
quantum theory encompasses atomic phenomena. Thus far no such picture has
emerged, although some intriguing lines of approach have been proposed. In
the remainder of this chapter we shall examine the regularities observed in
elementary particles and their apparent significance.
Table 13,2 is a listing in order of rest mass of the relatively stable elementary
particles we have thus far mentioned plus the ij meson, which we shall discuss
shortly. By relatively stable is meant that the half lives of the particles all greatly
exceed the time required for light to travel a distance equal to the "diameter"
Table 13.1.
HYPERON PROPERTIES,
Particle
Halt life, i
Decay
A"
2
»
T
a
2,184
1.7 x 10'"
A — * p + T '
 n + if
2,328
0.6 x lO" 1 "
V _ p + „o
» n + v'
2,342
1.1 X 10 "'
v _ „ + „
2,334
<10 ■"
2° ♦ A + t
2,585
IS x 10 '"
S * A + v~
2,573
2.0 X 10 '"
w_ A + Jr »
3,276
vHrw
« » A  K
ELEMENTARY PARTICLES 439
*• s
g
§
cr
o
1
I
a
2
II
II
+ +
I
o
s
js; ;£? «£ 2*
ill
= <F J:
& X4
d
9
I
I
s
i,
1
b
V
I
.S
i>i
»
IN
• I
M
I
+
a
I
fe
of an elementary particle. Tin's diameter is probably a little over 10 _ls m, and
the characteristic time required to traverse it at the speed of light is therefore
of the order of magnitude of 10" 23 s. Thus the particles in Tabic 13.2 are almost
all capable of traveling through space as distinct entities along paths of meas
urable length in such devices as bubble chambers.
A considerable Isody of experimental evidence also points to the existence of
many different "particles" whose lifetimes against decay are only about 10" 23 s.
What can lie meant by a particle which exists for so brief an interval? Indeed,
how can a time of JO 23 s even l>e measured? Such particles cannot be detected
by observing their formation and subsequent decay in a bubble chamber or other
instrument, but instead appear as resonant states in the interaction of more stable
{and hence more readily observable) particles. Resonant states occur in atoms
as energy levels; in Chap. 4 we reviewed the FranckHertz experiment, which
showed the existence of atomic energy levels through the occurrence of inelastic
electron scattering from atoms at certain energies only. An atom in a specific
excited state is not the same as that atom in its ground state or in another excited
state, but we do not usually speak of such an excited atom as though it were
a memljer of a special species only because the interaction that gives rise to
the excited state — the electromagnetic interaction — is well understood. A rather
different situation holds in the case of elementary particles, where the various
interactions involved are, except for the electromagnetic one, only partially
understood, and much of oiu information comes from the properties of the
resonances.
I^et us see what is involved in a resonance in the case of elementary particles.
An experiment is performed, for instance the Ixnnbardmcul of protons by ener
getic ir + mesons, and a certain reaction is studied, for instance
•7T+ + p — > TT + + p + 7T + + 1!~ + ff°
The effect of the interaction of the a* and the proton is the creation of three
new pions. In each such reaction the new mesons have a certain total energy
that consists of their rest energies plus their kinetic energies relative to their
center of mass. If we plot the number of events ohserved versus the total energy
of the new mesons in each event, we obtain a graph like that of Fig, 133,
Evidently there is a strong tendency for the total meson energy to be 785 MeV
and a somewhat weaker tendency for it to be 548 MeV. We can say that the
reaction exhibits resonances at 548 and 785 MeV or, equivalently, we can say
that this reaction proceeds via the creation of an intermediate particle which
can be either one whose mass is 548 MeV or one whose mass is 785 MeV, From
the graph we can even estimate the mean lifetimes of these intermediate particles,
which are known as the jj and w mesons, respectively. According to the un
certainty principle, the uncertainty in decay time of an unstable particle — which
ELEMENTARY PARTICLES
441
442
600 eoo
EFFECTIVE MASS. MeV
1.000
FIGURE 133 Resonant states In the reaction »* + p*w' + /? + r* +■ r + a"
occur at effective masses of 548 and 785 MeV. By effective mass is meant the
total energy, including mass energy, of the three new mesons relative to their cen
ter of mass,
is its mean lifetime t — will give rise to an uncertainty in the determination of
its energy — which is the width A£ at half maxim tun of die corresponding peak
in Fig, 133 — whose relationship is
13.5
tAE == ft
Hence the lifetimes of the resonances, or, just as well, the lifetimes of the i\ and
m mesons, can be established. The ij lifetime is sufficiently long for it to be
regarded as a relatively stable particle and it is included in Table 13.2, while
the « lifetime is too short by many orders of magnitude. We shall return to
the resonance particles later in this chapter.
The particles in Table 13.2 seem to fall naturally into four general categories.
In a class by itself is the photon, a stable particle with zero rest mass and imit
THE NUCLEUS
spin. If there is a graviton. a particle that is the quantum of the gravitational
field in the same sense that die photon is the quantum of the electromagnetic
field or the pion the quantum of the nuclear force field, il would be another
meml>er of this class. The gravitnu, its yet undetected, should be massless and
stable, and sbutild have a spin of 2. Us interaction with matter would be ex
tremely weak, and it is unlikely that present techniques are capable of verifying
its existence. (The zero mass of the graviton can be inferred from the unlimited
range of gravitational forces. As we saw in Sec. 13.2, the mutual forces Ixstween
two bodies can lie regarded as transmitted by the exchange of particles Ijetween
them. If energy conservation is to be preserved, the uncertainly principle
requires that the range of the force be inversely proportional to the mass of the
exchanging particles, and so gravitational forces can have an infinite range
only if the graviton mass is zero. A similar argument holds for the photon
mass.)
After the photon in Table 13.2 come the eneutrino and uneutrino, the
electron, and the inuon, all with spins of [ / 2 . These particles are jointly called
leplons. The w, K, and tj mesons, all widi spins of 0, are classed as mesons,
(Despite its name, the u meson has more in common with the other leptons than
with the it, K, and n mesons,) The heaviest particles, namely the nuclcons and
hyperons, comprise the baryons.
While this grouping is reasonable on the basis of mass and spin alone, there
ts hirther evidence in its favor. IjCt us introduce three new quantum numbers,
L, Af, and B as follows. We assign the number L = 1 to the electron and the
eneutrino, and L = — I to their auti particles; all other particles have L = 0.
We assign the number ,V/ = 1 to the ji meson and its neutrino, and Af ss — I
to their autiparticles; all other particles have M = 0, Finally, we assign B = 1
to all baryons, and B = —I to all an ti baryons; all other particles have B = 0.
The significance of these numbers is that, in every process of whatever kind lliat
involves elementary particles, die total values of L. Af, and B remain constant.
The classical conservation laws of energy, momentum, angular momentum, and
electric charge plus die new conservation laws of L, M, and B help us to deter
mine whether any given process is capable of taking place or not. An example
is the decay of the neutron.
*>>
p + + e" + » t
While /, = for the neutron and proton, L = 1 for the electron and — 1 for
the antineutrino, so that the total value of L before and after the decay is 0.
Similarly, B = 1 for both neutron and proton, so that the total value of B l)efore
and after the decay is 1 . The stability of the proton is a consequence of energy
and baryonnumber conservation; There are no baryons of smaller mass than
the proton, and so the proton cannot decay.
ELEMENTARY PARTICLES
443
13.6 STRANGENESS NUMBER
Despite the introduction of the quantum numbers L, M, and B, certain aspects
of elementaryparticle behavior still defied explanation. For instance, it was hard
to see why certain heavy particles decay into lighter ones together with the
emission of a gamma ray while others do not undergo apparently equally per
missible decays. Thus the S n baryon decays into a A n baryon and a gamma ray,
vo _> A fl + y
while the ^ + baryon is never observed to decay into a proton and a gamma
ray,
& A p' + y
Another peculiarity is based upon the general observation that physical proc
esses in nature that release large amounts of energy take place more rapidly
than processes that release small amounts. However, many strange particles
whose decay releases considerable energy have relatively long lifetimes, well over
a billion limes longer than theoretical calculations predict. A third odd feature
is that strange particles are never created singly, but always two or more at a
time. These and still other considerations led to the introduction of a quantity
called strangeness number S. Table 13.2 shows the values of S that are assigned
to the various elementary particles. We note that L, B, and S are for the
photon and w° and ij u mesons. Since these particles are also uncharged, there
is no way to distinguish between them and their antipartieles. For this reason
the photon and it" and ij" mesons are regarded as their own antipartieles. Before
we consider the interpretation of the strangeness iminlwr, we shall have to
examine the various kinds of particle interaction.
There are apparently four types of interaction between elementary particles
that, in principle, give rise to all the physical processes in the universe. The
feeblest of these is the gravitational interaction. Next is the socalled weak
interaction that is present between leptons and other leptons, mesons, or baryons
in addition to any electromagnetic forces that may exist. The weak interaction
is responsible for particle decays in which neutrinos are involved, notably beta
decays. Stronger than gravitational and weak interactions are the electro
magnetic interactions between all charged particles and also those with electric
or magnetic moments. Finally, strongest of all are the nuclear forces (usually
called simply strong forces when elementary particles are being discussed) that
are found between mesons, baryons, and mesons and baryons.
The relative strengths of the strong, electromagnetic, weak, and gravitational
interactions are in the ratios 1 : 10 a : 10 14 : 10 4 ". Of course, the distances
through which the corresponding forces act are very different. While the strong
force between nearby nucleons is many powers of 10 greater than the gravita
444
THE NUCLEUS
tionai force between them, when they are a meter apart the proportion is the
other way. The structure of nuclei is determined by the properties of the strong
interaction, while the structure of atoms is determined by those of the electro
magnetic interaction. Matter in bulk is electrically neutral, and the strong and
weak forces are severely limited in their range. Hence the gravitational inter
action, utterly insignificant on a small scale, becomes the dominant one on a
large scale. The role of the weak force in the structure of matter is apparently
that of a minor perturbation that sees to it that nuclei with inappropriate
neutron/ proton ratios undergo corrective beta decays.
t*t us now return to the strangeness number S. It is found that in all processes
involving strong and electromagnetic interactions the strangeness number is
conserved. The decay
vo , A<> + y
S = — 1 — 1
conserves S and is observed to occur, while the superficially similar decay
2 + A p + + Y
S =  1
does not conserve S and has never been observed. Strange particles are created
in highenergy nuclear collisions which involve strong interactions, and their
multiple appearance results from the necessity of conserving S. The relative
slowness with which all unstable elementary particles save the ir° meson and
tj° meson decay is accounted for if we assume that weak interactions are also
characteristic of mesons and baryons as well as leptons, though normally domi
nated by strong or electromagnetic interactions. With strong or electromagnetic
processes impossible except in the above cases owing to the lack of conservation
of S, only the weak interaction is available for processes in which the total value
of S changes. Events governed by weak interactions take place slowly, as borne
out by experiment. Even the weak interaction, however, is unable to permit
S to change by more than + 1 or  1 in a decay. Thus the 5" hyperon does
not decay directly into a neutron since
Z~ /* n" + 77
S = 2
but instead via the two steps
I * A + 7T
S= 2 1
A<> _> n o + ffO
S =  1
ELEMENTARY PARTICLES
445
A quantity called hypercharge, Y, has also !>een found useful in characterizing
particle families; it is conserved in strong interactions. Hypercharge is equal
to the sum of the strangeness and haryon numbers of the particle families;
13.6
V = S + B
For mesons the hypercharge is equal to the strangeness. The various hypercharge
assignments are listed in Table 13.2.
13.7 ISOTOPIC SPIN
It is obvious from Table 1 3.2 that there are a number of particle families each
of whose members has essentially the same mass and interaction properties but
different charge. These families are called nmltiplels. and it is natural to think
of the members of a multiple! as representing different charge states of a single
fundamental entity. It has proved useful to categorize each multiple! according
to the number of charge states it exhibits by a number / such that the multiplicity
of the state is given by 2/ + 1 . Thus the nucleoli multiplet is assigned I = } / 3 ,
and its 2 * '/, + 1 =2 states are the neutron and the proton. The ^ meson
umlliplct has I sb 1, and its 2*1 + 1=3 states are the t+, v~, and tt" mesons.
The ij meson has / = since it occurs in only a single state and 2 0 + 1 = 1.
There is evidently an analogy here with the splitting of an angularmomentum
state of quantum number I into 21 + 1 substates, and this has led to Ihe somewhat
misleading name of iialopic spin tfuontum number for I.
Pursuing the analogy with angular momentum, isotopic spin can be represented
by a vector I in '"isotopic spin space" whose component in any specified direction
is governed by a quantum number customarily denoted l 3 . The possible values
of /., are restricted to I, I — 1 (1 (/  1), /, so that / 3 is half
integral if I is half Integra I and integral or zero if / is integral. The isotopic
spin of the nueleon is I a '/ 2 . which means that / 3 can be either '^ or  1 /,; the
former is taken to represent the proton and the latter the neutron. In the case
of the v meson, I = I and /, = 1 corresponds to the tt ' meson, /., = to the
it" meson, and /., = — 1 to the ir~ meson. The values of / 3 for the other mesons
and baryons are assigned in a similar way.
The charge of a mesun or lwiryon is related to its haryon number B t its
strangeness number S, and the component I, of its isotopic spin by the formula
13.7
f'MH)
Each allowed orientation of the isotopic spin vector I hence is directly connected
to the charge of the particle thus represented. In the case of the nucleoli
446
THE NUCLEUS
multiplet, the proton has f 3 = '/ 2 , B = I, and S = 0, so that tf = e, while the
neutron has f 3 =  %, B = 1, and S = 0, so that q  (). In the ease of the w
meson multiplet, B = S = and the three values of 7 3 of I, 0, and 1 respec
tively yield q = e, 0, and e. Charge and baryon number B are conserved in
all interactions. Thus 7 3 must be conserved whenever S is conserved, namely
in strong and electromagnetic interactions. Only In weak interactions does the
total / :i change.
An additional conservation law is suggested by the observed charge inde
pendence of nuclear forces, which result from the strong interaction. Such
properties of a nucleus as its binding energy and pattern of energy levels change
when a neutron is substituted for a proton or vice versa only by amounts that
follow from purely electromagnetic considerations, implying that the strong
interaction itself does not depend upon electric charge. Now the difference
between a proton and a neutron in isotopic spin space lies only in the orientation
of their isotopic spin vectors, and so we can say ihul the charge independence
of the strong interaction means that this interaction is independent of orientation
in isotopic spin space. Angular momentum is likewise independent of orientation
in real space and it is conserved in all interactions, which might lead us to surmise
that isotopic spin is conserved in strong interactions. This surmise happens to
be correct; the isotopic spin quantum number / is found to be conserved in
strong, but not in weak or in electromagnetic, interactions. We shall return to
the relation l>ctween conservation principles and invariance with respect to
symmetry operations in the next section.
We note that, although /., is conserved in electromagnetic interactions, / itself
need not be. An example of a process in which / changes while 7 3 does not
is the decay of the w f> meson into two photons:
it" — » y + y
A w" meson has / = 1 and 7 3 = 0, while / is not defined for photons; there is
no component 7 ;J of isotopic spin on either side of the equation, which is consistent
with its conservation, although / has changed.
13.8 SYMMETRIES AND CONSERVATION PRINCIPLES
In the previous section the charge independence of the strong interaction was
expressed in terms of the isotropy of isotopic spin space. By analogy with angular
momentum, this symmetry was said to imply the conservation of isotopic spin
in strong interactions. It Ls a remarkable fact that all known symmetries in the
physical world lead directly to conservation laws, so that the relationship between
symmetry under rotations of I and the conservation of I is wholly plausible
ELEMENTARY PARTICLES
447
Let us survey some of these symmetryconservation relationships beti.
tinning our discussion of elementary particles.
What is meant by a "symmetry"? Formally, if rather vaguely, we might say
that a symmetry of a particular kind exists when a certain operation leaves
something unchanged. A candle is symmetric about a vertical axis because it
can be rotated about that axis without changing in appearance or any other
feature; it is also symmetric with respect to reflection in a mirror. Table 13.3
lists the principal symmetry operations which leave the laws of physics un
changed under some or all circumstances. The simplest symmetry operation is
translation in space, which means that the laws of physics do not depend upon
where we choose the origin of our coordinate system to be, By more advanced
methods than we are employing in ibis book, it is possible to show that the
mvariaiicc of the description oi nature to translations in space lias .is B COBSe
quence the conservation of linear momentum. Another simple symmetry opera
tion is translation in time, which means that the laws of physics do not depend
upon when we choose f = to l>e, and this invariance has as a consequence
the conservation of energy. Invariance under rotations in space, which means
that the laws of physics do not depend upon the orientation of the coordinate
Table 13.3.
SOME SYMMETRY OPERATIONS AND THEIR ASSOCIATED CONSERVATION PRINCIPLES.
Symmetry aeration
Conserved quantity
All interactions are dejiendent of:
Translation in space
Translation in time
flotation in .space
Ueetroinagnctic gauge transformation
Interchange of identical particles
Inversion nl space, time, and charge
?
?
?
Jhi afmng unit t'leetntma^wtie interactions
only are independent of:
Inversion of space
ileflecl ..ii of charge
The tlrrmg interaction only it independent
ofi
Charge
Linear momentum p
Energy £
lobular momentum I
Electric charge a
Type of statistical behavior
Product of charge parity, space parity, and
lime paril) CI' I
H.I I' Mill llllllllll'l ft
l/Cptnn numher f.
I ifilim number M
Parity P
Charge parity (.'. isotypic spin component
l v and strangeness S
Isotopic spin I
448
THE NUCLEUS
system in which they are expressed, has as a consequence the conservation of
angular momentum.
Conservation of electric charge is related to gauge transfonnatioas, which are
shifts in the zeros of the scalar and vector electromagnetic potentials V and A.
(As elaborated in electromagnetic theory, the electromagnetic field can be
described in terms of the potentials V and A instead of in terms of E and B,
where the two descriptions are related by the vector calculus formulas E = TV
an j b = V x A.) Gauge transformations leave E and B unaffected since they
are obtained by differentiating the potentials, and this invariance leads to charge
conservation.
The interchange of identical particles in a system is a type of symmetry
operation which leads to the preservation of the character of the wave function
of a system. The wave function may be symmetric under such an interchange,
in which case the particles do not ol>ey the exclusion principle and the system
follows BoseEinstein statistics, or it may be antisymmetric, in which case the
particles obey the exclusion principle and the system follows FermiDirac statis
tics. Conservation of statistics (or, equivalently, of wavefunction symmetry or
antisymmetry) signifies that no process occurring within an isolated system can
change the statistical behavior of that system. A system exhibiting BoseEinstein
behavior cannot spontaneously alter itself to exhibit FermiDirac statistical
behavior or vice versa. This conservation principle has applications in nuclear
physics, where it is found that nuclei that contain an odd number of nucleons
(odd mass number A) obey FermiDirac statistics while those with even A obey
BoseEinstein statistics; conservation of statistics is thus a further condition a
nuclear reaction must observe.
The conservations of the baryon number B and the lepton numbers L and
\1 are alone among the principal conservation principles in having no known
symmetries associated with them.
Apart from the charge independence of the strong interaction and its associated
coascrvation of isotopic spin, which we have already mentioned, the remaining
symmetry operations in Table 13.3 all involve purities of one kind or anotiier.
The term parity with no qualification refers to the behavior of a wave function
under an inversion in space. By inversion in space is meant the reflection of
spatial coordinates through the origin, with x replacing x, ij replacing y,
and a replacing z. If the sign of the wave function ^ does not change under
such an inversion,
W*y,z) = Mx. n.z)
and if is said to have eceri parity. If the sign of ^ changes,
uX*,t/,~) = tMx. f> )
ELEMENTARY PARTICLES
449
450
and ^ is said to have odd parity. Thus the function cos x has even parity since
cos x = cos ( — x), while sin x has odd parity since sin x = —sin (—x).
If we write
Mx,ti,z} = P$(x, y, x)
we can regard P as a quantum numter characterizing $ whose possible values
are + 1 (even parity) and  1 (odd parity). Every elementary particle has a
certain parity associated with it, and the parity of a system such as an atom
or a nucleus is the product of the parity of the wave function that descrilies
the coordinates of its constituent particles and the intrinsic parities of the particles
themselves. Since \\j/\ 2 is independent of P, the parity of a system is not a quantitv
that has an obvious physical consequence. However, it is found that the initial
parity of an isolated system does not change during whatever events occur within
it, which can be ascertained by comparing the parities of known final states of
a reaction or transformation with die parities of equally plausible final states
that are not observed to occur. A system of even parity retains even parity,
a system of odd parity retains odd parity: this principle is known as eonseivation
of parity.
The conservation of parity is an expression of the inversion symmetry of space,
that is, of the lack of dependence of the laws of physics upon whether a left
handed or a righthanded coordinate system is used to deserilx: phenomena. In
Sec, 13.1 it was noted that die neutrino has a lefthanded spin and the anti
neutrino a righthanded spin, so that there is a profound difference between die
mirror image of either particle and the particle itself. This asymmetry implies
that interactions in which neutrinos and antineulrinos participate— the weak
interactions — need not conserve parity, and indeed parity conservation is found
to hold true only in the strong and electromagnetic interactions. Historically
the fact that spatial inversion is not invariably a symmetry operation was
suggested by the failure of parity coaservation in the decay of the K+ meson,
and was later confirmed by experiments showing the specific handedness of p
and v.
Two other parities occur in Table 13.3, time parity T and charge parity C.
which respectively describe the l>ehavior of a wave function when t is replaced
by — t and when (/ is replaced by —q. The symmetry operation that corresponds
to the conservation of time parity is time reversal Time reversal symmetry
implies that the direction of increasing time is not significant, so that the reverse
of any process that can occur is also a process that can occur. In other words,
if symmetry under time reversal holds, it Is impossible to establish by viewing
it whether a motion picture of an event is being run forwards or backwards.
Although time parity '/'was long considered to be conserved in every interaction,
THE NUCLEUS
it was discovered in 1964 that the Kg meson can decay into a n* and a jt"
meson, which violates the conservation of T. The symmetry of phenomena under
time reversal thus has an ambiguous status at present. The symmetry operation
that corresponds to the coaservation of charge parity C is dmrge conjugation,
which is the replacement of every particle in a system by its antipartiele. Charge
paritv C, like space parity P, is not conserved in weak interactions. However,
despite the limited validities of the conservation of C, P, and T, there are good
theoretical reasons for believing that die product CPT of the charge, space, and
time parities of a system is invariably conserved. The conservation of CFF means
that for every process there is an antimatter mirrorimage counterpart that takes
place in reverse, and this particular symmetry seems to hold even though .its
component symmetries sometimes fail individually.
13.9 THEORIES OF ELEMENTARY PARTICLES
In addition to the particles listed in Table 13.2 there are, as mentioned earlier,
a great many "particles" of extremely brief lifetimes whose existence is revealed
by resonances in interactions involving their longerlived brethren. These reso
nant states are characterized by definite values of mass, charge, angular momen
tum, isotopic spin, parity, strangeness, and so on, and it is no more logical to
disqualify them as legitimate particles because their existences are so transient
Uian it Ls to coasidcr the neutron, say, as merely an unstable stale of the proton.
Of course, it is possible to make out an excellent case for supposing the latter,
and then to go on to generalize that all the various "elementary" particles are
actually excited states of a very few truly elementary particles, as yet unidentified;
this constitutes one line of attack toward a comprehensive theory of elementary
particles. On the other hand, if we accept the particles of Table 13,2 as legiti
mate, then it is consistent to include the resonant states as well, and to seek
a theoretical framework that embraces the entire collection of well over a
hundred entities.
A recent proposal attempts to account for the various elementary particles
in terms of another kind of particle called the quark. Three varieties of quark
are postulated, plus their an ti particles, and all elementary particles are supposed
to consist of combinations of quarks and antiquaries. The really revolutionary
thing about quarks is that two of them should have charges of  '/ 3 e and the
ibinl should have a charge of +%e According to this theory, each baryon Is
composed of three quarks, and each meson is composed of quarkantiquark pairs.
Despite much effort, no experimental evidence in support of the existence of
ELEMENTARY PARTICLES 451
quarks has been found thus far, but the ideas that underlie their prediction are
so persuasive that die hunt continues.
Several interesting and suggestive classification schemes have been devised for
the strongly interacting particles based upon the abstract theory of groups. One
of these schemes, the socalled eightfold way, collects isotopic spin multiplets
into supermuttiplers whose members have the same spin and parity but differ
in charge and hypereharge (Figs. 1.34 and 135). The scheme prescribes the
number of members each particular supermultiplet should have and also relates
mass differences among these members. The great triumph of the eightfold way
was its prediction of a previously unknown particle, the il~ hyperon, which was
subsequently searched for and finally discovered in 1 964. Other grouptheoretic
approaches have related the supermultiplets of the eightfold way to one another
and have attempted to incorporate relativistic considerations into the compre
hensive picture that is emerging.
The success of the eightfold way in organizing our knowledge of the strongly
interacting particles implies that the symmetry of its mathematical structure has
a counterpart in a symmetry in nature. The further we prol>e into nature, the
more hints we receive of a profound order that underlies the complications and
confusions of experience. But for all the elegance of the symmetries that have
lieen revealed, there still remaias the problem of the fundamental interactions
themselves, what they signify, and how they are related to one another and to
the properties of the particles through which they are manifested.
FIGURE 134 Supermultiplets of sppnVi baryons and spinD melons stable against decay by the strong
nuclear interaction. Arrows indicate possible transformations according to the eightfold way.
1
452 THE NUCLEUS
(+1 %) —
charge, e —  — 1
FIGURE 135 Baryon supermuttiplet whose members have spin % and (except !! ) are short
lived resonance particles; the I and 1 particles here are heavier and hane different spins Irom
the ones In Table 13.2. Arrows Indicate possible transformations according to the eightfold way.
The ti' particle was predicted from this scheme.
Problems
1. (a) Find the maxim in n kinetic energy of the electron emitted in the beta
decay of the free neutron, (b) What is the minimum binding energy that must
be contributed by a neutron to a nucleus so that the neutron docs not decay?
Compare this energy with the observed binding energies per nucleon in stable
nuclei.
2. Van der Waals forces arc limited to very short ranges and do not have an
inversesquare dependence on distance, yet nobody suggests that the exchange
of a special mesonlike particle is responsible for such forces. Why not?
3. What is the energy of each of the gamma rays produced when a n° meson
decays? Must they be equal?
ELEMENTARY PARTICLES 453
4. How much energy must a gammaray photon have if it is to materialize
into a neutronantineutron pair? Prove that such an event cannot occur in the
absence of another body without violating either the conservation of energy or
the conservation of linear momentum,
5. Why does a free neutron not decay into an electron and a positron? Into
a protonantiproton pair?
6. A •* meson whose kinetic energy is equal to its rest energy decays in flight.
Find the angle helween the two gammaray photons that are produced.
7. A proton of kinetic energy T Q collides with a stationary proton, and a
protonantiproton pair is produced. If the momentum of the Ixjmbarding proton
is shared equally by the four particles that emerge from the collision, find the
minimum value of 7" .
8. Trace the decay of the Z" particle into stable particles.
9. A it meson collides with a proton, and a neutron plus another particle arc
created. What is the other particle?
10. One theory of the evolution of the universe postulates that matter sponta
neously comes into being in free space, [f this matter were in the form of
neutronantineutron pairs, what conservation laws would be violated?
11. Which of the following reactions can occur? State the conservation laws
violated by the others.
(a) j) + p * n + p + !r +
;/«) P + p* p + a° + s»
(c) e* + e' > u + + w"
(rf) A" * 7T + + W"
(e) it + p » n + w°
12, The ir° meson has neither charge nor magnetic moment, which makes it
hard to understand how it ean decay into a pair of electromagnetic quanta. One
way to account for this process is to assume that the jr* first liecoines a "virtual"
unclean ant inuel eon pair, whose members then interact electromagnetically to
yield two photons whose energies total the mass energy of the tt°. How long
does the uncertainty principle allow the virtual nucleonantinucleon pair to
exist? ts this long enough for the process to be observed?
13. The interaction of one photon with another can be understood by assuming
that each photon can temporarily become a "virtual" electronpositron pair in
free space, and the respective pairs can then interact electromagnetically.
454
THE NUCLEUS
(a) How long does the uncertainty principle allow a virtual electronpositron
pair to exist if hv < 2m c 2 , where m,, is the electron rest mass? (b) If hi' > 2m n c 2 ,
can you use the notion of virtual electronpositron pairs to explain the role of
a nucleus in the production of an actual pair, apart from its function in assuring
the conservation of both energy and momentum?
ELEMENTARY PARTICLES 455
ANSWERS TO ODDNUMBERED PROBLEMS
Chapter 1
t.
3,
5.
7,
9.
II.
13.
15.
IT.
19.
21.
213 m
2.6 X 10 s m/s
fi ft; 2.6 ft
4.2 X 10 7 m/s
08c 0.988c; (lite; %J98Bc
4.2 x K)' m/s
1.H7 x 10* m/s: I.
H.9 X 10 M kg
0.291 MeV
2.7 X 10" kg
4.4 X 10" kg
, (de/di)
Hl X 10 s m/s
** ''  (i _ o7«yn
25. The results arc different because all
observers find the same value for Ihc speed
of light, whereas the speed of sound meas
ured by an observer depends upon bis own
motion relative to the medium in which
the sound waves propagate.
27. (a) 6, 1(1; (h) 10. 6. yes
Chapter 2
1.800 A
MOO A] 3.9 eV
2.83 X 10 ,n J
1.71 x Nr* photons/*
(«) B.H2 x 10 "' lb/in. 2
(b) 4.24 x ID" photons/ m*s
(<r) 3.9 X UP* watts; 1.2 X 10 38 pho
tons/s
(d) 1.41 X 10 la photons/m 1
II. 1.24 X 10* V
13. 3.14 A
15. 5 X H> ,a Hz
17. 0.015 A
19. 2.4 X 10' B Hz
21. 0.565 A
23. («) 2 X lO" 3 eV
(b) 2X10 » eV
(c) 3.5 X K) 1 " Hz; 7.7 X 10 3 H/.
Chapter 3
[. B.Hfi X 10 I2 m
1 .24 GeV; 616 MeV
eV
\ = 12.2'
'K&*')]
i/a
ft ti = w/2
11. 6.2%
13. 1.18 X 10 s m/s; 83 m/s
15. 1.05 X10~ M kgm/s; 1.16Xl0 7 m
(7.200 mi!); the original narrow wave
packet has spread out in 1 s to a much
wider one because the phase velocities of
the waves involved vary with k and a large
range of wave numbers was present in the
original packet.
17. Each atom in a solid is limited to a
i ei i. tin itofiinii iciiiiin <>\ space Hthamta
the assembly of atoms would not he a solid.
The uncertainty in the position of each
atom is therefore finite, and its momentum
and hence energy cannot be zero. There
is no restriction on the position of a mole
457
Cute in an ideal gas, ant) so the uncertainty
in its position i.s effectively infinite and its
momentum and hence energy can be zero.
Chapter 4
]. 10*
3. 0.878
7. 1.14 X K) "in
ft m/m = 1. 002
'1 ! = ^~ / ^T Forlhehydro
J 2*r V 4mynfl a '
gen atom, /= 6.6 x Iff**" 1 , winch is
comparable with the highest frequencies
in the hydrogen spectrum.
13. 920 A
15. 12 V
17. 8.2 X 10° rev
19. 1.05 X 10* K
21. 2,1 A
23. 1.04 MeV
(fc)
H He<_ .. .
r. = oo t = ll
n = 4 n = »
« = 3
n = 2
n ■ 1
n = 6
ii = 5
n = 4
n = 3
n = 2
ti i ijii
(c) 2.28 x 10 s in
Chapter 5
3. 2.1 MeV
5. 2.07 X 10'* eV
7. Classically f = V = £/2, where f and
V are averages over an entire period of
oscillation.
11
b. tl, f.
Chapter 6
5.
r = (3 ± i
ffig,
7.
68%; 25%
ft
p, 29%; <f.
18%;
ft
13";,
Chapter 7
1. 182
3. The alkali metals are the largest in each
period of the periodic table, since their
atomic structures consist of a single elec
tron outside closed inner shells thai shield
the electron from all but +e of nuclear
charge. There is then a regular decrease
in size within each period as the nuclear
chugs increases, which pulls the outer
electrons in closer to the nucleus. At the
end of each period there is a small increase
in si/.e due to the mutual repulsion of the
outer electrons.
5. 1.39 X 10* eV
7. 0.0283 A
ft state $. L A.
'S
3 P 2 112
2 3tz % 2 %
% 2 3 5
*«»/* % 5 %
II
There are no other al limed states.
13.
SB
r \f1
15.
5u B ;3
17.
18,5 T
tft
Cobalt (Z = 27); molybdenum
■'
= 42)
Chapter 8
f. 3.5 x 10 1 K
3.
5.
7.
ft
It.
13.
c»
1.27 A
2.23 A
No.
1.24 X 10" llz
Chapter 9
3. 2/\/™„
5. 1 .6 X I0" neutrons/m 3
7. (a) 1.00: 1.08:0.89:0.22:0.027
b\ yes; 15.33 K
458 ANSWERS TO ODDNUMBERED PROBLEMS
9. 1.00:2.3 X I0 IU :6.2 X 10" ,2 ;2.3 X
10'*
II. 5800 K
13. A fermion gas will exert the greatest
pressure because the Fermi distribution has
a larger proportion of high energy parti
cles than the other distributions; a bosun
gas will exert the least pressure tocMfee
the Bose distribution has a larger propor
tion of lowenergy particles than the
others.
15. 18.70 eV: 16.84 eV. The He atoms are
needed to maintain an inverted energy
population in the \e atoms by transferring
energy to them in collisions, which sup
plement the direct excitation of Ne atoms
by electron impact.
Chapter 10
1. {a) The van der Waals forces increase
the cohesive energy since they are attrac
tive, (b) The xeropiiiiit oscillations de
crease the cohesive energy since llicy
represent a mode of energy possession
present in a solid but not in individual
atoms or ions.
3. The heal lost by the expanding gas is
oiial to the work done against the attrac
tive van der Waals forces between its
molecules.
5. {a) In a metal, valence electrons can
1 1 1 n! unoccupied excited energy states in the
conduction bant! for am excitation energy',
however small.
(b) The energy gap in semiconductors is
small (< 1.5 eV), and so photons of visible
liuhl can excite valence electrons to. the
i (induction band although photons of in
frared light have insufficient energy for (bis
purpose.
(c) The energy gap is so large that pho
tons of visible light cannot gravida mug)
excitation energy from electrons in the
valence band to reach the conduction band.
7. ptvpe
9, fcT is a very small fraction of tp. and
so the electron energy distribution Is not
very temperature sensitive.
II. 1.9 eV; 7.6 x 10"* m
13. 7.29 eV; n = 8.4
17. 3.3 eV; 2.56 X 10* K; 1.08 X 10* m/s
19. 1 1 eV
21. 2
23. 50 A; the ionization energy of the
electron is 0.0O9 eV, which is much smaller
than the energy gap and not very far from
the 0.025 eV value of fcT at 20°C.
25. Because m'/m = 1.01 in copper.
Chapter 11
I . 8.83 cm
3. 19 percent H 1 ", 81 percent B"
5. 34.97 u
7. 7.98 MeV
ft 15.0 M>\
II, Nnclear forces cannot be strongly
chargedependent.
13. The nucleon kinetic energy thai cor
responds lo the momentum imcert:iiuiv
implied by an uncertainty in position of 2
bo is V2 MeV, which is entirely consistent
with a potential well 35 MeV deep.
Chapter 12
I. 1/4
3. 1,620 yr
5. 1.23 X 10* s'
7. 3.37
9. The mass of ;Be is not su Hi c i e n 1 1 y larger
ll.. in that of {LI to permit the creation of
a positron.
II. Hint: the 39th proton in !gY is nor
mally in a p„2 state, and the next higher
state open to this proton is a &, n si alt
13. 3.33 MeV
15. The neutron cross section decreases
ANSWERS TO ODDNUMBERED PROBLEMS 459
with increasing energy' because Lhe likeli
hood that a neutron will be captured
depends upon how much lime it spends
near a particular nucleus, and this is in
versely proportional to its speed. The
proton crass section is smaller at low ener
gies because of the repulsive force exerted
by the positive nuclear charge, which
provides a potential barrier the proton must
tunnel through.
17, 1.2 X 10" 6
19. 0.21
2f. (1.1%
Chapter 13
1. 0.78 MeV; L29 MeV, which is well
under the observed binding energies per
nucleon of stable nuclei.
3. 68 VleV; yes, in order that momentum
be conserved.
,5. This decay conserves neither haryon
number nor spin; this decay conserves
neither baryon on in her. spin, nor energy.
7 ,i,f«() MeV
9. A neutrino.
11. (a) and (e) can occur; (b) violates
conservation of li and spin; (c) violates
conservation of /,, M, and spin; (d) violates
conservation of rJ and spin.
13. 6.4 X HH* s; the strong electric field
of the micieiLS separates the electron and
positron far enough so that they cannot
recombine afterward to reconstitute the
photon.
460 ANSWERS TO ODDNUMBERED PROBLEMS
INDEX
AUmplinxi spectrum, 119, I'll
Acceptor level In semiconductor. 338
Acetylene molecule, 207
Wiuiidc elements, 2L5
Actinium decay series, 396
Wlmly of ttdfobOtOptt, 389. 392
Alpha decay, 396
theory of, UK
Alpha particle, 102
Ammonia molecule. 261, 264
Amorphous whd. 3 17
Angular frequency of wave. 78
AuguLr inotneiitttm:
atomic, IH.1. 222
molecular, 2711
nuclear, 384
♦.pin, 305
\iiiiiitn(r r .:. \':~
Antf particle, 112
Antisymmetric wave function, 211. 149
ami divalent handing. 249, 252
Atom:
lingular momentum of, 182, 222
Boh* model of iar Ruhr atmim 1 model.!
classical model of, 113
complex, irroclurc of, 213
energy level* of, (25. in I, 221
mass of, 381
Hutherbrd nodal of, 153
Thomson model of, 102
vector model of, 222
Atomic excitation, 131
Atomic number, 1 04
Atomic orhttritK. 254
Atomic rudll. 321
Atomic spec I rum {see Spectrum)
Auger effect, 237
Balmer s^rie* In hydrogen speelmm, 119, J2H
Band theory of solids, 334, 340* 352
Bant, the, 414
Barrier penetration, quantum theorv of, 157, 399
Baryon. 443
Benzene molecule, 26$
Beta decay, 369, 408
inverse, 4J I
Buttling energy:
of atomic electron. 214
nuclear, 372. 382
Black body:
properties of, 304
spectrum of, 306, 308
Bodycentered cobic crystal, 322
Bohr. .Niels. 101
Bohr atomic model. 121, 193
and correspondence principle. 133
effect of miclcar motion m. 129
limitations of, 139
and quantum theory of atom. 193
Bohr magneton, 139, 207
Bohr radius of hydrogen atom, 125
Boltzmann distribution:
of molecjW energies, 295
of molecular momenta, 297
Boltzmann factor, 299, 311
Bond
eovalent, 243, 325
hydrogen. 328
ionic, 243, 32 1
metallic 331
van tier Waals, 327
BoseEinstcin statistical distribution law, 288, 304
derivation of, 300
Boson, 212
Ikjundary surface diagram:
of atomic orbital, 255
of molecular orbital, 258
Braukert series in hydrogen spectrum, 120, 12S
Bragg planes in crystal, 57
Bragg reflection:
of de Brog.be waves, 345
of X ILiVM, V)
ikemsstrahluug, 51
HriHnuui y.one, 345
461
Carbon compound*:
hybrid orbital* In, 281. 285
saturated uiul uiisaturatetl, 267
Carbon cycle of unclear rvnecious, 424
Carbon dioxide molecule, vihruliuual modes of. 280
Carbon monoxide molecule:
rotational energy IcveN of. 27(1. 300
spectrum of, 279
vibrational energy levels uf, 875
Center of noes* coordinate system, 418
Qudwk.lt, janies. 365
Chain reaction, 423
rlurge < 'irmigatiuu. 451
Cohesive energy of crystal, 322
( ."uilixmiiv elastic and inelastic, 132
Complex euiuugate of wave function, 75
Compound nucleus 417
Gmupluil effect. 60
Curnpfmi wavelength. H2
Conduct ing wild, nature of, 352
Conservation principle* mid symmetry onr;itinnv
446
< iMirdination number of crystal* 322
Correspondence print :lp!c. 133
uiul harmonic oscillator, 161
Cosmic rays. 20. 437
L'ovdleul bond;
In crystal 326
In molecule, 243
and uncertainty principle, 246
Cross section. 109,413
Cryitit
covaletu, 325
tee, 329
ionic, 318
metallic, 331
mulct idur. 326
van der WmEs, 326
Crystal types, tabic of. 133
data*, the. 3511
t. v< lulrou resonance in solid, 356
DavlswnCenner experiment. 52
dc Brugjlc waves, 73
Bragg reflection of. 345
diffraction at, 62
group velocity of. 62
In bvditrgen dlum, 122
phase velocity uf. 76, 61
wavelength oC W
I )ecay constant of radioisotope, 390
iMocoli/cd electrons in bciixenc umletnlr.
Deuterium, 362
spectrum of, 130
as thermonuclear fuel, 427
Dcutcron:
binding energy of, 372
theory of, 374
Diamond:
bund unicture of. 337
crystal structure of, 320
Dims, P. A. at, 205. 431
Donor level in sc mi conductor. 338
Dopplrr effect ta light, 41
Doublet angular momentum Male, £29
Effective iua& of electron In crystal, 355
Eigcnfruiction. 149
Eigenvalue. 148
Eightfold way, 452
Einstein, Albert, 10. 47
Elastic ffflftfffffc 132
Electromagnetic field, Vicimm fluctuations m, 2,00,
229
delocjli/cd. in Iwn/cur uiulccule, 288
Dirac iheory of, 45]
muimclk moment of. 166. 205
as nuclear const! Incut. 91, 363
spin of, 205
Electron affinity, 320
Electron capture by nucleus 410
Electron configurations of elements, &I9
Electron "gas" in metal, 331
energy dlslrihuliun in. 342
Electron orbits in hydrogen atom. 113
Electron shells and MihvfirlK in atom, 213, 217
Eleciruuegulivity, 259
Electronic s>ecimui ni molecule, 251
Elementary particles:
mullkphl* of, 446,453
till tic of, 442
thcHirire of, 451
adiiiidc, 215
Elements:
electron configurations of, 219
EnnlxHlmu energies of, 320
lanlhanidc, 215
periodic faille of. 216
transition. 215
tratuunuiic, 423
Emission spcclruin. IIS, 131
Energy:
binding: of atomic electrom, 213
of ■IiIiImii. 372, 362
kinetic, relative to center of mass, 416
quantization of, and Schrodingcrs equation, 147
Mat, 30
therm unucl car, 424
uncertainty in uira.su or rncni of. Ml
Energy luind in crystal. 334, 340. 352
Energy levels;
atomic, 125. 131.221
of h. mi limit UM:i1ator, LOO
of helium atom, 232
of hydrogen atom. 125. ISO. 230
462
INDEX
Energy levcb:
of mercury atom, 234
motor, 366. 379, 364. 412. 419
of particle in a box. 151
of rotating molecule. 269. 209
of sodium atom, 230
uf vibrating molecule, 275
und Xray spectra. 235
Equivalence, principle of, 07
Ether. 3, 9
Ethylene molecule. 205
Exchange Korea, 433
Excitation nici.liJiUii.il* and meet rat lines. 131
Excited state of atom, 120
Exclusion principle. 210
repulsion due to, 243
Expectation value, 145
Facecentered cubic crystal. 322
Feruu, the, 370
FermiDime statistical distribution law, 256. 310
derivation ol. 308
Fermi energy. 332, 330
tabk of, 312
Fenni gan model of nucleus. 388
Fermi surface, 352
Permian. 212
Ferromagnetlsm, origin of, 222
Fine rtm c tlT B in spectral lines. 203, 207
Fission, nuclear, 420
Fluorescence. 261
E 'im bidden lurid In crystal. 336, 346
Fourier integral, SJi
Fourier transform. 86
Fraucli Hertz; experiment , 131
Fundamental interactions, 444
Fusion, nuclear, 424
i £•&!•■> transformation of coordinates, 23
Cumnui decay, 112
Gamma ray, 05
f^elgeiMursden experiment. 102
General theory of reUUvity, 07
Cuudsmit, S. A., 205
Gravitational red shift. 60.68
Cravitun, I 13
Ground state uf atom, 126
Group velocity of waves, St
Gvruiuugiifltc ratio:
of atomic eieetrott, 188
of electron spin, 207
1 1. ii moon oscillator:
quantum theory of. 103
wuve functions of, I hi
7.eru;poinl energy of. 100
Itaimatag Werner. 80
llcboin, energy levek of, 232
lirrinite polynomial. 168
Hole in crystal. 338
Elooke's law. 158
f hind's rule, 222, 228. 260
llvloui orbiiAl. 2C3, 205
Hydrogen:
lUstribiitiun uf mnfamnif speeds in. 290
specific heal of, 285
spectral series uf. LIU. 128
meet rum of. 220
Hydrogen atom:
Hohr model of. 121
Bohr radios of. 125
classical model of, 113
electron orbits in, 122
energy levels of, 125, 180. 230
probability density of electron in, 191
quantum numbers of, 170. 179
(.fuuntLuu Iheorv of. 173. 190
and uncertainty principle, 92
wave functions of. 176, 181
Hvchouru bond in MilUU, (IB
Hydrogen niolecubir Ion. 247
ElydpogH iiiob'colc. 252
1 lypercharge. 448
Hypadbn structure in \pettial lines, SS5, 371
Hyperon. 438
Ice, crystal structure of. 329
imjiact parami^rr in utotntc Matlcring, 105
Impurity M'liiivontinLtin . its
rmikilk c oliisni.i. 132
Inertfal frame of reference, 10
Insulating solid, nature of, 352
Ionic bond. 243, 321
Ionic crystal, 318
Ionization energy. 319
tafab of, 320
Ivniicr. 112
bofupe. 362
Isolopic spin. 446
It coupbii^; ol ani;iil.o nonMUU
atomic. 228
nuclear. 364
h , 69, 183
Half life. & l
Harmonic oscillator:
classical theory of, 158
energ)' levels of, 10T)
.438
Ijigrange's method ol undetermined multipliers. 293
I ..nub shift in hydrogen spectrum. 230
INDEX
463
Landi: £ fad or. 238
I jiilluiinlc dement*. 215
Laser. 311
LnttA ooatnetfODh rafaUvWk^ 17. 2fl
l.rptnn, 443
lifetime, menu, of <?sdled itate. 120
of radio.* it iijh.% 3U3
1. 1 rji liddrop model of nurleu*. 380
nim! lUick'ur fission. 120
and nuclear reactions ■< 1*4
[xiicnUFititgemltJ coutnicllon. 17
Lortml** lrumfonii.it icm of coordinate*. 22, 28
inverse* 2,7
iLS coupling of iiojTolm momenta.
atomic. 226* £32
nuclear. 384
Lyman *cri« in hydrogen spectrum* 120, 12&
Modelling constant, 324
uumbcr* in nuclear structure, 384
Mngm.1lc moment;
of atomic electron, 188
of deetmn spin. 2tf7
pf midem, 221
Magnetic fUHJiluiii number, ITS), ISl
M*wr, 314
atomic, 361
effective, of electron In crystal. 333
reduced: In hydrogen atom, 129
In molecular rotation. 289
in molecular vibration, £.74
relativity of. 30. 33
Muss defect of nucleus, 372
Mattieticrgy relatiousbip. 35. 37
Mass spectrometer, 301
Mm unit, atomic, 3d!
Maxwell Bottxmann vUtlshcal distribution law. 287,
293
derivation of, 289
Mean free putb. 4 14
Mendeleev. Dmitri. 213
Mercury, energy levoU of, 234
Mewn. 443
Maura decay and special relativity. 20
Meson theory of nuclear forces, 433
Metal, electron "gas" In. 331, 342
Meullfc bond 331
Mctaatablc state, 233
and the I.m:i. 312
Methane molecule. 261. 2o3
Mkhelsnn Murley experiment. 3, 7» 1ft
Molecular angular momctitmtu 270
Molecular bund, 243
Molecular energies in a gas. 295
Mob:cular orhital. 254
Imundary surface diagram of, 258
Molecule, 243
electronic spectrum of, 281
polar rovalcni, 239
rotational energy levels of, 289, 299
rotational spectrum of, 272
vibrational energy levels of L 275
vibrational sped mm of. 277
Momentum, angular (*»* Angular momentum)
Moseley\ law, 219
Mu&ftauer effect, 72
Multiple^ of elementary parliclev 4lfl. 452
Multiplicity of angular momentum state, 229
Uvea, 437
ntvpe tcmicnntluctor, 338
Neptunium, 423
Neptunium decay series, 394
Neutrino, 4li9
,u id until km it ri no, 432
riiran free juilh of, 416
and stellar energy. 425
Lypa of, 437
\cotron. 364
decay of, 365. 443
Normalized wave timet ton, 141
Nuclear fission, 373. 420
Nuclear forces:
inesoii thorny of. 433
saturation of, 368
strong and weak, 444
Nuclear fusion, 373, 424
Nuclear reactions, 417
Nucleon, 300
NncleiiA. 103
aliseuce of electron* from. 91, 363
angular momentum *talea of. 384
binding energy of, 372
compound, 417
conditions for stability of. 366
energy levels of, 366, 379, 384. 412, 419
Fermi gas model of, 388
liiiuitl drop model of. 380. 418. 420
magnetic moment of. 224, 364
radioactive decay of, 389
diupe of, 371
diell model of. 384
Mzoof, 112.370,407
spin of, 364
Nuclide, 362
Occupation index, 31 1
OroM
atomic, 254
hybrid, 203, 265
molecular, 254
Orbital quantum number, 179, ISO
QrthoJidiunu 232
Oxygen, distribution of molecular speeds In, 299
464
INDEX
ptype semiconductor. 338
Pair production, irt
Par jlit1 i. 232
Partly. 449
Particle in ,i bom
enerp'y levels nl, 151
cpianlum Iheory of. MB
wove Timet inns of. 154. 138
Particle diffraction. 82
Paschen scries in hyiiiuj!"* 1 spectrum. IW. 128
Putli. Mifllniitit;. 2 III. 4IIU
Periodic I.™ 215
Periodic iahle. 2T6
Phuid serins in hydrogen speelrum, 120. l2Ji
Phase space. 288
Phase velocity of wave, 8 1
Phosphorescence. 2Js1
nXuOllUlfcl effect, 43
mid e,ammarav absorption, 65
theory ul IT
Photon, 4fl
"maw" of, 60
Pion. 438
Planck's constant. 47
Planck's radiation formula. 308
Plutonium. 423
Polar molecule. 259. 327
Positron, 410. 432
E'riiidpal iniantum number, 180
Probability deiBily. 74
Proper lenglh, 17
Proper time, 12
Proton. 362
Proionproton cycle of nuclear reactions. 424
(Quanta, 47
Oii.Mihiiu eUvlrodynatiiics, 201)
9li,uiIu»i [urvrnuiiCf., 130
^Ill.Lllllllll UlUlilM'l
of Bukr orliil. 124
rh'clroll spin, 205
magnetic. JS4
nurlear. 3H4
orl.ilal. t8(l
of particle in a Ixjx. 151
prlnd]wl. I HO
nHallonal. 270
spin inagcwKc, 208
^ iUi.itjini.il, 275
r^iuuilum Iheory:
nf harrier ptrnclralion. 157, 31)9
of ueolerou. 374
of ii.uuiiijii oscillator, 163
of hydtiii;t:n hi in i>. liJO
of light. 47
of particle in a Lmo.. 14H
of radial I vc transit urns. 1 06
Qoark. the. 451
Radiative transitions. 198
allnsccd and forbidden, ISO
Kail ioai: live decay, 38'.)
Itatlioaclive series. 393
Hareearth elemcnLs, 215
of electron in hydrngen atom, 129
in molecular rolalion, 269
in molecular vibralion, 274
Relati vislu formulas, 37
Helatlvistic inass increase. 3(1, 33
Relativity;
general theory of, 87
special theory of. 10
Resonance particle, 440
Best energy. 36
RociniiMi. v\ illiulm, 51
ftllOll Ifll MfHM uiolecular spceti. 2SI7
Rotaliooal energy levels uf molecule. 2WI
relative >opulailnns of, 299
li'il.iliiiri.il cjoantum number, 270
ltrU.iEnni.il spectrum nf imilri ulf. 272
Kuliicrfiufi atomic mtxlel, 103
Kulhmionl scattering fonu id a. HI, 117
Etvillwrg. the 214
Itydberg constant, 120. 128
effect of nuclear morion on, 130
Saturated carltnn compound. 287
Schrodinger's equation, Ml
for deotcron, 375
for harmonic oscillator, 163
for hydrogen niotn, 175
for particle in a boa, 150
steady stale form. 147
limedependent fonu, 144
validity of. 144
Selection rules.
for atomic speclra. 108
for relational spectra, 272
for vibrational spectra, 278
Scoueornhnlor. 338. 352
Shell:
atomic electron. 213
closed, 217
Shell uickIcI of nucleus. 383
Shielding, electron, in atom, 218. 319
Kiiuullaneity and relativity, 15
Kinglel angular momentum slate:
of alum. 229
of helium. 232
of nucleus, 379
"nxlium
1 lion] ,1 fill lure of, 333
nui'j^ lt"*'b ni 2 i r s
Sodium chlnride crysial, 244
Solids, cryslallioe and amorphous, 317
Space rjuanti/ation of angular momentum. 185, 207
INDEX
465
Specific heal of hydrogen, 285
Spectral hmv
Rue structure In, 203. SOT
hypcrfiuc structure in, 225, 371
origin of, 127, 196
Zeemaii elf eel In. 189, 204
Spectral series. 119
Spectrum:
absorptlmi, 119, 131
hand, of molecule. 1 19
blade body. 306, 308
electronic. of molecule, 231
emission, 114, 131
»l helium. 232
of hydrogen, H9. 127.220
of mercury. 234
rotational, of molecule, 272
of sodium, 230
of sun, llil
vibrational, of mdeciile. 877
Xray. 235
Speed, rootmeansquare. of molecule, 2M7
Spherical polar coordinates, 174
Spin:
electron. 203
of elementary particles. 442
faatttpfcL MB
... iiiinio, 409, 432
nuclear. 364. 386
Sj LiuneliL' ininiihiiti number, £06
Spinurhit coupling:
in plom. 207
in nucleus, 384
Stars:
energy production in. 424
while dwarf, 371
Staimk'ol mechanics, 287
Statistical weight nf energy level. 299
Stefan Bolt /niumi law, 308
KlnrnOrluch espcrfment. 207
Mtaaaga I tn inula for logarithm ol a laetoria], 201
Strangeness number, 44 1
Strong intcractinu, 444
Subshcll
atomic ckclrcm, 213
closed, 217
order of (filing. 218
Symmetric wave funclion. 211, 449
and covaleni bonding, 249, 252
Symmetry operations and conservation principles,
44fi
Term symbol of atomic state. 229
TliciuiiDiilc emission of electrons. 46
Thermonuclear energy, 424
Thomson atomic model. 102
Thoi Emu decay series, 394
lime dilation, 12, 14. 27
Time reversal, 450
Transition elements. 215
Transfllnns, radiative: allowed and forbidden, 199
iniantuin theory of, 196
spontaneous and induced, 311
Triple! angular momentum state:
of atom, 229
of helium, 232
of nucleus. 379
Tritium. 362
Twin paradox, IS
Uhlcnbcck. C. E.. 205
Uncertainty principle, S9
and anuulai monicllhiui. 'IS, ISfi
and covalent lauuhng. 246
and energy measurement. 02
and meson theory of nuclear forces, 435
and particle In a box. 151. 150
and phase space. 286
and resonance particles, 44 1
and time Intervals, 92
Unsaturated carbon compound. 267
l' mold's theorem, 201
Uranium decay serfes, 309
Vacuum flo'tuatious fn electromagnetic field, 200.
229
Van dcr Waits crystal. 328
Van der Waals force. 327
Vector model of the atom. 222
Velocity:
group, Bl
phase. 81
Velocity addition, relatf visile. 28
Vibrational energy levels of molecule, 275
Vibrational spectrum of molecule, 277
Von l,uuc. Max. 52
Water molecule. 260. 264
vibrational states of, 250
Water molecules, hydrogen bonds between. 329
Wive equation. 141
Wave function, 74, 140
antisymmetric. 211, 449
complex conjugate of, 75
of deutcron. 379
of harmonic oscillator, 168
of hydrogen atom, IT6, IS]
nonnalUcd, 141
parity of, 449
of particle In a box. 151. 153. 156
symmetric, 211, 440
Wjvo group, 79
and uncertainly piinuple. 66
velocity of. 81
Wave iukoIht . 78
ol electron in crssial lattice, 345
Waveparticle duality. 85. 93
Willi interaction, 444
White dwarf star, 6S. 371
W ten's displacement law, 306
Work function of surface. 4ft
Xray spectre. 235
Xray spectrometer. 58
X rues, 51
absnrplion of, 65
X rays:
diffraction of, 56
scattering of, 66
Yukawa, llideki, 434
Zecman effect. 189
anumalous. 204
Zeropoint energy of harmonic oscillator, 160
j
466
INDEX
467
OTHER McGRAWHILL
INTERNATIONAL STUDENT EDITIONS
IN RELATED FIELDS
Beiser: PRINCIPLES OF MODERN PHYSICS
Bueche: PRINCIPLES OF PHYSICS, 2/e
Bueche: INTRODUCTION TO PHYSICS FOR SCIENTISTS
AND ENGINEERS
Green: NUCLEAR PHYSICS
Harris: INTRODUCTRY APPLIED PHYSICS, 2/e
Margenau: PHYSICS, 2/e
Morse: METHODS OF THEORETICAL PHYSICS, Parti
Morse: METHODS OF THEORETICAL PHYSICS, Pan2
Slater: MODERN PHYSICS
White: EXPERIMENTAL COLLEGE PHYSICS, 3/e
White: BASIC PHYSICS
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