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CONCEPTS OF
MODERN PHYSICS
Second Edition
Arthur Beiser
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CONCEPTS OF MODERN PHYSICS
McGRAW-HILL SERIES IN FUNDAMENTALS OF PHYSICS:
AN UNDERGRADUATE TEXTBOOK PROGRAM
E.
CONDON, EDITOR, UNIVERSITY OF COLORADO
U.
MEMBERS OF THE ADVISORY BOARD
D. Allan Bromley,
Arthur F. Kip,
!
Hugh D. Young,
CONCEPTS OF MODERN PHYSICS
Oiiirp.sili/
Ytilc
Second Edition
nittmity of California, Berkeley
f?rmwfftf
ftftiBrwn
t
'niversity
INTRODUCTORY TEXTS
Beiscr
Kip
•
Concepts of
Modem
luiHilmm-nliils of
Young
Arthur Beiser
Physics
I'.teitriciti/
and Magnetism
Young
Fundamentals of Mechanics and Seal
•
Fuiuliiuieiiliilfi
of Optics
tint!
Modem
I'liysics
UPPER-DIVISION TEXTS
Burger :mil ObsOfl
Beiser
.
Cohen
Longn
Trail!
Modem
INTERNATIONAL STUDENT EDITION
A Modern
Perspective
I'lit/xin
CoRetpti of \tolror Physics
frmilitiw
.
.
.
Physics of
-
i'.tcments (f
Waves
Mathenuitical Physics
nliit.i t>!
fundamentals of
Meyerhof
Rcif
Ctas&oal Mechanics:
Pnsperiirr.s of
Elmore and llculd
kraut
.
tllementtiry Particle Physics
Nuclear Physics
Fundamentals of Statistical ami Thermal Physics
and Potuilla Atomic Theory: An Introduction to Wave
McGRAW-HILL KOGAKUSHA, LTD.
Mec.hnoii.-i
Tokyo
New
Delhi
IXisseldorf
Panama
Johannesburg
Rio de Janeiro
London
Singapore
Mexico
Sydney
CONTENTS
Preface
ix
Chapter 3
List of Abbreviations
PART ONE
Chapter
1.1
BASIC CONCEPTS
The Michelson-Morlcy
3
Experi-
ment
Library of Congress Cataloging
in
Publication
Data
1.2
The
Concepts of modern physics.
(McGraw-Hill series
I.
in
Matter-Constitution,
fundamentals of physics)
2.
1973
530.1
72-7089
CONCEPTS OF MODERN PHYSICS
Copyright
©
1963, 1967, 1973 by McGraw-Hill, Inc.
may be reproduced, stored
in
a
All
rights reserved.
No part
retrieval system, or transmitted,
in
of
any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher.
IN
JAPAN
82
Diffraction of Particles
3.6
The Uncertainty
3.7
Applications of the Uncertainty
Principle
91
g
3.8
The Wave-particle Duality
93
Principle
17
20
formation
27
4.1
Velocity Addition
28
•4.2
Alpha-particle Scattering
30
•4.3
The Rutherford
12
1.11
Mass and Energy
Mass and Energy: Alternative
35
Derivation
Mass
Waves
Atomic Structure
101
101
105
Scattering
rn
37
4.5
Electron Orbits
113
39
4.6
Atomic Spectra
in
4.7
The Bohr Atom
121
4.8
Energy levels and Spectra
125
4.9
Nuclear Motion
129
2.3
X
2.4
X-Ray
2.5
The Compton
2.6
Pair Production
63
Gravitational Red Shift
68
•2.7
4
Atomic Models
wa-
The Photoelectric Effect
The Quantum TTiaory of
43
4.10
Atomic Excitation
47
4.11
The Correspondence
51
Effect
Chapter
Nuclear Dimensions
2.2
Diffraction
TWO THE ATOM
Formula
43
light
PART
4.4
Particle Properties of
Rays
96
22
The
Chapter 2
Problems
16
I.JO
Relativity of
76
3
Meson Decay
The Lorentz Transformation
The Inverse Lorentj; Trans-
Problems
PRINTED
The
length Contraction
2.1
this publication
79
3.5
Velocity
1.6
Dilation
Problems
Exclusive rights by McGraw-Hill Kogakusha, Ltd., for manufacture and export.
This
book cannot be re-exported from the country to which it is consigned by McGraw-Hill.
Phase and Croup Velocities
1.5
1.12
INTERNATIONAL STUDENT EDITION
3.4
73
74
The Twin Paradox
•1.9
ISBN 0-07-004363-4
3.3
Wave Function
De Broglie Wave
Waves
Time
Title.
QC173.B413
3.2
Broglie
1.4
"1.8
1.
De
1.3
•1.7
Quantum theory,
73
3.1
Special Theory of
Relativity
Beiser, Arthur,
Properties of
Particles
Special Relativity
I
Wave
131
Principle
Problems
133
135
56
60
70
Chapter 5
Quantum Mechanics
139
5.1
Introduction to Qiuui turn
Mechanics
139
5.2
The Wave Equation
140
5.3
1.4
5.5
df|H'tnlcnt Fortn
143
Expect sit ion Values
145
Sehrodinger'i Equation: Steady
Form
statc
5.8
Time-
Schrodingrr"-, Equation:
Tin
1
The
Particle in u Bus:
Wave
153
Particle in a
'['In'
•5.10
156
1
1.
milium
()s t
The Harmonic
ill,
158
ilni
Oscillator: Solu-
tion of Schrddioger's Equation
Problems
£43
Molecular Fun nation
I
163
Fleet ron Sharing
245
8.3
The
247
The
Molecular Ion
6V
252
Molecule
llj
8.5
Molecular
Mi
Hybrid
254
Orliitals
261
Orliitals
8.7
Carbon-carbon Bonds
265
8.8
Rotational Energy Levels
269
Vibrational Energy Levels
8.9
169
243
8.2
8.4
Nan rigid
Box
5.0
S.
Quantum Theory
Hydrogen Atom
of the
Chapter 9
173
6.1
Schrodinger's Equation
Hydrogen Atom
173
°(j.2
Separation of Variables
176
178
6.4
Quantum Numbers
Principal Quantum Number
6.5
Orbital
Quantum Number
Magnetic Quantuiii Number
ISO
"6.3
I'd
6.7
1
!n-
Normal Zeeman
for
the
Effect
£81
Mechanics
Statistical DfetrflnitfOQ
8.2
Phase Space
M ax well-Bol nim n
•ft3
t
"9.4
\m
Statistical
9,1
184
9.6
187
"9.7
Electron Probability Density
169
9.8
6.9
Radiative Transitions
im
•9.9
6. t()
Selection Rules
198
I
Uws
Many-electron Atoms
9.11
Rotutional Spectra
298
Rose-Einstein Distribution
300
Rluck-hody RadUlimi
3tM
Fcnni-Dirac Distribution
M)
The Laser
VI
Syn in let lies
327
222
10.5
331
2-2-2
10.6
226
•10.7
The Metallic Braid
The Rand Theory of
The Fermi Bong
228
•10.8
Electron-energy Distribution
*7.10 One-electron Spectra
229
•10.9
Rrtllouin
*7.1
232
"10.11) Origin y(
1
of
Alpha Decay
1 1.-
1
10
The
Crystalline
10.11
237
Problems
Conservation
447
Theories of Elementary Particles
•151
404
Answers to Odd-numliercd Problems
457
481
Solid State
Effect ive
317
and Amorphous
Solids
1
235
10 id
Principles
314
Hund's Rule
CONTENTS
13.8
+14
448
310
7.6
Problems
Isotoptc Spin
439
Number
311
325
7.12 X-ray Spectra
13.7
412
Van Dcr Waals Forces
i.i
Nuclear Transformations
Camma Decay
111.4
'
Chapter 12
12.8
295
215
Two-electron Spectra
389
Particles
411
The
1
Strangeness
Inverse Reta Decay
7.5
Coupling
13.6
438
397
12.7
318
ft
Systematic* of Elementary
293
Covalent Crystals
•7.9
Kaons and Ilyperons
13.5
Evaluation of Constant^
an Ideal
436
13.4
408
in
433
and Muons
383
Reta Decay
10.3
Coupling
380
Piraiv
Theory
213
1.S
13.3
12.8
El eel n )i
•7.8
Prohlems
379
*12.5
Gas
431
of Nuclear
Forces
Singlet States
The Liquid-drop Model
The Shell Model
Problems
7.1
Momentum
and
Meson Theory
431
377
399
317
Total Angular
11.10
Triplet
13.2
374
the Dculcrun
Barrier Penetration
Ionic Crystals
°7.7
1.8
Antiparticlcs
"12.4
Solids
Periodic Table
11.8
1
The Dcuteron
Ground Stale of
Elementary Particles
13.1
288
10.2
is
•1 1.7
Chapter 13
13.9
210
>i
372
"11.8
396
207
i i
Binding Energy
127
Alpha Decay
Spin -orbit Coupling
rat
370
11.5
424
12.3
The
1 1
Nuclear Sizes and Shapes
12.13 Thermonuclear Energy
Prohlems
287
7.3
lii>
11.4
423
393
7.2
i
386
420
12.12 Trausuranic Elements
389
10.1
in
Stable Nuclei
\lk [ear Fivsion
Radioactive Scries
<
'.
11.3
117
12.1
Radioactive Decay
203
(
384
413
The Compound Nucleus
12.2
203
i
361
Cross Section
12.10
12.1
Electron Spin
Exclusion Principle
Atomic Masses
The Neutron
289
Problems
7.1
361
12.9
287
>isl ri 1 1 1 1
9.10 Comparison of Results
200
Chapter 7
The Atomic Nucleus
lion
Molecular Energies
9.5
fl.8
Problems
272
1.1
11.2
1 1
THE NUCLEUS
8.10 Electronic Spectra of Molecules
Chapter 6
1
of Mole-
cules
140
Functions
The
5.8
The Physics
Chapter 8
Energy
Quantization
5.7
Chapter
146
Particle in » Box;
PART FOUR
PART THREE
PROPERTIES OF MATTER
333
339
342
Zones
344
Forhiddcn Bands
346
Mass
355
355
CONTENTS
VII
PREFACE
This book
that
is
intended for use with one-semester courses in modern physics
have elementary
and calculus
classical physics
and quantum theory are considered
first
standing the physics of atoms and nuclei.
as prerequisites.
Kelativity
framework
for under-
to provide a
The theory
of the
veloped with emphasis on quantum-mechanical notions, and
discussion of the properties of aggregates of atoms.
atom
is
is
then de-
followed by a
Finally atomic nuclei and
elementary particles are examined.
The balance here
more toward
deliberately leans
mental methods and practical applicatioas. because
student
is
of individual details.
However,
all
physical theories
by the sword of experiment, and a nmnlier of extended derivations
live or die
are included in order lo demonstrate exactly
related to actual measurements.
students responsible for the
Many
can be passed over
how an
abstract concept can
I
have indicated with
lightly without
on the contents of these sections are
lie
instructors will prefer not to hold their
more complicated (though not
matically difficult) discussions, and
that
modern physics by a conceptual
better served in his introduction to
framework than by a mass
ideas than toward experi-
believe that the beginning
I
also
loss
necessarily matheasterisks sections
of continuity; problems based
marked with
asterisks.
Other omis-
may well have teen
may be skipped when its contents
sions are also passible, of course. Relativity, For example,
covered
and Part 3
earlier,
will lie the subject of later
ion the type of course
into selected subjects,
in its entirety
work. Thus there
is
scope
for
an instructor to
fash-
he wishes, whether a general survey or a deeper inquiry
and to choose the level of treatment appropriate to his
audience.
An expanded
this series;
version of this book requiring no higher degree of mathematical
my
Perspectives of Modern Physics, an Upper Division Text in
other Upper Division Texts carry forward specific aspects of modem
preparation
is
physics in detail.
In preparing this edition of Concepts of
text has
Modern Physics much
of the original
LIST
OF ABBREVIATIONS
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
k
amp
angstrom
atm
atmosphere
b
barn
C
coulomb
Ci
cmie
d
day
eV
F
fm
electron volt
g
h
Hz
ampere
farad
fermi
gram
hour
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
xi
PREFACE
SPECIAL RELATIVITY
i
Our
modern physics begins with
study of
This
of relativity.
is
a consideration of the special theory
a logical starting point, since
concerned with measmeinent and
all
physics
is
urements depend upon the observer as
weD
as
upon what
is
ultimately
how
meas-
observed.
From
relativity involves an analysis of
there are intimate relationships
emerges a new mechanics in which
Iretween space and time, mass and energy. Without these relationships
relativity
to understand the microscopic
be impossible
elucidation
LI
is
modem
the central problem of
it
would
world within the atom whose
physics.
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 all-pervading elastic
medium
called the ether, and their successful explanation of diffraction
interference
phenomena
so familial- that
its
in
existence
and
terms Gf ether waves
made
was accepted without
question. Maxwell's develop-
the notion of the ether
ment of the electromagnetic theory of light in 1864 and Hertz's experimental
confirmation of it in 887 deprived the ether of most of its properties, but nobody
1
at the
lime seemed
willing to discard the
ether: thai light propagates relative to
Let us consider an example of
what
hmdamentai
some
idea represented by the
sort of universal
this idea
frame of reference.
implies with die help of a simple
analogy.
D which Hows with the speed v.
the
river
with the same speed V, Boal A
of
one
bank
Two boats start out from
opposite the starting point
directly
bank
other
on
the
crosses the river to a point
the distance D and then
for
downstream
heads
B
Bud then returns, white boal
Figure 1-1
is
a sketch of a river of width
returns to the starling point. Let us calculate the time required for each round
trip.
—
component
land
V as
From
net speed across the river.
its
Fig. 1-2
we
see that these
speeds are related by the formula
river
V =
+
V' 2
2
so that the actual
v2
A
speed with which boat
V'= VV 2 -
crosses the river
is
v2
rB
CDCD-
4
=3
Hence the time
speed
V,
the initial crossing
for
the total round-trip time tA
FIGURE
Boat A goes directly across the river and returns to
downstream for an identical distance and then returns.
1-1
its
starting point, while boat
B heads
We begin by considering Ixmt A.
(Fig. 1-2).
It
it
A
If
heads perpendicularly across Lhe
downstream from
its
goal
The
river,
on the opposite bank
-
should be exactly
—v
in
accomplish
this, its
and
in order to compensate
upstream component of velocity
order to cancel out the river current
v,
B
case of boat
relative to the shore
must therefore head somewhat upstream
for the current. In order to
1-2
Boat A must head upstream
in
divided by the
is
D/V\
twice
or
it
trip,
is
the
order to compensate for the river current.
%
somewhat
own
u of the river.
It
sum
V
to the
heads downstream,
t;
its
speed
of the river (Fig. 1-3),
D/(V + c). On its return
own speed V minus the
longer time D/{V — c) to travel
in the time
shoTe
therefore requires the
D
it
plus the speed
D downstream
travels the distance
As
different.
speed
however, B's speed relative
speed
leaving the
v2 /V 2
is its
upstream the distance
FIGURE
D
2.D/V
l.i
\/l
the current will carry
Lhe distance
is
Since the reverse crossing involves exactly die same amount of time,
to its starting point.
is its
The
total round-trip
time
H
t
is
of these times, namely,
.«
Using the
=
D
D
V+
V-v
v
common denominator (V
_
'
B
D(V ~
~
(V
v)
+
+
+ D(V +
v)(V
-
l)(V
—
v) for
both terms,
v)
o)
2DV
V*
-v"
•2D/V
1.2
1
which
The
1.3
Jl
4
is
_ oVV a
greater than
ratio
f^,
the corresponding round-trip time for the other boat.
between the times
= VT -
i;7
V
f.,
and tB
is
2
If we know the common speed Vof the two
we can determine the speed v of the river.
BASIC CONCEPTS
boats and measure the ratio t A /t 8
,
SPECIAL RELATIVITY
L
—
V
.
i
mirror
A
B
V
v
B
v
X
B
v
parallel light
from
FIGURE
1-3
The speed
river current while its
of boat
H downstream
speed upstream
is
relative to the shore is increased
mirror
B
single source
by the speed of the
reduced by the same amount.
half -silvered mirror
The
reasoning used in
problem of the passage of
we move
pervading space,
this
problem may be transferred
light
waves through the
through
with
it
at least the
speed of the earth's orbital motion about the sun;
our speed through the ether
of an observer
motion,
is
even greater
on the earth, the ether
we can
is
ether.
if
3
If
x
From
moving past the
there
|0*
the sun
(Fig. 1-4).
to the analogous
is
m/s
is
1-5).
One
of these light
viewing screen
(18.5 mi/s)
also in motion,
FIGURE
The MichetsonMortey experiment.
1-5
the point of view
To detect
earth.
this
use the pair of light lieams formed by a half-silvered mirror
instead of a pair of boats (Fig.
hypothetical
ether current
an ether
beams
is
directed to a
mirror along a path perpendicular to the ether current, while the other goes
is
1-4
Motions of the earth through a hypothetical ether
optical arrangement
such that both beams return to the same viewing screen. The purpose of the
clear glass plate
FIGURE
The
to a mirror along a path parallel to the ether current.
is
to ensure that
both beams pass through the same thicknesses
of air and glass.
If
at
the path lengths of the
the screen hi phase
of view.
and
The presence of an
two beams are exactly the same, they
will arrive
will interfere constructively to yield a bright Held
ether current
in
the direction shown, however, would
cause the beams to have different transit limes in going from the half-silvered
mirror to the screen, so that they would no longer arrive at the screen in phase
but would interfere destructively.
performed
in
In essence this
is
the famous experiment
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. 1-6). If either of the optical paths in the
the fringes appear to
of the
ran
move
waves succeed one another
tell
us nothing about
die apparatus
is
apparatus
is
across the screen as reinforcement
at
varied in length,
and cancellation
each point. The stationary apparatus, then,
any time difference between die two paths.
When
rotated by 90", however, the two padis change their orientations
beam formerly requiring
now requires tB and vice versa. If these times
move across the screen during the rotation.
relative to the hypothetical ether stream, so that the
the time
t
A for the
round
trip
are different, the fringes will
BASIC CONCEPTS
SPECIAL RELATIVITY
If
d corresponds
d
where X
FIGIWE
1-6
d we
t
Fringe pat-
tern observed
in
—
a fringes,
to the shifting of
nX
the wavelength of the light used.
is
Equating these two formulas
for
find that
Michel-
cAt
son-Morley experiment.
_ Dv 2
Ac 2
In the actual experiment Michelson
and Morley were able to make
D about 10 m
length through the use of multiple reflections, and the wavelength
10~ 10 m). The expected fringe
of the light they used was about 5,000 A (1 A
in effective
=
Let us calculate the fringe
From
Eqs. 1.1
ether drift
and
3
X
is
4
10
-
B
t
1
t>
expected on the basis of the ether theory.
each path when the apparatus
2D/V
- v 2 /V 2
the ether speed,
rn/s,
V
and
is
m X
10
tA
"5x
2D/V
y/1 - v 2/V 2
which we
= 0.2
shall take as the earth's orbital
the speed of light
c,
where c
= 3x
lf* w
speed of
m/s. Hence
Mr-
(3
mX
T
A
0.4 fringe.
To everybody's
much
fi
(3
X
10 s m/s) 2
fringe
shift of this
surprise,
iment was performed
smaller than
1.
extremely small compared with
According to the binomial theorem, when x
shift,
magnitude
(l±af « 1 ±
therefore express
the distance
The path
el
=
to
conshisions
nx
Af
quences.
to
no fringe
shift
whatever was found.
at different seasons of the
a good approximation as
cA(.
BASIC CONCEPTS
between the half-silvered mirror and each of the other
d corresponding to a time difference At is
difference
When
the exper-
year and in different locations,
were always
identical:
no motion through the ether was detected.
1,
result
First, it
of the
Michelson-Morley experiment had two conse-
rendered untenable the hypothesis of the ether by demon-
strating that the ether has
no measurable properties
what had once been a respected
idea.
Second,
is
the
it
— an ignominious end
suggested a
new
for
physical
same everywhere, regardless
any motion of source or ohserver.
1.2
is
amount
and when experiments of other kinds were tried for (he same purpose, the
of
D
the total shift should
readily observable, and therefore
to establishing directly the existence of
principle: the speed of light in free space
mirrors.
is
is
The negative
Here
therefore
the ether.
io-
We may
is
L0 J m/s} 2
X
Since both paths experience this fringe
2n or
V2
is
rotated by 90°
time difference between the two paths owing to the
Michelson and Morley looked forward
which
is
is
M=
Here
1.2 the
shift
shift in
We
THE SPECIAL THEORY OF RELATIVITY
mentioned
earlier the role of the ether as a universal
frame of reference
with respect to which light waves were supposed to propagate.
speak of "motion," of course,
we
really
mean "motion
Whenever we
relative to a
frame of
SPECIAL RELATIVITY
The frame
reference."
may be a road, the earth's
we must specify it.
of reference
surface, the sun,
the center of our galaxy; but in every case
Bermuda and
in
in exactly
both
in Perth, Australia,
fall
Stones dropped
"down," and yet the two move
opposite directions relative to the earth's center.
Which
the correct
is
This postulate follows directly from the results of the Mfcbelson-Morley experi-
ment and many
all
location of the frame of reference in this situation, the earth's surface or
its
center?
The answer
may be more
pervading
all
is
that all frames of reference are equally correct, aldiough
convenient to use
we
space,
a specific case.
in
could refer
Bermuda and Perth would escape from
then, implies that there
is
empty
or instrument observing
it.
to
and the inhabitants of
it,
their quandary.
one
were an ether
The absence of an
no universal frame of reference, since
general, electromagnetic waves)
transmitted through
motion
all
// there
the only
is
ether,
light (or, in
means whereby information can
Ik;
space. All motion exists solely relative to the person
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
first
implied by the absence oi a universal frame of reference.
The
sight these postulates hardly
the intuitive concepts of lime
constant velocity
drifts at the
instant that
H
uniformly in
all
one of them
went
in
The
off.
The
An
erated with respect to one another.
changing
Anybody who
round can verify
statement from his
had a profound influence on
only a brief glance
The
at
light
from the
postulate of special relativity, even though
with respect to the point where the
observers cannot detect which of them
is
flare
undergoing such a change
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. 1-7 unusual?
The boats are
a stone into the
at sea
Let us consider a more familiar
on a clear day and somel>ody on one
water when they arc abreast of each other.
A
of
them drops
circular pattern
1-7
Relativist*
phenomena
differ
from everyday experience.
at constant
general Uieory of relativity, proposed
all
A
<3D
A
has been in an elevator or on a merry-go-
own
physics,
and
experience.
we
shall
The
-<jDJ
special theory
concentrate on
it
with
the general theorv.
special theory of relativity
is
based upon two postulates. The
light
first
"CD
li
B
emitted by
flare
each person sees sphere
of light expand h)* about
states
may be expressed in equations having the same form
frames of reference moving at constant velocity with respect to one
that the laws of physics
in all
first
his position
observer in an isolated laboratory can
detect accelerations.
lias
The
At the
flare travels
problems involving frames of reference accel-
later, treats
this
is fired.
the moving one.
is
either boat must find a sphere of light expanding with hlmsefy
according to the
is
and so on
a low-lying fog present,
directions, according to the second postulate of special relativity.
An observer on
at its center,
FIGURE
by Einstein a decade
is
special theurv
frames of reference, which are frames of reference moving
velocity with respect to one another.
There
abreast of A, a flare
is
of relativity, develo|x:d by Albert Einstein in 1905, treats problems involving
inertial
v.
neither boat does the observer have any idea which
analog.
of relativity resulted from an analysis of the physical coii.se<|nenees
others.
seem radical. Actually they subvert almost
and space we form on the basis of our daily
experience. A simple example will illustrate this statement. In Fig. 1-7 we have
the two boats A and B once more, with boat A at rest in the water while boat B
At
himself
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,
could be determined from these differences which objects are "stationary" in space and which are "moving." But because there is no universal
frame of reference,
<3Dj
this distinction
-<3CP
-»i-<3D>
B
does not exist in nature; hence the above
pattern of ripples from
postulate.
The second
postulate of special relativity states that the speed of light in free
space has the same value for
10
A
A
it
BASIC CONCEPTS
each person sees pattern
in different
all
observers, regardless of their state of motion.
stone
dropped
in
water
place relative
to himself
SPECIAL RELATIVITY
11
of ripples spreads out, as at the bottom of Fig,
which appears different to
-7,
]
observers on each boat.
of
tiie
to the
Merely by observing whether or not he is at the center
pattern of ripples, each observer can tell whether he is moving relative
water or
not.
Water
on a boat moving through
it
a frame of reference, and an oliserver
in itself
is
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
not,
is
and wave speeds
water vary with the observer's motion while wave speeds of
light in
in
recording device
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,
predicted.
One
of the triumphs of
we
shall see that
modem
physics
is
many
light pulse
A
peculiar effects are
the experimental confirma-
tion of these effects.
photosensitive
surface
TIME DILATION
1.3
We
shall first use the postulates of special relativity to investigate
how
relative
A
it
clock moving with respect to an observer appears to tick
does when at
finds that the
rest
with respect to him. That
time interval between two events
the ground
would
find that the
quantity
which
is
r,„
When
less
someone
if
in the spacecraft
wc
is („,
interval has the longer duration
is
called the proper time of the interval
witnessed from the ground, the events
and end of the time interval occur
at different places,
Uut mark
and
l-S
A
f.
in
on
a photosensitive surface
signal
when
in
an
see
how
rime dilation comes about,
affects
shown
A
pulse of light
and an appropriate device
12
Fig.
what we measure. This clock
at each end.
of
in
is
is
is
how
relative
motion
up and down between the mirrors,
light strikes
it.
on the mirror which can be arranged to give an electric
The proper time t between ticks is
lt
2L
If
the stick
is
1
m
long,
3
X
2
called
consists of a stick / J(1 long with a mirror
reflected
of the light
consequence the
us examine the operation of the
inquire
trip
between
attached to one of the mirrors to give a "tick"
some kind each time the pulse of
BASIC CONCEPTS
let
IS and
round
1.4
time dilation.
To
to a
the light pulse arrives,)
The
the beginning
duration of the interval appears longer than the proper time. This effect
particularly simple clock
Each "tick" corresponds
simple clock.
rapidly than
in a spacecraft
determined by events dial occur at the soma place
observer's frame of reference,
the events.
same
is,
FIGURE
pulse from the lower mirror to the upper out and back.
motion affects measurements of time intervals and lengths.
(Such a device might he
and there are
one
is
1.5
X
m
10 s m/s
10s
=
ticks/s.
10" H s
0.67
X
Two
identical clocks of this kind are built, and
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
measured by an observer on the ground with an identical clock that
is
as
stationary
with respect to him. Each tick involves the passage of a pulse of light at speed
SPECIAL RELATIVITY
13
c from the lower mirror to the upper one mid hack.
passage the entire clock
spaceship
in the
is
in
During
this round-trip
motion, which means that the
pulse of light, as seen from the ground, actually follows a zigzag path (Fig,
On
way from
its
the vertical distance
between the
Since
mirrors.
same
Ebtaetfy the
v
w)
t
= 24V
2
c
-
v2
c 2 {l
the spaceship appears to tick at a slower rate than the
measurements of the clock on the ground
analysis holds for
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
(D^v + (ir
4(C * -
in
stationary one on the ground, as seen by an observer on the ground.
1-9).
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.
L,, is
The moving clock
of
He
(„.
too finds that
Vi - f
so the effect
W
-
him tick more slowly than
to
Our
v l /c-)
when
discussion has been based
bouncing back and
light pulse
and
reciprocal: every observer finds that clocks in
is
they are at
motion
relative
rest.
on a somewhat unusual clock
that
employs a
between two mirrors. Do the same con-
forth
more conventional clocks that use machinery— spring-controlled
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
clusions apply to
t
1.5
2/..A
=
VI But 2Lq/c
is
in Eq. 1.4,
v 2 /c 2
the time interval
*
n
l>etween ticks on the clock on the ground, as
flight,
and so
the disagreement between them could
used to determine the speed
lie
any other object—which contradicts the
Detailed calculations of what happens to
of the spaceship without reference to
1.6
t
principle that
=
Vl - bVc 3
Time
all
motion
dilation
conventional clocks in
For example, as
we
is
relative.
motion—as
shall learn in Sec. 1.10,
moving spaceship. Therefore
1-9
parallel to
A
light clock In a
spacecraft es seen by an observer at rest on the ground.
The mirrors are
ground— confirm
the mass of an object
is
this
answer.
greater
when
in motion, so that the period of an oscillating object must be greater in the
it is
FIGURE
seen from the
till
clocks at rest relative to one another behave
to all observers, regardless of
the
same
the
group of clocks or the observers.
any motion
at constant velocity of either
the direction of motion of the spacecraft.
The
that
to
relative character of lime has
seem
implications.
is,
is
a relative notion
which require simultaneity
The
that the total
in
right?
The
and not an absolute one, physical
events at different locations must be
principle of conservation of energy in
energy contenl of the universe
is
while an equal amount of energy
AE
its
elementary form states
constant, but
out a process in which a certain amount of energy
else
is
sees.
Because simultaneity
discarded.
Who
of course, meaningless: both observers are "right," since each simply
measures what he
theories
For example, events
one observer may not be simultaneous
another observer in relative motion, and vice versa.
question
<^>
many
to take place sitnullaneously to
AE
it
does not rule
vanishes at one point
spontaneously conies into being somewhere
with no actual transport of energy from one place to the other.
simultaneity
is relative,
some observers
Because
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
14
BASIC CONCEPTS
SPECIAL RELATIVITY
15
elsewhere,
has actually flotced from the
it
first location to the second (There
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
are
many ways
principle.
is
a relative quantity, not
the notions of time formed by
does not run backward to any olxserver,
ah"
everyday experience are incorrect. Time
for instance: a sequence of events that
occur somewhere at (,, t t ,
will
2
3
appear in the same order to all observers everywhere,
though not necessarily
with the same time intervals U - <„ t - t
between each pair
events.
.
,
3
2,
.
.
happens— more
it
speed of
light
is
finite
travel a distance L.
(and, as
we
.
.
of
.
Similarly, no distant olwerver, regardless of
his state of motion,
before
accelerated
and when
at
can see an event
precisely, before a nearby observer sees
it— since the
and signals reqnire the minimum period of time
L/c to
is no way to peer into the future,
although temporal
finally
it
events
may appear
different
comes
its
journey:
it
and the
fact that the spaceship
takes off,
During each
to a stop.
What B measured may
time.
when
when it
of these accelerations
inertia!
is
turns around,
A was
frames corresponding to
and return trips were different. The earthbound twin B, on the
was not accelerated and stayed in the same inertial frame all the
other hand,
therefore
— that A
lie
interpreted on the basis of special
—
younger when he comes back is correct.
Of course, A's life-span 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
relativity,
and
his conclusion
10 yr as far as
he
is
concerned.
is
What
has happened
is
that A's accelerations
and by applying the conclusions of general relativity
accelerated clocks we find that A is younger than B on his return by the
affected his
for
There
shall see, spatial) perspectives of past
various limes in
not in an inertial frame of reference,
the outward
Although time
upon the
resolution of the paradox depends
The
in
life
processes,
amount expected on the
exact
basis of fl's analysis using the formula for
time
dilation.
to different observers.
LENGTH CONTRACTION
1.5
1.4
THE TWIN PARADOX
Measurements of lengths
We
now
are
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 life-heartbeats,
respiration, and so
eonstitute biological clocks of reasonable
his
A
takes off
brother
B on
when he
the earth,
respect to him, a
20 yr old and travels at a speed of 0.99c. To
seems to 1% living more slowly, in fact at a rate
only
We
= 0.14 =
B
that
A
takes,
A
eats 7; for every tiiought that
A
thinks,
eats,
B
For
For every breath
goes.
yr have elapsed by B's reckoning,
A
B
to
B
thinks 7.
when
it
is
is
at rest
The length
purpose
and
it
L,,
called its projjer length.
light clock of the previous section to investigate the
this
with
we
Lorentz
imagine the clock oriented so that the light
At
(
=
arrives at the front mirror at
is
moving
the light pulse starts from the rear
/,.
The
pulse has traveled the distance
= L+
Ulj
Hence
Finally, after 70
1.7
returns
home, a man of only 30 while
B
then 90 yr old.
Where
of twin
is
A
the paradox? If
90 at
this
we examine
the situation from the point of view
B on the earth is in motion at 0.99c. Therefore we
be 30 yr old upon the return of the spaceship
while A is
in the spaceship,
might expect
B
to
time— the
precise opposite of
what was concluded
in the
where
L
the distance l>etween the mirrors as
is
The
at
t
pulse
is
at
then reflected by the front mirror and returns to the rear mirror
after traveling the distance c(<
-
(,),
where
preceding
c(t
BASIC CONCEPTS
measured by the observer
rest.
paragraph.
16
length L„
Lorentz-VitzCeraUl contraction. This
reach the front mirror, where from the diagram
14 percent
takes 7; for every meal that
frame
relative to the observer (Fig. 1-10).
C*,
as fast as
is
can use the
mirror,
(0.99c)Vc 2
its
as the
pulse travels back and forth parallel to the direction in which the clock
rf,
VI - cVc 2 = Vl -
phenomenon known
of an object in its rest
by relative
object in motion with respect to an observer always
contraction occurs only in the direction of the relative motion.
contraction.
is
A
The length L of an
ap|>ears to the observer to l>e shorter than
on-
regularity.
Twin
motion.
as well as of time intervals are affected
-
f,)
=L-
v(t
-
r,)
SPECIAL RELATIVITY
17
rear
(hint
mirror
(IS)
21^,/c
=
t
VI - oVe*
which
u
is
terms of
terms of Lq, the proper distance l>etwecn the mirrors, instead of
in
the distance as measured by an observer in relative motion.
/-,
two formulas must he equivalent, and hence
^>
-
L+
~ VI -
cVc-
L=
1.10
~<*H
liave
ILJc
2L/c
1
we
-
Vl
1^
v 2 /c 2
v?/c 2
Lorentz contraction
Because the relative velocity appears only as o8
t>f,-
in
The
in
Eq. 1.10, the Lorentz
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
contraction
I.
a reciprocal
length of an object
|/wI— -
is
The
Dit-IJ—*
to
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.
(,
as
determined from the ground,
and yet
us.
(
=
+
C
We
eliminate
/
/,
V
'l
with the help of Eq.
=
+
c
v
c
=
=
1.7 to find that
v)(c
—
light
is
C*
-
I
(186,000 mi/s)2
On
r=A
- V)
=
=
vz
-
"I
t-Vc*
I
lie
The
1.9
gives the time interval
t
between ticb of the moving clock
measured by an observer on the ground
We earlier fonnd another expression for
18
motion
the other band, a
body traveling
at 0.9 the
speed of
(0.9tf
0.436
43.6 percent
2L/c
19
Equation
results in a shortening in the direction of
shortened to
2Lc
=
it is
of 1,000 mi/s
99,9985 percent
of the length at rest.
V
it
A speed
0.9«W985
L
2Lc
+
(c
negligible for ordinary speeds, but
(],(HK)mi/s) 2
is
/
+
is
only
= ./!1.8
length any observer will find.
an important effect at speeds close to the speed of light.
FIGURE 1-10
the entire lime interval
maximum
relativistk' length contraction
seems enormous to
Hence
the
is
effect.
as
il
is
length at
ratio
rest,
Iwtwccu
a significant change.
/-
and L„
in Eq. 1.10
is
when
we might
the same as that in Eq, 1.3
applied to the times of travel of the two light beams, so that
be k-mpted to consider the Miehelson-Morley result solely as evidence for the
t,
contraction of the length of their apparatus in the direction of the earth's motion.
BASIC CONCEPTS
SPECIAL RELATIVITY
19
This interpretation was tested by Kennedy and Thorndike
using an interferometer with arms of unequal length.
which means
shift,
that these experiments
absence of an ether with
this
all
must
lie
experiment
in a similar
They
also
found no fringe
considered evidence for the
implies and not only for contractions of the
We
can resolve the meson paradox by using the
of relativity.
meson,
in
meson
which
to the
results of the special theory
examine the problem from the frame of reference of the
I<et us
lifetime
its
is
2
10~ fi
x
s.
In this case the distance from the
ground appears shortened by the factor
apparatus.
An
a
actual photograph of
somewhat
object
is
an object
different distortion,
viewed and the
very rapid relative motion would reveal
in
The reason for this effect is that light reaching
more distant parts of the object
ratio o/e.
the camera (or eye, for that matter) from the
was emitted
when
is
actually a composite, since the object
parts; the
was
camera "sees"
at different locations
the various elements of the single image that reaches the film
left
it.
This
supplements the Lorentz contraction by extending the apparent length
effect
of a
from the nearer
earlier than that corning
a picture that
moving object
in the direction of
body, such as a cube,
in shape,
may be
£«
depending upon the direction from which the
-
VI
v s /c*
is, while we, on the ground, measure the altitude at which the meson is
produced as yn 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
That
,
we
before decaying,
frame
find that the
phenomenon.
\/T= -cVe"
600
of a moving body
would
lie
itself, which
were no Lorentz contraction, the appearance
different from what it is at rest, but in another
in
(0.998c) 2
there
If
also
our reference
orientation as well as changed
in
again depending upon the position of the observer and the value of
a physical
in
motion. As a result, a three-dimensional
seen as rotated
e/c. This result must be distinguished from the Lorentz contraction
is
corresponding distance y„
is
f-
way.
c*
600
It is
rapidly
interesting to note that the
moving
objects
above approach
was not made
until 1959,
to the visual
54 years
appearance of
\/T- 0.996
-
after the publication
Will
of the special theory of relativity.
m
0.063
a
1.6
MESON DECAY
in greater detail later.
Now
For the moment our
into an electron
X
an average of 2
n mesons are created high
in the
interest lies in the fact that
10 -8 s after
atmosphere by
fast
it
comes
a
ft
into being.
have a typical speed of 2.994
But in
t
n
,
=
X
2
-8
10
s,
X
10 s m/s, which
the meson's
mean
is
Now
let
y
at
possible for the
mesons
to reach the
which they are actually formed.
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 (/,„
but
its
lifetime in our reference
frame has
l>een extended,
is
produced
owing
to the
relative motion, to the value
/
\/l
can travel a distance
-
Dt/c*
2X
of only
=
=
=
is
it
cosmic-ray particles
0.998 of the velocity of tight
lifetime, they
life-spans,
ground from the considerable altitudes
arriving at the earth from space, and reach sea level in profusion. Such mesons
c.
m
Hence, despite their brief
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
meson decays
9,500
1
-
10- 8
(0.998c) 2
««o
2.994
600
x
10"
m/s
x
2
X
I0" 6
s
2
in
X
IP" 6
S
0.063
while they are actually created at altitudes more than JO times greater than
20
BASIC CONCEPTS
this.
= 31.7X
10-*
s
SPECIAL RELATIVITY
21
almost 16 times greater than
a
meson whose speed
Vn
=
=
=
is
when
it is
at rest
with respect to
as.
In 31.7
X
I0" 6 s
0.998e can travel a distance
S coincided, measurements in the x direction
'
iu S'
bv the amount
x direction. That
X
9,500
m
10 H m/s
X
31.7
X
10" 6
s
moved
that S' has
in the
—
X
vt
1.12
if
=
tj
1.13
z'
=
z
in the
and
;/
and so
z directions,
THE IORENTZ TRANSFORMATION
Let us suppose that
some event
different
we
are in a frame of reference S
that occurs at the time
frame of reference
velocity o will find that the
nates x\
+x
y', z'.
S'
time
is
t
are
,t\
(In order to simplify
How
and
y, z.
find that the coordinates
An
observer located in a
moving with respect
the absence of any indication to the contrary in our everyday experience,
at the
our work,
we
time
shall
f and
has the coordi-
assume that v
are the measurements
we
further
assume that
=
('
1,14
*
to S at the constant
*, y, z,
is
in the
related to
t
W
s'.
we
which
same event occurs
direction, as in Fig. 1-11.)
f.
If
If
exceed those made
in S will
is
no relative motion
is
identical results.
In
x',
=
x'
1.11
same distance obtained before. The two points of view give
*1.7
of
made
which represents the distance
i"
2.994
There
the
vt,
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,
ij\
.r' :
and
z'
both systems
is
measured from the
instant
when
i.
simply differentiate
with respect to time:
are unaware of special relativity, the answer seems obvious enough.
in
we
»». — 6
is
the origins of S and
dt'
dy'
1.16
=
t>„
dt'
i
FIGURE
I-
11
tion with the
Frame
v moves
In
the
speed a relative to frame
r direc-
i
o;
1.17
dz'
=
dt'
S.
While the Galilean transformation and the velocity transformation
il
leads to
are both in accord with our intuitive expectations, they violate Irath of the
V
n
postulates of special relativity.
of physics in both the
of electricity
light e
The second
whether determined
light in the x direction in the
=
c
—
in
S or
l>e
S'.
a,
If
the Galilean
one frame into
postulate calls for the
S system to
their
same value of the
we measure
however,
the speed of
in the S'
Clearly a different transformation
postulates of special relativity are to
BASIC CONCEPTS
in
system
it
v
according to Eq. 1.15.
22
when
different forms
be
c'
»'
postulate calls for identical equations
used to convert quantities measured
is
equivalents in the other.
will
first
and magnetism assume very
transformation
speed of
The
S and S' frames of reference, but the fundamental equations
dilation
be
satisfied.
We
and length contraction to follow naturally from
is
required
if
the
would expect Iwth time
this
new
transformation.
SPECIAL RELATIVITY
23
A
reasonable guess as to the nature of the correct relationship between
x and
x' is
The second
postulate of relativity enables us to evaluate
according to our
1.18
k(x
k
vt)
is
a factor of proportionality that
is
may be a
but
-
common
set off at the
and
conditions,
initial
/'
and
origin of S
—
S' at
k.
function of
o.
The
does not depend upon either x or
choice of Eq. [18 follows from several
out.
in
Both observers must
find the
At the instant
S" are in the
same
(
=
0,
place,
then also. Suppose that a flare
/
=
I'
=
(),
each system proceed to measure the speed with which the
wham
t
=
x'
two frames of reference S and
ihe origins of the
same speed c
(Fig.
and the oKservers
light
1-7),
from
it
in
spreads
which means
that
the S frame
considerations:
1.23
1-
It
a single
2.
linear in
is
,v
and
event in frame
It is
so that a single event in frame S corresponds to
as it must.
X
=
Ct
.v',
S',
simple, and a simple solution to a problem should
always be explored
'
while in the S' frame
1.24
x'
—
ct'
first.
3.
U
has the possibility of reducing to Eq.
1.
11,
which we know
to
be correct
Substituting For
.v'
and
/'
Eq. 1.24 with the help of Eqs. I.IK and 1.22,
in
in ordinary mechanics.
Jt(jr
Because the equations of physics must have the same form in
both S and
we need only change the sign of o (in order to take into account the
terms of
in
=
x
1.19
and
.v'
motionl to write the corresponding equation for x
ami solving for
k(x'
+
same
,S"
in
both frames of reference since there
other than
in
the sign of
case of the Galilean transformation, there
i.s
nu
is
k
=
nothing to indicate that
+ -k
c
*-<4*W
y
=
+
.2,
ct
-tt-W
The time coordinates
t
and f, however, are not equal.
sulMtituting the value of x' given
x
=
from which we
1.22
<'
2
k (x
-
vt)
+
by Eq. LIS
We can see this
We obtain
by
into Eq. 1.19.
the
i
expression for x will be the
(^)>
postulate of special relativity.
BASIC CONCEPTS
same
as that given
provided that the quantity in the brackets equals
c
find that
=" +
first
II is
by Eq.
1.
1.23,
namely
.v
= ct,
Therefore
kvt'
Equations 1.18 and 1.20 to 1.22 constitute a coordinate
transformation that
24
cx
)
vkt
1
=
ki
- \~
(
c(
z,
1.21
satisfies
+
i
there might be differences Ixstween the corresponding
coordinates y, tf and
which are normal to the direction of r. Hence we again
take
>j'
+
-(^)<
vt')
z'
1.20
ckt
x,
ckt
difference between S and
in the
=
/':
ITie factor k must be the
As
tr)
S',
difference
in the direction of relative
-
=
1
-fe-')f
and
1.25
k
=
VI - »Vc2
SPECIAL RELATIVITY
25
Inserting the above value of k in Eqs. 1.18
and
transformation of measurements of an event
measurements made
1.26
have, for the complete
in S to the
S\ the equations
in
THE INVERSE LORENTZ TRANSFORMATION
•1.8
corresponding
In the
— vt
Vl - v2/c*
previous section the coordinates of the ends of the moving rod were
measured
x
=
X"
we
1.22,
made
L
though, Eq. 1.29
1.27
y
=
y
1.28
z'
=
z
chosen
Lorentz transformation
frame S
in the stationary
Eq. 1.26 to find
in
is
at the
terms of Lq and
v.
same time r, and it was easy to use
we want to examine time dilation,
If
not convenient, because
(,
and
f
2,
the start and
must be measured when the moving clock
time interval,
tive different positioas x l
and x2
In situations of this kind
.
it
is
is
finish
of the
at the respec-
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,
r =
1.29
primed and unprimed quantities
Vl - dVc !
replaced by
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
number of years
diat the
The
and
is
small compared with the velocity of light
relativists length contraction follows directly
An
x'3 ,
was not
is
Let us consider a rod lying along the
observer in
=
+ vt'
Vi - v2 /!?
1.31
y
=
y
until a
x' axis in the
and so the proper length of the rod
its
~
x2
Inverse Lorentz transformation
1.32
trans-
moving frame
ends to be
1.33
=
t
2
V\ - vVc
x\
is
xt
observer in S' finds that the time
In order to find
L = x2
—
frame S
time
we make
at the
t,
x v the length of the rod as measured in the stationary
use of Eq. 1.26.
We
=
have
1
VI -
n
v -/c*
~ vt
Vl
v*/c 2
*2
and
so
x?
— *j
= (4~*.)Vi-t>7c*
= L Vl- oVc 2
which
26
is
the
same
BASIC CONCEPTS
as Eq. 1.10.
x' in
the
moving frame
S'.
an observer in S will find
it
When
an
be
*,,
to
vx
~ Vl - eVc2
After a time interval of
the time
is
now
t
L =
is l'Y ,
where, from Eq. 1.33,
t
x;
is
x'
I«t us consider a clock at the point
=
Ml
and v
ft
from the Lorentz
frame determines the coordinates of
this
r
obvious
Lorentz transformation reduces to the Galilean transformation when
formation.
S'.
It
later that Einstein discovered their full significance. It
the relative velocity c
1.30
frames of reference in uniform relative
motion only when these transformation equations are used.
in Eqs. 1.26 to 1.29 are exchanged,
— <:
The observer
=
f
(to him), the
observer in the moving system finds that
t'
according to his clock. That
t
—
2
2
f
is,
j
however, measures the end of the same time interval to be
in S,
J
l
>
+
vx
c*
Vl - dVc2
SPECIAL RELATIVITY
27
him the duration of the
so to
interval
/
dt
= 1,-1,
~ 'i
VI - i'7r 2
1
Vl -
*~
or
we found
2
v /c 2
dt
+
dx'
6.
=
- H<*
v'l
as
tr
=
and so
t'a
/
sM.
df +
is
(If
earlier with the help of
a light-pulse clock.
v (If
dx'
+
1
dx
+
v
dt'
1
dx
o
1+^r
•1.9
+
v;
One
of the postulates of special relativity stales thai the speed of light c in free
space has the same value for
Bill
"common
moving
a car
sum
a
sense"
at
80
tells
ft/s,
all
us that,
we throw
if
the direction of
in
another frame S will have a speed of r
above
it is
a hall forward at
the hall's speed relative to the ground
"Common
postulate,
elsewhere, and
sense"
we must
that a ray of light
motion at velocity
c as measured in
no more
130
reliable as
S,
+
i
the
emitted
Similarly
in
c relative to
v„
1.35
= v'iVt
in science
+
i
contradicting the
a guide
vr =
1.36
S measures
its
three velocity
components
An
+
If
•
while
to
an observer
v:
»
dt
in S'
dY_
K=
df
=
c,
that
is, if
=
dz'
differentiating the inverse Lorentz transformation equations for
we
dy
dz
28
x, y,
and
l
obtain
dx
dx'
=
=
=
is
emitted
S,
in the
an observer
moving reference frame
in
frame S
will
measure
+
D
•9
df
c
t,
light
the velocity
(It
V'
v:
a ray of
motion relative to
S' in its direction of
dz
V.ss
df
c2
_zl
they are
—
=
dt
V^
2
2
v /c
vV't
observer in
to l»e
V —
• ~
ov;
v;VT 1
S'.
v 2 /c 2
than
addition.
Let us consider something moving relative to both S and
RelatMstic velocity transformation
vV'
from
ft/s,
turn to the Lorentz transformation equations for
scheme of velocity
the correct
is
+
its
5ft ft/s
is
i>
1.34
observers, regardless of their relative motion.
two speeds. Hence we would expect
of the
frame of reference H'
By
dt'
VELOCITY ADDITION
+
v
+ Jf
tr
c{c 4- v)
4 odf
C
=
4
c
ihj'
dz'
BASIC CONCEPTS
Both observers determine the same value for the speed of
light, as
they must.
SPECIAL RELATIVITY
29
The
relativistic velocity
iastance,
we
might imagine wishing
to the earth in 0.9c at a relative
to pass a
space ship whose speed with respect
speed of
0.5c.
According to conventional
mechanics our required speed relative to the earth would have
than the velocity of
v
=
V.
+
v;
=
speed
=
which
is less
.34,
be
1.4c,
=
however, with V^
kit
more
0.5c and
only
is
s
»
ff
0.5c
+
k-
0.9c
(0.9c){0.5e)
+
1
1
to
vV>
+
1
According to Eq.
light.
0.9c, the necessary
y
transformation has other peculiar consequences. For
0.9655c
than
We
c.
need go
traveling at 0.9c in order to pass
less
it
at
than
percent faster than a space ship
1
a relative speed of
0.5c.
THE RELATIVITY OF MASS
1.10
collision as seen
now we have been considering only the purely kinematical aspects
The dynamical consequences of relativity are at least
Until
special relativity.
from frame S
of
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
energy
is
conserved) between two particles
in the reference
frames S and S' which are
properties of A and
they are
in the
at rest.
+1
frame
the speed
S',
A
VA
B
as seen
at the
while
B was thrown
.
If
from
A as
When
of
§',
which kinetic
by observers
uniform relative motion. The
$' are oriented as in Fig. 1-12,
in
which
with S' moving
had been
instant,
in the
at rest in
A was
— if
thrown
frame S and particle B
in the
+y
collision as seen
from frame
S':
direction at
direction at the speed V^,
where
V
seen from S
is
exactly the
the two particles collide,
same
A
as die behavior
rebounds
in the
—y
FIGURE 1-12
An
elastic cplllslon is
observed
In
two different frames of reference,
A while B rebounds in the +y' direction at the speed
the particles are thrown from positions Y apart, an observer in S finds
,
that the collision occurs at y
30
in
vB
direction at the speed
V'B
same
Then,
Hence the behavior
of
a collision
B, as witnessed
direction with respect to S at the velocity v.
vA =
1.37
in
is,
B are identical when determined in reference frames
The frames S and
Before the collision, particle
in
A and
BASIC CONCEPTS
=
/2 Y and one
l
in S' finds that
it
occurs at y'
=
%Y.
SPECIAL RELATIVITY
31
The round-trip time Tn
for
A
measured
as
in
frame S
is
In the above example both
therefore
m
formula giving the mass
*"*V
1,38
and
it
is
same
the
H
v
moving
in S.
in S',
In order to obtain a
body measured while in motion in terms of
rest, we need only consider a similar example
m„ when measured at
which V", and V^ are very small. In this case an observer
A with Hie velocity v. make a glancing collision (since V'B
in S will see
<
v),
its
in
B approach
and then continue
In S
on.
'it-
are
of a
mass
for
A and 8
mA =
,
irtf,
and
If
momentum
is
conserved
in the S frame,
must be true
it
mn = m
that
and so
mA VA = mB VB
1.39
where
mA
mB
and
a* measured in the S /ra»i«.
A and
In S the
B,
speed
and V^ and
V'
w
is
VB
their velocities
found from
Relattvistic
Vl -
This mass increase
where
T
H
the time required for
is
to
make
In S', however, B's trip requires the time
'/;,,
its
round
trip as
measured
m,
in S.
is
mass
v 2 /c*
The mass of a body moving at
than its mass when at rest relative
=i
V.
1.40
m=
1.43
are the tnasses of
the speed v relative to an observer
to the
observer by
the factor 1/
is
larger
Vl -
o'/c*.
reciprocal; to an observer in S'
=m
and
where
m„ = m„
r=
1.41
/I
-
Measured from the
d»/c
the ground and
according to our previous
same event, they disagree
other frame requires to
Replacing
T
B
Prom
in
both frames see the
as to the length of time the particle
make
thrown from the
the collision and return.
Eq. 1.40 with
in
Although observers
results.
its
T we
,
have
ryi - dVc2
T
=
mass
its
h rocket ship
is
To somebody
momentum
is
\/l
VA
and
VB
in
Eq. 1.39,
we
see that
-
momentum
= m K Vl -
~
tt
T
rfiL \/l
»2 /c
This
still
OB
v 2 /c*
is
valid in special relativity just as in classical physics.
is
correct only in the form
is
ma v
1
2
- dVc
-!
not equivalent to saying that
original hypothesis
l>etween an observer and whatever he
32
_
us
was that A and B are identical when at rest with respect
to an observer; the difference between m and m therefore means thai measureA
B
ments of mass, like tho.se of space and time, depend upon the relative speed
Our
twin
on the rocket ship in Hight
rQ
conserved provided that
1.42
its
14
conservation of momentum
Inserting these expressions for
is
shorter than
defined as
However, Newton's second law of motion
A
is
plotted in Fig. 1-13.
Provided that
]
Eq. 1.38
greater.
in night
the ship on the ground also appears shorter and to have a greater mass. (The
1.43
effect is, of course, unobservably small for actual rocket speeds.) Equation
is
equivalent in terms of
earth,
BASIC CONCEPTS
is
observing.
F=
ina
dt
SPECIAL RELATIVITY
33
l.U
MASS AND ENERGY
relationship Einstein obtained from the postulates of special
mass and energy. This relationship can be derived directly
The most famous
concerns
done
from the definition of the kinetic energy T of a moving body as the work
That
is,
in bringing it from rest to its state of motion.
relativity
-£**
component of the applied force in the direction of the displacement ds and s is the distance over which the force acts. Using the relativistic
where F
is
the
form of the second law of motion
F=
rf(fflli)
~dT
becomes
the expression for kinetic energy
d(mv)
-[
FIGURE
1-13
The
relativity of
mm.
da
dt
hib
i;
J
even with
m
given by Eq. 1.43, because
d
\
and dm/elt does not vanish
resultant force
on a lx>dy
is
if
\y/l-
{
dm
dv
d{mc)
dy
Integrating by parts (/*
the speed of the
body
varies with lime.
T
The
always equal to the time rate of change of
=
light.
mass increases are
significant only at speeds
At a speed one-tenth that of
percent, but this increase
Only atomic
is
approaching that of
and
on have
so
in
ratio
e/m
this
its mass is smaller for fast electrons than
equation, like the others of special relativity, has been verified
by so many experiments
of physics.
34
BASIC CONCEPTS
that
it
is
now
- oVc 2
= mc 2 - m^c 2
suffi-
of the electron's charge to
slow ones;
v 2/c2
+ in^Vl -
i>
2
/c 2
- "hc
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
for
He 2
t/1
1.46
and
vdv
Vl VI -
mass increase amounts to only (1.5
over 100 percent at a speed nine-tenths that of light.
particles such as electrons, protons, mesons,
- fy dx),
its
light the
ciently high speeds for relativistic effects to be measurable,
xy
m„v £
=
momentum.
Relativistic
v 2 /c 2 I
Equation 1.46 states that the kinetic energy of a body
in its
mass consequent upon
speed of
1A7
relative
is
equal to the increase
motion multiplied by the square of the
light.
Eq nation
recognized as one of the basic formulas
its
1.46
may
mc 1 =
be rewritten
T+
m„c 2
SPECIAL RELATIVITY
35
If
we
body
mn c 2
interpret
is
is
mc 2
as the total energy
=
and T
at rest
0.
it
E
of the body,
it
follows that,
nevertheless possesses the energy jn c a
(1
called the rest energy
E
of a
l»dy whose mass
at rest
is
.
when
m„.
In the foregoing calculation relativity has
the
Accordingly
Equation
1.17 therefore Ixscomes
it
where we know by experience
speeds,
nevertheless important to keep in
iormulation of mechanics has
mna
E =
1.48
In addition to
between the
It is
2
Rest energy
kinetic, potential, electromagnetic,
its
familial- guises, then,
energy can manifest
itself as
mass,
unit of
above
thermal, and other
:ire
The conversion factor
is c\ so 1 kg of matter
1.49
mass (kg) and the unit of energy
(J)
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.
its
that the latter are perfectly valid.
mind
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.
the relative speed v
is
small
compared with
c,
energy must reduce to the familiar Y,m v 2 which has been verified
by experiment
low speeds. Let us see whether this is true. The binomial theorem of
algebra
()
us that
if
(1
basis in relativity, with classical
forms somewhat different from their original ones. The
some quantity x
is
much
we
so easy to derive thai
shall
£ = Vm^V' +
('
mnc
2
(
i- Ac
V
1.52
smaller than
1
.
=
1.54
\/l
-
physics,
is
i
;/
is
directly
with d*/«*
<
T
1
first
I
[1
+ (T/nw*)] 2
+
t:
from
T/m
lt
is
formula with the help of the binomial theorem,
much less than c,
36
BASIC CONCEPTS
permits us to find
r,
i:
c 2 the ratio between the kinetic and rest energies of a particle.
,
The
Tl*c equivalence of
*
An
mass and energy can be demonstrated
interesting derivation that
+ %m v v2
is
somewhat
in a
numlicr of
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
total
'Is
energy of a moving
energy of such a particle
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
is
"ferromagnetic radiation
c2
particles are customarily specified,
1.52, for Instance,
MASS AND ENERGY: ALTERNATIVE DERIVATION
different ways.
relativistic expression for the kinetic
particle reduces to the classical one.
E= m
Equation
this
= (1 + %^/c 2 )m v c 2 - m a c 2
= %m v 2
low speeds the
where the kinetic energies of moving
momentum mv.
nuclear and elementary-particle
in
— nup
term of
since
l)
b=/c»
1.12
Expanding the
=r -
used for the magnitude of the linear
rather than their velocities.
2
m,,r"~
1
1.53
The symbol
formula for kinetic energy
T = mc - m^c
-
1,
1
2
equations
2
p c-
1.50
T=
new
,
± *)» =; ± nx
relativistic
terice at
mechanics no
simply state them without proof:
These formulas are particularly useful
1
It is
known, the correct
is
the formula for kinetic
at
The
that, so far as
often convenient to express several of the relativistic formulas obtained
in
1.51
Since mass and energy are not independent entities, the separate
conservation
principles of energy and mass are properly a single one, the
principle of conservation of mass energy. Mass can lie created or
destroyed but when this
tells
test;
more than an approximation correct only under certain circumstances.
where
When
once again met an important
has yielded exactly the same results as those of ordinary mechanics at low
aiMl
is
momentum, and when
emitted, as in Fig. 1-14. This radiation carries energy
the emission occurs, the lx>x recoils in order that the
SPECIAL RELATIVITY
37
center of mass
initial
—
L
s
7burst of radiation
1
is
m
since
J-
much
is
L away;
lwx, a distance
emitted
s
when
true
is
=
in
<
= EL/Mc 2
vt
The time
smaller than M.
M).
means
this
that
t
During the lime
is
'/
issDciated with the energy
i,
of
in
same place
the
it
(%M -
new
that v
<
which
c,
displaced to the
is
is
end of the
left
In
.
right-hand end
its
L/c (assuming
the box
moves
Ixjx
to reach the opposite
=
t
After the box has stopped, the mass of
mass of
during which the
/
by the radiation
to the time required
,'((iial
was
2
M+m
its
left-hand
owing
end
is '/ :V/
2
— m
to the transfer of the
the radiation.
II
the renter of BOBS!
and the
mass
li
to
m
be
originally,
+
m)(%I.
s)
= /M + m){%L l
(
a)
or
center of mass
m = Ms
•— radiation
is
L
absorbed and box stops
Inserting the value of the displacement
Radiant energy possesses inertial mass.
FIGURE 1-14
total
momentum
at the
opposite end of the box,
transit, the
to
be
comes
compute
of the system
the
shall
l
ends to have the mass /3
ixjx.
a distance
is
to the
end
E
M each.
%L
The
center of mass
from each end.
carries the
A
if
We
the center of mass
is
lie
inassless
and
its
it
is
the principle of conservation
(M
-
Pbox
m)v
=^
starts to
when
move when
the radiation
which
the
the box
same
an amount of mass m.
When
now M — m, recoils with
of momentum,
the
'
A
does
is
t;
=
box
BASIC CONCEPTS
example, the
When
the finite speed of
elastic-
result that in
=
E/c 2
is
obtained.
An
go Ixrfore decaying
airplane
is
Hying
at
if its
speed
is
7
s
when measured at rest. How
when it is created?
How
fast
far
0.99c
300 m/s {672 mi/h).
How much
time must elapse
must a spacecraft travel relative
to the earth for
1 s?
each day on the
d on the earth?
is
" take
35
38
this specification; for
taken into account in a more elaborate calculation, however,
certain particle has a lifetime of 10
it
A
E
emitted and the entire
absorbed. Actually, of course, diere
Problems
*!
„
is
.
a perfectly rigid body:
travels witii the speed of light, will arrive at the right-hand
spacecraft to correspond to 2
E
is
before a clock in the airplane and one on the ground differ by
Pr«tl»Uon
recoil velocity of the
box
according to electromagnetic
c
and so the
is
end of the box before that end begins to move!
in
equal to E/c2
the radiation
no such thing as a rigid body that meets
radiation,
2-
=
box
Uiat the
is
therefore at the center
burst of electromagnetic radiation
momentum E/c
emitted, the box, whose mass
From
was absorbed.
it
consider the sides of the box to
by hypothesis, has associated with
v.
which
that the entire
box comes to a stop
is
we assumed
above derivation
In the
waves
theory, and,
velocity
at
must have transferred mass
of mass that must be transferred
The mass associated with an amount of energy £
of the
in
energy
is
momentum
the center of mass of the system
that has the
the radiation
absorbed
remain unchanged
For simplicity we
of the
If
is
which the radiation was
in
in space, the radiation
was emitted
it
amount
to
is
s.
the radiation
cancels the
During the time
to rest.
same location
from the end at which
shall
its
momentum
box has moved a distance
in the
When
of the system remain constant.
box, which then
still
s.
rocket ship leaves the earth at a speed of 0.98c.
for the
How much
time does
minute hand of a clock in the ship to make a complete revolution
measured by an observer on the earth?
SPECIAL RELATIVITY
39
An
5.
whose height on the earth
astronaut
exactly 6
is
as measured by an observer in the
A meter slick
6.
is
A rocket ship
7.
is
'
99
m
An
8.
of 2.9
'
10
s
x
m
s finds
H
m
fits!
is it
the
going
What
is
sees
it is
moving
spacecraft,
A and
H,
the speed with which he
for the
speed with which he
for the
the lalwratory system)
in the
approaching B?
is
speed with which he
is
—x
direction to
coming toward him from
What
(a)
approaching the
is
0.9r.
A
gains
of
What does a man on II measure
moon? For the speed with which
Dynamite
is
It is
possible for the electron
beam
X
does
this
A man
in Bight, his
is
When
mass
is
101
kg
as
he
is
How
fast
in a rocket ship
Win
determined by an observer on the ground.
in
order that
its
mass equal the
rest
mass
Find the speed of a
0.
'/ »ii(3
energy of 500
15.
1-MeV electron according
to classical
and
A man
energy of 500
16.
mass docs a proton gain when
it
is
it
is
so doing
in
explodes.
What
fraction
.-100
\V"/m 2 of surface
By how much does the mass
mean
(The
1
radius
of the
of the
orbit
earth's
is
m.
=
m<,/
vl—
vr/i", does not equal the kinetic
F = d(mv)/dt,
tit/ tit.
than his twin brother
Ins return
a year.
the distance light travels in
25.
makes a round
trip to (he
How much younger is he
who remained behind? (A light-year is
Light of frequency P
is
It is
emitted by
equal to 9.46
An
a source.
accelerated to a kinetic
accelerated to a kinetic
MeV?
10 15 m.)
X
observer moving
away
By considering the
v'.
source as a clock that ticks p times per second and gives off a pulse of light
show
that
r
+
1
t/
c/c
This constitutes the longitudinal do/ipler effect in light. (If the observer
ti'utiul
™
mass does an electron gain when
The total energy of a
it
leaves the earth in a rocket ship that
relativist ic
MeV?
How much
where
,
and
v, v,
V
How much
z
2
mechanics.
14.
J/kg when
Express the relativistic form of the second law of motion,
terms of m„,
with each tick,
of the proton?
13.
10"
from the source at the speed D measures a frequency of
must an electron move
0°C and
mass?
nearest star, 4 light-years distant, at a speed of 0.9c.
the speed of the rocket ship?
12.
the relative velocity
energy of a particle moving at relativistic speeds.
upon
has a mass of KM) kg on the ground.
X
when
this?
not
contradict special relativity?
1 1
is
speed relative
I0 11 m.)
Prove diat
22.
24.
Why
liberates about 5.4
energy content
density
at a
g/cnr1 when the sample
melts into water at
its initial
decrease in each second?
move
tube to
in a television picture
0°C
is its
perpendicular to the direction of the sun.
sun
in
10.
What
19.3
Solar energy reaches the earth at the rate of about
2).
approaching A?
across the screen at a speed faster than the speed of light.
total
its
23.
he
certain quantity of ice at
kg of mass. What was
1
moving
the $' frame
in
Cold has a density oF
(/»)
at rest relative to the observer.
does a
(b)
would determine.
is
1.5
approaching the moon? Pea
is
to S of c
20.
the speed of the object in the laboratory system?
two
Find the density p' that an observer
19.
in flight, its leogtfe
opposite directions at the respective speeds of 0.8c and 0,9c.
man on A measure
length appears
speed?
its
+ x direction at a speed (in
(he speed of an object
moon
When
is
its
per second?
in miles
long on the ground.
What
10 m/s.
A man on
9,
1(H)
observer moving in the
x
2,998
lie
is
low
I
his height
is
same spacecraft? By an observer on the earth?
an observer on the ground.
to
What
projected into space at so great a speed that
contracted to only 50 cm.
lying parallel to
ft is
the axis of a spacecraft moving at 0.9c relative to the earth.
the
the source at the relative speed o, the
above formula are interchanged.)
"""spending one
for
sound waves
Why
+
and
does
—
is
moving
signs in the radical
this result differ
from the
in air?
2 "particle
is
exactly twice
its
rest
energy. Find
its
spee<
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.
from
How much work must lie done in order
1.2 X 10 s m/s to 2.4 X 10 s m/s?
18.
(a)
17.
40
The
density of a substance
BASIC CONCEPTS
is
p
in
to increase the
speed of an electron
(In the preceding
problem the observer moves parallel to the direction of propagation.) Show that
'" the transverse
the S frame in which
it
is
at rest.
p'
=
doppler effect
c\/l
-
v 2 /c 2
SPECIAL RELATIVITY
41
27.
Twin A makes
while twin
B
How many
analyze
stays behind
own
a year by his
a round Irip at a speed of 0,8c to a star 4 light-years away,
does
reckoning,
B
earth.
Each twin sends the other a
{a) How many
signals does
A
signal
PARTICLE PROPERTIES
once
OF WAVES
send during the trip?
2
send? (h) Use the doppler effect formula of Prob. 25 to
this situation.
many does 8
on the
receive?
How many
Are these
signals does
A
receive during the trip?
results consistent
with those of part
How
(a)?
In
our everyday experience there
concepts of particle and wave.
spread out from
its
A
is
nothing mysterious or ambiguous about the
stone dropped into a lake and the ripples that
point of impact apparently have in
common
only the ability
carry energy and momentum from one place to another.
which mirrors the "physical reality" of our sense impressions, treats particles
and waves as separate components of that reality. The mechanics of particles
Classical physics,
to
and the optics of waves are traditionally independent disciplines, each with
Own chain of experiments and hypotheses.
The physical reality we perceive arises from phenomena
occur
that
microscopic world of atoms and molecules, electrons and nuclei, but
there are neither particles nor
waves
in
to
in the
in this
world
We
regard
our sense of these terms.
electrons as particles because they possess charge and mass
its
and behave according
the laws of particle mechanics in such familiar devices as television picture
tubes.
We
shall see,
however, that there
wave
preting a
moving electron
preting
as a particle manifestation.
it
as a
is
as
much evidence
manifestation as there
is
in favor
of inter-
in favor of inter-
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,
of
modern
the wave-parttcle duality
physics, and in this liook there are
is
central to an understanding
few arguments that do not draw
upon one or the other of these fundamental principles.
2.1
THE PHOTOELECTRIC EFFECT
Late in the nineteenth century a series of experiments revealed that electrons
are emitted
from a metal surface when
(ultraviolet light
phenomenon
42
BASIC CONCEPTS
is
is
required for
known
all
light of sufficiently high
but the alkali metals)
falls
frequency
upon
it.
This
as the photoelectric effect. Figure 2-1 illustrates the type
43
Let us consider violet light falling on a sodium surface,
10~ n
W/m* of electromagnetic energy is absorbed by
beam than
Now
is
apparatus that was employed
in
the
more precise of these experiments.
a good reflector of
sodium
atom
1
that the incident light
''
\\
in'-'
is
is
"
eV/s.
It
absorbed
in
if
FIGURE
2-2
is
to
voltage
call
..
Is
the
same
in~
the
10
among 10"-"
10"*" W, which
distributed
1
eV or
maximum
so of
energy that the photo-
possible time of I0"
upon some kind of resonance process
Photoelectron current
V
i.
1
receives energy at the average rate of
should therefore lake more than 10" s. or almost a year,
any single electron to accumulate the
we
light
thick and
!i
s,
an average
is
proportional to light Intensify lor
all
for all intensities of light of i given frequency
to explain
J0""; eV.
why some
retarding voltages
The
eitinc-
ft
An
circuit like that
frequency = V
= constant
he irradiated as
of the photoelcetrons that emerge from the irradiated surface
energy to reach the cathode despite
they constitute the current that
the retarding potential
V
is
its
negative polarity, and
measured by the ammeter
in the circuit.
As
increased, fewer and fewer electrons get to the
is
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
The
than 10
Even
lion
Shown schematically, with the metal plate whose surface
and the current
we assume
in a layer of
electron, according to electromagnetic theory, will have gained only
evacuated Lube contains two electrodes connected to an external
Some
less
for
Experimental observation of the photoelectric effect.
sufficient
atoms
electrons are found to have! In the
rAVWW
the anode.
-'
an apparatus like
the surface (a more intense
is
Hence each atom
:itoms.
—
have
if
1
uppermost layers of sodium atoms, the 10
o—
ni
required) of course, since sodium
there are about lu
area, so that,
in
evacuated tjuartz tube
2-1
is
electrons
^
FIGURE
this
in
be a detectable photoelectric current when
will
of Fig.
light
^
There
2-1.
that
5
I
ceases.
existence of the photoelectric effect ought not to
l>e
surprising; after all,
waves carry energy, and some of the energy absorbed by the metal may
somehow concentrate on individual electrons and reappear as kinetic energy.
light
When we
effect
look
more
closely at the data, however,
we
find that the photoelectric
can hardly he interpreted so simply.
One
of the features of the photoelectric effect that particularly puzzled its
discoverers
etectmns)
is
ts
that the
energy distribution in the emitted electrons (called phato-
independent of the intensity of the
more photoelcetrons than
electron energy
accuracy (about
is
a
weak one
the same (Fig, 2-2).
10" s),
of the
light,
A
strong light Ixmrn yields
same frequency, but the average
Also, within the limits of experimental
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.
44
BASIC CONCEPTS
there
is
RETARDING POTENTIAL
PARTICLE PROPERTIES OF WAVES
45
electrons acquire
mare energy than
have more than 10"
l
"
others, the fortunate electrons could hardly
of the observed energy.
Equally odd from the point of view of the wave theory
the fact that the
is
photoelectron energy depends upon the frequency of the light employed
At frequencies below a certain
(Fig. 2-3).
aacfe pint ict ilar metal,
critical
frequency characteristic of
no electrons whatever are emitted. Above
this threshold
frequency the photoelectrons have a range of energies from
maximum
value,
and
this
maximum energy
to a certain
increases linearly with increasing
frequency. High frequencies result in high-maximum photoelectron energies, low
low-maximum photoelectron
frequencies in
energies.
Thus a
faint blue light
produces electrons with more energy than those produced by a bright red
although the
Figure 2-4
quency
is
a plot of
It
is
light,
of them.
maximum
v of the incident light in
surface.
number
latter yields a greater
photoelectron energy TmKX versus the frea particular experiment that employed a sodium
clear that the relationship
we can
involves a proportionality, which
between
Tmtx and
the frequency c
express in the Form
=
hv
—
Maximum
photoelectron energy as a function ot the frequency of the incident light lor a
sodium surface.
where
k
2.1
2a
FIGURE
the threshold frequency below which no photocmission occurs
•>„ is
a constant. Significantly, the value of
is
h>
=
h
6.626
X
10- 34
and
li
J-s
\aahwttj$ the same, although v a varies with the particular metal being illuminated.
FIGURE
2-3
potential
is
The
V a
II,
extinction! voltage V'„
depends upon the frequency
the photoelectric current
is
the
same
9 of the light.
When
the retarding
for light of a given Intensity regardless of its
frequency.
THE QUANTUM THEORY OF LIGHT
2.2
light intensity
=
constant
'Hit
1
electromagnetic theory of light accounts so well for such a variety of
phenomena
theory
is
that
it
must contain some measure of
Einstein found that the paradox presented
3
truth.
completely at odds with the photoelectric
theoretical physicist
Max
Planck,
t
problem notorious at the time
to derive a
for
its
five
Planck was seeking to explain the
resistance to solution.
formula for the spectrum of
effect could l>e
years earlier by the
by bodies hot enough
characteristics of the radiation emitted
In JH05 Albert
by the photoelectric
understood only by taking seriously a notion proposed
German
Yet this well-founded
effect.
this radiation (that
to
be luminous,
Planck was able
is,
the relative
brightness of the various colors present) as a function of the temperature of the
"°&y that was in agreement
radiation
V„ (3)
V
^wgy
(2)
V„
RETAROING POTENTIAL
46
BASIC CONCEPTS
is
%vith
are called quanta.
particular frequency v of light
is
directly proportional to
little
bursts of energy.
These bursts of
Planck found that the quanta associated with a
(J)
£
experiment provided he assumed that the
emitted disconttrwouahj as
v.
all
have the same energy and that
That
this
energy
is.
PARTICLE PROPERTIES OF WAVES
47
E =
2.2
Quantum energy
/if
to
be the emission of electrons from such an object. Thermionic emission makes
possible the operation of such devices as television picture lubes, in
where
K
today
m
h
Wa
in
known
X
6.626
utewsUng problem and
L
its
that tlie electromagnetic
assume
to
W
emitted a quantum
at a time,
ii
empirical formula Eq.
2,1
light
may be
while in thermionic emission heat does
not
'"'
"
+
I'm™
ft,
means
Einstein's proposal
is
rewritten
When
2.2 eV.
=
10~ ,0
m)
as follows:
/ii'
u
*»
Tual
ht>„
a
=
=
quantum of the
the energy content of each
maximum
the
minimum energy needed
To convert
I
surface
called
is
its
this
u.oi-tl
1
[unction.
energy
.5.7
It is
easy to sec
all
energies
to
not
Tmax
all
:
electron
work function
+
eV=
hv
X
through the metal surface from
the
just
work
that
in
beneath
it,
and mora work
is
validity of this interpretation of the photoelectric effect
studies of thermionic emission.
required
when I
Long ago
it
became known
is
confirmed by
'''„,«
48
late in the nineteenth century the reason for this
BASIC CONCEPTS
I
that the presents
of a very hot object increases the electrical conductivity of the surrounding
and
always
to escape,
function of potassium
is
the
A =
angstrom
maximum energy in
(
t
1
Eq. 2.3,
electron volts,
M J-s X
10
3
X
we need
only compute the
is
HI-
10"
m/s
X
10 10
A/m
A
|W
J
X
10
5.7
x
X
IP" '"J
1.6
3.6
eV
<
volts,
we
recall that
3
1.6
J
=
10
1B
J
Hence the maximum photoelectron energy
must he done to take an electron
the electron originates deeper in the metal.
The
x
of surface
photoelectrons have the same energy, hut emerg
hi>„ is
is
and
energy
why
up
what
energy from joules to electron
=
with
From
3.500
=
maximum
by an electron
The work
surface,
of 3,500- A light. This
_ 6.63
an electron from the
There must he a mininmm energy required by an electron in order to escape
rum a metal surface, or else elect runs would pour out even in the absence
a
it
'»()
metal surface being ilhiiiiiuatcd
Quantum
e
In photoelectric
A
to dislodge
oi a particular
and
surfaces.
= ?£
photoelectron energy
The energy /w„ characteristic
Hence Eq. 2.3 states that
surfaces,
same
incident light
h»
the
~
already expressed
is
quantum energy /»
hv
on a potassium
falls
rm™ =
be interpreted
that the three terms of Eq. 2.3 are to
many
wavelength 3,500 A
ultraviolet light of
electron volts of the photoelectrons?
Photoelectric effect
'ti
expect
order to escape.
both cases the physical processes
so: in
Let us apply Eq. 2.3 to a specific situation.
be readily explained.
Since
light.
for
for the
in
involved in the emergence of an electron from a metal surface arc the same.
unit
2.3
work function
we should
metal, and
emission, photons of light provide the energy required
propagates continu-
Imt also propagates as individual quanta.
In terms of this hypothesis the photoelectric effect can
The
minimum energy can be determined
close to the photoelectric
energy radiated by a hot
ously through space as electromagnetic waves. Einstein proposed that
is
solution
tin-
must acquire a certain minimum energy
thai the electrons
This
UOtgOg intermittently. Planck did nol doubt that
met;)]
electrons evidently obtain their energy from the
thermal agitation of the particles constituting
9.
While he had
only
The emitted
of electrons.
details of this
which
filaments or specially coated cathodes at high temperature supply dense streams
10 •" J-s
examine BOOM uf the
shall
Chap,
object
as Planck's constant, has the value
air,
phenomenon was found!
The view
=
a
=
""
-
3.6
eV
1.4
eV
is
'»'„
-
2.2
eV
that light propagates as a series of
called photons)
is
directly
little
packets of energy (usually
opposed to the wave theory of
light.
provides the sole means of explaining a host of optical effects
The
latter,
— notably
diffi
which
action
PARTICLE PROPERTIES OF WAVES
49
—
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
and interference
is
1905 that Ught travels through space in the form of distinct photons, however,
elicited incredulity from his contemporaries. According to the wave theory, light
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 distr ibuted continWHBry tkrou^jhoat she- wave pattern. Accowbng to
waves spread out from a source
scries of localized concentrations of energy,
Of absorption
treats
j»,
it
each
phenomenon,
be capable
sufficiently small to
by a single electron. Curiously, the quantum theory of
strictly as a particle
light,
which
explicitly involves the light frequency
a wave concept.
strictly
The quantum theory of
electric effect.
It
is
strikingly successful in explaining the photo-
predicts correctly that the
wave theory
contrary to what the
the feeblest light can lead to the
wave
contrary to the
maximum
photoclectron energy
and
suggests,
it
its
part or
As
it
all
and
the inverse process also possible? That
Is
is,
can
moving electron be converted into a photon?
of the kinetic energy of a
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
Einstein.
In
1
895 Wilhelm Roentgen made the
unknown nature
trating radiation of
on matter. These
electric
A'
classic observation that a highly
produced when
is
were soon fomid
rays
and magnetic
fields, to
pene-
impinge
fast electrons
to travel in straight lines, even through
pass readily through opaque materials, to cause
able to explain
is
why even
immediate emission of photoeleetrons, again
The wave theory can give no reason why there
frequency such that, when light of tower frequency is
employed, no photoeleetrons are observed no matter
how
strong the light beam,
more penetrating
the original electrons, the
the
intensity,
theory.
should be a threshold
photoelectric effect provides convincing evidence that photons of light can
transfer energy to electrons.
phosphorescent substances to glow, and to expose photographic plates. The faster
light
should depend upon the frequency of the incident light and not upon
number of
Not long
the resulting
after this discovery
magnetic wav&s. After
all,
it
began
to lie suspected that
suddenly brought to
The absence
is
rest
given the
beam.
X rays are electro-
and a rapidly moving electron
certainly accelerated. Radiation
is
and the greater
electromagnetic theory predicts that an accelerated
electric charge will radiate electromagnetic waves,
circumstances
X rays,
electrons, the greater the intensity of die X-ray
produced under these
German name bremsntrahhing ("braking
radiation").
of any perceptible X-ray refraction in the early work could be
Something that follows naturally from the quantum theory.
Which theory are we to believe? A great many physical hypotheses have had
attributed to very short wavelengdis, below those in the ultraviolet range, since
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
line
to
for a single physical
from what
it is,
phenomenon. The
situation here
say, in the ease of relativistic versus
the tatter turns out to
lie
is
fundamentally different
Newtonian mechanics, where
an approximation of the former. There
is
no
way
of
wave dicory of light or vice versa.
wave
or a particle nature, never both
a
deriving the quantum theory of light from the
In a specific event light exhibits either
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
and the quantum theory of light
account for the observed manwaves
complement each other. Electromagnetic
for the observed manner
photons
account
ner in which light propagates, while
independently.
The wave theory
of light
transferred between light and matter.
in
which energy
to
regarding light as something that manifests
on occasion and
is
is
as a
wave
no longer something
and
we must
as the closest
50
The
theory, on the other hand, light spreads out from a source as a
quantum
the
in the
X RAYS
2.3
itself as
train the rest of the time.
that can
be visualized
in
We have iio alternative
a stream of discrete photons
The
"true nature'' of light
terms of everyday experience,
accept both wave and quantum theories, contradictions and
we can
BASIC CONCEPTS
get to a complete description of light.
all,
the refractive index of a substance decreases to unity (corresponding to straight-
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
in Fig. 2-5.
X
We
analyze
shall
this classic
rays are electromagnetic waves.
heading
in the
—z
are scattered by the carbon; this
set in vibration
by the
of unpolarized
X
rays
X
rays
the xy plane only.
electric vectors of the
xy plane.
A
The
scattered
electric vector in the
beam
The
of
X
and not
X rays and then
wave
is
carbon atoms are
reradiate.
perpendicular to
its
Because
direction
rays contains electric vectors that
X
ray that proceeds in the
y direction only, and so
carbon block
lie in
it is
is
+x
direction can have an
plane-polarized.
placed
in the
To demon-
path of die ray,
electrons in diis block are restricted to vibrate in the
direction and therefore reradiate
sively,
X
in the
target electrons dierefore are induced to vibrate in die
strate this polarization, another
as at the right.
beam
a
left
means that electrons
the electric vector in an electromagnetic
of propagation, the initial
sketched
on a small block of carbon. These
At the
direction impinges
is
experiment under the assumption that
X
t/
rays that propagate in the xz plane exclu-
at all in the y direction.
rays outside the xz plane confirms die
The observed absence of
wave character of X rays.
scattered
PARTICLE PROPERTIES OF WAVES
51
zero
intensity
*
evacuated -v
maximum
first
polarized
scattcrer
scattered ray
\
tube
intensity
second
r^
scattcrer
uupolarizcd
larKt-l
Xray
FIGURE 2
FIGURE 2-5
In
Barilla's
An Xray tube
1912 a method was devised for measuring the wavelengths of X rays. A
had been recognized as ideal, but. as we recall from
As was said
earlier, classical
electromagnetic theory predicts the production
when electrons are accelerated, thereby apparently accounting
emitted when fast electrons are stopped by the target of an X-ray
diffraction experiment
of bremsstrahlung
physical optics, the spacing between adjacent lines on a diffraction
grating must
be of the same order of magnitude as the wavelength of the
for the
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 A. He therefore
proposed
lhat crystals be used to diffract X rays, with their
regular lattices acting as a
data
barded by electrons at several different accelerating
kind of three-dimensional 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 were found, 0.13
to 0.48 A, 10-1 of those in visible light and hence
having quanta 10' times
wavelengths, indicating the enhanced production of
light tor satisfactory
results,
tiilw.
m
as
energetic.
We
shall consider
X-ray diffraction in Sec.
2.4.
For purposes of classification, electromagnetic radiations with wavelengths
fa approximate interval from 1(1 " to 10 * m (0.1 to 1«H> A) are today
X
is
exhibit
a diagram of an X-ray
lul>e.
filament through which an electric current
is
A cathode, healed by an adjacent
passed, supplies electrons copiously
by thermionic emission.
fa
The
X
is
The high potential difference V maintained between
cathode and a metallic target accelerates the electrons toward
the latter.
face of the target
rays that
is
at
emerge from the
an angle relative to the electron
evacuated to permit the electrons
BASIC CONCEPTS
lieain,
target pass through the side of the tulie.
to get to the target
unimpeded
two
when
tungsten and
molybdenum
targets are
The
potentials.
bom-
curves
distinctive features not explainable in terms of electromagnetic
theory:
1.
In the case of
at various specific
molybdenum
there are pronounced intensity peaks at certain
X
rays.
These peaks occur
wavelengths for each target material and originate
in re-
arrangements of the electron structures of the target atoms after having been
is
The important
thing to note at this point
the production of X rays of specific wavelengths, a decidedly nonclassical effect,
in addition to the
rays.
is
not satisfactory in certain important respects. Figures 2-7 and 2-8 show
the X-ray spectra that result
2.
Figure 2-6
rays
disturbed by the bombarding electrons.
in
con-
sidered as
X
However, the agreement between the classical theory and the experimental
I
52
6
Biperfnwnt to demonstrate Xray polarization.
The X
production of a continuous X-ray spectrum.
rays produced at a given accelerating potential
V vary
bul none has a wavelength shorter than a certain value A llUn
decreases X m)[1
molybdenum
.
At a particular
targets.
V,
A mln
is
the
same
Duane and Hunt have found
tional to V; their precise relationship
,
in
wavelength,
Increasing
V
for both the tungsten and
that
A m[n
is
inversely propor-
is
and the
The tube
2.4
\
rt
ni!n
_
—
1.24
X
10- 6
V-m
X-ray production
PARTICLE PROPERTIES OF WAVES
53
The second observation
is
quantum theory
readily understood in terms of the
Most of the electrons incident upon the target lose their kinetic
energy gradually in niurreroiLS collisions, their energy going simply into heat.
of radiation.
(This
is
the reason that the targets in X-ray tubes are normally of high-melting-
point metals, and an efficient
A few
electrons, though, lose
target atoms; this
FIGURE
2-7
X-ray spectra of tungsten at various accelerating potentials
means of cooling the
most or
the energy that
is
is
all
evolved as
except for the peaks mentioned in observation
A
photon energy
0,6
0.8
hr.
It is
X
1
rays.
X-ray production, then,
above, represents an inverse
''"max
is
being transformed into photon energy.
therefore logical to interpret the short wavelength limit
of Eq. 2.4 as corresponding to a
2.S
1.0
often employed.)
means a high frequency, and a high frequency means a high
short wavelength
\ min
O.i)
is
photoelectric effect. Instead of photon energy being transformed into electron
kinetic energy, electron kinetic energy
0.2
target
of their energy in single collisions with
=
maximum photon
energy hv mux , where
he
WAVELENGTH. A
Since work functions are only several electron volts while the accelerating
potentials in X-ray tubes are tens or hundreds of thousands of volts,
assume that the kinetic energy
7*
of the bombarding electrons
we may
is
7"=eV
2.6
When
the entire kinetic energy of an electron
is lost
to create a single photon,
then
hv„,„
2.7
= T
Substituting Eqs. 2.5 and 2.6 into 2.7,
Kmix
iw
FIGURE 2-e X-ray spectra of tungsten and molybdenum at 35 kV accel-
rt
erating potential.
X
=
= eV
mln
—
he
X
IP
1.6
=
is
see that
T
6.63
which
we
just the
1.24
X
V
-34
J-s
X
lO- 8
I0
X 3 X H)a m/s
C X V
,9
„
V-m
experimental relationship of Eq. 2.4.
It
is
therefore correct
to regard X-ray production as the inverse of the photoelectric effect.
0.2
0.4
0.6
WAVELENGTH. 8
54
BASIC CONCEPTS
0.8
1.0
A
conventional X-ray 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
scattered
waves
A
This
—
,,,,,,
X
5 X
1.24
=
2.5x
=
0.25
10- 6
V-m
V
10*
"m
10
incident waves
unscattcred waves
A
wavelength corresponds to the frequency
=>
^>
p~„ =
A n.ln
3 X 10" m/s
~ 2.5
X 10
_
»m
=
1.2
X
10
1B
Hz
FIGURE
2.4
of electromagnetic radiation by >
group ol atoms.
Incident plane
waves are
as spherical waves.
X-RAY DIFFRACTION
now
Lei us
another while
return to the question of
of electromagnetic waves.
of
The scattering
2-9
re emitted
A
how X
rays
may
he demonstrated to consist
crystal consists of a regular array of atoms,
each
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
the alternating electric field of an electromagnetic
polarization changes back and forth with the same
ol
wave
of frequency
frequency
electric dipole
is
dius created at the expense of
incoming wave, whose amplitude
in turn radiates electromagnetic
proceed
in all directions
exposed
to
is
some
accordingly decreased.
waves of frequency
»,
v.
An
e,
may
lie
in others
they will destructively interfere.
The atoms
thought of a* defining families of parallel planes, as
each family having a characteristic separation between
This analysis was suggested
in
1913 by W. L. Bragg,
in
its
scattered
FIGURE
2-
by
10
crystal
Two
atoms
sett ol
to
component
honor of
planes are called Bragg planes. Tin- conditions that must he
in a crystal
in Fig. 2-10,
whom
with
planes.
the above
fulfilled for radiation
undergo constructive interference may be obtained
Bragg planes
In a
NaCI
crystal.
the
oscillating
of the energy of the
The oscillating (Spate
and these secondary waves
except along the dipole axis. In an assembly of atoms
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. 2-9). The scattering process, then, involves
an atom
absorbing incident plane waves and reemitting
spherical waves of the same
frequency.
A monochromatic team
in
56
of
X
rays that falls upon a crystal will !>e scattered
owing to the regular arrangement of the atoms,
certain directions die scattered waves will
constructively interfere with one
in all directions
within
BASIC CONCEPTS
it,
but,
PARTICLE PROPERTIES OF WAVES
57
from a diagram
A
is
incident
spacing
next,
is
crystal at
The beam goes
rf.
A beam
like that in Fig. 2-11.
upon a
and each of them
an angle
past
atom
and whose paths
parallel
differ
and
B
for
which
condition upon
I
this
and
is
is
2dsin0
=
nA
n
=
1, 2, 3,
since ray II must travel the distance
n
is
in
random
in the
Con-
directions.
A, 2\, 3A,
and so on. That
is,
the
2d
I
and
II
in Fig. 2-11.
The
first
scattering angle be equal to the
The second condition
is
that
.
sin 6 farther than ray
1.
The
integer
the order of the scattered l>eam.
The schematic
is
detector
an integer. The only rays scattered by
common
that their
rays of wavelength
between those scattered rays diat are
angle of incidence & of the original beam.
2.8
beam
true are those labeled
II is
X
plane and atom 8
in the first
by exactly
path difference mast be nA, where n
A
A
scatters part of the
structive interference takes place only
containing
with a family of Bragg planes whose
shown
A
in Fig. 2-12.
angle &, and a detector
angle
is
also 0.
condition. As
Any X
is
we
can
collimated
beam
placed so that
in
of
X
is
by Eq.
2,8.
upon a
crystal at an
first
FIGURE
If the
spacing d between adjacent Bragg
determine the value of d? This
in Fig. 2-10,
in
cubic lattices similar to that of rock
As an example,
of adjacent atoms in a crystal of NaCl.
let
us
,V„
= 6.02 X
ber),
a simple task in the case of
is
X ray spectrometer.
2 12
Bragg
known, the X-ray wavelength A may he calculated.
whose atoms are arranged
(NaCl), illustrated
rays falls
records those rays whose scattering
varied, the detector will record intensity peaks corresponding
planes in the crystal
How
is
rays reaching the detector therefore obey the
to the orders predicted
crystals
collimators
design of an X-ray spectrometer based upon Bragg's analysis
of NaCl
is
atoms
—
is
salt
58.5
=
9.72
I
m
of
NaCl
„
=
2d
sin
8
If
d
is
X
6,02
two atoms
X
10 2(i molecules/kmol
10" 2a kg/molecule
in each
X
10 3
kg/m 3 and
,
so.
taking into account
NaCl "molecule," the number of atoms
in
is
=
=
path difference
kmol
NaCl has a density of 2.16
the presence of
3
kg
mKtC] -
58.5,
Crystalline
X-ray scattering front cubic crystal.
—
given by
which means there are 58.5 kg of NaCl per kilomole (kmol). Since there are
FIGURE 2-11
kmol of any substance (N is Avogadro's numNaCl "molecule" that is, of each Na + CI pair of
the mass of each
compute the separation
The molecular weight
10 2a molecules in a
atoms
2
.-- molecule
4.45
x
„.„,„,
10 3 -%X
X2.I6X
3
kg
m
1
9.72
X
10-- 6 kg/ molecule
10™ atoms/m 3
the distance between adjacent atoms in a crystal, there are
along any of the crystal axes and d~
d
3
=
d
=
=
=
n' 3
3
d
l
atoms/m
atoms/m;' in the entire crystal.
Hence
n
and
58
BASIC CONCEPTS
= {4.45 x lO 28)" 3 m
2.82 X l(r 10 m
2.82
A
PARTICLE PROPERTIES OF WAVES
59
2.5
THE COMPTON EFFECT
The quantum theory
for the
its
rest mass.
If this is tnie,
between photons and,
as billiard-ball collisions are treated in
Figure 2-13 shows
how such
then
it
should
lie
say, electrons in the
possible for
same manner
a collision might
to
be
!>e
represented, with an X-ray
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
are involved.
energy
the
If
T gained by
initial
the electron, though actually separate photons
photon has the frequency
scattered photon has the lower frequency v\
Loss
in
photon energy
29
hv
From
is
-
the previous chapter
hv'
we
=
2.10
c
elementary mechanics.
photon striking an electron {assumed
as the kinetic
momentum
its
of light postulates that photons behave like particles except
absence of any
to treat collisions
for a photon,
» associated
with
where
it,
the
Momentum,
unlike energy,
is
a vector quantity, incorporating direction as well
+
momentum must
a collision, of course,
photon and one perpendicular to
it
the scattered photon (Fig. 2-13),
The
momentum
scattered photon
and
hv'/c,
is
Initial
recall that
momentum =
—+
2.11
2 2
p c
and perpendicular to
its
total
energy
is
=
initial
and the
photon
initial
momentum
and
final
momentum
final
—
+
cos
n costf
momentum =
(!
momentum
final
=
sin
$
—
p
sin
(i
hv
is
between the directions of the
that
between the directions of the
andff
is
From
Ekjs. 2.9, 2.11,
that
and 2.12 we
shall
initial
now
initial
and scattered photons,
photon and the
recoil electron.
obtain a formula relating the wave-
length difference between initial and scattered photons with the angle
their directions,
The Compton .Heel.
^r^
incident photon
p
=
»*oc
i,
=
The
first
step
pc
hv'/c
is
sin
BASIC CONCEPTS
=
=
hv
—
hv' cos
hv' sin
is
eliminated, leaving
2.13
pV = {hv?
1
hetween
and 2.12 by c and rewrite them
-
$
(hv')
equate the two expressions for the
total
-f
as
$
+
K = T
scattered
electron
to multiply Eqs. 2.1
By squaring each of these equations and adding the new ones
we
4>
Iwth of which are readily measurable quantities.
pc cos
Next
60
momenta
photon direction
this direction
2.12
The angle §
FIGURE Z-13
hv/c, the
is
electron
c
pc
Since
E—
here are that of the original
plane containing the electron and
gain in electron energy
=T
Initial
=
we choose
in the
In the original
p.
be conserved in each of two
than two Inxlics participate in
be conserved in each of three mutually
perpendicular directions.) The directions
are respectively
so that, since the photon has no rest mass,
£
{When more
mutually perpendicular directions.
c
£= Vm„V
momentum must
as magnitude, and in the collision
2(hv)(lu>')cos<!>
together, the angle
2
energy of a particle
m^c*
E= Vm^V + p V
PARTICLE PROPERTIES OF WAVES
61
from the previous chapter
+m
{T
c 2) 2
p
2
c2
FIGURE 2-14 Experimenlal demonstration
of tho Compton effect-
give
to
V
= wi
+ p 2c 2
= T2 + Sm^T
X-ray spectrometer''
Since
T=
-
hv
fw'
we have
2-W
p
V = faf _
+
2{lw)(h»')
{hv'f
+ 2m
c 2 (/i*
-
hr')
unscattered
Substituting this value of
2ma c-(hv -
2.1S
This relationship
frequency.
is
™
c/
=
2(h>>)(hr'){l
1/A and v'/c
-
(1
we
-
cos
Xray
obtain
finally
expressed
terms of wavelength rather than
in
(1
-
X
ravs
path
patn oi
of -%
spectrometer
=
1/A',
and the wavelengths of the scattered
at a target,
angles
cos $)
0.
by Eq.
The
results,
shown
2.16, but at each angle the scattered
proportion having the
Com p ton
Equation 2.16 was derived by Arthur H. Cotnpton
describes,
It
which he was the
first
the early 1920s,
in
to observe,
is
effect
and the
known
constitutes very strong evidence in support of the
as the
quantum
initial
Equation 2.16 gives the change in wavelength expected for a photon that
4>
by a particle of
of the wavelength A of the incident photon.
Cwnpton wavelength
10 _lz m).
(2.4
X
that
can occur
lie
of
twice die
of the scattering particle,
From Eq.
2.16
is
0.024 A, while
to their larger rest masses, the
in
62
X
is
<f>
Compton wavelength
an electron
effect
we
will take place for
it is
rest
The
mass
m
quantity
which
for
it
;
h/m
is
c
is
is
predicted
not hard to understand.
that the scattering particle
many
shift
is
able to
move
In
freely,
of the electroas in matter are only loosely
lx»und to their parent atoms. Other electrons, however, are very tightly Iwund
and,
when
electron.
struck
In this
the resulting
is
by
a photon, the entire
atom
recoils instead of the single
to use in Eq. 2,16
event the value of »»
is
that of the entire
tens of thousands of times greater than that of
Compton
shift is
an electron, and
accordingly so minute as to be undetec table.
is
an electron
called the
is
0.024
A
2.6
PAIR PRODUCTION
note that the greatest wavelength change
=
180°,
h/niff.
when
the wavelength change will
Because the
Compton wavelength
considerably less for other particles owing
maximum wavelength change
wavelength
for visible light
in die
Compton
X
The experimental demonstration
As in Fig. 2-14, a beam of X rays
of the
rays of A
of a
is less
than 0.0 1 percent
= 1 A it is several percent.
Compton effect is straightforward.
single, known wavelength is directed
of the initial wavelength, while for
BASIC CONCEPTS
wavelength
rays also include a substantial
independent
0.048 A. Changes of this magnitude or less are readily observable only
rays, since the shift in
X
wavelength. This
we assumerl
deriving Eq. 2.16,
atom, which
theory of radiation.
scattered through the angle
/
,/
X rays are determined at various
in Fig. 2-15. exhibit the
a reasonable assumption since
phenomenon it
Compton effect.
\
cos*
-\=-2-(\ -COS*)
\'
/
,
AV
X'l
i
collimators
source of
<j>)
/
monochrnmntii
when
=
1\ _
1
U
h
2.16
=
simpler
)
since v/c
so,
hv')
in Eq. 2.13,
Dividing Eq. 2,15 by 2h 2 c 2
-r-(
and
2 2
p c
As
It
we have
is
seen, a
photon can give up
also possible for a
photon
(positive electron), a process in
all
or part of
its
energy hv to an electron.
to materialize into an electron
which electromagnetic energy
rest energy. No conservation principles are violated when an
pair is created near an atomic nucleus (Fig. 2-16). The sum
the electron (o
=
-e) and of the positron
(<7
=
+e)
is
is
and a positron
converted into
electron-posit run
of the charges of
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
help of the nucleus, which carries away enough photon
conserved with the
momentum
for the
PARTICLE PROPERTIES OF WAVES
63
<t>
= 0°
/
—jf electron
photon
WAVELENGTH
positron
X
FIGURE 216
Pair production.
process to occur but, because of
conserved
relatively
enormous mass, only a negligible
photon energy. (Energy and linear
fraction of the
!>e
its
if
pair production
were
to occur in
momentum
empty space,
could not
so
Ixith
does not
it
occur there.)
WAVELENGTH
The
energy «j u c 2 of an electron or positron
rest
production requires a photon energy of
at least
1
.02
is
0.51
MeV, and so
pair
MeV. Any additional photon
energy hecomes kinetic energy of the electron and positron. The corresponding
= 90"
maximum photon wavelength is 0.012 A.
wavelengths are called gamma rays, and
Electromagnetic waves with such
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
together and are annihilated to create a pair of photons.
photons are such as to conserve both energy and linear
WAVELENGTH
nucleus or other particle
Three processes
gamma
-AX-
*
At low photon energies
rays in matter.
(several cV)
Compton
scattering
is
X
and
the sole
definite thresholds for both the photoelectric effect
and electron pair production
(
1
.02
MeV). Both Compton scattering
and the photoelectric effect decrease in importance with increasing energy, as
shown in Fig. 2-17 for the case of a lead absorber, while the likelihood of pair
production increases. At high photon energies the dominant mechanism of
z
UJ
>
P
5
energy
UJ
a:
t
/
2-
15
Compton
loss
lead has
WAVELENGTH
FIGURE
135°
required for annihilation to take place.
in all are therefore responsible for the absorption of
mechanism, since there are
=
I
LU
is
come
The directions of the
momentum, and so no
its
is
pair production.
minimum
absorption coefficient
decrease
at
/i,
The curve
representing the total absorption in
about 2 MeV. The ordinate of the graph
which
in radiation intensity
is
is
the linear
equal to the ratio between the fractional
—dl/I and
the absorber thickness
tlx.
That
is,
scattering.
d±
=
—fidx
i
64
BASIC CONCEPTS
PARTICLE PROPERTIES OF WAVES
65
Does a photon possess gravitational mass
as well? Since the inertial
and gravita-
experimentally to be equal (this
tional masses of all material bodies are found
points
of Einstein's general theory
starting
of
the
is
one
equivalence
principle of
of relativity),
it
would seem worth looking
have the same
Let us consider a photon of frequency
of
mass
star's
M
and radius
surface
The
potential energy of the photon
its
total
,
_
4
e
10
PHOTON ENERGY, MeV
FIGURE 2-17
lion,
Linear absorption coefficients for photons
not to the likelihood of Interactions
in the
in lead.
These curves refer
to
energy absorp-
/
The
=
is
accordingly
c*R
V and
the
quantum energy
hv,
is
= hv- GMhf
the photon is beyond
At a large distance from the star, for instance at the earth,
same. The photon's
the
remains
energy
total
its
the star's gravitational field but
energy
now
is
entirely electromagnetic,
and
medium.
£
2.20
whose solution
emitted from the surface of a star
m on the
-4 -SB)
2.19
pl^ojoclectric effect
v
potential energy of a mass
GMhp
energy E, the sum of
E
2>^^Z.
R
The
(Fig. 2-18).
- CMm
V= and
whether photons
is
v=
2.18
into the question of
gravitational behavior as other particles.
=
hv
1
is
v-
FIGURE 2-18
from the
The frequency ot a photon emitted from the surf»ee
of a iter decreases as
It
moves away
star.
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
66
he
BASIC CONCEPTS
Photon "mass"
PARTICLE PROPERTIES OF WAVES
67
where
,' Is
photon
m
stars
the frequency of the arriving photon.
(The potential energy of the
the earth's gravitational field is negligible
compared
with that
— _=~ CMR = "41 X
A,>
c
Hence
field.)
10
-177T
2
in the
m
1.2
kg
9
X lO^kf
X 10 B m
Zz 10"^
»-»(i-£)
I
lere the gravitational ret! shift
which
is
dwarf
Sirius
~ 5.9 x
2.21
in the
A
would 1« :=0.5
wavelength 5,000 A
for light of
measurable under favorable circumstances. In the case of the white
B
"companion of
(the
M/R
predicted red
Sirius"), the
10~ 5 and the observed shift
X
6.C
is
ratio for Sirius B, these figures
!0~ 5
in
;
would seem
shift
is
Ac/k
view of the uncertainty
to confirm the attribution
of gravitational mass to photons.
I
he photon has a lower frequency
as
>s
.t
leaves the field of the star.
at the earth,
A photon
in
thus shifted toward the red end, and
this
corresponding to
its loss
in
phenomenon
is
accordingly
as the gravitational red shift. It
observed
energy
the visible region of the spectrum
known
must 1* distinguished from the doppler'red
afaifl
the spectra of distant galaxies due
to their apparent recession from
in
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
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
Eq2
G _
c
1
6.67
~
X 10" N-mVkg2
=
(3xl0«m/f
If
7Al
X
"
10 28
m/k S
Mm
C
'
=
S
r
Q
^
about 0.01
2.12
X
^
Ar^"'
for green
x
10-2"
HL x
kg
not actually
a
black hole in
about
is
1,
we
see from Eq. 2.10 that no photon
and would be invisible— a "Made
and
itself:
the mass
and
10 2a in
10 53 kg
why
black holes should
should be possible to detect them by virtue of the
it
and
ability
their gravitational effects
known
universe
and radius of the universe are believed
respectively
and,
since
on
may be
to
be
G/c* Si 10 -i" m/kg,
(CM/ C *fl) unlver8e =l.
Recently a gravitational frequency
v
is
has been detected in a laboratory
shift
frequency of
in
gamma rays after they had
A body of mass m that
a height
It
undetectable due to
gains
mgh
the change in mass
ht>'
For h
=
may be
hv
+
If
a
falling
photon of original frequency
=
=
its final
energy hv'
shift is
is
so small that
given by
hi"'h
1—
hv H
20 m,
A*
gn
=
shift
ingh
neglected),
-4+£)
_
_
~~1?~
A
of energy.
taken to have the ccaslaut mass hv/c- (the frequency
*» ^ solar radiation amounts to onlv
wavelength 5,000 A and
stages of their evolution called
while dwarfs that are composed of
atoms whose electron structures have
"collapsed, and such stars have
quite enonnous densities-lvpicallv ~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
BASIC CONCEPTS
CM/c-R >
star could not radiate
astronomical objects in their vicinity. Curiously, the
x 1U k g
em x m 8 m
the doppler broadening of the
spectral lines.
Ho«,v,.,, lhore is a class of stars
in the final
68
exist,
2.22
S raviEationaI red
Such a
combination of their light-alworhin;4
'•-' J
10-°
light of
it.
"fallen" through a height h near the earth's surface.
m
7.41
a star for which
hole" in space. There seems to be no fundamental reason
falls
'
is
experiment by measuring the change
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"
and
LSQ X Hr 3n kg, and
— = t~R =
there
can ever leave
2.2
x
9.8 in/s 2
(3
X
X
10 8
20
m
m/sf
10- I5
of Uiis magnitude
is just
and the
detectable,-
results
confirm Eq. 2.22.
PARTICLE PROPERTIES OF WAVES
69
Problems
What
X
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
2.
The
1.1
x
eV
to
be ejected?
threshold
frequency
energy of
1.5
10 15 Hz. Find the
electron volts)
when
A
13.
for
emission
photoelectric
frequency
1
.5
copper
in
is
of the photoelectrons (in jotiles and
x
10 15
Hz
is
a
What
2.3 eV.
is
What
maximum
kinetic energy of the photoelectrons
maximum wavelength
from sodium? What will
be
if
A
2,000
light falls
on
a sodium surface?
Find the wavelength and frequency of a 100-MeV photon.
4.
energy must a photon have
frequency
the
is
Prove that
16.
Find the energy of a 7,000-A photon.
6.
Under favorable circumstances the human eye can detect 10" IH
How many
of
it
How many
momentum
only
when photons
17.
A beam
of
strike
X
rays
X
bound
is
monochromatic with a wavelength
is
of area perpendicular to the direction of the light, (a) Find the
(in lb/in.
2
)
this light
can exert on the earth
s surface.
How many
consists exclusively of 6,000-A photons.
that part of the earth directly facing the sun in
radius of the earth's orbit
sun in watts, and
photons per
m3
a
is
how many
1.5
X
10"
in.
(Assume
(6)
1
,400
maximum
Assume
W/m 2
that sunlight
photons per m- arrive at
(c) The average
power output of the
each second?
What
is
the
photons per second does
What is the wavelength of the X
target? What is their frequency?
11.
it
emit? (d)
How many
X
An X-ray machine produces 0.1-A X
initial
70
distance
BASIC CONCEPTS
beam
is
the
X H) in Hz emerges from
19
1.2 X 10
Hz. How much
frequency was 1.5
to the electron?
and
rays.
of
initial
scattered through 90°. Find
is
x
frequency 3
its
new
0'''
1
1
\v cull ides
with an electron
frequency.
20.
Find the energy of an X-ray photon which can impart a
of
keV
.50
when 100-keV
What
21.
A monochromatic
electrons strike
accelerating voltage does
in calcite
is
3
X
energy
X-ray
beam whose wavelength
10
8
cm.
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 X-ray
23.
1
A,
As discussed
in
Chap.
1
2,
certain atomic nuclei emit photons in undergoing
from "excited" energy
photons constitute
gamma
the opposite direction, (a)
between adjacent atomic planes
maximum
to an electron.
rays.
states to their
When
The ||Co
"ground" or normal
a 5gFe
atom
is
9.5
X
I0" 26 kg.
states.
a nucleus emits a photon,
nucleus decays by
then emits a photon in losing 14.4 keV to reach
The
What
rays in the direct Ijcam?
An X-ray photon
wavelength of
rays emitted
employ?
12.
At 45° from the
rays have a wavelength of 0.022 A.
was imparted
kinetic energy
transitions
it
energy and
its
pressure
are there near the earth?
10.
all
of 6,(KX) A.)
Light from the sun arrives at the earth at the rate of about
9.
is
electrons.
scattered by free electrons.
An X-ray photon whose
18.
19.
the light
whose momentum
emit?
How many photons per second are emitted by a 10-W yellow lamp?
8.
of
to a free electron, so that the photoelectric effect can take place
wavelength of the
J of electro6,000-A photons does this represent?
radio transmitter operates at a frequency of 880 kHz.
photons per second does
momentum
have the
photon
X-ray
an
a collision with an electron with a frequency of
A 1,000-W
7.
lo
if it is
impossible for a photon to give up
is
it
direction the scattered
5.
magnetic energy.
74.55.
the
is
of light that will cause photoelectrons to be emitted
the
The
.
X lO-^kg-m/s?
1.1
function of sodium
kg/m 3
Find the distance between adjacent atoms.
10-MeV proton?
15.
The work
10 3
X
potassium chloride crystal has a density of 1.98
is
0.3-
be detected?
rays can
directed on a topper
surface.
3.
X
which scattered
How much
14.
between these planes and an incident beam of
the smallest angle
molecular weight of KC1
maximum energy
light of
is
rays at
By how much
is
its
K capture
ground
it
to jSgFe,
state.
These
recoils in
which
The mass
of
the photon energy reduced from
PARTICLE PROPERTIES OF WAVES
71
the
full 14.4
atom?
(ft)
keV available
as a result of
In certain crystals the
having
atoms are so
to share
energy with the recoiling
WAVE PROPERTIES OF
tightly Itound that the entire crystal
PARTICLES
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
recoils
energy reduced
'(c)
(ft)
in this situation if the excited -iJ-Fe
The essentially
means (hat it is
recoil-free emission of
nucleus
gamma
is
3
part of a I-g crystal?
rays in situations like that of
possible to construct a source of essentially monoencrgetic
and hence monochromatic photons. Such a source was used
in the experiment
paragraph of Sec. 2.7. What is the original frequency and
frequency of a 14.4-keV gamma-ray photon after it has fallen
20
described in the
the change in
last
m
near the earth's surface?
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.
24.
In retrospect
it
may seem odd
that
two decades passed between the discovery
1905 of the particle properties of waves and the speculation
in
particles
might exhibit wave behavior.
It
is
one
thing,
in
1924 that
however, to suggest a
revolutionary hypothesis to explain Otherwise mysterious data and quite another
advance an equally revolutionary hypothesis
to
experimental mandate.
The
latter is just
in
the absence of a strong
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
tion,
whereas the
hardly
any
earlier
stir
quantum theory of
despite
de Broglie waves was
its
striking
light of
empirical
demonstrated by 1927,
at the turn
and respectful atten-
of the century that de Broglie's notion received immediate
Planck and Einstein created
support.
The
existence
of
and the duality principle they
represent provided the starting point for Schrodinger's successful development
of
quantum mechanics
3.1
in the
previous year.
OE BROGLIE WAVES
A photon of
light of
frequency v has the
which can be expressed
P=
since A v
=
c.
in
momentum
terms of wavelength A as
\
The wavelength
of a photon
is
therefore specified by
its
momentum
according to the relation
3.1
72
A
=A
BASIC CONCEPTS
73
Drawing upon an
intuitive expectation that nature
is symmetric, de Brogiie
a completely general formula that applies to material
particles as well as to photons. The momentum
of a particle of mass
asserted that Eq. 3.1
m
velocity v
function
+
there at
(.
and
is
=
mis
and consequently
A
3.2
=
its
Born
is
itself,
De
tin:
The greater
in
is
the particle's
momentum,
Brogiie
waves
as
the shorter
its
wavelength.
In Eq. 3.2
is
we
speak of the wave function
shall
When an
and
experiment
made by Max
first
*
that describes a particle
does not mean that the particle
is
performed to detect electrons,
either foimd at a certain time
is
and place or
time
t
|4M
not; there
it is
time and
at that
2
.
wave function +,
all
the actual density of bodies at x, y, z
2
proportional to the corresponding value of |^|
is
thus
an experiment involves a great many identical bodies
if
described by the same
at the
by
itself is
for instance,
entirely possible
it is
20 percent chance that the electron be found
.
While the wavelength of the de Brogiie waves associated with a moving body
is
given by the simple formula
me
determining their amplitude
+
We
a formidable problem.
WAVE FUNCTION
this
this likelihood that is specified
it is
Alternatively,
3.2
presence.
its
a definite chance,
is
was
no such thing as 20 percent of an electron. However,
place,
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 appropriate 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?
there. This interpretation
it
being spread out in space,
for there to lie a
VI - oVc*
possibility of the body's
a big difference between the probability of an event and the event
is
Although
a whole electron
is
m=
proportional to the value of l+l*
somewhere, however, there
not actually
small, of delecting
spread out.
the relativistic mass
is
in 1926.
There
h
/
means the strong
2
presence, while a small value of t+l means the slight possibility of
however
de Brogiie wavelength
z at the time
x, y,
2
large value of l+l
A
As long as |+| 2
p
point
at the
is
as a function of position
and time usually presents
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
The
variable quantity characterizing de Brogiie waves
is
called the
wave
Chap.
func-
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.
+
conjugate
tion,
denoted
however, has no direct physical significance. There is a
simple reason
interpreted in terms of an experiment. The probability
P that
something be somewhere at a given time can have any value
between two
itself,
why + cannot he
we
Until then
6.
shall
In the event that a
wave
function
parts, the probability density
i {
*
assume that we have whatever knowledge of
*
required by the situation at hand.
is
V.
= \/— 1)
by
+
is
complex, with both real and
given by the product
is
^°
"V
of
+ and
i
in aginary
its
complex
obtained by replacing
The complex conjugate
of any function
is
—
in the function.
Every complex function
i
can be written
wherever
in the
appears
it
form
limits:
0,
corresponding to the certainty of
certainty of
its
absence, and
J,
corresponding
presence. (A probability of 0.2, for instance, signifies 20
a
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
=A + iB
its
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
2
|+j is known as probability density.
The probability of experimentally finding the laxly described
by the wave
74
+
to the
A
where
and B are
real functions.
** = A -
The complex conjugate
S*
*
of
^
is
iB
and so
*•* = A 2 since
2
i
= — 1.
Hence
i
2
B*
"fr* 4*
= A2 f
is
B-
always a positive real quantity.
BASIC CONCEPTS
WAVE PROPERTIES OF PARTICLES
75
DE BROGUE WAVE VELOCITY
3.3
(
=
1
=
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 velocity w,
we may
w=
to
apply the usual formula
vibrating string
p\
determine the value of
.«.
The wavelength A
FIGURE
is
just the
-
Wave motion.
3-1
de Broglie wavelength
III!
We
shall take the
E=
frequency
, to
be
that specified
by the quantum equation
lw
Hence
y
= A cos 2w
vt
h
where
or, since
E = me 2
A
is
the amplitude of the vibrations {that
on either side of the x
we have
Equation 3.4
—
v
mc-
is
as
3.3
i,
_
velocity
is
therefore
such a formula,
,.,\
_ mc2
h
so that
a wave
A complete
r.
however, should
What we want
c their
is,
iheir
maximum
tell
us what y
description of
is
at
wave motion
any point on the
let
us imagine that
starts to travel
we
down
shake the string
the string in the
+
x
at
,%
in a stretched
any time.
To obtain
when t = 0,
string at
a formula giving y as a function of both x
is
displacement
frequency.
us what the displacement of a single point on the string
a function of time
string,
The de Broglie wave
tells
and
axis)
and
=
t.
direction (Fig. 3-2).
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
h
me
x/w. Accordingly the displacement y of the string at v at any time t is exactly
he same as the value of y at x =
at the earlier time t — x/w. By simply
I
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
digress briefly to consider the notions
of plutse velocity
velocity is sometimes called wave
unexpected
zndgnmp
result,
we
replacing
t/
in
Eq. 3.4 with
in
t
terms of both
,v
and
t
—
x/w, then,
we have
the desired formula giving
/:
shall
velocity. (Phase
y
3.5
= A cos2wclr - —1
velocity.)
Let us begin by reviewing
elanty
we
shall consider
how waves
are described mathematically
a string stretched along the x
For
whose vibrations are
m the y direction, as in Fig. 3-1, 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
76
y
= A
axis
As a check,
we
Equation 3.5
y
note that Eq. 3.5 reduces to Eq, 3.4 at x
may be
=A
=
0.
rewritten
cas2fflcf
— —J
Since
cos
2-xt't
w=
v
BASIC CONCEPTS
WAVE PROPERTIES OF PARTICLES
77
becomes a vector k normal
In three dimensions k
*
t
replaced by the radius vector
=
r.
The
to the
scalar product k
•
wave
r is
fronts
and x
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
the probability that
it
moving body
reflects
be found at a particular place at a particular time.
It
formula reis clear that de Broglie waves cannot lw represented simply by a
the same
waves
with
all
of
sembling Eq. 3.10, which describes an indefinite series
amplitude A. Instead,
we
expect the
wave
representation of a
a wave packet, or wn\:e group,
like that
shown
correspond
to
constituent
waves have amplitudes upon which the likelihood of detecting the
Iwdy depends,
A familiar example of how wave groups come into being
When two
cies,
FIGURE 3-2
Wave propagation
sound waves of the same amplitude, but of
3.6
the average of the
two
=A
COs27r(j't
sounds have frequencies
- ij
of frequency
Equation 3.6
is
often
more convenient
to
apply then Eq.
3.5.
3.5.
We define the quantities angular frequency u and
is still
another
wove-
number
k by the Formulas
fluctuations occur a
of, say,
amplitude
=
to
Angular frequency
2trv
Wave number
3.S
is
the
wave group
wave speed
number of
2ir
rad in one complete wave. In
y
2.<nv
radians corresponding to a
= A cos (at —
BASIC CONCEPTS
kx)
the original
is
waves are the same, the speed with which
common wave speed. However,
identical with the
waves do not
proceed together, and the wave group has a speed different from that of the
it.
compute the speed tt with which a wave group travels,
a wave group arises from the combination of two waves
not difficult to
FIGURE 3-3
A wave
group.
The wave number is equal to
wave train m long, since there are
terms of a and k, Eq. 3,5 becomes
per second sweeps out
the
If
hear a fluctuating sound
varies with wavelength, the different individual
I«t us suppose that
circle v times
falls
illustrated in Fig. 3-4.
travels
the
It is
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
and
interference with one another results in the variation in amplitude that defines
waves that compose
3.9
will
rises
of times per second
A way of mathematically describing a wave group, then, is in terms of a
superposition of individual wave patterns, each of different wavelength, whose
if
78
its
number
440 and 442 Hz, we
the group shape. If the speeds of the
3.10
slightly different frequen-
441 Hz with two loudness peaks, called heats, per second. The
production of l>eats
Perhaps the most widely used description of a wave, however,
3.7
the case of beats.
hear has a frequency equal to
and
original frequencies
The amplitude
is
equal to the difference Iwtween the two original frequencies.
<j
form of Eq.
we
are produced simultaneously, the sound
periodically.
we have
moving body to
whose
in Fig. 3-3,
rad/s.
I
wave group
WAVE PROPERTIES OF PARTICLES
79
MVWWVWV
vwwwwwv
and
3.11
y
= 2A cos (w( —
A-*)
cos
(f-f')
Equation 3.11 represents a wave of angular frequency
+
that lias
superimposed upon
wave number
groups, as in
3.12
l
it
The effect of the modulation
3-4. The phase velocity w is
/2 dk.
I'ig.
o:
and wave number k
a modulation of angular frequency '/U/w
is
to
•-I
while die velocity u of the wave groups
Phase
velocity
Group
velocity
is
do
3.13
and of
produce successive wave
dk
In general,
number
depending upon the manner
in a particular
in which phase velocity varies with wave
medium, the group velocity may be greater than or less
than the phase velocity.
FIGURE
The production
3-4
If
the phase velocity
w
is
the same for
all
wavelengths,
of beaiv.
the group and phase velocities are the same.
The angular frequency and wave number
with the same amplitude
and
amount dk
aii
in
A
with a body of
rest
=
2,xt>
but differing by an amount <iw in angular frequency
We may
wave number.
represent the original waves by
a
3.14
mass
mu
of the
de Broglie waves associated
moving with the velocity v are
the formulas
;/!
j/jj
h
= A eos (oil —
= A cos [(« +
kx)
(/w)(
-
dk)x]
-f
(/c
11
The
and
resultant displacement
ij
t
.
ij
With the help of the
at
any time
and any position *
(
is
the
sum
of
y
l
identity
3.15
cos a
+
cos
ft
=
2 cos %{a
+
cos %(a
ft)
vl -
V 2 /r-
and
k-
2*
— ft)
Qmnv
and die
relation
cos{ — 9)
we
'Zirm^ti
eostf
find that
¥
Since «*« and
Both
- 9i + A
= 2A
tik
cos
&(&> +
f/«)f
-
(2fc
are small compared with
2w + rfu
2* + d*
80
=
BASIC CONCEPTS
+
t>
(tk)x\
and
fe
cos
w
</ (rf
2
f
-
respectively,
found
« and
k are functions of the velocity
o.
The phase
velocity
w
is,
as
we
earlier,
ilk x)
w =—
k
~ 2u
sr
2k
WAVE PROPERTIES OF PARTICLES
81
which exceeds
l!
<
Ixrth the velocity
of the lxidy v and the velocity of light
c,
since
C.
The group
velocity u of the de Broglie
waves associated with the body
electron gun
is
tha
elk
electron
detector
dw/dt
dk/dv
FIGURE
35
Tha
Damson Germer
experiment.
\V,\\
dw
2itm
'
dv
v>
- &f<?fn
ft(l
and
scattered
dk
dv
ft(l
- oVc2 3' 2
)
and so the group velocity
=
U
3.16
The de
beam
2irm u
is
was returned to the apparatus and the measurements resumed.
V
were very
Broglie
wave group associated with a moving body travels with the same
The phase velocity tt> of the de Broglie waves evidently
velocity as the body.
has no simple physical significance in
itself.
manifestation having no analog in the behavior of Newtonian particles
diffraction.
Thomson
in
In
1927 Davisson and Germer
in the
United States and G.
P.
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.
in
.3-5.
The energy
the primary beam, the angle at which they are incident
the position of the detector can
the scattered electrons will
dependence of
their intensity
of the primary electrons.
Germer
I
r>
all
emerge
method of plotting
to the distance of the
Two
is
the results
maxima
and minima were observed whose positions depended upon the electron energy!
Typical polar graphs of electron intensity after the accident are
THE DIFFRACTION OF PARTICLES
A wave
Now
from what had been found before the accident: instead of
a continuous variation of scattered electron intensity with angle, distinct
3-6; the
3.5
different
it
such that the intensity at any angle
to
mind immediately: what
is
much
as
X
This interpretation received support
R*»ultt ol the
Damson- Germer
waves were
rays are diffracted by planes of atoms
effect of heating a block of iiieke! at high
3-6
new
was baked?
Broglie's hypothesis suggested the interpretation that electron
in a crystal.
in Fig.
proportional
the reason for this
not appear until after the nickel target
being diffracted by the target,
FIGURC
shown
is
curve at that angle from the point of scattering.
come
and why did
effect,
De
questions
is
when
it
temperature
was realized
is
that the
to cause the
many
experiment.
of the electrons
upon the
target,
and
be varied. Classical physics predicts that
in all
directions with only a
upon scattering angle and even
less
moderate
upon the energy
Using a block of nickel as the target, Davisson and
verified these predictions.
the midst of their
work there occurred an accident
that allowed air to enter
and oxidize the metal surface. To reduce the oxide to pure nickel,
the target was baked in a high-temperature oven. After this treatment, the target
their apparatus
82
BASIC CONCEPTS
10
V
44
V
48
V
54
V
60
V
64
V
68
V
WAVE PROPERTIES OF PARTICLES
83
smalt individual crystals of which
targe crystal,
all
of
normally composed to form into a single
it is
whose atoms are arranged
Let ns sec whether
we
In
target,
and a sharp
Bragg planes shown
which can
in this family,
equation for
in Fig. 3-7 will
maxima
1m;
X
\/2
40
measured by X-ray
The electron wavelength
diffraction,
Here d
=
0,9
1
A
and 6
is
0.91 A.
The Bragg
n\
85°; assuming that
n
=
1,
the de Broglie wavelength
(i
1
Now we
X
= 1.66 X
= 1.66 A
is
= 2d sin
= 2 X 0.9 A X
= 1.65 A
54 eV
X
1-6
10~
X
li,
J/eV
1
therefore
is
me
4.0
=
X
is
i)
X of die diffracted electrons
'
both be 65°. The spacing of the planes
in the diffraction pattern
'2d sin
lO" 31 kg
X
9.1
lO" 2 kg-m/s
X
6.63
=
n\
is
occurred at any angle of 50° with the
beam. Tile angles of incidence and scattering relative to the family of
original
=
=
a particular determination, a beam of
54-eV elections was directed perpendicularly at the nickel
in die electron distribution
momentum mr
me = ^2mT
can verify that de Broglie waves arc responsible for
the findings of Davisson and Cermer.
maximum
the electron
in a regular lattice.
in excellent
65"
x
lO"*
1
J-s
10-- kg-m/s
'
10- u 'm
agreement with the observed wavelength. The Davisson-Cenner
experiment dins provides direct verification of de Broglie's
li\
wave
poliosis ol the
nature of moving bodies.
The analysis of the Davisson -Germer experiment
use de Broglie's formula
actually less straightforward
is
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 correto calculate the
energy of 54 eV
expected wavelength of the electrons.
is
small
and so we can ignore
compared with
its rest
relalivistic considerations.
The
mnc
energy
2
electron kinetic
of 5.1
X
10* eV,
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
Since
occurrence of maxima
restricts the
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 X-ray and electron
diffraction,
As
is
today a widely used tool for Investigating crystal structures,
in the case of
electromagnetic waves, the
wave and
moving bodies can never be simultaneously observed, so
mine which
FIGURE
is
3-7
The
diffraction of
de Broglie waves by the target
a
is
moving body
responsible for the results of Davisson and Germer.
properties.
die "correct" description. All
exhibits
Which
wave
we can
say
particle aspects of
that
is
properties and in other respects
set of properties
is
we cannot
thai in
it
deter-
some respects
exhibits particle
most conspicuous depends upon
how
the
de Broglie wavelength compares with the dimensions of the bodies involved;
the 1.66
single crystal
ol nickel
84
BASIC CONCEPTS
A wavelength
of a 54-eV electron
is
of the
as the lattice spacing in a nickel crystal, but the
moving
at
60 mi h
is
about 5
X
10
;is
ft,
same order
of
magnitude
wavelength of an automobile
far loo small to manifest itself.
WAVE PROPERTIES OF PARTICLES
85
THE UNCERTAINTY PRINCIPLE
3.6
The
fact that a
moving body must
Ik;
regarded as a de Broglie wave group rather
than as a localized entity suggests that there
accuracy with which
a de Broglie
If
the group
we can measure
wave group:
is
the particle
very narrow, as
found, hut the wavelength
wide group,
is
as in
its
is
a
fundamental
particle properties.
may
Imj
limit
to iho
Figure 3-Ha show
s
anywhere within the wave group.
in Fig. 3-8/j, the position of the particle
is
readily
impossible to establish.
At the other extreme, a
Fig. 3-Kc. permits a saHsfac lory wavelength estimate, but where
is
A
FIGURE
groups that result from the interference of wave trains hiving the same amplitudes
39 Wave
but different frequencies.
the particle located?
straightforward argument based upon the nature of
wave groups permits
us to relate the inherent uncertainty
A*
with the inherent uncertainty
simultaneous measurement of
Ap
in a
in
a measurement of particle position
its
momen-
From
are combined.
Eq. 3.11,
we
a calculation identical with the one used in obtaining
find that
tum.
The
simplest example of the formation of
Sec. 3.4,
where two wave
wave groups
trains slightly different in areolar
is
that
given
frequency
w
in
3.17
waves
which
+
2A
cos (wt
is
plotted in Fig. 3-9.
&k)x\
of the
same
FIGURE
3-8
represents.
The width of a wave group is a measure of the uncertainty in the
The narrower the wave group, (he greater the uncertainty In the
The width
348
/
- %\k x)
of each group
It is
is
evidently equal to half
reasonable to suppose that
this
width
Ax
=
is,
'/jA,,
The modulation wavelength
It
kx) cos ('/2 Aw
order of magnitude as the inherent uncertainty A.t in the position
of the group, that
*,
-
the wavelength X m of the modulation.
is
= A cos (wf - far)
* 2 = A [cos \*s + lu)t -(k +
*,
zz
and
propagation constant k were superposed to yield the series of groups shown
in
Fig. 3-4. Here let us consider the wave groups that arise
when the de Brogfie
%
+=
location of the particle
is
related to
its
propagation constant k m by
m
wavelength.
From
Eq, 3.17
km
we
see that the propagation constant of the modulation
is
= %\k
with the result that
A,„
—
2*f
%M
and
lx
~Tk
3.19
A moving body
but a single
corresponds to a single
wave group can
of trains of harmonic waves.
86
BASIC CONCEPTS
also
wave group,
be thought of
However, an
in
infinite
not a succession of them,
terms of the superposition
number
of
wave
trains of
WAVE PROPERTIES OF PARTICLES
87
wave numbers, and amplitudes
different frequencies,
is
required for an isolated
The de
momentum p
Broglie wavelength of a particle of
is
group of arbitrary shape.
At a certain time
the
I,
wave group
*
=
A
*(*) can be represented by the Fourier
integral
The wave number corresponding
3.20
*(*)
aa
where the function
to *{.r) vary with
g(k) cos
f
g(fc)
kx
describes how the amplitudes of the
wave number k. This fimetion
specifies the wave group just
is
waves
that contribute
waves is also included: only a single wave uumlier
wave numbers needed
Strictly speaking, the
A-
=
to
k
amplitudes
g(ftj
interval Afc.
As
of wave
m
oo,
is
is
2rrp
~h~
Hence an uncertainty At
with
in the
the particle results in
wave number
of the de Broglie
an uncertainly Ap
waves associated
in the particle's
momentum
ac-
cording to the formula
present in this ease, of course.
to represent a
but for a group whose length A.*
are appreciable have
=
called the Fourier transform
harmonic
Irani
wavelength
(Ik
and it
as completely as (*) does.
Figure 3-10 contains graphs of the Fourier transforms of a pulse and of a
wave
group. For comparison, the Fourier transform of an infinite train of
of +(x),
to diis
wave numbers
wave group extend
Skat
the waves
lie
within a
2iT
whose
finite
Since A.vAfc
Afc
I,
,
'
is finite,
a
1.
AA:«
Fig. 3- 10 indicates, the shorter die group, the broader the range
numbers needed
to describe it, and vice versa. The relationship
between
wave-number spread \k depends upon the shape of the
wave group and upon how Aa and AA are defined. The minimum value of die
Ap
A.v
3.22
>
I
and
/A.v
—
Uncertainty principle
Uie distance A.r anil the
product
Ax A*
which ease
and
its
Afc are
occurs
the group has the form of a gaussian fimetion. in
Fourier transform happens to be a gaussian function also.
taken as the standard deviations of the respective functions
gikl [hen A.v Afc
3.21
when
=
IxSk^
>/,.
In general, A.v Afc has
I
lie
If
-ffcc)
order of magnitude of
A*
and
I:
The sign
>
is
Equation 3.22
is
one form of the
1927.
unci-rtointij principle first
obtained by Werner
states that the product of the uncertainty
ft
neously both position anil
Fourier transforms for (a) a puis*, (b) a wa¥e
gfmpi and {c , 3n
requires a broader range of frequencies to describe it
than a dis-
matters so dial
A.v
is
the
•n
A.v in the
small,
corresponding to
we reduce Ap
If
3-fic,
If
we
arrange
the narrow wave group of
in
some way, corresponding
to
A.t will be large. These uncertainties are due
quantities involved.
The quantity
/i/2w appears quite often in
modern physics because, besides
connection with the uncertainty principle, Ii/2t also turns out to
unit of angular
"
symbol
(b)
with perfect accuracy.
not to inadequate apparatus but to the imprecise character in nature of the
WWWh
JL_
momentum
Ap will lw large.
wide wave group of Fig.
Fig. 3-SrJ,
A
BASIC CONCEPTS
that are conse-
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-
«
88
minima
the product A.v Ap.
ITeisenberg in
turbance of greater duration.
(a)
are irreducible
quences of the wave natures of mooing luufiea; any instrumental or statistical
uncertainties dial arise in the actual conduct of the measurement only augment
I
FIGURE 3-10 The wave functions and
infinite wave tram, A briel disturbance
e
\x and Ap
used because
It is
dierefore customary lo abbreviate h/2-z by the
ft:
ft
"I
momentum.
its
be the basic
=
-^-
=
J.
054
X
10- :il
J-s
2m
(c)
In the
remainder of
this
book we
shall use
ft
in
place of
ft/2ir.
WAVE PROPERTIES OF PARTtCLES
89
it
wavelength of die
"The uncertainty principle can be arrived at in a variety of ways. Let us obtain
L
by
our argument upon the particle nature of waves instead
wave nature of particles as we did above.
3.Z4
liasing
of
upon
light
being used. That
is,
~X
A.r
the
Suppose that we wish
at
measure the position and momentum of something
a certain moment. To accomplish this, we must prod it with something else
that
to
to carry the desired information
is
back to
us; that
we have
is.
to touch
The
shorter the wavelength, the smaller the uncertainty in the position of the
electron.
From
Ivqs.
3.23 and 3.24
if
we employ
light of short
wavelength
We
be a correto improve the accuracy of the position determination, there will
while light
determination,
momentum
sponding reduction tn the accuracy of the
in
of long wavelength will yield an accurate
with our
it
finger, illuminate it with light, or interact with it in some other way.
might be examining an electron with the help of light of wavelength A, as
Fig. 3-11. In this process photons of light strike the electron and bounce off
Each photon possesses the momentum h/\, and when
it.
electron, the electron's original
cannot be predicted, but
momentum
the photon
it
Ls
h/X.
momentum p
likely to
Hence
it
collides
with die
ehanged. The precise change
is
3.25
A.v
the act of
measurement introduces an
This result
is
Ap
>
momentum
value but an inaccurate
A.r into Eq, 3.23 yields
h
consistent with Eq. 3.22, since both
Ax and Ap here were
defined
rather pessimistically.
Arguments
preceding one, though superficially attractive, must as a
like the
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
rule be
3.23
in the
momentum
employ
of the electron.
The longer the wavelength
in "seeing" the electron, the smaller the
of the light
consequent uncertainty in
wc
this
trary,
indeterminacy
justification for the
Because light has wave properties,
tron's position
with
infinite
we cannot expect
An
to
determine the elec-
accuracy mider any circumstances, but
reasonably hope to keep the irreducible uncertainty
FIGURE 3-11
measurement process that introduces the indeterminacy
its
momentum.
electron cannol be observed without changing
Ax
in
its
we
might
position
is
f
viewer
"derivations" of this
way around
wave
is,
first,
that diey
the uncertainty principle,
view of the principle
familiar context than tiiat of
kind
The
that can
lie
show
it
and second,
appreciated in a more
packets.
APPLICATIONS OF THE UNCERTAINTY PRINCIPLE
so
is
minute— only
X
6.63
10~ :H J-s— 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
One
incidc nt
reflected
photon
As wc
in
terms of
this principle;
interesting question
is
we
shall consider several of
whether electrons are present
shall learn later, typical nuclei are less
'
Ap
original
than
If)
'
4
m
in
>
1
them
here.
atomic nuclei.
in radius.
electron to be confined within such a nucleus, the uncertainty in
may not exceed I0-1 m. The corresponding
mentum is
its
For an
position
uncertainly in the electron's
mo-
As
momentum
X 10' M J-s
10" tn
1.0,54
of electron
nal
momentum
lenrum
of electron
BASIC CONCEPTS
the con-
to
be understood
X
AxAp. On
momentum.
viewer
photon
in
inherent in the nature of a moving body.
is
impossible to imagine a
that they present a
3.7
rts
many
Planck's constant h
t-j
=
Substituting X
position value.
be of the same order of magnitude as
uncertainty of
90
clear that,
it is
\
\
>
1.1
x
lO-20 kg-m/s
WAVE PROPERTIES OF PARTICLES
91
If this is
ba
at
1.1
X
m
the uncertainty in the electron's
comparable
least
10"80 kg-iu
we may
c 2 , and
=
7*
to find
has
s
it
in
kinetic energy
the
momentum
itself
must
An electron whose momentum
7 many
limes greater than
its rest
The corresponding energy uncertainty
is
uj
III
T=
1.1
=
3.3
we
X 10-» kg-m/s X
X 10~ 12 J
obtain
3
X
20
I
CV
=
1.6
A mure
associated even with unstable atoms never have
energy, and
ft
realistic calculation
M.
>
It
is
about 5
in the position of its electron
uncertainty
An
in
=
X
10
may
"
in in radius,
and therefore the uncertainty
not exceed this figure.
The corresponding
urement and the uncertainty
is
2.1
X
As an example of the significance of Eq. 3,26
of light from an "excited" atom. Such an
AE =
this
order of magnitude
is
x HF" kg-in/s)3
X 9.1 X I0-" kg
Another form
10
is
»
an
|s
is
n
a
wave group
=
a wholly plausible figure.
We
might wish
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
II lis is
Af
92
BASIC CONCEPTS
radiates
is
10~ s
s.
uncertain by an amount
X
ol
I»- :i4 J-s
51
s
X UH»J
die light
is
uncertain by
1.6
x
10 7
Hz
the irreducible limit to the accuracy with
which we can determine the
frequency of the radiation emitted by an atom.
3.8
is
THE WAVE-PARTICLE DUALITY
many
wave can
Despite the abundance of experimental confirmation,
lo
A,=-L
it
ft
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
excess energy
AE
= —r-
j
oj the uncertaint) principle is sometimes
useful,
1.1
and the frequent)
to
-
its
f
1.054
(2.1
or about 15 eV. This
consider the radiation
A(
nnnretativistie
=
X
we can
divests itself of
between the excitation of an atom and the time
•2m
2.1
atom
by emitting one or more photons of characteristic frequency. The average period
10-
=
an energy meas-
in
10 -'kg-m/s
whose momentum is of
behavior, and its kinetic energy is
2
AE
M in the time at which the measurement was made
equal to or greater than H.
that elapses
is
electron
_
this to
Equation 3.26 states that the product of the uncertainty
Thus the photon energy
Ao
changes
h
a fraction of this
conclude that electrons cannot be present within nuclei.
ask how much energy an electron needs to be confined to
an atom.
now
The hvdmgcn atom
momentum
more than
we
1*1 us
A
M
x Wr* J.
the kinetic energy of the electron must be well over
to be a unclear constituent. Experiments indicate that the
electrons
MeV if it k
AE =
li>
AEAl >
IIV in/s
3.26
Since
ft
*>
)>c
c,
AE =
is
energy
accordingly ate the extreme relativists formula
Substituting for p and
7".
momentum,
magnitude.
how what we normally think
how what we normally think of as a
appreciate
in nl
of
EtS
a
particle can also
of us find
also
it
hard
be a particle
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
more concrete
end of Sec.
detect which of the
2.2
on a
Figure :3-!2
diffracted
show.-, an experimental arrangement in which
light that has hern
by a doable ditto detected on a "screen" that consists of
manv adjacent
photoelectric
cells.
the properties
we
The
photoelectric cells respond to photons, which have
associate with particles.
However, when we
all
is
SO low that, on the average, aoiy one photon at a time
The problem
is,
how can
itself?
rift?
In
other words,
This problem does not arise
out in spare, but
it
would seem
to
how
in the ease of
have meaning
behavior suggests that thev are localized
in
in
slits
be affected
can a photon interfere
very small regions
of
whose
and the screen. A photon that passes through one of the slits strikes
3-12).
a particle and gives it a certain impulse which enables us to detect it (Fig.
uncertainty
with
an
Provided that we can establish the position of the particle
lij that is less than half the space d between the slits, we can determine which
slit
the photon passed through. Therefore
But
we
are able to limit the uncertainty in the u coordinate of the struck
particle to Aw, the uncertainty
3.27
Ap v
in the
w component of
*P»
its
momentum
is
d
Aw
Since the collision introduces a change of
each photon contributing to a double
si
Ap v
in the
photon's
momentum means
a
momentum,
momentum. A change of
in the particle's
the same change must have occurred in the photon's
Ap u
slit
if
sp
To have meaning, every question or statement
in science must ultimately be
Here the relevant experiment is one that would
FIGURE
to the
waves, which arc spread
the case of photons,
reducible to an experiment.
3-1
2 Hypothetical e.periment to determine which
interference pattern has passed through
way
in the apparatus.
a photon that passes through one of the
by the presence of the other
mil,
is
we
its
introduce a cloud of small particles l>etween
the sbts
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
a particular photon passes through on
imagine that
l^et us
screen,
oasis.
slits
shift of
it
photoelectric cells ^
in the location
on the screen which the photon strikes; because p„ < p (the width
is small compared with the distance L), p,
p. and
~
of the diffraction pattern
we can
write
§.4Bu
p
particle after
i
^path of
photon before
collision
collision with
photon
The photon
momentum
p
path of
photon after
collision
=
A of the
light
by Eq.
3.1,
x
is
_ ^PiM*
From
Eq. 3.27
we have Ap„
screen position
number
of
photons
counted
BASIC CONCEPTS
related to the wavelength
and so
uncertain
94
is
'
3.28
S
=
> 2H/d, which means that
the shift in die photon's
is
—
t
•ad
WAVE PROPERTIES OF PARTICLES
95
The distance t^, between a maximum
pattern and an adjacent
to
minimum
(that
is,
a "bright line")
("dark line"!
in the interference
known from elementary
is
optics
Derive a formula expressing the de Broglie wavelength (in A) of an electron
terms of the potential difference V (in volts) through which it is accelerated.
5.
in
he
"
This distance
which
slit
"
is
almost the same as the
each photon passes through.
and dark
between the photons and the
ference can
lie
lines
minimum shift involved in establishing
What would otherwise lw a pattern of
becomes blurred owing
to the Interaction!
particles used to trace their paths. Thus no interobsenetk the price of determining the exact path of each
photon
the destruction of the interference pattern. If our interest
is in the wave aspects
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-
of a
sible to
demonstrate Imth aspects
simultaneous experiment. (Using photoelectric cells to detect an interference pattern is
not a simultaneous experiment
in this sense,
because there
through which
The
slit
is
in a
no way
in
which a photoelectric
a particular photon striking
original question of
out to be meaningless.
7"
and
its rest
how
it
cell
It
is
a photon can interfere with itself therefore turns
oi the uncertainly principle,
into the latter class in view
fall
whose own empirical
validity has
Show
that
been thoroughly
established.
that electromagnetic
"'„'"
bow
s
a photon of the same energy?
waves are a special case of de Broglie waves.
rest mass of the
having a characteristic frequency of vibration of i\ specified by the
2
The particle travels with the speed i? relative to an
relationship lit;, = m,f
iii
as
,
v
(
.
With the help of special relativity, show that the observer sees
2
progressive wave whose phase velocity is iv = c /t: and whose wavelength
observer.
h/mv, where
in
= m„/vl —
The
gravity.
waves
velocity of ocean
.
is
Vg\/2w, where g
a liquid surface
The
momentum
position and
termined.
its
\/2w/Xp, where S
is
the surface
1
A, what
is
the percentage of
its
ultimate resolving
momentum?
electron microscope uses -10-keV electrons.
thai this
is
Find
equal to the wavelength of the eleclrous.
the uncertainties in the velocities of an electron
Compare
is
of a 1-keV electron are simultaneously de-
located to within
If its position is
uncertainly in
An
the acceleration of
Find the group velocity of these waves.
tension and p the density of the liquid.
11.
is
Find the group velocity of these waves.
The velocity of ripples on
0.
1
v-/c 2
a
is
and a proton
confined in a 10-A box.
Find the dc Broglie wavelength of an electron whose speed
is
If)"
14.
m%
Find the de Broglie wavelength of a 1-McV proton.
Nuclear dimensions are of the order of in
whose de Broglie wavelength
of revealing details of nuclear structure,
is
(h)
•
'
10"
m
l5
Make
.
(a)
Find the energy
m and which
the
is
in
eV
thus capal.lv
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.
96
T>
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
13.
of an electron
If
h.
12.
Problems
3.
nttf?.
terms of
tloes tne particle
0.
power on the assumption
2.
energy
in
photons mast travel with the velocity c and that the
photon must be
9.
has passed.)
important to
apart the elements of the wave-particle duality
Assume
T.
can determine
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
1.
de Broglie wavelength of a particle
wavelength compare with the wavelength of
2d
alternating bright
is
kinetic energy
its
3.29
for the
Derive a formula
fi.
BASIC CONCEPTS
s What
determined with accuracies of one part in 10
wavelength
its
when
photon
the position of a l-A X-ray
Wavelengths can
l>e
.
is
the uncertainty in
is
simultaneously measured?
At a certain lime
15.
with an accuracy of
at
t
is
not
atul.
from
16.
ibis the
±10-"
particle as a
(a)
a measurement establishes the position of an electron
'
'
m. Find the uncertainty
uncertainty
in its
position
m, account for the difference
wave
is
10
in
1
in the electron's
momentum
slater. If the latter uncertainly
terms of the concept of a moving
packet.
How much
whose speed
t
±10
time
is
needed
to
measure the kinetic energy of an electron
m/s with an uncertainty of no more than
0.1
percent?
How
WAVE PROPERTIES OF PARTICLES
97
far will the electron
have traveled
calculations for a I-g insect
in this
whose speed
period of rime? (b)
is
the same.
Make the same
What do these sets of
figures indicate?
The atoms in a solid possess a certain minimum zero-point
energy even at
K, while no such restriction holds for the
molecules in an ideal gas. Use the
uncertainty principle to explain these statements.
1
.
.
18.
Verify
M,itf
98
>
that
the uncertainty
principle can be expressed in the form
the uncertainty in the angular momentum
of a Iwdy
the uncertainty in its angular position.
[Hint:
h,
where
1L
and A0
is
moving
in a circle.)
BASIC CONCEPTS
is
Consider a particle
ATOMIC STRUCTURE
Far
in
the past people began to suspect that matter, despite
its
4
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
form
until
a
atoms and molecules, the ultimate particles of matter
been amply demonstrated, and their own ultimate
and neutrons, have been
others to
ihis
come our
structure that
more concrete
over a century and a half ago; since then the existence of
little
is
identified
and studied as well. In
chief concern will
Ik-
responsible for nearly
shaped the world around
in its
common
forms, has
particles, electrons, protons,
chapter and
this
the structure of the atom, since
all
it
in
is
the properties of matter that have
us.
Every atom consists of a small nucleus of protons and neutrons with a mnntier
of electrons
some distance away.
circling the nucleus as planets
It
is
templing to think of the electrons as
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
which, despite
its
serious
to
atomic structure
in
1913 to obtain a model
inadequacies and subsequent
replacement
In
a
quantum-mechanical description of greater accuracy and usefulness, nevertheless
remains a convenient mental picture of the atom.
policy of this book to
we shall
go deepb
discuss Bohr's theory of
transition to the
more
abstract
While
into hypotheses that
tin:
it
is
have had
hydrogen atom hecause
it
not the general
to In
discarded,
provides a valuable
quantum theory of the atom. For
this reason
our account of the Bohr theory differs somewhat from the original one given
by
Uiilir,
4.1
W
though
all
the results are identical.
ATOMIC MODELS
bile the scientists of the
'-'leiucnts consist ol
selves.
nineteenth century accepted the idea that the chemical
atoms, thev knew virtually nothing about the atoms them-
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
to halancc the negative
charge of
its
thousands of times lighter than whole atoms;
charged constituent of atoms
When
charged matter
tins
suggests that the positively
what provide! them with nearly
is
Thomson proposed
J, J.
positively
Furthermore, electrons are
electrons.
all their
mass.
1898 that atoms are uniform spheres of posi-
in
tively charged matter in which electrons are embedded, his hypothesis then
seemed perfectly reasonable. Thomson's plum-pudding model of the atom— so
called from
4-1.
its
resemblance to Uiat raisin-studded delicacy -is sketched
Despite the importance of the problem,
experimental
test
we
compelled the abandonment of
shall see,
leaving in
its
].')
of the plum-pudding model
years
p^ved before a
in Fig.
was made. This experiment,
this
radioactive
definite
substance that
as
emits alpha
apparently plausible model,
place a concept of atomic structure incomprehensible
in
particles
the light
The most
a linger into
Marsdeu
in
it.
collimator
way
direct
to find out
what
is
inside a plain
to find out
wbaJ
is
inside
an atom. In
particles arc
plunge
employed
FIGURE 4-2
The Rutherford ««Hering experiment.
performed
as prolics the
spontaneously emitted by certain radioaetivc elements. Alpha
helium atoms
+ 2r; we
to
is
their classic experiment,
191] at the suggestion of Finest I'.titherford, the]
charge of
padding
technique not very different from that used by Ceiger and
a
fast ulfilm fuirtu-Us
zinc sulfide
screen
lead
of classical physics.
shall
that
have
lost
two
electrons, leaving
examine their origin and properties
in
them with
more
a
detail later
Ceiger and Marsden placed a sample of an alpha-particle-emitling substance
behind a lead screen that had a small hole in it. us in Fig. 1-2. SO that a narrow
Ixsam of alpha particles was produced. This beam WU directed at a thin eold
foil.
A moveable
zinc sulfide screen, which gives off a visible flash of light
by an alpha
struck
particle,
was placed on the other side of the
foil.
anticipated that most of the alpha particles would go right through the
the remainder
would
at
most suffer only
from the Thomson atomic model,
in
slight deflections.
is
weak
correct, only
through a thin metal
thern through
foil,
its
electric forces are exerted
and
foil,
while
This behavior follows
which the charges within an atom are
assumed to be uniformly distributed throughout
model
when
was
It
their initial
Thomson
volume.
If
on alpha
particles passing
the
momenta should be enough
to carry
with only minor departures from their original paths.
What Geiger and Marsden
particles indeed
FIGURE 41
Th » Thomson model
of lh«
atom.
large angles.
A
actually found was that, while most of the alpha
emerged without deviation, some were scattered through very
few were even scattered in the backward direction. Since alpha
heavy (over
more massive than electrons) and
it was clear that strong
be exerted upon them to cause such marked deflections. To explain
particles are relatively
7,(KX1 limes
those used in this experiment traveled at high speed,
had
forces
"
to
the results, Rutherford
tiny nucleus, in
trated,
with
as largely
fleet run
a thin
»"
-
positively
charged matin
foil.
its
which
electrons
empty space,
When
to picture
an atom as being composed of a
positive charge and nearly all of
some distance away
it is
easy to see
(Fig. 4-3).
why most
its
mass are concen-
Considering an atom
alpha particles go right through
an alpha particle approaches a nucleus, however,
intense electric field
The atomic
was forced
its
and
is
likely to
electrons, being so light,
it
encounters
be scattered through a considerable angle.
do not appreciably
affect the
motion of
incident alpha particles.
102
THE ATOM
ATOMIC STRUCTURE
103
/
e
e
X
«4.2
Bnlberford arrived at a formula, describing the scattering of alpha particles by
\
thin foils
\
G
L^-CJ
,
results.
\
FIGURE 4-3
model
oi the
The Rutherford
basis of his
atomic model, that agreed with the experimental
derivation of this formula both illustrates the application of fundatu a
novel setting and introduces certain notions, such as
an interaction, that are important
that of the crass section for
in
many other
atom.
aspects of
positive nucleus
T
on the
The
mental physical laws
+
_
!
electron
e
ALPHA-PARTICLE SCATTERING
modern
physics.
Kntherford began by assuming that the alpha particle atal the nucleus
it
\
e
Q
\
e
interacts with arc both small
/
/
are both positively charged)
is
Numerical estimates of electric-field 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
atom's surface (where
at the
hand,
is
we neglect
if
we assume
mi Kent rated
in a
the electrons completely, the
it is
a
maximum;
is
dec trie- field Intensity
V m. On the other
about I0
,:1
with Rutherford that the positive charge within a gold atom
small nucleus at its center, the elect ric-(ield intensity at the
V/m— a factor of 10 greater. Such a
can deflect or even reverse the direction of an energetic alpha
particle that comes near the nucleus, while the feebler field of
the Thomson
surface of the nucleus exceeds 10
strong
between alpha
21
Owing
particle
the only one acting;
it
and
and nucleus (which
that the nucleus
does not
move during
to the variation of the electrostatic force
with
I/r-,
is
so
their
where
the instantaneous separation between alpha particle and nucleus, the alpha
particle"*
if
is
be considered as point masses and charges:
massive compared with the alpha particle that
i
volume, and
to
that the electrostatic repulsive force
interaction.
its
enough
path
is
a hyperbola with the nucleus at the outer focus (Fig. 1-1".
is the minimum distance to which the alpha particle
The impact parameter b
would approach the nucleus
scattering angle,
of the alpha particle
task
is
there
were no force between them, and the
and the asymptotic direction
to find a relationship
FIGURE 4-4
if
the angle between the asymptotic direction of approach
is
between
/;
and
in
which
it
recedes.
Our
first
(J.
Rutherford scattering.
s
field
^ alpha particle
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
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
one clement
charges
element.
in
wen found
to
the nuclei of an element
We know now
is
today called the (domic number of the
target nucleus
that protons, each with a charge
+e, are responsible
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.
for the
104
THE ATOM
"
™
Ov 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
Ap from the initial value p, to the final value p
the alpha particle changes by
2
That
we have
momentum change
for the
.
is,
Ap
4.2
Ap = p2 -
4.1
Because the impulse J
Because the nucleus remains stationary during the passage of the alpha particle,
Ap,
by hypothesis, the alpha-particle kinetic energy remains constant; hence the
magnitude of its momentum also remains constant, and
4.3
=
Here v
is
its
From
Fig. 4-5
is
we
=1f*
2m<;sin—
between V and Ap along
J.-J
in Eij.
tin- pall,
8
sin
—
f-cosCKif
limits of integration will
to
Since
sin— (tt
-
8)
=
£
at
f
=
—
cos
and
I
2 sin
The quantity
— cos —
2
co respectively,
i=/.:
2
-hr-«/2
2
drf>/d< is just
the nucleus (this
is
/
</>,
-
we
note that the
0),
corresponding
and so
f
2tnm sin
4.4
and
=
=
the alpha
*,,
change the variable on the right-hand side from f to
change to - J^w - 8) and + )&(*
In
-
d
1.1.
mo
Ap
8
and
-4.2
2
sin
same direction as the momentum change
in the
the instantaneous angle
Inserting Eqs.
see that, according to the law of sines,
sin
is
/F*sa/FOM*«B
particle.
the alpha-particle velocity far from the nucleus.
df
F-"
magnitude U
where £
= mo
ft
—
sin
Pl
= JFdl
Pi
2mc
ae
"
i
eos«f>— d$
U
the angular velocity
evident from Kiu-
MS).
of the alpha particle about
elecAfbstatiC force exerted
The
(he nucleus on the alpha particle acts along the radius vector joining them,
FIGURE
45
Geometrical relallonthlps
in
Rutherford scattering.
so there
is
no torque on the alpha particle and
its
angular
momentum
l>\
and
row*8
is
constant Hence
=
mu'r
= mr —r-
M(r-ff)
=
from whJch
W(irpath of alpha particle
constant
/
/
i-**5^-—___ S_
-^i
LAj
we
see that
r1
"
'
alpha
fid,
7^— particle
mi h
vb
Substituting this expression for <*/<*$> in Eq, 4.4.
\
target nucleus
106
8
4.s
2»jr-/jsiii--
2
r
-
+fc
I
Ra co«^d^
,
'-<w-«/2
THE ATOM
ATOMIC STRUCTURE
107
As we
recall,
lier Z,
F
the electrostatic force exerted by the nucleus on the alpha
the nucleus is Ze, corresponding to the atomic man-
is
THE RUTHERFORD SCATTERING FORMULA
*4.3
The charge on
particle.
and
that
on the alpha particle
is
le.
Therefore
Equation
way
1
2Ze 2
4«? n
i"
F=
scattering angle.
and
indirect strategy
is
required.
Our
first
step
is
to
no
note that
h will be scattered through an angle of or inure, where is given in terms
b by Eq. 4.6. This means that an alpha particle that is initially directed
,ui> where within the area »fc* around a nucleus will be scattered through // or
to
-te-9)/2
AmMuPfa
sin
Ze a
I
=
f
J-fe-Sl/2
=
2 cos
of
cos
<j>
d$
more
—
(Fig. 4-6); the area
is
related to the impact parameter
a
cot
T=
—?3
more convenient
and velocity
it
tiic-
it
,
the
era?.? xectitnt
for the
and so here
is <r,
«r&a
must keep
mind
in
that the incident alpha particle
is
actually scattered Ivcfore
reaches the immediate vicinity of the nucleus and hence does not necessarily
pass within a distance b of
Now we
to specify the alpha-particle
energy T instead of
its
mass
consider a
I
that contains
The number of target nuclei per unit area
incident
4m T
2=^M
it.
of thickness
foil
upon an area
A
section for scatterings of
nt.
is
n atoms per unit volume.
and BO alpha-particle beam
therefore encounters nlA nuclei.
tl
or
more
is
the
number
The aggregate
cross
of target nuclei nf.\ multiplied
a
liy
the cross section o for such scattering per nucleus, or u/Ao.
/of
Figure 4-6
is
a schematic representation of Eq.
h increases
is
evident.
A very near
miss
is
-i.fi;
the rapid decrease in
incident alpha particles scattered by
(/
or
more
That
Hence the fraction
between ihe
the ratio
is
•'g^egalc cross section ittAo lor such scattering and the
as
total target
area A.
is.
required for a substantial deflection.
/=
FIGURE 4-6
ralleil
for cross section
'>
.separately; with ilus substitution,
rat
=
accordingly
is
b by the equation
We
2m
<nb'~
The general symbol
interaction.
47
scattering angle $
,
It is
An
is
observed
alpha particles approaching a target nucleus with an impact parameter from
.ill
The
cannot he directly confronted with experiment since there
1.6
of measuring the impact parameter corresponding to a particular
alpha particles scattered by
or
more
incident alpha particles
aggregate cross section
The scattering angle decreases with imreating impact
parameter.
target area
_
1 1
(An
A
b from Eq.
Substituting for
m
L
r
b
THE ATOM
^Lz<y*cot*f
\
—
m^
target nucleus
bi Ihe
•lie
urea
103
fS
= iri» s
4.fi.
lw„T7
above calculation
it
2
was assumed
thai the foil
cross sections of adjacent nuclei du not overlap
particle receives
its
entire deflection
I
ruin an
is
sufficiently thin so that
and
that a scattered alpha
encounter with a single nucleus.
ATOMIC STRUCTURE
109
beam of 7.7-MeV alpha
when incident upon a
Let us use Eq. 4 .8 to determine what fraction of a
particles
gold
foil
scattered through angles of more than 45°
is
X
3
energies and
human
hair
10" ~
foil
is
m
(These values are typical of the alpha-particle
thick.
thicknesses used by Gciger and Marsdcn; for comparison, a
about Kl
We
in diameter.)
volume
of gold atoms per unit
Atoms
m
'
X
tn the Rutherford experiment, particles
4-7
are detected that
have been scattered between S and
number
n, the
from the relationship
in the foil,
(atoms/kmol)
begin by finding
FIGURE
(mass/volume)
massAmol
Volume
\,t>
it;
where
Since
Nn is Avogadro's number, p the density of gold, and us its atomic weight.
Nu = 8.03 X 10* atoms/kmol, p = 1.93 X 10* kg/m 3 and w = 197, we
,
have
6.03
—
n
X
10 2(J
x
10
28
7,
of gold
atoms/kmol
X
1.93
X
10*
kg/m 3
2.9
197 kg/kmol
ss 5.91
The atomic number
1.23
x
vl
10
J,
/
=
and
7
X
tf
=
is
79, a kinetic energy of 7.7
45°; from these figures
we
A
MeV
is
thin
If a total
is
more
—only 0.007
quite transparent to alpha particles.
the
of
N
t
alpha particles strike the
number scattered
the screen at 0.
an actual experiment, a detector measures alpha particles scattered between
and + dO, as in Fig, 4-7. The fraction of incident alpha particles so scattered
In
is
d
equal to
10" 5
foil this
i
find that
of the incident alpha particles are scattered through 45° or
percent!
1
r "0
area as 4 irr sin ; cos
atoms/m 3
found by differentiating Eq. 4.8 with respect to
N(S)
which
is
is A/,
foil
df.
during the course of the experiment,
The number N(0) per
the quantity actually measured,
unit area striking
is
=
dS
an operation that yields
0,
into dB at
NwB ,(^eLV* crt | cse8 |. d »
\4m Q Tf
2
2
*
4.9
do
\4wr o r/
2
2
4-jtt
2
sin
— cos — d6
2
(The minus sign expresses the
fact that
/
decreases with increasing
experiment, a fluorescent screen was placed a distance
scattered alpha particles
Those alpha
of radius
r
were detected by means of the
particles scattered
whose width
nlS.
is
between
The zone
dS of the screen struck by these
f/S s=
=
=
(2vr sin
Tnr
1
THE ATOM
(I
is
+
from the
foil,
and the
of a sphere
and so the area
Equation 4.10
4irra sin
=
is
Rutherford scattering formula
(oVejVT 2
dO
sin* (0/2)
the Rutherford scattering formula.
According to Eq. 4.10, the number of alpha particles per unit area arriving
at the fluorescent
volume
sin 6
N(0)
4.10
they caused.
dO reach a zone
itself is rsintf,
2
tn the
screen a distance
proportional to the thickness
ff)(rd0)
n,
(
r
from the scattering
of the
foil,
foil
the nuuilier of
and the square of the atomic number
Z
of the
foil
should be directly
foil
2
particles
and
to sin 4 (0/2),
where 8
is
atoms per unit
atoms, and
"6 inversely proportional to the square of the kinetic energy
— cos — dO
2
110
particles
and
radius
r
scintillations
0.)
T
it
should
of the alpha
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
of the nuclear atom,
were
correct.
Rutherford
"discovery" of the nucleus. Figure 4-8 shows
is
how
among them
(he hypothesis
therefore credited with the
,V(ff)
varies wi(h 6.
minimum
distance to which the incident alpha particles approach the nuclei.
Rutherford scattering therefore permits us to determine an upper limit to nuclear
Let us compute the distance of closest approach
dimensions.
employed
energetic alpha particles
when
will liave its smallest r
4.4
NUCLEAR DIMENSIONS
say that the experimental data on the scattering of alpha particles by
thin foils verifies our
is
really
meant
is
fltctioKiatir potential energy,
and so
the most
part
it
1b
corresponding to
0,
Al the instant of closest
of the particle
entirely converted to
is
at thai instant
assumption Uiat atomic nuclei are point particles, what
that their dimensions are insignificant
compared with the
r=
4.11
2Ze 2
1
\~s
FIGURE 4-8
T
energy
initial kinetic
=
b
is
head-on approach followed by a 180° scattering.
a
approach the
When we
impact parameter
its
r„ of
An alpha
the early experiments.
in
,,
since the charge of the alpha particle
Rutherford nattering.
is 2<?
and
that of the nucleus
7,e.
I
lence
IZe 1
4*e«T
The
maxiitniui
/'
7.7
X
l/4w„
Since
=
found
10"
eV
9
x
10"
2
X
?>
alpha particles of natural origin
in
X
X
1.6
2
N-m /C
X
10 9
J/eV
=
1.2
X
>
10
MeV, which
is
2
J
2
N-mVC* X
1.2
= 3.8x
,B
10-
7.7
is
X
(1.6
X
10
"'
QZ
2
I0-'*J
10-'«Zm
The atomic number uf gold, a typical
foil
material,
is
Z =
79, so that
.Y<0)
r„ (
Au)
=
3.0
X
10-
M
The radius of the gold nucleus
'
n,
|
hi
been
the radius of the
more recent years
artificially
atom
m
is
therefore less than 3.0
10
" m, well under
as a whole.
particles of
accelerated,
X
and
formula does indeed eventually
it
much higher
energies than 7.7
MeV
have
haslieen found that the Rutherford scattering
fail
to agree with experiment.
We
shall discuss
experiments and the information (hey provide on actual nuclear dimensions
hi
Chap.
4-5
II.
ELECTRON ORBITS
:\'<l8Cn
>
0°
20°
40°
60°
80"
100'
120"
140 s
160°
180°
hu Rutherford model of the atom, so convincingly confirmed
greai distance In
112
THE ATOM
by experiment,
postulates a tiny, massive, positively charged nucleus surrounded at a relatively
enough electrons
to
render the atom, a> a whole, electrically
ATOMIC STRUCTURE
113
.
neutral.
Thomson
visualized the electrons in his
the positively charged matter that
The electrons in
there
is
fills
it,
model atom
as cml:>cddcd in
electron velocity c
The
Rutherford's model atom, however, cannot be stationary, because
4.13
I'
them
therefore related to
its
orbit radius r
by the formula
e
=
N/4l
nothing that can keep them in place against the electrostatic force
attracting
is
and thus as being unable to move.
to the nucleus. If the electrons are in
motion around the nucleus,
The
however, dynamically stable orbits (comparable with those of the planets about
total energy-
of the electron in a
/-.'
hydrogen atom
is
the
sum
of
its
kinetic
energy
the sun) are possible (Fig. 4-9).
Ijet us
electron
examine the
makes
it
classical
the simplest of
though
orbit for convenience,
shape.
The
r = /2 mt; 2
dynamics of the hydrogen atom, whose single
all
it
atoms.
We shall
l
assume a circular electron
might as reasonably be assumed
elliptical in
and
potential energy
its
centripetal force
Fc~
e2
V=
—
4we
The minus
:
holding the electron in an orbit r from the nucleus
is
provided by the electrostatic
1
sign signifies that the force
E=T+
F.=
2
e
—
2
4wc
is
Substituting for
F =F
£
ID 2
1
r
4ire
«;
l^f„r
from
J£q. 4.12,
=
8we
e2
4.12
— r direction.)
e2
_
2
for orbit stability
in the
is
V
nic2
r
between them, and the condition
on the electron
Eettoe
force
1
r
4TO
r
r
r
2
4.14
8OT(/
FIGURE 4-9
Fore* balance
the hydrogen atom
In
The
total
energy of an atomic electron
be bound to the nucleus.
much energy
too
to
E were
If
13.6
proton and an electron; that
eV
= 2,2 X
negative; this
necessary
is
if it is
to
would have
remain in a closed orbit about the nucleus.
Experiments indicate that 13.6 eV
Into a
is
greater than zero, the electron
10 -1B
J,
we can
is
required to separate a hydrogen atom
is, its
binding energy
E
—
is
13.6 eV,
Since
find the orbital radius of the electron In a
hydrogen atom from Eq. 4.14;
Swe^E
(1.6
8w
=
An atomic
5.3
X
X
8.85
10-
X
X
U>- I2
10
,!,
C)
F/m X (-2.2 X
Id
,s
j)
rn
radius of this order of magnitude agrees with estimates
made
in other
Ways.
114
THE ATOM
ATOMIC STRUCTURE
115
The above
analysis
is
and Coulomb's law of
in
il
a straightforward application of Newton's laws of motion
electric force
— both
pillars of classical physics
accord with the experimental observation that atoms are
no/
is
in
accord with electromagnetic theory
physics— which predicts
— another
— and
is
However,
stable.
pillar of classical
that accelerated electric charges radiate energy in [he
form of electromagnetic waves. An electron pursuing a curved path is accelerated and therefore should continuously lose energy, rapidly spiraliug into the
Whenever
nucleus (Fig. 4-10),
they have Iwen directly tested, the predictions
of electromagnetic theory have always agreed with experiment, yet atoms
not collapse. This contradiction can
thai are valid in the
mean only one
thing:
The laws
macroscopic world do not hold true
in
do
of phvsics
the Microscopic
world of the atom.
The reason
for the failure of classical physics to yield a
of atomic structure
is
that
it
approaches nature exclusively
concepts of "pure" particles and "pure" waves.
preceding chapters, particles ami waves have
many
in
meaningful analysis
terms of the abstract
As we learned
properties
in
in
two
the
atomic model, which combines classical and modern notions, accomplishes part
from the point of view of
of the latter task. Not until we consider the atom
daily lives, will
our
iti
the macroscopic world.
of the
phenomena under study
the particle behavior of
is
to
The
full
An
dismal failures
waves and the wave behavior of
understood. In the remainder of this chapter
lie
we
particles
shall see
if
how
thai the
likelv,
An atomic
electron
it
radiates energy
acceleration.
due to
vcrifv that a classical calculation ought to
To
we
X
2
be
at least
approximately correct,
note that the de Broglie wavelength of an alpha particle
R'
7
m/s
whose speed
is
is
~
h77
=
X
5
X
6.63
h
X
~
6\6
X
15
m
10
Kr*« kg
R)
M J-s
X
2
X
10 r
m/s
for
the
atom
As
ihe
Bohr
gets to a gold nucleus
we saw
m, which
We are therefore correct
matter entirely
is
wavelength ever
this
6 de Broglie wavelengths, and
in
thinking of the
atom
in
— requires a
in
the
terms of Rutheris
another
nonclassical approach.
ATOMIC SPECTRA
4.6
is
an alpha particle with
X W~ u
model, though the dynamics of the atomic electrons— which
ford's
The
is
'!
reasonable to regard the alpha particle as a classical particle
is
it
in Sec. 4.4, the closest
interaction.
its
not correct, and that the atom in reality does not
is
coincidence that the quantum-mechanical analysis of alpha-particle scattering
From thin foils results in precisely the same formula that Rutherford obtained.
should, classically, spiral rapidly Into the
nucleus as
formula
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
so
FIGURE 410
atom.
we made use of the same laws of physics that proved such
when applied to atomic stability. Is it not therefore possible, even
common, though
allowance must be made-
find a really successful theory of the
scattering formula
validity of classical physics decreases as the scale
decreases, and
we
interesting question arises at this point. In our derivation of the Rutherford
thesnialhiess of Hanek'5 constant renders the wave-particle duality imperceptible
in
intuitive notions acquired
quantum mechanics, which makes no compromise with
ability of the
among
its
Bohr theon
ol the
atom
to explain the origin of spectral lines
most spectacular accomplishments, and so
We
it is
appropriate to preface
atomic spectra.
"in exposition of the theory itself with a look at
have already incut ioued that healed solids emit radiation
wavelengths are present, though with different
intensities.
We
in
which
shall learn
all
in
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
<
'hap.
itself
9
or of the nature of the solid.
we
From
this fact
it
follows that,
when
a solid
are witnessing the collective behavior of a great
is
heated to incandescence,
n
m) interacting atoms rather than the characteristic behavior of the individual
atoms of a particular element.
116
THE ATOM
ATOMIC STRUCTURE
117
absorption spectrum
hydrogen
IHIWlll
FIGURE 4-13
The dark
lines in
of
sodium vapor
emission spectrum
of sodium vapor
the absorption spectrum of an element correspond to bright lines
in its
emission spactrum.
Itelii
spectroscopy
therefore a useful tool for analyzing the composition of an
is
unknown substance.
The spectrum of an excited molecular gas or vapor contains (Hinds which
consist of many separate lines very close together (Fig. 4-12), Bands owe their
origin to rotations
molecule, and
mercury
When
of the
line
7,000
A
6,000
red
orange
FIGURE 4.11
A
5,000
green
yellow
PDrtions of the Bm|jskln
^^
rf nydr()gen
A
hcIlum
4,000
blue
and vibrations of the atoms in an electronically excited
shall consider their interpretation in a later chapter.
light
is
passed through a gas,
wavelengths present in
found to absorb light of certain
The
resulting absolution
missing wavelengths (Fig. 4-13); emission spectra consist of bright lines
lines in the solar
spectrum occur
hecause the luminous part of the sun, which radiates almost exactly according
and mercury.
any object heated
to
5800 K,
is
surrounded by an
envelope of cooler gas which absorbs light of certain wavelengths only.
At
the other extreme, the atoms or
molecules in a rarefied gas are so
on the average that their only
mutual interactions occur during
far apart
occasional
collisions.
be
it is
emission spectrum.
on a dark background. The dark Fraunhofer
violet
to theoretical predictions for
to
its
spectrum consists of a bright background crossed by dark lines corresponding
to the
A
white
we
Under these circumstances we would
expect any emitted
radiation
characteristic of the individual
atoms or molecules present, an
expectation
hat IS realized experimentally.
When an atomic gas or vapor at somewhat
than atmospheric pressure is
suitably "excited," usually by the
passage of an
dectnc current through it, the emitted
radiation has a spectrum which
contains
<***
wavelengths only. Figure 4-1 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
Z
to
In the latter part of the nineteenth century
lengths present in atomic spectra
wavelengths
in
each
series
fall
it
was discovered that the wave-
into definite sets called spectral series.
The
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
by J.
J.
Balmer
in
first
such spectral series was found
1885 in the course of a study of the visible part of the hydrogen
spectrum.
Figure 4-14 shows the Balmer
FIGURE 4-14
The Balmer
series of hydrogen.
aeries.
The
tine
with the longest
*
exdted;
FIGURE 4-12
A portion
of the
band tpectrum
ol
PN.
Ill
I
H,
118
THE ATOM
ATOMIC STRUCTURE
119
1
is designated H
n the next, whose wavelength is 4.86.1 A,
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,
wavelength, 6,563 A,
is
designated
H
beyond which there arc no further separate
lines but only a faint
spectrum.
Maimer's formula for the wavelengths of this series
4.1S
x-
The
quantity R,
fi
-
(i
known
R =
1.097
=
1.097
limit corresponds to
as the
x
X
=
n
n
^)
= 3,
j J Pfund series
J Brackett series
50,000 k
20,000 A
continuous
10,000
A
r
Paschen series
is
5,000
A -
2,500
A
Bal iner series
Balmer
4. 5,
Rydberg constant, has the value
lOT
nr
tO"
3
J
corresponds to n
'ITie II„ line
X
,
,
fl
-
A"
1
—
the
3,
oo, so that
it
H
/}
Hue
to
n
=
2,000 A
4,
and so on. The
occurs at a wavelength of 4/8,
series
riGURE 4-15
The
spectral series of hydrogen.
agree-
in
ment with experiment.
The Bahner
series contains only those
the hydrogen spectrum.
infrared regions
fall
The
wavelengths
s-pectral lines of
in the visible
hydrogen
portion of
in the ultraviolet
inlo several other series. In the ultraviolet the
Lyman
A
1
n
2
= 2,
A -
1,250
A -
.w rirs
contains the vvavelenglhs specified by the formula
4.16
1,500
and
Lyman
3, 4,
n-7
Lvman series
In the infrared, three spectral series have been found
whose component
A
,000
lines
have the wavelengths specified by the formulas
—***••
H
i = (^-i)
17
Paschen
Brackett
4.7
THE BOHR ATOM
Pfund
We saw
The above
spectral series of
Fig 4-15; the Brackett
The value
'ITie
of
R
is
the
existence ul
hydrogen are plotted
series evidently overlaps the
same
120
THE ATOM
test for
terms of wavelength
Paschen and Pfund
in
series.
in Eqs. 4.15 to 4.19.
in
the spectra of
in Sec. 4-5 that die principles of classical physics are
the observed stability of the
to whir]
around the nucleus
hydrogen atom. The electron
to
keep from being pulled
electromagnetic energy continuously.
such remarkable regularities hi the hydrogen spectrum,
together with similar regularities
a definitive
in
more complex elements, poses
any theory of atomic structure.
phenomena,
Uptanarkm
this
it
atom
is
obliged
and yet must radiate
Because other apparently paradoxical
like the photoelectric effect
in terms of
into
incompatible with
in this
and the diffraction of electrons,
quantum concepts,
it is
find
appropriate to inquire whether
might not also be true lor the atom.
ATOMIC STRUCTURE
121
wave behavior of an
Let us start by examining the
a hydrogen nucleus.
X
The dc
electron
around
in orbit
Broglie wavelength of this electron
is
=
nit
where the electron speed v
is
that given
by Eq.
1.1-3:
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
1.6
=
33
This wavelength
X
lfl- 3 " J-s
10- ,9
JO"
1
C V
wave
The fact
joined on
8.85
X
10- 12
x
9J
same
F/m X
10
u
as the circumference of the electron orbit,
hydrogen atom corresponds to one complete
that the electron orbit in a
circumference provides the clue
If
we
hydrogen atom
we need
is
one electron wavelength
to construct
a theory of the atom.
consider the vibrations of a wire loop (Fig, 4-17),
fit
X
itself (Fig. 4-16).
in
wavelengths always
5.3
10~ 3 ' kg
m
10-"
orbit of the electron in a
electron
M*
—* X
m
exactly the
= 33 X
2wr
The
is
X
X
we
find that
their
an integral number of times into the loop's circumference
wave joins smoothly with the next. If the wire were perfectly rigid,
would continue indefinitely. Why are these the only vibrations
a wire loop? If a fractional number of wavelengths is placed around
so that each
these vibrations
possible in
the loop, as in Fig. 4-18, destructive interference will occur as the waves travel
around the bop, and the vibrations
behavior of electron waves
of a wire loop, then,
in the
we may
indefinitely without radiating
number of de
will die out rapidly.
hydrogen atom
as
By considering
the
THE ATOM
electron in 1 hydrogen atom corresponds to i complete electron de
analogous to the vibrations
energy provided that
its
orbit contains an integral
is the clue to understanding the atom. It combines !x>th the
and wave characters of the electron into a single statement, since the
electron wavelength is computed from the orbital speed required to balance the
122
FIGURE 4-16 The orbit of the
Broglie wave joined en itself.
postulate that an electron can circle a nucleus
Broglie wavelengths.
This postulate
particle
electron path
de Broglie electron wave
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
AD integral
number
orbit of radius r
of
to express the condition that
an electron orbit contain
de Broglie wavelengths. The circumference of a
is 27rr,
and so we may write the condition
circiilar
for orbit stability as
ATOMIC STRUCTURE
123
nX
4.21
where
t= 2T7Tn
n
-
1,
2. 3,
.
.
.
r designates the radius of the orbit that contains
n
integer n
is
number
called the ifuuntum
of the orbit.
n wavelengths. Tlie
Substituting lot A, the
electron wavelength given by Kq. 4.20. yields
nh
x /4**!?*
_
2.7TI-
e
and so the stable electron orbits are those whose
radii are
given In
2
n' h*E t>
4.22
FIGURE 4-17
n
The
=
!,
2, 3,
.
.
vibrations of a wire loop.
FIGURE 4-18
A fractional number
because destructive interference
The
=
a(
,
The other
=
2 wavelengths
circumference
—
so that the
-
\\
0.53
is
customarily called the Bohr rmliux of the
denoted by the symbol
is
X
5.3
10"
"
a,,:
in
A
given
radii are
in
terms of a„ by the formula
»"«„
%
4 wavelengths
4.8
wavelength t cannot persist
occur.
radius of the innermost orbit
hydrogen atom and
circumference
of
will
spacing between adjacent orbits increases progressively.
ENERGY LEVELS AND SPECTRA
The various permitted
energy
£n
is
given
K.
in
orbits involve different electron energies.
terms of the orbit radius
rn
by Eq.
The
electron
4. 14 as
-
-
8*V«
Substituting for
circumference
124
THE ATOM
=
r
n
from Eq. 4.22, we see that
Energy levels
S wavelengths
ATOMIC STRUCTURE
125
The
energies specified by Eq. 4,23 are called the energy
The presence
level's
of the hydrogen
atom and are plotted
in Fig. 4-19. These levels are all negative, signifying thai
the electron does not have enough energy to escape from the atom.
The lowest
energy level £,
E2>
&). £4*
•
•
is
called the
a
oe,
atom.
Ex -
state of the atom,
are called excited states.
the corresponding energy
n
ground
and the higher
As the quantum number n
an excited state drops to a lower state, the
in
in certain specific
increases,
En
approaches closer and closer to 0; in the limit of
and the electron is no longer bound to the nucleus to form an
{A positive energy for a nucleus-electron combination means that the
is not bound to the nucleus and has no
quantum conditions to fulfill;
atomic model?
energy
energy
The jump
levels.
is
atom suggests
when an
electron
emitted as a single
exist in
an atom except
of an electron from one level to
another, with the difference in energy between the levels being given off
once in a photon rather than in some more gradual manner,
at
this
fits
all
with
in well
model. If the quantum number of the initial (higher energy) state is n and
quantum number of the final (lower energy) state is n we are asserting that
t
the
,
f
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
both emission and absorption; do these spectra follow from our
lost
photon of light. According to "ur model, electrons cannot
levels
electron
line spectra in
of definite, discrete energy levels in the hydrogen
a connection with line spectra. Let as tentatively assert that,
energy
Initial
where
The
»
—
final
energy
=
photon energy
the frequency of the emitted photon.
is
initial
and
final states
of a hydrogen atom that correspond to the quantum
numbers n and nf have, from Eq.
(
4.23, the energies
free electron
= as
n = 5
n
r
n = 4
n = 3
excited states
v.
r>
=
—
energy, J
-0.87
energy,
x
lO"
-1.36
x
10- ,B
-2.42
X
10
1 *1
1!>
eV
-0.54
-0.85
Hence the energy difference I>etween these
-1.51
states
is
8*o
2
-5.43 x 10-
3.40
me4
&
The frequency
FIGURE
4.
19
Energy tenets of the
v
2
hs
\nj~~nj)
'
of the photon released in
this transition is
K -E,
hydrogen atom.
4.35
8e
In terms of
3
2
/i
\n/'
n
photon wavelength
2
{
/
X, since
we have
4.26
ground
126
state
THE ATOM
n
=
1
-21.76
X
10-'
-
Ki.fi
me
1
n
fk^ch 3 \nf
_ _l
n
2
t
Hydrogen spectrum
)
ATOMIC STRUCTURE
127
=E =
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 must always
7
be greater than the final quantum number n in each case, in order that there
f
be an excess of energy to be given off as a photon, the calculated formulas for
the
first
five series
are
1
StjWVl1
X
'
n,
=
2:
1-
nf
=
3:
I=
»,
=
4:
=
2, 3, 4,
Lyman
>»^
( V
J_\
n
-
3, 4. 5,
B aimer
( 1
J_\
n
=
4, 5, 6,
Paschen
ti
=
5, 6, 7,
Brackett
n
=
6, 7, 8,
Pfyrtd
W\3
2
A
8k
1_
me A ( I
8efdfi\$
1_
n
energy
me*
A
n2 /
me'1
(
n2 I
J_\
n2 /
J
1
\
FIGURE 4-20
These sequences are identical
earlier.
to n
r
=
The Lyman
2;
form with the empirical spectral
in
still
4;
and the Pfund
series
series discussed
the Balmer series corresponds
1;
=
3;
f
-
however. The
final
step
is
-
In the preceding analysis,
5.
The value
of this constant term
me 4
9.1
&j,W
8
=
X
(8.85
1.097
X
to
X
low energy
states as proved,
compare the value of the constant term in the
the Rydberg constant R of the empirical equations
to
above equations with that of
4.15 to 4.19.
from high
x
JO" 12
10*0!
1Q- 31 kg
F/m) 2
X
3
is
X
X
(1.6
10 s
X
1(1" IB
m/s
X
(6.63
indeed the same
spectral lines are related to atomic energy levels.
THE ATOM
much
is
hydrogen nucleus
(a single proton)
very close to the nucleus liecause the nuclear mass is
Because the nucleus and the
greater than that of the electron (Fig, 4-21).
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,
system of this kind
is
equivalent tu a single particle of mass m' that revolves
around the position of the heavier
H)~-» J-sf
that the
in
particle.
most mechanics textbooks; see Sec.
8,8.)
(This equivalence
If
m
is
is
demonstrated
the electron mass
and
M
'
essentially that
128
center of mass, which
A
x
we assumed
remains stationary while the orbital electron revolves around it. What must
actually happen is that both nucleus and electron revolve around their common
C) 1
as fi! This theory of the hydrogen atom, which is
developed by Bohr in 1913, therefore agrees both qualitatively
and quantitatively with experiment. Figure 4-2(1 shows schematically how
is
NUCLEAR MOTION
4.9
the Brackett series corre-
corresponds to n,
cannot consider our assertion that the line spectrum of hydrogen
originates in electron transitions
which
=
corresponds to n
f
the Paschen series corresponds to n
sponds to n {
We
series
Spectral lines originate in transitions between energy levels.
the nuclear mass, the m'
4.27
m +
is
given by
iW
The quantity m' is called
is less than in. To correct
the reduced mass of the electron because
for the motion of the nucleus in the
its
value
hydrogen atom.
ATOMIC STRUCTURE
129
H a line of deuterium,
gen.
The
that
of hydrogen
is
for
example, has a wavelength of 6,561 A, while
6.563 A: a small but definite difference, sufficient for the
identification of deuterium.
ATOMIC EXCITATION
4.10
There are two principal mechanisms
above
is
Both the electron and nucleus ot a hydrogen atom revolve around a
common
center ol
mait.
to
do
is
to
— e.
of mass m' and charge
m'e* /
4.28
imagine that the electron
The energy
1
is
levels of the
replaced by a particle
atom therefore become
\
electric discharge
and atomic
motion of the nucleus,
all
the energy levels of hydrogen are changed
by the fraction
=
0.99945
The use
En
,
being smaller
in absolute value,
of Eq. 4.28 in place of 4,23 removes a small
waveRydberg constant H to eight significant figures without
nuclear motion is 1.0973731 X 10 T m -1 ; the correction lowers it
The value
1.0967758
of the
X lO'm"
1
.
The notion of reduced mass played an important
deuterium, an isotope of hydrogen whose atomic mass
that of ordinary
in the nucleus.
rium are
130
all
THE ATOM
is
mechanism
is
signs
and mercury-vapor lamps are
between electrodes
in
involved
For example, a photon of wavelength
in the
n
=
=
2 state drops to the n
it
up
when an atom
absorbs a photon
the right amount to raise the atom to a higher energy
jnsl
A by
a hydrogen
to the
n
=
2
1
,217
A
1 state;
atom
is
emitted
when a hydrogen
the absorption of a photon
initially in the
n
=
1
state will
This process explains the origin of
state.
lietween energy levels are absorbed
spectral lines of hydrogen and the actual experimentally determined
to
Neon
a strong electric field applied
When white light, which contains all wavelengths, is passed
through hydrogen gas, photons of those wavelengths that correspond to transitions
less negative.
correcting for
by emitting one or more photons. To produce an
an electric field is established which acceler-
absorption spectra.
1,837
but definite discrepancy between the predicted wavelengths of the various
lengths.
whose energy
therefore bring
an increase of 0.055 percent since the energies
are therefore
of light
of wavelength 1,217
1,836
~
energy
ground
ions until their kinetic energies are sufficient to excite
how
different excitation
atom
M+m
m
its
the case of mercury vapor.
level.
M
joint kinetic
will return to
radiation of
a gas-filled tube leads to the emission of the characteristic spectral
bluish light
and
neon
case
of
that gas, which happens to be reddish light in the
A
to
s
to collide with.
familiar examples of
in
Owing
-8
way
excited in this
in a rarefied gas,
atoms they happen
we need
An atom
an average of 10
ates electrons
then, all
can excite an atom to an energy level
One mechanism is a
it to radiate.
with another particle during which part of their
absorbed by the atom.
state in
FIGURE 4-21
ground
its
collision
that
thereby enabling
state,
hydrogen owing
to the presence of
part in the discovery of
is
almost exactly double
a neutron as well
as a
proton
Because of the greater nuclear mass, the spectral lines of deute-
shifted slightly to wavelengths shorter than those of ordinary hydro-
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
atomic energy
in 1914,
levels
These experiments provided a
do indeed
exist and,
direct demonstration thai
furthermore, that these levels are the
same as those suggested by observations of
line spectra.
ATOMIC STRUCTURE
131
of
Franck and Hertz bombarded the vapors of various
elements with electrons
known energy, using an apparatus like that shown in Fig. 4-22. 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
If kinetic
atoms
its
energy
is
loses
electrons arrive at the plate and
conserved
in a collision
In the vapor, the electron
original one. Because
is
so
rises
A
is
(Fig. 4-22).
much
off in a direction different
from
heavier than an electron, the latter
in the process. After a certain critical electron
reached, however, die plate current drops abruptly. The
interpretation
is
this effect is that
or all of
its
FIGURE 4-23
Results of the
Franck-Hertz experiment, showing
A /
critical potentials,
an electron colliding with one of the atoms gives up some
kinetic energy in exciting the
Such a
state.
/
in-
lwtween an electron and one of the
merely lxmiices
an atom
i
V
almost no kinetic energy
energy
of
through the ammeter. As the accelerating potential
i
more and more
creased,
collision
which kinetic energy
is
is
atom
to an
energy level above
its
ground
called trtdosik, in contrast to an elastic collision in
conserved.
The
critical electron
energy corresponds
to
the excitation energy of the atom.
Then,
as die accelerating potential
increases, since the electrons
an
Vis
now have
inelastic collision to reach the plate.
current
i
occurs, which
is
raised further, the plate current again
sufficient
left after
ACCELERATING POTENTIAL, V
experiencing
Eventually another sharp drop in plate
interpreted as arising
higher energy level in another atom.
energy
from the excitation of the same
As Fig. 4-23 indicates, a
potentials for a particular atomic species
is
obtained
in
this
.series
of critical
way. Thus the
highest potentials result from several inelastic collisions
and are multiples of
the lower ones.
mercury—and a photon of 2,536-A light has an energy of just
The Franck-Hertz experiments were performed shortly after llohr an-
spectral line of
eV.
1.9
nounced
his theory of the
hydrogen atom, and they provided independent
confirmation of his basic ideas.
To check the interpretation of critical
potentials as being due to discrete atomic
Frauds and Hertz observed the emission spectra of vapors (luring
electron bombardment, hi die case of mercury vapor,
for example, they found
energy
that a
levels,
minimum
electron energy of 4.H
eV was required
to excite the 2,536-A
4.11
The principles of quantum
in
FIGURE A-22
Apparatus lor the Frant*. Hertz experiment.
filament
THE CORRESPONDENCE PRINCIPLE
i
physics, so different from those of classical physics
he microscopic world that
lies
beyond the reach of our
senses,
theless yield results identical with those of classical physics in
grid
plate
experiment indicates the latter
fundamental requirement
theory of radiation, and
Is
satisfied also
According
to
l
is
lie
to
be
satisfied
valid.
We
lie
must never-
domain where
have alreadv seen that
by the theory of
wave theory
I
of matter;
relativity, the
we
shall
now show
Hint
it
by Bohr's theory of the atom.
electromagnetic theory, an electron moving
radiates electromagnetic
waves whose frequencies are equal
in
a circular orbit
to its
frequency of
(evolution and to harmonics (that is, integral multiples) of that frequencv.
a hydrogen atom die electron's;
speed is
V4iTC
132
this
quantum
In
mf
THE ATOM
ATOMIC STRUCTURE
133
according
Eq. 4.13, where r
to
/of
of revolution
_
the electron
the radius of
is
its
orbit.
Hence
ihe frequency
so that
is
4.30
electron speed
orbit circumference
3
Se^h
fc
When p —
W
3
/
the frequency v of the radiation
1,
exactly the
is
same
as the frequent)'
of rotation/of the orbital electron given in Eq. 4.29. Harmonics of this frequency
2wr
are radiated
2-7T
The
radius
V^wfQmr3
given in terms of
is
its
n
Eq. 4,22 as
=
2, 3, 4,
,
.
,
.
Hence both quantum and
2,
classical pictures
Eq. 4.29 predicts a radiation frequency that
10,000, the discrepancy
is
only about 0.01 percent.
The requirement that quantum physics give the same results as
=
r.
n 2hh n
very large
in the limit of
from that given by Eq. 4.25 by almost 300 percent, while when
differs
quantum Bomber n by
=
When n =
quantum numbers.
of a stable orbit
rn
when p
hydrogen atom make identical predictions
of the
classical physics
quantum numbers was called by Bohr the correspondence
has played an important role in the development of the quantum
in the limit of large
principle.
It
theory of matter.
and so the frequency of revolution
is
Problems
Under what circumstances should the Bohr atom behave
electron orbit
quantum
is
we might
so large that
classically?
expect to be able to measure
be entirely inconspicuous. An orbit
effects should
example, meets this specification;
its
1
cm
quantum number is very close to n
it
If
the
directly,
10,000,
*
and, while hydrogen atoms so grotesquely large do not actually occur because
their energies
would be only
are not prohibited in theory.
atom
will radiate?
What
me* (
n f th energy
1
1
below the ionization energy, they
does the Bohr theory predict that such an
According to Eq.
Hjth energy level to the
^
infinitesimal ly
4.25, a
hydrogen atom dropping from the
level emits a
photon whose frequency
X
of 2.6
across, for
=
A 5-MeV alpha
'1.
10" 13 m.
particle
approaches a gold nucleus with an impact parameter
Through what angle
will it
be scattered?
What is the impact parameter of a 5-MeV alpha
when it approaches a gold nucleus?
2.
What
* 3.
foil
3
x
fraction of a
I0" T
m
thick
is
beam
of
7.7-MeV alpha
particle scattered
particles incident
upon
by 10"
a gold
scattered by less than 1"?
is
What
*4.
\
foil
3
X
fraction of a l>eam of
I0" T
m
thick
is
7,7-MeV alpha
scattered by 90° or
particles incident
upon a gold
more?
* 5.
quantum number n and n — p (where p
quantum number n r With this substitution,
Let us write n for the
initial
{
3,
.
.
.)
for the final
=
1,
2,
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.
A beam
6.
me 4
I
8e u2n 3
me
|>-p) 2
1
inp
ite^h*
n 2 (n
of
8.3-MeV alpha
angles exceeding about 60°.
— pi
— pf
of
2
x
a,
2np
(n
134
THE ATOM
and
n, are
— p'2
— pf
both very
21
inp
~
n2
is
directed at an
aluminum
foil.
It is
If
large,
n
is
much
greater than p, and
the alpha particle
is
assumed
to
have a radius
10~ 15 m, find the radius of the aluminum nucleus.
Determine the distance of
Now, when
particles
found that the Rutherford scattering formula ceases to be obeyed at scattering
J
closest
approach of 1-MeV protons incident upon
gold nuclei.
8-
Find the distance of closest approach of 8-MeV protons incident upon gold
nuclei.
ATOMIC STRUCTURE
135
The derivation
9.
was made
of the Rutherford scattering formula
nonrelativisti-
approximation by computing the mass ratio between an 8-MeV
eally. Justih this
alpha particle and an alpha particle at
rest.
be
Kind
frequency of rotation of the electron
iht?
in the classical
model
The
and
Ii
total
charge
Thomson model
a sphere corresponds to the
in this
from the center of
electric-field intensity at a distance r
charged sphere of radius
of the
this
H when r < /{. Such
atom. Show that an electron
its
center and derive a
motion. Evaluate the frequency of the electron
the hydrogen atom and compare
illations for the case of
uniformly
a
3
(J is ()r/45re
()
sphere executes simple harmonic motion about
formula for the frequency of
with the frequencies
it
A
21.
hydrogen from the n
=
fl
state to the n
=
3
is
from the n
=
HI state to
How much
11.
to the transition
i
energy
its
ground
hydrogen atom goes
a
state.
A
22.
required to remove an electron
is
in
the n
=
2 slate
a hydrogen atom?
A beam
ji'
meson
=
(in
207
of the Balnier series
is
to
bombards a sample of hydrogen.
l>e
A
How many
alum make
dropping
about H)~ M
is
=
1
it
Through what
A
positronium atom
and an electron,
—
3
—»
n
—
2
the
first
is
The average
to the
n
=
emits a photon
19.
20.
I
state?
going
=
{a)
form a
Calculate the angular
ol
of Bohr's
first
THE ATOM
postulate
ii
"
momentum
is
Ha
(b)
line,
is
litjtlrogenic
is
+ Ze
Compare
atom, which
is
and which contains
Sketch the energy levels of the He* ion and compare them
H
atom,
(c)
An
election joins a bare helium nucleus
Find the wavelength of the photon emitted in
ion.
die electron
26'.
(The average lifetime of an
the
atomic state
;i
Find the energy
state.
the wavelength of the photon emitted in the
or Li* + whose nuclear charge
He 1
He 4
assumed
to
have had no kinetic energy when
Use the uncertainty principle
hydrogen atom
is
N
10
energy
bytfeogep
s.
is
If
the wavelength
5.000 A,
find the
it
this process
combined with
to
determine ihe ground-state radius
terms of the
momentum
r,.
Add
an electron must have
this kinetic
potential energy of an electron the distance
wiih respect to
by Eq.
r,
r,
for
J.22 with
which E
=
l
of
r,
from
if
confined to
energy to the electrostatic
a proton,
and
different late
the resulting expression for the total electron energy
it
r
following way. First find a formula for the electron
is
a
minimum. Compare the
result
I:
to
liiul
with that given
1.
atom?
about the nucleus of an electron
that the angular
in
in Ihe
a region of linear dimension
the value of
the binding energ)
ground
Derive a formula for ihe energy levels of a
an ion such as
2 state of a hydrogen
line.
eijiial
its
the ionization energy in positronium with that in h dr ogen.
y
kinetic
nth orbit of a hydrogen atom, and show from
136
in
At what temperature will the average molecular kinetic energy in gaseous
hydrogen
of a titanium atom.
the nucleus.
of the spectral line associated with the decay of this state
width of the
form a "mesne
In
Bohr orbit of such an atom.
transition in positronium with that of the
line
s.)
lifetime of an excited
Hn
apart in wavelength will the
a system consisting of a positron (positive electron)
Compare
(a)
a single electron, (b)
if
state.
revolutions does an electron in the n
Ixjfore
excited state
)H,
4 stale to the n
far
can be captured by a proton
first
meson is in the n — 2 state
when the mesic atom drops to
emitted?
Find the recoil speed of a hydrogen atom after
=
r)
(i~
radiated
to
from the n
m
Find the radius of the
atom."
25.
of electrons
How
spectrum observed.
its
with the energy levels of the
17.
is
the two kinds of hydrogen be?
lines of
ii
potential difference must the electrons have been accelerated
HI
momentum
approximately three times more massive than ordinary hydrogen, b
excited and
24.
15.
quantization
this
of a system in one
State.
Find the wavelength of the photon emitted when
13.
ron
8 that
momentum
mixture of ordinary hydrogen and tritium, a hydrogen isotope whose
nucleus
23.
Find the wavelength of the spectral line corresponding
12.
I
shall see in (-hap.
of the angular
quantized in a somewhat different way.)
of the spectral lines of hydrogen.
in
the
frequency?
11.
<ist
was
We
as yet.
component
particular direction, while the magnitude of the total angular
of
ft"
had not been proposed
of the
hydrogen atom. In what region of the spectrum are electromagnetic waves
this
in units of
work, since the hypothesis of de Broglie waves
rule holds only for the
10.
momentum
(In fact, the quantization of angular
nti.
starting point of Bohr's original
this that
momentum
in the
an alternate expression
of such an
atom must
ATOMIC STRUCTURE
137
QUANTUM MECHANICS
The Bohr theory of the atom, discussed
for certain
in the
previous chapter,
experimental data in a convincing manner, but
it
is
5
able to account
has a
number of
severe limitations. While the Bohr theory correctly predicts the spectral series
of hydrogen,
extended
each;
it
hydrogen isotopes, and hydrogenic atoms,
to treat the spectra of
can give no explanation of why certain spectral
than others (that
is,
why
certain transitions
probabilities of occurrence);
and
slightly.
it
individual
These objections
for the theory
to the
many
differ
it
does not permit us to obtain what a
atom should make
an understanding of
possible:
endow macroscopic aggregates
and chemical properties we observe.
to
Bohr theory are not put forward
was one of those seminal achievements
in
an unfriendly way,
that transform scientific
thought, but rather to emphasize that an approach to atomic
greater generality
is
more intense
have greater
whose wavelengths
atoms interact with one another
of matter with the physical
levels
cannot account for the observation that
And, perhaps most important,
really successful theory of the
incapable of being
lines are
between energy
spectral lines actually consist of several separate lines
hew
it is
complex atoms having two or more electrons
required.
Such an approach was developed
phenomena
in
by Erwin Schrodinger, Werner Heisenberg, and others under the apt
quantum
to
it
of
1925-1926
name
of
quantum mechanics
meclianics. By
problems involving nuclei, atoms, molecules, and matter in the solid state made
possible to understand a vast body of otherwise-puzzling data and
a vital
—
attribute of
5.1
the early 1930s the application of
any theory
— led to predictions of remarkable accuracy.
INTRODUCTION TO QUANTUM MECHANICS
Hie fundamental difference between Newtonian mechanics and quantum mechanics 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, acceler139
be measured. This assumption is, of course, completely valid
our everyday experience, and Newtonian met banks provides the "correct"
explanation far llic behavior of moving bodies in the sense that the values il
ation, etc.. vim
in
predicts for observable magnitudes agree with the
measured values of those
maunitudrs.
mean anything:
between olwervable magni-
i,
defined,
tudes, but tile uncertainty principle radically alters the definition of
'observable
if
>k
is
II
is
loo, consists of relationships
magnitude"
in
position and
momentum
the atomic realm.
According
ol a particle
to the uncertainly principle, the
cannot be accurately measured
tainable values at every instant.
The
quantities
mechanics explores are probabilities. Instead of
if
X
10
> m, quantum mechanics
we conduct
the
same
to have definite, ascerwhose relationships quantum
a suitable
states that this
experiment, most
lie
atom
is
experiment
is
a
X
then
/'
only a single
sel,
A wave function
cerned
warn function
is
the
3,
the quantity with which
interpretation, the square of
its
*
of a body.
and quantum me
we
bility
quantum mechanics
is
is
to
+
itself
absolute magnitude j+[ 2 (or
of experimentally finding the body there
determine
'I'
for a
at
we
is
++*
140
space must be
if
«i'
is
complex)
proportional to the proba-
that lime.
body when
consider the actual calculation of +,
THE ATOM
finite
— the body
is
If l^l"'
its
somewhere, alter
1
obeys Eq. 5.2
that
The problem
<>l
freedom of motion
how
that
all.
If
this
+
at all times
Kverv acceptable
it
and
its
H"
const. ml,
done.
is
must be single-valued, since
partial derivatives
•
A
t,
/'
can have only
further condition that
•
+
*!'
must obey
continuous every-
"\'/<)z l>e
i/.
-
where.
s
equation, which
same sense
second law of motion
equation of Newtonian mechanics,
is
we
let
tackle .Sehrodinger's equation,
fcr*
quantum me-
the fundamental equation of
is
that the
wave equation
a
ils
is
the fundamental
in the variable
+. Before
review the general wave equation
Wave equation
~
e»
i
fi
wlueli governs a
wave whose
direction with the speed
o.
establish certain
variable quantity
In the case of a
I^plaeeinenl of ibe string from the
me
in
a stretched string,
the ease ol
ij
is
is
a
r/
is
sound wave.
the
1/
is
either the electric or
derived
in
textbooks of
textbooks of electricity and magnetism
electromagnetic waves.
Solutions to the
of
axis; in
thai propagates in the x
ij
in
The wave equation
mechanics for mechanical waves and
•or
.v
is
wave
pressure difference; in the case of a light wave,
die magnetic field magnitude.
we can
somewhere
said to Ix: normalized.
is
has no physical
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
finding
true thai
one value at a particular place and lime.
con-
limited by the action of external forces.
liven liefore
P.
Normalization
Besides being normalizable,
5.3
is
=
shall shortly sec exactly
v/mWtJigrr
quantum mechanics
While
evaluated at a particular point at a particular time
P of
wave function can be normalized by multiplying by an appropriate
it.
THE WAVE EQUATION
Chap.
l>e
the mathematical statement that the particle exists
is
chanics in the
in
to the probability
(IV=l
I'dV
f
is
chanics represents our best effort to date in formulating
As mentioned
be equal
|4'|-
by +, rather than merely be proportional to
must
it
i
\^\
j
HI" 11 m.
individual atoms that departures from average behavior are uniiotieeable.
Instead of two sets of physical principles, one for the macroscopic universe and
5.2
it
integral be a finite quantity
its
X
5.2
many
is
that
is
awl
complex because of the way
to descrilx; properly a real body.
a different value, eitl.er
found will be 5.3
tersiott of tpiantaia nirriinnies. The certainties
Newtonian mechanics are illusory, and their agreement with
consequence of the fact thai macroscopic bodies consist of so
one lor the microscopic universe, there
integral obviously cannot be oc
since
nothing but an approximate
proclaimed In
possibility left
usually convenient to have
equal
In
and the
aiwavs exactly
At first glance quantum mechanics seems a poor substitute for Newtonian
mechanics, hut closer inspection reveals a striking fact: Xnutonuin nieehimif.
It
and so the only
the particle described
is
exist,
j^l- cannot be negative or
the most pmbiibh- radios;
is
trials will yield
larger or smaller, but the value most likely to
example, that the
asserting, for
radius of the electron's orbit in a ground-state hydrogen
5,3
at
Newtonian mechanics Imth are assumed
time, while in
does not
0, the particle
is
still
Quantum mechanics,
l+l-dV
J
S.1
wave equation may be of many
waves thai can occur
— a single traveling pulse,
kinds, reflecting the variety
a train of
waves of constant
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 wavelengths, a standing
wave
in a string fastened at
both ends, and so on.
AH
solutions
must be of the form
SCHRODINGER'S EQUATION: TIMEDEPENDENT FORM
5.3
quantum mechanics the wave function * corresjionds to the wave variable tj
wave motion in general. However, +, unlike t/, is not itself a measurable
In
of
quantity and
5.4
'I
where
F
=
''('
4> is
*i)
any function that can
is
lie differentiated.
+x
represent waves traveling in the
represent waves traveling in the
Our
interest here
particle that
is
in the
—x
direction,
and the solutions F(t
—
+
x/v)
x/v)
When we
of a "free particle,"
namely a
This equivalent corresponds to the general
undamped (that
is,
constant amplitude A), monochromatic
(constant angular frequency w) harmonic
waves
in the
+*
replace
which
total
a
complex quantity, with both
= cos —
e~'*
is
in the
and v by Xr, we obtain
we already know what f and A are in terms
momentum p of the particle being described by *.
=
he
real
and imaginary
that
of the
Since
2irHp
=
h.
=
?lEE
parts.
+ = Ae
5,10
y
5.7
= A cos aft - — I
is
a mathematical description of the
unrestricted particle of tola] energy
iA sin ult
direction, just as Eq. 5.5
- —1
is
/.
and
wave equivalent of an
p moving in the +x
momentum
a mathematical description of, for example, a harmonic
wave moving freely along a stretched string.
The expression for the wave function <fr given by Eq. 5.10
displacement
real part of Eq. 5.7 has significance in the case of
where
<"*><£<-«"'
sin
i
Equation 5.10
ij
waves
represents the displacement of the string from
(Fig. 5-1); in this case the imaginary part
is
in a stretched
freely
its
normal position
moving
particles, while
motion of a particle
is
Wxis
In the
xy plane traveling (n the
+1
wc
is
correct only for
are most interested in situations where the
subject to various restrictions.
An important
concern,
discarded as irrelevant.
for
example,
is
an electron Ixmnd
What we must now do
FIGURE
2t,v
convenient since
E=
Eq. 5.5 can be written in the form
5-1
assume
we have
5.6
string,
above formula by
shall
and
Because
Only the
w
energy E and
\
is
reason
by
* = Ae -3'""-"'"
5.9
direction,
J»-MI~i/»)
— Ae
=
In this formula y
this
direction
Ae- M> -* /pi
* =
5.8
not under the influence of any forces and therefore pursues a
is
solution of Eq. 5.3 for
y
solutions F(t
.t
we
For
therefore be complex.
direction.
wave equivalent
straight path at constant speed.
5.5
The
may
specified in the
direction along a stretched string lying
on the i
which
We
we
!>egin
an atom by the electric
field
of
its
nucleus.
obtain the fundamental differential equation for
is
can then solve
to
4',
in a specific situation.
by differentiating Eq. 5.10 twice with respect
to x, yielding
= -£- +
5.11
ft*
and once with respect to
3l
ff
142
THE ATOM
= Atoswll -i/o)
(.
yielding
*
ft
A speeds small compared with
I
that of light, the total energy
E
of a particle
QUANTUM MECHANICS
143
is
the
sum
of
its
p-/2m and
kinetic energy
potential energy V,
its
general a function of position x and time
in
V
where
is
t
qilorcd. In other words, Schrodinger's equation cannot be derived from
principles," but represents a
<:
In practice,
5.13
=
/;
+ v
principle
first
Schrodinger's equation has turned out to Iw completely accurate
To be
predicting the results of experiments.
in
sure,
we must keep
Eq. 5.18 can be used only for nonrelativistic problems,
Multiplying both sides of this equation by the wave function
formulation
'I'.
FA'
5.14
From
= *-— +
we
Kqs. 5.11 and 5.12
p»* =
s.i6
-ft 2
see that
is
when
in
particle speeds
comparable with
accord with experiment within
its
that of light are
range of applica-
are entitled to regard Schrodinger's equation as representing a success-
same sense
But for
all
its
as the postulates
of special relativity or statistical
mechanics: None of these can be derived from
some other
a fundamental generalization neither
less valid
principle,
and each
is
than the empirical data
it
based upon.
is
It
is
more nor
worth noting
in this
connection that .Schrodinger's equation does not represent an increase in the
—
<
it
success, this equation remains a postulate in the
3/
and
required
in mind that
and a more elaborate
postulate concerning certain aspects of the physical world.
ful
r
we
bility,
\'*l'
2m
is
Because
involved.
"first
itself.
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-
\-
and p-1'
Substftttting these expressions for !'A'
into Eq. 5.14,
we
obtain
ties
relates are understood to
it
be averages rather than definite values,
Time-dependent
Equation 5.17
is
Schrodinger's equation
dx 2
2m
3<
in
the time-dependent
form of
one dimension
In three
Schrddktg/sr's equation.
dimensions the time-dependent form of Schrodinger's equation
is
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/ 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
r
2m
9f
where the
<
I
that
may be
wave
finiction
V
on
present
is
p
+
+
V
J
form of probabilities and not specific numbers.
some function of
the
particle's
Once Vis known,
polential-energy tun. linn V.
solved for the
+
dtf-
energy
particle's potential
lions
+
&*
x,
;/,
motion
~,
will
and
ailed
Schrodinger's equation
of the particle, from
which
its
Am
/.
Ihe
may be
probability density
ma) be determined lor a specified v, ;/, :, /.
The manner in which Schrodinger's equation was obtained
'I'-
wave Function of a
freely
Ncliriidingcr's equation
energy
that
no
V=
vary
in
a priori
moving
particle deserves attention,
to print- that
=
ibis
V(.v,r/ .».()]
extension
postulate Sehriklinger's equation, solve
compare
The
it
is
is
entirely plausible, but there
correct.
All
embodied
in
we
can do
the postulate must be discarded and
THE ATOM
is
is
valid:
if llie)
If
is
to
and
for a variety of physical Situations,
Schrodinger's equation
this
they
disagree,
some other approach would have
to
he
let
the value of
is
us calculate the expectation value (x> of the position of
1"
is
Hie distribution,
__
described by the
if
results,
we shall
the average position x of a
such a way that there
The average position in
is
obtain
particles
the procedure clear,
axis in
J
wave function ^(x.i).
we determined experimental l\ ihe
described by the same wave function at some
X axis that
we would
and then averaged the
make
What
(
t
,v
many
positions of a great
llie
extension of
the results of the calculations with the results of experiments.
agree, the postulate
144
starling from
from the special case of an unrestricted particle potential
time and space [V
an example,
idsiiiin
constant) to the general case of a particle subject to arbitrary forces
way
Ys
a particle confined to the
first
answer
number
are
;V,
this
case
a slight!) different question:
of particles distributed along the
particles at x,,
is
N2
particles at x.„
and
the same as the center of mass of
and SO
+ X2 x2 + A a +
.Y, + V + N +
2
a
'
,V,x,
:l
S Y
.v
•••
r
2M
QUANTUM MECHANICS
145
When we
are dealing wilh a single particle,
by the probability
of particles at i,
dx
at
This probability
x,.
=
/>
2
i*,,
we mnsl
the particle
P, that
replace the numlier
found
Ije
in
N
t
an interval
is
= A e-HB/KH
5,22
dx
That
wave function evaluated
where +, Ls the particle
tution and changing the summations
at
to integrals,
value of the position of the single particle
.r
=
we
Making
.*,.
this substi-
see that the expectation
+
is,
is
the product of a time-dependent function e "*'*
dependent function $. As
particles acted
happens, the time variations of
it
f
<*>
5,19
j
If
+
is
x\*]
we
m
and
and therefore has the value
=
|*
;
|
if
a
|+]
and the x
is
axis
somewhere between
= - oo
x
is
and x
=
dH
Equation 5.23
mensions
Ls
it
as that
=
must be evaluated
Steady -state
°
Schrodinger's equation
form of Schrodinger's
the steady-state
is
Bfy
3ty
5.24
if
2m
3ty
Steady-state
in
In general, Schrodinger's steady-state equation
values of the energy E.
any mathematical
Expectation value
at a particular
obtain a
What
is
meant by
Uwt may
difficulties
time
wave function ^
since
(
+
is itself
a function of
function— namely, that
SCHRODINGER'S EQUATION: STEADY-STATE FORM
A
lie
situations the potential energy of a particle does not
In a great
many
upon time
explicitly; the forces that act
THE ATOM
all
upon
this is true,
reference to
of an unrestricted particle
it,
and hence
may
l>e
vary with the
may be
the one-dimensional wave
Schrodinger's equation
We note that
/.
V,
depend
written
this
can
solved only for certain
lie
statement has nothing to do with
present, but
is
something much more
Schrodinger's equation for a given system means to
that not only obeys the equation
it
and
fulfills
and whatever boundary
the requirements for an acceptable
derivatives be continuous,
its
finite,
and
wave
single-
If there is no such wave function, the system cannot exist in a
steady
Thus energy quantization appears in wave mechanics as a natural element
and energy quantization
phenomenon
in solutions
string of length
/.
world
ts
revealed as a
manner in which energy quantization
of Schrodinger's equal ion
that
propagating indefinitely
and
in the physical
characteristic of all stable systems.
familiar and quite close analogy to the
occurs
by removing
three dimensions
valued.
f.
state.
When
To "solve"
conditions there are, but also
of the theory,
+
In three di-
etiuation.
followed aliove can be used to obtain the ex-
G(x) varies with time, because <C(*}> in any event
position of the particle only.
one dimension
Schrodinger's equation
universal
146
^=
E
TF< "
located at the center of mass (so to speak) of
J"c(*))*|*rfx
This formula holds even
function
2m
cut out, the balance point will he at (*>.
(C{x)>
simplified
+
is
5.21
exponential factor,
is
fundamental,
5.5
common
in
pectation value <C(x)> of any quantity [for instance, potential energy V(x}\ that
by a wave function +,
is a function of the position x of a particle described
result
dividing through by the
a?
fxWfdx
The same procedure
The
so,
oo,
plotted versus x on a graph and the area enclosed by the curve
is
V
dx 2
equal to
Hence
1.
This formula states that <x>
2
an
of Eq. 5.22 into the time-dependent
find that
2m
the probability that the particle exists
<*>
functions of
as that of
|*|* dx
a normalized wave function, the denominator of Eq. 5.19
5.20
all
dx
—»
=
*
Substituting the
form of Schrodinger's equation,
2
and a position-
'
upon by stationary forces have the same form
unrestricted particle.
is
1
is
in
fixed at
is
both ends.
with standing waves
one direction, waves are traveling
—x directions simultaneously
its
in
both the
+x
subject to the condition that the displacement
V always be zero at both ends of the string.
the displacement must, with
in a stretched
Here, instead of a single wave
derivatives,
An
acceptable function
y(x,t) for
obey the same requirements of
QUANTUM MECHANICS
147
continuity, finiteness, ami single-valuedness as
Since
1/
and, in addition, must
4-
The
represents a directly measurable quantity.
real
lie
only solutions of the
wave
equation
1
dx
come from
called eigenfuiwthiiif;. (These terms
E„
3^y
=
32tt% 2H 2
=
n
(*)
which the wave-
lengths arc given by
An
rr
n
+
in
=
(),
I, 2,
3,
.
.
.
is
can
Fig. 5-2.
It
is
the combination of the
wave equation ami
on the nature of its solution that leads ps
only for certain wavelengths A„.
exist
to
I
he
found
to
in stable
hydrogen atom, we
systems
total
momentum
angular
is
shall find that die
momentum
E = Vt(t+
t
F1
Of
energy that
L. In the case
eigenvalues of the magnitude of
are specified by
l)fi
/
=
course, a dynamical variable
urements of
G
made on
result but instead
5-2
these
wave functions
hydrogen atom.
be quantized
die total angular
why
Chap, 6
shall see in
are the only ones that yield acceptable
conclude thai
The values of energy En for which Schrodinger's steady-stale equation can
be solved are called eigenvalues and the corresponding wave functions ii are
FIGURE
2, 3,
important example of a dynamical variable other than
1
restrictions placed
y(x,t)
E
for the electron in the
2L
=
particular values of
of the
shown
1,
we
are an example of a set of eigenvalues;
as
German Eigtmwm, meaning
2
that are in accord with these various limitations are those in
<V.
the
"proper or characteristic value," and Eigtinfunktion, or "proper or characteristic
function.") The discrete energy levels of the hydrogen atom
Standing waves In a stretched siring fastened at both ends.
<G>
<>,
(n-
2
1,
G may
1)
not be quantized. In this case meas-
a numlier of identical systems will not yield a unique
a spread of values whose average
the expectation value
is
= f'Gh!,\?dx
\=2L
In the
that
hydrogen atom, the electron's position
we must
is
not quantized, for instance, so
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
it
fact that
experiments performed on hydrogen atoms always show that
contains one whole electron, not 27 percent of an electron in a certain region
and 73 percent elsewhere; the probability
is
although
space, the election itself
5,6
this probability is
THE PARTICLE
IN
smeared nut
in
one of finding the electron, and
in its
simpler steady-state form, usually
requires sophisticated mathematical techniques.
traditionally
reality,
is
the theoretical structure
we must
understanding of
explore
its
modem
mathematical background
ri
148
THE ATOM
+
I
whose
As we
this
reason the studv of
advanced students
for
results are closest to
shall see,
if
we
to
its
experimental
are to achieve any
even a relatively limited
sufficient for us to follow the trains of
have led quantum mechanics
who
However, since quantum me-
mediods and applications
physics.
is
For
been reserved
have the required proficiency in mathematics.
chanics
not.
A BOX: ENERGY QUANTIZATION
To solve Schrodi tiger's equation, even
quantum mechanics has
is
thought that
greatest achievements.
QUANTUM MECHANICS
149
Our firsl problem using Schrodinger's equation is that of a particle bouncing
back and forth l>etween the walls of a box (Fig, .5-3}. Our interest in this
problem
how Schrodinger's equation is solved when the motion of
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
is
threefold: to see
a particle
V =
since
2
total derivative tl'^/dx'
(The
there.
derivative d^/Sx2 because
<p is
same
the
is
as the partial
a function of x only in this problem.) Equation
5.25 has the two possible solutions
is
5.26
=A
tfr
f2m~E
sin
2
V
with
ft"
those of Newtonian mechanics.
We may specify the particle's motion by saying that
along the x axis between x =
and x = L by infinitely
it is
does not lose energy
stays constant.
potential energy
is
From
V
a constant—say
have an
wave
infinite
function
when
hard walls.
collides with such walls, so that
the formal point of view of
its
A
for
of energy,
<
it
and x
cannot
>
/.,
energy
quantum mechanics,
exist outside the box,
Our
task
is
to find
and so
what +
the
is
its
within
= and x = L.
Within the box Schrodinger's equation becomes
the box, namely, Ixjtween x
d 2^
B cos
particle
total
is infinite on both sides of the Iwx, while
V
convenience— on the inside. Since the particle cannot
for x
2,i,l
f2m~E
=
y
5.27
which we can verify by substitution back into Eq. 5.25;
A and
solution.
of the particle
amount
is
it
restricted to traveling
These
ii
solutions are subject to the important
for x
=
the particle because
it
for
x
=
and
=
Since sin
^
will
be
=
jc
2f"£
llm'E
2m
V
Since cost)
=
does not vanish
at
L.
0, the first solution
at
ft
sum
their
/..
,
L
^—
2
5.28
x
boundary condition
=
0.
Hence we conclude
always yields y
only
when
=
v, 2ff, .Jw,
,
=
nir
Eq. 5.28
it is
=
at
(1
x
=
that
=
li
0, as required,
(1.
but
„
it
.
.
=
.
1,
2, 3,
all 0,
clear that the energy of the particle can have only certain
which are the eigenvalues mentioned
values,
=
^
that
the second solution cannot describe
1,
This result comes about because the sines of the angles », 2v, 3^, ... are
From
a
also
is
are coast ants to be evaluated.
previous
in the
.section.
These
eigenvalues, constituting the energy levels of the system, are
E =
5.29
The
FIGURE
A
5-3
box of width
I
.
particle confined to a
A
uMPs2
n
2mL'
=
Particle in a
I, 2, 3,
integer n corresponding to the energy level K„
called
ts
its
box
quantum number.
Iwx cannot have an arbitrary energy: the fact of
particle confined to a
confinement leads to restrictioas on
its
wave function
that permit
it
to
its
have only
those energies specified by Eq. 5.29.
It is
significant that the particle cannot
function
ij,
would have
to
the particle cannot be present there.
for the
energy of a trapped particle,
of definite values,
classical
is
THE ATOM
if it
did, the
the box. and tlm
exclusion of
E—
like the limitation of
all
energies, including zero, are
principle provides confirmation that
Because the particle
150
The
in
wave
means
that
as a possible value
E
to a discrete set
a quantum-mechanical result that has no counterpart in
mechanics, where
The uncertainty
have zero energy;
be zero everywhere
is
trapped
in the box, the
E=
presumed
is
uncertainty in
possible.
not admissible.
its
position
is
QUANTUM MECHANICS
151
=
Ax
L, the width of the box.
The uncertainty
in its
momentum must
therefore
be
700
which
to
/;,
E =
not compatible with
is
We note that
0.
is,
since the particle energy here
is
the
momentum
corresponding
suo
~
^00
—
3C0
—
entirely kinetic,
nh
which
in
it
Why
are
accord with the uncertainty principle.
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
we
door can have any speed, and therefore any energy,
choose to give
it,
including zero. In order to assure ourselves that Eq. 5.29 does not conflict with
FIGURE
5-4
Energy
confined to a box
1
levels of
A
an electron
wide.
our direct observations while providing unique insights on a microscopic scale,
we
compote
shall
widu and
In case
|2:
I
;i
the permitted energy levels of (1) an electron in a Ijox
lOaa W 'kk-.
=9.1 X W"*1 kg and L =
10-g marble
m
we have
1
A
w
!>m
a
in
I
A =
10" w m, so that the
permitted electron energies are
E" ~
=
=
n2
X
X
2
6.0
iff
:,1
9.1
x
X (1-054 X lQ-^J-s) 2
X 10 kgX (10- ,o m) 2
2
10- 18 n z
J
Z
3H« eV
The minimum energy the electron can have is 38 eV, corresponding to n =
The sequence of energy levels continues with E^ ~ 152 eV, E, = 342 eV, E A
60S eV, and SO on
make
These energy
(Fig. 5-4).
levels are sufficiently far apart to
the quantization of electron energy in such a box conspicuous
box act n ally did
In case 2
f
=
if
such a
so very close together, then, that there
exist.
we have
m=
10 g
=
I0~s kg and L
=
10
cm =
10
-1
m, so
that the
Hence
permitted marble energies are
2
IL
=
1
n %<x'
2
=
5.5
X
X
X
X
J-s)'
X (lO^ni)2
(1.054
10" a kg
lO-'Mfi 2
5.7
J
The minimum energy the marble can have
n
=
and
1.
is
A marble
with
this kinetic
is
5.5
X
-84
lO
J,
corresponding to
X 10~ 31 m/s
energy has a speed of only 3.3
therefore experimentally indistinguishable from a stationary marble.
reasonable speed a marble might have
energy level of quantum number n
152
this
10- 34
THE ATOM
=
is,
say,
10 3u
!
—
no way
to
determine whether the marble
in the
domain of everyday experience quantum effects are imperceptible;
in this domain of Newtonian mechanics.
accounts for the success
THE PARTICLE
IN
In the previous section
whose energy
is
A
/3 m/s which corresponds to the
The permissible energy levels are
l
is
can take on only those energies predicted by Eq, 5,29 or any energy whatever.
V
=b
E
A
A BOX: WAVE FUNCTIONS
we
found that the wave function of a particle
in a
b
is
sin
/
V
ft
2
QUANTUM MECHANICS
153
The normalized wave
Since the possible energies arc
densities li^l 2 \$ 2 2
,
,
and
as well as positive,
\$J-
=
*;
\
2mL3
at
a given x
case
substituting E„ for
E
a
is
functions i^,
|^ 3
is
2
j
and
^
together with the probability
While ^„ may
always positive and, since ^„
equal to the probability
=
=
if.,,
are plotted in Fig. 5-5.
P of
is
l>e
normalized,
finding the particle there.
negative
its
value
In every
m
at x
and x
L, the boundaries of the box. At a particular
point in the box the probability of the particle being present may
be verv
yields
ji/J
quantum numbers. For instance, |^,p has its maximum
middle of the box, while \^\ 2 =
there: a particle in the
of n = 1 is most likely to be in the middle of the box, while
different for different
#„
5.30
A
a*
-jr-
sin
for the cigenfunclions
value of
to the
corresponding
energy eigenvalues
En
lowest energy level
we
easy to verify that these eigenfunctions meet all the requirements
of
function
single-valued
have discussed: for each quantum number n, & n is a
over
of
integral
^Jand t>4> /Bx are continuous. Furthermore, the
x, and
$n
space
n
finite, as
is
(since the particle,
we can
see by integrating
by hypothesis,
\4>
nr
dx from x
confined within these
is
=
to x
=
L
limits!:
_
To normalize y we must
a
2
is
never there!
for the particle Ixn'ng
Classical physics,
anywhere
in the
box.
The wave functions shown in Fig. 5-5 resemble the possible vibrations of a
string fixed at both ends, such as those of the stretched string of
Fig. 5-2. This
is a consequence of the fact that waves in
a stretched string and the wave
when
identical restrictions are placed
same form,
upon each kind of wave, the formal
results are identical.
farx\
FIGURE
If |^„i
/'.
assign a value to
A
such that
the particle between x and
Pdx of finding
proportional to
5.32
same probability
5-5
Wave
functions
and
probability densities of a particle confined to a box with rigid walls.
»A»f
5.31
probability
.
of course, predicts the
so that,
<-m*
L
a particle in the next higher state of n
representing a moving particle are descriljed by equations of the
f\^dx
f\^?dx =
in the
.
It is
all
%L
2
is
to equal
P,
then
it
at
+
|^„i
2
is
eoutif to the
dk, rather than merely
must be true that
f$J*dx=l
since
j
is
Pdx =
the mathematical
Comparing
1
way
Eqs. 5.31
and
a box are normalized
5.33
of stating that the particle exists
5.32,
wc
see that the
wave
somewhere
at all limes.
functions of a particle in
if
-A
The normalized wave
/2
functions of the particle are therefore
nxx
*
154
THE ATOM
=
x
=L
*=0
*=L
QUANTUM MECHANICS
155
THE PARTICLE
5.8
IN
A NONRIGID BOX
either
— that
As we
the particle will "leak" out.
Chap,
shall see in
the
12,
quantum-mechanical prediction that particles always have some chance of
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
ease of a vibrating string
at
is
a finite quantity; the corresponding situation in the
would involve an imperfect attachment of the
each end, so that the ends can move
to treat,
and we
shall
wheu we examine
look at a particle in a nonrigid box
in
Chap.
The
When
string
shall take
another
are therefore
the theory of the
deulemn
particle's
energy
is
smaller than the value of
a definite probability Chat
it
be found outside
V outside
itl
the box, there
In other words,
even
though the particle does not have enough energy to break through the walls
of the box according to
"common
them. This peculiar situation
is
sense,"
the uncertainty
Iv
may
nevertheless
somehow
penetrate
readily understandable in terms of the uncertainty
Because the uncertainty Sp
principle.
it
in its position
in
a particle's
mo men turn
related to
is
hy the formula
consequence:
V.
Such a
to
penetrate
above
less
to
V.
than
it is
its
now
is
V outside
energy
walls,
energy
its
the
kinetic energy inside,
which
it
infinite uncertainty in particle
In the optics of light waves,
reaches a region where
momentiuu outside
infinite
amount
implying that
of energy
V
=
if its
is
never there,
the box
A
is
the price of
particle requires an
momentum is to have an infinite uncertainty,
If V instead has a finite value outside
sc outside the box.
the box, then, there
is
some pmlrability
— not
necessarily great, but not zero
is
Wave
5-6
functions and probability densities of a particle confined to a box with nonrigid walls.
and
it
wave
its
A Av
it
etc.)
and
p
in
The
the case of sound waves,
its first
derivative
t>e
it
effect
is
if
V
is,
in the
THE ATOM
wave
to
is
all
we
the reason
types of waves,
case of electromagnetic waves,
wave height h
in the case of
wave
we saw
water waves,
function
\p
representing
reflection occurs at the boundaries
\p is
we
in
above, decreases in waveif
V
increases.
In either
between the regions. What
are discussing the motion of a single
related to the probability of finding the particle in a particular
if
is,
\p
means
we
shoot
that there
many
is
a chance that the particle
particles at a
box with nonrigid
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
the fact that they have insufficient energy to penetrate
156
This
prediction complements the "leaking" out of particles trapped
=L
light
a Tegion of different index
common
decreases and increases in wavelength
some
be reflected. That
wails,
x
4-
from the requirement that the
to follow
£
has a different potential energy, as
place, the partial reflection of
—
so
continuous at the boundary where the wavelength
does "some" reflection mean when
will
x
value
always
a moving particle. The wave function of a particle encountering a region
particle? Since
z
is
change takes place.
length
^^-^^
V,
box according
when a
readily observed that
is
may be shown mathematically
situation
#,
in the
—
means longer wavelength, and
wavelength changes (that
variable (electric-field intensity
pressure
which
J
that exceeds
may have any
£ since V =
just
Exactly the same considerations apply to the
FIGURE
E
levels.
has another
has a longer wavelength outside the box than inside.
see our reflections in shop windows.
definitely establishing that the particle
finite
always has enough energy
of refraction), reflection as well as transmission occurs.
an
lower energy
energy outside the box, E
Less energy
our original specification.
to
box be
not quantized but
is
particle's kinetic
box
into the
fit
have an energy
possible for a particle to
and
can
of the box with rigid walls,
not trapped inside the box, since
However, the
its
in the case
momenta and hence
that the potential
particle
wave function ^ n does
particle wavelengths that
somewhat longer than
The condition
few wave hmctions for a particle in such a box are shown in
The wave functions -^ n now do not equal zero outside the box. Even
The
corresponding to lower particle
11.)
though the
the confining box has nonrigid walls, the particle
not equal zero at the walls.
first
Fig. 5-6.
ts still
(We
simply present the result here.
the
fits
observed behavior of those radioactive nuclei that emit alpha particles.
difficult
is
infinite in the real
world, our original rigid-walled box has no physical counterpart) exactly
more
This problem
slightly.
escaping from confinement (since potential energies are never
*
=
x
=L
predictions are unique with
its
in
walls.
the box despite
Both of these
quantum mechanics and do not correspond
to
any
QUANTUM MECHANICS
157
numerous atomic
l>ehavior expected in classical physics. Their confirmation in
and nuclear experiments supports the
Fix)
UL/ + 2W), / + eU-'L,,-
= F^, +
1
=
quantum-mechanical ap-
validity of the
proach.
Since x
=
of x-, x3
,
W
is
.
.
,
the equilibrium position,
THE HARMONIC OSCILLATOR
Harmonic motion occurs when
a system of
lie
floating in a liquid, a diatomic molecule,
some kind
an equilib-
vibrates alxmt
in
a crystal lattice
— there are
countless examples in both the macroscopic and the microscopic realms.
condition for harmonic motion to occur
acts to return the system to
is
the presence of a restoring force that
equilibrium configuration
its
the inertia of the masses involved causes
the system oscillates indefiiutely
if
them
when
it
is
disturbed;
to overshoot equilibrium,
no dissipativc processes are
x from
its
m
is
linear; that
is,
/•'
is
and
also present.
In the special case of simple harmonic motion, the restoring force
particle of mass
The
F on
which
is
Hooke's law when {dF/dx)r=0
F=
V(x)
5.39
= -
is
f
plotted in Fig. 5-7.
F(x)
If
back and forth between x
law of motion,
F =
small
is
for
is
is
negative, as of course
it
any
According to the second
E
dx
= kj xdx=
'/aitx
2
the energy of the oscillator
= —A
and x
==
is
E, the particle vibrates
+A, where E and A
arc related by
m V2 kA-.
ma, and so here
rf»x
5-7 The potential enerpy of a harmonic oscillator is proportional to r, where t
ment from the equilibrium position. The amplitude A of the motion Is determined by the
dl 2
of the oscillator,
FIGURE
— kx =
#+ m
-*-*
5.36
is
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
and
customarily called Hooke's law.
is
when x
a
proportional to the particle's displacement
-far
This relationship
x the values
restoring force.
equilibrium position, so that
5.35
for small
the third and higher terms of
*-(£L
an object supported by a spring or
an atom
and since
Hence
therefore the second one.
rium configuration. The system may
8,
x,
'Ihe only term of significance
the series can be neglected.
5.9
Fr - Q =
are very small compared with
=
which
classically
Is
the displace-
total
energy
f.
can have any value.
<>
(It*
There are various ways
X
5,37
— A
COS
to write the solution to Eq. 5.36,
(llivt
+
a convenient one being
<M
where
V
5.38
is
is
2w
= % he'
Vm
the frequency of the oscillations, A
is
their amplitude,
and
the phase constant,
<#>,
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
law for small displacements
which
is
we
reduce to Hooke's
note that any force
a function of X can be expressed in a Maclaurin's series about the
equilibrium position x
158
x.
To appreciate this point
THE ATOM
=
as
QUANTUM MECHANICS
159
—
Even before we make a detailed
tum-mechanical modifications to
we can
calculation
anticipate three quanFirst, there will not
this classical picture.
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
beyond the
The
—A
limits of
it
is
and go
in
and +A.
actual results agree with these expectations.
harmonic oscillator whose
classical
The energy
frequency of oscillation
is
levels of
s>
a
*=— A x=+A
x=-A
x=+A
x=-A
*=+A
x=-A
s=+A
(given by
Eq. 5.38) turn out to be given by the formula
(-
+
n
5.40
e»
The energy
of a harmonic oscillator
-
levels here are
in a
D»
evenly spaced
Energy levels of
harmonic oscillator
=
0, 1, 2,
is
thus quantized in steps of
(Fig. 5-8), unlike the
box whose spacing diverges.
We
energy
when n
note that,
£o
which
is
= fJ»
the lowest value the energy of the oscillator can have. Tins value
surroundings would approach an energy of
temperature approaches
The wave
shown
in
E — E and
tt
x= — A x=+A
is
equilibrium with
not
E=
as the
K.
functions corresponding to the
oscillator are
^
Zero-point energy
called the sero-jwint energy because a harmonic oscillator
its
The energy
= 0,
x
5.41
ftp.
levels of a particle
in Fig. 5-9.
In
first six
energy levels of a harmonic
each case the range
FIGURE
5-8
to
which a
particle
Energy levels of a harmonic
oscillator
according to quantum mechanics.
i=— A x=+A
FIGURE
The
5-9
first sii
harmonic-oscillator
wave functions. The vertical lines show the
same energy would vibrate.
limits
—A and
T.l between which a classical oscillator with the
oscillating classically with the
same
cated; evidently the particle
is
regions
—
in
total
energy E„ would be confined
other words, to exceed the amplitude
with an exponentially decreasing probability,
in a
It
box with
is
classical
the
160
THE ATOM
is
indi-
able to penetrate into classically forbidden
A determined by
just as tn
the energy
the situation of a particle
rtonrigid walls.
interesting
harmonic
and instructive to compare the probability densities of
oscillator
and a quantum-mechanical harmonic
same energy. The upper graph of Fig. 5-10 shows
a
oscillator of
this density for the classical
QUANTUM MECHANICS
161
oscillator:
The
probability
at the end-points of
equilibrium position (x
behavior
state of
at x
=
is
n
finding the particle at a given position
=
where
0),
it
moves slowly, and
it
moves
=
0.
As shown, the probability density
less
and
off
=
this position.
marked with increasing
less
coricsponds to n
on either side of
10,
and
it
is
clear that
approximately the general character of the
n:
<^ ln
i/>
has
maximum
this
value
disagreement
The lower graph of Fig. 5-JO
2
when averaged over z has
classical probability
quantum numbers, quantum physics
greatest
near the
lowest energy
its
its
However,
example of the correspondence principle mentioned
of large
2
is
least
Exactly the opposite
rapidly.
manifested by a quantum -mechanical oscillator in
and drops
becomes
P of
motion, where
its
P.
This
is
another
in Sec. 4.11: In the limit
yields the
same
results as classical
physics.
It
*=— A
might be objected
although
that,
smoothed
out, nevertheless i^ l0
However,
this objection has
a
|
meaning only
if
P when
does indeed approach
fluctuates rapidly with r
the smaller the spacing of the peaks
*=+A
2
l^jnl
whereas P does
not.
the fluctuations are observable, and
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 *
Thus the
= —A
magnitude with increasing
to
resemble each other more and more the larger the value of
n.
classical
also decrease
and quantum pictures begin
in
n,
in
agreement
with the correspondence principle, although they are radically different
for
small n.
r
*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
It is
is,
with
V = yz fcr 2
.
convenient to simplify Eq. 5.42 by introducing the dimensionless quantities
(»'
11/2
x
= -A
FIGURE 5 10
oscillator.
The
Probability densities lor the n
x
=
probability densities for classical
= +A
and n = 10 stales at a quantum -mechanical harmonic
harmonic oscillators with the same energies are shown
»
=
I2mnv
5.43
In white.
162
THE ATOM
QUANTUM MECHANICS
163
We
and
2£
hv
is
able to write
= f(y)e-* 1 <*
5.48
M
where v
now
i
ft
5.44
are
,
where
the classical frequency of the oscillation given by Eq. &38. In making
we have essentially clone is change the units in which
and E are expressed from meters and joules, respectively, to appropriate
/((/) is
a function of y that remains to be found.
we
Eq. 5.48 in Eq. 5.45
By inserting the ^ of
obtain
these substitutions, what
X
tlimcnsionless units.
which
+ ("-^ = "
7J
dy
5.45
|^
549
Jn terms of y and « Sehrodinger's equation becomes
The
2
is
2
4
We begin the solution of Eq. 5.45 by finding the asymptotic form that £ must
as y —* ±tx>. If any wave function
is to represent an actual particle
the differential equation that
f(y)
if
localized in space,
5.50
that
2
J
)^|
be a
<ii/
finite,
to
expanded
/ ol>eys.
in
a power series
in y,
is
to
namely
+ A 2 y 2 + A3 y3 +
y
t
S A„y
=»
no n vanishing quantity. Let us rewrite Eq. 5.45 as
and then
t>e
=A +A
value must approach zero as y approaches infinity in order
its
- 1,/=n
(a
standard procedure for solving differential equations like Eq. 5.49
assume that /( y) can
have
+
n
determine the values of the coefficients A,. Differentiating /yields
follows:
=
A,
=
2 nKy'"
|
3-(^-«»=fl
dy
+
+ 3A a ,f +
2A.z y
.
.
SB
d^j,
t
= (9
i£
dtj
dH/dtf
As y
—*
^
oo, y~
SM
..
hm
>i
wo
-«K>
=
By multiplying
1
this
1
equation by y vve obtain
« and we have
d^/dy
—
\/
2
=
,
I
yty
5.51
=
^ nA
a
y"
n-0
A unction
I
5.47
t£
v
that satisfies Eq. 5.46
is
The second
^ = *****
derivative of
/ with
respect to y
is
since
rfV.
lii
1
(/-•*
dy 2
=
lim (if
-
\)e-""
n = f/V
:
;>
x
tj-ns
=
Equation 5.47
164
THE ATOM
is
the required asymptotic form of
2
"(*>
-
VAnl}"--
if-.
QUANTUM MECHANICS
165
1
which
equal to
is
Since
d2f
"
J
n=H
e'
we can
(That the latter two series are indeed equal can
We now
terms of each.)
first
verified
lie
express e y
2 Kn + 2
substitute Eqs. 5.50 to 5.52 in Eq. 5.49 to obtain
+ lK*S "
)("
(
2»
+
~
+
^
/2
^n =
+
+ ML + _JL_
2^-2!
]
it=n,2, (....
+
(n
and
+
2}(n
=
l)A n+2
(2,i
+
-
1
2 3 -3!
2 n/2 [
—
1
G)'
the condition that
W
X
=
n=Q,2,4....
«)A„
so
5.54
-^nt2
Tin's recursion
of
Hence we have
values of n.
all
-1—
I
S
=
In order for this equation to hold for all values of y, the quantity in brackets
must be zero for
S+
a power series as
in
" a )A n]tf" = °
1
+
lT
by working out the
e
5.53
= 1+
A
+
2n
—
{n
+
-
1
2)(„
The
a
+
is
find the coefficients
A2
A3 A5 A7
obtain the sequence of coefficients
,
A2
,
A.,,
At
,
,
A n and Aj here.)
A 4 A e .... and
,
,
A„ we
from
A we
starting
»
solution
its
Starting from
S (-
—*
oo;
only
if
Because
if/
—>
f(tj)
as
is
y
—*
oo
n ->
+
1
2
-2"«(£+l)(£)!
+
y
,
In the limit of
function.
2-(f),
"^rtwft+£\,
necessary for us to inquire into the behavior of
It is
as y
2-(f)l
BL
... in terms
a second-order differential equation,
has two arbitrary constants, winch are
obtain the other sequence
is
])
formula enables us to
and A,. (Since Eq. 5,49
coefficients of y" here
between successive
ratio
)
becomes
oo this ratio
can ifbti physically acceptable wave
multiplied by e~ v/2
,
f
will
meet
this
lim
requirement
1+* _
'
provided that
Thus successive
lim /{y)<e*-' /2
those in the
we
(As
be
shall see,
in the limit
A
suitable
it is
than
way
to
express the latter in
and
to
From
examine the
'
ev
n
it— x
THE ATOM
n
smaller / must
f{ij)e
v'f-
There
compare the asymptotic behaviors of f(u) and e v /2 is to
a power series (/ is already in the form of a power series)
is
series for
docs not vanish as
a simple
way out
e*'"'
1/
power
-*
between successive
2
coefficients of
we
can
tell
each series as n
by inspection
that
—*
oo.
$
In other words, if/
is
factor.
of
an
infinite series,
2
n
A" +2 ~
it
is
2n
(n
+
will
all
go
1
2)(n
rapidly,
If
y
—*
than
From
A„ are zero
for values
oo because of the e~ v
a polynomial with a finite
-n
+ 1)
tess rapidly
which means that
the series representing / termi-
the coefficients
to zero as
acceptable.
+
/ decrease
more
oo.
of this dilemma.
nates at a certain value of n, so that
of n higher than this one,
series for
/2 instead of
'~
ratio
VA
how much
.)
the recursion formula of Eq. 5.54
lim
166
unnecessary for us to specify just
coefficients in the
power
number
*
n
of terms Instead
the recursion formula
"
QUANTUM MECHANICS
167
it is
clear that
-
«
5.55
if
The
+
2n
1
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
for
-
in the
polynomial, while
powers of
polynomial
if
We
apj>ear.
;/
«
odd,
is
it
must be true
that
A„
-
1.
where
^=
Verify that
+
solutions of the
all
wave equation
n.)
v2 di?
dx 2
must be of the form y
2.
2n
5-9 and
the
1
=
in Figs.
Problems
and only odd
The condition that a = 2ri + 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
«.
are listed in Table 5.1, and the
-
shall see the result later in this section,
tabulated for various values of
is
Hn (y)
Ilermite polynomials
six
first
corresponding wave functions ^„ are those that were plotted
5-10 of the preceding section.
=
F((
± x/v)
as asserted in Sec. 5.2.
+,(x,0 and %{x,t) are lx>th solutions of Sehrodinger's equation for a given
If
I
potential V(x),
show
that the linear combination
or
*-(•+!)*
5.56
This
is
the formula that
=
«t
0,
1. 2,
En
was given
as Eq. 5,40 in the preceding section.
For each choice of the parameter «„ there
Each function
either
a different wave function ^„.
{called a iiennite polynomial) in
-'
which
coefficient
,
and a numerical
3.
ftyjdy=\
(This result
also a solution.
in
is
=0,1,2,...
wave function
+n =
(2f.y* (2"n\)-^lUy)e-"^
a nucleus
is
in
Chap.
3.)
Ik>x 10"
'"'
(The
in across.
of this order of magnitude.)
results as classical physics in the limit of large
that, as
»
'Hie general formula for the nth
Davisson-Germer experiment discussed
According to the correspondence principle, quantum theory should give the
same
n
X
5.57
is
Find the lowest energy of a neutron confined to a
size of
4.
-
and « a are arbitrary coastants
for instance in the
needed for ^„ to meet the normalization condition
is
a,
is
consists of a polynomial ll (ij)
n
or even powers of y, the exponential factor e _,, /2
odd
which
accord with the empirical observation of the interference of dc Broglie waves,
—*
n
and x
+
dx
oo, the probability of finding a particle
is
independent of
x,
which
is
quantum numlsers. Show
trapped in a irox Iwtween
the classical expectation.
is
5.
Find the zero-point
6.
An
in
pendulum whose period
electron volts of a
important property of the eigenfunctions of a system
ttrthogpnal to
is
I
s.
that they are
is
one another, which means that
Table 5.1.
SOME HERMITE POLYNOMIALS.
f
"M
W
m dV = Q
n*m
K,
Verify this relationship for the eigenfunelions of a particle in a onc-dimcnsional
1
3
1
Sjc
4
168
1
%
3
V--
2
S
1
lBy*
32y s
THE ATOM
-
2
5
12y
7
- 4%>
leOy3
+ 12
+
120y
9
a
&>
y»
y>
w»
w»
"&h*
box with the help of the relationship
*7.
Show
that the expectation values
sin
=
<T> and
{e
<
energies of a harmonic oscillator are given by
in the
Mow
n
=
docs
state.
(This
this result
is
true for
all states
compare with the
iB
—
e"'*)/2t.
V) of
<T)
the kinetic and potential
= (V) =
/2 when
l£t)
of a harmonic oscillator, in
classical values of 1*
it
is
fact.)
and V?
QUANTUM MECHANICS
169
'8.
'
Use the fact that «
Ecj.
5.50 arc
9,
Show
all
>
E
(since
>
(!)
zero for negative values of
that the
first
to
show
that the coefficients ,\„ of
n.
three harmonic-oscillator
wave
functions are normalized
solutions of Sehrodinger's equation.
10.
According to elementary
oscillator of
mass
in,
classical physics, the total
frequency
v,
and amplitude
A
is
energy of a harmonic-
2^-A'J f'-m.
Use the un-
certainty principle to verify that the lowest possible energy of the oscillator
lw/2 by assuming that Ax
1
1
Which
of the
wave
=
is
A.
functions
shown
in Fig. 5-
1
1
might conceivably have
physical significance?
FIGURE 5-11
170
THE ATOM
(Continued)
QUANTUM MECHANICS
171
QUANTUM THEORY OF
THE HYDROGEN
ATOM
6
The quantum-mechanical 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,
related matters as
this
theory has
how atoms
made
interact with
it
possible for us to understand such
one another
the origin of the periodic table of the elements,
to
form stable molecules,
and why
solids arc
endowed
with their characteristic electrical, magnetic, and mechanical properties,
topics
we
on the quantum theory of the
results
6.1
V
may be
interpreted in terms of familiar concepts.
SCHRODINGER'S EQUATION FOR THE HYDROGEN ATOM
hydrogen atom
consists of a proton, a particle of electric charge
—e which
electron, a particle of charge
For the sake of convenience vve
the electron
moving about
proton's electric
is
Is
shall consider the
in its vicinity
proton
to lie stationary,
with
hut prevented from escaping by the
simply a matte* of replacing the electron mass
must use for the hydrogen atom,
6.2
and an
(As in the Bohr theory, the correction for proton motion
field.
d^d.
d%
2»i
ox*
r)j/-
dz*
nl
V
here
is
m
liii
by the reduced mass
net is ions,
which
is
in'.}
what we
is
r^
potential energy
+ e.
1,836 limes lighter than the proton.
Schrodinger's equation for the electron in three
The
all
moment we shall concentrate
hydrogen atom and how its lormal mathematical
For the
shall explore in later chapters.
the electrostatic potential energy
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
There are two alternatives: we can express V in terms
of the cartesian coordinates x, y, z by replacing r by y/x2 + 2 + z 2 or we
can
y
On
directly into Eq. 6.1.
,
express Schrodinger's equation in terms of the spherical polar coordinates
r. S,
defined in Fig, 6-1. As it happens, owing to the symmetry of
the physical
<>
situation,
The
doing the
latter
makes the problem considerably easier
spherical polar coordinates
r.
0,
^
of the point
!'
shown
to solve.
in Fig. 6-1
whose center
the surface of a sphere
O, lines of constant zenith angle
at
is
we
are like parallels of latitude on a globe (but
a point
not the
is
same
as
the latitude of the equator
=
latitude;
its
6°),
is
and
note that the value of
lines of constant
meridians of longitude (here the definitions coincide
+z
taken as the
have
axis
and the +.v
of
90° at the equator, for instance, hut
axis is at
=
<p
azimuth angle
if
arc like
<>
the axis of the globe
is
0°).
becomes
In spherical polar coordinates Schrodinger's equation
the following interpretation:
r
(i
= length of radius vector from
= Vx 2 + if + z 2
= angle between radius vector
= zenith angle
origin
O
to point
P
1
-
6.3
r
2
dr\
drf
i
s,n
r-'-sinW
—r
+z
axis
r
l
cos"
V^ +
c>
=
y
2
+
Spherical
z2
2
sin
=
azimuth angle
=
tan
-1
equation by
r
2
sin
2
we
tf,
2
vw-
=
o
ft
/y
V and
multiplying the entire
obtain
polar
angle I>etween the projection of the radius vector in the xy plane and the + x axis, measured
in the direction
dU
2m
E +
<V -zr(
I
and
Substituting Eq. 6.2 for the potential energy
=
I
coordinates
sin
6.4
2mr 2 S m 2 & f_e*_
shown
H-
Equation 6.4
-2-
is
\4<ne t) r
+
A, ^ Q
Hydroger> atom
/
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
what
to see just
When
equation completely specifies the behavior of the electron. In order
Eq. 6.4
this
is
behavior
solved,
it
to describe the electron in
is,
we must
solve Eq. 6.4 for ^.
quantum numbers are required
a hydrogen atom, in place of the single quantum
I
urns out that three
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
RGUHE
6-1
Spherical polar coordinate!.
model, the electron's motion
that varies as
*-y
it
moves
THE ATOM
one quantum number
enough
to specify the state of a particle in a one-dimensional l>ox.
A
particle in a three-dimensional box needs three
x,
y,
now
quantum numbers
for its
there are three sets of boundary conditions that the
wave function i£ must obey: ^ mast to at the walls of the box in
and z directions independently. In a hydrogen atom the electron's
is
restricted
by the inverse-square electric
field
of by the walls of a box, but nevertheless the electron
dimensions, and
govern
174
One quantum number
just as
is
motion
<f>
such an electron,
to specify the state of
the
r
basically one-dimensional, since the only quantity
position in a definite orbit.
enough
particle's
— s'm$cos<t>
U — r sin & sin
~ — TCOS$
is
is
description, since
i
is its
its
wave
it
is
is
of the nucleus instead
free to
move
in three
accordingly not surprising that three quantum numbers
function also.
QUANTUM THEORY OF THE HYDROGEN ATOM
175
}
The
three
quantum numbers revealed by the solution
of Eq. 6.4, together with
their possible values, are as follows:
Principal
—
Orbital
=s f
quantum number
quantum number
Magnetic quantum niimlier
The
=
r»
=
1,
2, 3,
.
.
=
n
±1, ±2,
0,
.
—
.
.
,
3rV
R
1
virtue of writing Schrodinger's equation in spherical polar coordinates for
procedure
is thai
in this
is
to look for solutions in
form
it
may be
which the wave function
form of a product of three different functions:
8(0), which depends
we assume
6.5
The
third
term of Eq. 6.6
terms are functions of
readily separated
upon
alone; and
<1>(0),
is
a function of azimuth angle
and 6 only.
l>cl
+)
=
dr\
dcscriljes
The
dr)
how
the
6.8
,
,,\
<£
_
n
f
only, while the other
+
8
how i£
varies with zenith angle
the nucleus, with r and
with azimudi angle
<J>
tj>
constant.
and
$
of the electron varies along
constant.
The
The
function
4>(<£}
constant.
$ = R8#
write
W
R
r2
m, 2 for the
dR\,
dr /
dr \
more simply
as
V
+h
)
|
9^
~*"3^
~
am equal to the same constant,
As we
shall see,
it
is
convenient
is
therefore
'
sulwtitute
±±(
6.9
e2
\4frf. r
right
hand
side of Eq. 6.7, divide die entire
equation by sin 2 0, and rearrange the various terms,
W Um
2
2
find
dmt
A
/_e _
ft
wc
nr
1
function H(fl)
1
describes
how
i/>
sin a
varies
at the nucleus,
with
Again
we have
call
1(1
+
1),
ff
8sin0
90
H£)
an equation in which different variables appear on each
side,
we
shall
once more
8
same
constant. This constant
for reasoas that will Ix: apparent later.
The equal ions
and R are therefore
6.10
sin
3r
dr
7)0
80
8siuffd»
Filiations 6.8, 6.10,
6,1?
dt*
2
6.11
d
a* 2
THE ATOM
2
for the functions
see that
/
2
differential equation for the function 4>
reqiu'ring that Ixith sides Ix: equal to the
we may
2mr2 sin*
along a meridian on a sphere centered
along a parallel on a sphere centered
Eq. 6.5, which
30 }
30 l
- m2
"
*
*
which depends upon $ alone. That
function
.
<***
1
r alone;
Hydrogen atom wave function
wave
m, 2 The
to call this constant
^(r, 6, £) has the
which depends upon
R(r),
/l(r)8(tf )*(<*.)
function R(r) describes
From
e2
/
We t
2
us rearrange Eq. 6.6 to read
Thix equation can Ik correct only if both sides of it
When we
a radius vector from the nucleus, with
176
r
ft
since they are functions of different variables.
that
flr, 8,
and
2mr z sin z g
* 3^
+
into three independent equations, each involving only a single coordinate.
The
BB
+ J_ * +
momentum.
the problem of the hydrogen atom
we
W\
SEPARATION OF VARIABLES
The
r
8
a2
R
*6.2
hydrogen
and the magnetic quantum number m, governs die direction of the
nucleus,
at
+
±(
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
is,
dr/
principal
angular
for the
.
= 0, 1, 2
m,
ilence, when we substitute flQG for ^ in Schrodinger's equation
atom and divide the entire equation by H8<!>, we find thai
*
and 6.11 arc usually wrillen
2
d4>
2
+
m, 2^
a
QUANTUM THEORY OF THE HYDROGEN ATOM
177
Hence
meridian plane.
6.13
Mil'/
must
it
true that *(£)
l>e
=
+
<l>(£
2-a),
or
,lil
= Ae ,m
Ae im t*
( 'lW"
'
aT,
6.14
which can only happen when
Each of these
We
variable.
equation for
is
an ordinary differential equation for a single function of a single
have therefore accomplished our task of simplifying Schrodinger's
the hydrogen atom, which began as a partial differential equation
for a function
i£
±3,
.
.
.).
The
constant
wi (
in, is
or a positive or negative integer (±1,
is
known
as the
hydrogen atom.
The
equation 6.13 for
differential
Ixgmdre
purpose, the important thing about these functions
the constant
QUANTUM NUMBERS
'6.3
The
first
of the alwve equations, Eq.
6.15
where
<t>($)
A
m,.
=
ft 12. is
wave
space.
the constant of integration.
function iff-musl olie\
From
Fig. 6-2
=
Nil
0,
1
We
have already stated that one of the
conditions a wave function— and hence *, which is a component of the complete
is
it
is
is
that
it
evident that
have
a single
$ and
tj>
+
value at a given point in
2tt lx>th identify
the
same
The constant I
The solution
only
when £
where n
and
$.
+
known
of the
as the orbital
is
bound
form
in the
quantum number.
is
of the
fi(r)
also complicated, l>eing in terms of poly-
,__
me*
an integer.
We
one of the negative values En (signifying that
atom) specified by
to the
(±\
recognize that this
2v.
is
precisely the
same formula
for
both identify the
Another condition that must be obeyed
as the principal
in
quantum number, must
may be
order to solve Eq, 6,14
l>e
,(n-
Hence we may tabulate the
three
is
that n,
equal to or greater than
expressed as a condition on
i=0, 1.2
/
in the
I
+
1
form
1}
quantum numbers
n,
/,
and
m
together with
their permissible values as follows:
n
6.17
/
m.
It is
=
=
1, 2, 3,
0,
1,
.
.
.
2
(n
=0, ±1, ±2,
worth noting again
how
-
1}
inevitably
Principal
quantum number
Orbital
quantum number
Magnetic quantum number
:J
quantum numbers appear
mechanical theories of particles trapped
THE ATOM
;
equation, Eq. 6.14, for the radial part
final
positive or has
is
is
This requirement
178
m
the energy levels of the hydrogen atom that Bohr obtained
known
*-y
absolute value of
nomials called the associated Laguerre functioiu. Equation 6.14 can be solved
6.1S
?.
is
when
that they exist only
m,\, the
±1, ±2, .... ±t
hydrogen-atom wave function ^
the electron
FIGURE 5-2 The angles
same meridian plane.
is
This requirement can be expressed as a condition on
readily solved, with the result
Ae*'"'"
functions. For our present
an integer equal to or greater than
/ is
complicated solution in
0(ff) has a rather
terms of polynomials called the associated
of three variables.
±2,
magnetic quantum number of the
in
quantum-
in a particular region of space.
QUANTUM THEORY OF THE HYDROGEN ATOM
179
To
exhibit ihc
we may
m,
dependence of H, B, and
write for the electron
upon
wave Function
<l>
the
quantum numbers
>i,
I,
4
to
n
6.18
*
The wave
and
2.
= WlmPm,
I
functions H, W, and
<J>
together with
\l
are given in Table
6.
1
for
n
=
+
8
L
<*>
1
I
3.
S
I*
I
PRINCIPAL
6.4
T
i
QUANTUM NUMBER
a
°
s*^^*-
k
==
M
i
interesting to consider the interpretation of the
It is
numbers
in
Chap.
4,
in
terms of the classical model of the atom. This model, as we saw
corresponds exactly to planetary motion in the solar svstetn except
that the inverse-square force
M
hydrogen-atom quantum
holding the electron to the nucleus
ft
I
S
a"
5.
—
<1
M
s
M
-.'
na
"c
™0
A- i-
1$
•$ L*
oe
5
-=
-
4
j
CO
3
is
electrical
f
Two quantities :ire conserved— \h&t is, maintain a
rimes— in planetary motion, as Newton was able to slum
rather than gravitational.
constant value at
Iron
all
t
Kepler's three empirical laws. These are the scalar total energy and the
vector angular momentum of each planet.
<
i
SasstcaBy the total energy can have any value whatever, but it must, of course,
if the planet is to be trapped permanently in the
solar system. In
be negative
the quantum-mechanical theory of the hydrogen
also
a constant, but while
may have any
ettetg)
is
c
i
it
can have are
»»<•'
*„ =
positive value whatever, the only
.specified
by the formula
total
H
*
(J_\
a
it
yields
quantum number
an energy
worked out from Schrocliu^i's
restriction identical in form.
» fur any of the planets turns out to be so
the separation of permitted energy levels
is
far too small to
immense
that
be observable. For
reason classical physics provides an adequate description of planetary motion
hut fails within the atom.
The quantization of electron energy in the hydrogen atom
scribed by the principal quantum number n,
\is
(6.14)
180
THE ATOM
a
*
*
differential
j
H
-
H
s
3
is
quantum nmnber
o*
^
a
k
i«
IS
l»
M
«
ee
3
is
ce
a
•I
1
therefore de-
B
I
a
-I
? -i^ ?|- ^I« -\%
"?i«
?|-
^h ?r
_b
_i^
^ii|
i
I
/
*
BS
o
i
QUANTUM NUMBER
Interpretation of the orbital
examine the
_
^
1a
=i-
o
2-
The
^
;
mi
UJ
ORBITAL
:
However, the
this
6.5
-
a
of planetary motion can also be
equation, and
I*
#
>
? „
M
1
The theory
§
i
1
w
&
VI
negative vaJues
(6.16)
it
atom the electron
II
a bit less obvious.
Let
equation for the radial part R{r) of the wave function
i*„L|«i|«|i|
-i|
^1'
t
_i|„
3
p
o
z
2
o
o
H(*%hm&+*hm*-<
MoJiMe-tnpjKesrt
'i
This equation
(hat
is, its
solely
is
concerned with the radial aspect of the electron's motion,
motion toward or away from the nucleus; yet
of E, the total electron energy, in
it.
The
total
kinetic energy of orbital motion, which should
energy
/-;
we
notice the presence
includes the electron's
have nothing
l2
to
do with
its
radial
m(i +
i)
2nir-
or
6.21
=
/.
V/(/
+
Electron angular
1) ft
momentum
motion.
may he removed by the following argument. The kinetic
two parts, TraliiHi due to its motion toward or away
and TnrMM due to its motion around the nucleus. The potential
Our
This contradiction
energy
'/'
of the electron has
from the nucleus,
V
energy
the electrostatic energy
is
Kq. 6.21.
**W
the total energy of the electron
T
+
Ttitiilial
= Zradial +
T„nrliifjil
radii nl
=
In
the
ont,
last
we
E
in
2m
/ ,,dfi\
two terms
shall
in
have what
we
Eq. 6,14
we
obtain, after a slight rearrangement.
r
H 2 lll +
1)1
momentum
+
r exclusively.
We
is
to
is
I
/,
of the electron
we may
=
«
B»art*w'
write for the orbital kinetic energy
L
t?
2mn
2,6
I
—
=
is
his
0,
182
THE ATOM
that
the separation
into
angular-
discrete
whose
orbital
quantum number
is
2 has
x
l)ft"
M J-s
10
p
to
=
/
(I
1
2
s
p
d
momentum
of the earth
is
2.7
by a
states
x
l()
letter,
,0
J-sl
with
s corre-
and so on according to the following scheme:
1,
3
f
I
5
6,
g
h
i.
Angular-momentum
states
peculiar code originated in the empirical classification of spectra into series
and fundamental which occurred before the
theory of the atom was developed.
Thus an
momentum,
momentum
a
p
slate has the angular
The combination of the
total
momentum
atomic
In this notation a state in
slates.
n state
is
one with no angular
\/2«~, etc.
quantum number with the
orbital angular
letter that represents
provides a convenient and widely used notation for
winch n
=
2,
I
=
is
which n = 4, / = 2 is a 4d state. Table
designations of atomic states in hydrogen through n = 6, / = 5.
example, and one
Hence, from Eq. 6.20,
by
quoit-
experimentally observed. For example, an electron
lie
called sharp, principal, diffuse,
L
and
momentum.
customary to specify aiigular-inomeutum
/a wll; "orbllal
momentum
specified
;i
so large
cannot
contrast the orbital angular
sponding
energy of the electron
/-
10- 'J-s
X
1.054
= \/2(2+
s \/6ft
=
I
Since the angular
momenta
hoth conserved
I)
2mr-
—
is
therefore require that
ll is
'orbital
momentum
momentum
want: a differential equation for H(r) that involves
6.20
orbital kinetic
=
matter, any other body)
lot that
(he angular
B)
H*K1
/i/2w
tiiouieiiliiiu slates
the square brackets of this equation cancel each other
functions of the radius vector
The
1)
macroscopic planetary motion, once again, the quantum number describing
L
If
-
(n
thus the natural unit of angular
or,
d
1,2
+ v
angular
Inserting this expression for
/
The quantity
is
is
1
0.
Like total energy K, angular
ft
B»
quantum number
that, since the orbital
is
the electron can have only those particular angular
—
tizetl.
Hence
=
I
of the electron
V=
interpretation of this result
restricted to the values
is
in
a 2s
slate, for
b'.2
gives the
QUANTUM THEORY OF THE HYDROGEN ATOM
183
Table 6.2.
THE SYMBOLIC DESIGNATION OF ATOMIC STATES
IN
the direction of
HYDROGEN.
L
This phenomenon
i
(
p
=
1
=
/
ii
1
J
=
I
U3
2
1
I*
2
2j
%,
Fl
3
3*
ii
=-
4
U
Si
3p
«d
Hd
•/
= S
n = 8
*
*
5/
%
Si
6p
ad
6/
(i-
n
n
6.6
=
("
If
9
of
=
=
=
n
/i
4
;
L
we
direction
L,
=
The
possible values of m, for a given value of
in
to describe
it
eh
autiparallel to B, since
completely requires that
magnitude. (The vector L,
we
recall,
which the rotational motion lakes place, and
its
is
ils
a minute current loop and lias
Hence an atomic electron that
is
with an external magnetic
field
B,
a
+/ through
to
sense
is
/.
The space quantization
is
shown
;
is
always smaller than the magnitude
to the
magnetic
+
1 )
ft
of
in Fig. 6-4.
We
of the orbital angular
momentum of the hydrogen atom
must regard an atom characterized by a certain value
plane
given bv the right-hand
field like that
possesses angular
\/i{l
direction be specified
perpendicular
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. 6-3.)
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
dipole.
range from
momentum.
FIGURE
6-4
rule:
nucleus
/
/,
The orbital quantum number / determines the magnitude of the electron's angular
momentum. Angular momentum, however, like linear momentum, is a vector
and so
Space quantization
m,fi
— so that the number of possible orientations of the angular-momentum 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, Lt may lie 2ft, ft, 0, —ft,
or — 2ft; and so on. We note that L can never be aligned exactly parallel or
the total angular
as well as its
component
is
.V
MAGNETIC QUANTUM NUMBER
quantity,
in the field direction.
often referred to as space quantization.
the magnetic-field direction be parallel to the ; axis, the
let
in this
6,22
by determining the component of L
is
Space quantization
of orbital
angular
momentum.
of a magnetic
momentum
The magnetic quantum number
m
interacts
t
f
=2
specifies
fl.
thumb
in direction of
momentum
FIGURE 6-3 The right-hand
lor angular momentum.
angular*
vector
lule
= Veft
hand in direction
of rotational motion
fingers of right
184
THE ATOM
QUANTUM THEORY OF THE HYDROGEN ATOM
185
of
jii,
as standing ready to
L relative
to
assume a certain orientation of
an external magnetic
the event
field in
In the absence of an external magnetic
Hence
entirely arbitrary.
direction
we choose
is
m,ti;
field,
it
angular
its
finds itself in
momentum
such a
field.
the direction of the t axis
is
must be true that the component of L in any
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 meanit
ingful as the direction of a
momenta
of the angular
U
by
is
magnetic
L
can never point
out a cone in space such that
phenomenon
and L„
j»s
is
t>e in
electron \s
the xy plane at
if
is
impossible
since in reality only one
definite values
any
L were
all
momentum component
which of coarse
in
and
|£|
if it
/. is
;
if
m,H.
L were
The
>
|L_|,
r^y
reason for the latter
fixed in space, so that
to l>e part of a
L
the electron
lie
Lt
1=2
confined to a
This can occur only
p. in the r direction
component Lt of
m, values.
closely related
is
the z direction, the electron would
in
times (Fig, 6-5«).
is
is
(he
hydrogen atom. However,
together with
is
if
infinitely uncertain,
its
magnitude
6-5/)),
The
The uncertainty
principal prohibit* the
I.
is
a built-in uncertainty, as
it
were,
L is constantly changing, as
Lt and Ly are 0, although Ls always
in the electron's
in Fig. 6-fl,
and
z coordinate.
so the
average
has the specific value mfl.
L have
not limited to a single plane
angular-momentum vector
and there
direction of
values of
Fig.
6.7
FIGURE 6-5
axis.
components
specific direction z but instead traces
had definite values, (he electron would
For instance,
definite plane.
have to
Lt
their
quantized? The answer
projection
its
by
L
the uncertainty principle:
well as
this line the
of the IT atoms are determined
only one component of
to the fact that
and along
field,
The angular-momentum vector L
FIGURE 6 6
processes constantly about the a
from having a definite
direction in space.
In
THE NORMAL ZEEMAN EFFECT
an external magnetic
Vm
energy
that depends
and the orientation of
The torque
t
t
=
field B,
upon
this
a magnetic dipole has an amount of potential
lioth the
moment with
on a magnetic dipole
magnitude
/i
of
its
magnetic moment
respect to the field (Fig. 6-7).
in a
magnetic
field
of flux density
B
is
fiB sin
L = VW + l)fi
FIGURE 6
7
moment
at the angle
r.
A magnetic
a magnetic held
dipole of
relative to
li
QUANTUM THEORY OF THE HYDROGEN ATOM
187
where
is
dipole
to
To
it.
the angle between
perpendicular to the
is
p and
B.
The
torque
and zero when
field,
calculate the potential energy
Vm we
,
it
must
maximum when
is
a
is
parallel or antiparallcl
first
the
that
establish a reference
Vm is zero by definition. (Since only clianges in potential
energy are ever experimentally observed, the choice of a reference configuration
configuration at which
is
arbitrary.)
It is
The
work
to lire externa!
Vm =
convenient to set
perpendicular to B.
—
when
90°, that
when
is,
potential energy at any other orientation of
must be done to rotate the dipole from
that
ihe angle 6 characterizing that orientation.
ll,
L
is in the opposite direction to
While the alwve expression for the
magnetic moment of an orbital electron has been obtained
by a classical calculation, quantum mechanics yields the same
result. The magnetic potential energy
of an atom in a magnetic field is therefore
,i
u, is
fe)'B
6 25
equal
ji is
-
y
cos
90° to
which
Hence
is
From
a function of l>oth
we
Fig. 6-5
B and
9.
see that the angle 9 Iietwcen
L
and the z direction can
have only the values specified by the relation
cosfl
8
sin
= — //Bcosff
6.23
When
This
fi
is
itself
points in the
while the permitted values of
same direction
as B, then,
V,,,
=
-fiB,
its
minimum
with an external magnetic
its
respect to the field determine the extent of the magnetic contribution to the
energy of the atom when
total
of a current loop
6.26
it
a magnetic
is in
The magnetic moment
field.
v
-
—<£)>
where
icv
,
i
in
is
.1
=
iA
the current
fi
The
linear
and
A
the area
is
~e), and
its
r
Is
it
An
encloses.
magnetic moment
is
fa a Magnetic
electron which makes )
equivalent to a correal of
m
since
me
Bohr magneton;
v of the electron
is
2WIT,
and
so
angular
its
momentum
is
up
J**ral Hw»
upon
that of 0.
into several subslates
more or
9 27
x
M
10
I/tesla
when
the
A
state of total
atom
is
in a
slightly less than the
depends upon
quantum number
magnetic
field,
and
energy of the state
in
''"to
This phenomenon leads to a "splitting" of
individual
separate lines when atoms radiate in a
magnetic field, with
of spectral lines by a magnetic
2w in t'f2
the formulas for magnetic
moment
— te>
(i
and angular momentum L
Electron magnetic
an orbital electron. The quantity
mass of ihe electron only,
188
is
then, the energy of a particular atomic state
Zeeman, who
first
field
is
observed
confirmation of space quantization;
called the
it
it is
in 1896.
is
called
its
{— e/2m), which
gymmagnetiv
moment
involves the charge and
ratio.
The minus
sign
means
Zm««„
field.
The
splitting
rffhi alter the Dutch
The Zeeman
effect
further discussed in Chap.
shows that
for
value
he spacing of the lines dependent upon the
magnitude of the
physicist
6.24
its
tbe absence of the field.
L = mcr
Comparing
field,
their energies are slightly
I
=
called the
the value of m, as well as
u breaks
therefore
= — evzr*
speed
is
(T).
circular nihil of radius
electronic charge
Magnetic energy
is
The quantity cti/2m
/t
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
field.
moment of the orbital electron in a hydrogen atom depends
momentum L, both the magnitude of I- and its orientation with
angular
£i
value.
a natural consequence of the fact that a magnetic dipole tends to align
Since the magnetic
upon
=
dll
is
a vivid
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, 6-8. We see it implies
that, if a suitable experiment
THE ATOM
QUANTUM THEORY OF THE HYDROGEN ATOM
189
how ^
describes
varies with
probability density
Bohr electron
l+l
where
orbit
=
a
6.27
product of
it
lading
about the
FIGURE
The Bohr model of the hydrogen
5-8
in a
The
spherical polar coordinate system.
r
z
its
probability density
at all.
of the hydrogen
*
°">
(J>
This imprecision
nature of the electron. Second,
around die nucleus
is
in
electron
wave
of course, a consequence of the
wave
we
cannot even think of the electron as moving
any conventional sense since the probability density
independent of lime and
The
is,
R
—
of the
2
l^l
,
wave
describes
dV
there
is
function
\p
in a
hydrogen atom
is
the orbital and total
and
so that, since
from the nucleus
how
tf.
*=
190
THE ATOM
symmetrical
as at another.
ci
'!>,
not only varies with
R
versus r for
while
R
it is
Is, 2,s\
maximum
a
is
zero at
r
=
2w, 3«, 3p, and
3d
= 0— that
at
at r
is,
for states that possess
of the electron at the point
it
r,
0,
£
proportional to
is
in the infinitesimal
volume element
in spherical polar coordinates
hydrogen atom somewhere between
r
and
r
+
dr
is
2:7
r2
|
R \*dr f
|0|
2
sin
d0
f
|<l>
»d$
=
quantum numbers have
is
r
2
R\-dr
plotted in Fig. 6-10 for the same states whose radial functions
curves are quite different as a
in Fig. 6-9; the
P is not a maximum at the
maximum at a finite distance from
We
rule.
immediately that
nucleus for s states, as R
but has
it.
its
note
itself
Interestingly enough, the
is,
most
I;
probable value of r for a la electron
varies with $
/
is
are normalized functions, the actual numerical probability
<l»
IWdr =
electron
have the values
density
so that the electron has
it is in,
function, in contrast to
Evidently
F[r)dr of finding the electron in a
Bohr model.
state electron in the
describes
a constant that does
is
<£,
= r'^mOdrdOd^
</V
R were shown
when
a measure of the likelihood
given by
(r)
varies with r
the values n and
Now
Ivf-j^jV.
Equation 6.28
how $
state
one angle
but the actual probability of finding
vary considerably from place to place.
where
= R ol
at
for all * states,
itself
6.28
R
quantum
W?
may
is
by the
a complex quantity.
is
a different way for each combination of quantum numliers n
in
momentum.
The pwlmhiiity density
atom modifies the straightforward
First, no definite values for r, 0, or
S'ven, but only the relative probabilities for finding the electron
at
,le
various locations.
The
angular
(where n
prediction of the Bohr model in two ways.
which
|'l'|-,
Figure 6-9 contains graphs of
the nucleus
is »»,.
to l>e replaced
is
the function
if
Hence the electron's probability
slaues of the hydrogen atom.
The quantum toeory
quantum number
written
lie
electron at a particular azimuth angle
radial part
/.
the magnetic
complex conjugate
axis regardless of the
but does so
and
were performed, the electron would always be found a distance
of r = n'an
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.
when
therefore
same chance of being found
the
atom
lln-'
depend upon
not
<J>
may
\Rf\&f\<
and
The a/imulhal
i)l
2
understood that the square of any function
is
it
|i£>|
and m,; and
'KW
when
the magnetic and orbital quantum numbers
are the
is
I.5«
same
in
,
is
exactly «„, the orbital radios of a ground-
However, die average value of
which seems puzzling
at
first
sight because the
r for
a
Is
-
energy levels
both the quanlum-mechanical and Bohr atomic models. This
apparent discrepancy
is
removed when we
depends upon 1/r rather than upon
value of 1/r for a
l.s
electron
is
recall
that
r directly, and, as
it
the electron energy
happens, the average
exactly l/a^.
QUANTUM THEORY OF THE HYDROGEN ATOM
191
densities as functions of r
plotted
is
picture of
lufrj*,
2
l^l
When
axis.
2
not
is
l^f
and B are shown
Since
dV.)
2
\$>,
for several
this
\
we
many
1
1
2p
lobe patterns
manner
which adjacent atoms
in
notion once more
in
Chap.
in a
molecule
H.
also reveals qaairttan-mecliardcai stales wftt a
to those of the
tribution for a
The pronounced
of the stales turn out to be significant in chemistry since
shall refer to this
study of Fig, 6
resemblance
a three-dimensional
done, the probability densities for s states are evidently
is
these patterns help determine the
interact;
<;>,
obtained by rotating a particular representation about a vertical
spherically symmetric, while others are not.
characteristic of
atomic states. (The quantity
independent of
is
remaikable
Bohr model. The electron probability-density
state with in,
=
±1,
for instance,
is
like
a doughnut
dis-
in the
equatorial plane centered at the nucleus, and calculation shows the most probable
distance of the electron from the nucleus to
Bohr orbit
for
3d
same
for the
states with
m,
a
total
lie
4%— precisely
the radius of the
quantum number. Similar correspondences
±2, if states with m,
=
±3, and
exist
so on: in every case
the angular
momentum possible for that energy level, and in every case
momentum vector as near the z axis as possible so that the probability
density
as close as possible lo the equatorial
the highest angular
lx:
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 6-10 The probability of finding
•>• from the nucleus for
tween r and r
5"n
the electron
the
quantum
in a
hydrogen atom
a distance be-
at
states of Fig. 6-9.
15h„
10«o
FIGURE 6-9 The variation with distance from trie nucleus or ihe radial part of
electron wave function In hydrogen tor various quantum
states. The quantity
« = IP fme' = 0.S3 A is the radius of the first Bohr
orbit.
the
r.
The function
except
is
I
=
m,
a constant
=
<%
varies with zenith angle
(),
which are a
in fact),
stales.
which means
electron probability density
\ip\
2
for all
quantum numbers
The probability
that, since
.
and m,
density |9j* for an s state
|<t>|
2
is
also
a constant, the
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. 6-11, in which electron
probability
192
5u n
10u n
15d
20.i
(
,
25,.,,
THE ATOM
QUANTUM THEORY OF THE HYDROGEN ATOM
193
FIGURE 6-11 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
We
theory of the hydrogen atom,
Bohr was obliged to postulate
art'
now
state
when,
at
nhflflghtg Inuii dih-
Let ns formulate a definite problem: an atom
—
(
0,
an excitation process of some kind
atom emits
enefgy
ground
in its
is
beam
(a
with other particles) begins to act upon
say, or collisions
find that the
coruddM an rlcrlnin
n. a position to
state to another.
of radiation,
Subsequently
it.
radiation corresponding to a transition
we
from an excited
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
state of
of the radiation?
The wave
+
function
of an electron capable of existing in states n or
m may be
written
- n^a,
""
b constant
in time doe, not
radiate, whil, if it ve
relative to the nucleus
oscillates, electromagnetic
wave* „,-
recjuency
the
is
j.
a, that of the oscillation"
component of electron motion
The tune-dependent wave
the
For
^^StjX
^
()f an e]eclr0]1
n
the product of a
time-independent
*„ and a hme-varying function whose frequency
is
U) „b er
n and energy
^
wham a*a
b*b
,
is
wav
Z
=
1.
At
I
ft+ m
the probability that the electron
is
probability that
i„ the , direction only
fiction
+ = «+„ +
633
cmiL w
=
in state
is
it
0, o
=
excited state, a
—
1
is
—
and b
and b
While the electron
m. Of course,
=
n and b*h the
when
=
the electron
and
1
//
=
it
is
+
in the
once more.
is no radiation, but when it is in the
when both a and b have uonvanishinu
produced. Substituting the composite wave
we obtain for the average electron position
n (that
rn to
/-:.
electromagnetic waves are
function of Eq. 6.33 into Eq. 6.31,
Hence
by hypothesis;
and ultimately a
1;
state
in
is
must always be true that a*a
in eitiier state, there
midst of the tratisiliun from
values),
it
is,
% = l^C^.'*"
6.29
and
6.30
+;
The average
v,
=
6.34
,£«.+(«,,/»»
position of such an electron
the expectation value
(Sec. 5.4) of
is
namely.
i
Here, as before,
we
let
a* a
=
constants, according to Eq. 6.32,
ones capable of contributing
1
a'
and b*b
and
to a
wave
<*>
=
functions of Eqs. 6.29 and
6.30,
J
x^^eiHVitl-HE./AHi
6.35
<*>
=
«a
winch
* comta*
The dec ron does not
in
time since *„ and
mechanics predicts that
tins
196
agrees
THE ATOM
wdh
oscillate,
„ atom
«
dx
are,
,)
The
+
first
and
last
integrals are
and third integrals are the only
time variation in (x).
we may expand
*
j x^„ dx + b*af" x^;e
In ibe case of a finite
only.
b2
so the second
Willi the help of Eqs. 6.29 to 6.31
Inserting the
=
a'bf x4%****</ mi
bound system of two
4f
+t
Eq. 6.34 to obtain
»V'
,1
V
J1
<r (,fi" /
jr**J**tk + b z
i
states,
which
is
what
'11 '
f*
dx
x^m dx
we have
here,
by definition, functions of
position
and no radiation occur, This
in a specific
quantum
state docs
observation, though not with
classical physics
"2
q!,,,,,,,
„t
and so
we
can combine the time-varying terms of Eq. 6.35 into the single term
—»
QUANTUM THEORY OF THE HYDROGEN ATOM
197
1
Now
*"+£-'*=
2 OOSl9
'
Hence Eq. 6.36
simplifies to
—x
proportional to it. Transitions
not be zero, since the intensity of the radiation is
Iramilions, while those for
allowed
for which this integral is finite are called
which
it is
zero are called forfridden &mwi*fon#,
In the case of the
which contains the time- varying factor
The
the initial and
and magnetic quantum numbers of die
the
V
a
hill
final slate
are n,
z coordinate, the
E — ±»
F
-iS
m„
/,
are
n', V,
m't respectively
and the coordinate u represents either
condition for an allowed transition
is
^V, m
(
>*
°
example, the radiation referred to is that which would
axis. Since the
he produced by an ordinary dipole antenna lying along the x
can be evalu6.39
known,
Eq.
are
atom
hydrogen
for the
wave functions'^
When
expression for <x), the average position of
the electron,
«sS,,,.„,
J*
is
u
is
taken as
ii
=
i
ated for
<*>=^J xW»dx+b*fxrm t m dx
6.38
and
6.39
h
and the
x, y,
initial state
cos 2irvt
electron's position therefore oscillates sinu.snidally
at the frequency
6.37
to specify
total, orbital,
final stales involved in a radiative transition. If the
and those of the
=
hydrogen atom, three quantum numbers are needed
a
x,
.t,
for
mi
=
tj,
and u
=
z for
more
pairs of states differing in one or
all
found that the only transitions that
quantum numbers. When this is done,
changes by +
can occur are those in which the orbital quantum number
or changes by
change
not
- 1 and the magnetic quantum number m, does
it is
i
+
2«*fc cos
%m
[ x^„, dx
-1;
or
When
die electron
¥
state n or state m. the probabilities
or o» respecand the expectation value of the electron's position
is constant
the electron is undergoing a transition
between these states, its position
is in
tively are zero,
When
in other words, the condition for
6.40
A/
6.4i
Am,
an
allowed transition
is
I
or
+
1
that
= ±1
Selection rules
—
(1,
±1
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
as the beats
we might
existed in both the n
tively
EJh
imagine
and
and EJh.
m
to l>e
slates,
produced
whose
if
is
The change in total quantiun number n is not restricted. Equations
6.4
are known as the selection rute.s for allowed transitions.
characteristic frequencies are respec-
let
a
us refer again to Fig. 6-11. There
2p
state to a
we
sec thai, for instance, a transition
U state involves a change from one
behaves like
another such that the oscillating charge during the transitions
2s state to
from
a
transition
a
an electric dipole antenna. On the other hand,
SELECTION RULES
l.v
change from a spherically symmetric probability-density
another spherically symmetric distribution, which means thai the
state involves a
distribution to
oscillations that take place are like those of a
It
was not necessary
for us to
know
the values of the probabilities a
factions of time, nor the electron wave functions
fc, and
imne v. We must know these quantities, however,
if we
likelihood for a given transition to occur.
an atom in an excited state
198
THE ATOM
from
probability-density distribution
to
a
6.10
and
selection rules,
In order to get an intuitive idea of the physical basis for these
the same frecmency
the electron simultaneously
6.40
to radiate
is
*
and b
wish to compute the
The general condition necessary
that the integral
as
in order to deter-
for
expands and contracts. Oscillations of
this
charged sphere
kind do
thai alternately
not lead to the radiation of
rk'ctroniagnetic waves.
The selection
rule requiring that
/
change by
±1
if
an atom
an emitted photon carries off angular momentum
between the angular momenta of the atom's initial and final
lhat
is
to radiate
means
equal to the difference
states.
The
classical
QUANTUM THEORY OF THE HYDROGEN ATOM
199
analog of a photon with angular
momentum
inagneric wave, so that this notion
The preceding analysis
is
is
atom
is
based on a mixture
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
and indeed
v,
the observed radiation has this frequency.
hydrogen atom between rand
that P dr has
not unique with <|iianlum theory.
of radiative transitions in an
and quantum concepts. As we have
of classical
a circularly polarized elect ro-
However,
concepts are not always reliable guides to atomic processes, and a
deeper
treatment is required. Such a treatment, called quantum rkctrodtjnamics.
modifies 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 eleetric-dipole
values of
r at
The wave
6.
r
+ drhom
the nucleus
which
these
P dr
is
value for a Is electron at
maxima
=
r
s
r'^RJ"
dr.
Verify
aa the Bohr radius.
,
Find the
for a 2s electron.
occur.
function for a hydrogen
atom
in a
with direction
state varies
2p
with distance from the nucleus. In the case of a 2u electron for which
where does P have its maximum in the % direction? In the xy plane?
as well as
m =
0,
,
The
7.
probability of finding an atomic electron
outside a sphere of radius
is Il(r)
radiation
which would be the
maximum
According to Fig. 6-10, Pdr has two maxima
5.
classical
in all directions except that of the electron's motion,
its
f
r„
whose
wave
radial
centered on the nucleus
function
is
|fl(r)|V(/r
classical
prediction.
The wave function R (r) of Prob, 3 corresponds to the groiuid state of a hydrogen
state,
atom, and «„ there is the radius of the Bohr orbit corresponding to that
hydrogen
in
a
electron
ground-state
(a) Calculate the probability of finding a
1(J
quantum electrodynamics provides an explanation for the mechanism 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
In addition,
E
about the
and B
that
fluctuations occur even
classically,
tions"
E=B=
and analogous
oscillator) that
0.
would be expected on purely
when electromagnetic waves
It
is
classical grounds.
Such
are absent
diese fluctuations (often called
and when,
"vacuum fluctua-
sense to the zero-point vibrations of a harmonic
in a
at a distance greater than «„
atom
atom
a ground-state hydrogen
niergy. According to classical
the distance 2a
is
from the nucleus,
(b)
2«„ from the nucleus,
When
the electron in
energy
all its
is
potential
physics, the electron therefore cannot ever exceed
from the nucleus. Find the probability dial
r
> 2a
for the
electron in a ground-state hydrogen atom.
induce the "spontaneous" emission of photons by atoms in
exeited
Unsold's theorem stales that, for any value of the orbital
8.
states.
L the probability densities
= +/
m,
summed
over
all
possible states
yield a constant independent of angles
or
tf>;
quantum number
from m,
that
— —/
to
is,
Problems
1.
"2.
Verify that
Show
Eq.t. 6.1
and
V
6.3 are equivalent
]8p|«|>|
2
=
constant
that
Ibis theorem means that every closed subshell
\/io
&sd0}^-~r-€iaj^9 -
I)
for
4
is
a solution of Eq. 6.13 and that
it is
Show
«iofo
is
a solution of
4,
In Sec. 6.8
I
an atomic electron
=
Im|.
-r/a„
3/2'
6.14
and
that
it
is
it
is
=
2 with the help of Table
in p, d,
and /
a
6.1.
L and
the
maximum
value of
Lt
states.
P
in
=s
selection rule for transitions l>elween states in a
±
1
.
(a) Justify this rule
on
of finding the electron
for
!>)
.1
With
1.
n
the help of the
= 2—» n =
1
wave
harmonic
classical grounds, (b) Verify
wave functions that the transitions n =
possible for a harmonic oscillator while n
normalized.
stated that the probability
The
10.
is
1
THE ATOM
(),/=], and
(Sec. 7.5) has
Verify Unsold's theorem
that
°
200
=
/
Find the percentage difference between
9.
normalized.
fur
°3.
atom or ion
spherically symmetric distribution of electric charge.
1
-* n
=
1
=
-» n
and n
=
3
is
oscillator
from the relevant
=
1
-» n
=
functions listed in Table 6.1 verify that 11
transitions in the
2 are
prohibited.
— ±1
hydrogen atom.
QUANTUM THEORY OF THE HYDROGEN ATOM
201
MANY-ELECTRON ATOMS
7
Despite the accuracy with which the quantum theory accounts for certain of
and despite the elegance and
the properties of the hydrogen atom,
simplicity of this theory,
it
cannot approach a complete description of
essential
atom
this
or of other atoms without the further hypothesis of electron spin and the exclusion
principle associated with
principle
it.
In this chapter
wc
shall
be introduced to the role
atomic phenomena and to the reason why the exclusion
of electron spin in
the key to understanding the structures of complex atomic systems.
is
ELECTRON SPIN
7.1
Let us !>egin by citing two of the most conspicuous shortcomings of the theory
developed
in the
The
preceding chapter.
first is
spectral lines actually consist of
two separate
An example
is
which
of this /me structure
arises
from
transitions
the
first
line of the
between the n
alums. Here the theoretical prediction
while in reality there are two lines
1.4
=
Maimer
3 and n
=
for a single line of
is
A
the experi mental fact that
apart
many
lines that are very close together.
— a small
series of
hydrogen,
2 levels in hydrogen
wavelength
effect,
fi.563 A,
but a conspicuous
failure for the theory.
The second major discrepancy between the simple quantum theory of the atom
in the Zceman effect. In Sec. 6.7 we saw that
hydrogen atom of magnetic quantum number m, has the magnetic energy
and the experimental data occurs
;i
Vm =
7.1
when
21 -f
I
is
is
—
it is
1
located in a magnetic field of Hux density B.
values of
split into 2/
in
1,
m.—B
a magnetic
+1 through
+
1
field.
to
—
f,
Now
m, can have the
quantum number
energy by {cH/2m)B when the atom
so a state of given orbital
substates differing in
However, because changes
in
m, are restricted to -im,
a given spectral line that arises from a transition
between two
=
0,
states of
203
only three components, as shown
different
/ is
Y.tnnun
efftvt, then, consists
into three
split into
in Fig. 7-1.
The normal
of the splitting of a spectral line of frequency
no magnetic
components whose frequencies are
HUH s
magnetic
eh B
1
2m
°
e
h
°
V
4wm
Normal Zeeman
7.2
'*
=
""
+
eH B
I^T
=
While the normal Zeeman
effect
'°
4ffwi
effect
is
indeed observed
elements under certain circumstances, more often
it
is
in the spectra of a
not: four,
six,
few
or even
\
more components may appear, and even when three components are present
their spacing
are
shown
FIGURE
may
in Fig.
not agree with Eq. 7.2. Several anomalous
patterns
FIGURE 7-2
magnetic
Magnetic
field
field present
no magnetic
Reld
magnetic
field
present
expected splitting
The normal and anomalous Zeeman effects
—
m,=
m,=
m,=
= 2-
pii/
K#>
Zeeman
Coudsmit and G.
effect, S. A.
the electron possesses an intrinsic angular
ntf
1
expected splitting
in various spectral lines.
In an effort to account for Ixith fine structure in spectra! lines
effect.
lous
No
present
field
7-2 together with the predictions of Eq. 7.2,
The normal leeman
7>1
Zeeman
field
i>
angular
a
momentum
E.
momentum
this
a certain magnetic
1
a classical picture of an electron as a charged sphere spinning
i
2
hv +
-?r\(*i+tf)
-2n\
on
in
mind was
The
its axis.
momentum, and because he electron is negatively
moment u, opposite in direction to its angular
The notion of electron spin proved to be successful in
rotation involves angular
charged,
1925 that
momentum,
angular
moment. What GoudsmH and Uhlenbeck had
l
—
in
independent of any orbital
might have and, associated with
it
and the anoma-
Uhlenbeck proposed
I
has a magnetic
it
momentum
vector L,.
*
hv a
explaining not only line structure and the anomalous
Zeeman
effect but a
wide
variety of other atomic effects as well.
(If
course, the idea that electrons are spinning charged spheres
is
hardlv
faj
accord with quantum mechanics, but in 1928 Dirac was able to show on the
m, =
to, =
m = ~I
basis of a relalivistie quantum-theoretical
treatment that particles having ibe
charge and mass of the electron must have
just the intrinsic
1
1=
1-
angular
momentum
and magnetic moment attributed to them by Coudsmit and Uhlenbeck.
'
t
Am,
= —1
Am —
Ami =
the electron.
t
("°~4JFW
v
The quantum number S is used to describe the
The only value a can have is s = %;
1
("°
+
4W
Is
gfoen
in
204
THE ATOM
Spectrum with magnetic
7.3
shall see
The magnitude
below,
may
=
\A(*
+
momentum
of
this restriction follows
from
also lie obtained empirically
from
S of the angular
terms of the spin quantum number
S
Spectrum without
magnetic field
we
Dirac *s theory and, as
spectral data.
spin angular
momentum due
.v
to electron spin
by the formula
\)H
= -*—«
field present
MANY-ELECTRON ATOMS
205
which
the
is
momentum
same formula
in
as that giving the
magnitude L of the
orbital angular
M)iu magnetic
terms of the orbital quantum number h
7
L=
Vl(l
+
1)
+
1
momentum
and m„
orientations in a magnetic field frotn
vector can have the 2s
= - \/2
(Fig, 7-1).
%=
4
so that
'n„fr
ft
The space quantization of electron spin is described by the spin magnetic
quantum number m„. Just as the orbital angular-momentum vector can have
the 21
quantum numl)er,
+
I
=
The component St
of an electron along a magnetic
field in
+1
to
-
/,
the spin angular-
2 orientations specified by m,
of the spin angular
the z direction
is
= +%
ratio characteristic of electron spin
The gyromagnetic
is
almost exactly twice
this ratio as equal
that characteristic of electron orbital motion. Thus, taking
related
to its spin angular
is
to 2, the spin magnetic moment p,t of an electron
momentum S by
momentum
determined by the
7.5
m
The
passible
components
any
of ft along
axis,
say the a axis, are therefore limited
to
M„=^ 2m
7.6
recognize the quantity (eft/2m) as the Bohr magneton.
Space quantization was first explicitly demonstrated by
We
W. Gertach
m=
They directed
in 1921.
through a set of collimating
Vi
A
in Fig, 7-4.
its
Stern
and
inhomogeneous magnetic
field, as
shown
beam after
magnetic moment
photographic plate recorded the configuration of the
passage through the
of a silver
into an
slits
O.
a Iream of neutral silver atoms from an oven
atom
is
In
field.
its
normal
due to the spin of one of
state, the entire
its
electrons. In a uniform magnetic
such a dipole would merely experience a torque tending to align it with
field. In an inhomogeneous field, however, each pole" of the dipole is subject
field,
*
dm
FIGURE
7-3
The two possibl* orienangular-momentum
to
,
a force of different magnitude, and therefore there
is
a resultant force on
tations of the spin
vector.
the dipole that varies
orientations should
in a
with
its
be present
broad trace on
beam
in
m = — V4
7.2
The
split into
two
field.
distinct
formed
in
by space quantization.
SPIN-ORBIT COUPLING
fine-structure doubling of spectral lines
electrons.
may be
explained on the basis of
momenta
of atomic
This spin-orbit coupling can be understood in terms of a straight-
forward classical model.
netic field because, in
THE ATOM
result tm-ivly
Stem and Cerlaeh found, however, that the
parts, corresponding to the two opposite spin
I magnetic interaction between the spin and orbital angular
206
Classically, all
a beam of atoms, which would
orientations in the magnetic field that are permitted
-Virt
field.
the photographic plate instead of the thin line
the absence of any magnetic
initial
orientation relative to the
its
An electron revolving about a proton finds itself in a mag-
own frame
of reference, the proton
is
circling alxnit
it.
MANY-ELECTRON ATOMS
207
quantum
state (except 8 states) into
two separate substates and, consequently,
two component
the splitting of every spectral line into
assignment of s
m %
fine-structure doubling.
The
The
what should
fact that
twin states imposes the condition that the St
magnet pole
spin
angular-momentum vector S must
+
2s
To check whether the observed
single states are in fact
lie
possible orientations of the
1
Hence
total 2.
fine structure in spectral lines
we must compute
obtain.
field
field
A
corresponds to
the magnitude
An
the magnetic field experienced by an atomic electron.
inhomogeii MBS
magnetic
+
=2
1
the energy shifts predicted by Eq. 7.8,
**
lines.
the only one that conforms to the observed
is
estimate
circular wire loop of radius r that carries the current
i
is
B of
easy to
has a magnetic
of (lux density
field off
fttrg
2r
classical
pattern
An
at its center.
orbital electron, say in
a hydrogen atom, "sees"
by a proton of charge +e,
/"times each second
a resulting
for
itself
circled
flux
density of
-"
m, so that
actual
photographic
pattern
plate
B
FIGURE
= Hfe
The SternGerlKh eiperimenl.
7-4
In the ground-state
This magnetic
to
then acts
field
upon the
produce a kind of internal Zeeman
dipole of
moment u
magnetic
in a
Vm = —pBcos$
where
the angle between u and B.
is
effect.
density
The quantity
1
=
is,
in general.
is
moment
the
which
a very strong magnetic
=
^
2m
1«
u
Hence the magnetic energy
Vm
component
of the election
is
X
9.27
a,.
=
V"
5.3
X
10
field.
The value
of the Bohr
magneton
is
J/T
of such an electron
from Eq.
is,
7.8,
2m
S 9.27 X
ci
2m
B
Hie wavelength
for
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
in the
THE ATOM
=^-B
2m
7S 1.2
V = ±
7.8
208
=
10 15 and r
B=: 1ST
find that
energy
x
lence, letting
u cos
we
B
/icostf
of p parallel to B, which in the case of the spin magnetic
is tisi .
6.8
electron's
field of flux
7.7
own spin magnetic moment
The energy Vm of a magnetic
/=
Bohr atom
absence of spin-orbit coupling.
The
result
is
the splitting of every
shift
X
10
-''
J/T
X
13
T
10"*"J
corresponding to such a change
a spectral line of unperturbed wavelength 6.5B"} A
observed splitting of the line originating
However, the
ilux
in
the
,
n
in
energy
is
about 2
A
somewhat more than the
=
3
—*
n
—
2 transition.
density of the magnetic field at orbits of higher order
is
less
than for ground-slate orbits, which accounts for the discrepancy.
MANY ELECTRON ATOMS
209
THE EXCLUSION PRINCIPLE
7.3
y(I, 2, 3,
normal configuration of a hydrogen atom, the electron
In the
quantum
Are
What
state.
sioned perhaps as circling
Many
by
just
and
1 1
structures differ
10,
9,
and the
neon,
controls
alkali
this
in
the
hypothesis unlikely.
one electron:
One example
for instance, the
is
the great
elements having atomic
are respectively the halogen gas fluorine, the inert gas
metal sodium. Sinn- the electronic Structure of an atom
interactions with other atoms,
its
n) of a system of n particles can l*j approximately expressed
,
,
That
particles.
t£{3),
of the individual
>£(.'0
.
•
is,
hard
is
it
why
understand
to
the
=
«l,2,3,...,n)
7.9
We
+<l)iKS)iK3)...Mii)
shall use this result to investigate the kinds of
wave
functions that can be
used to describe a system of two identical particles.
chemical behavior exhibited by certain elements whose atomic
(inference in
numbers
lowest
same quantum slate, to Im: envithe nucleus crowded together in a single Bohr orbit?
evidence make
lines of
in its
,
are the normal configurations of more complex atoms?
92 electrons of a uranium atom
all
is
.
product of the wave functions *( 1 ), ^(2),
as the
us suppose that
l^et
in state h.
one of the
particles
in
is
Because the particles are identical,
2
the probability density
the
one
we
require that
|«^|
of the system
a replacing the one
in state
if
quantum
it
should
state
a and the other
make no
difference in
the particles are exchanged, with
b and vice
in state
Symbolically,
versa.
chemical properties of the elements should change so abruptly with a small
change
atomic number
in
same quantum
if all
the electrons in an
atom
exist
together in the
state.
the electronic configurations of atoms having more than one electron.
no two electrons in an atom can exist
Each electron in an atom must have a different set
exclusion principle states that
in
quantum
state.
of
numbers
n,
Pauli
/,
w»
by
a
atom from
possible to determine the various states of an
quantum numbers of these states can be
and from
states
number
of lines are
that
in the
element
correspond to transitions to
same direction
Thus no
in
to give a total spin
although transitions are observed to and from the other ground-state
configuration, in which the spins are in opposite directions to give a total spin
quantum numbers of hath electrons would be it —
while in the slate known to exist one of the electrons
other m, = — '/,. Pauli showed that every unobserved
of 0. In the absent state the
/
=
0,
m =
=
t
has
hi,
0, wi j
'/j
=
and the
atomic state involves two or mora electrons with identical quantum numbers,
Before
we
is
a statement ol this empirical finding.
explore the role of the exclusion principle
structures, let us look into
its
i"
7.11
and still
l>e
r, '/,
210
o.
It is
THE ATOM
wave
in
the
* of the electron in a hydrogen
three separate wave functions, each
function
expressed as the product of
possible
=
*<2 t l)
= -W2)
fulfill F.q. 7.
quantity, and so
Symmetric
*(1,2)
Antisymmetric
it
The wave function
10.
can
of the system
is
not
itself
a measurable
altered in sign by the exchange of the particles.
l>e
functions that are unaffected
by an exchange
metric, while those that reverse sign
of particles arc said to l>e
upon such an exchange are
Wave
sym-
said to
be
antisymmetric.
If
particle
Hie system
in state
is
I
a and particle 2
according to Eq.
is,
in state b,
is
the
wave
function of
7.9,
*,=WI)W2)
7.13,
while
if
* which is a function
to show in an analogous waj
of
one of the three coordinates
lliat
fcMBmplete wave
particle 2
7.13b
=
*n
is
in stale
a and particle
1
is
in state h,
at
that C
correct al any
is
function
any moment whether
'''(uivalently,
we can say
whose wave function
function
is
*n
description
of
.
the
wave
function
is
*.(W)
knowing
i
describing that part of
*(2,1)
or
Because the two particles are in fact indistinguishable,
determining atomic
quantum-mechanical implications. \\V saw
|ue\ ions chapter that the complete
atom can
can be
particles,
I
'/_,,
and the exclusion principle
exchanged
is
spectrum, and the
inferred. In the spectra of every
miwmg
It
helium to or from the ground-state configuration
transitions are observed in
I,
its
having certain combinations of quantum numbers.
which the spins of both electrons arc
of
study of atomic spectra.
*(2,1), representing the
given by either
7.12
led to the exclusion principle
but hydrogen a
1*2(2,1)
His
the same
quantum
m,.
mi,,
=
1,2)
Hence the wave function
1925 Wolfgang Pauli discovered the fundamental principle that governs
In
C-,
7.10
is
moment
C, or
is
the
we have no way
* u describes ihe system. The
same as the likelihood that ii lr
is
correcl.
lhat the system spends half the time in the configuration
y,
and the other
half in the configuration
Therefore a linear combination of *, and
the
of
likelihood
system.
There are two
whose wave
u is the proper
such combinations possible
\fi
MANY-ELECTRON ATOMS
21]
the symmetric one
™
=
*s
ELECTRON CONFIGURATIONS
7.4
~k [U]) U2)
+ * a{2) wl)1
Two
and the antisymmetric one
basic rules determine the electron structures of many-electron atoms:
1.
A system
2.
Only one electron can
Before
\/2
of particles
we apply
factor 1/
y2
is
required to normalize
and 2 leaves ^ s unaffected, while
uhi-v
and
<frB
tyA
.
Exchanging particles
reverses the sign of
it
$A
,
Doth
i£
s and $ 4
7.10.
l",t|.
There are a number of important distinctions between the behavior of particles
systems whose wave functions are symmetric and that of particles in systems
in
whose wave functions arc antisymmetric. The most obvious is that, in the former
case, both particles
and 2 can simultaneously exist in the same state, with a = b,
1
while
in
the latter case,
cannot be
if
we
a
set
same quantum
in the
=
b,
slate.
we
find that
Comparing
s> i
this
=
0:
the two particles
conclusion with Pauli's
empirical exclusion principle, which states that no two electrons
in
an atom can
quantum
While the various electrons
1
total
its
energy
is
a
minimum.
any particular quantum state
exist in
in
an atom.
these rules to actual atoms, let us examine the variation of
electron energy with
The
when
stable
is
state.
in a
complex atom certainly
interact directly with
one another, much about atomic structure can be understood by simply considering 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.
have the same
All the electrons that
total
quantum number n
on the average, roughly the same distance from the nucleus. These electrons
are,
therefore interact with virtually the
same
electric field
and have
similar energies.
conventional to speak of such electrons as occupying the same atomic
It is
Shells are
denoted by capital
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.
shell.
according to the following scheme:
letters
be
The
of
results of various
Y2 have wave
experiments show thai
obey the exclusion principle when they are
move
different
in a
which have
common
quantum
state.
force Held, each
shall
in
'/j,
is,
when
of the system must be in a
are often referred to as Fermi
learn in
Chap.
statistical distribution
9, the
behavior of
law discovered by
Fermi and Dirac.
hawm
gates ol them
because the
was discovered by Hose and
and helium atoms are Base
'['here
wave
Photons, alpha particles,
is
wave
make
it
THE ATOM
It
useful to classify particles according to the
functions rather than simply according to whether or not
they oliey the exclusion principle.
5
I
U
V
O
.
.
.
Atomic shells
.
.
.
In a complex atom the degree to which the
n.
full
nuclear charge
shielded from a given electron by intervening shells of other electrons varies
with
its
to lie
found near the nucleus (where
probability-density distribution.
than one of higher
I
(see Fig. 6-1
higher binding energy) for
it.
1),
The
it is
An
electron of small
/ is
more
likely
poorly shielded by the other electrons)
which
results in a
lower
total
energy
(that
electrons in each shell accordingly increase
in energy with increasing /. This effect is illustrated in Fig. 7-5, 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
mbshelt.
the
particles,
upon
that
law that describes aggre-
functions lwsides that expressed in the exclusion principle.
these consequences that
nature of their
212
Einstein.
4
The energy of an electron in a particular shell also depends to a certain extent
its orbital quantum number /, though this dependence is not so great as
All the electrons in a subshell
/ in
a
shell are said to
occupy the same
have almost identical energies, since
dependence of electron energy upon m, and
The occupancy
are other important consequences of the symmetry or antisymmetry
nl particle
is
statistical distribution
3
upon
is,
Particles whose spins are
or an integer have wave functions that arc symmetric 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
2
a spin
as well as electrons,
the saime system; that
member
Particles of spio
fermiom because, as we
aggregates of them is governed by a
particles or
and neutrons
J
K
functions that are antisymmetric to an exchange of any pair
of them. Such particles, which include protons
they
all particles
n at
of the various subshelts in an
mt
is
atom
is
comparatively minor.
usually expressed with
Ihe help of the notation introduced in the previous chapter for the various
quantum
is
states of the
identified
'" its orbital
hydrogen atom. As indicated
in
Table
6.2,
each subshell
quantum number n followed by the letter corresponding
quantum number /. A superscript after the letter indicates the
by
its
total
MANY-ELECTRON ATOMS
213
r
THE PERIODIC TABLE
7.5
When
the elements arc listed in order of atomic number, elements with similar
chemical and physical properties recur
known as
observation,
about a century ago.
the periodic law,
A
This empirical
regular intervals.
at
was
first
formulated by Dmitri Mendeleev
tabular arrangement of the elements exhibiting this
recurrence of properties
is
called a periodic table,
simplest form of periodic table; though
Table
7.1
more elaborate periodic
devised to exhibit the periodic law in finer detail. Table
7,
1
is
is
perhaps the
tables
have been
adequate for our
purposes.
Kleinents with similar properties form the groups
Table
in
Thus group
7.1.
I
which are extremely active chemically and
Group VII
of
—
1
all
columns
of
+1.
nnmnetals that have valences
gaseous state. Croup VIII consists
in the
of the inert gases, elements so inactive that not only
compounds with other elements, but
as vertical
alkali metals, all
of which have valences of
consists of the halogens, volatile, active
and form diatomic molecules
shown
hydrogen plus the
consists of
their
do they almost never form
atoms do not join together
into
molecules like the atoms of oilier gases.
a
The horizontal rows in Table 7.1 are called periods. Across each period is
more or less steady transition from an active metal through less active metals
and weakly active nonmetaJs to highly active nonmetals and
gas.
finally to
an
inert
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
the reverse
A
the
is
metals
is
accompanied by greater chemical
series of transition elements
group
II
and group
III
;
in
each period after the third between
transition elements are metals with
13.6 sV
The binding energies
in
ol atomtc electroni tn Ry,
period 6 are virtually indistinguishable
as the
(1 Rjj
=;
1
Rydberg =
kmthantde elements
The notion of
of sodium
is
ls a 2s
which means
2
2p
214
=
3,
THE ATOM
f
a
= 0)
Let us see
The
V
how
which
is
just a
subshell contains
I
one
electron.
contain
and the
similar
found in period
fits
group of
and arc known
closely
related
7.
perfectly into the pattern of
mirror of the atomic structures of the elements.
this pattern arises.
quantum number n and
I
A
number of electrons that
A subshell is characterized by a certain total
quantum number /, where
exclusion principle places definite limits on the
can occupy a given subshell.
= 1, = 0) and 2s (n = 2, = 0) subshells
tlie 2p (n = 2,1= 1) subshell contains six electrons,
that the Is (n
two electroas each,
3s (n
For example, the electron configuration
written
is
in their properties,
electron shells and subshells
the periodic table,
of electrons in that subshell.
(or rare earths).
metals, the actinide elements,
ground-slate energy of H atom.)
number
one another but no pronounced resem-
blance to the elements in the major groups. Fifteen of the transition elements
|
ATOMIC NUMBER
FIGURE 7-5
while
The
appears
elements.
a considerable chemical resemblance to
H Li B|N|FNe
He Be C O
activity,
true in the halogens.
t
There are 21
orbital
= (\l,2,...,(n-l)
+
1
different values of the
magnetic quantum number m, for any
MANY ELECTRON ATOMS
215
E
„
...
a.
lj
1 i
°
?
a
.g
—
Sig
n
e
—
>
= U.|
"1
a
g
>
*o|
=
<8 M
sn
S
e4
«8
££§
/,
8
-,
=
lJU
sa| 3-g £<§
8
since
S£0 §j|
8*|
and two possible values of the spin magnetic quantum number hi, +
(
for any »t
Hence each snbshell caj) contain a maximum of 2(2/ +
a
I!
h
»u| ,-8
EL
i
=
a.
a B
* e
«
-
»k3
3
E
a.
8
1^
=
««l
(9
2
maximum
shell a
+
2(2/
1)
g|g
1-T
<i
s
R
^i *s|
B»3
«8
? = S
2
8
gas
c
*|
8-^
"5 4=
a
'Be—
S[S|
Ml
i||
5fi|
sdS
?£f
x |[1+
{2»
An atomic
A
3
+
5
+
-
1)]
(/
=
1) six
Rj-i
closed s subshell
electrons, a closed
•
+
2(n
-
1)
+
lj
•
-
+ 2n -
1]
since
-
PZ
tlie
*
moment,
readily detached.
lie
gases— and the
_.8 "*1
l~
t.
8
if?
pfi
f
at fc
electron,
B5|
*8
which
&£|
is
i
i
.1
*
(I
=
quota of electrons
is
electrons, a closed
2) ten electrons,
said to
subshell
p
and so on.
is
large relative to the negative charge of the
it
atom containing only closed
does not attract other electrons, and
Such atoms
we
shells
electrons cannot
expect to be passive chemically, like the
inert gases all turn out to
have
a single electron in their
relatively far
its
closed-shell electron con-
outermost shells tend to lose
from the nucleus and
all
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
1
two
0) holds
.
be accounted
Sdg
+
is
is shielded by the inner
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
a,
8
x
**I
--*
=
snbshell
positive nuclear charge
electrons from
sag
%[1
is
2n*
inner shielding electrons (Fig. 7-6). Since an
B*| s£§
*2gj
=
of electrons in the nth shell
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,
3
'-
(1
</
a>38
a.
members
of the various groups of the periodic table
may
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
element to exhibit
cn
+
lliis
—
-
2[1
figurations or their equivalents.
as
1
=
•if
8*!!
n>3 5
+
shell or subshell that contains its full
Those atoms with bul
- = !
5
5?
ao| **8
<m
™
+
maximum number
so thai the
he closed.
inert
3
s
3
s
-8
M
i «i
"
a
I)],
2
*(;|
fc
t-
-
has no dipole
(3
+
quantity in the brackets contains n terms whose average value
"ITie
»«1 *"3
- -
U.
jjj
a.
2[1
So|
t- J«
8
*2|
tflj Sfl|
M
_
c
3 "5
a *l
s
=
(-0
(2n
o
electrons
IN
X 1
=
1)
of
sag
^
8
$5* S*|
go
_
= 8
8p"|
o>
% and - %)
.
(
and each
i»
(9
±1.±2,...,±(
0,
complex atoms, discussed
this effect is potassium,
in the previous section.
whose outermost electron
The
is
first
in a
4s
n
MANY-ELECTRON ATOMS
217
Table 7.2.
ELECTRON CONFIGURATIONS OF THE ELEMENTS,
L
Z>
1
li
2 lie
3
2
1
2
K
2
2
1
6C
7N
2
2
2
2
2
3
SO
2
2
4
9P
2
2
5
2
10
Ne
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
2
2
6
2
ISA
2
2
K
2
2
6
e
2
5
8
6
20 C»
2
e
2
21
19
a
St-
2
22 Ti
2
2
b
2
23V
2
2
1
2
24 Cr
I
2
6
2
23 \ln
2
2
6
2
26 Fe
2
2
6
Co
2
2
6
28 Ni
The
difference in binding energy
at Fig. 7.5 will
be instructive.
The order
which electron
in
Is.
as
we
atoms
2
2
31
Ca
Ce
2
2
2
2
All of
3
2
6
6
5
|
5
2
2
6
e
2
2
7
2
8
2
2
e
e
e
10
I
2
8
10
2
2
6
e
10
2
1
2
10
2
2
6
6
2
e
10
2
2
10
2
4
chromium
35 Br
2
2
6
2
10
2
present at
36 Kr
2
2
6
2
10
2
37 Hl>
2
2
6
2
10
2
38 Sr
2
2
e
2
10
2
5
6
e
e
is
fip, 7s,
Y
40 Zr
is
6d
similarities in
W
1
The
2
6
6
addition of 4/ electrons
has virtually no effect on the chemical properties of the lanthanide elements,
THE ATOM
Zn
2
2
and actinides are easy to understand
i
2
the lanlhanides have the same 5s 5p
configurations but have incomplete 4/ subshells.
218
30
I
2
2
The remarkable
the lanthanidcs
basis of this sequence.
2
39
subshells are filled in
can see from Table 7.2 and Fig. 7-7.
among
2
6
e
2
connection another glance
2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4/, 5d,
chemical behavior
on the
tiiis
2
Cu
2
2
and copper. In both of these elements an additional 3d electron
4/
5j>
1
34 Se
not very great, as can be seen in the configurations of
4rf
2
33 As
electrons
the expense of a vacancy in the 4s subshell. In
4(1
between 3d and 4s
3d
is
2
29
32
4i
2
3
instead of a
substate.
2
2
2
27
3,1
2
2
it
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,
3) shell.
even though the outer electrons In both cases are In the M(»
3p
1
Be
.-,
FIGURE 7-6
2|>
1
1
Na
M
3t
2
2
6
2
a
a
e
e
e
e
10
2
1
2
2
2
6
2
n
10
2
2
2
1
2
II
Nb
2
2
(i
2
6
10
2
4
1
12
Mo
2
2
6
2
10
2
6
5
1
43 Tc
2
2
6
2
6
6
10
2
6
5
2
fill
2
2
G
2
6
10
2
B
7
1
45 Rh
46 Pd
2
2
6
2
6
10
2
6
8
1
2
2
6
2
10
2
6
in
47 Ag
2
2
a
2
10
2
1
2
2
6
2
10
2
e
6
10
CI
10
2
49 In
2
2
6
2
10
2
10
2
1
50
Sri
2
2
2
10
2
10
2
2
51 Sb
2
2
e
a
6
6
6
6
6
e
10
2
10
2
3
!
1
IS
2
6
5d
5/
6,
6p
6il
7
{Continued)
Table 7.2
K
U
1.
2P
3./
4,
4|<
4,1
b
3p
2
6
2
10
2
2
4
2
6
2
10
2
6
6
III
10
2
5
2
6
2
10
2
li
10
2
6
10
2
6
10
2
6
10
2
10
2
10
2
10
2
2
3
2
10
6
6
8
8
6
6
8
52 Te
2
53!
2
34 Xc
2
2
2
6
2
59 Pi
2
2
6
2
60
V.I
2
2
6
2
fil
Pm
2
2
6
2
62
Km
2
2
6
2
6
6
6
6
e
6
6
6
6
6
6
63
l-.,i
2
2
6
2
6
64
Cd
2
2
6
2
6,5
Tb
2
2
6
66
Dy
2
2
6
55 Cs
2
2
6
2
58
ll..
2
2
6
2
57 La
2
2
6
2
Cc
P
4)
3;
2i
58
,v
3.
li
«/
U
5/
6.
6,,
Q
6,1
7.
)'
1
10
2
10
2
6
6
6
6
10
s
6
10
4
2
10
2
10
5
2
10
2
10
8
2
10
2
6
e
6
10
7
2
6
10
2
6
10
7
2
6
2
6
10
2
e
10
9
2
6
2
2
6
[0
2
6
10
10
2
6
2
10
2
10
11
2
6
10
2
10
12
2
6
2
10
2
8
6
8
2
10
13
2
6
2
10
2
J_
2
2
1
2
2
)'
2
f
2
2
2
2
1
»
67 l[o
2
2
2
2
6
6
2
68 Er
69 Tin
2
2
6
2
e
e
6
2
6
10
2
6
10
14
2
6
2
e
10
2
8
10
14
2
6
1
2
2
6
10
2
10
14
2
6
2
2
2
e
10
2
11
2
3
2
e
10
2
8
8
10
2
10
14
2
6
4
2
5
6
2
which arc determined by the outer
2
6s 26p fi 7s 2 configurations, and differ only in the
2
Til
Vh
2
2
71
Lu
2
2
72
llf
2
2
73 Ta
2
2
6
6
6
6
74 VV
2
2
6
75 Re
2
2
6
2
6
10
2
6
10
14
2
a
76 Os
2
2
6
2
10
2
14
2
6
Ir
2
2
6
2
10
2
7fi
PI
2
2
a
2
6
10
2
6
8
6
10
77
6
6
79 An
2
2
6
2
10
2
ll
SO lig
2
2
e
2
10
2
81 Tl
2
2
6
2
10
2
82 Fh
2
2
6
2
10
2
8
8
6
S3 Bi
2
2
6
2
10
2
84
2
2
6
2
10
2
85 At
2
2
6
2
10
2
Mi Ki,
2
2
6
2
10
2
ll
87 Ft
2
2
6
2
10
2
li
88
Eta
2
2
g
2
10
2
89
Ac
2
2
6
2
10
2
91 Pa
2
2
6
2
92
2
2
6
2
Np
2
2
6
2
94 Pu
2
2
B
2
e
e
6
6
e
6
6
6
6
6
6
e
6
6
6
6
1'.
90TTi
!Ki
2
2
6
2
10
2
10
2
10
2
10
2
10
2
14
2
6
7
2
14
2
6
a
I
10
14
2
6
10
t
10
14
2
6
10
2
10
14
2
2
t
14
2
10
2
2
e
10
14
2
10
2
3
I
8
6
10
14
2
8
6
6
6
10
10
10
2
4
3s
10
14
2
10
2
to
14
2
6
10
2
5
6
10
14
1
6
10
2
to
14
2
6
to
2
to
14
2
6
10
2
ii
10
14
2
6
to
2
2
6
2
8
10
3
2
10
14
2
8
10
4
2
8
6
10
14
2
6
to
5
2
6
10
14
2
6
10
6
2
10
14
2
6
10
7
2
8
6
10
8
2
10
10
2
6
10
11
2
6
6
6
6
6
6
6
6
6
2
2
6
2
6
10
2
2
2
B
2
10
2
97
Ilk
2
2
6
2
10
2
1
10
14
2
98 Cf
2
2
6
2
10
2
li
10
14
2
I
2
6
o
6
e
6
6
10
2
li
10
14
ft
6
2
6
10
2
2
6
2
6
10
2
1(12
\„
2
2
6
2
6
10
2
103
Lw
2
2
6
2
i;
to
2
6
6
6
6
2
6
6
8
2
in
10
14
2
to
14
2
10
14
2
10
14
2
6
6
6
of
quantum
S
states In
7
fl
an atom. Not to
scale.
electrons. Similarly, all of the actinides
10
12
2
10
13
2
10
14
2
10
14
2
irregularities in the binding energies of
=
2)
and neon (Z
argon (Z
=
18) has only
Helium (Z
"it
and 3p
subshells.
=
10)
full
numbers of
their
have
5/ and
Bel
the
I
2
and so the 4s subshell
is filled first
3d
in its
is
2
outer 4s electrons that
2
I
2
1
2
1
2
t
2.
1
2
1
2
1
2
2
2
and
L
shells respectively,
M shell, corresponding to
suhshell
is
not
3d
filled
next
electrons, as
make
is
closed
simply that
we have
said,
potassium and calcium. As the 3d subshell
successively heavier transition elements, there are
2
filled in
in
K
contain closed
8 electrons
The reason
atomic electrons are also re-
outer shells in the heavier inert gases.
4s electrons have higher binding energies than
1
14
I
2
10
These
14
Am
The sequence
4
3
sponsible for the lack of completely
10
!M>
2
(1
2
1
electrons.
10
95
2
FIGURE 7-7
10
e
6
8
8
e
=
n
10
8
6
6
99E
100 Fm
1111
Md
2
possible chemical activity.
still
Not
one or two
krypton
another inert gas reached, and here a similarly incomplete outer shell
occurs with only the 4s and 4p subshells filled. Following krypton is rubidium
(Z
=
36)
=
37),
until
is
which skips both the 4d and 4/ subshells to have a 5s electron. The
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
I
Z
next inert gas
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
2
properties of the elements in terms of their electron configurations,
2
can be quantitatively understood by similar reasoning.
many more
2
1
2
MANY-ELECTRON ATOMS
221
and magnetic moment of a closed shell are
ions He 4 Be + Mg + , B + \ Al ++ and so on.
HUND'S RULE
7.6
,
In general, the electrons
spins— whenever
magnetism
in
possible.
an atom remain unpaired— that
This principle is called Hund's
and nickel
of iron, cobalt,
subshells are only partially occupied,
is
instance, five of the six
has a large resultant magnetic moment.
The
origin of Hund's rule
lies in
lower the energy of the atom.
moments
We
shall
whose
more separated
ment, having
less
energy,
In iron, for
to cancel out.
so that each iron atom
in
of the orbital angular
determined by
orbital quant' im
l=
7.18
if
off.
and
TOTAL ANGULAR
Each electron
spin angular
mentum
in
this arrange-
7.M
S,
momentum L and a certain
Like
J of the atom.
all
angular momenta, J
is
to the total
angular mo-
VJ(J
+
7.17
/,
Mj n
z
component
of total atomic angular
7.22
is
which provides a more intuitively accessible framework for understanding angular-momentum considerations than does a purely quantummechanical approach.
a single electron.
Atoms
of the elements in group
hydrogen, lithium, sodium, and so
on— are
momentum
I
is
of the periodic table-
THE ATOM
momentum
sole value
+
S
is
determined by
1
/2 ) according
to the
l)ft
to the
is
determined by the magnetic
formula
they must be added vectorially to yield the total
S
rtij
for the
quantum numbers
that describe
/(
\//{/
=
+
1) ft
mfi
numbers,
it is
relationships
among
the various angular-momentum
simplest to start with the Z components of the vectors
quantum
J,
L,
and
Since }t L^, and S^ are scalar quantities,
S.
,
ffijfi
=
nijft
±
m,h~
sad
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 = /2 and as a
r, " 'ili
m, must l>e half-integral. The possible values of m also range from
+/
f
through
to -/in integral steps, and so, for any value of i.
l
,
electrons outside closed inner shells (except for hydrogen, which has no inner
222
determined by the magnetic
for a single electron, so that
7.24
provided by
of this kind since they have single
and the exclusion principle assures
+
customary to use the symbols /and
/=
7,25
consider an atom whose total angular
is
formula
m.ft
= L+
7.23
of the atom,
electrons)
J
To obtain the
.
first
to the
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
Let us
along the z axis
fi
L and S are vectors,
momentum J:
momentum
and a component Jt in the z direction given by
=
(
V«(s
=
S,
angular
It
Total atomic angular
1)«
L
quantum number m, according
J and }t
/=
of an atomic electron
according to the formula
quantized, with a magni-
tude given by
7.16
=
S
Because
both of which contribute
/
while the component St of S along the z axis
spin
MOMENTUM
an atom has a certain orbital angular
momentum
of
quantum number s (which has the
7.21
*7.7
category are the
by wave
Electroas with parallel spins
they paired
in this
momentum L
number
Similarly the magnitude S of the spin angular
formula
space than
m
Lt =
the same subshcll with the same
in
Lt
quantum number m, according
the spin
Electrons
its
while (he component
7.19
Also
Vi(iTT)H
examine other consequences
the more stable one.
is
The magnitude L
apart the electrons in an atom are, the
spatial distributions are different.
are therefore
is
rule; their 3cf
spin must have different m, values and accordingly are described
functions
ferro-
the mutual repulsion of atomic electrons.
this repulsion, the farther
Because of
parallel
The
connection with molecular (wilding.
in
i)
have
the electrons in these subshells do not
electrons have parallel spins,
3d
of Hund's rule in Chap,
a consequence of Hund's
and
pair off to permit their spin magnetic
is,
rule.
zero.
,
,
that the total angular
momentum
i
MANY-ELECTRON ATOMS
223
7.26
j
Like
trip j is
= l±S
always
half- integral.
Because of the simultaneous quantization of
certain specific relative orientations.
This
of a one-electron atom, there are only
of these corresponds to
<
that J
L,
j
= +
/
s,
two
so that /
is
L, and S they can have only
J,
a general conclusion;
in
the ease
relative orientations possible.
>
«.,
and the other
L and
Figure 7-8 shows the two ways in which
to
/
-
/
—
One
s,
so
S can combine to
form J when / = 1 Evidently the orbital and spin angular-momentum vectors
can never be exactly parallel or antiparallel to each other or to the total angular.
momentum
vector.
as a result exert torques
S interact magnetically, as
on each other.
momentum
If
there
conserved
is
we saw
magnitude and
and
I
around the direction of
B
is
I.
and S precess about
cone traced out by L
J
of the
atom.
However,
if
L and S
there
of B
B
is
an
cone traced out by
what gives
rise to the
anomalous Zeeman
effect, since different orientations
the
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
momenta and magnetic moments. These
to the total
atomic angular
contributions are small because nuclear
magnetic moments are
7-8
The
tvro
ways
In
which
I.
and S can be added to form
J
when
I
=
1, t
=
!*.
—lO -3
atom
the magnitude of electronic moments,
and they
lead to the hi/perfine .structure of spectra! lines with typical spacings between
components of
FIGURE
S
while
present, then J precesses about the direction
S continue precessing about J, as in Fig. 7-1 (J. The precession of J about
external magnetic field
L and
their resultant J (Fig. 7-9).
and
direction,
he effect of the internal torques can only be the precession of
in
orbital
no external magnetic
the total angular
is
The
according to the vector model
in Sec. 7.2,
field,
J
7-9
spin angular- momentum vectors
The angular momenta L and
and
FIGURE
— I0~ 2 A
as
compared with
typical fine-structure spacings of
several angstroms.
FIGURE 7-10 In the
presence ol an enter"ill magnetic tie Id B,
the total angular-
momentum
vector J
Precesses about fl according to the vector
model of the atom.
cone traced
out by
,=! + ,= 3/2
J
,=/-»= 1/2
i- the
224
atom
THE ATOM
MANY-ELECTRON ATOMS
225
LS COUPLING
*7.8
When more
momenta
than one electron contributes orbital and spin angular
momentum
to the total angular
J of an atom, J
the vector
is still
sum of
these
momenta. Because the electrons involved interact with one another,
the manner in which their individual momenta L, and S, add together to form
follows certain definite patterns depending upon the eireum stances. The usual
J
individual
pattern for
but the heaviest atoms
all
momenta L
that the orbital angular
is
s
(
L and
resultant
the spin angular
we
examine the reasons
shall
The momenta L and S
later in this section.
may be summarized
for this behavior
This scheme, called LS
J.
as follows:
L=2
L=3
then interact magnetically via the
momentum
spin-orbit effect to form a total angular
coupling,
together independently
momenta 8, are coupled
into another single resultant S;
When
FIGURE 7-11
combine
to
form
I.
=
I,
1, i,
4
J
As
usual, L, S,
numbers being
L,
7.29
j,
S
Lt
and
S,,
,
\A.(L
= ML
M L Ms
,
+
i)
Both L and
M
L
with
=
/
2
2.
L
These correspond
1
—
l
2.
The
that
=
S 2 to
226
1.
S,
We
spin
=
when
values between L
trito
a resultant
1
and
it
remaias true
in general that ht|
to contribute
/
=
2
is
differ-
not spherically symmetric
and the electron has no orbital angular
anyway.) Because of the asymmetrical distributions
between the electrons
an atom
in
certain relative orientations are stable. These stable configurations correspond
is
is
quantum numbers are
involved and integral or
11
if
an
consider two electrons, one with
in
which L, and
/,
=
and the other
1
can be combined into
L*,
quantized according to Eq. 7.28, as shown in Fig. 7-11,
1,2,
and
Sj
+
3, since all values of
Sa as
,
the vector
s
is
always
+
in Fig. 7-11, that
+ S
and L
sum
—
to
a
L
=
L are possible from
l
/2 so
,
I,
that there are
correspond
to
S
=
+
l^
two
and
is
S, so
1).
The quantum number
here J can be
11,
1, 2,
3,
J can have
or 4.
+
\/L(L
1)
momentum
angular
total orbital
energy
understand
if
that
is
quantized according to the formula
is
usually such that the configuration
ft.
The coupling between
of lowest
involved.
quantum number
sum
momenta
into a resultant S.
vary with the relative orientations of their angular-momentum vectors, and only
note that L, and Lj can never be exactly parallel to L, nor S t and
except
THE ATOM
is
to L
possibilities for the
momenta
of their charge densities, the electrostatic forces
1) ft
There are three ways
a single vector
f
let us
can
The origins of these
forces are interesting. The coupling Iwtwccn orbital angular momenta can be
understood by reference to Fig. 6-12, which shows how the electron probability
density |^| 2 varies in space for different quantum states in hydrogen. The corre-
momentum
an odd number of electrons
if
I.,
sponding patterns for electrons in more complex atoms will be somewhat
n
are always integers or 0, while the other
As an example,
all
,
1) ft
even munl>er of electrons
S
quantum
:
half-integral
to
quantized, with the respective
and Mj. Hence
which L, and
in
existence to the relative strengths of the electrostatic
except for s states. (In the latter case
7.33
7.32
}c are
s=o
S=l
and 1. = 2, t, = W, there are three ways
which S, and Bj can combine to form S.
fi
= \/S(S +
S, = MsR
/= \/J(J +
! = Mj fi
7.31
its
L=l
forces that couple the individual orbital angular
ent, of course, but
S
7.30
in
the individual spin angular
L, S, J,
l=
7.28
LS coupling
= W,
and two ways
The LS scheme owes
e 2S
=L+
S
7.27
8
'li
of the various electrons are coupled together electrostatically into a single
is
the various Lj
the one for which
we imagine two
L
is
a maximum. This effect
electrons in the
same Bohr
orbit.
is
easy to
Because the
electrons repel each other electrostatically, they tend to revolve around the
nucleus in the same direction, which maximizes
directions to
minimize
/,,
L
If
they revolved
the electrons would pass each other
more
in
opposite
frequently,
leading to a higher energy for the system.
The
origin of the strong coupling
because
it is
direct interaction
he noted,
is
between electron spins
a purely quantum-mechanical effect with no
between the
insignificant
intrinsic electron
and not responsible
is
harder to visualize
classical analog.
magnetic moments,
for the coupling
it
(The
should
between electron
MANY-ELECTRON ATOMS
227
The
spin angular momenta.)
basic idea
complete wave function
that the
is
^{1,2, ...,n) of a system of n electrons
is
the product of a
and a spin function
h{1, 2, .... n) that describes the coordinates of the electrons
As
n) that describes the orientations of their spins.
$(1,2
7,3, the complete function ^(1,
that u(l, 2
n)
is
2,
.
.
,
,
must
n)
not independent of
s{l, 2,
l>e
.
.
we saw
in Sec.
antisymmetric, which means
A change
n).
,
.
wave function
to
=
I
1,
d
to
accompanied by a change
which means a change
one
total spin
angular-momentum
in the spatial electronic configuration of the
in the
atom
s electrostatic potential
momentum
angular
a
weak magnetic
This situation
force.
momenta
the spin
S,
is
what
energy. To go from
.
.
,
SB which requires only
,
described
when
it
is
said (hat
are strongly coupled together electrostatically.
The S always combine into a ground-state
maximum. This is an example of Hund's rule;
i
with parallel spins have different
which means
functions,
is
.
,
m
that there
configuration in which S
as
mentioned
is
a
earlier, electrons
values and are described by different
t
is
wave
a greater average separation in space of
the eleclroas and accordingly a lower total energy.
Although
wc
S that makes / a
minimum
S
A
superscript
combination of
1
2
P
D
jj
L and
COUPLING
—
L
and
1;
by 2L
+
The
the St into
is
accordingly slower than the precession of
spin-orbit interactions
comparable
the L, and between the S
\
similar
LS
coupling,
in the
I
lie
directly to
situation referred to as
/
and S around
jj
effect in
total angular
coupling since each
earlier.
magnetic
/...
2
the letter
which
is
{
P
for instance)
number
the
used to indicate the
is
of different possible orientations
in
=
which S
l
/2
<
=
L
*>
P
three-halves")
a "d J **
%
l*° r
ground
/
=
state of the
outside closed n
V?)
momentum
sodium atom
1
and n
=
2
arises
of this electron
from a single outer
used as a prefix:
is
described by 3 2 S t/2 since
is
,
=
an electron with n
=
atom
of the
quantum numltcr n
3,
/
=
0,
and *
For consistency
shells.
=
'/
it is
2
only a single possibility for
J since
L
=
elec-
conventional
denote the above state by 3 2 S, /;! with the superscript 2 indicating
is
its
(and hence
a doublet,
0.
J.)
produce
l>egins to
break down.
fields (typically
— 10 T),
atomic spectra. In the limit of the
momenta
form the angular
manner described
to
H
'7,10
ONE-ELECTRON SPECTRA
magnitude to the electrostatic ones between
in strong external
which produces the Paschen-Back
failure of
in
and the IS coupling scheme
,
f
breakdown occurs
add together
L
heavy atoms the nuclear charge becomes great enough
G
an electronic configuration
even though there
form J in light atoms, and dominate the situation even when a
moderate external magnetic field is applied. (In the latter case the precession
B
6...
historical reasons, these designations are called term symbols.
thus the
to
in
5
script after the letter, so that a a P3/2 state {read as "doublet
L
However,
4
which S > L, the multiplicity is given
angular-momentum quantum number J is used as a sub-
total
to
of J around
to its total orbital
so on. (In a configuration in
1.)
another vector S are stronger than the magnetic spin-orbit forces that couple
and S
I-
In the even! that the angular
L and
p
used
L and S and hence the numtwr of different possible values of J. The multiis 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 = y2> 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
results in the lowest energy.
electrostatic forces that couple the Lj into a single vector
= 0,
plicity
tronic configuration has
The
atom according
I
letters is
of
electron, the principal
•7.9
3
number Iwfore
multiplicity of the state,
refers to
shall not try to justify this conclusion, the
states are
corresponding to
and so on. A similar scheme using capital
2,
=
atom,
S to a different one involves altering the
momenta S„ S 2
s
tie
structure of the atom, and therefore a strong electrostatic force, besides altering
the directions of the spin angular
=
with
letter,
to designate the entire electronic state of an
L
orientations of the electron spin
/
angular-momentum
that individual orbital
angidar-m omen turn quantum munl>er L as follows;
in the relative
vectors must therefore
we saw
In Sec, 6.4
customarily described by a lowercase
J,
of the individual electrons
momentum
J, is
J of the entire atom, a
described by a
quantum number
We
are
now
a position to understand the chief features of the spectra of the
Before
we examine some
be mentioned thai further complications
representative examples,
exist
lictween electrons and
6.10).
vacuum
it
should
which have not been considered
here, for instance those that originate in relativistic effects
and
in the
coupling
fluctuations in the electromagnetic field (see Sec.
These additional factors
*ul .states
Hence
in
various elements.
split certain
energy states into closely spaced
and therefore represent other sources of
fine structure in spectral lines.
Figure 7-12 shows the various states of the hydrogen atom classified by their
7.34
h =U +
s
(
jj
coupling
lotal
quantum number n and
orbital
angular-momentum quantum number
selection rule for allowed transitions here
238
THE ATOM
is
if
= ±1,
which
is
/.
The
illustrated
by
MANY-ELECTRON ATOMS
229
excitation
that the lowest
D
eV
energy,
n
13.0
one
correspond
will
exclusion principle. Figure 7-13
= o»S
=3
instead of n
=
1
shown, there
/,
that
is,
of
fi-11
because of the
indeed agreement for
levels also
for the states of highest angular
a
S
n
the states of highest
only refer to Fig.
Un=
to
the energy-level diagram for sodium and, by
comparison with the hydrogen
To understand the reason
10
is
is
momentum.
for the discrepancies at lower values of
to see
how
/,
we
need
the probability for finding the electron in
hydrogen atom varies with distance from the nucleus. The smaller the value
/
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
similar,
is
and accordingly we expect the outer electron
excitation
energy,
D
eV
F
hydrogen
5.13 -m
7S
7p
6p
Ss
/
-Sp
0-"
n=
/
4d
'*'-M
—
1
FIGURE 7-12 Energy-level diagram
mora prominent spectral lines. The
lor
hydrogen showing the origins of soma of the
n = 2 and » - 3 levels and
FIGURE 7-13
detailed structures of the
the transitions that lead to the various components of the H
.
Una are pictured
level
In the
dium.
diagram
Energy,
6/
6d
5d
5/
—
n
"
=6
=5
-»=4
4,
-.=>
43
3-
for so.
The energy
levels of
hydrogen are
included for com-
the transitions shown.
diagram of
energy, but the same
The
indicate
latter effect
is
The
state.
—>
n
=
is
"Lamb
we assume
+c
2 and n
=
different
j
same n and
in
a simple
is
its
3 levels are
separated in
/,
and was
rt
=2
first
% 2 S l/2 state relative to the 2 z Pl/2
the
HB
spectral
line
(ti
=
1
shells,
inner core completely shield
and
+ \i)e
so,
of
acted upon by an effective nuclear charge
hydrogen atom. Hence
sodium
parison.
but with different
/
n and
has a single 3s electron outside closed inner
that the 10 electrons in
just as in the
THE ATOM
=
omitted
is
seven closely spaced components.
that the energy levels of
230
ti
states of small
shift" of the
nuclear charge, the outer electron
of
of the
same n and
true of states of the
most marked for
detail that
various separations eoaspire to split
2) into
The sodium atom
if
substates of the
all
established in 1947 in the
3
some of the
this kind, the detailed structures
pictured; not only arc
I.
To
will
we
expect, as a
be the same
as those
first
approximation,
of hydrogen except
MANY- ELECTRON ATOMS
231
in
is
a
a sodium atom to penetrate the core oF inner electrons most often
an
in
d
s state, lass often
and so on. The
state,
when
less
it
in a
is
p
state,
are displaced
its total
it,
and
often
less
shielded an outer electron
charge, the greater the average force acting on
more negative)
still
is
when
from the
nuclear
/
in
is.
excitation
singlet states
triplet states
(parahelium)
(orthohelium)
energy, cV
~
24.6
the
sodium
their equivalents in hydrogen, as in Fig. 7-13,
there are pronounced differences in energy Iretween states of the
different
full
it
in
is
it
smaller (that
ilu-
energy. For this reason the states of small
downward from
when
and
1
same n but
I.
20
TWO-ELECTRON SPECTRA
*7.11
A
single electron
is
responsible for the energy levels of both hydrogen and
sodium. However, there are two 1$ electrons
it
interesting to consider the effect of
is
To do
behavior of the helium atom.
this
in
LS coupling on
wc
15
the ground slate of helium, and
first
and
the properties
note the selection rules for
allowed transitions under LS coupling:
735
AL =
7.36
AJ
7.37
AS a
±1
=0, ±1
0,
to
LS selection rules
When only a single electron
is
the only possibility.
J
=
0, so that J
The helium
=
is
AL =
involved,
is
-» J
=
is
energy-level diagram
an excited
in
shown
is
are coupled,
it
is
:£l
has
5
The
in Fig. 7-14.
is
but because the angular
state,
=
initial state
AL = Af
prohibited.
represent configurations in which one electron
is
prohibited and
Furthermore, J must change when the
in
its
ground
momenta
state
of the
various levels
and the other
two electrons
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
which the
states in
and
spins of the
parallel (to give
transitions can occur
S
=
1).
iparallel spins) constitute
from transitions
spins) constitute ortltohelium.
in
one
and
I
are, respectively,
lei (to
AS =
triplet states,
set or the other.
0,
give
pamhelium and those in
=
or gaseous helium
THE ATOM
is
in singlet
triplet states (parallel
lose excitation
The lowest
Energy-level diagram for helium showing the division Into singlet (paraheilum) and
triplet
(orthohelium) states.
There
is
no
1 -'S
state.
no allowed
and become one of orthohelium; ordinary
therefore a mixture of both.
FIGURE 7-14
0)
and the helium
Helium atoms
An orthohelium atom can
S
energy
a collision and become one of parallel ium, while a parallel ium atom can gain
excitation energy in a collision
232
singlet states
states
in
which
Becan.se of the selection rule
between
arises
il
triplet states,
two electrons arc an ti para
spectrum
.in
and
called
liecause, in the absence of collisions,
an atom
in
one of them
excitation energy for a relatively long time (a second or more)
before radiating.
its
The second obvious
Hie lowest triplet state
state
liquid
the
triplet states are
metastabk
can retain
is
peculiarity in Fig, 7-14
is 2-\S,
is
the absence of the
although the lowest singlet state
is
PS".
PS
state.
The PS
missing as a consequence of the exclusion principle, since in this state
two electrons would have
parallel spins
and therefore
identical sets of
MANY-ELECTRON ATOMS
233
quantum
numlicrs. Third, the energy difference between the ground state
and
_»
tj>
is
is
relatively large,
which
closed-shell electrons discussed earfier in this chapter.
of helium
atom
—
is
binding of
reflects the tight
The
ionization energy
— the work that must be done to remove an electron from a helium
24,6 eV, the highest of any element.
an example, and
is
(
the lowest excited state
To be
ultraviolet.
(Table
7.2},
We
is
that of mercury,
LS coupling
is
so heavy
of angular
which has
of 78 electrons in closed shells or suljshells
expect a division into singlet and triplet states as
but because the atom
in the
shall consider
W8 might
also expect signs of a
momenta. As
Fig. 7-15 reveals,
in
helium,
breakdown
both of these
expectations are realized, and several prominent lines in the mercury spectrum
arise
from transitions that violate the AS
FIGURE 7-15
ground
state,
EnorgyJevel diagram for mercury.
and the designation
In
of the levels in the
=
selection rule.
each excited
level
The
one outer electron
transition
Is In
responsible for the strong 2,537-A line in the
necessarily very high, since the three
mean
a
that the transition probability
P, states
is
are the lowest of the triplet set
and therefore tend to be highly populated in excited mercury vapor. The
is transitions, respectively, violate the rules that forbid
3p _> ic^ antj 3pa
u
_
()
The last energy-level diagram we
two electrons outside an inner core
is
sure, this does not
transitions
=
AS
from
J
=
=
to J
and that
and hence are considerably
0,
3
The P and P2
3
transition.
an atom can
of collisions,
LS coupling
is
AJ
or
to
±1,
as well as violating
3
occur than the
P,
states are therefore metastable and, in the
persist in either of
them
also respoasible for the
-» 'S
absence
for a relatively long time.
mercury that leads
spin-orbit interaction in
The strong
limit
less likely to
to the partial failure of
wide spacing of the elements of the
3P
triplets.
the
diagram corresponds to (he state of the other
elec-
X-RAY SPECTRA
7,12
tron.
Chap. 2 we learned
In
electrons exhibit
is
fast
to a minimum
The continuous X-ray
wavelengths down
wavelength inversely proportional to the electron energy.
spectrum
by
wavelengths characteristic of the target mate-
at
in addition to a continuous distribution of
ria]
•6/
that the X-ray spectra of targets l>onibarded
narrow spikes
the result of the inverse photoelectric effect, with electron kinetic
Hie
energy being transformed into photon energy hf.
other hand, has
its
discrete spectrum, on the
origin in electronic transitions within
atoms
have been
that
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
spectrum, as
The
is
lie in
or near the visible part of the electromagnetic
evident from the diagram in the back endpapers of this book.
inner electrons of heavier elements arc a quite different matter, because
these electrons experience
all
or
much
of the
complete shielding by intervening electron
tightly
full
nuclear charge without nearly
shells
and
bound. In sodium, for example, only 5.13 eV
in
is
consequence are very
needed
to
remove the
outermost 3s electron, while the corresponding figures for the inner ones are
3]
eV
for
each 2p electron, 63
Is electron.
eV
for
each 2a electron, and 1,041 eV for each
Transitions that involve the inner electrons in an
atom are what
give rise to discrete X-ray spectra because of the high photon energies involved.
Figure 7-16 shows the energy levels (not to scale) of a heavy atom classed
by
total
quantum number
states within a shell are
shells.
n;
energy differences between angular -momentum
minor compared with the energy differences between
Let us consider what happens when an energetic electron
strikes the
atom
MANY-ELECTRON ATOMS
235
arising in transitions
—
L
longer-wavelength
!
from the L,
,
11=4
Ma
My
S
U
n
=3
series
when an
M
spikes in the X-ray spectrum of
of
St
its
K
,
N,
,
.
.
electron
K
levels to the
when an L
electron
is
Similarly the
level.
knocked out of the
knocked out, and so on. The two
is
molybdenum
in Fig, 2-8 are the
Ka
and Kp
lines
series.
An atom with a missing inner electron can also lose excitation energy by the
Auger effect without emitting an X-ray photon. In the Auger effect an outer-shell
electron is ejected from the atom at the same time that another outer-shell
electron drops to the incomplete inner shell; the ejected electron carries off the
Ly
t«
M
atom, the
Afcty
N
M
series originates
h
atom's excitation energy instead of a photon doing
h
r
effect represents
'
n
=2
actually
comes
an internal photoelectric
into being within the atom.
effect,
In a sense the Auger
this.
although the photon never
The Auger
process
is
competitive
with X-ray emission in most atoms, but the resulting electrons are usually
absorbed
in the target material
while the
X
emerge
rays readily
to
be detected.
Problems
1.
Ka
Ky
K<
%
If
atoms could contain electrons with principal quantum numbers up to and
=
including n
Kj
2.
The
6,
how many
elements would there he?
ionization energies of the elements of atomic
are very nearly equal.
Why
should
this
numbers
2(1
through 29
be so when considerable variations
exist
in the ionization energies of other consecutive sequences of elements?
3.
on
The atomic radius of an element can be determined from measurements made
which it is a constituent. The results are shown in Fig. H)-'3,
crystals of
Account
4.
Many
are given
1
I
n
FIGURE 7-16
The
=
for the general trend of the variation of radius
years ago
ZTNe)
origin of X-ray spectra.
Z{Ar)
and knocks oul one of the K-shell electrons. (The K electron could also be
elevated to one of the unfilled upper quantum states of the atom, but ihe
Z(Xe)
difference between the energy needed to
Z(Rn)
electron completely
heavier atoms.)
is
the
236
K
shell
An atom
with a missing
in
K
in the
sodium and
electron gives
K
Z(Kr)
and that needed to remove the
the form of an X-ray photon
drops into the "hole"
series of lines in the
THE ATOM
this
insignificant, only 11.2 percent in
erable excitation energy
an outer
do
shell.
up most of
when an
As indicated
still
its
less in
was pointed out that the atomic numbers of the rare gases
by the following scheme:
Z(He)
l
it
with atomic number.
=
=
=
=
=
=
2(1*)
2(I 2
2(l a
2(1*
2(1*
2(1*
=2
+ 2*} = 10
+ 2 2 + 2 Z = 18
+ 2* + 2* + 3s = 36
+ 2* + 2* + 3* + 3 2 = 54
+ 2* + 2* + 3* + 3* + 4 2 =
Kxplain the origin of this scheme
)
)
)
)
in
86
terms of atomic theory.
consid-
electron from
in Fig, 7-36,
X-ray spectrum of an element consists of wavelengths
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.
MANY-ELECTRON ATOMS
237
Mow does the agreement
fi.
and the theory of
pendent
entities within
A sample
7.
number
Show
effect
I
atoms?
of a certain element
placed in a magnetic
is
if
CDS
—
the angle
Zeeman
effect
tl
occur only in atoms with an even
+1) -
i(j
17.
The
S ()
*
Pz 'D3ri
,
,
,
and
f'
5,
allowed
Its
state
corresponding to 3?3/2
18,
any?
state
Why
2s electrons and two 2p electrons outside a
Show
would you think the
—*
its
a
=
"
Si /2 state
is
where R
is
the
be regarded
charge
the ground state?
is
3s electrons outside
filled
inner shells. Find
state,
has
two 3s electrons and one 3p electron outside
filled
in the
Zct\{Z
in
sodium (which gives
two
lines,
corresponding to 3P1/2
Find the term symbol of
The magnetic moment
fij
of an
its
ground
sodium atom
atom
-
its
where nB
=
VJ(J
eft"/2m
is
^ =]
+
l)firfi fl
line of
an clement of atomic
that
each
is
—
I)
L
electron in an
the iMtule g factor,
2
K
electrons are present,
was used by Moseley
numbers of the elements from
J(J
+
1)
-
L(L
2J(J
(a)
Derive
parallel to J contribute to
that
is
many
tic t
in
a
+ 1) +
+ 1)
this result
S(S
their X-ray spectra.
This propor-
referred to as Moseley "s law.]
20,
Explain
why
has a
Ka
X-ray line of wavelength 1.785 A? Of wavelength
the X-ray spectra of elements of nearby atomic numbers are
may
differ
+
1)
with the help of the law of cosines
weak magnetic
/z,.
field
(b)
B
components of pL and
Consider an atom that obeys IS coupling
in
which the coupling
substates are there for a given value of J?
ween
[The
in 1913 to establish
the Bohr magneton and
starting from the fact that, averaged over time, only the
us
atom may
hydrogenic atom whose effective nuclear
considerably.
,
is
.
orbital motion.
qualitatively very similar, while the optical spectra of these elements
=
A
3S 1/2
if
as the single electron in a
which IS coupling holds has the
in
as a result of
K„ X-ray
Rydberg constant. Assume
What element
A?
state.
magnitude
ixj
5,890
—»
4
0.712
19.
*14.
A
reduced by the presence of whatever
the atomic
tionality
inner shells.
- 3S transition
given by
,
:i
ground
The aluminum atom
* 13.
+1)
1)
3S 1/Z and 5,896
proportionality between * and (Z
the term symbol of
S(S
filled
What are the tenn symbols of the other
P
would you think the P state is the ground state?
The magnesium atom has two
*12.
3P
that the frequency of the
is
3
is
.
Why
-
+ lWs +
spin-orbit effect splits the
number Z
The lithium atom has one 2s electron outside a filled inner shell. Its ground
2
is
S t/:! What are the term symbols of the other allowed states, if any?
*I1.
1)
is 8,
slates:
//a/2 .
ground
states, if
correspond to each of the following
J values that
The carbon atom has two
10.
inner shell.
+
and S in Fig. 7-8
the yellow light of sodium-vapor highway lamps) into
by the outer electron
S, L,
1(1
L
Use these wavelengths to calculate the effective magnetic induction experienced
of electrons?
Find the
*9.
between the directions of
2Vf((
rise to
does the normal
that,
field of flux density
A?
Why
8,
ma Zeeman
How far apart are the Zeeman components of a spectral line of wavelength
0.3 T.
4,500
lierween observations of the hid
tend to confirm the existence of electrons as inde-
(his effect
What
is
is
preserved.
How
the energy difference
different substates?
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
* 15.
.
field?
238
THE ATOM
MANY- ELECTRON ATOMS
239
THE PHYSICS
What
ihe nature of the forces that
is
OF MOLECULES
bond atoms together
This question, of fundamental importance to the chemist,
to the physicist,
whose theory of the atom cannot
a satisfactory answer.
classical
analog
is
to
form molecules?
less
correct unless
quantum theory
do so
to
hardly
of the
important
it
provides
atom not only
partly in tonus of an effect thai has
further testimony to the
power
of this approach.
MOLECULAR FORMATION
8.1
A molecule
is
a stable
thai a molecule
up
ability of the
bonding but
to explain chemical
no
The
lie
is
8
arrangement of two or more atoms. By "stable"
must he given energy from an outside source
into its constituent atoms,
energy of the joint system
atoms.
If
is
the interactions
less
in
is
meant
order to break
in other words, a molecule exists because the
than that of the system of separate noninteracting
among
energy, a molecule can be formed;
a certain group of atoms reduce their total
if
the interactions increase their total energy,
the atoms repel one another.
Let us coasider what happens
together.
when two atoms
are brought closer and closer
Three extreme situations may occur:
A covalent Ixmd is formed. One or more pairs of electrons are shared by
two atoms. As these electrons circulate lw;tween the atoms, they spend more
1.
the
time lietween the atoms than elsewhere, which produces an attractive force.
An example
is
two protons
2.
H2
,
the hydrogen molecule,
whose electrons belong jointly
An ionic Iwid is formed. One or more electrons from one atom may
to the other,
and the
resulting positive
and negative ions
attract
An example is NaCl, where the liond exists Iwtween Na* and CI
between Na and CI atoms (Fig. o-lfc).
3.
to the
(Fig. ti-lu).
No bond
is
formed.
When
the electron structures of
transfer
each other.
ions
and not
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
interacting electrons are thereby forced into higher
occupied
in the separate
as fleeing as far
To
away from one another
of the
visualize this effect,
as possible to
system, which leads to a repulsive force between the
exclusion principle can
some
may have much more energy than
we may regard the electrons
atoms, the system
before and be unstable.
If
state.
energy states than they
lie
avoid forming a single
nuclei,
(Even when the
FIGURE 8
2
model
Scale
of
NaCI
crystal.
olieyed with no increase in energy, there will be an
electrostatic repulsive force
between the various electrons;
this
significant factor than the exclusion principle in influencing
a
is
much
less
bond formation,
however.)
A molecule
Ionic tends usually do not result in the formation of molecules.
is
to
an electrically neutral aggregate of atoms that
be experimentally observable
as a particle.
constitute gaseous hydrogen each coasist of
entitled to regard
salt (N'aCl)
arranged
them as molecules.
Thus the individual units
that
two hydrogen atoms, and we are
the other hand, the crystals of rock
in a certain definite structure (Fig. 8-2),
be of almost any
size.
Na*
ions
off into discrete
may
ion and one CI
There are always equal numbers of Na + and CI
ions in rock salt, so that the formula
However, these
do not pair
ion; rock salt crystals
NaCI
correctly represents
form molecules rather than crystals only
its
in the
Hj
In
held together strongly enough
are aggregates of sodium and chlorine ions which, although invariably
molecules consisting of one
Fact
On
is
8-1
(a)
(b) Ionic bonding.
is
purely covalent and
in
NaCI
purely ionic, hut in
is
it
atom
An example
attracts the shared electrons
argument can be made
more
the
is
MCI molecule, where
H atom. A
strongly than the
for thinking of the ionic
Iwnd
as
gaseous
state.
Sodium and chlorine combine
electrostatically.
case of the covalent bond.
8.2
The
ELECTRON SHARING
simplest possible molecular system
single electron
a general
share an electron and
to
is l\
2
the hydrogen molecular ion, in
'
,
bonds two protons. Before
in detail, let us look in
why
way
into
how
we
it is
consider the bond
possible for
we
Chap. 5
enough energy
A\
OH
+ 17
cr...
extends l>eyond
such sharing should lead to a lower total energy
phenomenon
discussed the
of quantum-mechanical barrier
(«)
does
exist.
the
its
if
the wall
is
infinitely strong
held around a proton
is
same probability
lie
how
is
is
the
wave function wholly
in effect
to a
neighboring proton more
a certain proI>ability that an electron trapped in one box
through the wall and get into the other box, and once there
that the electron
I'M
Only
parent proton. In quantum physics, however, such a mechanism
There
is
for tunneling back. This situation
it
hie*
can be described bv saving
shared by the protons.
sure, the likelihood that
high potential
PROPERTIES OF MATTER
it.
hydrogen atom can transfer spontaneously
will tunnel
To
244
does not have
wave function
a Ijox for an electron,
and two nearby protons correspond to a pair of boxes with a wall between them
(Fig. 8-3). There is no mechanism in classical physics by which the electron
distant than
m\
it
to break through the wall because the particle's
The electric
inside the box.
in a
o
in H.,~
two protons
and hence to a stable system.
In
Na"
strong
no more than an extreme
penetration: a particle can "leak" out of a box even though
o
the CI
composition.
chemically by the transfer of electrons from sodium atoms to chlorine atoms; the resulting Ions attract
.
many
in
Covalent bonding. The shared electrons spend more time on the average between their
parent nuclei and therefore lead to an attractive force,
bond
electrons to an unequal extent.
which a
FIGURE
the
other molecules an intermediate type of bond occurs in which the atoms share
an electron will pass through the region of
energy— the "wall"— between two protons depends strongly upon
far apart the
protons are.
If
the pro ton- proton distance
is 1
A, the electron
THE PHYSICS OF MOLECULES
245
An
and hence kinetic energy.
electron shared by two protons
than one belonging to a single proton, which means that
The
total energy' of the electron in 11./
+
in 11 4- II
is
is
it
has
H2
ought to
*
The preceding arguments
There
trostatic forces.
is
that both types of
lh'
in
l)e stable.
are quantum-mechanical ones, while
in
we normally
terms of elec-
a very important theorem, independently proved
by Feynman and by Hellmann and named
after them,
approach always yield identical
Feynman-IIelhnann theorem,
which
results.
states in essence
According to the
the electron probability distribution
if
in
a mole-
known, the calculation of the system energy can proceed classically and
lead to the same conclusions as a purely quantum-mechanical calculation.
is
The Fcynmati-Helhnann theorem
molecule
not an obvious one, because treating a
is
terms of electrostatic forces does not explicitly take into account
in
quantum treatment involves
wave function i£ has
electron kinetic energy, while a
energy; nevertheless, once die electron
way
either
of proceeding
may be
the total electron
lieen determined,
used.
THE IV MOLECULAR ION
8.3
What we would
from
since
separation
can
energy.
therefore less than that of the electron
tend to consider the interactions between charged particles
will
confined
and provided the magnitude of the proton-proton repulsion
,
not too great,
cule
is less
less kinetic
exist,
i£
know
like to
we
wave
the
function
$ of the electron
in
H2+
,
can calculate the energy of the system as a function of the
of the protons.
ft
and
is
we can
also
If
we
E(R) has a minimum,
will
know
thai a
bond
determine the bond energy and the equilibrium spacing
of the protons.
FIGURE 8-3
protoni.
The
Instead of solving Schrodinger's equation fort£, which
(i) Potential energy of an electron in the electric field o! two nearby
lotal enafgy of a ground-slate electron In the hydrogen atom Is indi-
(o) Two nearby protoni correspond quantum-methanfeally to
a pair of
boxes separated by a wall.
cated,
cated procedure,
we
predict what
when
with «„, the
may
lie
regarded as going from one proton to the other about everv 10- !3 s,
that we can legitimately consider the electron as being shared by
Ls
both. If the proton-proton 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
called
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
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
that
the region to which
we
restrict a particle, die greater
PROPERTIES OF MATTER
must be
its
momentum
R, the distance between the protons,
Bohr
is
large
hydrogen atom.
orbit in the
compared
In this event
l.v wave function of the hydrogen
wave function around proton a is
near each proton must closely rasemble the
atom, as pictured
0.53 A,
a lengthy and compli-
is
an intuitive approach. Let as begin bv trying to
radiiLs of the smallest
which means
is
246
4>
i£
shall use
We
know what ^
l>e
the electron
Ls
U
where the
and that around proton h
4>„
also
imagined to
The
in Fig. 8-4
fused together.
now
in the
wave function
amplitude
looks like
is
ls
called
when
Here the
li
tf-
6
.
is 0,
that
situation
is
is,
when
the protons are
that of the lie
1
ion, since
presence of a single nucleus whose charge
of He* has ihe
same form
at the origin, as in Fig. 8-4e.
as that of
II
is
+ 2e
but with a greater
^ is going to be something
when R is comparable with a ir
Evidently
like the
wave function sketched
There
an enhanced likelihood of finding the electron in the region between
is
the protons,
in Fig. 8-4rf
which we have spoken of
in terms of sharing of the electron
by
THE PHYSICS OF MOLECULES
247
n
contours of
electron probability
"0
Thus there
the protons.
A*.
on the average an excess of negative charge between
is
the protons, and this attracts
them
the magnitude of this attraction
+'
We
together.
enough
is
have still to establish whether
overcome the mutual repulsion
to
of the protons.
(«)
The combination of ^a and ^ 6
and
does not affect
/;
^
in Fig. 8-4
(sec Sec. 7.3).
could have an antisymmetric combination of
\J/u
there
a node
is
between a and
where
/;
symmetric, since exchanging a
is
However,
t^
also conceivable that
it is
=
,
Now
between the protons.
likelihood of finding the electron
we
and ^6 as in Fig. 8-5, Here
n
0, which implies a diminished
t^
there
is
on the
average a deficiency of negative charge between the protons, and in consequence
/&
With only repulsive
a repulsive force.
An
function
t^
as
H
R —
He + when R =
(6)
(Fig. 8-5e),
*+
However,
0.
tyA does not become the
Obviously
0.
which has a node
state while the Is state
is
ij>
A
dm® approach
the ground state,
it
is
A
U—
is
Ha+
in the
wave
our inference from the shapes of the
case there
in the
state of
ll
wave
*
2
function of
function of
He'
is
I
ie^
an excited
antisymmetric state ought
state,
which agrees with
$A and C K that in the former
a repulsive force and in the latter an attractive one.
line of reasoning similar to the
rieal state.
wave
Is
wave
2;j
2p
symmetric
functions
preceding one enables us to estimate
the total energy of the H;,* system varies with R.
«.
the
at the origin. Since the
have more energy than when
to
bonding cannot occur.
forces acting,
interesting question concerns the behavior of the antisymmetric
When R
is
large, the electron
energy
We
how
consider the syitnuct
first
Es mast lie
— 13.6-eV energy
the
V
of the hydrogen atom, while the electrostatic potential energy
of the protons.
(c)
1^-1
V„
8.1
falls to
force.)
*s
is
+•
•
r-*-j
+
Za
=
4™„R
R — oo. (Vp is a positive quantity, corresponding to a repulsive
When R = 0, the electron energy must equal that of the He + ion, which
(J
as
or 4 limes that of the
result
is
Es = -54,4 eV when R =
and
<rf)
^,i
Vj,
Es can
its
II
atom.
(See Prob. 25 of Chap. 4; the
same
Hence
obtained from the quantum theory of one-electron atoms.)
when R -*
0,
V,,
are sketched in Fig. 8-6 as functions of
/{;
the shape of ihc curve for
Also,
0,
-» oo
as l/R.
only be approximated without a detailed calculation, but
value for both
R =
t)
R
and
=
oo and, of course,
V
Both
Es
we do have
<»l«ys Eq. 8.1.
fi
The
A
total
energy
and the potential energy
which corresponds
R=0
•
2+
needed
to break 11./
of the hydrogen
FIGURE 8-4 The combination
wave (unction
^
of two hydrogen-atom 1,
H2 +
wave (unctions
to
form the symmetric H
2
is
the
of the protons.
molecular
Evidently
state.
-which indicate a
R
sum of
This
the electron energy
Esiotai
result
H+H
— 2.65-eV
atom plus the
;
is
E8
has a minimum,
confirmed by the
bond energy of
of 1,06 A. By "bond energy"
+
into
the total energy of
equilibrium separation
(»)
Vp
to a stable
experimental data on
I
of the system
/-.j,""" 1
2.(55
eV and an
meant the energy
H 2 + is the -13.6 eV
is
bond energy, or
— 16.3 eV
in
all.
+
THE PHYSICS OF MOLECULES
249
iffn
contours of
electron probability
_^
()
4><\
— b—
*
(6)
FIGURE 8-6
for the
Electronic, proton repulsion,
and
symmetric and antisymmetric states
,
total
energy
In
H,' as a function of nuclear separation N
The antisymmetric
state has no
minimum
in its total en-
ergy.
1
11
This energy
is
ft
is
enough
—*
oo,
.
at
two hydrogen-atom
1.
wave functions
to
form tha antisymmetric
2
;
hence with Z
=
2 and n
think that the electron energy
a small dip at intermediate distances.
to yield a
that of the
is
of the ground-state hydrogen atom.
we might
minimum
as indicated in Fig. 8-6,
FIGURE 9-5 Tha combination
wave function * A
EA when R —
proportional lo Z-/n
— 13.6 eV
to the
also as
there
same way
the case of the antisymmetric state, the analysis proceeds in the
except that the electron energy
in the total
and so
is
=
2
for the
no bond
is
state of
it is
just
EA —* —
He +
.
equal
13.6
eV
constant, hut actually
However, the dip
energy curve
in tins state
Since
2p
is
not nearly
antisymmetric
state,
formed.
H* +
THE PHYSICS OF MOLECULES
251
8,4
THE H, MOLECULE
arc acceptable.
The H 2 molecule
According
{that
is,
lie
contains
two electrons instead of the
to the exclusion principle,
described by the same
single electron of
is
)
their spins are
With two electrons to contribute to the bond, H ought
to be more
a
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
The
latter repulsion Wakens
2
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
Hence
must
the coordinate
'
.
,
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
H2
£
was formulated
in
terms of the symmetry
and antisymmetry of wave functions, and it was
concluded
electrons arc always described by antisymmetric
wave functions
that systems of
(that
is,
by wave
I
lowever,
functions that reverse sign
we have just
On
=
is
said that
upon the exchange of any pair of electrons).
the bound state in H, corresponds to Ixjth
by a symmetrical wave function
above conclusion.
^ which seems
electrons being
function
we may
it
H./
is
shown
is
due
when
for
we may
when
parallel
the electrons are exchanged.
two electrons whose
spins are anti-
express this by writing
H z molecule has no exact solution.
The
when
In fact, only
other molecular systems must be
molecule are
the electrons have their spins parallel
The difference between
their spins arc antiparallel.
state in a system
all
results of a detailed analysis of the II
Z
to the exclusion principle,
quantum
4-
an exact solution possible, and
in Fig. 8-7 for the case
the case
two electrons whose spins are
<h
Schrodinger's equation for the
for
for
reverses sign
wave function
must be symmetric, and
m=
^
express this by writing
the spins of the two electrons are antiparallel, their spin
if
antisymmetric since
the coordinate
parallel
parallel, their spin function
f.
the other hand,
function
Hence
wave
treated approximately.
,
In See. 7.3 the exclusion principle
descriliod
*TT
two electrons are
does not change sign when the electrons are exchanged.
antisymmetric:
lie
antiparallel.
valid for
it
11.,".
both electrons can share the same orbital
wave function fntmi provided
the spins of the
If
symmetric since
the
which prevents two electrons
and
two on v es
in the
same
from having the same spin and therefore leads to a
dominating repulsion when the spins are
parallel.
to contradict the
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
FIGURE
8-7
The
variation of the eneigy of the
system H
•+
H
with their distances apart
when
the electron spins are parallel and antiparallel.
the electrons and a spin
describes the orientations of their spins.
The exclusion
principle requires diat the complete wave
function
I
unction
412} which
+(1,2)
= iHUHU)
be antisymmetric to an exchange of Iwth
coordinates and
itself, and what we have teen
calling a molecular
orbital
is
spins, not tijl 2)
the
same
as
#|
by
2)
An antisymmetric complete wave function
+„ can result from the combination
wave function + and an antisymmetric spin function
sA or from the combination of
an antisymmetric coordinate wave function
*
of a symmetric coordinate
fl
and a symmetric spin function
fc That
and
+ =
is,
only
H + II,
spins antiparallel
-6'
****
B
1
2
NUCLEAR SEPARATION
252
3
H,
A
PROPERTIES OF MATTER
THE PHYSICS OF MOLECULES
253
n
orbital
Covalcnt bonding in molecules other than
H.,,
in I
diatomic as well as polyatomic,
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
is
I
MOLECULAR ORBITALS
8.5
usually a
*
1,2,3
Ps
2,3.4,.,.
I
±1
Py
2,3,4,...
1
±1
P«
2,3,4..,,
1
o
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
are
become predominant while
the inner shells of
atoms
its
relatively far apart.
Direct evidence in support of the idea that only
the valence electrons are involved in chemical bonding is available
from the
still
X-ray spectra that arise from transitions
to inner-shell electron states;
that these spectra are virtually independent of
molecules or
how
the atoms are
it is
found
combined
in
solids.
In discussing
chemical bonding
il
is
helpful to be able to visualize the dig
tributions in space of the various atomic orbitals.
The
those of hydrogen.
which qualitatively resemble
pictures in Fig. 6-11 are limited to
and hence are not suitable
for this purpose.
It is
two dimensioas
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
each lobe of such a drawing, even though what is being represented is
W*.
Figure 8-8 contains boundary-surface diagrams for s,
p, and d orbitals. These
diagrams show |0«j 2 in each case; for the corresponding radial probability
in
densities )R\1
,
Fig. 6-10
can
lie
The
consulted.
of coarse, equal to the product of
|0«l>|
a
and
total probability density
2
|fi|
tyf
is,
.
In a munber of cases the orbitals shown in Fig. 8-8 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
for iw,
pt
= +1
(The factor 1/
orbital
and m,
V§
is
is
the result of adding together the
I
=
1
wave function
= — 1:
required to normalize
ty p,-)
Similarly the p„ orbital
is
given
by
254
PROPERTIES OF MATTER
FIGURE 8 8 Boundary surface diagrams
wave function In each region.
sl(n of the
for ( p.
and d atomic
orbitals.
The + and - signs
refer to the
orbital
n
„i,
I
The p. orbital, however, is identical with the I = I,m, = wave function. The
wave functions which are combined to form the (/„, d vt dlv and d,t_*« orbitais
are indicated in Fig. 8-8. The interactions between two atoms that yield a
,
dty
.1,4,5,
...
i2
2
,
covalent bond between them have, as a consequence, different probability-density
distributions for the electrons that participate in the
characteristic of the
atoms when alone
understand
easiest to
bond than those which are
and these new
in space,
terms of the orbitais shown
in
When two atoms come
and the
together, their orbitais overlap
between them
either an increased electron probability density
Inmding molecular orbital or a decreased concentration that
<*=
3.4A...
±1
2
Ixmding molecular
of
In the previous section
orbital.
two hydrogen atoms could combine
or the antibonding orbital
$A
he
result will
that signifies a
an tmtt-
signifies
we saw how
the Is orbitais
form either the bonding orbital
to
In the terminology of molecular physics,
.
and $A as a lstr°
are imagined to combine
referred to as a Is o orbital
atomic orbitais that
distributions are
in Fig. 8-8.
The "Is"
orbital.
to
$s
is
the
identifies
form the molecular
i£ s
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
<k
2
3,4,5,...
±1
\ti
momentum
=
where X
of a molecule
1,2
0,
m
those for which A
1
is
quantized, and
is
restricted to the values
Molecular states for which A
by
w, those for
=
which A
order. Finally, an antibonding orbital
is
2
by
6,
=
and
are denoted by u,
on
so
in alphabetical
labeled with an asterisk, as
in
lso° for
the antibonding II, orbital y_ r
Figure 8-9 contains boundary-surface diagrams that show the formation of o
and
77
molecular orbitais from
molecules.
s
and p atomic
orbitais in
homonuclear diatomic
Evidently a orbitais show rotational symmetry about the bond
while n orbitais change sign upon a 180° rotation aliout the bond
the lobes of
d-,i
3,4,5.
molecular
A
ps
orbitais arc
same in such molecules, so
them are not equally shared by both atoms. 1J1I
that the
trons in
is
II
atom
that there
of
l.i
is
Is
is
:£S
is
a
good example
a single valence electron.
form a a
1
valence electron
total nuclear
of this effect.
and that of the Li atom
is
The
is
l.v
J
2\.
is
The normal
II
88
it
is
5,4
-t-e (in
charge of
eV
configuration
in
each case
and the 2s orbital
occupied by the two valence
Li the core of
+ 3e),
two
Is electrons shields
but in Li a valence electron
in Li.)
Hence
is
+ 2p
on a
of the
on the average
H. (The respective ionization
energies reflect this difference, with the ionization energy of
while
bonding elec-
In both atoms the effective nuclear charge acting
several times farther from the nucleus than in
FIGURE
orbitais.
the simplest hetero-
which means
Is orbital of
wilding orbital in Lill that
electrons (Fig. 8-10).
2
form v molecular
the atomic orbitais are not the
of the
3,4,5,...
these atomic orbitais form o
axis,
orbitais both
heteronuclear diatomic molecule consists of two unlike atoms. In general,
nuclear molecule and
ds %_^
on the Iwnd
The px and p v
orbitais.
axis,
Since
axis.
H
being 13.6
the electrons in the a bonding orbital of
eV
LiH
(Continued.)
THE PHYSICS OF MOLECULES
257
o
Li
electrons
from (he
In Lift
occupy
U
of the LI
jO
;iiilil)ini(liim
H
favor the
P:
nucleus,
and there
is
a partial separation of charge
If
there were a complete separation of charge in LiH, as there
would
ionic.
bond mil;
when
it
is
In a hctcroiuiclear
form
a
example
F,
bonding
consist of
atom is called its electronegativity. In the LiH
more electronegative than Li.
molecule the atomic orbitals that are imagined to combine
part of an
H
is
molecular orbital
IIF,
is
where the
There are two
of
F each
by two
b
may be
of different character in
atomic orbital of
possibilities, as in Fig. h- II
contain one electron (Table
8.1),
and we may
HF
electrons,
regard
covalent bond. 'Hie electron structure of the
p.ir
FIGURE 8-11
F
.in! iliunit iiij!
2P,
Bonding and antibonding molecular orbitals
FIGURE 8-9 Boundary surface diagrams showing the formation of molecular orbitals
from
atomic orbitals in homonuclear diatomic molecules. The
aiis is along the internuelear
.-
258
p,!:
and p,v*
orbital* eicept that they are rotated
PROPERTIES OF MATTER
»
and
,,
axis of the
through 90'.
,i,r
orhHals are the
i
same as
,
II joins
a
bonding spo molecular
the spa orbital in IIF
HF molecule
In
An
is
orbital
and the 2p. orbital
as t>eing held together
shown
is
occupied
by a single
in Fig. 8-12,
HF.
HF
U
*fw
Q
each atom.
with the 2p. orbital of
Is orbital of 11
H
OO Q - O-O
II
p.*
The ;V r and
NaCl,
and the hond would
and an antibonding spn° orbital. Since the
the nj plane.
in
Instead the
molecule, for instance,
to
Is
is
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
px
the
an Li* ion and a H~ ion,
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 Oa spend .50
percent of the time near each nucleus. Molecules whose bonds are neither purely
the molecule
r>-a
".
in
molecule.
:iiiiiliiin<liiib:
the
SJK7
orbital
atom.
be purely
cute in each case, and the plane of the paper
La
orbital of the
H atom and the 2t
*tr
''x
LiH
a •
molecular orbital formed
GO
P,
H
The bonding
FIGURE 8 10
bonding
2P,
U
HF
spa
THE PHYSICS OF MOLECULES
259
Table
8.
1
ATOMIC STRUCTURES OF
tion, ol electrons.
The
AND SECOND- PED(00 ELEMENTS
Hund's rule (See 76 .n.«.
1
FIRST-
According to
*„«.
"""el" B ipln
»
*«
fine!
l»ent
shape of the water molecule
that the
U orbital
can join the
Atomic
Element
number
Hydrogen
11
Helium
He
S
Lithium
l.i
3
Iti
4
Beryllium
Boron
The y and z
Occupancy
T
w
C
e
Nitrogen
\
7
Oxygen
O
8
Fluorine
F
9
Neon
\,
to
:
1
plausibly
-'2,
2»;
Ij-2*-'2rj'
!
1
jom with .
orlntals to
n
r
n
Ti
Dumber Z =
A
sma
y
'
positive charge
-
;
«
2TS
11
P^
and 90°, respectively, which we may ascribe
atoms around the larger atoms S (atomic
16)
=
and Se (Z
34).
n
t
T
n
u
r
T
T
ti
t
I
orbital with the Is orbital of
u
u
U
n
ti
H
atom
«*»**»
—'>'
,
?
Ti
it
poarfHe
h„
O-H
.
-O-H
in
I
to
H
atom. The bonding molecular orbitals
should therefore be centered along the
bonds 90° apart. As
in
I
2 0. the actual
case 107.5°,
owing
hydrides of the larger atoms
P (Z =
this
x,
(/,
and
bond angles
to repulsions
15)
in
=
NIL,
N— II
Nil, are somewhat greater
among the H
and As (Z
in
t axes (Fig. 8-14) with
atoms.
The similar
33) exhibit the smaller bond
angles of 94° and 90°, respectively, again in consequence of the reduced mutual
among
the
more
distant
H
atoms.
s
be^LTH
"PP-te
an
,
accord!
*E ckS
«"
I
the two
atoms. In support
u
u
u
n
u
and each
-
II
the otherwise similar
i
linear molecule, II
1
in
(Fig. 8-13),
actually found
U
"« "»***
"
are 92°
bond angles
is
u
'"" US far Rpart
"6
t° In reality, however, the
the O atom,
water molecule
O -II will, an angle of 104.5° between
P
form a spa landing orbital
similar
repulsions
2 t
to
the fact that the
H a Se
we
atom
of an II
attributed to the mutual repulsion of the
is
8.1
are only singly occupied, so that each
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
form lading molecular orbitals
make
H*Ofe an example. Offhand we might
expect a
oxygen is more electronegative than
hydrogen
From Table
easy to explain.
O
to the greater separation of the II
than 90°, in
to
lit'
molecules II,S and
Ti
S
Cordon
may
is
orbitals in
axes are 90° apart, and the larger 104.5° angle that
of the latter idea
u
B
of orbital;
structure
2p v and 2n.
8.6
HYBRID ORBITALS
sides rf
to
bonds.
The
straightforward
are explained
is
way
atom has two electrons
II
FIGURE 8-12
FIGURE 8-13
HF.
H 0. The bond angle
Valence atomic orbital* In
the atomic orbitals shown as overlapping form a a bonding
molecular
in
which the stapes
of the IL.0
and NTL, molecules
a conspicuous failure in the case of methane, CH,. A carbon
in its 2a- orbital
Valence atomic orbitals
is
and one each
in
its
2p t and 2p u
orbitals.
in
actually 104.5'.
orbrtal.
260
PROPERTIES OF MATTER
THE PHYSICS OF MOLECULES
261
The correct explanation for CH 4
which can occur when the iv and
together
of Imth
in
its
energy.
and
2s
2-l>
more
an atom
in
a molecule are close
atom can contribute a
atomic orbitals
orbitals can occur follows
In
2j> states of
In this case the
the resulting bonds are
FIGURE 8-14 Valence atom!; orbilals
NH The bond angle* are actually
based on a phenomenon called hybridization
is
linear
each molecular orbital
to
combination
if
in this
way
stable than otherwise. Thai such composite atomic
from the nature of Schrodiiiger's equation, which
a partial differential equation.
The
2s
i;>
and 2p wave functions of an atom are
,.
Iwth solutions of the same equation
107.5*,
if
the corresponding energies are the same,
and a linear combination of solutions of a
itself
(is
a solution.
more
In
bound) than an electron in a 2p
lightly
no tendency for hybrid atomic
atom
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
bonds are exactly equivalent
whose
C—
We cannot consider carbon as an isolated exception, a
which a fortuitous combination of circumstances leads to
BFand
BC1.
Clearly,
what
is
happening
in
carbon and toron
sort of freak
CH,
atom
in
that the 2s orbitals, despite
—
less energy— more stability
than
enter into the formation of molecular orbitals and
thereby permit the 2s electrons to contribute to the bonds formed by C and
2p
A C atom
B with other atoms.
while a
B atom
has three n
=
has four n
=
2 electrons
2 electrons in
all
is
accordingly
and p
when an
orbitals to the molec-
stronger than the bonds that the s and
lo,
rise to is greater
when
than that which pure orbilals would
the s
and p
levels of an
atom are close
bonding. These four orbitals are hybrids of one 2s and three 2p orbitals, and
we may
consider each one as a combination of
combination
therefore called a
is
:l
.v/)
hybrid.
Its
is
in all
and forms CH,,,
and forms BF3 respectively
,
its
ability to
for the
FIGURE 815
tour
jjj
3
In
promote a 2* electron
to
tji
1
and %p. This particular
in Fig. 8-15,
Evidently a sp 3
strongly concentrated in a single direction, which accounts for
produce an exceptionally strong bond
need
'/,s
configuration can be visualized
terms of boundary-surface diagrams as shown
hybrid orbital
-
somehow
orbilals,
in
s
and there
the other hand,
In CH.,, then, carbon has four equivalent hybrid orbitals which participate
in
in
is
On
even though the p parts of the hybrids
the isolated atom. Hybrid orbitals thus occur when the
bonding energy they give
CH,,
instead of
toing occupied by electron pairs and having
the
had higher energies
always
energy
together.
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
may be
by themselves would lead
produce, which happens in practice
one another.
to
orbitals
a 2s orbital has
orbital,
orbitals to occur.
ular orbitals, the resulting bonds
p
in
a certain molecule contributes superposed
in
is
less
partial differential equation
an isolated atom, an electron
hybridization,
an
<
orbital
lo a
and three
2p
,.
—strong enough to compensate
slate.
orbitals In (he
same atom combine
to form
hybrid orbHais.
sharing four and three electrons with their partners.
The
easiest
way
two 2* electrons
now one
in
to explain the existence of CH,,
C
electron each in the 2s,
C
(to yield
atom, but
CH 4
resulting molecule
to the
is
to
assume
that
one of the
vacant 2p. orbital, so that there
is
2pr %p v and 2u r orbitals and four bonds can
an electron from a 2s to a 2j- state means increasing the
be formed. To raise
energy of the
bonds
"promoted"
is
)
in
it is
,
,
reasonable to suppose that the formation of four
place of two (to yield
more than enough
to
CH 2
)
lowers the energy of
compensate
for this.
I
he
The foregoing
Q
«p-
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,
262
all
four bonds are found to be identical.
PROPERTIES OF MATTER
THE PHYSICS OF MOLECULES
263
It
inusl
mind
kept in
l>e
even when
it is
in
that hybrid orbitals
an excited
do not
an isolated atom,
exist in
but arise while that atom
state,
is
interacting with
others to form a molecule.
Figure
a representation of the
fi-16 is
C
of this molecule that consists of a
has
B
atoms
at alternate
comers.
A
CB4
atom
molecule. Also
in the
shown
is
a model
center of a unit cube which
.)
triangle with the
C atom
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
H bonds is 0, from the law of cosines {tr =
+
er — 2hccas0) we have
at
C—
= -
COS0
FIGURE 8-17 Valence atomic orbitals in th* ammonia (NH molecule based on the assumption of
-v/j
hybridization in the N orbitals. One of the sp J
orbitals is occupied by two N electrons and does
&
1
not contribute to bonding.
-iV- -c 2
a*
2bc
2- -%-
2X%
=
%.
of
which
is
what
The bond
determined experimentally.
is
angles of 104.5° in
to lire tetrahedral
90° expected
provides a
if
way
HaO
p
orbitals in the
to explain
how
O
and
orbitals, leaving
FIGURE 8-16
atom and
N
the repulsions
molecules that were spoken of earlier can
orbital description of bonding. In
NII 3
p
orbitals, the
system.
and 107.5°
angle of 109.5° that occurs in
only
in
1
sp-
NH3
are evidently closer
hybrid landing than to the
atoms are involved. This
among
the
H
atoms
in
fact
these
incorporated into the molecularthere arc three doubly occupied bonding
l>e
one unshared pair of electrons
that
we
earlier
supposed to be
Opposing
The tetrahedraf methane (CHJ molecuto. The overlapping ,,-'
hybrid
H atoms form bonding molecular orbital*.
orbitals of the
this gain in
molecular
stability
is
the need to
instead
for the
promote the
pair of nonbonding 2$ electrons to the higher-energy sp 3 hybrid state without
any contribution by them
NH 3
angle in
.vp
to the
bonding process (unlike the case of Cll, where
four sp 3 orbitals participate in bonds).
all
3
as (he result of a
hybrid orbitals
orbitals
tn
N
Hence we ma)' regard the 107.5° bond
compromise lietween the two extremes of four
with one of them nonbondin» and three 2p bonding
and one 2s nonbonding (but low-energy)
orbital.
Figure 8-17
is
a repre-
sentation of die NIL, molecule on the basis of up 3 hybridization, which
compared with
1* orbitals of the lout
N. If there were spP hybrid orbitals furnished by H
more separated bonds would mean a lower energy
in the 2s orbital of
109.5°
C
Fig.
In H./), since there are
in
may
lie
8-14 which was drawn on the basis of p orbitals.
two Ixmding
orbitals
O, the tendency to form hybrid sp 3 orbitals
and two nonbonding orbitals
is less
than in
NH a
where there
and only one nonbonding orbital. The smaller bond
agreement with this conclusion. The structure of the H a O
are three bonding orbitals
angle in
molecule
8.7
H2
is
is
in
further discussed in Sec. 10.4.
CARBON-CARBON BONDS
Two
other types of hybrid orbital in addition to
.sp
In sp
2
a pure p orbital and the other
hybridization, one valence electron
three are in hybrid orbitals that are
two valence electrons are
orbitals that are
l
/2
Fthyleue, CgH*,
.v
and
in
%p
is
%v and %p
in
3
can occur
in carl>on
atoms.
in character. In .vp hybridization,
pure p orbitals and the other two are
in
hybrid
in character.
an example of sp 2 hybridization
in which the two carbon
atoms are joined by two bonds. Figure 8-18 contains a boundary -surface diagram
264
PROPERTIES OF MATTER
is
THE PHYSICS OF MOLECULES
265
>^
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
each
C
atoms
is
2j)„ orbitals in
Itonds i>etween the carbon
The conventional
{Fig. 8- 19).
atom form v bonds, so
a o.
one of the three
that
bond and the others are
struetnral formula of acetylene
tt,
and x^ bonds
is
=C—
In both ethylene
Acetylene
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
m
as ethane,
II
II
H—C—C—
Ethane
[
H
in
which
all
II
the l»onds are formed from ip8 hybrid orbitals in
Carbon compounds with double and
because they can add other atoms to
>°\
FIGURE 8-19
FIGURE 8-18
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 between the C atoms and between each C atom and two H atoms, (c) Side view, showing th*
(a)
between
ijj
The acetylene
+ HCI
H
th.-
carbon atoms.
be unsaturated
to
11
H— C-C-ll
H
(C.H..) molecule.
hybrid orbitals and two
bonds are said
their molecules in such reactions as
II
H
(O
triple
CI
There are three bonds between the
bonds between pure p, and p,
C
one i bond
orbitals.
;
>
pure
i>,
orbitals that
form a - bond between the C atoms.
showing ihe three
2
sp'
hybrid orbitals, which are 120° apart in the plane of the
paper, and the pure pT 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
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
the other
other
266
is
a
it
bond. The conventional structural formula of ethylene
PROPERTIES OF MATTER
is
accordingly
THE PHYSICS OF MOLECULES
267
ROTATIONAL ENERGY LEVELS
8.8
Molecular energy stales arise from the rotation of a molecule as a whole and
from the vibrations of
from changes
0>
constituent atoms relative to one another as well as
its
electronic configuration. Rotational states are separated
in its
quite small energy intervals
<©
transitions
of 0.1
mm
between these
to
3
eV
is
typical),
states are in the
and the spectra
eV
typical),
is
with wavelengths of 10,000
to 0.1
between the energy
treatment here will
The
its
x
in,
the
and
in.,
/
where
The overlaps between the .,,,• hybrid orbitals in .he
C
H atoms lead to „ bonds, <b) Each C atom
electron. (e> The bonding - molecular
orbitals formed by
r,
=
and
n^r, 2
t2
(a)
l
"'" / "'= <
compound such as methane or ethane only single bonds
--
(
,;Hv
ii"'
sw
atoms
I
an
arranged
In
f| ;1
i
m
8.3
=
r
x
orbitals.
Of
the-
one 2p,
plane of the ring.
orbital per
The total
of six
atoms of
The moment
of inertia
center of mass and perpen-
is
a
"*j*j
1
and 2 respectively from the center
in.,r.,
moment
of inertia
may be
written
m »u \
+ m.,1
v
=
8.4
llt'tt
5
ei bonding by v/r' hybrid
one forms a a bonding orbital
atom and the other two form a bondim- orbilals
three sp» orhitab per
C atom,
with the I.v orbital oj aa H
with the corresponding ap* orbitals of the
Hits leaves
+
its
as consisting of
.
\«ii
ring.
we conclude
main ideas apply
{
(
are present,
hexagonal
Becau.se the carbon-carbon bonds in the
benzene ring are I20« apart,
that die basic structure of the moleculeis ihe result
may picture such a molecule
H apart, as in Fig. S 21
are the distances of atoms
by definition, the
1,1
For simplicity the
bill tin:
Since
of mass.
sl« ,,, atomic orbitals constitute a
continuous electron probability distribution around
the
molecule that contains si* detocalized electrons.
In a saturated
We
a distance
dicular lo a line joining the atoms
orbitals of the
A
spectra,
as well.
of this molecule alxiut an axis passing through
8.2
The benzene molecule.
restricted to diatomic molecules,
its
lowest energy levels of a diatomic molecule arise from rotation about
center of mass.
masses
w
B-20
l>e
more complicated ones
(u
£-
HGURE
ultraviolet regions.
including bond lengths, force constants, and Iwind angles.
^—
Oh
atoms with each other and with the .
has a pure
f orbital occupied by one
levels of valence electrons
and
detailed picture of a particular molecule can often be obtained from
H
9
H
in the infrared region
mm. Molecular electronic states have higher
of several electron volts and spectra in the visible
'\Q
by
from
microwave region with wavelengths
and vibrational spectra are
A
energies, with typical separations
w
that arise
cm. Vibrational slates are separated by somewhat larger energy
1
intervals (0.1
10
(
C
center
of moss
atoms on either side (Fig 8-20)
C
atom, which has lobes above and below the
2», orbitals i„ the molecule combine to produce
FIGURE 8
tate
about
2]
Its
A
diatomic molecule can ro-
center of mass.
bonding * orbitals which take the form of
a continuous electron prohabihlv
distnhi.tm,. .move and below the
plane of the ring.
The
to the
molecule as a whole and not
to
six
electrons belong
any particular pair ofatoms: these electrons
are debvafizetl.
268
PROPERTIES OF MATTER
THE PHYSICS OF MOLECULES
269
,
.
where
S.5
=—
in
JU+
^,=
Reduced mass
+ m.
Ht]
the reduced mass of the molecule as mentioned in Sec, 4.9,
states that the rotation of a
diatomic molecule
R
= 7.61
=
away.
where to
as
we know.
8,7
!u
If
we denote
=
y/jtj
momentum is always quantized in nature,
rotational quantum number by /, we have here
the
/
=
0, 1,
of a rotating molecule
is
%/w 2 and
/.
+
1) ft
2,3,
.
,
' 21
1.46
Rotational energy levels
11
Let us see what sorts of energies and angular velocities are involved in molecThe carbon monoxide (CO) molecule has a bond length R of 1.13 A
ular rotation.
and the masses of the
-26
kg.
C and
O
X
atoms are respectively 1.99
The reduced mass m'
of the
CO
molecule
is
10" 2S
kg and
I0-
when AT
molecule when /
=
1
is
The
its
IP' 23
,o
J
kg-m 2
considering only rotation alxntt an axis perpendicular
What about
rotations about
its
whose
nucleus,
is
that the
radius
principal contribution to the
is
as in Fig, H-21
symmetry
the axis of
mass of an atom
only
moment
— 10
"'
is
— eud-over-end
itself?
The
+ m„
_
"
=
1.99
X
2.66
1.99
+
2.66
of the radius of the
atom
1.1-1
X
10
10--" kg
symmetry axis theretore comes from its electrons, which are concentrated
whose radius about the axis is roughly half the bond length R but
a region
whose
only about %/Ajaoa of the total molecular mass.
'/. rotation
allowed rotational energy levels are proportional to
total
mass
is
symmetry
a\is
rotations.
Hence energies
this
M kg
itself.
of inertia of a diatomic molecule about
must involve energies
— M)'
1
of at least several
Since the
a!
unit
the
times the Ej values for end-over-end
eV would be
involved in any rotation
Since bond energies are of
about the symmetry axis of a diatomic molecule.
X
reason
almost entirely concen-
1
Hi,
C 2,6 x
are in excited rotational
10" rad/s
the latter can be neglected
in
therefore
X
3.23
X
we have been
far
trated in
lfl
CO
ads of symmetry of a diatomic molecule,
rotations.
I)* 2
8.S
10
velocity of the
-I
so its energy levels are specified
Thus
12
CO
.
to the
+
J
eV
*
2E
=
J(J
«kg-m2
10" 23
10
2
J-s)
the molecules in a sample of
all
The angular
states.
= y2 tu s
Ej
X
10- 4
2X7.61 X
The energy
by
2.66
5.07
X
X
X
lO" 3-'
not a great deal of energy, and at room temperature,
is
10" 2 eV, nearly
angular velocity. Angular
is its
ft*
T
of the molecule has the magnitude
This
L =
S.G
1.46
HA
equivalent to the rotation of
is
a single particle of mass m' about an axis located a distance of
The angular momentum L
Equation
X
(1.054
_
~
is
Qft-
2f
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
and
its
moment
/
of inertia
molecules that have electric dipole moments can absorb or emit electromagnetic
/ is
= m'R 2
photons
=
as
=
1.14
1.46
X I0-» kg X (1.13 X
X 10 » kg-m 2
10-'°m) 2
(Fig. 8-16)
in
The
lowest rotational energy level corresponds to J
CO
270
PROPERTIES OF MATTER
in
=
1
,
and
for this level in
such transitions, which means that nonpolar diatomic molecules such
H, and symmetric polyatomic molecules such
do not exhibit rotational spectra,
molecules
like
Ha
Furthermore, even
transitions
between
,
in
as
OOj (0=C=0)
[Trausi lions
and CI 1
between rotational states
CO,, and CH, can take place during
collisions,
however.)
molecules that possess permanent dipole moments, not
rotational states involve radiation.
As
in the case of
all
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
1
=
/
=3
J
=
2
l
=
\
1
=
4
&J=±l
s.9
In practice, rotational spectra are
transition that
number
/ in
is
found involves
always obtained
change from some
a
rigid molecule, the frequency of die
''j-j+l
I
is
the
7+1.
is
FIGURE 8-22
U+
of inertia for end-over-end rotations.
The spectrum of
The
lines, as in Fig. H-22,
it
corresponds to
ascertained from the sequence of lines; from these data the
of inertia of the molecule can be readily calculated,
quencies of any two successive lines
may be used
to
CO,
frequency of 1.153
'r„
=
for instance, the /
X
lit" 11/.
= ^-(/+
-» /
=
1
moment
(Alternatively, the fre-
determine
trometer used does not record the lowest- frequency lines
sequence.) In
Energy levels and spectrum of
Rotational spectra
1)
frequency of each line can be measured, and the transition
Ix:
levels
In the case of a
molecular rotation.
n
moment
energy
quantum
h
a rigid molecule therefore consists of equally spaced
can often
of
%
*Wi
h
=
8.10
where
IE
photon absorbed
rotational
absorption, so that each
initial state
number
the next higher state of tjuantum
—
in
I if
the spec-
in a particular, spectral
absorption line occurs at a
rotational
Hence
I
I
I
I
spectrum
I)
2ffi>
1.054
~
-
2ir
1
X
.46
X
1.153
X
HI" 3
X
of the
'
J-s
10"
minimum
\/7/m'
=
1.13 A. This
earlier in this section
CO
is
very nearly a parabola. In
this region,
then,
V = V + V2 k{R - R a f
8.ii
fi co is
is
s-'
10"10 kg-m-
Since the reduced mass of the
of this curve, which corresponds to the normal configuration
of the molecule, the shape of the curve
the
way
in
X
2,i
bead length
which the bond length for CO quoted
molecule
is
1.14
l»""
kg, the
where R
{,
is
the equilibrium separation of the atoms.
gives rise to this potential energy
was determined
may be found by
The interatomic
force that
differentiating V:
*~~lR
8.9
VIBRATIONAL ENERGY LEVELS
When soilicicntly excited, a molecule can
we shall only consider diatomic molecules.
8.12
vibrate as well as rotate.
Figure 8-23 shows
how
As
licfnre.
the potential
energy of a molecule varies with the mternuclear distance R. In the neighlwrhood
272
PROPERTIES OF MATTER
=
-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
substituted for m:
Two -body
8.14
2s-
oscillator
in
When the harmonic-oscillator problem is solved quantum mechanically, as was
done
Chap.
in
energy of the oscillator
5, the
found to
is
lie restricted to
the
values
8.
is
where
u,
The
((.•+
V2
=
(>,
2, 3,
,
1
/»<„
)
the vibrational
f
parabolic approximation
=
/-.'„
quantum number, may have
.
.
lowest vibrational state (e
=
value of 0; as discussed in ("hap.
principle, because
to
if
would
position
its
it
=
5, this result is in
and
its
be infinite— and a particle with E
momentum.
energy
0) has the finite
^''"o" ,mt tne classical
accord with the uncertainty
were
stationary, the uncertainty in
momentum
uncertainty would then have
the oscillating particle
l>c
the values
=
cannot have an
infinitely uncertain
view of Eq. 8.14 the vibrational energy levels of a diatomic
In
molecule are specified by
The
FIGURE 8-23
potential energy of a diatomic molecule si a function of intern uclear distance.
Classically, the frequency of
of force constant k
a vibrating body of mass
m
connected to a spring
1
-et
is
:
187
and, as
III
1.14
What we have
situation of
in
the case of a diatomic molecule
two bodies of masses m, and
in.,
In the absence of external forces the linear
constant,
of their
and the
I
mass
momentum
is
somewhat
vibrational energy levels.
X
we
found
10 2a kg.
the same
times.
given by Eq.
S.I.'3
PROPERTIES OF MATTER
CO molecule and the spacing
CO
force constant k of the bond in
— not an exceptional figure for an ordinary spring)
reduced mass of the
of vibration
CO
molecule
is
m'
=
therefore
is
m'
N/m
X lO^kg
X K> 13 Hz
187
2v
V
= 2.04
and both reach the extremes of
The frequency
1..14
of oscillation of such
with the reduced mass
FIGURE 8-24
274
lb/in.
The frequency
cannot affect the mot ion
in'
of Eq. 8.5
The separation AE lietween
force constant k
"'o^MimmmKT
1(1
in Sec. 8.8, the
2*
of the system remains
M-:
1
is
The
different
reason m, and nt, vibrate hack and forth relative
in opposite directions,
heir respective motions at
a two-body oscillator
this
the
Vibrational energy levels
joined by a spring, as in Fig. 8,24.
oscillations of the bodies therefore
center of mass. For
to their center of
is
its
N/m (which
k
1
8.13
+ V2 )HJ—
m
(v
us calculate the frequency of vibration of the
between
is
=
Ev
A
m
=
=
/;,,,,
6M
8.44
the vibrational energy levels in
- EF = hv n
X 10- M J-s X
X IlH eV
2.04
X
10
,:,
s
CO
is
'
Iwo-botfy oscillator.
which
is
Because
considerably
AE
more than
the spacing between
> kf for vibrational
states in a
sample
its
at
rotational energy levels.
room temperature, Most
THE PHYSICS OF MOLECULES
275
of
the molecules in such a sample exist in the
zero-point energies.
This situation
=
state with (inly their
very different from that diaraeterislie of
is
where the much smaller energies mean (hat the majority of
the molecules in a room-temperature 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 potential-energy curve becomes less and less valid
rotational states,
with increasing energy. As a
of high B
shown
levels
The
is less
result, the
spacing between adjacent energy levels
than the spacing Iwtween adjacent levels of low
in Fig. .S-25.
This diagram also shows the
finer
between vibrational
the harmonic oscillator approximation.
oscillating dipole
radiation of the
whose frequency
is
is
states
is
.
The
only emit
to (c
The potential energy
ing vibrational and rotational energy
At
= ±1
easy to understand.
in
An
y
all
quanta of frequency *„ have the energy
of a diatomic molecule as a function of interatomic distance,
levels.
its
energy increases from
AE =
+ Y2 —
l)hv
a time,
at
hi> u
Hence
.
in
+
{t
l
/2 )rW u to [v
which case
its
=
the selection rule At)
between adjacent molecules
AE =
+ %+
l)ht>
inhibit rotation.
,
and
(u
+
it
can
y»)hv„
±1,
liquids
in
a time, in
hi\, at
energy decreases from
Pure vibrational spectra arc observed only
where
interactions
Because the excitation energies
involved in molecular rotation are considerably smaller than those involved
moving molecules
vibration, the freely
in a gas
in
or vapor nearly always are
The spectra of such molecules do
show isolated lines corresponding to each vibrational transition, but instead
large number of closely spaced lines due to transitions between the various
rotating, regardless of their vibrational state.
a
rotational states of
show-
one vibrational
level
and the rotational
In spectra obtained using a spectrometer
states of the other.
with inadequate resolution, the
lines
appear as a broad streak called a vibration-rotation band.
To a
FIGURE 8-25
oscillating dipole accordingly can only alvsorb
which case
not
levels.
can only absorb or emit electromagnetic
isi'
same frequency, and
This rule
which
structure in the vibrational
caused by the simultaneous excitation of rotational
selection rule for transitions
c,
/w>
first
approximation, the vibrations and rotations of a molecule take place
independently of each other, and
we
can also ignore the effects of centrifugal
and anharnionicity. Under these circumstances the energy
diatomic molecule arc specified by
distortion
a
+ /(/+l»|J-
S.17
=
Figure 8.26 shows the /
the e
=
and o
=
0, 1, 2, 3,
and 4
levels of a diatomic
A/
=
—
•
r;
-
transitions
1
= -
From
molecule for
vibrational states, together with the spectra! lines in
I
absorption that are consistent with the selection rules Ac
lite D
levels of
fall
into
two
= +
categories, the
and A/ = ±1.
P branch in which
1
I (that is, / -> } ) and the R branch in which A/
+ 1 (/ -» J I).
Eq, 8.17 the frequencies of the spectral lines in each branch are given
=
1
+
by
''p
=
h
=
^/5 + W-
u
-
i
f-(i,j
ft
'-*' +
11
Iwl
ft
8. IB
vibrational energy levels
/=
*^>-fcr
1.2,3..
I'
branch
and
rotational energy levels
8.19
—
^IJt
=
i'o
+
I
(/
'"11.,/
+
!)
J
=
0,
1,
2,
H branch
2<nl
276
PROPERTIES OF MATTER
THE PHYSICS OF MOLECULES
277
There
;=4
J
1
no line
at v
v
=
=
because transitions for which SJ
i>„
diatomic molecules. The spacing between the lines
R
branch
&»
is
=
ascertained from
=3
I
is
in
microwave
hence the moment of
fi/2ir/;
its
are forbidden
f)
inertia of a
P and
the
molecule can be
infrared vibration-rotation spectrum as well as from
—>
=
Figure 8-27 shows the v
pure- rotation spectrum.
t
its
=
1
l
vibration-rotation absorption band in
J
=
both the
in
=2
A
CO.
molecule that consists of many atoms
may have a
large
number
of different
1
normal modes of vibration. Some of these modes involve the entire molecule,
J=l
-J
.,
but others involve only groups of atoms
1
1
independently of the
1
teristic vibrational
rest of the
whose vibrations occur more or
molecule. Thus the
frequency of
1.1
X 10 M Hz. The
X
10 1
'1
—OH
Hz and
the
— NH
Z
group has a
frequency of
1 .0
carlx>n-earl>on
group depends upon the number of bonds between the
the
—£ —C—
group vibrates
vibrates at about 5.0
6.7
X
lO 13
Hz
(Figs.
X
I0
13
at
characteristic vibrational
about 3.3
Hz, and the
8-28 and 8-29).
{As
X
10
13
less
group has a charac-
Hz, the
frequency of a
C
U=C^
atoms:
group
— C^C —
group vibrates
we would
expect, the greater the
at
about
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
FIGURE 8-27
The
lines are identified
1
=4
/
=3
>v =
J
=2
J
=
}
=
1
-J
FIGURE
There
no Una
26
AJ = +
P branch
K
The
8-
is
&} = -l
rotational structure of the
at r
a
>-„{1he
t,)
-
r
I
ir
1
r
=
—
c
by the value
=
>
tool in
vibratkjnrotalion absorption band fn
determining molecular
CO undw
high resolution.
The
of 1 in the Initial rotational state.
[
l
I
:m cl>
transitions In
branch} because of the selection rule li
• diatomic molecule.
6.1
6.2
6.3
6.4
6.5
6.6
6.7 x
10" Hz
= ^L
FREQUENCY
278
PROPERTIES OF MATTER
THE PHYSICS OF MOLECULES
279
E*
'
E*
ELECTRONIC SPECTRA OF MOLECULES
8.10
*
The
its
D°
0.4527 eV
"10.1915 eV
.4658
«fV
energies of rotation and vibration in a molecule are clue to the motion of
atomic nuclei, since the nuclei contain essentially
The molecular
corresponding to
levels
much
is
all
of the molecule's mass.
electrons also can be excited to higher energy levels than those
I
In-
ground
state of the molecule,
(hough the spacing of these
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
electronic state (see Fig, 4-12),
a dipole
moment change always accompanies
symmetric stretching
FIGURE S-28
The normal mode*
of vibration of the
HX
asymmetric stretching
molecule and the energy levels ot each mode.
each
a change in the electronic con-
figuration of a molecule. Therefore hoi noun clear molecules, such as
symmetric bending
in
All molecules exhibit electronic spectra, since
1
1,
and
N2
,
which have neither rotational nor vibrational spectra becawBB they lack permanent dipole moments, nevertheless have electronic spectra which possess rotalional
and
vibrational line structures that permit their
moments
of inertia and
Ixind force constants to l>e ascertained.
structures.
be either
An example
CH 3CO— SIX
is
or
thioaeetie acid,
whose structure might conceivably
CH 3CS —OH. The
infrared absorption spectrum of
Electronic excitation in a polyatomic molecule often leads to a change in
shape, which can be determined from the rotational fine structure in
spectrum.
The
origin of such changes
lies
thioacetic acid contains lines at frequencies equal to the vibrational frequencies
of the
or
jC=0
and
— SH
groups, but no lines corresponding to the
—OH groups, so the former alternative
is
/C=S
the correct one.
I'uiiclioiis ol electrons in different states,
hinds involve sp hybrid orbitals
that, in
levels of
each mode.
band
in the different characters of the
wave
which
lead in
UBTespandfogjIy different
types of Ixmd. For example, a possible electronic transition in a molecule whose
involve pure
FIGURE 8 29 The norma! modes of vibration of the CO. molecule and the energy
The symmetric bending mode can occur In two perpendicular planes.
its
its
is
p
From
orbitals.
,
p
orbitals
is
is
which the Ixmds
to a higher-energy state in
a molecule such as Bell 2 the Ixmd angle
180° and the molecule
of pure
is
the sketches earlier in this chapter
linear (II
in the ease of
we
— Be — H), while the Ixmd angle
bent (H — Be).
90° and the molecule
can see
sp hybridization
in the case
is
11
There arc various ways
in
which a molecule
energy and return to
its
ground
lose
:>
o£e>—r^-0«
T
eV
c
o
-O-
PROPERTIES OF MATTER
o
—
c
O* *~-o
tile
O*
ground
may
some of
downward
vibrational level in the
upper electronic
is
give up
asymmetric stretching
its
Another
it
of course, simply
absorl>eei.
possibility
is
thereby
fluorescence;
vibrational energy in collisions with oilier
radiative transition originates from a lower
State (Fig. 8-30).
Fluorescent radiation
therefore of lower frequency than that of the absorbed radiation.
In molecular spectra, as in
symmetric stretching
The molecule may,
slate in a single step.
molecules, so that the
the molecule
o
an excited electronic state can
emit a phi>ton of the same frequency as that of the photon
returning to
symmetric bending
280
11.2912
.0.1649 eV
0.827 cV
state.
in
atomic spectra, radiative transitions !>elween
electronic states of different total spin are prohibited (see Sec. 7-11). Figure 8-3
THE PHYSICS OF MOLECULES
281
ultimately reaches the v
Mnglet state
it
is
is
=
level.
impossible to occur but that
.Such transitions accordingly
excited state
A
radiative transition
"forbidden" by the selection
phorescmt radiation may
l>e
it
rules,
which
from a
realty
triplet to a
means not
that
has only a minute likelihood of doing so.
have very long half
lives,
and the resulting phos-
emitted minutes or even hours after the
initial
absorption.
FIGURE 8-31
vibrational transition
The
origin ol
phosphorescence. The
final transition Is
delayed because
ft
violates the s*.
lection rules lor electronic transitions.
C3
singlet excited state
triplet excited
state
ground
state
vibrational transition
1^^— forbidden
REPRESENTATIVE COORDINATE
FIGURE 8-30
The
which the molecule
in
a photon and
elevated to a singlet excited state.
is
can undergo radiationless
and there
is
in its singlet (S
tratisitions to a
have about the same energy
state,
singlet
ground
state
origin of fluorescence,
shows a situation
to
transition
as
=
ground state absorbs
0)
In collisions the molecule
lower vibrational level that
one of the
levels in the triplet {S
then a certain probability for a
shift to
may happen
—
1
)
excited
the triplet state to
occur. Further collisions in the triplet state bring the molecule's energy below
that of the crossover point, so that
282
PROPERTIES OF MATTER
it
is
now trapped
in the triplet state
and
REPRESENTATIVE COORDINATE
THE PHYSICS OF MOLECULES
283
Problems
The bond between
11.
the hydrogen
N/m.
has a force constant of 516
At what temperature would the average kinetic energy of the molecule*
a hydrogen sample be equal to their binding energy?
1.
in
in
1
2.
Although the molecule He 2
ion
He,
1
unstable and docs not occur, the molecular
and has a bond energy about equal to that of H, +
stable
is
is
.
The hydrogen
2.
a 'H 3S CI molecule
a HCI molecule will be vibrating
room temperature?
isotope deuterium has an atomic mass approximately twice
Does
that of ordinary hydrogen.
Explain
How
or K, ?
of vibration of the molecule.
does
in
Is it likely that
excited vibrational state at
its first
and chlorine atoms
this affect
H2
or
HD have the greater zero-point energy?
the binding energies of the two molecules?
this observation.
The
13.
Which would you expect
3.
The lowest bond
have the highest ixmd energy,
to
F,, ¥./,
ionization energy of II,
is
l.->.7
eV and
that of
H
13.6.
is
Why
arc they
so different?
I2
C
1B
and
=
1.1(12 X
y
c
al
The observed molar
1
rotational absorption line occurs at 1,153
16°
Hz
in
?
C
ia
X
10" Hz
sion) in
in
O. Find the mass number of the unknown
a gas molecule contributes
gas, this
hydrogen gas
curve
is
—
is,
kcal/kmol
1
volume
is
(The temperature scale
is
at constant
each mode of energy posses-
K
to the specific heal of the
interpreted as indicating thai only translalional motion, with
is possible for hydrogen molecules at very low temperAt higher temperatures the specific heat rises to -~5 kcal/kmol K,
atures.
6.
Calculate the energies of the four lowest rotational energy states of the
and
D2
molecules, where
The
specific heat of
three degrees of freedom,
carbon isotope.
7.
966 N/m, Find the frequency
plotted in Fig. 8-32 versus absolute temperature.
logarithmic.) Since each degree of freedom (that
—
The/ =
5.
is
energy:"
14.
The
4.
force constant of the 'H 1!, F molecule
I)
represents the deuterium
rotational spectrum of
HCI
atom
H2
indicating that two
more degrees
temperatures the specific heat
fll.
is
of freedom are available,
—7 kcal/kmol
K, indicating
and
two
at
still
higher
further degrees
The additional pairs of degrees of freedom represent, respectively,
which can take place about two independent axes perpendicular to the
symmetry of the U 2 molecule, and vibration, in which the two degrees
of freedom.
contaias the following wavelengths:
rotation,
12.(13
9.60
8.04
6.89
6.04
If
10 5
X
X
X
x
m
axis of
m
HH m
10~ n m
x 10 s m
I0- S
of freedom correspond to the kinetic and potential
by the molecule,
in
is
equal to the
an HCI molecule. (The mass of
:i3
Cl
5.81
is
X W~M
FIGURE 8-32
Molar specific heat
of
hydrogen
at
modes of energy possession
Verify this interpretation of Fig. 8-32 by calculating the
which kT
temperatures
the isotopes involved are }H and ffCt, find the distance between the hvdrogen
and chlorine nuclei
(a)
minimum
rotational energy
and
to the
at constant volume.
kg.
c„
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
7
of the spectral line associated with a transition between these states
6
8.
between the rotational frequencies of the
diate
9.
A
""'Hg^'CI molecule emits a 4.4-cm photon
transition
from /
(The masses of
10
10.
=
" ,H,
I
to /
Hg and
=
as
0.
interme-
states.
when
it
undergoes a rotational
Find the interatomic distance
Cl are, respectively, 3,32
X
in this
molecule.
10 25 kg and 5.81
x
^kg.'
Assume
4
I
*
3
2
that the
H2
1
molecule behaves exactly
with a force constant of 573 N'/ni and
corresponding to
284
is
,1
its
-
a harmonic oscillator
find (he vibrational
4.5-eV dissociation energy.
PROPERTIES OF MATTER
like
quantum number
100
200
500
1000
2000
5000
-*T
TEMPERATURE, K
THE PHYSICS OF MOLECULES
285
minimum
vibrational energy a
constant of the bond in H..
is
H„ molecule can have. Assume that the force
N/m and that the H atoms are 7.42 X 10~" m
STATISTICAL
573
MECHANICS
apart. (At these temperatures, approximately half the molecules are rotating or
9
some are in higher states than J = 1
coasidering only two degrees of rotational freedom in
vibrating, respectively, though in each case
= I.) (b) To justify
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
molecule with } = 1 and t =
make per
2
or v
the
1
vibration?
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,
concerned with the actual motions or interactions of individual
investigates instead their most probable behavior.
cannot help us determine the
life
While
is
DOt
particles, but
statistical
history of a particular particle,
mechanics
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
many phenomena
so
in the physical
world involve assemblies of
value oJ a Statistical rather than deterministic approach
generality of
its
clear.
particles, the
Owing
to the
mechanics can be applied with equal
problems (such as that of molecules in a gas) and qiiantum-
arguments,
facility to classical
is
statistical
incchanical problems (such as those of free electrons in a metal or photons in
a
lx>x),
9.1
and
it is
one of the most powerful
tools of the theoretical physicist
STATISTICAL DISTRIBUTION LAWS
We shall
use statistical mechanics to determine the most probable
way in which
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
a fixed total
amount of energy
is
distributed
,
assumed
to interact with
an extent
sufficient to establish
sufficient to result in
We
1.
286
PROPERTIES OF MATTER
(or
with the walls of their container) to
thermal equilibrium in the assembly but not
any correlation between the motions of individual
particles.
shall consider assemblies of three kinds of particles:
Identical particles of
distinguished.
u
one another
any spin that are
The molecules
eQ-Bohmatm
distribution
sufficiently
widely separated to be
of a gas are particles of this kind,
and the Max-
law holds for them.
287
2.
Identical particles of
Such
from another.
or integral spin that cannot be distinguished one
dx dp t
do not obey the exclusion principle, and the
dy dp u
particles
Base-Einstein distribution law holds for them.
bosom; and we
Photons are Rose particles, or
shall use the Bosc- Einstein distribution
trum of radiation from
a
cfo
we
lenee
I
black body,
we
Electrons are Fermi particles, or fennions, and
Fenni-Dirae distribution law
to explain the
dPl
ft
ft
ft
law to explain the specsee that
3. Identical particles of spin )' that cannot be distinguished one from another.
2
Such particles obey the exclusion principle, and Ihe Fermi-Dime distribution low
holds for them.
>
>
>
shall use the
behavior of the free electrons
in a
A
1
ft-
"point" in phase space
order of
where
metal.
>
t
ft
3
.
We
A more
PHASE SPACE
volume
>
h3
The state of a system
instant
of particles
is
completely specified classically
momentum of each of its
and momentum are vectors with
the position and
if
known. Since position
we must know
at a particular
constituent particles are
three
components apiece,
fc
ft
for
each
The
tj,
pr
.
Pl) ,
3
position of a particle
is
It
a point having the coordinates
convenient to generalize
is
x, y,
this
imagining a six-dimensional space in which a point has the
six
z
in
ordinary
conception bv
coordinates
z,
,
mechanics
in
a geometrical framework, thereby permitting a simpler and niore
A
point in phase space corresponds to a particular position
momentum, while
only.
Thus every
and
a point in ordinary space corresponds to a particular position
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
much
precision
our knowledge of
its
It is
it
among
the task of
how
tf
p? instead of being
statistical
mechanics to
the particles constituting
the cells in phase space.
phase space can have no
infinitesimal size in
violates the uncertainty principle, the notion of
momentum, and
momentum
space alone
the position of a particle
an unlimited uncertainty
vice versa.
what we mean by a "point"
phase space. Let us divide phase space into tiny sLx-ditncnsional
cells
whose
dpx dp y dpz 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
=
MAXWELL-BOITZMANN DISTRIBUTION
l^t us consider an assembly of
e., 6o,
....
tj,
...
,
,
.
dx dy dz dp x dp y
dp,.
cell in
to
PROPERTIES OF MATTER
molecules whose energies are limited to
These energies may represent either discrete quantum
.
may conespond
phase space
know
is
stales
more than one
a given energy. What we would like
to
the most probable distribution of molecules
among the various possible
energies.
W of different
in
ways
of statistical mechanics
is
that the greater the
number
which die molecules can be arranged among the
in
phase space to yield a particular distribution of molecules
among
cells
the different
more probable is the distribution. The most probable distriOur first step, then,
which
is a maximum.
a general expression for W. We assume dial each cell in phase space
energy
levels, the
bution
is
W
therefore the one for
is
to find
is
equally likely to
justification for
and, according to the uncertainty principle,
;V
or average energies within a sequence of energy intervals, and
A fundamental premise
sides are dx, dy, dz,
288
.
we can in principle determine
as we like merely by accepting
perfectly acceptable:
*9.3
phase space.
'Hie uncertainty principle compels us to elaborate
in
in
,
,
.
,
straightforward method of analysis than an equivalent one wholly abstract in
character.
px p
x.
pz pv ps This combined position and momentum space is called phase
space. The notion of phase space is introduced to enable us to develop statistical
tj,
of h k
the system distribute themselves
with as
of the
which does not contradict the uncertainty-principle argument since
and
determine the state of a system by investigating
is
three-dimensional space.
x, y, z,
a point of infinitesimal size in either position space or
particle.
is
In general, each cell in a phase space consisting of k coordinates
.
While the notion of a point of
&
whose minimum volume
phase space as being located some-
detailed analysis shows that each cell in phase space actually has the
lr\
momenta occupies a volume
six quantities,
z,
in
itself.
physical significance, since
x,
actually a cell
such a cell centered at some location
in
precisely at the point
9.2
is
must think of a particle
arrived at with
it
l>e
(as in
its
occupied;
this
assumption
is
plausible, but the ultimate
the case of Schrodinger's equation)
help agree with experimental
is
that the conclusions
results.
STATISTICAL MECHANICS
289
If there are
g
with the energy f„ the number of ways
cells
(
molecule can have the energy
molecules can have the energy
umlcc
nli-s
can have the energy
molecules ran
\
all
of factors of the fonn
e,
e{
2
is
g,
,
number
total
and the
in
which one
in
which two
ways
of
mimlicr of ways that
total
among
the various micrgies
is
[Ik-
What we now mast do
product
most probable, that
is
first
step
is
{gxTAfkTAgs)" 3
-(^"'(feNgy)"*
is,
determine
is
which
just
which distribution of the molecules
distribution yields the largest value of
to obtain a suitable analytic
We
number.
M
A'!
n^btgln,!
n,
namely,
(g,)"',
w=
9.4
(&)"' Hence the number of ways in which
is
distributed
In-
The
t, is g,.
W. Our
approximation for the factorial of a large
note that, since
•
l)(n
-
the natural logarithm of n!
is
ti!
=
n(n
-
2)
.
+
In
.
(4) (3) (2)
.
subject to the condition thai
=
in.
9.2
n.
+
+
Ha
=
+
n,
A'
Inn!
Equation 9.1 does not equal W, however, since
the possible permutations of the molecules
The
.V
total
number
molecules can he arranged
=
the different energy levels.
X
4
in A'! different sequences.
and
a, b, c,
X
3
x
2
d.
=
1
Hie value
of 4!
=
2
In
+
is >V!; in
other words,
abdc
bade
cadb
dacb
acbd
bead
chad
dhac
tin lb
beda
rbdii
dlmi
adbc
udch
bdac
cdtib
dcab
l/tlrtt
alba
del hi
As an example,
we might
than one molecule
is
in
among themselves has no significance
a, b,
and c happen
to
be
in level
/,
it
an energy
level,
in this situation.
=
n
and we can
=
In n
I
In n!
Equation 9.5
For instance,
does not matter here whether
=
the fact that n
}
3.
Thus the
n,
contribute n,! irrelevant permutations. If there are
if
by merely integrating
In
is
In n!
become
n from n
—
I
r»
is
=
n
In
known
r*
—
dn
n
that
—
+
«
1
>
n
1,
n
we may
>
neglect the
1
in the
above
Stirling's
1
result,
formula
as Stirling's formula.
natural logarithm of Eq. 9.4
molecules
is
!
.
.
L
In
W=
In A'!
- X
In
n
!
(
+ X
n, In g,
molecules in the
molecules
ri
in level
n, molecules in level 2, and so on, there are n,!n !n
irrelevant permuta2
3
tions. What we want is the total number of possible permutations Ar divided
1,
find Inn!
area under the stepped curve
we enumerate
as abc, acb, bat, bat; cab, or cbtr, these six distributions are equivalent,
is
+ tan
however, permuting than
all
care about
n In
we are assuming
we obtain
so
9.5
them
we
1)
n:
Because
and
since
ith level
The
a plot of In n versus n.
indistinguishable,
to
is
The
When more
-
very large, the stepped curve and the smooth curve of In n
is
=
dahc
ln{n
is
n
In ni
cnbd
+
+
4
When
and there are indeed 24 ways of arranging them:
hurt!
3
Figure 9-1
24
abctt
In
must take into account
of permutations possible for A' molecules
have four molecules,
4!
among
we
.
FIGURE
9-J
The area under
the stepped curve
is
In
pit
!
by the
total
number
of irrelevant ones, or
Whan
ts
very large, the
smooth curve
Is
a good approx-
imation of the stepped curve,
V!
and
9.3
«,».,!
a
The
total
number
of
ways
the possible energy levels
290
In n!
grating
PROPERTIES OF MATTER
in
is
which the
.V
molecules can
Ik-
distributed
=
In
can be found by inten from n = 1 to
n.
among
the product of Eqs. 9.1 and 9.3:
STATISTICAL MECHANICS
291
formula enables us
Stirling's
W = JVln N - N -
In
Since
write this expression as
to
Sn,=
n, In n,
+ 2
n,
+ 2
n, ln
ln.i
l
While we have an equation
l
+ 2n,
Ing,
W rather than for W
for In
itself, this Is
no handicap
since
W)'mai
In W .
m„ = "
"max
(In
\
The
condition for a distribution to be the mast probable one
changes dn,
any of the
in
is
that small
not affect the value of W.
B,*s
(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
corresponding to a change in
97
Wmla =
8 In
N is
since ,V In
-v
=
n,
n ,S
In
n
S In W, from Eq. 9.6
is
(
- 2
f
we
W
see that
Because
changes
2
Sn,
9.11
2
rjfin,
In
n,&i,
+2
In gjSn,
tlie
8
In
n
=2
+
«n 2
= tiSrif +
Sn 3
e 2 8n,2
+
+
=
•
£ fin
3
+
3
-
=
t
We
these expressions to Eq. 9.8.
2(-ln
9.12
In
n,
+
lngj
-
a
obtain
-
=
/SfJSn,
8
each of the separate equations added together to give Eq. 9.12, the variation
fin,
is
effectively an independent variable.
— In
from which
Sn,
for
lie
In order for Eq. 9.12 to hold, then,
each value of
Hence
t.
we
+
n,
In g,
—
o
(it
t
=
obtain the Maxwell-Bolt/.mann distribution law:
= & e ~° e
"i
—
Maxwell-Boltzmann
fit,
distribution law
This formula gives the number of molecules n, that have the energy
mimlwr of molecules is coastant, die sum 2 5Hj of all the
number of molecules in each energy level must lie (1, which means
total
in the
+
Sn,
=
— 8n,
(
of molecules in each energy level are not
To incorporate the above conditions on the various 8n into Eq, 9.8 we make
use of ^grange's method of undetermined multipliers, which is simply a convenient 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
9.13
t
number
the
the quantity in parentheses must
and so
2n
=
9.10
Now
constant.
8 In
6n
n, of
Sn.,, ... in
independent of one another but must obey the relationships
6
.V,
liiH'=iVlnW-^n
96
2
variations Sn,,
of the
number
a and p.
We
of cells in
phase space g that have the energy
s
must now evaluate &, «, and
e(
e,
in terms
and the constants
/S.
that
2
Hence Eq.
nfi In n,
=
9.7 liecomes
Energy quantization
-2 In RfSn, +
9.8
2
In g,6*n,
=
in a gas,
1 1
While Eq.
molecules
must be
9.8
among
distribution.
fulfilled
the energy levels,
We
must
EVALUATION OF CONSTANTS
*9.4
by the most probable distribution of the
it does not by itself completely specify
this
is
therefore
=
ri[
total
n2
+
n3
+
...
of molecules in a sample
molecules whose energies
=N
In
n(t)
de
is
usually very large.
convenient to consider a continuous distribution of molecular
more
also take into account the conservation of particles
+
number
energies rather than the discrete set
9.14
2n,
(9.2)
and the
inconspicuous in the Iranslational motion of the molecules
is
=
lie
e,,
between
<?
t
If
z , c3
and
e
+
dr,
n{()tk
is
the
number of
Eq. 9. 13 becomes
g(e)e~"e~&' de
terms of molecular momentum, since
and the conservation of energy
2n, ej
9.9
where E
292
is
=
iijfj
+
n 2t2
+
n,E,
+
•-=£
the total energy of the assembly of molecules.
PROPERTIES OF MATTER
f
=
2>u
we have
In consequence the
9.13
n{p) dp
=
g(p)e
•**"** dp
STATISTICAL MECHANICS
293
The quantity
g(p)
a molecule has a
volume
is
equal to the
number
of cells in phase space in which
momentum between p and
+
u
dp.
Since each
cell
specified
has the
p
=
ffSS)'<lxdydzdpz
is
the phase-space
<lp
ll
dpl
we
volume occupied by
particles with the
2
=
2mi
the total energy
— m
dp
and
of molecules. Since
ffr
/2lHF
can write Eq. 9.18
in the
form
2AyJ3/a
=
»1{e) ffr
9.19
E of the assembly
Ve^ e
- '*'
df
momenta. Here
The
fffdxdydz= V
V
we compute
ft,
,
where the numerator
volume occupied by the gas
the
is
= 4*p
ff P! dl\<'P:
tl
2
where 4wp dp
2
is
in
total
energy
E= f
ordinary position space, and
?n(f)df
2
the
e^V**
f"
shell of radius
p and thickness dp
3 JV
9.30
2
^dp^-^pl
9,16
is
<Ip
volume of a spherical
momentum space. Hence
in
find
1
ft-
gp) dp
where
To
/J
where we have made use of the
definite integral
and
n tP) dP
9.17
We
are
now
=
4wV»V D e-'",:/2m
P
According
able to find e~". Since
x 3/2 e -** dx
I
<{
j£
4a
to the kinetic
an ideal gas (which
f
We
find
n {p)dp
by integrating
N=
=N
teq,
is
4-r»-«u -«
47re-°V
E=
9.21
9,17 that
*e-a9*/2m
Jj,
where
J;
is
a
theory of gases, the total energy
what we have been considering)
E
of A* molecules ul
at the absolute
temperature
£2 NkT
=
1.380
X
10" 23 J/molecule-degree
Equations 9.20 and 9.21 agree
where we have made use of the
A
V
Bollzmann's constant
*n
Jt
I
fn
a
7/ is
—-— Jr" p
It"
3
=
4
V
definite integral
9.22
li
=
if
w
nJ
lence
9.5
MOLECULAR ENERGIES
Now
that the parameters «
IN
and
lioll/.mann distribution law in
AN IDEAL GAS
ft
have been evaluated, we can write the
its final
form,
and
,/
9.18
294
n(p)
dp
=
4ttA'
PROPERTIES OF MATTER
/^-V
'!
p2«-AP,/2-. dp
9.23
n(t) df
=
2wA'
VeV''""*
Boltzmann distribution
of energies
STATISTICAL MECHANICS
295
This equation gives the
in a
sample of an
temperature
is T.
many
is,
e
anil t
+
tie
N molecules and whose absolute
The Boltzmann energy
terms of kT. The curve
while there
number of molecules with energies between
ideal gas that contains a total of
is
distribution is plotted in Tig. 9-2 in
not symmetrica! because the lower limit to r'is
e =
in principle,
times greater than
no upper
kT
is
According to Eq. 9.20, the
This average energy
limit (although the likelihood of energies
small).
total
energy
£
of an assembly of
V molecules
is
3 JV
e-
We
2/f
The average energy
t
per molecule
Ls
E/N, so
same
for all molecules at .100 K, regardless of their
l
2
dp
n{p)
the
is
P
2m
2
find that
9.25
that
the
= —dp = me dv
m
tie
£=
is
mass. The Boltzmann distributions of molecular momenta and speeds can be
obtained From Ec|. 9.23 by noting dial
number
Boltzmann distribution
^jfi 2e-pl/2nif dp
=
of
momenta between p and p
of molecules having
+
momenta
dp, and
2ft
.{*
9.24
At 300 K, which
e
=
is
6.21
Average molecular energy
approximately room temperature,
X
10" Z1 J/molecule
n(v)tlr
9.26
\/2wjYm' ,/a
.,
__,,.„.„
l '' 2kT
v^e- mr
=
'
Boltzmann distribution
of speeds
,
tk
[vkTff*
number
the
is
of molecules
was
Formula, which
The speed
= 54s eV/inolecule
IMS9,
in
is
=
9. 27
yjv*
=
The
last
ft-3.
%AT
of a molecule with the average energy of
+
plotted in Fig.
having speeds between p and v
obtained by Maxwell
first
dv.
is
3*f
Rms speed
in
FIGURE 9-2
Maiwell-Bottimann enargy distribution.
since
%mv- = %kT.
This speed
is
denoted r nils lweause
it
is
the square root
of the average of the squared molecular speeds— the rtnit-iiiean-xquare 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
speeds being considered.
rms
m>
\i
that
governs the molecular
For a Bolt/.tnann distributinn,
si
fad the rms speed
is
about
percent greater than the arithmetical average
speed.
Because the Boll/inann distribution oF speeds
probable speed v p
is
Ls
smaller than cidier F or v Tta3
.
not symmetrical, the most
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
!
2kT
Most probable speed
9.28
Molecular speeds
in
a gas vary considerably on either side of Dp,
shows the distribution of molecular speeds
296
PROPERTIES OF MATTER
in
Figure 9-4
oxygen at 73 K (-200°C),
in
oxygen
STATISTICAL MECHANICS
297
800
1.600
1.200
MOLECULAR SPEED, m/s
k
»
*
FIGURE 9-4
o
L «
Op
FIGURE 9-3
hydrogen
s_
=
root-mean-square speed
= average speed =
=
which
simply Eq. 9.13 with
is
273
K
~\j2kTfm
bility that a
factor
Maxwell Boltimonn velocity distribution
(0°C). and in hydrogen at
increases with temperature
molecular speeds
at
273
K
in
K.
The most probable molecular speed
and decreases with molecular mass.
oxygen at 73
molecular speeds
273
K
speeds
In
oxygen
st
73
K. In
oxygen
273
at
are on the whole
less
Accordingly
than at 273 K, and
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.)
In
% replacing e -" and
E^kT
replacing
0w
.
(
quantum
state of
energy E be occupied
at the
The
the multiplicity (or statistical weight) of the level,
g,,
have the same energy
states that
is
the
£,.
relative populations of atomic energy levels can be treated in the
As
we know, more
rotational
quantum
than one rotational state
niimlier
/.
L, in any specified direction of the angular
in
multiples of ti from }h through
is,
there are 2/
+
I
may correspond
The degeneracy
in
arises
I
to
a particular
momentum L may have
to —Jli, for a total of 2/
possible orientations of
L
fence an energy level whose rotational quantum
(The
same way.)
because the component
+
1
any value
possible values.
relative to the specified (z)
direction, with each of these orientations constituting a separate
ROTATIONAL SPECTRA
T. The
number of
temperature
t
Let us apply Eq, 9.29 to the rotational energy levels of a molecule.
That
9.6
and
K.
factor e~ K,/kT, often called the liollzmonn factor, expresses the relative proba-
quantum
at
of molecular
K.
\f3kTfm
-ifikT/wm
= most probable speed =
The distribution*
273
at
number
is
quantum
state.
/ has a statistical
weight of
A
continuous distribution of energies occurs only in the translational motions
of molecules.
As we saw
in
Chap.
8,
quantized, with only certain specific energies £, l>eing possible.
distribution
9.29
298
law
n
(
=
for
modes of energy
n„ g,
e-V* r
gr
molecular rotations and vibrations are
possession of the latter sort
The Bolt/.mann
may
!>e
For a
rigid
= 2/+l
diatomic molecule,
written
4 = /(/ +
%
PROPERTIES OF MATTER
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/
Here the quantity »
In Sec.
1.46
x
we
tt.8
is
is
the
number
found that the
of molecules in the /
moment
=
rotational state.
of inertia of the
CO
molecule is
For a sample of carbon monoxide gas at room temperature
20° C)
H)-10 kg-m 2
(293 K, which
HpflHW+liM/MW
I)
.
2
fr
(1.054
2f*T
~
2
=
0.00941
X
1.46
X
10"« kg-m*
X 10-"
X 1.38 X
.
4
6
8
10
12
14
16
ROTATIONAL QUANTUM NUMBER, /
18
20
W
,-
|-s,i
10" 23
J/K
X
293
K
£
L0
and so
>ij
= (21+
J)„ 0<r o.<w!MiJtf+i>
Figure 9-5 contains graphs of the statistical weight
2J
e -o.omm ju* B
lhc reUuivc p0p(ll uj0|1 tlj/H() for
md
off.
The/ =
and about
the
/=
The
as
+ I, the Boltzmann factor
CQ a( 2{r(; a| n fmMam
.
7 rotational energy level
many molecules
29 level as are
in a
in the /
=
sample of
()
,
evidently the most highly populated,
is
CO
at
room temperature
are in
level.
intensities of the rotational lines in a
molecular spectrum are proportional
energy levels. Figure 8-27
to the relative populations of the various
rotational
the initial rotational level.
/
=
*§.7
7, as
The P and
ft
branches both have their maxima
10
8
6
shows the vibration-rotation band of CO for the
o =
-n> = 1 vibrational
transition under high resolution; lines are identified
according to the / value of
12
14
20
16
ROTATIONAL QUANTUM NUMBER,/
(h)
at
expected.
BOSE-EINSTEIN DISTRIBUTION
The basic distinction between Maxwell-Boltzmann
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
Bose-Kimlein statistics,
as before, all quantum states are assumed to
have equal probabilities of occupancy, 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
6
8
10
12
14
16
4
ROTATIONAL QUANTUM NUMBER, /
lie
distributed in
g cells.
PROPERTIES OF MATTER
20
(c)
f
300
18
FIGURE 9-5
The
multiplicities (a),
rotational energy levels of the
CO
Boltzmann factors
molecule
at
20' C.
(bl.
and
relative population* (c) of the
To
we
carry out the required enumeration,
objects placed in a line (Fig. 9-6).
We
regarded as partitions separating a total of
therefore representing n
and
=
n,
contains
20;
two
particles,
+
".
ft
particles, the
do not
(g,
-
(it,
1)!
In the picture
20 particles into 12
+
g,
-
colls.
The
g
+
-
ft
"
"t«g.
=
f
first
-
and are
I
irrelevant.
12
partitions
among them-
Hence there are
In
9.32
W=2
In
—
n
r»
W as
+
I(n,
ft)
As before, the condition that
\n(n
+&)-«, In n, -
{
this distribution
small changes Sn, in any of the individual n
YV of 8
change
in In
may be
written
W occurs when
In
-
1)!
-
ft]
be the most probable one
's
f
In (ft
not affect the value of
is
W.
that
If
a
changes by on,, the above condition
ii,
1)!
W
The number of ways
W
in
which the
W
particles can be distributed
is
the
product
("i
II
n
permits us to rewrite In
Hence,
W=
=
In nl
cell
possible distinguishably different arrangements of the n, indistinguishable
particles among the
ft cells.
3.31
formula
Stirling's
I
l>e
among
among
possible permutations
1)1
permutations of theft
affect the distribution
(n,
-
with the entire series
g, intervals,
objects, but of these the n,! permutations of the n, particles
1
themselves and the
selves
ft
g,
of the objects can
1
second none, the third one particle, the fourth three
and so on. There are
-
-
particles arranged in g, cells.
t
partitions separate the
J 1
+
consider a series of n,
note thai
+
ft
-
if
Wof Eq. 9.32
the
9.33
In
ft
Wnm = V
[In (n,
where we have made use of the
S In n
I)'
=
maximum,
represents a
+
ft)
-
In
nj Sn
t
=
fact that
- Sn
n
»|l(ft *" 1)'
of the
numbers of
We now
each energy.
+
(«i
so that
(it,
+
ft
ft)
-
arrangements of particles among the states having
assume that
As
in Sec. 9.3
1)
>
2
can be replaced by
W=2
(n,
+
ft),
and take the natural logarithm
[In (n,
S e,
+& _
)!
ra „,!
_
fa
(
& _
i)i]
series of n, Indistinguishable particles separated
by
tt
-
1 partitions Into g, cells.
•
fc
2
1
2
The
Eq. 9.33.
/.
—a
and the
latter
by
— /?
and adding
is
+
ft)
-
In n,
-
a
-
fc,] Sn,
in
=
brackets must vanish for each
Hence
•
In
1
form
=
result
[hi (n,
in the
Since the 8n,'s are indepetident. the quantity
"'
+
fe
-
-
a
fr,
=
()
particle
partition
1
number of indistinguishable particles
number of partitions = g^ — 1 == 11
number of cells = g = 12
t
302
S»,
by multiplying the former equation by
value of
I
=
and the conservation of energy, expressed
2
A
Sn,
I
to
FIGURE 9-6
incorporate the conservation of particles, expressed in the
form
of both sides of Eq, 9.31 to give
In
we
distinct
PROPERTIES OF MATTER
— » = 20
.
+
ii-
=
<?«$•
and
(
9.34
s*"
-
1
STATISTICAL MECHANICS
303
Substituting for
(9.22»
we
from Kq. 9.22,
fi
arrive Hi the liase-Kifintein tlistrifwtimi law:
935
=
n.
n
ft
e n e" nT
—
/
A
""IF
Bose-Einstein distribution taw
1
FIGURE 9-7
identical to
from the
ferent
faces
Surfaces
1!
and
I
I'
each other and are
and
are
dif
identical pair of sur-
II'.
BLACK-BODY RADIATION
9.8
v
Every substance emits elect roinagnette
radiation, the character «r
upon the nature and temperature of the substanee.
We
which depends
<d
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
all frequencies are present; the atoms in a solid are so
close
ous spectra in whieh
together that their mutual interactions result in
states indistinguishable
The
radiation.
This
the
same
rate as
one that absorbs
a
body
is
It
is
it
all radiation incident
upon
it,
regardless of frequency.
Such
show experimentally that a black body is a twlter emitter of
than anything else. The experiment, illustrated in Fig 9-7, involves
identical pairs of dissimilar surfaces.
1'
at the rate of g,
W/ni 2 while
surfaces
II
and, since
I
a.,
<
al a faster rate
No temperature
difference
At a given temperature the surfaces
I!'.
II
,
and IF radiate
and F absorb some fraction
at
tin-
<
1.
Hence
—
The
1
>
e%.
A black body at
point of introducing the idealized black lx>dy in a discussion of thermal
we can now disregard the
precise nature of whatever
is
since
black ixxiics behave identically.
1ms
a given temperature radiates energy
all
that
is
radiating,
In the lalwratory a black
body can
approximated by a hollow object with a very small hole leading
(Fig. 9-8).
Any
radiation striking the hole enters the cavity,
where
to
it
its
Interior
is
trapped
observed
is
and
1'
different rate
a, or the radiation falling
1, e,
than any other Ijody.
radiation
called a black htuhj.
and
=
<%,
ability to absorb
its
between surfaces
a
its
easy to
radiation
two
closely related to
at a constant temperature is in
surroundings and must absorb energy from llieni
emits energy. It b convenient to consider as an ideal body
thermal equilibrium with
at
arc not, so that
II'
e,
is
be expected, since a body
to
is
and
from a continuous band of pennilted energies.
a Ixjdy to radiate
ability of
II
multitude of adjacent quantum
radiate
rz
.
The
on them, while
and IF absorb some other
rate-
proportion] to
to 0,6,.
rt,e
Because F and
fraction «... Hence V absorbs energy from II at
and IF absorbs energy from L at a rate proportional
remain at the same temperature, it must l>e true that
a,
11'
FIGURE 9 8 A hole
the wan of a hollow
object
is
In
an excellent
approximation of a black
and
body.
a,
The
ability of a Ixxly to
radiation.
304
«,,
I>et
emit radiation
us suppose that
PROPERTIES OF MATTER
I
is
proportions] to
and F are black
its
ability to absorb
lx>dies, so that a,
=
I,
while
STATISTICAL MECHANICS
305
by reflection
Ixick
and
forth until
it is
absorbed.
emitting and absorbing radiation, and
(black-body mdittliott) that
we
it
is
The
are interested.
Experimentally
we
can sample
black-body radiation simply by inspecting what emerges from the hole. The
results agree with our everyday experience; a black Ixxly radiates move when
at
hot than
when
and the spectrum of a hot black body has its peak
a higher frequency than the peak in the spectrum of a cooler one. We recall
it is
it is
cold,
the familiar behavior of an iron bar as
temperatures: at
first
it
lxjenmes "white hot."
is
heated
progressively higher
to
glows dull red, then bright orange-red, and eventually
The spectrum
of black-body radiation
is
shown
in
Fig.
two temperatures.
9-9 for
The
it
Our
cavity walls are constantly
in the properties of this radiation
model of a black Ixxly
theoretical
some opaque
version, namely, a cavity in
and
V,
same
will lx* the
some volume
contains a large nmnlx*r of indistinguishable photons of various
it
particles that follow the Bose-Einstein distribution law.
The number
momentum between p and p + dp
which a photon can have a
of states
is
equal
may
twice the number of cells in phase space within which such a photon
to
fre-
Photons do not obey the exclusion principle, and so they are Bose
quencies.
g(p) in
as the laboratory
material. This cavity has
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
exist.
clockwise and circularly counter-clockwise). Hence, using the argument that led
to Eq. 9.16,
principles of classical physics are unable to account for the observed
black-body spectrum. In
that led
Max
nomenon.
1'Ianck in
fact,
1900
it
was
this particular failure of classical phvsics
to suggest that light emission
is
a
We shall use quantum-statistical mechanics to derive the Planck radia-
tion formula,
which predicts the same spectrum
as that
g(p)
=
dp
^
quantum phe-
momentum
Since the
of a photon
p
is
=
hv/c,
found by experiment.
'"
=
2
p dp
and
...
9.36
g(t')
=
dv
8wV
—
t— f
We
we
4
(/f
cJ
T=lSWt
must now evaluate the I-igrangian multiplier a
in
Eq, 9.35,
To do
ma\
gas molecules or electrons, photons
destrmei!.
lie ere.ifci! .mil
;iitil
the total radiant energy within the cavity must remain constant, the
of photons that incorporate this energy can change. For instance,
of energy
of
FIGURE
tra.
9 9
The
energy
In
hi'
energy
6 x 10" Hz
visible light
FREQUENCY,
306
PROPERTIES OF MATTER
v
two photons
Hence
2hi>.
2
the radiation depends
on,
/
can express by letting «
=
Substituting Eq. 9.36 for g, and
hi>
which
we
between
4 x 10 M
while
number
can be emitted simultaneously with the absorption of a single photon
distribution law (Eq. 9.35),
M
2 x 10
si.
Black. body spec-
spectral distribution of
only upon the temperature at the
body.
this,
note that the numlier of photons in the cavity need not be conserved. Unlike
v
and
p>
+
we
pa
fo
= &VV
The corresponding
volume
t ,
it
2 fin, as 0.
= in the Bose-Einstein
multiplies
and
find that the
T
n(i') di'
t
letting
o
number of photons with frequencies
dp in the radiation within a cavity of volume
are at the absolute temperature
9.37
since
for
-
between
walls
1
spectral energy density viv) dv,
in radiation
V whose
is
v aixl v
+
dv
in
which
frequency,
is
is
the energy per unit
given by
STATISTICAL MECHANICS
307
dv
f(c)
/ic»j(c)
=
dv
Both Wien's displacement law and the Stefan-Boll/.mann law are evident in
qualitative fashion in Fig. 9-9; the
p-
fcrft
9.38
Equation 9.38
Two
Planck radiation formula
_
in the various
curves
shift to
higher
temperature.
|
the Planck radiation fannida, which agrees with experiment.
is
interesting results can
be obtained from the Planck radiation formula.
wavelength whose energy density
In find the
in
dv
e hr/kT
c3
maxima
frequencies and the total areas underneath them increase rapidly with rising
terms of wavelength and
is
greatest,
we
FERMI-DIRAC DISTRIBUTION
"9.9
express Kq. 9.3fS
Fermi-Dirac
set
statistics
apply
to indistinguishable particles
which are governed
by the exclusion principle. Our derivation of the Fenni-Dirae distribution law
=
will therefore parallel that of the Base- Einstein distribution
()
dX
=
and then solve for A
\ max
,
We
each
cell (that
If
there are
obtain
filled
he
=
and
g, cells
—
(g,
ways, but the n
1.965
quantum
is,
The &
permutations of the
!
(
occupied by
lie
having the same energy
are vacant.
n,)
can
state)
cells
r,
and
which
is
more conveniently expressed
among themselves
as
The number
he
n„„lT —
v
is
law except that
n, particles,
in
among themselves
filled cells
—
now
most one particle.
can be rearranged
since the particles are indistinguishable and the (g,
vacant cells
at
n,)!
ii,
g
i
t
cells are
different
are irrelevant
permutations of the
are irrelevant since the cells are not occupied.
of distinguishable arrangements of the particles
among
the cells
therefore
496Sfc
= £wS98 X
9.39
10
3
ft!
n>K
-
n l'(gi
Equation 9.39
known
is
as Wien's displacement tan:.
It
quantitatively expresses
the empirical fact that the peak in the black-body spectrum shifts to progressively
shorter wavelengths (higher frequencies! as the temperature
Another
we can
result
the cavity. This
is
obtain from Eq. 9.3S
the integral of
the:
is
is
all
probability
W of the entire distribution of particles
i
w thin
i
U'
9.41
t
I
f(i»)
frequencies,
dv
In
_
far8**
'
I5c 3/r<
=
-*
which
In
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
T\
a conclusion
embodied
W=
Stirling's
al*
also proportional to
rtl
v
[In
g
!
-
(
In n,!
e=oT'
The value
of Stefan's constant a
a
308
=
5.67
X
W
PROPERTIES OF MATTER
*
sides.
-
In
(ft
-
n,)t]
formula
=
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,)]
is
in the Stefun-Hultzniann late:
For
this distribution to represent
maximum
of the individual n/s must not alter
9.40
the product
ft]
=11
Taking the natural logarithm of both
=
is
increased.
the total energy density
cnerg) tieusih over
The
"() !
9.43
« 1"
As before,
we
WBM =
2 [-In
n,
+
probability, small changes
fin,
in
any
W. Hence
In (g,
-
n,)]
5n,
=
is
VV/m 2 K'
take into account the conservation of particles and of energy
by
adding
STATISTICAL MECHANICS
309
— «2
6n
(
=
jw-j
9.47
Occupation index
and
called the occupation index of a state of energy
therefore the average number
The occupation index docs not
f,, is
of particles in each of the stales of that energy.
to Eq. 9.43, with the result that
S — In
9.44
[
+
ri(
In (g (
—
depend upon how the energy
n t)
—
«
—
fit,]
5n,
for this reason
=
it
levels of a system of particles are distributed,
provides a convenient
way
and
of comparing the essential natures
of the three distribution laws.
Since the Bn,'s are independent, the quantity in brackets must vanish Tor each
value of
i,
and so
In
The Maxwell-Boltzmann occupation index
\/e for each increase in
the factor
parameter
&~
"'
-
- fc =
«
!)
«, the ratio
energy levels
t
,
and
r^
f
(
is
a pure exponential, dropping by
While
of kT.
J\t t )
depends upon the
between the occupation indices/^) and
/?<',-)
of the
two
does not;
n,
i*-
n,1
9.45
-
=
1
= eV"
_ e Ur r,\/kT
9,48
This formula
ft
eV" +
is
useful because
distributions resemble the
1
us to
Substituting
when/(r)<
—
1
term
yields the Fermi-Dirac distribution law,
causes
above
ft
.,
a
1
e e"
/kT
+
Fermi-Dirac distribution (aw
1
The most important application
free-electron theory of metals,
m
photon gas, a
of the Fermi-Dirac distribution law
which we
shall
examine
in
is
in the
the next chapter.
it
to exceed the latter.
then permits
it
in
a
(
=-a
e
1,
signifying one
=
ft
e « e </kT
_
Bose Einstein
1
ft
ffe 'i'
formulas n,
is
of states that
kr
+
Fermi-Dirac
number of particles whose energy is
have the same energy e,. The quantity
the
PROPERTIES OF MATTER
and
g, is
kT,
whereas when
e,
< kT
former occupation index
which
is
index never goes
a consequence of
At high
rapidly near a certain critical
is
sufficiently small at all
energies for the
be unimportant, and the Fermi-Dirac
dis-
Maxwell-Boltzmann one.
THE LASER
Three kinds of transition involving electromagnetic radiation can occur between
in
an atom,
of
atom is initially in state i,
light whose energy is /ic = E,
If
the atom
If
the
There
fi
>
dropping
is initially in
a photon of energy
1
t
for the
The Fermi-Dirac occupation
tribution lie-comes similar to the
Max welt- Boltzmann
/kT
e'>
e
energy known as the Fermi energy.
two energy levels
«,
and the Bosc-Einstein occupation index
virtually all the lower energy states are filled, with the occupation index
9.11
310
and
the obedience of Fermi particles to the exclusion principle. At low temperatures
three statistical distribution laws are as follows:
number
distribution,
particle per state at most,
temperatures the occupation index
COMPARISON OF RESULTS
n
0,
denominator of the formula
in the
effects of the exclusion principle to
In these
the Bose-Einstein and Fermi-Dirac
approaches the Maxwell-Boltzmann one when
kT
the
The
1,
Maxwell-Boltzmann
determine the relative degrees of occupancy of two quantum states
In the case of a
9.10
factor
simple way.
P
9.46
Bolkmann
fa)
is
lower one
it
can be raised to state
—
hi'
£,.
this is
i
This process
the upper state
/,
it
is
/
/
(Fig. 9-10),
by absorbing a photon
called induced absorption.
con drop to state
i
by emitting
spontaneous emission.
also a third possibility,
the
of energy
hi>;
and an upper one
a
induced emission,
in
which an incident photon
causes a transition from the upper state to the lower one. Induced
STATISTICAL MECHANICS
311
'
a
\.
iV it iS i.
J
hi-
AW*
h»
WW*
hi'
I
3
\AA/-»-
t,r
*b*
ill
W
hf
w
induced
spontaneous
absorption
emission
« g
S-5.9
I*?J1
ggsgs
induced
emission
2«
2 "3
.so. &•%!
-e
FIGURE 9-10 Transitions between Iwo energy
(annus emission, and induced emission.
atom can occur by Induced
levels In an
absorption, soon
(0
emission involves DO novel concepts.
An analogy
is
harmonic
a
instance a pendulum, which has a sinusoidal force applied to
is
same
the
as
its
natural period of vibration.
If
it
oscillator, for
whose period
the applied force
is
exactly in
phase with the pendulum swings, the amplitude of the latter increases;
corresponds lo induced absorption of energy. However,
w
this
the applied force
if
.3 >
c—
is
&'£
180° out of phase with the pendulum swings, the amplitude of the latter decrctiM v
Since
this
hi- is
corresponds to induced emission of energy.
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.
of frequency
state
t>
upon a system
and an excited state
will lie little
in
is hi>.
Suppose we shim;
IU —«
> a
U
M
light
which the energy difference between the ground
With the upper
State largely unoccupied, there
stimulated emission, and the chief events dial occur will he absorp-
tion ot incident
photons by atoms
in the
ground
1 -
and the subsequent sponta-
state
.2
neous random rcradiation of photons of the same frequency, (A certain proportion of excited
atoms
will give
up
their energies in cotlisioas.)
2
g-H.1
S3
2
c
II
—
E
S
**
CJ
a
"4
~0
U)
Certain atomic systems can sustain inverted energy populations, with an upper
slale
occupied
three-level system in
that the transition from
=
(E s
upper
—
E[,)//i.
state.)
the ground state.
which the intermediate
it
lo the
The system can be "pumped"
i''
men
to a greater extenl
ground
lo the
slate
upper
Atoms
in slale 2
s
is
met as table, which means
—
forbidden In selection rules.
2 by radiation of frequency
have lifetimes of about 10
the ground state) almost at once.
I
is
state
I
% B
(Electron impacts are another wa\ to raise the system to the
emission via an allowed transition, so they
over
slale
w
Figure 9-11 shows a
fall
and
against spontaneous
to the metastable slate
Metastable states
against spontaneous emission,
" s
it is
E
may have
1
W
(or to
&"§ s
lifetimes of well
therefore possible to continue
tu>
pmnping until there is a higher population in state
If now we direct radiation of frequency v = (£, —
induced emission of pholous of
312
PROPERTIES OF MATTER
this
I
than there
£,,)/&
is
in slate 0.
on the system, Ihe
frequency will exceed their absorption since
w
03
o a -O
-c
more atoms
are in the higher state,
the net result will be an output of
and
radiation of frequency f that exceeds the input. This
(microwave amplification by stimulated emission of
is
the principle of the
radiation)
and the
mascr
laser (fight
amplification by stimulated emission of radiation).
The
radiated waves from spontaneous emission are, as might be expected,
random phase
incoherent, with
and time since there
relationships in space
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
is
mascr or
laser to
produce a completely coherent beam.
A
typical laser
a gas-filled tulic or a transparent solid that has mirrors at both ends, one of
them
partially transmitting to allow
pumping
light of frequency v
some
The
of the light produced to emerge.
directed at the active
is
medium from
the sides
of the tube, while the baek-and- forth traversal of the trapped light stimulate
emissions of frequency P thai maintain the emerging
beam
A wide
collimated.
variety of outsets and lasers have been devised: usually, the required inverted
energy distribution
is
obtained
less direct 1>
Can
(b)
=
the populations of the J
what temperature does
The N 2
N—O bond
molecule
8.
is
14
N
atom
is
The mass
2,32
10~ 26 kg.
X
bond length of
of the 18
(a)
300 K.
9.
The temperature
the relative
3,
numbers
and 4 energy
of the sun's chromosphere
of
levels.
If so, at
is
is
K?
1.126'
2.66
is
A and
X
an
1CT 2G kg
the quantum
number
(b) Plot rij/n,, versus
approximately 5000 K. Find
=
1, 2,
to take into account the multiplicity of
each
hydrogen atoms
Be sure
atom
What
of the most populated rotational energy level at 3(X*
J at
ever be equal?
states
chromosphere
in the
in the n
level.
10.
the tungsten filament of a light bulb
If
2900 K,
light
find the
is
equivalent to a black body at
percentage of the emitted radiant energy in the form of visible
with frequencies between 4
X
10 14
and 7
X
10 14 Hz.
than by the straightforward meeha-
sun
8
directly overhead.
is
of the earth's orbit
is
ture of the sun on the assumption that
surface temperature of the sun
Problems
is
equal
The problem
2.
1
Verify that the most probable speed of a molecule of an ideal gas
W/m 2
when the
The sun's radias is 6.96 X 10 m and the mean radius
1.49 X 10" m. From these data find the surface tempera-
Sunlight arrives at the earth at the rate of about 1,400
1.
1
msni described above.
1.
=3
N— N
linear with an
length of 1.191 A.
and that of the
2 and /
occur?
this
is
it
radiates like a black body. (The actual
slightly less than this value.)
of the black-body spectrum
was examined
at die
end of the
nineteenth century by Rayleigh and Jeans, using classical physics, since the notion
to
2.
V2JtT/m.
of electromagnetic quanta
Verify that the average speed of a molecule of an ideal gas
\/SkT/irm.
e(y)dv
3.
Find the average value of l/v
4.
What
in a gas
allying Maxwell-Boltzmann
proportion of the molecules of an ideal gas have components of
velocity in any particular direction greater than twice the most probable speed?
A
flux of
10
l
-
neutrons/
reactor. If these neutrons
sponding
6.
to
T
=
2
emerges each second from a port
have a Maxwell-Boltzmann
The frequency
of vibration of the
= 0.
v — 2
(b)
Can
the populations of the
what temperature does
7.
The moment
this
of inertia of the
PROPERTIES OF MATTER
molecule
and v
is
in the
1.32
X I0 N Hz.
and 4 vibrational
=
beam.
states at
3 states ever be equal?
to the
0, 1, 2, 3,
(«)
Find
5000 K.
If so, at
is
4,64
X
10 -48 kg-nf. (a) Find
and 4 rotational
=
Hirv'^kT
unknown. They obtained the formula
eb>
states at
300 K.
is it
13.
impossible for a formula with this dependence on frequency to
Show that, in the
B ay leigh -Jeans formula,
(fo)
limit of
i>
—*
At the same temperature, will a gas of
particles that
(
obey Bose-Einstein
Fermi-Dirac
statistics) exert
statistics),
0, the
Planck radiation law reduces
classical molecules,
a gas of bosom
or a gas of fennions (particles that
the greatest pressure?
The
least pressure?
Why?
14.
Derive the Stefan-Boltzmann law
in the following
way. Consider a
Camot
engine that consists of a cylinder and piston whose inside surfaces are perfect
reflectors
H, molecule
=
Why
be correct?
oliey
occur?
the relative populations of the /
314
H2
1, 2, 3,
a nuclear
energy distribution corre-
300 K, calculate the density of neutrons
the relative populations of the v
in
as yet
statistics.
(a)
5.
was
equal to
is
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
STATISTICAt MECHANICS
315
expansion during which the temperature drops by (IT and the pressure drops
by dp, an isothermal compression
p
—
al the
dp; and an adiabatic compression
temperature
to the original
T — dT and
THE SOLID STATE
pressure
temperature, pressure, and
volume. The pressure exerted by radiation of energy density u in a container
with reflecting walls
1
_
dW
is
n/3, and the efficiency of
Caraol engines
—
is
the work done by the engine during the entire cycle.
dT)/T, where
Q is
diagram and show
= aT\
that u
is
dW/Q =
the heat input during the isothermal expansion
efficiency of this particular engine in tenns of u
15.
all
(7"
where a
In a continuous helium-neon laser.
He
is
and
Calculate the
and T with the help of a p-V
a constant.
and Ne atoms are pumped to meta-
2tl.fi
and 20.88 eV above their ground states by electron
Some of the excited 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 fifi80-A photon is emitted in an allowed transition to
stable states respectively
1
impact.
I
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
10
A
solid consists of atoms, ioas, or
proximity
molecules packed closely together, and their
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
is
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.
atoms needed?
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 three-dimensional patterns. The
presence of long-range order
is thus the defining property of a crystal.
Other
long-range order in the arrangements of their constituent particles
properly be regarded as supercooled liquids whose stiffness is due to
solids lack
and may
an exceptionally high viscosity. Class, pitch, and many plastics are examples
of such amorphous ("without form") solids.
Amorphous solids do exhibit short-range order
The distinction between the two kinds of order
(rioxide (B
2
3 ),
which can occur
each case every boron atom
sent a short-range order.
hexagonal arrays, as
is
in
is
however.
nicely exhibited in boron
both crystalline and amorphous forms.
In
surrounded by three oxygen atoms, which repre-
In a B,,0.{ crystal the
in Fig. 10-1
in their structures,
.
which
is
oxygen atoms are present
in
a long-range ordering, while amorphous
BjOg, a vitreous or "glassy" substance, lacks this additional regularity. A conspicuous example of short-range 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
a
316
PROPERTIES OF MATTER
move.
Ihe analogy between an amorphous
means of better understanding lioth
solid
and a
liquid
states of matter.
is
worth pursuing
as
Liquids are usually
317
•
respectively.
boron a lorn
come
In an ionic crystal these ions
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
oxygen atom
of the cohesive forces present by the action of the exclusion principle, which
occupancy of higher energy
requires the
when
states
electron shells of different
atoms overlap and mesh together.
Table
how
10.
1
contains the ionization energies of the elements, and Fig. 10-2 shows
these energies vary with atomic number.
It
is
ionization energies of the elements vary as they do.
any of the
The
w
(a)
FIGURE
Two-dimensional representation of B ;0,. <a> Amorphous BjO.,
10-1
(0) Crystalline
order,
GO.
are both
fluids,
also
have much
packing
is
common. The
in
which suggests
and gases
ihe two
liquids
is
become
and
Furthermore, X-ray diffraction indicates that
at
any
many
liquids
instant, quite simitar to those of
solids
usually close to
that the
degree of
an inference supported by the compressibilities of these
similar,
range structures
critical point
density of a given liquid
that of the corresponding solid for instance,
has a single
I
.s
the
the
electron outside a closed subshell.
electrons in the inner shells partially shield the outer electron from the
+ Ze, so that the effective charge holding the outer
just +e rather than +Ze. Relatively little work must
atom
is
electron to
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
However, from a microscopic point of view,
indistinguishable.
short-range
eithlbrti only
solids; after all, liquids
and at temperatures above the
metals of group
why
For instance, an atom of
nuclear charge
exhibits king-range order as well.
regarded as resembling gases more closely than
alkali
not hard to see
have
stales.
(Fig, 10-3)
and the weaker
is
the electrostatic force on
energy generally decreases as
energy from
nuclear charge while the
There are two electrons
this is
we go down any group. The
across
left to right
it;
any period
number
in the
is
accounted
why
the ionization
increase in ionization
for
by the increase
in
of inner shielding electrons stays constant.
common
inner shell of period 2 elements, and
the effective nuclear charge acting on the outer electrons of these atoms
is
definite short-
amorphous
solids
except
that the groupings of liquid molecules are continually shifting.
FIGURE 10-2
The
variation of Ionization energy with atomic
number,
30,
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
vary
amorphous
in strength.
solid lacks long-range order, the
When
the solid
is
bonds between
its
molecules
heated, the weakest bonds rupture at lower
temperatures than the others, so that
it
softens gradually.
In a crystalline solid
between long-range and short-range order (or no order at all)
involves the breaking of bonds whose strengths are more or less identical, and
the transition
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
318
to the latter,
PROPERTIES OF MATTER
and they thereupon l>ecome positive and negative ions
30
40
50
60
70
80
90
ATOMIC NUMBER
THE SOLID STATE
319
Table 10.1.
IONIZATION ENERGIES OF THE ELEMENTS.
In
2
II.-
84.6
i.'i.n
3
4
S
6
7
Li
Be
li
f:
N
5.4
0.3
N.3
113
8
*
M
F
Mb
I4.S 1.3.6 17-1 2l.fi
II
12
1.1
14
15
16
IT
IS
N»
Mg
Al
Si
P
S
CI!
Ar
5.1
7.0
rut
s.i
n.o
32
io.
i
W
i
15.8
19
20
21
22
23
24
25
26
27
28
28
:»0
31
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
55
50
72
73
74
75
70
77
78
79
80
61
82
83
84
85
88
Cs
l)»
111
H,
W
Ha
Os
It
Pi
An
lit:
Tl
I'h
Hi
I'o
Al
Rn
3,9
3.2
5,5
T.'i
Mi
7.8
8.7
9.2
8.0
\>.2
0.1
7.4
7..1
S.4
-
10.7
87
88
Fr
Ri
—
5.3
III.
I
20
10
58
50
00
La
Co
Pr
Nd
">.(>
(>.')
is
[>..)
Atomic
radii of
(he elements.
01
Pin
02
03
04
65
67
60
68
68
70
electron.
71
SO
81
\c
Hi
_
7.11
therefore
70
corresponding to different crystal
+(z
—
82
93
Pa
I!
Np
—
6.1
—
2)e.
etc.,
atoms
is
Kit
Ccl
Til
l>v
11,.
Ec
Tin
VI.
I.u
electron affinities
5.7
62
6.7
B.8
—
6,1
5.8
6.2
5.0
increase going from left lo right across any period.
94
90
97
98
99
Pii
Am Cm
HL
U
5.1
6.0
____
The outer
held to
US
— — —
100
K*
its
ID!
Mil
I'm
electron in a lithium
102
atom
In general.
is
quite
difficult,
The experimental determi-
and those
for only a
few elements
are accurately known.
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
\u
ionic
beryllium, boron,
parent atom by effective charges of
+ 2e,
-f 3e,
Table 10.2.
ELECTRON AFFINITIES OF THE HALOGENS.
most electrons, are halogen atoms, which tend to complete their outer p suhshells
by picking up an additional electron each.
defined as the energy released
halogens.
l.w
N'n
held to the atom
is
in
103
etc.
when an
element. The greater the electron
PROPERTIES OF MATTER
affinities of the
decrease going down any group of the periodic table and
5.0
At the other extreme from alkali metal atoms, which tend to lose their outer-
320
radii,
Sm
by an effective charge of +e, while each outer electron
is
Several have two
Table 10.2 shows the electron
nation of electron affinities
68
+4e,
60
i tinctures.
57
carbon,
50
ATOMIC NUMBER
FIGURE 103
1
40
30
affinity,
The
limine
3,45
electron affinity of an element
Clilurmc
3.61
an atom of each
Bniiiiiiii"
3.36
Iodine
3.06
electron
the
in electron volts.
is
more
added
tightly
to
bound
is
!')
the added
THE SOLID STATE
321
ion. Sodium, with an ionization energy of 5.14 eV,
and chlorine, with an electron affinity of -1.61 eV,
When
a
Na +
the attractive electrostatic force
i
same
ion and a Cl~ ion are in the
nniliiion that a stable
is
an example of the former
is
an example of the
fatter.
and are free
move,
vicinity
to
between them brings them together.
molecule of NaCI result
is
simply that the
total
The
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
In general, in an ionic crystal each ion
opposite sign as can
fit
closely,
which leads
sizes of the ions involved therefore
Two common
and
is
is
met by NaCI.
surrounded by as many ions of the
to
maximum
shown
types of structure found in ionic crystals are
In a sodium chloride crystal, the ions of either kind
10-5.
of as being located at the comers and
with the
of culies,
Na
+
and CI
The
stability.
relative
govern the type of structure that occurs.
at the
in Figs. 10-4
may be
Each
ion thus has six
nearest neighbors of the other kind, a consequence of the considerable difference
in the sizes of the
cubic.
ion
is
A
different
+
Na and CI"
arrangement
is
ions.
Such a structure
found
in
located at the center of a cube at
is
cesium chloride
called focfrcentered
crystals,
whose corners are
ions of the other
Each ion has eight nearest neighbors of the other land in such a Imdijcenlered vulrir structure, which results when the participating ions are compara-
The
is
of a
The body-centered cubic structure
model of CsCI crystal.
CsCI
crystal.
The coordination
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
even though molecules as such do not
The
where each
kind.
ble in
(a)
(b) Scale
thought
centers of the faces of an assembly
assemblies interleaved.
FIGURE 10-5
number is 8
NaCI.
The
crystal)
exist in the crystal
is
the
Na +
ion
principal contribution to the cohesive energy of an ionic crystal
electrostatic potential
in
sodium chloride
in the case of a
Its
energy
KOUlomb of the ions.
v*
us consider an
l.et
nearest neighbors are six CI" ioas, each one the distance r away.
potential energy of the
Na +
ion due to these
six
CI
ions
is
therefore
size.
cohesive energy of an ionic crystal
by the formation of the
crystal
is
the energy that
would be
tilierated
4 7rf l) r
from individual neutral atoms. Cohesive energy
The
FIGURE 10-4
(a)
The Face-centared cubic structure of 3 NaCI
is 6. (6) Scale model
ber of nearest neighbors about each Ion)
crystal.
of
NaCI
next nearest neighbors are 12
Na*
ions,
each one the distance \/2
Tha coordination number (num-
since the diagonal of a square r long on a side
crystal.
of the Na* ion due to the 12
V,
Na*
ions
is
v2
r.
The
r
away
potential energy
is
12e 2
= +
4KF a \/2r
When
the
summation
infinite size, the result
Veouio
™b
is
continued over
all
the
+
and
—
ions in a crystal of
is
-"w(
6
"^i
+ '")
= - 1.748
•k7f„r
10.1
=:
— «-
2
Coulomb energy
of ionic crystal
»"<„ r
(b)
THE SOLID STATE
322
PROPERTIES OF MATTER
323
(This result holds for the potential energy of a Cl~ ion as well, of course.) The
quantity a is called the Mtith-tun« 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
that of cesium chloride (Fig. 10-5), for instance, have a
=
1.763,
structures like that of zinc blende (one form of 7.nS) have a
crystal Structures
The
have Madelung constants that
lie
between
m
1.638.
and
1.0
A
really precise
n
=
like
and those with
a
9,
knowledge of n
V would change by
In
an NaCI
=
1.748 and n
=
only
4ire
r \
nf
x
It*"
N-mVC a X
9
=
=
r"
which corresponds
positive,
is
and the dependence on r~ n {where n
to a repulsive interaction,
a large number) corresponds to a short-
is
range force that increases rapidly with decreasing intemuclear distance
total potential
ions
V
energy
of each ion due to
its
interactions with
all
V=
r.
The
+ K,
B
ate'
r
when
=
r
*
8
=
r n-
V
is
a
minimum by
definition,
and the
v=
io.s
It
is
=
to
X
10
lfl
C) 2
/
_
IV
m
more than
once, only half
<>1
tins potential
the energy needed to transfer an electron from
a CI atom to yield
a
Na + — CI"
ion pair. This electron transfer
the difference between the +5.14-eV ionization energy of
is
Na and Aw
thus
=
(-3.99
+
0.77)
eV/atom
=
-3.22 eV/atom
empirical ligure for the cohesive energy of an ionic crystal can be obtained
its
exchange energy. The
heat of vaporization, dissociation energy, and electron
result for
Nat
Tl
is :?.2-S
Most ionic
"'
agreement with the
eV. in close
force varies quite sharply with
to the strength of the
They are
I
wauls between their
usually brittle as well,
since the slipping of atoms past one another that accounts for the ductility of
is
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.
result
r:
owing
1
^L(i-l)
The average
solids are hard,
constituent ions, and have high melting points.
ii
energy
isnsi), which means
At the equilibrium ion spacing, the
mutual repulsion due to the exclusion principle
(as distinct
10.3
COVALENT CRYSTALS
that the repulsive
the ions are "hard" rather than "soft" and
strongly resist (wing packed too tightly.
PROPERTIES OF MATTER
(1.60
1
possible to evaluate the exponent n from the observed compressibilities
of ionic crystals.
K)- "
1
J
not count each ion
I:
eobetfvs
metals
total potential
x
X
calculated value.
r
(j
10'"
X
1.748
-7.97 eV
from measurements of
B-l
r
'0
•1tti
is
and
nB
B -
10.4
is
Hence
r„.
'''r= r„
<*Ve
-1.27
We must also take into account
Na atom
An
ae-
we may
per atom
Ai the equilibrium separation r of the ions,
=
2.8J A, Since
— 3,61-eV 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
ijmtiiH'i*
4ro r
is
of the crystal.
energy
10.3
so (dV/dr)
Because
a
V.CMiHiHiib
ions
energy, or —3.99 eV, represents the contribution per ion to the cohesive energy
the other
therefore
is
between
1.8.
niniKiv.'
Vrepi|hlve
11 percent.
10 instead of
energy of an ion of either sign
2.S1
sign of
=
n
Simple
the form
The
if
percent.
1
9, the potential
potential energy contribution of the repulsive forces due to the action
10.2
324
evidently not essential;
is
crystal, the equilibrium distance ru
of the exclusion principle can l)e expressed to a fair degree of approximation
in
by about
repulsion lietweeu like ions) decreases the potential energy
from the electrostatic
The cohesive
forces in covalent crystals arise from the presence of electrons
between adjacent atoms. Each atom participating
an electron to the
bond and
in a covalent
bond contributes
these electrons are shared by imth atoms rather
THE SOLID STATE
325
always possible to
classify a particular crystal as Ixjing
AgCl, whose structure
is
wholly ionic or covalent:
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.
VAN DER WAALS FORCES
10.4
AM atoms and
argon
—
exhibit
Wauls
liquids
forces.
molecules
—even
inert-gas
atoms such
as those of
helium and
weak, short-range attractions for one another due to van der
These forces arc responsible
and the freezing of
for the condensation of gases into
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
FIGURE 10-6 (a) The
of diamond crystal
The coordination number
tetrahedral structure of diamond.
is
(b) Scale
4.
model
so
on
also arise
two molecules
from van der Waals
r
apart
is
forces.
The van der Waals attraction between
proportional to r~ 7
,
so that
it
is
significant only for
molecules very close together.
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 10-6 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. 8-16).
is
silicon,
germanium, and
silicon carbide; in
surrounded by four atoms of the other kind
as that of
diamond.
in
All covalent crystals arc
permanent
is
the
SiC each atom
same tetrahedral structure
hard (diamond
is
all
(called polar molecules) possess
is
the
H2
molecule, in which
atom makes
that
end of the
Such
as in Fig. 10-7,
A
is
and
polar molecule
a permanent dipole
in this orientation the
is
molecules strongly attract each other.
also able to attract molecules
moment. The process
is
which do not normally have
illustrated in Fig. 10-fS: the electric
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.
ordinary liquids, behavior which reflects
Cohesive energies of 3 to 5 eV/atom arc
the strength of the covalent bonds.
which
many molecules
moments. An example
molecule more negative than the end where the hydrogen atoms arc.
the hardest
the industrial abrasive carborundum), have high
melting points, and are insoluble in
typical of covalent crystals,
electric dtpole
the concentration of electrons around the oxygen
field
substance known, and SiC
begin by noting that
molecules tend to align themselves so diat ends of opposite sign are adjacent,
Purely covalent crystals are relatively few in number. In addition to diamond,
some examples are
We
the
same order of magnitude as the cohesive
FIGURE 10-7
Polar molecules attract each other.
energies in tonic crystals.
There are several ways
to ascertain
whether the bonds
pound of an element from group I or
group VI or VII exhibits ionic bonding
coordination
number of
in a
given nonmctallic,
predominantly ionic or covalent. In general, a com-
iioiiiiiolecular crystal are
11
of the periodic table with one from
in the solid state.
the crystal, which
is
the
Another guide
is
the
mimtwr of nearest neighlmrs
A high coordination numlier suggests an tonic
how an atom can form purely covalent Irauds with
about each constituent particle.
crystal, since
six
it is
other atoms
eight other
hard to see
(as in
atoms
coordination
a face-centered cubic structure like that of NaCl) or with
(as in
number of
a body-centered cubic structure like that of CsCl).
4,
however, as
in the
with an exclusively covalent character. To
326
PROPERTIES OF MATTER
diamond
Ix; sure, as
structure,
is
A
compatible
with molecules,
it is
tiot
THE SOLID STATE
327
key factor here
the small effective size of the poorly shielded proton, since
is
electric forces vary as
-.
r
Water molecules are exceptionally prone
four pairs of electrons around the
outward
as
atoms are
at
two of these
Kach
state,
FIGURE 10
and
large
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.)
at
am/
average, the electrons themselves are in constant motion and
tlte
giiY-n instant
one part or another
of the
(Fig. 10-10).
molecule has an excess of them.
Hydrogen
which accordingly exhibit localized positive
two vertexes
exhibit
somewhat more
diffuse negative
in
two of these bonds the central molecule provides the bridging
other two the attached molecules provide them. In the liquid
the hydrogen bonds between adjacent H.,0 molecules are continually being
to
thermal agitation, but even so at any instant the
in definite clusters.
and stable and constitute
The
In the solid state, these clusters are
ice crystals.
characteristic hexagonal pattern (Fig. 10-1
1)
of an ice crystal arises from
the tetrahedral arrangement of the four hydrogen bonds each
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
hybrid orhitals that project
in the
molecules are combined
The
vertexes,
broken and reformed owing
Polar molecules attract potarizable molecules.
8
&•{)'*
molecule can therefore form hydrogen bonds with four other
ll 2
H a O molecules;
protons,
form hydrogen bonds because the
though toward the vertexes of a tetrahedron
charges, while the other
charges.
to
O atom occupy
crystals
H2
molecule can
With only four nearest neighbors around each molecule, ice
have extremely open structures, which is the reason for the exceptionally
participate
in.
low density of
ice.
in the liquid state,
Because the molecular clusters are smaller and
less stable
water molecules on the average are packed more closely
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
rj
always having opposite sign (Fig. 10-9) and so always causing an attractive force.
This kind of force
suggested
its
is
named
Dutch
after the
existence nearly a century ago to explain observed departures from
the ideal-gas law; the explanation of the actual
is
more
van der Waals, who
physicist
mechanism of the
force, of course,
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
and
points
eV/atom
little
in solid
hydrogen (mp
- 183
An
8
o
mechanical strength.
argon (melting point
— 25S)°C),
and
0.1
Cohesive energies are low, only 0.08
— 1W)*C),
eV/molecule
(M)l
eV/molecule
in solid
in solid
methane, C1I, imp
FIGURE 10-9
On the average, nonpolar molecules
have symmetric charge distributions, but at any Instant the distributions are asymmetric.
especially strong type of van der
fluctua-
tions in the charge distributions of nearby molecules
are coordinated as shown, which leads to an
between them whose magnitude
1/r.
tive force
tional to
C).
The
is
atir.it-
propor-
++ ++
-
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.
molecule with a localized positive charge which can
link
tration of negative charge elsewhere in another molecule of the
328
PROPERTIES OF MATTER
The
result is a
up with the concen-
same
kind.
The
o o
THE SOUD STATE
329
together than are ice molecules, and water has the higher density: hence ice
4 "C as large
floats. The density of water increases from 0°C to a maximum at
clusters of
in the
molecules are broken up into smaller ones that occupy
aggregate; only past
manifest
10.5
H zO
4°C
less
space
does the normal thermal expansion of a liquid
a decreasing density with increasing temperature.
itself in
THE METALLIC BONO
The underlying theme of the modern theory of metals is that the valence electrons
3
of the atoms comprising a metal are
hybrid orbital^
common
a kind of "gas" of free electrons pervades
to the entire aggregate, so that
it.
The
interaction
between
electron gas and the positive metal ions leads to a strong cohesive force.
3.
this
The
presence of such free electrons accounts nicely for the high electrical and thermal
conductivities, opacity, surface luster,
To be
sure,
interior
no electrons
with
total
any
in
freedom.
other particles present, and
solid,
All of
when
and other unique properties of metals.
even a metal, are able
to
move about
its
them are influenced to some extent by the
the theory of metals
is
refined to include these
complications, there emerges a comprehensive picture that
is
in excellent
accord
with experiment.
Some
FIGURE
10-
10
In
an
HO
molecule, the four pairs of valence electrons around
insight into the ability of metal
atom and one each by the H atoms) oc
the oxygen atom (sin contributed by the
cupy lour *j/ hybrid orbital* that form a tetrahedral pattern. Each H.0 molecule
can form hydrogen bonds with four other HO molecules.
Iwlh members of group
\.i
of
H.O molecules.
1
of the periodic table.
electrons with opposite spins, the
be present. The
Top view of an ice crystal, showing the open hexagonal arrangement
Each molecule has four nearest neighbors to which It is attached by hydrogen bonds.
to lx>nd together to
form crystals
covalent Irond. Let us compare the bonding processes in hydrogen and in lithium,
L
FIGURE 10-11
atoms
can be gained by viewing the metallic bond as an unsaturated
of unlimited size
1
2
molecule
is
of
K
contains two
electrons that can
therefore saturated, since the exclusion principle
requires that any additional electrons
attachment of further
A H a molecule
maximum number
be
in states
of higher energy and the stable
H atoms cannot occur unless their electrons are in
Superficially lithium might seem obliged to behave
in a similar
Is states.
way, having the
2
electron configuration ls 2s. There are, however, six unfilled 2j> states in every
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
Li
no limit to the
of Li atoms that can join together in this way, since
number
lithium forms body-centered cubic crystals (Fig, 10-5) in which each
eighl nearest iit-iuhlmrs.
into bonds,
of
each l»nd
two electrons
being saturated;
330
PROPERTIES OF MATTER
With
only one electron per
atom has
to enter
involves one-fourth of an electron on the average instead
as in ordinary covalent bonds.
this
atom available
is
true of the bonds
Hence the bonds
are far Iroin
in other metals as well.
THE SOLID STATE
331
One consequence
of the unsaturated nature of the metallic
that the properties of a
bond
on the proportion of each kind of atom, provided
atomic proportions found
tion, in contrast to the specific
covalent solids such as
the fact
13
=
composi-
in its
in ionic solids
and
in
SiC
'-
only spends a short time between
it
cannot remember
it
is
move on
just as likely to
The valence
at all.
am
pair
fit
Speak) which of the two ions
(so to
t
E
to a liond that
it
The
Li* ions.
its
_c
£
||
= £
1
3
=
-
S
.=
£
11
o
-s
a
ill
-i
and
>
parent ion
manner quite
electrons in a metal therefore liehave in a
IS
« a
£
fi
I
B
electron
really belongs to,
does not involve
J
=
« 4
°& u
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
I
1}
*1
1 a
Thus
their sizes are similar.
an alloy often vary smoothly with changes
the characteristics of
is
mixture of different metal atoms do not depend critically
a
2
similar to that of molecules in a gas.
As
in the case of
energy
any other
why
note that, because of the proximity of the ions, each valence electron
Hence the
an isolated atom.
crystal than in the atom,
1 5
exist as separate
and
it
would be
if
potential energy of the electron
it
is
this
it
potential energy
The
increased.
is
reduced
in
1
l?3
C u <
=
i
** w
is less in
i
the
11E
is
S
Whereas
5
J:
"5
the electron
a metallic crystal, the electron kinetic energy
free electrons in
U
is
JB
is
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
f
2
lithium as an example, only eight valence electrons in an entire Li crystal could
occupy n
=
2 quantum
states of such great
happens
are
all
is
less
states,
slightly altered
that consists of as
with the
rest
being forced into higher and higher
energy as to disrupt the entire structure.
The valence energy
dramatic.
by
their interactions,
many
valence energy levels in
to
The
,
is
L
l>cing
a positive quantity,
its
I'cnni
4,72 eV, and the average
kinetic energy of the free electrons in metallic lithium
is
2.6 eV. Since electron
increase in the metal over
what
it
was
.
(V,
in separate
atoms leads
to a repulsion.
Metallic bonding occurs
when
the attraction between the positive metal ions
ci
-
S3
J
.
-i
1
332
PROPERTIES OF MATTER
a
a
i
I
atoms
free electrons accord-
some maximum n F called the
energy; the Fermi energy in lithium, for example,
is
actually
and an energy hand comes into
the atoms in the crystal.
ingly range in kinetic energy from
kinetic energy
What
levels of the various metal
closely spaced energy levels as the total muiilicr of
all
i
'<
an
is
decrease in potential energy that
another factor to be considered, however.
is
-
I
Iwlonged
responsible for the metallic Imud.
There
.'i
5
reduction in energy occurs in a metallic crvstal,
this
the average closer to one nucleus or another than
to
—
metal atoms cohere liecause their collective
lower when they are bound together than when they
is
atoms. To understand
we
solid,
5
•I
•/.'
1
"3
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
For
elements are nearly
this reason the metallic
of the periodic table.
Some elements
both metallic and covalent
the metal "white tin" exists
crystals.
whose
Gray
and white
tin
Found
is
groups
in the first three
are right on the tjordcr line
Tin
tin
exists
and may form
Above 13.2°C
Below
same
as
that
the
a notable example.
whose structure
is
-10
are quite different substances; they have
the respective densities of 5.8 and 7.3
semiconductor whereas white
all
atoms each have six nearest neighbors.
13.2°C the covalent solid "gray tin"
of diamond.
in the potential energy.
commensurate drop
g/cm 3
,
for instance,
and gray
tin is
a
has the typically high electric conductivity of
tin
a metal.
10.6
THE BAND THEORY OF SOLIDS
The atoms
almost every crystalline
in
solid,
whether a metal or
not, are so close
together that their valence electrons constitute a single system of electrons
common
The
to the entire crystal.
exclusion principle
is
-20
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
many
Since there are as
composed of myriad separate
levels very close together.
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
extent to
of energy bands, the gaps that
which
the)' are
filled
behavior of a solid but also
There are two ways
to look into
to consider the origin of
what happens
occur between them, and the
have important bearing on other of
to the
energy
brought closer and closer together to fonn a
in this
may
by electrons not only determine the electrical
energy bands. The simplest
levels of isolated
solid.
atoms
-30-
properties.
its
as
We shall introduce
is
they are
the subject
way, and then examine some of the consequences of the notion of energy
bands.
t-aler in the
restrict ions
chapter
we
shall
imposed by the periodicity of a
crystal lattice
5
10
15
INTERMUCLEAR DISTANCE. A
on the motion of
electrons, a more powerful approach that provides the basis of
modem
3.67
analyze energy bands in terms of the
much
of the
FIGURE 10-12
The energy levels of sodium atoms become bands as their intemuclesr
The observed internuclear distance In solid sodium is 3.67 A.
distance decreases.
theory of
solids.
Figure 10-12 shows the energy levels in sodium plotted versus intcrnuclear
distance.
The
3.1
into a band; the
334
level
2p
is
the
firs!
occupied level
in the
sodium atom to broaden
level does not begin to spread out until a quite small
PROPERTIES OF MATTER
THE SOLID STATE
335
solid
I
overlapping
energy bands
sodium
sodium atom
3a
FIGURE 10-13 The energy bands
)
In a solid
may
overlap,
FIGURE 10 15
Energy levels
the corresponding situation
scale).
See
Fig.
in
the sodium
in solid
atom and
sodium (not to
2p-
1012.
2s-
Is
inlemuclear separation. This behavior
sub-shells of
sodium atoms interact
average energies
The
as
actual
it
in
the
3/j
and
an electron
The
which case
in a solid
in
bands drop at
in solid
first,
sodium
is
implying attractive
indicated, and
can possess only those energies that
bands
may
may
lx>th
Figure 1(1-15
by
its
outer
within these energy
overlap, as in Fig. HI- 13, in
The
is
shell.
po.ssess.
Such
intervals
electrical behavior of a crystalline solid
energy-band structure and by
how
is
these bands arc
solid sodium.
A
sodium atom has a single 3s electron
This means that the 3s band in a sodium crystal
is
occupied, since each level in the band, like each level in the atom,
When
that are onty partially filled.
Figure 10-16 is a simplified diagram of the energy bands of diamond There
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
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
it
field.
An
electron
moving through
an electric Held
is
set
only half
it
loses
undergoes a
it
gained from any electric
intensity
needed
to cause
and
the collision.
a current to flow in sodium. Diamond
a
very poor conductor of electricity and
a
gap separates the top of a
solid
lorbidden band in silicon,
is
however
is
only
Ll'eV wide. At low temperatures
if
energy bands.
Energy bands
In
therefore
resembbng that of diamond, and, as in diamond,
energy band from a vacant higher band. The
A forbidden band separates non overlap ping
FIGURE 10-16
is
accordingly classed as an insulator.
filled
forbidden
FIGURE 10-14
field in
An electric-field intensity of (i X 10s V/m is necessary if an electron is to gain
eV in a path length of I0" 8 m, well over 10 ln times greater than the electric
able to
up across a piece of
of the energy
collision
— I0~"m,
6
licit!
is
much
Silicon has a crystal structure
band
a crystal
with an imperfection in the crystal lattice an average of everv
a simplified diagram of the energy levels of a sodium atom
contain two electrons.
and the moving electrons constitute an electric current. Sodium is therefore a
good conductor of electricity, as are odier crystalline solids with energy bands
energy, since
by electrons.
and the energy bands of
in its
an atom, and
sodium, electrons easily acquire additional energy while remaining in their
original energy band. The additional energy is in the form of kinetic energy,
is
Ihem represent energies which their elections can not
filled
fall
in
not overlap (Fig. 10-14), and the intervals between
arc called forbiiitlen bonds.
normally
forces.
corresponds,
electrons have a continuous distribution of permitted energies.
In other solids the
determined
it
The
energy.
a solid correspond to the energy levels
various energy bands in a sobd
its
order in which the electron
atoms are brought together.
minimum average
should, to a situation of
The energy bands
bunds.
3s
intemudear distance
reflects the
as the
-
diamond
li
n
fleV
(not to scale).
28
Is
336
PROPERTIES OF MATTER
THE SOLID STATE
337
silicon
to
jump
diamond
better than
is little
a small proportion of
its
conductor, but at room temperature
as a
1
electrons have sufficient kinetic energy of thermal origin
the forbidden hand and enter the energy band above
are sufficient to permit a limited
These electrons
it.
when an
of current to flow
amount
J
FIGURE 1018
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
five electrons in their
atoms have
few arsenic atoms
outermost
shells,
atom replaces a
silicon
atom
are incorporated in eovalent bonds with
requires
23 and
its
The
nearest neighbors.
energy to be detached and move about
little
When
its
fifth
is
semiconductor.
donw keels, and
called an n-type semiconductor because electric current
in
the substance
carried by
is
it
alternatively incorporate gallium atoms in a silicon crystal, a different
we
effect occurs.
Gallium atoms have only three electrons
configuration
is
2
4p,
*
THE FERMI ENERGY
10.7
We
shall
DMtal,
now
in their
outer
shells,
electron structure of the crystal.
enter a hole, hut as
an electric
electrons
here
is
does
so, it
electron needs relatively
leaves a
new
hole in
its
little
energy to
former locution.
the energy
10.6
move toward the anode by
successively
filling holes.
r.j
is
",
-
of positive charges since they
substance of
as zinc,
this
kind
is
move toward
termed acceptor
we see
flow of current
is
the negative electrode.
letels, just
holes.)
In the energy-band
that the presence of gallium provides energy levels,
above the highest
behind them,
occupy these
levels leave
winch permit
electric current to flow.
in the
filled
baud.
formerly
Any
filled
law
for the
number
of electrons
n with
t
1
more convenient
to consider a continuous distribution of electron energies
than the discrete distribution of Eq. 10.6, so that the distribution law becomes
When
called a p-type semiconductor. (Certain metals, such
conduct current primarily by the motion of
diagram of Fig. 10-18
metal has a Fermi -Dirac distribution of energies
distribution
a
eV JkT +
g(ej(fe
n(e) dt is
eV /kr +
a trace of gallium,
The
in a
The Fermi-Dirac
(Sec. 9.9).
10.7
applied across a silicon crystal containing
closely into the properties of the free electrons in a
whose
conveniently described with reference to the holes, whose behavior
like that
A
field is
it
An
more
look
Electrons are Fermi particles since they obey the exclusion principle,
and hence the electron gas
their presence leaves vacancies called holes in the
and
filled
band
ft is
4.v
band
1
negative charges.
If
forbidden
electron
in the crystal (see Probs.
provides ener^v levels
conduction to take place. Such levels are termed
jj-type
I
^^
levels
an
electrons
As shown in Fig. 10-17, die presence of arsenic as an impurity
just below the band which electrons must occupy for
24).
acceptor
impurity y
in
Arsenic
in a silicon crystal.
a silicon crystal, four of
in
acceptor levels
while silicon atoms have four,
3
1
2
{These shells have the configurations 4s 4p and Zs'^p' respectively.)
arsenic
trace of gallium In a
the normally forbidden band, producing a
field is applied.
of impurity. Let us incorporate a
A
silicon crystal provides
electric
empty
band
electrons thai
To
find g(r) de, the
energies between
number
and
f
+
e
1
of
quantum
we
de,
gas involved in black-body radiation.
are
two
possible spin states, tn,
bling the
number
states available to electrons
with
use the same reasoning as for the photon
The correspondence is exact because
= +% and ms = — %,
there
for electrons, thus dou-
of available phase-space 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
baud, vacancies
momentum we found
in Sec. 9,8
that
gC»)
SirVp 2 dp
dP =
1 empty
J
donor
impurity
^
«
FIGURE 1017
r
forbidden
_
,
band
levels
f
PROPERTIES OF MATTER
A
trace of arsenic in a
silicon crystal provides
donor levels
p" dp
In
=
Ji.fype
semiconductor.
with the result that
filled
bund
(2m%) l/2 de
the normally forbidden band, producing
an
]
338
For nonrelativistic electrons
band
io.S
g
,vj
e) de
=
8\/2 W Vm 3/2
-
,,
hJ
,_
e 1/2 de
,
THE SOLID STATE
339
t>,
where
principle:
it
T>0K
r = ok
The next step is to evaluate the parameter «. In order to do this, we consider
9.10.
the condition of the electron gas at low temperatures. As observed in Sec.
7" is small is 1 from r until near the Fermi energy
the occupation index when
10
0. This situation reflects the effect of the exclusion
tan contain more than one electron, and so the minimum
drops rapidly to
no
states
o.s
one in which the lowest states are
energy configuration of an electron gas is
filled and the remaining ones are empty. If
we
set
FIGURE 1019
10.9
kl
the occupation index becomes
10.10
n
in
is
e{ '
•&**
+
accord with the exclusion principle.
f{t)
At
T=
K,
If
when
'/
2
e
=
Fermi energy
rf
by
filling
up
-
*
N
C'r
g(e) dt
more
to
to the
*> by
free electrons,
=
0.
The
we
can calculate
in
its
3/i
of electrons that
340
fF
_ M.
(
tr
that
have
this
X„
=
Avogadro's number
p
=
density of copper
it
=
atomic mass of copper
=
IP 2 " atoms/kmol
X
energy,
X
6.02
8.94
X
10-"
I0
;
*
atoms/kmol
kg/m a
kg/kmol
8.'3.5
6.02
X
S-04
X
I0 a
kg/m 3
63.5 kg/kmol
=
=
8.5
X
10-* atoms
8.5
X
I0-
S
m
;
cleetrons/m 3
/<*•
The corresponding Fermi energy
/2
3/2
3
PROPERTIES OF MATTER
=
=
so that
_
~
i
2m \HvVt
(mass/ volume)
mass/ kmol
''
3 -v \ 2/
is
free
iere
can
'''
and
10.12
It
10 13
=«
16\/2^Vw-'
X
(atoms/ kmol)
order of
highest energy state to be filled will
definition.
)
h3
shells.
atom contributes one
A>
Substituting Eq. 10.8 for g(e)de yields
N=
electron
The electron density tj = A'/V is accordingly equal
number of copper atoms per unit volume, which is given by
Atoms
I
<
10.11
1
energy states with these electrons
The number
number
of states g(t?
have the same energy is equal to the
Hence
electron.
since each state is limited to one
then have the energy f
The
2p*3a i3p e3d i'>4s;
Sl
F
its
increasing energy starting from
copper.
la t2s
each atom has a single 4s electron outside closed inner
Volume
i
in
is
electron to the electron gas.
a particular metal sample contains
.
is.
atom
therefore reasonable to assume that each copper
fF
t
As the temperature increases, the occupation index changes from
and more gradually, as in Fig. 10- 19. At all temperatures
f(e) z=
higher
t
F is independent of
being considered.
is
configuration of the ground state of the copper
<
f > eF
when
when
1
at a
the density of free electrons; hence
is
Let us use Eq, 10.12 to calculate the Fermi energy
I
that
=
=
Fermi Durac distribution at absolute zero and
lor a
the dimensions of whatever metal sample
g(t)
The formula
The quantity A'/V
"^
t
The occupation indei
temperature.
Fermi energy
_
"
=
=
(6.63
2
X
9.11
1.13
X
7.04
eV
X
X
10
10- 31
10- |S
^
is,
from Eq. 10.12,
l-sf
/3
kg/electron I
X
8.5
X
10** cleetrons/nv' V
8n
^a
}
J
THE SOLID STATE
341
=
At absolute zero, T
eV
By
copper.
in
K, there
would be electrons with energies of up
to 7.04
contrast, all the molecules in an ideal gas al absolute zero
would have zero energy. Because of
electron gas in a metal
is
common
energies of several
its
decidedly nnnelassical behavior, the
said to l>e degenerate.
Table
10.-4
gives
l
he Fermi
metals.
ELECTRON-ENERGY DISTRIBUTION
*10.8
We may now
substitute for a
and
g(c)
cfe
in
Eq. 10.7 to obtain a formula for
number of electrons in an electron gas having energies between some value
and e + de. This formula is
the
e
n{e)de
10.14
If
we
(SySr Vm A/ yh3
=
y
/s
de
express the numerator of Eq. 10.14 in terms of the Fermi energy
<^.,
we
obtain
.
.
.
n[e) d,
10.15
Equation 10.15
=
———-f
f&ffifc,-****
-
de
plotted in Fig. 10-20 for the temperatures
is
T=
0,
300, and
1200 K.
It is
To do
Distribution of electron energies in 3 metal at various temperatures.
FIGURE 10.20
:
Since at
energy
t
T = K all the
F we may let
electrons have energies less than or equal to the Fermi
,
interacting to determine the average electron energy at absolute zero.
we
this,
first
obtain the total energy
V
at
K, which
is
and
I '„
=
r'r
I
fn(e) de
-
10.16
fjfc.
Table 10.4.
The average
SOME FERMI ENERGIES.
electron energy
electrons present
which
*,,
is
this total
energv dh ided by ihe number of
yields
Fermi energy, eV
Metal
342
.V,
10.17
~<>
=
f
F
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
of
Silver
Ag
5.51
of copper would have to
Cold
Au
5.54
to
PROPERTIES OF MATTER
^
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
1
eV
is
11,600 K; this means that,
l>e at
if
free electrons Ijehaved classically, a
a temperature of about 50,000
have the same average energy they actually have
at
K
for
lis
sample
electrons
K.
THE SOLID STATE
343
The considerable amount
in the electron
10.5.
The
of kinetic energy possessed by the valence electrons
gas of a metal represents a repulsive influence, as noted in Sec.
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
The atoms
higher energy states required by the exclusion principle.
a metallic solid,
in
however, are closer together because of their bonds than they
result the valence electrons are, on the average, closer
would be otherwise. As a
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
the added electron kinetic energy
is
to a net
Unbound low-energv
lattice
utlli u,
and such electrons are
(Sec. 2.4) or electrons in
a
«.
More energetic
same way
diffracted in precisely the
l>eani (Sec. 3.5) directed at the crystal
electrons,
as
X
ravs
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,
taken into account.
It is
customary to treat the situation of electron waves
A by the wave number k introduced
•10.9
spacing
such as those with the Kermi energy in a metal, have wavelengths comparable
when
cohesive force even
electrons can travel freely through a crystal since their
wavelengths are long relative to the
BRILLOU1N ZONES
=
i
k
10.20
in Sec. 3,3,
hv replacing
in a crystal
where
—
25T
Wave number
A
We now turn to a more detailed examination of how allowed and forbidden bands
in
a solid originate. The fundamental idea
in a region of periodically
is
that an electron in a crystal
moves
varying potential (Fig. 10-21) rather than one of
constant potential, and as a result diffraction effects occur that limit the electron
to certain ranges of
this
way
momenta
that correspond to the allowed
of thinking the interactions
behavior indirectly through the
among
the
will
in Set:. 10.fi
An
bring
we can
of k
describe the
a
Figure
momentum
p
A
Free electron
=
A„
for
The potential energy of in electron
11-22
1
sin
by means of a vector
the
wave
train
as the particle,
Bragg's formula in terms
k.
6
shows Bragg
we can
equal to n-n/u.
I
FIGURE 10 21
train
in
same direction
in the
Similarly, reflection
us consider
them
first
in
the
.r
direction, IL,
from the horizontal rows occurs when
those electrons whose
to avoid diffraction.
through the
lattice in
Uk
is
less
any
direction.
wave numbers are sufficiently small
move freelv
than tr/o, the electron can
When
k
=
<n/a,
they are prevented from
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
ruining in the x or
km
lattice.
nx/a.
jet
in a periodic array of positive ions.
positive
a two-dimensional square
reflection in
express the Bragg condition by saying that reflection from the
rows of ions occurs when the component of k
vertical
is
is
=
wave
moves
train
=
k
10,21
be used here, rather than a formal treatment based on Sehriidi tiger's
Broglic wavelength of a free electron of
x
wave
is
Evidently
10.18
equal to the numlier of radians per meter
is
Since the
describes.
intuitive
equation.
The de
it
atoms influence valence-electron
lattice of the crystal these interactions
about, instead of directly as in the approach described
approach
energy bauds. In
The wave number
;/
directions by diffraction.
the lattice.
^nnnnr
The region
is
in fc-spacc that low-fc
electrons can occupy without teing diffracted
called the first lirillauin ^i»u?and
/one
first
is
also
shown;
it
is
shown
in Fig. 10-23.
contains electrons with k
zone yet which have
sufficiently small
diffraction by the diagonal sets of
344
PROPERTIES OF MATTER
The second
Brillouin
do not
into the
that
fit
propagation constants to avoid
atomic planes
contains electrons with k values from <n/a to
> t/u
in Fig. 10-22.
2w/«
The second zone
moving in the
for electrons
THE SOLID STATE
345
E=
10.22
2»i
It)
the case of an electron in a crystal for which k
no interaction with the
positive ions
and
lattice,
an electron depends upon
10.22
I£ci.
is
increasing k the constant-energy contour lines
more
together and also
k
Mr
_
a
K=
_
sin
E
straightforward.
9
The
in Jt-spaee, the closer
more
anil
varies with
k2
.
distorted.
The
is
to
A\ as in Fig.
become
The reason
is
to die
10-25.
With
progressively closer
for the
reason for the second
closer an electron
it
ir/a, there is practically
Since the energy of such
the contour lines of constant energy in a two-
A-,
dimensional it-space are simply circles of constant
merely that
<
valid.
boundary
is
first
effect
is
almost equally
of a Brillouiu
being diffracted by the actual crystal
zone
lattice.
But
ksinB
H7T
FIGURE 10 23
a
The
first
and second
Brillouin
tones of a two-dimensional square
lattice.
*..
r
a
I
I.
U^
FIGURE 10-22
"
\
and
Bragg
reflection
"ij directions,
from the
with
vertical
ili<-
raws
when
A,
=
Further Brillouin /ones can he constructed
same manner. The extension
of this analysis to actual three-dimensional
"10.10
The
significance of the Brillouin zones
its
in Fig.
k*
=-Ia
*-+f
^
10-24.
ORIGIN OF FORBIDDEN BANDS
energies of the electrons in each zone.
to
shown
°
-second Brillouin
zone
possible range ol k values narrowing as the
structures leads to the Brilbuin zones
Brillouin zone
iir/n.
diagonal directions arc approached.
in the
momentum p
E=
and hence
346
of ions occurs
+
first
to its
becomes apparent when we examine the
The energy of a free electron
is
t
k
V
—
N
related
by
2m
wave numher
PROPERTIES OF MATTER
k by
THE SOLID STATE
347
(a)
second Brillouin zone
First
Brillouin /xmc
(b)
FIGURE 10-25
10-24
First and second Brillouin zones in (a) lace-centered cubic structure and
body ctniered cubic structure (tee Figs. 10-4 and 10-5)
FIGURE
(o)
in particle
Energy contours
ol
positive ions that
occupy the
lattice points,
and the
FIGURE 10-26
B varies wtth
more
stronger the interaction, the
Figure 10-26 shows
E
increases
how E
the election's energy
more slowly than
the
second zone.
first
There
is
the
first
and second
Brillouin
rones of a
Electron energy E versus
2
ti'
k~/2m,
is
a definite
As
the free-particle figure.
first
Brillouin
gap between the
wave number
k In the k, direction.
The dashed
line
shows how
affected.
varies with k in the x direction.
has two values, the lower lielougiug to the
to the
In
lor a free electron.
It
E
electron volts
terms the diffraction occurs by virtue of the interaction of the electron
with the periodic arniv
17/a,
in
hypothetical square lattice.
I;
approaches
At k
=
w/a.
zone and the higher
possible energies
allowed energies
in
and second brillouin zones which corresponds to the forbidden band
spoken of
earlier.
The same
pattern continues as successively higher Brillonii,
•/ones are reached.
boundary of a Brillouin /one follows from
the limiting values of k correspond to standing waves rather than
The energy
the fact that
discontinuity at the
traveling waves. For clarity
we shall consider electrons moving in
the extension of the argument to any other direction
348
PROPERTIES OF MATTER
is
the X direction;
straightforward.
When
-47T -37T -27T
a
a
a
-IT
a
u
a
a
a
THE SOLID STATE
349
=
it
±ir/fl, as
we have
seen, the
waves are Bragg-reflected back and forth, and
waves whose
nCEJ
so the only solutions of Sehrodinger's equation consist of standing
wavelength
is
equal to the periodicity of the
for these standing
10.3.1
jf>,
waves
=A
sin
for
n a*
1,
lattice.
There are two
possibilities
namely
—
liirbiddcn
a
band
10.24
ip.,
= A cos
—
irx
a
The
z
|tf-,|
probability densities
has
1^1' has
to
its
its
minima
maxima
a
|^,|
and
2
|if<
2
|
are plotted in Fig. 10-27.
at the lattice points
occupied by the positive
at the lattice points. Since the
an electron wave function
i£>
is
,
FIGURE 10-27
Distributions ot the probability densities
V',
in
'
the case of
while
and |y,
in the case of
^
it is
15
10
ions,
E,eV
charge density corresponding
e#\ 2 the charge density
concentrated between the positive ions while
Evidently
^,
is
concentrated
The distributions
FIGURE 10-28
line Is
o! electron energies in the Brlllouin
Fig. 10-25.
lones of
The potential energy of an electron in
midway Itctwccn each pair of ions and least at
at the positive ions.
'.
ions
The dashed
the distribution predicted by the free-electron theory.
is
greatest
so the electron energies
v., are different.
E and
{
situation for
in Fig. 10-25.
eV) the curve
is
At low energies
is
small and
and
that corresponds to
(in this
hypolhetical
almost exactly the same as thai of Fig.
10-20 based on the free-electron theory, which
energies k
^
-±^/a, and accordingly
no electron can have an energy between £, and E^.
Figure 10-28 shows the distribution of electron energies
E < —2
the ions themselves,
E, associated with the standing waves
No other solutions are possible when k =
die Rrillouin /.ones pictured
a lattice of positive
is
not surprising since
the electrons in a periodic lattice
at
tow
then do behave like
With increasing energy, however, the number of available energy
states goes beyond that of the free-electron theory owing to the distortion of
the energy contours by the lattice: there are more different k values for each
free electrons.
=
energy.
Then, when k
the
zone, and energies higher than about 4
first
^.v/a.
I
he energy contours reach the boundaries of
eV
(in this particular
model)
are forbidden for electrons in the k r and kv directions although permitted
other directions.
available energy states liccome restricted
/one, and n{E)
and n(E)
m
0.
falls.
The
more and more
Finally, at approximately
6%
lowest possible energy in the second /.one
gap between the possible energies
350
PROPERTIES OF MATTER
Italic! is
to the corners of the
eV, there are no more states
than 10 eV, and another curve similar in shape to the
forbidden
in
As the energy goes farther and farther lieyond 4 eV, the
about
-5
eV
in the
two zones
is
first
is
somewhat
begins,
less
(fere the
about 3 eV, and so die
wide.
THE SOLID STATE
351
Although there must be an energy gap between successive Brillouin zones
any given direction, the various gaps may overlap
directions so that there
no forbidden hand
is
E
10-29 contains graphs of
a forbidden band
and
in
pe: nitted energies in other
in the crystal as
a whole. Figure
versus k for three directions in a crystal that has
a crystal whose allowed bands overlap sufficiently to
in
avoid having a forbidden band.
The
its
electrical Ijehavior of a solid
There are two available energy
Structural unit in the crystal.
atom
in a
molecular
if
depends upon the degree of occupancy of
energy bands as well as upon the nature of the band structure, as noted
states (one for
each spin)
(By "structural unit"
in
in this
each band
context
earlier.
each
for
meant an
is
metal or covalent elemental solid Such as diamond, a molecule
A solid will he an
must have an even number of valence
and an ion pair
solid,
two conditions are met:
per structural unit, and
(1) It
(2)
an ionic
in
The reason
kT.
highest-energy band
the electrons
lie
for condition (1)
completely
lie
able to
electrons
the band that contains the highest-energy electrons
must be separated from the allowed band above
compared with
in a
insulator
solid.)
filled,
cross the gap
it
by an energy gap large
is
that
it
and the reason for
ensures that the
(2)
to reach unfilled slates.
is
that
none of
Thus diamond,
with four valence electrons per atom, solid hydrogen, with two valence electrons
CI" ion pair,
per II,, molecule, and NaCI, with eight valence electrons per Ha4
—
all
insulators. Figure 10-30o
shows
violation of either (or both) of the
above
have wide forbidden bands in addition and are
the energy contours of a hypothetical insulator.
A
conductor
conditions.
is
characterized by
Thus the
alkali metals,
its
with an odd number of valence electrons per
one per atom), are conductors, as are such divalent metals
structural unit (namely
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
as
the forbidden
is
band
in
an insulator
is
narrow or the amount of overlap
in a
small, the electrical conductivity falls in the semiconductor region,
metal
and
it is
not reallv correct to speak of the substance as either a metal or a nonmetal.
The boundary between
sional fc-space
is
Experiments indicate
/inc.
filled
and empty electron energy
states in three-dimen-
called the Fenni surface.
and cadmium
is
that the conductivity of the divalent metals beryllium,
largely
This unexpected finding
is
due
to positive
charge carriers, not to electrons.
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
play a minor role.
352
PROPERTIES OF MATTER
in the
upper band
FIGURE 1029
£ versus
den band, while
In
k
curves
for
three directions
(b) the allowed energy
in
two
crystals.
bands overlap and there
Is
In (a) there is a forbid-
no forbidden band.
THE SOLID STATE
353
EFFECTIVE MASS
10.11
An
electron in a crystal interacts with the crystal lattice, and because of this
interaction
its
response to external forces
of a free electron. There
7.0nC
IllHllillai icv
is
is
same
not, in general, the
nothing unusual alxnit
this
phenomenon
What
subject to constraints behaves like a free particle.
is
as that
— no particle
unusual
is
that the
deviations of an electron in a crystal from free-electron behavior under the
influence of external forces can
all
he incorporated into the simple statement
mass of such an electron
that the effective
The most important
Sections 10,7 and 10.8 can
incorporated
l>e
merely by replacing the electron mass
m
a
M„
N/V
where
mass
is
=
ft
2m'
/ 3JV
in
in a
actual mass.
its
theory of metals discussed
the
more
metal
is
in
band theory
realistic
effective mass
m"
at
given by
V'
WW
Fermi energy
the density of valence electrons. Table 10.5
m°/m
ratios
a
not the same as
by the average
the Fermi surface. Thus the Fermi energy
io.
is
results of the free-electron
is
a
list
of effective
in metals.
Table 10.5.
EFFECTIVE
MASS RATIOS »• u, AT THE
IN SOME METALS.
FERMI SURFACE
mVm
Metal
Lithium
J
lt'T\
I
111
I
Ml
l.i
1.2
Ht
1.6
Sodium
Va
1.2
Aluminum
Al
0.97
Cnlwlt
C«
N
Nickel
Ni
28
Copper
Cu
1.01
/in?
Zn
0.85
Silver
Ag
Platinum
ft
0,99
13
Problems
vacant i-iuthv levels
Fermi
level
)
.
i«)
What
is
the effect
on the cohesive energy of
van der Waals forces and
(b)
ionic and eovalent crystals of
zero-point oscillations of the ions and atoms
about their equilibrium positions?
L occupied
energy levels
THE SOLID STATE
FIGURE 10-30
Electron energy contours and Fermi levels in three types ot solid: (a) Insulator; (b)
valent metal; (c) divalent metal.
Energies are
in
electron votts.
mono-
355
The van dor Waals
2.
energy of about 6
X
cV
'
cannot
at
lie
atoms leads
to
it
binding
an equilibrium separation of about 3 A,
show
the uncertainty principle to
He
between two
attraction
Id
that, at
ordinary pressures
(<25
Nat
are
'1
Use
«««--%+
atm), solid
V5
exist.
The Joulr- Thomson effect
when it passes slowly from a
refers to the
3.
plug. Since the expansion
F.xplain the
full
in
effect
empty one through a porous
container, no mechanical work is done.
terms of the van der Waals attraction
in
13.
temperature a gas undergoes
container to an
into a rigid
is
Joule-Thomson
drop
The
inu spaefngs and melting points of the sodium halides are as follows:
N»F
Imi
v|)LH'iji!;,
Mult ins
I" "'
1
\
1
2.3
*C
'
KG
is
find the
in
of solids:
and
2.8
2.0
3.2
14.
740
060
inn spacing
is
On
per ion pair.
that
obtained
in
in (a)
eV and
4.34
sign
(b)
structure
On
3.14 A.
is
the electron affinity
KG
constant for the
is
1.74S
the basis of these
The observed cohesive energy
the assumption that the difference l>etvveen
is
due
to the exclusion- principle repulsion,
the formula Br~" for the potential energy arising from
Repeat Proh. 13 for LiCl,
is
in
which the Madclung constant
2.57 A, and the observed cohesive energy
ionization energy of Li
is
is
6.8
is
1.748. the
eV per
ion pair.
5.4 eV.
these quantities with halogen atomic number.
metals are opaque to light of all wavelengths,
(b)
light,
(c)
potential energy V{x) of a pair of atoms in a solid that arc displaced
separation at
K may be written V(*)
as8
tor
Most
3
—
ex*,
=
their equilibrium
8
where the anharmonic terms —far and
— ex'
in
diamond
it
is
-
represent, respectively,
the asymmetry introduced by the repulsive forces between the atoms and the
leveling off of the attractive forces at large displacements.
eV and
in silicon is ].]
transparency of these substances to visible
The
by x from
Semiconductors
insulators are transparent lo visible tight.
The energy gap
vS
this source,
801
arc transparent to infrared light although opaque to visible
6.
eV
exponent n
Mat
notion of energy bands to explain the lullowing optical properties
(o) All
2
The Madehmg
3.61 cV.
6.42
NaBr
15.
I'se the
5.
V5
24
j
ionization energy of potassium
is
NaCI
The
Explain the regular variation
6
data only, compute the cohesive energy of KC1.
this figure
The
4.
S
and the distance between ions of opposite
of
between molecules.
(a)
of chlorine
'
At a temperature
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
T
6 eV. Discuss the
,
light.
is
A
7.
small proportion of indium
is
incorporated
in
a
germanium
crystal.
Is
the crystal an rt-type or a jj-type semiconductor?
A
8.
small proportion of antimony
is
ineoq>orated in a germanium crystal.
x
What
is
obey Fermi
the connection
statistics
between the
and the
J*
fact that the free electrons in a
fact that the photoelectric effect
is
xe- v ' kT
- X
metal
Show
virtually tem-
<lx
___
Is
the crystal an n-type or a jj-type semiconductor?
9.
f
=
e-
VJkT
dx
that, for small displacements,
change
when
its
copper
is
in length of a solid
x =; 3hfcT/4fl a
.
(This
temperature changes
is
is
the reason that the
proportional to AT.)
po rat ii re - nde pe nden t ?
i
°
10.
(a)
How much
of these atoms? (b)
if
their total energy
energy
required to form a
is
What must
is
to
K' and
I
ion pair from a pair
the separation be lietwcen a K'
and an
1" ion
{a)
How much
pair of these atoms? (b)
ion
12.
356
if
their total
Show
energy
that the
is
required lo form a Li' and Br" ion pair from a
What must
is
first
PROPERTIES OF MATTER
iu
copper are
7.04 eV.
(a)
Approximately what percentage
in excited states at
room temperature?
{!>)
zero?
lie
energy
The Fermi energy
At the melting point of copper, 1083"C?
'
1 1.
16.
of the free electrons in
to
five
the separation be between a
l.i'
and a Hi
be zero?
terms
in the series for the
Madclung constant of
17.
The Fermi energy
in silver
*
in
speed of an election with
this
8.
The density
of aluminum
(a) What is the average energy of
What temperature is necessary for the
gas to have this value? (c) What is the
5.51 eV.
(b)
an ideal
average molecular energy
1
is
K?
the free electrons in silver at
energy?
is
2.70
g/em 3 and
its
atomic weight
is
26,97,
The
THE SOLID STATE
357
aluminum is given in Table 7,2 (note: the energy difference
ween 3* and 3p electrons is very small), and the effective mass of an electron
aluminum is 0.97 m r Calculate the Fermi energy in aluminum.
electronic structure of
lie!
in
.
The
*19.
density of zinc
electronic structure of zinc
electron in zinc
to
why
Kxplain
20.
0.85
is
m
7.13
is
r.
is
g/cm a and
atomic weight
its
65.4.
is
Calculate the Fermi energy in
make
the free electrons in a metal
/.inc.
only a minor contribution
specific heat,
its
Find the ratio between the kinetic energies of an electron
"21.
dimensional square lattice which has
=
kx
Draw
two
=
=
k^
it
in
a two-
la and an electron which has
whose
the third Brillouin zone of the two-dimensional square lattice
shown
Brillouin /onus arc
Phosphorus
23.
Jt,
s 0.
7T la, Jr„
°22.
first
The
given in Table 7,2, and the effective mass of an
present in a
is
10-23.
in Fig.
germanium sample. Assume
that one of
its live
valence electrons revolves in a Bohr orbit around each P" ion in the germanium
lattice,
in'*
effective mass
How
0.65 eV.
1
ft,
Repeal Prob. 23
mass
ol
an electron
is
sample
about 0.31
Bohr
ilief-elnc
orbit of the electron.
germanium
effective
The
that contains arsenic.
mr
,
effective
the dielectric constant of silicon
made
calculation of the Fenni energy in copper
obtained was approximately correct.
The
H.ITm, and the
in silicon is 1.1 eV.
take into account the difference between
26.
i<
first
the valence and conduction bands in
for a silicon
in silicon
and the energy gap
The
"25.
the electron
does the ionization energy of the above electron compare with
24.
12,
>!
find the radius of the
and with kT at room temperature?
this energy'
is
is
The energy gap l)Ctweeu
''
is
tile
if
germanium
constant of
mr
in Sec, 10.7 did not
and m°, and yet the n F value
Why?
mass m° of a current carrier in a semiconductor can be
directly determined by
means of a cyclotron resonance experiment
in
which the
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
carriers (whether electrons or holes)
i
Is
I
requeue)
V is
an equation for
equal to the frequency of tv\ olulion
rr in
terms of »i°,
and maximum absorption
(c)
Find the
speed
358
is
3
maximum
X
is
e,
and
B.
PROPERTIES OF MATTER
v
r
of the carrier, (a)
In a certain experiment,
found to occur at
orbital radius of a
H) 4 m/s.
(ft)
v
=
1.1
charge carrier
X
H)
in this
1
"
Hz.
Derive
B =
0.1
T
Find m'.
experiment whose
THE ATOMIC NUCLEUS
11
Thus
far
we have regarded
the atomic nucleus solely as a point mass that possesses
The behavior
positive charge.
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
For instance, the elements
of things.
possess multiple electric charges,
from insignificance
in the
grand scheme
because of the ability of nuclei
exist
and explaining
this ability
is
to
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-
And, of course, the mundane applications of nuclear energy are
formations.
familiar enough.
11.1
ATOMIC MASSES
The nucleus of an atom
contains nearly
on nuclear properties can
lie
all its
mass, and a good deal of information
knowledge of atomic masses. An
inferred from a
instrument used to measure atomic masses
called a nuiits spectrometer;
is
spectrometers and techniques are capable of precisions of better than
in
modern
1
part
108.
Atomic masses
refer to the masses of neutral atoms, not of stripped nuclei.
ThaH the masses of the
orbital electrons
and
energies are incorporated in the figures,
expressed
mass units
in
carbon atom
is,
by
(u)
the mass equivalent of their binding
Atomic masses are conventionally
such that the mass of the most abundant type of
definition, exactly 12.(HX)
.
.
.
u.
The value
of a mass unit
is,
to five significant figures,
I
and
its
u
=
l.fifiOi
X
energy equivalent
Not long
after the
early in this century,
is
10
"kg
931.48 MeV.
development of methods
it
was discovered
that not
for determining
all
atomic masses
of the atoms of a particular
361
element have the same mass. The different varieties of the same element
called
its
Another widely used term, nuclide,
isotopes.
species of nucleus; thus each isotope of ;m clement
The atomic masses
stamen I, which
listed in
Table
7.
1
refer to the
is
a nuclide.
murage atomic mass
the quantity of primary interest to chemists.
is
sire
refers to a particular
of each
Table 11.1
contains the atomic masses and relative abundances of the five stable isotopes
The
of zinc.
relative
is
65.38
individual masses range from 63.92914 u to 69.92535
abundances range from
u,
which
The average mass
Twenty elements are
single nuclide each; beryllium, Huorine, sodium,
and alumi-
are examples.
It
is
masses more than double that of 30 hydrogen atoms.
and the
percent lo 48.89 percent
accordingly the atomic mass of /inc.
is
composed of only a
num
0.fi2
ii,
therefore tempting to regard all nuclei as consisting of protons
somehow bound together. However, a closer look rotes out
nuclei—
hydrogen
nuclide mass is invariably greater than the mass of a number
since
a
this notion,
equal to its atomic number Z— and the nuclear charge of
atoms
hydrogen
of
atomic number of /.inc is 30, but its isotopes all have
The
is
-f-Ze.
an atom
mass.
Another
possibility
nucleus,
+ 2e,
whose mass
would be
and Mliutit respectively. (Tritium nuclei, called Iritom, are unstable and decay
adioaetively into an isotope of helium.) The nucleus of the lightest isotope is
may be present in nuclei
some of the protons. Thus the helium
to mind. Perhaps electrons
four times that of the proton though
is
regarded as being
This explanation
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 tUtutatjut n
comes
which neutralize the positive charge of
composed
its
of four protons and
charge
two
is
only
electrons.
buttressed by the fact that certain radioactive nuclei sponta-
is
neouslv emit electrons, a
easy to account for
if
phenomenon
called Ixtta decay,
whose occurence
is
electrons are present in nuclei.
Despite the superficial attraction of the hypothesis of nuclear electrons, how-
number
ever, there are a
of strong arguments against
it:
i
the proton, whose mass of
m„=
=
is,
1.
X
it
electron
also contains.
The
proton, like the electron,
is
an elemen-
tary particle rather than a composite ol other particles. (The notion of elementary particle
An
considered in some detail in Ghap.
is
interesting regularity
is
apparent
in listings
1
anSHfra
.007825
.is
as massive.
the
u.
For example, the deuterium atom
is
approximately twice as
hydrogen atom, and the tritium atom approximately three times
The masses
momentum
of the zinc isotopes listed in Table 11.1 further illustrate
must
electron kinetic energy
is
21
MeV. (This
level of
figure
an electron
calculation must be
13.)
of nuclide masses: the values
are always very close to being integral multiples of the mass of the hydrogen
atom.
—10 M m
across.
To confine an electron
X
10' 27 kg
within experimental error, equal to the mass of the entire atom minus the
mass of the electron
Nuclei are only
size.
in
to so small a region requires, by the uncertainly principle, an uncertainty
10~*°
The
in
Sec.
3.7.
calculated
was
kg-m/s,
as
1.1
momentum
of
>
its
\p
1.0072766 u
1.6725
Nuclear
l>e at least
may
in a
made
also
minimum value of Ap. The
momentum of 1.1 X 10" kg-m/s
as large as the
that corresponds to a
lw obtained by calculating the lowest energy
box of nuclear dimensions; since
relativistically.)
beta decay have energies of only 2 or 3
T
>m
t)
c
2
,
the latter
However, the electrons emitted during
MeV— an
order of magnitude smaller
than the energies they must have had within the nucleus
if
they were to have
existed there.
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
i0
kg-m/s. T < «i[,<? 2 and its kinetic energy can be calculated
l.l X 10"
We
'
this pattern, lieing
quite near to 64, 66, 67, 68, and 70 times the hydrogen-atom
,
classically.
Table 11.1.
PROPERTIES OF THE STABLE ISOTOPES OF ZINC, / =
We
T= Pi
2m
30.
(1.1
Mass number
of Isotope
have
Atomic mass,
U
Relative abundance.
64
B.1.929I4
48,89
SO
05,92005
27.81
87
00.02715
4.11
68
07.92436
tN.se
70
09.92535
0.62
%
X
10- 2 " kg-m/s)"
X 1.67 X Hr-* 7 kg
= 3.6 X 10- " j
= 0.23 MeV
2
The presence
of protons with such kinetic energies in a nucleus
is
entirely
plausible.
362
THE NUCLEUS
THE ATOMIC NUCLEUS
363
Suclear spin. Protons and electrons are Fermi particles with spins of %,
angular momenta of ,-Ji. Thus nuclei with an even iiumlrer of protons
2.
that
is,
much
through as
gamma
as several centimeters of lead without being absorbed suggested
rays of unpreccdentedly short wavelength.
]
plus electrons should have integral spins, while those with an
protons plus electrons should have half-integral spias.
obeyed The
fact that the denteron,
hydrogen, has an atomic number of
1
which
is
odd munlier of
This prediction
and a mass number of
2,
would he
preted as implying the presence of two protons and one electron.
upon the orientations of the
be %> Yf
something
3.
1.5
—%.
—%
<"
is
not
the nucleus of an isotope of
particles, the nuclear spin of
However, the observed spin
'\\\
inter-
Depending
should therefore
of the denteron
is
J,
10"-' that of
of the
the
The proton has a magnetic moment only about
electron, so that nuclear magnetic moments ought to Iw
same order of magnitude as that of the electron if electrons are present
However, the observed magnetic moments of nuclei arc comparable
its piopertics in detail.
In one such experiment Irene Curie and
observed that when the radiation struck a slab of paraffin, a hydrogen-
P. Joliot
X
prising:
is
no reason why shorter-wavelength
observed that the forces that act between
It
is
therefore hard to see why,
if
with
its
nucleus.
That
is,
in
how can
an atom interact onlv elechalf the electrons in an
escape the strong binding of the other half? Furthermore,
are scattered by nuclei, they hehave as though acted
forces, while the nuclear scattering of fast
electrostatic influences that can
The
upon
when
solely
fast
by
atom
electrons
electrostatic
protons reveals departures from
Ik ascribed only
needed
is
a
Compton
effect the
to a specifically nuclear force.
difficulties of the
nuclear electron hypothesis were
no serious
alternative.
the proton in atomic nuclei
The problem
came
known
to light,
some time
but there seemed
for
of the mysterious ingredient Iwsides
was not solved
until 1932.
minimum
initial
it
was calculated
not very sur-
gamma-ray photon energy of 53 MeV, This seemed peculiar
known at the time had more than a
The peculiarity liecame even more
small fraction
striking
nucleus to
—
is equivalent
to only 10.7 MeV
one-fifth the energy needed by a
gamma-fay 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 now-mysterious radiation emitted by beryllium
when bon
barded by alpha particles. lie assumed that the radiation consisted of neutral
particles
whose mass
approximately the same as that of the proton.
is
electrical neutrality of these particles,
Tor their ability to
one
at rest
penetrate matter readily. Their mass accounted nicely for
whose mass
is
,i
moving particle colliding head-on with
same can
the
transfer
all
of
its
kinetic energy to the
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,
latter.
of
also
be knocked mil of appropriate absorbers In the bcrvllium
and the measurements tnade of the energies of these nuclei
mn
fact,
Chadwick arrived at
widi alpha particles from a sample of polonium and found that radiation was
note that
emitted which was able to penetrate matter readily,
radioactively into a proton, an electron, and an antiueutrino: the half
it
consisted of
gamma
rays,
(Camilla rays are electromagnetic
waves of extremely short wavelength.) The
THE NUCLEUS
ability of the radiation to pass
in well
iiii p from an analysis of observed proton and nitrogen nuclei recoil
Before
free
fit
the neutron-mass figure
energies;
naturally, that
The
which were called iwulmws, accounted
The composition of atomic nuclei was finally understood in 1932. Two years
earlier the German physicists W. Bothe and II. Becker had liombarded beryllium
tained that the radiation did not coasist of charged particles and assumed, quite
when
presumed reaction of an alpha particle and a beryllium
yield a carbon nucleus would result in a mass decrease of ().(H 144 u,
with the neutron hypothesis. In
Bothe and Becker ascer-
and there
rays cannot give energy to protoas
minimum gamma-ray photon energy £ = hr
T lo a proton can be calculated. The result
which
radiation,
THE NEUTRON
this is
collisions,
that the
and nitrogen could
11.2
gamma
glance
Compton
energy
to transfer the kinetic
die observed proton recoil cnci»ics:
Iwfore the correct explanation for nuclear masses
to l>e
per particle.
electrons can interact strongly enough with
protons to form nuclei, the orbital electrons
trostatically
MeV
first
Curie and Joliot found proton recoil energies of up to about 5.3 MeV. From
Fq. 2.15 for the
of this considerable energy.
It is
At
in similar processes.
with that of the proton, not with that of the electron, a discrepancy that cannot
he understood if electrons arc nuclear constituents,
Elect mn- nuclear interaction.
out.
rays can give energy to electrons in
because no nuclear radiation
4.
364
were knocked
rich substance, protons
in nuclei.
nuclear particles lead to binding energies of the order of 8
Other physicists became
and a number of experiments were performed to
determine
cannot be reconciled with the hypothesis of unclear electrons.
that
MttgaeUe moment.
x
interested in this radiation,
no other mass gave
we
it
neutron
as
good agreement with the experimental
consider the role of the neutron in nuclear structure,
is
is
not a stable particle outside nuclei.
The
we
data.
should
free neutron decays
life
of the
I0A min.
Immediately after
its
discovery the neutron was recognized as the missing
ingredient in atomic nuclei.
Its
mass of
THE ATOMIC NUCLEUS
365
m =
1.0086654(1
n
which
is
spin of
%
10- 2T kg
more than
slightly
all
X
1.6748
in perfectly
fit
thai of the proton,
assumed that nuclei are composed
The
its
electrical neutrality,
and
with the observed properties of nuclei when
its
it is
and protons.
solely of neutrons
Following teniis and symlrols arc widely used to describe a nucleus:
Z = atomic number = number
of protons
= number of neutrons
A = Z + .V = mass number = total number of neutrons and protons
.V
=
neutron number
The term nucleon refers to both protons and neutrons, so that the mass number
Nuclides are identified
is the number of nucleoos in a particular nucleus.
A
according to the scheme
where
X
is
Dumber 75
Mm
i'
the chemical symlxj] of the species. Thus the arsenic isotope of mass
is
denoted by
the atomic
gen, which
is
number
a proton,
of arsenic
is
is
33. Similarly a nucleus of ordinary hydro-
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
explains die existence of isotopes: the isotopes of an element
numbers
charge
is
all
of protons hut have different numl>ers of neutrons.
what
is
immediately
contain the same
Since
its
nuclear
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.
Not
10
STABLE NUCLEI
11.3
all
<
20) contain approximately equal
nuclei.
energy
366
is
evident from Fig. 11-1, which
The tendency
levels,
whose
THE NUCLEUS
for .V to equal
Z
is
In general,
numbers of neutrons and
protons, while in heavier nuclei the proportion of neutrons
greater. Tin's
30
40
50
60
70
80
90
PROTON NUMBER (Z)
combinations of neutrons and protons form stable nuclei.
light nuclei (A
20
FIGURE 11-1 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.
>
•
becomes progressively
a plot of
N versus
Z
for stable
follows from the existence of nuclear
origin and properties
we
shall
examine
shortly.
Nucleons,
THE ATOMIC NUCLEUS
367
which have spins of
V,.
prim
oliev the exclusion
As a
iplc.
result,
each nuclear
energy level can contain two neutrons of opposite spins and two protons of
opposite spins. Energy levels in nuclei are
in
atoms
mum
A nucleus with,
stability.
inner levels will have
in
the
same
say, three neutrons anil
situation, since in the
'i(B
Figure
shows how
1-2
1
isotope while permitting
The preceding argument
more than
nuclei with
produce only attractive
11-1 departs
nuclei .V
all
four nucleons
energy levels
therefore maxi-
one proton outside tiled
only part of the story.
is
forces,
is
is
stability
curve
Z
decreases by 2
into the lowest available
'jjC to exist.
.
7 &
UJ
CD
negative beta decay
Protons are positively
This repulsion becomes SO greai
K) protons or so that
A. hut
fit
accounts for the absence of a stable
this notion
an excess of neutrons, which
/
Z
iV decreases
o
*-
.V
—
7,
never smaller;
line as
'j-B is
7.
increases.
Even
N
Z
positive beta
decreases by
decreases by
decay
1
t
er.
Z decreases
required for stability; thus the curve of Fig.
more and more from the
may exceed
just as
former case one of the neutrons must go inlo
charged and repel one another electrostatically
in
sequence,
more energy than one with two neutrons and two protons
a higher level while in the latter case
level.
filled in
minimum energy and
are, to achieve configurations of
by
1
in light
stable, for instance, but not
Iff;
Nuclear forces are limited
in range,
and as a
only with Iheir nearest neighbors. This effect
of nuclear forces.
is
referred to as the saturation
Because the coulomb repulsion of the protons
throughout the entire nucleus, Ihere
is
appreciable
represented by the bismuth isotope
is
"gBi, which
All nuclei
the heaviest stable nut
is
a limit to the ability of neutrons to prevent
the disruption of a large nucleus. This limit
is
nucleons interact strongly
result
lick*.
with Z
spontaneously transform themselves into lighter ones through the emission of
one or more alpha
FIGURE
1
1-2
ple limits the
particles,
which are
II
le
Simplified energy 'evel diagrams of stable boron
occupancy
of
each
level to
nuclei.
PROTON NUMBER
> 83 and A > 2(19
(7.)
FIGURE 113
Alpha and beta decays permit en unstable nucleus to (each
consists of
two protons and two neutrons, an alpha decay reduces the Z and
a stable configuration.
Since an alpha particle
and carbon Isotopes. The exclusion
princi-
two neutrons of opposite spin and two protons of apposite
the .V of the original nucleus by
two each.
If
the resulting daughter nucleus
spin.
has either too smalt or loo large a neutron/proton ratio for stability,
decay to a more appropriate configuration.
Q neutron
m proton
is
In
it
mav
beta
negative beta decay, a neutron
transformed into a proton and an electron:
u
The
—* p + e~
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
?»
368
i
THE NUCLEUS
>>B
12,
decay increases
13,.
to
it.
Figure
1
1-3
be achieved. Kadioactivity
shows how alpha and beta decays enable
is
considered in more detail
in
Chap.
stability
12.
THE ATOMIC NUCLEUS
369
11.4
NUCLEAR SIZES AND SHAPES
1
The Rutherford
scattering experiment provided the
of finite size. In that experiment, as
we saw
in
Chap.
manner
deflected by a target nucleus in a
is
evidence that nuclei an:
firsl
provided die distance between them exceeds about 10 "
tions the predictions of
Coulomb's law
For smaller separa-
in.
Coulomb's law are not obeyed because the nucleus no
=
1
fin
=
I()-
m
ls
Hence we can write
an incident alpha particle
4,
consistent with
fermi
fl= l,2A 1/a
11.2
From
for nuclear radii.
fin
this
formula
we
find that the radius of the *|C nucleus
is
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
fi
with a nucleus only through electrical forces while a neutron interacts only
Similarly, the radius of the 'J|Ag nucleus
is
a nucleus and neutron scattering provides
information on the distribution of nuclear matter. In both cases the de liroglie
wavelength of the particle must
study {see Frob. 3 of Chap.
lie
smaller than the radius of the nucleus wider
of nuclear matter.
P
number of nucleons
proportional to the
A. If a nuclear radius
proportional
R
n.i
lo
\.
it
contains,
R. the corresponding
is
T]u, relationship
A
tx fl
r
is
iisualls
_
is its mass number
%srH 3 and so R is
which
volume
-
:l
is
expressed
inVGtSe form as
in
,/;s
Nuclear
of
Ru
radii
/{„=
is
1.2
x
10
2
for all nuclei.
indefiniteness in fl„
We
m
Ls
a consequence, not just of experimental error, but
of die characters oi the various experiments: electrons and neutrons interact
differently with nuclei.
from electron
scattering,
The value
of
R„
which implies
is
when
slightly smaller
that nuclear matter
it is
deduced
and nuclear charge
are not identically distributed throughout a nucleus.
As we saw
is
in
X
10" kg/m
Certain
shells
the earlier part of this book, the angstrom unit
(1
A
=
"'
may
known as "white dwarfs," are composed of atoms
have collapsed owing to enormous pressure, and the
approach that of pure nuclear matter.
stars,
If
m)
a convenient unit of length for expressing distances in the atomic realm. For
How
can nuclear shapes
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
guished from that due to the magnetic
10
electrons
:i
have been assuming that nuclei are spherical.
be determined?
The
six
12.nuXl.66x 10-"kg/u
%* X(2.7x lO^m)3
densities of such stars
ls
and binding energies of the
This figure— equivalent to 3 billion tons per cubic inch!— is essentially the same
whose electron
The value
we can compute the density
^C, whose atomic mass is 12.0 u, we have
=
MeV -
111" eV) and neutrons of
MeV to over 1 CeV (1 CeV = 0<K)
MeV and up. In every case it is found that the volume of a nucleus is directly
J
and that of the *§§U nucleus
be neglected here)
Actual experiments on the sizes of nuclei have employed electrons of several
20
5,7 rrn
nuclear sizes as well as masses,
In the case of
for the nuclear density (the masses
3),
hundred
is
7.4 fin.
Xow that we know
through specifically nuclear forces. Thus election scattering provides information
in
X(12)" 3 fm=:2.7fm
a Favored technique.
still
Fast electrons and neutrons are ideal for this purpose, since an electron interacts
on the distribution of charge
= 1.2
moment
of the nucleus, but
be distinwhen this
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
is
!
example, the radius of the hydrogen atom
CO
molecule are 1.13 A apart, and die
are 2.81
A
370
THE NUCLEUS
0.53 A, the
Na + and
is
an appropriate unit of length:
C
and
O
atoms
CI'" ions in crystalline
Nuclei are so small diat the fermi
apart.
of the angstrom,
is
(fin),
in a
XaCI
only 10 -9 the size
oblate spheroids, but the difference
-20
percent and
is
between major and minor axes never exceeds
usually much less. For almost all purposes it is sufficient
regard nuclei as being spherical; nevertheless the departures from sphericity,
small as they are, furnish valuable information on nuclear structure.
to
THE ATOMIC NUCLEUS
371
11.5
\
BINDING ENERGY
mass than the ABU of the moans of
Stable iitnni Invariably has a smaller
2.014l()2ii
while the mass of
T
"Vd^™ +
which
is
'»-.
=
=
+
1-0OT825U
LOOSOflSti
2.016490 V
O.0023N8 n greater. Since a
nucleus
detiteritiin
— called a deuteron—K
composed of a proton and a neutron, and Iwth \U and
electrons,
it is
of a proton
its
atom JH, For example, has a mass of
a hydrogen atom (}ll) plus thai of a neutron is
Tlic deuterium
constituent purlitk-s.
evident that the
and a neutron
to
mass
tiefcrt
form a
of 0.002388 n
dci ileum.
V
j'H
is
have single
orbital
related to the binding
mass of 0.002388
11
is
equal
100
to
200
150
250
MASS NUMBER, A
0.002388 u
X
931
MeV/u =
2.2.3
MeV
FIGURE
When
a deuteron
of energy
is
is
formed from a
free proton
MeV
liberated. Conversely, 2.23
and neutron, then, 2.23
MeV
gamma-ray photon must have an energy
MeV
of at least 2.23
to disrupt
energy and
is
equivalent of the mass defect in a nucleus
a measure of the stability of the nucleus.
from the action of the forces
from them,
forces.
called
to
form
binding
nuclei,
jusl
remove electrons
Binding energies range
MeV for the deuteron, which is the smallest compound
MeV for '-$|Bi, the heaviest stable nucleus,
from 2.23
to 1,640
its
Binding energies arise
which must be provided
from the action of electrostatic
arise
is
that hold uucleons together to
as ionization energies of atoms,
mass number. The peak
st
V
=A
corresponds
The binding energy per nucleoli, arrived at by dividing the total binding energy
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
of a nucleus by the
Fig.
a deuteron (Fig. 11-4).
The energy
Binding energy per nucleon as a function of
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
15
1
to the !He nucleus.
nucleus, up
a
1-5.
1
.
The curve
maximum
rises steeply at first
MeV
of 8.79
A=
at
and then drops slowly to about 7.6
and then more gradually
until
it
reaches
56, corresponding lo the iron nucleus fgFc,
MeV
mass numbers. Evidently
at the highest
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
ones or
if
process
is
light nuclei
known
if
heavy nuclei can somehow
can somehow be joined
as nuclear fission
and the
lie
split into lighter
form heavier ones. The former
to
latter as nuclear fusion,
and both
indeed occur under proper circumstances and do evolve energy as predicted.
FIGURE 11-4 The binding energy at the deuteron is 2.23 MeV. which li confirmed by experiments that
show that a gamma-ray p hoi on with a minimum energy of 2.23 MaV can split a deuteron into a tree neutron and a free proton.
Nuclear binding energies are strikingly
it
is
helpful to convert the figures from
say kcal/kg. Since
that
1
MeV =
3.83
and each nucleoli
tic nil'
/WW
2.2.1
MeV
photon
mi
X
in a
10
1.60
IT
X
kcal.
10- ,n
J
J
=
2.39
is
equal to 1.66
1
nucleus has a mass of very nearly
X
1
MeV
1
nucleon
o
neutron
A
_
IP" 17 kcal
X 10~"
-i v
~ -Ail
x
1.66 X 10-2- kg
3.83
u.
X
10
find
^ kg,
Hence
U \u> kcal
kg
binding energy of 8 MeV/nucleon, a typical value,
to 1.85
is
THE NUCLEUS
we
10-" kcal,
x
10" kcal/kg. By
is
therefore equivalent
contrast, the heat of vaporization of water
540 kcal/kg, and even the heat of combustion of gasoline, 1.13
372
magnitude,
familiar units,
unit
and
One mass
their
more
to
proton
i
®
eV =
1
To appreciate
large.
MeV/ nucleon
x KH
is
a
mere
kcal/kg,
10 million times smaller.
THE ATOMIC NUCLEUS
373
THE DEUTERON
•11.6
The unique short-range forces that hind nucleonsso securely into nuclei
by
is still
and in consequence the
compared with the theory oi
primitive as
Let us choose the particle
it
atomic structure. However, even without a satisfactory understanding of nuclear
been made
forces, considerable progress has
in recent years in interpreting the
we
properties and behavior of nuclei in terms of detailed models, and
examine some of the concepts embodied
is
in
is
2.23
is
shall
what can
in'.
and a neutron.
can be obtained either
»i + m.
p
gamma rays
and
.
that only
can disrupt dcuterons into their constituent nucleons.
If
Schrodinger's equation.
is known for an
V can be found and
a force law
corresponding potential energy function
or from photo-
with hv
>
2.23
Chap. 6 we
In
the deuteron in as
The
interaction, the
much
not possible to discuss
it is
We
negative and
is
note from Fig.
1 1
-6 that E, the total
energy of the
the
necessary to consider the effects of nuclear motion, and
it is still
Chaps. 4 and
In this
way
center of mass
tin;
is
by replacing the electron mass
ft
A
similar procedure
m'=
hi
did
a
common
in'
moving
According to Kq.
4,27, the
is
in
"
in.
we
reduced mass
even more appropriate here, since
is
reduced mass of a neutron-proton system
we
its
problem of a proton and an electron moving about
neutron and proton masses are almost the same.
and so
by
replaced by the problem of a single particle of mass
about a fixed point.
11.4
mf
+
replace the
"
in.
m
of Eq. 11.3 with
in'
as given alnive.
FIGURE
1
1-6
The actual potential energy of either proton or neutron In a deuteron and the square-well
this potential energy at functions of the distance between proton and neutron.
approximation to
quantitative detail as the hydrogen atom.
actual potential energy
V
of the deuteron, that
the potential energy
is.
of either nueleon with respect to the other, depends upon the distance
the centers of the neutron and proton more or
in Fig. 11-6.
is
substituted into
Our understanding of nuclear forces is less complete than
our understanding of coulomb forces, however, and so
we imagine
same as the binding energy of the deuteron.
In analyzing the hydrogen atom, where one particle is much heavier than the
neutron,
The simplest nucleus containing more
figure that
be the neutron, so that
in question tn
l>c
analyzed another two-body 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.
(-2)
chapter. Before
instructive to see
m dcula on
which show
a0
of the proton. (The opposite choice would, of course,
in the force field
yield identical results.)
this in
MeV, a
mass between
disintegration experiments
MeV
it
the deuteron, which consists of a proton
The deuteron binding energy
From the discrepancy
in this
moving
other,
revealed by even a very general approach.
than one uucleoo
models
in these
looking intq any of these theories, though,
r- sin
constitute
as well understood as electrical forces,
theory of nuclear structure
i
Unfortunately nuclear forces are
known.
far the strongest class of forces
nowhere near
&H9
11,3
less as
(The repulsive "core" perhaps 0,4
shown by the
X W~a m
r
between
solid line
in radius expresses
mesh together more than a certain amount.) We
by the "square well" shown as a dashed line in the
the inability of nucleons to
shall
approximate
figure.
this V(r)
This approximation means that
neutron and proton to be zero
when
we
consider the nuclear force between
they .are more than
r„
apart,
a constant magnitude, leading to the constant potential energy
are closer together than
r„.
Thus the parameters Vn and
r
and
to
- V when
,
have
they
and
the square-well potential
itself is
A
is
;si:e
r
alone, and therefore,
easiest to
examine the problem
a function of
it is
Rg. fill
nates Sehrodinger's equation for a particle of mass
374
THE NUCLEUS
actual potential
kinetic energy
T
potential energy
V
energy of nueleon
square-well approximation
V
as in the case of other central-force potentials,
in a spherical polar-coordinate system
E
MeV
representative of the short-range
character of the interaction.
square-well potential means that
energy
-2.23
are representative
of the strength and range, respectively, of the interaction holding the deuteron
together,
total
In spherical polar coordi-
m
is,
with
ft
=
fc/2w,
THE ATOMIC NUCLEUS
375
We now
of radial
assume that the solution of Eq. 11.3 can be written as the prod net
and angular
115
functions,
=
+</,*,)
u.ii
We
ii
fl(l)©(ff)*fc)
As before, the function R(r) describes
radius vector from the nucleus, with $
if,
with
r
angle
<ji
coaslant;
<f>
<*
the
wave
function
^
and the function
wave
recall that the radial
must
=
and
function
inside the well
«j
fi
discarded
lie
is
is
given by
if fi
is
fi
=
which means
»/r,
not to be infinite
at
r
=
0.
just
varies along a
constant; the function 8(0 ) describes
along a meridian on a sphere centered at
varies with zenith angle
and
how
and
+ B sin ar
cos ar
that the cosine solution
Hence A
how
= A
1
=
r
11.12
0,
II
=
|
B
Outside the well
how ^ varies with azimuth
= 0, with r and 9 constant.
ar
'.ill
=
V*
and so
*{$>) describes
along a parallel on a sphere centered
at r
2
<i
2m'
«„
_,
11.13
Although angular mutton can occur
lor the
moment
in radial motion, that
is
proton about their center of mass.
square-well potential, uur interest
in a
is,
If there
neutron and
in oscillations of the
is
no angular motion,
both constant and their derivatives are zero. With
dip/riS
=
<->
and
d^ift/d^2
=
The
energy
total
are
<t>
0,
11.14
W^f)***-**-"
11.6
A
further simplification can be
made by
is
=
fi*
=
u(r)
In terms of the
d 2 u,
letting
P.
dr
tR{t)
new
function u the
The
wave equation becomes
«„
Because
it,
fi
two
Because
different solutions to Eq.
wave equation
Inside the well the
2m'
or, if
we
1.8,
ti
t
for r
^
ra
and
i/
([
outside
for r
>
r
.
o
2m'
= ^-(E
+ VJ
2
rf
2
+
(We note from
can write
is
+ De "
1
—*
as r
—
oo,
we conclude
that
D=
0.
Hence
=
Ce'
bf
GROUND STATE OF THE DEUTERON
and
where
11.18
a2 ">
*
box of Chap.
THE NUCLEUS
"
have expressions
it
remains
to
for u (and
hence for
>£)
match these expressions and
it is
in the region for
both inside and outside the
their
necessary that Ixith u and du/dr
which they are defined. At
r
=
first
derivatives at the
!>e
continuous every-
rn , then.
5,
fi
sin
ara
»
Ce "*
and
Fig. 11-6 that, since |V
are positive.) Equation
in a
376
1.15
Ce~ bT
well boundary since
w,
ir*
a
bound to
becomes simply
11.10
2
1
must be true that u
un
We now
(P
it
it
ill?
*11.7
let
fl
is
outside the well
well,
11.9
it
is
v )Ul =
^? + -p-<£ +
1
a negative quantity since
o
2
V has two different values, V = - Vu inside the well and V =
there are
=
we
- ft* =
solution of Eq.
n.16
dr-
is
~(-E)
a positive quantity, and
11.15
11.7
of the neutron
Eq.
becomes
11.3
E
the proton. Therefore
1 1
.
10
is
the
|
>
|£|,
the quantities
E + V and
same as the wave equation
and again the solution
is
hence
for a particle
11.19
du,
du u
dr
dr
«Bcosflr„
= —bCe~ br
>
THE ATOMIC NUCLEUS
377
By dh iding Eq.
by Eq. 11.19
1.18
!
we
eliminate the coefficients B and
C
and
HI,
obtain the transcendental equation
tun
11.20
= ——
rir
approximation
first
Equation 11.20 cannot
solved analytically
lie
.
but
can be solved either
it
graphically or numerically to any desired degree of accuracy.
+
\Z2m\E
where
VW(-E)/fi
—£
the binding energy of the deuteron and
is
Since
potential well.
we might assume
that
tan urn
a
becomes
Since tan
j
V
|
>
a/b
i
is
note that
=m
V„)/ft
11.21
h
We
E|, in
order to obtain a
V;,
first
the depth of the
is
crude approximation
FIGURE 117
so large that
The wave function
«(r) ol either proton or neutron In a deuteron.
oo
and therefore representative of the range of nuclear
infinite at
=
htt/2, in this approxi-
77/2, w, .377/2
mation the ground state of the deuteron, for which »
=
1,
correspond to
two nucleons due
1
;
forces,
Vn
the depth of the
,
What we now
to nuclear forces, and
ask
is
whether
— E, the binding energy of the deuteron.
this relationship
corresponds to reality in the sense
leads to a reasonable value for
that a reasonable choice of r„
V
(There
.
in our simple model that can point to a unique value for either
(In fact, this
Is
the only hound state of the deuteron.)
V2m'{£
+ vy
°~
vSS%
r
ft
since
we
are assuming that
E
is
Hence
a choice for r„
11.22
2 fm. Substituting for the
l>e
and expressing energies
MeV
in
known
r
()
or
is
V
nothing
.)
quantities in Eq.
Such
1
1.23
yields
*
"~2
negligible
compared with V
,
,
1.0
.
X
™ MeV-m*
10
,
1.9
X
HI'
14
MeV-m
and
and
v„ ss
might
r
n
fwtween
well and therefore representative of the interaction potential energy
=
so, for r
V„
Hm'rr -
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. 11-7; a more
detailed calculation shows that ar zz 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. 1.22, and when this neglect is remedied, the l>etter approximation
This
is
2
fm
=
x
2
u
m,
= 35 MeV
a plausible figure for
of our model
10
— nucleons
V
()
that
from which
,
we
conclude that the basic features
maintain their identities
in the
(using together, strong nuclear forces with a short range
angular dependence
— are
nucleus instead of
and
relatively
minor
valid.
1
(,
1
11.23
V'„
=
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
obtained.
Equation
378
a binding energy of
1 1
THE NUCLEUS
.23
is
a relationship
among
rtl ,
the radius of the potential well
Among
— E.
To go
further
the most significant such data
we
is
is
a stable system with
require additional experimental data.
that
concerning angular momentum,
THE ATOMIC NUCLEUS
379
which was ignored
in the
preceding analysis of the deuteron when
that the neutron-proton potential
is
a function of
momentum
correct; in general, angular
dot's
r
was assumed
it
The
alone.
latter
not
is
play a significant role in nuclear
structure, although certain aspects of this structure are virtually independent
In the case of the deuteron, for example, the proton and neutron interact
of
it.
in
such a manner that binding occurs only when their spins are parallel to produce
when
a triplet state, not
their spins arc ant [parallel to
produce a
ving/*-/ State.
Evidently the neutron-proton lone ilepemls on the relative orientation of the
spins
and
The
weaker when the
is
makes
independence of nuclear
a
rli
The
forces.
neutron from occurring
nucleoli's in
possible to see
it
why
diprntons and di neutrons
despite the observed stability of the deuteron antl the charge-
exist
in
a triplet stale, since with parallel spins both
each system would be
in identical
quantum
distinguishable particles even with parallel spins.
is
in
in
packed together into the smallest volume,
is
contact with
(Fig.
it
as
we suppose
is
X
1
1-S).
Hence each
14L7 or 6L'.
If all
interior nucleoli in a nucleus
A
has a
nucleons in a nucleus were in
its
binding energy of the nucleus would he
interior, the total
EV = 6AU
H.Z4
Equation
1 1
.'24
often written simply as
is
states.
No such
restriction
consists are
it
However, while
a
EB =
11.25
The energy
£',,
a
is
t
A
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
The number of such nucleons depends
therefore have fewer than 12 neighbors.
upon the surface area of the nucleus
in question.
A
nucleus of radius
R
has an
area of
diproton
principle in a singlet state, the singlet nuclear force
not sufficiently strong to produce binding
— and, indeed, diprotons and dineu-
Hence
trons have never been found.
is
the
THE LIQUID-DROP MODEL
number
of nucleons with fewer than the
maximum number
of bonds
proportional to A'i/3 , reducing the total binding energy by
E,
11.26
11.9
size
binding energy of 12
exclusion principle prevents a diproton or
applies to the deuteron, since the neutron and proton of which
or dineutron could occur
same
the case of nucleons within a nucleus, each interior sphere has 12 other spheres
spins are anliparallel.
difference between the triplet and singlet potentials, together with the
Pauli exclusion principle,
do not
of the
The
= -«jA 2/;!
negative energy
Et
is
called the surface energy of a nucleus;
it
is
most
significant for the lighter nuclei since a greater fraction of their nucleons are
While the
attractive forces that nucleons exert
their range
is
so small that
each particle
upon one another are very
in a
strong,
on the surface. Because natural systems always tend
nucleus interacts solely with
Its
tions of
nearest neighbors. This situation
the
is
same
as that of
atoms
in a solid,
minimum
to evolve
toward configura-
potential energy, nuclei lend toward configurations of maxi-
which
ideally vibrate about fixed positions in a crystal lattice, or that of molecules in
a liquid,
which
ideally are free to
The
molecular distance.
move about while maintaining
a fixed inter-
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
9>
of nuclear behavior.
Let us see
how
FIGUREll-3
the picture of a nucleus as a drop of liquid accounts for the
observed variation of binding energy per nucleoli with mass number.
start
shall
sphere
li
In
packed assem-
each
Interior
contact with twelve others.
by assuming that the energy associated with each nucleoli- micleon bond
has some value
involved, but
is
I';
this
energy
is
really negative, since attractive forces are
usually written as positive [>ecause binding energy
a positive quantity for convenience.
Because each liond energy
two nucleons, each has a binding energy of %['.
380
We
In a lightly
bly ot Identical spheres,
THE NUCLEUS
When
V
is
is
considered
shared by
an assembly of spheres
THE ATOMIC NUCLEUS
381
mum
binding energy, {We recall that binding energy is the mass-energy difference between a nucleus and the same numbers of free neutrons and protons.)
Hence a nucleus should
and
in
exhibit the
same surface-tension
the absence of external forces
it
effects as a liquid drop,
should be spherical since a sphere has
the least surface area for a given volume.
The
electrostatic repulsion
contributes toward decreasing
nucleus
into a
Z(Z
-
is
between each pair of protons in a nucleus also
its binding energy. The coulomb energy 2*. of a
the work that must be done to bring together
volume equal
l)/2, the
number
£,
Z{Z
= -a
1
The coulomb energy
protons from infinity
and co u tomb energies.
is
proportional to
of proton pairs in a nucleus containing
inversely proportional to the nuclear radius
11,27
Z
Hence Ec
that of the nucleus.
to
-
=
/i,,A
Z protons, and
1/3
:
1)
1/3
-10
negative because
is
li
i
FIGURE 11-9 The binding energy per nucleon is ttic sum of the volume, surface,
arises
it
from a force that opposes
nuclear stability.
The total binding energy Eb
and coulomb energies:
Efc
= £v +
= a,A
11.26
The binding energy
+
E,
a 2A
2/3
E,
—
of a nucleus
is
the
sum
of
its
volume, surface,
THE SHELL MODEL
11.10
The
—a
per nucleon
is
Z(Z
model
is
that the constituents of a nucleus
interact only with their nearest neighbors, like the molecules of a liquid.
-
1)
A 1/3
however,
extensive experimental evidence for the contrary hypothesis that the
nucleoli.-,
a good deal of empirical support for Ibis assumption. There
is
in a nucleus interact primarily with a general force field raider than directly
therefore
with one another. This latter situation resembles that of electrons
E,
11.29
Z{Z
=
ai
" ^TTa- "
"3
-
A V3
1
shown
in Fig. 11-5.
it
and
in Fig. 11-9 versus
A, together with
close to the empirical curve ot
the analogy of a nucleus with a liquid
/..
\
drop has some
we may
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,
and neutrons are especially stable, as are
nuclei with even numbers of protons and neutrons. Thus such nuclei as |He.
l
curve.
'gC, and gO appear as peaks on the empirical binding energy per nucleon
These peaks imply that the energy states of neutrons and protons in a nucleus
nuclei with equal
numbers
certain
quantum
slates are
in
an atom,
permitted and no more than two electrons,
particles, can occupy each state, N'ucleons are also Fermi
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"
which are Fermi
be encouraged to see what further aspects of nuclear
can illuminate. We shall do this in Chap. 12 in connection with
validity at least,
liehavior
Hence
where only
1)
Each of the terms of Eq. 1.29 is plotted
their sum, E^/A. The latter is reasonably
There
also,
is
1
basic assumption of the liquid-drop
of protons
are almast identical and that each state can be occupied by two particles of
particles,
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
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
of situation
among lighter ones, the evolution of heavier and heavier
became retarded when each relatively inert nucleus was formed; this
arose from reactions
nuclei
accounts for their abundance.
opposite spin, as discussed in Sec. 11.3.
THE ATOMIC NUCLEUS
382
THE NUCLEUS
383
Other evidence
also points
up the
numbers
significance of the
spin-orbit coupling
2, 8, 20, 28,
villi
spin-orbit coupling
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
50, 82,
A
from sphericity.
shaped
spherical nucleus has no quadrupote
2j
pumpkin
like a
are found to have zero
4s5d-
quadrupole moments and hence are spherical, while other nuclei arc distorted
6g"
moment.
has a negative
in
N
Nuclei of magic
and
Z
*^
«£9/2
7i-
model of the nucleus
shell
A
used that corresponds to a square well
4p-
MeV deep with rounded corners, so that there is a more realistic gradual
from V = V to V =
than the sudden change of the pure square-well
5i"
potential energy function
we
potential
in
used
in
is
14
4PI/2
2
4
6
this
kind
is
then solved, and
it
12C
.12
82
22
50
8
28
IS
ill
8
10
system occur characterized by quantum numbers
significance
is
same
«,
/,
sets of states in a
""*-
and m, whose
atomic
3s-
nucleus since
4d-
as in the analogous case of stationary states of
Neutrons and protons occupy separate
6ft-
found that stationary
is
states of the
electrons.
14
treating the deuteron, Schrodinger's equation for a particle
a potential well of
the
total
nucleons
6
10
*- 7i
13/2
about 50
change
shell
an attempt to account for the existence of
is
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.
1
12
shape.
The
+
16
4
2
8
moment, while one
moment and one shaped
like a football has a positive
nutieons nucleons
per
per level
__
6h 11/2
Ad.3/2
AJt,-.,':
„.*<•« " %7/s
the latter interact electrically as well as through the specifically nuclear charge.
12
2
4
6
8
In order to obtain a series of energy levels that leads to the observed magic
numbers,
tude
is
it is
merely necessary to assume a spin-orbit interaction whose magni-
such that the consequent splitting of energy levels into subtevels
for large
I,
that
is,
for large orbital angular
momenta.
coupling holds only for the very lightest nuclei,
sarily small in their
normal configurations. In
the intrinsic spin angular
momenta
is
which the
in
this
It
scheme, as
/
and the
S,
a total angular
in
values are neces-
we saw
in
and
L
6
4
-4/ 7/!
total spin
f
\/J(J
+
1) ft.
holds, the heavier nuclei exhibit
first
_„__-
2s.
3d 3/2
1/2
3d'
After a transition region
of each particle are
t
10
4/-
7,
L; S and
momentum J of magnitude
In this case the S,
Chap.
momenta L are separately coupled
L are then coupled to form
momentum
which an intermediate coupling scheme
coupling.
3p-
S ( of the particles concerned (the neutrons
orbital angular
together into a total orbital
Sge/2
*Pl/2
4 'S/2
large
assumed that L$
form one group and the protons another) are coupled together into a
momentum
is
3d 5/2
fj
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
Xk
2 J>l/2
2p-
2p,V2
for the great majority of nuclei.
When
an appropriate strength
energy levels of either
The
levels are designated
letter that indicates I for
384
THE NUCLEUS
is
assumed
for the spin-orbit interaction, the
class of nucleoli fall into the
by a
sequence shown
prefix equal to the total
each particle
in that level
in Fig.
1
quantum number
Is.
1/2
1-10.
n,
a
FIGURE 1110
Sequence
of
nuclecn energy levels according to the shell model (not to scale).
according to the usual pattern
THE ATOMIC NUCLEUS
385
p, d,f, g,
(s,
equal to
.
correspond respectively to
.
.
The
j.
in the
in
.
.
,).
4
allowed orientations of J
1
The number
.
(
and protons
a nucleus with
2,
(i,
and 44; hence
12, 8, 22, 32,
when
sublevels are present
filled
in
shells arc
in
sublevels
liorne out
is
by the
and thereby
In essence, then, the exclusion
in initially.
the independent-particle approach to nuclear structure.
justifies
Both the liquid-drop and
shell
models of the nucleus
much
ways, able to account for
that
known
is
are, in their
very different
Recently
of nuclear liehavior,
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
The
the endeavor.
particles (spin
odd numbers of neutrons and protons ("odd-odd" nucleus) contains
filled
was
resulting
spherical shape of
possibility of a nucleus
model includes the
The
vibrating and rotating as a whole.
there are even numl>ers
stability
it
addition
a nucleus ("even-even" nucleus). At the other extreme.
The
state
I
in
account for several nuclear phenomena
to
unfilled sublevels for lx)lh kinds of particle.
conferred by
+
same
principle prevents nucleon-uucleon collisions even in a tightly packed nucleus,
energy gaps
20, 28, 50, 82, or I2K neutrons or protons in a nucleus.
magic numbers. Since each energy sublcvel can contain two
up and spin down), only
of neutrons
into 2/
;
Larftic
of available nuclear states in each nuclear shell
ascending order of energy,
filled when there arc 2, 8,
The shell model is able
to
ami a subscript
spacing of the levels at intervals that are consistent with the notion
of separate shells.
is.
in exactly the
0, J, 2. 3, 4,
spin-orbit interaction splits each state of given
substates, since there are 2/
appear
=
/
situation
is
complicated by the non-
but even-even nuclei and the centrifugal distortion experi-
all
enced by a rotating nucleus; the detailed theory
is
consistent with the spacing
of excited nuclear levels inferred from the gamma-ray spectra of nuclei and in
other ways.
we
expect to he
160 stable even-even
fact that
nuclides arc known, as against only four stable odd-odd nuclides:
-]\, !*Li,
'JJB,
Problems
and 'IN.
Further evidence
in
favor of the shell model
angidar momenta. In even-even nuclei,
all
is its
ability to predict total nuclear
the protons and neutrons should pair
cancel out one another's spin and orbital angular momenta. Thus
even-even nuclei ought to have zero nuclear angular momenta, as observed. In
off so as to
even-odd (even Z, odd N) and odd-even (odd Z, even
spin of the single "extra" nucieon should be
momentum
jV) nuclei,
the half-integral
combined with the
integral angular
of the rest of the nucleus for a half-integral total angular
momentum,
1
mass
2.
is
6.01513
A beam
uniform magnetic field of dux density 0.2 T.
(1
3.
the nuclenns in a nucleus are so close together
and
interact so strongly thai
how can these same
of each other in a common force
the nucleus can be considered as analogous to a liquid drop,
nucleons be regarded as moving independently
field as
required by die shell model?
mutually exclusive, since a nucleoli
would seem that the points of view are
moving alxiut in a liquid-drop nucleus must
It
closer look shows that there
is
nucleus, the neutrons and protons
fill
ol increasing
1
1-2).
energy
in
way
such a
In a collision, energy
is
no contradiction.
In the
ground
the energy levels available to
all
them
in
order
it is
possible for
if
the exclusion principle
is
so such an energy
violated.
two nucleons of the same kind to merely exchange
tive energies, but such a collision
THE NUCLEUS
filled,
is
(The §Li atomic
in
Ordinary boron
is
radii of the
the magnetic
eV enters a
move perpendicular to
What
path of the '°B (10.013
u)
and ",H
field.
a mixture of the
u.
The
ions
l
|JB
and 'JB isotopes and has a composite
percentage of each isotope
is
present in ordinary
boron?
4.
Show
density.
that the nuclear density of
[H
is
10 M times greater than
(Assume the atom to have the radius of the
298 MeV. Find
5.
The binding energy
6.
The mass
7.
Find the average binding energy per nucieon
as to ol>ey the exclusion principle (see Fig,
transferred from one nucieon to another, leaving
the available levels of lower energy are already
transfer can take place only
field.
of ffCl
is
its
first
Bohr
mass in
its
atomic
orbit.)
u.
state of a
the former in a state of reduced energy and the latter in one of increased energy.
But
1.009 u) isotopes
Find the
atomic weight of 10.82
surely undergo frequent collisions with other nucleons.
A
the magnetic
of singly charged lx>ron ions with energies of 1,000
half-integral spins should yield integral total angular
It
in
ions
u.)
the field direction.
momenta. Both of these
The
of flux density 0.08 T.
Find the radius of their path
direction.
eV enters a uniform
move perpendicular to the field
of singly charged ions of § I A with energies of 4(X)
field
and odd-odd nuclei each have an extra neutron and an extra proton whose
prediction are experimental!) confirmed
386
A lieam
magnetic
Of
course,
'|Oatom
8.
9.
19.9924
jj{]Ne is
u.
Find
its
binding energy
in '!jO (the
MeV.
in
mass of the neutral
15.9H49u).
How much
energy
of the neutral '?N
their respec-
hardly significant since the system remains
is
of
How much
atom
energy
is
is
is
required to remove one proton from 'fO? (The mass
15.0001 u; that of the neutral
'jjO
atom
required to remove one neutron from
!
is
15.0030
u.)
gO?
THE ATOMIC NUCLEUS
387
Compare
tO,
minimum energy a gamma-ray photon must possess if it is
an alpha particle into a triton and a proton with that it mast
to disintegrate an alpha particle into a file nucleus
and a neutron.
the
NUCLEAR TRANSFORMATIONS
to disintegrate
posses;
if it is
(The atomic masses of fU and pie are respectively 3.01605 u and
3.01603
1
Show
1.
apart
is
u.i
that the electrostatic potential energy of two protons 1.7
X 10" l5
of the correct order of magnitude to account for the
difference in binding
m
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
oliey Bose-Einstein statistics while
13.
is
Show
alx>ul
35
that the results of Sec.
MeV
deep and 2
fin in
all
have spins of l/2 Why do iHe atoms
ol>ey Fermi-Dirac statistics?
.
|He atoms
1
1.7— a
potential well for the deuteron that
radius— are consistent with the uncertainty
principle.
14.
Calculate the approximate value of
^
Eq.
1
1.27 using
whatever assump-
seem appropriate.
15.
According to the Fermi gas model of the nucleus, its protons and
neutrons
box of nuclear dimensions and fill the lowest available
quantum
to the extent permitted
protons have spins of
%
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.
in
tions
exist in a
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-
The
principal aspects of radioactivity
12.1
states
RADIOACTIVE DECAY
by the exclusion
principle. Since both neutrons and
they are Fermi particles and must obey Fermi-Dirac
Perhaps no single phenomenon has played so significant a role
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-
of lx>th atomic and nuclear physics as radioactivity.
tons are present, (h) Find the Fermi energy for such a nucleus in
which R,
1.2 fm.
(beta particle), or a photon
statistics,
and
nuclear reactions are discussed in this chapter.
(a)
=
radioactive decay spontaneously emits a
tion
{gamma
A
^He nucleus (alpha
ray),
thereby ridding
energy or achieving a configuration that
is
in the
development
nucleus undergoing
particle),
itself
an electron
of nuclear excita-
or will lead to one of greater
stability.
Tile activity of a
nuclei of
its
a certain time
12.1
sample of any radioactive material
constituent atoms decay.
in the
sample,
its
If
activity
X
Ls
R
is
is
the rate at which the
number
given by
the
of nuclei present at
R=-M
dt
The minus
sit>n
i.s
inserted to
make R
a positive quantity, since di\/di
is,
of course,
While the natural units for activity are disintegrations
customary to express ft in terms of the curie (Ci) anil its
intrinsically negative.
per second,
it
is
submultiples, the millicurie (mCi) and miaocurie
Ci
=
3.70
mc =
H>- 3 Ci
3.70
=
10- 6 Ci
=
=
1
1
1
388
THE NUCLEUS
fie
3.70
X
X
X
(fiCi).
By
10 10 dismtegrations/s
1<)
7
dismtegrations/s
lO" disintegrations/!!
definition.
12
Experimental measurements on the activities of radioactive samples Indicate
Figure 12-1 is a graph
that, in every ease, they fall off exponentially with time.
of
fi
versus
when
icg&l dless of
We
for a typical radioisotope.
t
note that in every 5-h period,
what
it
was
of the isotope
is
5
the period starts, the activity drops to half of
at the start of the period.
Accordingly the half
Every radioisotope has a characteristic half
m
T
life
li.
some have half lives of
range up to billions of years.
life;
millionth of a second, others have half lives that
the observations plotted in Fig. 12-1 began, the activity of the sample
After another 3 h, fl again decreased
R„. Five h later it decreased to 0.5fl
When
was
.
((
by a factor of 2
its initial
That
to 0.25fl„.
value after an
is,
interval of
the activity of the sample
2Tlri
.
of 5 h. corresponding to a total interval of 3T,
illustrated in Fig. 12-
The behavior
1
fl
information alxiut the time variation of activity
=R
fl
12.2
where
e
became
indicates that
WW
only 0.25
lapse of another half life
With the
,j;{0.25R u ), or 0.I25R,,.
l
we can express our empirical
in
the form
""
X, called the tlvanj constant, has
a different value for each radioisotope.
The connection between decay constant A and half life Th 2 is easy to establish.
After a half life lias elapsed, that is, when t = T1/2 the activity fl drops to %R
FIGURE
12.
The
1
activity of a radioisotope
decreases exponentially with time
.
(I
,
by
If
Hence
definition.
the .sample
fraction of
R
'/.fl,,
e^
T \/t
= fl,,*-*'
= /v "*
=2
bility for
it
large
is
— that
is,
if
many
life
5
of
— the
actual
to the
proba-
nuclei are present
span will be very close
any individual nucleus to decay. The statement that
sotope has a half
••
enough
that decays in a certain time
h, then, signifies that
a certain radioi-
every nucleus of
this isotope
This does not mean
has a 50 percent chance of decaying in any 5-h period.
a 100 percent probability of decaying in 10 h; a nucleus does not have a
Taking natural logarithms of
XT, /2
_
=
_
,1/2_
12.3
lioth sides of this et|uation,
and
X
decay probability per unit time
2
Half
life
X
5-h interval the probability
constant of the radioisotope whose half
X
=
life is
5 h
is
therefore
is
20
is
for the
'1/2
0.693
X 3,600 s/ta
= 3.85 X 10- 5 s"'
h,
and so on, liecause
nuclei, the
number dX
X and
nuclei
VIA
dN= -XXdt
every
that
decay
from the assumption
dt.
in a
(It is
If
the probability that any
a sample contains
time dt
is
V
uu-
the product of the
the probability A dt that each will decay in
number of
in
decay of each nucleus of a given
the probability per unit time, A
nucleus will undergo decay in a time interval
decayed
memory,
actually does decay.
30 percent.
X per unit time
of a ((instant probability
isotope. Since X
0.693
it
activity law of Eq. 12.2 follows directly
The empirical
The decay
constant until
to 87.3 percent in 15 h, to 93.75 percent in
0.693
"
is
implies a 75 percent probability of decay in 10 h, which increases
A half life of 5 h
ln2
In
its
dt.
That
is.
5 h
The fact
that radioactive
evidence that
this
decay follows the exponential law of Eq.
phenomenon
is
statistical in nature:
12.2 is strong
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
where the minus
12.4
can
sign
is
required because
\"
decreases with increasing
f.
Equation
rewritten
lie
dN = -\dt
N
NUCLEAR TRANSFORMATIONS
391
and each side can now be integrated:
rf'
f'
v
=
lnJV— lnN
A
Equation 12.5
time
J
is
and the number
Nn
Thus the
number
undecayed nuclei
at
=
t
Since the activity of a radioactive sample
N of undccayed nuclei
activity of the
R
at the
and
so
1
gm
x
10M aloms/kmol
is
0.
defined as
It is
dt
as
its
worth keeping
mean
sample
6.69
10 21 atoms
x
is
=\N
= 7.83 X
= 5.23 X
= 141 Ci
dN
=
fl
6.025
=
decay probability per unit time X of the isotope involved
of
X
-At
a formula that gives the
in terms of the
10- 5 kmol
X
1.1 1
JV=-,V e-*'
12.5
of any isotope contains Avogadro's numlier of atoms,
of -$Sr contains
dK
V
fr*
One kmol
TO" 10
10
12
X
6.69
X
1021 s" 1
s-'
mind thai the half life of a radioisotope is not the same
The mean lifetime of an isotope is the reciprocal of its
in
lifetime T.
decay probability per unit time:
we
see that, from Eq. 12.5,
= kN
R
e~ M
This agrees with the empirical activity law
R„
or, in general,
Hence
if
= XNU
f
1 -
12.8
if
T
R = KN
12.6
'-*
12.7
is
half
Evidently the decay constant X of a radioisotope
is
the
same
i-
"
Tu2
—
-
A
0.693
nearly half again
life is
5 h
is
"
I
447'
'
Mean
u2
more than
7'
l/2 .
The mean
lifetime
lifetime of an isotope
whose
7.2 h.
as the probability
per unit lime for the decay of a nucleus of that isotope.
Equation 12.6 permits
if
we know
its
its
to calculate the activity of a radioisotope
mass, atomic mass, and decay constant.
us determine the activity of a 1-gm sample of jgSr,
decay
is
28
yr.
The decay constant
of
^Sr
whose
sample
As an example,
half
life
12.2
against beta
Most of the radioactive elements found
is
series*
A
=
of a nucleus
0.693
X
7.83
10 T s/yr
kg
90 kg/kmol
A
12.9
where n
THE NUCLEUS
The reason
by
=
4.
all ul-
that there are exactly
Thus the nuclides whose mass numl>ers are
all
number
given by
4rj
10- 1 "*- 1
A kmol of an isotope has a mass very nearly equal to the mass number of
isotope expressed in kilograms. Hence 1 gm of jgSr contains
392
members of four radioactive
four such series follows from the fact that alpha decay reduces the mass
28yr X3.I6 X
ltl- a
in nature are
with each series consisting of a succession of daughter products
timately derived from a single parent nuclide.
(1.693
"1/2
=
RADIOACTIVE SERIES
let
that
is
an integer, can decay into one another
in
number. Radioactive nuclides whose mass numbers
he tnemljers of the 4n
scries.
The members
descending order of mass
otiey Eq. 12.9 are said to
of the 4n
+
I
series
have mass
numbers specified by
=
1.1
1
X
H)" a kmol
12.10
A = 4n +
I
NUCLEAR TRANSFORMATIONS
393
Table 12.1.
FOUR RADIOACTIVE SERIES.
Mass numbers
Thorium
)M
v
Half
life,
-Vj-Tli
1.39
2.S5
X
X
10'"
:N>
10"
H
Neptunium
>M'I.
4»
4-
*n
+1
I'rrmium
,,-,.
4.51
X
10"
>gft
+
Actinium
•go
7.07
x
10*
=!gpb
tn
1
3
FIGURE
12-3
decay series
decay of
The neptunium
- 4n r 1). The
i-\
'!,',B\
may proceed
either
by alpha emission and then beta
and mcmticrs of the 4n
+
+
2 and 4n
.1
series
have mass numbers specified
emission or
In
the reverse order.
>
respectively In
12.11
A = 4n +
2
12.12
,4
=ln +
3
The meml>ers
of eaeli of these series, too, can decay into
one another
in
descend80
ing order of mass number.
Table
12.1
is
a
list
of the
names
with the estimated age
series are not
found
in
(—
is
half life of
neptunium
is
so short
They have, however,
nature today.
given in Sec. 12.12.
compared
members
10"' yr) of the universe that the
the laboratory by the neutron
cussion
The
SB
92
four important radioactive scries, their
of
parent nuclides and the half lives of these parents, and the stable daughters which
are end products of the series,
84
f>een
of this
produced
bombardment of other heavy nuclei; a
The sequences of alpha and lieta decays
in
from parent to stable end product
Some
nuclides
chain branches
12-2
sion
each
series are
shown
in Figs. 12-2 to 12-5.
'^jjBi,
a
member
"iifPo
that the
decav
of the thorium scries, has a
n'li.'S
and a 33.7 percent chance of alpha
that lead
series {A
The thorium decay
either
them. Tims
in
by beta or alpha emission, so
brief dis-
arias (i = 4»). The decay at
may proceed
at
either
percent chance of beta decaying into
FIGURE
FIGURE
may decay
Br
by alpha emis-
and then beta emission or
In
-'i'Bi
12-4
4n
The uranium decay
2). The decay of
-
may proceed
either by alpha
,
^
>1
emission and then beta emission or
In the
reverse order.
130
lha reverse order.
fc—
—
394
THE NUCLEUS
•
a
decay
decay
NUCLEAR TRANSFORMATIONS
395
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,
Q
the kinetic energy
This
nucleus.
is
Qwhere m,
FIGURE 12-5 The actinium decay
series (\ = 4n
3). The decays
of ; Ac and ^/Bi may proceed ei-
I
*
fher by alpha emission and then
beta emission or
in
the reverse
and
mn
released
(m,
various particles are emitted bv a
heaw
- m, - m
the mass of the
is
when
given by
initial
m
nucleus.
We
the alpha-particle mass.
find that
f
the mass of the
final
nucleus.
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
order.
a proton
if
is
to
MeV would somehow have
MeV if a jjHc nucleus
be emitted and 9.6
The observed disintegration energies
in
Ls
to
be furnished
to
be emitted.
alpha decay agree with the corresponding
predicted values based upon the nuclear masses involved.
The kinetic energy Ta of the emitted alpha particle is never quite equal to
the disintegration energy
because, since momentum must be conserved, the
Q
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,
2
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
decaying into
2
lead to
^1
J$T1.
Ta
related to
J
I».
Several alpha-radioactive nuclides whose atomic numbers are
less
Q and
Q=
5.587
MeV
While a heavy nucleus can,
while
Because the attractive forces l)etwecu nudeons arc of short range, the
approximately proportional to
contains.
it
The
its
total
mass number
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 short-range nuclear forces that hold them together
are barely able to counterbalance the mutual repulsion of their protons.
decay occurs
in
Alpha
such nuclei as a means of increasing their stability by reducing
Why
are alpha particles almost invariably emitted rather than, say, individual
protons or r]He nuclei?
the alpha particle.
The answer
To escape from
and the alpha-particle mass
396
THE NUCLEUS
from the nucleus. Figure 12-6
is
follows from the high binding energy of
a nucleus, a particle
must have kinetic energy,
sufficiently smaller than that of
its
constituent
=
5.486 MeV.
how an
its
bulk by alpha
alpha particle can actually escape
energy V of an alpha
from the center of a certain heaw nucleus.
The height of the potential barrier is about 25 MeV, which is equal to the work
particle as a function of
thai
its
is
a plot of the potential
distance
r
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.
may therefore regard an alpha particle in such
We
a
nucleus as being inside a box whose walls require an energy of 25
MeV
to
be mm
4 to
their size.
7;,
in principle, spontaneously reduce
decay, there remains the problem of
A, the number of nudeons
of the original nucleus
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.
is
number A
The mass numbers
ALPHA DECAY
binding energy in a nucleus
the mass
v
A
than 82 are
found in nature, though they are not very abundant.
12.3
is
by
mounted However, decay alpha particles have energies that range from
9 MeV, depending upon the particular nuclide involved— 16 to 21 MeV
short of die energy needed for escape.
Although alpha decay
is
quantum mechanics provides
inexplicable on the basis of classical arguments,
a straightforward explanation. In fact, the theory
of alpha decay developed independently in 192K by
Carnow and
by
Curney and
NUCLEAR TRANSFORMATIONS
397
Condon
chanics.
WW
greeted as an especially striking confirmation of quantum me-
In the following
two
sections
we
shall
show how even a
treatment of the problem of the escape of an alpha particle from a nucleus gives
agreement with experiment.
results in
The
1.
2.
alpha particle
Such a particle
Es
may
in constant
An
12-7
may pass through
motion and
is
a
even a perfect
re-
it
If
the surface
is
and
suffi-
cienlly thin.
heavy nucleus;
contained in the nucleus by
the surrounding potential harrier;
'3.
There is a small
the harrier (despite
thin
wave pene-
incident
flecting surface 'or a short distance
an entity within
exist as
FIGURE
trates the surface of
basic notions of this theory are:
An
Thick mirror
simplified
Tola) reflection
Partial reflection
—but definite — likelihood that the particle nay pass through
its
height) each lime a collision with
it
occurs.
11ms the decay probability per unit lime \ can be expressed
when-
R
as
is
i
the alpha-particle velocity
is
when
it
the nuclear radius. Typical values for r
eventually leaves the nucleus anil
,mt.\
B might
l«2x
Id 7
m/s and
K)"'" 1 in respectively, so that
A
=
vf
=
where
i'
is
the
number
of times per second an alpha parlit
strikes the potential barrier
aroimd
it
and F
is
le
within a nucleus
the probability that the particle
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 " yr to escape from some nuclei!
Since V > /;, classical physics predicts a transnu'ssioii probability /' of zero.
1
will
he transmitted through the
one alpha particle
exists as
harrier. If
such
in a
we suppose
nucleus and that
that at
it
any moment only
moves hack and
forth
In ijuantuiu mechanics a
along a nuclear diameter,
result
is
a m null
known: a
c
I
lor
/'.
paiticlc
The
is
regarded
it
ils
a wave, anil the
optical analog of this effect
wave ondergoing reHcclion from even
light
theless penetrates
2H
moving alpha
mi definite value
a perfect
is
well
mirror never-
with an exponential!) decreasing amplilude before reversing
direction iFig. 12-7'.
FIGURE
12-6
The
potential energy of in alpha particle
n
a function of
its
distance from the center of a
nucleus.
•12.4
BARRIER PENETRATION
ol a beam of particles of kinetic energy '/'incident from
on a potential harrier of height V and width /., as in Fig 12-8. On
both sides ol the barrier V = 0, which means that no forces act upon the particles
Let us consider the case
the
potential energy of alpha particle
left
there
4ire i
in these regions Sehrodingcr's
•
-V
2
.
equation for the particles
is
m
12.13
alpha particle
cannot escape
alpha particle cannot enter (classical!*)
"'""*
(classically)
T
and
^%a =
f
"T"
—
12.14
'-^ni
.
kinetic
2 »<
energy of
alpha particle
r
.
Let us assume that
K.15
12.16
398
THE NUCLEUS
trV
""
^,
s
Ae**
+
Be
tll
=
fie""
+
Fe- ,ax
NUCLEAR TRANSFORMATIONS
399
-
[]]
II
<h+
FIGURE 12-9
Vm+
tl/
—
-
*n+
Schematic
representation of barrier
FIGURE 12-8
A beam
penetration.
of
particles can "leak" through
—
a finite barrier.
,
exponential
*
— ^u-
*,_
sinusoidal
*
=
x=L
and
=
=
^111
ia.21
The various terms
are .solutions to Eqs. 12.13 and 12.14 respectively.
solutions are not hard to interpret.
is
wave
a
of amplitude
A
As shown schematically
incident from the
left
on the
That
Ee
in these
in Fig. 12-8,
harrier.
"rSll+
iat
Ae tar
By
and ^ lu hack into
substituting ^,
their respective differential equations,
we
find that
is,
;
ia.17
u/
l+
2n,r
= Ae ial
12.22
This wave corresponds to the incident
is
their probability density.
If
v
is
beam
of particles in the sense that
2
\4>t+\
the group velocity of the wave, which equals
12.18
is
I^i+I'p
12.19
^t-
=
and
is
On
^=
At x
=
the incident
between
uV,
+
+
12-9).
Hence
V
L) there can be only a
by hypothesis, there
THE NUCLEUS
is
nothing
11
region
/'
=
()
solution
its
because the particle cannot
result
probability density in reexist inside the barrier; let
is.
Schrodinger's equation for the particles
B^n
ta«
in
AA'
what the quantum-mechanical
wave
12.Z5
direction, since,
could reflect the wave. Hence
F=
40Q
+x
=
2
probability density in region III and
its
12.24
>
IT '"'
Classically
In region
Its
III that
I.
ns see
the far side of the barrier (x
traveling in the
p-
Be'""
(Fig.
the ratio
W,!
gion
wave
is
wave
partially reflected, with
representing the reflected
12.20
evident that the transmission probability P for a particle to pass through
12.23
the (lux of particles that arrive at the barrier.
strikes the barrier
It is
the barrier
the particle velocity,
+
2m
(T
- V
fi*-
is
^=°
is
<£
H
=
Ce ib*
+ De-""
where
/2„,(T-V)
ft
2
NUCLEAR TRANSFORMATIONS
401
>
V
Since
we may
imaginary and
T.
h
IS
= -ih
Is
define a
new wave number
V
by
12.32*
^iri
x
V -
2tn
3*
T)
dx
Substituting £,,
lei ice
C„ - Gs •*
iz.28
The
^ n and ^„, from
+
De*<
A + H=C+ D
12-33
^„ + = Ce
"-*»
nonoscillatory
an exponentially decreasing wave function thai corresponds to a
liarricr part
Within
the
barrier.
disturbance moving to the right through the
disturbance
lie
12.30
is
-
*n
reflected,
is
and
wave
thick harrier,
iiifiuileb
The
are reflected.
not at
tion
it
originally had.
P
Its
$ nl
=
0,
which implies
>£,„.
I.
Figure 12-8
and
11,
wave function
where. With
same value but
will
its
*1
tyl
12.31b
dx
at the
in visualizing the
derivative
same
= ^u
_
rty
u
x
dx
right-hand wall
THE NUCLEUS
probability
P, is
of
A/E, which we require
found by replacing
i
by
-
i
to
compute the transmission
wherever
it
occurs in A/E:
be vni
an
frafi
Let us assume that the potential harrier
wave
i^,,
\}>
lv
functions in
12.39
boundary conditions.
ty/dx must be continuous every-
mean
thai, at
each wall
means
high relative to (he kinetic energy
is
that // ">
tt
and
("-£)
=b I
V a
a
Let us also assume that the barrier
ated (>etween
.v
=
and x
=
is
L; this
wide enough
means
for t£„ to l>e severely attenu-
that b'L
>
1
and
functions inside and outside must not only have the
slope, so that they
at the left-hand wall of the barrier
12.31a
readily solved to yield
however,
certain Ixmndary conditions to
a schematic representation of the
both $ and
The complex conjugate
of an incident particle; this
is
also the
may be
with
it.
we must apply
wave
12.36
that ail the incident particles
reference to Fig. 12-H, these conditions
of the harrier, the
to
iaEe' aI'
In the limit of
P,
which may help
111
\$ discussed earlier,
402
its
111
left-hand wall, and a harrier of finite width therefore permits a
of the initial lieain to pass through
redans
and
and
into region
reflection process takes place within the barrier,
In order to calculate
and
emerge
not reflected there will
continues muvitig unimpeded in the +.v direction.
it
Ee*' L
oscillate,
is
T
+ De iL =
b'D
function that corresponds to the reflected
|
the same kinetic energy
= -b'C +
-aCe- b L + aDe hL a
12.36
and therefore does not represent a moving
2
zero. There
particle of positive kinetic energy, the probability density |i£ n is not
at the
panicle
A
the barrier.
is a Suite probability of finding a particle within
end of the barrier that
iaB
left.
Even though £„ does not
as
12-35
Equations 12.33
disturbance moving to the
far
Ce- bL
'
f>'
an exponentially decreasing
-
iaA
is
I
Eqs. 12.15, 12.2S, and 12.21 into the aliove
,
equations yields
U'iui
12.29
of
=L
12.32 b
•>*
12.27
I
=
^11
=
match up
perfectly,
lien...
12,40
Hence
e"'
L
>
Eqs. 12.37
e- VL
and 12.38 may
lie
approximated by
12.11
and
12.42
(*)*-(*-D*~
NUCLEAR TRANSFORMATIONS
403
(A/E) and (A/E)* yields
\iultiplviiig
(i)ft)'-(i**)and so
tlic
transmission probability
=
i2.4 3
[
L
4
,
+
/" is
J «-»
if,,
2
(ft'/a)
7
-
J
Since from the definitions of a (Eq. 12.22) and
ft'
(Eq. 12.27)
T and V
the variation in the coefficient of the exponential of Eq. 12.43 with
is
negligible compared with the variation in ihc exponential
efficient,
furthermore,
a
far
co-
From unity, and so
p~ e -2bl.
12.44
is
never
is
The
itself.
good approximation
for the transmission probability.
We shall
find
it
conven-
ient to write Eq. 12.44 as
In/'- -2ft7.
12.45
FIGURE
Equation 12.45
is
derived for a rectangular potential barrier, while an alpha
particle inside a nucleus
We
7'.
is
= -2
= —
2ft'/.,
the radius of the nucleus and
R
Beyond R the
able to
I'
I
P
move
V(x)
I
THE NUCLEUS
b'[x)
t/.r
=
by
ft'(.t)
y= /
kinetic energy of the alpha particle
is
27,t?
Now
the electrostatic potential energy of an alpha particle at a distance x from
the center of a nucleus of charge 7.e (that
(lx
its
freely (Fig. 12-10).
in Fig. 12-6.
is
Alpha decay from the point at view of wave mechanic!.
the alpha-particle charge of 2e).
the distance from
tef rt
404
faced with a barrier of varying height, as
= -2
In
where R„
is
is
tnust therefore replace In
12.46
V=
12- 10
THEORY OF ALPHA DECAY
12.5
=
center at which
positive,
and
2m ( v - r
We
is,
Ze
is
the nuclear charge minus
therefore have
>
L\- /|Zei _ T
\ 1/2
\
ft
2
\4Tre„x
/
/
it
and, since
T = V when
x
=
R,
H¥nf-')'
NUCLEAR TRANSFORMATIONS
405
I
To
lence
biP
= -2
express Eq. 12,49 in terms of
b'(x)dx
In
J*
A =
logi,,
z
1S.47
=-(^)'""h-'(T)"
Because the potential barrier
is
relatively wide,
"
logarithms,
l°gu>A
-®"
R >
fi () ,
=
,0
('-^)'i
and
Figure 12-1
cos
=
loKio
1
is
+
(^~) +
°- 4343 ( 2 - 97Z1/i!
!
29ZU2
a plot of log [0 A versus
The
V
/2
ZT~ ,n
V" ~ 3-H5Zr-"2)
"
for a
1.72ZT-'"
number
—1.72
We can
use the
slope predicted throughout the entire range of decay constants.
what
is
namely,
,nP= - 2
fi
(ir)
U- 2 lT)
The
result
is just
about
obtained from nuclear scattering experiments like that of Rutherford,
—10 fm
in
such very heavy nuclei.
independent means for determining nuclear
with the result that
of alpha-radio-
straight line fitted to the experimental data has the
position of the line to determine R„, the nuclear radius.
/I
note that
0.4343
g!«(2R")
2
active nuclides.
Replacing
we
and so
-fen; ("-')"**
=
e
common
This approach constitutes an
sizes.
J
by
a
R = 2Ze
47r Eo
we
r
obtain
1218
^=f{-%Y'*"'«°"'-ik(fY'
The
ZT -"'
result of evaluating the various constants in Eq. 12.48 is
InP
=
2.97Z> /2
fi
,/a
-
3.95ZT
1 '*
FIGURE 1211
MeV, R (the nuclear
where T (the alpha-particle kinetic energy) is
-13
number of the
is
the
atomic
=
Z
10
m), and
radius) is expressed in fm(l fin
given
A,
by
constant
nucleus minus the alpha particle. The decay
Experimental
verifica-
tion of the theory of alpha decay.
expressed in
A
=
-10
vP
= -?-P
w
may
12.49
406
therefore be written
hi
A
THE NUCLEUS
= In(^) +
2.97Z I/2
R
,/S
-
3.95Z7-" 2
Alpha decay
NUCLEAR TRANSFORMATIONS
407
The quantum-mechanical
analysis of alpha- particle emission,
complete accord with the observed data,
is
significant
which
on two grounds.
is
in
First,
makes understandable the enormous variation in half life with disintegration
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
1 1
energy.
its
half
10 24 greater, the disintegration energy of a HoTh (4.05
life is
MeV)
is
only
FIGURE 12-12
about half that of £fPo (8.95 MeV)—behavior predicted by Eq. 12.19.
2
electron s
(mm
Energy spectrum of
the beta deciy of *ijBt.
energy equivalent
of mass lost by
decaying nucleus
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.
the probability
is
much more
not
than zero, but
it
is
In
quantum mechanics
0.4
not identically equal to
0.6
o.s
ELECTRON ENERGY. MeV
zero.
An experiment
first
performed
in
1927 showed that
this
hypothesis
In the experiment a sample of a fueta- radioactive nuclide
BETA DECAY
12.6
and the heat evolved
rimeter,
after a given
is
number of decays
not correct.
is
placed
is
in
a calo-
measured. The
evolved heat divided by the number of decays gives the average energy per
Beta decay,
like
alpha decay,
a means whereby a nucleus can alter
is
ratio to achieve greater stability. Beta decay,
problem
to the physicist
obvious difficulty
we have
is
who
its
Z/iV
however, presents a rather different
seeks to understand natural phenomena.
The most
decay a nucleus emits an electron, while, as
that in beta
seen in the previous chapter, there are strong arguments against the
presence of electrons in nuclei. Since beta decay
is
essentially the
is
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 rmM characteristic of
disposed of
if
the nuclide. Figure 12-12 shows the energy spectrum of the electrons emitted
in the
beta decay of 2|" Bi; here
Tmax =1.17 MeV.
In every case the
maximum
energy
is
=m
c
t»
carried off by the decay electron
equal to the energy equivalent of the mass
difference l>etween the parent and daughter nuclei.
an emitted electron found with an energy of Tmtx
Only seldom, however,
is
.
was suspected at one time that the "missing" energy was lost during collisions
between the emitted electron and the atomic electrons surrounding the nucleus.
It
THE NUCLEUS
be 0.35
away indeed from the Tmax value of 1.17 MeV. The conclusion is that
the observed continuous spectra represent the actual energy distributions of the
electrons emitted by [>eta-radioaetive 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 '/ of the electron, proton, and neutron.
2
Beta decay involves the conversion of a nuclear neutron into a proton:
n—*p +
e~
Since the spin of each of the particles involved
place
if
spin (and hence angular
%
is
emitted
in
if
momentum)
removed.
off
It
was supposed
is
is
%.
this reaction
cannot take
to be conserved.
an uncharged particle of small or zero mass
beta decay together with the electron, the energy,
momentum, and angular-momentum
is
to
but far
and spin
+ 'nu
was found
very close to the 0,39-MeV average of the spectrum in Fig. 12-12
In 1930 Pauli proposed that
^miut
408
MeV, which
spontaneous
conversion of a nuclear neutron into a proton and electron, this difficulty
In the case of 2 ^Bi the average evolved energy
decay.
discrepancies discussed above
would be
that this particle, later christened the neutrino, carries
Tmix and the actual electron kinetic
away negligible kinetic energy) and, in so doing,
an energy equal to the difference between
energy (the
has a
recoil nucleus carries
momentum
exactly balancing those of the electron
daughter nucleus. Subsequently
it
was found
that there are
and the recoiling
two kinds of neutrino
NUCLEAR TRANSFORMATIONS
409
involved in beta decay, the neutrino
T).
We shall discuss the distinction
decay
an antineutrino that
it is
—*
n
12.50
+
p
+
e
is
itself
(symbol
and the antineutrino (symlxil
w)
lietween them in Chap. 13. In ordinary beta
The beta decay of
emitted:
Beta decay
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
Tm!a is observed to be equal (within experimental error) to the
value calculated from the parent-daughter mass difference; the neutrino mass
mass because
is
now
believed to be zero.
detected until recently
The reason neutrinos were not experimentally
that their interaction
is
INVERSE BETA DECAY
12.7
with matter
is
extremely feeble.
Lacking charge and mass, and not electromagnetic in nature as is the photon,
unimpeded through vast amounts of matter. A neutrino
p —» n
12.53
is
on the average
to pass through over l(K) light-tjeurx of solid iron
before interacting!
The
only interaction with matter a neutrino can experience
through a process called inverse
Ijeta
decay, which
Positive electrons, usually called positrons,
years later were found to
lie
we
shall consider shortly.
were discovered
in
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:
n
12.51
+
e+
+
p
12.54
is
+
essentially the
12.55
p
same
is
+
c
is
equivalent to
its
emission
c
as the beta
decay of Eq. 12.53. Similarly, the absorption
equivalent to the emission of a neutrino, so that the reaction
n
also involves the
+
n
e~
of an antineutrino
+
e"
same physical process
called inverse beta
tiecaij, is
as that of Eq, 12.53,
interesting because
it
This latter reaction,
provides a method for estab-
lishing the actual existence of neutrinos.
Starting in 1953, a series of experiments
and others to detect the immense
occur
in
A
a nuclear reactor.
in solution
trinos.
Positron emission
v
+
Because the absorption of an electron by a nucleus
1932 and two
spontaneously emitted by certain nuclei.
e+
+
scheme
of a positron, the electron capture reaction
the neutrino can pass
would have
a proton within a nucleus follows the
were begun by
flux of
F.
Reines, C, L.
Cowan,
neutrinos from the beta decays that
tank of water containing a
cadmium compound
supplied the protons which were to interact with the incident neu-
Surrounding the tank were gamma-ray 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 gamma-ray detectors
While a neutron outside a nucleus can undergo negative beta decay
because
its
mass
be transformed
greater than that of the proton, the lighter proton cannot
into a neutron except within a nucleus.
daughter nucleus of lower atomic number
to a
ber
is
A
into a proton
Z
Positron emission leads
while leaving the mass num-
unchanged.
is
the
capture. In electron capture a nucleus absorbs one of
phenomenon
its
of electron
inner orbital electrons,
with the result that a nuclear proton becomes a neutron and a neutrino
12.52
Thus the hindamental reaction
p
+
er
formed neutron migrated through the solution
it
is
n
+
v
in electron
capture
is
is
Electron capture
competitive with positron emission since both processes lead
same nuclear transformation. Electron capture occurs more often than
positron emission in heavy elements Ijccause the electron orbits in such elements
to the
released about 8
MeV of excitation energy divided among three or
the positron-electron annihilation.
sequence of photons at the detector
is
from
above
a sure sign that the reaction of Eq. 12.55
regarded as experimentally established.
Inverse beta decay
trinos interact
with the nucleus. Since almost the only unstable nuclei found in nature are of
n
high Z, positron emission was not discovered until several decades after electron
In principle, then, the arrival of the
To avoid any uncertainty, the experiment was performed with
the reactor alternately on and off, and the expected variation in the frequency
of neutrino-capture events was observed. Thus the existence of the neutrino may
p
radii; the closer
four photons,
after those
has occurred.
proximity of the electrons promotes their interaction
have smaller
few microseconds,
was captured by a cadmium nucleus. The new, heavier cadmium nucleus then
l)c
Electron capture
Meanwhile the newly
until, after a
which were picked up by the detectors several microseconds
Closely connected with positron emission
emitted.
responded to the resulting pair of 0.5I-MeV photons.
The
is
the sole
known means whereby
neutrinos and antineu-
with matter:
+
+
v
n
c
p
+
+
e
+
e~
probability for these reactions
is
almost vanishingly small;
this is
why
emission had been established.
410
THE NUCLEUS
NUCLEAR TRANSFORMATIONS
411
liberated,
incident
target
neutrinos travel freely through space and matter indefinitely, constituting a kind
particles
nucleus
neutrinos are able to traverse freely such vast amounts of matter.
Once
geometrical
of independent universe within the universe of other particles.
cross section
Nuclei can
nucleus
is
its
An
atoms can.
exist in states of definite energies, just as
denoted by an asterisk after
an excited
in
interaction
cross section
GAMMA DECAY
12.8
excited
usual symbol; thus {JgSr* refers to ||Sr
Excited nuclei return to their ground states by emitting
state.
photons whose energies correspond to the energy differences between the various
initial
and
range
in
A
is
energy up to several MeV, and are traditionally called
in Fig. 12-13,
of the decay
states of f|Al.
which pictures the beta decay of
and
9.5 min,
is
The
resulting
decays to reach the ground
As an alternative
to
The photons emitted by
the transitions involved.
nuclei
gamma
rni/s.
its
ground
aroiuid
it.
to
it
may
interact
^Al" nucleus then tmdergoes one
up
its
it is
half
or
two
gamma
an excited nucleus
in
some cases may
return
excitation energy to one of the orbital electrons
While we can think of
by an atomic electron,
The
state.
gamma decay,
state by giving
^Mg to gAl.
take place to either of the two excited
this process,
which
is
known
conversion, as a kind of photoelectric effect in which a nuclear photon
in better
accord with experiment
to
as internal
is
absorbed
regard internal
conversion as representing a direct transfer of excitation energy from a nucleus
to
particles will
simple example of the relationship between energy levels and decay schemes
shown
life
only these
final states in
an electron. The emitted electron has a kinetic energy equal
to the lost nuclear
FIGURE 12-14
to, or larger
The concept
of cross faction.
The
interaction cross section
may be
smaller than, equal
than the geometrical cross section.
excitation energy minus the binding energy of the electron in the atom.
Most excited nuclei have very short
few remain excited
is
called an isomer of the
fJSr* has a half
life
half lives against
for as long as several hours.
same nucleus
of 2.8 h
and
is
in its
A
ground
gamma
decay, but a
12.9
CROSS SECTION
long-lived excited nucleus
state.
The
excited nucleus
Nuclear reactions,
of utilizing
accordingly an isomer of f£Sr.
this
like
chemical reactions, provide both information and a means
atomic nuclei has come from experiments
in
cles collide with stationary target nuclei.
A
probability that a
particle
we know
about
which energetic bombarding
parti-
information in a practical way. Most of what
bombarding
employs the idea of
particle will
way to express the
interact in a certain way with a target
very convenient
cross section that
was introduced
connection with the Rutherford scattering experiment.
FIGURE 12-13
1.015
MeV
gamma em is lion
each target particle as presenting a certain area, called
Successive beta anal
fn
the decay of J'Mg
I
incident particles, as in Fig. 12-14.
Any
in
What we do
its
Chap. 4
is
in
visualize
cross section, to the
incident particle that
is
directed
at
tlii*
;ai.
0.834
MeV
area interacts with the target particle.
greater the likelihood of an interaction.
y\
\y
412
THE NUCLEUS
'I*he
the greater the cross section, the
interaction cross section of a target
particle varies with the nature of the process involved
the incident particle;
s*
Hence
it
may
be greater or
less
and with the energy of
than the geometrical cross section
of the particle.
NUCLEAR TRANSFORMATIONS
413
is
Suppose
we
dx
12-15),
(Fig.
have a slab of some material whose area
nAdx
a total of
If
nuclei in the slab, since
in a
some particular
the nuclei in the slab
all
n atoms/m
A and whose
thickness
area
is
bombarding beam, the number
its
nAa
tl.X
volume
Each nucleus has
Adx.
is
interaction, so that the aggregate cross
If there
dx.
are
N incident
N-dN
particles
that interact with nuclei in the slab
is
N incident
particles
particles
emerge
from slab
by
therefore specified
aggregate cross section
Interacting particles
target area
incident particles
,l\
nAadx
_
N
tr=
A
=
12.56
na dx
cross section/atom
proportion of incident particles that interact with nuclei
we must
thickness,
is
To
valid only for a slab of infinitesimal thickness.
is
integrate tlN/N.
If
dN/N
= no- dx
Cross section
we assume
capable of only a single interaction, d\' particles
between cross section and beam
IS
The
relationship
FIGURE 13-16
The
cross section lor neutron absorption
FIGURE
Equation 12.56
=A
the material contains n atoms per unit volume, there are
a cross section of a for
section of
is
12-
intensity.
find the
a stab of finite
in
each incident particle
that
may be
thought of as being
removed from the beam in passing through the first dx of the
must introduce a minus sign in Eq. 12.56, which becomes
Hence we
slab.
;:Cd varies with neutron energy.
10.000 r
Denoting the
initial
number
of incident particles
C
-* (IN
r"
dN
In ,V
—
by
<VU ,
we have
dx
-
l.ooo
= — nax
In JV„
N = N e-""
1257
The number
of surviving particles
N decreases exponentially with increasing stab
100
thickness
jr.
While cross
bam
The
sections,
and customary
nient
is
which are
to express
areas, should lie expressed in irr,
them
in
hams
(b),
where
1
b
=
it is
10~ 38
conve-
m2
.
The
of the order of magnitude of the geometrical cross section of a nucleus.
cross sections for most nuclear reactioas
incident particle. Figure 12-16 shows
how
depend upon the energy of the
the neutron-absorption cross section
of '.JgCd varies with neutron energy; the narrow peak at 0.176
eV
is
associated
0.001
with a specific energy level in the resulting
The mean
can travel
bility
/
free path
0.01
0.1
1.0
100
10
1,000
10,000
nucleus.
of a particle in a material
I
in the material
that
"JCd
NEUTRON ENERGY, eV
is
the average distance
it
The probaslab ir thick
before interacting with a target nucleus.
an incident particle will undergo an interaction in a
NUCLEAR TRANSFORMATIONS
414
THE NUCLEUS
415
of the material
is
/=
iz.58
no
of times II
must traverse the
it
slab before interacting
is
therefore
=
//
on the average.
unimpeded through the universe. There are already
neutrinos than atoms in the universe, and their number continues to
by
these neutrinos carry
—
Many
mean
nuclear reactions actually involve two separate stages.
nucleus, called a comjxwrul nucleus,
Hence
free path.
-!-
Mean
no
free path
The
cross section for the interaction of a neutrino with matter has been found
a
to be approximately 10 -47
Let us use Eq. 12.60 to find the mean free path
m
is,
sum
of their mass numbers.
formed, since
its
brought into
it
of iron
is
55.9, so that the
mass of
The compound nucleus
by the incident particle
this,
Table 12.2 shows
signifies
by amounts equal
mFt = 55.9 u/atom X 1.66 X
= 9.3 X 10" M kg/atom
7.8
is
to
X
10 3
HI' 27 kg/u
them.) While
J
original particles
has no
"memory"
,
the
number
of atoms per
m
six
reactions
an excited
in
iron is
Compound
re-
how
it
was
is
shared
among
all
of them.
To
whose product
state;
is
compound
the
compound
A
given
illustrate
nucleus 'fN'.
nuclei are invariably excited
to at least the binding energies of the incident particles in
|N and '^C
that these reactions
:l
an
and the sum
of
therefore be formed in a variety of ways.
arc beta- radioactive with such short half lives as to
preclude the detailed study of their reactions
kg/m 3
first,
form a new
uuclcons are mixed together regardless of origin and the energy
compound nucleus may
(The asterisk
on the average,
Since the density of iron
In the
whose atomic and mass numbers are
numbers of the
of the atomic
combine
.
The atomic mass
of neutrinos in solid iron.
an iron atom
increase.
forever in the sense of
lost
incident particle strikes a target nucleus and the two
definition, the
=
flux
more
THE COMPOUND NUCLEUS
12.10
spectively the
/
—
is—apparently
far
Accordingly the average distance the particle travels before
no
is,
sun and other
in the
being unavailable for conversion into other forms.
is
HAx =
which
produced
is
!
MO Aa
interacting
neutrinos
practically
The energy
12.59
flux of
course of the nuclear reactions that occur within them, and this
A.\
moves
The number
An immense
in solid iron'
stars in the
nuclei
to
form
l
|\*, there
Ls
no doubt
can occur.
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
n
7.8
=
X
X
9.3
=
X
8.4
103
long relative to the 10~ 21
3
kg/m
—
10~ 2fi kg/atom
10 as
of several
s
or so required for a nuclear particle with an energy
MeV to pass through a nucleus. A given compound nucleus may decay
atoms/m 3
Table 12.2.
The mean
free path for neutrinos in iron
is
therefore
NUCLEAR REACTIONS WHOSE PRODUCT
NUCLEU5
no
A
1.2
X
X
If)
1 ''
1
K) 18
atoms/m 3
x
lO"" m 2
1.2
=
THE NUCLEUS
.
The exc itation e nergies
IS
THE COMPOUND
gi ve n a re c alcu lat ed
I
rom
incident particle will
X
m
x
H) ]a
15
m/light-year
10
add
to the excitation
energy of
its
reaction
by an amount depending upon the dynamics of the reaction.
m
m, and so the mean free path turns out
9.46
416
10* 8
x
8.4
light-year (the distance light travels in free space in a year)
9.46
X*
the masses of the particles involved; the kinetic energy of an
1
=
' ;
=
to
is
equal to
be
130 light-years
NUCLEAR TRANSFORMATIONS
417
in
one or more different ways, depending upon
with an excitation energy
of, say,
MeV
12
excitation energy
its
1
.
Thus 'jN*
can decay via the reactions
i\*
'|C
+ ;n
'IC
+ fH
=
TlBh -
|(.«,
12.61
or simply einit one or
more gamma
rays
whose energies
MeV, but
12
total
does not have enough energy
mode
them. Usually a particular decay
to liberate
compound nucleus in a specific
The formation and decay of a compound nucleus
favored by a
is
The
rH,)V
liquid,
Such a drop of liquid
evaporate one or more molecules, thereby cooling down.
when
statistical fluctuations in the
a
compound
nucleus persists in
its
to
have a
heat of
to the
The
compound
nucleus
analysis of the reaction that occurs
one
at rest
is
eventually
will
The
moving center of mass. Thus we can regard Ttm as
relative motion of the particles. When the particles
Similarly,
interval
in nicely
fits
when
a
with
between the
always
less
if
a particle of mass
of mass i»2 as
of mass
is
m, and speed
viewed by an observer
7j, h
.
Information about the excited states of nuclei can be gained from nuclear
may l>e detected by
The presence
—
V)
reaction, as in the neutron-capture reaction of Fig. 12-16.
FIGURE 12-17
Laboratory and center -of
mi si
In most nuclear reactions, c
t;
is
incident
momenta
(Fig.
upon a stationary
system before
center of moss
<
in the laboratory, the
speed
1
2-
V of the
(h) Motion in the center-of-mass coordinate system before collision.
center
and so a
nonrelalivistic treatment
energy
center of mass
—V
THE NUCLEUS
\
— V ^?
o
A completely
inelastic collision as
is
is
satisfactory.
that of the incident particle
m i»
seen In laboratory and center-of-mass coordinate systems.
laboratory
Ceater-of-mats
coordinate system
coordinate system
o
o
before
2
In the center-of-mass system, lx>th particles arc
418
c
collision
energy:
m2
/-\
!
1 7).
i«i
In the laboratory system, the total kinetic
total kinetic
\ V= m tm
l
+ ms f
c,
called
collision.
-r
m.
particle
only:
/2
is
coordinate syitemi.
(a) Motion in the laboratory coordinate
(c)
Tub =
Such a peak
= m^V
\fii,
l
of an excited state
a peak in the cross section versus energy curve of a particular
moving nucleon or nucleus
defined by the condition
m,(«
collide, the
,
than
reactions as well as from radioactive decay.
this picture.
greatly simplified by the use of a coordinate system
center of mass, the particles have equal and opposite
Hence
+
the kinetic
maximum amount of kinetic energy that can be
compound nucleus while still conserving momentum is Tem which
moving with the center of mass of the colliding particles. To an observer located
at the
their
converted to excitation energy
evaporation
For escape.
is
sufficiently large fraction of
The time
the excitation energy to leave the nucleus.
strikes another
laboratory system minus
mass
the kinetic energy '/^{m,
excited state until a particular nucleoli or
group of nucleons momentarily happeas
formation and decay of a
of the
in the
to the center of
of the resulting
energy distribution within the
drop cause a particular molecule to have enough energy
energy of the particles relative
energy
energy of the
is
process occurs
2
has an interesting inter-
with the binding energy of the emitted particles corresponding
vaporization of the liquid molecules.
total kinetic
total kinetic
analogous to a drop of hot
is
mi.
excited state.
pretation on the basis of the liquid-drop nuclear model described in Chap. 11.
In terms of this model, an excited nucleus
m2 )V 2
it
cannot decay hy the emission of a triton (^H) or a lielium-3 (|He) particle since
it
+
moving and contribute
to the
after
collision
o
o
NUCLEAR TRANSFORMATIONS
419
oCooo
a resonance by analogy with ordinary acoustic or ac circuit resonances: a
pound nucleus
more
is
likely to
exactly matches one of
its
combe formed when the excitation energy provided
energy levels than
if
the excitation energy has
some
other value.
The
\E&t >
mean
tainty
12-16 has a resonance at 0.176 eV
reaction of Fig.
half-maximum)
is
r
=
0.115
The uncertainty
enables us to relate the level width
ft
The width
lifetime t of the state.
A£
eV.
in the
energy of the
state,
I"
I"
whose width
principle
in
the
(at
time
form
of an excited state with the
oscillations of a liquid drop.
evidently correspond"; to die uncer-
and the mean
lifetime t corresponds to
when the state will decay, in the present example
emission of a gamma ray. The mean lifetime of an excited state is defined
tension
is
the uncertainty At in the time
it
by the
distortion
in
The
FIGURE 12-18
general as
adequate to do both, and the nucleus vibrates back and forth until
eventually loses
excitation energy
its
sufficiently great,
is
bring back together the
splits into
two
parts.
by gamma decay.
however, the surface tension
now widely
If
is
the degree of
not adequate to
separated groups of protoas, and the nucleus
This picture of
fission is illustrated in Fig. 12-19,
ft
12 62
Mean
lifetime of excited state
The new
fission
In the case of the above reaction, the level width of 0,1 15
lifetime for the
compound
eV
implies a
mean
nucleus of
UKH X UH« J-S
X 1.60 X 10- ia J/eV
5.73
X
10- *
fragments are of unequal size
(Fig. 32-20), and,
Usually
because heavy nuclei have
a greater neutron/proton ratio than lighter ones, they contain an excess of
neutrons.
To reduce
fragments
as
this excess,
two or three neutrons are emitted by the
soon as they are formed, and subsequent beta decays bring their
neutron/ proton ratios to stable values.
O.llSeV
=
nuclei that result from fission are called fission fragments.
A heavy
s
(5
MeV
nucleus undergoes
fission
when
it
acquires enough excitation energy
or so) to oscillate violently. Certain nuclei, notably
2
(gU, are sufficiently
excited by the mere absorption of an additional neutron to split
in
Other
two.
nuclei, notably ^U (which composes 99.3 percent of natural uranium, with
12.11
NUCLEAR FISSION
binding energy released when another neutron
splits into
two
successively
is
fission, in
becomes
a prolate
a prolate spheroid again,
lie
analyzed with the
which a heavy nucleus (A
When a liquid
ways. A simple one
lighter ones.
oscillate in a variety of
drop
is
is
shown
it
may
in Fig. 12-18: the
drop
suitably excited,
spheroid a sphere, an oblate spheroid,
and so
on.
The
> — 230}
restoring force of
its
a sphere,
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
like a liquid
drop when
in
an excited
tension,
and so can vibrate
only by reaction with fast
neutrons whose
Fission can occur after excitation
absorbed, and undergo
is
kinetic energies exceed about
by other means
fission
1
MeV.
besides neutron capture, for
by gamma-ray or proton bombardment. Some nuclides are so unstable
be capable of spontaneous fission, but they are more likely to undergo alpha
instance,
as to
decay before
The most
this takes place.
striking aspect of nuclear fission
evolved. This energy
is
readily computed.
the magnitude of the energy
is
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
state, they also are subject lo disruptive
forces due to the mutual electrostatic repulsion of their protons.
is
g|U
composing the remainder), require more excitation energy for fission than the
Another type of nuclear-reaction phenomenon that can
help of the liquid-drop model
a
When
a nucleus
FIGURE
12 19
Nuclear
fist ion
according to the liquid drop modal.
distorted from a spherical shape, the short-range restoring force of surface
tension must cope with the latter long-range repulsive force as well as with the
inertia of the nuclear matter.
420
THE NUCLEUS
If
the degree of distortion
is
small, the surface
NUCLEAR TRANSFORMATIONS
421
sequence of
the consequent evolution of additional neutrons, a self-sustaining
to occur
chain
reaction
a
for
such
fissions is, in principle, possible. The condition
in
an assembly of fissionable material
during each
simple: at least one neutron produced
must, on the average, initiate another
fission
neutrons initiate
is
fissions,
the reaction will slow
down and
fission.
stop;
if
If
too few
precisely one
energy will be released at a constant
the frequency of fissioas
rate (which is the case in a nuclear reactor); and if
will occur (which
explosion
increases, the energy release will be so rapid that an
neutron per
is
causes another
fission
These
the case in an atomic bomb).
critical, critical,
12.12
and
fission,
situations are respectively called sub-
supercritical.
TRANSURANIC ELEMENTS
have such
Elements of atomic number greater than 98, which is that of uranium,
into being,
vuiiverse
came
the
when
formed
been
they
had
that,
short half lives
Such transuranic elements may lie
by the bombardment of certain heavy nuclides with
2
2
which beta-decays
neutrons. Thus $)U may absorb a neutron to become jjgU,
neptunium:
element
transuranic
the
of
isotope
an
ss 23 min) into "IJjNp,
(T
they would have disappeared long ago.
produced in the laboratory
1/2
*gfU
+
«gu
Jn
+
This neptunium isotope
of 2.3
life
d
110
120
130
MASS NUMBER
FIGURE 12-20
200
MeV
The
distribution of
for the
mass numbers
the fragments from the fission of =£U.
in
«gP«
combustion of coal and
oil,
liberate only a
few
electron volts per individual reaction, and even nuclear reactions (other than
fission) liberate
that
is
no more than several million electron
volts.
Most of the energy
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
rrcn^ni/ed
422
ihai,
THE NUCLEUS
after the discovery of nuclear fission in
because a neutron can induce
^
l
8Pu +
Pliitonium alpha-decays into
240 or so nucleons involved! Ordinary chemical reactioas, such
as those that participate in the
*IN P
170
160
150
itself radioactive,
fission in
;i
it
was
suitable nucleus
WH4
1939
undergoing beta decay with a half
into an isotope of the transuranic element plutoniwn:
0.001
100
is
e"
«r
«gU
with a half
life
of 24,000 yr:
>SU
a + IHe
like 2 jgU, is fissionable and can be used in
is chemically different from uranium;
Plutonium
nuclear reactors and weapons.
2
neutron irradiation is more easily
after
jgU
its separation from the remaining
"jgU
from
the much more abundant ^|U
accomplished than the separation of
It is
in
interesting to note that
"gPu,
natural uranium.
=
99) have half-lives too short for
can l>e identified by chemical
they
their isolation in weighable quantities, though
number yet discovered
atomic
highest
means. The transuranic element of the
Transuranic elements past einsteinum (Z
has
Z =
105.
NUCLEAR TRANSFORMATIONS
423
THERMONUCLEAR ENERGY
12.13
The
basic exothermic reaction in stars— and
hence the source of nearly all 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
energy
in the
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 proton-proton cycle
+
}H
JH -* «H
e+
+
+
is
9
the formation of deuterons by the direct combination of two
protons accompanied by the emission of a positron. A cleuteron may then join with
a proton
to
form a ;]He nucleus:
ill
Finally two
+ *H
|He
£He
The
+
-» PBe
y
nuclei react to produce a jjHe nucleus plus
-I-
|He -• ^He
+
+
}H
two protons:
JH
is {&m)c\ where Am is the difference between
the mass
and the mass of an alpha particle plus two positrons; it buns
be 24.7 MeV. Figure 12-21 shows the entire sequence.
total
evolved energy
of four protons
out to
The carbon cycle proceeds
following way:
in the
FIGURE 12-21
ill
fS
iH
i
+
'§C
>?N
fN
1C +
»*N +
»|C -»
+ »N
-*
l
place
e
+
+
JH +
i|
N
-» >JC
y
Carbon cycle
is
estimated to be 2
for occurrence.
result again
is
four protons, with the evolution of 24.7
MeV;
two positrons from
the initial "gC acts as a kind of
reappears at
its end (Fig. 12-22).
Self-sustaining fusion reactions can occur only under
conditions of extreme
temperature and pressure, to ensure that the participating nuclei
have
it
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
424
THE NUCLEUS
is
X
It)
8
one of the two nuclear reaction sequences that take
K, the proton-proton cycle has the greater probability
meet these specifications,
in the sun,
whose
interior
temperature
is more efficient at high temperamore efficient at low temperatures. Hence
In general, the carbon cycle
while the proton-proton cycle
JHe
the formation of an alpha particle and
catalyst for the process, since
This
p
shirs
The net
cycle.
the evolution of energy.
tures,
+
The proton-proton
the sun and that involve the combination of tour hydrogen nuclei to form a helium nucleus with
v
p+y
'JO-* '»N + «» +
in
is
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
called fliermonuclear energy, particularly
control.
On the earth
when
is
often
the fusion takes place under man's
neither the proton-proton 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
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
contain
deuterium. Such a source is the oceans and seas of the world, which
15
more
about 0.015 percent deuterium— a total of perhaps 10 tons! In addition, a
a
target
bombarding
merely
reactions
than
fusion
promoting
efficient means of
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
all involve
reactions that occur in the target. Current approaches to this problem
mixtures
deuterium-tritium
or
deuterium
of
very hot plasmas (fully ionized gases)
temperature
of
high
purpose
fields.
The
which are contained by strong magnetic
to ensure that the individual *II
is
together and
and ?H nuclei have enough energy
A
react despite their electrostatic repulsion.
magnetic
to
come
field is
used
other material which
as a container to keep the reactive gas from contacting any
will
cool it down or contaminate it; there is little likelihood that the wall
might
melt since the gas. though
at a
temperature of several million degrees K, actually
have a high energy density. While nuclear- fusion reactors present more
they
severe practical difficulties than fission reactors, there is little doubt that
does
iidI
become
will eventually
a reality.
Problems
(The masses in u of neutral atoms of nuclides mentioned below
are: {H, 1.007825;
m
7.0169; $B,
3.016050; ilUe, 3.01fi030; ^le, 4.002603; JLI, 7.0160; jBe,
15.9949;
'gO,
1.0031;
"X,
i§C,
13.0034;
'?
12.0000;
»gC.
10.0129;
B, 12.0)44;
I
P
The
'£0, 16.9994.
are listed
1.
in
Table
neutron mass
is
1.008665
u.
Atomic masses of the elements
7.1.)
Tritium gfij has a half
life
of 12.5 yr against beta decay.
What
fraction
after 25 yr?
of a sample of pure tritium "ill remain undeeayed
FIGURE 12-22
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
?H
-> pie + },n
?H -* JH + {H
+ fH
+
+ 3.3 MeV
+ 4.0 MeV
2.
The
half life of
a sample of
One g
life
of radium.
4.
The mass of
of
THE NUCLEUS
is
15 h.
How
long does
it
take for 93.75 percent of
isotope to decay?
radium has an activity of
3.
a millicurie of *JJPb
1
is
Ci.
3
X
From
this fact
10" " kg.
decay constant of J|Pb. (Assume atomic mass
z
Probs. 4 to
426
this
ffXa
in
determine the half
From
this fact find the
u equal to mass number
in
6.)
NUCLEAR TRANSFORMATIONS
427
The half life
5.
of
^U against alpha decay
grations per second occur in
g
1
is
4.5
X
109
yr.
How many disinte-
Neutron capture
of 23JU?
The potassium isotope JgK undergoes beta decay with a half life of .83 X 10"
Find the number of licta decays that occur per second in I g of pure ^K.
6.
1
yr.
A 5.78-MeV alpha particle is emitted in the decay of radium. If the diameter
-M m, how many alpha-particle de Broglie
is 2 X 10
7.
of the radium nucleus
wavelengths
inside the nucleus?
fit
Calculate the
8.
]
of
maximum energy
of the electrons emitted in the beta decay
fB.
ENERGY
Why
does
decay by electron capture tastead of by positron
emission? Note that ]Be contains one more atomic electron than does jLi.
9.
.{Be invariably
Neutron and proton
FIGURE 12-23
capture cross section! wary differently
with particle energy.
Proton capture
10.
Positron emission resembles electron emission in
all
respects except that
the shapes of their respective energy spectra are different: there are many
low-energy electrons emitted, but few tow-energy positrons. Thus the average
electron energy in beta decay
energy
11.
is
0ATmhs Can
about
about 0.3rmax> whereas the average positron
is
you suggest a simple reason
.
Determine the ground and lowest excited
with the help of Fig. 11-10. Use
!I Y together
with the
fact,
noted
this
states of the
for this difference?
39th proton
in Sec. 6.10, that radiative transitions
states with very different angular
in ";V
information to explain the isomerism of
momenta
between
are extremely improbable.
ENERGY
12.
Find the minimum energy
have
in
in the laboratory
system that a neutron must
order to initiate the reaction
16.
+
£n
Find the
13.
in
x
$) +
2,20
MeV
-» l|C
+ pie
When
nucleus
Ls
a neutron
usually
or alpha particle.
minimum energy
in
is
more
absorbed by a target nucleus, the resulting compound
emit a
likely to
gamma
ray than a proton, deuteron,
Why?
the laboratory system that a proton must have
order to initiate the reaction
17.
There are approximately 6
X
10 28
atoms/m 3
in solid
of 0.5-MeV neutrons is directed at an aluminum foil 0.
cross section for neutrons of this energy in aluminum
1
p
14.
+
+
2.22
MeV
-» p
Find the minimum energy
must have
in
The
+
p
+
n
in the laboratory
system that an alpha particle
order to initiate the reaction
JHc
15.
(I
+
'{N
+
1.18
cross sections for
MeV
-
18.
+
jH
Why
manner shown
in Fig.
12-2-3.
does the neutron cross section decrease with increasing energy whereas
the proton cross section increases?
THE NUCLEUS
2
X
If
10~ 31
the capture
m2
,
find the
density of
l
"B
is
2.5
X
I0 3
kg/m 3 The capture
.
about 4,<XX) b for "thermal" neutrons, that
is,
cross section of
l
|,'B
neutrons in thermal equilibrium
How thick a layer of 'j|B
is
required to absorb
99 percent of an incident beam of thermal neutrons?
19.
The
density of iron
section of iron
is
428
The
with matter at room temperature.
comparable neutron- and proton-induced nuclear
reactions vary with energy in approximately the
is
fraction of incident neutrons that are captured.
is
>|0
aluminum. A beam
mm thick.
is
is
about 2.5
about 8
b.
absorbed by a sheet of iron
What
1
cm
X
H> 3
kg/m 3
.
The neutron-capture cross
beam of neutrons
fraction of an incident
thick?
NUCLEAR TRANSFORMATIONS
429
20.
The
cross section of iron for neutron capture
is
2.5 b.
What
is
the
mean
ELEMENTARY PARTICLES
free path of neutrons in iron?
21.
The
fission of
-$|U releases approximately 200 MeV.
the original mass of
22.
2
jj§U
+
What
percentage of
13
n disappears?
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?
While nuclei are apparently composed solely of protons and neutrons, several
emitted by nuclei
score other elementary particles have heen observed to be
under appropriate circumstances. These particles, christened "straiiijc particles"
soon after their discovery about two decades ago, bring the total number merely
discern order in this
of relatively stable elementary particles to over .30. To
multiplicity of particles has not proved to
regularities in
be an easy
task.
While certain
elementary-particle properties have been established, and while
well
such particles as the electron, the neutrino, and the * meson arc relatively
wide
found
lias
yet
particles
understood, no comprehensive theory of elemental?
conclude our survey of modern physics with this topic,
a reminder that there remains much to Ix.' learned about the natural
acceptance.
then, as
It is fitting
to
world.
13.1
The
ANTI PARTICLES
electron
is
the only elementary particle for which a satisfactory theory
is
)2S by P. A. M. Dirac, who obtained
known. This theory was developed in
particle
in
an electromagnetic field that incorpocharged
for
a
equation
wave
a
When
the observed mass and charge of
relativity.
special
of
results
rated the
l
l
the intrinsic
the electron are inserted in the appropriate solutions of this equation,
is, spin H,< and its
(that
found
to
be
Is
the
of
electron
momentum
angular
%H
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
well as negative electrons should
exist.
At
first it
its
prediction that positive as
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 itradiation at the earth's surface.
430
THE NUCLEUS
Positive electrons, as
mentioned
earlier, are
431
The
usually called pttsitrom.
materialization of an electron-positron 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
is
positron
often spoken of as the antipartide of the electron, since
is
able to undergo mutual annihilation with an electron.
elementary particles except for the photon and the
and
«r*
autiparttele counterparts: the latter constitute their
All other
it
known
V mesons also have
own
ant part teles.
i
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
The
spin and magnetic
its
distinction lietween the neutrino
interesting one.
of
its
moment
The
spin of the neutrino
motion; viewed from behind, as
clockwise.
The
direction as
its
is
also opposite to that of (he particle.
and the antineutrino
is
opposite
in
a particularly
in Fig. 13-1, the neutrino spins
spin of the antineutrino, on the other hand,
direction of motion;
is
direction to the direction
viewed from behind,
it
counter-
in the
same
spins clockwise.
Thus
is
the neutrino moves through space in the manner of a left-handed
screw, while
the antineutrino does so in the maimer of a right-handed screw.
Prior to 1956
had been universally assumed
it
that neutrinos could
assumption had roots going
all
if
we
is
in
we
a mirror,
cannot ideally distinguish which object or
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
the interchange of right and
is
left,
must occur with equal probability with right and
doctrine
is
1U56
its
applicability to neutrinos
and processes
theoretical discrepancies
would
different handedness, even
in a mirror.
had never been
l^e and C. N. Yang suggested
In that year T. D.
be reflected
all objects
interchanged. This plausible
indeed experimentally valid for nuclear and electromagnetic inter-
actions, but until
tested.
and so
left
l>e
though
removed
it
meant
if
actually
that several serious
neutrinos and anti neutrinos have
that neither particle could therefore
Experiments performed soon after their proposal showed
unequivocally that neutrinos and antineutrinos are distinguishable, having
handed and right-handed spins
of right-left
symmetry
in neutrinos
zero, thereby resolving
We
respectively.
what had
can occur only
might note
if
the neutrino mass
exact ly
is
liecn the very difficult experimental
left-
absence
that the
problem
of measuring the neutrino mass.
way back to Leibniz, Newton's
calculus. The argument may be
MESON THEORY OF NUCLEAR FORCES
13.2
the
contemporary and an independent inventor of
stated as follows:
process
be either
left-handed or right-handed, implying that, since no difference was
possible
l>etwcen them except one of spin direction, the neutrino and antineutrino
are
identical. This
both directly and
observe an object or a physical process of some kind
If
nuclear forces were exclusively attractive, a nucleus would be stable only
were
so small (about
its
size
all
the others.
2 fm
The binding energy per nucleon would then be proportional
A, the uumt)cr of nuclcons present. In
proportional to
all
nuclei;
A
and the binding energy per nucleoli
lie
component
a repulsive
to
nuclear volumes are found to be
is
roughly the same for
each nucleon interacts only with a small number of
Thus there must
bors.
fact,
if
each nucleon interacted with
in radius) that
nuclei from collapsing, as indicated in Fig. 11-6,
in
its
nearest neigh-
nuclear forces that keeps
which means
that these forces
are not analogous to the "ordinary" gravitational and electrical forces.
^^^^•^
neutrino
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
FIGURE 131
Neutrinos and antineutrinos have apposite elec-
force which arises because of the possibility that the electron can
shift
trons af spin.
of the protons to the other.
/
system
force
is
is
symmetric or antisymmetric
either attractive or repulsive.
between nuclcons
force as well.
o:
antineutrino
432
THE NUCLEUS
for the particle exchange, the
It is
to be, at least in part,
wave function
is
of the
exchange
tempting to consider the interaction
a consequence of some kind of exchange
For instance, exchange forces provide an explanation
stability of the triplet state of the deuteron,
which
from one
Depending on whether the wave amotion
since the spins are parallel,
described by an antisymmetric
which
and the
wave
is
descril>ed
for the
by a symmetric
instability of the singlet state,
function.
Since the nudeons in
ELEMENTARY PARTICLES
433
,i
nucleus are
quantum
different
all in
states (by the exclusion principle), both
would occur, and a mixture of an
attractive and repulsive exchange forces
"ordinary" attractive nuclear force and such exchange forces
way
a general
in
The
many
for a great
next question
is,
is
able to account
nuclear properties.
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
However, calculations based on beta-decay 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,
a neutron.
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
13-2
Attractive and repulsive
forces can bath
ariic
From
According
to
due
to particle
exchange
change.
produce between micleons arc of the correct order
lie
meson theory
the
of nuclear forces,
all
neutral or curry cither charge, and the sole difference
protons
The
is
nueleons consist of
surrounded by a "cloud" of one or more mesons.
identical cores
supposed
forces that act
to lie in the
composition of their respective meson clouds.
result of the
exchange of neutral mesons (designated
between them. The force lietween a neutron and a proton
exchange of charged mesons
tt~
meson and
ii
-*
is
Mesons may
between neutrons and
between one neutron And another and between one proton
and another are the
^* and
57"
is
rr")
the result of the
converted into a proton;
p +
attractive force
yields the
same
effect as a repulsive force
it
tt
by the proton the neutron was interacting with
into a neutron;
it
p
+
it
-
-> n
converts
proton emits a ** meson whose absorption by a neutron
into a proton:
it
of particles
no simple mathematical way
between two bodies can lead
of
demons! rating how the exchange
to attractive
and repulsive
rough analogy may make the process intuitively meaningful.
two boys exchanging basketballs (Fig.
they each
b'3-2).
move backward, and when
backward momentum
THE NUCLEUS
increases.
l>e
equivalent
between them.
of physics refer exclusively to experimentally measurable quantities,
and the uncertainty principle limits the accuracy with which certain combinations 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
is
the boys snatch
—
p —* n + it*
n + 7T —» p
While there
If
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
In the reverse process, a
between the boys.
the basketballs from each other's hands, however, the result will
while the absorption of the
converts
due to particle exchange
between them. Thus a neutron emits
>
to an attractive force acting
434
repulsive force
particle ex-
magnitude.
ol
a
FIGURE
Thus
If
they throw the
forces, a
method
interacting with so soon afterward that even in principle
impossible to determine whether or not any mass change actually has been
involved.
Since the uncertainty principle
may be
written
Let us imagine
lralls
at
each other,
they catch the balls thrown at them, their
this
it is
it is
of exchanging the basketballs
13.1
an event
IE \t >
in
ft
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
that nuclear forces
we assume
if
of light
ft
a
meson
have
a
maximum
range
R
of about
1
.7
fm, so
travels lietween nuclei at approximately the speed
the time interval At during which
in flight
it is
Tire emission of a
so short, in fact, that
its
existence
is
was
7
X
10" 1T
s.
The
lifetime of the
not established until 1050,
into lighter
mesons called
ii
mesons
i
meson
9
of mass »i T represents the nonconservalion of
These neutrinos are not the same as those involved in beta decay, which is why
was estabtheir symbols are ^ and Fy The existence of two classes of neutrino
protons,
and
high-energy
lished in 1962. A metal target was bombarded with
AE = m^c 2
13.3
tt" is
and that of the neutral pion
Charged pions almost invariably decay
(or m jo m) and MllUlUMa
is
*«£
11.2
X 10" s,
1.8
of energy. According to Eq. 13.1 this can occur
if
AE At >
ft;
that
is,
if
pions were created
Inverse reactions traceable to the neutrinos
profusion.
in
Hence
from the decay of these pious produced unions only, and no electrons.
decay.
v.
h
lieta
these neutrinos must lie somehow different from those associated
Hence the minimum meson mass
is
specified
by
The
neutral pion decays into a pair of
w° -> 7
+
gamma
rays:
y
13.4
The
>
which
is
X
1.9
about 200
mg
,
H>- 2S kg
that
is,
less,
it*
264
and
in,.
rest
is
the antiparticle of the
it
shares only with the photon and the
particle, a distinction
200 electron masses.
masses of 273 m,, while that of the «*
have
tt"
tr"
The
Whereas the existence oF pions
predicted
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
many
energy must be supplied to a nucleoli so that
/a
energy.
Thus
at least
mTc 2
of energy, about 140
MeV,
is
required.
To
furnish
MeV are
I
therefore required to produce free pions, and such particles are found
nature only in the diffuse stream of cosmic radiation that bombards the earth.
lence the discovery of the pion had to await the development of sufficiently
sensitive
and precise methods of investigating cosmic-ray interactions.
More
anti-
meson.
were
unions even today represent
—*
e+
—»
e~
+ r. +
+h +
As with electrons, the
particle.
There
is
'-,-
''.
positive- charge state of the union represents the anti-
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 cosmic-ray particles at sea level are muons from
the decay of pions created in nuclear collisions caused by fast primary cosmic-ray
atomic nuclei, siuce nearly all the other particles in the cosmic-ray stream either
necessary particle energies, and the profusion of pions thai were created with
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
their help
lack of any function.
recently high-energy accelerators were placed in operation; they yielded the
could be studied readily.
The second
reason for the lag between the prediction and experimental
discovery of the pion
436
slightly
own
so readily understandable that they
years licfore their actual discovery,
emission of a pion conserves
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
in
+
enough
a stationary nucleoli with this
ij°
is
.
ii"
its
tf" is its
iintiiieutrino pairs:
/i
to the belated discovery of the free pion. First,
,
something of a puzzle. Their physical properties are known quite accurately.
spin,
Positive and negative unions have the same rest mass, 207 m„, and the same
u
neutrinol
electrons
and
10
s
into
1.5
of
a
half
life
X
Both decay with
pious.
Two factors contributed
is
w + and the
THE NUCLEUS
is its
instability: the half life
of the charged pion
is
only
Only
in
its
mass and
instability
muon differ
muon is poet)
does the
from the electron, leading to the hypothesis that the
a kind of "heavy electron" rather than a unique entity. Other evidence, which
significantly
ELEMENTARY PARTICLES
437
we
shall
examine
later in this chapter,
is
less
unflattering to the
mnon, although
mass. (A, 2, E, and
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
xi,
and omega.)
of
all
decay directly into electrons and neutrinos.
half lives,
it
is still
hyperons
ft are,
respectively, the
Creek
capital letters lambda, sigpui.
mean
All are unstable with extremely brief
is
%
except that of the
ft
The
lifetimes.
hyperon, which
and decay schemes of various hyperons are given
is
spin
%. The masses,
Table
in
13.1,
Like pions and kaons (but unlike inuons), hyperons exhibit definite interactions
with nuclei. The A" hyperon
13.4
KAONS AND HYPERONS
the
Pions and inuons do not exhaust the
list
of
known
particles with masses interme-
between those of the electron and proton. A third class of mesons, called
(or toons), has Iwen discovered whose members may decay in
a variety
of ways. Charged kaons have rest masses of 96ftn
spins of Q, and half lives
A
is
even able to act as
A" hyperon
nucleus containing a bound
is
w meson
decays, of course, with the resulting nucleoli and
with the parent nucleus or emerging from
A
nuclear constituent.
a
called a hyperfragpient; eventually
either reacting
entirely.
it
diate
K
mesons
r,
of 8
X
10" 9
s.
The following decay schemes are
possible for
K+
SYSTEMATICS OF ELEMENTARY PARTICLES
13.5
mesons, in order
of relative probability:
Despite the multiplicity of elementary particles and the diversity of their properties,
'
J»*
-» w~
-» JT +
-*
ifl
-* w°
-+ V+
+
»V
+ it"
+ + + V~
is
possible to discern an underlying order in their behavior.
more than
+
+
e*
4-
+
+
+ V° +
fi
but
vt
it
may indeed be
a single theoretical picture
can encompass elementary-particle 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
that
*-„
57°
we
elementary particles and their apparent significance.
following decay modes are
particles
The
known
for neutral
K
mesons, again in order of relative
probability:
a
listing in
By
+
v~ +
relatively stable
A"
meson, which
we
shall discuss
light to travel a distance
equal to the "diameter"
life,
Decay
i
2,184
1.7
x
10-'"
A —* p + T
2,328
0.6
x
lO" 1 "
V-
1.1
X
10
"'
v-
2,342
K mesons exhibit varying degrees of specifically nuclear interactions.
and K° mesons interact only weakly with nuclei, while their antiparticle
»
2,334
T
2,585
they pass,
counterparts are readily scattered and absorbed by nuclei in their paths.
Elementary particles heavier than protons are called hijpemm. The known
and
ft
n
a
2,573
<10 "
IS x 10 '"
'"
2.0 X 10
3,276
vHr-w
+
'
if
_ p + „o
-» n
2-
THE NUCLEUS
ij
that the half lives of the particles all greatly
-
*
into four classes, A, 2, 2,
meant
Halt
which
fall
is
regularities observed in
mass of the relatively stable elementary
thus far mentioned plus the
Particle
In addition to their electromagnetic interaction with matter through
hyperons
rest
examine the
HYPERON PROPERTIES,
';-
ir+
order of
shall
Table 13.1.
7T
-»
is
we have
17*
•',
K+
Table 13,2
exceed the time required for
+W
-» n* + w°
K z -* w" + e T +
— IT* + e +
—
+ M + + r.
-* ff+
+r + 1
-» ** + ir + -"
The
the remainder of this chapter
shortly.
,
fact
the order found in atomic spectra constitutes a theory of the atom,
does provide hope that there
There are apparently two distinct varieties of neutral K mesons, the A", and
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.
K<
The
5T
K 3°.
438
it
of this order does not constitute a theory of elementary particles, however, any
+
v'
_ „ + „2° - A + t
S-
-*
A
+
w_ A +
«- -» A
-
v~
Jr
»
K
hyperons, in order of increasing
ELEMENTARY PARTICLES
439
of an elementary particle. Tin's diameter
*•
probably a
is
the characteristic time required to traverse
s
23
of the order of magnitude of 10"
M
all
s.
it
Thus the
at the
little
over 10 _ls m, and
speed of
light is therefore
particles in Tabic 13.2 are almost
capable of traveling through space as distinct entities along paths of meas-
I
urable length in such devices as bubble chambers.
A
+
+
considerable Isody of experimental evidence also points to the existence of
10" 23
many different "particles" whose lifetimes against decay are only about
What can lie meant by a particle which exists for so brief an interval?
+
how
can a time of JO -23
by observing
s
even
their formation
l>e
s.
Indeed,
measured? Such particles cannot be detected
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
i,
II
as
1
V
energy
Chap. 4
levels; in
we
reviewed the Franck-Hertz experiment, which
showed the existence of atomic energy
levels
through the occurrence of inelastic
An atom
electron scattering from atoms at certain energies only.
excited state
o
I I
I.S
s
but
state,
a I
is
not the same as that atom
we do
-s
II
ground
in a specific
state or in another excited
not usually speak of such an excited atom as though
— the electromagnetic interaction —
different situation holds in the case of
elementary
is
well understood.
particles,
were
it
a memljer of a special species only because the interaction that gives
the excited state
g
in its
A
rise to
rather
where the various
interactions involved are, except for the electromagnetic one, only partially
§
cr
I
js;
;£?
much
understood, and
«£ 2*
comes from the properties of the
of oiu information
resonances.
I^et
o
i>i
b
us see
what
An experiment
getic
»
ir
•I
+
is
involved in a resonance in the case of elementary particles.
performed, for instance the Ixnnbardmcul of protons by ener-
+ mesons,
•7T+
IN
is
and a certain reaction
p
—>
TT
+
+p+
7T
+
+
studied, for instance
is
+
1!~
ff°
fe
The
new
effect of the interaction of the -a*
pions.
In each such reaction the
and the proton
is
new mesons have
the creation of three
a certain total energy
that consists of their rest energies plus their kinetic energies relative to their
If we plot the number of events ohserved versus the total energy
new mesons in each event, we obtain a graph like that of Fig, 13-3,
center of mass.
of the
Evidently there
1
is
a strong tendency for the total
and a somewhat weaker tendency
I
for
it
to
reaction exhibits resonances at 548 and 785
meson energy
be 548 MeV.
MeV
or,
We
to
be 785
equivalently,
we can
that this reaction proceeds via the creation of an intermediate particle
a
can be either one whose mass
ill
=
&
X4
<F
J:
is
548
MeV or one whose mass
is
MeV
can say that the
say
which
785 MeV, From
we can even estimate the mean lifetimes of these intermediate particles,
which are known as the
and w mesons, respectively. According to the unthe graph
jj
d
|2
9
certainty principle, the uncertainty in decay time of an unstable particle
— which
ELEMENTARY PARTICLES
441
spin.
field
there
If
is
a graviton. a particle that
the same sense that die photon
field in
meml>er of
and
stable,
The
this class.
sbutild
tremely weak, and
its
quantum
or the pion the
the quantum of the gravitational
is
the
quantum
of the nuclear force
gravitnu,
have a spin of
it
is
of the electromagnetic
field, il
Us interaction with matter would be ex-
2.
unlikely that present techniques are capable of verifying
is
existence. (The zero mass of the graviton can be inferred
we saw
range of gravitational forces. As
two bodies can
them.
lie
in Sec. 13.2, the
from the unlimited
mutual forces Ixstween
regarded as transmitted by the exchange of particles Ijetween
energy conservation
If
would be another
yet undetected, should be massless and
its
be preserved, the uncertainly principle
to
is
requires that the range of the force be inversely proportional to the mass of the
exchanging particles, and so gravitational forces can have an
only
the graviton mass
if
A
zero.
is
similar
argument holds
infinite
for the
range
photon
mass.)
After the photon in Table 13.2
come
the e-neutrino and u-neutrino, the
with spins of /2 These particles are jointly called
mesons, all widi spins of 0, are classed as mesons,
electron,
and the inuon,
leplons.
The
(Despite
its
name, the u meson has more
with the
it,
K, and n mesons,) The heaviest particles, namely the nuclcons and
w,
K, and
all
tj
[
.
in
common
with the other leptons than
hyperons, comprise the baryons.
While
ts
L, Af,
600
eoo
EFFECTIVE MASS. MeV
FIGURE 13-3
total
Resonant states
In the reaction
»*
1.000
and B as
reasonable on the basis of mass and spin alone, there
is
its
We
follows.
= —
favor.
I
+ p-*w' +
+
/?
r*
IjCt
new quantum numbers,
us introduce three
assign the
number L
to their auti particles; all
=
to the electron
1
+ r- + a"
The
mass,
significance of these
=
to the
1
numbers
is
ji
meson and
that, in
neutrino,
its
its
mean
in Fig,
tAE
Hence
— will
give
rise to
is
an uncertainty
at half
maxim tun
peak
==
is
The
ij
lifetime
too short by
many
it
is
is
sufficiently long for
it
i\
particles in Table 13.2
In a class
442
by
itself is
THE NUCLEUS
classical conservation
electric
ss
0.
I
lliat
and B remain constant.
momentum, angular momentum, and
B help us to detercapable of taking place or not. An example
laws of energy,
charge plus die new conservation laws of L, M, and
is
the decay of the neutron.
to
be
included in Table 13.2, while
orders of magnitude.
We
seem
*>-> p +
and
shall return to
the resonance particles later in this chapter.
The
L =
ft
the lifetimes of the resonances, or, just as well, the lifetimes of the
lifetime
The
mine whether any given process
regarded as a relatively stable particle and
«
the determination of
is
mesons, can be established.
the
in
of die corresponding
is
13.5
m
lifetime t
—which the width A£
13-3 — whose relationship
energy
and Af
every process of whatever kind
involves elementary particles, die total values of L. Af,
is its
and the
other particles have
—
We assign the number
=
=
B
1
0, Finally, we assign
to their autiparticles; all other particles have M
=
0.
to all baryons, and B = —I to all an ti baryons; all other particles have B
,V/
at effective
ter of
grouping
e-neutrino, and L
masses of 548 and 785 MeV. By effective mass is meant the
energy, including mass energy, of the three new mesons relative to their cen-
occur
this
hirther evidence in
to fall naturally into four general categories.
the photon, a stable particle with zero rest mass and imit
While
/,
=
+
e"
+
»t
for the neutron
L = for the electron and — for
L before and after the decay is 0.
and proton, so that the total value of B l)efore
and proton,
1
1
the antineutrino, so that the total value of
Similarly,
and
B =
after the
1
for both neutron
decay
is
1
.
The
stability of the
proton
is
a consequence of energy
and baryon-number conservation; There are no baryons of smaller mass than
the proton,
and so the proton cannot decay.
ELEMENTARY PARTICLES
443
Despite the introduction of the quantum numbers L, M, and B, certain aspects
of elementary-particle behavior
why
to see
gamma
defied explanation.
For instance,
it
was hard
Thus the S baryon decays
into a
n
A baryon and
a
gamma
A +
fl
baryon
never observed
+
-A p'
Another peculiarity
to
decay into a proton and a
gamma
is
minor
now
perturbation that sees to
it
is
However, many strange
have relatively long
number S.
It is
found that
in all processes
A
These and
to the various
number
elementary
no way
ij
u
it"
over
and
S.
Table 13.2 shows the values of S that are assigned
particles.
We
note that L, B, and S are
between them and
ij"
their antipartieles.
mesons are regarded
as their
own
for the
For
this reason
antipartieles. Before
consider the interpretation of the strangeness iminlwr,
we
shall
There are apparently four types of interaction between elementary
have
to
give
rise to all the
particles
physical processes in the universe.
the gravitational interaction.
Next
The
weak
feeblest of these
is
interaction that
present between leptons and other leptons, mesons, or baryons
is
addition to any electromagnetic forces that
may
exist.
is
the so-called
The weak
interaction
responsible for particle decays in which neutrinos are involved, notably beta
decays.
S
all
charged particles and also those with electric
or magnetic moments. Finally, strongest of
called simply strong forces
when elementary
all
in
observed to occur, while the superficially similar decay
is
2 + -A p + +
= -
Y
1
high-energy nuclear collisions which involve strong interactions, and their
multiple appearance results from the necessity of conserving
slowness with which
tj°
meson decay
characteristic of
:
:
of
S,
is
S.
unstable elementary particles save the
accounted
for if
we assume
mesons and baryons
that
weak
as well as leptons,
The relative
meson and
ir°
interactions are also
though normally domi-
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~
S
-/*
n"
+
77
= -2
but instead via the two steps
I- -* A
S= -2
-1
A<>
_> n o
+
7T-
+
ffO
:
through which the corresponding forces act are very different. While the strong
between nearby nucleons
is
all
nated by strong or electromagnetic interactions. With strong or electromagnetic
processes impossible except in the above cases owing to the lack of conservation
between mesons, baryons, and mesons and baryons.
relative strengths of the strong, electromagnetic, weak, and gravitational
-4
interactions are in the ratios 1 10 -a 10 -14 10 ". Of course, the distances
THE NUCLEUS
y
1
are the nuclear forces (usually
particles are being discussed) that
The
force
+
A<>
—
1
does not conserve S and has never been observed. Strange particles are created
Stronger than gravitational and weak interactions are the electro-
magnetic interactions between
are found
= —
conserves S and
at a
examine the various kinds of particle interaction.
that, in principle,
S
odd feature
two or more
mesons. Since these particles are also uncharged, there
to distinguish
the photon and
is
other considerations led to the introduction of a quantity
still
photon and w° and
third
number
particles
lifetimes, well
that strange particles are never created singly, but always
time.
apparently
inappropriate
that nuclei with
The decay
vo -,
amounts of energy take place more rapidly
releases considerable energy
called strangeness
444
scale,
force in the structure of matter
return to the strangeness
based upon the general observation that physical proc-
a billion limes longer than theoretical calculations predict.
is
t*t us
conserved.
y
than processes that release small amounts.
whose decay
in
weak
role of the
neutron/ proton ratios undergo corrective beta decays.
esses in nature that release large
we
The
involving strong and electromagnetic interactions the strangeness
&
is
on a small
action, utterly insignificant
ray,
y
is
Hence the gravitational interbecomes the dominant one on a
forces are severely limited in their range.
large scale.
ray,
is
weak
that of a
vo _>
^+
ray while others do not undergo apparently equally pern
missible decays.
while the
still
certain heavy particles decay into lighter ones together with the
emission of a
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 electromagnetic interaction. Matter in bulk is electrically neutral, and the strong and
tionai force
STRANGENESS NUMBER
13.6
many powers
of 10 greater than the gravita-
S
= -
1
ELEMENTARY PARTICLES
445
A
quantity called hypercharge,
Y,
has also !>een found useful in characterizing
= '/2 B = I, and S = 0, so that = e, while the
=
=
In the ease of the w
B
1, and S = 0, so that q neutron has f3
%,
=
=
values
of
7
of
I,
three
0, and -1 respecand
the
S
meson multiplet, B
3
number
B
are conserved in
=
baryon
Charge
and
e, 0, and -e.
tively yield q
multiplet, the proton has f 3
tf
,
().
particle families;
sum
to the
it
conserved
is
Hypercharge
in strong interactions.
and haryon numbers of
of the strangeness
equal
is
the particle families;
V = S + B
13.6
all
For mesons the hypercharge
in strong
The
equal to the strangeness.
is
Thus
interactions.
and
7
assignments are listed in Table 13.2.
/:i
An
additional conservation law
of whose
members
members
of the
fundamental
to the
3.2 that there are a
2
its
*
'/,
+
=2
1
umlliplct has I sb 1,
ij
There
meson has
/
1
and
=
since
name
occurs
it
+
into 21
I
in
1
I in '"isotopic
spin space"
of
/.,
are restricted to
integral
if I is
of the
it"
is
substates,
and
momentum,
and integral or zero
is I
a
'/
2.
which means
v meson,
/.,
=
= —
I
and
I
1
The charge
strangeness
of a
number
/-,
=
to the ir~
and baryons are assigned
mesun or
S,
1
of their isotopic spin vectors,
/
3
it is
space
we
only in the orientation
can say ihul the charge independence
is
independent of orientation
momentum is likewise
independent of orientation
that this interaction
Angular
lies
the difference
,
which might lead
conserved
in all interactions,
conserved
in strong interactions.
us to surmise
that isotopic spin
is
be correct; the isotopic spin quantum number
strong, but not in
weak
somewhat
We note
that,
is
This surmise happens to
found
to
be conserved
We
or in electromagnetic, interactions.
in
shall return to
and invariance with respect
the relation l>ctween conservation principles
symmetry operations
/
to
in the next section.
although
/.,
is
conserved
in
electromagnetic interactions,
need not be. An example of a process in which
meson into two photons:
is the decay of the w
/
changes while
73
/ itself
does not
f>
any specified direction
l3
.
The
possible values
-/, so that
1),
if /
space and
and so
means
of the strong interaction
Now
electric charge.
in isotopic spin
between a proton and a neutron
in real
/3
I.
in
can be either
/ 3 is half-
The
integral.
is
corresponds to the
tt
meson. The values of
in a similar
'^
or
isotopic
-
1
/,;
it"
A
—»
w" meson has
no component
the
with
lwiryon
'
/3
meson,
/.,
=
for the other
its
7
;J
y
/
+
=
y
1
and
73
=
0,
while
/ is
not defined for photons; there
of isotopic spin on either side of the equation, which
conservation, although
/
is
is
consistent
has changed.
to the
mesons
13.8
way.
is
and the component
related to
I,
of
its
its
haryon number
isotopic spin
B
t
In the previous section the
charge independence of the strong interaction was
expressed in terms of the isotropy of isotopic spin space.
momentum,
this
symmetry was
in strong interactions.
It Ls
said to
By analogy with angular
imply the conservation of isotopic spin
a remarkable fact that all
known symmetries
in the
physical world lead directly to conservation laws, so that the relationship between
isotopic spin vector
charge of the particle thus represented.
THE NUCLEUS
SYMMETRIES AND CONSERVATION PRINCIPLES
its
by the formula
f--'MH)
Each allowed orientation of the
to the
that
-
depend upon
}
isotopic spin can be represented
-(/
(1
1
=
I
angular-momentum
this has led to Ihe
whose component
Such
taken to represent the proton and the latter the neutron. In the case
meson, and
13.7
—
half- Integra I
spin of the nueleon
former
I
I,
assigned
and tt" mesons.
and 2 -0 + 1 = 1.
only a single state
governed by a quantum number customarily denoted
is
is
the t+, v~,
of iialopic spin tfuontum number for
Pursuing the analogy with angular
by a vector
interaction itself does not
such that the multiplicity
and the proton. The ^ meson
1=3 states are
2*1 +
its
/
Thus the nucleoli multiplet
.
states are the neutron
quantum number
misleading
446
by a number
its
from the strong interaction.
result
substituted for a proton or vice versa only by amounts that
in isotopic spin space.
exhibits
it
evidently an analogy here with the splitting of an
is
state of
+
is
suggested by the observed charge inde-
binding energy and pattern of energy levels change
natural to think
it is
has proved useful to categorize each multiple! according
It
given by 2/
is
of particle families each
of a multiple! as representing different charge states of a single
entity.
interactions does the
follow from purely electromagnetic considerations, implying that the strong
number
has essentially the same mass and interaction properties but
number of charge states
of the state
The
1
These families are called nmltiplels. and
different charge.
and
when a neutron
is
which
forces,
properties of a nucleus as
ISOTOPIC SPIN
obvious from Table
In
conserved, namely
is
weak
change.
pendence of nuclear
It is
Only
electromagnetic interactions.
various hypercharge
total
13.7
must be conserved whenever S
3
I
hence
is
directly connected
symmetry under
rotations of
I
and the conservation of
I
is
wholly plausible
In the case of the nucleoli
ELEMENTARY PARTICLES
447
Let us survey some of these symmetry-conservation relationships
conservation of
system in which they are expressed, has as a consequence the
beti.
angular
tinning our discussion of elementary particles.
What
that a
meant by a "symmetry"? Formally,
is
symmetry of a
when
particular kind exists
something unchanged. A candle
is
rather vaguely,
if
feature;
lists
it is
also
symmetric with respect
in
because
a vertical axis
(As elaborated in
Table 13.3
the principal symmetry operations which leave the laws of physics un-
changed under some or
circumstances.
all
The
simplest symmetry operation
which means
mvariaiicc of the description
oi
space
in
is
which means
translation in time,
upon when we choose
=
f
the conservation of energy.
that the laws of physics
to
=Vx
an j b
.is
l>e,
that the laws of physics
and
B COBSe-
do not depend
invariance has as a consequence
this
Invariance under rotations
space, which
in
A
instead of in terms of
means
do not depend upon the orientation of the coordinate
A.)
of identical particles in a system
The interchange
is
The wave
function
symmetry
wave function
a type of
operation which leads to the preservation of the character of the
may be symmetric under such an
interchange,
principle and the system
in which case the particles do not ol>ey the exclusion
in which case the
antisymmetric,
follows Bose-Einstein statistics, or it may be
statisFermi-Dirac
follows
particles obey the exclusion principle and the system
tics.
statistics (or, equivalently, of
Conservation of
antisymmetry)
signifies that
statistical
wave-function symmetry or
no process occurring within an isolated system can
behavior of that system.
A system
exhibiting Bose-Einstein
behavior cannot spontaneously alter itself to exhibit Fermi-Dirac
behavior or vice versa. This conservation principle has applications
statistical
in
nuclear
of nucleons
physics, where it is found that nuclei that contain an odd number
with even A obey
(odd mass number A) obey Fermi-Dirac statistics while those
further condition a
statistics; conservation of statistics is thus a
Table 13.3.
PRINCIPLES.
Bose-Einstein
Conserved quantity
Symmetry aeration
and B,
conservation.
change the
SOME SYMMETRY OPERATIONS AND THEIR ASSOCIATED CONSERVATION
E
leads to charge
are obtained by differentiating the potentials, and this invariance
of a system.
lias
quence the conservation of linear momentum. Another simple symmetry operation
where the two
and
by the vector calculus formulas E = -TV
leave E and B unaffected since they
transformations
Gauge
descriptions are related
that the laws of physics
nature to translations
V
described in terms of the potentials
is
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
translation in space,
shifts in the zeros of the scalar
it
appearance or any other
to reflection in a mirror.
is related to gauge transfonnatioas, which are
and vector electromagnetic potentials V and A.
electromagnetic theory, the electromagnetic field can be
say
a certain operation leaves
symmetric about
can be rotated about that axis without changing
we might
momentum.
Conservation of electric charge
nuclear reaction must observe.
The conservations of the baryon number B and the lepton numbers L and
in having no known
\1 are alone among the principal conservation principles
All interactions are dejiendent of:
momentum p
Energy £
lobular momentum
Linear
Translation in space
Translation in time
flotation in .space
Ueetroinagnctic gauge transformation
Electric charge
Interchange of identical particles
Type
Inversion
nl
space, time,
and charge
symmetries associated with them.
I
Apart from the charge independence of the strong interaction and its associated
coascrvation of isotopic spin, which we have already mentioned, the remaining
a
of statistical behavior
Product of charge parity, space parity, and
Jhi-
afmng
H.I
?
l/Cptnn
numher
f.
?
I
i-filim
number
M
llllllllll'l
ft
spatial coordinates
and -a replacing
unit t'leetntma^wtie interactions
only are independent
..ii
Parity
Table 13.3
involve purities of one kind or anotiier.
all
through the origin, with
z.
If
the sign of the
-x
replacing
x,
-ij replacing
y,
wave function ^ does not change under
P
Charge parity
of charge
l
The tlrrmg interaction only
in
such an inversion,
of:
Inversion of space
ileflecl
Mill
?
I'
symmetry operations
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
lime paril) CI'
it
(.'.
isotypic spin
component
W*y,z)
= M-x.
-n.-z)
v and strangeness S
and
independent
if-
is
said to
have
eceri parity.
If
the sign of
^
changes,
ofi
Charge
Isotopic spin
I
uX*,t/,~)
= -tM-x.
-f> --)
ELEMENTARY PARTICLES
448
THE NUCLEUS
449
and ^
=
cos x
If
said to
is
cos
we
(
have odd
— x),
Thus the function cos x has even parity since
x has odd parity since sin x = —sin (—x).
parity.
while sin
it
meson can decay into a -n* and a jt"
T. The symmetry of phenomena under
status
at present. The symmetry operation
thus has an ambiguous
was discovered
1964 that the Kg
meson, which violates the conservation of
time reversal
write
in
that corresponds to the coaservation of charge parity
Mx,ti,z}
we can
are
+
=
P$(-x, -y, -x)
which
regard P as a quantum numter characterizing
1
(even parity) and
-
is
Every elementary particle has a
and the parity of a system such as an atom
it,
the product of the parity of the
wave
the replacement of every
paritv C, like space parity
possible values
(odd parity).
1
certain parity associated with
or a nucleus
$ whose
is
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
particle in a
C
system by
is
its
dmrge
conjugation,
antipartiele.
Charge
P, is
not conserved in weak interactions. However,
despite the limited validities of the conservation of C,
P,
and
T,
there are
good
and
theoretical reasons for believing that die product CPT of the charge, space,
means
of
CFF
conservation
The
conserved.
time parities of a system is invariably
that for every process there
place in reverse, and this
an antimatter mirror-image counterpart that takes
particular symmetry seems to hold even though .its
is
component symmetries sometimes
fail
individually.
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 eonsei-vation
of parity.
The conservation
that
is,
of parity
is
an expression of the inversion symmetry of space,
of the lack of dependence of the laws of physics
upon whether a
is
a profound difference between die
mirror image of either particle and the particle
that interactions in
interactions
to
This asymmetry implies
itself.
which neutrinos and antineulrinos participate— the weak
— need not conserve parity, and indeed parity conservation
hold true only
in the strong
the fact that spatial inversion
and electromagnetic
is
interactions.
is
earlier,
In addition to the particles listed in Table 13.2 there are, as mentioned
is revealed
a great many "particles" of extremely brief lifetimes whose existence
found
Uian
Of
to coasidcr the neutron, say, as
it Ls
course,
and then
it is
go on
to
this constitutes
particles.
On
mate, then
and
a theoretical
other parities occur in Table 13.3, time parity
T and charge parity C.
wave function when t is replaced
to the conservation of time parity
is
time reversal
implies that the direction of increasing time
of any process that can occur
if
symmetry under time
is
is
it Is
that corresponds
reversal
symmetry
not significant, so that the reverse
also a process that
reversal holds,
Time
can occur. In other words,
impossible to establish by viewing
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,
it
THE NUCLEUS
make
possible to
reso-
momen-
hundred
one
few truly elementary
line of attack
the other hand,
is
it
merely an unstable stale of the proton.
out an excellent case for supposing the latter,
to generalize that all the various
Historically
v.
These
tum, isotopic spin, parity, strangeness, and so on, and it is no more logical to
transient
disqualify them as legitimate particles because their existences are so
actually excited states of a very
not invariably a symmetry operation was
which respectively describe the l>ehavior of a
by — t and when (/ is replaced by —q. The symmetry operation
in interactions involving their longer-lived brethren.
nant states are characterized by definite values of mass, charge, angular
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
Two
450
THEORIES OF ELEMENTARY PARTICLES
by resonances
left-
handed or a right-handed coordinate system is used to deserilx: phenomena. In
Sec, 13.1 it was noted that die neutrino has a left-handed spin and the antineutrino a right-handed spin, so that there
13.9
if
"elementary" particles are
particles, as yet unidentified;
toward a comprehensive theory of elementary
we
accept the particles of Table 13,2 as
legiti-
consistent to include the resonant states as well, and to seek
framework
that
embraces the entire collection of well over a
entities.
recent proposal attempts to account for the various elementary particles
in terms of another kind of particle called the quark. Three varieties of quark
supposed
are postulated, plus their an ti particles, and all elementary particles are
A
and antiquaries. The really revolutionary
them
should have charges of - '/3 e and the
two of
to consist of combinations of quarks
thing about quarks
ibinl should
is
that
have a charge of
+%e- According
composed of three quarks, and each meson
Despite
much
effort,
is
to this theory,
each baryon
composed of quark-antiquark
no experimental evidence
in
Is
pairs.
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 so-called eightfold way, collects isotopic spin multiplets
into supermuttiplers whose members have the same spin and parity but differ
in
charge and hypereharge
(Figs. 1.3-4 and 13-5). The scheme prescribes the
members each particular supermultiplet should have and also relates
mass differences among these members. The great triumph of the eightfold way
number
was
its
of
prediction of a previously
subsequently searched for and
unknown
particle, the il~
discovered
finally
in
1
hyperon, which was
964. Other group-theoretic
approaches have related the supermultiplets of the eightfold way to one another
and have attempted
to incorporate relativistic considerations into the
hensive picture that
is
The
success of the eightfold
way
in organizing
interacting particles implies that the
a counterpart in a
more
hints
we
compre-
emerging.
symmetry
symmetry of
in nature.
The
our knowledge of the strongly
mathematical structure has
its
further
we
prol>e into nature, the
receive of a profound order that underlies the complications and
confusions of experience.
lieen revealed, there
still
But for
all
the elegance of the symmetries that have
%)
(+1
—
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.
—-
charge, e
—
1
%
and (except !! ) are shortBaryon supermuttiplet whose members have spin
particles; the I and 1 particles here are heavier and hane different spins Irom
Table 13.2. Arrows Indicate possible transformations according to the eightfold way.
FIGURE 13-5
lived
resonance
the ones
FIGURE 13-4
Supermultiplets of sppn-Vi baryons and spin-D
nuclear interaction.
melons
stable against decay by the strong
The
ti'
In
particle
was predicted from
this
scheme.
Arrows indicate possible transformations according to the eightfold way.
Problems
Find the maxim in n kinetic energy of the electron emitted in the beta
that must
decay of the free neutron, (b) What is the minimum binding energy
not decay?
docs
neutron
the
so
that
nucleus
a
neutron
to
a
by
be contributed
Compare this energy with the observed binding energies per nucleon in stable
1.
(a)
nuclei.
Van der Waals forces arc limited to very short ranges and do not have an
exchange
inverse-square dependence on distance, yet nobody suggests that the
2.
of a special meson-like particle
3.
-1
452
What
is
is
responsible for such forces.
the energy of each of the
gamma
rays
Why
produced when a
not?
-n°
meson
decays? Must they be equal?
THE NUCLEUS
ELEMENTARY PARTICLES
453
How much
4.
energy must a gamma-ray photon have
if
it
is
to materialize
in the
absence of another body without violating either the conservation of energy or
momentum,
the conservation of linear
Why
5.
long does the uncertainty principle allow a virtual electron-positron
2
2m n c 2 ,
(b) If hi'
pair to exist if hv < 2m c , where m,, is the electron rest mass?
(a)
Prove that such an event cannot occur
into a neutron-antineutron pair?
How
>
can you use the notion of virtual electron-positron pairs to explain the role of
a nucleus in the production of an actual pair, apart from
does a free neutron not decay into an electron and a positron? Into
the conservation of both energy and
its
function
in
assuring
momentum?
a proton-antiproton pair?
A
6.
•* meson whose kinetic energy
is
equal to
rest
its
energy decays
in flight.
Find the angle helween the two gamma-ray photons that are produced.
A proton of
7.
TQ
kinetic energy
proton-antiproton pair
is
produced.
collides with a stationary proton,
If
the
momentum
and a
of the Ixjmbarding proton
shared equally by the four particles that emerge from the collision, find the
is
minimum
value of
7"
.
8.
Trace the decay of the Z" particle into stable
9.
A
it
created.
One
10.
meson
What
collides with a proton,
particles.
and a neutron plus another
particle
arc-
the other particle?
is
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
neutron-antineutron pairs, what conservation laws would be violated?
Which
11.
of the following reactions can occur? State the conservation laws
violated by the others.
+p
(a)
j)
;/«)
P +
(c)
e*
(rf)
A" -*
12,
The
p-* p
+
+
it-
(e)
-* n
e' ->
+
+
W"
p -» n
+
7T
w°
meson has neither charge nor magnetic moment, which makes
ir°
hard to understand how
way
+ p + !r +
+ a° + s»
u + + w"
it
to account for this process
is
unclean- ant inuel eon pair, whose
to
assume
that the jr*
members then
two photons whose energies
yield
it
ean decay into a pair of electromagnetic quanta. One
total the
first
liecoines a "virtual"
interact electromagnetically to
mass energy of the
tt°.
How
long
does the uncertainty principle allow the virtual nucleon-antinucleon pair to
exist?
13.
ts this
The
long enough for the process to be observed?
interaction of one photon with another can
that each photon can temporarily
free space,
454
become a
be understood by assuming
"virtual" electron-positron pair in
and the respective pairs can then interact electromagnetically.
THE NUCLEUS
ELEMENTARY PARTICLES
455
ANSWERS TO ODD-NUMBERED PROBLEMS
Chapter
II.
1.24
X
13.
3.14
A
15.
5
17.
0.015
19.
2.4
21.
0.565
A
23.
(«)
2
X
(b)
2X10 »
(c)
3.5
t.
213
m
3,
2.6
X
5.
fi
7,
4.2
9.
II.
13.
15.
IT.
19.
21.
**
25.
10 s
2.6
ft;
m/s
m/s
0.988c;
x
4.2
K)'
x 10* m/s:
H.9 X 10- M kg
0.291 MeV
2.7 X 10" kg
4.4 X 10" kg
1.H7
,
-
I. H-l
X
s
10
m/s
10' B
X
B.Hfi
[.
(de/di)
.24
1
=
\
results arc different
because
Hz
lO" 3
X
X
10
eV
ft
I2
= w/2
ti
own
13.
1.18
X
which
15.
1.05
X10~ M
motion relative to the
medium
in
the sound waves propagate.
1(1;
10. 6.
(h)
-i/a
eV
'K&*')]
6.2%
ured by an observer depends upon bis
MeV
12.2'
11.
(a) 6,
103 H/.
X
m
whereas the speed of sound meas-
of light,
eV
1
K) " Hz; 7.7
GeV; 616
all
observers find the same value for Ihc speed
27.
Hz
A
Chapter 3
_ o7«yn
(i
The
%J98Bc
(lite;
m/s
10*
H> ,a
X
ft
10 7
X
08c
''
V
1
mi!);
(7.200
10 s m/s;
the
83 m/s
narrow wave
original
packet has spread out
yes
1.16Xl0 7 m
kg-m/s;
in
1
to a
s
much
wider one because the phase velocities of
Chapter 2
1.800
the waves involved vary with k and a large
range of wave numbers was present
A
MOO A]
X
2.83
3.9
10
,n
(b)
(<r)
tons/s
(d) 1.41
X
10 la
Each atom
17.
J
x Nr* photons/*
B.H2 x 10- "' lb/in. 2
4.24 x ID" photons/ m*-s
38
3.9 X UP* watts; 1.2 X 10
1.71
(«)
in the
original packet.
eV
photons/m 1
i
ei
i. tin
in
a solid
itofiinii- iciiiiin
<>\
is
limited to a
space
Hthamta
the assembly of atoms would not he a solid.
The
pho-
atom
uncertainty in the position of each
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
effectively infinite
i.s
and
Chapter 7
its
energy can be zero.
The alkali metals are the largest
3.
in
each
period of the periodic table, since their
Chapter 4
atomic structures consist of a single elec-
10*
tron outside closed inner shells thai shield
3.
0.878
the electron from
7.
1.14
ft
m/m =
].
X
charge. There
"in
K)
chugs
^T-
= ^~ /
!
'1
J
Forlhehydro'
V 4mynfl a
/= 6.6 x Iff**"1
2*r
gen atom,
,
winch
is
comparable with the highest frequencies
V
15.
12
17.
8.2
ft
2,1
1.04
which
increases,
pulls
the outer
At the
X
1.39
10-*
L
112
state
$.
%
3tz
%
2
2
3
%
*«»/*
5
%
5
%
=
=
=
r.
n
«
oo
=
n
4
n
n
=
=
»
3
ii
=
5
2
n
n
n
n
1
(c)
2.28
x
10- s
=
=
=
.
..
t =
ll
There are no other
SB
r
II
13.
4
i
ijii
18,5
Cobalt
7.29 eV; n
17.
3.3 eV;
=
27);
others.
18.70 eV: 16.84 eV.
15.
molybdenum
42)
7.
V
5.
eV
= V=
£/2, where
f and
are averages over an entire period of
b.
K
7.
ft
Chapter 6
3.
458
±
7.
68%; 25%
ft
p,
(3
29%;
<f.
i
5.
ffig,
7.
18%;
50 A; the ionization energy of the
is
0.0O9 eV, which
m'/m =
Because
25.
8.83
forces increase
7.
7.98
MeV
ft
15.0
M>\
represent
energy
cohesive
a
mode
smaller
1.01 in copper.
cm
the cohesive energy since they are attrac(b) The xero-piiiiit oscillations detive,
the
much
Chapter 11
34.97 u
The van der Waals
is
than the energy gap and not very far from
the 0.025 eV value of fcT at 20°C.
I
of energy
since
llicy
possession
H
1
",
B"
81 percent
Nnclear forces cannot be strongly
II,
charge-dependent.
The nucleon
13.
responds lo the
kinetic energy thai cor-
momentum
implied by an uncertainty
imcert:iiuiv
in position of
2
The
ions.
heal
lost
by the expanding gas
is
work done against the attrac
van der Waals forces between its
o|iial to the
1
{a)
1
n!
In a metal, valence electrons can
unoccupied excited energy states in the
1.27
A
2.23
bo
V2 MeV, which
is
entirely consistent
is
with a potential well 35
MeV
deep.
A
1.24
am
excitation energy',
i
13";,
ft
ANSWERS TO ODD-NUMBERED PROBLEMS
1
.6
(a)
b\
X
neutrons/m3
1.00: 1.08:0.89:0.22:0.027
yes;
15.33
K
X
10* s-'
1.23
3.37
can excite valence electrons to. the
(induction band although photons of in-
9.
The mass of ;Be
(c)
I0-"
1,620 yr
7.
(<
1.5 eV),
and
semiconductors
The energy gap
is
so large that pho-
tons of visible light cannot gravida
mug)
not su Hi c e n 1 y larger
i
1
a positron.
Hint: the 39th proton in !gY
mally in a
state
p„2
open
state,
is
nor-
and the next higher
to this proton
is
a &,
n
si
alt
MeV
electrons in the
13.
3.33
valence band to reach the conduction band.
15.
The neutron
excitation energy from
is
ll.. in that of {LI to permit the creation of
II.
purpose.
2/\/™„
1/4
so photons of visible
The energy gap
in
liuhl
llz
I.
3.
5.
small
X 10"
Chapter 12
is
(b)
No.
for
small.
frared light have insufficient energy for (bis
r
23.
1
5.
however
Chapter 9
=
Ne atoms
Chapter 10
1
f.
5.
1
2
which sup-
the direct excitation of
conduction bant!
13.
tl,
x
10 1
c»
It.
oscillation.
11
3.5
3.
2.1
f
19.
19 percent
{a)
m/s
21.
3.
1.
10*
molecules.
f.
Classically
in collisions,
by electron impact.
5.
5.
are
needed to maintain an inverted energy
population in the \e atoms by transferring
tive
in
Chapter 5
MeV
2.07 X 10-'*
The He atoms
not
eV
electron
Chapter 8
3.
the
than
particles
low-energy
of
3.
=
bosun
gas will exert the least pressure tocMfee
the Bose distribution has a larger propor-
atoms or
(Z
x 10"* m
= 8.4
2.56 X 10* K; 1.08 X
1.9
13.
tion
Is
eV; 7.6
II.
cles than the other distributions; a
and
tp.
energy distribution
very temperature sensitive.
present in a solid but not in individual
T
17.
tft
'
2
states.
5uB ;3
15.
t-i
limed
\f1
6
3
al
a very small fraction of
is
a larger proportion of high -energy parti
crease
He<_
fcT
A.
(fc)
H
p-tvpe
9,
so the electron
fermion gas will exert the greatest
A
13.
7.
pressure because the Fermi distribution has
plement
3P
2
2
MeV
X
10" ,2 ;2.3
5800 K
II.
energy to them
eV
0.0283 A
'S
A
21.
23.
then a regular decrease
is
end of each period there is a small increase
in si/.e due to the mutual repulsion of the
7.
X 10° rev
1.05 X 10* K
19.
nuclear
of
electrons in closer to the nucleus.
5.
920 A
13.
+e
but
outer electrons.
the hydrogen spectrum.
in
all
within each period as the nuclear
in size
002
1.
X
10'*
182
1.
momentum and hence
I0- IU :6.2
X
1.00:2.3
9.
cross section decreases
ANSWERS TO ODD-NUMBERED PROBLEMS
459
with increasing energy' because Lhe
hood that a neutron
will
likeli-
be captured
depends upon how much lime
it
spends
near a particular nucleus, and this
versely
proportional
proton crass section
gies
is
to
its
speed.
is in-
The
smaller at low ener-
because of the repulsive force exerted
3.
68 VleV; yes, in order that
,5.
number nor
spin;
7
,i,f«()
9.
A
decay conserves
MeV
neutrino.
11.
conservation of
10" 6
this
neither baryon on in her. spin, nor energy.
provides a potential barrier the proton must
X
INDEX
This decay conserves neither haryon
by the positive nuclear charge, which
tunnel through.
momentum
be conserved.
(a)
and
can occur;
(e)
li
conservation of
/,,
conservation of
rJ
and
M, and spin;
and
(b)
violates
spin; (c) violates
(d) violates
17,
1.2
19.
0.21
13.
2f.
(1.1%
of the micieiLS separates the electron and
AUmplinxi spectrum,
positron far enough so that they cannot
Acceptor level
recombine afterward to reconstitute the
Acetylene molecule, 207
Chapter 13
1.
0.78
MeV; L29 MeV, which
6.4
spin.
X HH* s; the strong electric field
119,
is
well
photon.
Actinium decay
nucleon of stable nuclei.
Alpha decay, 396
inverse, 4J
392
of ttdfobOtOptt, 389.
theory
Alpha
396
series,
Wlmly
Black body:
properties
molecule. 261, 264
Amorphous whd. 3 17
AuguLr
spectrum
Bohr.
molecular, 2711
nuclear, 384
.pin,
\iiiiiit-n(r
r
.:.
and quantum theory of atom. 193
-149
ami divalent handing. 249, 252
Atom:
momentum
182,
of,
of
222
1
model
complex, irroclurc
energy level*
Thomson model
vector model
221
of,
of,
153
102
222
{see
311
hydrogen. 328
32
metallic 331
van
tier
Waals, 327
Bose-Einstcin statistical distribution law, 288, 304
of,
300
Boson, 212
1
Atomic spec rum
Auger effect, 237
momenta, 297
factor, 299,
derivation
excitation, 131
Atomic number, 04
Atomic orhttritK. 254
Atomic rudll. 321
I
Boltzmann
ionic, 243,
of,
energies, 295
of molecular
eovalent, 243, 325
213
381
Hutherbrd nodal
Atomic
of,
(25. in I,
of,
distribution:
molecjW
Bond
113
of,
Bohr radius of hydrogen atom, 125
Boltzmann
Boh* model of iar Ruhr atmim model.!
of,
motion m. 129
Bohr magneton, 139, 207
Antisymmetric wave function, 211.
mass
101
limitations of, 139
\':~
Antf particle, 112
classical
.Niels.
effect of miclcar
305
lingular
306, 308
Bohr atomic model. 121, 193
and correspondence principle. 133
222
IH.1.
304
of,
of,
Body-centered cobic crystal, 322
wave. 78
of
inotneiitttm:
atomic,
214
382
nuclear, 372.
Angular frequency
399
I
of atomic electron.
particle, 102
Ammonia
157,
Buttling energy:
UK
of,
of,
Baryon. 443
Benzene molecule, 26$
Beta decay, 369, 408
Wiuiidc elements, 2L5
under the observed binding energies per
quantum theorv
Barrier penetration,
I'll
semiconductor. 338
In
Ikjundary- surface diagram:
of atomic orbital, 255
Spectrum)
of molecular orbital,
258
Braukert series in hydrogen spectrum, 120, 12S
Bragg planes
in crystal,
57
Bragg reflection:
of de Brog.be waves,
Balmer
s^rie* In
Band theory
of solids, 334, 340* 352
Bant, the, 414
460
ANSWERS TO ODD-NUMBERED PROBLEMS
hydrogen speelmm, 119, J2H
of
X
ILiVM,
345
V)
ikemsstrahluug, 51
HriHnuui y.one, 345
461
hybrid orbital*
285
281.
In,
saturated uiul uiisaturatetl, 267
of,
of.
280
Dims,
Donor
Dopplrr
338
effect ta light, 41
llcboin, energy levek of,
275
Hole
07
of,
in crystal.
338
Elooke's law. 158
Effective iua& of electron In crystal,
Qudwk.lt, janies. 365
Eigcnfruiction. 149
Ethylene molecule. 205
Chain reaction, 423
rlurge 'irmigatiuu. 451
Eigenvalue. 148
Exchange
Eightfold way, 452
Excitation
Einstein, Albert, 10. 47
Excited state of atom, 120
specific heal of,
Exclusion principle. 210
spectral series
Cohesive energy of
( ."uilixmiiv
crystal,
and
elastic
Complex euiuugate
322
inelastic,
of
wave
132
function, 75
Elastic
Compound nucleus 417
Gmupluil
of,
352
number of
crystal*
322
Correspondence print:lp!c. 133
of,
Electron
320
affinity,
tabk
Electron capture by nucleus 410
437
Electron configurations of elements, &I9
rays. 20.
342
hydrogen atom. 113
energy dlslrihuliun
molecule, 243
and uncertainty
Electron orbits in
principle,
246
Ferromagnetlsm, origin
Elementary
325
329
ionic,
till tic
of,
bidden lurid
of,
adiiiidc,
van der WmEs, 326
Crystal types, tabic
133
EnnlxHlmu energies
lulrou resonance in solid,
356
faille of.
transition.
DavlswnCenner experiment. 52
Bragg reflection
of.
345
diffraction at, 62
group velocity
In bvditrgen
uf. 76,
wavelength oC
W
in bciixenc
Deuterium, 362
of,
130
INDEX
of,
374
of,
372
427
iiiob'colc.
346
Ion.
(IB
247
252
structure in \pettial lines, SS5, 371
Hyperon. 438
,
131
Impurity
Gamma
oliisni.i.
ray,
05
Cravitun,
416
and Schrodingcrs equation, 147
of.
S. A.,
Ionic crystal,
I
205
352
Ionic bond. 243, 321
tafab of,
362
in uira.su or rncni
its
frame of reference, 10
General theory of reUUvity, 07
in crystal.
.
132
Insulating solid, nature of,
318
Ionization energy.
319
320
Ivniicr. -112
shift.
60.68
13
bofupe. 362
Isolopic spin.
446
Ground state uf atom, 126
Group velocity of waves, St
Gvruiuugiifltc ratio:
therm unucl car, 424
uncertainty
M'liiivontinL-tin
c
f^elgei-Mursden experiment. 102
Gravitational red
of,
329
Ice, crystal structure of.
Fusion, nuclear, 424
Energy:
Energy
fuel,
Dcutcron:
binding energy
experiment
Cuudsmit,
Energy luind
thermonuclear
theory
umletnlr.
MilUU,
imjiact parami^rr in utotntc M-atlcring, 105
Mat, 30
iMocoli/cd electrons
in
SJi
£•&!•> transformation of coordinates, 23
Cumnui decay, 112
215
quantization
)ecay constant of radioisotope, 390
190
i
kinetic, relative to center of mass,
61
191
lypercharge. 448
Inertfal
320
216
of IiIiImii. 372,
dlum, 122
in,
179
Fundamental interactions, 444
Emission spcclruin. IIS, 131
62
phase velocity
spectrum
of,
binding: of atomic electrom, 213
of.
ElydpogH
rmikilk
tratuunuiic, 423
dc Brugjlc waves, 73
of, 170.
functions of. 176, 181
Hypadbn
In crystal. 336,
219
of,
lanlhanidc, 215
periodic
230
Fourier transform. 86
215
electron configurations
data*, the. 3511
as
207
Elements:
of.
of, 125, 180.
442
Fraucli- Hertz;
I
lines. 203,
420
Fourier integral,
446,453
113
122
Hydrogen niolecubir
1
thcHirire of, 451
mulct idur. 326
v<
222
particles:
mullkph-l*
318
metallic, 331
t.
of,
259
E -'im
tee,
wave
Hvchouru bond
Electronic s|>ecimui ni molecule, 251
covaletu,
of,
in,
and uncertainty principle, 92
Fluorescence. 261
Cryitit
model
(.fuuntLuu Iheorv of. 173.
Fermi surface, 352
Permian. 212
atom, 213, 217
125
quantum numbers
Fine rtmctlTB in spectral
in
121
of.
energy levels
of, 3-12
Fission, nuclear,
Eleciruuegulivity,
of.
Bohr radios
probability density of electron
in.
Electron shells and MihvfirlK
Cross section. 109,413
Hohr model
classical
310
Fenni gan model of nucleus. 388
Electron "gas" in metal, 331
326
290
LIU. 128
uf.
electron orbits
statistical distribution law, 256.
derivation ol. 308
Fermi energy. 332, 330
205
oscillator, 161
harmonic
L'ovdleul bond;
In
spin
Face-centered cubic crystal. 322
Fermi-Dime
in.
285
meet rum of. 220
Hydrogen atom:
Feruu, the, 370
205
166.
of.
mnfamnif speeds
lUstribiitiun uf
lines. 131
of, 4-5]
muimclk moment
as nuclear const! Incut. 91, 363
In crystal
Hydrogen:
and -meet rat
Expectation value, 145
Dirac iheory
446
iMirdination
433
nici.liJiUii.il*
repulsion due to, 243
2,00,
delocjli/cd. in Iw-n/cur uiulccule, 288
Conservation principle* mid symmetry o|nr;itinnv
uiul
fluctuations m,
229
Conduct ing wild, nature
Cosmic
Vicimm
field,
llvloui orbiiAl. 2C3, 205
60
effect.
Curnpfmi wavelength. H2
<
132
ffflftfffffc
Electromagnetic
Korea,
260
hind's rule, 222, 228.
f
Center -of -noes* coordinate system, 418
<
232
polynomial. 168
lirrinite
9
Ether. 3,
355
100
of sodium atom, 230
Equivalence, principle
875
of.
Itaimatag Werner. 80
und X-ray spectra. 235
279
vibrational energy levels uf,
462
I
energy
of rotating molecule. 269. 209
uf vibrating molecule,
Doublet angular- momentum Male, £29
oscillator:
7.eru;poinl
of particle in a box. 151
level in sc mi conductor.
moon
quantum theory of. 103
wuve functions of, hi
412. 419
motor, 366. 379, 364.
320
A. at, 205. 431
P.
1. ii
1
mercury atom, 234
of
337
of.
crystal structure of,
Carbon cycle of unclear rvnecious, 424
Carbon dioxide molecule, vihruliuual modes
Carbon monoxide molecule:
rotational energy IcveN of. 27(1. 300
spectrum
Energy levcb:
Diamond:
bund unicture
Carbon compound*:
It
coupbii^; ol ani;iil.o
of atomic eieetrott, 188
atomic. 228
of electron spin, 207
nuclear. 364
nonMUU
Ml
334, 340. 352
levels;
h
69, 183
,
atomic, 125. 131.221
Half
of h. mi limit UM:i1|ator, LOO
Harmonic
life.
&
.438
l
oscillator:
158
of
helium atom, 232
classical theory of,
of
hydrogen atom. 125. ISO. 230
energ)' levels of, 10T)
Ijigrange's
I
method
..nub shift in
ol undetermined multipliers.
293
hydrogen spectrum. 230
INDEX
463
Landi:
I
Molecule, 243
£ fad or. 238
dement*. 215
LnttA ooatnetfODh rafaUvWk^
l.rptnn,
rotational energy levels
17. 2fl
model
lid-drop
<
of,
Mu&ftauer
1*4
2-8
wove
momentum
Multiplicity of angular
4-lfl.
Paschen scries
452
state,
hydrogen spectrum*
120, 12&
uumb-cr* in nuclear structure, 384
Mngm.1lc moment;
Resonance
Phase velocity of wave, 8
Best energy.
Phosphorescence.
RociniiMi.
394
416
and
425
Lypa
decay
spin. 2tf7
of,
22-1
Nuclear
of, 365.
443
fission,
373. 420
Nuclear forces:
433
reduced: In hydrogen atom, 129
In molecular rotation. 289
molecular vibration,
£.74
33
relativity of. 30.
Matti-eticrgy relatiousbip. 35.
37
368
293
Mercury, energy levoU
234
Maura decay and
special relativity.
of nuclear forces,
Metal, electron "gas"
In.
20
433
331, 342
harmonic
oscillator,
hydrogen
niotn, 175
*talea of.
energy levels
of.
366
of, 366, 379, 384. 412,
of,
419
^Ill.Lllllllll
drop model
of.
380. 418. 420
of.
224, 364
389
371
model
of.
orhital.
in a gas.
rh'clroll spin,
205
Mzoof, 112.370,407
spin of, 364
295
Scoueornhnlor. 338. 352
nurlear. 3H4
Shell:
258
Shielding, electron, in atom, 218.
spin inagcwKc, 208
Kiiuullaneity
iUi.itjini.il,
of
ii.uuiiijii
299
INDEX
momentum
slate:
229
helium. 232
nucleus, 379
"nxlium
1
lion]
,1 fill
lure of,
nui'j^ lt"*'b
47
Sodium chlnride
14H
of radial vc transit urns.
I
of
of
oscillator, 163
of particle in a Lmo..
319
relativity, 15
of alum.
of hydtiii;t:n hi in i>. liJO
of light.
and
Kinglel angular
275
of ueolerou. 374
In,
383
nHallonal. 270
Qoark. the. 451
464
closed, 217
Shell uickIcI of nucleus.
HO
nf harrier ptrnclralion. 157, 31)9
hybrid, 203, 265
Oxygen, distribution of molecular speeds
I
278
atomic electron. 213
t8(l
r^iuuilum Iheory:
QrthoJidiunu 232
of,
for vibrational spectra,
magnetic. JS4
prlnd]wl.
atomic, 254
atomic speclra. 108
272
for relational spectra,
of particle in a Ixjx. 151
Orbital quantum number, 179, ISO
254
Imundary surface diagram
124
orl.ilal.
384
for
UlUlilM'l
Bukr
orliil.
of
validity of. 144
Selection rules.
9li,uiIu»i [urvrnuiiCf., 130
388
magnetic moment
of,
lime-dependent fonu, 144
(Quanta, 47
Oii.Mihiiu eUvlrodynatiiics, 201)
conditions for stability
163
steady stale form. 147
372
of,
Ml
375
for particle in a boa, 150
384
molecular, 254
Molecular bund, 243
Mob:cular
iniantum number, 180
for
OroM
1ft
morion on, 130
effect of nuclear
327
for
^
Methane molecule. 261. 2o3
Mkhelsnn Murley experiment. 3, 7»
Molecular angular momctitmtu 270
the 214
Proion-proton cycle of nuclear reactions. 424
Occupation index, 31
312
Molecular energies
Itydberg constant, 120. 128
Proton. 362
Mctaatablc state, 233
I.m:i.
Etvillwrg.
Plutonium. 423
NncleiiA. 103
Nuclide, 362
Meullfc bond 331
and the
Kulhmionl scattering fonu id a. HI, 117
Planck's radiation formula. 308
Nucleon, 300
momentum
272
imilri ulf.
atomic mtxlel, 103
for deotcron,
diell
Meson theory
spectrum nf
Schrodinger's equation,
diupe
Mewn. 443
ltrU.iEnni.il
Proper time, 12
radioactive decay of,
of,
cjoantum number, 270
Nuclear reactions, 417
liiiuitl
free putb. 4 14
299
li'il.iliiiri.il
Saturated carltnn compound. 287
289
Mendeleev. Dmitri. 213
Rotaliooal energy levels uf molecule. 2WI
Proper lenglh, 17
Fermi gas model
of,
51
uiolecular spceti. 2SI7
Planck's constant. 47
E'riiidpal
compound, 417
unit, atomic, 3d!
Maxwell- Bottxmann vUtlshcal distribution law. 287,
v\ illiulm,
Ifll MfHM
Nuclear fusion, 373, 424
binding energy
Mm
36
Probability deiBily. 74
angular
Mass spectrometer, 301
ftllOll
440
particle,
Kuliicrfiufi
aliseuce of electron* from. 91, 363
Muss defect of nucleus, 372
87
Positron, 410. 432
strong and weak, 444
333
60
438
Polar molecule. 259.
saturation of,
effective, of electron In crystal.
4fl
of,
of,
relative |>opulailnns of,
IT
ul-
"maw"
Pion.
IS-l
ITS),
atomic, 361
Mean
Photon,
437
inesoii thorny of.
derivation
theory
Normalized wave timet ton, 141
M*wr, 314
in
432
stellar energy.
2Js1
nXuOllUlfcl effect, 43
mid e,amma-rav absorption, 65
riiran free juilh of,
33
special theory of. 10
Phase space. 288
\cotron. 364
of atomic electron, 188
number,
general theory
l2Ji
Neptunium, 423
Neptunium decay
series,
3(1,
Relativity;
hydrogen speelrum, 120.
in
274
37
Helatlvistic inass increase.
4IIU
III.
Relati vislu- formulas,
n-tvpe tcmicnntluctor, 338
,u id until km it ri no,
Modelling constant, 324
f|UHJiluiii
spectrum. IW. 128
215
Periodic I.™
molecular rolalion, 269
in
in molecular vibralion,
in hyiiiuj!"* 1
Putli. Mifllniitit;. 2
229
Neutrino, 4li9
Magnetic
Timet inns of. 154. 138
Phuid serins
nuclear. 384
pf
of electron in hydrngen atom, 129
MB
of.
Periodic iahle. 2T6
atomic. 226* £32
deetmn
midem,
151
nl,
Particle diffraction. 82
72
effect,
coupling of iiojTolm momenta.
of
bom
,i
cpianlum Iheory
Angular momentum)
(*»*
Uvea, 437
2,7
in
38'.)
393
Hare-earth elemcnLs, 215
Itatlioaclive series.
449
enerp'y levels
Multiple^ of elementary parliclev
Lortml** lrumfonii.it icm of coordinate*. 22,
Lyman *cri«
and forbidden, ISO
allnsccd
Kail ioai: live decay,
232
i.
Particle in
277
Moseley\ law, 219
[xiicnU-FititgemltJ coutnicllon. 17
inverse*
Partly.
272
mm of.
Momentum, angular
380
nim! lUick'ur fission. 120
and nuclear reactions
Par jlit-1
289, 299
irt
L
sped
vibrational
of nurleu*.
of,
vibrational energy levels of 275
3U3
of radio.* it iijh.%
iLS
spectrum
rotational
443
lifetime, menu, of <?sdled itate. -120
1. 1 rji
Pair production,
polar rovalcni, 239
Laser. 311
Radiative transitions. 198
p-type semiconductor. 338
electronic spectrum of, 281
jiilluiinlc
1
06
ni
2
333
i-
r
crysial,
Solids, cryslallioe
244
and amorphous, 317
Space rjuanti/ation of angular momentum.
185,
207
INDEX
465
Specific heal of hydrogen, 285
Time
Spectral hm-v
Transition elements. 215
Rue structure
hypcrfiuc structure
225, 371
in,
origin of, 127, 196
Zeemaii
elf eel In. 189,
ol
204
Wave-particle duality. 85. 93
momentum
of nucleus.
star, 6S.
W ten's displacement
state:
Work
345
rays:
56
diffraction of,
scattering of,
66
371
law,
function of surface.
306
Yukawa,
llideki,
Zecman
effect.
434
4ft
232
379
X-ray spectre. 235
Tritium. 362
X-ray spectrometer. 58
Twin paradox,
blade -body. 306, 308
X
78
White dwarf
of helium,
19
.
Willi interaction, 444
Spectrum:
absorptlmi, 119, 131
and forbidden, 199
iukoIht
electron in crssial lattice,
spontaneous and induced, 311
Triple! angular
Spectral series. 119
1
Wave
iniantuin theory of, 196
of atom, 229
hand, of molecule.
450
Transfllnns, radiative: allowed
SOT
203.
In,
reversal,
IS
X
electronic. of molecule, 231
absnrplion
189
anumalous. 204
rues, 51
of,
65
Zero-point energy of harmonic oscillator, 160
emission, 114, 131
of sun,
230
877
X-ray. 235
and resonance
44
particles,
92
Unsaturated carbon compound. 267
and time
Spherical polar coordinates, 174
Spin:
Intervals,
mold's theorem, 201
electron. 203
l'
of elementary particles. 442
Uranium decay
309
serfes,
MB
faatttpfcL
iiiinio,
409, 432
nuclear. 364.
Vacuum
386
LiuneliL' ininiihiiti
Van dcr Waits
Van der Waals
plom. 207
in nucleus,
flo'-tuatious fn
electromagnetic
field,
200.
229
number, £06
Spin-urhit coupling:
in
ISfi
and meson theory of nuclear forces, 435
and particle In a box. 151. 150
and phase space. 286
llil
Speed, root-mean-square. of molecule, 2M7
S|j
'IS,
and covalent lauuhng. 246
and energy measurement. 02
272
vibrational, of mdeciile.
...
205
and anuulai monicllhiui.
mercury. 234
rotational, of molecule,
of sodium,
E..
Uncertainty principle, S9
H9. 127.220
of hydrogen,
of
Uhlcnbcck. C.
232
»l helium.
328
327
crystal.
force.
Vector model of the atom. 222
384
Velocity:
Stars:
energy production
in.
group, Bl
424
Velocity addition, relatf visile. 28
Staimk'ol mechanics, 287
Vibrational energy levels of molecule, 275
weight nf energy level. 299
Statistical
j
phase. 81
while dwarf, 371
Stefan- Bolt /niumi law,
Vibrational spectrum of molecule, 277
308
KlnrnOrluch espcrfment. 207
Mtaaaga tn inula for logarithm
I
Von
ol
a
laetoria],
l,uuc.
Max. 52
201
Strangeness number, 44
Water molecule. 260. 264
vibrational states of, 250
Strong intcractinu, 444
Subshcll
atomic ck-clrcm, 213
Water molecules, hydrogen bonds between. 329
closed, 217
Wive
Wave
order of
(filing.
218
44fi
function, 74, 140
antisymmetric. 211, 449
Symmetric wave funclion. 211, 449
and covaleni bonding, 249, 252
Symmetry operations and conservation
equation. 141
complex conjugate
principles,
of deutcron.
of,
75
379
of
harmonic
of
hydrogen atom,
oscillator,
IT6,
168
IS]
nonnalUcd, 141
Term symbol
of atomic state.
Thermonuclear energy, 424
Thomson atomic model.
Thoi Emu decay
lime
466
INDEX
394
14. 27
series,
dilation, 12,
102
parity
229
TliciuiiDiilc emission of electrons.
46
of,
449
of particle In a box. 151. 153. 156
symmetric, 211, 440
Wjvo
group, 79
and uncertainly piinuple. 66
velocity of. 81
467
^
8
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