Introduction to Magnetism and Magnetic Materials
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Introduction to Magnetism
and Magnetic Materials
Introduction to Magnetism
and Magnetic Materials
David Jiles
Ames Laboratory,
US Department of Energy
and
Department of Materials Science
and Engineering, Iowa State University,
Ames, Iowa, USA
SPRINGER-SCffiNCE+BUSINESS MEDIA, B.V.
First edition 1991
© 1 9 9 1 David Jiles
Originally published by Chapman and Hall in 1991
Typeset in 10/12 pt Times by
Thomson Press (India) Ltd, New Delhi
All rights reserved. No part of this publication may be reproduced or
transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording or otherwise, or stored in any retrieval system of any
nature, without the written permission of the copyright holder and the
publisher, application for which shall be made to the publisher.
British Library Cataloguing in Publication Data
Jiles, David
Introduction to magnetism and magnetic materials.
1. Magnetic materials. Properties
I. Title
538.3
ISBN 978-0-412-38640-4
ISBN 978-1-4615-3868-4 (eBook)
DOI 10.1007/978-1-4615-3868-4
Library of Congress Cataloging-in-Publication Data
Jiles, David.
Introduction to magnetism and magnetic materials / David Jiles.—
1st ed.
p. cm.
Includes bibliographical references and index.
1. Magnetism. 2. Magnetic materials. I. Title.
QC753.2.J55
1990
90-41506
538-dc20
CIP
Few subjects in science are more difficult to understand than magnetism.
Encyclopedia Britannica, Fifteenth Edition 1989.
Preface
Over the years there have been a number of excellent textbooks on the subject
of magnetism. Among these we must include Bozorth's Ferromagnetism (1950),
Chikazumi's Physics of Magnetism (1964) and Cullity's Introduction to Magnetic
Materials (1972). However at present there is no up to date general textbook on
magnetism. This book will, I hope, satisfy this need. It is a book for the newcomer
to magnetism, and so I anticipate it will be useful as a text for final-year
undergraduate courses in magnetism and magnetic materials or for graduate
courses. I would also hope that it will be useful to the researcher who, for one
reason or another, is beginning a study of magnetism and needs an introductory
general text. In this case the extensive references to the literature of magnetism
given in the text should prove useful in enabling the reader to gain rapid access
to the most important papers on the subject. For the expert there are of course
already numerous excellent specialist works, of which the most significant is
Wohlfarth's four-volume series Ferromagnetic Materials.
The book was conceived as a whole and deals with the fundamentals of
magnetism in Chapters 1 to 11, and the principal applications in Chapters 12 to
16. The approach which I have taken is to consider magnetic phenomena first
on an everyday macroscopic scale, which should be familiar to most readers,
and then gradually to progress to smaller-scale phenomena in the search for
explanations of observations on the larger scale. In this way I hope that the
book will be of interest to a wider audience consisting of physicists, materials
scientists and electrical engineers. One advantage of this approach is that it is
possible to introduce the subject from an appeal to the reader's experience rather
than through abstract concepts. It is also easier to maintain the reader's interest
if he does not find himself immediately confronted with difficult concepts when
he first opens the book at chapter one.
Whereas physicists are likely to be mainly interested in the microscopic
phenomena discussed in Chapters 9, 10 and 11, the materials scientists and
metallurgists are more interested in the domain processes and how these are
affected by microstructure, as described in Chapters 6, 7 and 8. Electrical engineers
are probably more familiar with field calculations and modelling of magnetic
properties in Chapters 1, 2 and 5. Each of these groups should be interested in
the applications of our subject since it is the applications which sustain it. There
Vl11
Preface
is a strong demand today for scientists and engineers with skills in magnetism
because of applications in magnetic recording, permanent magnets, electrical
steels, soft magnetic materials and materials evaluation and measurements; not
because it is a deeply interesting and difficult subject - which it undoubtedly is.
The choice of units in magnetism presents a continual problem which those
not experienced in the subject will find difficult to comprehend. In research
journals papers are primarily written in CGS (Gaussian) units. This system has
the advantage that the permeability of free space is unity and that the unit of
magnetic field, the oersted, has a very convenient size for practical applications.
In electromagnetism the Sommerfeld system of units has been adopted widely.
This has the advantage of being completely compatible with the SI unit system
but suffers from a serious disadvantage because the permeability of free space,
which has the cumbersome value of 12.56 x 10 - 7 henry jmetre, enters into many
of the equations. This value has no real significance, being merely the result of
the choice of our definition of units, specifically the ampere. In this book I have
nevertheless chosen the Sommerfeld unit convention because it is the unit system
recommended by the International Union of Pure and Applied Physics, and
because this is the unit system for the future. However it was not practicable to
convert every diagram taken from research journals and monographs into this
unit system. Nor was it desirable since the practitioner of magnetism must learn
to be adept in both unit systems. Therefore many of the figures given later in
the book remain in their original units. Conversion factors are given in section
1.2.6. so that the reader becomes immediately familiar with these alternative units.
Finally I would like to take this opportunity to acknowledge the advice and
assistance given to me by many friends and colleagues while writing this book.
In particular thanks go to D. L. Atherton who persuaded me to write it, S. B.
Palmer, F. J. Friedlaender and C. D. Graham Jr for reading the entire text, and
D. K. Finnemore, R. D. Greenough, K. A. Gschneidner Jr, W. Lord, B. Lograsso,
K. J. Overshott, J. Mallinson, R. W. McCallum, A. J. Moses and E. Williams for
advice on particular chapters.
DJ.
Ames, Iowa
Foreword
As you study the intricate subject of magnetism in this book you will find that
the journey begins at a familiar level with electric currents passing through wires,
compass needles rotating in magnetic fields and bar magnets attracting or
repelling each other. As the journey progresses though, in order to understand our
observations, we must soon peel back the surface and begin to delve into the
materials, to look at ever increasing magnification at smaller and smaller details
to explain what is happening. This process takes us from bulk magnets (1023_1026
atoms) down to the domain scale (10 12 _10 18 atoms) and then down to the scale
of a domain wall (10 3 -1 0 2 atoms). In critical phenomena one is often concerned
with the behaviour of even smaller numbers (10 atoms or less) in a localized
array. Then comes the question of how the magnetic moment of a single atom
arises. We must go inside the atom to find the answer by looking at the behaviour
of a single electron orbiting a nucleus. The next question is why the magnetic
moments of neighbouring atoms are aligned. In order to answer this we must
go even further and consider the quantum mechanical exchange interaction
between two electrons on neighbouring atoms. This then marks the limit of our
journey into the fundamentals of our subject. Subsequently we must ask how can
this knowledge be used to our benefit. In Chapters 12 through 16 we look at the
most significant applications of magnetism. It is no surprise that apart from
superconductors these applications deal exclusively with ferromagnetism.
Ferromagnetism is easily the most important technological branch of magnetism
and most scientific studies, even of other forms of magnetism, are ultimately
designed to help further our understanding of ferromagnetism so that we can
both fabricate new magnetic materials with improved properties and make better
use of existing materials.
Finally I have adopted an unusual format for the book in which each section
is introduced by a question, which the following discussion attempts to answer.
Many have said they found this useful in focusing attention on the subject matter
at hand since it is then clear what is the objective of each section. I have decided
therefore to retain this format from my original notes, realizing that it is unusual
in a textbook but hoping that it proves helpful to the reader.
Contents
Preface
Foreword
VB
ix
1 Magnetic Fields
1.1 The Magnetic Field
1.2 Magnetic Induction
1.3 Magnetic Field Calculations
Examples and Exercises
1
1
6
12
26
2 Magnetization and Magnetic Moment
2.1 Magnetic Moment and Magnetization
2.2 Permeability and Susceptibility of Various Materials
2.3 Magnetic Circuits and Demagnetizing Field
Examples and Exercises
27
27
32
36
44
3 Magnetic Measurements
47
47
53
60
67
3.1 Induction Methods
3.2 Methods Depending on Changes in Material Properties
3.3 Other Methods
Examples and Exercises
4 Magnetic Materials
4.1 Important Magnetic Properties of Ferromagnets
4.2 Different Types of Ferromagnets Materials
for Applications
4.3 Paramagnetism and Diamagnetism
Examples and Exercises
69
69
5 Magnetic Properties
5.1 Hysteresis and Related Properties
5.2 The Barkhausen Effect and Related Phenomena
5.3 Magnetostriction
Examples and Exercises
89
89
97
98
105
74
81
86
XlI
Contents
6 Magnetic Domains
6.1 Development of Domain Theory
6.2 Energy Considerations and Domain Patterns
Examples and Exercises
7 Domain Walls
7.1 Properties of Domain Boundaries
7.2 Domain-Wall Motion
Examples and Exercises
8 Domain Processes
8.1 Reversible and Irreversible Domain Processes
8.2 Determination of Magnetization Curves from Pinning Models
8.3 Theory of Ferromagnetic Hysteresis
Examples and Exercises
9 Magnetic Order and Critical Phenomena
9.1 Theories of Diamagnetism and Paramagnetism
9.2 Theories of Ordered Magnetism
9.3 Magnetic Structure
Examples and Exercises
10 Electronic Magnetic Moments
10.1 The Classical Model of Electronic Magnetic Moments
107
107
118
125
127
127
137
144
147
147
156
165
174
177
177
188
197
217
219
219
The Quantum Mechanical Model of Electronic Magnetic
Moments
10.3 Magnetic Properties of Free Atoms
Examples and Exercises
221
237
245
Quantum Theory of Magnetism
11.1 Electron-Electron Interactions
11.2 The Localized Electron Theory
11.3 The Itinerant Electron Theory
Examples and Exercises
247
247
254
261
268
10.2
11
12 Soft Magnetic Materials
12.1 Properties and Uses of Soft Magnetic Materials
12.2 Materials for a.c. Applications
12.3 Materials for d.c. Applications
13
Hard Magnetic Materials
13.1 Properties and Applications
13.2 Permanent Magnet Materials
269
269
272
290
299
299
311
Contents
14
Magnetic Recording
14.1 Magnetic Recording Media
14.2 The Recording Process and Applications of Magnetic
Recording
Xlll
323
323
334
15 Superconductivity
15.1 Basic Properties of Superconductors
15.2 Applications of Superconductors
345
345
358
16 Magnetic Methods for Materials Evaluation
16.1 Methods for Evaluation of Intrinsic Properties
16.2 Methods for Detection of Flaws and other Inhomogeneities
16.3 Conclusions
365
365
377
394
Solutions
399
Index
425
Acknowledgements
I am grateful to the authors and publishers for permission to reproduce the
following figures which appear in this book.
1.2
P. Ruth (1969) Introduction to Field and Particle, Butterworths, London.
1.7
G. V. Brown and L. Flax (1964) Journal of Applied Physics, 35, 1764.
2.3
F. W. Sears (1951) Electricity and Magnetism, Addison Wesley, Reading,
Mass.
2.5
1. A. Osborne (1945) Physical Review, 67, 351.
3.1
D. O. Smith (1956) Review of Scientific Instruments, 27, 261.
3.2, 3.6, 3. 7
T. R. McGuire and P. 1. Flanders (1969) in Magnetism and Metallurgy (eds
A. E. Berkowitz and E. Kneller), Academic Press, New York.
3.5
S. Chikazumi (1964) Physics of Magnetism" John Wiley, New York.
3.9
B. I. Bleaney and B. Bleaney (1976) Electricity and Magnetism, Oxford
University Press, Oxford.
4.4
J. Fidler, J. Bernardi and P. Skalicky (1987) High Performance Permanent
Magnet Materials, (eds S. G. Sankar, 1. F. Herbst and N. C. Koon),
Materials Research Society.
4.5
C. Kittel (1986) Introduction to Solid State Physics, 6th edn, Wiley, New
York. R. 1. Elliott and A. F. Gibson (1974), An Introduction to Solid State
Physics and its Applications, MacMillan, London.
4.7,5.1
A. E. E. McKenzie (1971) A Second Course of Electricity, 2nd edn,
Cambridge University Press.
XVI
Acknowledgements
5.7
E. W. Lee (1955) Rep. Prog. Phys. 18, 184. A. E. Clark and H. T. Savage
(1983) Journal of Magnetism and Magnetics Materials, 31,849.
6.4
H. J. Williams, R. J. Bozorth and W. Shockley (1940) Physical Review,
75, 155.
6.6
C. Kittel (1986) Introduction to Solid State Physics, 6th edn, Wiley,
New York.
6.7
R. M. Bozorth (1951) Ferromagnetism, Van Nostrand, New York.
6.8
C. Kittel and J. K. Galt (1956) Solid State Physics, 3, 437.
7.1
C. Kittel (1949) Reviews of Modern Physics, 21, 541.
7.4
R. W. Deblois and C. D. Graham (1958) Journal of Applied Physics, 29, 931.
7.6
S. Chikazumi (1964) Physics of Magnetism, Wiley, New York.
8.2
M. Kersten (1938) Probleme der Technische Magnetisierungs Kurve, SpringerVerlag, Berlin.
8.5
K. Hosclitz (1951) Ferromagnetic Properties of Metals and Alloys, Oxford
University Press, Oxford.
8.8,8.9
J. Degaugue, B. Astie, J. L. Porteseil and R. Vergne (1982) Journal of
Magnetism and M agnetic Material, 26, 261.
8.10
A Globus, P. Duplex and M. Guyot (1971) IEEE Transactions on Magnetics,
7,617. A. Globus and P. Duplex (1966) IEEE Transactions on Magnetics, 2,
441.
8.12
S. Chikazumi (1964) Physics of Magnetism, Wiley, New York.
9.2,9.3
C. Kittel (1986) Introduction to Solid State Physics, 6th edn, Wiley,
New York.
9.5
P. Weiss and R. Forrer (1929) Annalen der Physik, 12, 279.
9.6
D. H. Martin (1967) Magnetism in Solids, Illife Books, London.
9.7,9.9,9.10
G. E. Bacon (1975) Neutron Diffraction, 3rd edn, Clarendon Press,
Oxford.
Acknowledgements
XVll
9.8
J. Crangle (1977) The Magnetic Properties of Solids, Edward Arnold, London.
9.11
G. L. Squires (1954) Proceedings of the Physical Society of London, A67, 248.
9.12
D. Cribier, B. Jacrot and G. Parette (1962) Journal of the Physical Society of
Japan, 17-BIII, 67.
9.14,9.15,9.16
B. D. Cullity (1972) Introduction to Magnetic Materials, Addison-Wesley,
Reading, Mass.
9.17
C. Kittel (1986) Introduction to Solid State Physics, 6th edn, Wiley,
New York.
9.19
J. A. Hofman, A. Pashkin, K. J. Tauer and R. J. Weiss (1956) Journal of
Physics and Chemistry of Solids, 1, 45.
9.20
D. H. Martin (1967) Magnetism in Solids, Illife Books, London.
9.21
S.B. Palmer and C. Isci (1978) Journal of Physics F. Metal Physics, 8, 247.
D. C. Jiles and S. B. Palmer (1980) Journal of Physics F. Metal Physics, 10,
2857.
9.22
R. D. Greenough and C. Isci (1978) Journal of Magnetism and Magnetic
Materials, 8, 43.
10.7,10.8
H. Semat (1972) Introduction to Atomic and Nuclear Physics, 5th edn, Holt,
Rinehart and Winston, New York.
11.1
D. H. Martin (1967) Magnetism in Solids, Illife Books, London.
11.3
J. Crangle (1977) The Magnetic Properties of Solids, Edwards Arnold,
London.
11.4
W. E. Henry (1952) Physical Review, 88, 559.
11.8
B. D. Cullity (1972) Introduction to Magnetic Materials, Addison-Wesley,
Reading, Mass.
11.9,12.1
R. M. Bozorth (1951) Ferromagnetism, Van Nostrand, New York.
12.2,12.3,12.13,12.19
G. Y. Chin and J. H. Wernick (1980) in Ferromagnetic Materials, Vol. 2,
(ed. E. P. Wohlfarth), North Holland.
xviii
Acknowledgements
12.4
M. F. Litmann (1971) IEEE Transactions on Magnetics, 7, 48.
12.5,12.6
M. F. Litmann (1967) Journal of Applied Physics, 38, 1104.
12.7
E. Adams (1962) Journal of Applied Physics, 33,1214.
12.8,12.9,12.10,12.11
C. Heck (1972) Magnetic Materials and their Applications, Crane, Russak
and Company, New York.
12.14,12.16,12.1
F. E. Luborsky (1980) in Ferromagnetic Materials, Vol. 1, (ed. E. P. Wohlfarth), North Holland.
12.15,12.17
Reproduced by permission of Allied Signal Company, Morristown, New
Jersey.
12.20,12.21
1. H. Swisher and E. O. Fuchs (1970) Journal of the Iron and Steel Institute,
August.
12.24
G. W. Elman (1935) Electrical Engineering, 54, 1292.
13.5,13.6,13.7,13.8,13.9
D.1. Craik (1971) Structure and Properties of Magnetic Materials, Pion,
London.
13.10,13.11
R. J. Parker and R. J. Studders (1962) Permanent Magnets and their
Applications, Wiley, New York.
13.12,13.13,13.14
R. A. McCurrie (1982) in Ferromagnetic Materials, Vol. 3 (ed. E. P.
Wohlfarth), North Holland.
13.16,13.17
M. McCaig and A. G. Clegg (1987) Permanent Magnets in Theory and
Practice, 2nd edn, Pen tech Press, London.
14.1,14.2,14.3,14.4,14.8
R. M. White (1985) Introduction to Magnetic Recording, IEEE Press,
New York.
14.5
C. D. Mee (1964) The Physics of Magnetic Recording, North Holland.
15.2
H. Kamerlingh-Onnes Akad (1911) Wertenschappen, 14, 113.
15.3
C. Kittel (1986) Introduction to Solid State Physics, 6th edn, Wiley,
New York.
15.4
U. Essmann and H. Trauble (1968) Journal of Applied Physics, 39, 4052.
Acknowledgements
XIX
15.7
IEEE Spectrum, May 1988.
15.8,15.9
J. Clarke, Physics Today, March 1986.
16.2
R. S. Tebble, I. C. Skidmore and W. D. Corner (1950) Proceedings of the
Physical Society, A63, 739.
16.3
G. A. Matzkanin, R. E. Beissner and C. M. Teller, Southwest Research
Institute, Report No. NTIAC-79-2.
16.6
R. A. Langman (1981) NDT International, 14,255.
16.9,16.10
H. Kwun (1986) Materials Evaluation, 44, 1560.
16.15
R. E. Beissner, G A. Matzkanin and C. M. Teller, Southwest Research
Institute, Report No. NTIAC-80-1.
16.18
W. E. Lord and J. H. Hwang (1975) Journal of Testing and Evaluation, 3, 21.
16.20
T. R. Schmidt (1984) Materials Evaluation, 42, 225.
16.21
D. L. Atherton and S. Sullivan (1986) Materials Evaluation, 44, 1544.
Glossary of Symbols
A
A
a
B
Bg
Bix)
BR
B.
Pl' P2' P3
c
c
X
D
D
d
E
E
e
Magnetic vector potential
Area
Helmholz energy
Distance
Lattice spacing
Radius of coil or solenoid
Mean field constant
Direction cosines of magnetic vector with respect to the applied field
Magnetic induction
Magnetic induction in air gap
Brillouin function of x
Remanent magnetic induction
Saturation magnetic induction
Direction cosines of direction of measurement with respect to the
applied field
Capacitance
Curie constant
Velocity of light
Susceptibility
Electric displacement
Diameter of solenoid
Diameter
Electric field strength
Energy
Electronic charge
Spontaneous strain within a domain
Anisotropy energy
Exchange energy
Fermi energy
XXII
EH
EHall
E,oss
Ep
E(J
E
Eo
Epin
1'/
F
F
F
F
f
G
g
y
H
h
He
Her
Hd
He
Heff
Hg
Glossary of symbols
Magnetic field energy (Zeeman energy)
Hall field
Energy loss
Potential energy
Stress energy
Permittivity
Permittivity of free space
Domain wall pinning energy
Magnetomotive force
Field factor
Force
Force
Magnetomotive force
Frequency
Current factor
Gibbs free energy
Spectroscopic splitting factor
Lande splitting factor
Gyromagnetic ratio
Domain wall energy per unit area
Magnetic field strength
Planck's constant
Coercivity
Critical field
Remanent coercivity
Demagnetizing field
Weiss mean field
Effective magnetic field
Magnetic field in air gap
I
Magnetic polarization
Intensity of magnetization
Current
J
Current density
Total atomic angular momentum quantum number
Exchange constant
Total electronic angular momentum quantum number
Coupling between nearest-neighbour magnetic moments
Exchange integral for an electron on an atom with electrons on
several nearest neighbours
Exchange integral; exchange interaction between two electrons
J
j
"
J atom
J ex
Glossary of symbols
K
Anisotropy constant
Kundt's constant
Pinning coefficient in hysteresis equation
Boltzmann's constant
First anisotropy constant for uniaxial system
Second anisotropy constant for uniaxial system
First anisotropy constant for cubic system
Second anisotropy constant for cubic system
L
Inductance
Length
Length of solenoid
Atomic orbital angular momentum
Electronic orbit length
Length
Orbital angular momentum quantum number
Domain wall thickness
Langevin function of x
Wavelength
Magnetostriction
Filling factor for solenoid
Penetration depth in superconductor
Penetration depth
Transverse magnetostriction
Saturation magneto stricti on
Spontaneous bulk magnetostriction
M
m
m
Magnetization
Magnetic moment
Mass
Momentum
Anhysteretic magnetization
Electronic mass
Orbital magnetic quantum number
Orbital magnetic moment of electron
Remanent magnetization
Saturation magnetization (spontaneous magnetization at 0 K)
Spontaneous magnetization within a domain
Spin magnetic moment of electron
Spin magnetic quantum number
Total magnetic moment of atom
Permeability
Rayleigh coefficients
Bohr magneton
Man
me
m1
mo
MR
Mo
M.
m.
m.
IntO!
J..l
J..l,V
J..lB
XXlll
XXIV
Glossary of symbols
Jlo
Permeability of free space
N
ns
v
Number of turns of solenoid
Number of atoms per unit volume
Number of turns per unit length on solenoid
Principal quantum number
Demagnetizing factor
Avogadro's number
Number density of paired electrons
Frequency
OJ
Angular frequency
P
'"
Pressure
Magnetic pole strength
Angular momentum operator
Orbital angular momentum
Spin angular momentum of electron
Total angular momentum of electron
Magnetic flux
Angle
Spin wavefunction
Total wavefunction
Electron wavefunction
Q
Electric charge
R
Resistance
Radius vector
Radius
Electronic orbit radius
Magnetic reluctance
Density
Resistivity
n
Nd
No
p
Po
Ps
Ptot
<I>
4>
'P
r
Rm
p
S
S
s
(J
T
t
Tc
Atomic spin angular momentum
Entropy
Electronic spin angular momentum quantum number
Conductivity
Stress
Temperature
Time
Thickness
Curie temperature
Critical temperature
Glossary of symbols
to
()
!
!max
U
u
Orbital period of electron
Angle
Torque
Orbital period
Maximum torque
Internal energy
Unit vector
v
Potential difference
Volume
Verdet's constant
Velocity
W
W.
WH
Power
Atomic weight
Hysteresis loss
x
Distance along x-axis
y
Distance along y-axis
Z
Impedance
Atomic number
Distance along z-axis
Number of nearest-neighbour atoms
V
z
Convention for crystallographic directions and planes:
[100J denotes specific directions;
<100) denotes family of equivalent directions;
(100) denotes specific planes;
{100} denotes family of equivalent directions.
xxv
Abbreviations for SI Units
Quantity
Symbol
Units
Length
Mass
Time
Frequency
m
kg
s
Hz
metre
kilogram
second
hertz
Force
Pressure
Energy
Power
N
Pa
J
W
newton
pascal
joule
watt
Electric charge
Electric current
Electric potential
Resistance
Capacitance
C
F
coulomb
ampere
volt
ohm
farad
H
Wb
T
henry
weber
tesla
Inductance
Magnetic flux
Magnetic induction
A
V
n
Values of Selected
Physical Constants
Avogadro's number
Boltzmann's constant
Gas constant
Planck's constant
Velocity of light in empty space
Permittivity of empty space
Permeability of empty space
Atomic mass unit
Properties of electrons
Electronic charge
Electronic rest mass
Charge to mass ratio
Electron volt
Properties of protons
Proton charge
Rest mass
Gyromagnetic ratio of proton
Magnetic constants
Bohr magneton
Nuclear magneton
Magnetic flux quantum
No = 6.022 x 10 26 atoms/kgmole
kB = 1.381 X 10- 23 J/K
R = 8.314J/mole K
h = 6.626
X
c = 2.998
= 8.854
110 = 1.257
X
10- 34 J s
hl2n = 1.054 x 10- 34 J s
Eo
a.m.u.
X
X
10 8 mls
10- 12 F/m
10- 6 Him
= 1.661 x 1O- 27 kg
e = - 1.602 X 10- 19 C
me = 9.109 x 10- 31 kg
elme = 1.759 x 10- 11 C/kg
eV = 1.602 x 10- 19 J
ep = 1.602 x 10 - 19 C
mp= 1.673 x 1O- 27 kg
YP = 2.675 X 10 8 HzIT
I1B = 9.274 X 10- 24 A m 2 (= J/T)
= 1.165 x 10- 29 Jm/A
I1N = 5.051 X 10- 27 A m 2 ( = J/T)
<1>0 = 2.067 X 10- 15 Wb ( = (V s)
1
Magnetic Fields
In this chapter we will clarify our ideas about what is meant by a 'magnetic field'
and then show that it is always the result of electrical charge in motion. This will be
followed by a discussion of the concept of magnetic induction or 'flux density' and
its relation to the magnetic field. We will look at the various unit conventions
currently in use in magnetism and finally discuss methods for calculating magnetic
fields.
1.1 THE MAGNETIC FIELD
What do we mean by 'magnetic field'?
One of the most fundamental ideas in magnetism is the concept of the magnetic
field. When a field is generated in a volume of space it means that there is a change
in energy of that volume, and furthermore that there is an energy gradient so that a
force is produced which can be detected by the acceleration of an electric charge
moving in the field, by the force on a current-carrying conductor, by the torque on
a magnetic dipole such as a bar magnet or even by a reorientation of spins on
electrons within certain types of atoms. The torque on a compass needle, which is
an example of a magnetic dipole, is probably the most familiar property of a
magnetic field.
1.1.1
Generation of a magnetic field
What causes magnetic fields in the first place?
A magnetic field is produced whenever there is electrical charge in motion. This can
be due to an electrical current flowing in a conductor for example, as was first
discovered by Oersted in 1819 [1]. A magnetic field is also produced by a
permanent magnet. In this case there is no conventional electric current, but there
are the orbital motions and spins of electrons (the so called 'Amperian currents')
within the permanent magnet material which lead to a magnetization within the
material and a magnetic field outside. The magnetic field exerts a force on both
current-carrying conductors and permanent magnets.
2 Magnetic fields
1.1.2 Definition of magnetic field strength H
What is the unit of magnetic field strength?
There are a number of ways in which the magnetic field strength H can be defined.
In accordance with the ideas developed here we wish to emphasize the connection
between the magnetic field H and the generating electrical current. We shall
therefore define the unit of magnetic field strength, the ampere per metre, in terms
of the generating current. The simplest definition is as follows.
The ampere per metre
The ampere per metre is the field strength produced by an infinitely long solenoid
containing n turns per metre of coil and carrying a current of lin amperes.
Since infinitely long solenoids are hypothetical a more practical alternative
definition is to define the magnetic field strength in terms of the current passing
through unit length of a conductor. A current of 1 ampere passing through a
straight 1 metre length of conductor generates a tangential field strength of 1/4n
amperes per metre at a radial distance of 1 metre. These two definitions are
equivalent provided the Biot-Savart law holds.
For the time being we will take the viewpoint that the magnetic field H is solely
determined by the size and distribution of currents producing it and is independent
of the material medium. This will allow us to draw a distinction between magnetic
field and induction. However we shall see in section 2.3.3 that this assumption
needs to be modified under certain circumstances, particularly when
demagnetizing fields are encountered in magnetic materials.
1.1.3 The Biot-Savart law
Is there any way we can calculate the magnetic field strength generated by an electric
current?
The Biot-Savart law, which enables us to calculate the magnetic field H generated
by an electrical current, is one of the fundamental laws of electromagnetism. It is a
statement of experimental observation rather than a theoretical prediction. In its
usual form the law gives the field contribution generated by a current flowing in an
elementary length of conductor,
JH=~iMxu,
4nr
where i is the current flowing in an elemental length M of a conductor, r is
the radial distance, u is a unit vector along the radial direction and JH is the
contribution to the magnetic field at r due to the current element i M.
This form is known as the Biot-Savart Law (1820) although it was also
The magnetic fields
3
discovered independently in a different form by Ampere in the same year. For
steady currents it is equivalent to Ampere's circuital law. It is not really capable of
direct proof, but is justified by experimental measurements. Notice in particular
that it is an inverse square law.
Example 1.1 Field due to a long conductor. Determine the magnetic field H at
some point P distant a metres from an infinitely long conductor carrying a current
ofi amps. Calculate therefore the field at a distance of 10cm from the conductor
when it carries a current of 0.1 A.
Using the Biot-Savart law the contribution (jR to the field at the point P, as
shown in Fig. 1.1, due to a current element i (jl at an angle a is given by
(jR = 4 1 2 i (jlsin (90 -a).
nr
We can write (jl = rda/cosa = a (ja/cos 2 a
(jH = i cos a(ja.
4na
Element of conductor
P
a
Field point at which
H is measured
Linear current
carrying conductor
Current i amps
(a)
Element of conductor
To P
Current i amps
( b)
Fig. 1.1 Magnetic field due to a long conductor carrying electric current.
4 Magnetic fields
Now integrating the expression from C( = - n/2 to C( = n/2 to obtain the total H
H=
J1t 12 _i_ cos C( dC(
-1t12 4na
H=-
i
2na
amps/metre.
Therefore if a = 0.1 m and i = 0.1 A the field is 1/2n amps/metre, or
H = 0.159
amps/metre.
The direction of this magnetic field is such that it circulates the conductor
obeying the right-hand rule. That is if we look along the conductor in the direction
of the conventional current the magnetic field circulates in a clockwise direction.
1.1.4 Field patterns around current-carrying conductors
What do these 'fields' look like?
The magnetic field patterns, detected by magnetic powder, around a bar magnet
(magnetic dipole), a straight conductor, a single circular loop and a solenoid are
shown in Figs. 1.2 (a-e). The field circulates around a single current-carrying
conductor in a direction given by the right-hand corkscrew rule. The fields around
a single current loop and a solenoid are similar to those around a bar magnet.
In a bar magnet the field emerges from one end of the magnet, conventionally
known as the 'north pole' and passes through the air making a return path to the
other end of the bar magnet, known conventionally as the 'south pole'. We can
think of the 'north pole' of a magnet as a source of magnetic field H while a 'south
pole' behaves as a field sink. Whether such poles have any real existence is
debatable. At present the convention is to assume that such poles are fictitious,
although the concept of the magnetic pole is very useful to those working with
magnetic materials. The matter is discussed again in section 2.1.1.
Notice that the magnetic field produced by a bar magnet is not identical to that
of a solenoid. In particular the magnetic field lines within the bar magnet run in the
opposite direction to the field lines within the solenoid. We shall look at this again
in sections 2.3.1 and 2.3.2. It can be explained because the bar magnet has a
magnetization M while the solenoid does not, and this magnetization leads to the
generation of a magnetic dipole which acts as a source and sink for magnetic field.
1.1.5 Ampere's circuital law
How can we calculate the strength of a magnetic field generated by an electrical
current?
Ampere deduced that a magnetic field is produced by electrical charge in motion
when he read of Oersted's discovery of the effect of an electric current on a compass
The magnetic fields
{ol
5
(b1
{el
(.1
Fig. 1.2 Magnetic field patterns in various situations obtained by using iron filings; (a) a bar
magnet; (b) a straight conductor carrying an electric current; (c) a perspective view of (b); (d) a
single circular loop of conductor carrying a current; and (e) a solenoid with an air core.
needle. This was a rather remarkable conclusion considering that until then
magnetic fields were known to be generated only by permanent magnets and the
earth and in neither case was the presence of electrical charge in motion obvious.
According to Ampere the magnetic field generated by an electrical circuit
depended on the shape of the circuit (i.e. the conduction path) and the current
carried. By assuming that each circuit is made up of an infinite number of current
elements each contributing to the field, and by summing or integrating these
6 Magnetic fields
contributions at a point to determine the field, Ampere arrived at the result
[2]
Ni=f
H·dl
closed path
'
where N is the number of current-carrying conductors, each carrying a current i
amps. This is the source of the magnetic field H. I is simply a line vector. The total
current Ni equals the line integral of H around a closed path containing the
current. We should note that this equation is only true for steady currents.
Ampere's law and the Biot-Savart law can be shown to be equivalent. Consider
the field due to a steady current flowing in a long current-carrying conductor. By
the Biot-Savart rule the field at a radial distance r from the conductor is
H=_i_
2nr'
while from Ampere's circuital theorem
f
H·d/= i
and integrating along a closed path around the conductor at a distance r leads to
f
H .d1 =2nrH=i
i
H=2nr'
(Furthermore Ampere's law, which we have really used to define H above, can be
shown to be equivalent to one of Maxwell's equations of electromagnetism,
specifically V x H = Jr, where Jr is the current density of conventional electrical
currents.)
1.2
MAGNETIC INDUCTION
How does a medium respond to a magnetic field?
When a magnetic field H has been generated in a medium by a current, in
accordance with Ampere's law, the response of the medium is its magnetic
induction B, also sometimes called the flux density. All media will respond with
some induction and, as we shall see, the relation between magnetic induction and
magnetic field is a property called the permeability of the medium. For our
purposes we shall also consider free space to be a medium since a magnetic
induction is produced by the presence of a magnetic field in free space.
Magnetic induction
7
1.2.1 Magnetic flux «I>
How can we demonstrate the presence of a magnetic field?
Whenever a magnetic field is present in free space there will be a magnetic flux <1>.
This magnetic flux is measured in units of webers and its rate of change can be
measured since it generates an e.m.f. in a closed circuit of conductor through which
the flux passes. Small magnetic particles such as iron filings align themselves along
the direction of the magnetic flux as shown in Fig. 1.2. We can consider the
magnetic flux to be caused by the presence of a magnetic field in a medium. We
shall see in the next chapter that the amount of flux generated by a given field
strength depends on the properties of the medium and varies from one medium to
another.
The weber
The weber is the amount of magnetic flux which when reduced uniformly to zero in
one second produces an e.m.f. of one volt in a one-turn coil of conductor through
which the flux passes.
1.2.2 Definition of magnetic induction B
What is the unit of magnetic induction?
The flux density in webers/metre 2 is also known as the magnetic induction Band
consequently a flux density of one weber per square metre is identical to a magnetic
induction of one tesla. The magnetic induction is most usefully described in terms
of the force on a moving electric charge or electric current. If the induction is
constant then we can define the tesla as follows.
The tesla
A magnetic induction B of 1 tesla generates a force of 1 newton per metre on a
conductor carrying a current of 1 ampere perpendicular to the direction of the
induction.
This definition can be shown to be equivalent to the older definition of the tesla
as the couple exerted in newtons per metre on a small current loop when its axis is
normal to the field, divided by the product of the loop current and surface area. We
shall see in the next chapter that there are two contributions to the magnetic
induction, one from the magnetic field Hand one from the magnetization M of the
medium.
There is often some confusion between the concept of the magnetic field Hand
the magnetic induction B, and since a clear idea of the important difference
between these two is essential to the development of the subject presented here a
discussion of the difference is called for. In many media B is a linear function of H.
8 Magnetic fields
In particular in free space we can write
B=J.loH,
where J.lo is the permeability offree space which is a universal constant. In the unit
convention adopted in this book H is measured in amps/metre and B is measured
in tesla (= Vs/m 2 ), the units of J.lo are therefore (volt second)/(ampmetre), also
known as henries/metre, and its value is J.lo = 4n x 10- 7 H/m. If the value of B in
free space is known, then H in free space is immediately known from this
relationship.
However in other media, particularly ferromagnets and ferrimagnets, B is no
longer a linear function of H, nor is it even a single-valued function of H. In these
materials the distinction becomes readily apparent and important. A simple
measurement of the BH loop of a ferromagnet should be all that is necessary to
convince anyone of this. Finally Hand B are still related by the permeability of the
medium J.l through the equation,
B=J.lH
but now of course J.l is not necessarily a constant. We shall see shortly that in
paramagnets and diamagnets J.l is constant over a considerable range of values of
H. However in ferromagnets J.l varies rapidly with H.
All of this means that a field H in amps per metre gives rise to a magnetic
induction or flux density B in tesla in a medium with permeability J.l measured in
henries per metre.
1.2.3
Force per unit length on a current-carrying conductor in a magnetic field
How does the presence of a magnetic induction affect the passage of an electric
current?
The unit of magnetic induction has been defined in terms of the force exerted on a
current carrying conductor. This will now be generalized to obtain the force F on a
current-carrying conductor in a magnetic induction B. The force per metre on a
conductor carrying a current i in the direction of the unit vector I caused by a
magnetic induction B is
F=ilxB
and hence in free space
F= J.loil x H.
Therefore if two long wires are arranged parallel at a distance of a metres apart
and carry currents of il and i2 amps, the force per metre exerted by one wire on the
other is
J.lo ..
F =-2
11 / 2 ,
na
Magnetic induction 9
1.2.4 Lines of magnetic induction
How can we visualize the magnetic induction?
The lines of magnetic induction are a geometrical abstraction which help us to
visualize the direction and strength of a magnetic field. The direction of the
induction can be examined by using a small compass needle (magnetic dipole) or a
fine magnetic powder such as iron filings. These show that the magnetic induction
around a single linear current-carrying conductor are coaxial with the conductor
and follow the right-hand, or corkscrew, rule. While in a solenoid the lines are
uniform within the solenoid but form a closed return path outside the solenoid.
The lines of induction around a bar magnet are very similar to those around a
solenoid since both act as magnetic dipoles.
The lines of induction always form a closed path since we have no direct evidence
that isolated magnetic poles exist. This means that through any closed surface the
amount of flux entering is equal to the amount of flux leaving. That is the
divergence of B is always zero.
f
B·dA =0.
closed surface
We sometimes say that B is 'solenoidal' which is the same as saying that the lines
of B form closed paths. (The above equation is equivalent to another of Maxwell's
equations of electromagnetism, specifically V' B = 0.)
1.2.5 Electromagnetic induction
Can the magnetic field generate an electrical current or voltage in return?
When the magnetic flux linking an electric circuit changes an e.mJ. is induced and
this phenomenon is called electromagnetic induction. Faraday and Lenz were two
of the early investigators of this effect and from their work we have the two laws of
induction.
Faraday's law
The voltage induced in an electrical circuit is proportional to the rate of change of
magnetic flux linking the circuit.
Lenz's law
The induced voltage is in a direction which opposes the flux change producing it.
The phenomenon of electromagnetic induction can be used to determine the
magnetic flux <1>. The unit of magnetic flux is the weber which has been chosen so
10
Magnetic fields
that the rate of change of flux linking a circuit is equal to the induced e.m.f. in volts.
d<l>
V=-N-
dt '
where <I> is the magnetic flux passing through a coil of N turns and d<l>/dt the rate of
change of flux.
Since the magnetic induction is the flux density,
B=~
A
we can rewrite the law of electromagnetic induction as
This is an important result since it tells us that an electrical current can be
generated by a time-dependent magnetic induction.
Example 1.2. Electromagnetic Induction. What is the voltage induced in a 50
turn coil of area 1 cm 2 when the magnetic induction linking it changes uniformly
from 3 tesla to zero in 0.01 seconds?
dB
dt
V=-NA-
(50)(1 x 10- 4 )(3)
0.01
= 1.5 volts.
1.2.6 The magnetic dipole
What is the most elementary unit of magnetism?
As shown above in Ampere's theorem a current in an electrical circuit generates a
field. A circular loop of a conductor carrying an electric current is the simplest
circuit which can generate a magnetic field. Such a current loop can be considered
the most elementary unit of magnetism.
If a current loop has area A and carries a current i, then its magnetic dipole
moment is m = iA. The units of magnetic moment in the convention which we are
adopting are amp metre 2 .
The torque on a magnetic dipole of moment m in a magnetic induction B is then
simply
T=m x
B
and hence in free space
T= 110m
x H.
Magnetic induction
11
This means that the magnetic induction B tries to align the dipole so that the
moment m lies parallel to the induction. Alternatively we can consider that B tries
to align the current loop such that the field produced by the current loop is parallel
to it.
If no frictional forces are operating the work done by the turning force will be
conserved. This gives rise to the following expression for the energy of the dipole
moment m in the presence of a magnetic induction B
E= -m'B
and hence in free space
E= -J.Lom·H.
The current loop is known as the magnetic dipole for historical reasons, since the
field produced by such a loop is identical in form to the field produced by
calculation from two hypothetical magnetic poles of strength p separated by a
distance I, the dipole moment of such an arrangement being
m=pl.
We will see in a later chapter how important the concept of magnetic dipole
moment is in the case of magnetic materials. For in that case the electrical 'current'
is caused by the motion of electrons within the solid, particularly the spins of
unpaired electrons, which generate a magnetic moment even in the absence of a
conventional current.
1.2.7 Unit systems in magnetism
What unit systems are currently used to measure the various magnetic quantities?
There are currently three systems of units in widespread use in magnetism and
several other systems of units which are variants of these. The three unit systems
are the Gaussian or CGS system and two MKS unit systems, the Sommerfeld
convention and the Kennelly convention. Each of these unit systems has certain
advantages and disadvantages. The SI system of units was adopted at the 11th
General Congress on Weights and Measures (1960). The Sommerfeld convention
was subsequently the one accepted for magnetic measurements by the International Union for Pure and Applied Physics (IUPAP), and therefore this system
has slowly been adopted by the magnetism community. This is the system of units
used in this book.
Conversion Factors
1 oersted = (1000/4n) A/m = 79.58 A/m
1 gauss = 10- 4 tesla
1 emu/cm - 3 = 1000 A/m.
To give some idea ofthe sizes of these units, the Earth's magnetic field is typically
12
Magnetic fields
Table 1.1 Principal unit systems currently used in magnetism
Quantity
Field
Induction
Magnetization
Intensity of
magnetization
Flux
Moment
Pole strength
Field equation
Energy of moment
(in free space)
Torque on moment
(in free space)
H
B
M
/
<I>
m
p
SI
SI
(Sommerfeld)
(Kennelly)
Aim
Aim
tesla
tesla
Aim
EMU
(Gaussian)
oersteds
gauss
emulcc
weber
Am 2
Am
tesla
weber
weber metre
weber
maxwell
emu
emulcm
B=J-lo(H+M)
B=J-loH+/
B=H+4nM
E = - J-lomoH
E= -moH
E= -moH
1'= J-lom x H
1'=mxH
r=mxH
Note: The intensity of magnetization / used in the Kennelly system of units is merely an
alternative measure ofthe magnetization M, in which tesla is used instead of Aim. Under all
circumstances therefore / = J-loM.
H = 56 Aim (0.7 Oe), B = 0.7 X 10- 4 tesla. The saturation magnetization of iron is
Mo = 1.7 X 106 Aim. Remanence of iron is typically 0.8 x 106 Aim. The magnetic
field generated by a large laboratory electromagnet is H = 1.6 X 10 6 Aim,
B= 2 tesla.
1.3
MAGNETIC FIELD CALCULATIONS
How are magnetic fields of known strength usually produced?
Magnetic fields are usually produced by solenoids or electromagnets. A solenoid is
made by winding a large number of turns of insulated copper wire, or a similar
electrical conductor, in a helical fashion on an insulated tube known as a 'former'.
Solenoids are often cylindrical in shape. An electromagnet is made in a similar way
except that the windings are made on a soft ferromagnetic material, such as soft
iron. The ferromagnetic core of an electromagnet generates a higher magnetic
induction B than a solenoid for the same magnetic field H.
In view of the widespread use of solenoids of various forms to produce magnetic
fields we shall take some time to examine the field strengths produced by a number
of different configurations.
1.301
Field at the centre of a long thin solenoid
What is the simplest way to produce a uniform magnetic field?
The simplest way to produce a uniform magnetic field is in a long thin solenoid. If
the solenoid has N turns wound on a former of length L and carries a current i
Magnetic field calculations
13
Magnetic field lines araund a single loop
of current carrying conductor
®®
Convent ion for f ind ing wh ich
end of a solenoid acts os 0
north po le (f ield source) and
sou t h pole (f ield s ink)
Magnetic f e
i ld li nes
around a solenoid
Fig. 1.3 Magnetic field lines around a solenoid.
amperes the field inside it will be
H
Ni
.
=L=
nl,
where n is defined as the number of turns per unit length.
The magnetic field lines in and around a solenoid are shown in Fig. 1.3. A
practical method of making an 'infinite' solenoid is to make the solenoid toroidal in
shape. This ensures that the field is uniform throughout the length of the solenoid.
The magnetic field is then
N
H=-2 i,
nr
where N is the total number ofturns, r is the radius of the toroid and i is the current
flowing in the windings in amperes.
14
Magnetic fields
81 Element of conductor
Current = i amps
Field point at which
H is measured
Single loop of
current carrying
conductor
Fig. 1.4 The magnetic field due to a single circular coil carrying an electric current.
1.3.2
Field due to a circular coil
What is the field strength produced by the simplest form of coil geometry the singleturn circular coil?
Field at the centre of a circular coil
The Biot-Savart law can be used to determine the magnetic field Hat the centre of
a circular coil of one turn with radius a metres, carrying a current of i amperes as
shown in Fig. 1.4. We divide the coil into elements of arc length <51 each of which
contributes <5H to the field at the centre of the coil. Since by Biot-Savart
<5H = (1/4nr2)i M x u
H=~iMsne
4nr
and "LM = 2na, while dl is perpendicular to u, so () = 90°, and hence sin () = 1,
and r= a
H=~
2a
amps/metre.
Field on the axis of a circular coil
The previous calculation of the field at the centre of a circular coil can be
generalized to obtain an exact expression for the magnetic field on the axis of a
circular coil. Using the situation depicted in Fig. 1.4 and applying the Biot-Savart
rule, the field at the general point P is
1
dH=--2idlxu
4nr
where u is a unit vector along the r direction. We can make the substitution
a
r=-.-,
smcx
which gives
dH =
~(sin2
4na
cx)id/ xu.
Magnetic field calculations
15
The component of the field along the axis, which by symmetry will be the only
resultant, is dHaxial = dH sin a
dHaxial =
~(sin34na
a)idl x u.
J
Integrating round the coil, dl = 2na and remembering dl is perpendicular to u
i . 3
H=-sm a
2a
or, equivalently,
ia z
H=-;~
2(a Z + XZ)3/Z .
This can be expressed in the form of a series in x and by symmetry all terms of
odd order must have zero coefficients so the form of the dipole field becomes
H = Ho(1
+ czx z + C4 X 4 + C6 X 6 + ... ),
where Ho = i/2a is the field at the centre of the coil, and the coefficients have the
values Cz = - 3/2a z, C4 = 15/8a4 , and C6 = - 105/48a 6 .
Example 1.3 If a coil of 100 turns and diameter 10 cm carries a current of 0.1 A,
calculate the magnetic field at a distance of 50 cm along the axis of the coil.
When i = 0.1 A, a = 5 cm, x = 50 cm and the coil has 100 turns
H=
(100)(0.0S?(0.1)
2[(0.0S? + (0.5)zJ 3 / Z
0.025
2(0.253)3/2
0.025
0.254
H = 0.098
amps per metre.
Off-axis field of a circular coil
As shown in the above derivations a simple analytical expression can be obtained
for the magnetic field along the axis of a single loop of conductor carrying a current
by using the Biot-Savart law. However, in the vast majority of cases there is no
closed-form analytic solution for the field generated by a current-carrying
conductor. Those that can produce closed-form analytic solutions are only the
very simplest types of situation.
To give an example, there is no closed-form analytic solution for the off-axis field
of a single circular loop of conductor carrying a steady current, except for the farfield dipole field which varies with l/r 3 . This comes at first as somewhat of a
16
Magnetic fields
surprise given the extreme simplicity of the situation. However the example does
show how very limited are the situations which yield analytical solutions.
In the case of the off-axis field of the single circular loop the analysis leads to an
elliptic integral which has no exact solution. From the Biot-Savart law the
magnetic field contribution at any point dB due to a current element i dl is
dB= idl x u
4nr2 '
where r is the distance from the coil.
dB =
idlsin{}
4n(x2 + a2)'
where now a is also a function of {} instead of being a constant. In the case ofthe offaxis field the field strength can be calculated from this equation by computer using
numerical techniques.
1.3.3 Field due to two coaxial coils
Which simple coil configurations produce: (i) a constant magnetic field, or (ii) a
constant field gradient?
In superposition
Often when it is necessary to produce a uniform field over a large volume of space a
pair of Helmholz coils is used. This consists of two flat coaxial coils, each
Fig. 1.5 Two coaxial coils configured as a Helmholtz pair with the separation between the
coils equal to their common radius.
Magnetic field calculations
17
containing N turns, with the current flowing in the same sense in each coil as shown
in Fig. 1.5. The separation d of the coils in a Helmholz pair is equal to their
common radius a.
The axial component of the magnetic field on the axis of the two coils can be
calculated from the Biot-Savart law. Since the field on the axis of a single coil of N
turns and radius a carrying a current i at a distance x from the plane of the coil is
_(Ni)( X2)-1.5
H- -
2a
1+~
a2
Ifwe define one coil at location x = 0 and the other at location x = a the field at the
centre of two such coils wound in superposition is
_(Ni)[(
X2)-1.5 + ( 1 +--;:;--(a_x)2)-1.5]
1 +~
H-
2a
a2
a2
and since for the Helmholz coils x = al2 at the point on the axis midway between
the coils, this gives the axial component of the magnetic field at the midpoint as
H=
(~:)
[(1.25)-1.5
(0.8)2/3
+ (1.25)-1.5J
Ni
a
0.7155 Ni
a
and by symmetry the radial component on the axis must be zero.
In fact if a series expansion is made for the axial component of H in terms of the
distance x along the axis from the centre of the coils, as was given for the single coil
in section 1.3.2 it is found that the term in x 2 disappears when the coil separation d
equals the coil radius a, so that the fourth-order correction term becomes the most
significant. The series expansion for the field in terms of x is then,
H = HoO
+ C4X4 + C6X6 + ... ).
This results in a small value of dHldx at the centre of the coils, and consequently a
very uniform field along the axis as x is varied close to zero, which is shown in
Fig. 1.6 for three different values of coil radius a. In addition, the axial component
of the magnetic field close to the centre of a pair of Helmholz coils is only very
weakly dependent on the radial distance z from the axis. This means that the
magnetic field strength His maintained fairly constant over a large volume of space
between the Helmholz coils.
The useful region of uniform field between a Helmholz pair can be increased by
making the coil spacing slightly larger than a12, although this leads to a slight
reduction in field strength over this region.
18
Magnetic fields
H(kA/m)
1.0
a=1m
a= .75m
a = 1.5m
-1.0
1.0
Distance x from centre of coils in meters
Fig. 1.6 Axial component ofthe magnetic field Has a function of position along the axis of a
pair of Helmholtz coils for various coil radii. The calculation is for an N = 100 turn coil
carrying a current i = 10 A with coil radii a = 0.5 m, 0.75 m, 1 m and 1.5 m. The arrows mark
the location of the coils in each case.
In opposition
If the current in one of the the two coaxial coils described above is reversed then the
magnetic fields generated by the two coils will be in opposition. This is a specific
example of a quadrupole field, so called because the form of the field obtained is
similar to that obtained from two magnetic dipoles aligned coaxially and
antiparallel.
Under these conditions the magnetic field along the axis of the pair of coils is
given by
Such a configuration generates a uniform field gradient which can be useful for
applying a constant force to a sample. See, for example, section 3.3.2.
Magnetic field calculations
19
1.3.4 Field due to a thin solenoid of finite length
What field strength is produced in the more practical case of a solenoid of limited
length?
So far the field of an infinite solenoid has been considered. Now solenoids of finite
length will be considered. A thin solenoid is one in which the inner and outer
diameters of the coil windings are equal. So for example a solenoid consisting of
one layer of windings would be considered a thin solenoid.
The field of a long thin solenoid has already been calculated in section 1.3.1. The
field on the axis of a thin solenoid of finite length has an analytical solution. If Lis
the length of the solenoid, D the diameter, i the current in the windings and x the
distance from the centre of the solenoid, then the field at x is given by
H=
( Ni) [
L
(L + 2x)
(L - 2x)
]
2[D2 + (L + 2x)2r/2 + 2[D 2 + (L _ 2X)2] 1/2 .
At the centre of the solenoid x = 0 and hence
H=
(:i)
~2)1/
LL2 +
Finally, for a long solenoid, L» D and (L2 + D2)1/2 = L so that the result from
section 1.3.1 is a limiting case
Ni
H=-=ni.
L
The fields generated by solenoids are of course dipole fields.
The field calculations for thin solenoids, that is solenoids with L» D, at least
along the axis, are relatively straightforward and yield analytical solutions as
shown. A useful result to remember is that the field at the end of a solenoid is half
the value of the field at the centre. The field in the middle 50% of a solenoid is also
known to be very uniform.
1.3.5 Field due to a thick solenoid of finite length
What coil corifigurations are used to produce higher field strengths?
In order to produce higher field strengths from a solenoid it eventually becomes
more effective to increase the number of windings per unit length NIL than to
increase the coil current. This is because the louIe heating is proportional to i2
whereas the field is proportional to i. Consequently if the current is doubled in a
coil of fixed resistance the louIe heating is quadrupled while if the number of
windings is doubled the louIe heating is only doubled. Both methods will result in a
doubling of the Hfield. Therefore solenoids are often wound with several layers of
windings and hence are no longer 'thin'. That is the radii ofthe inner and the outer
20 Magnetic fields
windings are no longer equal or even close to being equal. In the case of these thick
solenoids the calculation ofthe magnetic field is more complicated than for the thin
solenoids.
Let L be the length of the solenoid, al the radius of the inner windings and az the
radius of the outer windings. Two parameters ct and 13 can then be defined which
describe the geometrical properties of the solenoid
and
L
13=-2
al .
The field generated by such a solenoid is then a function of ct and 13 and the coil
current i. If Ho is the field at the centre of such a coil then as shown by Montgomery
[3]
Ho = F(ct, f3)f(i, ai' az),
where F(ct,f3), known as the field factor, is
F(ct, 13) = f3[arcsinh(ct/f3) - arcsinh(l/f3)]
and the current factor f(i, ai' a2 ) is
.
Ni a l
f(z, ai' a2 ) = -~.
L (a 2 - a l )
The expression for the field can also be written in the slightly different but
equivalent form
H _ Ni F(ct, 13)
0 - a l 2f3(ct - 1)"
The above equations for the field at the centre of a thick solenoid are totally
general and can be shown to reduce to the simpler more familiar expression in
the limiting case a 2 = al when
so that as
L~O,
Ho~Ni/2a
and as
L~o,HNi/.
1.3.6 Optimization of solenoid geometry for a given power rating
How can the maximum field be produced for a given power consumption?
Often it is necessary to consider the limitations imposed by available power
supplies when designing a solenoid coil which will give maximum field. Clearly the
Magnetic field calculations
21
impedance or resistance of the solenoid should be chosen to operate as close to the
current and voltage limitations of the supply as possible. That is the optimum
resistance R opt should be
A formula has been given originally by Fabry [4] and later by Cock croft [5]
which shows the maximum field attainable by a solenoid operated at a given power
rating. The equation of the field at the centre of such a solenoid is
H= G(a,p}(WA/pa 1)0.5
where once again all the geometrical terms have been collected together in a single
expression G(a, p} known as the 'geometrical factor'. This equation indicates the
geometry of the windings needed to obtain the maximum field for a given power
supplied W. Here A is the filling factor which is defined as
A=~
~ot
where Va is the active volume of windings and Vtot is the total volume of windings, p
is the specific resistivity of the material used for the coil in ohm metres, and a 1 is the
radius of the inner windings.
The value of G(a, {3) varies from zero up to 0.179. Since this is dependent
solely on geometry it is possible to define an optimum shape of solenoid
which is independent of power considerations. The maximum value of G(a, {3)
occurs at a = 3, {3 = 2, which indicates that a solenoid of this geometry produces
the highest field for a given power consumption. The exact expression for
G(a, {3) is
G(a,{3) = [{3/2n(a 2 -1)]0.5[arcsinh(a/{3) - arcsinh(l/{3)]'
From the above equations it is clear that the relationship between the geometry
factor and the field factor is,
F(a, {3) = G(a, {3)[2n(a 2
F(a, {3) = G(a, {3)[nL(a~
-
1){3]0.5
- ai/ai)]0.5
The general equation for the field of a solenoid given above can be rewritten in
terms of the available power W = Vi, assuming that the filling factor A, the
resistivity of the coil material p, and internal radius a 1 remain constant.
H o = G(a,{3)(W),/pa 1 )0.5,
H o = G(a, {3)i(RA/ pa 1 )0.5
H o = G(a, {3)V(A/Rpa 1 )0.5
22
Magnetic fields
1.3.7 General formula for the field of a solenoid
How can the field strength generated by a solenoid be determined in more general
cases?
Not surprisingly there is no general analytic formula for the magnetic field from a
solenoid at a general point in space. However there are some methods of
calculation available. The most obvious method is by the straight-forward
procedure of using either Ampere's law or the Biot-Savart law, as in section 1.3.2.
This leads to a solution containing an elliptic integral which can then be solved
numerically.
However there are also some quicker methods which can be used such as that
developed by Brown and Flax [6] and by Hart [7]. In the former method any
desired solenoid of finite dimensions can be treated as the superposition of four
semi-infinite solenoids as shown in Fig. 1.7. The field at any point is the vector sum
of the contributions of the four component solenoids.
The advantage of considering four semi-infinite solenoids with no cylindrical
hole is that the field contribution of such a solenoid can be expressed in terms
of only two variables: the axial distance beyond the end of the solenoid and the
radial distance from the unique axis. Therefore one table can be
provided for the field at these two reduced coordinates for all semi-infinite
CDc
I
"-.;t,~:
I
I I
:Z4
11
I I
I
I
I
I
I I
I I
I I
I I
: I
I
I I
: :
I
f ;IELD
JOINT
'-8
Fig. 1.7 Method of superposing four semi-infinite solenoids to obtain the mathematical
equivalent of one finite solenoid, after Brown and Flax [6]. Curved arrows indicate the
direction of current flow, z1", . , Z4 represent the distances along the axis, r 1" •. , r 4 indicate
the radii of the solenoids.
Magnetic field calculations
23
solenoids. The field can then be calculated by vector summation of the four
contributions.
This method allows rapid and simple calculation of the field of any finite
solenoid. Further details of the application ofthis method can be obtained from the
original paper [6].
1.3.8 Field calculations using numerical methods
How can magnetic field strengths be calculated in more complicated situations,for
example over an entire volume?
Although it has been shown above how the Biot-Savart law can be applied to
determine the magnetic field H in various simple situations it has also been
demonstrated that there is not always an analytic solution (e.g. the off-axis field of a
single circular current loop). In more general cases therefore it is necessary to resort
to numerical techniques in order to obtain a solution.
In most cases the problem amounts to solving either Laplace's or Poisson's
equation over a finite region of space, known as the spatial domain, which may be
three dimensional or two dimensional. In most cases the calculation is made over a
two-dimensional spatial domain, the third domain being assumed to be effectively
infinite in extent in comparison with the other two dimensions. The Laplace's or
Poisson's equations cannot be solved without an appropriate set of boundary
conditions, even though the current density J is known over the entire spatial
domain.
These numerical methods for calculating the magnetic field are often used to
determine the magnetic field in the air gap of an electrical machine. In this case
there are no field sources in the gap and so Laplace's equation applies.
V 2 A =0
where A is the magnetic potential. In cases where field sources occur within the
region of interest then the source distributions must be known and included in
the calculation. The problem then becomes solving Poisson's equation with the
appropriate boundary conditions.
V 2 A = - f.lo J,
the only difference between this and the previous case being the presence of field
sources, in the form of the current density J. In two-dimensional problems the
vector potential A given here reduces to a scalar equivalent which simplifies the
equations and enables a more rapid solution to be obtained.
There are a number of general numerical methods which can be used to solve the
equation for the magnetic field H. Here we shall consider finite-difference, finiteelement and boundary-element techniques. At present much research effort is
being devoted to the boundary-element method, although the finite-element
method which has been in use for over twenty years is well established and a
24
Magnetic fields
number of software packages are available. The finite-difference method which has
been in use since the 1940s, and actually traces its origins back to Gauss, is the
oldest method, and has largely been superseded by the finite-element method since
about 1970.
It should also be realized that there are a number of analytical methods which
have been developed for calculation of magnetic fields. Some examples of these are
series solutions, conformal mappings and variational formulations. The problem is
that these methods suffer from a lack of generality which restricts their use to only
the simplest situations. Often they are restricted to steady-state conditions and
many are applicable to two dimensions only.
A review of progress in electromagnetic field computation by Trowbridge [8]
covers the period 1962-88. This discusses all three main methods and gives a
selection of the most important references during the period. The finite-difference
method of calculating magnetic fields was the principal numerical tool from the
early days of the digital computer in the 1940s until about 1970. The method has
been described by Adey and Brebbia [9] and by Chari and Silvester [10]. It is a
technique for the solution of differential equations in which each derivative
appearing in the differential equation is replaced by its finite-difference
approximation at regularly spaced intervals over the volume of space of interest.
This means that a continuous region of space must be replaced by a regular grid of
discrete points at which the field values are calculated. The process is known as
discretization.
In order to ease the computation a regularly spaced orthogonal or polar grid is
used for discretization, although in principle the use of curvilinear grids is possible.
However as a result of the practical restriction of grids the use of finite differences in
the case of complicated geometries has encountered difficulties. Also in situations
with large field gradients in order to obtain sufficient accuracy the number of nodes
needs to be increased over the whole volume in order to maintain a regular grid
(not just over the region of high gradient) and this results in a rapid increase in
computation time and memory requirements. Despite these difficulties the finitedifference method has been successfully implemented for field calculations and
some excellent examples of its use have been reported [11].
In the finite-element method the spatial domain is divided into triangle-shaped
elements and the field values are computed at the three nodes of each element. The
sizes ofthe elements can be varied over the region of interest (unlike the grid in the
finite-difference calculations) so that more elements can be included in regions
where the field gradient is large. Furthermore the elements used to discretize the
spatial domain need not be triangular, they can be of any polygonal shape,
however triangles remain the most popular form of element.
An introductory survey of the finite-element technique for the non-specialist has
been given by Owen and Hinton [12J, while an excellent and more detailed review
has been given by Silvester and Ferrari [13]. The method first came to attention in
1965 in the work of Winslow [14J and began to be used on a regular basis for field
calculations from about 1968. From this time onwards it gradually supplanted the
older finite-difference method.
Further reading
25
The advantages of numerical techniques such as the finite- element method over
analytical methods of field computation were demonstrated in the case of leakage
fields by Hwang and Lord [15]. This was the first successful attempt to use
numerical field calculations for the determination of fields in the vicinity of defects
in materials. The implementation of both finite-difference and finite-element
methods to two-dimensional nonlinear magnetic problems have been compared
by Demerdash and Nehl [16]. They concluded that the finite-element technique
was superior in that it required less computation time and less memory. However
although this conclusion that finite elements seem to be preferable is on the whole
true, the relative performance of various numerical methods is highly dependent on
the nature of the specific problem under consideration.
Boundary-element techniques for field calculations are the most recent
development. The general method has been discussed by Brebbia and Walker [17J
and its application to the problem of magnetic field calculations in particular by
Lean and Wexler [18]. A comparison of integral and differential equation methods
has been made by Simkin [19]. A comprehensive survey of these techniques is
given in the book by Hoole [20J, which provides an up-to-date guide to the whole
subject of numerical methods of field calculation.
REFERENCES
1. Oersted, H. C. (1820) Experiment on the effects of a current on the magnetic needle,
Annals of Philosophy, 16.
2. Ampere, A. M. (1958) Theorie Mathematique des Phenomenes Electrodynamiques
Uniquement Diduite de I'Experience, reprinted by Blanchard, Paris.
3. Montgomery, D. B. (1980) Solenoid Magnet Design, Robert Krieger, New York.
4. Fabry, M. P. A. (1898) Eclairage Electrique, 17, 133.
5. Cockcroft, J. D. (1928) Phil. Trans. Roy. Soc., 227, 317.
6. Brown, G. V. and Flax, 1. (1964) J. Appl. Phys., 35, 1764.
7. Hart, P. J. (1967) Universal Tablesfor Magnetic Fields of Filamentary and Distributed
Circular Currents, Elsevier, New York.
8. Trowbridge, C. W. (1988) IEEE Trans. Mag., 24, 13.
9. Adey, R. A. and Brebbia, C. A. (1983) Basic Computational Techniques for Engineers,
Pen tech Press, London.
10. Chari, M. V. K. and Silvester, P. P. (1980) Finite Elements in Electrical and Magnetic
Field Problems, Wiley, New York.
11. Fuchs, E. F. and Erdelyi, E. A. (1973) IEEE Trans. PAS, 92, 583.
12. Owen, D. R. J. and Hinton, E. (1980) A Simple Guide to Finite Elements, Pineridge Press.
13. Silvester, P. P. and Ferrari, R. L. (1983) Finite Elements for Electrical Engineers,
Cambridge University Press, Cambridge.
14. Winslow, A. M. (1965) Magnetic Field Calculation in an Irregular Mesh, Lawrence
Radiation Laboratory, Report UCRL-7784-T.
15. Hwang, J. H. and Lord, W. (1975) J. Test and Eval., 3, 21.
16. Demerdash, N. A. and Nehl, T. W. (1976) IEEE Trans. Mag., 12, 1036.
17. Brebbia, C. A. and Walker, S. (1980) Boundary Element Techniques in Engineering,
Newnes-Butterworths, London.
18. Lean, M. H. and Wexler, A. (1982) IEEE Trans. Mag., 18, 331.
19. Simkin, J. (1982) IEEE Trans. Mag. 18,401.
20. Hoole, S. R. H. (1989) Computer Aided Analysis and Design of Electromagnetic Devices,
Elsevier, New York.
26 Magnetic fields
FURTHER READING
Bennet, G. A. (1968) Electricity and Modem Physics, Arnold, Cambridge, London, Ch. 5.
Cendes, Z. (ed.) (1986) Computational Electromagnetics, North Holland Publishing,
Amsterdam.
Chari:M. V. K. and Silvester, P. P. (eds) (1980) Finite Elements in Electrical and Magnetic
Field Problems, Wiley, New York.
Grover, F. W. (1946) Inductance Calculations, Dover, New York.
Hoole, S. R. H. (1989) Computer Aided Analysis and Design of Electromagnetic Devices,
Elsevier, New York.
Lorrain, P. and Corson, D. R. (1978) Electromagnetism, W. H. Freeman, San Fancisco.
McKenzie, A. E. E. (1961) A Second Course of ElectriCity, 2nd edn, Cambridge University
Press, Ch 3 and 4, Cambridge.
Reitz, 1. R. and Milford, F.1. (1980) Foundations of Electromagnetic Theory, 3rd edn,
Addison-Wesley, Ch. 8, Reading, Mass.
EXAMPLES AND EXERCISES
Example 1.4 Magnetic field at the centre of a long solenoid. Prove that the
magnetic field H at the centre of a 'long' solenoid is H = ni, where n is the number of
turns per metre and i is the current flowing in the coils in amps.
Example 1.5 Force on a current-carrying conductor
(a) Calculate the force per unit length between two parallel current-carrying
conductors 1 metre apart when each carries a current of 1 amp.
(b) Find the force exerted on a straight current-carrying conductor of length
3.5 cm carrying a current of 5 amps and situated at right angles to a magnetic field
of 160kA/m.
Example 1.6 Torque on a current-loop dipole. Find the torque on a circular coil
of area 4 cm 2 containing 100 turns when a current of 1 milliampere flows through it
and the coil is in a field of magnetic induction 0.2 tesla.
2
Magnetization and Magnetic Moment
We now consider the effect that a magnetic material has on the magnetic induction
B when a field passes through it. This is represented by the magnetization.
Materials can alter the magnetic induction either by making it larger, as in the case
of paramagnets and ferromagnets, or by making it smaller as in diamagnets. The
relative permeability of the material indicates how it changes the magnetic
induction compared with the induction that would be observed in free space.
2.1
MAGNETIC MOMENT AND MAGNETIZATION
How do we measure the response of a material to a magnetic field?
When going on to consider magnetic materials it is first necessary to define
quantities which represent the response of these materials to the field. These
quantities are magnetic moment and magnetization. Once that has been done we
can consider another property, the susceptibility, which is closely related to the
permea bili ty.
2.1.1
Magnetic moment
Can we use the torque on a specimen in afield ofknown strength to define its magnetic
properties?
In the previous chapter we have defined the magnetic moment m of a current loop
dipole and shown that the torque on the dipole in the presence of a magnetic field in
free space is given by 'l' = m x B. Therefore the magnetic moment can be expressed
as the maximum torque on a magnetic dipole 'l'max divided by B.
'tmax
m=B
and hence in free space
28
Magnetization and magnetic moment
where the magnetic moment m in the convention we are using is measured in amp
metre 2 • This formula applies equally to a current loop or to a bar magnet.
The unit oj magnetic moment
A magnetic moment of 1 ampere metre 2 experiences a maximum torque of 1
newton metre when oriented perpendicular to a magnetic induction of 1 tesla.
In the case of a current loop, as in Fig. 2.1, m = iA, where i is the current flowing
and A is the cross-sectional area of the loop. In the case of a bar magnet, as in
Fig. 2.2, m = pi where p is the pole strength in amp metres and i is the dipole length
in metres. The 'pole strength' is an archaic term arising from the more traditional
eGS treatment of magnetism, in which pole strength was defined in terms of the
_0
b
F
(a)
\-- axis of coil
F
\
\
current up
B
current -""''''--down
\
\
(b)
F
\
\
\
Fig. 2.1 The torque on a current loop in an external magnetic field; (a) side view, and (b) top
view. If the loop is free to rotate the torque turns the loop until its plane is normal to the field
direction.
Magnetic moment and magnetization
29
F
B
F
Fig. 2.2 The torque on a bar magnet in an external magnetic field. If the bar is free to rotate
the torque turns the bar until its plane is parallel to the field direction.
magnetic flux <I> emanating from a single magnetic pole [1]. In the Sommerfeld
convention this is given by p = <I>/flo so that
<1>1
m=-,
flo
where <I> is the flux in webers passing through the current loop or bar magnet dipole
and I is the dipole length.
The magnetic moment vector m in a bar magnet tends to align itself with B under
the action of the torque as shown in Fig. 2.2. For this reason a bar magnet in the
field generated by a second bar magnet experiences a torque which aligns it parallel
to the local direction B. It is this force which gives rise to the most widely
recognized phenomenon in magnetism that unlike magnetic poles attract each
other while like magnetic poles repel each other. However our difficulty in
explaining the exact meaning of magnetic 'poles' remains.
As discussed in section 1.2.3 the basic unit of magnetism is the magnetic dipole.
The basic unit of electricity is the electric charge. There have been searches for the
magnetic monopole which has a theoretical value, according to some authors [2J,
of p = 3.29 x 10 - 9 A m (floP = 4.136 x 10 - 15 Wb) and, if it exists, would be the
magnetic analogue of electric charge. The discovery of the monopole would have
important consequences but the search has so far been inconclusive [3,4].
However if the magnetic monopole is discovered this wi1llead to fundamental
changes in our understanding of magnetism. In particular the much maligned
concept of the magnetic pole, which is a useful concept for those working in
magnetic materials, would become more acceptable. Maxwell's equations of
electromagnetism would also need to be altered to allow the divergence of B to be
non-zero. A recent work on the subject has been written by Carrigan and Trower
[5].
2.1.2
Magnetization M
How are the magnetic properties of the material and the magnetic induction B
related?
30 Magnetization and magnetic moment
We can define a new quantity M, the magnetization, as the magnetic moment per
unit volume of a solid.
m
M=V·
From the relationship between magnetic moment m and flux given in section
2.1.1 a simple relationship between M and B can be found. A bar magnet with flux
density <I> at the centre, dipole length 1 and with cross-sectional area A has a
magnetic moment m given by m = <1>1/Ilo. The magnetization M is therefore given
by M = m/AI. Hence,
Ilo
In this case there are no conventional external electric currents present to
generate an external magnetic field and so B = lloM. We see therefore that the
magnetization M and magnetic field H contribute to the magnetic induction in a
similar way. If both magnetization and magnetic field are present then their
contributions can be summed.
2.1.3 Relation between H, M and B
Can we define a universal equation relating these three magnetic quantities, field,
induction and magnetization?
We have seen that the magnetic induction B consists of two contributions: one
from the magnetic field, the other from the magnetization. The magnetic induction
in free space is lloH, while in the convention which we are following the
contribution to the induction from the magnetization of a material is lloM. The
magnetic induction is then simply the vector sum of these,
B=llo(H+M),
where B is in tesla, and Hand M are in amps per metre. The above equation which
relates these three basic magnetic quantities is true under all circumstances.
The magnetic field H is generated by electrical currents outside the material
either from a solenoid or electromagnet, or from a permanent magnet. The
magnetization is generated by the resultant (uncompensated) spin and orbital
angular momentum of electrons within the solid. The origin of the net angular
momentum of the electrons requires further development of ideas before it can be
explained. Discussion is therefore deferred until Chapters 9, 10 and 11.
Magnetic moment and magnetization
31
A related quantity, the magnetic polarization or intensity of magnetization I is
used in the Kennelly convention. This is defined by
1=f.1oM .
Although it is not often employed when the Sommerfeld system is used, it is a
useful unit. Measuring the magnetic polarization I of a material in tesla is often
more convenient than measuring the magnetization Min A/m. Crangle [6] has
remarked that since the Sommerfeld and Kennelly systems are not mutually
exclusive this unit can easily be incorporated into the IUPAP system without
contradictions.
In the SI system of units [7,8] M is usually measured in amps per metre (the
Sommerfeld convention, which we are using) but you will sometimes find it
measured, as indicated above, in tesla (the Kennelly convention). This means that
the torque equation in free space is different in the two conventions by a factor
of f.1o, being f = m x H in the Kennelly convention but f = m x B = f.1om x H in the
Sommerfeld convention. Similarly, the magnetic moment in the Sommerfeld
convention is measured in amp metre 2 , whereas in the Kennelly convention it is
measured in weber metre. The relative merits of the two conventions have been
discussed at length [9, 10, 11], and each has its own advantages and disadvantages.
In the Sommerfeld convention the definition of susceptibility is useful, but in the
Kennelly system the susceptibility is an awkward unit. However, in the Kennelly
system the unit of magnetic polarization is more convenient than the unit of
magnetization in the Sommerfeld convention. Also the energy of a magnetic
moment in a field and the torque on a magnetic moment in a field are simpler in the
Kennelly system because f.1o does not enter the equations.
2.1.4
Saturation magnetization
Is there a limit to the magnetization that a given material can reach?
If a material has n elementary atomic magnetic dipoles per unit volume each of
magnetic moment m then the magnetic moment per unit volume of the material
when all these moments are aligned parallel is termed the saturation magnetization
Mo. This is equal to the product of nand m.
A distinction can be made between technical saturation Ms and complete
saturation Mo. In order to fully understand this distinction a discussion of domain
processes must first be presented. At this stage we shall merely note that technical
saturation magnetization is achieved when a material is converted to a single
magnetic domain, but at higher fields the magnetization increases very slowly
beyond technical saturation. This slow increase of magnetization at high fields is
due to an increase in the spontaneous magnetization within a single domain, as
discussed in section 6.2.5 and is known as forced magnetization.
32 Magnetization and magnetic moment
2.2 PERMEABILITY AND SUSCEPTIBILITY OF VARIOUS MATERIALS
Can the various types of magnetic materials be classified from their bulk magnetic
properties?
The different types of magnetic materials are usually classified on the basis of their
susceptibility or permeability. Therefore we must define these related properties
precisely before going on to describe the differences between ferromagnetic,
paramagnetic and diamagnetic materials.
2.2.1 Permeability and susceptibility
How can we represent the response of a magnetic material to a magnetic field?
We can now make a general statement for the permeability Jl and susceptibility X.
The permeability is defined as
B
H
Jl=-
and the susceptibility is defined as
M
x=H
and the differential permeability and susceptibility are defined as,
,
dB
11 = dH
,
dM
X = dH·
Since Band M mayor may not be linear functions of H, depending on the type of
material or medium, it should be noted here that permeability and susceptibility
mayor may not be constant.
Sometimes you will find the term relative permeability used, particularly in SI
units. The relative permeability of a medium, denoted Jlr' is given by
Jl
Jlr =-,
Jlo
where Jlo is of course the permeability of free space 110 = 4n x 10- 7 henry/metre.
The relative permeability of free space is 1. The relative permeability is closely
related to the susceptibility and the following equation is always true
I1r = X + 1.
Other commonly encountered properties are the initial permeability l1in and the
initial susceptibility Xin. These are the values of the respective quantities at the
Permeability and susceptibility of various materials
33
origin of the initial magnetization curve
In general physicists and materials scientists are more interested in magnetization and susceptibility, while engineers who work mainly with ferromagnets are
usually more concerned with magnetic induction and permeability.
Example 2.1 Permeability of and magnetic induction in iron. In a magnetic field of
400 amp/metre the relative permeability of a piece of soft iron is 3000. Calculate the
magnetic induction in the iron at this field strength.
B=J.lo{H+M)
=J.loJ.lrH
={4n x 10- 7 )(3000)400
= 0.48n tesla.
2.2.2 Diamagnets, paramagnets and ferromagnets
How are the different types of magnetic materials classified?
These various different type of magnetic materials are classified according to their
bulk susceptibility. The first group are materials for which X is small and negative
X ~ -10- 5 • These materials are called diamagnetic, their magnetic response
opposes the applied magnetic field. Examples of diamagnets are copper, silver,
gold, bismuth and berylium. Superconductors form another group of diamagnets
for which X ~ - 1.
A second group of materials for which X is small and positive and typically
X ~ 10- 3 to 10- 5 are the paramagnets. The magnetization of paramagnets is weak
but aligned parallel with the direction of the magnetic field. Examples of
paramagnets are aluminum, platinum and manganese.
The most widely recognized magnetic materials are the ferromagnetic solids for
which the susceptibility is positive, much greater than 1, and typically can have
values X ~ 50 to 10000. Examples of these materials are iron, cobalt and nickel and
several rare earth metals and their alloys.
2.2.3 Susceptibilities of diamagnetic and paramagnetic materials
What are typical values of J.l and X in diamagnets and paramagnets?
At constant temperature and for relatively low values of magnetic field H, the
magnetic susceptibilities of diamagnets and paramagnets are constant. Under
34 Magnetization and magnetic moment
these conditions the materials are called 'linear', that is M is proportional to H.
Consequently, it is possible to write
M=XH
B= fio(1 + X)H
= fiofir H
=/lH.
Clearly then fir is slightly greater than one in the paramagnets and slightly less
than one in the diamagnets. At the same time X is slightly more than zero in the
paramagnets and slightly less than zero in the diamagnets.
The linear result is a useful one in that it allows us to write a proportional
relation between Band H in these materials at low fields. However we shall see
later in the classical Langevin theories of diamagnetism and paramagnetism that
the linear approximation no longer holds at higher fields and in fact paramagnets
exhibits saturation of magnetization at very high fields.
2.2.4 Values of fl, and X for various materials
What values of permeability and susceptibility do various metals have?
Table 2.1
Susceptibilities and permeabilities of various elements
f1r
X
Diamagnets
Bi
Be
Ag
Au
Ge
eu
Paramagnets
fJ-Sn
W
Al
Pt
Mn
-1.31
-1.85
-2.02
-2.74
-0.56
-0.77
x 10- 6
x 10- 6
x 10- 6
x 10- 6
x 10- 6
x 10- 6
0.99983
0.99998
0.99997
0.99996
0.99999
0.99999
0.19
6.18
1.65
21.04
66.10
x 10- 6
X 10- 6
x 10- 6
x 10- 6
x 10- 6
1.00000
1.00008
1.00002
1.00026
1.00083
In ferromagnets neither X nor fir have a constant value. Both permeability and
susceptibility in ferromagnets are strongly affected by the prevailing magnetic field
H and the previous history ofthe material. For example the change in permeability
of annealed iron along its initial magnetization curve is shown in Fig. 2.3.
Permeability and susceptibility of various materials
B
35
~
(him)
(wi"")
1.6
1.4
7
1.2
--6
CI:l
"" 1.0
:~
~
~ ...
0.8
::<
0.6
}J.
L'S
H
0.4
--0- __
2
0.2
0
0
200
400
GOO
Magnetic intensity H
800
1000
Fig. 2.3 Initial magnetization curve and permeability along the same curve for annealed
iron.
2.2.5
Other types of magnetic materials
Are there other types of magnetic materials which the above classification fails to
identify?
There are some other types of magnetic materials apart from the three classes of
diamagnets, paramagnets and ferromagnets given above. These other materials are
all very closely related to ferromagnets because they are magnetically ordered, as
explained in Chapter 6. They are ferrimagnets, antiferromagnets, helimagnets and
superparamagnets. All were discovered many years after the three classical groups
of magnetic materials discussed above. From bulk magnetic measurements the
ferrimagnets are indistinguishable from ferromagnets, while the antiferromagnets
and helimagnets were for many years mistaken for paramagnets. These magnetic
types will be discussed in chapter 9.
Example 2.2 Flux differences between air core and iron core. Calculate the
magnetic induction B and flux <I> at the centre of a toroidal solenoid with mean
circumference of 50cm and cross-sectional area of 2.0cm 2 , wound with 800 turns
36
Magnetization and magnetic moment
of wire carrying 1.0 amp, (a) when the solenoid has an air core and (b) when the
solenoid has a soft iron core of relative permeability 1000.
The magnetic field H will be given by
H=ni
= 1600 amp/metre.
(a) In air B = lloH
B = (4n x 10- 7 )1600
= 2.0 x 10 - 3 tesla.
<I> = BA
<I> = (2.0 x 10- 3 ) (2.0 x 10- 4 )
= 4.0 x 10 -
7
webers.
(b) In iron with Ilr = 1000
B= IlrlloH
= (1000) (2.0 x 10- 3) tesla
= 2.0 tesla.
<I>
= BA
<I>
= (2.0) (2.0 x 10- 4 )
= 4.0 x 10- 4 webers.
2.3
MAGNETIC CIRCUITS AND DEMAGNETIZING FIELD
How does a magnetic material alter the field and flux density in its vicinity?
In a given magnetic field H the presence of a magnetic material affects the magnetic
induction B due to its permeability 11 as indicated in the previous chapter. For
determining the magnetic flux in magnetic circuits a useful concept is the
reluctance R which is the magnetic analogue of electrical resistance. Furthermore,
if the magnetic material has finite length the generation of 'magnetic poles' near its
ends gives rise to a magnetic field opposing the applied field. This opposing field is
called the demagnetization field and its strength is dependent on the geometry and
magnetization M of the material.
2.3.1
Flux lines around a bar magnet
How does the presence of a magnetic material alter the flux lines in its vicinity?
The magnetic flux lines around a bar magnet can be mapped using a small dipole
such as a compass needle or iron filings. The flux lines are continuous throughout
the material and have the form shown in Fig. 2.4. The flux lines have a similar form
Magnetic circuits and demagnetizing field
37
(a)
B
Hd
M
..
--
} h,,;do<h,
magnet
(b)
Fig. 2.4 (a) Magnetic field Hboth inside and outside a bar magnet, (b) Magnetic induction B
both inside and outside a bar magnet. Notice in particular that the magnetic field and
induction lines are identical outside the material, but inside they are quite different (they
even point in opposite directions).
to the flux lines in and around a current loop dipole such as a single turn of currentcarrying conductor or short solenoid.
2.3.2
Field lines around a bar magnet
Are the magnetic field lines identical to the magnetic flux lines?
The field lines around a bar magnet are the same as the flux lines outside the
material as shown in Fig. 2.4, since B = f.1,oH in free space. However inside the
38
Magnetization and magnetic moment
material they are different and in fact Band H point in different directions because
of the magnetization of the material M. This can be proved using Ampere's
circuital law. We can view the magnetization as being the effect of aligning the
magnetic dipoles within the material which creates magnetic 'poles' near the ends
of a finite specimen.
The magnetization vector M inside a magnetized ferromagnet points from the
'south pole' to the 'north pole' since this is the convention adopted for the definition
of magnetic moment for a magnetic dipole. The magnetic field H always points
from a 'north pole' to a 'south pole'.
2.3.3 Demagnetizing fields
Does the shape of a specimen have any effect on the H field?
In view of the fact that the magnetization M and the magnetic field H point in
opposite directions inside a magnetized material of finite dimensions, due to the
presence of the magnetic dipole moment, it is possible to define a demagnetizing
field Hd which is present whenever magnetic poles are created in a material. This
demagnetizing field can be detected during hysteresis measurements on finitelength samples when the applied field is reduced to zero but the measured field is
negative due to the remanent magnetization.
The demagnetizing field depends on two factors only. These are the
magnetization in the material (i.e. the pole strength) and the shape of the specimen
(i.e. the pole separation which is determined by sample geometry). The
demagnetizing field is proportional to the magnetization and is given by the
expression
Hd=Nd M ,
where N d is a demagnetizing factor which is calculated solely from the sample
geometry. It should be remembered that the numerical value of N d depends on the
units used for M and H d. In the unit convention used in this book both M and Hd
are measured in amps/metre so that N d is simply a dimensionless number.
2.3.4
Demagnetizing factors
What value does the demagnetizing factor have in different cases?
Exact analytic solutions for N d can only be obtained in the case of second-order
shapes, that is spheres and ellipsoids [12]. However tables or charts of
approximate calculated demagnetizing factors are available for solids of various
shapes, as shown in Fig. 2.5.
Fig. 2.5 Calculated demagnetizing factors for ellipsoids and cylinders. Note that the values
of the demagnetizing factor are dependent on the permeability as well as on the shape.
10
tIT~
.~
A
l
2
•
, •
(b)
100
a
.
, •
A
l
H
•
..
1000
Z
~AJI".LC
•
l
10 L.ONG
"I~
~
~
l
z
ICI"
Z
\~
(-..",
r-",L'I'SOIOS
\~
._.:>'\
.
CYL
}ol.fol
•
•
loA
GO
" \\
..
I\~
l
~
z
\\
1\
•
•
•
,
,\\
•
~
l
~
Z
z
1.0
. ••
l
10
. "
z
.1=
a
100
01 _ _"" II..TlO,
(a)
..
S •
• ..
..
l
m. ~
z
'I
'\
•
1000
•
100
z
" ,.AIIA"'....... TO L.OHG AXIS
A
.
..
•S
..
· -'" ,l.!.
t---
z~
0.1
Z
c·
g
u
~
"Ii ~
"~'\
~
"~
•
••
.
~ ;::
~
1
...z
,\ \
\
c
~I'OL"f
~\
X
041
•
I'
,,-,
C'tLINDCIIS:
•
.•
.\
I~g
~'
160
CD
l
z
l
.,. •
~\
"-
l'\ \\ ~
z
I
,
~
OILAT~-
.\
:I
"£LU~SO',
Z
'\.
\\ l\ ~
~ 1,\ ~
10
z
DIWIMSIONA&.. RAtIO,
• • •• •
m = lH;f.r~?S
"-
100
Z
l
..
40
Magnetization and magnetic moment
Table 2.2 Demagnetizing factors for various
simple geometries
Geometry
Nd
Toroid
Long cylinder
Cylinder
Cylinder
Cylinder
Cylinder
Cylinder
Sphere
0
0
0.00617
0.0172
0.02
0.040
0.27
0.333
lid = 20
lid = 10
lid = 8
lid = 5
lld= 1
2.3.5 Field correction due to demagnetizing field
How do we make a correction to the field if we have specimens offinite dimensions?
We have shown above how the demagnetizing field arises in a sample of given
shape with magnetization M in zero field. When dealing with samples of finite
dimensions in an applied magnetic field Bapp it is necessary to make some
demagnetizing field correction to determine the exact internal field in the solid Bin'
2.3.6 Magnetic circuits and reluctance
How do we calculate magnetic field or flux in situations where we have an air gap or
two materials with different magnetic properties?
Situations in which a magnetic flux path is interrupted by an air gap are of practical
importance because they occur in engineering applications of permanent magnets,
electric motors and generators and in materials testing. The problems encountered
here are more complicated than in calculating the flux in a single material, however
the ideas of demagnetizing fields together with some generalizations of the
equations relating magnetic flux to magnetic field can be employed to provide
solutions.
The magnet engineer is often in the situation of calculating the magnetic flux in
magnetic circuits [13,14] with a combination of an iron and air core. Consider for
example the situation where a ring of iron is wound with N turns of a solenoid
which carries a current i as shown in Fig. 2.6 (a). In this case the magnetic field will
be NijL where Lis the average length of the ring and from this, and a knowledge of
the permeability, the magnetic flux or flux density passing in the ring can be
Magnetic circuits and demagnetizing field
(a)
41
Field Coil
(N turns carrying
a current i)
Demagnetizing
Field
Total
Field
Hd~( .
H
(b)
Field Coil
(N turns carrying
a current i)
Magnetic Circuits: (a) closed, and (b) open.
Fig. 2.6 Magnetic flux path in (a) an iron toroid forming a closed magnetic path, and (b)
an iron toroid with an air gap. The flux lines are shown as concentric circles within the
iron. The air gap increases the reluctance of the magnetic circuit and reduces the flux
density in the iron as well as in the gap.
calculated.
B=/-lo(H+M)
and from Ampere's law, in this simple situation H = NijL, and therefore
B=/-lo(~
+M)
42
Magnetization and magnetic moment
and consequently
Ni=(~
-M)L
BL
Il
We can define here the magnetomotive force I'f which for a solenoid is N i where
N is the number of turns and i is the current flowing in the solenoid. From this we
can formulate a general equation relating the magnetic flux <1>, the magnetic
reluctance Rm and the magnetomotive force I'f as follows,
which may be considered as a magnetic analogue of Ohm's law. If the iron ring has
cross-sectional area A metre 2 and permeability Il with N turns of solenoid on a
length L we can derive an expression for the reluctance. Starting from the
relationship between flux, magnetic induction and magnetic field
<I> = BA
= pHA
=1l(~)A
Ni
(LillA) .
From our definitions above it follows that the term LillA is the magnetic
reluctance of the path. This is analogous to electrical resistance in an electrical
circuit, which means that magnetic reluctances in series in a magnetic circuit may
be added.
A saw slot can be introduced in the ring as shown in Fig. 2.6 (b) to provide an air
gap. If the air gap is small there will be little leakage of the flux at the gap, but a
single equation B = IlH can no longer apply since the permeability of air and the
iron ring are very different.
Ignoring demagnetizing effects for the present calculation and starting from
Ampere's circuital law
Ni=fHdl
closed path
and for a two-component magnetic circuit consisting of an iron ring and an air gap
Ni = HiLi + HaLa,
Magnetic circuits and demagnetizing field
43
La
where L j is the path length in the iron and is the path length in the air and Hi and
Ha are correspondingly the respective fields in the iron and in the air.
For the ring with the air gap the flux density in the gap equals the flux density in
the iron, but the magnetization in the air gap is necessarily zero. Therefore we can
write a similar equation for the magnetic induction.
. (B- - M)L +BLa
Nl=
j
fta
fto
,/L
j
= U\/lj
La)
+ fta
.
It is clear from this that there is a discontinuity in H across the air gap while at
the same time there is continuity in B. Rewriting the equation in terms of flux <I>
leads to
Therefore from the above definition of magnetic reluctance as the ratio of flux to
magnetomotive force, the reluctance of this magnetic circuit with an air gap is
Lj
La
Rm=--+--·
ftjAj
ftaAa
The equation for the flux passing through the toroid with an air gap is then
Ni
The magnetic flux in the ring is reduced when the air gap is introduced because it
requires more energy to drive the same flux across the air gap than through an
equal volume of the iron due to the much lower permeability of the air.
2.3.7 Magnetic field calculations in magnetic materials
How can the magnetic field be calculated in more complex situations such as those
with both magnetic materials and air gaps of different shapes?
At the end of Chapter 1 we showed how a variety of numerical techniques could be
used to calculate the magnetic field in free space. This was a particularly simple
situation. In the presence of magnetic materials the situation becomes more
difficult because the magnetization of the material needs to be taken into account.
Nevertheless the same general techniques, such as finite-element analysis [15] can
be used successfully.
44
Magnetization and magnetic moment
The effect of hysteresis adds a further complication to the problem since the
magnetization M of the material then depends not only on magnetic field H, but
also on the field history. Even in the cases where hysteresis occurs methods have
been devised for calculation of magnetic fields in devices [16] which take hysteresis
into account.
REFERENCES
1. Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading
Mass., p. 5.
2. Dirac, P. A. M. (1931) Proc. Roy. Soc. Lond., A133, 60.
3. Fleischer, R. L., Hart, H. R., Jacobs, I. S., Price, P. B., Schwarz, W. M. and Woods, R. T.
(1970) J. Appl. Phys., 41, 958.
4. Trower, W. P. (1983) IEEE Trans. Mag., 19, 2061.
5. Carrigan, R. A. and Trower, W. P. (1983) Magnetic Monopoles, Plenum, New York.
6. Crangle, J. (1977) The Magnetic Properties of Solids, Edward Arnold, London, Ch. 1,
pp.13-15.
7. Vigoureux, P. (1971) Units and Standards in Electromagnetism, Wykeham Press,
London.
8. Hopkins, R. A. (1975) The International (SI) Metric System, 3rd edn, AMJ Publishers
Tarzana, California.
9. Giacolletto, L. J. (1974) IEEE Trans. Mag., 10, 1134.
10. Graham, C. D. (1976) IEEE Trans. Mag., 12, 822.
1l. Brown, W. F. (1984) IEEE Trans. Mag., 20, 112.
12. Osborne, J. A. (1945) Phys. Rev., 67, 351.
13. Proceedings of the IEEE Workshop on applied magnetics (1972) Washington DC.
Published by IEEE, New York.
14. Proceedings of the IEEE Workshop on applied magnetics (1975) Milwaukee,
Wisconsin. Published by IEEE, New York.
15. Preis, K., Stogner, H. and Richter, K. R. (1981) IEEE Trans. Mag., 17, 3396.
16. Saito, Y. (1982) IEEE Trans. Mag., 18, 546.
FURTHER READING
Bennet, G. A. (1968) Electricity and Modern Physics, Arnold, London, Ch. 6.
Bradley, F. N. (1971) Materialsfor Magnetic Functions, Hayden, New York.
Lorrain, P. and Corson, D. R. (1978) Electromagnetism, W. H. Freeman, San Francisco.
McKenzie, A. E. E. (1961) A Second Course of Electricity, 2nd edn, Cambridge University
Press. Cambridge, Ch. 5 and 6.
Reitz, J. R. and Milford, F. J. (1980) Foundations of Electromagnetic Theory, 3rd edn,
Addison-Wesley, Reading, Mass., Ch. 9.
Rieger, H. (1978) The Magnetic Circuit, Heyden and Son, London, Siemens
Aktiengesellschaft, Berlin and Munich.
EXAMPLES AND EXERCISES
Example 2.3 Demagnetizing field calculation. How strong is the magnetic field
needed to magnetize an iron sphere to its saturation magnetization (Ms = 1.69
Examples and exercises 45
x 106 amps/metre) assuming that the field needed to overcome the demagnetizing
field is much greater than the field needed to saturate the material in toroidal form.
Example 2.4 Demagnetizing effects at different field strengths. A specimen of
Terfenol has a length to diameter ratio of 8: 1. At a field strength of H = 80 kA/m the
magnetic induction is 0.9 tesla, while at a field strength of H = 160 kA/m the
induction is 1.1 tesla. Calculate the internal magnetic field H in in each case and
compare with the applied field.
If we consider the fractional error in the observation of the uncorrected field
what do you conclude about this error at higher fields? Do demagnetizing effects
become more or less of a problem as the applied field increases?
Example 2.5 Flux density in an iron ring with and without an air gap. An iron
toroid has a mean path length of 0.5 m with a cross-sectional area of 2 x 10- 4 m 2 .
The number of turns on the coil wound on the toroid isN = 800 and this carries a
current of 1 A.lfthe relative permeability of the iron is 1500 estimate the flux in the
iron ring. An air gap of length 0.0005 m is then cut in the ring. Estimate the flux
under these conditions. Calculate the current needed to restore the flux to its
original value in the uncut toroid.
3
Magnetic Measurements
This chapter is concerned with the various means of measuring the magneti{; field,
magnetic induction or magnetization. There are several methods available and
these divide broadly into two categories offield measurement: those which depend
on magnetic induction using coils and those which depend on measuring changes
in various properties of materials caused by the presence of a magnetic field.
Finally, the measurements of magnetization are either force measurements such as
in the torque magnetometer or gradiometer measurements which measure the
difference in magnetic induction with and without the sample present.
3.1
INDUCTION METHODS
How can the strength of an external field be measured from the e.mf generated in an
electrical circuit due to a change in flux linking the circuit?
The induction methods of measuring magnetic flux are all dependent on Faraday's
law of electromagnetic induction which has been discussed in section 1.2.2. This
states that the e.mJ. induced in a circuit is equal to the rate of change of flux linking
the circuit.
If A is the cross-sectional area ofthe coil and N is the number ofturns the magnetic
induction is then B= cJ)/A
dB
V=-NAdt'
Note that these coil methods measure the magnetic flux passing through the coil
cJ), and from a knowledge of the cross-sectional area A the magnetic induction B
can be found. The induced voltage is increased if B is increased while H is
maintained constant by inserting a high-permeability core into the coil. In free
space, of course, B = {loH and so
48
Magnetic measurements
3.1.1 Stationary-coil methods
How can the rate of change offield be found using the e.mf generated in a stationary
coil?
A stationary-coil method can only measure the rate of change of magnetic
induction by measuring the induced voltage. Such devices do have applications but
if magnetic-induction measurements are required it is necessary to include some
form of time integration [1]. A number of integrating voltmeter/fluxmeter devices
are now commercially available. These measure the magnetic induction from the
relation
B= - ;Af Vdt.
These instruments can be highly sensitive and are used extensively in hysteresis
graphs for the measurement of the magnetic properties of soft magnetic materials
[2]. These instruments work well but in setting them up it is necessary to pay
attention to the problem of drift, which can be adjusted using an offset voltage
control. If this is not done the fluxmeter will continue to integrate a small
out-of-balance voltage with time, giving the impression of a linearly varying
magnetic induction. Under present capabilities this can be a problem for
high-sensitivity measurement when the magnetic flux needs to be measured to
better than about 10- 10 webers (0.01 maxwell).
3.1.2
Moving-coil (extraction) method
How is the magnetic induction measured when a search coil is placed in a magnetic
field and rapidly removed?
As we know from the Faraday law of electromagnetic induction (section1.2.2)
the induced e.m.f. in a coil resulting from a change in flux linking the coil is given by
dB
V= -NA dt .
Integrating this gives
f
Vdt
= - NA(Bf - Bi)
where Bi is the initial magnetic induction and Bf is the final magnetic induction.
Therefore if a search coil is moved from the location where the field strength needs
to be measured (e.g. between the poles of an electromagnet) to a region of 'zero' field
(e.g. outside the electromagnet) JV dt will be proportional to the magnetic
induction.
A ballistic (moving-coil) galvanometer can be used to measure the induction
I nduction methods
49
since, providing the time of oscillation of the ballistic galvanometer is long
compared with the voltage pulse from the search coil, the angular deflection 4> is
proportional to SV dt. The device needs calibrating in order to determine the
coefficient of proportionality.
4> = constant x
f
V dt
= constant x N A(Br - BJ
The deflection of the galvanometer is therefore proportional to the change in
magnetic induction. After calibration this method is accurate to better than 1%.
3.1.3 Rotating-coil method
Can we make use of a rotating coil to generate the necessary induced e.mf. in a static
field?
In order to obtain a measurement of the magnetic induction it is also possible
to use various moving-coil instruments. The simplest of these is the rotating coil
which rotates at a fixed angular velocity ru. Under these conditions the flux linking the coil is
B(t) = Bcos rut
and the voltage generated is
dB
V= -NAdt
=
dH
-/loNA-
= -
dt
/loNAruHsin rut.
Therefore the amplitude of the voltage generated by the rotating coil is
proportional to the magnetic induction and therefore the amplitude can be used to
measure B or H in free space. The signal can be read directly as an a.c. voltage or
converted to a d.c. voltage which is proportional to the amplitude. Typical
inductions for this instrument range from 10 tesla down to 10 - 7 tesla. The
electrical connections to the rotating coil include slip rings which are a source of
error in dealing with small voltages. The precision is of the order of one part in 104 •
3.1.4 Vibrating-coil magnetometer
How can the linear displacement of a coil in a magnetic field be used to measure the
strength of the field?
50 Magnetic measurements
~
\
RADIAL MAGNE TIC
FIELD
B
,
~-
BAKELITE
( SPACER
THOMSON BALL
BUSHING ---.,.
(
STAINLESS STEEL
COUPLI NG ROO
~ATlNG
COIL
Fig. 3.1 Schematic diagram of a vibrating-coil magnetometer.
The vibrating-coil magnetometer [3,4] is based on the same principles as the
previous technique, but is used primarily as a method of determining the
magnetization M. The arrangement is shown in Fig. 3. 1. The coil vibrates between
the sample and a region of free space and thereby acts as a gradiometer by
measuring the difference in induction in the two positions. While surrounding the
sample the magnetic induction is
I nduction methods
51
whereas the induction linking the coil when it has moved away from the sample is
Bo = l1oH.
The change in induction is then simply
IlB= 110M.
The method therefore depends on the flux change caused when the coil is
removed from the specimen
f
Vdt = - NAl1 0 M
and consequently the output of the vibrating coil magnetometer is independent of
H, but is dependent on M. This device is subject to noise caused by variation of the
magnetic field H with time and is rarely used in situations in which it is possible to
vibrate the sample instead of the coil as in the vibrating-sample magnetometer
(section3.1.5).
60 HZ
1800 RPM
SYN.MOTOR
PICK UP
FIELO]
~L
n
IA
COMPENSATING COIL
Fig. 3.2 Schematic diagram of a vibrating-sample magnetometer.
52
Magnetic measurements
3.1.5 Vibrating-sample magnetometer (VSM)
If the specimen is moved instead of the coil how can the induced voltage be used to
determine the magnetization of the specimen?
The vibrating-sample magnetometer (VSM) is identical in principle to the
vibrating-coil magnetometer except that the sample is moved instead of the coil.
The VSM was first described by Foner [5J and has now almost completely
superseded the vibrating-coil device.
A VSM is really a gradiometer measuring the difference in magnetic induction
between a region of space with and without the specimen. It therefore gives a direct
measure of the magnetization M.
A schematic of a typical VSM is shown in Fig.3.2. The specimen in general has to
be rather short to fit between poles ofthe electromagnet. The method is therefore in
most cases not well suited to the determination of the magnetization curve or
hysteresis loop because of the demagnetizing effects associated with the short
specimen. However it is well suited for the determination of the saturation
magnetization Ms.
The detected signal, being an a.c. signal of fixed frequency, is measured using a
lock-in amplifier. A reference signal is provided for the lock-in amplifier as shown
in Fig.3.2 by using a permanent magnet and a reference pick-up coil. Magnetic
moments as small as 5 x 104 A m 2 (5 x 10- 5 emu) are measurable with a VSM. Its
accuracy is better than 2%.
3.1.6 Fluxgate magnetometers
How can the nonlinear magnetization characteristics of ferromagnets be used to
determine external field strength?
Fluxgate magnetometers, also known as saturable-core magnetometers, were first
developed in the 1930s for measurement of the Earth's magnetic field. There are
several different modes of operation of fluxgates; we consider here the simplest
type. The simple fluxgate consists of one or two cores of high-permeability
material, surrounded by a drive coil and one or two sense coils as shown in Fig.3.3
Reviews of fluxgate magnetometry have been given by Primdahl [6J and by
Gordon and Brown [7]. The range of field strengths measurable by fluxgates is
down to 10- 4 A/m (10- 6 Oe). They are mainly used for measuring inductions in
the range 10- 10 to 10- 7 tesla, corresponding to magnetic fields in the range 10- 4
to 10 - 1 A/m in free space.
The drive coil is excited with a periodic waveform which saturates the
magnetization. The measured field is a d.c. or quasi-d.c. field by comparison. When
the measured field is zero therefore the voltage induced in the pick-up coil is
necessarily symmetrical, but ifthere is an external field component along the axis of
the core, then the voltage in the secondary coil becomes asymmetric. The degree of
Methods depending on changes in material properties
H
H
~
~
53
H
H
~
Fig. 3.3 Diagram of various forms offluxgate magnetometer. These include the single-core
configuration, the Vacquier gradiometer configuration with a single secondary coil, the
Forster gradiometer configuration with separate secondary coils and the Aschenbrenner
and Goubau configuration with a toroidal core.
asymmetry of this voltage, as measured by the second harmonic component of the
induced signal, can then be used to measure the strength of the external field.
The most usual configuration used for fluxgates is the two-core arrangement
with the coils wound in opposition (the gradiometer configuration). In this case the
voltage in the sense coils is by virtue of symmetry identically zero when the external
field is zero. When the external field is non-zero this will lead to an apparent
asymmetry in the magnetization curves of the two cores and hence a net voltage
observed at twice the drive frequency.
Fluxgates can only measure the component offield strength parallel to the coils,
since any field perpendicular to this direction does not affect symmetry.
3.2 METHODS DEPENDING ON CHANGES IN MATERIAL
PROPERTIES
How can magnetic field strengths be determined from changes in material properties?
Whereas in the previous sections the measurement of magnetic field was dependent
on the change in flux linking a circuit, in the following sections the measurement
depends on changes in the properties of materials under the action of a magnetic
field.
3.2.1
Hall effect magnetometers
How does the presence of a magnetic field alter the motion of charge carriers?
54 Magnetic measurements
The Hall effect magnetometers are perhaps the most versatile and widely used form
of magnetometer. The range of fields measurable by these devices is typically
0.4 A/m up to 4 X 106 A/m (equivalently 5 x 10- 3 Oe up to 5 X 104 Oe). Accuracy
of measurements is typically 1%.
When a magnetic field is applied to a conducting material carrying an electric
current, there is a transverse Lorentz force on the charge carriers given by
F=/loev x H,
as discussed in section 1.2.3. The expression here is for the force on a single charge e
with velocity v in a field of strength H. Since the force on a charge e can be
expressed as,
F=eE
where E is the electric field we can consider that the force is due to an equivalent
electric field E Hall , known as the Hall field.
EHall =/loV x H
and hence to a Hall e.m.f. VHall which is in the direction perpendicular to the plane
containing i and H. The Hall e.m.f. therefore depends linearly on the magnetic field
H if the current is kept constant and this provides a very convenient measure of
magnetic field H, as the following analysis shows.
If the electric current passes in the x direction and the magnetic field passes in the
y direction of a slab of semiconductor of dimensions lx, ly, lz' the Hall e.m.f. will be
along the z-axis as shown in Fig. 3.4.
H
Fig. 3.4 Generation of a Hall e.m.f. in a slab of conducting material. The electric current
density J passes along the x-axis, the magnetic field Hlies along the y-axis, and the Hall field
EHall is generated along the z-axis. The sign of the Hall e.m.f. depends on the sign of the
charge carriers.
Methods depending on changes in material properties
55
If n is the number of charge carriers per unit volume, then the current density will be
J= nev
J
ne
V=-,
and so the Hall field is
H
EHall =lloJxne
and by replacing line by the term R H , which is called the Hall constant
EHall = lloRHJ x H
and since the electric field E in volts/metre can be expressed by the form E = V /lx
where V is the potential difference over a distance lx,
Since the current density J = i/lylz
.
VHall = lloRHl
X -~.
H
Ix ly lz
The value of the Hall coefficient RH is typically 10- 11 m 3 per coulomb.
The Hall effect magnetometers can be fabricated with very small active areas,
down to 1.0 x 10 - 2 cm 2 which can therefore be used to measure the magnetic field
with high spatial resolution. Another important factor is that unlike coils, which
measure the flux linkage and therefore need to be scaled appropriately for their
cross-sectional area in order to determine the magnetic induction, the Hall
magnetometers measure the field strength directly.
The only difficulties with Hall probes arise from deviations from linearly at
higher fields and from the temperature dependence of the response. Most
commercial Hall probes are made of InSb.
Table 3.1 Values of the Hall coefficient
for various materials
Material
Li
In
Sb
Bi
-1.7 xlO- 12
+ 1.59 X 10- 12
-1.98 X 10- 11
-5.4 X 10- 9
56 Magnetic measurements
3.2.2
Magnetoresistors
How does the presence of a magnetic field alter the resistance of a material?
Magnetoresistance is the change in electrical resistance of a material when
subjected to a magnetic field. Once the variation of resistance with field is known
then resistance measurements can be made in order to determine the field strength.
Generally the resistance increases when a field is applied but is nonlinear. The main
advantage of this method is that very small probes can be fabricated to measure the
field at a point. Magnetoresistive probes are particularly useful for field
measurements at low temperatures.
In all materials the effect of magnetic field on resistance is greater when the field
is perpendicular to the direction of current flow. In ferromagnetic materials the
change in resistance can be quite large, typically I1R/R = 2% at saturation in nickel
and 0.3% at saturation in iron.
The earliest application of this effect as a field-measuring device was the use of
bismuth in which the magneto resistance increases by 150% in a field of 9.5 x
10 5 A/m (1.2 T). More recently materials which are rather more sensitive have
been found such as the eutectic compound of InSb-NiSb which undergoes a
300% change in magneto resistance in a field of 2J x 10 5 A/m (OJ T). Unfortunately
the effects of temperature on the resistance of this material are also large and
this limits its applications.
Despite these drawbacks the measurement of resistance is fairly simple and
hence the method does have great merit in situations where the temperature can be
well controlled (e.g. cryogenic applications) and where the range of flux density is
limited to less than 16 T but greater than 2 T. The accuracy of field measurements
made with this method is about 1%.
There are a number of thin-film magneto resistive devices available
commercially which are capable of a resolution of 10- 5 A/m (10- 7 Oe). These
devices look very much like strain gauges being flat plates of typically 5 mm
x 5 mm with two terminals attached.
3.2.3
Magnetostrictive devices
Can the change in length of a ferromagnet in the presence of a field be used to
measure that field?
When a specimen of magnetic material is subjected to a magnetic field changes in
shape of the specimen occur [8]. This phenomenon is known as magneto stricti on
and it is most often demonstrated by measuring the fractional change in length 111/1
of a specimen as it is magnetized. The effect is quite small in most materials but in
ferromagnets it is typically of the order of 111/1 = 10- 6 which is measurable by
resistive strain gauges or by optical techniques. Recent materials with much higher
magnetostrictions have been discovered [9], with 111/1 up to 2500 X 10- 6 •
Methods depending on changes in material properties
57
The magnetostriction and magneto resistance are closely related, both being
generated by the spin orbit coupling so that the electron distribution at each ionic
site is rotated. This change in electron distribution alters the scattering undergone
by the conduction electrons (magneto resistance). The rotation of the moments also
leads to a change in the ionic spacing (magnetostriction).
If the magnetostriction of a material as a function of field is known this can be
used as a measurement of the magnetic field. High-resolution measurements of
magnetostriction down to dl/l ~ 10- 10 can now be made with magnetostrictive
amorphous ferromagnetic materials [10]. The drawback with this is that the
magnetostriction is nonlinear, and furthermore exhibits hysteresis.
3.2.4
Magneto-optic methods
How can the changes in optical properties of media under the action of an external
field be used to determine field strength or magnetization?
The two principal magneto-optic effects are the Faraday effect, which occurs when
light is transmitted through a transparent medium in the presence of a magnetic
field along the direction of propagation of the light, and the Kerr effect, which
occurs when light is reflected from a ferromagnetic medium. Both involve rotation
of the angle of polarization oflineady polarized light [11]. Another phenomenon,
the Cotton-Mouton effect, is related to the Faraday effect, and occurs when the
magnetic field is perpendicular to the direction of propagation of light transmitted
in a medium. However the magnitude of the rotation observed in the CottonMouton effect is much smaller than in the Faraday effect.
The Faraday effect is easily adapted as a technique for measurement of field
strength since the rotation of polarization of light passing through a transparent
paramagnetic material (such as MgCe(P0 4)2) can give a measure of the local
magnetic field. The rotation of the plane of polarization is given by
(J= VHt,
where Vis the Verdet constant (V = 0.001 to 0.1 minute/A, or equivalently 0.1 to
10 minute/Oe cm), H is the field strength and t is the thickness of the specimen or
more precisely the path length of light in the material.
In ferromagnetic or ferrimagnetic materials the angle of rotation (J can also be
related to the magnetization M by
(J=KMt,
where K is Kundt's constant, M is the magnetization of the material and t is the
path length. K is typically up to 350 X 106 degrees/tesla m (equal to 350
degrees/gauss cm.). Both the Faraday effect and the Cotton-Mouton effect can be
used for domain observation. However these techniques are limited to thin sections
of ferromagnetic materials in order that sufficient light is transmitted.
The Kerr effect is used to observe domain patterns in ferromagnetic materials
58
Magnetic measurements
and is discussed in more detail in Chapter 6. A beam of linearly polarized light is
incident on the surface of the specimen. It is important that the beam is not normal
to the surface of the ferromagnet or ferrimagnet since there must be a component of
the magnetization in the surface of the specimen parallel to the direction of
propagation of the light and it is well known that surface domains are oriented with
their magnetizations in the plane of the surface. The magnetization within a
domain rotates the plane of polarization of the beam that is reflected from the
surface through an angle 8 which is related to the magnetization M by
8= KrM.
Typical rotations are 9 minutes of arc from saturated nickel and 20 minutes of arc
from saturated iron and cobalt. This technique can only be used for the
determination of field strength in situations where the magnetization can be
directly related to the field.
3.2.5
Thin-film magnetometers
Can the presence of magnetic anisotropy be used to measure external field strength?
Magnetic fields can be measured using thin-film magnetometers [12]. Thin films
which are in the range 200-5000 angstroms thick are usually fabricated for these
purposes from a non-magnetostrictive alloy such as Ni- 20% Fe. They have uniaxial
anisotropy with the easy axis parallel to the direction of applied field during
deposition.
Magnetization measurements are made along the hard axis while the field being
measured is applied along the easy axis. Since the applied field effectively alters the
anisotropy this leads to changes in the magnetization characteristics. The film is
used as the inductor in the frequency-controlling circuit of an oscillator. The
output frequency is then a function ofthe external field. The field ranges over which
the thin-film devices are useful are typically from 10- 7 tesla to 10- 3 tesla [13].
3.2.6
Magnetic resonance methods
How can we utilize the magnetic properties of elementary particles to determine field
strength?
Resonance magnetometers include all magnetic field measurement techniques
based on electron spin resonance, nuclear magnetic resonance and proton
precession. A review of these has been given by Seiden [14]. The sensitivities of
these instruments can be of the order of 10- 14 tesla (10- 10 gauss). These methods
have the advantage of measuring the total magnetic field in a region of space. That
is they are not dependent on the orientation of the field for measurement, as are
most other techniques which can only measure the component of magnetization
along a given direction.
The discrete energy levels of electrons in materials are changed by the presence of
Methods depending on changes in material properties
59
a magnetic field. This was first discovered by Zeeman [15,16]. In electron spin
resonance (ESR) electrons can be excited from one state to another by highfrequency radiation and hence will exhibit absorption or resonance at those
characteristic frequencies.
The resonance frequency Vo can be used as a measure of the magnetic field
strength since it is related to the field by the expression
wo= 2nv o=yB
=
YJioH,
where "I is a constant known as the gyromagnetic ratio. The value of "I for a free
electron (electron paramagnetic spin resonance) is 1.76 x 1011 Hz/T and so by this
method very weak fields may be measured. (The value of "I is related to the
fundamental properties of the electron as we shall discuss in Chapter 10, in
particular "I = - JiBg(2n/h) where jiB is the Bohr magneton, g is the Lande splitting
factor of the electron and h is Planck's constant.)
The nuclear magnetic resonance (NMR) technique depends on resonance of the
magnetic moment of the nucleus in an d. field in a similar way to ESR. The energy
levels of the nucleus are quantized and are altered by the presence of a magnetic
field as in ESR. Therefore resonant absorption is observed when the r.f. energy
equals the difference in energy level between these quantized states. The resonant
frequency is therefore a measure of the field strength.
Once again, as in ESR, the resonant frequency is proportional to the field,
however the gyromagnetic ratio of the particular nucleus Yo must be used in the
equation
Wo
=
YnJiOH.
The material used as the medium in this method need not be very special, in fact
water is often used. In this case the nuclei are protons for which 'Yn = 2.68 x
108 Hz/T. Values of'Y for various nuclei and for free electrons are given in Table 3.2.
The experimental set-up for these techniques for field measurement involves two
mutually orthogonal pairs of coils: the d. coils which emit the resonant radio
frequency and the receiver coils which indicate when resonance occurs. The
Table 3.2 Values of the gyro magnetic ratio y for various nuclei
and free electrons
Particle
IH (proton)
2D (deuteron)
7Li
Free electron
y
(rads. sec. -1 T- 1 )
(Hz. r l )
2.6753 x 10 8
4.1064 x 10 7
1.0396 x 10 8
1.762 x 1011
42.579
6.536
16.546
28.043
y/2n
60
Magnetic measurements
resonance methods typically have a precision of 1 part in 106 . This means that
highly accurate field measurements can be made under the right conditions, but
because of the high precision there is a need for a highly homogeneous field
throughout the specimen being used.
In practice, when these methods are used for field measurement, as distinct from
materials property measurements, there are two ways of deducing the field. In one
method the field is calculated from the resonance frequency with y already known;
in the other method the field is determined from an amplitude measurement with w
fixed and y known. Under small-amplitude field modulations I1H of the quasi-d.c.
(,static') field close to the resonance Ho the voltage response of the pick-up coil
I1V is
I1V= KI1H,
where K is a constant of the material, which must be known. In this way I1H can be
measured to a high degree of accuracy [17].
A closely related technique based on proton precession is widely used by
geophysicists for precise measurements of the local strength of the Earth's field.
Proton precession magnetometers are used for flux densities in the range 10- 9 to
10 - 4 tesla. A container of water is placed in a coil perpendicular to the field to be
measured. This coil is then pulsed to align the magnetic moments of the protons.
When it is switched off the nuclear moments of the protons precess about the field
being measured. This precession generates an e.m.f. in the coil at the precession
frequency. Measurement of this frequency enables the field strength to be
calculated.
3.3 OTHER METHODS
In this section we look at three additional techniques: two older methods which
depend on the force on a magnetic dipole in a field and one much more recent
method which depends on the quantization of flux in a superconducting circuit
containing a weak link.
3.3.1 Torque magnetometers
How can the magnetic moment or magnetization be found from the torque exerted by
a known external field?
The torque magnetometer is used mainly for anisotropy measurements on short
specimens. It is based on the fact that a magnetic dipole m in an external magnetic
field H in free space experiences a torque r
r=J1om x H
as described in section 1.2.3. The torque must be measured in a uniform magnetic
field. The specimen is aligned so that its magnetization lies in the plane of rotation
Other methods
61
of the field (if the field is to be rotated) or the sample is rotated in the plane defined
by the magnetic field H and the magnetization M.
A restoring torque is used to maintain the specimen in position. In some
instruments, notably the earlier ones, this restoring torque was provided by
twisting a torsion fibre. The angle ¢ through which the torsion fibre is twisted is
dependent on the length of the fibre and its shear modulus as well as the torque. ¢ is
therefore proportional to the turning force on the specimen.
¢ = constant x r
= constant x flom x H.
If rx is the angle between m and H
¢ = constant x flomH sin rx.
The instrument is calibrated using a specimen of known anisotropy, and
consequently since flo, H and the constant are known, the angle rx can be measured
and the magnetic moment m can be calculated.
A schematic diagram of a torque magnetometer is shown in Fig. 3.5. In this
Fig. 3.5 Schematic diagram of a torque magnetometer.
62
Magnetic measurements
instrument the restoring torque is provided by the torque coil in the presence of a
field from the permanent magnet, and the torque is proportional to the current in
the torque coil.
3.3.2 Susceptibility balances
How can the magnetization be found from the force exerted on a specimen by a
known field?
Two types of force-balance methods have been devised for determining the
magnetization M or equivalently the susceptibility X. These are the analytical
balance and the torsion balance. Both depend on measuring the linear force on a
sample suspended in a magnetic field gradient.
AUTOMATIC
BALANCE
in
sealed
volume
SEAL TO REMOVE SAMPLE
THERMOCOUPLE",
ANO
HEATER LEADS
l
SUDING ARRANGEMENT
TO REMOVE SAMPLE
DEWAR
HEATER WIRES
I'H'!-f+--SAMPLE
Fig. 3.6 Schematic diagram of an analytical balance.
Other methods
63
The specimen is suspended using a long string in a magnetic field with a constant
field gradient. The force on the specimen of volume V and magnetization M is then
dB
= -f.1oVM~
dx
which is obtained by differentiating the equation in section 1.2.3. Since we can
write the susceptibility as
X=M/B
this leads to
The force on the specimen is therefore proportional to its susceptibility.
This method can usually only be used on a short specimen, so that the whole
specimen can be located between the pole pieces of an electromagnet. In addition
since the field must have a constant field gradient, the spatial extent of the specimen
perpendicular to the field must also be small.
SPUT
CATHODE
PHOTOCELL
DC
'-------.__...,------!AMPLIFIER
MAGNET
Fig. 3.7 Schematic diagram of a torsion balance.
64
Magnetic measurements
In the analytical balance method in order to measure the force the specimen is
suspended by a string from one arm of an analytical balance (hence the name of the
method). In zero field the weight of the specimen is counterbalanced and when the
field is switched on the force on the specimen can be measured directly. A diagram
of the experimental arrangement for this method is shown in Fig. 3.6. The origin of
the method is attributed to Faraday.
The torsion balance is shown in Fig. 3.7 and is a variation on the same linear
force method. It can be designed to have a very small restoring torque and
therefore a high sensitivity, however it cannot support as large a mass as the
analytical balance, and therefore is limited to specimens of a few grams.
3.3.3 SQUIDS
What is a SQUID and how can it be used to measure a magnetic field?
At present, SQUIDS (superconducting quantum interference devices) provide the
ultimate in resolution for field measurements. The SQUID consists of a
superconducting ring with a small insulating layer known as the 'weak link', as
shown in Fig. 3.8. The weak link is also known as a Josephson junction. The
resolution of these devices is down to 10- 14 tesla (10- 10 gauss) [18]. The flux
passing through the ring is quantized once the ring has gone superconducting but
the weak link enables the flux trapped in the ring to change by discrete amounts.
Changes in the pick-up voltage occur as the flux is incremented in amounts of LlcI>
= 2.067 x 10- 15 Wb. The device can thereby be used to measure very small
changes in flux. In fact it can be used to count the changes in flux quanta in the ring.
With no weak link flux cannot enter the ring, as we know from
superconductivity, and so the field passing through the ring remains at the value it
was at when the ring became superconducting. If the link is very thick, so that no
Poorly conducting
"weak link"
Superconducting
circuit
Superconducting
current Is
-----Fig. 3.8 A Josephson junction device, which consists of a superconductor with a poorly
conducting 'weak link', ABeD.
Other methods
65
supercurrent can flow, then the flux in the ring will be exactly that which is expected
from the applied field. The presence of the weakly superconducting link typically
restricts the value of the supercurrent flowing in the ring to less than 10 - 5 A.
Therefore with a weak link magnetic flux can enter the ring. The supercurrent in
the weak link tries to oppose the entry of flux but because it is limited by the weak
link it cannot achieve this entirely as the flux is increased. It therefore becomes a
periodic function of the flux threading the superconducting ring.
The relation between the flux density in the ring and the flux density due to the
applied field is
<I> = <l>a + Lis
where <I> is the flux density in the ring, <l>a is the flux due to the applied field, L is the
inductance of the ring and Is is the supercurrent which produces a flux of<l>s = LIs.
In the Josephson junction the supercurrent Is in the ring is related to the critical
current Ie determined by the properties of the weak link.
Is=Ie sin8,
where 8 is the phase difference of the electron wavefunctions across the weak link.
Therefore
<I> = <l>a + LI e sin 8.
In a completely superconducting ring the flux is an integral number offlux quanta.
/
o
/
2
Fig. 3.9 The relation between cp, the flux in the ring, and cp., the flux due to the applied field,
in a SQUID magnetometer. (See also section 15.2.4).
66
Magnetic measurements
Therefore if <1>0 is the flux quantum of 2.067 x 10 - 15 Wb
<1> = N<1>o
with the weak link the phase angle 8 across the link depends on the flux in the
following way
8 = 2nN - 2n(<1>/<1>0}.
Since N is an integer
sin 8 = sin( - 2n<1>/<1>0)
= -
sin(2n<1>/<1>0)
and therefore
<1> = <1>. - LIe sin (2n<1>/<1>0)
and the relation between <1> and <1>. is given in Fig. 3.9. From this graph we see that
the SQUID counts flux quanta of the applied field in units of 2.067 x 10- 15 Wb.
Each time sin (2n<l>/<1>0) = 8 then <1>/<1>0 and <1>J<I> become equal, however at values
in between when the flux is not an integral multiple of 2.067 x 10- 15 Wb, they are
not identical. If a loop of wire or a coil is placed around the superconducting ring
then a voltage pulse is induced in the coil at each quantum jump, and this pulse can
be used to measure the applied field.
The SQUID is clearly a very highly sensitive device and is therefore really best
suited to measuring very small changes in magnetic field.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
DeMott, E. G. (1970) IEEE Trans. Mag., 6, 269.
Gordon, D. I., Brown, R. E. and Haben, J. F. (1972) IEEE Trans. Mag., 8, 48.
Smith, D. O. (1956) Rev. Sci. Inst., 27, 261.
Fert, C. and Gautier, P. (1951) c.R. Acad. Sci., Paris, 233, 148.
Foner, S. (1959) Rev. Sci. Inst., 30, 548.
Primdahl, H. (1970) IEEE Trans. Mag., 6, 376.
Gordon, D.1. and Brown, R. E. (1972) IEEE Trans. Mag., 8, 76.
Cullity, B. D. (1971) J. Metals, 23, 35.
Clark, A. E. and Abbundi, R. (1977) IEEE Trans. Mag., 13, 1519.
Wun Fogle, M., Savage, H. T. and Clark, A. E. (1987) Sensors and actuators 12, 323.
Craik, D. 1. and Tebble, R. S. (1965) Ferromagnetism and Ferromagnetic Domains, North
Holland, Amsterdam.
Irons, H. R. and Schwee, L. 1. (1972) IEEE Trans. Mag., 8, 61.
Bader, C. 1. and Fussel, R. L. (1965) Proceedings 1965 Intermag Conference, New York.
Seiden, P. E. (1969) in Magnetism and Metallurgy, Vol. 1 (eds. A. E. Berkowitz and E.
Kneller); Academic Press, New York.
Zeeman, P. (1897) Phil. Mag., 43, 226.
Hertzberg, G. (1944) Atomic Spectra and Atomic Structure, 2nd ed, Dover, New York,
p.96.
17. Hartman, F. (1972) IEEE Trans. Mag., 8, 66.
18. Webb, W. (1972) IEEE Trans. Mag., 8, 51.
Examples and exercises
67
FURTHER READING
Berkowitz, A. E. and Kneller, E. (1969) Magnetism and Metallurgy, Vol. 1, Academic Press,
Ch.4.
Crangle,1. (1977) The Magnetic Properties of Solids, Edward Arnold, London.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading, Mass.
Janicke, 1. (1975) Magnetic measurements in IEEE Workshop on Applied Magnetics.
Milwaukee. Wisconsin, IEEE, New York.
EXAMPLES AND EXERCISES
Torque magnetometer. Explain how to measure the strength of a
magnetic field H using a magnetic dipole of known strength. Define the magnetic
Example 3.1
moment of a coil and a bar magnet and show that m = <p1/#0' A magnetized rod of
magnetic moment 0.318 A m2 is suspended horizontally in a magnetic field of
14 A/m. What is the torque required to keep it at an angle of (a) 90° (b) 30° to the
field direction?
Example 3.2 Magnetic resonance. Describe the phenomenon of electron spin
resonance and explain briefly why it occurs. Explain what is meant by:
(a) Bohr magneton;
(b) Gyromagnetic ratio;
(c) Lande splitting factor.
A dilute paramagnetic material has a ground state S = 1. Calculate the
gyro magnetic ratio y and hence determine the expected resonance frequency in a
field of 0.796 x 10 5 A/m (equivalent to a free-space magnetic induction of 0.1 tesla).
Example 3.3 Induction coil method. The initial magnetization curve of a
specimen of iron oflength 20 cm and diameter 0.5 cm is measured using a ballistic
galvanometer. The galvanometer has a sensitivity of 0.17 x 10 - 4 weber turns per
mm, and the solenoid used to generate the field gives an H field of 400 A/m for each
ampere flowing in the coil.
Table 3.3
i (Amp)
1.5
3.1
4.9
8.5
11.0
12.7
d(mm)
24.0
49.2
77.6
103.7
107.5
109.1
68
Magnetic measurements
A 40-turn induction coil is wound on the specimen to measure the flux passing
through it. The current reversed in the solenoid i and the deflection d were as in
Table 3.3.
(a) Find the demagnetization factor N d •
(b) Plot the initial magnetizing curves B against Happlied and B against H.
(c) Find the true and apparent permeability at B = 1.0 tesla.
4
Magnetic Materials
The macroscopic behaviour of magnetic materials can be classified using a few
magnetic parameters. We look at the most significant of these, give some
definitions and show how the most important class of magnetic materials, the
ferromagnets, can be classified on this basis. We then survey the main uses of
ferro magnets and indicate how the macroscopic properties determine the
suitability of a material for a given application.
4.1
IMPORTANT MAGNETIC PROPERTIES OF FERRO MAGNETS
Which magnetic materials are the most significant for applications?
By far the most important class of magnetic materials is the ferro magnets. We
can make this statement unreservedly both from the practical and the theoretical
viewpoints. The applications which these materials find are very diverse and have
been discussed in the two most significant ferromagnetic materials reference texts
by Heck [IJ and Wohlfarth [2]. In engineering applications the ferromagnets are
used because of their high permeabilities which enable high magnetic inductions to
be obtained with only modest magnetic fields, their ability to retain magnetization
and thereby act as a source offield and of course the torque on a magnetic dipole in
a field can be used in electric motors. It is perhaps somewhat surprising that the few
ferromagnetic elements in the periodic table, iron, cobalt, nickel and several of the
lanthanides are so technologically vital.
At this stage we are still considering the materials on a macroscopic scale and
consequently all that we have really discussed so far is the definition that
ferromagnets have in general very large values of relative permeability Jlr and
susceptibility x. These are important quantities, and later we shall go on to
consider what the special factors are that cause such high permeabilities. However
we will now pause to consider some of the more characteristic features of
ferromagnets which are noticeable on the everyday, macroscopic scale.
4.1.1
Permeability
What is the most important single property of ferromagnets for applications?
70 Magnetic materials
By far the most important single property of ferromagnets is their high relative
permeabilities. The permeability of a ferromagnet is not constant as a function of
magnetic field in the way that the permeability of a paramagnet is. Instead, in order
to characterize the properties of a given ferromagnetic material it is necessary to
measure the magnetic induction B as a function of H over a continuous range of H
to obtain a hysteresis curve.
We can still make some comments about permeabilities, however. For
ferro magnets the initial relative permeabilities usually lie in the range 10 to 105 .
The highest values occur for special alloys such as permalloy and supermalloy,
which are nickel-iron alloys. These materials are useful as flux concentrators.
Permanent magnet materials do not have such high permeabilities, but their
applications depend on their retentivity which is the next most important property.
4.1.2 Retentivity
What is the most characteristic property of ferromagnets?
It is well known that ferromagnets can be magnetized. That is to say that once
exposed to a magnetic field they retain their magnetization even when the field is
removed. Probably this is the most widely recognized property of ferromagnets
since we have all spent time magnetizing pieces of iron using a permanent magnet.
The retention of magnetization distinguishes ferromagnets from paramagnets
which although they acquire a magnetic moment in an applied field H, cannot
maintain the magnetization after the field is removed.
4.1.3 Hysteresis
How can we best represent the magnetic properties of ferromagnets?
The most common way to represent the bulk magnetic properties of a
ferromagnetic material is by a plot of magnetic induction B for various field
strengths H. Alternatively plots of magnetization M against H are used, but these
contain the same information since B = JJ.o(H + M). Hysteresis in iron was first
observed by Warburg [3]. The term hysteresis, meaning to lag behind, was
introduced by Ewing [4] who was the first to systematically investigate it. A typical
hysteresis loop is shown in Fig. 4.1.
The suitability of ferromagnetic materials for applications is determined
principally from characteristics shown by their hysteresis loops. Therefore
materials for transformer applications need to have high permeability and low
hysteresis losses because of the need for efficient conversion of electrical energy.
Materials for electromagnets need to have low remanence and coercivity in order
to ensure that the magnetization can easily be reduced to zero as needed.
Permanent magnet materials need high remanence and coercivity in order to
retain the magnetization as much as possible.
Important magnetic properties oj Jerromagnets
71
B (Tes!a)
100
200
H(Amp/m)
Fig. 4.1 A typical hysteresis loop of a ferromagnetic material.
4.1.4 Saturation magnetization
Is there an upper limit to the magnetization of a ferromagnet?
From the hysteresis plot it can be seen that the ferromagnet in its initial state is not
magnetized. Application of a field H causes the magnetic induction to increase in
the field direction. If H is increased indefinitely the magnetization eventually
reaches saturation at a value which we shall designate Mo. This represents a
condition where all the magnetic dipoles within the material are aligned in the
direction of the magnetic field H. The saturation magnetization is dependent only
on the magnitude of the atomic magnetic moments m and the number of atoms per
unit volume n.
Mo=nm.
Table 4.1 Saturation magnetization of various ferromagnets
Material
(10 6 A/m)
Iron
Cobalt
Nickel
78 Permalloy (78% Ni, 22% Fe)
Supermalloy (80% Ni, 15% Fe, 5% Mo)
Metglas 2605 (Fe so B2o)
Metglas 2615 (FesoP 16C3Bl)
Permendur (50% Co, 50% Fe)
1.71
1.42
0.48
0.86
0.63
1.27
1.36
1.91
72
Magnetic materials
Mo is therefore dependent only on the materials present in a specimen, it is not
structure sensitive. Some typical values of saturation magnetization for different
materials are shown in Table 4.1.
4.1.5
Remanence
What happens to the magnetic induction of a ferromagnet when the magnetic field is
switched off?
When the field is reduced to zero after magnetizing a magnetic material the
remaining magnetic induction is called the remanent induction BR and the
remaining magnetization is called the remanent magnetization MR'
BR = JloMR
A convention seems to be emerging in which a distinction is drawn between the
remanence and the remanent induction or magnetization. The remanence is used
to describe the value of either the remaining induction or magnetization when the
field has been removed after the magnetic material has been magnetized to
saturation. The remanent induction or magnetization is used to describe the
remaining induction or magnetization when the field has been removed after
magnetizing to an arbitrary level. The remanence therefore becomes the upper
limit for all remanent inductions or magnetizations.
4.1.6 Coercivity
How is the magnetic induction reduced to zero?
The magnetic induction can be reduced to zero by applying a reverse magnetic field
of strength He. This field strength is known as the coercivity. It is strongly
dependent on the condition of the sample, being affected by such factors as heat
treatment or deformation.
As with the remanence a distinction is drawn by some authors between the
coercive field (or coercive force), which is the magnetic field needed to reduce the
magnetization to zero from an arbitrary level, and the coercivity which is the
magnetic field needed to reduce the magnetization to zero from saturation [5]. In
this nomenclature the coercivity becomes an upper limit for all values of coercive
force.
The intrinsic coercivity, denoted H ei , is defined as the field strength at which
the magnetization M is reduced to zero. In soft magnetic materials He and Hei
are so close in value that usually no distinction is made. However in hard magnetic
materials there is a clear difference between them, with Hei always being larger than
He·
Important magnetic properties oj Jerromagnets
73
4.1.7 Differential permeability
How useful is permeability when considering ferromagnets?
We should note in passing that the permeability /l is not a particularly useful
parameter for characterization of ferromagnets, since by virtue of the hysteresis
loop almost any value of /l can be obtained including /l = 00 at the remanence
B = BR , H = 0 and /l = 0 at the coercivity B = 0, H = He.
The differential permeability /l' = dB/dH is a more useful quantity although we
should remember that this also varies with field. The maximum differential
permeability /l;"ax, which usually occurs at the coercive point H = He' B = 0, and
the initial differential permeability /l;n which is the slope of the initial
magnetization curve at the origin, are much more useful since it is possible to relate
them to other material properties such as the number and strength of pinning sites
[6J and applied stress [7].
4.1.8
Curie temperature
What happens if a ferromagnet is heated?
All ferromagnets when heated to sufficiently high temperatures become
paramagnetic. The transition temperature from ferromagnetic to paramagnetic
behaviour is called the Curie temperature. At this temperature the permeability of
the material drops suddenly and both coercivity and remanence become zero. This
property of ferromagnets was known long before the work of Curie. In fact the
existence of a transition temperature was first reported by Gilbert [8J who was the
author of the first treatise on magnetism.
Table 4.2 Curie
materials
temperatures
of various
Material
Curie temperature
Iron
Nickel
Cobalt
Gadolinium
Terfenol
Nd 2 Fe 14 B
Alnico
SmCo s
Hard ferrites
Barium ferrite
770°C
358°C
1130°C
20°C
380-430°C
312°C
850°C
nooe
400-700 o e
450 0 e
74
Magnetic materials
The reasons for the sudden transition from ferromagnetism to paramagnetism
will be discussed in detail in a later chapter. At this stage however we are interested
in this merely as an empirical observation based on the macroscopic magnetic
properties that the permeability suddenly decreases at a characteristic
temperature.
4.2
DIFFERENT TYPES OF FERROMAGNETIC MATERIALS FOR
APPLICATIONS
Where do ferromagnetic materials find their main uses?
We now consider the different applications offerromagnets such as in permanent
magnets, electrical motors, magnetic recording, power generation and inductors.
The objective of this section is to give a concise summary of the types of magnetic
materials and their uses before treating the subject in detail in Chapters 12, 13 and
14 and before discussing the underlying mechanisms behind the observed
macroscopic magnetic properties of these materials in the following chapters. A
review of the development and role of magnetic materials in science and
technology has been given by Enz [9].
4.2.1
Classification of hard and soft magnetic materials
Is there a simple method of classlfzcation of the various ferromagnetic materials?
We can make a simple classification of ferromagnetic materials on the basis of their
coercivity. Coercivity is a structure-sensitive magnetic property, which is to say
that it can be altered by subjecting the specimen to different thermal and
mechanical treatments, in a way that for example saturation magnetization
cannot.
It has been noticed in the past that iron and steel specimens that were
mechanically hard also had high coercivity, while those that were soft had low
coercivity. Therefore the terms 'hard' and 'soft' were used to distinguish
ferromagnets on the basis of their coercivity. Broadly 'hard' magnetic materials
were those with coercivities above 10 kAjm(125 Oe) while 'soft' magnetic
materials were those with coercivities of below 1 kAjm(12.5 Oe).
Some values of relative permeability and coercive force for various materials are
given in Fig. 4.2.
4.2.2
Electromagnets
Where do soft magnetic materials find uses?
Soft magnetic materials find applications in electromagnets, motors, transformers
and relays. The properties of various soft magnetic materials in use at present have
been discussed by Chin and Wernick [10]. The criteria for consideration of
Different types of ferromagnetic materials
I
75
I
• Metglas (1976)
106
I
i
105 f--- Supermalloy (1947)
I
I
• Sendust (1936)
I
104
I
• 78 Permalloy (1923)
I
I
Hypersil (1934)
I
Iron
Nickel
I
I
10
• 5% W (1885)
KS (192,3) •• MK (1931)
Alnico V (1940)
I
I
I
.1
Ferro-platinum (1936) •
Sm-Co (1967)
•
Nd-Fe-B (1984)
10-1
I
10-3 10-2
I
I
10- 1
Fig. 4.2 Relative permeabilities and coercive forces (in Oe) of various ferromagnetic
materials.
materials for electromagnets are that the core material should have high
permeability, to enable high magnetic induction to be achieved, while having a low
coercivity so that the induction can easily be reversed.
In electromagnets soft iron is used almost exclusively. Its coercivity is typically
80 A/m (1 Oe), as can be seen from Fig. 4.2, and this when coupled with its high
saturation magnetization of 1.7 x 106 A/m make it the ideal material.
Electromagnets are used in the laboratory for generating high magnetic fields. A
typical laboratory electromagnet is capable of generating fields of up to 2.0 tesla
without any special configuration and, with small air gaps, fields of up to 2.5 tesla
can be produced. Sometimes the pole tips of the electromagnet are made from a
cobalt-iron alloy which has a higher saturation magnetization in order to achieve
slightly higher fields in the air gap (Ms = 1.95 x 106 A/m for an alloy of 35% Co,
65% Fe, compared with Ms = 1.7 X 106 A/m for iron.) The material most often used
for these pole tips is an alloy containing 49% Fe, 49% Co, 2% V. For magnetic
inductions above 3 tesla electromagnets are not very useful because the iron cannot
contribute much additional field. Therefore for higher field strengths either watercooled iron-free magnets, known as Bitter magnets, or superconducting magnets
are used.
76
Magnetic materials
4.2.3 Transformers
What are the characteristics offerromagnetic materials needed in power generation?
At first sight it would appear that the requirements for transformer materials were
identical to those for electromagnets. However this is not quite true. Transformers
operate under a.c. conditions and therefore although high permeability of the core
material is desirable it is also necessary to reduce the eddy current losses by
employing as Iowa conductivity material as possible.
The material that is used exclusively for transformer cores is grain-oriented
silicon-iron. This contains about 3-4% by weight silicon to reduce conductivity.
The material is usually hot-rolled then cold-worked twice followed by an anneal to
improve the grain orientation, increasing permeability along the rolling direction.
One of the most important parameters for transformer steels is the total core loss
at line frequency of 50 or 60 Hz. Engineers usually measure this in watts per
kilogram. Losses decrease with increasing silicon content but the material also
becomes more brittle.
Table 4.3 below shows the electrical core losses in watts per kilogram at a
frequency of 60 Hz for sheets of the various materials of thickness 0.3-0.5 mm.
These thicknesses are equal to or less than the skin depth b in the materials, which
at 60 Hz is typically b = 0.3-0.7 mm.
In recent years there has been an attempt to develop amorphous metals [11] for
use in electromagnetic devices. These alloys such as Metglas, have found
applications in some smaller devices, but have not been successful in replacing
silicon-iron in transformers except in some cases where distribution transformers
have been required in locations where fuel costs are high. Several thousand of these
Metglas transformers have been built and sold, however this remains a very small
fraction of the market for transformers. There does not seem to be any real
likelihood oflarge-scale adoption of Metglas as a transformer core material either
in view of recent Japanese efforts which have continued to improve the properties
of silicon-iron [12].
Table 4.3 Core losses of selected soft magnetic materials
Core loss at 1.5 T and 60 Hz
Material
Low-carbon steel
Non-oriented silicon-iron
Grain-oriented silicon-iron
80-Permalloy (80% Ni, 20% Fe)
Metglas
(Wjkg)
2.8
0.9
OJ
~2+
0.2-0.3*
* At 1.4 tesla since loss increases rapidly above this.
+ At saturation magnetization of 0.86 MAjm (1.1 tesla).
Different types of ferromagnetic materials
4.2.4
77
Electromagnetic relays
What magnetic properties are useful in magnetic switches and control devices?
A relay is a form of electromagnet which can be used as a switch for opening or
closing an electrical circuit. The control circuit which opens and closes the switch
and the operating circuit are electrically isolated from each other. Consequently a
very weak current in the control circuit can be used to control a much larger
current in the operating circuit.
The control circuit of the relay consists of a coil with a magnetizable core and a
movable component called the armature, which is used to make or break the
circuit. The yoke and core materials of relays have much the same requirements as
electromagnets, that is low coercivity, low remanence and high magnetic
induction. This leads in addition to low core loss and high permeability. Relay
materials are almost always iron or iron-based alloys such as Fe-Si or Fe-Ni.
Unalloyed iron is the most frequently used material for relays. The addition of
silicon to iron reduces the coercivity from typically 100 amps per metre to a few
amps per metre. The addition of nickel to iron can reduce the coercivity to as low as
1 amp per metre.
4.2.5
Magnetic recording materials
What are the desirable characteristics of recording media?
Magnetic recording materials have some characteristics in common with
permanent magnets in that to be useful they need to have a relatively high remanence and a sufficiently high coercivity to prevent unanticipated demagnetization resulting in the loss of information stored on the magnetic tape or
disk. Magnetic recording can either be analog, as in audio recording of signals
on magnetic tape or digital recording, as used in the storage of information of data
on magnetic disks and tapes for computer applications. A review of magnetic
recording media has been given by Bate [13].
The most widely used magnetic recording material is y-Fe 2 0 3 (gamma ferric
oxide) although both chromium dioxide and cobalt-doped y-Fe 2 0 3 are also used.
Gamma ferric oxide is used in both equiaxed and acicular form. Equiaxed gamma
ferric oxide particles used for magnetic recording have diameters of 0.05-0.3 /lm.
Magnetic recording tapes contain small needle-shaped particles of one of these
oxides. The particles are embedded in a flexible binding material and at present the
needles lie in the plane of the tape.
The needle-shaped particles are aligned by a magnetic field during the
fabrication process. The final tapes of y-Fe 2 0 3 have coercivities typically of 2024 kA/m, and the acicular particles have lengths ranging from 0.1-0.7 /lm [13],
with length-to-diameter ratios from 3:1 to 10:1. Tapes made from Cr0 2 have
coercivities of 36-44kA/m. The chromium dioxide particles have dimensions
ranging from 0.5 x 0.03/lm to 0.2 x 0.02/lm which are significantly smaller than
78
Magnetic materials
M(emu/(c)
eoo
200
H(Oe)
Fig. 4.3 Hysteresis loop for a typical metallic magnetic recording material.
the typical sizes of gamma iron oxide particles used in recording tapes. In all cases
the ferromagnetic particles used in magnetic recording are too small to contain a
domain wall and we therefore have single-domain particles.
Attempts are being made to develop 'perpendicular recording' media in which
the needles lie perpendicular to the plane. The advantages of this are that it may be
possible to increase the information storage density. Research into perpendicular
recording media is continuing [14], in particular much attention is being directed
towards CoCr layers for this purpose. However so far the development of these
media has encountered difficulties, among which is the fact that the material does
not perform as well as was expected.
The hysteresis loops which are desirable for magnetic recording materials are
generally square loops, with high remanence, moderately high coercivity and rapid
switching from one state to the other, as shown in Fig. 4.3, which is the hysteresis
loop of a metallic magnetic recording medium. In this case the coercivity 56 kA/m
and remanence 0.9 x 106 Aim are substantially higher than for y-Fe 2 0 3 particles.
4.2.6
Permanent magnets
Where do we use ferromagnetic materials that remain permanently magnetized such
as alnico, neodymium-iron-boron or samarium-cobalt?
Permanent magnets are one of the three most important classes of magnetic
materials, the others being electrical steels and magnetic recording media.
Different types of ferromagnetic materials
79
Permanent magnets find applications in electrical motors and generators,
loudspeakers, television tubes, moving-coil meters, magnetic suspension devices
and clamps [15,16].
Clearly the application determines the choice of the magnetic material based on
its hysteresis characteristics. The properties of these materials are usually
represented by the 'demagnetization curve' which is the portion of the hysteresis
curve in the second quadrant as the magnetization is reduced from saturation. The
demagnetization curves for some permanent magnet materials are shown in
Fig. 4.4. It is important to realize that the final magnetic properties depend as
much on the metallurgical treatment and processing of the material as on its
chemical composition.
In recent years a permanent magnet material based on neodymium-iron-boron
has been discovered [17]. This has superior magnetic properties for many
applications when compared with its predecessor samarium-cobalt [18]. For
example its coercivity can be as high as 1.12 x 106 Aim (14 000 Oe) compared with
0.72 x 106 Aim (90000e) for samarium-cobalt.
In addition to the coercivity another parameter of prime importance to
permanent magnet users is the maximum energy product BHmaX" This is obtained
by finding the maximum value of the product IBHI in the second, or
demagnetizing, quadrant of the hysteresis loop. It represents the magnetic energy
stored in a permanent magnet material. We will discuss its significance to
permanent magnet users in detail in Chapter 14. Generally the maximum energy
product by itself does not give sufficient information for permanent magnet users
to decide on the suitability of a material for a particular application, but it is one
parameter which is widely quoted when comparing various permanent magnet
materials.
8(T)
1.2
/
/
I
/
1.0
3/
0.0
I I
I
II
0.6
2 ,- "
"
,/1
I
I
0.4
I
I
0.2
II
/
/
,
a
,/ b
///
o~
-1000-800 -600 -400 -200
a
H (kA/m)
Fig. 4.4 Second quadrant magnetization curves of specimens of samarium-cobalt (1 and 2)
and neodymium-iron-boron (3).
80
Magnetic materials
For many years the maximum energy product was in the range of 50 x
10 3 J m - 3 (a few megagauss-oersted). Development of samarium-cobalt permanent magnets raised this to about 160 x 10 3 J m - 3 (20 megagauss-oersted) and
lately in the neodymium-iron-boron magnets energy products of typically
320 x 10 3 J m - 3 (40 megagauss-oersted) have been achieved.
In most applications the stability of the permanent magnet is an important
consideration and therefore the material must be operated sufficiently far from its
Curie point since the spontaneous magnetization decreases rapidly with
temperature above about 75% of the Curie temperature. This is one of the
problems that has arisen with neodymium-iron-boron magnets for higher
temperature applications.
4.2.7 Inductance cores: soft ferrites
What additional properties are needed for high-frequency applications of ferromagnets?
Soft magnetic materials are also used as cores for induction coils. They enhance the
flux density inside the coil and thereby improve inductance. When inductors are
required to operate at high frequencies then, due to the skin depth, only nonconducting or finely laminated magnetic materials can be used. This usually means
soft ferrites which are magnetic materials with high electrical resistivity and high
permeability which for many years were thought to be ferromagnets. This was
because their bulk magnetic properties are very similarly to ferromagnets. It is now
known that these materials are different from ferromagnets and this difference is
discussed in Chapter 9. Ferrite-cored inductors are used extensively in frequencyselective circuits, so that the resonant frequency of the circuit ensures that it only
responds to the given frequency.
Another application of soft ferrites is in antennae for radio receivers. These have
an internal ferrite-cored antenna consisting of a short solenoidal coil of N turns
enclosing an area A. When this is oriented with its axis parallel to the magnetic field
vector of the radio wave signal being received the induced e.mJ. in the coil is
E = EoJlr(2nAN)/.Ie,
where Jlr is the relative permeability of the ferrite rod,.Ie the wavelength of the radio
waves and Eo is the strength of the electric field of the received signal in free space.
Typical values of Jlr for these applications are Jlr:::::: 100 to 1000.
Soft ferrites came into commercial production in 1948. They consist of a
compound oxide consisting of iron oxide (Fe 2 0 3 ) together with other oxides such
as manganese, nickel or magnesium which have a complicated chemical
composition. For example nickel ferrite has the composition NiO· Fe 2 0 3 . In their
final form they are usually a brown-coloured ceramic. Their saturation
magnetization is typically Ms = 0.2 X 106 A/m (Bs = 0.25 tesla), with coercivities of
the order of 8 A/m (0.1 Oe) and maximum permeability Jlr = 1500.
Paramagnetism and diamagnetism
81
4.2.8 Ceramic magnets: hard ferrites
Which permanent magnet materials should be used where the demagnetizing effects
are large?
The hard ferrites, also known as ceramic magnets, are in widespread use in motors,
generators and other rotating machines, loudspeakers and various holding or
clamping devices. According to Heck [1] half the West German magnet
production in 1963 was in the form of barium ferrite ceramic magnets.
They are usually made from barium or strontium ferrite. They are very cheap to
produce and can be powdered and included in a plastic binder to form the so called
'plastic magnets' which can be formed easily into any desirable shape. They have
very high coercivity, typically 200 kAjm so that the can be usefully used in the form
of short magnets even though the demagnetizing effects are large.
4.3 PARAMAGNETISM AND DIAMAGNETISM
What uses do paramagnets and diamagnets find?
Paramagnets do not find nearly as many applications as ferromagnets and
therefore our discussion at this stage will be somewhat limited. The description of
paramagnetism is however of vital importance in the understanding of magnetism.
The reason for this is that paramagnetism is a much simpler phenomenon to
describe than ferromagnetism and quite reasonable theories of paramagnetism
have been developed on the basis of very simple models and these simple theories
give good agreement with experimental observation. In the limiting case the
atomic magnetic moments of paramagnets can be treated as non-interacting (i.e.
'dilute paramagnetism'), an approximation which simplifies the modelling greatly.
Diamagnets generally do not find many applications which depend on their
magnetic properties either, except for the special case of the superconductors,
which are perfect diamagnets with X = - 1.
4.3.1
Paramagnets
How do paramagnets differ fundamentally from ferromagnets?
The study of paramagnetism allows us to investigate the atomic magnetic
moments of atoms almost in isolation, since unlike ferromagnetism
paramagnetism is not a cooperative phenomenon. Solid-state physicists are
therefore more familiar with the underlying theories of paramagnetism such as the
temperature dependence of paramagnetic susceptibility, and its description using
the classical expression the Langevin function (see Section 9.1.5) or its quantum
mechanical analog the Brillouin function (see Section 11.2.2). Materials
exhibiting paramagnetism are usually atoms and molecules with an odd number of
82
Magnetic materials
electrons so that there is an unpaired electron spin, giving rise to a net magnetic
moment. These include atoms and ions with partially filled inner shells, such as
transition elements. Some elements with even numbers of electrons are
paramagnetic.
Examples of paramagnetic materials are platinum, aluminum, oxygen, various
salts of the transition metals such as chlorides, sulphates and carbonates of
manganese, chromium, iron and copper, in which the paramagnetic moments
reside on the Cr 3 +, Mn 2+, Fe 2+ and Cu 2 + respectively, and hydrated salts such as
potassium-chromium alum KCr(S04h·12H20. These salts obey the Curie law,
which states that the susceptibility Xis inversely proportional to the temperature T,
because the magnetic moments are localized on the metal ions, while the presence
of the water molecules in the hydrated salts ensures that the interactions between
these electrons on neighbouring metal ions are weak.
Salts and oxides of rare earth (lanthanide) elements are strongly paramagnetic.
In these solids the magnetic properties are determined by highly localized 4f
electrons. These are closely bound to the nucleus, and are effectively shielded by the
outer electrons from the magnetic field at the ionic site caused by the other atoms in
the crystal lattice, that is the crystal field. Rare earth metals are also paramagnetic
for the same reasons, however if the temperature is reduced many of them exhibit
ordered states such as ferromagnetism.
All ferromagnetic metals such as cobalt, iron and nickel become paramagnetic
above their Curie points, as do the antiferromagnetic metals chromium and
manganese above their transition temperatures of 35°C and - 173°C, respectively.
Paramagnetic metals which do not exhibit a ferromagnetic state include all the
alkali metals (sodium series), and the alkaline earth metals (calcium series) with the
exception of berylium. The 3d, 4d and 5d transition metals are all paramagnetic
with the exception of copper, zinc, silver, cadmium, aluminum and mercury which
are diamagnetic. The elements oxygen, aluminum and tin are also paramagnetic.
4.3.2
Temperature dependence of paramagnetic susceptibility
How does the susceptibility vary with environmental Jactors such as temperature?
In many paramagnets the susceptibility is inversely proportional to temperature.
This dependence is known as the Curie law
C
x=-T
where T is the temperature in kelvin and C is a constant known as the Curie
constant. In other paramagnets the susceptibility is independent of temperature.
Two theories have evolved to deal with these two types of paramagnetism: the
localized moment model which leads to the Curie law, and the conduction band
electron model due to Pauli which leads to temperature independent and rather
weaker susceptibility. The dependence of the susceptibility on temperature of some
paramagnetic solids is shown in Fig. 4.5.
Paramagnetism and diamagnetism
83
12r-,~
40
8
",
E 30
u
"-
"0
~
E
><
0
",
0
t!
::!..
'c::l
20
c:
4
~
~
o
200
><
10
400
200
T(K)
300
Temperature, K
(b)
(a)
Fig. 4.5 Temperature dependence of the reciprocal magnetic susceptibility of some magnetic
materials; (a) manganese compounds after de Haas et al.; and (b) Gd(C 2 H sS0 4 h'9H 2 0
after Jackson and Kamerlingh Onnes.
4.3.3
Field dependence of paramagnetic susceptibility
What effect does a magnetic field have on the susceptibility of a paramagnet?
In paramagnets subjected to magnetic fields other than very high fields the
magnetization Mis proportional to the field H. That is the susceptibility is virtually
constant and lies in the range 10- 3 to 10- 5. In most cases the spins are not coupled
or are only weakly coupled so that they can be considered independent to a good
approximation. The reason for this is the thermal energy is sufficiently great to
cause random alignment of the moments in zero field. When a field is applied the
atomic moments begin to align, but the fraction deflected into the field direction
remains small for all practical field strengths.
The variation of the magnetization of a typical paramagnet with temperature
and field is shown in Fig. 4.6. At moderate to strong fields the susceptibility is still
constant and saturation only occurs at very high field strengths. The dependence
can be expressed classically using the Langevin function
flomH) M/nm = coth ( -k---
BT
(kB
T ),
---flo mH
84
Magnetic materials
M
Nm
1·0
0·5
0.0
L -_ _ _ _ _ _ _ _ _ _ _ _ __
o
2
3
4
JlomH
kT
a=--
Fig. 4.6 Variation of magnetization of a typical paramagnet with temperature and
magnetic field using the classical Langevin equation.
where n is the number of atoms per unit volume, m is the magnetic moment per
atom, kB is Boltzmann's constant and T is the temperature in kelvin. This leads to
the following approximate expression for susceptibility, which works well at high
temperatures
Jlonm2
X= 3kB T
as described in Chapter 9, where Jlonm 2/3k B is the Curie constant C. A more
accurate expression is obtained using the Brillouin function as described in
Chapter 11.
4.3.4 Applications of paramagnets
Where do paramagnets find uses?
There are very few applications of paramagnetic materials on the basis of their
magnetic properties. Their use occurs primarily in the scientific study of magnetism
since they help in our understanding of the much more important phenomenon of
ferromagnetism by allowing us to study the electronic properties of materials with
net atomic magnetic moments in the absence of strong cooperative effects.
There is an increasing use of paramagnetic materials in electron spin resonance
(ESR) for the purpose of measuring magnetic fields in which the magnetic
properties of the material are already well characterized (rather than studying the
resonance of the material to determine its electronic energy states).
One other application is in the production of very low temperatures. The use of
paramagnetic salts to achieve ultra-low temperatures was first suggested by Debye
[19] and Giauque [20]. A paramagnetic salt is magnetized isothermally and then
cooled to as Iowa temperature as possible by conventional cryogenic means, for
References
85
example by using liquid helium at reduced pressure. It is then thermally isolated
and adiabatically demagnetized whereupon the temperature drops even further.
By this means temperatures down to the millikelvin range can be achieved.
4.3.5
Diamagnets
How do diamagnets differ fundamentally from paramagnets and ferromagnets?
Elements without permanent atomic electronic magnetic moments are unable to
exhibit paramagnetism or ferromagnetism. These atoms have filled electron shells
and therefore no net magnetic moment. When subjected to a magnetic field their
induced magnetization opposes the applied field, in the manner described by
Lenz's law, and so they have negative susceptibility.
The dependence of the magnetization on applied field in diamagnets, that is the
susceptibility, is according to the classical Langevin theory of diamagnetism
(Section 9.1.2) given by
where n is the number of atoms per unit volume, Z is the number of electrons per
atom, e is the electronic charge, me is the electronic mass and r2 >is the root mean
square atomic radius, which is typically 10- 21 m 2 • Diamagnetic susceptibility is
substantially independent of temperature.
<
4.3.6 Superconductors
How are superconductors classified among magnetic materials?
Superconductors are diamagnets which find many applications, however they are
a unique class of diamagnet in which the susceptibility is caused by macroscopic
currents circulating in the material which oppose the applied field rather than
changes in the orbital motion of closely bound electrons. They therefore represent
a very special case. Clearly their susceptibility is temperature dependent since
above their critical temperature they are no longer perfect diamagnets. These
materials will be dealt with separately in Chapter 12.
REFERENCES
1. Heck, C. (1974) Magnetic Materials and their Applications, Crane and Russak & Co.,
New York.
2. Wohlfarth, E. P. (ed.) (1980 and 1982) Ferromagnetic Materials, Three-volume series,
North Holland, Amsterdam.
3. Warbug, E. (1981) Ann. Physik, 13, 141.
4. Ewing, 1. A. (1900) Magnetic Induction in Iron and other Metals, 3rd edn, The
Electrician Publishing Co., London, 1900, and (1882) Proc. Roy. Soc., 220, 39.
86
Magnetic materials
5. Chen, C. W. (1977) Magnetism and Metallurgy of Soft Magnetic Materials, North
Holland, Amsterdam.
6. Hilzinger, H. R. and Kronmuller, H. (1977) Physica, 86-88B, 1365.
7. Jiles, D. c., Garikepati, P. and Chang, T. T. (1988) IEEE Trans. Mag., 24, 2922.
8. Gilbert, W. (1958), De Magnete (Trans. On the magnet) (1600) Republished by Dover,
New York.
9. Enz, U. (1982) Magnetism and magnetic materials: Historical developments and
present role in industry and technology, in Ferromagnetic Materials, Vo!. 3, (ed. E. P.
Wohlfarth), North Holland, Amsterdam.
10. Chin, G. Y. and Wernick, 1. H. (1980) Soft magnetic metallic materials, in
Ferromagnetic Materials, Vo!' 2 (ed. E. P. Wohlfarth), North Holland, Amsterdam.
11. Luborsky, F. E. (1980) Amorphous ferromagnets, in Ferromagnetic Materials, Vo!'1
(ed. E. P. Wohlfarth), North Holland, Amsterdam.
12. Shiozaki, M. (1987) Recent trends in non oriented and grain oriented electrical steel
sheets in Japan. Proceedings of the Sixth Annual Conference on Properties and
Applications of Magnetic Materials, Illinois Institute of Technology, Chicago.
13. Bate, G. (1980) Recording materials, in Ferromagnetic Materials, Vo!' 2 (ed. E. P.
Wohlfarth), North Holland, Amsterdam.
14. Bernards, 1. P. c., Schrauwen, C. P. G., Luitjens, S. B., Zieren, V. and de Bie, R. W.
(1987) IEEE Trans. Mag., 23, 125.
15. McCaig, M. (1977) Permanent Magnets in Theory and Practice, Wiley, New York.
16. Moskowitz, L. R. (1976) Permanent Magnet Design and Application Handbook, Cahner,
Boston.
17. Croat,1. 1., Herbst, 1. F., Lee, R. W. and Pinkerton, F. E. (1984) J. Appl. Phys., 55, 2078.
18. Strnat, K., Hoffer, G., Olson, 1., Ostertag, W. and Becker, 1. 1. (1967) J. Appl. Phys., 38,
1001.
19. Debye, P. (1926) Ann. Physik, 81, 1154.
20. Giauque, W. F. (1927) Am. Chern. Soc. J., 49, 1864.
21. Fidler, 1., Bernard, 1. and Skalicky, P. (1987) in High Performance Permanent Magnet
Materials (ed. S. G. Sankar, J. F. Herbst and N. C. Koon) Materials Research Society,
New York, p. 181.
FURTHER READING
Bozorth, R. M. (1951) Ferromagnetism, Van Nostrand, New York.
Heck, C. (1974) Magnetic Materials and their Applications, Crane Russak and Co. New
York.
Hummel, R. (1985) Electronic Properties of Materials, Springer-Verlag, Berlin, Ch. 17.
Jorgensen, F. (1988) The Complete Handbook of Magnetic Recording, 3rd edn, TAB
publishers, Pennsylvania.
McCaig, M. (1977) Permanent Magnets in Theory and Practice, Wiley, New York.
Snelling, E. C. (1988) Soft Ferrites, Properties and Applications, 2nd edn, Butterworths,
London.
EXAMPLES AND EXERCISES
Example 4.1 Properties of ferromagnets. Explain the difference between a
ferromagnet and a paramagnet on the basis of macroscopic measurements. What is
the saturation magnetization and how does it differ as a material is heated through
its Curie point?
What criteria are used to distinguish between hard and soft magnetic materials?
Examples and exercises
87
In deciding on a material for use in (a) an electromagnet (b) a transformer what
magnetic properties would you take into consideration?
Example 4.2 Use of initial magnetization curve to find flux in core. Explain the
meaning of hysteresis loss, coercivity and remanence. How do the magnitudes of
these quantities determine the suitability of a ferromagnet for applications?
Explain what is meant by the initial magnetization curve.
The initial magnetization curve for a specimen of steel is given in Fig. 4.7 below,
a toroid of this steel has a mean circumference of 40 cm and a cross-sectional area
of 4 cm 2 . It is wound with a coil of 400 turns. What will be the total flux in the
material for currents of 0.1, 0.2, OJ, 0.4 and 0.5 amps?
1.4
1.2
1.0
0.8
0.6
0.4
0.2
H(omp.lm.)
o~-
100
200
300
400
500
Fig. 4.7 Initial magnetization curve for a low-carbon steel.
Example 4.3 Calculation of atomic magnetic moment. The saturation magnetization of iron is 1.7 x 106 A/m. If the density of iron is 7970 kg/m 3 and Avogadro's
number is 6.025 x 10 26 per kilogram atom calculate the magnetic moment per
iron atom in amp metre 2 and in Bohr magnetons. (1 Bohr magneton /1B =
9.27 X 10- 24 joule/tesla or 1.16 x 10- 29 joule/ampmetre- 1 . Relative atomic mass
of iron = 56.)
5
Magnetic Properties
We will now look at the causes of hysteresis in ferro magnets and how the variation
of magnetization with magnetic field can be quantified in restricted cases such as at
low field and in the approach to saturation. High-resolution measurements of the
variation of M with H indicate that there are discontinuities. These are known as
the Barkhausen effect. We will also consider the change in length of a ferromagnet
as it is magnetized, that is the magnetostriction, and discuss anisotropy in relation
to magnetostriction.
Which are the most important macroscopic magnetic properties of ferromagnets?
We have shown in the previous chapter that most of the important macroscopic
magnetic properties of ferromagnets can be represented on a B, H plot or hyste~i
loop. From this the magnetization can be calculated at every point on the
hysteresis curve using the totally general formula B = f.Lo(H + M). As the
magnetization curve is traversed there are discontinuous, irreversible changes in
magnetization known as the Barkhausen effect after their discoverer. In recent
years the Barkhausen effect has become an important tool for investigating the
microstructural properties of ferromagnetic materials.
One important bulk property of interest which is not contained in the hysteresis
plot is the magnetostriction. This is the change in length of a material either as a
result of a magnetic order (spontaneous magnetostriction) or under the action of
a magnetic field (field-induced magnetostriction). This will also be discussed in
the chapter.
5.1
HYSTERESIS AND RELATED PROPERTIES
What information can be obtained from the hysteresis curve?
From the hysteresis curve such as the one shown in Fig. 5.1 we can define a number
of magnetic properties of the material which can be used to characterize the
hysteresis loop. A question immediately arises: how many degrees of freedom are
there in a hysteresis loop? Or to put the question another way: how many
parameters are needed to characterize a hysteresis loop? Clearly there is no general
answer to this but for the commonly encountered sigmoid-shaped hysteresis loop
90
Magnetic properties
MIMA/m)
1.6
Fe-C 0·25 Wt~.
5
5
H (kA/mJ
1.6
Fig. 5.1 Typical sigmoid-shaped hysteresis curve of a specimen of iron containing 0.25% by
wt carbon.
such as the one in Fig. 5.1 we can start to enumerate the important properties and
thereby make an estimate.
5.1.1 Parametric characterization of hysteresis
Which are the parameters that can be used to define hysteresis?
First of all the saturation magnetization Mo will give us an upper limit to the
magnetization that can be achieved. At temperatures well below the Curie point
the technical saturation Ms can be used instead. The width ofthe loop across the H
axis which is twice the coercivity He is also an independent parameter since this can
be altered by heat treatment and hence is not dependent on Ms. The height of the
curve along the B axis, that is the remanence BR , is also an independent parameter
since it is not wholly dependent on Ms and He. The orientation of the whole
hysteresis curve, which can be expressed as /J'max the slope of the curve at the
coercive point, can change independently of the other parameters.
Hysteresis and related properties
91
Table 5.1 Magnetic properties of various high-permeability ferromagnetic materials.
Relative permeability at a magnetic induction of2 tesla IIZT' maximum relative permeability
IImax> saturation magnetic induction Bs' d.c. hysteresis loss WH and coercivity He
Material
112T
Purified iron
Iron
Carbonyl iron
Cold rolled steel
Iron-4% silicon
45 Permalloy
Hipernik
Monimax
Sinimax
78 Permalloy
Mumetal
Supermalloy
Permendur
2V Permendur
Hiperco
Ferroxcube
5000
200
55
180
500
2500
4500
2000
3000
8000
20000
100000
800
800
650
1000
Bs
WH
lImax
(tesla)
(J/m3)
He
(A/m)
180000
5000
132
2000
7000
25000
70000
35000
35000
100000
100000
800000
5000
4500
10000
1500
2.15
2.15
2.15
2.1
1.97
1.6
1.6
1.5
30
500
4
80
350
120
22
144
40
24
4
8
1.1
1.07
0.65
0.8
2.45
2.4
2.42
2.5
20
1200
600
4
4
0.16
160
160
80
8
The hysteresis loss WH may also be an independent parameter as may the initial
permeability .u;n. Finally the curvature of the sides of the hysteresis loop, which
although not immediately obvious as an independent parameter, is clearly not
dependent on such factors as coercivity or maximum differential permeability. This
parameter emerges more clearly from a consideration of anhysteretic
magnetization given below, which requires at least two independent parameters in
addition to Ms in order to characterize it.
From the above simplistic considerations we may expect to be able to
characterize the bulk magnetic properties of a material in terms of perhaps five or
six independent parameters. In fact we often find that when the magnetic
properties of ferromagnetic materials are displayed in tabular form the properties
are represented in terms of coercivity, remanence, hysteresis loss, initial
permeability, maximum permeability and saturation magnetization or saturation
magnetic induction, as in Table 5.1.
5.1.2
Causes of hysteresis
What are the underlying mechanisms behind hysteresis?
It is well known that if a specimen of iron or steel is subjected to cold working the
hysteresis loss and the coercivity increase (Fig. 5.2). It is also well known that the
addition of other non-magnetic elements to iron such as carbon in making steel,
92
Magnetic properties
1.5
....... ----:;:::
~
,,'
,,
Hardened,'
~
......
0
,
,-"
CD
-~
-1.5
I
ste~'
~-,it
L-_=~I
-5
"
...--
,
,,
I
.
Soft
iron
_ _ _----'
o
5
H(kA/m)
Fig. 5.2 Dependence of the hysteresis loop of iron or steel on hardness caused by the
addition of carbon or other non-magnetic material or by cold working.
increases the hysteresis loss and coercivity. These empirical facts were known long
before theories of hysteresis were suggested.
From these results it would appear that 'imperfections', whether in the form of
dislocations or impurity elements in the metal, cause an increase in the energy lost
during the magnetization process, in the form of a kind of internal friction. It is
these 'imperfections' which give rise to hysteresis.
Another mechanism which gives rise to hysteresis is caused by magnetocrystalline anisotropy. Ferromagnetic materials with higher anisotropy have
greater hysteresis. This is well known by those working with permanent magnets.
In an anisotropic solid certain crystallographic axes are favoured by the magnetic
moments which will prefer to lie along these directions as this leads to a lower
energy. The magnetic moments can be dislodged from the direction they are
occupying by application of a magnetic field but when this occurs they jump to
crystallographically equivalent axes which are closer to the field direction, and
hence oflower energy. This results in discontinuous and irreversible rotation of the
magnetic moments which leads to a kind of switching action.
In order to discuss this second process properly we need some additional
background knowledge of domain processes which have yet to be discussed
(Chapter 8). Therefore we will defer the discussion until after the necessary
background material has been presented.
5.1.3
Anhysteretic, or hysteresis-free, magnetization
What happens in the case of a material without defects such as dislocations or
impurity elements?
Hysteresis and related properties
93
M/MS
1.0
H kA/M
t
-7
I
7
-1.0
Fig. 5.3 Anhysteretic magnetization curve. This is antisymmetric with respect to the
magnetic field. The differential susceptibility is greatest at the origin and decreases
monotonically with increasing field. The curve has no hysteresis and is completely
reversible.
Ifwe accept the hypothesis that it is the imperfections, whatever their nature, which
cause hysteresis, then we must also ask ourselves what the magnetization curve
would look like if the material were devoid of all imperfections. The answer is that,
ignoring anisotropic effects for the moment, it would be hysteresis free. That is the
magnetic induction would be a single-valued function of the magnetic field H. The
magnetization curve would therefore be reversible.
We can briefly speculate on the form of such a curve before presenting a simple
model for it. Suppose we consider a plot of magnetization against field for such an
ideal case. Since the magnetization of a ferromagnet saturates it is clear that as H
increases so M tends towards Ms. Furthermore, we would expect that at first the
magnetization would change fairly rapidly with H but as H increased the rate of
change would decrease, since this is in the nature of physical systems which
saturate. So we would expect M to be a monotonically increasing function of H
while dM/dH was monotonically decreasing. This would give the S-shaped curve
of Fig. 5.3.
5.1.4
The Frohlich-Kennelly relation
Can we find a simple equation for the anhysteretic?
94 Magnetic properties
A quantitative relationship between magnetization M and magnetic field H is
clearly highly desirable since any such equation provides a means oftelling how the
magnetization or magnetic induction of a material will change with field. An
empirical relationship between M and H along the anhysteretic magnetization
curve was suggested by Frohlich [1 J and later in a different, but equivalent, form by
Kennelly [2].
The Frohlich equation for the anhysteretic magnetization is
aH
M= l+PH'
where alP = Ms since as H --+ 00 the magnetization must tend to Ms·
Independently Kennelly arrived at an expression for the high-field susceptibility
as the magnetization approached saturation. If Kennelly's expression is converted
to SI units it becomes
1
--=a+bH,
fl- flo
which can easily be shown to be equivalent to the Frohlich equation in which
floa = 1/a and flob = Pia = 1/Ms.
This equation can also be rewritten in the form of a series
M=Ms[l- aMsIH+(aMs/H)2 ... ].
This is the form of eq uation used by Weiss [3J for finding Ms from magnetization
curves by extrapolation, using only the terms up to 1/H. It is also of interest in
section 5.1.7 when we compare it with the law of approach to saturation given
much later by Becker and Doring [4].
5.1.5 Measurement of anhysteretic magnetization
If the anhysteretic is so important how do we measure it?
Elimination of all defects within a material is not usually practicable, however
there is a way of reaching the anhysteretic magnetization by other means. This is by
cycling the magnetization by applying an alternating field of gradually decreasing
amplitude superimposed on the d.c. field of interest. As the a.c. field is cycled the
hysteresis is gradually removed and the magnetization converges on the
anhysteretic value for the prevailing d.c. field strength. This procedure can be
thought of as 'shaking' the magnetization so that it overcomes the internal
frictional forces, and anisotropic or switching hysteresis effects, and reaches its true
equilibrium value. The same effect can also be brought about by stress cycling
although this is generally more difficult to achieve.
In magnetic recording (Chapter 14) the use of field cycling is well known as a
method of reaching the anhysteretic magnetization [5]. The anhysteretic
susceptibility at the origin is typically an order of magnitude greater that the
Hysteresis and related properties
95
normal d.c. susceptibility at the origin. The anhysteretic magnetization also varies
linearly with field at low values which is one of the prime considerations for
magnetic recording.
5.1.6 Low-field behaviour: the Rayleigh law
Is there a simple equation for the initial magnetization curve?
We now go on to consider the initial magnetization curve, that is the variation of
magnetization with field obtained when a d.c. field is first applied to a
demagnetized ferromagnet. It was noticed by Rayleigh [6] that in the low-field
region of the initial magnetization curve the permeability could be represented by
an equation of the form
jl(H) = jl(O) + vH,
which leads to the following parabolic dependence of B on H along the initial
magnetization curve
B(H) = jl(O)H + VH2.
According to Rayleigh the term jl(O)H represented the reversible change in
magnetic induction while the term VH2 represented the irreversible change in
magnetic induction. Furthermore, Rayleigh indicated that low-amplitude hysteresis loops could be represented by parabolic curves which have a reversible
differential permeability at the loop tips which is equal to jl(O) as shown in Fig. 5.4.
- H-~t.:_,'
+H
-8
Fig. 5.4 Hysteresis loops of low-field amplitude in the Rayleigh region.
96
Magnetic properties
It follows from this assumption and the Rayleigh law that in the low-field region
the small-amplitude hysteresis loops can be described by an equation of the form
B = [11(0) + vHm]H ± (v/2)(Hm2 - H2),
where Hm is the maximum field at the loop tip. Low-amplitude hysteresis loops for
which this parabolic relation applies are known as Rayleigh loops.
This leads to two further results of interest: expressions for the hysteresis loss
WH and the remanence BR • In SI units these are
WH =
=
and
fHodB
(4/3)vHm 3
BR = (v/2)Hm 2.
It must be remembered of course that these relations only hold true in the lowfield region. As Hm is increased the parabolic relation breaks down. In order to
model the hysteresis behaviour over a wider range of H it is necessary to gain
further insight into the microscopic mechanisms occurring within the material.
501.7 High-field behaviour: the law of approach to saturation
Is there an equation for the magnetization at high fields?
In the high-field region the magnetization approaches saturation. The first attempt
to give an equation describing the behaviour in this region was Lamont's law
[7,p.484],
dM
X= - = constant x (Ms - M),
dH
which states simply that at high fields the susceptibility is proportional to the
displacement from saturation. This was later shown to be equivalent to the
Frohlich-Kennelly relation. This is of interest in relation to the development of the
hysteresis model in section 8.2
Later work indicated that the high-field behaviour can be modelled by the law of
approach to saturation as given by Becker and Doring [4] and Bozorth [7, p. 484].
This is expressed in the form of a series,
M=M s
(1-~
H
H2
..
)+kH
where the final term kH represents the forced magnetization, that is the field
induced increase in spontaneous magnetization, which is a very small contribution
except at high fields. Typically Hhas to be ofthe order of 10 5 or 106 Oe before this
last term becomes significant.
It is interesting to note that this law which was only derived at high
The Barkhausen effect and related phenomena
97
magnetizations is also very similar to the series form of the Frohlich-Kennelly
relation. The reason for this is that at high fields the initial magnetization curve,
the upper and lower branches of the hysteresis loop and the anhysteretic
magnetization approach each other asymptotically.
5.2 THE BARKHAUSEN EFFECT AND RELATED PHENOMENA
5.2.1
The Barkhausen effect
Does the magnetization change smoothly with magnetic field?
The Barkhausen effect is the phenomenon of discontinuous changes in the flux
density B within a ferromagnet as the magnetic field H is changed continuously.
This was first observed in 1919 [8] when a secondary coil was wound on a piece of
iron and connected to an amplifier and loudspeaker. As the H field was increased
smoothly a series of clicks were heard over the loudspeaker which were due to
small voltage pulses induced in the secondary coil. These voltages were caused
through the law of electromagnetic induction by small changes in flux density
through the coil arising from discontinuous changes in magnetization M and
hence in the induction B.
If the initial magnetization curve, which looks to be a smooth variation of Bwith
H under normal circumstances, is greatly magnified, then the discontinuous
changes in B which constitute the Barkhausen effect can be observed directly as in
Fig. 5.5. At first these discontinuities in induction were attributed to sudden
discontinuous rotation of the direction of magnetization within a domain, a
mechanism known as domain rotation, but it is now known that discontinuous
domain boundary motion is the most significant factor contributing to
Barkhausen emissions [9]. Nevertheless both of these mechanisms do occur and
contribute to the Barkhausen effect.
B
~-
H
Fig. 5.5 Barkausen discontinuities along the initial magnetization curve observed by
amplifying the magnetization.
98
Magnetic properties
The Barkhausen emissions are greatly affected by changes in the microstructure
of the material and also by stress. Therefore Barkhausen measurements have found
an important role in materials evaluation as discussed by Matzkanin, Beissner and
Teller [10].
5.2.2 Magnetoacoustic emission
What other effects occur as a result of sudden discontinuous domain-wall motion?
Magnetoacoustic emission, also known sometimes as the acoustic Barkhausen
effect, is closely related to the magnetic Barkhausen effect described above.
Magnetoacoustic emission consists of bursts of low-level acoustic energy generated by sudden discontinuous changes in magnetization involving localized
strains or magnetostriction (section 5.3). These can be detected by a broad-band
ultrasonic transducer. They are caused by microscopic magnetostrictive pulses as
the domain walls move, and in non-magnetostrictive materials such as Fe-20%Ni
are almost non existent. Therefore the effect depends on both sudden discontinuous domain processes and magneto stricti on.
Magnetoacoustic emission was first observed by Lord [11]. Subsequently
investigations were carried out by Ono and Shibata [12] and others [13]. Because
the effect depends on magnetostriction it cannot be generated by 1800 domain-wall
motion or rotation, as these involve no change in magnetostriction. These 180
domain boundaries exist between neighbouring domains in which the magnetization vectors point in exactly opposite directions. The relative number density of
180 and non-180° domain walls is affected by the application of uniaxial stress.
Therefore the method has been suggested as a means of detecting stress in
ferromagnetic materials and this has been shown to be viable by Buttle, Scruby,
Briggs and Jakubovics [14].
0
0
5.3
MAGNETO STRICTI ON
Do the dimensions of a specimen change when it is magnetized?
The magnetization of a ferromagnetic material is in nearly all cases accompanied
by changes in dimensions. The resulting strain is called the magnetostriction A.
From a phenomenological viewpoint there are really two main types of
magnetostriction to consider: spontaneous magneto stricti on arising from the
ordering of magnetic moments into domains at the Curie temperature; and fieldinduced magnetostriction. These are manifestations of the same effect but can be
usefully treated as distinct.
In both cases the magnetostriction is simply defined as A, the fractional change in
length.
A = dill
M agnetostriction
99
The existence of magnetostriction was first discovered by Joule [15,16].
Spontaneous magnetostriction within domains arises from the creation of
domains as the temperature of the ferromagnet passes through the Curie (or
ordering) temperature. Field-induced magnetostriction arises when domains that
have spontaneous magneto stricti on are reoriented under the action of a magnetic
field. Magnetostriction can be measured by using resistive strain gauges or by
optical techniques.
We will first consider the magnetostriction of a hypothetical isotropic solid,
since this leads to the simplest mathematical results.
5.3.1 Spontaneous magnetostriction in isotropic materials
Does the length of a specimen change when it becomes ferromagnetic?
When a ferromagnetic material is cooled through its Curie point the previously
disordered magnetic moments, which had completely random alignment above the
Curie point, become ordered over volumes containing large numbers (typically
a) Disordered
b) Ordered
& unal i gned
c) Ordered & aligned
Fig. 5.6 Schematic diagram illustrating the magneto stricti on in: (a) the disordered
(paramagnetic) regime; (b) the ferromagnetic regime demagnetized; and (c) the
ferromagnetic regime, magnetized to saturation.
100
Magnetic properties
10 12 _10 15 ) of atoms. These volumes in which all moments lie parallel are called
domains and can be observed under a microscope. The direction of spontaneous
magnetization Ms varies from domain to domain throughout the material to
ensure that the bulk magnetization is zero.
The transition to ferromagnetism is said to cause the onset of 'long-range order'
of the atomic moments. By this we mean of course long-range compared with
atomic dimensions, since the range is still microscopic, and three or four orders of
magnitude smaller than the range of magnetic order imposed when the material is
magnetically saturated.
Let us therefore consider spherical volumes of unstrained solid within the solid
above the Curie temperature and hence in the disordered phase as shown in
Fig. 5.6(a). When the material becomes ferromagnetic at the Curie point,
spontaneous magnetization appears within the domains and with it an associated
spontaneous strain e or magnetostriction Ao, along a particular direction, as shown
in Fig. 5.6(b).
For the present isotropic case the amplitudes of these spontaneous magnetostrictions are independent of crystallographic direction. Within each 'isotropic' domain the strain varies with angle e from the direction of spontaneous
magnetization according to the relation
e( e) = e cos 2 e.
The average deformation throughout the solid due to the onset of spontaneous
magnetostriction can then be obtained by integration assuming that the domains
are oriented at random so that any particular direction is equally likely.
"/2
Ao =
J
e cos 2 esin ede
-,,/2
= el3.
This then is the spontaneous magnetostnctlOn caused by ordering of the
magnetic moments at the onset of ferromagnetism. We should note that since we
have assumed an isotropic material the domains are arranged with equal
probability in any direction and therefore the strain is equivalent in all directions.
Therefore in this case although the sample undergoes changes in dimensions its
shape remains the same.
5.3.2 Saturation magnetostriction As
What is the maximum change in length when a ferromagnet is magnetized?
Next we will consider saturation magnetostriction which is the fractional change in
length between a demagnetized ferromagnetic specimen and the same specimen in
a magnetic field sufficiently strong to saturate the magnetization along the field
direction. In this case there will be a change of shape since the applied field
generates a preferred direction.
Magnetostriction
101
Using the very simple model above we cause the transition from the ordered but
demagnetized state to the ordered saturated state by application of a magnetic
field. In the saturated state of course the magnetic moments in the domains are all
aligned parallel to the field and hence the strains are parallel as shown in Fig. 5.6(c).
As = e - ,10
=~e.
This gives us a method of measuring the spontaneous strain e within a material
due to magnetic ordering along a particular direction, by measuring As,
5.3.3
Technical saturation and forced magnetostriction
Can the saturation increase even after the magnetization has reached technical
saturation?
As discussed in section 2.1.4 technical saturation of magnetization occurs when all
magnetic domains within a material have been aligned in the same direction to
form a single-domain specimen. However if the magnetic field is increased further
there is still a very slow rise in M and this process is called forced magnetization.
Similar behaviour is observed in the magnetostriction. Technical saturation
magnetostriction is reached when the specimen has been converted to a single
domain. However a very slow increase in magnetostriction, called forced
magnetostriction, is observed as the field is increased further. Forced
magnetostriction is a very small effect, being appreciable only at fields of the order
of 800 kA/m (10 000 Oe).
The phenomenon is caused by an increase in the ordering of individual atomic
magnetic moments within the single domain which is treated in detail in Chapter 9.
This is the same mechanism which leads to an increase in the spontaneous
magnetization within a domain. The statistical ordering of magnetic moments
within a domain is highly temperature dependent [17] and consequently so is the
magnitude of the forced magnetostriction.
5.3.4
Magnetostriction at an angle
(J
to the magnetic field
How does the saturation magnetostriction vary as a function of angle?
Since we are still considering a completely isotropic medium, we can write down an
equation for the saturation magnetostriction As(O) at any angle 0 to the field
direction.
As(O) = !As(cos 2 0 - t),
where As is the saturation magnetostriction along the direction of magnetization.
This leads to an explanation of some of the magneto stricti on measurements
which you will often find in the literature in which the magneto stricti on with the
102
Magnetic properties
field parallel to a given direction Asli' and the magneto stricti on with the field
perpendicular to the given direction ),s1-' are measured and the difference taken.
The difference between them gives the spontaneous strain within a single domain.
ASII - AS-L = As + As/2 = JAs = e
5.3.5 Anisotropic materials
Are the magnetic properties identical in all directions in a crystal?
Although nickel comes fairly close to having isotropic properties such as
magneto striction, the reality is that all solids are anisotropic to some degree, and
therefore the magnetostriction needs to be defined in relation to the crystal axis
along which the magnetization lies. An extensive review of magneto stricti on in
anisotropic materials has been given by Lee [18].
The magnetostrictions or spontaneous strains are defined along each of the
principal axes of the crystal. For cubic materials there are two independent
magneto stricti on constants A100 and A111 as shown in Table 5.2.
The saturation magnetostriction in cubic materials is then given by a generalized
version of the equation for isotropic materials above,
As = JA 1 oo((X1 2 f31 2 + (X22f32 2 + (X3 2f33 2 -
t)
+ 3A111((X1(X2f31f32 + (X2(X3f32f33 + (X3(X1f33f31)'
where Aloo is the saturation magnetostriction measured along the (100) directions
and A111 is the saturation magnetostriction along the (111) directions. The
spontaneous strains along these axes when the material cools through its Curie
point are clearly e111 = (3/2)A111 and e100 = (3/2)A 100 .
In this equation 131,132,133 are the direction cosines, relative to the field direction,
in which the saturation magnetostriction is measured, while (Xl' (X2, (X3 are the
direction cosines, relative to the field direction, of the axis along which the
magnetic moments are saturated. These equations of course only apply to the
magnetostriction within a domain. The anisotropic magnetostriction equations
for cubic and other crystal classes have been given by Lee [19].
Usually we will wish to know the saturation magneto stricti on in the same
Table 5.2
Magnetostriction coefficients of cubic materials
Material
Iron
Nickel
Terfenol
21
-46
-21
-24
90
1600
M agnetostriction
103
direction as the field in which case the above expression reduces to
As = A100 + 3(A111 - A100)(0(/0(/ + 0(/0(/ + 0(/0(/).
Once again as in the case of isotropic materials the two constants A100 and All1
can be determined by saturating the magneto stricti on along the axis of interest and
then at right angles. The difference in strain remains (3/2)A100 and (3/2)A111
depending on the axis chosen.
The behaviour of the magnetostriction of an assembly of domains, a polycrystal
for example, can only be calculated by averaging the effects. This is not possible in
general and therefore it is assumed that the material consists of a large number of
domains and hence that the strain is uniform in all directions. In a randomly
oriented polycrystalline cubic material (i.e. one in which there is no preferred grain
orientation) this formula simplifies further to become,
As = ~A10
+ -tA 1l1 .
5.3.6 Field-induced magnetostriction
How does the length of a ferromagnet change with magnetic field?
The field-induced magneto stricti on is the variation of Awith H or B and is often the
most interesting feature of the magnetostrictive properties to the materials
scientist. However the variations A(H) or A(B) are very structure sensitive so that it
is not possible to give any general formula for the relation of magnetostriction to
field. This may at first seem strange when we have already been able to give
equations for the saturation magnetostriction.
The magnetostrictions of polycrystalline iron, cobalt and nickel are shown in
Fig. 5.7 which indicates some of the problems immediately, since for example the
bulk magnetostriction of iron actually changes sign from positive to negative as H
or B is increased. Lee [19] has reported on the magneto stricti on curves of
polycrystalline ferromagnets.
There is one case at least where a fairly simple solution occurs. If the magnetic
field is applied in a direction perpendicular to the easy axis in a single crystal with
uniaxial anisotropy, or perpendicular to the axis in which the moments have been
completely aligned in a polycrystalline material such as nickel under extreme
tension or terfenol under compression.
In this case magnetization takes place entirely by rotation of magnetization. So
we may make the substitution
M=Mscose,
e
which gives the magnetization along the field axis in terms of the angle which the
saturation magnetization within the domains makes with this axis. The
magneto stricti on along the field axis is given by
A= ~AsCOS2
e
104 Magnetic properties
120
10
1.4
0
iD -10
Fe
-< -20
Co
Q
-30
-40
1.2
1.0
rfj
b .8
0 100
300
500
700
H(Oel
-<
.6
(aj
o. 6.9MPa
b. 12.4MPa
c. 17.9MPa
d. 24.1 MPa
.4
.2
500
(b)
1000
1500
H(Oel
Fig. 5.7 Dependence of the bulk magnetostriction on applied magnetic field in: (a) iron,
nickel and cobalt; and (b) a highly magnetostrictive rare earth-iron alloy
Tbo.27DYo.73Fel.95
and substituting for cos 2 e leaves
which gives the variation of the observed magnetostriction with magnetization M.
5.3.7 Transverse magnetostriction
When the length of a ferromagnet changes what happens to the cross section?
Between the demagnetized state and saturation magnetization the volume of a
ferromagnet remains fairly constant (although there is a very small volume
magnetostriction in most materials). Therefore there is a transverse
magnetostriction of one half the longitudinal magnetostriction and opposite in
sign.
At = - ,1/2.
REFERENCES
1. Frohlich, O. (1881) Electrotech Z., 2, 134.
2. Kennelly, A. E. (1891) Trans. Am. lEE., 8, 485.
Examples and exercises
105
Weiss, P. (1910) J. Phys., 9, 373.
Becker, R. and Doring, W. (1938) Ferromagnetismus, Springer, Berlin.
Mee, C. D. (1964) The Physics of Magnetic Recording, North Holland, Amsterdam.
Rayleigh, Lord (1887) Phil. Mag., 23, 225.
Bozorth, R. M. (1951) Ferromagnetism, Van Nostrand, New York.
Barkhausen, H. (1919) Phys. Z., 29, 401.
Schroeder, K. and McClure, J. C. (1976) The Barkhausen effect, CRC Critical Reviews of
Solid State Science, 6, 45.
10. Matzkanin, G. A., Beissner, R. E. and Teller, C. M. (1979) The Barkhausen effect and its
application to NDE, SWRI Report No. NTIAC-79-2, Southwest Research Institute,
San Antorio, Texas.
11. Lord, A. E. (1975) Acoustic emission, in Physical Acoustics, Vol. XI (ed. W. P. Mason
and R. N. Thurston), Academic Press, New York.
12. Ono, K. and Shibata, M. (1980) Materials Evaluation, 38, 55.
13. Kusanagi, H., Kimura, H. and Sasaki, H. (1979) J. Appl. Phys., SO, 1989.
14. Buttle, D. J., Scruby, C. B., Briggs, G. A. D. and Jakubovics, J. P. (1987) Proc. Roy. Soc.
Lond., A414, 469.
15. Joule, J. P. (1842) Ann. Electr. Magn. Chem., 8, 219.
16. Joule, J. P. (1847) Phil. Mag., 30, 76.
17. Kittel, C. (1987) Introduction of Solid State Physics, 6th edn, Wiley, p. 248, New York.
18. Lee, E. W. (1955) Magnetostriction and magnetomechanical effects, Reports on Prog. in
Phys., 18, 184.
19. Lee, E. W. (1958) Magnetostriction curves of polycrystalline ferromagnets, Proc. Phys.
Soc., 72, 249.
3.
4.
5.
6.
7.
8.
9.
FURTHER READING
Cullity, B. D. (1971) Fundamentals of Magnetostriction, Int. J. Magnetism, 1, 323.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading. Mass.
Heck, C. (1974) Magnetic Materials and their Applications, Crane Russak & Co., New York,
Ch.4.
EXAMPLES AND EXERCISES
Example 5.1 Determination of Rayleigh coefficients at low fields. Using the
results for the initial magnetization of iron given in Table 5.3, plot the initial
magnetization curve. Estimate the extent of the Rayleigh region. Calculate the
Rayleigh coefficients and use them to determine the remanence that would be
observed if the field were reduced to zero after reaching a maximum value of
(a) 10 Aim (b) 20 Aim. Find the hysteresis loss when the field is given one complete
cycle at an amplitude of 20 Aim.
Example 5.2 Magnetostriction of terfenol without stress. In polycrystalline terfenol which is not subjected to stress and does not have a preferred orientation, the
domains are aligned randomly. If .1111 = 1600 X 10- 6 and .1100 = 90 X 10- 6
calculate the saturation magnetostriction along the field direction on the basis of
initial random alignment of domains.
For the same specimen of terfenol calculate the 'saturation magneto stricti on'
All - A.L obtained by applying a saturating field at right angles to a given direction
106
Magnetic properties
Table 5.3 Magnetic properties of annealed
iron
H
(Aim)
(T)
0
5
10
15
20
40
50
60
80
100
150
200
500
1000
10000
0
0.0019
0.0042
0.0069
0.010
0.Q28
0.043
0.095
0.45
0.67
1.01
1.18
1.44
1.58
1.72
B
and setting the strain at an arbitrary value of zero, then rotating the field into the
direction of interest. How does this value of magnetostriction compare with the
saturation magneto stricti on obtained by the first method?
Example 5.3 Magnetostriction ofterfenol under compressive load. By applying a
uniaxial compressive stress to terfenol the moments can also be made to line up
perpendicular to the stress axis. Using the saturation magnetostriction calculated
above for the non-oriented terfenol determine how the magnetostriction of stressed terfenol varies with magnetization (assume the stress is sufficient to align all
domains at right angles to the stress axis) and hence find the magnetostriction at
M = Ms/2 and at M = Ms. How does this compare with All - A1- which was
calculated in exercise 5.2?
Discuss how the magneto stricti on along the field axis would be changed if
instead of having a non-oriented sample we had one with the (111) directions
aligned preferentially along the field axis.
6
Magnetic Domains
We now consider the organization of the magnetic moments within ferromagnets.
Two questions arise: are the magnetic moments permanent or field induced, and
are they randomly aligned or ordered? It is shown that the moments are permanent
and are aligned parallel within volumes containing large numbers of atoms, but
these 'magnetic domains' are still on the microscopic scale in most cases.
6.1
DEVELOPMENT OF DOMAIN THEORY
What microscopic theories are needed to account for the observed macroscopic
properties of ferromagnets?
On the macroscopic scale the bulk magnetization Mis clearly field induced. On the
microscopic scale we need to find how the magnetization varies within a
ferromagnet. In the demagnetized state for example is the magnetization
everywhere zero or are there large local values of magnetic moment which sum to
zero on the macroscopic scale? If there are large local magnetic moments we need
to find how these are arranged and what happens to them when subjected to a field.
6.1.1
Atomic magnetic moments
Do the atoms ofaferromagnet have permanent magnetic moments or are the moments
induced by the magnetic field?
Since atoms are the units from which solids are composed it is reasonable to
suppose that when a ferromagnetic solid is magnetized there is a net magnetic
moment per atom. For example in Chapter 4 we have calculated the net magnetic
moment per atom in saturated iron. There are two possible origins for the atomic
magnetic moments in ferromagnets. The material could already have small atomic
magnetic moments within the solid which are randomly aligned (or at least give a
zero vector sum over the whole solid) in the demagnetized state but which became
ordered (or aligned) under the action of a magnetic field. This was first suggested by
Weber [1]. Alternatively the atomic magnetic moments may not exist at all in the
demagnetized state but could be induced on the application of a magnetic field as
suggested by Poisson [2].
108
Magnetic domains
The existence of saturation magnetization and remanence support the former
idea, and in fact it has been established beyond doubt that in ferromagnets
permanent magnetic moments exist on the atomic scale and that they do not rely
on the presence of a field for their existence. The origin of the atomic moments was
first suggested by Ampere [3] who with extraordinary insight, suggested that they
were due to 'electrical currents continually circulating within the atom'. This was
some seventy-five years before 1. 1. Thomson discovered the electron and at a time
when it was not known whether charge separation existed within an atom or even
whether atoms existed.
We should note that both paramagnets and ferromagnets have permanent
atomic magnetic moments. Next it is necessary to distinguish between them on a
microscopic scale knowing that on the macroscopic scale the permeabilities of
ferromagnets are much higher.
6.1.2 Magnetic order in ferromagnets
Are theferromagnets already in an ordered state before a magnetic field is applied or
is the order induced by the field?
Ewing [4] followed the earlier ideas of Weber in explaining the difference between
a magnetized and demagnetized ferromagnet as due to the atomic moments (or
'molecular magnets' as they called them in those days) being randomly oriented in
demagnetized iron but aligned in the magnetized material. Ewing was particularly
interested in explaining hysteresis on the basis of interactions between the atomic
dipole moments of the type envisaged by Weber. As we shall see this was inevitably
to fail because he did not realize that the demagnetized iron was actually already in
an ordered state with large numbers of atomic magnetic moments aligned locally in
parallel.
6.1.3 Permeability of ferromagnets
Can the properties of ferromagnets be explained best by assuming that a magnetic
field rearranges existing ordered-volume magnetic moments or by assuming that the
field aligns disordered (randomly oriented) atomic magnetic moments?
One of the problems in the field of magnetism that needed to be addressed was the
very large permeabilities and susceptibilities offerromagnets. In their original state
the bulk magnetization of ferro magnets is zero, but on application of a magnetic
field they become 'magnetically polarized' that is they acquire a magnetization.
However the magnetizations of ferromagnets are mostly orders of magnitude
greater than the field strengths which produce them.
There are two possible explanations: either the atomic magnetic moments are
randomly oriented on the interatomic scale and the field gradually aligns them as
in the case of paramagnets. Alternatively, the moments could be aligned on the
Development of domain theory
109
interatomic scale but at some larger scale the magnetizations of whole aligned
regions, known as domains, could be randomly aligned from one domain to the
next.
The properties of ferromagnets can be explained if long-range magnetic order
exists within the solid but the volumes, or domains, containing the magnetic
moments are randomly aligned in the demagnetized state. Magnetization is then
simply the process of rearranging these volumes so that their magnetizations are
aligned parallel. The paramagnets, which also have permanent atomic magnetic
moments, can then be distinguished from the ferromagnets because the
paramagnets do not exhibit long-range order such as is found in ferromagnetic
domains. In fact in paramagnets the atomic magnetic moments are randomly
aligned in the absence of a field due to the thermal, or Boltzmann, energy.
6.1.4 Weiss domain theory
If the atomic magnetic moments are ordered then how do we explain the demagnetized
state?
Some years after Ewing's work one of the most important advances in the
understanding of ferromagnetism was made by Weiss in two papers in 1906 and
1907 [5,6]. In these papers Weiss built on the earlier work of Ampere, Weber and
Ewing and suggested the existence of magnetic domains in ferromagnets, in which
the atomic magnetic moments were aligned parallel over much larger volumes of
the solid than had previously been suspected. In these domains large numbers of
atomic moments, typically 10 12 to 10 15 , are aligned parallel so that the
magnetization within the domain is almost saturated. However the direction of
alignment varies from domain to domain in a more or less random manner,
although certain crystallographic axes are preferred by the magnetic moments,
which in the absence of a magnetic field will align along one of these equivalent
'magnetic easy axes'.
The immediate consequences of this were (a) atomic magnetic moments were in
permanent existence (Weber's hypothesis), (b) the atomic moments were ordered
(aligned) even in the demagnetized state, (c) it was the domains only which were
randomly aligned in the demagnetized state, and (d) the magnetization process
consisted of reorienting the domains so that either more domains were aligned with
the field, or the volumes of domains aligned with the field were greater than the
volume of domains aligned against the field.
6.1.5 Weiss mean field theory
What is the underlying cause of the alignment of atomic magnetic moments?
If the atomic moments are aligned within the domains of ferromagnets it is
necessary to explain this ordering and if possible to explain why when a
110
Magnetic domains
ferromagnet is heated up it eventually undergoes a transition to a paramagnet at
the Curie temperature.
In order to explain these observations Weiss further developed the statistical
thermodynamic ideas of Boltzmann and Langevin as they applied to magnetic
materials. Some years previously Langevin [7J had produced a theory of
paramagnetism based on classical Boltzmann statistics. Weiss used the Langevin
model of paramagnetism and added an extra term, the so called Weiss mean field,
which was in effect an interatomic interaction which caused neighbouring atomic
magnetic moments to align parallel because the energy was lower if they did so.
In the original Weiss theory the mean (or 'molecular') field was proportional to
the bulk magnetization M so that
He=rx.M,
where rx. is the mean field constant. This can be proved to be equivalent to assuming
that each atomic moment interacts equally with every other atomic moment within
the solid. This was found to be a viable assumption in the paramagnetic phase
because due to the homogeneous distribution of magnetic moment directions the
local value of magnetization, obtained by considering a microscopic volume of the
material surrounding a given atomic magnetic moment, is equal to the bulk
magnetization.
However in the ferromagnetic phase the magnetization is locally inhomogeneous on a scale larger than the domain size due to the variation in the direction
of magnetization from domain to domain. Therefore subsequent authors preferred
to apply the idea of a Weiss mean field only within a domain, arguing that the
interaction between the atomic moments decayed with distance and that therefore
such an interaction was unlikely to extend beyond the domain. It is generally
considered that the Weiss field is a good approximation to the real situation within
a given domain because within the domain the magnetization is homogeneous and
has a known value Ms. So the interaction field which is responsible for the ordering
of moments within domains can be expressed as
where Ms is the spontaneous magnetization within the domain which has been
discussed in section 2.1.4; it is equal to the saturation magnetization at 0 K but
decreases as the temperature is increased, becoming zero at the Curie point.
Subsequent models, such as the Ising model [8J applied to ferromagnets, have
been based on interaction fields only between nearest neighbours. This gives rise to
ordering of moments within a domain as shown in Fig. 6.1. When rx. > 0 the
ordering is parallel, leading to ferromagnetism. When rx. < 0 the ordering is
antiparallelleading to simple antiferromagnetism. At this state we should note that
a number of different types of magnetic order are possible depending on the nature
ofthe interaction parameter rx.. Some ofthese configurations are shown in Fig. 6.2.
In Chapter 9 we will discuss the Langevin and Weiss theories in detail, showing
Development of domain theory
111
(a)
(b)
Fig. 6.1 Ordered arrangement of a linear chain of atomic magnetic moments when 0( > 0
leading to ferromagnetism as shown in (a), and 0( < 0 leading to simple antiferromagnetism,
as shown in (b).
r
1
1111
Simple ferromagnet
1
I\I\I
Canted antiferromagnet
!1!1
~B
Simple antiferromagnet
l
I I I
l
l
Fernmagnet
Helical spin array
Fig. 6.2 Examples of different types of magnetic order using a linear array of localized
moments. These include ferromagnetism, simple antiferromagnetism, ferrimagnetism and
helical antiferromagnetism.
derivation of the equations and how the Curie temperature is determined by the
mean field parameter. We will also leave the explanation of the Weiss mean field
until later, although we should note in passing that it cannot be explained in
classical terms and depends entirely on quantum mechanical considerations. In
quantum mechanics it is known as the exchange interaction and we shall refer to it
as such occasionally.
Example 6.1 The equivalent field strength of the Weiss mean field. Suppose the
field experienced by any magnetic moment m i within a domain due to its
interaction with any other moment mj is He = (Xijmj" (a) Find the field experienced
by this moment as a result of its interactions with all other moments. (b) The
energy of the moment as a result of this interaction. (c) If each moment interacts
equally with all other moments within the domain show that this is equivalent
112 Magnetic domains
to a mean field. (d) Calculate the field strength if the mean field parameter in
iron is iY. = 400 in SI units (=400 x 4n in CGS units).
If the interaction with one moment is
He = iY.ijmj
then the interaction with all moments is simply the sum over the moments within
the domain,
He = LiY.ijmj
and the energy of the moment will be
E = - Jiom i•He
= - Jiom/LiY.ijmj.
If the interactions with all moments are equal then all the iY. ij are equal. Let these
be iY.
He = iY.Lmj
the vector sum over all the moments within a domain gives the spontaneous
magnetization Ms
which is equivalent to a mean field.
It is known that Mo = 1.7 X 106 A/m, and at room temperature the spontaneous
magnetization Ms in iron may be taken as almost equal to the saturation
magnetization Mo. If iY. = 400 in iron it follows that the mean field is
He=iY.Ms
=(400)(1.7 x 106)A/m
= 6.8 x 10 8 A/m
which is equivalent to a magnetic induction of 855 tesla.
Notice in particular the size of this field. Remember that a standard laboratory
electromagnet generates an induction of typically 2 tesla. The expected field based
on simple classical dipole interactions between the moments in a solid is of the
order of 8 x 104 A/m or equivalently 0.1 tesla. The exchange field is therefore
several orders of magnitude greater than expected classically.
6.1.6 Energy states of different arrangements of moments
Ifbar magnets in a linear chain prefer to align antiparallel why do atomic magnetic
moments align parallel?
Consider the two configurations of magnetic moments in Fig. 6.3. It can easily be
shown that for iY. > 0 that the configuration with all moments aligned parallel is a
lower energy stete than the configuration with one moment antiparallel.
Development of domain theory
113
I I I II I I I I I I I
---l
la
f-
(a)
(b)
Fig. 6.3 Two possible configurations of a linear array of magnetic moments. When IX> 0 the
parallel alignment is the ground state.
If we only consider the exchange energy of the six-moment system shown in
Fig. 6.3 the energy of any moment mi is simply
E i = - Ilom;" r. riijm j
with the mean field approximation
The total energy is therefore
E = - IlOrir.m i • r.m j'
When all moments are parallel
E = -ll ori(6m)(5m)
= - 30 m2llori.
With one moment antiparallel
E = -ll ori(5m3m - m5m)
= - 10 m2llori.
Therefore as a result of the positive exchange interaction the energy is lower
when all moments are aligned parallel within the domain and hence the aligned
state is preferred.
6.1.7 Early observational evidence of domains
If these magnetic domains exist how can we see them?
There were two important experimental observations in the years after Weiss'
work which served to confirm the essential correctness of his theories. There have
subsequently been innumerable observations of ferromagnetic domains, both
direct and indirect. The first confirmation was the indirect detection of domains by
the Barkhausen effect [9], in which the reorientation of domains caused discrete
changes in magnetic induction within a ferromagnet which could be detected by
suitable amplification of the signals from a search coil wound around the specimen.
The Barkhausen effect leads to discontinuities in other bulk properties if they are
114
Magnetic domains
measured accurately enough. Recent discontinuities in magneto resistance have
been reported, for example [10J, and of course acoustic emissions can be generated
as discussed in section 5.2.2.
The second confirmation was by direct observations of domain patterns on the
surfaces of ferromagnetic materials made by Bitter in [11]. He used a very fine
magnetic powder suspended in a carrier liquid which was spread on the surface of
the material. Patterns were observed in the particle accumulations when viewed
under a microscope. The particles accumulate at positions where the magnetic field
gradient is greatest. This occurs where domain walls intersect the surface. It seems
likely that the idea of doing this had come from the magnetic particle inspection
techniques of Hoke and DeForest [12J which had been developed a few years
before.
More recently colloidal solutions of ferromagnetic particles in a carrier liquid
('ferrofluids') have been used. The particles are usually Fe 3 0 4 and the
commercially available ferrofluids usually have to be diluted for the best
observation of domains. In order to produce optimum surface conditions for
domain observations the surface of the material should be electropolished to
remove strains which would otherwise reduce the size of domains. Some Bitter
patterns produced by magnetic colloids such as ferrofluid on the surface of iron
from the work of Williams, Bozorth and Shockley [13J are shown in Fig. 6.4.
Although the Bitter method is a mature technique, and consequently not many
new developments are reported now, one new variation of this method has evolved
(a)
(b)
Fig. 6.4 (a) Magnetic domains in the surface of iron observed using the Bitter method
(magnification x 120), after Williams et al. [13], (b) interpretation of domain pattern in (a)
showing the direction of the spontaneous magentizations within the domains.
Development of domain theory
115
recently [14]. In this the conventional Bitter colloid pattern was observed in
polarized light in order to study stray field-induced birefringence on the surfaces of
magnetic materials. Results on garnets were reported.
6.1.8
Techniques for domain observation
What other methods are available for looking at domains?
Several other techniques are regularly used for the observation of domains. Two of
these are related optical methods the Faraday and Kerr effects, in which the axis of
polarization of a linearly polarized light beam is rotated by the action of a magnetic
field.
The rotation of the direction of polarization of a polarized light beam reflected
from the surface of a magnetic material is known as the Kerr effect [15]. The angle
of rotation of the axis of polarization is dependent upon the magnitude and
direction of the magnetization M at the surface of the material. This is determined
by the domain configuration in the surface and hence an image of the domain
structure at the surface can be formed. One of the difficulties with the Kerr effect is
that the angle of rotation is usually very small so that there is little contrast between
the different domains.
There are three types of Kerr effect which can be observed depending on the
relative orientation of the magnetization with respect to the plane of incidence of
the light beam and the plane of the reflecting surface. These are known as the polar,
longitudinal and transverse Kerr effects [16] and are shown in Fig. 6.5. In the polar
Kerr effect, Fig.6.5(a), the domain magnetization M has a component
perpendicular to the surface of the specimen. In this case the angle of rotation is
largest and may be up to 20 minutes of arc. However the demagnetizing energy
strongly favours the alignment of magnetization in the plane of the surface, so
unless there is sufficient anisotropy to maintain a component of M perpendicular
to the surface this method can not be used. In the longitudinal Kerr effect,
Fig. 6.S(b), the domain magnetization M lies in the plane of incidence and also in
the plane of the surface. In this case the rotation of the direction of polarization is
much smaller, typically being up to 4 minutes of arc. The maximum rotation in the
longitudinal Kerr effect is obtained at an angle of incidence of 60°. In the transverse
Kerr effect, Fig. 6.S(c), Mlies in the surface plane but is perpendicular to the plane
of incidence. This leads to an effective rotation of the direction of polarization, as
described originally by Ingersoll [17]. This is comparable in magnitude to the
rotation observed in the longitudinal Kerr effect.
Hubert and coworkers [18] have developed a new technique based on the Kerr
effect which uses digital image processing based on a combination oflongitudinal
and transverse Kerr effect measurements. This enables quantitative determination
of the magnetization and its direction in the domain patterns of soft magnetic
materials.
The Faraday effect [19], which is less useful for domain observations, is similar
116
Magnetic domains
POLAR EFFECT
z
a)
y
o
LONGITUDINAL
EFFECT
(b)
x
TRANSVERSE EFFECT
(c)
Fig. 6.5 Arrangement for polar, longitudinal and transverse Kerr effect observation of
surface domain structures showing the relative orientation of the magnetization M with
respect to the incident linearly polarized light beam. The plane of incidence is the X -Z plane
and the surface is in the X - Y plane.
except that the rotation of the axis of polarization is caused during transmission of
a polarized light beam through a ferromagnetic solid. This can therefore only be
applied to a thin transparent sample of a ferromagnetic material and is hence
restricted to thin slices of ferromagnetic oxide or to metal films. In both cases the
resulting beam is analysed using a second polarizing filter which then reveals the
location of various domains as either light or dark regions in the polarized image.
A combination of magneto-optic Faraday and Kerr effects with the Bitter
pattern technique has been reported by Hartmann [20]. This method provides a
high contrast image of the surface domain pattern and can be used for both
transparent and opaque magnetic materials. It is known as the interference
contrast colloid technique.
The most recent development in the general area of magneto optic methods for
domain observations is the laser magneto-optic microscope (LAMOM) of Argyle
and Herman [21]. This is a highly sophisticated application of the Kerr effect
which has been used for domain studies in read and write heads for magnetic
recording devices [22].
Another method of domain observation is transmission electron microscopy,
TEM [23], also known as Lorentz microscopy, in which the specimens are usually
in the form of thin films. The transmitted electrons are deflected by the local
magnetic field gradients in the material and this can be used to produce an image of
the domain structure [24,25]. The angular deflection of the electrons in
Development of domain theory
117
ferromagnetic materials such as iron is typically 0.01°. In order to obtain images
normal bright-field imaging conditions cannot be used. Instead the image must be
either under- or over-focused, since as the focus is changed the images of domains
with different magnetizations move in different directions. The domain walls then
either appear as dark or bright lines. The TEM pictures obtained under these
conditions reveal only the domain walls.
The force on an electron as it passes through the material is
F= -f.1oev x M,
where M is the local magnetization and v is the velocity. For 100 ke V electrons
the maximum thickness of the ferromagnetic material that can usefully be
observed is about 200 nm. This method is capable of a spatial resolution of 5 nm
however one problem is that the electron beam uses a strong magnetic field to
focus it and therefore a ferromagnetic specimen cannot be placed in the usual
location in the microscope without some magnetic shielding, otherwise the
domain structure will be disturbed. An alternative is to move the specimen from
the normal position inside the objective lens.
Scanning electron microscopy (SEM) can also be used for domain imaging [26].
This has the added advantage over TEM that it can be used to image domains in
thick specimens. Two methods are used in SEM, one which depends on the
deflection of incident electrons after they enter the material, so that the number of
back scattered electrons at a given angle is dependent on the direction of
magnetization within a domain [27]. The second method depends on the
deflection of secondary electrons by stray magnetic fields near the specimen surface
[28].
X-ray topography is another technique of domain imaging [29]. It is well known
that the diffraction of X-rays by a solid is dependent on the lattice spacing. In
ferromagnets because of spontaneous magnetostriction the lattice parameter is
dependent upon the direction of magnetization within the domain. This means
that domains with magnetic moments aligned at angles other than 180 0 can be
distinguished on the basis of Bragg diffraction of X-rays. This method can be used
on thick specimens but the diffraction of X-rays is affected by dislocations and
grain boundaries and therefore the method is best suited to fairly pure singlecrystal materials with few dislocations or other defects.
Another method which has been reported in recent years is neutron diffraction
topography [30]. The neutron carries a magnetic moment but no charge and
therefore can be affected by a magnetic field or the magnetization within a domain.
In this method a beam of polarized neutrons passes through a specimen and the
angle of rotation of polarization is detected. The results are then used to
reconstruct an image of the domain pattern in the material.
A very recent technique for imaging the domains is the atomic force method. At
this time there is little information available although there have been brief reports
in conference proceedings [31,32].
118
6.2
6.2.1
Magnetic domains
ENERGY CONSIDERATIONS AND DOMAIN PATTERNS
Existence of domains as a result of energy minimization
Why do domains exist at all? Surely the moments should simply be aligned throughout
the solid?
Since Weiss has shown that there exists an interaction field between the atomic
moments within a ferromagnet which causes alignment of the magnetic moments,
we are left once again with the question why a ferromagnet is not spontaneously
magnetized, or if you wish, why the domains themselves are not aligned
throughout the whole volume of the solid.
Clearly Weiss, in postulating the existence of domains, had sought to provide
an empirical explanation of why the mean field did not lead to spontaneous magnetization of the material. It was left to Landau and Lifschitz
in 1935 [33J to show that the existence of domains is a consequence of energy
minimization. A single domain specimen has associated with it a large magnetostatic energy, but the breakup of the magnetization into localized regions
(domains), providing for flux closure at the ends of the specimen, reduces the
magnetostatic energy. Providing that the decrease in magneto static energy is
greater than the energy needed to form magnetic domain walls then multi-domain
specimens will arise.
6.2.2 Magnetostatic energy of single-domain specimens
What is the self energy of a single-domain particle?
The energy per unit volume of a dipole of magnetization M in a magnetic field His
given by
f
E= -flo H ' dM,
as given earlier in section 1.2.4. When it is subjected only to its own demagnetizing
field H d, which is of course generated by M anyway, we can put Hd = - N dM in the
integral where N d is the demagnetizing factor so that the energy becomes
E=
-
f
flONd M·dM
E=flo N M2
2
d
•
The calculation of the energy of a multi-domain specimen is more complex and
we shall consider it later, but we can note immediately that if M can be reduced by
the emergence of domains the magnetostatic self energy will be reduced. The size of
domains is determined by another factor which we have not considered yet: the
domain wall energy.
Energy considerations and domain patterns
119
6.2.3 Domain patterns and configurations
How do the domains arise from a single-domain specimen as the magnetic field is
reduced?
Many direct observations of domain patterns have been made by Bitter pattern
techniques or by other methods such as the magneto-optic Kerr and Faraday
effects. Figure 6.6 shows diagrammatically the emergence of domains as a sample
which is originally in the saturated condition is demagnetized. The lower diagram
shows a closure domain at the end of a single crystal of iron. Closure domains
usually emerge fairly early in the demagnetizing process since they provide return
paths for the magnetic flux within the solid. They are nucleated by defects including
the boundary of the material. They are also usually the last domains to be swept
out at higher fields.
y
,I , I'
t ,
t
,
I
I
'lti'it
, I
,
,
I
,
,
,
0r"y
, ,I ,:
:
f:'lti'
,, ,I ,I
I
I
I
1• ...' .I.,,
,
~
(8)
(C)
(0)
(E)
Fig. 6.6 Changing domain patterns as a sample of single-crystal iron is demagnetized (after
Coleman, Scott and Isin).
120
Magnetic domains
6.2.4 Magnetization process in terms of domain theory
What changes occur within domains as a magnetic field is applied and gradually
increased?
Since according to the domain theory the atomic magnetic moments are ordered
even in the demagnetized state in a ferromagnet, the difference between the
demagnetized state and the magnetized state must be due to the configuration of
the domains.
When a magnetic field is applied to a demagnetized ferromagnetic material the
changes in magnetic induction Bwhen traced on the B, H plane generate the initial
magnetization curve. At low fields the first domain process occurs which is a
growth of domains which are aligned favourably with respect to the field according
to the minimization of the field energy E = - JloMs' H and a consequent reduction
(a)
(b)
(c)
(d)
Fig. 6.7 Domain processes occurring as a material is magnetized to saturation; from the
demagnetized state (a) to partial magnetization (b) by domain wall movement; from partial
magnetization to the knee of the magnetization curve (c) by irreversible rotation of domain
magnetization, from the knee of the magnetization curve to technical saturation (d) by
reversible rotation.
Energy considerations and domain patterns
121
in size of domains which are aligned in directions opposing the field, as shown in
Fig. 6.7.
At moderate field strengths a second mechanism becomes significant; this is
domain rotation, in which the atomic magnetic moments within an unfavourably
aligned domain overcome the anisotropy energy and suddenly rotate from their
original direction of magnetization into one of the crystallographic 'easy' axes
which is nearest to the field direction.
The final domain process which occurs at high fields is coherent rotation. In this
process the magnetic moments, which are all aligned along the preferred magnetic
crystallographic easy axes lying close to the field direction, are gradually rotated
into the field direction as the magnitude of the field is increased. This results in a
single-domain sample.
6.2.5 Technical saturation magnetization
Why is the magnetization within the domains not equal in magnitude to the saturation
magnetization?
,
-
•
t
..-
t
"
"
/
,/
/
,
\
\
\
I
I
I
I
/
I
I
"\
\
;'
I
I
I
I
...
•
(h)
(al
V Xl
"- "-
•
~,'
"-
""-
\
"
\
\
\
\
\
"-
+ + + +
+ + + +
+ + + +
(e)
+ + +
(d)
Fig. 6.8 Alignment of individual magnetic moments within a domain at various
temperatures: (a) above the Curie point showing random alignment; (b) below the Curie
point; (c) at low temperatures in which the magnetic moments precess about the field
direction in low-level excited states; and (d) perfect alignment at 0 K where there is no
thermal energy for precession.
122
Magnetic domains
When all domains have been aligned with their spontaneous magnetization
vectors parallel to the field the material consists of a single magnetic domain and is
said to have reached technical saturation magnetization. If the magnetic field is
increased further it is noticed however that the magnetization does continue to
increase very slowly. This is due to an increase in the spontaneous magnetization
Ms within the domain as the atomic magnetic moments within the single domain
which are not perfectly aligned with the field because the thermal activation, are
brought into complete alignment.
The spontaneous magnetization is temperature dependent as we have noted
earlier. At 0 K it is equal to the saturation magnetization but decays to zero at
the Curie point. At temperatures above 0 K the individual magnetic moments
have thermal energy which causes them to precess about the field direction as
shown in Fig. 6.8. The precession becomes greater as the temperature increases.
It is the precession which causes the spontaneous magnetization to be smaller
than the saturation magnetization. Ultimately when all of the magnetic moments
within the domain are completely aligned parallel due to a very high magnetic field
the magnetization reaches Mo.
6.2.6
Domain rotation and anisotropy
How does anisotropy affect the ability of domains to change the direction of their
magnetization?
Although the domain growth process is not much affected by anisotropy, both the
incoherent (irreversible) rotation and coherent (reversible) rotation are determined
principally by the magnetocrystalline anisotropy. The rotation ofthe domains can
be considered as a competition between the anisotropy energy Ea(l}, 4» and the field
energy EH giving a total energy E tot of
E tot = Ea(f}, 4»
+ EH
where
EH = - l1 oM s ·H.
Ms is the magnetization within a domain, which is often replaced by the saturation
magnetization Mo of the material for the purpose of calculation.
6.2.7
Hexagonal anisotropy
What is the simplest representation of hexagonal (i.e. uniaxial) anisotropy?
In this case the anisotropy can be represented by the one-constant approximation
Ea = Kul sin 2 4>,
where 4> is the angle of the magnetization with respect to the unique axis, which for
K > 0 is the easy axis, while for K < 0 it is the hard axis. In the case of single-crystal
cobalt which is hexagonal Kul = 4.1 X 10 5 11m 3 •
Energy considerations and domain patterns
123
Example 6.2 Anisotropy 'field'. Calculate the anisotropy field for a cobalt crystal
whose magnetization is close to the hexagonal axis. Use Ms = Mo = 1.42
X 106 A/m and Ku1 = 4.1 X 10 5J/m 3. Assume that for small angles Ea = Ku1 ¢2
and that EH = constant + (1/2)lloH¢2.
The anisotropy energy for a uniaxial material such as cobalt is given by
Ea = Ku1 sin 2 ¢
and for small angles ¢ we can put sin 2 ¢ : : : : ¢2,
Ea = KU1¢2.
If a magnetic field is applied along the unique axis then the field energy is given by,
EH =
- lloMsH cos ¢
and for small angles ¢, cos ¢ = 1 - ¢2/2
EH = - lloMsH(l - ¢2/2).
Removing the constant term, which has no bearing on the equilibrium condition
EH = llo M sH ¢2
2
Equating these energies will give the magnetic field along the unique axis which
is equivalent to the anisotropy
K
_llo M sH
u1 2
and hence
Substituting in the values for cobalt
H=
105 )
(4n x 10- 7 )(1.42 x 106 )
H = 4.6
6.2.8
(8.2
X
X
10 5 A/m.
Cubic anisotropy
What is the simplest representation of cubic anisotropy?
In the case of cubic anisotropy a one-constant anisotropy equation can also be
used as a first approximation
Ea = K1 (cos 2 81 cos 2 82
+ cos 2 82 cos 2 83 + cos 2 83 cos 2 8d,
where 81, 82 , 83 are angles which the magnetization makes relative to the three
crystal axes.
124
Magnetic domains
Table 6.1
Anisotropy constants for various ferromagnetic materials
Material
Iron
Nickel
Cobalt
SmCo s
Kl
Cubic
Cubic
Hexagonal
Hexagonal
Kul
K2
(lOS J/m 3 )
0.480
-0.045
0.050
0.023
4.1
1.0
1100
So, for example, if we consider the anisotropy in, say, the (001) plane only, we will
have 83 = n12, and consequently cos8 3 = 0
E(OOI)
= K 1 (cos 2 8 1 cos 2 (
2 ),
Once we have established that we are only considering moments in this plane,
81 = 90° - 82
E(OOI) = (K d4) sin 2 28,
where now 8 = 0 is the direction of the magnetic easy axis within the plane (001)
when K 1 > 0, that is the 100) directions, as in the case of iron for which K 1 =
4.8 x 104 Jim 3 . When K 1 < 0 the magnetic easy axes are (111) as in nickel for
which Kl = -4.5 X 10 3 J/m 3 .
<
REFERENCES
1. Weber, W. (1852) Pogg. Ann., LXXXVII, 167.
2. Poisson, S. D. (1893) in Magnetic Induction in Iron and Other Methods (ed. J. A. Ewing),
Electrician Publishing Company, London, p. 282.
3. Ampere, A. M. (1827) Theorie Mathematique des Phenomenes Electrodynamiques
Uniquement Deduite de I'Experience, Reprinted by Blanchard, Paris (1958).
4. Ewing, 1. A. (1893) Magnetic Induction in Iron and Other Metals, The Electrician
Publishing Company, London.
5. Weiss, P. (1906) Compt. Rend., 143, 1136.
6. Weiss, P. (1907) J. Phys., 6, 661.
7. Langevin, P. (1905) Ann. Chern. et Phys., 5, 70.
8. Ising, E. (1925) Z. Phys., 31, 253.
9. Barkhausen, H. (1919) Physik Z., 20, 401.
10. Tsang, C. and Decker, S. K. (1981) J. Appl. Phys., 52, 2465.
11. Bitter, F. (1931) Phys. Rev., 38, 1903.
12. Betz, C. E. (1967) Principles of Magnetic Particle Testing, Magnaflux Corporation,
Chicago, p. 48.
13. Williams, H. 1., Bozorth, R. M. and Shockley, W. (1949) Phys. Rev., 75, 155.
14. Jones, G. A. and Puchalska, I. (1979) Phys. Stat. Sol., A51, 549.
15. Kerr, 1. (1876) Rep. Brit. Assoc., 5.
16. Carey, R. and Isaac, E. D. (1966) Magnetic Domains and Techniques for their
Observation, Academic Press, Ch. 5. London.
17. Ingersoll, L. P. (1912) Phys. Rev., 35, 315.
Examples and exercises
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
125
Rave, W. Schafer, R. and Hubert, A. (1987) J. Mag. Mag. Mater., 65, 7.
Faraday, M. (1846) Phil. Trans. Roy. Soc., 136, 1.
Hartmann, U. (1987) Appl. Phys. Letts., 51, 374.
Argyle, B. and Herman, D. (1986) IEEE Trans. Mag., MAG-22, 772.
Herman, D. and Argyle, B. (1987) J. Appl. Phys., 61, 4200.
Hale, M. E., Fuller, H. W. and Rubinstein, H. (1959) J. Appl. Phys., 30, 789, and (1960) J.
Appl. Phys., 31, 238.
Hwang, c., Laughlin, D. E., Mitchell, P. V., Layadi, A., Mountfield, K., Snyder, J. E. and
Artman, J. O. (1986) J. Mag. Mag. Mater., 54-7,1676.
Lodder, J. C. (1983) Thin Solid Films, 101, 61.
Mayer, L. (1957) J. Appl. Phys., 28, 975.
Koike, K, Matsuyama, H., Todokoro, H. and Hayakawa, K. (1985) Jap. J. Appl. Phys.,
24, 1078.
Unguris, J., Hembree, G., Celiotta, R. J. and Pierce, D. T. (1986) J. Mag. Mag. Mater.,
54-7,1629.
Polcarova, M. and Lang, A. R. (1962) Appl. Phys. Letts., 1, 13.
Baruchel, J., Palmer, S. B. and Schlenker, M. (1981) J. de Phys., 42, 1279.
Saenz, J. J. and Garcia, N. (1987) J. Appl. Phys., 62, 4293.
Wickramsinghe, H. K and Martin, Y. (1988) J. Appl. Phys., 63, 2948.
Landau, L. D. and Lifschitz, E. M. (1935) Physik Z. Sowjetunion, 8, 153.
FURTHER READING
Argyle, B. and Herman, D. (1986) IEEE Trans. Mag., MAG-22, 772.
Carey, R. and Isaac, E. D. (1966) Magnetic Domains and Techniquesfor their Observation,
Academic Press, London.
Craik, D. J. and Tebble, R. S. (1965) Ferromagnetism and Ferromagnetic Domains, North
Holland, Amsterdam.
Crangle, J. (1977) The Magnetic Properties of Solids, Arnold, London, Ch. 6.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading, Mass.,
Ch.7.
Kittel, C. and Galt, J. K (1956) Ferromagnetic domains, Solid State Physics, 3, 437.
EXAMPLES AND EXERCISES
Example 6.3 Effects of anisotropy on rotation of magnetization. Derive an
expression for the magnetization of a specimen with cubic anisotropy when a field
is applied along the (110) direction and the easy axes are the (100) directions. You
may assume that the material is a perfect single crystal so that the domain wall
motion all occurs at negligibly small fields, leaving the magnetizations within
domains aligned along either of the (100) directions closest to the field direction.
Also you may use the one-constant approximation for the anisotropy.
Example 6.4 Critical fields as determined by anisotropy. A magnetic field is
applied in the base plane of cobalt. The easy direction is the unique axis. Using a
one-constant model for the anisotropy with Kul = 4.1 x lOs J/m 3 and Ms = 1.42
X 10 6 Aim calculate the field needed to rotate the magnetic moments (a) into the
base plane (i.e. perpendicular to the unique axis) and (b) 45° from the hexagonal
axis.
126 Magnetic domains
Example 6.5 Stress-induced anisotropy. If the saturation magnetostriction of a
randomly oriented rod of polycrystalline terfenol is As = 1067 X 10- 6 , and
assuming the anisotropy constant is Kl = - 2 X 104 J/m 3 , *determine the stress
needed completely to align the moments in the easy plane perpendicular to the
stress.
You may use the one-constant approximation to the cubic anisotropy and since
the material has no preferred orientation in the bulk sense, you may use the
expression EiT = - (3/2)AsCTcos 2 () for the stress-induced anisotropy, where CT is the
stress and () is the angle from the stress axis.
*The anisotropy constant of this alloy varies significantly with temperature and alloy composition.
Therefore this value, although in the correct range, should not be assumed to be generally valid for
terfenol.
7
Domain Walls
In this chapter we shall investigate the local magnetic properties in the vicinity of
domain boundaries. These regions are called domain walls and have properties
which are very different from the rest of the domain. Most of the magnetic changes
under the action of weak and moderate magnetic fields occur at the domain walls
and hence an understanding of domain-wall behaviour is essential to describing
the magnetizing process.
7.1
PROPERTIES OF DOMAIN BOUNDARIES
How do the magnetic moments behave in the domain boundaries?
Once we have accepted the idea of domains within ferromagnets there arises the
question of how the moments change direction in the vicinity of the domain
I
N
N
i
I
I
I
I
I
N
Fig. 7.1 Alignment of individual magnetic moments within a 180 domain wall, after Kittel
(1949).
0
128
Domain walls
boundary. There are two possibilities, either the domain boundary is infinitesimal
in width with nearest-neighbour moments belonging either to one domain or the
other. Alternatively there could be a transition region in which the magnetic
moments realign between the domains and therefore belong to neither domain, as
for example in Fig. 7.1.
7.1.1
Bloch walls
How thick are the domain walls?
The existence of these transition layers between domains, in which the magnetic
moments undergo a reorientation, was first suggested by Bloch [1]. The transition
layers are commonly referred to as domain walls or Bloch walls, although we
should note immediately that not all domain walls are necessarily Bloch walls. The
total angular displacement across a domain wall is often 180° or 90°, particularly in
cubic materials because of the anisotropy and, as we shall see, the change in
direction of the moments takes place gradually over many atomic planes.
7.1.2 Domain wall energy
Is there any additional energy associated with the domain wall?
We may define the domain wall energy most conveniently as the difference in
energy ofthe magnetic moments when they are part of the wall and when they are
within the main body ofthe domain. This is usually expressed as the energy per unit
area of the wall.
It is found by considering the changes in energy of the moments in the domain
wall caused by Weiss-type interaction coupling between the atomic magnetic
moments (or exchange interaction energy as it is now known because of its
quantum mechanical origin) and the anisotropy. The anisotropy tends to make the
domain walls thinner because anisotropy energy is lowest with all moments
aligned along crystallographically equivalent axes. The exchange energy tends to
make the walls thicker since the exchange energy in a ferromagnet is minimized
when neighbouring moments are aligned parallel.
Exchange energy
If we consider only the interactions between the magnetic moments these give rise
to the interaction or exchange energy
where He is the interaction field. In the Weiss mean field model we have said that He
is proportional to the magnetization within the domain, that is M s' which at
Properties of domain boundaries
129
temperatures well below the Curie temperature is almost equal to Mo,
conseq uen tly
He = IY.Ms
where IY. is the mean field parameter. If we consider this in terms of an interaction
with the individual magnetic moments m, then, since Ms = Nm, where N is the
number of atoms per unit volume
He = IY.Nm.
In order to consider the domain walls we now wish to go on and look at
situations where only the nearest neighbour interactions are significant. We need
to do this because the directions of the magnetic moments within a domain wall
vary with position and hence the mean field approximation is not valid. We
consider nearest neighbour interactions in order to provide a tractable solution for
the domain wall energy.
The energy per moment due to this interaction is then
Eex = -floIY.Nmi·m j
where m i and mj are neighbouring moments.
Once we allow the moments to have different orientations in the domain wall the
mean field interaction as defined above becomes meaningless. We therefore will
replace it with an analogous interaction" for use between nearest neighbours
only. This will be defined such that the interaction energy per moment becomes
where z is the number of nearest neighbours and the magnetic moments m i and mj
are in units of A m 2. (The approach used here in defining the exchange interaction
in terms of magnetic moment will be useful in introducing the quantum mechanical
exchange in terms of electron spins in section 7.1.4).
In this case the interaction energy per moment becomes dependent on the turn
angle between neighbouring moments. If ¢ is the angle between the neighbouring
moments m i and mj the interaction energy per moment becomes
Eex =
- floZ" m 2 cos¢.
If we consider a linear chain, each moment has two nearest neighbours. Making
the substitution cos¢ = 1 - ¢2/2 for small ¢, the energy per moment becomes
Eex = flo" m 2(¢2 - 2).
The extra interaction energy due to the rotation of the angle of moments within
the wall is therefore the sum of the individual interaction energies over the number
of moments in the wall,
Eex = flo" m 2¢2n,
where n is the number of moments in the wall.
130 Domain walls
For a 180 0 domain wall of thickness n lattice parameters, each of size a, <p = n/n,
and the energy per unit area is then
Eex =
flo/m 2 n 2
2
na
From this result [2] it is clear that the interaction energy is minimized when <p is
very small, corresponding to a very wide domain wall. Therefore the exchange
interaction energy favours wide domain walls.
Anisotropy energy
The anisotropy of a cubic crystal expressed using the one constant approximation,
gives the energy of the pth moment in the wall as
Ea = (Kd4)sin22p<p
and summing this over the width of the domain wall gives [3] the energy per
unit area as
Ea = K1na,
where a is the lattice spacing and n is the number oflayers of atoms in the domain
wall. If Id is the thickness of the wall Id = na.
Ea=Kl/d·
Therefore the anisotropy energy is minimized for a very thin wall.
Energy per unit area of the wall
If the anisotropy and exchange energies are summed this gives
Etot =
flo/m 2 n 2
I
da
+ Kl/d
which is the domain wall energy per unit area.
If the anisotropy is the dominant term then energy is minimized at small Id'
whereas if the exchange is dominant then energy is minimized at large Id . The
domain wall thickness is determined by competing influence of these two factors.
7.1.3
Width of domain walls
How wide are the domain walls and what determines the width?
The width of the domain walls in a ferromagnet is determined by minimizing
energy of the wall with respect to its width. Therefore differentiating the energy
with respect to ld to find the equilibrium, remembering that <p = n/n and ld = na
dEtot = - flo/m 2 n 2 K = 0
dId
Va + 1 .
Properties of domain boundaries
131
This leads to
llofm 2 n 2
K 1a
This is the width of the domain wall which increases with the exchange energy
but decreases with anisotropy. The anisotropy favours alignment of moments only
along the crystallographic easy axes, such as the (100) axes in iron or the (111) axes
in nickel.
Example 7.1 Domain wall thickness in iron. Determine the domain wall width of a
1800 domain wall in iron. For iron K1 = 4.8 X 104 J/m 3 , a = 2.5 A, m = 2.14 Bohr
magnetons (= 1.98 X 10- 23 Am2) and J = 1.9 X 10- 21 Joules (f = 386 X 10 28 ).
If ld is the domain wall thickness then, from the above,
V = (Ilof m 2 n 2IK 1a).
Substituting in the various values
12 _
d
-
V=
(4n x 10- 7 )(386 x 10 28 )(1.98 x 10- 23Vn 2
(4.8 x 104 )(2.5 x 10- 10)
15.59 x 10- 16 m2
Hence,
ld = 3.95 x 1O- 8 m = 395A
=
160 lattice parameters (or atomic layers).
This is an approximate value for the domain wall widths in iron. Actually the
width depends on the type of domain wall. Typical values of domain wall
thicknesses in iron, nickel and cobalt along different crystallographic directions
have been given by Lilley [4].
7.1.4 Interaction energy in terms of electron spin and quantum mechanical
exchange energy
Can the interaction energy be expressed in terms of the exchange interaction between
two electrons on neighbouring atomic sites?
There is an alternative expression for the wall energy and width that you will
encounter more frequently. This is in terms of the quantum mechanical exchange
constant J [5], which is really an alternative representation of our nearestneighbour interaction f, and which also determines the Weiss mean-field coupling
ex where such an approximation is appropriate. The wall energy is then expressed in
132 Domain walls
terms of the exchange constant and the spin on the electrons in the atom S. The spin
on one electron s is always 1/2, which corresponds to a magnetic moment of one
Bohr magneton or 9.27 x 10- 24 Am 2 • In iron each atom has a net spin of S = 1,
which gives 2 Bohr magnetons per atom [6].
In practice the magnetic moments on the atoms are not always integral multiples
of a Bohr magneton because there is a contribution to the magnetic moment from
the electron orbital motion and indeed the magnetic electrons in solids such as
iron, cobalt and nickel may not be localized at the atomic sites which can also lead
to non-integral values of magnetic moment as discussed in Chapter 10. In iron the
moment corresponds closely to a spin of S = 1. However in nickel this is quite
seriously in error since if we take S = 1/2 this indicates a moment per atom of
0.93 J1.B' whereas the true value is 0.6 J1.B. Therefore the use of the quantum
mechanical exchange interaction J, apart from making first principles calculations
for very simple systems, is limited. Practical estimates of the value of J can be made
using the Curie temperature, the low-temperature saturation magnetization or the
specific heat [7,8].
The relation between the classical nearest-neighbour interaction, ,I, and the
quantum mechanical exchange energy, J, is
J1.o,l m 2 = JS 2,
and consequently,
J
,I = 43.5
X
10- 53
and where appropriate, using the mean-field model
J1.oNrxm2 = zJS 2,
where z is the number of nearest-neighbour atoms, S is the total electron spins per
atom and J is the exchange constant (also known as the exchange integral).
For iron, assuming S=1, the exchange constant J=l1.9xlO- 3 eV, or
1.9 x 10- 21 J between each pair of nearest-neighbour atoms, this gives rise to an
exchange energy per atom of Jatom = 8J since iron has z = 8 nearest neighbours
with which it interacts in the b.c.c. crystal structure.
J atom = 95.2
X
10- 3 eV
= 15.2 x 10- 21 J/atom
and a volume energy of N J atom' where N is the number of atoms per unit volume.
(N = 8.58 x 1028 atoms per m 3 in iron.)
J v = 1.30 x 109 Jm- 3 .
The mean-field parameter rx can be related to the exchange coupling since within
the body of a domain we can use the mean-field approximation
J1.oN rxm 2 = zJS 2
Properties of domain boundaries
133
and in the case of iron we have S = 1.
(X
(8)(1.9 x 10- 21 )
ll oN(1.98 X 10- 23 )2
= ------=-=-""""""
15.2 X 10- 21
3.92 x 10 46 lloN
3.88 x 10 25
(8.58 x 10 28 )(4n x 10- 7 )
3.88 x 10 25
108 X 10 21
= 360.
The interaction parameter
f
is given by
llofm 2 =JS 2
1.9 x 10- 21
= (4n x 10 7)(3.92 x 10 46)
f
f = 3.86 X 1030 (dimensionless),
a rather cumbersome number to deal with. In most cases when these types of
interactions are considered the quantum mechanical exchange energy is used. The
drawback, as already mentioned, is that the magnetic moments per atom, whether
localized on the atomic sites or itinerant, are seldom integral numbers of electron
spins.
Example 7.2 Domain wall energy. Estimate the domain wall energy of a 1800
domain wall in iron. For iron K1 =4.8 x 104 J/m 3 , a=2.5A, m=2.14Bohr
magnetons, J = 1.9 X 10- 21 J (f = 3.86 x 1030 ).
Using the equation,
where the wall thickness is 4 x 10 - 8 m
E = (4n x 10- 7)(386 x 1028 )(1.98 x 1O- 23 )2n 2 (48
tot
(4 x 10 8)(2.5 x 10 10)
+ .
= 1.87
= 3.8
X
X
10- 3 + 1.92
10- 3 Jm- 2 •
X
10- 3
10- 8)
104)(4
X
X
134 Domain walls
Table 7.1 Magnetic properties of iron, cobalt and nickel
Magnetic moment per atom [6]
at 0 K (in Bohr magnetons)
Saturation magnetization (in 106 A m - 1)
at OK
at 300K
Curie temperature [6]
(in DC)
(in K)
Co
2.22
1.72
0.62
1.74
1.43
1.40
0.52
0.48
1.71
Exchange energy J [3, 7]
(in J)
(in meV)
Ni
Fe
2.5 x 10- 21
0.015
770
1043
3.2 X 10- 21
0.020
1131
1404
358
631
Anisotropy energy K1 [8,9,10,11] (in Jm- 3)
300K
OK
4.8 x 104
5.7 x 104
Lattice spacing [12] (in nm)
a
c
0.29
Domain wall thickness [3,4]
(in nm)
(in lattice parameters)
40
138
Domain wall energy [6,11]
(in J/m 2 )
3 x 10- 3
45
68
X
104
X
104
0.25
0.41
15
36
8xlO- 3
-0.5
-5.7
X
104
X
104
0.35
100
285
1 X 10- 3
Note: Domain wall thicknesses and energies are approximate values only, since they will depend on the
crystallographic direction of the moments in the domains on either side of the wall. For further details
consult the paper by Lilley [4].
7.1.5 180° and non-180° domain walls
What different types of domain wall exist and how may we classify them?
Generally domain walls can be classified into 180° and non-180° walls. The 180°
domain walls occur in virtually all materials and are distinct from all other non180° walls in that they are not affected by stress [13]. In 180° walls the directions of
magnetization in the neighbouring domains are anti parallel and consequently the
moments in the two domains always lie in equivalent crystallographic directions.
In cubic materials with K > 0 the non-180° walls are all 90°, so that the direction
of the moments in neighbouring domains are at right angles. Therefore in iron the
easy axes are all the <100> directions, and domain walls between the [100] and
[IOO] directions are 180° walls, while those between [100] and [010] are 90°
Properties of domain boundaries
135
<
walls. In nickel for which K < 0 the easy axes are all the 111) directions.
Consequently the non-180° domain walls will be either 71 ° or 109°.
Often all non-180° domain walls are incorrectly referred to collectively as '90°
walls' to distinguish the stress-sensitive from the non-stress sensitive walls.
7.1.6 Effects of stress on 180° and non-180° domain walls
Why are 90° domain walls stress sensitive but 180° domain walls are not?
Suppose for example a uniaxial stress is applied along the [100J direction in
iron. If this is a tensile stress it will make that particular [100J direction lower
in energy than the [OlOJ and the [OOlJ directions which were degenerate in
energy in the unstressed state. However the energy of the [TOOJ direction will
be reduced by the same amount. Consequently, a 180° wall separating domains
aligned along these two directions will not be affected by stress.
Consider now a 90° wall between domains along the [100J and [OlOJ
directions in iron which are energetically equal. Application of tensile stress along
the [l00J direction causes the [100J domain to be energetically favoured, so
the 90° wall will move under the action of such a stress to increase the volume
of the [100J domain at the expense of the [OlOJ domain. It can be seen therefore
that the 90° domain walls are stress-sensitive.
7.1.8
Closure domains
Where do these different types of domain wall occur?
Closure domains occur more often in cubic materials than in hexagonal materials
because the cubic anisotropy ensures that the directions at right angles to the
J
I
J
Fig. 7.2 Closure domains at the surface in a material with cubic crystal structure, in this case
grain-oriented silicon-iron.
136 Domain walls
magnetisation in a given domain are also magnetically easy axes. Therefore it is
energetically favourable to have 90° as well as 180° domain walls. In a material
such as cobalt in which the unique axis is the easy direction it is difficult to form 90°
walls since the magnetic moments in one of the domains must then lie in the
hexagonal basal plane which is energetically unfavourable.
One example of where 90° domain walls occur is in the closure domains of grainoriented silicon iron, Fig. 7.2. Of course there are many other instances of 90° walls,
and they certainly are not dependent on high anisotropy, in fact high anisotropy
impedes the formation of closure domains as in the case of hexagonal materials
such as cobalt. The domain boundaries between the neighbouring longitudinal
domains are 180° walls, while those between the closure domains at the end of the
material and the main longitudinal domains are 90° walls.
7.1.8 Neel walls
Are there other types of domain wall apart from Bloch walls?
If a specimen is in the form of a thin film the ferromagnetic domains can extend
across the whole width of the specimen. In this case the Bloch wall, which would
have its magnetization normal to the plane of the material, as shown in Fig. 7.3,
causes a large demagnetization energy, whereas the Neel wall [14], in which the
moments rotate within the plane of the specimen, results in a lower energy.
Neel walls do not occur in bulk specimens because they generate a rather high
demagnetization energy within the volume of the domain wall. It is only in thin
films that this energy becomes lower than the demagnetization energy of the Bloch
wall.
7.1.9 Antiferromagnetic domain walls
What about domain walls in other ordered magnetic materials?
Bloch wall
Neel wall
Fig. 7.3 A conventional Bloch wall and Neel wall in a thin film offerromagnetic material.
The Bloch wall requires that some ofthe magnetic moments be oriented normal to the plane
ofthe film. This leads to a demagnetization energy associated with the Bloch wall. The Neel
wall has all moments oriented in the plane. The Neel wall is energetically favoured once the
film thickness decreases below a certain critical value.
Domain walls motion
137
Magnetic domain walls occur in all forms of ordered magnetic materials and so
there are correspondingly many different forms of domain walls. Domain walls in
simple antiferromagnets and ferrimagnets are similar to domain walls in
ferromagnets. However the helical antiferromagnet such as occurs in dysprosium
and terbium presents a very interesting case. Domains in these materials were first
suggested by Palmer [15] in which it was suggested that the domains consisted of
helices of different chirality. These were subsequently observed experimentally
[16]. The domain walls are transition regions between helical domains and are
regions in which the neighbouring moments are aligned nearly parallel, or at least
the turn angle between successive moments ¢ is very small compared to within the
domain.
Generally the two senses of helix are energetically equal so that either may occur.
Application of a field perpendicular to the unique axis of the helix can cause
favourably oriented domain walls to grow. Therefore the helical to ferromagnetic
transition, which can be induced by application of a field perpendicular to the
unique axis, is caused by growth of domain walls in this type of magnet.
7.2 DOMAIN WALL MOTION
What changes occur at the domain walls when the magnetic field increases?
In discussing the magnetizing process we have mentioned the rotation of magnetic
moments within a domain as a result of the competing effects of anisotropy and
field energy. We have also touched very briefly on the idea of the growth of
favourably oriented domains without really explaining what is meant by this.
In fact on application of a field it is the moments within domain walls which can
most easily be rotated. This is because the resulting directions of the moments
within the walls are a fine balance between the exchange and anisotropy energies.
So a change in the field energy E = - !lorn· H can alter this balance causing the
moments to rotate. Within the main body of the domain the moments are locked
into a particular direction by the exchange interaction so that an applied field can
not immediately alter the balance of the energy in favour of another direction.
7.2.1
Effect of magnetic field on the energy balance in domain walls
Why do magnetic changes under the action of weak and moderate fields occur at
domain boundaries?
Consider for example Fig, 7.1. If a weak field is applied in the 'up' direction the
moments within the 'down' domain will not change direction because they are at
the bottom of a deep energy well caused by their mutual interactions through the
exchange field. However in the domain wall the energy introduced by the magnetic
field can just tip the balance slightly in favour of the 'up' direction.
The net result is that the moments within the wall rotate slightly into the field
138 Domain walls
direction as the field is increased. To an observer it therefore appears as though
the domain wall moves towards the right. This process is called domain-wall
motion, although in truth there is no translational motion at all. The movement
is more like that of a wave, and in fact Kittel [17J and others have considered
the movement of Bloch walls to be an example of the motion of a soliton, that
is a solitary wave.
In discussing domain-wall motion it is conventional to treat the domain wall as
an entity in itself and to discuss its motion through the material in terms similar to
those used for interfaces such as elastic membranes.
7.2.2 Domain walls as elastic membranes
Is there a simple model that we can use to envisage the behaviour of domain walls?
Since domain walls have an energy associated with them which is proportional to
the area the walls behave in such a way as to minimize their area. We may therefore
consider them to be analogous to the surfaces of liquids where the domain-wall
energy is equivalent to the surface tension of the liquid. This analogy works well in
trying to understand the behaviour of domain walls, particularly the reversible
bowing of domain walls [18,19J under the action of a magnetic field, and their
tendency to become attached to non-magnetic particles, impurities or secondphase materials within the solid [20J and to regions of inhomogeneous microstrain
[21,22].
7.2.3 Forces on domain walls
What is the force exerted on a domain wall by a magnetic field?
If a magnetic field is applied to a ferromagnetic material this results first in the
movement of domain walls, in such a way that domains aligned favourably with
the field direction grow at the expense of those aligned unfavourably. The energy
per unit volume of such a domain when subjected to a field His
EH
= -floMs· H .
Consequently the energy change caused by displacement of a 1800 domain wall
through a distance x is
E = - 2floAMs' Hx,
where A is the area of the wall. Therefore the force per unit area on such a wall is
F= -
(l/A)( :~)
=2floM s· H.
Domain walls motion
139
7.2.4 Planar displacement of rigid, high-energy domain walls: the potential
approximation
How do walls with high surface energy compared to pinning energy behave?
Walls with high surface energy tend to remain planar. Planar domain-wall motion
has been considered by Kersten [23,24]. The movement of planar domain walls
under the action of a magnetic field in a specimen of high-purity iron are shown in
Fig. 7.4. The energy supplied by a magnetic field H to a ferromagnet is given by
~E
= -
f
J10 H·dM,
and consequently if a 180 0 domain wall with unit cross sectional area is moved
through a distance dx, the change in magnetization is
dM=2Msdx.
Therefore the change in field energy is given by,
~EH
f
= -
2J1oMs'H dx
= -
2J1oMs' Hx,
I
(y
I
-H
I=y
I
H
<!)
H=O
I
<0
1 %=-1
I
H-
H
Fig. 7.4 Translational motion of planar domain walls in high-purity iron.
140 Domain walls
where x is the displacement of the wall. We assume in this analysis that the H field is
parallel to the direction of magnetization in one of the domains and antiparallel to
the other domain.
If the domain wall is subject to a potential energy Ep within the material then the
total energy is, [25,26]
Etot = Ep - 2J.loMs· Hx.
The potential Ep will vary throughout the solid since the defects within the solid
provide minima in Ep , while regions of inhomogeneous micros tress, associated
with dislocations, can provide either energy minima or maxima depending on the
sign of the stress and the magnetostriction coefficient. In general the form of this
potential will be irregular as shown in Fig. 7.5, where the energy is plotted against
wall displacement.
The displacement x of the 1800 wall can then be calculated from the condition
dE tot = 0
dx
dE p
= dx - 2J.loM s·H.
Specific solutions may be attempted if the form of Ep is known or can be
approximated. As an example consider the simplest form of potential
Ep=tax 2 •
In this case the solution for x is
2J.loMs·H
x=---a
This refers to a small displacement, and hence to a completely reversible process
since if the field is removed then x returns to zero. As might be expected the wall
displacement increases as the potential well becomes flatter and decreases as it
becomes steeper.
Fig. 7.5 Potential energy seen by domain
wall as a function of position. This vari-
o
ation of energy with displacement is used
in models based on the rigid wall approximation.
Domain walls motion
141
7.2.5 Magnetization and initial susceptibility in the rigid wall approximation
What value of initial reversible susceptibility is expected on the basis of the rigid wall
model?
This enables us to calculate the initial reversible change in magnetization. Suppose
x = corresponds to the demagnetized state M = 0, then the magnetization is
given by
°
M = 2MsA cos Ox,
where A is the cross-sectional area of the wall, and 0 is the angle between the
magnetization vector in the domain and the direction of displacement. Substituting for x
4J.1o ABMs2 cos 2 0
M=------
a
and consequently the initial reversible susceptibility is
dM
Xin= dB
4J.1o AMs2 cos 2 0
a
In a cubic material such as iron the average of cos 2 0 over the three easy axes is
1/3, so for a multidomain specimen,
Xin =
4J.1o AMs2
3a
The initial reversible susceptibility calculated on the basis of the rigid wall
approximation therefore depends on the saturation magnetization and on the
potential seen by the wall as expressed in this simple case by the parameter a. As the
potential well becomes steeper so the initial susceptibility decreases.
7.2.6
Bending of flexible, low-energy domain walls: the wall bowing
approximation
How do walls with low surface energy compared to pinning energy behave?
Walls with low surface energy show a tendency to bend. In practice domain walls
exhibit both reversible bending and reversible translation under the action of a
magnetic field. Wall bending is shown in Fig. 7.6. Of course the amount of bending
that the walls undergo depends on many factors, including the field strength, but
most particularly on the domain-wall energy [27-31]. Walls with high energy do
not bend easily while those with a low energy are more flexible.
142 Domain walls
,,-, ---
.... ....,
r,
H
I
I
I
h
I
I
Mst /~ ___ _
/
/
/
/
/
Fig. 7.6 Bending of a domain wall under the action of a field. The domain wall is pinned at
the boundaries and expands in the manner of an elastic membrane as described by Kersten
and Neel [27,29].
To give some indication the wall energies, based on a one-constant anisotropy
model calculation, of 180 0 domain walls in iron are about 2.9 ergs/em 2 , in nickel
0.7 ergs/cm 2 and in cobalt 7.6 ergs/cm 2 • Clearly the domain walls in cobalt are
more rigid than the domain walls in nickel. However if a specimen is very pure and
free of defects the domain walls can still move in a planar manner.
Now let us suppose that a 180 domain wall extends throughout a single grain of
a relatively pure specimen. Therefore the grain boundary is the principal
impediment to domain-wall motion. The domain wall will be attached to the grain
boundary in much the same way that it becomes attached to any other defect in the
material. If a field is applied along the direction of one of the domains that domain
will grow by domain-wall motion. However as the domain wall is attached to the
grain boundary it will first move by bending.
The force per unit area on the domain wall is, as before,
0
F=2Jl.oMs·H.
The difference in wall energy caused by bending will be,
E = y[A(H) - A(O)]
where y is the wall energy, A(O) is the area at zero magnetic field, that is before
deformation, and A(H) is the area under a field H.
Assuming for simplicity a cylindrical deformation, this leads to an expression for
the force per unit area on the wall in terms of its radius of curvature
Y
F=-,
r
as in the expression for excess pressure across an elastic membrane such as a liquid
interface where y is the surface tension.
References
7.2.7
143
Magnetization and initial susceptibility in the flexible approximation
What initial reversible susceptibility is expected on the basis of the wall-bending
model?
Once again it is possible to calculate the initial reversible susceptibility. The change
in magnetization as a result of bending of the domain wall is
M=2MsdV,
where d V = (2/3)lhx, x being the displacement of the wall at its centre, and hand 1
the spatial extent of the undeformed wall.
M=1M slhx.
Since for small x, x ~ [2/8r
M=~sPh
6
r
and at equilibrium the force due to the field must equal the force due to the surface
tension of the wall. Equating the pressures,
"I
-=
r
2floMsH.
This allows a substitution to be made for r, since now r = y/(2floMsH)
M = floM, 2 H1 3 h
3"1
The initial susceptibility is thus
floMs 2[3h
Xin
=
3"1
This is the reversible susceptibility, which depends on the saturation magnetization
and on the domain wall surface energy "I. As the domain wall surface energy
increases so the wall becomes more rigid and the initial susceptibility decreases.
We see therefore that two types of wall motion can occur: wall displacement and
wall bending. The strength of the domain wall pinning and the surface energy of the
wall determine which of these occurs in a particular case.
REFERENCES
1. Bloch, F. (1932) Z. Physik, 74, 295.
2. Kittel, C. and Galt, 1. K. (1956) Ferromagnetic domain theory, Solid State Physics, 3,
437.
3. Chen, C. W. (1977) Magnetism and Metallurgy of Soft Magnetic Materials, North
Holland, p. 84.
144 Domain walls
4. Lilley, B. A. (1950) Phil. Mag., 41, 792.
5. Kittel, C. (1949) Revs. Mod. Phys., 21, 541.
6. Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading,
Mass., p. 617.
7. HolTmann, 1. A., Paskin, A., Tauer, K. 1. and Weiss, R. 1. (1956)J. Phys. Chern. Sol., 1,45.
8. Graham, C. D. (1958) Phys. Rev., 112, 1117.
9. Sato, H. and Chandrasekar, B. S. (1957) J. Phys. Chern. Sol., 1, 228.
10. Pauthenet, R., Barnier, Y. and Rimet, G. (1962) Proceedings of the International
Conference on Magnetism and Crystallography, Kyoto, 1961, J. Phys. Soc. Japan, 17
(Sup. B-1), 309.
11. Chikazumi, S. (1964) Physics of Magnetism, Wiley, New York, p. 130.
11. Kittel, C. (1986) Introduction to Solid State Physics, 6th edn, Wiley, New York, p. 23.
13. Chikazumi, S. (1964) Physics of Magnetism, Wiley, New York, p. 192.
14. Neel, L. (1955) Comptes Rendu, 241, 533.
15. Palmer, S. B. (1975) J. Phys. F., 5, 2370.
16. Palmer, S. B., Baruchel, 1., Drillat, A., Patterson, C. and Fort, D. (1986) J. Mag. Mag.
Mater., 53-7, 1626.
17. Kittel, C. (1986) Introduction to Solid State Physics, 6th Edn, Wiley, New York.
18. Kersten, M. (1956) Z. Angewandte Phys., 8, 496.
19. Neel, L. (1944) Cahiers de Phys., 25, 21.
20. Kersten, M. (1943) Grundlagen einer Theorie der Ferromagnetischen Hysterese und der
Koecitivkraft, Hirzel, Leipzig.
21. Becker, R. (1932) Phys. Zeits., 33, 905.
22. Kondorsky, E. (1937) Phys. Z. Sovietunion, 11, 597.
23. Kersten, M. (1938) Phys. Zeits., 39, 860.
24. Kersten, M. (1938) in Problem der Technischen Magnetisierungskurve (ed. R. Becker),
Springer, Berlin, p. 42.
25. Hoselitz, K. (1952) Ferromagnetic Properties of Metals and Alloys, Oxford, Ch. 2, p. 79.
26. Chikazumi, S. (1964) Physics of Magnetism, Wiley, p. 267.
27. Neel, L. (1946) Annales Univ. Grenoble, 22, 299.
28. Neel, L. (1944) Cahiers de Phys., 25, 21.
29. Kersten, M. (1956) Z. Angewandte Phys. 7, 313.
30. Kersten, M. (1956) Z. Angewandte Phys. 8, 382.
31. Kersten, M. (1956) Z. Angewandte Phys. 8, 496.
FURTHER READING
Chikazumi, S. (1964) Physics of Magnetism, John Wiley, New York, Ch. 9.
Crangle,1. (1977) The Magnetic Properties of Solids, Arnold, London, Ch. 6.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading, Mass.,
Ch.9.
Kittel, C. (1949) Physical theory of ferromagnetic domains, Revs. Mod. Phys., 21, 541.
Kittel, C. and Galt, 1. K. (1956) Ferromagnetic domain theory, Solid State Physics, 3, 437.
EXAMPLES AND EXERCISES
Critical dimensions for single-domain particles in nickel. Calculate
the magnetostatic energy of a small spherical particle of nickel of radius rand
magnetized to saturation Ms = 5.1 X 10 5 Aim. (You can assume the magneto static
Example 7.3
Examples and exercises
145
energy of a sphere with magnetization M and volume V is Ems = 110M2 VJ6 and that
this energy is halved when the particle is divided into two domains.) The energy
required to form a Bloch wall is y = 0.7 x 10- 3 J m - 2 in nickel. Hence find the
critical radius for single-domain particles which will be such that the reduction in
magnetostatic energy obtained by division into two domains is less than the energy
needed to form a wall.
Example 7.4 Calculation of wall energy and thickness from anisotropy energy,
saturation magnetization and exchange energy. Find the domain-wall energy y, wall
thickness I and critical dimension rc for single-domain particles for a material with
anisotropy energy K = 33 X 104 J m - 3, saturation magnetization Ms = 0.38
x 106 A/m, lattice spacing 3 x 10 - 10 m and exchange energy J = 3 x 10 - 21 1. You
may assume this material has a spin of S = 1/2, or magnetic moment per atom of
one Bohr magneton.
Example 7.5 Estimation of domain spacing in cobalt. Estimate the domain
spacing d in a specimen of single-crystal cobalt in the form of a plate of infinite
per unit area is 1.7 x 10- 7 M/d. The domain-wall energy for cobalt is
7.6 x 10- 3 Jm- 2 and Ms = 1.42 X 106 A/m.
Calculate the domain spacing when 1= 0.01 m.
8
Domain Processes
In this chapter we will look at the behaviour of domains under the action of a
changing magnetic field. We are interested principally in the changes which occur
in the domain structure and the mechanisms by which these are brought about. We
will look at reversible and irreversible changes in magnetization and describe them
in terms of the two main mechanisms: domain-wall motion and rotation. Finally
we will use some of these ideas to formulate a model of hysteresis.
8.1
REVERSIBLE AND IRREVERSIBLE DOMAIN PROCESSES
Are the changes in magnetization reversible or irreversible or both?
The changes in magnetization arising from the application of a magnetic field to a
ferromagnet can be either reversible or irreversible depending on the domain
processes involved. A reversible change in magnetization is one in which after
application and removal of a magnetic field, the magnetization returns to its
original value. In ferromagnetic materials this only occurs for small field
increments.
More often both reversible and irreversible changes occur together, so that on
removal of the field the magnetization does not return to its initial value. If the
magnetic field is cycled under these conditions hysteresis is observed in M. The
next task is to interpret such changes in magnetization in terms of domain
mechanisms so that both reversible and irreversible changes can be explained.
8.1.1
Domain rotation and wall motion
What mechanisms are available for interpreting and describing the changes in
magnetization with field?
The domain mechanisms which we have discussed so far are rotation and wall
motion. Both of these processes can be manifested as either reversible or
irreversible mechanisms, and the transition from reversible to irreversible is in both
cases dependent on the amplitude of the magnetic field.
Wall motion incorporates two distinct effects: bowing of domain walls and
148
Domain processes
translation. Domain-wall bowing is a reversible process at low field amplitudes.
The domain wall expands like an elastic membrane under the action of the
magnetic field. When the field is removed the wall returns to its original position.
Wall bowing becomes irreversible once the domain wall is sufficiently deformed
that the expansion continues without further increase of field. The bending of the
domain wall which begins as reversible can also become irreversible if during this
process the wall encounters further pinning sites which prevent it relaxing once the
field is removed.
The translation of domain walls is usually irreversible unless the material is
sufficiently pure that the domain wall can exist in a region of the material that is free
from defects. The displacement of planar walls can be modelled using a potential
energy which fluctuates as a function of distance, as in Fig. 8.1. There are two
possible origins for the short-range fluctuations in potential experienced by the
domain walls. These are short-range variations in strain associated with
dislocations within the material and the presence of particles of a second phase
within the matrix material, for example the presence of carbides in steel. There is no
general way of treating these energy fluctuations since they are random in nature,
however we can consider a simple case of a sinusoidal variation of potential energy,
as in Example 8.1, which will have many of the essential features of the real solid.
Rotation of magnetic moments within a domain has been discussed in
Chapter 6. At low field amplitudes the direction of alignment of the magnetic
moments which corresponds to minimum energy is displaced slightly from the
crystallographic easy axes towards the field direction. This results in a reversible
rotation of the magnetic moments within a domain.
At intermediate to high field amplitudes there is an irreversible mechanism
within the domain when the moments rotate from their original easy axis to the
easy axis closest to the field direction. This occurs when the field energy overcomes
the anisotropy energy. In this case once the magnetic moments within the domain
Xo
x
Fig. 8.1 Magnetic energy potential as a function of distance seen by a magnetic domain
wall.
Reversible and irreversible domain processes
149
have rotated into a different easy axis the moments remain within the potential well
surrounding this easy axis if the field is reduced.
At high fields the energy minimum of the easy axis closest to the field is perturbed
by the field energy until the minimum lies in the field directipn. This results in a
reversible rotation of the moments into the field direction and hence a reversible
change in magnetization at high fields. Finally at very high fields there is a
reversible change in which the magnetic moments within the specimen, which is
already a single domain, are aligned more closely with the field direction. This
occurs because the individual magnetic moments precess about the field direction
due to thermal energy. This precession does not give any net moment in other
directions but does reduce the component of magnetization along the field
direction. As the field strength increases the angle of precession is reduced.
Similarly if the temperature is lowered the angle of precession is reduced due to a
reduction in thermal energy.
8.1.2 The strain theory: pinning of domain walls by strains
What causes domain wall motion to be irreversible?
Even before Bloch's theoretical prediction of the existence of a finite transition
region between domains, which he called domain walls, it was realized that the
principal domain process occuring at low fields was caused by domain boundary
motion. Early experiments on domain boundary motion were reported by Sixtus
and Tonks [1]. In view of the hysteresis loss, most of which occurs at low fields, it
was necessary to explain the loss mechanism in terms of wall motion.
One of the earliest suggestions was by Becker [2J that the domain walls were
impeded in their movements by regions of inhomogeneous strain which interacted
via the magnetostriction with the magnetic moments to provide local energy
barriers which the domain walls needed to overcome. An excellent survey of this
early work has been given by Hoselitz [3]. The first calculations were by Becker
and Kersten [4J and by Kondorsky [5]. Becker and Kersten determined the initial
susceptibility and the coercivity as a function of internal stress. The results of
Kersten's investigations are shown in Fig. 8.2.
The strain theory of coercivity was treated in detail by Becker and Doring [6J
and by Kersten [7]. During the time in which the effects of stress on magnetization
were being investigated, the existence of dislocations within crystals was first
suggested by Orowan, Polanyi and Taylor [8,9,10]. Dislocations have an
associated local stress field which gives rise to highly inhomogeneous micros trains
within a solid [11].
Through the magnetoelastic coupling the dislocations pin domain walls, and
therefore the higher the dislocation density within a ferromagnet the greater the
impedance to domain wall motion. This explains why cold-worked specimens have
higher coercivity and lower initial susceptibility than the same material in a wellannealed state, a fact that can be exploited for non-destructive evaluation of the
150 Domain processes
CII
o
r-~
u
J:
o
20
10
30
OJ_ kg/mm 2
Fig. 8.2 Variation of coercivity of nickel with internal stress calculated by Kersten: (a)
recrystallized nickel, and (b) hard-drawn nickel. This result confirmed the strain theory of
coercivity.
mechanical state of a ferromagnetic material such as iron or steel using magnetic
measurements.
Example 8.1 Initial susceptibility as a result of planar wall displacement in a
sinusoidal potential. Suppose the stress (j within a material can be represented as a
function of distance by
as shown in Fig. 8.3, where I is the periodicity of the stresses and (j 0 the maximum
amplitude.
The stress energy per unit volume Eo is
Reversible and irreversible domain processes
151
I
(j'
Fig. 8.3 Idealized variation of internal microstress with displacement, approximated by a
sinusoidal function.
where As is the magnetostriction coefficient. Substituting for a in this equation and
differentiating, for a domain wall of unit area
a
dE
dx = (3n)
T Asaosm. (2nx)
-1- .
The energy per unit volume of a domain under the action of a field His
EH =
- /loMs •H.
Therefore the change in field energy caused by moving a 180 0 domain wall of
unit area through a distance x will be
EH =
- 2/loMs' Hx
dEH
dx = - 2/loMs . H.
The total energy will be the sum of the field energy and the stress energy
Etot = EH + Ea
and differentiating with respect to x to find equilibrium
~ dE
tot = (3n)
. (2nx)
T Asa 0 sm
-[- - 2/loMsH cos e= 0
At equilibrium
3n) Asa 0 sm
. (2nx)
2/loMsH cos e = ( T
-1- ,
which can be solved for the displacement x. For small x we can make the
approximation sin 2nx/l ~ 2nx/l
2/loMsH cos e=
6A saon2 x
[2
[2 /loMsH cos e
x=----3n 2 Asa o
152 Domain processes
The initial susceptibility can be obtained by noting that for a 180 domain wall the
change in magnetization is
0
M = 2AMsx cos e
dM = (dM)( dX)
dx
dB
dH
2AF fJoMs 2 COS 2 e
3n 2 AsO"O
M2F
= constant x ~.
AsO"O
Therefore the initial susceptibility decreases with AsO"~
separation of stress maxima I.
and increases with the
8.1.3 The inclusion theory: pinning of domain walls by impurities
What other factors cause irreversibility in domain wall motion?
Isolated regions of second-phase materials with magnetic properties different from
those of the matrix material are known as magnetic inclusions. These reduce the
i
i
s
s
i
(a)
(b)
(c)
Fig. 8.4 (a) Free-pole distribution on a naked inclusion, (b) spike domain attached to an
inclusion and (c) reduction of magneto static energy associated with an inclusion when
intersected by a domain wall.
Reversible and irreversible domain processes
153
I02r-~'",
• Fe-C
o Fe-N
.to
6
Fe-Cu
Fe-Au
-
-
(b)
101
Volume fraction of precipitate
Fig. 8.5 (a) Coercivity of various steels as a function of the total volume fraction of
inclusions. Experimental results of Kersten and calculations by Nee! (1946). (b) Increase in
coercivity of iron caused by precipitates of interstitial or substitutional solutes. Theoretical
calculations are represented by the two dashed lines: (a) due to Kersten and (b) due to Nee!.
154 Domain processes
energy of domain walls when the domain walls intersect them and consequently
the domain walls are attracted to the inclusions which effectively impede wall
motion.
The inclusion theory of domain-wall pinning was suggested by Kersten [12, 13].
He assumed that the magnetic domain walls move in a planar manner through the
solid and that the energy of the walls is reduced when they intersect inclusions. The
inclusions themselves may take many forms such as insoluble second-phase
material which appears if the solubility limit has been exceeded, they may be oxides
or carbides, or they may be pores, voids, cracks or other mechanical
inhomogeneities. Well-known examples of magnetic inclusions are cementite
(Fe 3 C) particles in iron and steels.
Neel [14] criticised the assumption of planar wall motion and further indicated
that Kersten's interpretation ignored the fact that the magnetic free poles
associated with a defect such as a void or crack would be a greater source of energy.
So that a naked inclusion totally enclosed within the body of a domain would have
free poles attached, as shown in Fig. 8.4(a), with an attendant magnetostatic energy
of 2nl1oMs 2 r3/9 but when the wall bisects the defect there occurs a change in
distribution of the free poles on the defect as shown in Fig. 8.4(c) which results in a
reduction of the magneto static energy to nl10Ms 2 r3 /9. Neel's model became known
as the disperse field theory (or the variable field theory).
The energy reduction of the domain wall using the simple elastic membrane
model is the reduction in area of the wall times the wall energy per unit area. This
can be calculated as nr2y, where y is the domain-wall energy. For a 111m diameter
non-magnetic inclusion in iron with y = 1.5 erg/cm 2 the energy reduction due to
free poles is about 140 times the energy reduction due to wall area. Figure 8.5
shows the results of a comparison of calculated and observed coercivities in steels
by Neel [15].
An attempt to describe the effect of carbide inclusions on the coercivity of iron
was made by Dijkstra and Wert [16]. In their model the domain wall was
considered to be an elastic surface, so that the difference in energy between the wall
when it intersected the defect and when it was free was determined simply from the
difference in area of the wall. In reality an inclusion would not be entirely free from
spike domains when within the body of a domain as shown in Fig. 8.4(b). These
spike domains themselves lower the magneto static energy of the inclusion.
Nevertheless the energy of the inclusion and spike domains is still reduced when
intersected by a domain wall and the energy reduction is still considerably higher
than the energy obtained from reducing the wall area.
8.1.4
Critical field when a domain wall is strongly pinned
What is the field strength needed to break a domain wall away from a pinning site in
the situation where the wall bends?
Kersten [17] modified the assumptions of his previous model and calculated the
initial susceptibility of a solid with flexible domain walls. Consider for example the
Reversible and irreversible domain processes
..... -...
......
155
,
'\
\
\
r
\
\
I
I
tel
I
I
/
/
1
/
/
....
----
...- ,,-
/'
Fig. 8.6 Reversible and irreversible expansion of a domain wall according to Kersten.
case where the pinning strength of the defect sites is relatively high and the wall
energy is relatively low. In this case the domain wall will bend under the action of a
field. The process will be reversible until the radius of curvature reaches a critical
value, after which the wall expands discontinuously and irreversibly.
Consider the situation depicted in Fig. 8.6. The radius of curvature r of the
domain wall as it bends under the influence of an applied field H is such that the
excess pressure on the wall y/r exactly balances the force per unit area due to the
field. For a 180° wall
y
- = 2JloMsH cos
r
e,
e
where is the angle between the direction of the H field and the magnetization
within one of the domains. The critical condition arises when the radius of
curvature of the wall reaches 1/2 where 1is the distance between the pinning sites.
Y- H c180 = - JloM.l cos
e
and similarly for a 90° wall
H
90
C
=
yJ2
JloMsl cos e
Kersten used these equations to determine the coercivity of nickel-iron alloys.
This tells us that for strong pinning the critical field, and hence the coercivity, are
dependent on the number density of pinning sites, expressed through the coefficient
I, and the domain-wall energy y.
8.1.5
Critical field when a domain wall is weakly pinned
What is the field strength needed to break a domain wall away from its pinning site
before it bends?
156 Domain processes
In this case the walls move in a planar manner because they break away from the
pinning sites before they have a chance to bend. The walls experience a certain
potential as a result of the distribution of defects and dislocations. The force
exerted on unit area of a 1800 domain wall by a field H is given by
e,
F = 2110MsH cos
where is the angle between the magnetization and the magnetic field.
If the potential energy is given by
e
Ep = Ep(x)
and the maximum slope of this is [dEp(x)/dx]max> then at the critical field we must
have the maximum force per unit area exerted on the wall,
Fmax = [dEp(X)] .
dx max
So the critical field, beyond which the domain wall breaks away from its pinning
site is given by
_
H
eri! -
H .=
en!
Fmax
2110Mscos e
1
[dEp(X)]
2110Ms cos e dx max
In the case of a sinusoidal variation of the potential energy of
. (21rX)
Ep(x) = Emax sm -[and therefore
[ dEp(X)]
dx max
this leads to
- Il10Ms cos e'
In this case therefore the critical field and hence the coercivity are dependent on
the maximum pinning energy Emax and the number density of pinning sites,
expressed via the separation I.
8.2 DETERMINATION OF MAGNETIZATION CURVES FROM
PINNING MODELS
Can realistic models of the magnetization process be devised from the concept of
domain-wall pinning?
Determination of magnetization curves from pinning models
157
One of the most difficult problems in the field of magnetism is to describe the
magnetization curves of a ferromagnet in terms of materials properties. The
complexity of domain-wall interactions with randomly distributed structural
features is further compounded by the possibility of magnetization changes by
domain rotation. Therefore attempts at deriving the underlying theory have dealt
only with the simplest situations.
8.2.1
Effects of microstructural features on magnetization
How do dislocations and other defects affect coercivity and initial susceptibility?
The investigations of Becker, Kersten and Neel have served to clarify the processes
taking place on the domain level when a domain wall interacts with defects. To be
useful such a theory must be related to the observed bulk properties. As we have
shown calculations were made by Kersten of the initial susceptibility based
on a single Bloch wall model. The next development in understanding the
magnetization process was a generalization of the earlier models to include the
interaction of domain walls with defects such as dislocations. Dislocations have an
associated stress field which impedes the motion of magnetic domain walls. The
suggestion that dislocations affect the coercive force was first made by Vicena [18].
Subsequently there were a number of papers published by Kronmuller and
coworkers on this subject.
Seeger et al. [19J considered the influence oflattice defects on the magnetization
curve. They noted that the internal stresses in the earlier work of Becker were of an
unspecified origin and were characterized by an average stress amplitude. Their
theoretical calculations, based on a rigid wall approximation, indicated that the
product of coercivity and initial susceptibility should be a constant.
XinHe = constant.
In addition it was shown that in the rigid wall approximation the coercivity was
dependent on the square root of dislocation density p which in an f.c.c. material is
proportional to applied stress.
He
vi p = constant.
Kronmuller [20J derived a statistical theory of Rayleigh's law in the low-field
magnetization region based on a rigid wall approximation model of domain-wall
motion. The Rayleigh law (section 5.1.6.) gives the relation between magnetization
M and magnetic field H as
M = Xin H + VH2,
where v is the Rayleigh constant and Xin is the initial susceptibility. The coefficients
were shown to be determined by the domain wall potential, and it was found that
they were related to the dislocation density p by the following
Xin
J p = constant
vp = constant.
158 Domain processes
Hilzinger and Kronmuller [21] also investigated theoretically the coercive field
of hard ferromagnetic materials. In particular they found that the interaction of
domain walls in hard magnetic materials with defects lead to the experimentally
observed temperature dependence of coercivity of SmCo s. They have shown that
high coercivities can be obtained by (a) pinning of domain walls by defects and
(b) elimination of domain walls by using single-domain particles. For multidomain
materials the first mechanism is the most important and it is known that in many
materials the most significant mechanism for pinning is the interaction between
dislocations and domain walls. However in SmCo s they considered that point
defects are the most important pinning sites because the domain walls in this
material are so thin, being typically 30 A in width.
Further investigation of Bloch wall pinning in rare-earth cobalt alloys [22]
suggested that antiphase boundaries are the dominant mechanism determining
coercive force in these materials. (Antiphase boundaries occur when the A and B
sublattices within an ordered superlattice become out of phase, with the type of
atoms which were on sublattice A switching to sublattice B and vice versa. The
nearest-neighbour coordination of the superlattice is therefore disrupted locally.)
Hilzinger and Kronmuller [23] developed and generalized these theories further
for rigid (planar) domain-wall motion, considering the pinning of Bloch walls by
randomly distributed defects. As they have indicated, the theories of domain-wall
pinning basically still fall into two categories: potential theories with rigid domain
walls and wall bowing theories with flexible walls. The problem of domain wall
defect interactions has been treated previously for dislocations, point defects, and
antiphase boundaries. They have shown that the coercivity depends upon defect
density, the interaction (or pinning force) and the area and flexibility of domain
walls. These concepts have also been considered by Jiles and Atherton [24].
In the case of potential theories Hilzinger and Kronmuller have shown further
that the coervice force He, depends on the square root of the total domain wall area
AB as well as the defect density p
He~
Jp
JAB·
In the case of domain-wall bowing models the coercive field was found to be
dependent on p2/3,
according to Labusch [25]. The question of how much domain wall bowing takes
place is still slightly controversial. In Kronmuller's statistical theory of Rayleigh's
law the domain walls were assumed to remain planar, but the papers of Kersten
[17] and Dietze [26J assumed that bowing took place. It seems that the amount of
domain-wall bowing that occurs depends on the strength of the pinning sites and
the domain wall surface energy. Chikazumi [27, p. 193J has considered that the
domain walls remain planar along the axis of magnetization within the domain,
such as the [I OOJ direction in iron, in order to avoid the appearance of free poles at
the curved portion of the walls, but the walls are free to bend in other directions.
Determination of magnetization curves from pinning models
159
/
8
Fig. 8.7 Domain wall in iron showing its planar and bowing aspects in different
crystallographic planes.
For example magnetizations along the [100] and [TOO] direction on either side
of a domain wall lead to a planar wall when viewed from either the (001) or (010)
planes. But when viewed from the (100) plane the wall can bend, as in Fig. 8.7.
Finally after considering only planar walls in their earlier papers Hilzinger and
Kronmuller [28] considered the pinning of curved domain walls by randomly
distributed defects. The two theories, the potential theory for planar walls and the
bowing theory for flexible walls, were shown to be merely limiting cases of a more
general theory. In this more general theory the coercive force was found to depend
on pl/2 for weak defect domain wall interactions and on p2/3 for strong defect
domain wall interactions. So the two models provided limiting ranges of a single
curve. The result was important because it brought together the two principal wall
mechanisms under a unified theory of domain-wall motion.
8.2.2 Domain-wall defect interactions in metals
What do experimental observations of domain-wall defect interactions show?
The problem of describing the magnetization process in terms of Bloch wall
motion has also been discussed in a series of papers by Porteseil, Astie and
coworkers. In these papers a number of observations of domain-wall motion were
made and the results were interpreted in terms of theoretical models. Porteseil and
Vergne in a theoretical paper [29] calculated the magnetization curve of a
polycrystal using a single isolated Bloch wall model. They investigated conditions
under which a single irreversible event remained independent and when it lead to
an avalanche effect with numerous subsequent irreversible wall jumps or
Barkhausen discontinuities.
Astie, Degauque, Porteseil and Vergne [30] studied the influence of dislocation
structures on the magnetic and magnetomechanical properties of high-purity
iron. The dislocation structures were formed by strain hardening of polycrystalline
iron. They criticized earlier work on domain-wall defect interactions because
earlier studies failed to make precise observations of lattice defects. Their
160 Domain processes
investigation involved the dependence of the initial susceptibility Xim the Rayleigh
coefficient v and the coercivity He on the strain hardening /1(J = (J - (J 0, where
(Jo is the yield strength and (J is the maximum stress applied. The authors
concluded that there were three distinct regions on the strain hardening curve
between /1(J = 0 and /1(J = 100 MPa and these were identified both from transmission electron microscopy (TEM) and from the magnetic parameters.
I
60
::I
I
I
-
r-
'<11
0
I
40 r- 0
•
• •
or
-
E
<11
0
0
20 -
A·
B
(a)
0
I
I
0.1
0.2
I
I
0.3
0.4
-
0
d(mm)
N
'<11
o
::I
~
E
.0
(b)
o
0.1
0.2
0.3
0.4
d(mm)
Fig. 8.8 Dependence of Rayleigh coefficient v on the grain diameter d in high-purity iron
after Degauque, Astie, Porteseil and Vergne [32].
Determination of magnetization curves from pinning models
161
Interestingly in the low stress region « 30 MPa) the coercivity and initial
permeability were almost independent of stress. However at higher stresses, which
resulted in larger dislocation densities in the form of tangles of dislocations
separated by relatively large distances with lower dislocation densities, He
increased rapidly, while Xin was less affected. Further studies of the interaction of
magnetic domain walls with defects in high-purity iron were reported by
Degauque and Astie [31]. The evolution of domain structures under an applied
field was studied using high-voltage electron microscopy (HVEM). Once again it
was remarked that there have been many theoretical studies on the subject of
domain-wall defect interactions, but very few direct observations of these
interactions.
The effect of grain size on the magnetic properties of high-purity iron was
reported by Degauque, Astie, Porteseil and Vergne [32]. The coercivity He' the
initial susceptibility Xin and the Rayleigh coefficient v were studied together with
the remanent induction BR • The study concluded that the initial susceptibility was
independent of grain size but the Rayleigh coefficient was proportional to grain
diameter d
vld = constant,
as shown in Fig. 8.8. They also found that coercivity was linearly dependent on lid
as shown in Fig. 8.9.
Hed = constant.
The potential model for domain-wall motion was used by Astie, Degauque,
Porteseil and Vergne [33J to interpret results obtained on polycrystalline iron.
From the work of Hilzinger and Kronmuller reported above it was known that the
theory predicted
HeXin = constant
and
vHe 2 = constant.
However it was found from experimental measurements on polycrystalline iron
in which the density of defects was altered, that these values did change in
contradiction of the predictions of the potential model of Hilzinger and
Kronmuller. The authors found that the initial susceptibility Xin was independent
of grain size in iron. This is because Xin is determined by the short-range reversible
motion of domain walls in low fields. Under these conditions the dislocations are
the predominant factor contributing to Xin. The Rayleigh coefficient v was however
found to be dependent on grain size d and increased with grain size, whereas the
coercivity decreased with grain size.
In summary the predictions of invariance for HeXin and vHe 2 follow from the
model only ifthe assumption is valid that the potential curve V(x) is only altered by
effectively scaling both V and x maintaining a similar shape of potential. However
Astie et al. indicate that such an assumption is unlikely to be valid in many cases,
162 Domain processes
0
I
1.0
I
I
/
/
0
I
0.5
A•
B0
OL-1o'2~304.J
I/d(mm-')
Fig. 8.9 Dependence of the coercivity
He on the grain diameter d in highpurity iron after Degauque, Astie, Porteseil and Vergne [32].
and as observed in their results and those of Jiles et al. [34J these products do not
remain constant.
8.2.3
Magnetization processes in materials with few defects
Which factors irifluence the magnetization curve in materials with few defects?
The investigation of domain-wall motion in materials such as ferrites, garnets and
spinels, which are relatively free of defects compared with iron, steel and nickel, has
been conducted by Globus et al. in a series of papers. The effects of grain
boundaries on the bulk magnetization curves of these materials proved to be the
most significant microstructural factor, and in fact the domain-wall motion in
these materials can be modelled with relatively few parameters, due to the simplicity
of the situation.
In the Globus model which was suggested in the earliest paper of this series [35J,
it was assumed that domain walls are pinned at the grain boundaries. The model
assumes that the magnetic domain walls are fixed by the pinning sites on the grain
boundaries and that the walls deform by bending like an elastic membrane under
the action of a magnetic field as was first described by Kersten. Therefore in
experimental studies Globus and Duplex [36] prepared specimens which had only
one variable parameter: the grain size. They investigated ferrites such as nickel
Determination of magnetization curves from pinning models
163
ferrite NiO· Fe 2 0 3 , spinels such as nickel-zinc ferrite NixZn1-xO' Fe 2 0 3 and
yttrium-iron-garnet 5Fe 2 0 3' 3Y 203' Precautions were taken to avoid other
features such as non-magnetic inclusion, pores and dislocations within the grains,
since the authors acknowledged that these strongly influence the domain-wall
motion and hence the bulk magnetization curve.
From the model it was possible to calculate the wall curvature which corresponded to the initial susceptibility. It was found that Xin depended linearly on the grain
diameter d as shown in Fig. 8.10. They also concluded that the initial susceptibility
of these materials is due almost entirely to wall motion. From the results it was
therefore possible to separate the influence of wall motion (proportional to d) and
rotation (independent of d) on the initial susceptibility. These contributions can be
found in Fig. 8.10(b)from the slope of the line, which is due to wall pinning, and the
intercept at d = 0, which is due to rotation of the magnetic moments. The
rotational contribution to Xin is, according to Globus and Duplex [37], dependent
on anisotropy but independent of structure, whereas the wall motion contribution
70
200
60
50
150
0.02
40
_u
,
u
~
NiFe204
::I...
100
0.17
0.13
0.12
1fi€=~
30
0.11
0.02
0.03
20
~=-O.l6
50
0.20
0.14
0.16
r========o
10
spin susceptibility
0.34
0.26
5
Om microns
0
10
5
Om ,um
Fig. 8.10 Dependence of the initial susceptibility Xin on grain diameter d in yttrium-irongarnet and nickel ferrite, after Globus and Duplex [36 and 39]. © (1966) IEEE.
164 Domain processes
is highly structure-sensitive. Subsequently, Globus and Duplex [38] applied the
same model to spinels and garnets.
Globus, Duplex and Guyot [39] have considered the magnetization process in
yttrium-iron-ganet in terms of domain-wall processes. In their work the reversible
component of magnetization was considered to be due to wall bulging, while the
irreversible component of magnetization was due to domain-wall displacement,
since the fields used were well below those required for domain rotation. In this
paper an explicit formula for the susceptibility in terms of domain-wall bending
was given.
The extended Globus-Guyot model [40] included both wall bulging and
displacement. From this they provided a model for coercivity and remanence. In
the extended model the authors determined a critical field strength beyond which
wall translation, and hence irreversible processes, begin.
Further work by Guyot and Globus [41, 42] attempted to relate hysteresis
loss to the creation and annihilation of domain-wall surface area and the energy
lost by continual pinning and unpinning of magnetic domain walls. They concluded that the losses in these materials cannot be explained solely on the basis
of pinning losses. The first loss term is of course a solid friction term due to the
pinning and unpinning of domain walls. The second contribution comes from
the variations in the magnetic domain surface area. Both of these lead to
hysteresis loss.
From this work the authors concluded that the magnetization curves of these
materials all had the same general form which was dependent upon grain size
d, saturation magnetization Ms and the magnetic anisotropy field H eff. In general
the critical field for wall motion was much smaller than the anisotropy field
(Her!Heff = 0.003) so that domain rotation mechanisms could not playa significant
role.
Finally Globus [43J has summarized earlier work giving a universal curve for
the initial magnetization and hysteresis loss of spinels, ferrites and garnets. This
universal curve depends upon M., the anisotropy K and the grain size d. The model
seems to work well for these materials because they have very few intragranular
defects and therefore the factors influencing the magnetization curve are relatively
few and can be expressed in an analytic form of the magnetization curve. In metals
such as iron and nickel the presence of other defects such as non-magnetic
inclusions and dislocations playa significant role, and consequently the whole
mechanism is more complicated.
8.2.4 The Barkhausen effect and domain-wall motion
How can other magnetic phenomena be interpreted in terms of wall motion?
The related phenomena of the Barkhausen effect and magnetoacoustic emission
are both due to discontinuous irreversible changes in magnetization. These
irreversible changes can occur as a result of irreversible domain-wall motion, either
Theory offerromagnetic hysteresis
165
as the unpinning of planar or non-planar domain walls from their pinning sites, or
as the discontinuous expansion of a domain wall once its curvature has exceeded
the critical value.
Barkhausen emissions can also result from the discontinuous rotation of
moments within a domain from one of the easy axes into the easy axis aligned
closest to the field direction. However in practice the contribution from this type of
process is less significant.
The Barkhausen effect can be caused by the motion of either 180° or non-180°
walls. However magnetoacoustic emission is only caused by the discontinuous
motion of non-180° walls, or the irreversible rotation of domains through angles
other than 180°. The reason for this is that to generate acoustic pulses there must be
generation of stresses. No stresses are generated as a result of 180° wall motion or
rotation because the strain along a particular axis is independent ofthe direction of
magnetic moments if they lie along the axis.
8.2.5 Magnetostriction and domain-wall motion
Are there differences in the contribution to magnetostriction from different types of
domain wall?
There is no change in the bulk magnetostriction as a result of 180° wall motion or
rotation. So in the case of terfenol and other highly magnetostrictive materials it is
important to arrange for as much non-180° activity as possible to optimize the
performance of the material, that is to maximize the bulk magnetostriction. This is
achieved by inducing stress anisotropy via an applied compressive stress to align
the domains at right angles to the field direction.
8.3
THEORY OF FERROMAGNETIC HYSTERESIS
How can the bulk magnetic properties offerromagnets be described with a minimum
number of parameters?
It would obviously be very useful to be able to describe hysteresis mathematically
in order to model magnetic properties of ferromagnetic materials. Therefore we
will now consider how the ideas developed so far can be brought together to
provide a theoretical model of hysteresis in ferromagnets. A few models are in use
at present; these include the Preisach model [44] which is widely used in the
magnetic recording industry for describing the magnetization characteristics of
recording tapes, and the Stoner-Wohlfarth model [45] of rotational hysteresis,
which really only applies to single-domain particles, but has been used for
modelling properties of hard magnetic materials. We will consider a recent model
of hysteresis [24] based on domain-wall motion which, as we have mentioned
previously, is the principal cause of hysteresis in multidomain specimens.
166
Domain processes
8.3.1
Energy loss through wall pinning
Can we describe the energy loss in pinning of a domain wall?
Consider a 180 0 domain wall of area A between two domains aligned parallel and
antiparallel to a magnetic field H. If the wall moves through a distance dx under the
action of a field the change in energy due to this movement is
dE= -2J1 oM s·HAdx.
Suppose the pinning energy of a given pinning site is J1of." for a 180 0 domain wall,
and that the pinning energy is proportional to the change in energy per unit volume
caused by moving the wall.
f. pin = tf.,,(1 - cos¢),
where ¢ is the angle between the moments in the neighbouring domains. This gives
us an expression for the pinning energy of a given site as a function of angle ¢.
Clearly when ¢ = 0 the pinning energy must go to zero since the wall no longer
exists. In the case of a 1800 wall f. pin = f.". If there are n pinning sites per unit volume
the energy lost by moving a domain wall is
dElos s = J1 onf."A dx,
where A is the cross-sectional area of the wall. The change in magnetization will be
dM= 2MsAdx.
Therefore
d
_J1onf."dM
Eloss2M
s
Replacing (nf.,,/2Ms) by the constant k gives
dElos s = J1okdM.
8.3.2
Irreversible magnetization changes
If we know the energy loss can we write down an energy equation for the
magnetization process?
Suppose the change in energy of a ferromagnet is manifested either as a change in
magnetization or as hysteresis loss. Then we can write the energy equation as
follows
(
Energy supplied to)
matenal
=
(Change in. )
magnetostatIc
energy
+
(HystereSiS)
loss.
In the case where there is no hysteresis loss, then the change in magneto static
energy must equal the total energy supplied. When there is no hysteresis the
Theory of ferromagnetic hysteresis
167
magnetization follows the anhysteretic curve Man(H)
f
f
Ilo Man (H) dH = Ilo M(H)dH + Ilo
f (;~J
~)dH
and consequently
Man(H) = M(H) +
dM
dH
k( :~)
Man(H) - M(H)
k
This simple result states that the rate of change of magnetization M with field H
is proportional to the displacement of the magnetization from the anhysteretic
magnetization. That is the bulk magnetization M experiences a harmonic potential
about Man {H}.
In fact the situation is a little more complex than this in reality due to the
coupling between the magnetic domains which is of the form envisaged by Weiss,
so that the effective field becomes He = H + aM resulting in the following equation,
which we should note represents only the irreversible component of magnetization
dMirr
dH
8.3.3
Man(H) - Mirr(H)
k - a[Man(H) - Mirr(H)]
Reversible magnetization changes
Can we incorporate reversible changes in magnetization into the differential equation
of hysteresis?
There is of course also a reversible component of magnetization due to reversible
domain wall bowing, reversible translation and reversible rotation. In the model
this has the form
M rev = c(Man - M irr }
and since we must have either reversible or irreversible changes in magnetization,
the total magnetization M tot is given by
In fact the above equation is not really very helpful since magnetization changes
which begin as reversible can become locked in and end up as irreversible. A much
more useful expression however is the expression for the change in magnetization
with field. In this case we are more justified in distinguishing between a reversible
component of susceptibility and an irreversible component of susceptibility
M/Ms
M/Ms
1.0
1.0
MA/m
- 1000 A/m
- 2000 A/m
- .001
- .1
MA/m
a - 1000 A/m
k - 2000 A/m
alpha - .003
Ms - 1.7
a
k
alpha
C
Ms -
1.7
c - .1
H (kA/m)
-10
(c)
10
-1.0
H (kA/m)
-10
10
-1.0
(d)
Fig.8.11 Theoretical hysteresis loops calculated using the equations derived in the theory of hysteresis by Jiles and Atherton.
M/MS
1.0
M/Ms
1.0
Ms -
Ms - 1.7 MAim
a - 1000 Aim
k - 0 Aim
alpha - .001
c - .1
1.7
MAim
a - 1000 Aim
1000 Aim
.001
c - .1
k -
alpha -
H (kA/m)
H (kA/m)
-10
10
-10
-1.0
-1.0
(a)
10
(b)
170 Domain processes
Removing the subscript and assuming that whenever we talk about a change in
magnetization without further qualification, we mean the total magnetization
dM =
dH
k
~Man
-
a(Man
~ir)
+ c(dMan _
M irr)
°
dH
dMirr ).
dH
It is clear from this model that if k -+ then M -+ Man(H) which conforms with
earlier comments that if there are no pinning sites then the magnetization will
follow the anhysteretic magnetization curve. Solutions of this differential equation
for various values of the parameters are shown in Fig. 8.11.
8.3.4 Relationship between hysteresis coefficient and measured magnetic
properties
How can these theoretical parameters be calculated from the conventional
magnetization curve?
It is clearly important from the viewpoint of applications to be able to calculate the
values of these various parameters governing hysteresis from a measured
magnetization curve [46]. From the equations it is easily shown that the initial
susceptibility of the normal magnetization curve is given by
Xin = cXan'
where Xan is the differential susceptibility of the anhysteretic curve at the origin.
This concept agrees well with Rayleigh's idea that Xin represents the reversible
component of magnetization at the origin of the initial magnetization curve.
In the case where k is constant and the reversible component is negligible, that is
c = 0, we have the following very simple solution for k in terms of the coercivity He
and the slope X'nc of the hysteresis loop at He
k = Man(HJ(rx +
when c =1=
°
-+-).
XHc
this becomes
k = Man(He)
1(1: + (1~~
c)
)' 1
- c XHc - c
dM,.(H)I·
J
dH
In the case of soft magnetic materials which have a low coercivity we can make
some approximations which lead to an interesting relation between k and He. For
low coercivity materials the slope of the hysteresis loop at the coercive point is
approximately equal to the slope of the an hysteretic curve at the origin. Setting
these equal,
X~n
Ms
3a - rxMs
= X~c
Man(He}
k - rxMan(HJ
Theory of ferromagnetic hysteresis
171
which leads to the following equation for k,
3a
k = -Man(He).
Ms
Secondly, the slope of the anhysteretic curve at the origin is quite linear and
therefore, for small He we can write
Man(He) =
X~nHe
=
Ca ~MJHe
and substituting this result into the expression for k gives
He
k=
1-
(a;:)
This brings us to the important result that the coercivity of a soft ferromagnetic
material is determined principally by the pinning of domain-wall motion. In fact
k ~ He for soft magnetic materials.
8.3.5 Effects of microstructure and deformation on hysteresis
How does the hysteresis depend on the details of the material microstructure?
Changes in microstructure, in the form of additional magnetic inclusions such as
second-phase particles with different magnetic properties from those of the matrix
material, cause changes in the hysteresis properties by introducing more pinning
sites that impede domain-wall motion and thereby lead to increased coercivity and
hysteresis loss [47,48,49]. The same is also true of dislocations when their number
density is increased by plastic deformation, either in tension or compression [50].
So for example the addition of carbon in the form of iron carbide particles increases
coercivity and hysteresis loss. Cold working of the material has a similar effect.
The effect of these pinning sites is expressed via the coefficient k in the theory of
ferromagnetic hysteresis. Clearly as their numbers increase k will increase
proportionally and this results in an increase in the coercivity He as given in the
equations above.
In the low field limit the coercivity is He = k and consequently the coercivity is
proportional to the product of number density and average pinning energy per site.
8.3.6 Effects of stress on bulk magnetization
How does applied stress alter the hysteresis properties?
Following classical thermodynamics of reversible systems the Gibbs energy is
G = V - TS
+ tcr A,
where Ais the bulk magnetostriction, cr is the stress, U is the internal energy, T is the
172 Domain processes
thermodynamic temperature and S is the entropy. The Helmholz energy is
A=G+l1oHM,
where H is the field and M is the magnetization. The internal energy U due to
magnetization is
U =tal1oM2.
The total effective field is given by [51]
assuming the material is under a constant stress (J we can write the effective field as
3
(d)')r'
H eff =H+aM+1:(J dM
where H is the magnetic field, aM is the mean field coupling to the magnetization
and Hu is an equivalent stress field [51].
3
(d)')
Hu=1:(J dM
T'
This can be used to determine the reversible magnetization under the action of
stress. So for example the anhysteretic magnetization curve, which is a reversible
magnetization curve, can be determined by adding the stress equivalent field Hu to
the sum of the true field H and the internal field due to coupling between the
magnetic moments aM. Using the Frohlich-Kennelly relation with a = 0 (the
Frohlich-Kennelly equation does not contain any coupling to the magnetization)
this would become under the action of a stress (J
This result tells us how the anhysteretic magnetization curve is altered under the
action of a constant stress (J for a material with magneto stricti on l We should note
that this analysis applies only to a reversible process. In the case of irreversible
processes the thermodynamic relationships become more complicated.
REFERENCES
1. Sixtus, K. 1. and Tonks, L. (1931) Phys. Rev., 37, 930.
2. Becker, R. (1932) Phys. Zeits., 33, 905.
3. Hoselitz, K. (1952) Ferromagnetic Properties of Metals and Alloys, Oxford University
Press, Oxford.
4. Becker, R. and Kersten, M. (1930) Z. Phys., 64, 660.
Further reading
173
5. Kondorsky, E. (1937) Physik. Z. Sowjetunion, 11, 597.
6. Becker, R. and Doring, W. (1938) Ferromagnetismus, Springer-Verlag, Berlin.
7. Kersten, M. (1938) in Problems of the Technical Magnetisation Curve (ed. R. Becker),
Springer, Berlin.
8. Orowan, E. (1934) Z. Phys., 89, 605.
9. Taylor, G. I. (1934) Proc. Roy. Soc. Lond., 145A, 362.
10. Polanyi, M. (1934) Z. Phys., 89, 660.
11. Hull, D. and Bacon, D.1. (1964) Introduction to Dislocations, 3rd edn, Pergamon
Oxford.
12. Kersten, M. (1943) Underlying Theory of Ferromagnetic Hysteresis and Coercivity,
Hirzel, Leipzig.
13. Kersten, M. (1943) Phys. Z., 44, 63.
14. Neel, L. (1944) Cahiers de Phys., 25, 21.
15. Neel, L. (1946) Basis of a new general theory of the coercive field, Ann. Univ. Grenoble,
22,299.
16. Dijkstra, L. 1. and Wert, C. (1950) Phys. Rev., 79, 979.
17. Kersten, M. (1956) Z. Angew. Phys., 7, 313; 8, 382; 8, 496.
18. Vicena, F. (1955) Czech. J. Phys., 5, 480.
19. Seeger, A., Kronmuller, H., Rieger, H. and Trauble, H. (1964) J. Appl. Phys., 35, 740.
20. Kronmuller, H. (1970) Z. Angew. Phys., 30, 9.
21. Kronmuller, H. and Hilzinger, H. R. (1973) Int. J. Magnetism, 5, 27.
22. Hilzinger, H. R, and Kronmuller, H. (1975) Phys. Letts., 51A, 59.
23. Hilzinger, H. R. and Kronmuller, H. (1976) J. Mag. Mag. Mater., 2,11.
24. Jiles, D. C. and Atherton, D. L. (1986) J. Mag. Mag. Mater., 61, 48.
25. Labusch, R. (1969) Cryst. Latt. Def, 1, 1.
26. Dietze, H. D. (1957) Z. Phys. 149, 276.
27. Chikazumi, S. (1964) Physics of Magnetism, Wiley, New York.
28. Hilzinger, H. R. and Kronmuller, H. (1977) Physica, 86-8B, 1365.
29. Porteseil, 1. L. and Vergne, R. (1979) J. de Phys. (1981) 40, 871.
30. Astie, B., Degauque, 1., Porteseil, 1. L. and Vergne, R. (1981) IEEE Trans. Mag., 17,
2929.
31. Degauque,1. and Astie, B. (1982) Phys. Stat. Sol., A74, 201.
32. Degauque, 1., Astie, B., Porteseil, 1. L. and Vergne, R. (1982) J. Mag. Mag. Mater., 26,
261.
33. Astie, B., Degauque, 1., Porteseil, 1. L. and Vergne, R. (1982) J. Mag. Mag., Mater., 28,
149.
34. Jiles, D. c., Chang, T. T., Hougen, D. R. and Ranjan, R. (1988) J. Appl. Phys., 64, 3620.
35. Globus, A. (1962) Comptes Rendus Acad. Seances, 255, 1709.
36. Globus, A. and Duplex, P. (1966) IEEE Trans. Mag., 2, 441.
37. Globus, A. and Duplex, P. (1969) Phys. Stat. Sol., 31, 765.
38. Globus, A. and Duplex, P. (1970) Phys. Stat. Sol., A3, 53.
39. Globus, A. Duplex, P. and Guyot, M. (1971) IEEE Trans. Mag., 7, 617.
40. Globus, A. and Guyot, M. (1972) Phys. Stat. Sol., B52, 427.
41. Guyot, M. and Globus, A. (1973) Phys. Stat. Sol., B59, 447.
42. Guyot, M. and Globus, A. (1977) J. de Phys., 38, Cl-157.
43. Globus, A. (1977) Physica, 86-8B, 943.
44. Preisach, F. (1935) Z. Phys., 94, 277.
45. Stoner, E. C. and Wohlfarth, E. P. (1948) Phil. Trans. Roy. Soc., A240, 599.
46. Jiles, D. C. and Thoelke, 1. B. (1989) Proceedings of the 1989 Intermag Conference,
IEEE Trans. Mag., 25, 3928, 1989.
47. Leslie, W. C. and Stevens, D. W. (1964) Trans. ASM, 57, 261.
48. English, A. T. (1967) Acta. Metall., 15, 1573.
49. Jiles, D. C. (1988) J. Phys. D. (Appl. Phys.), 21, 1186.
174
Domain processes
50. Jiles, D. C. (1988) J. Phys. D. (Appl. Phys.), 21, 1196.
51. Sablik, M. 1., Kwun, H., Burkhardt, G. L. and Jiles, D. C. (1988) J. Appl. Phys., 63,3930.
FURTHER READING
Becker, R. and Doring, W. (1939) Ferromagnetismus, Springer-Verlag, Berlin.
Chikazumi, S. (1964) Physics of Magnetism, Wiley, Ch. 11,13,14.
Crangle,1. (1977) The Magnetic Properties of Solids, Arnold, Ch. 6.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading, Mass,
Ch.9.
Hoselitz, K. (1952) Ferromagnetic Properties of Metals and Alloys, Oxford, Ch. 2.
Stoner, E. C. (1948) Phil. Trans. Roy. Soc., A240, 599.
EXAMPLES AND EXERCISES
Example 8.2 Magnetostatic energy associated with a void. Show that the
magneto static energy of a spherical void enclosed entirely within a domain as
shown in Fig. 8.4 is 2/JoMs 2 nr 3/9 where Ms is the saturation magnetization of the
material and r is the radius of the void.
Assuming that the reduction in energy when a 1800 domain wall intersects such a
void is /JoMs 2 nr 3 /9, calculate the energy reduction when a domain wall in iron
in tersects spherical voids of r = 5 x 10 - 8 m and r = 10 - 6 m. Compare this to the
z
1
I
0
,
I
,
,
Q
I
0
I
~
X
V
/"
'"
•
I
.
A •
.. ...
)
0
.. .. .
0
...
Fig. 8.12 Domain walls within a ferromagnetic material stabilized by inclusions distributed
within the material. In this case the inclusions are voids.
Examples and exercises
175
difference in wall energy caused by the intersection with the void, assuming the
domain-wall energy in iron is 2 x 10- 311m 2 and Ms = 1.7 X 106 Aim.
Example 8.3 Reduction of domain-wall energy by voids. If the reduction in
energy when a domain wall intersects a void is given by ll oM/nr 3 /9, estimate the
reduction in energy per unit volume in a material which has N voids per unit
volume each of radius r. (Assume all domain walls form parallel planes of
separation d within a cube of unit volume as shown in Fig. 8.12.)
If the wall energy is to be completely compensated for by the decrease in energy
associated with the voids estimate the number density of voids of radius r which
can generate domain walls in a crysta1.
Calculate this number for iron with Ms = 1.7 X 106 Aim, r = 0.01 mm and a wall
energy of 2 x 10- 3 11m 3.
Example 8.4 Effect of stress on anhysteretic susceptibility. Derive an expression
for the susceptibility of a 180 domain wall moving in a linear stress field a = ax 2 ,
where a is a constant. Assume there is no change in wall energy as it moves in the
0
stress field.
Determine the anhysteretic susceptibility at the origin X~n
(a) of a magnetic
material under a compressive stress of - 20 MPa, if the zero stress value of X~n (0) is
1000, and the low-field magneto stricti on increases with magnetization M
according to A = b(a)W, where b(a) = 1.8 x 10- 12 (A/m)- 2 when a = - 20 MPa.
The saturation magnetization of this material Ms = 0.9 X 106 Aim.
9
Magnetic Order and Critical Phenomena
In this chapter we discuss theories of magnetic behaviour of materials, particularly
the alignment of magnetic moments within the material. These theories can
provide quite useful phenomenological models of the magnetic properties
including order-disorder transitions such as occur at the Curie temperature. These
models assume each atom in a paramagnet or ferromagnet has a fixed magnetic
moment but make no assumptions about the electronic structure of the atoms or
origin of the atomic magnetic moment.
9.1
THEORIES OF PARAMAGNETISM AND DIAMAGNETISM
What atomic scale theories do we have to account for the properties of diamagnets
and paramagnets?
Diamagnets are solids with no permanent net magnetic moment per atom.
Diamagnetic susceptibility arises from the realignment of electron orbitals under
the action of a magnetic field. Therefore all materials exhibit a diamagnetic
susceptibility, although not all are classified as diamagnets. Some materials have a
net magnetic moment per atom, due to an unpaired electron spin in each atom
which leads to paramagnetism or even to ordered magnetic states such as
ferromagnetism. In either case the paramagnetic or ferromagnetic susceptibility is
much greater than the diamagnetic susceptibility and therefore is the dominant
effect.
Paramagnetism occurs at higher temperatures in all materials which have a net
magnetic moment. The atomic magnetic moments are randomly oriented but can
be aligned by a magnetic field.
9.1.1 Diamagnetism
What causes the negative susceptibility observed in some materials?
The magnetic moments associated with atoms in magnetic materials have three
origins. These are the electron spins, the electron orbital motion and the change in
orbital motion of the electrons caused by an applied magnetic field. Only the
178
Magnetic order and critical phenomena
change in orbital motion gives rise to a diamagnetic susceptibility. Diamagnetism
leads to a very weak magnetization which opposes the applied field. The
diamagnetic susceptibility is therefore negative and has an order of magnitude of
10- 5 or 10- 6 . It is also found to be independent of temperature. Most elements in
the periodic table are diamagnetic, we mention as examples copper, gold, silver and
bismuth.
9.1.2
Langevin theory of diamagnetism
How can we explain the negative susceptibility ofdiamagnets in terms ofthe motion of
electrons?
The susceptibility of diamagnets was first explained by Langevin [1]. In this work
he applied some of the earlier ideas of Ampere, Weber and Lenz on the effect of a
magnetic field on a current-carrying conductor, to the motion of an electron within
an atom. An electron in orbit about an atomic nucleus can be compared with
current passing through a loop of conductor and it will therefore have an orbital
magnetic moment mo since as we already know from Chapter 1 electric charge in
closed loop motion generates a magnetic moment.
In the case of a current loop the magnetic moment is
mo = iA,
where iis the current and A is the area of the loop. For an electron in orbital motion
eA
mo = - ,
T
where e is the charge on the electron and r is the orbital period. If we have circular
orbital of area A = nr2 and r = 2nr/v, where v is the instantaneous tangential
velocity of the electron and r is the radius of the orbital
evr
mO =2'
This is the magnetic moment obtained as a result of the orbital motion of an
electron. In the absence of a magnetic field the orbital moments of paired electrons
within an atom will cancel. An applied magnetic field H will accelerate or
decelerate the orbital motion of the electron and thereby contribute to a change in
the orbital magnetic moment. Once a magnetic field has been applied the
perturbation of the electron velocity can be determined. The change in magnetic
flux through the current loop described by the electron about the nucleus gives rise
to an emf Ve in the current loop. This leads to an electric field E given by
E= Ve
L
=
-(l)~,
Theories of paramagnetism and diamagnetism
179
where L is the orbit length ( = 2nr) and Ve is the induced voltage.
E=
(-=-!) d(BA)
L
dt
=( -LA)~.
The acceleration of the electron is
dv
a=-
dt
eE
,
me
where F = eE is the force on the electron due to the field E and me is the mass of the
electron.
eE
me
dv
-
-
dt
-(~:L)
=
= -
er )dB
( 2me
dt
-(~:).
=
Now integrating from zero field strength to an arbitrary field strength H gives
I
V
2
dv = _ (f.loer)
2me
VI
V2 - VI
= -
fH dH
Jo
( f.loer)
2me H.
The change in magnetic moments arising from this is
~mo=()V2-I
~mo
f.l oe2r2H
4me
= - -.,---
This result only applies in the case where the magnetic field is perpendicular to
the plane of motion of the electron. In the case where the field H is in the plane of
motion the net change is zero. In the general case therefore we have the projection
180 Magnetic order and critical phenomena
R of the orbit radius r on a plane normal to the field
R = rsin8,
where 8 = 0 corresponds to a field in the plane of orbit and 8 = nl2 corresponds to a
field perpendicular to the plane.
flmo = -
f(:2) 2
(~:)
sin 8dA,
where now A is the area of a hemisphere and dA = 2nR 2 sin8d8, as shown in
Fig. 9.1. The average value of R2 is
<R2)
=(~)r2
and consequently
where now the radius r can have any orientation. If we consider Z outer electrons in
H
Fig. 9.1 Unit sphere defining the parameters R, () and A used to describe the orbital motion
of an electron about the nucleus.
Theories of paramagnetism and diamagnetism
181
the atom then the change in magnetic moment per atom is
floZe 2r2H
l!mo= - - - - 6m e
and of course if we wish to convert this to a bulk magnetization
l!M =
_
(N oP) (floZe2r2 H)
Wa
6me
where No = Avogadro's number, P is the density and Wa the relative atomic mass.
M
x=H
= _
(N op) (floZe 2r2).
Wa
6m e
This deduction tells us that in the case of a diamagnet, which has no net atomic
magnetic moment in the absence of a field, the action of a magnetic field causes
changes in the velocity of electrons in the atom in such a way that the induced
moment opposes the field producing it. It is also clear that the above expression for
diamagnetic susceptibility is independent of temperature, which is in accordance
with experimental observations.
The result that is often quoted as confirmation of the Langevin model of
diamagnetism is the susceptibility of carbon, for which
No = 6.02
X
10- 29
P = 2220kg/m 3
e=1.6xl0- 19 C
Z=6
<R2) =(0.7 x 1O- 10 )2m2.
The expected value of susceptibility on the basis of the Langevin model is
X=
-
18.85
X
10- 6 ,
X=
-
13.82
X
10- 6 •
the true value is
9.1.3
Paramagnetism
How can we explain the paramagnetic susceptibility of solids which have a permanent
magnetic moment per atom?
Both the electron spin and the orbital angular momentum give contributions to the
magnetization which lead to positive susceptibility. The susceptibilities of
182
Magnetic order and critical phenomena
paramagnets are typically of the order of X ~ 10 - 3 to 10 - 5, and at low fields Mis
proportional to H, although deviations from proportionality occur at very high
fields where the magnetization begins to saturate. Examples of paramagnets are
aluminum, platinum and manganese above its Neel temperature of 100 K.
There are a number of possible explanations of paramagnetic behaviour in
solids. These range from the localized moments model of Langevin [1], in which
the non interacting electronic magnetic moments on the atomic sites are randomly
oriented as a result of their thermal energy, to the Van Vleck model [2] oflocalized
moment paramagnetism, which leads to a temperature-independent susceptibility
under certain circumstances. Finally, there is the Pauli paramagnetism model [3],
which depends on the weak spin paramagnetism of the conduction-band electrons
in metals. In this model the conduction electrons are considered essentially to be
free and so non-localized. The Pauli model also leads to a temperatureindependent paramagnetic susceptibility.
9.1.4
Curie's law
How does the susceptibility of a paramagnet depend on temperature?
The susceptibilities of a number of paramagnetic solids were measured over a wide
temperature range by Curie [4]. In this he found that the susceptibility varied
inversely with temperature, as shown in Fig. 9.2.
c
X=Y'
where C is a constant known as the Curie constant.
The materials which obey this law are materials in which the magnetic moments
are localized at the atomic or ionic sites. These can be considered to be 'dilute'
magnetic materials, in which the magnetic atoms are surrounded by a number of
non-magnetic atoms. Hydrated salts of transition metals such as CuS0 4·SH 2 0
and CrK(S04)·12H 2 0 obey the Curie law.
9.1.5 Langevin theory of paramagnetism
If the electronic magnetic moments are localized on the atom how does the
susceptibility depend on magnetic field and temperature?
In materials with unpaired electrons, and consequently in which the orbital
magnetic moments are not balanced, there is a net permanent magnetic moment
per atom. If this net atomic magnetic moment is m, which will be the vector sum of
the spin ms and orbital mo components, then the energy of the moment in a
magnetic field H will be
Theories of paramagnetism and diamagnetism
183
(h)
40
III
E
30
u
.......
(5
E
III
o
Susceptibility X
~20
(a)
c:
:J
c:
10
T
X=~
100
200
Temperature,K
Curie Law
300
Fig. 9.2 Temperature dependence of paramagnetic susceptibility. Left-hand diagram shows
a schematic of the variation of X with T. The right-hand diagram shows the variation of ilx
with temperature for the paramagnetic salt Gd(C 2 H s 'S0 4 ), 9H 2 0. The circles are
experimental points; the straight line is the Curie law prediction.
Thermal energy tends to randomize the alignment of the moments. Langevin [1]
supposed that the moments are non-interacting in which case we can use classical
Boltzmann statistics to express the probability of any given electron occupying an
energy state E. If kB T is the thermal energy
p(E) = exp( - ElkBT).
We evaluate the probability function for the case of an isotropic material. The
number of moments lying between angles 0 and 0 + dO is dn which will be
proportional to the surface area dA
dA
=
2nr 2 sin 0 dO
dn = C2nr 2 sin 0 dO,
where C is a normalizing constant which gives the total number of moments per
unit area. Now incorporating the probability of occupation of any given state
.
dn = C2n sm 0 dO exp
(110mB
cos 0)
kB T
.
184 Magnetic order and critical phenomena
Integrating this expression over a hemisphere gives the resultant total number of
moments per unit volume N
mB) dO
r
N = 2nC ,,0 sin 0 exp (11 ~B T
1t
c~ 2. J: Singex:(tr)d9·
The magnetization is then given by the expression
M=
f:
mcosOdn
M = Nm
i
i
1t
•
cosOsmOexp
0
1t
o
If we put x = cos 0, dx =
(110 mB cos
kBT
0) dO
0
•
0 exp (110 mB
sm
k cos ) dO
BT
- sin OdO and integrate this gives
kB T- ]
110mB)
M=Nm [ coth ( - -kBT
110mB
M=
Nm!l(I1~:).
This is the Langevin equation for the magnetization of a paramagnet. !I!(x) is
called the Langevin function and always lies in the range - 1 < !I!(x) < 1. The
Langevin function can be expressed as an infinite power series in l1omB/k BT. In
most cases l1omB/k BT « 1, so that the expression for M becomes equal to the first
term in the series
NI1om 2 B
M=--3kB T
which leads immediately to the Curie law, since
M
B
X=-
which is Curie's law. This demonstrates that the Langevin model leads to a
paramagnetic susceptibility which varies inversely with the temperature.
Theories of paramagnetism and diamagnetism
9.1.6
185
Curie-Weiss law
Is there a more general law for the dependence of paramagnetic susceptibility on
temperature?
It was found that the susceptibilities of a number of paramagnetic metals obey a
modified or generalized law known as the Curie-Weiss law [5]. In metals such as
nickel and the lanthanides it was found to vary as shown in Fig. 9.3, which can be
represented at least in the paramagnetic region by an equation of the form
c
where C is again the Curie constant and Tc is another constant with dimensions of
temperature.
Tc can be either positive, negative or zero. Tc = 0 corresponds of course to the
earlier Curie law. For materials that undergo a paramagnetic to ferromagnetic
transition Tc > 0 corresponds to the Curie temperature. For materials that
undergo a paramagnetic to antiferromagnetic transition the term Tc is less than
zero, although in practice the transition temperature between the paramagnetic
and antiferromagnetic phases occurs at a positive temperature TN known as the
Neel temperature.
It should be remembered that the susceptibility only follows the Curie-Weiss
law in the paramagnetic region. Once the material becomes ordered the
(b)
3
(a)
X
v
2
b
)(
~
.3-
0
TC
c
X = T-Tc
Curie-Weiss Low
(T> Tc )
350
400
450
500
Temperature inoC
Fig. 9.3 Variation of X with temperature for paramagnetic materials which undergo a
transition to ferromagnetism at the Curie temperature Te. Left-hand diagram shows a
schematic of the variation of X with T. The right-hand diagram shows the variation of
Ilx with temperature for nickel.
186 Magnetic order and critical phenomena
susceptibility behaves in a very complicated way and no longer has a unique value
for a given field strength.
9.1.7 Weiss theory of paramagnetism
What does the Curie- Weiss law tell us about the interactions between the individual
electronic magnetic moments?
Weiss [6] showed that the variation of paramagnetic susceptibility with
temperature of materials which obeyed the Curie-Weiss law could be explained if
the individual atomic magnetic moments interacted with each other via an
interaction field He which Weiss called the 'molecular field' but more accurately
should be called the atomic field.
Since in paramagnets the magnetization is locally homogeneous the magnetic
moment per unit volume will be everywhere equal to the bulk magnetization M
(unlike in ferromagnets where the local magnetic moment per unit volume, the
'spontaneous magnetization', is unrelated to the bulk magnetization because ofthe
existence of domains). Therefore interactions between any individual magnetic
moment and other moments within a localized volume can be expressed as an
interaction between the given moment and the bulk magnetization M.
This is represented as an interaction field He which can be written as
He = rxM,
where so far we have made no assumptions about the nature of rx. The total
magnetic field experienced by a magnetic moment then becomes
Htot =H+He
and hence
Htot=H+aM.
Consider the paramagnetic susceptibility of a material in which such a field
operates. This is a perturbation of the Langevin model, so a Curie-type law should
still be obeyed providing that the orientation of the magnetic moments is in
thermal equilibrium and obeys Boltzmann statistics.
C
M
The susceptibility is still given by
M
x=H
and substituting H = H tot - aM leads to
C
x=--T-rxC
Theories of paramagnetism and diamagnetism
187
C
X=y_ T'
c
This is the Curie-Weiss law. The derivation shows that a paramagnetic solid
with localized but interacting atomic moments will have a susceptibility that obeys
the Curie-Weiss law. The critical temperature Tc is known as the Curie
temperature and marks the boundary between the paramagnetic and
ferromagnetic states of the material.
9.1.8
Consequences of the Weiss theory
What is the magnetization equation for a material with a Weiss interaction field?
Having established the concept of an interatomic coupling it is possible to provide
an equation for the paramagnetic magnetization as a function of the applied
magnetic field using a perturbation of the Langevin function. In this case the
energy of a magnetic moment within a magnetic field needs to be modified slightly
to
E=
-
f.1.om(H + Q(M)
and consequently the magnetization as a function of field becomes
M = Ms{Coth[f.1.0m(H + Q(M)] _ __kB~
kB T
__ }.
f.1.om(H + Q(M)
This means that the paramagnetic susceptibility is greater in the case of the
interacting moment system. It is also apparent however that at the critical
temperature Tc there arises a discontinuity in the function on the right-hand side of
the equation. So that below Tc the behaviour is very different.
9.1.9
Critique of the Langevin-Weiss theory
In what way does the classical Langevin-Weiss theory of paramagnetism fail?
The Langevin model requires that the magnetic moments be localized on the
atomic sites. The model does not apply to a material in which the moments are not
localized and such materials should not follow the Curie or Curie-Weiss law.
Although a number of paramagnetic materials obey the Curie law most metals do
not. In these cases the susceptibility is independent of temperature.
The Langevin theory does not work for most metals because the magnetic
electrons are usually the outer electrons of the atom which are not localized at the
atomic core. The unpaired electrons must exist in unfilled shells and for many
elements this means that the magnetic electrons are the outer electrons which are
only loosely bound. These are unlikely to remain localized at the atomic sites.
188
Magnetic order and critical phenomena
Despite this the Curie-Weiss law still works very well for some metals such as
nickel for which it is unlikely that the magnetic electrons are tied to the ionic sites.
One possible explanation is that the electrons do migrate but that they spend a
large amount of their time close to the ionic sites which results in behaviour that is
similar to that predicted by a localized model. Also in the rare earth metals and
their alloys and compounds, the 4f electrons which determine the magnetic
properties, are closely bound to the atomic core. Hence these materials do exhibit
the Curie-Weiss type behaviour.
9.2
THEORIES OF ORDERED MAGNETISM
What types of ordered magnetic structures exist and how do they differ?
There are a number of different types of magnetic order in solids including
ferromagnetism, antiferromagnetism, ferrimagnetism and helimagnetism. Some
materials, such as the heavy rare earths, exhibit more than one ordered magnetic
state. These ordered states undergo transitions at critical temperatures so that
every solid that exhibits one of these types of magnetic order will become
paramagnetic at higher temperatures. For example in a ferromagnet the Curie
point is the transition temperature above which the material becomes
paramagnetic and below which an ordered ferromagnetic state exists. The Neel
point is the temperature below which an ordered antiferromagnetic state exists.
Some solids such as terbium, dysprosium and holmium have both Curie and Neel
temperatures.
9.2.1
Ferromagnetism
What causes the transition from paramagnetism to ferromagnetism?
In ferromagnetic solids at temperatures well below the Curie temperature the
magnetic moments within domains are aligned parallel. This can be explained
phenomenologically by the Weiss interaction field which was originally suggested
in order to explain the dependence of paramagnetic susceptibility on temperature
in certain materials.
Examples of ferromagnetic elements are the three familiar transition metal
ferromagnets iron, Tc = 770°C, nickel, Tc = 358°C, and cobalt, Tc = 1131 0c.
Several of the rare earth metals also exhibit ferromagnetism including gadolinium,
Tc = 293 K, dysprosium Tc = 85 K, terbium Tc = 219 K, holmium Tc = 19 K,
erbium Tc = 19.5 K and thulium Tc = 32 K.
The alignment of magnetic moments in various ordered ferromagnetic solids is
shown in Fig. 9.4. As the temperature of a ferromagnet is increased the thermal
energy increases while the interaction energy is unaffected. At a critical temperature the randomizing effect of thermal energy overcomes the aligning effect of the
Theories of ordered magnetism
189
Fe
~I
---f--~{I
1/
Ni
I
Co
Fig. 9.4 Crystallographic alignment of the magnetic moments in various ferromagnetic
solids.
interaction energy and above this temperature the magnetic state becomes
disordered.
9.2.2 Weiss theory of ferromagnetism
How can the Weiss interaction be used to explain magnetic order inferromagnets?
If the unpaired electronic magnetic moments which are responsible for the
magnetic properties are localized on the atomic sites then we can consider an
190
Magnetic order and critical phenomena
interaction between the unpaired moments of the form discussed above in
section 9.1.7. This interaction, which Weiss introduced to explain the
paramagnetic susceptibilities of certain materials, leads to the existence of a critical
temperature below which the thermal energy of the electronic moments is
insufficient to cause random paramagnetic alignment. This means that the effective
field He can be used to explain the alignment of magnetic moments within domains
for temperatures below Te.
There are a number of possible variations on the theme of the interatomic
interaction or exchange field. We will look at two of these: the mean-field
approximation, which was used successfully in the paramagnetic region, and a
nearest-neighbour-only coupling which is more appropriate in the ferromagnetic
regime. We will begin by considering the interaction between just two magnetic
moments.
Suppose that any atomic magnetic moment mi experiences an effective field Heij
due to another moment mj.lf we assume that this field is also in the direction of mj
we can write
Heij = ,1ijmj'
The total exchange interaction field at the moment mi will then simply be the
vector sum of all the interactions with other moments
9.2.3
Mean-field approximation
Is there a simple explanation of the Weiss interaction?
So far we have made no assumption about the form of ,1ij' We can show that if the
interactions between all moments are identical and hence independent of
displacement between the moments, then all of the ,1ij are equal. Let these be a as
we have used before for the mean-field interaction in section 6.1.5. Hence
He=a
L: mj
all j
so that within a domain
He = a(Ms - m i )
The interaction energy of the moment under these conditions is
Ee =
-
= -
/-lomi·He
/-loami· Ms·
This was the original formulation of the Weiss theory. In essence the mean-field
Theories of ordered magnetism
191
approximation is not very realistic because each moment does not interact equally
with all others in the domain. However for moments within the body of a domain it
works out quite well in practice simply because each moment will experience the
same exchange field as its neighbours, and this will be in the direction of the
spontaneous magnetization Ms in the domain. Therefore if the mean-field
parameter is treated completely empirically it can give a reasonable model of the
behaviour of the moments.
Ifwe consider the case of a zero external field, then the only field operating within
a domain will be the Weiss field
When considering the magnetic moments within the body of a domain if we
apply the mean-field model the interaction field will be proportional to the
spontaneous magnetization Ms within a domain. Following an analogous
argument to that given by Langevin for paramagnetism then if there are no
constraints on the possible direction of m we arrive at
Ms
M0
=
coth (JlOmCXMs ) _ kB T .
kB T
JlomcxMs
The solution of this equation leads to perfect alignment of magnetic moments
1.0
o
~
0
~
O.B
~
,\p
Ko
0' 0.6
\.
\
~-...
~
~
0.4
0.2
o
o
0.2
0.4
0.6
O.B
1.0
Fig. 9.5 Variation of the spontaneous magnetization Ms of nickel within a domain as a
function of temperature according to he Weiss mean-field model, after Weiss and Forrer
(1929).
192
Magnetic order and critical phenomena
within a domain as the temperature approaches absolute zero. As T increases the
spontaneous magnetization within a domain decreases as shown in Fig. 9.5. At a
finite temperature which corresponds to the Curie point the spontaneous
magnetization tends rapidly to zero representing the transition from
ferromagnetism to paramagnetism. The above expression can be generalized to
include the effects of a magnetic field H, so that the energy of a moment within a
domain becomes
E = - llom·(H + aM,)
which leads immediately to the following magnetization within a domain
This equation is not encountered very often however because aMs» H in
ferromagnets (e.g. in iron Ms = 1.7 X 106 A/m, and so aMs can be up to 6.8
x 108 A/m, while H will rarely exceed 2 x 106 A/m). Consequently within the body
of a domain the action of the H field is not very significant when compared with the
interaction field.
However it is well known that moderate magnetic fields (H ~ 8 x 10 3 A/m) can
cause significant changes in the bulk magnetization Min ferromagnets. Therefore
these changes occur principally at the domain boundaries where the exchange
interaction is competing with the anisotropy energy to give an energy balance.
Under these conditions the additional field energy can just tip the balance and
result in changes in the direction of magnetic moments within the domain wall.
This mechanism manifests itself as domain-wall motion, as described in Chapter 8.
The magnetic moments in the domain wall, being on the periphery of the domain
do not couple to the spontaneous magnetization of the domains although the
Weiss-type coupling is still strong between nearest neighbours. The net interaction
field per moment is different in this case because the domains on either side are
aligned in different directions. Therefore the energy is very finely balanced in the
domain wall and slight perturbations due to an applied magnetic field can cause
changes in the direction of alignment of the moments which would not be possible
within the body of the domain.
9.2.4
Nearest-neighbour interactions
Can the Weiss model be interpreted on the basis of localized interactions only?
Another variation of the Weiss model which provides mathematically tractable
solutions is the nearest-neighbour approximation [7] in which the electronic
moments interact only with those of its z nearest neighbours. So for a simple cubic
lattice z = 6, for body-centred cubic z = 8, for face-centred cubic z = 12 and for a
hexagonal lattice z = 12. The nearest-neighbour approach is particularly useful for
Theories of ordered magnetism
193
considering magnetic moments in the domain walls since in this case the moments
do not couple to the magnetization within the body of the domain simply because
they lie between domains with different magnetic directions and the direction of
magnetization changes within the wall.
In this approximation we can write the exchange interaction field as
L
He =
/ijm j.
nearest
neighbours
We assume that each nearest-neighbour interaction is identical and equal to /,
where once again / = 0 corresponds to the non-interacting limit described by
Langevin theory. When / is non-zero it is usually convenient to consider that each
moment interacts equally with each of its nearest neighbours. (In order to arrive at
the same order of magnitUde of the exchange field as the mean-field approximation
we need to have Nr:x:::::;, z/.)
L
He=
/mj
nearest
neighbours
=/
L
mj'
nearest
neighbours
,,<
On the basis of this nearest-neighbour interaction we find that / > 0
corresponds to ferromagnetic alignment while
0 corresponds to
antiferromagnetic alignment. This can easily be seen by considering the
configuration of moments which leads to a minimum in the interaction energy.
Ee = - /lom/
L
mj
nearest
neighbours
and summing over the z nearest neighbours
Ee =
-
/loz/ m 2 .
Having established the existence ofthe Weiss interaction it is possible to provide
a description of ferromagnets which is similar to the Langevin model of
paramagnetism. Such a model is strictly only correct for ferromagnets in which the
moments are localized on the atomic cores. Thus it applies to the lanthanide series
because the 4f electrons which determine the magnetic properties, are tightly
bound to the nuclei. The model also works reasonably well for nickel, which obeys
the Curie-Weiss law.
9.2.5
Curie temperature on the basis of the mean-field model
How does the Weiss interaction explain the existence of a critical temperature?
In the original papers by Weiss [5,6] it was shown that the existence of an internal
or atomic field proportional to the magnetization M led to a modified form of
194 Magnetic order and critical phenomena
Curie's law known as the Curie-Weiss law
C
X= T-Tc
From the Curie-Weiss law the Curie constant is given by
NJ-lom2
c= 3kB
and the Curie temperature Tc = (XC is therefore given by
T = J-loN(Xm 2
c
3k B
We see from this that it is possible to determine the mean-field coupling or Weiss
constant (X from the Curie temperature providing the magnetic moment per atom is
known.
Similarly for a nearest-neighbour coupling the interaction parameter ,I can be
found from the Curie temperature using the equation
T = J-loz,lm2
c
3k B
'
9.2.6 Antiferromagnetism
Is it also possible to explain antiferromagnetic order by a Weiss interaction?
Simple antiferromagnetism [8] in which nearest-neighbour moments are aligned
anti parallel can also be interpreted on the basis of the Weiss model. There are two
ways of considering this. One is to divide the material into two sublattices A and B,
with the moments on one sublattice interacting with the moments on the other
sublattice with a negative coupling coefficient, but interacting with the moments
on their own sublattice with a positive coupling coefficient. This ensures that the
magnetic moments on the two sublattices point in different directions. Another
way to envisage the problem is on the basis of nearest-neighbour interactions.
With a negative interaction between nearest neighbours this leads to simple
an tiferromagnetism.
The Curie-Weiss law also applies to antiferromagnets above their ordering
temperatures. However the sign of the constant term Tc in the denominator is
positive so that the law becomes
C
X= T+ Tc'
It therefore appears superficially as though the critical temperature in this case is
below 0 K. In fact the plot of t/x against temperature for these antiferromagnets
does appear to be a straight line intercepting the temperature axis at - ~, but at a
temperature above 0 K known as the Neel temperature the materials undergo an
Theories of ordered magnetism 195
2.0 r----,...----;---..,---,
(b)
rtl
I
~
U
~
(,!)
z 1.0
'0
rtl
\
\
-8
(a)
X
\
x
-1><
\
\
o
TN
T
C
X=T+'8
(T> TN)
/
/
/
250
300
350
Temperature in oK
Fig.9.6 The left-hand diagram is the schematic variation of X with temperature in the
paramagnetic regime of materials which undergo a transformation to antiferromagnetism.
The right-hand diagram is i/x versus T data for terbium.
order-disorder transition after which they cease to obey the Curie-Weiss law, as
shown in Fig. 9.6.
Examples of antiferromagnetic materials are chromium, below its Neel
temperature of 37 DC, and manganese below its Neel temperature of 100 K.
9.2.7 Ferrimagnetism
How can the properties of ferrites be explained?
Ferrimagnetism is a particular case of antiferromagnetism in which the magnetic
moments on the A and B sublattices while still pointing in opposite directions have
different magnitudes. Ferrimagnetic order was first suggested by Neel in 1948 [9]
to explain the behaviour of ferrites. The ferrimagnets behave on a macroscopic
scale very much like ferromagnets so that it was not realized for many years that
there was a distinction. They have a spontaneous magnetization below the Curie
196 Magnetic order and critical phenomena
temperature and are organized into domains. They also exhibit hysteresis and
saturation in their magnetization curves.
The most familiar ferrimagnet is Fe 3 0 4 , although other magnetic ferrites with
the general formula MO· Fe Z 0 3 where M is a transition metal such as manganese,
nickel, cobalt, zinc or magnesium are widely used. These ferrites are cubic and have
the 'spinel' crystal structure referred to in Chapter 8.
Another class of ferrites is made up of the hexagonal ferrites such as barium
ferrite BaO·6(Fe z0 3 ) and strontium ferrite SrO·6(Fe z0 3 ). These are magnetically
hard and have been extensively used as permanent-magnet materials. They have a
high anisotropy with the moments lying along the c axis. Their critical
temperatures are typically in the range Tc = 500-800 dc.
A third group of ferrimagnets are the garnets which have the chemical formula
5Fe z0 3 ·3R z0 3 where R is a rare earth ion. The best known of these is yttriumiron garnet, 5Fe z0 3· 3Y Z03. These materials have a complicated cubic crystal
structure. Their order-disorder transition temperatures are around 550°C.
Another ferrimagnetic material is gamma iron oxide y-Fe 2 0 3 which is widely
used as a magnetic recording medium. This is obtained by oxidizing magnetite
Fe 3 0 4 . Above 400 DC this transforms to rhombohedral alpha iron oxide, or
haematite, which is a canted antiferromagnet.
9.2.8 Helimagnetism
Are there other types of ordered magnetic materials?
So far we have discussed the situations where the nearest-neighbour interaction is
positive (ferromagnetic) and negative (antiferromagnetic). There is a more general
case in which we consider nearest- and next-nearest-neighbour interactions.
As an example we will look at the case of dysprosium as treated by Enz [10] and
later by Nicklow [11]. In the base plane the moments are all aligned
ferromagnetically, however successive base planes have their moments inclined at
an angle 8 to the moments in the next base plane. This gives a helical magnetic
structure.
If /il is the interaction between nearest-neighbour planes and /i2 the interaction
between next-nearest-neighbour planes, the total exchange energy Eex becomes
Eex = - LL/ijcos(8i + j
i
-
8i )m 2 •
j
If the turn angle between moments in successive base planes is 8
j8 t = 8i + j
and if fil
=
f
l'
and fi2
=
f
2
-
8i
for all i, then
Magnetic structure
197
If we assume that the interaction from further than two planes away is negligible by
comparison
At equilibrium
= Nm 2 (2f 1 sin 8t + 4f 2 sin 28J
cos
8
t
f1
= - 4f2'
This gives the value of the turn angle 8t between successive base planes which
leads to the minimum exchange energy. For ferromagnetic alignment we need
cos 8t = 1, or f 1 = - 4f 2> while for simple antiparallel antiferromagnetic
alignment we need cos 8t = - 1, f 1 = 4f 2'
Examples of this form of helimagnetic order occur in terbium, dysprosium and
holmium.
9.3
MAGNETIC STRUCTURE
How is the magnetic structure within a domain determined?
The magnetic structure of materials is usually deduced from neutron diffraction
and magnetization/susceptibility measurements. The use of neutron diffraction for
investigating the structure of magnetic materials has been discussed by Bacon [12]
and more recently by Lovesey [13]. The first material to be studied in this way was
MnO in 1949 and this was followed by other antiferromagnetic oxides. The
determination of the magnetic structure of the 3d series metals iron, nickel and
cobalt by neutron diffraction was first made by Koehler and coworkers at Oak
Ridge National Laboratory beginning in 1951. The magnetic structures of the 4f
series metals, the lanthanides, was also studied by the same group using neutron
diffraction in an extensive research program in the 1950s and 1960s. The results of
this were summarized by Koehler [14] in a review of the magnetic properties of the
rare earths.
9.3.1 Neutron diffraction
How do neutrons interact with magnetic materials?
Although the various different types of magnetic order described in sections 9.1
and 9.2 can be inferred from measurements of magnetic anisotropy and
susceptibility, their existence has only been directly verified by neutron diffraction.
In the older and perhaps more familiar technique of X-ray diffraction a beam of X-
198
Magnetic order and critical phenomena
rays is diffracted by the distribution of electric charge at the periodic lattice sites in
the solid. This is known as Bragg reflection. Neutrons however are diffracted both
by the distribution of the nuclei on the lattice sites and by the magnetic moments
associated with the electron distribution on each atom. This leads to both Bragg
peaks and magnetic peaks in the resulting neutron diffraction spectrum.
Neutrons have a net magnetic moment of 5.4 x 10- 4 Bohr magnetons
(= 5.0 X 10- 27 Am2) but have no electric charge. This presence of magnetic
moment in the absence of charge is itself an anomaly, but we shall not discuss the
problem here. Nevertheless this combination of properties means that neutrons
can pass relatively easily through a solid since they are not influenced by the
localized electric charge distribution. The neutrons do interact with the nuclei to a
greater or lesser extent depending on the type of nuclei present as shown in
Table 9.1. This gives rise to a nuclear scattering component in the total neutron
diffraction spectrum. For the wavelengths used in neutron diffraction the nuclei act
as point scatterers and the nuclear scattering spectrum is therefore isotropic.
The neutrons necessary for neutron diffraction studies of this type must have
wavelengths comparable with the atomic dimensions, which are typically 0.1 nm.
Neutrons with de Broglie wavelengths of this order of magnitude are produced in a
nuclear reactor as thermal neutrons at a temperature of about 300 K and hence an
energy of 4 x 10- 21 J (25 meV) and a corresponding wavelength of 0.18 nm.
The neutron diffraction spectra of magnetic materials contain at least three
different contributions which can be used to examine different properties of the
materials. There is elastic neutron scattering which has two components: the first of
these is the nuclear (or Bragg type) diffraction peaks which are determined by the
periodicity of the lattice, and the neutron scattering cross-section of the nucleus;
the second is the magnetic scattering which is determined by the magnetic order in
the solid and the magnetic scattering cross-section. Finally there is inelastic
Table 9.1 Comparison of nuclear and magnetic scattering amplitudes of various atoms
(after Bacon [12])
Atom
or ion
Nuclear
scattering
amplitude
(10- 12 cm)
Effective
spin quantum
number S
Magnetic scattering amplitude
p(10- 12 cm)
(sin OJ). = 0.25 A-I)
0=0
Cr 2 +
MnH
Fe
Fe 2 +
Fe 3 +
Co
Co2+
Ni
NiH
0.35
-0.37
0.96
0.96
0.96
0.28
0.28
1.03
1.03
2
5/2
1.11
2
5/2
0.87
2.2
0.3
1.0
1.08
1.35
0.60
1.08
1.35
0.47
1.21
0.16
0.54
0.45
0.57
0.35
0.45
0.57
0.27
0.51
0.10
0.23
Magnetic structure
199
800
~40 c
~
.s
200
t
(llO)
Scattering angle
Fig. 9.7 Neutron diffraction spectrum ofiron showing the intensity of diffracted peaks as a
function of scattering angle. The peaks have been indexed with the corresponding
crystallographic directions (after Shull et ai. [51]).
neutron scattering which results in the creation or annihilation of a magnon (spin
wave excitation) and from such measurements it is possible to study the spin wave
spectrum of the solid.
The experimental arrangement for neutron diffraction investigations is similar
to that used in X-ray diffraction. A collimated, monochromatic (i.e. single-energy)
and in some cases polarized beam of neutrons is directed on to the specimen. A
neutron detector can be moved around the specimen to any angle in order to
measure the angular dependence of the intensity of the diffracted beam. Nuclear
scattering patterns have the same critieria as for X-rays and so the Bragg angle f}
can be defined as sin f} = A/21 G 1where A is the de Broglie wavelength of the
neutrons, and 1 GI is the magnitude of the reciprocal lattice vector. The magnetic
reflections are superimposed on the Bragg reflection spectrum. Generally both are
referenced to the crystallographic unit cell, as indicated in Fig. 9.7 which is taken
from the work of Shull et al. [15].
9.3.2 Elastic neutron scattering
What can elastic neutron scattering tell us about the magnetic structure ofa material?
The idea of using neutrons for investigating directly the magnetic structure of
materials was suggested first by Bloch [16J and later in more detail by Halpern and
200
Magnetic order and critical phenomena
Johnson [17]. Elastic neutron scattering gives two types of diffraction peaks. One
group is due to nuclear scattering and these are isotropic, that is they do not
depend on the scattering angle, and they also persist above the magnetic ordering
temperature as shown in Fig. 9.8. The second group, the magnetic scattering
spectrum, is caused by the presence of localized magnetic moments in the solid.
This spectrum is anisotropic and also depends on temperature, the intensity of the
peaks decreasing with increasing temperature until at the magnetic ordering
7
6
...§
4K
x
Nuclear
...
5
I
C'I
I
C'I
•
II)
CD
I I I I
•. ChS!:~
1111111
"t:~N
I II I I I I
~
~
I I I
4
Nuclear + magnetic
3
2
1
I)~
11)11)
M •
II)
~
CD ... CDm ...
11111111
1111
Magnetic
I I
Or-~L
3
2
lOOK
Nuclear
Superlattice
II)
CD
I I
1
o
20
40
Scattering angle 29 (degrees)
Fig. 9.8 Neutron diffraction spectra for TbIn 3 above and below the magnetic ordering
temperature (after Crangle [18]).
Magnetic structure
201
temperature the spectrum disappears to become a diffuse background due to
paramagnetic scattering.
The cross-sections for neutron-electron and neutron-nuclear scattering are
usually comparable in magnitude, with either being larger depending on the
particular case. This means that the spectral peaks due to Bragg diffraction and
magnetic scattering are usually comparable in magnitude. The two contributions
can however be distinguished for example by the application of a magnetic field
(since the nuclear component remains isotropic) or by raising the temperature
above the magnetic ordering temperature (whereupon the magnetic component
becomes diffuse due to paramagnetic scattering). Once the magnetic scattering
contribution to the spectrum has been isolated the distribution, direction and
ordering of the magnetic moments within the solid can be determined.
Paramagnetic scattering
In the case of a paramagnet the magnetic scattering is diffuse and appears as a
contribution throughout the background which decreases in intensity with
increasing angle of scatter e. Paramagnetic scattering is found by subtracting all
other contributions from the spectrum, providing that the magnetic contribution is
sufficiently large compared with the other contributions to render this calculation
accurate enough.
Ferromagnetic scattering
The Bragg and magnetic scattering spectra have peaks in the same locations when
neutrons are diffracted by a ferromagnet. Therefore the two components of the
spectrum will be completely superposed as shown in Fig. 9.9(a). The magnetic
contribution however decreases with temperature and therefore the two
contributions can be distinguished by making measurements above and below the
Curie point.
Simple antiferromagnetic scattering
In the case of simple antiferromagnetism the magnetic moments on neighbouring
atoms point in opposite directions. This leads to a doubling of the crystallographic
repeat distance for magnetic moments and hence to a halving ofthe repeat distance
in the reciprocal lattice. Therefore additional magnetic peaks appear midway
between the nuclear scattering peaks in the spectrum as shown in Fig. 9.9(b). The
magnetic peaks occur at sin evalues which are half those of the expected nuclear
diffraction peaks. The scattering of neutrons by antiferromagnets is discussed in
detail by Bacon [12 (p. 208)].
Helical antiferromagnetic scattering
In helimagnetic materials the diffraction peaks consist of the central nuclear peaks
each accompanied by a pair of satellite peaks due to the magnetic scattering as
(a)
.~
~L-28
.5
~
=
:;
u
Z
(b)
p
Q
28
Fig. 9.9 Neutron diffraction spectra: (a) for a ferromagnetic solid showing superposition of
nuclear and magnetic peaks; and (b) for a simple antiferromagnetic solid in which the
nuclear and magnetic peaks occur at different locations (after Bacon [12]).
200
(OOO)±
(929)
(101)
?;-ISO
.~
~
.=§ 100
...
(103)
::I
I
(002)
II
Z
(002)O~-
(11 2)
(110)
50
I
(101)(11O)"t
(002)+
1,101)+ (103)II
(004)I
(004)
(D2~04)+
1
II (103)+(112)+
28
Fig. 9.10 Neutron diffraction spectrum for a helical antiferromagnetic structure (after
Bacon [12]).
Magnetic structure
203
shown in Fig. 9.10. The displacement of the satellite peaks from the nuclear peaks
can be used to determine both the direction of the helical axis and the magnitude of
the turn angle between successive helical planes.
The technique of neutron diffraction finds its greatest use in the investigation of
antiferromagnetic (including helical antiferromagnetic) and ferrimagnetic
structures. The case of ferromagnetic materals is fairly trivial by comparison since all moments lie parallel within a domain, and therefore little
further information can be obtained, since the crystallographic easy axes can be
determined from anisotropic susceptibility measurements. However information
about the spin wave spectrum can be obtained from inelastic neutron scattering
data on ferromagnets as they approach the Curie point.
9.3.3
Inelastic neutron scattering
How can we study higher energy states (magnetic excitations) in the magnetic
structure?
The thermal fluctuations of the individual atomic magnetic moments in a
magnetically ordered solid become particularly prevalent as the magnetic
ordering temperature is approached. These spin waves can be studied by inelastic
neutron scattering. It was observed for example by Squires [19] that the magnetic
scattering cross-section for iron has a peak at the Curie point as shown in Fig. 9.11
which is caused by inelastic scattering of neutrons by spin clusters. This scattering
becomes greater as the temperature is increased towards the Curie temperature as
the spin fluctuations become greater. A futher example for nickel [20] is given in
Fig. 9.12.
-e.
19~-.,
II>
E
,g 15
~
II>
II>
eu
~
________
10~
o
~
__________
500
Temperature (. K)
~
__- J
1000
Fig. 9.11 Total scattering cross-section for inelastic scattering of neutrons by iron as a
function of temperature (after Cribier et al. [19]).
204
Magnetic order and critical phenomena
1500
1000
II>
8
...
8.
I
!!l
c
::s
0
U
500
o
300
Temperature (0C)
Fig. 9.12 Total scattering cross-section for inelastic scattering of neutrons by nickel as a
function of temperature (after Cribier et al. [20J).
A more detailed discussion of neutron diffraction by magnetic solids is beyond
the intended scope of this book. However those interested in this area of
magnetism are referred to the excellent, recent, two-volume work by Lovesey [13]
which provides a comprehensive summary of the subject.
9.3.4
Magnetic order in various solids
What examples do we have of these many different types of magnetic order?
The ordering of the magnetic moments within domains in various rare earth metals
is shown in Fig. 9.13. For the cubic metals such as iron and nickel the magnetic
moments align preferentially along the 100> and 111 > axes respectively
[21,22]. The magnetization curves along the various crystal axes in iron are
shown in Fig. 9.14, and in nickel in Fig. 9.15. From these it is clear that the initial
and low-field susceptibility of iron is highest along the 100> axes and in nickel
is highest along the 111 > axes. Conversely it is more difficult to magnetize iron
along the <111 ) axes and nickel along the <100) axes. Cobalt has a hexagonal
crystal lattice. In this case the moments are aligned along the unique axis [0001]
which is the easy direction [23], while the [1010] axis in the base plane is the
hard axis. The magnetization curves along these directions are shown in Fig. 9.16.
<
<
<
<
,....12.5 ~-,.19
05 - r - 90 - -293 -- -229-:: -\'178.5- -132 - -85- -56-221':"
~
c!::>
C!:>
13
C!:>
<3
C!5)
(?)G-
r-<;?o-
13
c::>
d:> G
BdJ<3
[!]
EJc!:JG
r7
....-h.
/?,.
fy~gG)<7J
ILl
13
[?J
Ce
Pr
8m
Nd
\...:J
~
0
0
~
0
(::)
1-25-
C¥Q
I-&;- 0
Gd>cb~
Tb
Qf-40-
r--.cb
1-85- ~
G
Gd
~53.-
8d>
c!:J <3 G
Eu
CU
<9d>c!:)
Oy
Ho
Tm
Er
Fig. 9.13 Magnetic ordering of magnetic moments within the domains of various rare earth
metals under zero applied magnetic field.
Easy
<111 >
400
E
~
300
:::l
E
Q)
~
200-
100
-
oL--L__
o
L-~
100
__
~
200
__
300
__~
~-L
400
__~
500
__
600
H(Oc)
Fig. 9.14 Magnetization curves for iron along the three axes (100), (110) and (111).
1800
Hard
1200
E
~
:J
1000
E
Q)
~
-
800
600
400
-
200
0
0
200
400
600
1000
800
H(Oc)
Fig. 9.15 Magnetization curves for nickel along the three axes <100), <110) and <111).
1400
1200
1000
~
E
:J
E
Q)
~
800
600
400
'200
o
1000
2000
3000
4000
5000
6000
7000
8000
9000 10,000
H(Oc)
Fig. 9.16 Magnetization curves for cobalt along the unique axis <0001) and the base plane
<WID).
Magnetic structure
207
\
\
\~
\
\
\
unit cell
Fig. 9.17 Simple antiferromagnetism in manganese.
Simple antiparallel antiferromagnetism occurs in chromium and manganese
and these ordered structures are shown in Fig. 9.17 [24,25]. Antiferromagnetism is
actually far more commonly occurring than ferromagnetism, but so far no use has
been found for this type of magnetic order in solids.
In the rare earth metals the magnetic ordering can be much more complex. In
gadolinium [26-29J, which exhibits the simplest magnetic structure among the
rare earths, the magnetic moments are aligned along the c axis when the
temperature lies between the Curie point of 293 K and the spin-reorientation
temperature of 240 K. Below 240 K the moments deviate from the (001) axis and
the angular deviation increases continuously as the temperature is reduced. In this
temperature range we speak of gadolinium as a 'canted' ferromagnet.
Terbium [30J and dysprosium [31,32, 33J, the next two elements in the periodic
table, behave very differently from gadolinium. At the Neel temperature,
Tn = 230K for terbium and Tn = 180K for dysprosium, which is the ordering
temperature for antiferromagnets, they both form base-plane helical
antiferromagnets. In terbium the easy axis is the [1010J direction (the b axis) while
in dysprosium it is the [1000J direction (or a axis). At a lower temperature of 219 K
in terbium or 85 K in dysprosium (which are the Curie temperatures) they form
base-plane ferromagnets. In terbium the domains have their axes along the b axes
while in dysprosium the moments align along the a axes.
Holmium has an ordering temperature of 132 K, below which it forms a baseplane antiferromagnet [34J that is similar to dysprosium, with an easy axis along
the [1000J direction. This helical structure is retained down to 20 K, the Curie
208
Magnetic order and critical phenomena
point, below which there appears a ferromagnetic c axis component, while the
base plane continues to exhibit helical antiferromagnetic order.
Erbium has a Neel point of 85 K, below which the magnetization is sinusoidally
modulated along the c axis [34,35,36]. At 53 K a further transition occurs when the
base plane orders helically, and the c axis structure begins to change to a squarewave modulation of four moments up followed by four moments down. This
square-wave modulation is never completed, however. At the Curie point of 20 K
the c axis components become ferromagnetic, while the base plane continues its
helical order. This leads to a 'conical' ferromagnetic structure of the type observed
also in holmium.
Ferrimagnetism is exhibited by a number of transition metal oxides known as
ferrites. These fall into three classes: the cubic ferrites, also known as spinels, such
as NiO'Fe203' which are soft magnetic materials with the exception of cobalt
ferrite which is hard; the hexagonal ferrites such as BaO'6(Fe 20 3) which are hard
magnetic materials used to make ceramic magnets; and the garnets such as
yttrium-iron garnet 5Fe203'3Y203'
9.3.5 Excited states and spin waves
How does temperature affect the alignment of magnetic moments within a domain?
We have noted above in section 6.2.5 that the spontaneous magnetization within a
domain Ms is only equal to the saturation magnetization Mo = Nm at absolute
zero of temperature (0 K). The reason for this is that the thermal energy at
temperatures above 0 K causes some misalignment of the directions of magnetic
moments within the domains. We shall now explain how this misalignment occurs.
Consider the ground state of a six-moment linear ferromagnet as shown in
Fig. 9.l8(a). The ground state is clearly when all six moments are aligned parallel.
On the basis of nearest-neighbour-only interaction this has a ground energy of Eo
given by
I II "I II J " I
-lal-
-lal--
(a)
(b)
(c)
Fig. 9.18 A six-moment linear ferromagnetic chain: (a) ground state, (b) a state with one
moment antiparallel, and (c) a state with each successive moment at an angle () to its
neighbours.
Magnetic structure
209
There are several possible candidates for the lowest-level excited state. One
possible candidate is the configuration with one moment pointing anti parallel to
the rest as shown in Fig. 9.18(b). This will have an energy of
E1 = - 2110/ m 2
=
Eo + 8110/ m 2
and in fact for a linear chain with any number of moments greater than 2 the energy
of the system with one moment anti parallel is always 8110/ m 2 above the ground
state.
Another candidate excited state is one in which each moment is aligned at an
angle to the direction of the previous moment as shown in Fig. 9.18(c). In this case
the energy of the system is
e
E(e) =
=
-
10110/m 2 cos e
Eo + 101l0/m2(1 - cos e).
This allows for much lower-energy excited states which in the classical
approximation form a continuum of allowed energy states above the ground state,
each with gradually increasing values of e.
The spontaneous magnetization within a domain therefore decreases as the
temperature increases above 0 K.1t can be shown from an analysis of the spin-wave
structure [37] that
kBT
=Mo[ l-c ( 1l0/ m2
)3/2J
'
where 110/m 2 is the nearest-neighbour exchange constant in joules and c is a
constant. c = 0.118 for a simple cubic lattice, c = 0.059 for a body-centred cubic
lattice, and c = 0.029 for a face-centred cubic lattice.
9.3.6 Critical behaviour at the ordering temperature
What happens to other bulk properties of a ferromagnet at the Curie temperature?
The bulk properties of magnetic materials show anomalous behaviour in the
vicinity of the transition temperatures such as the Curie and Neel points. The
anomalous behaviour is due to coupling between the particular bulk property and
the magnetic structure. The effects are known as critical phenomena. We have
already remarked that the bulk susceptibility has anomalous behaviour close to Te,
for example. Other properties such as the specific heat, elastic moduli,
magneto stricti on, magnetoresistance and thermal expansion all reveal critical
behaviour at the magnetic transition temperatures.
210
Magnetic order and critical phenomena
9.3.7 Susceptibility anomalies
How does the susceptibility behave at the ordering temperature?
We have shown that the susceptibility of ferro magnets, ferrimagnets and
antiferromagnets behaves in an anomalous way as the temperature is reduced in
the paramagnetic regime towards the critical temperature Te. In many cases these
materials obey the Curie-Weiss law and this leads to a dependence of susceptibility
on temperature of the form X = C/(T - Te) in the paramagnetic regime. The
susceptibility therefore starts to become very large as Te is approached from above.
Work on the magnetization curves and susceptibility of a number of rare earth
metals which obey the Curie-Weiss law was performed by Legvold, Spedding and
coworkers in a series of papers [38-42].
9.3.8
Specific heat anomalies
How does the specific heat behave at the ordering temperature?
The specific heats of materials which undergo order-disorder transitions show
lambda-type anomalies at the critical temperature. Some examples are shown in
Fig. 9.19 from the work of Hofmann et al. [43]. Measurements of the heat
capacities of several heavy rare earth elements were made by Spedding et al. [4448].
The specific heat of a magnetic material has a magnetic component Cm given by
C= (_1
)(dEdT
/.loP
ex )
m
where Eex is the total exchange self energy of the material per unit volume.
Eex =
-
/.lo.! M/
C = _2(.!Ms) dMs.
m
p
dT
The dependence of Ms on temperature is known from the equation, given in
section 9.2.3,
kBTM] '
dMs/dT=Mo-dd [ coth (/.lomaMs)
k
T
BT
/.loma s
The variation of Cm with temperature is plotted for nickel in Fig. 9.20. The
behaviour gives a remarkably good fit to the experimental data for such a simple
model. The experimental results show some broadening of the lambda anomaly in
the paramagnetic region.
------------------------------------------4
Fig. 9.19 Examples of specific heat anomalies in nickel, iron and gadolinium close to their
ordering temperatures (after Hofmann et al. (1956)).
10
I
Q)
"0
E
I
Ol
Q)
'0
"iii
()
TEMPERATURE IN K
Fe
IQ) 10
"0
E
I
Ol
Q)
'0
"iii 5
()
~
E
U
0
1,500
TEMPERATURE IN K
2,000
8
Gd
IQ) 6
"0
E
I 4
Ol
Q)
'0
"iii
()
~
2
E
U
0
100
200
300
TEMPERATURE IN K
500
212
Magnetic order and critical phenomena
3.0
2.0
,,
~
:::J
0
~
:::J
«
()
?;
u
I
E
7
\
\
,
\
\
I
1.0
I
\
/
\
/
o-~
0.2
--
.., .... ..,"'"
0.4
,.
,. ,. "
0.6
TITc
/
\
0.8
,
" ...
1.0
1.2
Fig. 9.20 Specific heat anomaly for nickel at its Curie point compared with the theoretical
prediction.
9.3.9 Elastic constant anomalies
How do the elastic properties behave at the ordering temperature?
The elastic constants of materials show critical behaviour close to magnetic phase
transitions such as the Curie point as shown in Fig. 9.21. Magnetoelastic anomalies are known to occur in rare earth metals as a result of the very strong
magnetoelastic coupling. These have been thoroughly investigated by Palmer a:J.d
coworkers in a series of papers [49-55] and Moran and Luthy [56,57]. Some of
the theoretical aspects have been addressed by Tachiki and Maekawa [58].
effect in
The dependence of elastic modulus on magnetization, the so called ~E
iron is well known and has been reported by Bozorth [59].
9.3.10 Thermal expansion anomalies
How does the thermal expansion behave at the ordering temperature?
The thermal expansion and magnetostriction also undergo unusual behaviour at
phase transitions such as at the Curie and Neel points as shown in Fig. 9.22. This is
Magnetic structure
213
(a)
x 10"
82
80~
~
~ 78
u
76
74
72
o
50
150
100
200
250
300
T(K)
(b)
7.38
°o o
o
o
°o o
7.26
o
0 00
o
o
o
°'b
7.14
o
o
o
00
7.02'------'-...,..I_--l.-......._--l.._-.!.._....l..-o°':'.-O..'_....l...._...l'
200
225
250)
275
300
325
T(K)
Fig.9.21 Critical behaviour of the elastic constants of dysprosium (above) and gadolinium
(below).
because there is the sudden appearance of spontaneous magnetostriction at the
order-disorder transition temperature. Anomalous thermal expansion and
magnetostriction have been investigated by Bozorth and Wakiyama [60] and by
Greenough and coworkers in a number of papers [61-63].
214
Magnetic order and critical phenomena
5.0
4.0
X
10- 3
,
II
"""
I,
.,
""
I,
3.0
II
""
..
"""
,
2.0
1.0
Tc
0r-~
100
-1.0
Fig. 9.22 Anomalous behaviour of the c-axis thermal expansion of dysprosium at its Neel
temperature of 180 K and Curie temperature of 85 K.
9.3.11 The Ising model
Is there a thermodynamic model which can be used to describe the behaviour of
magnetic materials as a function of temperature and field?
The idea of a Weiss-type coupling between electronic moments can be
incorporated into a simple classical model which describes the effects of
temperature and magnetic field on the magnetization. The Ising model [64]
provides a useful mathematical description of critical phenomena and is
particularly relevent to magnetic materials. It has the unique distinction of being
Magnetic structure
215
the only model of a second-order phase transition which has so far yielded a
mathematical solution.
In the Ising model as applied to magnetic materials [65,66] the solid is divided
into 'cells' each of which contains one magnetic moment or spin. Each cell is
allowed a limited number of possible states and in the simplest models only two
states are allowed which correspond to 'spin up' and 'spin down'. At first sight this
restriction may appear to be unrealistic, because classically the spins can point in
any direction. But in quantum mechanics the spins are restricted to certain in
directions and in particular the directions of the spins can be referenced parallel
and anti parallel to an applied magnetic field H to which they are restricted in
quantum mechanics. Therefore this actually results in a quite reasonable
approximation of reality.
One drawback however is that in the simple Ising model described here the
restriction on orientation of the spins does not allow spin waves. The lowest excited
energy state that can be produced is therefore when one spin is anti parallel to the
rest and consequently there exists an energy gap of 8flo'/ m2 between the ground
state and the first excited state.
The Ising model has then two imposed restrictions and these are (a) that the
spins have two states and (b) that there are interactions between the spins which
correlate their orientations, but the interactions are restricted to nearestneighbour, or nearest-neighbour and second-nearest-neighbour, or else all
moments interact equally (which is equivalent to the mean-field approximation).
As before the magnetic interaction has the form
Eex = - flo L ,/mimj.
We can define an order parameter for the Ising model YJ given by
(p+ - p-)
YJ=---(p+
+ p-)
such that when YJ = 0 the system is paramagnetic and when YJ =F 0 the system is
ferromagnetic. A phase transition therefore occurs when YJ reaches zero.
A mathematical treatment of the probable configuration of magnetic moments
on the atomic sites in the two-dimensional Ising lattice has shown that an orderdisorder transition occurs at a Curie temperature of
T
C
=
0. 88E ex
kB
0.88 Js 2
kB
0.88flo'/ m 2
kB
where Eex is the nearest-neighbour coupling energy between the spins. Similarly, as
shown above for the mean-field model the specific heat has a lambda anomaly.
216
Magnetic order and critical phenomena
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15.
16.
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18.
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20.
Langevin, P. (1905) Annales de Chem. et. Phys., 5, 70.
Van Vleck, 1. H. (1932) The Theory of Electric and Magnetic Susceptibilities, Oxford.
Pauli, W. (1926) Z. Phys., 41, 81.
Curie, P. (1895) Ann. Chem. Phys., 5, 289.
Weiss, P. (1906) Compt. Rend., 143, 1136.
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Enz, U. (1960) Physica, 26, 698.
Nicklow, R. M., Wakabayashi, N., Wilkinson, M. K. and Read, R. E. (1971) Phys. Rev.
Letts., 26, 140.
Becon, G. E. (1975) Neutron Diffraction, 3rd edn. Clarendon Press, Oxford.
Lovesey, S. W. (1984) Theory of Neutron Scattering from Condensed Matter, Vo!' 2:
Polarization Effects and Magnetic Scattering, Clarendon Press, Oxford.
Koehler, W. C. (1965) J. Appl. Phys., 36, 1078.
Shull, C. G., Wollan, E. O. and Koehler, W. C. (1951) Phys. Rev., 84, 912.
Bloch, F. (1936) Phys. Rev., 50, 259.
Halpern, O. and Johnson, M. H. (1939) Phys. Rev., 55, 898.
Crangle, 1. (1977) The Magnetic Properties of Solids, Edward Arnold, London.
Squires, G. L. (1954) Proc. Phys. Soc. Lond. A67, 248.
Cribier, D., Jacrot B. and Parette, G. (1962) J. Phys. Soc. Jap. 17 (Supp!. BIll Proceedings
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
of the International Conference on Magnetism and Crystallography, Kyoto), 67.
Honda, K. and Kaya, (1926) S. Sci. Reps. Tohuku Univ. 15, 721.
Kaya, S. (1928) Sci. Reps. Tohuku Univ. 17, 639.
Kaya, S. (1928) Sci. Reps. Tohuku Univ. 17, 1157.
Corliss, L. M., Hastings, J. M. and Weiss, R. J. (1959) Phys. Rev. Letts., 3, 211.
Shull, C. G. and Wilkinson, M. K. (1953) Revs. Mod. Phys. 25, 100.
Cable, J. W. and Wollan, E. O. (1968) Phys. Rev., 165, 733.
Graham, C. D. (1963) J. Appl. Phys., 34, 1341.
Corner, W. D., Roe, W. C. and Taylor, K. N. R. (1962) Proc. Phys. Soc., 80, 927.
Corner, W. D. and Tanner, B. K. (1976) J. Phys. c., 9, 627.
Koehler, W. c., Child, H. R., Wollan, E. O. and Cable, 1. W. (1963) J. Appl. Phys., 34,
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
1335.
31. Bly, P. H., Corner, W. D. and Taylor, K. N. R. (1969) J. Appl. Phys., 40, 4787.
32. Landry, P. C. (1967) Phys. Rev., 156, 578.
33. Wilkinson, M. K., Koehler, W. c., Wollan, E. O. and Cable, 1. W. (1961) J. Appl. Phys.,
32,48S.
34. Koehler, W. c., Cable, 1. W., Wollan, E. O. and Wilkinson, M. K. (1962) J. Phys. Soc.
Jap., 17, (Supp!. BIll Proceedings of the International Conference on Magnetism and
Crystallography, Kyoto), 32.
35. Rhyne, 1. 1., Foner, S., McNiff, E. 1. and Dodo, R. (1968) J. Appl. Phys., 39, 892.
36. Cable,1. W., Wollan, E. 0., Koehler, W. C. and Wilkinson, M. K. (1961) J. Appl. Phys.,
32,49S.
37. Bloch, F. (1930) Z. Phys. 61, 206.
38. Nigh, H. E., Legvold, S. and Spedding, F. H. (1963) Phys. Rev., 132, 1092.
39. Hegland, D. E., Legvold, S. and Spedding, F. H. (1963) Phys. Rev., 131, 158.
40. Behrendt, D. R., Legvold, S. and Spedding, F. H. (1958) 109, 1544.
41. Strandburg, D. L., Legvold, S. and Spedding, F. H. (1962) Phys. Rev., 127.
42. Green, R. W., Legvold, S. and Spedding, F. H. (1961) Phys. Rev., 122, 827.
Examples and exercises
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
217
Hofmann,1. A., Paskin, A., Tauer, K. 1. and Weiss, R. 1. (1956) J. Phys. Chern. Sol., 1, 45.
Jennings, L. D., Stanton, R. M. and Spedding, F. H. (1957) J. Chern. Phys., 27, 909.
Skochdopole, R. E., Griffel, M. and Spedding, F. H. (1955) J. Chern. Phys., 23, 2258.
Griffel, M., Skochdopole, R. E. and Spedding, F. H. (1954) Phys. Rev., 93, 657.
Griffel, M., Skochdopole, R. E. and Spedding, F. H. (1956) J. Chern. Phys., 25, 75.
Gerstein, B. c., Griffel, M., Jennings, L. D., Miller, R. E., Skochdopole, R. E. and
Spedding, F. H. (1957) J. Chern. Phys., 27, 394.
Palmer, S. B. and Lee, E. W. (1972) Proc. Roy. Soc., A327, 519.
Palmer, S. B., Lee, E. W. and Islam, M. N. (1974) Proc. Roy. Soc., A338, 341.
Isci, C. and Palmer, S. B. (1977) J. Phys. Chern. Sol., 38, 1253.
Isci, C. and Palmer, S. B. (1978) J. Phys. F., 8, 247.
Jiles, D. C., Blackie, G. N. and Palmer, S. B. (1981) J. Mag. Mag Mater., 24, 75.
Jiles, D. C. and Palmer, S. B. (1980) J. Phys. F., to, 2857.
Jiles, D. C. and Palmer, S. B. (1981) J. Phys. F., 11, 45.
Luthi, B., Moran, T. 1. and Pollina, R. 1. (1970) J. Phys. Chern. Soc., 31, 1741.
Moran, T. 1. and Luthi, B. (1970) J. Phys. Chern. Sol., 31, 1735.
Tachiki, M. and Maekawa, S. (1974) Prog. Theor. Phys. Jap., 51, 1.
Bozorth, R. M. (1951) Ferromagnetism, Van Nostrand, New York.
Bozorth, R. M. and Wakiyama, T. (1963) J. Phys. Soc. Jap., 18, 97.
Greenough, R. D. and Isci, C. and Palmer, S. B. (1977) Physica, 86-88B, 61.
Greenough, R. D. and Isci, C. (1978) J. Mag. Mag. Mater, 8, 43.
Greenough, R. D. (1979) J. Phys. c., 12, 1113.
Ising, E. (1925) Z. Phys. 31, 253.
Stanley, H. E. (1971) Introduction to Phase Transition and Critical Phenomena,
Clarendon Press, Oxford.
Green, H. S. and Hurst, C. A. (1964) Order-Disorder Phenomena, Wiley Interscience,
London.
FURTHER READING
Bacon, G. E. (1975) Neutron Diffraction, 3rd edn, Clarendon Press, Oxford.
Chikazumi, S. (1964) Physics of Magnetism, Wiley, New York, Ch. 4 and 5.
Coqblin, B. (1977) The Electronic Structure of Rare Earth Solids, Wiley, New York.
Cullity, B. D. (1977) Introduction to Magnetic Materials, Addison-Wesley, Reading, Mass.,
Ch. 3,4 and 5.
Elliott, R. J. (1972) Magnetic Properties of Rare Earth Metals, Plenum London.
Lovesey, S. W. (1984) Theory of Neutron Scattering from Condensed Matter, Clarendon
Press, Oxford.
Mackintosh, A. R. (1977) The Magnetism of Rare Earth Metals, Physics Today, June, 23.
EXAMPLES AND EXERCISES
Example 9.1
Paramagnetic susceptibility of oxygen. Calculate the susceptibility
at O°C of a paramagnetic gas of molecular mass 32 with a magnetic moment of
3 Bohr magnetons per molecule (S = 3/2).
Example 9.2 Magnetic mean interaction field for iron. Derive the relationship
between the Weiss 'molecular' field of a ferromagnet and the Curie temperature.
Calculate the value of this field for iron which has a Curie temperature of 770°C
and an effective magnetic moment per atom of 2.2 Bohr magnetons.
218
Magnetic order and critical phenomena
Example 9.3 Critical behaviour of spontaneous magnetization. In the mean-field
approximation for a system with two possible microstates (e.g. magnetic moments
either parallel or antiparallel to a unique axis) the spontaneous magnetization
within a domain is given by Ms = Mo tanh [JlomaMJkBTJ. Show that at just
below the Curie point the spontaneous magnetization varies with .J(Tc - T).
10
Electronic Magnetic Moments
In this chapter we discuss the prime cause of the magnetic moment within an
individual atom. We wi11look at the properties of electrons which are of central
importance to magnetism and in particular the origin of the electron's magnetic
moment which is a result of its angular momentum. We also look at how the
magnetic properties of the electrons lead to differences in the available energy
states in the presence of a magnetic field. Finally we show how the magnetic
moments ofthe electrons are combined to give the magnetic moment of the atom.
10.1
THE CLASSICAL MODEL OF ELECTRONIC MAGNETIC
MOMENTS
Why do electrons have a magnetic moment?
In the classical model the angular momentum of the electrons can be used to
determine the magnetic moments of the electrons by invoking the concept of
electrical charge in motion. It is known from Chapter 1 for example that a current
loop behaves as a magnetic dipole and has an associated magnetic moment. There
are two contributions to the electronic magnetic moment: an orbital magnetic
moment due to orbital angular momentum, and a spin magnetic moment due to
electron spin.
10.1.1
Electron orbital magnetic moment
How does the angular momentum of an electron lead to a net magnetic moment?
We can envisage an electron moving in an orbit about an atomic nucleus, as shown
in Fig. 10.1, with orbital area A and period r. This would then be equivalent to a
current i given by
. e
1=-.
r
From the earlier definitions, in sections 2.1.1 and 9.1.2, this gives an orbital
220 Electronic magnetic moments
Fig. 10.1 Classical model of an electron in an elliptical orbit around a nucleus.
magnetic moment mo
mo=iA
eA
The angular momentum of such an orbital Po will be
2dlj>
Po =me r dr'
where me is the mass of an electron and r is the radius
We can therefore write down the orbital magnetic moment mo of the electron in
terms of the orbital angular momentum Po as
mo= -(2:.)PO
remembering that the magnetic moment vector points in the opposite direction to
the angular momentum vector because the electron has negative charge.
10.1.2
Electron spin magnetic moment
How does the electron spin contribute to the magnetic moment?
The electronic spin angular momentum Ps also generates a spin magnetic moment
ms. In this case the relation is
eps
ms= - - .
me
Notice that for a given angular momentum the spin gives twice the magnetic
moment of the orbit.
The quantum mechanical model of electronic magnetic moments
221
10.1.3 Total electronic magnetic moment
How can we combine the spin and orbital angular momentum contributions to get the
total electon magnetic moment?
If we consider the total magnetic moment per electron as the vector sum of the
orbital and spin magnetic moments
=_
(_e
)2P (_e )po.
2me
2me
s_
These terms on the right-hand side can then be combined to give
mtot = - g
(2:
e
)Ptot,
where now Ptot is the total angular momentum of the electron. The term 'g' is called
the Lande splitting factor, which has a value of g = 2 for spin-only components and
g = 1 for orbital-only components of magnetic moment. The value of the splitting
factor must therefore lie between 1 and 2 depending on the relative sizes of the
contributions from the spin and orbit to the total angular momentum.
10.2 THE QUANTUM MECHANICAL MODEL OF ELECTRONIC
MAGNETIC MOMENTS
How are the above definitions modified as a result of quantum mechanics?
We have shown abov~
how the angular momentum of electrons leads to a net
magnetic moment. However in the discussion we have so far allowed all vaues ofPo
and Ps' In fact this is not realistic, the possible values of angular momentum are
restricted by quantum mechanics, and consequently the magnetic moments are
also quantized. We begin by defining the quantum numbers needed to fully
describe electrons within atoms, then look at the restrictions on magnetic moment,
and finally look at experimental evidence confirming the quantization of magnetic
moments, specifically the Zeeman effect and the Stern-Gerlach experiment.
We need to use four quantum numbers in order to define uniquely each electron
in an atom. These are the principal quantum number n, defined by Bohr, the
angular momentum quantum number I defined by Sommerfeld, the orbital
magnetic quantum number ml and the spin magnetic quantum number ms.
10.2.1 Principal quantum number n
What energies can electrons have within the atom?
This n is the quantum number introduced by Bohr in his theory of the atom [1]. It
determines the shell of the electron, and hence its energy. In Bohr's original
222
Electronic magnetic moments
G
n=l
k=l
n=2
k=2
n=3
k=3
Fig. 10.2 Electron orbitals with the same energy but with different angular momentum
described in the Bohr-Sommerfeld theory.
formulation it also defined the orbital angular momentum but in later models
several different orbitals with different angular momenta, but degenerate energy,
were found to exist for a given n, as shown in Fig. 10.2.
The energy En of an electron with principal quantum number n is then,
Z2 m e4
En = - 8n2h;BO 2'
where
n = 1,2,3, ... ,
and Z is the atomic number, me the mass of the electron, e the electronic charge and
the permittivity of free space. The maximum number of electrons permitted in
the nth shell is 2n 2 •
Bo
10.2.2 Orbital angular momentum quantum number 1
What values oforbital angular momentum are possible for electrons within the atom?
This 1is synonymous with the quantum number k introduced in the Sommerfeld
theory of the atom [2] except that 1= k - 1. It was needed to define the orbital
The quantum mechanical model of electronic magnetic moments
223
angular momentum Po of the electron once it was realized that one energy level
could have more than one allowed value of angular momentum. It is a measure of
the orbital angular momentum of the electron when multiplied by (h/2n). It also
gives a measure of the eccentricity of the electron orbit.
It can have value of 1= 0,1,2,3 ... (n - 1) where n is the principal quantum
number.
10.2.3
Spin quantum number s
What values of spin angular momentum can electrons have within the atom?
Electrons have spin angular momentum which can be represented by the spin
quantum number s. The value of s is always 1/2 for an electron. The angular
momentum due to spin sh/2n is therefore always a multiple of h/4n
The total angular momentum quantum number j of an electron is not an
independent quantum number since it is determined by I and s. However it is a
useful quantity to define, particularly since it gives a measure of the total magnetic
moment of an electron. It is the vector sum of the spin and orbital angular
momenta and is necessarily quantized
p.=j(~)
10.2.4
J
2n
Magnetic quantum numbers ml and ms
What orientations can the spin and orbit angular momentum vectors of an electron
take when subjected to a magnetic field?
The angular momentum vector precesses about the direction of an applied field as
shown in Fig. 10.3. This was predicted by Larmor's theorem [3J which states that
the effect of a magnetic field on an electron moving in an orbit is to impose on the
electron a precession motion about the direction of the magnetic field with an
angular velocity given classically by w = 1l0(e/2me)H. The component of the
angular momentum along the field axis is mocos e, where e is the angle of
precession. The component perpendicular to this direction is time averaged to
zero.
224 Electronic magnetic moments
H
H
;--....
-,
\
.....
I
,
,"
/
,,,
\
....
-2
-3
I
"(
_-
\
/
......... . , .
~
"'--'
I
I
Fig. 10.3 (a) Precession of the orbital angular momentum vector 1about the magnetic field
axis. (b) Quantization of the allowed projection of angular momentum 1 along the field
direction.
The component of angular momentum along the axis of a magnetic field is
restricted to discrete values by quantum mechanics. The magnetic quantum
number ml which represents these discrete values arises from the solution of
Schroedinger's equation for a single electron atom. It gives the component Iz of the
orbital angular momentum 1along the z-axis of a coordinate system
I=m{2:}
z
where the z-axis is defined as the axis perpendicular to the plane in which the
electron orbit lies.
The values of ml are restricted to ml = -/, ... , - 2, - 1, 0, 1, 2, ... , + I. The
H
j
ms = 1/2
ms =-1/2
Fig. 10.4 Quantization of the allowed projection of the spin s along the magnetic field
direction.
The quantum mechanical model of electronic magnetic moments
225
physical significance of this number is that when a magnetic field is applied to an
atom the electron's orbital angular momentum I can only have certain values of the
component parallel to the magnetic field direction, and these are given by the
magnetic quantum number mI.
Electron spins are constrained to lie either parallel or anti parallel to a magnetic
field. The orientation of the electron can be represented by the spin magnetic
quantum number ms which is always constrained to have the values ms = + t or
- t as shown in Fig. 10.4.
10.2.5 Quantized angular momentum and magnetic moments
What restrictions does quantum theory impose on the allowed values of the electronic
magnetic moment?
If we express the orbital angular momentum in units of hj2rc then the equation of
section 10.1.1 becomes
mo = _
(~)2rcpo
-(~)I
4rcm e
=
4rcm e
h
'
where 1is the orbital angular momentum quantum number, since by the quantum
theory we expect the angular momentum of an electron to be an integral multiple
of hj2rc (remembering that from Bohr's atomic theory Po = nhj2rc and from
Sommerfeld's theory Po = khj2rc).
This means that on this basis we expect the orbital contribution to the magnetic
moment to be an integral multiple of ehj4rcm e • This quantity is known as the Bohr
magneton, designated I1B' which has a value of 9.27 x 10- 24 Am 2 (Am2 is
equivalent to ljT).
where I must be an integer.
The spin on an electron is t, and so if we suppose Ps is an integer multiple of
(hj2rc) then Ps = shj2rc, where s is the spin quantum number. The magnetic moment
ms due to the spin is then
m
s
=
_~(2rcps)
2rcm e
h
Replacing 2rcp./h by the spin quantum number s
ms=
-2(~)S.
4rcm e
.
226 Electronic magnetic moments
H
Fig. 10.5 Projection of the total angular momentum j along the magnetic field direction.
Again replacing (eh/4nme) by JJ.B the Bohr magneton
ms = - 2JJ.Bs,
where now because of the quantization of angular momentum 2s = 4nPs/h. Since s
must be +! or -! this means that the spin magnetic moment of an electron is one
Bohr magnet on.
The total magnetic moment of an electron can be expressed in terms of multiples
of the total angular momentum (h/2n)j of the electron.
__ g(~)2npto 4
mtot -
h
nme
p
2n -tot )
= -gJJ.B (h
mtot = -gJJ.si,
where j is the total angular momentum quantum number, and for a particular
electron
= - (JJ.BI + 2JJ. Bs).
The projection mj of the total angular momentum along the direction of a
magnetic field is also quantized as shown in Fig. 10.5.
10.2.6 The Vector model of the atom
Do we have a simple model that can be used to interpret the electronic properties ofan
atom?
We know that an electron has a total angular momentumj which arises from its
spin angular momentum s and its orbital angular momentum I. In the vector model
The quantum mechanical model of electronic magnetic moments
227
of the atom the total angular momentum of an electron is simply the vector sum of
its spin and orbital angular momenta.
j=l+s
as indicated in Fig. 1O.6(a). Of course the vector sum must always be half integral.
For any given value of 1the total angular momentumj can only have two possible
values, 1± s, depending on whether s is parallel or anti parallel to I. An exception is
the case 1=0 when j must be l
For an atom with a single electron in an outer unfilled shell, and with all other
shells filled, these values become the angular momenta of the atom. For a
multielectron atom, that is one with more than one electron in an unfilled shell, we
use the terms J, Land S to designate the total, orbital and spin angular momenta of
the atom respectively where L is the vector sum of all the orbital components mZ, S
is the vector sum of all spin components ms and J is the vector sum of total angular
momenta of the electrons which can be calculated in at least two ways. For an atom
with completely filled shells J = 0, L = 0 and S = o.
We shall see shortly the assumption that the angular momentum is an integral
multiple of the orbital quantum number I and the spin quantum number s is
not quite valid. More precise values according to wave mechanics are Po 2 =
(h/211ll(1 + 1) and P; = (h/2n)2s(s + 1). The respective magnetic moments become
mo = ,uBJ[I(1 + 1)] and ms = ,uBJ[S(S + 1)]. Similarly the total angular
s =1/2
!
s= 1/2
j=5/2
l=2
1=2
j
(0)
=3/2
( b)
Fig. 10.6 (a) Vector addition of the components of angular momentum I and s to form the
total angular momentum j according to the semi-classical vector model of the atom. (b)
Vector addition of the components of angular momentum I and s to form the total angular
momentumj.
228
Electronic magnetic moments
momentum needs a correction from wave mechanics because Ptot i= (h/2n)j, but is in
fact given by Ptot = (h/2n)J [j(j + 1)]. This has been shown for example by Sherwin
[4].
10.2.7 Wave mechanical corrections to angular momentum of electrons
Why is the angular momentum of an electron not simply the quantum number
multiplied by (h/2n), as expected on the 'old quantum theory,?
We may now consider very briefly the quantum theory of angular momentum.
This will be needed to provide the expectation value of the angular momentum
Po = h/2n,Jl(l + 1) for the following discussion of the quantum theories of ferro-
n=1
1=0
o
5
00=0.529
10
15
r/o o
r2"1~
1=1 1= 0
Fig. 10.7 Probability functions 1jJ*1jJ for an electron as a function of radial distance from
the nucleus in a simple atom such as hydrogen.
The quantum mechanical model of electronic magnetic moments
229
magnetism and paramagnetism. The treatment given here is abbreviated and only
intended to be a guide to the derivation.
Consider a very simple case of a single electron orbiting an ionic core. We are
interested only in the angular momentum of this one electron. We could
approximate the situation with the wave function of the single electron in a
hydrogen atom. In this case the wave function can be represented by the expression
'I'(r, e,(/» = R(r)0(e)<I>(¢)
e
always assuming that the dependences on r, and ¢ are independent of each other
and hence can be separated. The solutions of the Schroedinger equation in this case
are shown in Figs. 10.7 and 10.8. These show the possible electron states.
If <Po 2 ) is the operator for the orbital angular momentum squared, and Po 2
is the expectation value of the same quantity, then <Po 2 ) 'I' (r, ¢) = P02 'I' (r, ¢).
Replacing <Po 2 ) by (h/2n)2 [2) and Po 2 by (h/2n)2 F gives
e,
<
e,
where the first term on the left-hand side is the expression for <F). If we suppose
<I>(¢) = exp(im¢)
d 2 <1>(¢ )
-~2=
d¢
-m 2 exp(im¢)
where m is an integer, the operator for <[2) is modified, and separating the terms in
the wave function, the previous equation becomes
It can be shown by a rather lengthy proof that this equation only has well
behaved solutions when = 1(1 + 1) for 1=0,1,2,3, ... . The expected values of the
angular momentum are therefore
r
The result of this is that an electron with orbital angular momentum quantum
number [will have an angular momentum of (h/2n)J[I(1 + I)J and not (h/2n)1 as
might have been expected on the basis of the old quantum theory.
The same argument holds for the spin angular momentum (h/2n)J[(s(s + l)J
and the total angular momentum (h/2n)J[j(j + 1)]. The vector addition of these
quantities is then shown in Fig. 1O.6(b).
f"'I
+t
II
f
+1
II
f
+t
II
f
'"
o
II
f
o
II
f
2s
3p
4d
m, = ±)
ml
ml
= ±2
Sf
ml
= ±3
= 0.
m, =
0
ml = ±)
m, = ±2
Fig. 10.8 Probability density functions in real space for various electronic levels in hydrogen.
232
Electronic magnetic moments
10.2.8 The normal Zeeman effect
What effect does a magnetic field have on the energy levels ofelectrons within atoms?
The energy levels of electrons within the atom are altered by the presence of a
magnetic field. This is shown in the optical spectra for example, where the
emissions from the atoms in the form of light quanta are the result of electrons
moving to lower energy levels. These spectra are altered by the presence of a
magnetic field as shown in the Zeeman effect [5,6].
The normal Zeeman effect is exhibited by atoms in which the net spin angular
momentum is zero. A spectral line of frequency Vo in zero field is split into two lines
displaced symmetrically about the original zero-field line at frequencies ofvo + ~v
and Vo - ~v.
The displacement in frequency ~v is proportional to the magnetic
field strength. When viewed perpendicular to the field direction all three lines are
observed, but when viewed along the direction of the field only the two displaced
spectral lines are observed.
The normal Zeeman effect occurs in calcium (the singlet at A= 422.7 nm),
magnesium and cadmium (the singlet at A= 643.8 nm), for example. The
displacement in energy is 0.93 x 10- 23 J or 0.58 x 10- 4 eV in magneticinductions
of 1 T, corresponding in the optical region to a shift in wavelength of typically
0.01 nm.
Under the action of a magnetic field the allowed changes in the electron energy
levels ~EH
= Eo ± EH are quantized and are given by
~EH
= - fJ,o~mH
where ~m
is the difference in the component of magnetic moment along the field
direction. This can take only the values given by ml (eh/4nm e ) where ml = -I, -I
+ 1, -I + 2, 0, 1- 1, I
~EH
=
4:~e
ml (
)
fJ,0H.
Therefore a P state, which has I = 1 splits into three levels, ml = - 1,0, + 1, while
a D state which has 1= 2 splits into five levels, ml = - 2, - 1, 0, 1,2.
There is also a selection rule which governs allowable transitions between states
with different values of ml [7]. This rule, which is found empirically states that an
electron undergoing a transition from different energy levels can not change its
magnetic quantum number by more than 1. This rule ensures that there are no
more than three available transition energies in the normal Zeeman effect.
~ml
=
°or ±
The shift in energy of the spectral lines is
hv - hvo =
(4:~Jf,OH.
1.
The quantum mechanical model oj electronic magnetic moments 233
By measuring the change in frequency the ratio of charge to mass of an electron
can be found from
P-oH.
v- Vo = (_e_)
4nme
Experimental measurements of the change in frequency yield the value
ejme = 1.7587 x 1011 Cjkg.
Notice that the expression for the change in spectral frequency does not depend
on Planck's constant. In fact an explanation of the normal Zeeman effect of
spectral frequency can be given by classical physics [8].
An example of the splitting of the electron energy levels of the D states (1 = 2) and
the P states (l = 1) in cadmium is shown in Fig. 10.9. In the absence of a net spin the
P levels have three states corresponding to ml = 1,0, - 1 which are degenerate in
zero field. The D levels have five states. Once a magnetic field is applied
the degeneracy is lifed, and on application of the selection rule this results in
three allowed transition energies. The shift in spectral lines is also shown in
Fig. 10.9.
magnetic field off
magnetic field on
(B=O)
ED
o state ---r---
t
~Em
t
(B=B)
-
2=m,
1
o
-1
(/= 2)
-2
energy
( hvo+ ehB)
2m
(hv o)
1 =m,
o
P state - - - ' - - (/ = 1)
~m,=
-1
vo
-1
eB
vo- 4nm
~m,=
~m,=O
Vo
+1
eB
vo+4nm
Fig. 10.9 Splitting of the electronic D and P states in the normal Zeeman effect.
234
Electronic magnetic moments
We should note that the quantization of the various states with m, =
-1, ... ,0, ... , + 1remains even when the field is removed (that is, it is not the field
which induces the quantization), but in zero field the various states are degenerate
(i.e. have the same energy) and therefore they all contribute to the same spectral
line. All electronic transitions for which dl = ± 1 lead to the same Zeeman effect,
that is the splitting of one spectral line into three.
10.2.9 The anomalous Zeeman effect
Why do most atoms not exhibit the simple 'normal' Zeeman effect?
The spectra of most atoms do not show the simple normal Zeeman effect. In these
cases the spectral lines split into more than three levels described above. The
resulting behaviour is called the anomalous Zeeman effect, even though it occurs
more often than the 'normal' Zeeman effect.
These spectra cannot be accounted for on the basis of quantization of the orbital
angular momentum alone under the action of a magnetic field, and therefore it is
necessary to look for further explanations. In order to account for the anomalous
Zeeman effect Goudsmit and Uhlenbeck [9] suggested that the electrons have a
spin angular momentum which leads to a magnetic moment associated with the
spm.
The spectra of most atoms exhibit fine structure even in the absence of a
magnetic field and hence the name anomalous Zeeman effect since it does not need
a magnetic field to cause splitting of the energy states. When such zero-field fine
structure is present it is indicative that the element will exhibit the anomalous
Zeeman effect under the action of a field also. The energy splitting between states
with spin up and spin down in the presence of a magnetic field H is
using f.-lB = eh/4nm e
and ms = 2Sf.-lB
dEs = f.-lomsH.
Therefore since any energy state can be occupied by electrons in degenerate spinup spin-down states it will be split into two, that is it has the degeneracy lifted, by a
magnetic field.
A well-known example of the anomalous Zeeman effect occurs in the sodium
The quantum mechanical model of electronic magnetic moments
235
spectrum. The zero-field transitions can be resolved into a doublet with
wavelengths 589.0 nm and 589.6 nm. Each of these lines itself exhibits fine structure
and can be resolved further into two or three lines. The doublet is due to the 3P-3S
transition as shown in Fig. 10.10. Application of a magnetic field splits the P states
into four new levels (compare with P level splitting into two new levels in Fig. 10.9)
and the S state into two new levels compared with none in the normal Zeeman
effect.
The diagram of Fig. 10.10 is kept relatively simple because the P state (/ = 1) is
split by the normal Zeeman into only two levels which are each further split into
two by the spin. If we had looked at a D state (l = 2) for example there would have
been a total of ten levels in an applied field, five as a result of the normal Zeeman
splitting (m( = 2,1,0, - 1, - 2) each of which are further split into two by the spin
splitting (ms = + t or - t).
1
2
.
3
2
iP'/2
-....,....-+----<
1
_2--,._ _....,._ 1
----+- ~
Ii
"
1
3
I
--'-....1._---<
/...I...-I..---+-+---.....oI.......I--t--t-t-
/
325112
-f------------
-
--
1"---....1-..1-------..-.........
..0-
-1
2
I
I
C1
1[
1[
I
!
1
C1
!
C1
I
1[
!
C1
C1
1[
C1
I
I
J
J
Zeeman components
Fig. 10.10 Splitting of the electronic D and P states in the anomalous Zeeman effect.
236
Electronic magnetic moments
10.2.10 The Stern-Gerlach experiment
How do we know that the electronic angular momentum is quantized?
In zero field of course all of the values of m] represent degenerate energy levels,
although the quantization is still present. In the presence of a field the component
of angular momentum along the field direction is determined by Larmor
precession of the angular momentum as shown in Fig. 1OJ. The total orbital
angular momentum remains the same for each of these electrons but by precessing
at an angle to the field direction the component perpendicular to the field direction
averages to zero. The quantization of the component of the angular momentum, or
magnetic moment, along the direction of a magnetic field represented by the
magnetic quantum numbers was first confirmed by the experiment of Stern and
Gerlach [10].
Stern and Gerlach tested the space quantization of the angular momentum using
atoms of silver in which J = S = The only contribution to the magnetic
moment comes from the spin of a single electron which according to theory should
be quantized with value + or - along the field direction. The arrangement
of the experiment is shown in Fig. 10.11. Silver atoms leaving an oven at high
speeds, were collimated by slits and passed through an inhomogeneous magnetic
field generated by the pole pieces shown. They were stopped by a photographic
plate where their final positions were recorded. It was found that the locations of
the atoms arriving on the photographic plate were not continuous but instead were
in two lines corresponding to silver in the two allowed spin states of ms = +
and
t
t.
t
-to
10J
t
MAGNETIC PROPERTIES OF FREE ATOMS
What determines the magnetic moment of an atom?
The magnetic properties of an atom are determined principally by the magnetic
moments of its electrons. We have shown in the previous chapter that the magnetic
Fig. 10.11 Experimental arrangement in the Stern-Gerlach experiment.
Magnetic properties of free atoms
237
moments of these electrons can be calculated from the sum of orbital and spin
angular momentum. The net magnetic moment of a filled electron shell is zero so
that only the unfilled shell needs to be considered. This simplifies the problem of
calculation greatly.
The angular momentum of the atom can be found by summing vectorially the
spin and orbital angular momenta of its electrons. This can be done in two ways,
either the values of j for each electron can be found and the vector sum of these
calculated to give J. Alternatively the orbital angular momentum L of all the
electrons can be found by vector sum and the spin angular momentum S of all the
electrons can be found. Land S can then be added vectorially to provide the J for
the atom. These two methods do not give identical answers.
10.3.1
Magnetic moment of a closed shell of electrons
Does a closed shell of electrons have any net magnetic moment?
The total J, orbital L and spin S angular momenta of a closed shell of electrons are
always zero. This result enables us to simplify calculations of the net magnetic
moment of an atom by considering only the contributions due to the partially filled
shells.
Example 10.1. To demonstrate that a closed shell of electrons has zero angular
momentum consider the Zn 2 + ion which has ten electrons in its 3d shell. For the 3d
series I = 2, the electron configuration is
-1
-2
2
1
0
-1
1
1
1
1
"2
1
-2
1
-2
-2
t
t
t
t
m1
2 1 0
ms
1
"2
"2
"2
2
"2
s
i i i
i
i
t
1
"2
1
Consequently L
10.3.2
= Lm] = 0 and
-2
1
S=Lms=O.
Atomic magnetic moment
How can we calculate the magnetic moment of an atom from a knowledge of its
electron structure?
We can find the magnetic moment of an atom from its total angular momentum.
The vector model of the atom is a simple semi-classical model which allows us to
calculate the total angular momentum J from the orbital and spin angular
momenta of the electrons belonging to an atom. The model is semi-classical
because although the sum J = L + S is made in the classical vector manner the
actual values of Land S are calculated from quantum mechanics.
238
Electronic magnetic moments
Corrections are made to the model as a result of quantum mechanics in order
to find the magnitude of the vectors IL I, IS I and IJI in terms of the quantum
numbers L, Sand J. Fortunately the correction is very simple and is the same
as the correction needed for individual electrons as discussed in the previous
chapter.
We should mention that there is a very small contribution to the total angular
momentum of the atom from the nucleus due to nuclear spin. This contribution is
about 10- 3 of the spin of an electron. The nuclear magnetic moment is measured in
units of nuclear magnetons, denoted J.1N. The value of J.1N is 5.05 X 10- 27 Am2,
compared with 9.27 x 10- 24 Am 2 for the Bohr magneton.
10.3.3
Atomic orbital angular momentum L
How does the atomic orbital angular momentum depend on the electronic angular
momenta?
The atomic orbital angular momentum, denoted by the quantum number L, is the
vector sum of the orbital angular momenta of the electrons within the atom.
An example of the vector summation for an atom with two electrons in its unfilled
shell with quantum numbers 1= 1 and 1= 2 is shown in Fig. 10.12.
The magnitude of the orbital angular momentum vector IL I in terms of the
orbital angular momentum quantum number L is
ILl =
J[L(L + 1)]
which is identical in form to the relation between III and the orbital angular
momentum quantum number 1 of a single electron.
10.3.4
Atomic spin angular momentum S
How does the atomic spin depend on the electron spins?
The spin angular momentum S for an atom can be found in a similar way using the
vector model. It is the vector sum of the spin angular momenta of the electrons.
S=LSi.
The summation process is indicated in Fig. 10.13(a) for an atom with three
electrons in its unfilled shell and in Fig. 10. 13(b) for an atom with four electrons in
its unfilled shell.
The magnitude of the spin angular momentum vector ISI in terms of the spin
Magnetic properties of free atoms
239
1=1
L=3
1=2
Fig. 10.12 Vector addition of the orbital
angular moment un in a two-electron system.
5=1
I
S=2'
I
S='2
I
5='2
5=i
li
I
5='2
s=t
I
S=2'
S=2'
1
5=t
I
5='2
I
S='2
5=2
I s=t
5='!
I
S='2
I
S='2
5=1
lti"=* 5=0
S=2
I
.1 S='2
5=2
(b)
(0)
Fig. 10.13 Vector addition of electron spins for: (a) a three-electron system; and (b) a fourelectron system.
quantum number S is according to quantum mechanics,
lSI = J[S(S + 1)]
which is again identical to the relation for a single electron.
10.3.5 Hund's rules: occupancy of available electron states
How do electrons decide which values of s and I to take?
There is a set of empirical rules which determine the occupancy of the available
electronic states within an atom. These rules identify how the possible electronic
states are filled and can be used therefore to calculate L, Sand J for an atom from
the electron configuration in its unfilled shell.
The Hund rules [11J can be applied to electrons in a particular shell to determine
the ground state of the atom. The three rules apply to the atomic spin S, the atomic
240
Electronic magnetic moments
orbital angular momentum L and the atomic angular momentum J respectively.
Electrons occupy available states as follows:
1. such that the maximum total atomic spin S = L:ms is obtained without violating
the Pauli exclusion principle;
2. such that the maximum value of total atomic orbital angular momentum
L = Lml is obtained, while remaining consistent with the given value of S; and
3. the total atomic angular momentum J is equal to IL - SI when the shell is less
than half full, and is equal to IL + SI when the shell is more than halffull. When
the shell is exactly half full L = 0 so that J = S.
This means that the electrons will occupy states with all spins parallel within a
shell as far as is possible. They will also start by occupying the state with the largest
orbital angular momentum followed by the state with the next largest orbital
angular momentum, and so on.
The next two examples show how Hund's rules are applied in specific cases to
calculate L, Sand J.
Example 10.2. The Sm 3 + ion which has 5 electrons in its 4f shell (n = 4, I = 3), must
have the electrons arranged with the following configuration.
m1 3 2 1 0
1
ms "21 "21 "21 "2
occupancy s
i i i i
-1
1
"2
-2 -3
1
"2
1
"2
i
This gives S = L ms = ~ and L = L: m1 = 5. The shell is less than half full so that J
=L-S~.
10.3.6
Total atomic angular momentum J
How is the total atomic angular momentum related to the spin and orbit components?
The coupling between the orbital angular momentum of the atom L and the spin
angular momentum S gives the total atomic angular momentum J. This
summation can be obtained in two ways, by independently summing the orbital
and spin moments of the unpaired electrons
This form of coupling is known as Russell-Saunders coupling [12]. In this case the
I vectors of individual electrons are coupled together (I-I) and the vectors are
coupled togethers (s-s). These couplings are stronger than the coupling between
the resultant Land S vectors.
Another method of obtaining J is by summing the total angular momenta of
the electrons, which are obtained for each electron individually by summing I and s
Magnetic properties of free atoms
241
first
J=
L)i = I(li + sJ
This is the so called j~ coupling. This type of coupling occurs if there is a strong
spin orbit coupling (l~s)
for each electron. Then the coupling between I and s for a
particular electron is stronger than the coupling between I and I for separate
individual electrons.
In these calculations the vectors which are strongly coupled together must
always be added first. In practice the most common form of coupling is the first
(or I~s)
coupling is the most appropriate form of coupling
kind. Rusel~Sandr
to consider for magnetism.
10.3.7 Russell-Saunders coupling
How do the spin and orbital angular momenta combine to form the total angular
momentum of an atom?
Rusel~Sandr
coupling, in which the spin~
and orbit ~orbit
couplings are
strongest applies in most cases. It occurs in all of the light elements and situations
in which the multiplet splitting is small compared with the energy difference of
levels having the same electron structure but different values of L.
In this coupling it is assumed that when several electrons are present in an
unfilled shell the orbital angular momenta are so strongly coupled with each other
that states with a different total L have a different energy. Similarly it is assumed
that states with a different total S have a significant difference in energy. The
resultant vectors Land S are then less strongly coupled with one another.
5=1
J=3
L=2
LO{)2
15 ='
L=2
tJ=1
5=3/2
5=312
J=7/2lfo"
L=2
L=2
~=3/2
5=3/2
J=5/2 L=2
L =2
J= 3/2
tJ =I/2
Fig. 10.14 Vector addition of atomic orbital angular momentum L and atomic spin angular
momentum S to give the total atomic angular momentum J.
242
Electronic magnetic moments
In Russell-Saunders coupling the orbital momentum vectors combine to form a
total orbital angular momentum L while the spin momentum vectors combine
independently to form a total spin momentum vector S. This means that the
calculated S is not affected by the value of L.
This leads to a total angular momentum J of the atom which is simply the vector
sum of the two non-interacting momenta Land S.
as shown in Fig. 10.14. There are certain restrictions on this however. J must be an
integer if S is an integer, and J must be a half integer if S is half integral. RussellSaunders coupling applies to iron and the rare earth atoms.
10.3.8
The j-j coupling
Are there other ways of coupling to form the total angular momentum of an atom?
The alternative form of coupling, spin-orbit coupling, in which the orbital and spin
angular momenta are dependent on each other is calledj-j coupling. This form of
coupling is more applicable to very heavy atoms. It assumes there is a strong spinorbit (1- s) coupling for each electron and that this coupling is stronger than the 1-1
or s-s coupling between electrons. Since the strongly coupled vectors are summed
first these give a resultant j for each electron. The j vectors are then summed to
obtain J for the whole atom.
10.3.9
Quenching of the orbital angular momentum
Why do the magnetic moments of the 3d series elements imply that L = 0 in all cases?
In the 3d series of elements it has been found experimentally that J = S as shown in
Table 10.1. This implies that L = 0 throughout the 3d series, which is a surprising
result if we only consider the properties of electrons in isolated atoms. In these
cases we speak of the orbital angular momentum being 'quenched'. The
phenomenon has been discussed by Morrish [13] and Kittel [14]. Here we shall
merely note that under certain circumstances the plane of the orbit can move about
and this can average to zero over the whole atom. This results in a net zero value of
L. Notice from Table 10.2 that this does not occur in the 4f series which is the other
principal group of magnetic materials.
Example 10.3. The Fez + ion has 6 electrons in its 3d shell. Since 1= 2 for this series
the electron configuration is
m1 2
ms I1 I1
occupancy s
i i
0
-1
-2
2
1
0
-1
1
1
1
1
1
1
1
I
I
I
i
i
i 1
I
I
-I
-I
-2
1
-I
Magnetic properties of free atoms
243
Table 10.1 Magnetic moments of isolated ions of the 3d transition metal series
Ion
Configuration
Calculated moment
9J[J(J+ 1)] 2)[S(S+ I)J
Measured moment
TP+, y4+
yH
Cr3+, y3+
Mn 3+,Cr 3+
Fe 3+,Mn 2+
Fe2+
C0 2 +
Ni2+
Cu 2+
3d l
3d 2
3d 3
3d4
3d 5
3d 6
3d?
3d B
3d 9
1.55
1.63
0.77
0
5.92
6.70
6.63
5.59
3.55
1.8
2.8
3.8
4.9
5.9
5.4
4.8
3.2
1.9
1.73
2.83
3.87
4.90
5.92
4.90
3.87
2.83
1.73
Table 10.2 Magnetic moments of isolated ions of the 4f transition metal series
Ion
Configuration
Calculated moment
gJ[J(J+ 1)]
Ce 3+
Pr 3+
Nd H
Pm H
Sm H
Eu H
Gd H
Tb 3+
Dy3+
HOH
Er3+
Tm H
Yb H
4fl 5s 2 p6
4f25s 2p6
4f35s 2 p6
4f45s 2p 6
4f5 5s 2p6
4f65s 2p6
4r5s 2p 6
4fB5s 2p 6
4f95s 2p6
4flO5s 2p6
4fl1 5s 2p6
4fl25s 2p6
4fl35s 2 p6
2.54
3.58
3.62
2.68
0.84
0
7.94
9.72
10.63
10.60
9.59
7.57
4.54
Measured moment
2.4
3.5
3.5
1.5
3.4
8.0
9.5
10.6
10.4
9.5
7.3
4.5
This leads to S = 2 and L = 2. Since the orbital angular momentum is quenched
in these elements J = S and therefore the total magnetic moment is
m = 2~BJ[S(
+ 1)]
which corresponds to 4.90 Bohr magnetons for the isolated Fe 2 + atom. This agrees
reasonably well with the observed value of 5.36 Bohr magnetons. The expected
value without quenching of the orbital angular momentum is 6.7 Bohr magnetons
which is clearly in serious error.
We should note that these calculations apply only to the isolated paramagnetic
(i.e. non-interacting) atoms or ions. When a large number of atoms are brought
together in a solid the electron energy levels are drastically altered and the
244
Electronic magnetic moments
magnetic moment per atom is in most cases significantly different from the values
calculated above.
10.3.10 Electronic behaviour in strong magnetic fields
I s the coupling between the spin and orbital angular momenta different in a strong
magnetic field?
As the magnetic field is increased and the field splitting becomes greater than the
multiplet splitting the anomalous Zeeman effect changes over to a normal Zeeman
effect. The reason for this is that the precessional velocity of J about the field axis
becomes greater than the precession of the Sand L vectors about J. Therefore this
is described better as independent precessions of Sand L about the field direction,
that is the L-S coupling breaks down. We speak of Land S being decoupled by the
magnetic field. This transition is known as the Paschen-Back effect [15J and only
occurs in high magnetic fields.
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4.
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6.
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11.
12.
13.
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Kittel, C. (1986) Introduction to Solid State Physics, 6th edn, Wiley, New York.
Paschen, F. and Back, E. (1913) Ann. Physik., 40, 960.
FURTHER READING
Chikazumi, S. (1964) Physics of Magnetism, Wiley, New York, Ch. 3.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading, Mass,
Ch. 3.
Hertzberg, G. (1937) Atomic Spectra and Atomic Structure, Prentice Hall, New York.
Semat, H. (1972) Introduction to Atomic and Nuelear Physics, Holt, Rinehart and Winston,
New York, Ch. 8 and 9.
Sherwin, C. W. (1959). Introduction to Quantum Mechanics, Holt, Rinehart and Winston,
New York.
Examples and exercises
245
Weidner, R. and Sells, R. L. (1968) Elementary Modern Physics, 2nd edn, Allyn and Bacon,
Boston.
EXAMPLE AND EXERCISES
Example lOA Orbital and spin angular momentum of an electron (a) Explain the
significance of the four quantum numbers n, 1, s and mi' Describe how to calculate
the orbital, spin and total angular momentum of an electron from its quantum
numbers using the 'classical' vector model of the atom. Why do the values of Is
and j differ from the classically expected results when calculated using quantum
mechanics?
(b) Using vector diagrams determine the different values for the total orbital
angular momentum of a two-electron system. (i.e. atom) for which 11 = 3 and 12 = 2.
What are the possible values for L, for S and for J of the atom?
(c) Find the values of L, Sand J for the Co2+ ion.
Example 10.5 The Zeeman effect. Calculate the normal Zeeman effect separation in energy levels in cadmium when subjected to magnetic field strengths of
1.6 x 106 , 0.4 X 106 and 0.064 x 106 A/m (equivalent to a free space magnetic
induction of 2.0, 0.5 and 0.1 tesla). The wavelength of the main 6 1 D 2-5 1 P 1
transition is A= 643.8 nm, which is a spectral line at the red end of the visible
spectrum. Calculate the shift in wavelength.
Example 10.5 The Zeeman Effect. Calculate the normal Zeeman effect
separation in energy levels in cadmium when subjected to magnetic field strengths
of (equivalent to a free space magnetic induction of 0.1, 0.5 and 2.0 tesla). The
wavelength of the main 6 1 D z-5 1 P 1 transition is )0 = 643.8 nm, which is a spectral
line at the red of the visible spectrum. Calculate the shift in wavelength.
Draw an energy level diagram showing the 6 1 D2 and 51 P 1 states before and after
application of a magnetic field and explain why the original singlet is split only into
a triplet and not into a larger number of spectral lines.
Example 10.6 Determination of atomic angular momentum. Using vector diagrams determine the possible values of J for an atom with
(a)
(b)
(c)
(d)
L = 2, S = 3
L = 3, S = 2
L=3, S=5/2
L=2, S=5/2
What are the values of L, Sand J for the ground state of the paramagnetic carbon
atom?
11
Quantum Theory of Magnetism
In the previous chapter we discussed the origin ofthe atomic magnetic moment. In
this chapter we look at how these magnetic moments interact to give cooperative
magnetic phenomena such as ferromagnetism and paramagnetism in solids. We
then consider theories of magnetism based on two mutually exclusive models: the
localized moment model and the band or itinerant electron model. The localized
model works well for the rare earth metals, while the itinerant electron model
describes the magnetic properties of the 3d transition metals and their alloys quite
well.
11.1
ELECTRON-ELECTRON INTERACTIONS
The exchange interaction is obviously crucial to an understanding of ordered
magnetic states, but where does it come from?
We have seen that the behaviour of the susceptibility of many ferromagnets above
their Curie temperature follows a Curie-Weiss law and that this implies the
existence of an internal field which has been called the Weiss molecular field, or
more accurately the atomic field. The field originates from quantum mechanical
interactions. A discussion of these together with a derivation of the exchange
interaction from considering overlap of the electron wave functions has been given
in detail by Martin [1].
11.1.1
Wavefunctions of a two-electron system
How do we form the wavefunction for a system consisting of two electrons?
The wave function of two electrons can be represented as 'I'(r1,r Z) where r1 is the
coordinate of the first electron and rz is the coordinate of the second electron.
Then we may consider possible representations of this wave function using linear
combinations of the separate wave functions of the individual electrons tfJ(r d and
tfJ(r 2)'
248
Quantum theory of magnetism
There are four possibilities
'P(r 1,r2) = l/Ia(rdl/lb(r2)
'P(r 1,r2) = l/Ia(r2)l/Ib(rd
'P(r1>r 2) = l/Ia(rdl/lb(r2) + l/Ia(r 2)l/Ib(r 1)
'P(r 1,r2) = l/Ia(rdl/lb(r2) -l/Ia(r2)l/Ib(r 1),
where l/Ia(r1) represents the wave function of electron 1 on atom a, and l/Ib(r2)
represents the wave function of electron 2 on atom b and so on. We need a solution
such that the 'observed' properties '1'*'1' are unaltered by interchanging electrons,
while the electrons remain distinct, that is 'P(r 1,r2) must be antisymmetric. This is
an important point; if the wave function is symmetric and '1'*'1' is identical then the
wavefunctions of the two electrons are identical and hence they have the same four
quantum numbers, which is not allowed.
In both cases 1 and 2 above the interchange of electrons, r1 for r2 alters the value
of '1'*'1' and hence alters the electron distribution at all points. Therefore these
cannot be solutions of the wave function. Another possibility is
'P(r 1,r2) = l/Ia(r 1)l/Ib(r2) + l/Ia(r2)l/Ib(rd·
In this case the function remains symmetric after interchanging the electrons and
'1'*'1' is also unaltered. However according to the Pauli exclusion principle the
probability of finding both electrons in an identical state (that is with the same set
of quantum numbers) is zero.
If the wavefunctions of the separate electrons are otherwise identical the Pauli
principle can only be satisfied among this group of candidate wave functions by
'P(r 1,Yz) = l/Ia(r 1)l/Ib(r 2) -l/Ia(r 2)l/Ib(r 1)·
See for example Fig. 11.1.
Fig. 11.1 Wave function for a two-electron system using a linear combination of atomic
orbitals as in the Heitler-London approximation [2], IjJ is antisymmetric, while 1jJ*1jJ is
symmetric.
Electron-electron interactions
249
11.1.2 Heider-London approximation
What is the total energy of two interacting electrons according to quantum theory?
Do we find a result that is just a modification ofclassical theory or something radically
different?
The Heilter-London approximation [2] is merely a method of obtaining the
orbital wave function of two electrons by assuming that it can be approximated by a
linear combination of the atomic orbital wave functions of two electrons localized
on the two atomic sites.
Consider the orbital functions and in particular the energy for such a system of
two atoms with one electron each. Heitler and London calculated the energy of this
system by evaluating the integral
E=
ff
'¥(r1,r2)(H)'¥*(r1,r2)dr1dr2
and the Hamiltonian <H) must contain the separate Hamiltonians <H 1) and
(H 2) for each electron and an interaction Hamiltonian H 12)
<
<H) = (H 1 )
+ (H 2 ) + <H12)'
Of course <H 1) contains only the coordinate r 1 and <H 2) contains only the
coordinate r 2 •
<H1 )'¥.(rd = E.'¥(r1)
<H 2) '¥ /3(r 2) = Eb'¥(r2)'
The solution of the total energy equation leads to
E = E.
ff
'¥*(r1)'¥(r1)dr1
+ Eb
ff
'¥*(r2)'¥(r2)dr2
+ (1 ~
0(2)
ff
+ (1 ~
0(2)
f f t/t.*(r1)t/tb*(r2)(H12)t/tb(r1)t/t.(r2)dr1dr2·
t/t.*(r1)t/tb*(r2)<H12)t/t.(rdt/tb(r2)dr1dr2
The last energy term is obtained by exchanging the electrons 1 and 2 between the
atoms a and b and so is called the 'exchange energy'.
11.1.3 Exchange interaction
What is the cause of the extra energy term?
We can write this expression for the energy obtained from evaluating these
integrals in the form,
250
Quantum theory of magnetism
where [1/(1 ± (X2)] is a normalizing constant. In this equation we have on the righthand side the energy of the electrons when they belong to their separate atoms Ea
and Eb , plus some additional energy which results from their interactions. Qis the
coulomb electrostatic energy. J is an energy which arises from exchange of the
electrons, as can be seen from the form of the wave function. We have already
encountered this exchange constant in Chapter 6. Here it is italicized to distinguish
it from J the atomic angular momentum. It is immediately clear that the existence
of the term J depends on the necessity of an antisymmetric wave function above,
since if 'P(r1' r2) were symmetric then J would automatically become zero.
11.1.4 Wave function including electron spin
What is the form of the two-electron wavefunction
if we include electron spin?
We have shown from the very simple quantum mechanical treatment above that
there exists an exchange energy which has no classical analog. It is now
necessary to show that the ordering of the electron spins arises from this. Suppose
we write the spin wave functions as /P1 (s) and /P2(S). Then the spin state for two
electrons can be represented, using the same approximation as before, as a linear
combination of the individual spin wave functions.
/p(s 1, S2) = /Pa(S 1)/Pb(S2) - /Pa(S2)/Pb(S 1)
/P(Sl, S2) = /Pa(Sl)/Pb(S2) + /Pa(S2)/Pb(Sl)
/P(Sl, S2) = /Pa(Sl)/Pb(S2)
/p(S 1, S2) = /Pa(S2)/Pb(S 1)·
The total wave function of the two electrons system with spins can then be
represented as
'PTo! = 'P(rt>r 2)/p(Sl,S2)
which as we already know must be antisymmetric with respect to interchange of
the electrons. So if 'P(r l' r2) is anti symmetric then /P(Sl, S2) must be symmetric, and
vice versa, in order to maintain the anti symmetric nature of the total electronic
wave function 'PTo!.
Symmetric and anti symmetric /p's correspond to parallel and anti parallel spins.
Therefore the exchange energy as derived above can be considered as really the
interaction between the spins, since it is the spins which maintain the total wave
function 'PTo! anti symmetric. In other words the spin can be used to distinguish
between two electrons which are otherwise identical because the spins can be used
to satisfy the Pauli principle when the other three quantum numbers of the two
electrons are identical. For parallel spins the exchange energy is then simply
[1/(1 ± (X2)] (- J). This means that a positive J corresponds to parallel spin alignment, and hence to ferromagnetic ordering. A negative J leads to antiparallel
alignment.
Electron-electron interactions
11.1.5
251
Exchange energy in terms of electron spin
Can we separate the exchange energy and make it dependent only on spin?
From the above discussion the exchange energy dictates a lower energy and hence
ferromagnetic order when J > 0, and antiferromagnetic alignment when J < O. The
model has only discussed two localized electrons on neighbouring atoms and so
has some obvious limitations when we wish to apply it to a solid. The exchange
Hamiltonian can however now be written simply in terms of the spins on two
electrons
<H)
= - 2JijSi·Sj
which leads to the Heisenberg model of ferromagnetism [3J which was also
proposed by Dirac [4].
11.1.6
The Heisenberg model of ferromagnetism
How might the direct exchange energy between two neighbouring electrons account
Jor long-range magnetic order in Jerromagnets and ant(ferromagnets?
The Heisenberg model of ferromagnetism, like the classical Weiss model [5J,
is another local moment theory which considers the quantum mechanical
exchange interaction between two neighbouring electrons with overlapping wave
functions. The idea of direct exchange between electrons on neighbouring atoms
first occurred in the Heitler-London treatment of electron orbitals. Heisenberg
was the first to include the electron spins in the wave function and then apply the
same Heitler-London approximation to obtain the total wave function of a twoelectron system. The energy integral was evaluated but now including spin and this
showed that the relative orientations of the spins of two interacting electrons can
be changed only at the expense of changing the spatial distribution of the charge.
If two electron wave functions overlap then the Pauli exclusion principle [6J
applies to the region of overlap. Since no two electrons can have the same set of
quantum numbers, when the orbital wave function is symmetrical the spins must
be antisymmetric and vice versa. This leads immediately to a correlation between
the spins on the two electrons, which is all that is needed to cause a magnetically
ordered state. The correlation can be expressed in the form of a magnetic field
(although strictly it is not of magnetic origin but is rather electrostatic) or as an
energy. The interaction energy is proportional to the dot product of the spins.
Here J > 0 gives ferromagnetism and J < 0 gives antiferromagnetism.
When considering a solid it is then necessary to sum the exchange over all the
electrons which can contribute to this energy so that
<H) =
-
2'i. 'i. Jil;" Sj.
252
Quantum theory of magnetism
In many cases we are only interested in nearest-neighbour interactions and this
simplifies the Heisenberg Hamiltonian considerably.
<H)=-2J
II
Si'Sj,
nearest
neighbours
where it is assumed that the exchange integral J is identical for all nearestneighbour pairs. The exchange integral J has no classical analogue, so it is difficult
to understand it from a classical viewpoint. However a brief consideration of the
Pauli principle tells us that two electrons with like spins cannot approach one
another closely. This leads to an effective energy which influences the alignment of
the moments.
Empirical values of the exchange interaction J for various ferromagnetic metals
have been calculated from specific heat measurements and from spin wave
considerations. Some of these have been reported by Hofmann et al. [7] which
indicate that for iron and nickel J ~ 0.01 to 0.02 eV and for gadolinium
J~0.2eV
Wohlfarth [8] had encountered problems with first principles calculation of J,
obtaining the wrong sign when using spherical wave functions. Later Stuart and
Marshall [9] made detailed calculations of the direct Heisenberg exchange in iron.
They obtained a value of 6.8 x 10 - 3 eV for two neighbouring localized electron
interactions, and on this calculation a similar result would be obtained for both
cobalt and nickel. This is about a third of the value obtained from experimental
measurements. Watson and Freeman [10,11] refined the calculations of Stuart
and Marshall to include additional terms neglected in the earlier work and with
improved orbital wave functions. They confirmed that the magnitude of J is in
serious error, being much smaller than is required to explain magnetic order in
these 3d metals, but furthermore they found that J should be negative.
Finally in the rare earth metals the magnetic 4f electrons which determine the
magnetic properties are highly localized so that there is no significant overlap and
hence no direct exchange mechanism of the Heisenberg type can occur, although
these metals show a range of different magnetic order. Therefore an additional
exchange mechanism involving indirect exchange needs to be invoked.
It seems therefore that although the Heisenberg model is a useful concept the
interactions between electrons in real solids is not the simple direct Heisenberg
exchange.
11.1. 7 Exchange interactions between electrons in filled shells
In order to determine the type of magnetic order is it then necessary to find the
exchange energy between every electron in one atom and every electron in a
neighbouring atom?
It is easy to show on the basis of the Heisenberg model that the exchange energy
between electrons in a filled shell of one atom and electrons in a filled shell of a
Electron-electron interactions
253
neighbouring atom is zero.
2J L LSi'Sj
= - 2JLSiLSj
Eex =
-
and in a filled shell LSi = O. The exchange energy is consequently zero. This is a
useful result in that we only need to take into account interactions from electrons in
partially filled shells which simplifies the analysis considerably.
11.1.8 The Bethe-Slater curve
Can we make any simple and verifiable predictions of magnetic order in materials
based on the quantum theory of exchange interactions?
The magnetic behaviour of the 3d elements chromium, manganese, iron, cobalt
and nickel was of interest to the early investigators of the quantum theory of
magnetism. Slater [12,13] published values ofthe interatomic distances rab and the
radii of the incompletely filled d subshell r d of some transition elements. It was
found that the values of r ab/r d seemed to correlate with the sign of the exchange
interaction. For large values of rab/rd the exchange was positive while for small
values it was negative.
Bethe [14] made some calculations of the exchange integral in order to obtain J
as a function of interatomic spacing and radius of d orbitals. He found the
exchange integral for the electrons on two one electron atoms is given by
where rab is the distance between the atomic cores, r 21 is the distance between the
two electrons, ra2 and r b1 are the distances between the electrons and the respective
nuclei. Evaluation of this integral was made using the results of Slater and it was
found that the exchange integral J becomes positive for small r 12 and small rab.1t
also become positive for large ra2 and r b1 . The exchange integral can therefore be
plotted against the ratio rab/r d where rd is the radius of the 3d orbital. This gives the
Bethe-Slater curve as shown in Fig. 11.2 which correctly separates the
ferromagnetic 3d elements such as iron, cobalt and nickel from the
antiferromagnetic 3d elements chromium and manganese.
This means that if two atoms of the same kind are brought closer together
without altering the radius of their 3d shells then r ab/r d will decrease. When r ab/rdis
large then J is small and positive. As the ratio is decreased J at first increases and
then after reaching a maximum decreases and finally becomes negative, indicating
antiferromagnetic order at small values of rab' The exact nature of the exchange
interaction is therefore dependent on the interatomic and interelectronic spacing.
Subsequently Slatter [15], Wohlfarth and Stuart and Marshall [9] expressed
dissatisfaction with the Bethe-Slater model after difficulties with the sign and
254
Quantum theory of magnetism
+
o
Fig. 11.2 The Bethe-Slater curve representing the variation of the exchange integral J with
interatomic spacing r ab and radius of unfilled d shell rd' The positions of various magnetic
elements on this curve are indicated. The rare earth elements lie to the right of nickel on the
curve.
magnitude of J. Finally Herring [16] showed that the Heitler-London method of
calculating the spin coupling is ultimately unreliable.
Therefore although the whole approach of Heitler-London, Heisenberg and
Bethe still provides a useful conceptual framework for discussing the magnetic
interactions of electrons, the method seems to be ultimately inadequate and we
await a better description which can give more accurate values of J from first
principles.
11.2 THE LOCALIZED ELECTRON THEORY
Can the magnetic properties of a solid be described in terms of the properties of
electrons localized at the atomic sites?
We now go on to look at the magnetic properties in solids rather than isolated
atoms. The most natural extension of the above discussion of atomic magnetism is
to consider that the magnetic moments of atoms in a solid are due to electrons
localized at the ionic sites. This means that we can deal with the magnetic
properties of solids as merely a perturbation of the magnetic properties of the
individual atoms. We shall see how far this proves to be correct in the two main
groups of magnetic elements the 3d and 4f series.
11.2.1 Atomic magnetic moment due to localized electrons
How is the atomic magnetic moment determined from the angular momentum of the
atom?
We can apply the above ideas to calculate the magnetic moment of an atom if we
assume that the electrons in the unfilled shell which contribute to the magnetic
moment are all localized at the ionic site. The magnetic moment of an atom is
determined on this basis from the total angular momentum of the isolated atom
Electron-electron interactions
255
(hj2n)J which is obtained as a vector sum of the orbital and spin angular momenta
of electrons in unfilled shells.
m=
- gJiBJ,
where jiB is the Bohr magneton and g is the Lande splitting factor which is equal to
g = 1+
J(J + 1) + S(S + 1) - L(L + 1)
2J(J + 1)
.
The magnetic moment can equally well be expressed as
as discussed in Chapter 3 where y is the gyromagnetic ratio and h is Planck's
constant.
We should note however that the values of J will not always be the same in the
solid as in the free atoms discussed above. In the 3d series metals there are
significant differences, even allowing for the quenching of the orbital angular
momentum, but in the 4f or lanthanide series there is good agreement between the
magnetic moments in the solid and the isolated atoms.
11.2.2
Quantum theory of paramagnetism
How do all these ideas combine to provide an explanation of paramagnetism, and how
difJerent is this theory from the classical models?
Single-electron atoms
By single-electron atoms we mean atoms with only one unpaired electron
contributing to the magnetic properties. A good example is nickel. If we consider
the energy of a single magnetic moment m under the action of a magnetic field H,
the energy is given by
E = - Jiom·H.
We have now an expression for the magnetic moment of an isolated, and hence
paramagnetic atom
m= gJiBJ
E=
-
JiogJiBJ· H.
If we consider only the spins on the electrons and ignore the orbital contribution
then g = 2 and J = S = s = Hor a single-electron spin. As the spin can only have the
values ms = ± 1this means that, in the case of a single-electron atom, the electronic
magnetic moment, as a result of quantization, can align only either parallel (ms = 1)
or antiparallel (ms = - 1) to the field direction.
An atom with one unpaired electron therefore has two possible energy states in
the magnetic field. The estimation of the magnetization using Boltzmann statistics
256
Quantum theory of magnetism
leads to
M = Nm tanh (j.,tomH/kBT)
= N gJ j.,tB tanh (j.,togJ j.,tBH/kB/T),
where M is the bulk magnetization or magnetic moment per unit volume, N is the
number of atoms per unit volume and m is the magnetic moment per atom.
Solutions of this equation as a function of H/T are shown as a special case of the
Brillouin function with J = 1in Fig. 11.3.
For values of j.,tomH/kBT« 1 this leads to the approximation
M = Nm 2 j.,toH/k BT
Nm 2 j.,to
x=-----:--kBT
which of course is the Curie law.
1.0
0.8
~0.6
><
c:
o
u
c:
:;::
tL
c:
·5 0.4
.2
.~
CD
0.2
o
2
3
4
5
x
Fig. 11.3 The value of the Brillouin function BJ(x) for various values of J (!,~
and infinity)
and as a function of x. In the quantum theory of paramagnetism, for example
x = j1ogJj1BHjkBT.
Electron-electron interaction
257
Multielectron atoms
By multielectron atoms we mean simply atoms with more than one electron
contributing to the magnetic properties. Nickel is by this definition a singleelectron atom, while iron is a multielectron atom.
7.00
r1"T-C:ZP~V?'
I
6.00 I-~"T+.t
5.00
S=t(Fe3+)
4.00
M
Me
3.00
S=t(Cr3+)
I
2.00
0 1.30° K
6 2.00° K
x 3.00° K
o
1.00
4.21° K
.00 1.-.............1.-.1.-...1.-...1.--'--'--'--'--'-........................................_ -....
o
10
20
30
40
H
104
A
x---47t m -deg
T
( X
103~)
deg
Fig. 11.4 Comparison of theory and experiment for the magnetization curves of three
paramagnetic salts containing Gd H , Fe H and Cr H ions, respectively, after Henry [19J.
These salts are potassium chromium alum, ferric ammonium alum and gadolinium sulphate
octahydrate.
258
Quantum theory of magnetism
In these more complex situations there are 2J + 1 energy levels and so the
expression for the magnetization is
M
=
N J
g J1B
B (gJJ1BJ10H)
J
kBT
'
where Bix) is the Brillouin function [17] which is defined as
Bix) = (2J2; 1) ) coth (2J ~
I)X) - (2~
) coth (;J ).
The above equations for the single-electron atoms are then seen to be merely
special or restricted cases of this general function.
The equation for the magnetization of the multiple-electron atom is the
quantum analogue of the Langevin function [18] given in Chapter 9. In the
limiting case when J ---+ 00, that is when there are no quantum mechanical
restrictions on the allowed directions and values of magnetic moment in the atom
we arrive at the classical Langevin expression for magnetization.
Solutions for the paramagnetic magnetization equation are shown for different
values of S in Fig. 11.4 together with experimental values for three paramagnetic
salts as determined by Henry [19].
Curie's law
The Curie law of paramagnetic susceptibilities can be derived from the quantum
theory of paramagnetism. For J1omH/kBT« 1 the Brillouin function becomes
BJ{p,omH/k BT) = J1omH/3kB T
NJ1om2H
M=--'--3kB T
M NJ1om2
x= H= 3kB T
N J1og 2 J1/ J(J
+ 1)
3kB T
which is the Curie law, with
We see from this discussion that the quantum theory of paramagnetism provides
qualitatively similar results to the classical theories, however the classical Langevin
function needs to be replaced by its quantum mechanical analogue, the Brillouin
function which leads to different numerical results.
Electron-electron interactions
259
11.2.3 Quantum theory of ferromagnetism
How can we extend the above theories to a quantum theory offerromagnetism and
how different is this from the classical models?
The quantum theory of ferromagnetism is derived from the quantum theory of
paramagnetism in much the same way as the classical Weiss theory of
ferromagnetism was derived from the Langevin theory of paramagnetism. A
perturbation in the form of an interaction, or exchange coupling, is introduced
into the quantum theory of paramagnetism so that the electrons on neighbouring atoms interact with one another. The energy of an electron in a magnetic
field therefore becomes, as before
E = - J1om(H + aM),
where aM represents the interaction of the electron moment with those other
electrons close by.
Magnetization
In the case of an atom with one electron this leads via statistical thermodynamics to
M = N gJ J1B tanh [J10gJ J1B~
+ aM) ]
and, for multielectron atoms, to
M
=
N J
g J1B
B
J
[gJ J1BJ1o(H
+ aM) ]
kBT
where Blx) is the Brillouin function.
The Curie- Weiss law
At higher temperatures this system will be paramagnetic, so that M will be uniform
throughout the material, and hence
M
=
N J1og 2J1B 2 J(J
+ 1)(H + aM)
3kB T
NJ10g 2J1 B2J(J + 1)/3kBT
M
X = H = 1 - aNJ1og 2J1B2 J(J + 1)/3kBT
x=
-~
N J1og 2J1B 2 J(J + 1)
3k BT - aN J10g 2J1B2 J(J + 1)'
-~
which is the Curie-Weiss law in which
C=NJ10g 2J1 B2J(J + 1)
Tc=aNJ1og 2J1 B2J(J + 1)/3kBT.
260
Quantum theory of magnetism
We see therefore that the quantum theory of ferromagnetism also leads to
qualitatively similar results to the classical theory.
In the ferromagnetic regime the magnetization M will not be uniform
throughout the solid. Since any individual magnetic moment is likely to be more
affected by other moments nearby than by distant magnetic moments, the
moments will couple to the spontaneous magnetization within the domain Ms
rather than the bulk magnetization M when in the ferromagnetic state. Therefore a
nearest-neighbour model, with possible extension to next and higher order
neighbours, is more appropriate in the ferromagnetic regime.
11.2.4 Temperature dependence of the spontaneous magnetization
within a domain
How does the magnetization within a domain vary with temperature according to the
quantum theory of ferromagnetism?
The spontaneous magnetization within a domain Ms is determined by solving the
ferromagnetic Brillouin function in the absence of a field
M =N J
g JiB
s
B (JiOgJ JiBrxMS)
J
kBT
.
0.8
J=1/2
J=I
Classical
J =co
0.6
Ms
Me
0.4
o
o Fe
o Ce,Ni
TI
0.2
0.4
0.6
0.8
1.0
Fig. 11.5 Temperature dependence of the spontaneous magnetization within a domain for
iron, cobalt and nickel, compared with calculations based on the ferromagnetic Brillouin
function.
The itinerant electron theory
In the case of nickel we can use the S =
261
t solution so that
Ms = NgJ /lB tanh (
/logJ /lBIXMs)
kBT
.
Solutions of this are shown in Fig. 11.5. It is seen from this that the spontaneous
magnetization is only weakly dependent on temperature until it reaches 0.75 Te.
Above that it decreases rapidly towards zero at the Curie temperature.
11.2.5 Critique of the local moment model
How well does this model work and how realistic is it?
The local moment model works well in that it does provide relatively simple
mathematical functions for the dependence of magnetization on magnetic field and
temperature. It is quite realistic for the lanthanides with their closely bound 4f
electrons which determine the magnetic properties of the solid. In most other cases
the magnetic properties of paramagnets and the other main group of ferro magnets
the 3d series, the 'magnetic' electrons are outer electrons which are relatively free to
move through the solid. Therefore the localized model, despite its successes, is not a
realistic model for these cases.
11.3
THE ITINERANT ELECTRON THEORY
What happens
if the 'magnetic' electrons are not localized at the atomic sites?
If the magnetic electrons are in unfilled shells then in a number of metals these
magnetic electrons are unlikely to be localized as described above. The unfilled
shells are in most cases among the furthest removed from the nucleus and it is these
electrons which are most easily removed. This means that alternative models must
be sought for these metals in order to provide a realistic theory.
11.3.1
The magnetism of electrons in energy bands
What are the magnetic properties of electrons in energy bands?
So far we have only considered the magnetic properties of solids in terms of
localized magnetic moments which behave as if they were attached to the atomic
cores in the material. Thus we have talked about atomic magnetic moments.
However we have come to realize that these magnetic moments are really due to
the angular momentum of unpaired electrons in unfilled shells. With the exception
of the lanthanides the unpaired 'magnetic' electrons are usually outer electrons and
so are unlikely to be closely bound to the atoms.
Metals such as the alkali metal series lithium, sodium, potassium, rubidium and
caesium all show temperature independent paramagnetism for example which can
not be explained by the local moment model. In this case the paramagnetism is due
to the outer electrons which behave as a free electron gas.
262 Quantum theory of magnetism
This is true of the 3d transition elements iron, nickel and cobalt. We therefore
need to find a theoretical description of magnetism due to itinerant electrons in
these cases.
11.3.2 Pauli paramagnetism of 'free' electrons
Can paramagnetism be described simply on the basis of changes in population of
nearly free electrons in bands?
The Langevin theory of paramagnetism and its quantum mechanical analogue
work for dilute paramagnets such as the hydrated salts of transition metal ions.
However it is well known that the paramagnetic susceptibility of most metals is
independent of temperature and hence does not follow the classical Langevin
theory. In addition the paramagnetism of most metals is considerably weaker than
would be expected on the basis of the localized model.
The reason that the paramagnetic susceptibility is so much lower is that the
electrons are in general not free to rotate into the field direction as required by the
Langevin model. This is because, as a result ofthe Pauli exclusion principle [6], the
electron states needed for reorientation are already occupied by other electrons.
Only those electrons within an energy kB T of the Fermi level are able to change
H=O
H>O
E
E
N
N(E}
Spin Moments
Parollel To H
Spin Moments
Antiporollel To H
(a)
N(E}
fLomH
Spin Moments
Parallel To H
Spin Moments
Antiparallel To H
(b)
Fig. 11.6 Pauli paramagnetism for nearly free electrons in a magnetic field H. In (a) the field
is zero so that both up and down spin sub-bands have identical energy. In (b) the field causes
an energy difference between sub-bands with spins aligned parallel and anti parallel to the
field.
The itinerant electron theory
263
orientation. We therefore need to consider an alternative model of paramagnetism
due to the band electrons as conceived by Pauli [20].
Consider therefore the parabolic distribution of nearly free electrons as shown in
Fig. 11.6. Only the fraction T ITf of the total number of electrons can contribute to
the magnetization where T is the thermodynamic temperature and Tf is the Fermi
temperature, defined such that Ef = kBTf • Therefore
M=(N~;H)
NJi om 2H
3kB Tf •
The number density of electrons parallel to the field is
f
~ tf
N+=
t
f(e)D(e
+ JiH)de
f(e)D(e)de
+ tJiomHD(ef )
and the number density of those antiparallel to the field
N- =
t f f(e)D(e -
JioH)de
~ t f f(e) D(e) de -
!JiomHD(e f )·
The magnetization is then given by
M=m(N+ -N_)
=
m2/JoHD(ec)
3Nm 2JioH
2kB Tf
This equation gives the Pauli spin magnetization of the paramagnetic conduction band electrons. The susceptibility is then
3Nm 2Jio
X= 2k T
B f
which is temperature independent since Tf is a constant.
11.3.3
Band theory of ferromagnetism
How do electrons in bands behave when there is an exchange interaction which causes
alignment of the spins?
The band theory of ferromagnetism is a simple extension of the band theory of
paramagnetism by the introduction of an exchange coupling between the
264 Quantum theory of magnetism
electrons. The simplest case is to consider the electrons to be entirely free, that is a
parabolic energy distribution. This simplifying assumption does not alter the main
conclusions of the theory. The band theory of ferromagnetism was first proposed
by Stoner [21J and then independently by Slater [22].
Since magnetic moments can only arise from unpaired electrons it is
immediately clear that a completely filled energy band cannot contribute a
magnetic moment since in such an energy band all electron spins will be paired
giving L = 0 and S = O. In a partially filled energy band it is possible to have an
imbalance of spins leading to a net magnetic moment per atom. This arises
because the exchange energy removes the degeneracy of the spin up and spin
down half bands as shown in Fig. 11.7. The larger the exchange energy the greater
the difference in energy between these two half bands. Electrons fill up the band
by occupying the lowest energy levels first. If the half bands are split as in
Fig. l1.7(a) the electrons can begin to occupy the spin down half band before
the spin up half band is full. This usually leads to a non integral number of
magnetic moments per atom. A larger exchange splitting can lead to a separation
between the half bands as in Fig. l1.7(b). In this case the spin up band must be
filled before electrons can enter the spin down band. This leads to an integral
number of magnetic moments per atom.
For example consider the situation in Fig. 11.8, where 10 electron states exist
Spins up
E Spins down
Spins up
E Spins down
tt
N(E)
N(E)
(a)
N(E)
N(E)
(b)
Fig. 11.7 (a) Schematic band structure density of states, showing exchange splitting of the
spin up and spin down half bands, with remaining overlap between the two half bands
leading to a magnetic moment with a non integral number of Bohr magnetons per atom.
(b) Schematic band structure density of states with a larger exchange splitting leading to
complete energy separation of the half bands and to a magnetic moment with an integral
number of Bohr magnetons per atom.
The itinerant electron theory
Spin up Spin down
265
Spin up Spin down
t
E
JlH =
0
JlH = 2/10
= 0.2 JlB/atom
(a)
lb)
Fig. 11.8 Diagram representing the occupation of allowed energy states in an electron
band: (a) balanced numbers of spin up and spin down electrons; (b) unbalanced numbers of
spin up and spin down electrons due to the exchange interaction, leading to a net magnetic
moment per atom.
in close proximity in an electronic band. A spin imbalance of 2 can be created
by flipping one of the moments into the opposite direction. This then gives rise
to a magnetic moment of 0.2 JIB per atom. We now see the possibility of atomic
magnetic moments that are non-integral multiples of the Bohr magneton.
11.3.4 Magnetic properties of 3d band electrons
Can the band theory provide a satisfactory description of the magnetic properties of
the 3d metals?
In the transition metals iron, nickel and cobalt which are the three ferromagnetic
elements with electronic structure for which the band theory should apply, the
magnetic properties are due to the 3d band electrons. Of course there is also a 4s
electron band but this contains two paired electrons and so does not affect
magnetic properties.
The 3d band can hold up to 10 electrons (5 up and 5 down) and it is here that we
must concentrate attention in order to explain the observed properties. The
exchange interaction is responsible for creating the imbalance in spin up and spin
down states. In the absence of the exchange energy the spin imbalance would be an
excited state but this does not require too much energy in the 3d band because of
the high density of states and therefore a positive exchange interaction can be
sufficient to cause the alignment resulting in a spin imbalance and a net magnetic
moment per atom.
If we suppose that in these metals the exchange interaction causes 5 of the 3d
electrons to align 'up' and the remainder 'down', we arrive at an equation which
approximates observed magnetic moments in these elements very well. Let n be the
266
Quantum theory of magnetism
number of 3d + 4s electrons per atom, x be the number of 4s electrons per atom,
and n - x the number of 3d electrons per atom
m
= [5 -
(n - x -
5)] fiB
= [10 - (n - x)] fiB'
We can approximate observed magnetic properties by the assumption that x
=0.6.
m= (10.6 - n)fiB
which for nickel with n = 10, cobalt with n = 9 and iron n = 8 gives the following
= 0.6 fiB
Ni:
m
Co:
m=1.6fiB
Fe:
m = 2.6 fiB'
These results are quite close to the known values.
In this way the band theory can account for the non-integral atomic magnetic
moments in these metals, a result that is more difficult to justify on the localized
moments model.
11.3.5 The Slater-Pauling curve
How well does the itinerant electron model describe the magnetic properties of 3d
alloys?
The above argument can be used to explain the moment per atom of several of the
3d transition metals from manganese to copper. The Slater-Pauling curve [23,24]
gives the magnetic moments of these 3d metals and their alloys from the premises of
the itinerant electron theory. This is shown in Fig. 11.9. Most alloys fall on to a
locus consisting of two straight lines beginning at chromium with 0 Bohr
magnetons rising to 2.5 Bohr magnetons between iron and cobalt and dropping to
o Bohr magnetons again between nickel and copper. The metals in this range
chromium, manganese, iron, cobalt, nickel and copper have total numbers of
electrons ranging from 24 to 29, while the number of 3d electrons ranges from 5 in
chromium to 10 in copper.
The interpretation ofthe results in Fig. 11.9 is in terms of the rigid band model. It
is considered that the alloy metals share a common 3d band to which both
elements contribute electrons. The maximum magnetic moment occurs at a point
between iron and cobalt. It appears from this model that, as expected, the 3d and 4s
electrons are responsible for the magnetic properties of these metals and alloys, and
that they are relatively free. Therefore it is a reasonable assumption that they are
shared between the ions in a common 3d band.
Pauling [24] has suggested that the 3d band is broken into two parts: the upper
part capable of holding 4.8 electrons (2.4 up and 2.4 down) and the lower part
capable of holding 5.2 electrons (2.6 up and 2.6 down). This means that as electrons
are removed beginning with zinc for example the magnetic moment is increased by
Further reading
3.0
FCC
'"c0
2.5 t-+:~ir1
Ie
Fe-V
. + Fe- Cr
0 Fe-Ni(1)
•
o
Qi
c
OJ
'"
E
2.0
I-+.,;~
t::.
"
+-~.,_i
"l
o
1:
0
III
.=
'0
1.5
...
o
C
Q>
E
0
[>
1.0 1 - - - i r - - - . l•."-+trJf1~I'O
0
•
x
E
<>
'E
0
;(
267
Fe-Co
Ni-Co
Ni-Cu
Ni- Zn
Ni-V
Ni-Cr
Ni-Mn
Co-Cr
Co-Mn
SONI5OCo- V
SONi5OCo- Cr
Fe-Ni(2)
Pure metals
0.5
a
Cr
24
Mn
25
Fe
26
Co
27
Ni
28
Cu
29
Fig. 11.9 The Slater-Pauling curve which gives the net magnetic moment per atom as a
function of the number of 3d electrons per atom. The various 3d elements are shown at the
relevant points on the x-axis, but the curve is more general, being primarily of interest in
explaining the magnetic moments of various intra-3d alloys.
depletion of spin down electrons in the upper part of the band until a magnetic
moment of 2.4 Bohr magnetons is reached between iron and cobalt. Then the
moment begins to decrease again towards chromium because the removal of
further electrons results in a reduction in spin up electrons from the upper part of
the 3d band.
11.3.6
Critique of the itinerant electron model
What are the strengths and weaknesses of the itinerant electron model?
The itinerant electron model has had successes and these include the ability to
explain non-integral values of atomic magnetic moments and to predict some
aspects of the magnetic behaviour of the 3d series metals and alloys. Furthermore
the localized moment model, which is the main alternative, is certainly open to the
objection that it is unrealistic in so far as the magnetic electrons of most atoms, with
the exception of the lanthanides, are relatively free being in the outer shells.
The drawback of the itinerant electron theory is that is extremely difficult to
make fundamental calculations based on it. Unlike the local moment theory which
lends itself readily to simple models such as the Weiss [25] and Heisenberg [26]
models of ferromagnetism, the itinerant electron theory does not provide any
simple model from which first principles calculations can be made. Therefore
although the current opinion is that the itinerant theory is intrinsically closer to
reality in most cases, interpretations of magnetic properties are still more often
made on the basis of the localized moment model.
268
Quantum theory of magnetism
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
Martin, D. H. (1967) Magnetism in Solids, Illife, London, Ch. 5.
Heitler, W. and London, F. (1927) Z. Phys., 44, 455.
Heisenberg, W. (1928) Z. Phys., 49, 619.
Dirac, P. A. M. (1929) Proc. Roy. Soc., A123, 714.
Weiss, P. (1907) J. de Phys., 6, 661.
Pauli, W. (1925) Z. Phys., 31, 765.
Hofmann, 1., Paskin, A., Tauer, K. J. and Weiss, R. J. (1956) J. Phys. Chem. Sol.,l, 45.
Wohlfarth, E. P. (1949) Nature, 163, 57.
Stuart, R. and Marshall, W. (1960) Phys. Rev., 120, 353.
Watson, R. E. and Freeman, A. 1. (1961) Phys. Rev., 123, 2027.
Watson, R. E. and Freeman, A. 1. (1962) J. Phys. Soc. Jap., 17, B-1, 40.
Slater, 1. C. (1930) Phys. Rev., 35, 509.
Slater, 1. C. (1930) Phys. Rev., 36, 57.
Bethe, H. (1933) Handb. d. Phys., 24, 595.
Slater, 1. C. (1936) Phys. Rev., 49, 537; 49, 931.
Herring, C. (1962) Rev. Mod. Phys., 34, 631.
Brillouin, L. (1927) J. de Phys. Radium, 8, 74.
Langevin, P. (1905) Ann. de Chem. et Phys., 5, 70.
Henry, W. E. (1952) Phys. Rev., 88, 559.
Pauli, W. (1926) Z. Phys., 41, 81.
Stoner, E. C. (1933) Phil. Mag., 15, 1080.
Slater, 1. C. (1936) Phys. Rev., 49, 537.
Slater, 1. C. (1937) J. Appl. Phys., 8, 385.
Pauling, L. (1938) Phys. Rev., 54, 899.
Weiss, P. (1907) J. de Phys., 6, 661.
Heisenberg, W. (1928) Z. Phys., 49, 619.
FURTHER READING
Chikazumi, S. (1964) Physics of Magnetism, Ch. 3, Wiley, New York.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Ch. 3, Addison-Wesley, Reading,
Mass.
Morrish, A. H. (1965) The Physical Principles of Magnetism, Wiley, New York.
EXAMPLES AND EXERCISES
Example 11.1 The exchange interaction. Explain the origin of the exchange
interaction J. Does this have a classical analogue? If not how can it be interpreted
classically? Since the exchange interaction is only invoked to prevent two electrons
occupying the same energy levels with the same set of quantum numbers it would
seem that the exchange interaction should lead only to antiferromagnetism by
ensuring that the 'exchanged' electrons have antiparallel spins. Explain why the
exchange interaction does not lead only to antiferromagnetism.
Example 11.2 Magnetic moment of dysprosium ions. Dy3 + has nine electrons in
its 4f shell. What are the values of L, Sand J? Calculate the susceptibility of a salt
containing 1 g mole of Dy 3 + at 4 K.
Example 11.3 Paramagnetism of S = 1 system. Find the magnetization as a
function of magnetic field and temperature of a system with S = 1, moment m and
concentration N atoms per unit volume. Show that the limit J.1omH « kB T leads to
M ~ (2J.1oNm2H)j3kBT.
12
Soft Magnetic Materials
Applications of soft ferromagnetic materials are almost exclusively associated with
electrical circuits in which the magnetic material is used to amplify the flux
generated by the electric currents. In this chapter we will consider both a.c. and d.c.
applications of soft magnetic materials. These each require somewhat different
material properties, although in general high permeability and low coercivity are
needed in all cases.
12.1
PROPERTIES AND USES OF SOFT MAGNETIC MATERIALS
Soft ferromagnetic materials find extensive applications as a result of their ability
to enhance the flux produced by an electrical current. Consequently the uses of soft
materials are closely connected with electrical applications such as electrical power
generation and transmission, receipt of radio signals, microwaves, inductors, relays
and electromagnets. The available range of magnetic properties of materials is
continually being increased and here we shall give some indication of the present
status.
12.1.1 Permeability
Permeability is the most important parameter for soft magnetic materials since it
indicates how much magnetic induction is generated by the material in a given
magnetic field. In general the higher the permeability the better for these materials.
Initial permeabilities range from Ilr = 100000 [1] in materials such as permalloy
down to as low as Ilr = 1.1 in some of the cobalt-platinum permanent magnets. It is
well known that initial permeability and coercivity have in broad terms a
reciprocal relationship, so that materials with high coercivity necessarily have a
low initial permeability and vice versa.
12.1.2 Coercivity
Coercivity is the parameter which is used to distinguish hard and soft magnetic
materials. Traditionally a material with a coercivity of less than 1000 A/m is
270
Soft magnetic materials
considered magnetically 'soft'. A material with a coerclVlty of greater than
10000 A/m is considered magnetically 'hard'. Low coercivities are achieved in
nickel alloys such as permalloy in which the coercivity can be as low as 0.4 A/m [2].
In some ofthe recent permanent magnet materials intrinsic coercivities in the range
1.2 x 106 A/m are commonly observed.
12.1.3 Saturation magnetization
The highest saturation magnetization available is 2.43 T which is achieved in an
iron-cobalt alloy containing 35% cobalt. The possible values of saturation
magnetization then range downward continuously to effectively zero.
12.1.4 Hysteresis loss
The hysteresis loss is the area enclosed by the d.c. hysteresis loop on the B, H plane
as discussed in section 5.1. It represents the energy expended during one cycle of the
hysteresis loop. The hysteresis loss increases as the maximum magnetic field
reached during the cycle increases. Clearly for a.c. applications in which energy
dissipation should be minimized the hysteresis loss should be as low as possible.
This loss is closely related to the coercivity so that processing of materials to reduce
coercivity also reduces the hysteresis loss. However in a.c. applications the
hysteresis loss is not the only dissipative or loss mechanism.
12.1.5
Conductivity and a.c. electrical losses
One of the most important parameters of a magnetic material for a.c. applications
is its electrical loss. The electrical losses are shown for various materials in Table
12.1 at 50 Hz and an amplitude of excitation of 1 T [3].
The electrical losses depend on the frequency of excitation v, the amplitude of
magnetic induction Bmax, the hysteresis loss WH , the sheet thickness t (due to the
Table 12.1 Total electrical losses of various soft magnetic
materials with sheet thicknesses t between 0.2 mm and 0.5 mm at
a frequency of 50 Hz and an induction amplitude of 1 T [3, p. 46]
Total losses
Material
Commercial iron
Si-Fe hot rolled
Si-Fe cold rolled, grain oriented
50% Ni-Fe
65% Ni-Fe
~ot
(W/kg)
5-10
1-3
0.3-0.6
0.2
0.06
Properties and uses of soft magnetic materials
penetration depth of the a.c. magnetic field) and the eddy current dissipation
271
Wee'
In addition there is usually a discrepancy between the measured loss and the loss
expected from the sum of hysteresis and eddy current losses and this is usually
referred to as the anomalous loss Wa' The total electrical power loss ~ot
can be
expressed as the sum of these components
where the hysteresis loss and eddy current loss are frequency dependent. The
hysteresis loss increases linearly with frequency while the eddy current loss
increases with the frequency squared.
The eddy current loss, at frequencies which are low enough for the inductive
effects of the eddy currents to be neglected, is given by the general equation
W
ee
=
1[2 B2
max
d2 v2
pf3
where d is the cross sectional dimension, p is the bulk resistivity in n· m and f3 is a
coefficient which has different values for different geometrics. For example f3 = 6
for laminations, in which d represents the thickness t. For cylinders f3 = 16 and d
represents the diameter. For spheres f3 = 20 and d represents the diameter.
Stephenson [4] has shown that for low alloy, non oriented electrical steel sheets the
above equation, which reduces to
1.644B!ax t2v2
Wee = ----'----"-'--
pD
where D is the density in kilograms/m 3 , gives the eddy current loss in watts per
kilogram. The anomalous loss at 60 Hz and an induction amplitude of 1.5 to 1.7 T
was found to be dependent on t 2 / p, so that
kt 2
Wa= Wao +-,
p
where Wao and k are empirical constants. From the empirical Steinmetz law [5] the
d.c. hysteresis loss is known to depend on induction amplitude Bmax and frequency
v according to the relation
where 1] is a material constant and n is an exponent which lies in the range 1.6-2.0.
The total losses can be reduced ifthe conductivity of the material is reduced. This
is exploited in transformer material such as silicon iron in which the silicon is
added principally to reduce the conductivity, although it has an adverse effect on
the permeability but does reduce the coercivity. The losses in Ni-Fe alloys are
lower than for silicon-iron, and this is also used in a.c. applications such as
inductance coils and transformers, but silicon-iron has a higher saturation
magnetization.
272
Soft magnetic materials
12.1.6 Electromagnets and relays
The most important d.c. uses which soft magnetic materials find are in electromagnets and relays. An electromagnet is any device in which a magnetic field
H is generated by an electric current and the resulting flux density B is increased by
the use of a high-permeability core. The simplest example is a solenoid carrying a
d.c. current wound around a ferromagnetic core. In electromagnets soft iron is still
the most widely used material because it is relatively cheap and can produce high
magnetic flux densities. It is often alloyed with a small amount of carbon
( < 0.02 wt%) without seriously impairing its magnetic properties for this
application. Also the alloy Fe-35%Co is used in electromagnet pole tips to increase
the saturation magnetization.
A relay is a special form of d.c. electromagnet with a moving armature which
operates a switch. This can be used for example to open and close electrical circuits
and is therefore important as a control device.
12.1.7 Transformers, motors and generators
A transformer is a device which can transfer electrical energy from one electrical
circuit to another although the two circuits are not connected electrically. This is
achieved by a magnetic flux which links the two circuits through an inductance
coil in each circuit. The two coils are connected by a high-permeability magnetic
core. The main consideration for the purpose of this chapter is the choice of a
suitable material for the magnetic core of the transformer. Transformers have
also been described in section 4.2.3. The material used for transformer cores is
almost exclusively grain-oriented silicon-iron, although small cores still use
non-oriented silicon-iron. High-frequency transformers use cobalt-iron although
this represents only a small volume of the total transformer market.
Generators are devices for converting mechanical energy into electrical energy.
Motors are devices for converting electrical energy into mechanical energy. Both
are constructed from high-induction, high-permeability magnetic materials. The
most common material used for these applications is non-oriented silicon-iron but
many smaller motors use silicon-free low-carbon steels.
12.2
MATERIALS FOR A.C. APPLICATIONS
We will first look at uses in power generation and transmission since these are
easily the most important areas in which soft ferromagnetic materials are
employed. The desirable properties here are, as in most soft magnetic materials
applications, high permeability and saturation magnetization with low coercivity
and power loss. In addition suitable mechanical properties are a consideration
since silicon-iron becomes very brittle at high silicon content. Generally it is not
possible to have all of these properties in a single material so it is necessary to
decide which is the most crucial property in any given situation. For high power
applications silicon-iron is widely used. Non-oriented silicon-iron is the material
w
~2-
o
N 20~-
ti
~
u
!
~
1a~-
_-l
16
~
~
14-~_
ci.
r
I-
:>
in~
It:
1.~:!rt=J
lO W
0
0.2
0.4
0.6
Qa
1.0
1.2
1.4
1.6
1.a
2.0
PERCENT OF ALLOYING ELEMENT IN IRON
Fig. 12.1 Variation of the electrical resistivity of iron with the addition of different alloying
elements.
1.80
1
1.60
1.40
~
1.20
.J
.....
~
- 1.00
en
en
o
.J
w
.80
u
1\
~I-
\
\
3.0
~.7T
~T
.60
.40
.20
o
4.0
3.5
0::
o
-
I
T
60Hz; 0.35mm
1900 1910
~
"'-
'\
""
go
.&
-
""'-
-
.tr
2.5 ~
2.0
en
en
g
\oJ
It:
1.5 0
u
HIGH INDUCTION ........
1.0
'"
""-
1920 1930 1940 1950
-
1960
0.5
1970 198)
Fig. 12.2 Reduction of core loss of 0.35 mm thick silicon-iron at 60 Hz from the year 1900
until 1975.
274
Soft magnetic materials
of choice in motors and generators while grain-oriented silicon-iron is used for
transformers.
12.2.1 Iron-silicon alloys
In electrical power generation and transmISSIOn the greatest demand is for
transformer cores. In this area silicon-iron is used to the exclusion of all others.
This is also known as 'electrical steel' or silicon steel both of which are misnomers
since these materials are not really steels.
In the power industry the electrical voltage is almost always low-frequency a.c.
at 50 or 60 Hz. This leads to an alternating flux in the cores of the electromagnetic
24~-,
12
8
>'
X
oell
-4
-8
-12
7
Si (wt. %)
Fig. 12.3 Dependence of the magneto stricti on coefficients A100 and A111 of silicon-iron on
silicon content. The lower curves in the range 4-7% Si are for slowly cooled samples.
Materials for A.C. applications
275
devices and consequently to the generation of 'eddy currents' if the material is an
electrical conductor. Eddy currents reduce the efficiency of transformers because
some of the energy is lost through eddy current dissipation.
There are several ways in which the properties of pure iron can be improved in
order to make it more suitable for transformer cores at low frequencies and these
have been discussed in a review paper by Littman [6]. First the resistivity can be
increased so that the eddy current losses become less. This is achieved by the
alloying of silicon with iron. The variation of resistivity of silcon~r
with silicon
content is shown in Fig. 12.1. Iron containing 3% silicon has four times the
resistivity of pure iron [7]. Over the years there have been substantial
improvements in the core losses of silicon iron. These improvements for 0.35 mm
thick sheets are shown in Fig. 12.2.
Silicon of course is a very cheap material which is an important consideration
800
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(BY
5
6
WEIGHT)
70
n
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0
z
Fig. 12.4 Various magnetic and electrical properties of silicon-iron as a function of silicon
content. © 1971 IEEE.
276
Soft magnetic materials
when so much transformer iron is needed. It has two main beneficial effects on the
properties of the alloy. The conductivity is reduced as silicon is added and the
magnetostriction is reduced as shown in Fig. 12.3. For a.c. applications this
reduction of magneto stricti on is an additional advantage since the cyclic stresses
resulting from magnetostrictive strains at 50 or 60 Hz produce acoustic noise.
Therefore any reduction of magneto stricti on is desirable, particularly if it arises as
a result of modifying the material to suit other unrelated requirements. A third
benefit caused by the addition of silicon is that it reduces the anisotropy (Fig. 12.4)
of the alloy leading to an increase in permeability in the non-oriented silicon-iron.
It is also beneficial to laminate the cores in such a way that the laminations run
parallel to the magnetic field direction. This does not interfere with the magnetic
flux path but does reduce the eddy current losses, by only allowing the eddy
currents to exist in a narrow layer of material. The dependence of core losses at
60 Hz on sheet thickness is shown in Fig. 12.5. Furthermore the coating of
laminations with an insulating material also improves the eddy current losses by
preventing current passing from one layer to the next. The thickness of the
laminations is comparable with the skin depth at 50 or 60 Hz, which is typically
0.3-0.7mm [8], for optimum performance.
1.8
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NEAR "UNORIENTED"
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THICKNESS - MILS
Fig. 12.5 Dependence of core loss on sheet thickness in 3.15% silicon-iron.
Materials for A.C. applications
277
1.4
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Fig. 12.6 Dependence of core loss on grain size in 3.15% silicon-iron sheets.
Since in these applications the magnetic flux only passes in one direction along
the laminations it is advantageous to ensure that the permeability is highest along
this direction. Therefore techniques have been devised for producing grainoriented silicon-iron by hot and cold rolling and annealing which result in the
[OOlJ direction lying along the length ofthe laminations. The 001> directions are
the magnetically easy axes. The addition of silicon to iron increases the grain size so
that the grain diameters in 3% silicon-iron can be as large as 10 mm. The core
losses of 3.15% silicon-iron vary with grain size as shown in Fig. 12.6 in which a
minimum in core loss occurs at a grain diameter of between 0.5 and 1.0 mm.
There are nevertheless some disadvantages involved with the addition of silicon
to iron. At higher silicon contents the alloy becomes very brittle and this makes a
practical limitation on the level of silicon that can be added without the material
becoming too brittle to use. This limit is about 4% and most silicon-iron
transformer material has a composition of 3-4% Si, although material with a
silicon content of 6% with adequate mechanical properties for transformer
applications has recently been developed. Another disadvantage resulting from the
addition of silicon to iron is the reduction of saturation induction.
<
12.2.2 Iron-aluminum alloys
The properties of aluminum-iron are very similar to those of silicon-iron and
since aluminum is rather more expensive than silicon these alloys are unlikely to
replace silicon-iron in applications where they compete. Furthermore the
278
Soft magnetic materials
103X 60 , - - - - - - r - - - , - - - - r - - - - r - - - r - - - - ,
(0) ANNEALED WITHOUT FIELD
(b) ANNEALED
50
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12
WEIGHT PERCENT ALUMINUM
Fig. 12.7 Maximum permeability of iron-aluminium alloys as a function of aluminium
content after two different types of anneal. The field anneal was in a magnetic field of
136 A/m. Specimens were 0.35 mm thick laminations.
presence of Al z0 3 particles in iron-aluminum alloys causes rapid wear of
punching dies which is disadvantageous. To date therefore the binary aluminumiron alloys have made little commercial impact.
The magnetic properties of some aluminum-iron alloys are shown in Figs. 12.7
and 12.8. Alloys of up to 17% Al are ferromagnetic but at higher aluminum
contents the alloys become paramagnetic. Often aluminum is used as an addition
in silicon-iron because it promotes grain growth, which can lead to lower losses.
Furthermore the addition of aluminum produces higher resistivity with less danger
from brittleness. Therefore ternary alloys of iron, silicon and aluminum are used in
electrical steels for special applications.
12.2.3 Nickel-iron alloys (Permalloy)
These alloys are the most versatile of all soft magnetic materials for
electromagnetic applications. Only the alloys with above 30% nickel are widely
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sheets of iron-16% aluminium.
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Fig. 12.9 Initial permeability of iron-nickel alloys: 1, slow cooled; and 2, normal permalloy
treatment.
H
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Fig. 12.10 Saturation magnetic induction in iron-nickel alloys.
10
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100
Materials for A.C. applications
281
used because at lower nickel contents there is a lattice transformation which occurs
as a function of temperature. This transformation exhibits temperature hysteresis
and hence there is no well-defined Curie temperature. As a result of this
complication the alloys in this range are not widely used.
Three groups of these alloys are commonly encountered [9]. These have nickel
contents close to 80%, 50% or in the range 30~4%.
The permeability is highest for
the alloys close to 80% Ni, as shown in Fig. 12.9. The saturation magnetization is
highest in the vicinity of 50% Ni (Fig. 12.10). The electrical resistivity is highest in
the 30% Ni range (Fig. 12.11). These are the three magnetic properties which are of
most interest in soft magnetic material applications and so the alloys used are often
close to one of these compositions depending on the specific application.
The main uses of these alloys are in inductance coils and transformers,
particularly power supply transformers. They are used at audio frequencies as
transformer cores and also for much higher frequency applications. They can be
made with low, or even zero, magnetostriction (Fig. 12.12). Some of the high
permeability alloys, Mumetal and Supermalloy, have relative permeabilities of up
80
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NICKEL CONTENT
Fig. 12.11 Electrical resistance in iron-nickel alloys.
............
90%,100
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282
Soft magnetic materials
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100
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Fig. 12.12 Dependence of magnetostriction coefficients A1 00 and A111 with nickel content in
iron-nickel alloys.
to 3 X 10 5 and coercivities down to 0.4 A/m. They also have low anisotropy and
this contributes to their high permeability in the polycrystalline form.
Permalloy and Mumetal, both Ni-Fe alloys with close to 80% Ni, are used in
magnetic screening because of very high permeability. Some of these alloys can
reach an induction of 0.6 T in a field as low as 1.6 A/m, corresponding to a relative
permeability of ttr = 3 X 10 5 . In these applications the low saturation of the
material is not a disadvantage since the magnetic screen can always carry more
magnetic flux if it is made thicker.
Alloy additions to the basic iron-nickel alloy and processing allow the magnetic
properties of these alloys to be varied within wide limits. Cold working by rolling
gives rise to high permeability perpendicular to the field as in Isoperm, a 50%-50%
Fe-Ni alloys. Invar is a 64% Fe 36% Ni alloy with zero thermal expansion.
High-quality transformers are often made of this material. Relative
permeabilities of up to 100000 are attainable with coercivities in the range 0.16800 A/m (0.002 to 100e) and these can be adjusted with precision by suitable
processing of the material. The core losses of two commercial Ni-Fe alloys are
shown in Fig. 12.13.
This alloy system is also used in some magnetic memory devices and amplifiers.
For high-frequency applications of up to 100kHz the alloy can be used in the form
of powdered cores in which each particle is by virtue of its nature electrically
insulated from others and therefore the bulk conductivity of the material is low.
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FLUX DENSITY - GAUSS
Fig. 12.13 Core losses against peak magnetic induction and frequency for 0.35 mm thick
laminations of two commercial nickel-iron alloys; Permalloy 80 and Alloy 48.
284
Soft magnetic materials
12.2.4 Amorphous metals (metallic glasses)
These materials have only been developed in the last 20 years. They are produced
by rapid cooling (quenching) of magnetic alloys consisting of iron, nickel and/or
cobalt together with one or more of the following elements: phosphorus, silicon,
boron and sometimes carbon [10]. Even in the as cast condition these alloys have
very soft magnetic properties but the rapidly quenched material has even better
properties for soft magnetic material applications. The molten metal is sprayed in a
continuous jet under high pressure on to a rapidly moving cold surface such as the
surface of a rotating metal wheel. The material is produced in the form of a thin
ribbon.
As a result of the rapid cooling the materials do not form a crystalline state but
instead produce a solid with only short-range order with otherwise random
microstructural properties. These materials can be considered as a random
packing of spheres. They are also known as 'metallic glasses' because of this
random structure. The materials produced in this way have large internal strains,
which as we know lead to high coercivity and low permeability. This can be altered
by annealing the material at intermediate temperatures in order to relieve the
strains without leading to recrystallization.
o
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Fig. 12.14 Variation of coercivity of a number of amorphous alloys with chemical
composition.
Materials for A.C. applications
285
About ten years ago amorphous alloys were considered to have great promise
and a large effort was expended to further develop them. The main interest arose
from the low coercivities, as shown in Fig. 12.14., which were an order of
magnitude smaller than silicon-iron while the permeability was about an order of
magnitude greater. The hysteresis loops of the Fe so B2o material (Metglas 2605CO)
at different frequencies are shown in Fig. 12.15. Core losses are also very low as
indicated in Figs. 12.16 and 12.17. Such properties were perceived to be a distinct
advantage, however certain disadvantages turned out to be even more significant.
One of the disadvantages is the low saturation magnetization of the alloys, as
shown in Fig. 12.18, which limits their use in heavy current engineering when
compared for example with silicon-iron. Secondly at higher flux densities their
core losses begin to increase rapidly. Therefore their general applicability has
fallen far short of expectations.
There is a better market for these alloys in rather low-current applications and
specialized small-device applications in which transformers are needed with only
moderate flux densities where the amorphous alloys can compete successfully with
the nickel-iron alloys such as Permalloy. These alloys are now being produced in
large quantities and are finding specific uses in pulsed power transformers and in
magnetic sensors and magnetostrictive transducers. Also because of their unique
metallurgical properties they continue to provide scientific interest for researchers
in magnetism.
Amorphous alloys with widely differing magnetic hysteresis properties can be
produced by annealing in the presence of a magnetic field. They can be used in low-
MAGNETIZING FORCE - H - AMPERES PER METER
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Fig. 12.15 Upper two quadrants of the hysteresis loops of Metglas 2605CO at different
frequencies.
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Fig. 12.16 Dependence of core loss on magnetic induction and frequency for various
amorphous alloys. All sample thicknesses were in the range 25-50/tm.
MAGNETIC INDUCTION· B • KILOGAUSSES
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1.8
MAGNETIC INDUCTION· B • TESLAS
Fig. 12.17 Comparison of core losses for different soft magnetic materials as a function of
peak magnetic induction.
288
~
Soft magnetic materials
10r-___
5
400
600
800
TEMPERATURE. K
Fig. 12.18 Temperature dependence of the saturation magnetic induction for different
amorphous alloys. Allied Chemical Metglas alloys are as follows: 2605 is FesoB 10; 2615 is
FesoP 16C3B; 2826 is Fe 4o Ni 4o P 14B6; 2826A is Fe 31 Ni 36 Cr 14P 12B6; and 2826B is
Fe 19 Ni49 P 14B6Si l'
power transformers which are needed in communications equipment for example.
The magnetic properties of a range of amorphous alloys is shown in Table 12.2.
12.2.5 High-frequency applications: soft ferrites
For high-frequency applications the conductivity of metals limits their use and so
we must turn to magnetic insulators. These materials must of course exhibit the
usual properties associated with soft ferro magnets: high permeability, low
coercivity and high saturation magnetization. In these applications soft ferrites are
widely used. Ferrites are ceramic magnetic solids which first appeared
commercially in 1945. They are ferrimagnetic but on the bulk scale behave in much
the same way as ferromagnets. The cubic or soft ferrites all have the general
chemical formula MO, Fe Z0 3 , where M is a transition metal such as nickel, iron,
manganese, magnesium or zinc. The most familiar of these is Fe 3 0 4 . The ferrite
289
Materials for A.C. applications
Table 12.2 Magnetic properties of amorphous alloys under d.c. conditions
Annealed
As cast
He
I1max
He
(10 3)
(Aim)
0.51
100
3.2
0.77
300
4.8
0.45
58
1.6
0.71
275
Toroid
4.6
0.54
46
0.88
0.70
310
Strip
Strip
Toroid
1.04
1.44
4.96
0.36
0.41
0.4
190
300
96
0.48
0.48
4.0
0.63
0.95
0.42
700
2000
130
Alloy
Shape
(Aim)
Metglas #2605
Feso B20
Metglas #2826
Fe40 Ni 4o P14B6
Metglas #2826
Fe29 Ni44P14B6 Si2
Fe4.7Co70.3SilSB10
(Fe.sNi.2bSisB14
Metglas #2615
FeSOP16C3B
Toroid
6.4
Toroid
Mr/Ms
MrlMs
I1max
(10 3)
CoO, Fe z0 3 although of the same general type is nevertheless a hard ferrite rather
than a soft ferrite. The magnetic garnets were discovered by Bertaut and Forret
[11]. Yttrium-iron garnet is the best-known example of these.
Soft ferrites can be further classified into the 'non-microwave ferrites' [12J for
frequencies from audio up to SOO MHz and 'microwave ferrites' for frequencies
from 100 MHz to SOOGHz [13]. Microwave ferrites, such as yttrium-iron garnet,
are used as waveguides for electromagnetic radiation and in devices such as phase
shifters.
Soft ferrites are also used in frequency selective circuits in electronic equipment,
for example in telephone signal transmitters and receivers. Manganese-zinc ferrite,
which is sold under the commercial name of Ferroxcube, is widely used for
applications at frequencies of up to 10 MHz, while beyond that frequency nickelzinc ferrites are preferred because they have lower conductivity. Another area
where ferrites find wide application is in antennae for radio receivers. Almost all
radio receivers using amplitude modulation of signals are now provided with
ferrite rod antennae. Other applications include waveguides and wave shaping for
example in pulse-compression systems.
The permeability of these materials does not change much with frequency up to
a critical frequency but then decays rapidly with increasing frequency. The critical
frequency ofthese materials varies between 10 MHz and 100 MHz. The saturation
magnetization of ferrites is typically O.S T, which is low compared with iron and
cobalt alloys.
For very high-frequency applications, beyond 100 MHz, there are other
materials such as the hexagonal ferrites which have special properties which make
them suitable for use at these frequencies. These materials are uniaxial with
magnetic moments confined to the hexagonal base plane.
290 Soft magnetic materials
12.3
MATERIALS FOR D.C. APPLICATIONS
In d.c. applications the need for a low conductivity does not arise and so there are
fewer constraints on the type of material suitable for particular applications. These
applications generally require low coercivity and high permeability. High
permeability is best achieved through high saturation magnetization and this
means that alloys of iron and cobalt are widely used. A review of soft magnetic
materials for d.c. applications has been given by Chin and Wernick [14].
12.3.1 Iron and low-carbon steels (soft iron)
These were the original materials for transformers, motors and generators but have
been superseded by silicon-iron both in its oriented form for transformers and in
its non-oriented form for motors and generators.
Soft iron is used as a core material for d.c. electromagnets such as laboratory
electromagnets for which it remains the best material. The prime concern is merely
to obtain either high fields and or very uniform magnetic fields. Iron with low levels
of impurities such as carbon (0.05%) and nitrogen has a coercivity of about 80 A/m
(1 Oe) and a maximum relative permeability of the order of 10000. By annealing in
hydrogen the impurities can be removed and this results in a reduction in
coercivity to 4A/m (0.05 Oe) and an increase in maximum relative permeability to
about 100000 as shown in Table 12.3. The highest relative permeability obtained
Table 12.3 Magnetic properties of various high-purity forms of iron
Saturation
induction
B,
Cast magnetic
ingot iron
Magnetic ingot
iron (2mm sheet)
Electromagnet
iron (2mm sheet)
Ingot iron
(vacuum melted)
Electrol ytic
iron (annealed)
Electrolytic iron
(vacuum melted
Coercivity
(T)
He
(A/m)
2.15
Maximum
relative
permeability
Relative
permeability
at
80A/m
800A/m
68
3500
1500
2.15
89
1800
1575
2.15
81.6
2750
1575
Ilmax
24.8
21000
18.4
41500
7.2
61000
4.0
100000
and annealed)
Puron
(H2 treated)
2.16
Materials for D.C. applications
291
18r---------------------------,
15
o HIGH-PURITY STEEL,
Hz ANNEAL AT 13000(;
6
SAE 1010 STEEL,
He + 10 % H2 ANNEAL
AT 680~C
- - TYPICAL SPECIFICATION
CURVE FOR INTERMED~T
GRADE STEEL
3
1b!:-.L~'=27
Fig. 12.19 Initial magnetization curves
for two high-purity steels, after annealing in a reducing atmosphere of hydrogen and a typical intermediate-grade
steel, after Swisher (1969) [15] and
Swisher and Fuchs (1970) [16].
for pure iron is 1.5 x 106 , however the problem with this from a commercial
viewpoint is that it is too expensive for many applications.
In most applications ultra-high-purity iron is unnecessary. A typical commercial
soft iron for electromagnet applications will therefore contain about 0.02% C,
0.035% Mn, 0.025% S, 0.015% P and 0.002% Si in the form of impurities. The
magnetization curves of high-purity iron and a commercial soft iron after
annealing in hydrogen to remove impurities are shown in Fig. 12.19.
For electromagnets the principle question that arises is what field is necessary to
produce an induction of 1.0 Tor 1.5 T. For the commercial soft iron given above
the values are typically 200 Aim and 700 Aim, respectively.
Any form of mechanical deformation will result in a deterioration of the
magnetic properties of soft iron for electromagnet applications. The internal
stresses produced by such cold working can be removed by annealing at
temperature between 725°C and 900°C providing the material does not suffer
oxidation during the anneal which would also result in impaired magnetic
properties. The usual procedure now is to anneal in a hydrogen atmosphere which
has the additional advantage of removing some of the impurities. The variation of
coercivity with impurity nitrogen and carbon content is shown in Figs. 12.20 and
12.21.
292
Soft magnetic materials
2.4r-------------------------------------,
(J)
8
2.2
~
(J)
Q:
!oJ
o
2.0
!oJ
u
Q:
o
IL.
~
u
1.8
Q:
!oJ
o
U
...J
ct
1.6
~
Z
1.4
1.2~-
o
40
80
120
160
200
240
PPM NITROGEN (COMPUTED)
Fig. 12.20 Dependence of coercivity on nitrogen content in high-purity iron, after Swisher
(1969) and Swisher and Fuchs (1970).
12.3.2 Iron-nickel alloys (Permalloy)
Iron and nickel form a number of commercially important alloys, most of which
are in the 'Permalloy' range with nickel contents above 35%. For d.c. applications
the iron-nickel alloys are very versatile since by suitable alloying they can be
produced with a wide range of properties. Among these it is possible to produce for
example an alloy with zero magneto stricti on (19%Fe-81%Ni).
The low coercivity of these alloys can be seen from Table 12.4 and this has made
them ideal for relays with a short release time. Both 30% Fe-50% Ni and Mumetal
are used for the cores. However the low saturation magnetization of these alloys
has meant that they are not widely used in relays in general.
The nickel-iron alloys in general have very high permeabilities, as shown in
Tables 12.4 and 12.5. The maximum permeability of the polycrystalline alloy is
expected to occur when the anisotropy and magneto stricti on are small. The value
of the anisotropy constant K 1 is zero at a nickel content of78%. The addition of 5%
copper to Permalloy produces the alloy known as Mumetal, although these days
commercial M umetal also contains 2% Cr. Its magnetic properties are no better
Materials for D.C. applications
293
2.5.-----------------------------------.
(I)
o
w
~ 2.0
a:
w
9
w
~ 1.5
o
u...
w
~
~ 1.0
w
oU
...J
~
~
0.5
z
o o~-
60
120
PPM CARBON
180
240
Fig 12.21 Dependence of coercivity on carbon content in high-purity iron, after Swisher
(1969) and Swisher and Fuchs (1970).
Table 12.4 Selected magnetic properties of materials used for relays
Saturation
induction
Coercivity
B,
He
Remanence
Br
(T)
(A/m)
(T)
2.05
2.15
2.15
2.15
2.15
60-140
24-120
16-40
8-24
6-10
0.8
0.9
4000-15000
4000-12000
5000-20000
2.1
2.0
2.0
30-120
12-120
5-90
0.9-1.45
0.8-1.2
0.8-1.2
6000-14000
15 000-60 000
5000-300000
1.3
8-24
5-14
1-8
0.8
0.8-1.2
0.5-0.75
Relative
permeability
f.1max
Unalloyed iron
Auto machining iron
Open smelted iron
Vacuum smelted iron
Carbonyl iron
Carbonyl iron
(critically stretched)
4300-8000
2200-7500
30000
40000
0.8
0.8
Silicon steels
Fe-l% Si
Fe-2.5% Si
Fe-4% Si
Nickel Steels
Fe-36% Ni
Fe-50% Ni
Fe-78% Ni
1.55
0.7
Table 12.5 Selected magnetic properties of different soft magnetic materials
Relative
permeability
Composition
Coercivity
He
I1m.x
(Aim)
5000
7000
80
40
2.15
1.97
40000
8
2.0
100000
4
1.08
70000
4
1.60
0.16
0.79
4
0.65
160
2.45
10000
80
2.42
60000
16
2.40
l1i
Iron
150
100% Fe
Silicon-iron
96% Fe
500
(Non-oriented)
4%Si
Silicon-iron
1500
97% Fe
(Grain-oriented)
3%Si
78 Permalloy
8000
78%Ni
22% Fe
Hipernik
4000
50%Ni
50% Fe
Supermalloy
79%Ni
100000
16% Fe, 5% Mo
Mumetal
77% Ni, 16% Fe
20000
5% Cu,2%Cr
Permendur
50% Fe
800
50% Co
Hiperco
650
64% Fe
35% Co, 0.5% Cr
Supermendur
49% Fe
49%Co,2%V
Saturation
induction
Bs
(T)
1000000
100000
5000
240
220
~
~
:J
E 200
cu
\
\
I/)
:::!E
\
180
20
40
60
80
100
Weight Percent Go
Fig. 12.22 Variation of saturation magnetization with composition in iron-cobalt alloys,
after Weiss and Forrer (1929).
Materials for D.C. applications
295
than permalloy but this metal is rather more ductile than permalloy and is
therefore used in the form of thin sheets in magnetic shielding in order to prevent
stray magnetic fields from affecting sensitive components.
Addition of cobalt to iron-nickel gives the so-called 'perminvar' ternary alloys
which have constant permeability and zero hysteresis losses at low fields of up to
200A/m.
- --
r---------------------________________-.
5X105
..........
4
3
",-
t
Ie
~
~
.E'
FeCo
~
2
~
.....-
z
~
_ x
IJ'I
.;
8
>-
... \
".
~
if
o
\
\
,
x
0
a::
~
..
\
Z
"'
\
-1
I-
IJ'I
,
\
a::
u.
-0
---x
WATER QUENCHED; DISORDERED
20·C/HR. 1 ORDERED
,
\
\
\
-2
\
\
\
~
-3
_4~-L
o
10
20
30
40
WEIGHT PERCENT COBALT IN IRON
50
60
Fig. 12.23 Variation of the first anisotropy constant with composition in iron-cobalt
alloys, after Hall (1960).
296
Soft magnetic materials
12.3.3 Iron-cobalt alloys (Permendur)
Cobalt is the only element which when alloyed with iron causes an increase in
saturation magnetization (Fig. 12.22) and Curie temperature and so the cobaltiron alloys are of some interest. These alloys have low anisotropy (Fig. 12.23) and
high permeability (Fig. 12.24). They have found some applications in both a.c. and
d.c. devices but the high cost of cobalt has been a limiting factor. Nickel and
niobium are also now used as alloying constituents in iron-cobalt alloys.
The highest saturation magnetization occurs in 65% iron-35% cobalt alloys in
which it reaches 1.95 MAjm. These binary alloys are brittle, but improved
mechanical properties can be obtained by alloying with vanadium. Permendur
which has a composition of 49% Fe, 49% Co, 2% V known as vanadium permendur
has a saturation magnetization which is close to the maximum 1.95 MAjm and
also has a permeability which remains constant with Hover a wide range of fields.
The rather expensive cost of cobalt has also prevented the use of these alloys in
1
1~O
JlO
f\ 1'1
1200
\
1000
1/ /
800
J
600
400
200
oV
~ \
/
~
·4 .2"4
~
~
VI
17
~
\
~
~
~\
-~8
i'-..
-0-6
-0·4
0·2
o
0·2
0-10
06
0-1
H
!!.
m
4
•
Fig. 12.24 Reversible permeability of permendur as a function of field strength measured
with a small superimposed cyclic field of amplitude 0.34 A/m at a frequency of 200 Hz.
Further reading
297
relay armatures where they would otherwise be the ideal material in view of their
high saturation magnetization which would give rise to a large attractive force with
which to operate the moving parts of the relay.
The ternary alloys are used in magnetic amplifiers and in some switching and
memory-storage cores. They are used in diaphragms of high-quality telephone
receivers, where the high value of reversible permeability at high flux density is
important, and as the pole pieces of servomotors in the aircraft industry where high
flux density is crucial. These alloys were also used previously to a lesser extent as
magnetostrictive transducers, however there are now far superior magnetostrictive
transducer materials available.
REFERENCES
1. Elmen, G. W. (1935; 1936) Magnetic alloys of iron, nickel and cobalt, Elect. Engng, 54,
1292; 54, 887.
2. lakubovics,1. (1987) Magnetism and Magnetic Materials, Institute of Metals, London,
p.97.
3. Boll, R. (1978) Vacuumschmelze Handbook ofSoftM agnetic Materials, Heyden, London.
4. Stephenson, E. T. (1985) J. Appl. Phys., 57, 4226.
5. Steinmetz, C. (1891) Electrician, 26, 261.
6. Littmann, M. F. (1971) IEEE Trans. Mag., 7, 48.
7. Chen, C. W. (1977) Magnetism and Metallurgy of Soft Magnetic Materials, North
Holland Publishing Company, Amsterdam p. 276.
8. Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading,
Mass., p. 494.
9. Heck, C. (1974) Magnetic Materials and their Applications, Crane and Russak, New
York, p. 392.
10. Luborsky, F. E. (1980) Amorphous ferromagnets, in Ferromagnetic Materials, Vol. 1
(ed. E. P. Wohlfarth), North Holland, Amsterdam.
11. Bertraut, F. and Forret, F. (1956) Comptes Rend. Hebd. Seance. Acad. Sci., 242, 382.
12. Slick, P. I. (1980) Ferrites for non-microwave applications, in Ferromagnetic Materials,
(ed. E. P. Wohlfarth) North Holland, Amsterdam.
13. Nicolas, 1. (1980) Microwave ferrites, in Ferromagnetic Materials, Vol. 2 (ed. E. P.
Wohlfarth) North Holland, Amsterdam.
14. Chin, G. Y. and Wernick, 1. H. (1980) Soft magnetic materials, in Ferromagnetic
materials, (ed. E. P. Wohlfarth) North Holland, Amsterdam.
15. Swisher, 1. H., English, A. T. and Stoffers, R. C. (1969) Trans. A. S. M., 62, 257.
16. Swisher, 1. H. and Fuchs, E. O. (1970) J. Iron. Steel. Inst. August, 777.
FURTHER READING
Boll, R. (ed.) (1978) Vacuumschmelze Handbook of Soft Magnetic Materials, Heyden,
London.
Chin. G. Y. and Wernick, 1. H. (1980) Soft magnetic materials, in Feromagnetic materials,
Vol.2 (ed. E. P. Wohlfarth), North Holland, Ch. 2.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading, Mass.,
Ch.13.
Heck, C. (1972) Magnetic Materials and their Applications, Crane Russak & Co. New York
Ch.1O-13.
298
Soft magnetic materials
Hummel, R. E. (1985) Electronic Properties of Materials, Springer-Verlag, Ch. 17 Berlin.
Luborsky, F. E. (1980) Amorphous ferromagnets, in Ferromagnetic Materials, Vol. 1
(ed. E. P. Wohlforth), North Holland, Ch. 6.
Siemens Akgt (1987) Siemens Ferrites: Soft Magnetic Materials Data Book, Munich.
Snelling, E. C. (1988) Soft Ferrites, Properties and Applications, 2nd edn, Butterworths,
London.
Valvo Inc. (1987) Ferroxcube Datenbuch, Huthig Verlag GmbH, Heidelberg.
13
Hard Magnetic Materials
In this chapter we consider the properties of ferro magnets that make them useful as
permanent magnets. A number of different materials are now available and
improvements in properties such as coercivity and maximum energy product
continue to be made, as in the recent discovery of neodymium-iron-boron
permanent magnets. There is a wide range of applications of permanent magnets
from heavy current engineering such as in electrical motors and generators to very
small-scale uses for example control devices for electron beams and moving-coil
meters, and intermediate power range applications such as microphones and
loudspeakers.
13.1
PROPERTIES AND APPLICATIONS
A permanent magnet is a passive device used for generating a magnetic field. That
is to say it does not need an electric current flowing in a coil or solenoid to maintain
the field. The energy needed to maintain the magnetic field has been stored
previously when the permanent magnet was 'charged' (i.e. magnetized initially to a
high field strength and then to remanence when the applied field was removed).
Permanent magnets are used for field generation in a variety of situations in which
it is difficult to provide electrical power, as in portable equipment, or where
geometrical constraints such as space restrictions dictate their use rather than
electromagnets.
When ferromagnetic materials are used as permanent magnets they must
operate in conditions where at best they are subject to their own demagnetizing
field and at worst can be subjected to various demagnetizing effects of other
magnetic materials or magnetic fields in their vicinity. It is therefore essential that
they are not easily demagnetized.
In addition a permanent magnet is only of use if it has a relatively high
magnetization when removed from the applied magnetic field. Therefore a high
remanence is also desirable and this inevitably means a high saturation
magnetization. However most of the other properties that were considered
desirable in soft magnetic materials, such as high permeability and low
conductivity, are irrelevant in hard magnetic materials.
300
13.1.1
Hard magnetic materials
Coercivity
Since the permanent magnets operate without an applied field their ability to resist
demagnetization is an important quality and consequently a high coercivity is
desirable. Over the years there has been continual progress in the discovery of new
permanent magnet materials with higher coercivities [1]. The coercivity is used to
distinguish between hard and soft magnetic materials. The hard magnetic
materials are classified rather arbitrarily to have coercivities above 10 kAjm
(1250e). Recent permanent magnet materials have coercivities two orders of
magnitude greater than this. For example the intrinsic coercivity is typically
1.1 MAjm (140000e) in neodymium-iron-boron, 0.69MAjm (87000e) in
samarium-cobalt and 56 kAjm (7000e) in Alnico [2].
Unlike soft magnetic materials, in which B is approximately equal to JloM, in
permanent magnets the magnetization is not simply an approximate linear
function of the flux density because the values of magnetic field H used in the
permanent magnets are generally much larger than in soft magnetic materials. The
result is that the coercivity can be defined as either the field at which the
magnetization is zero, the intrinsic coercivity mHc, or the field at which the
magnetic flux density in the material is zero BHc. These quantities have quite
different values in hard magnetic materials, and the greater the difference the better
the material is as a permanent magnet. It should be noted that mHc is always
greater than BHc.
13.1.2 Remanence
No matter what the coercivity of the permanent magnet is it will be oflittle use if the
remanent magnetization is low. Therefore a high remanence combined with a high
coercivity is essential. The remanence MR is the maximum residual magnetization
which can be obtained only in a closed-loop configuration in which there is no
demagnetizing field. Since all permanent magnets must be operated in an 'open
circuit' configuration to be of any use, the residual magnetization at which the
permanent magnet operates in open circuit will always be below the remanence
value MR. The remanence in neodymium-iron-boron is for example typically
MR = 1.05 MAjm (1050emujcc), BR = 1.3 T (13 kG).
13.1.3 Saturation magnetization
The remanence is of course dependent on the saturation magnetization, and for
this reason the saturation magnetization of a permanent magnet should be large.
While this condition is necessary it is not sufficient since the squareness ratio
MJMs must also be as close to 1 as can be achieved in order to ensure a large
remanence. The saturation magnetization in neodymium-iron-boron is
1.27MAjm [3J, in samarium-cobalt it is 0.768MAjm [4,p. 80J and in Alnico
alloys typically 0.87-0.95 MAjm [5, p. 574].
Properties and applications
301
13.1.4 Energy product
One parameter that is often of interest to permanent magnet manufacturers and
users is the maximum energy product, which is the maximum value of BH obtained
in the second quadrant, as shown in Fig. 13.1. This is clearly closely related to the
total hysteresis loss or area enclosed by the hysteresis loop. The maximum energy
product is the maximum amount of useful work that can be performed by the
magnet.
The improvement in the maximum energy product of various permanent
magnet materials over the years are shown in Fig. 13.2 [6]. We can estimate the
ultimate limits of the maximum energy product that may be achieved in the future.
For a material with a remanence MR and a very square hysteresis loop the
coercivity sHe can never exceed MR' Therefore the maximum energy product is
lloMR2/4. Clearly remanence can never be greater than the saturation
magnetization, and the largest saturation known is 1.95 x 106 Aim in some cobaltiron alloys. Therefore it has been suggested [7] that the maximum energy product
cannot exceed 1.19 x 106 J/m 3 (150MGOe).
8 IT)
0.4
0.3
30
Fig. 13.1 Second quadrant 'demagnetization curve' of the ceramic magnet barium ferrite
(left half of the diagram) and energy product as a function of induction (right half of the
diagram).
302
Hard magnetic materials
1000
I
I
/
500
1~f
200
(BH)mox 100
/J
50
r
/
5
10/
-----l
/f-J
//
20
10
/~
12/
/
//
//
frr-
/
f-
/
/
/
2 1
1
1880
/
f5
J.
3
/
-
~
I
I
1900 1920 1940 1960
1980
Fig. 13.2 Improvements in maximum energy product (BH}max in kJ/m 3 over the
years 1880 to 1980: 1, carbon steel; 2, tungsten steel; 3, cobalt steel; 4, Fe-Ni-AI alloy;
5, Ticonal II; 6, Ticonal G; 7, Ticonal GG; 8, Ticonal XX; 9, SmCo 5 ; 10, (SmPr)C0 5 ; 11,
Sm z(Co o . s5 F 0.11 Mn O. 04 )17; and 12, Nd zFe 14 B.
A high energy product results from high coercivity and remanence, however it is
certainly not adequate merely to choose the material with the highest energy
product for any given application since the optimum operating conditions, which
are affected by geometrical considerations, may dictate that other properties are
more important. In order to understand this it is necessary to consider the
complete demagnetization curve of the permanent magnet.
In most cases in industry the maximum energy product is measured in nonstandard units of megaGauss-Oersted (MG Oe). The conversion factors between
MGOe, Jjm 3 and ergsjcc are as follows
1 MGOe = (106j4n)ergsjcc = 7.96kJjm 3 •
13.1.5 Demagnetization curve
The maximum energy product by itself gives insufficient information about the
properties of a permanent magnet. A more useful way of displaying the magnetic
Properties and applications
303
properties of a permanent magnet is to plot the portion of the hysteresis loop in the
second quadrant, that is from the remanence to the coercivity. This is known as the
demagnetization curve and is the information which is always used in order to
decide on the suitability of a permanent magnet for particular applications. Such a
curve contains information about the maximum energy product, but also contains
additional information for the designer if the permanent magnet cannot be
operated at its optimum condition.
This demagnetization curve indicates that magnetization under various
demagnetizing fields. The strength of the demagnetizing field of a permanent
magnet in open-circuit configuration depends on the shape of the permanent
magnet. It is therefore immediately apparent that the choice of material is
dependent as much on its shape as on the intrinsic material properties.
The operating conditions for a permanent magnet are determined by the
demagnetizing field as shown in Fig. 13.3. If the geometry is known, that is either
the length of the air gap or the total length of magnet that can be used, the designer
must select a material with the largest possible value of BH on the load line. The
load line is the locus of possible operating points as dictated by the demagnetizing
factor for the particular geometry of specimen. This is discussed in the following
section.
13.1.6
Permanent magnet circuit design
The operating point of a permanent magnet is the point of intersection of the load
line with the demagnetization curve. The slope of the load line is determined by the
demagnetizing factor and hence by the shape of the magnet. Consider for example
a permanent magnet with length to diameter ratio 1:d which has a demagnetizing
factor N d' Under these circumstances it is fairly easy to show [5, p. 560] that the
demagnetizing field Hd is related to the induction B by
Hd= - (
Nd
Ilo(l- N d)
) B.
Hence a load line of slope - llo(1 - N d)/N d passing through the origin on the B, H
plane gives the locus of possible operating conditions for a magnet with the given
geometry, as shown in Fig. 13.3. The slope of this load line BIlloHd = - (1 - N d)/N d
is called the permeance coefficient. It is often one ofthe parameters supplied to the
designer because it contains information on the shape of the permanent magnet
required and therefore is important in determining the suitability of a material for a
particular application.
The field is the gap of a permanent magnet is given by, [5, p. 561].
Hg = J(HmBm Vm),
Ilo Vg
where Hm is the field inside the magnet, Bm is the flux density inside the magnet, Vm
304
Hard magnetic materials
12
14
16 1a 20
25
40
\ \
\ \ \Energy\ product
\
BH (M G Oe)
10",
a
6
4
2
10
a""
a
E
:r:
....
co
6"",
6
C
Q)
g
0
Q)
()
'"til
~
Qi
()
en
Ol
.2
4 ...........
4
c:
~
CO
til
Q)
E
Q;
a.
2
2
1000
aoo
600
400
H
200
o
(Oe)
Fig. 13.3 Demagnetization curves, load lines and energy product values in the second
quadrant of the magnetization curve. Two load lines are shown, one for a short specimen
with I: d = 1: 1, giving a permeance coefficient of 2 (N d = 0.23) and the other for a longer
specimen with I: d = 7: 1, giving a permeance coefficient of 40 (N d = 0.244).
is the volume of the magnet and Vg is the volume of the gap. The important result
here is that the maximum field is obtained by operating the permanent magnet at
maximum energy product (BHmax ).
Combining these two results it is clear that the most appropriate material for a
given application is the one with the largest value of BH along the load line dictated
by geometrical constraints of the given application. Conversely given a particular
material for an application it should be shaped such that its load line passes
through (BH}max in order to optimize its performance.
For example if a short permanent magnet is required the demagnetizing field
may be large so that the operating conditions are far from the (BH}max point. Under
these circumstances the material with the largest coercivity may be the best. If a
long permanent magnet is used then the demagnetizing field may be quite low so
that the magnet operates closer to the remanence. Under this condition the
Properties and applications
305
material with the largest remanence may be the best. These two conditions are
depicted by the load lines in Fig. 13.3.
13.1.7 Stoner-Wohlfarth model of rotational hysteresis
The Stoner-Wohlfarth model [8] describes the magnetization curves of an
aggregation of single-domain particles with uniaxial anisotropy either as a result of
particle shape or from the magnetocrystaIIine anisotropy. If the anisotropy energy,
whether due to crystalline anisotropy or shape, can be represented as a single
constant expression, then
Ean =
-Ksin 2 e.
When the magnetization is oriented at an angle eto the easy direction, as shown in
Fig. 13.4 this will give rise to a torque of
= 2K sin ecos e.
The torque produced by a field H will be dependent on the angle ¢ between the
magnetization and the field direction as shown in Fig. 13.4
'LH=f.10H x Ms
=f.1oHMssin¢
and when the torque produced by the field H equals the torque due to the
anisotropy we will have equilibrium.
'LH
= 'Lan
/loHMs sin </J = 2K sin ecos e.
Fig. 13.4 Shape anisotropy of an ellipsoidal single-domain particle assumed to have neither
crystal or stress anisotropy. The particle has higher demagnetizing factor N d along the short
axis than along the long axis. This leads to shape anisotropy.
306
Hard magnetic materials
HK
M
H
H
(b)
(a)
Fig. 13.5 (a) A spherical single-domain particle with anisotropy field HK and with the
magnetic field H perpendicular to the easy axis. (b) Magnetization curve obtained for the
situation depicted in (a).
The field strength Hs needed to saturate the magnetization in a poly crystalline
specimen is therefore the field needed to overcome the anisotropy and rotate the
magnetic moments from the easy axes at 90° to the field direction into the field
direction. This field strength is easily shown from the above equations to be
2K
Hs=--·
/loMs
If H is perpendicular to the anisotropy field this gives completely reversible
changes in magnetization as shown in Fig. 13.5. If H is antiparallel to the
anisotropy field there arises irreversible switching of magnetization as soon as H
exceeds 2Kj/loM" as shown in Fig. 13.6. If H is at some arbitrary angle () to the
anisotropy field then the behaviour is partly reversible and partly irreversible as
shown in Fig. 13.7. In these cases whenever () is greater than 45° it is found that
increases with () while 'an decreases. A discontinuity occurs in the magnetization at
a critical field He where
'H
K
He=--·
/loMs
When various different possible combinations of domain direction are
investigated this forms a domain distribution and we obtain the kind of curves
shown in Fig. 13.8. Stoner and Wohlfarth considered a random assembly of such
single-domain particles, each with uniaxial anisotropy. From their calculations the
composite hysteresis loop of Fig. 13.9 was obtained.
The Stoner-Wohlfarth model has been used by permanent magnet producers to
Properties and applications
307
M
Ms
HK
}I
H
(a)
(b)
Fig. 13.6 (a) A spherical single-domain particle with anisotropy field HK and with the
magnetic field H parallel to the easy axis. (b) Magnetization curve obtained for the situation
depicted in (a).
M
K/Ms
H
H
(a)
(b)
Fig. 13.7 (a) A spherical single-domain particle with anisotropy field HK and with the
magnetic field Hat an arbitrary angle to the easy axis. (b) Magnetization curve obtained for
the situation depicted in (a).
indicate ways in which improved properties can be obtained essentially by
increasing the anisotropy. Despite the wide use of this theory there are questions
about its general validity for most real permanent magnet materials. One of the
most serious weaknesses of the theory is that it makes no provision for interactions
between the single-domain particles. In addition real permanent magnet materials
-1
o
0.10.90
80 45
cos cp
o I------+---f--f--,L--E-+---il--+----I----I 0
M/Ms
-1
45 80
0.10.90
-1
-1
o
h
Fig. 13.8 Magnetization curves obtained on the Stoner-Wohlfarth model for various
angles between the direction of the magnetic field and the easy axis.
I ·0 r-.,T:~
M/MS
0·5~-4_+
0~-4
-0·5~4+
- I ·0 1::::::::==..1._ _-L_ _.l-_---lL-_--L_ _-l
-1,0
-0,5
1·0
I· 5
-1'5
0·5
o
H
Fig. 13.9 Composite hysteresis loop obtained from summing the elementary magnetization
curves for a particular distribution of easy axes (i.e. magnetic texture) with respect to the field
direction. In this case the distribution of easy axes is random.
Properties and applications
309
are in no sense arrays of isolated single-domain particles, although particulate
media used in magnetic recording give a rather more valid approximation to the
model as discussed in the next chapter.
Recently therefore there has been a shift in emphasis towards explaining
properties of these materials in terms of domain-wall motion which is a more valid
physical model. The problem is that domain-wall motion is much more difficult to
describe theoretically than is domain rotation. At present there is considerable
debate over the details of the magnetization mechanism in hard magnetic materials
such as neodymium-iron-boron, particularly over the cause of the high coercivity
ofthese materials. Chikazumi [9] has indicated that domain-wall mechanisms are
probably the dominant factor in permanent magnet materials such as samariumcobalt and neodymium-iron-boron. Some authors have argued that the high
coercivity is due to difficulty in nucleating domains after the material has been
magnetized to near saturation [10]. Others suggest that the high coercivity is due
principally to domain-wall pinning [11]. Current evidence seems to favour the
strong pinning mechanism [12].
It is certainly true that higher anisotropy will lead to improved properties for
permanent magnet materials [13], however this result is not unique to the StonerWohlfarth model it is far more general and applies just as well to domain-wall
motion models. Therefore the essential feature of the Stoner-Wohlfarth model
that has proved so useful is rather more general than the model itself.
The idea of producing a very fine microstructure for permanent magnets is also
well established. This has little to do with the anisotropy (shape or otherwise), but a
lot to do with the creation of a large number of impediments to domain-wall
motion, both through the particle or grain sizes and through the presence of
localized residual strains which pin domain walls and lead to higher coercivity.
13.1.8
Applications
Permanent magnets generate stable magnetic fields without continuous
expenditure of electrical energy. This is an advantage in certain circumstances.
There are many different applications of permanent magnets [14,15] and as we
have seen considerations of geometry lead to many different materials
requirements. Therefore there remains a wide range of permanent magnet
materials available commercially. The main applications of permanent magnets
are in electric motors, generators, loudspeakers, moving-coil meters, magnetic
separators, control devices for electron beams such as in TV sets, frictionless
bearings and magnetic levitation systems, and various forms of holding magnets
such as door catches.
Electric motors, in which electrical energy is converted into mechanical energy,
and electric generators in which mechanical energy is converted into electrical
energy are the most important single application of permanent magnets. The
recent neodymium-iron-boron permanent magnet material was developed by
General Motors for use in the starter motors of their cars and trucks. The size of
31 0 Hard magnetic materials
motors can be reduced greatly by the use of stronger permanent magnet materials
and this is often an important consideration which outweighs the additional cost of
the high-performance magnets. Neodymium-iron-boron permanent magnets are
also now being used in magnetic resonance imaging systems [16]. This application
requires a very high field homogeneity, typically 5 parts per million over a volume
0.6 m in diameter [17]. Previously such fields had to be produced using
superconducting magnet technology as discussed in section 15.2.5.
13.1.9
Stability of permanent magnets
It is imporant to know under what conditions a permanent magnet will perform to
its design specifications. Two problems may arise (a) temporary effects due to
operating at temperatures beyond those for which the material was designed, and
(b) permanent deterioration of the magnetic properties caused by exposure to very
high fields (demagnetization) or by alteration of microstructure caused by
exposure to elevated temperatures (ageing).
The temporary or reversible changes in the magnetic properties with
temperature are caused by the reduction of spontaneous magnetization within the
domains as the temperature is raised. This becomes more significant the closer the
temperature is to the Curie point. The permanent changes which occur as a result
of exposure to elevated temperatures are caused by acceleration of the ageing
Table 13.1 Important magnetic properties of selected permanent magnet materials
Material
Composition
Steel
36Co Steel
99% Fe,
36% Co,
5.75% Cr,
12% AI,
3%Cu,
8% AI,
24% Co,
Alnico 2
Alnico 5
Alnico DG
Ba Ferrite
PtCo
Remalloy
Vicalloy
Samarium-cobalt
Neodymiumiron-boron
l%C
3.75%W
0.8%C
26%Ni
63% Fe
15%Ni
3%Cu,
50% Fe
8% AI,
15%Ni
24% Co,
3%Cu,
50% Fe
BaO'6Fe 2 0 3
77%Pt,
23% Co
12% Co,
17%Mo
71%Fe
13% V,
52% Co
35% Fe
SmCO s
Nd 2 Fe 14B
Remanence
(T)
Coercivity
(kA/m)
(BBjrnax
0.9
0.96
4
18.25
1.59
7.42
0.7
52
13.5
1.2
57.6
40
1.31
56
52
0.395
0.645
1.0
192
344
18.4
28
76
9
1.0
36
24
0.9
696
160
1.3
1120
320
(kJ/m3)
Permanent magnet materials
311
process. Many permanent magnet materials exist in a metastable metallurgical
state so that a phase transformation does occur but at room temperature this
proceeds very slowly. At higher temperature the transformation proceeds more
rapidly. Other factors such as mechanical treatments, corrosion and radiation
effects can alter the properties of permanent magnets. Thes have been discussed in
detail by McCaig [18].
13.2
PERMANENT MAGNET MATERIALS
In this section we look at the various different materials which have been used as
permanent magnets. Materials that were considered 'hard' magnetic materials in
the past are in many instances not recognized as hard materials today because of
the great improvement of magnetic properties such as coercivity and maximum
energy product which have taken place in the last fifty years. The magnetic
properties of various permanent magnet materials are shown in Table 13.1.
A summary of the properties of various permanent magnet materials has been
given by Becker et ai. [19] which covers developments up to 1968. A later review
paper by Buschow [20] considered recent permanent magnet materials, concentrating principally on the rare earth-iron and rare earth-cobalt magnet
materials.
13.2.1
Magnetite or 'lodestone'
This material Fe 3 0 4 which is a naturally occurring oxide of iron was the first
'permanent magnet' material to be recognized. Today it is not even considered to
be a hard magnetic material.
13.2.2
Permanent magnet steels
The addition of carbon to iron has long been known to increase coercivity and
hysteresis loss, and this has been discussed in earlier chapters. The first
commercially produced permanent magnets were high-carbon steels containing
about 1% carbon. These were also mechanically hard while the low-carbon steels
and iron were mechanically soft. Hence the classification 'hard' and 'soft', which,
for those involved in magnetism, later came to be a measure of coercivity rather
than of mechanical properties.
Later permanent magnet steels were made with the addition of tungsten and
chromium which improved the coercivity compared with the carbon steels. Later
still came the cobalt steels. In these materials the improved magnetic properties
arose from the presence of second-phase particles which impeded the motion of
domain walls, thereby leading to higher coercivity and maximum energy product.
These permanent magnet steels have coercivities of up to 20 kAjm and
maximum product of up to 7 kJjm 3 • The magnetic properties of some chromium
and cobalt steels are shown in Fig. 13.10 [12].
312
Hard magnetic materials
40
30
60
80
200
\ \ \ \ \\\\\\\\\\
Energy product BH (M G Oe)
E 20
J:
"-
Formed steel magnets
1 3.5% chromium
2 17% cobalt
3 36% cobalt
E
III
10
C
QI
'u
Q1
o
o
8
'"'":::J
0
01
E
10
6
QI
-"
III
o
c:
o
4
QI
E
QI
a..
2
250 200 150
100 50
0
H (Oe)
Fig. 13.10 Demagnetization curves of three permanent magnet steels.
13.2.3 Alnico alloys
The Alnico alloys were developed in the 1930s. The discovery of their important
magnetic properties was quite fortuitous during the development of a new type of
steel for other purposes. The Alnico alloys consist mainly of iron, cobalt, nickel
and aluminum with small amounts of other metals such as copper [22]. These
constituents form a finely intermixed two-phase alloy consisting of a strongly
magnetic C(l phase (Fe-Co) and a very weakly magnetic C(2 phase (Ni-AI) which
provides pinning sites to restrain the motion of the magnetic domain walls.
The magnetic properties of the alloy are improved by suitable heat treatment
involving quenching followed by tempering at 700°C. They are also improved by
annealing in a magnetic field. This raises the coercivity and maximum energy
product. The magnetic properties of these alloys were superior to other materials
available at that time as a result of the formation oflong rod-shaped grains of ironcobalt which give rise to shape anisotropy. The particles are also embedded in a
nickel-aluminum matrix which impeded domain-wall motion. One disadvantage
of these alloys is that they are very hard and brittle and therefore can only be
shaped by casting or by pressing and sintering of metal powder.
15
E
\
40
\
16
10~
14
z
E
r
C
CII
·u
--r
"-
III
-
30
20
I II
/"'~I
////1//////_1 /
I I
ri 12
L-/.-I-H-H 10
~
CII
0
U
CII
Co)
c:
0
5-1
CII
E
If
~
-
c.:>
8 .:II!
III
---- ----- ----- ---------
6
----
.......-~I
~
./
/
/
/
I
-'4
2
1400 1300 1200 1100
1000 900
800
700
600
500
400
300
H (Oe)
Fig. 13.11 Demagnetization curves of various forms of Alnico, After Parker and Studders (1962) [21].
200
100
00
0·8
Isotropic Alnicos 1,2,3.4
'P
"-'
CXI
...:>;en
0·4 c:CI)
"0
><
:J
0·2 u.
~-o
50
60
40
30
Applied field,H
o
10
20
(kAm- 1)
Fig. 13.12 Demagnetization curves of the isotropic forms of Alnico; Alnicos 1, 2, 3 and 4.
BH
80
.,-
.,-
(kJm- 3)
40
60
1-4
1-2
'"
B/H
1-0
"P
~
,/
0-8
a
,/
CXI
...>-en
0-6 c:CI)
"0
0-4
Alnico 5
><
:J
u.
0-2
80
70
60
20
1
(kAm)
Applied field, H
50
40
30
10
0
0
Fig. 13.13 Demagnetization curves of oriented forms of Alnico 5: (a) equiaxed, (b) grainoriented and (c) single-crystal.
Permanent magnet materials
315
1'2
Alnicos 6,7,8
lCC
0'4
-12
-8
-4
H (10 4 Am- 1)
Fig. 13.14 Demagnetization curves of the anisotropic forms of Alnico: Alnicos 6, 7 and 8.
Alnico alloys have remanences in the range 50-130 kA/m with maximum energy
products of 50-75 kJ/m 3 . These alloys represent a mature technology and no
significant improvements in their magnetic properties have occurred in the last
twenty years. Typical magnetic properties of some Alnico alloys are shown in
Figs. 13.11 to 13.14.
13.2.4 Hard ferrites
These materials, also known as ceramic magnets, were developed in the 1950s as a
result of the Stoner-Wohlfarth theory which indicated that the coercivity of a
system of single-domain particles was proportional to the anisotropy. The theory
thus provided a direction for the permanent magnet industry by indicating the
types of materials which should prove to be good permanent magnets.
This led the permanent magnet manufacturers to try to develop highly
anisotropic materials in the form of aggregations of single-domain particles. The
cause of the anisotropy could be either crystalline or small-particle shape effects.
However it has been found that the coercivities of real materials have always been
much smaller than the theoretical predictions of Stoner and Wohlfarth since
mechanisms other than coherent domain rotation are almost always available and
these can take place at lower coercivities.
The hard hexagonal ferrites in widespread use are usually either barium or
strontium ferrite (BaO·6Fe 2 0 3 or SrO·6Fe 2 0 3 ). These materials are relatively
cheap to produce and commercially remain the most important of the permanent
316 Hard magnetic materials
1.5
~
1.25 ........
1.0 ......
-l
;J
1.75
'"
2.5
3.0
""
\
4.0 5.0 6.0 8.0 10.0 30.0
\
q
\\\
q
\
q
\
q
\
\
3
0.75-- ~
'u
~
~
'"'"0
:J
2
0.50- u
~
10
~
III
~
10 0 ~
10 q- q- rt'l rt'l N N ~ ~ 00
5.0
~ 4.0
0.25 - § 3.0 I:=-_~
If 2.0 E=:;z1~
1.0 E
~32SEi=:acIL1-.
00.5
3.23.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 104 1.2 1.0 0.8 0.6 0.4 0.2
0>
.2
:i:
c:
!Xl
0
H (kOel
Fig. 13.15 Demagnetization curves for: 1, isotropic barium ferrite; and 2, 5, and 6,
anisotropic barium ferrite permanent magnets.
magnet materials. The ferrites are often used to produce the so called 'plastIc
magnets' by embedding the ferrite in a flexible plastic matrix.
The coercivities are larger than the Alnicos for example, being typically 150250 kA/m, but their maximum energy product is low, being typically 20 kJ/m3. The
magnetic properties of some barium ferrite magnets are shown in Fig. 13.15.
Reviews of the properties of the hard ferrites have been given by Stablein [23] and
by Kojima [24].
13.2.5 Platinum-cobalt
This permanent magnet material was developed in the late 1950s. Although its
magnetic properties were an improvement over other materials that were available
at the time, its cost made it impractical for any except the smallest magnets. Other
cheaper magnets with superior properties are now available so this permanent
magnet material is no longer in use. It has coercivities of typically, 400 kA/m and a
maximum energy product of, typically, 80kJ/m 3.
13.2.6 Samarium-cobalt
Samarium-cobalt permanent magnet material was developed in the late 1960s as a
result of a concerted research effort to identify new permanent magnet materials
based on alloys of the rare earths with the 3d transition series ferromagnets, iron,
Permanent magnet materials
317
BIT)
or
J(T)
1.2
1.0
10kG
0.8
0.6
5kG
0.4
0.2
lJoH IT) 2.0
1.8
H=1600kAm- 1
1.6
1.4
1.2
1.0
0.8
800kAm- 1
0.6
0.4
0.2
Fig. 13.16 Second quadrant demagnetization curves for four specimens of samarium cobalt
permanent magnet material. The broken lines are curves of M against H, the full lines
are curves of B against H. R2 = sintered SmCo s, R3 = sintered Sm ZCo 17 , R5 = bonded
SmCo s, R6 = bonded Sm ZCo 17 . (From M. McCaig and A. G. Clegg [31].)
cobalt and nickel [25]. It was found that the cobalt-rare earth alloys had higher
anisotropies than the nickel- or iron-rare earth alloys. Furthermore the alloys
with the light rare earths generally had higher saturation magnetizations. The first
alloy to be developed was SmCo s which has a saturation magnetization of
780kAjm, a coercivity typically of RHc = 760kAjm (mHc = 3 MAjm) with a
maximum energy product of 150-200 kJjm 3 . This was followed by Sm 2 Co 17 ,
which has RHc = 800kAjm and maximum energy product of 260kJjm 3 . The
demagnetizing curves of four typical samples of samarium-cobalt permanent
magnet material are shown in Fig. 13.16. The samarium-cobalt alloys have been
discussed in detail by Nesbitt and Wernick [26].
13.2.7 Neodymium-iron-boron
This material was discovered in the early 1980s largely because of the intervention
of economic circumstances. At that time due to problems with the supply of cobalt
there was a need for a new permanent magnet material to replace samariumcobalt even though the properties of the samarium-cobalt were adequate for the
applications for which the new material was to be used. There had been some
attempts to develop neodymium-iron materials which were known to have large
coercivities, but the properties of these alloys were not sufficiently reproducible.
From this research the addition of a small amount of boron was found to improve
the properties dramatically.
318
Hard magnetic materials
The main neodymium-iron-boron alloy developed contained the Nd 2 Fe 14B
phase [27,28], which has greater coercivity and energy product than samariumcobalt. It is the presence of this very hard magnetic phase which leads to improved
magnetic properties. Demagnetization curves of three specimens of the material
are shown in Fig. 13.17. The magnetic properties of the neodymium-iron-boron
material are very sensitive to the metallurgical processing. Two principal methods
of production have been devised. The material is produced either by powdering
and sintering, as in samarium-cobalt, or by rapidly quenching from the melt.
In the powder sintering method developed by Sagawa et al. [27] the constituents
are induction melted in an alumina crucible under an inert atmosphere, of argon
for example, to prevent oxidation. The alloy is then milled into a powder with
particles of diameter 311m. The particles are aligned in a 800 kA/m field, compacted
under 200 MPa pressure and then sintered in an argon atmosphere at
temperatures in the range 1050-1150°C. This process is followed by a postsintering anneal.
The rapid quenching process, known as 'magnequench' was developed by Croat
et al. [28]. Constituents are arc melted together and then 'melt spun' in an argon
atmosphere by ejecting the molten alloy through a hole in the quartz crucible on to
a rapidly rotating substrate where the metal cools rapidly to form ribbons. This
8(T)
or
J IT)
1.4
1J0Hk
1-10 Hk /' - -
~-I,
/--
/
I
//
I
I
I
1.8
H= 1600kAm- 1
1.6
/
1.4
0.6
SkG
--1--
0.4
I
I
1.2
1.0
10kG
0.8
I
I
I
I
1.2
1.0
I
I
//
1J0H (T) 2.0
N1
-+ - - - -
14kG
0.2
0.8
0.6
0.4
0.2
o
800kAm- 1
Fig. 13.17 Second quadrant demagnetization curves for three specimens of neodymiumiron-boron permanent magnet material. The broken lines are curves of M against H, the
full lines are curves of B against H. N1 = sintered NdFeB, N2 = high coercivity NdFeB,
MQ1 = bonded 'magnequench' NdFeB'. (From M. McCaig and A. G. Clegg [31].)
References
319
Table 13.2 Magnetic properties of domains and domain walls in
permanent magnet materials samarium-cobalt and neodymium-ironboron
Nd 2Fe 14 B
SmCo s
Sm 2Co 17
Domain wall
surface
energy
(mJ/m2)
Domain wall
width
(nm)
Single-domain
particle
diameter
(11 m)
30
85
43
5.2
5.1
10.0
0.26
1.6
0.66
gives a fine-grained microstructure of the equilibrium Nd 2 Fe 14 B phase. Particle
sizes are in the range 20-80 nm. The ribbons then undergo one of two further
processes: either they are bonded with epoxy to form a 'bonded magnet' with
intermediate maximum energy product typically of 72 kJ1m 3; or the ribbon
fragments are vacuum hot pressed and vacuum die upset to form aligned magnets
with high maximum energy products, typically of 320 kJ/m 3 .
The principle advantage ofthese alloys compared with samarium-cobalt is that
the alloy constituents, iron and neodymium, are cheaper than samarium and
cobalt. One disadvantage of the alloy is its rather low Curie point of around 300500°C. This means that the magnetic properties are rather more sensitive to
temperature than are those of samarium-cobalt (Tc = 720°C) and are not suitable
for some higher temperature applications. Typical coercivities of these permanent
magnet alloys are in the range 1100 kA/m with maximum energy product of 300350kJ/m 3 .
Sagawa et al. [29] have reviewed the development of neodymium--iron-boron
permanent magnet materials over the period 1984-7. A recent summary of
research and development covering this another rare earth permanent magnet
materials has been given by Strnat [30]. Finally the most comprehensive treatment
of all aspects of permanent magnets and their applications can be found in the
revision of McCaig's text by Clegg [31].
The domain wall surface energies, domain wall widths and single-domain
particles diameters of the three most widely used rare earth transition metal alloy
permanent magnets are given in Table 13.2 which is taken from the review by
Sagawa et al. [29]. These may be compared with the values for iron, nickel and
cobalt in Table 7.1 whence it is seen that the permanent magnets have much higher
surface energies and correspondingly thinner domain walls than the soft magnetic
materials.
REFERENCES
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320
Hard magnetic materials
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Amsterdam.
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26. Nesbitt, E. A. and Wernick, 1. H. (1973) Rare Earth Permanent Magnets, Academic
Press, New York.
27. Sagawa, M., Fujimura, S., Togawa, N., Yamamoto, H. and Matsuura, Y. (1984) J. Appl.
Phys., 55, 2083.
28. Croat, 1. 1., Herbst, 1. F., Lee, R. W. and Pinkerton, F. E. (1984) J. Appl. Phys., 55, 2078.
29. Sagawa, M., Hirosawa, S., Yamamoto, H., Fujimura, S. and Matsuura, Y. (1987) Jap. J.
Appl. Phys., 26, 785.
30. Strnat, K. 1. (1987) IEEE Trans. Mag., 23, 2094.
31. McCaig, M. and Clegg, A. G. (1987) Permanent Magnets in Theory and Practice, 2nd
edn, Pen tech Press, London.
FURTHER READING
Bradley, F. N. (1971) Materialfor Magnetic Functions, Hayden, New York.
Cullity, B. D. (1972) Introduction to Magnetic Materials, Addison-Wesley, Reading, Mass,
Ch.14.
Further reading
321
Hadfield, D. (1962) Permanent Magnets and Magnetism, Iliffe Books, London.
Hummel, R. (1985) The Electronic Properties of Solids, Springer-Verlag, Berlin.
Jakubovics,1. (1987) Magnetism and Magnetic Materials, Institute of Metals, London.
Kojima, H. (1987) Fundamental properties of hexagonal ferrites with magneto plum bite
structure, in Ferromagnetic Materials, Vol. 3 (ed. E. P. Wohlfarth) Amsterdam.
McCaig, M. and Clegg, A. G. (1987) Permanent Magnets in Theory and Practice, 2nd edn,
Pentech Press, London.
Nesbitt, E. A. and Wernick, 1. H. (1973) Rare Earth Permanent Magnets, Academic Press,
New York.
Stablein, H. (1982) Hard ferrites, in Ferromagnetic Materials, Vol. 3 (ed. E. P. Wohlfarth)
Amsterdam.
14
Magnetic Recording
In this chapter we look at the various magnetic methods available for information
storage. The most important of these are magnetic tape recording, which is widely
used for both audio and video, and magnetic disk recording which is vital for the
permanent storage of computer information. We also look at the magnetic
recording process.
14.1
MAGNETIC RECORDING MEDIA
The hysteresis of magnetization versus magnetic field in ferro magnets and
ferrimagnets can be used to good effect in magnetic recording. Without hysteresis
the magnetic state of the material in zero field would be independent of the field
that it had last experienced. However in hysteretic systems the remanent
magnetization is a kind of memory of the last field maximum, both in magnitude
and direction, experienced by the magnetic material. Therefore data, either in
digital form for computers and related devices, or analog signals as in sound
recording, can be stored in the form of magnetic 'imprint' on magnetic media.
Of course to make this of any practical use it must be possible to store large
amounts of data in as small a space as possible. So we notice that in the recording
industry there is a continual desire to increase the recording density of storage
media.
The information must also be able to be retrieved with a minimum of distortion,
that is it must not be easily erased or changed by the presence of extraneous
magnetic fields since the information should be capable of being stored
permanently. Nor should it be altered by the reading process since it is usually
necessary to reread the data many times without loss of information. Furthermore
it should be written and read with minimal power requirements.
The magnetic recording media must have high saturation magnetization to give
as large a signal as possible during the reading process. The coercivity must be
sufficient to prevent erasure, but small enough to allow the material to be reused
for recording. Coercivities in the range of 20-100 kAjm are common for magnetic
recording tapes and disks.
324 Magnetic recording
14.1.1
History and background of magnetic recording
Analog magnetic recording of the human voice was first demonstrated by
Poulsen [1], a Danish engineer, as long ago as 1898. In his device, called the
'telegraphone', acoustic signals were recorded on a ferromagnetic wire using an
electromagnet connected to a microphone. However the reproduction was very
weak due to the absence of an amplifier.
By 1920 with the development of amplifiers the signals from the magnetic
medium could be recreated more strongly and the sound reproduction was easily
audible. However there was also a low signal to noise ratio, due to the nonlinear
nature of the recording process, which still meant that the quality of the sound was
not good.
The a.c. biasing method of recording was devised in 1921 [2] and resulted in
much better signal to noise ratios because the recorded magnetization could be
made linearly dependent on the signal level. However it was not exploited properly
for another twenty years [3].
Magnetic tape was invented in 1927 in both the USA, using a paper tape coated
with dried ferrimagnetic liquid, and in Germany using a tape containing iron
powder. Oxide tapes were developed in 1947 by 3M Corporation and audio
recorders became available in 1948. Video recording began in 1956. Digital
recording for storage of computer information was developed by IBM and the first
magnetic disk drive became available in 1955.
Since the 1950s there has been a growth in digital magnetic recording for the
storage of computer data which, together with the consumer demand for audio
analog magnetic recording, particularly recording of music, form commercially
the most important areas of the magnetic recording industry. The industry is
currently worth about $80 billion a year.
14.1.2 Magnetic tapes
Magnetic tapes are the most widely used recording medium for audio and video
signals, and also to a lesser extent for computer data. Magnetic recording tapes
usually consist of a coating of fine particles of ferrimagnetic material (usually
gamma ferric oxide) on a flexible, non-magnetic, plastic substrate or base of
thickness about 25 pm. The particles are typically of dimensions 0.25-0.75 J.lm in
length by 0.05-0.15 J.lm in width [4, p. 34,5, p. 105] and are single domains which
can relatively easily be magnetized parallel to their long axes. Their saturation
magnetization is 370 kA/m (370 emu/cc) and the Curie temperature is in the
vicinity of 600°C.
Magnetic tapes are magnetically anisotropic. Gamma ferric oxide tapes usually
have larger particle sizes than do chromium dioxide tapes in which the particle size
is typically 0.4 x 0.05 J.lm. [5, p. 105]. At present the particles are aligned in the
plane of the tape. In order to align the single-domain particles the tapes are placed
in a magnetic field oriented in the plane of the tape. The field is applied before the
Magnetic recording media
325
solvent which carries the magnetic particles evaporates and leaves the dry binder
which carries the magnetic particles. The tape is then heated to completely dry the
coating and is rolled or squeezed to densify the coating. Efforts have been made to
develop cobalt-chromium tapes for so-called perpendicular recording media in
which the long axes of the particles are aligned at right angles to the tape surface.
This should in principle allow higher recording densities to be achieved although
several difficulties have been encountered in the development of perpendicular
recording including flying height of the head [6J and noise during the reading
process [7].
14.1.3 Magnetic disks
The principles of recording on magnetic disks are almost identical to those of
recording on magnetic tapes and these will be discussed shortly. Magnetic
recording disks come in two categories: floppy disks and hard disks. The materials
used as a magnetic recording medium on disk are also broadly similar to those
used on tapes. Floppy disks are made in the same way as tapes, and are usually
either 5i" or 31" diameter. The floppies are widely used in microcomputers. Hard
disks were previously more often associated with large computers, however there is
an increasing demand for hard disks in the microcomputer industry now because
of their great storage capabilities.
One ofthe advantages of disks over tapes is that access time is much shorter on
disks. This is mainly because the reading heads can be moved quickly to the right
sector of the disk, whereas in tape recording it is necessary to rewind the tape to
find the data. In disk recording the access time can also be improved by rotating the
disk at a higher angular velocity. This by itself can bring problems of additional
wear on floppy disks but not on hard disks. The reason for this is that on floppy
disks the read/write head is in contact with the disk during the reading and writing
process, but in hard disks, such as the Winchester disk systems, there is no direct
contact during reading or writing.
In magnetic tape recording the contact between tape and the read and write
heads is a crucial factor in determining performance but in that case actual contact
is acceptable because of the relatively low number of replay head/magnetic tape
passes expected. The heads are even contoured to improve contact with tape. In
magnetic floppy disks the head also rides in contact with the disk. On hard disks
the read/write head is not in contact with the disk. In order to optimize the
performance of hard disks while ensuring that there is no direct contact head and
disk an air bearing is used. In this way the head can be maintained close to, but not
actually in direct contact with the disk. This is the so-called 'flying head'. The air
flow is caused by the relative velocity between the disk and head and this maintains
a small gap. When this arrangement fails, as it does ocassionally, we encounter the
so called 'disk head crash' which usually results in some damage in the form oflost
data.
In Winchester disk drives [8] the load applied to the heads is so small that the
READ/WRITE GAP
IGLASS) ,
Fig. 14.1 Winchester read/write head showing the air bearing which enables the heads to
start and stop in contact with the disk. © 1985 IEEE.
Magnetic recording media
327
heads can start and stop in contact with the disk. The Winchester head air bearing
is shown in Fig. 14.1. In other earlier systems the heads were gradually lowered on
to the disk once it was spinning.
14.1.4 Materials for magnetic recording media
Gamma ferric oxide
The most widely used magnetic recording material is gamma ferric oxide (y- Fe 2 0 3 )
which has been used in magnetic tapes since 1937. Gamma ferric oxide is not a
commonly occurring form of Fe 2 0 3 but is produced by oxidation of specially
prepared Fe 3 0 4 . The coercivity of these tapes is in the range 20-30 kA/m
(250-3750e) [4, p. 34, 5, p. 104]. The particle size is typically a few tenths of a
micrometre with a length-to-diameter ratio of anything from 10: 1 to 3: 1. The shape
anisotropy of the particles of course also determines their magnetization
characteristics such as coercivity.
Saturation magnetization of the gamma ferric oxide is 370 kA/m (370 emu/cc)
while the Curie temperature is about 600°C which is sufficiently high to avoid
undue temperature dependence of the properties of the medium while operating
under normal conditions in the vicinity of room temperature.
Chromium dioxide
Chromium dioxide was also popular as a high-performance material for audio
recording, before the cobalt-doped surface modification process was invented, in
order to produce a magnetic recording material with higher coercivity than
gamma ferric oxide [4, p. 39, 5, p. 110]. Chromium dioxide has a coercivity of 4080kA/m and can usefully be employed with a rather smaller particle size of 0.4 by
0.05 p,m and hence higher recording densities are possible. Its saturation
magnetization is slightly higher than gamma ferric oxide at 500 kA/m (500 emu/cc),
but it has a rather low Curie temperature of 128°C which makes its performance
more temperature sensitive, a factor which is a distinct disadvantage. It is also
more expensive than iron oxide which reduces its commercial attraction. It has
been replaced by the cobalt-doped gamma ferric oxide as a high-performance
recording material.
Cobalt surface-modified gamma ferric oxide
Cobalt surface-modified gamma ferric oxide was discovered in Japan and is now
used as a magnetic recording medium because it has a higher coercivity than
gamma ferric oxide [4, p. 38, 5, p. 108]. The cobalt accumulates preferentially in
the surface ofthe tape to a depth of about 30 A. The addition of cobalt increases the
anisotropy of the material leading to higher coercivity. The disadvantage is that it
lowers the Curie point leading to greater temperature sensitivity.
328
Magnetic recording
The cobalt is added to the ferric oxide at the last stage of processing before it is
coated on to the substrate. Most video tape now contains cobalt surface-modified
ferric oxide which has a coercivity of 48 kA/m (6000e).
Hexagonal ferrites
Hexagonal ferrites have much higher coercivities than any of the above and are
used for more specialized applications such as credit cards where there is less
likelihood of a need for rerecording, but where it is imperative that there is little
chance of demagnetization by unanticipated exposure to low and moderate
external magnetic fields.
Ferromagnetic powders
Iron powder is also used as a recording medium [5, p. 111]. This has higher
saturation magnetization than the oxide particulate media described above and so
can be used in thinner coatings. The coercivity of these fine particles is typically
120 kA/m. The production of the iron particle tapes is a modification of the
production process for iron oxide tapes in which the oxide is finally reduced to
metallic iron under a hydrogen atmosphere at 300°C. However these tapes also
need a surface coating of tin to prevent sintering whereby the particles coalesce and
are no longer single domains. Typical magnetic properties are saturation
magnetization 1700 kA/m (1700 emu/cc), and coercivity 120 kA/m (1450 Oe).
Thin metallic films
Metallic films are also coming under consideration for recording tapes because of
their high saturation magnetization and remanence. They can be used in the form
of very thin coatings since the leakage magnetic fields, which are used in the
reading process, are proportional to the remanent magnetization on the tape. The
higher saturation magnetization therefore ensures that these leakage fields are
rather larger than for similar thin films of other materials. The pick up voltage in
the read head is proportional to the magnetic field from the tape. Thinner
recording media allow higher recording densities since the rate of change of field
with distance dH/dx along the tape can be made larger.
Their disadvantage is that they do not wear very well and so their lifetimes are
relatively short. Particulate media in a plastic binder have lubricants built in. Thin
metallic films need surface coatings of lubricant and corrosion inhibitor. Metal
evaporated films of nickel-cobalt alloy have been tried with varying degrees of
success as have sputtered films of gamma ferric oxide.
Thin metallic films have now become the principal medium for rigid disk
recording media. Aluminum disks are coated with a paramagnetic layer of nickelphosphorus, then covered with a magnetic recording medium such as cobaltnickel-phosphorus, cobalt-nickel-chromium or cobalt-chromium-tantalum.
Magnetic recording media
329
This is then coated with a non-magnetic protective layer of carbon to provide
lubrication and improve wear. These films have typical coercivities of 60100 kA/m (750-12500e) and saturation magnetization of (toOO emu/cc)
toOOkA/m.
Perpendicular recording media
Perpendicular recording media have been pioneered by Iwasaki in Japan [9].
These media offer higher recording density but so far seem to suffer from other
problems which have prevented them becoming viable, such as the need for a very
small head-to-medium distance and noise problems in the reading process. The
material which has been used for this is a sputtered cobalt-chromium film
containing greater than 18% Cr, which forms columns about 100-200nm in
diameter normal to the surface of the substrate. In addition, due to the nature of the
growth of these films, the magnetic moments remain perpendicular to the plane of
the film, unlike the previous materials in which the magnetic moments lie in the
plane of the material. When the moments are perpendicular to the plane it is
believed that the transitions between neighbouring 'bits' become much sharper
leading to an increased recording density. Another material that has been tried is
oriented barium ferrite.
In principle perpendicular recording has been shown to be possible, however it
has yet to become more than merely of scientific interest. In practice there are
problems with mechanical failure of the medium on floppy disks and also the small
head-to-medium separation needed is less than present technology is capable of
handling [to].
14.1.5 Bubble domain devices
Bubble domains are cylinder-shaped domains with their magnetization vector
perpendicular to the plane of a thin ferromagnetic film. Until quite recently the
potential uses of bubble domain devices for providing compact data storage for
computers were thought to be sufficient to make them competitive with
semiconductor memory devices. Their access time is relatively slow compared with
other magnetic storage techniques, and much slower than semiconductor memory.
However their storage capacity is very large compared with the semiconductor
devices. A number of companies were actively engaged in research and
development of bubble memory devices in the 1970s. Among these were Bell
Telephones, Texas Instruments, Motorola and particularly Intel, but few are now
active in this area.
Bubble domains are produced in thin ferrimagnetic films with high uniaxial
anisotropy such that the easy axis is perpendicular to the plane of the film. This
means that only two types of domain can be produced: those with magnetization
'up' and those with magnetization 'down', as shown in Fig. 14.2. There are many
criteria for the development of workable bubble memory devices. Among these are
330 Magnetic recording
b
--
c ~
-
-
_-
--,
1
1
1
-I
-~_I
-
~
I
-
1
-I
.... (
- 0 --- -
....
....
MAGNETIC RELD
1
....
~
SMAL~XTERN
-- ;:.~- ~ -~.:
• .--
-
-
-
:
-I
1
LARGER EXTERNAL
MAGNETIC FIELD
Fig. 14.2 Magnetic bubble domain patterns in a thin layer of magnetic garnet. The relative
numbers of 'magnetization up' domains (white) and 'magnetization down' domains (black)
are dependent on the strength of the magnetic field. The domains are observed by the
magneto-optic Faraday effect. © 1985 IEEE.
the need for a magnetocrystalline anisotropy much greater than the magneto static
energy to give stable magnetic bubbles which prefer to align along the magnetic
easy axis rather than in the plane of the film, which would be preferred by the
demagnetizing field. Another criterion concerns the film thickness which must not
be much greater than the bubble diameter.
The materials used in bubble domain devices are rare earth iron oxides R· Fe0 3
(garnets) and their derivatives such as substituted yttrium-iron garnet (EUY)3
(GaFe)5012. These superseded the orthoferrites which had high bubble mobility
but large bubble size, and hexagonal ferrites, which had small bubble size but low
mobility.
Once the bubble domains are created by nearly saturating the magnetization
normal to the film they are very stable and can be moved around within the film
without being destroyed. They even repel each other which is a useful property
since it inhibits coalescence of bubbles. There are limits to the field strengths under
Magnetic recording media
331
which the bubbles are stable however, and if this is exceeded (collapse field) the
material merely becomes saturated when all bubbles disappear, or if the lower limit
is exceeded (stripe-out field) the bubbles grow to form stripes and again are no
longer useful. The bubbles are moved along a 'track', which is usually made by
evaporating Ni-Fe on to a non-conducting, non-magnetic substrate. The bubbles
are moved by using current pulses to generate local magnetic fields or by a rotating
magnetic field.
The problem with bubble memories seems to be that they are just too slow
successfully to compete with either magnetic tape or disk or semiconductor
memory. The high-capacity tape, disk and semiconductor memories that have
become available over the last few years have been able to provide sufficient
storage capacity for nearly all applications at much faster speeds and the bubble
memory chips are rather bulky by comparison because they need magnetic field
coils. The bubble domain chips are also more expensive than magnetic tapes and
disks and therefore they are not commercially competitive for mass storage of data.
14.1.6 Magneto-optic recording devices
Another area of interest in magnetic recording is that of magneto-optic devices.
These make use of the Faraday and Kerr effects in which the direction of
polarization oflight is rotated in the presence of a magnetic field (see sections 3.2.4
and 6.1.8). In this way two oppositely magnetized regions on a magnetic medium
can be distinguished. The advantage of magneto-optical disks is that the storage
density can be more than 1000 times greater than for floppy disks [11, 12], while
access time for magneto-optic disks are 40-100 ms which are about ten times faster
than for floppy disks but are not yet competitive with access times for Winchester
disks which are typically 20-60ms [12]. The total storage capacity of a 5.25"
magneto-optic disk is currently 600 MBytes. This compares with the top capacity
5.25" Winchester disks which can store 300 MBytes, and the typical 5.25" floppy
which can store about 350 kBytes.
The recording of information depends on thermomagnetic magnetization in
which an intense light source such as a focused laser beam is used to heat a small
region of a thin film of ferrimagnetic material above its 'compensation' or Curie
point and then it is allowed to cool again. If the material is exposed to a reverse
magnetic field throughout the process (i.e. is operating in the second quadrant of its
magnetic hysteresis loop) then we know from earlier discussion of the anhysteretic
magnetization that its optimum magnetic energy state corresponds to
magnetization in the opposite direction.
As it cools through the Curie point the magnetization taken by the region
exposed to the laser beam will be the an hysteretic magnetization under the
prevailing field, which will be in the third quadrant. This means that the regions
which have been exposed to the laser beam will be magnetized in the opposite
direction, as shown in Fig. 14.3.
The subsequent reading of magnetic information on the medium depends on the
1
Write
beam
(a)
l7r:'~ __ •.
~
:r.<l_'~
Read
beam
I
__ -"_' _ .
Before
HB~
D~M
tM
Thermomagnetic recording.
Inc;ldentand
reflected beams IE Ill)
beam
(b)
\ I
Dielectric overlayer \
Thin samplelayar
Intermediate
V
~
Magnetooptical
ra.d lallon IE J. x)
I
Cancellation
,Addllion
}
Z
tell
B
Fig. 14.3 The magneto-optic recording and reading process. In the recording process a laser beam heats a small region of the
material so that its magnetization can more easily be changed by a magnetic field. In the reading process the direction of
polarization of a light beam is altered by the magnetization of these regions (the Kerr effect) and this is used to determine the
direction of magnetization in the region of interest. © 1985 IEEE.
Magnetic recording media
333
magneto-optic Kerr effect. A polarized laser beam of weaker intensity than that
used for writing is reflected from the surface of the magnetic recording medium, as
in Fig. 14.4 and is then passed through a polarized analyser before being detected.
The presence or absence of the reverse domains can then represent either '0' or '1'.
The film can later be wiped clean by saturating the magnetization in the original
direction.
It should be noted that for purposes of detection this technique works best in
perpendicularly magnetized media. Signal to noise ratios are comparable with
conventional magnetic disk recording. Magneto-optic disks made by Philips have
a 500 Athick magnetic coating on a transparent 3 mm thick plastic substrate. Their
method uses a 3 mW laser with a spot size of 2 x 5 micrometres reading the disk by
the Faraday effect. Whereas the recording densities of these magneto-optic disks
are about two or three times greater than that of high-performance conventional
magnetic hard disk drives, the access time is generally about twice as long.
R
WAITE LASER
READ LASER
OUTPUT.DETECTOR ARRAV
...----
HOLES IN DISK MEDIUM
Fig. 14.4 A simplified diagram showing the conventional optical recording method using a
laser to burn holes in the optical disk. This is a non-erasable process unlike magneto-optic
recording. © 1985 IEEE.
334
Magnetic recording
14.2 THE RECORDING PROCESS AND APPLICATIONS OF
MAGNETIC RECORDING
The recording process involves the mechanism by which a magnetic imprint is left
on the magnetic medium and the mechanism by which this imprint is read from the
medium and the original information, whether an audio signal or some digital
data, is recreated. A recent detailed review which discusses the fundamentals of the
magnetic recording process has been given by Bertram [13].
The writing process is the means of transferring electrical impulses in a coil
wound on an electromagnet (the writing head) into magnetic patterns on the
storage medium. The reading process is the inverse of this mechanism. The reading
process is quite well understood since it requires no knowledge of the
magnetization characteristics of the medium. Only the remanent magnetization of
the medium determines the response. However the writing process, which involves
the effect of an applied field on the magnetization of a magnetic medium, is by
comparison still rather poorly understood. This is because it is difficult accurately
to model the dependence of magnetization of the medium on the magnetic field
even when the field is completely uniform, and in these cases the field is not even
uniform.
14.2.1
Recording heads
Recording heads are small electromagnets in the form of a ferromagnetic core with
a very small pole gap of typical width OJ 11m, as shown in Fig. 14.5. The recording
head material must have high saturation magnetization in order to leave a large
imprint (high magnetization) on the tape but it must also have low remanence to
ensure that there is no writing when the current in the coil is zero. Further it is also
clear that a low coercivity is desirable. Recording heads are constructed of
magnetically soft material. Many of the soft magnetic materials described in
Chapter 12 are suitable for the cores of recording heads; these include soft ferrites,
AI-Fe, AI-Fe-Si, Permalloy, and amorphous cobalt-zirconium.
The magnetic tape or disk passes the head where the fringing field causes a
realignment of the magnetization within the single-domain articles. The
magnetization in the tape is then a record of the strength of the field in the gap of
the recording head at the time that the tape passed it. In reading mode the passage
of the tape causes a variation in flux density in the reading head which is then
converted into a voltage in the coil wound on the reading head. The signal is then
amplified and, in the case of audio recording, used to activate a loudspeaker.
The magnetic field in the gap of the write head, which is the main region of
interest to the magnetic recording engineer in the writing process, can be
determined by the finite-element techniques described briefly in Chapter 1. This
fringing field is shown in Fig. 14.6. With present technology which uses parallel
recording media the tape responds to the component of the fringing field which is
parallel to the tape surface. However for perpendicular recording the fringing field
Area Ag
Gap
Length
Fig. 14.5 Magnetic recording heads used in tape recording.
336
Magnetic recording
Fig. 14.6 The magnetic fringing field in the gap of a recording head.
needs to be normal to the plane of the tape or disk and this leads to differences in
the design of the read and write heads.
14.2.2 Writing bead efficiency
Head efficiency is the ratio of magnetomotive force obtained across the head gap to
the magnetomotive force supplied by the energizing coil. This is determined from
consideration of the magnetic circuit formed by the magnetic core and air gap of
the head.
In the air gap
In the core
where flr is the relative permeability of the core and here He is the field in the core.
The magnetomotive force of the driving coil is also the magnetomotive force
across the whole magnetic circuit. If Ig is the length of the gap and Ie is the length of
the ferromagnetic core
Recording process and applications of magnetic recording
337
The efficiency of the core, being simply the ratio of magnetomotive force across
the gap Hg/g to the magnetomotive force supplied from the coil Ni is then
Hg/g
(He/e + Hgl g )
1'/=----"--"--
Hglg
1'/= Ni
The efficiency can also be expressed in terms of the reluctances of the magnetic
paths Re in the core and Rg in the gap.
( Ig
Ag
Ie)
+ f.1rAe
where Ac is the cross-sectional area of the core and Ag is the cross-sectional area of
the gap. This means that a large gap field, and hence a large fringing field, requires a
large permeability f.1r in the core and a large ratio Ae/Ag.
Fig. 14.7 Vertical and horizontal components of the magnetic field in the vicinity of a
Karlqvist head.
338
Magnetic recording
The head gap together with the saturation magnetization of the head material
determines the field in the gap, as indicated from the discussion of air gaps in
magnetic circuits in Chapter 2. This determines the fringing field in the vicinity of
the gap and consequently the maximum coercivity of the recording medium that
can be used with the head.
Karlqvist heads [14] are an idealization often used for calculating the field in the
vicinity of the gap. In Karlqvist heads the pole gap is small compared with the pole
tip lengths, such as in the head depicted in Fig. 14.7. These provide a relatively
simple geometrical situation for determining the magnetic field in the gap. The
heads on a Winchester disk drive are a good example of Karlqvist heads.
14.2.3 The writing process
We now consider the process of magnetizing the recording media. Specifically we
wish to know how the medium responds to an applied field in the head gap. In
general this is a rather difficult problem which is not very well understood although
some empirically based models can be used to good effect. One problem that arises
as the medium passes close to the head is that the magnetic fields at different depths
in the magnetic medium are different, as shown in Fig. 14.8. Secondly, as the
medium passes the gap the field it experiences changes with time.
To give an example of the writing process we will suppose that a given region of
the recording medium passing the head begins at the positive remanence point on
its hysteresis curve, as shown in Fig. 14.9. Then as it passes the gap with a negative
field the material passes down the second quadrant of the hysteresis loop to its
coercive point - He for example. As it passes the head the magnetic field it
experiences from the head gap decreases to zero and the material magnetization
passes along a recoil minor hysteresis loop to H = 0, ending with a small positive
remanence.
This means that even where the magnetization had been reduced to zero at the
coercive point the magnetization will actually finally increase again to give a
positive remanent magnetization. Therefore to result in a completely demagnetized
state M = 0 the field experienced by the medium must be greater than - He' a
point referred to as the remanent coercivity Her. A very square hysteresis loop in
which the recoil minor loops are very flat (dM/dH) ~ 0 is therefore desirable.
During the writing process the time-varying current in the writing head coil
changes thereby altering the field in the gap. This causes localized changes in the
magnetization in the recording medium which passes the write head at a constant
speed. It has proved very difficult to determine the magnetization of the tape in two
dimensions and therefore theoretical models have only limited usefulness in the
predictions of tape magnetization.
Models for the magnetization of the recording medium usually make use of
some very simplistic approximations to the magnetization characteristics of the
medium such as assuming M is a single-valued function of field. One such model is
the Williams-Comstock [15] model. This is a one-dimensional model which
HORIZONTAl FIELD STRENGTH
..
..
.. .. .. ... ... ....,..
,
~
..
.... .,.
..
... ... ... '. .
;r
,.Y
;f
}I
~
JI
~
;4
~
~
~
~
.A
....
",
A
~
...
.;r
---
--
'4
......
.....
....
'll
- ' " . - . - . . - . ......
'l&
It
"
II
READ WRITE GAP
FERRITE CORE
II
~
~
FERRITE CORE
HORIZONTAL FIELD STRENGTH
.. .... .. ....
... ... .. .
..
.. ... ... ... ..
;r
.....
...
Jr
;r
~
;/I
~
~
~
If
'"
"
.;r
~
,,;r
.A
PERMAlJ.OY CORE
... ... ... ...
...
....
.. .
..,. .
-4
---00
.A
......
READ WRITE GAP
\0
II
~
PERMAlLOY CORE
Fig. 14.8 Variation of the magnetic field strength in both direction and magnitude close to
(a) a Winchester head and (b) a thin film head. © 1985 IEEE.
340
Magnetic recording
M
H
Fig. 14.9 Recoil minor loops during the writing process for different regions of a magnetic
recording medium as it passes a recording head.
employs the arctangent function. The magnetization M(x) in the tape can be
expressed as a function of distance in response to a step function change in field in
the gap by the equation
M(x) =
e~R
) Arctan (:).
where x is the distance along the tape or disk, a is an adjustable parameter which is
determined by the rate of change of magnetization with distance and MR is the
remanent magnetization.
14.2.4 Recording density
The recording density in a medium depends on the magnetic properties of the
medium and the characteristics of the writing head. The recording density is
determined by the product of bits per inch (BPI) and the number of tracks per inch
(TPI). The maximum attainable BPI can be measured by the parameter a, known
as the transition length, which is the minimum distance along the tape that is
needed to reverse completely the magnetization from saturation remanence in one
direction to saturation remanence in the other direction.
The transition length in which a signal can be made to change is dependent on
dM/dx in the recording medium. This can be expressed as the product
(dM/dH) (dH/dx), where (dM/dH) is a property of the medium, specifically the
Recording process and applications of magnetic recording
341
slope of the hysteresis curve, while (dH/dx) is a property of the writing head. If we
make the approximation that the slope of the hysteresis loop is constant then for a
fixed field gradient in the head gap we have the following expression for the
transition length a
a
C~R)
=----,-----,-
(~)
C~R)
(~)}
This can be verified from the Williams-Comstock equation above. The
transition length a is therefore made smaller for large field gradient dH/dx and
large dM/dH on the major hysteresis loop, that is for a square hysteresis loop. Of
course there are other factors which have not been taken into account in this simple
analysis, such as the demagnetizing field in the tape and the spatial variation in the
transition region. However these do not alter the basic conclusion about the
desirability of high field gradient in the gap and square hysteresis loop materials.
Notice that while large dM/dH on the major hysteresis loop is desirable this should
coincide with small dM/dH on the recoil minor loops.
14.2.5 a.c. bias recording
When signals are recorded in analog form, such as in audio recording, it is
advantageous to have the recorded signal proportional to the amplitude of the
input signal. The recorded signal is the remanent magnetization on a region of the
magnetic tape, while the input signal voltage is converted to an applied field in the
recording head gap. Because the initial part of the magnetization curve is nonlinear
the remanent magnetization on the tape would be a nonlinear function of the
applied field if the d.c. initial magnetization curve of the tape were used. However
by a.c. biasing, which produces the anhysteretic magnetization curve, the variation
of remanence with applied field is linear at low fields, and this produces a more
desirable recording characteristic which also improves the signal to noise ratio.
14.2.6 The reading process
The reading process in magnetic recording is very well understood. The tape or
disk passes below the read head and causes a fluctuation in the flux density in the
magnetic core of the read head. The fringing fields from the tapes can be dealt with
using simple models.
F or example, consider the situation shown in Fig. 14.10. As the tape passes near
342
Magnetic recording
Fig. 14.10 Schematic diagram of magnetic flux capture by the high-permeability read head
during the reading process.
to the reading head the stray field associated with the magnetic imprint on the
medium enters the reading head. At the gap this field passes through the coil giving
a voltage which, as we have shown earlier, is proportional to - dB/dt, the rate of
change of magnetic induction linking the coil. Therefore the voltage in the reading
head will be dependent on the stray magnetic induction emanating from the tape
which is collected by the head and passed through the coil.
The reading head efficiency is defined as the ratio of the tape flux entering the
reading head that actually passes through the sensing coil.
14.2.7
Various types of recording devices
The most common form of magnetic recorders are audio recorders. These have
traditionally been analog recording devices which use a.c. bias recording, that is
they make use ofthe linearity of the anhysteretic remanent magnetization curve to
avoid distortion ofthe reproduced signal. By this method it is possible to make the
magnetization imprinted on the recording medium proportional to the amplitude
of the signal. In audio recording, particularly music, any distortion of the signal is
undesirable. Therefore the reading and recording processes take place relatively
slowly. The typical tape velocity in professional audio recording is 40cm/s [13],
while on audio cassettes it is less than 5 cm/s [13,16].
Recording process and applications of magnetic recording
343
Video recorders use frequency modulation in which the signal S(t) imprinted on
the tape is related to the original input signal f(t) by
f
S(t) = cos ( OJt + 2nf3 f(t)dt),
where OJ is the carrier frequency and f3 is the modulation index. The video signals
range from 30 Hz to 7 MHz so that tape velocities are relatively fast and may be up
to 500 cm/s [13].
Digital recorders are in most cases peripheral devices for computers, whether
disks or tapes. Lately, however, digital recording of music has also become
available. In digital recording it is only necessary to distinguish between '0' and'!'
so these devices can function with much lower signal to noise ratios than are
acceptable in analog recording. Furthermore since in digital recording the
actual level of the signal is not really crucial, providing that a '0' and'!' can be
distinguished, the reading and writing process is very fast. However, even though
the signal to noise ratio can be relatively small in digital recording the tolerable
error rate is also very low.
14.2.8 The Preisach model
In the magnetic recording industry the magnetic properties of the medium are
usually represented using a model for magnetization as a function of field that was
devised in the 1930s by Preisach in Germany [17]. This model really does not give
much physical insight into the magnetic properties of materials, being a essence
merely a complicated curve-fitting procedure, but it can be used to give reasonable
mathematical representations of hysteresis curves once the curves are already
known. It has been found useful for modelling the magnetic properties of the
recording media and is quite widely used in the recording industry [18].
The essential idea of the Preisach model is that the observed bulk magnetic
hysteresis loop of a material is due to a summation of more elementary hysteresis
loops of domains with differing switching fields (coercivities). These domains can
only have two states within the confines of the model, with magnetization parallel
or anti parallel to a given direction. The model relies on a density function called the
Preisach function which is defined on a plane described by the positive and
negative fields H + and H _. This function is used to determine how many domains
switch their orientation from + to -, or vice versa, as the field is swept between
limiting values of magnetic field H.
The model works fairly well for weak interactions between domains such as
occur in these media, which are simply aggregates of single-domain particles, and
because the magnetic moments within the elongated single-domain particles can
only have magnetic moments along one axis leading to a magnetization either
parallel or antiparallel to the long axis of the particles.
344
Magnetic recording
14.2.9 Stoner-Wohlfarth theory
The Stoner-Wohlfarth theory [19] which was discussed in the previous chapter
has far more relevance to particulate recording media than to hard ferromagnetic
alloys. In the particulate media the isolated single-domain particles are
deliberately created on the tapes or disks and these are clearly well-suited for the
application of Stoner-Wohlfarth. The model has therefore found appropriate
applications in determining the magnetization characteristics of fine-particle
recording meda [20].
REFERENCES
1. Poulsen, V. (c. 1899) Danish patent No. 2653; (1905) US patent No. 789,336.
2. Carlson, W. L. and Carpenter, G. W. (1921) referenced in White, R. M. (1984)
Introduction to Magnetic Recording, IEEE, p. 1.
3. Holmes, L. C. and Clark, D. L. (1945) Electronics, 18, 126.
4. Mallinson, J. C. (1987) The Foundations of Magnetic Recording, Academic Press, San
Diego.
5. Camras, M. Magnetic Recording Handbook, Van Nostrand, New York.
6. Jeanniot, D. and Bull, S. A. (1988) IEEE Trans. Mag., 24, 2476.
7. deBie, R. W., Luitjens, S. B., Zieren, V., Schrauwen, C. P. G. and Bernards, J. P. C.
(1987) IEEE Trans. Mag., 23, 2091.
8. Harker, J. M. (1981) A quarter century of disk file innovation, IBM J. Res. Develop., 25,
677.
9. Ouchi, K and Iwasaki, S. (1987) IEEE Trans. Mag., 23, 180.
10. Bonnebat, C. (1987) IEEE Trans. Mag., 23, 9.
11. White, R. M. (1983) IEEE Spectrum, 20(8), 32.
12. Freese, R. P. (1988) IEEE Spectrum, 25(2), 41.
13. Bertram, H. N. (1986) Proceedings IEEE, 74, 1512.
14. Karlqvist, O. (1954) Trans. Roy. Inst. Techn. (Stockholm), 86.
15. Williams, M. L. and Comstock, R. L. (1971) AlP Con! Proc. No.5, 758.
16. Crangie, J. (1977) The Magnetic Properties of Solids, Edward Arnold, London.
17. Preisach, F. (1935) Zeit. Phys., 94, 277.
18. Della Torre, E. and Kadar, G. (1988) J. Appl. Phys., 63, 3004.
19. Stoner, E. C. and Wohlfarth, E. P. (1948) Phil. Trans. Roy. Soc. Lond., A240, 599.
20. Chantrell, R. W., O'Grady, K., Bradbury, A., Charles, S. W. and Hopkins, N. (1987)
IEEE Trans. Mag., 23, 204.
FURTHER READING
Bate, G. (1980) Recording materials, in Ferromagnetic Materials, Vol. 2 (ed. E. P.
Wohlfarth), North Holland Amsterdam.
Camras, M. (1988) Magnetic Recording Handbook, Van Nostrand, New York.
Hoagland, A. S. (1983) Digital Magnetic Recording, Robert Krieger, Orlando, Florida.
Jorgensen, F. (1980) Handbook of Magnetic Recording, TAB Books, Blue Ridge Summit,
Pennsylvania.
Mallinson, J. C. (1987) The Foundations ofMagnetic Recording, Academic Press, San Diego.
Mee, C. D. (1964) The Physics of Magnetic Recording, North Holland, Amsterdam.
White, R. M. (1980) Disc storage technology, Scientific American, August.
White, R. M. (1983) Magnetic disks: Storage densities on the rise, IEEE Spectrum, August.
White, R. M. (1984) Introduction to Magnetic Recording, IEEE Press, New York.
15
Superconductivity
Superconductivity is not really part of magnetism but the two subjects do have
considerable common ground, for example in the unusual magnetic properties
arising from the Meissner effect and in the generation of high magnetic fields using
superconducting coils. It is a subject in its own right and consequently we can only
present a summary here which, while omitting much of the detail, gives a guide to
the main results and ideas with suggestions for further reading.
15.1
BASIC PROPERTIES OF SUPERCONDUCTORS
Electrical resistivity in solids is such a familiar property that it came as rather a
surprise when zero d.c. resistivity was discovered in mercury at low temperature by
Kamerlingh Onnes in 1911 [1]. This phenomenon was named superconductivity.
Since the original discovery a number of elements and many alloys have been
found to exhibit this state. The elements known to exhibit superconductivity
together with their transition temperatures are shown in Fig. 15.1.
It is of course well known that the resistivity of metals increases with
temperature due to the scattering of electrons by various means, including the ionic
sites which vibrate with greater amplitude as the temperature increases (phonon
scattering). As the temperature decreases towards absolute zero the resistivity
decreases continuously due to the reduction in phonon scattering, but does not
reach zero because of impurity and defect scattering which leaves a residual
resistance even when resistivity is extrapolated to absolute zero of temperature.
This continuous reduction of resistivity with decreasing temperature does not
lead to superconductivity. In superconductivity, as the temperature decreases
there occurs a rapid or in some cases nearly discontinuous reduction in resistivity.
It is this which marks the onset of the superconducting state and it appears to be
the result of a totally different conduction mechanism rather than merely a limiting
case of the normal conduction mechanism.
One model that was proposed for understanding the properties of
superconductors was the two-carrier model of Gorter and Casimir [2] and
this is still a useful concept. The model assumes there are two types of charge
carner:
346
Superconductivity
'"iii
'Li
"iHe'"
4Be
5B
"Na "Mg
ec
7N
'0
13AI "Si
up
.es
"F .oNe
.7CI
"A
1.20
.IK :><lCa 2·CS 22"Jl
039
nRb 31Sr sty
55CS seBa 57La
'nV 24Cr 25Mn 21Fe nco
5·3
21Ni 2·CU 30Zn 31Ga 32Ge "As " Se "Br HKr
088 109
40Zr 41Nb 42Mo 43"fc "Ru .5Rh ··Pd <JAg "Cd "In
50Sn 51Sb 5ZTe
06
3·7
93
095
8
049
056
34
53
1 S4Xe
72Hf 73"fa 7·W TIRe 1I0s nlr 71Pt 71Au IOHg "TI 12Pb alBi "Po "At "Rn
4·5 001 1-7 066 014
p.t15 2·4 72
04·9 ' 01
{36
{3-~
17Fr "Ra "Ac 10Th "Pa I2U
1·37
1·4 a
oS!:
{31·80
Fig. 15.1 Periodic table showing elements which exhibit superconductivity.
1. 'superconducting' carriers condensed into an ordered state with zero entropy
which are not scattered by collisions with the lattice; and
2. 'normal' carriers which are scattered by collisions with the lattice.
It must be remembered that although the d.c. resistivity in these materials is zero
in the superconducting state, the a.c. losses remain finite due to eddy currents, but
at low frequencies these losses are very small (see also section 15.1.4).
15.1.1
Critical temperature
The most characteristic feature of the superconductor is the complete
disappearance of d.c. electrical resistivity at a critical temperature Te. In zero field
this is a second-order transition. The results of Fig. 15.2 are the original data of
Kamerlingh Onnes for the resistance of mercury which can be seen to exhibit a
sudden change at close to 4.2 K. In nearly perfect single crystals the transition is
very sharp, occurring over a range of less than a millikelvin in some cases.
The resistance of superconductors is so low that persistent electrical currents
have been found to circulate in a superconducting loop for several years without
diminution. This absence of detectable decay of current has indicated an upper
limit to the resistivity of 1O- 25 !lm which may be compared with the lowest
resistivity of pure copper at low temperatures of 1O- 12 !l m. It is also found that
Basic properties of superconductors
347
0·15
0·10
0·05
0·00 '--_::::.-I.-_ _I.-_ _.l....-_ _.l....-_----l
4·22
4·26
4·28
4·30
4·32
T(K)
Fig. 15.2 Original data of Kamerlingh Onnes showing the superconduction transition in
mercury.
there is no correlation between the normal electrical resistivity of the
superconductors above their critical temperature and their performance below the
critical temperature. In fact superconductors have relatively high resistivity at
room temperature, for example lead which has p ~ 20 X to- 8 Om.
Until quite recently the highest known critical temperature was 23.2 K observed
in Nb 3 Ge, but in 1986 there began a series of developments in the ceramic
Table 15.1
Date
Table showing the sudden progress in high 7;, superconductors in 1986-8
Development
La-Ba-Cu-O superconducts
at 30K
Dec. '86 Confirmation of IBM result
La-Ba-Cu-O under pressure
superconducts at 40 K
La-Sr-Cu-O superconducts
at 36K
Jan. '87 La-Ba-Cu-O superconducts
at 70K
Feb. '87 Y-Ba-Cu-O superconducts
at 95K
May. '87 Critical current densities in
excess of 10 5 A/cm 2 in
Y-Ba-Cu-O
Nov. '87 Critical current density of
7000A/cm 2 in bulk
Y-Ba-Cu-O
Jan. '88 Bi-Sr-Ca-Cu-O superconducts
at 106K
Feb. '88 Th-Ba-Ca-Cu-O superconducts
at 106K
Mar. '88 Th-Ba-Ca-Cu-O superconducts
at 125K
Apr. '86
Researchers
Institution
Bednorz and Muller
IBM, Zurich
Tanaka and Kitazawa
Chu
Tokyo University
Houston University
Cava
AT&T Bell Labs
Zhao
Chu and Wu
Chinese Academy of
Sciences
Houston University
Chaudhari
IBM Yorktown Heights
Jin
AT&T Bell Labs
Maeda
NRIM Tsukuba, Japan
Herman and Sheng
University of Arkansas
Parkin
IBM Almaden
348
Superconductivity
superconductors [3,4] based on cuprate materials with a perovskite structure,
such as lanthanum-barium-copper oxide. This lead to the highest known critical
temperature being raised to 125 K fo Th2Ba2Ca2Cu301o by early 1988,
Table 15.1.
15.1.2 Critical field
A few years after the discovery of superconductivity it was found that the
superconducting state can be destroyed by application of a magnetic field above a
certain strength. The value of the critical field depends on the temperature and on
the material. In type I superconductors (see section 15.1.3) it can be accurately
represented by the simple equation
He = Ho[1- (T/Te)2]
Table 15.2 Critical temperature and critical
strengths for the superconducting elements
Ho
Element
Aluminum
Cadmium
Gallium
Indium
Iradium
Lanthanum
Lead
Mercury
Molybdenum
Niobium
Osmium
Rhenium
Ruthenium
Tantalum
Technetium
Thallium
Thorium
Tin
Titanium
Tungsten
Uranium
Vanadium
Zinc
Zirconium
(kA/m)
1.2
0.5
1.1
3.4
0.1
7.9
2.4
4.1
22
1.6
CI.
4.8
f3
4.9
7.2
64
CI.
4.2
4.0
27
f3
33
0.9
9.3
Type II
0.7
1.7
5
16
0.5
4.5
66
5.3
8.2
2.4
14
1.4
13
24
3.7
0.4
0.01
CI.
f3
0.6
1.8
5.3
0.9
0.8
Type II
4.2
3.7
field
Basic properties of superconductors
349
which means that the critical field becomes zero at the critical temperature Te.
Values Ho for the elements are shown in Table 15.2. It must also be remembered
that the critical field strength can be caused by the current carried by a
superconductor as well as by an externally applied field.
15.1.3 Type I and type II superconductors
There are two main classifications of superconductors based on the nature of their
transition to the normal state when a magnetic field is applied. The type I
superconductors follow the Meissner-Ochsenfeld [5] behaviour up to their
critical field He. Type II superconductors undergo a transition region from the type
I superconducting state at a critical field Hel to the 'mixed' state, also known as the
'vortex' state, in which the applied field does partially penetrate the material. At a
higher field He2 the material undergoes a further transition from the 'mixed' state
to the normal state.
Type I superconductors exclude the magnetic field until the superconductivity is
spontaneously destroyed at the critical field He as shown in Fig. 15.3(a). These
superconductors have generally rather low critical fields and are not found very
useful for applications.
Type II superconductors exclude the field entirely up to the field H el . Beyond
this field the flux partly penetrates the material (the vortex state) but some
superconducting volume remains until the critical field He2 is reached, when the
flux penetrates completely and the superconducting state is destroyed
(Fig. 15.3(b)). These are the superconductors used for applications such as
superconducting solenoids.
Type I
Type II
/
/
/
/1
I
I
I
I
I
I
I
.",~-rNomal
He
Applied magnetic field (a)
HcI
state
He
Applied magnetic field (b)
Fig. 15.3 Magnetization function of applied field in superconductors. (a) A type I
superconductor which exhibits perfect diamagnetism via the Meissner effect, up to its
critical field strength He. At field strengths above He the material is a normal conductor. (b) A
type II superconductor which exhibits perfect diamagnetism up to the field strengths Hel
whereupon the field begins to penetrate the material. The material is in the 'vortex' state
between Hel and H e2 . At field strengths above He2 the material is a normal conductor.
350 Superconductivity
Fig. 15.4 Mixed state in a type II superconductor. The dark regions are normal material
within a matrix of superconducting material (the light regions).
In the intermediate 'mixed' state, of type II superconductors the material
consists of tubes or bundles of normal (i.e. non-superconducting) metal, known as
vortices, carrying magnetic flux lying inside a matrix of superconducting material
in which there is no flux penetration. Figure 15.4. shows a photograph of the end of
a type II superconductor in its vortex state. The dark regions represent the normal
metal where the magnetic flux emerges from the surface. The radius of the vortices
is equal to the penetration depth. Each vortex carries one quantum of magnetic
flux B = h/2e = 2.067 x 10 - 15 Wb.
Pure metals tend to be type I superconductors while alloys tend to be type II.
Type I can be changed to type II by the addition of a small amount of an alloying
element. For many years it was thought that the type II superconductors were an
imperfect form of type I caused by the presence of impurities in the metal, but it is
now understood that the type II superconductors are a different class of materials
with fundamentally different properties. The theory of type II superconductors has
been developed by Abrikosov [6] Ginzburg and Landau [7] and Gorkov [8].
15.1.4 Flux pinning in type II superconductors
In type II superconductors the vortices in the normal material through which the
magnetic flux penetrates are pinned by microstructural defects and grain
boundaries. So long as the vortices remain pinned the material exhibits
superconductivity. However as the magnetic field is increased the vortices pile up
and eventually overcome the pinning forces. They then swirl through the material
and provide resistance to the passage of electrons causing loss of the
Magnetic field, teslas
Temperature, kelvins
60
50
20
25
30
35
107
Critical
current
density,
Alcm2
1Q4
103
Fig. 15.5 Limiting region of the superconducting state in temperature, current density and magnetic field parameter space.
A: Y -Ba-Cu-O (bulk material)
B: Y -Ba-Cu-O (thin films)
C: Niobium-titanium
D: Niobium-tin
352
Superconductivity
superconducting state. A moving vortex causes a time-varying magnetic field
which by Faraday's law generates a voltage and consequently power loss.
15.1.5 Critical current density
When an electrical current is passing in a conductor it generates a magnetic field as
described in Chapter 1. When the surface magnetic field generated by this current
in a superconductor exceeds the critical field He the type I superconductor
undergoes a transition to the normal state. This is known as the Silsbee effect [9].
The current density at which the material goes normal is called the critical current
density, which for superconducting wire is dependent on the radius of the
superconducting specimen. In a type II superconductor when the surface field
exceeds Hel the material undergoes a transition to its mixed state. In this state the
magnetic field penetrates the material and it is no longer useful to consider the
concept of a surface field.
The three parameters temperature, magnetic field and current density form a
parameter space within which the material is superconducting, as shown in Fig.
15.5. The critical current density depends on both temperature and applied
magnetic field. It is clear from Fig. 15.5 that the further the temperature is below Te
the higher the critical current density and magnetic field a superconductor can
sustain before going 'normal'.
15.1.6 The Meissner effect: diamagnetism in superconductors
The type I superconductors are perfect diamagnets if subjected to a magn~tic
field
that is lower than the critical field He. This means that once the material makes the
transition to its superconducting state the magnetic flux inside becomes zero, an
observation first made by Meissner and Ochsenfeld in 1933 [5] and depicted in
Fig. 15.6. Type II superconductors are also perfect diamagnets below their lower
Fig. 15.6 The Meissner effect - flux exclusion in a superconductor cooled in a constant
magnetic field. The solid sphere on the left is in the normal state in which the magnetic flux
can still penetrate. As it is cooled it passes into the superconducting state, whereupon the
magnetic flux is immediately expelled.
Basic properties of superconductors
353
critical field H el .
so that
M=-H
and hence
X= -1
J.l=O.
It should be noted here however that there is no magnetization in the
superconductor in the sense that there is magnetization in a ferromagnet. The flux
density inside is zero due to the existence of surface currents which set up a totally
opposing magnetic field, as in Lenz's law, because the resistivity is zero. This
prevents the flux in the specimen from being non-zero no matter what the
magnitude of the external field is, providing it is below the critical field strength,
since the surface current will always exactly balance the external field.
We should note that this result is not a consequence of the zero resistivity of the
superconductor which would lead only to the conclusion that the flux in the
superconductor remains constant once it has passed into the superconducting
state.
An important result here is that the Meissner effect suggests that the state of a
superconductor can be described using equilibrium thermodynamics (i.e. is not
path dependent). Clearly if the flux in the material was trapped this would result in
a path-dependent state because the flux then depends on what the prevailing field
was on entering the superconducting state. Note that the magnetic field does not
penetrate the bulk of the material, but it does actually penetrate a certain depth of
the surface of the superconductor, a distance known as the penetration depth A..
In type II superconductors the situation is slightly different. The magnetic field is
excluded, with a susceptibility of - I, up to the field strength H el ; beyond this field
strength the flux does partially penetrate the bulk of the material and the material
is therefore no longer perfectly diamagnetic.
15.1.7
Flux trapping
Another consequence ofthe Meissner effect occurs ifflux passes through a toroid of
superconductor when we observe the phenomenon of flux trapping. If a magnetic
field passes through a solenoid or toroid of material in its normal state and this
material is then cooled into a superconducting state the magnetic field is pushed
out of the superconductor (the Meissner effect) but not out ofthe hole in the toroid.
Instead the flux passing through the solenoid or toroid becomes trapped and
remains at the value it had at the moment of transition to the superconducting
state.
We know that
v x E= -dB/dt
354 Superconductivity
and further that
pJ=E,
where E is the electric field, p is the resistivity and I the current density. Since I is
finite and p is zero this gives zero electric field E inside a superconductor, so that B
must remain constant.
15.1.8 The London equations: surface currents
Classical electromagnetism can be used to describe some of the bulk properties of
superconductors in terms of surface currents flowing in the superconductor. The
zero value of the flux density inside a superconductor is a consequence of the
induced surface currents generated by an applied field which, in accordance with
Lenz's law, oppose the field producing them. Such surface currents of course flow in
a surface layer of finite thickness which means that the applied field does penetrate
the solid to a certain depth, and this depth is called the penetration depth, denoted
A. The field strength decays exponentially with depth x according to the equation
H=Hoe- X / A•
The Meissner effect and the field penetration depth cannot be determined from
the Maxwell equations alone. Two additional equations are needed which were
proposed by London and London [10]. These are
B
V x J s = -A
and
dIs
dt
-
E
A'
-
where A = m/(n se2 ), m is the electronic mass, e the electronic charge, ns the number
density of paired electrons, E is the electric field and Is is the superconducting
current density.
The penetration depth A can then be shown to be
A-J(~)_m
-
110
-
llon se2
)
'
As the number of superconducting electrons decreases on approaching Tc so the
penetration depth increases.
15.1.9 Coherence length
We have mentioned above that the superconducting state is one of long-range
order. The understanding of superconductivity is dependent on the 'coherence
length', that is the long-range correlation of the behaviour of electrons. We may
Basic properties of superconductors
355
Table 15.3 The coherence length and penetration
depth at absolute zero of temperature for various
metals
Metal
Pippard
coherence
length
(10-6 cm)
Sn
Al
Pb
Cd
Nb
23
160
8.3
76
3.8
London
penetration
depth AL
(10- 16 cm)
eo
3.4
1.6
3.7
11
3.9
Adeo
0.16
0.01
0.45
0.14
1.02
think of this as in some ways analogous to the ordering of two electron spins in
ferromagnetism. The coherence length is the distance over which these electron
states can be correlated. In fact the paired, or correlated, electrons can be
thousands of lattice spacings apart (i.e. ~ ~ 10 - 6 metre).
In the London equations which have been given above it was assumed that the
number density of superconducting electrons was identical to the number density
of total electrons. This is only true at the absolute zero of temperature, T = O. At
higher temperatures ns < n and so the penetration depth A given by the above
equation becomes larger as the temperature rises. However experimental results on
the temperature dependence of Aindicated that the penetration depth was greater
than could be explained even on this basis. These larger than expected skin depths
which were observed in superconductors were termed 'anomalous'. An
explanation was given by Pippard [11] who showed that when the coherence
length was greater than the skin depth predicted by the classical London equations
the observed skin depth would be greater than expected by the London theory.
This coherence length is known as the Pippard coherence length. It can therefore
be considered to be the minimum spatial extent of a transition layer between a
normal and a superconducting region in the material. In type I superconductors A
is small compared with ~ while in type II superconductors the opposite is true. The
values of these parameters for various metals are shown in Table 15.3.
e
15.1.10 Ginzburg-Landau theory
The Ginzburg-Landau theory [7] is a quantum mechanical alternative to the
classical London theory which we shall only mention in passing. It does have the
benefit of including some known facts about the electronic behaviour of these
materials, but ultimately also includes ad hoc assumptions whose only justification
is that they correctly describe the behaviour in zero field.
In this theory the concept of long-range order of electron pairs is central in
determining the equilibrium state of a superconductor in an external magnetic
356
Superconductivity
field. An order parameter t/I is introduced such that It/ll equals the density of
superconducting electrons pairs. At any temperature T this is dependent on the
location inside the material, that is to say it is not constant. This can then be used to
predict the properties of surface layers and the boundary between the normal and
superconducting regions of the solid. The equilibrium state is found by minimizing
the Gibbs free energy G which is a function of both Band t/I. The Gibbs function
minimum contains the coherence length over which t/I slowly varies.
The theory was able to determine the effect offield on penetration depth, and to
predict the behaviour of type II superconductors with their two critical fields Bel
and B e2 .
15.1.11 The Bardeen-Cooper-Schrieffer (BCS) theory
The two outstanding properties of a superconductor are perfect conductivity and
perfect diamagnetism. The Bardeen-Cooper-Schrieffer theory [12] is able to
explain these and in addition accounts for the variation of all the thermodynamic
variables including the dependence of critical field on temperature below Te.
The isotope effect [13] was discovered in 1950, in which it was found that the
critical temperature Te of mercury depended on the isotopic mass such that Te was
proportional to M-a where a =!. In other elements the exponent is not always
but Te does remain dependent on the isotopic mass. Clearly the lattice structure,
chemical and electronic properties are independent of the isotopic mass. Therefore
some other effects such as the vibration or deformation of the lattice must playa
role in determining the superconducting properties. The only explanation
remaining is that it is the lattice-electron interactions.
The BCS theory was able to show the existence of an attractive force between
two electrons due to the polarization of the lattice during deformation. The
interaction which binds the electrons is therefore indirect, being communicated
through the lattice. An attractive force between electrons instead of the usual
Coulomb repulsive force results which gives rise to loosely bound electron pairs
with wave vectors k and - k and with antiparallel spins. Therefore each pair has a
total wavevector ofO and total spin ofO while having a mass of2m* and charge 2e.
The zero wavevector of these electrons in the form of Cooper pairs corresponds
to very long wavelengths which are not scattered by the lattice and this explains
why they experience no electrical resistance, whereas the normal conducting
electrons with k :::::: kr, where kr is the wave vector at the Fermi surface, are easily
scattered by the lattice. The penetration depth and coherence length emerged as a
natural consequence of the theory. The London equations and the Meissner effect
also emerged naturally from the theory.
In the original BCS theory in which other interactions are ignored the exponent
in the isotopic dependence of the critical temperature was found to be t. However
if the Coulomb interaction is also included this gives rise to other values of the
exponent.
t
Basic properties of superconductors
357
15.1.12 Energy gap
In a superconductor there is an energy gap of E ~ 3.5 kB Te caused by the fact
that the superconducting Cooper pairs have a lower energy than two unpaired
electrons. This gap only exists below Te, separating the superconducting electrons
below the gap from the normal electrons above the gap. The conductivity of the
superconducting electrons effectively 'short circuits' the conductivity of the normal
electrons above the gap. The origin of this gap is however entirely different from
that in insulators and semiconductors. In semiconductors the gap arises from
Bragg scattering of the electrons by the lattice. In superconductors the energy gap
is determined by electron-electron interactions, not by electron-lattice interactions. The energy gap decreases continuously to zero as the critical temperature
is approached.
15.1.13 Thermal conductivity
In normal metals there is a close relation between the electrical and thermal
conductivities since both are due almost entirely to electronic conduction. This is
expressed as the Wiedemann-Franz ratio and can be explained on the basis of a free
electron model of metals. In superconductors the situation is different because the
Cooper pairs which cause the electrical superconductivity, cannot interact with
thermal phonons (lattice vibrations) because of their very long wavelength and
hence can not transfer heat. Only the unpaired electrons interact with the lattice
and as the temperature is reduced these become fewer in number. Therefore the
thermal conductivity in the superconductors is below that in the normal state.
There are some exceptions to this statement however, notably lead, in which
phonons are responsible for a substantial fraction of the thermal conductivity and
so thermal conductivity rises at the critical temperature because there are fewer
normal electrons available to scatter the thermal phonons.
The number of unpaired electrons decreases so markedly with temperature Tin
these materials that phonon conduction becomes the dominant mechanism for
heat transport, and the conductivity therefore follows at T3 dependence. This
means of course that the thermal conductivity is much less than the normal state in
which it is proportional to T.
15.1.14 Flux quantization
When flux posses through a toroid of superconductor this flux will be quantized in
units of h/2e, where h is Planck's constant and e is the electronic charge. This
happens because the electron pairs cannot be scattered and so the flow of a
superconducting current leads to long-range phase coherence. This means that no
matter what the frequency with which the electron pairs circulate the toroid they
must remain in phase and hence the phase around the inner circumference must be
358
Superconductivity
2nn, where n is an integer. The resulting flux density is
<I> = n(h/2e) = n<l>o.
This can be used in superconducting devices such as SQUIDS to detect fields
with resolution down to a flux quantum. The value of the flux quantum is
<I> = 2.067
X
to- 15 Wb.
Dirac [14J has shown that the flux quantum is related to the flux generated
by a magnetic monopole, the ultimate unit of magnetic 'charge', which has a
predicted pole strength of 3.29 x to- 9 Am and emits a total flux of 4.136
x to- 15 Wb.
15.2 APPLICATIONS OF SUPERCONDUCTORS
In order to make superconductors more useful it is obviously desirable to increase
the critical temperature and the critical current density. This has been a constant
goal of researchers in the field but for many years the critical temperatures
advanced very slowly so that even by 1986 the highest known critical temperature
was 23.2 K in niobium-germanium.
15.2.1 Development of improved superconducting materials
Properties
In October 1986 Bednorz and Muller [3J reported that lanthanum-bariumcopper oxide became superconducting at 30 K and this opened a new era in superconductivity. Within months critical temperatures of the new class of ceramic
superconductors had been found as high as 95 Kin yttrium-barium-copper oxide
(YBCO) [4]. This meant that the critical temperature was above the liquid
nitrogen point of 77 K and so it was no longer necessary to cool with expensive
liquid helium at 4.2 K in order to observe superconductivity. This was a significant
advance, which meant that many uses which were impractical with earlier
superconductors came within the realm offeasibility although there are still many
problems to be overcome, particularly in the fabriation of these materials.
Unfortunately this advance also led inevitably to unjustifiable and exaggerated
speculations about applications to power transmission, cheaper domestic
electricity and heavy current engineering which at present remain beyond the
technological horizon.
One unfortunate property of these materials is their brittleness which does make
the production of wires a problem. However the materials are quite flexible before
being given their final heat treatment at 900 °C and so can be formed into wire
before the material processing ends. However once heat treated they can no longer
be formed into other shapes. Another problem is that the critical current density in
these ceramics (1500 A/cm 2 at 4.2 K) is quite inferior to that in materials such as
Applications of superconductors
359
niobium-tin (10 MA/cm 2 at 4.2 K). At 77 K this critical current density has
dropped to 400 A/cm 2 . Even the recent bismuth and thallium superconductors
discovered in 1988 [15,16J which have been higher critical temperatures have
comparable critical current densities in the bulk.
The critical magnetic fields in these ceramics are however very high, higher than
has been possible to measure at this time, and estimates have been made that they
are as high as 300 T. Since the critical current density is usually linked to the critical
field this property of the ceramic superconductors has been puzzling.
Conductivity mechanism
One of the crucial questions regarding the conductivity mechanisms in these
superconductors appears to have been solved. The unit cell of YBCO is shown in
Fig. 15.7. For a long time it was not known whether the superconductivity
proceeded via a two-dimensional mechanism (in planes of copper and oxygen) or a
one-dimensional mechanism (in chains). This remained unanswered until two
newer materials with critical temperatures above 100 K were found, bismuth-
o
Oxygen
•
Copper
,...
........
..: E
""e
-..,
"'>Co
'l'
c:
u'"'
r-
a·axla 3.83
angstroma
-,v
b-aXI,3.88
angatroma
Fig. 15.7 The unit cell in the crystal lattice of the yttrium-barium-copper oxide.
© 1988 IEEE.
360 Superconductivity
strontium-ca1cium-copper oxide and the most recent superconductor thalliumbarium-ca1cium-copper oxide which has Tc = 125 K, these have planes of copper
and oxygen but not chains.
Theories of the conduction mechanism in these superconductors lie outside the
BCS theory that has been so successful in earlier materials. Most theories
concentrate on the copper-oxygen bonds. The amount of oxygen present seems to
affect the ability of these materials to superconduct by changing the valence state of
the copper. These materials are not subject to the isotope effect which seems to rule
out lattice vibrations (phonons) which are central to the earlier theories.
The yttrium-barium-copper oxide is highly anisotropic being able to conduct
100 times more current along two directions in the crystal lattice than along the
third. This gives partial explanation of why the sputtered single-crystal thin films
have superior properties to the bulk of the material. The other reason is that it
appears that fewer grain boundaries help to raise the critical current density. This is
a result that is contrary to observations in other superconductors in which flux
pinning by grain boundaries and other microstructural defects keeps the vortices
in place ensuring that the superconducting electrons meet no resistance. Recent
results announced by Tanaka and Itozaki of Sumitomo [17J have reported a
critical current density of 3.5 MA/cm 2 in epitaxially grown thin-film YBCO
deposited on a magnesium oxide substrate.
Applications
Electronic applications for the new superconductors look relatively promising.
The material can be laid down in thin films for Josephsonjunction devices and chip
interconnections. Thicker films will need to be used for connecting devices on
printed circuit boards. The sputtered films have shown critical current densities at
77 K as high as 4 MA/cm 2 which is 2 to 3 orders of magnitude greater than that
obtained in the bulk material. It must be remembered however that the resistance
of copper itself is very low at 77 K and not much advantage may be gained here by
replacing copper with superconducting material.
Applications using bulk material are proving much more difficult. In fact there
has been much less progress towards useful materials here. The highest critical
current density obtained to date is 17 kA/cm 2 in zero magnetic field by Jin et al. at
AT & T [18J, although in view of the effort being expended on improving the
critical current density it is likely that this will soon be raised.
15.2.2 Superconducting solenoids and magnets
Superconductors have found important applications in the generation of high
magnetic fields using superconducting solenoids. In superconducting wires very
high currents can be obtained with no Joule heating and furthermore once started
such currents will persist almost indefinitely without any power input. The
problem was of course to find a material with a sufficiently high critical field, since
otherwise the actual generation of the field would destroy the superconductivity.
Applications of superconductors
361
This was not achieved until the 1960s when niobium-zirconium, niobiumtitanium and niobium-tin were discovered to have superconducting states with
very high critical fields. Niobium-tin (NbSn 3) was found to remain superconducting even at fields of 17 x 106 A/m and this paved the way for the development of
superconducting magnets. Niobium-tin can carry currents of up to 10 5 A/cm 2 and
this material remains the most widely used material for construction of superconducting magnets.
Fields of up to 12 X 106 A/m (150kOe, B= 150kG, 15T) are obtained in
commercially available superconducting magnets. This should be compared with
the 2 x 106 A/m (25 kOe, B = 2.5T) fields available from a typical laboratory
electromagnet. Applications of these superconducting magnets were found in highenergy physics such as in bubble chambers and in fusion research, in magnetic
detectors (SQUIDS) and in medical applications such as magnetic resonance
imaging (MRI).
15.2.3 Josephson junction devices
If two pieces of superconductor are separated by a weak link such as a constriction
or a very thin insulating layer superconducting electrons can tunnel through the
link. This effect was first discovered by Josephson [19]. The critical current density
can be affected by the presence of a small field so the state of the junction can easily
be controlled by varying the current in a nearby wire to generate a critical magnetic
field.
The junction therefore has two states which can be used as a switch to direct
current across the barrier when its resistance is zero but along a different path when
the resistance ofthe barrier is finite, that is when it is driven 'normal'. The switching
speed is of the order of picoseconds. These junctions have found widespread use in
SQUIDS.
15.2.4 SQUIDS
Superconducting quantum interference devices are high-resolution magnetometers which rely on the Josephson effect. The first type of SQUID was the
two-junction d.c. SQUID (1964), later the one-junction SQUID, the r.f. SQUID,
appeared (1970).
In the d.c. SQUID two Josephson junctions connected in parallel as shown in
Fig. 15.8 demonstrate quantum interference. As the magnetic flux cD threading the
superconducting loop changes the critical current of the two junctions oscillates
with a period equal to that which corresponds to the flux quantum. When the flux
through the loop changes the voltage-current characteristic oscillates smoothly as
shown in Fig. 15.9. The SQUID is therefore a highly sensitive flux-to-voltage
converter.
In general for practical field strength or induction measurements one usually
needs a dynamic range somewhat greater than fractions of a flux quantum.
Therefore the SQUID is often used as a null detector in a feedback circuit in which
362 Superconductivity
R
R
1v
j
Fig. 15.8 Schematic diagram of the connection of two Josephson junctions in parallel to
form a SQUID.
n-l
CURRENT I
n
n+l
FLUX <1>/<1>0 (number of quanta)
Fig. 15.9 Current-voltage characteristics of a SQUID. The voltage is an oscillating
function of the flux threading the circuit. The current-voltage characteristics are for a
SQUID with fluxes n</1o and (n + t)</1o. The variation of voltage across the SQUID with flux
linking the circuit is shown on the right. Notice that the voltage is a sinusoidal function of
the flux linking the circuit (see section 3.3.3).
any change in voltage across the SQUID is amplified and converted to a current
through a coil coupled to the SQUID to produce an equal and opposite flux. This
allows large magnetic fields to be measured to an accuracy of much less than a flux
quantum.
Further reading
363
15.2.5 Magnetic resonance imaging
One of the most interesting developments in the medical field in recent years has
been magnetic resonance imaging (MRI) [20]. This needs large, 1 m diameter,
magnets capable of generation high-intensity uniform 2 T fields. This has been one
of the most important markets for superconducting magnets.
The MRI image is a representation of the spatial distribution of nuclear
magnetic resonance signals which can identify the location of elements at different
points within a test specimen, for example a human body. Field strengths of 0.152.0 T are used and the fields must be very uniform, typically 0.1 p.p.m. over a 10 cm
diameter sphere. The first MRI systems became operational in 1979.
It should be noted that as a result of recent developments superconducting
magnets are no longer essential to the operation of magnetic resonance imaging
systems. Permanent magnet based MRI systems have now been developed using
the new neodymium-iron-boron magnets, as discussed in sections 13.1.8 and
13.2.7. Nevertheless the superconducting magnet based systems are far more
prevalent.
REFERENCES
1. Kamerlingh Onnes, H. (1911) Akad. van Wertenschappen (Amsterdam), 14, 113.
2. Gorter, C. J. and Casimir, H. B. G. (1934) Physica, 1, 305.
3. Bednorz, J. G. and Muller, A. (1986) Z. Phys., B64, 189.
4. Wu, M. K., Ashburn, J. R., Torng, C. J., Hor, P. H., Meng, R. L., Gao, L., Huang,
Z. J.,Wang, Y. Q. and Chu, C. W. (1987) Phys. Rev. Letts., 58, 908.
5. Meissner, W. and Ochsenfeld, R. (1933) Naturwiss., 21, 787.
6. Abrikosov, A. A. (1957) Sov. Phys. JETP, 5,1174.
7. Ginzburg, V. L. and Landau, L. D. (1950) Sov. Phys. JETP, 20, 1064.
8. Gorkov, L. P. (1959) Sov. Phys. JETP, 9, 1364.
9. Silsbee, F. B. (1916) J. Wash. Acad. Sci., 6,597.
10. London, F. and London, H. (1935) Proc. Roy. Soc. Land., A149, 71.
11. Pippard, A. B. (1953) Proc. Roy. Soc., A216, 547.
12. Bardeen, J., Copper, L. N. and Schrieffer, J. R. (1957) Phys. Rev., 108, 1175.
13. Maxwell, E. (1950) Phys. Rev., 78, 477.
14. Dirac, P. A. M. (1931) Proc. Roy. Soc. Land., A133, 60.
15. Sheng, Z. Z. and Herman, A. M. (1988) Nature, 332, 138.
16. Maeda, M., Tanaka, Y., Fukutomi, M. and Asano, T. (1988) J ap. J. Appl. Phys., 27, L209.
17. Tanaka, S. and Itozaki, H. (1988) Jap. J. Appl. Phys. Letts., 27, L622.
18. Jin, S., Tieffel, T. H., Sherwood, R. C., Davis, M. E., Van Dover, R. B., Kammlott, G. W.,
Fastnacht, R. A. and Keith, H. D. (1988) Appl. Phys. Letts., 52, 2074.
19. Josephson, B. D. (1962) Phys. Letts., 1,251.
20. Schwall, R. E. (1987) IEEE Trans. Mag., 23, 1287.
FURTHER READING
Ani! Khurana (1987) Superconductivity seen above the boiling point of liquid nitrogen,
Physics Today, April, 17.
Bleaney, B. I. and Bleaney, B. (1976) Electricity and Magnetism, 3rd edn, Oxford University
Press, Oxford Ch. 13.
364
Superconductivity
de Bruyn Ouboter, R. (1987) IEEE Trans. Mag., 23, 355.
Kittel, C. (1986) Introduction to Solid State Physics, 6th edn, Wiley Ch. 12.
Pippard, A. B. (1987) IEEE Trans. Mag., 23, 371.
Rose-Innes, A. C. and Rhoderick, E. H. (1978) Introduction to Superconductivity, 2nd edn,
Pergamon, Oxford.
Rosenberg, H. M. (1978) The Solid State, 2nd edn, Oxford University Press, Oxford Ch. 14.
Tilley, D. R. and Tilley, J. (1974) Superfluidity and Superconductivity, Van Nostrand
Reinhold, London.
Tinkham, M. (1975) Introduction to Superconductivity, McGraw-Hill, New York.
Superconductivity, Special issue, Physics Today, March (1986).
Superconductivity: Special report, IEEE Spectrum, May (1988).
16
Magnetic Methods for Materials
Evaluation
In this chapter we look at applications of magnetism to engineering problems and
in particular to the non destructive evaluation (NDE) of materials properties.
Magnetic methods can be used to solve two main classes of problem: detection of
defects and evaluation of intrinsic properties such as residual stress. The subject of
magnetic NDE has received little attention in the past due to the complexity of the
magnetic response of materials. It is now one of the fastest developing fields in
NDE.
16.1
METHODS FOR EVALUATION OF INTRINSIC PROPERTIES
More iron and steel is produced each year than any other metal. In 1986 world
production of crude steel was 622 x 106 tons and of pig iron and ferro alloys 484
x 106 tons. So economically steel must be considered one of the most important
industrial commodities. It is of course under widespread use as a constructional
material on large-scale projects such as pipelines, railroads and bridges while also
being used for the fabrication of high-strength components. Consequently there is
a growing need for nondestructive inspection of steel structures both for the
detection of corrosion, cracks and other defects and for the evaluation of stresses,
elastic and plastic deformation and the likelihood offailure due to creep or fatigue.
A number of nondestructive testing techniques have appeared over the years,
but today the subject ofNDE has assumed a vital role, as more industries become
aware of the potential benefits of plant life extension, the cost effectiveness of only
retiring defective components (retirement for cause) and the possibilities of
avoiding potentially catastrophic failures by monitoring the condition of
structures both for defects and the presence of high levels of stress.
While various NDE techniques may be used on steels the magnetic methods are
unique because they utilize the inherent ferromagnetic properties of the steel. They
can be used for nondestructive evaluation of a wide range of material properties
from cracks to residual strain. In general the changes in magnetic properties that
are observed are easily measurable and unlike ultrasonic methods do not need
high-resolution electronics for their use. Nevertheless the magnetic methods have
not yet fully been exploited when compared for example with ultrasound.
366
Magnetic methods for materials evaluation
Probably this is because other techniques can be applied to a wide variety of
materials and previously there was more incentive for their development. Now
however, as the limitations of other techniques become apparent, as for example in
the important area of detection and prediction of failure such as fatigue or
thermomechanical degradation (creep damage), attention has focused on the
capabilities of magnetic methods applied to steels.
16.1.1 The magnetic Barkhausen effect (MBE)
The Barkhausen effect [1] was discovered in 1919, but it was many years before its
potential as an NDE tool was realized. It is now one of the most popular magnetic
NDE methods for investigating intrinsic properties of magnetic materials such as
grain size, heat treatment, strain and other mechanical properties such as hardness.
The Barkhausen effect consists of discontinuous changes in flux density, known
as Barkhausen jumps, as shown in Fig. 16.1. These are caused by sudden
irreversible motion of magnetic domain walls when they break away from pinning
sites as a result of changes in magnetic field H. The Barkhausen spectrum, which is
the number of events against pulse height, as shown in Fig. 16.2, is dependent on
the number density and nature of pinning sites within the material. These may be
grain boundaries, dislocations or precipitates of a second phase with different
magnetic properties from the matrix material, such as iron carbide in steels. Most
Barkhausen activity occurs close to the coercive field He. Double peaks in the
count rate can occur, as shown in Fig. 16.3. Also the location and size of the peaks
can shift as a result of changes in the defect distribution, Fig. 16.3, where results are
for two specimens of the same material with different defect distributions.
The first attempt to use the magnetic Barkhausen effect to determine stress was
reported by Leep in 1967, but the method really only began to gain acceptance after
the work of Pasley who showed distinct variations in Barkhausen signal amplitude
B
NORMAL MAGNETIZATION CURVE
~-H
HIGH RESOLUTION REVEALS
DISCONTINUOUS FLUX
CHANGES AS
FIELD IS INCREASED
Fig. 16.1 Discontinuous changes in flux density B as the magnetic field H is changed.
Methods for evaluation of intrinsic properties
367
1500
::E
<J 1000
'tJ
.....
Z
'tJ
500
°o~-L=Ct
2D
3D
4D
6M (E.M.U.)
Borkhousen Pulse Amplitude
Fig. 16.2 Magnetic Barkhausen spectrum (pulse height distribution) after Tebble et al. [2].
.gl~
"I:I:
"0"0
QI
0
0:
-"
:I
0
u
i:
:I
0
u
"
en
"
en
0
0
QI
:I
~
QI
0
0:
.s::
0
al
QI
:I
.s::
"'...0"
al
Fig. 16.3 Barkhausen count rate as function of magnetic field H for specimens with two
different defect distributions.
with applied and residual stress. As stress increased in tension the peak
Barkhausen amplitude in steel was found to increase while in compression it was
found to decrease. Subsequently there were a number of investigations in Finland
by Tiitto and co-workers [3,4]. Tiitto investigated the effects of elastic and plastic
strain on the MBE in silicon-iron and the microstructural dependence of MBE in
steels. He was also able to show that MBE could be used to determine grain size in
steels. Sundstrom and Torronen reported that MBE can be used for the
determination of microstructure, mechanical and electrical properties while
Karjalainen and Moilanen investigated the effects of plastic deformation and
368
Magnetic methods for materials evaluation
fatigue on MBE. Otala and Saynajakangas devised an MBE instrument for grain
size determination.
The effect of tensile and cyclic loading on the RMS value of MBE signals in mild
steel was the subject of an investigation by Karjalainen et al. [5]. They found that
residual strains in unloaded specimens could be identified from MBE. But changes
occurring under cyclic loading (fatigue cycling) were very complex so that the
Barkhausen signals could not simply be related to the applied load. However
subsequent investigations by Ruuskanen and Kettunen were able to demonstrate
that the median Barkhausen pulse amplitude could be used to assess whether the
applied stress amplitude was above or below the fatigue limit.
Lomaev has reviewed the literature relating to nondestructive evaluation
applications of MBE [6]. In these papers he identified five mechanisms by which
MBE is caused: (a) discontinuous, irreversible domain-wall motion; (b)
discontinuous rotation within a domain; (c) appearance and disappearance ofNeel
peaks; (d) inversion of magnetization in single-domain particles; and (e)
displacement of Bloch or Neellines in two 180 0 walls with oppositely directed
magnetizations. The first of these mechanisms has been studied most intensively
and is often, incorrectly, quoted as the sole mechanism for generation of MBE. It is
interesting to note that in the early years of MBE the effect was attributed to the
second mechanism, irreversible domain rotation.
A number of other papers have appeared on MBE in the Soviet literature,
usually in DeJectoskopia (Soviet Journal ofNDT). Klyuev et al. have reported that
on the basis of their results there does not seem to be any single unambiguous
correlation between their MBE measurements and the parameters of magnetic
hysteresis. This is an interesting and very singificant result since it shows that the
magnetic Barkhausen effect provides independent nondestructive information on
the state of a material from that gained by bulk magnetic properties such as
hysteresis.
Filinov et al. have shown that MBE can be used to probe surface plastic
deformation of steel components by using different magnetization frequencies.
Such a technique can be used for evaluation of a variety of different types of surface
condition such as case hardening or surface decarburization. A combination of
MBE at different frequencies and hysteresis measurements has been used by
Mayos et al. [7] for the determination of surface decarburization in steels. By this
method different depths of the material were inspected to investigate changes in
magnetic properties. Segalini et al. have used MBE for evaluation of heat
treatment and microstructure of constructional steels.
In Germany Theiner and co-workers have used MBE in conjunction with
incremental permeability and ultrasonic measurements for the evaluation of stress
[8,9]. As they have noted, all ferromagnetic NDE methods are sensitive to both
mechanical stress and the microstructure of the material. In order to determine
stress it is therefore necessary to use two or three independent measurement
parameters. They found that Barkhausen effect, incremental permeability X-ray
and hardness measurements were successful in estimating residual stress. As might
Methods for evaluation of intrinsic properties
369
have been anticipated, they found changes in the density of dislocations affected
the MBE signals. They also found that MBE could be used to distinguish between
microstructures which cannot be distinguished on the basis of optical microscopy.
In the United States a large amont of work on MBE has been conducted at
Southwest Research Institute by Matzkanin, Beissner and co-workers. Much of
this work has been summarized in 'The Barkhausen effect and its applications to
NDE' [10J by Matzkanin, Beissner and Teller which remains the most
comprehensive work on the subject to date. Other reviews are also available, the
most notable being by McClure and Schroeder [11].
16.1.2 Magneto-acoustic emission (MAE)
Magneto-acoustic emission is an effect which is very closely related to the magnetic
Barkhausen effect. It is caused by microscopic changes in strain due to
magnetostriction when discontinuous irreversible domain-wall motion of non1800 domain walls occurs. It therefore arises in ferromagnetic materials when
subjected to a time-dependent field (Fig. 16.4). The acoustic emissions may be
detected by a piezoelectric transducer bonded on to the test part. The amplitude of
MAE depends on the spontaneous magnetostriction, being zero if AO = 0 and
H
+Hmax
LINEAR
VARIATION
OF MAGNETIC
FIELD H
WITH TIME
+Bmax
VARIATION OF
MAGNETIC INDUCTION B
-B max
MAGNETO-ACOUSTIC
EMISSIONS
Time ---.
Fig. 16.4 Schematic diagram showing magnetic field H with time, variation in flux density
over the same period and the emergence of magneto-acoustic emission pulses as flux density
changes.
370 Magnetic methods for materials evaluation
increasing with Ao. The amplitude of emissions is also a function of the frequency
and amplitude of the driving field.
It is clearly apparent that MAE must change with applied stress, since stress
alters the magnetocrystalline anisotropy. This results in a change in the relative
numbers of 180° and non-180° domain walls. Since 180 0 domain walls do not
contribute to MAE, the amplitude of emissions and the total number of emissions
will change with stress.
Despite its close relation to the magnetic Barkhausen effect, MAE has a much
shorter history. It was first reported by Lord [12] during magnetization of nickel.
Its significance for NDE was realized by Kusanagi et ai. [13] who were first to
demonstrate the effect of stress on MAE. Shortly afterwards Ono and Shibata [14]
reported MAE results on a number of steels. Their results indicated that the
method could be used to determine the amount of prior cold work and differences
in heat treatment.
Burkhardt et ai. [15] have also investigated the dependence on MAE on the
mechanical and thermal treatment of steels. They found that MAE was very
sensitive to the amount of plastic deformation. Theiner and Willems [16] used
MAE in conjunction with other independent measurements such as incremental
permeability, MBE and magnetostriction. Their results showed that the MAE
amplitude decreased with the mechanical hardness of steels but increased with
tempering.
Edwards and Palmer [17] have recently shown that MAE signals are affected
not only by stress and frequency of field but also by factors such as sample shape.
Ranjan et ai. [18] have used MAE and MBE for the determination of grain size in
decarburized steels. They used two types of measurement, the MAE peak height
and the total number of emissions, both of which were found to increase with grain
SIze.
A related phenomenon has also been reported by Higgins and Carpenter [19].
Acoustic and magnetic Barkhausen emissions due to domain-wall motion were
observed in ferromagnetic materials when the applied stress was changed without
changing the magnetic field. This can easily be understood in terms of the domainwall pinning model. The magnetic Barkhausen emission were also observed under
dynamic stress by Jiles and Atherton during investigations of magnetomechanical
effects, although they were are not reported in the paper [20]. This phenomenon of
magnetomechanical emissions has received little attention but would appear to
have significant implications for the detection of dynamic stresses in steels.
16.1.3 Magnetic hysteresis
All ferromagnetic materials exhibit hysteresis in the variation of flux density B with
magnetic field H (Fig. 16.5). The hysteretic properties such as permeability,
coercivity, remanence and hysteresis loss are known to be sensitive to such factors
as stress, strain, grain size, heat treatment and the presence of precipitates of a
second phase, such as iron carbide in steels. In addition, the measurement of
Methods for evaluation of intrinsic properties
B(T)
f-L'max
371
ANHYSTERETIC
CURVE
Br
f-Lin
Fig. 16.5 Typical hysteresis loop of a
ferromagnetic material.
hysteresis yields a number of independent parameters, each of which changes to
some degree with stress, strain and microstructure. Since it has been remarked
earlier that several independent parameters are needed in general to separate
effects of mechanical treatment from microstructure, hysteresis measurement
would seem to be ideally suited for the determination of intrinsic properties of
steels because the several independent parameters can be obtained from one
measurement.
Despite this economy of means, there are certain difficulties that need to be
overcome with the magnetic hysteresis method. Firstly, the problem of
demagnetizing effects due to finite geometries needs to be addressed, since results
which may appear to be due to changes in sample properties can be caused by
geometrical effects. Second, until recently it has proved impossible adequately to
model hysteresis in ferromagnets so that it has been difficult to interpret changes in
the hysteresis characteristics in terms offundamental changes in sample properties.
Of course this has also been true of other magnetic methods, such as the
Barkhausen effect, and has not prevented their use for NDE.
Evaluation of the condition of magnetic steel components has been one area
where NDE via hysteresis has had great success. Numerous applications have been
reported in the literature. Mikheev and co-workers at the Urals Science Centre
have made many investigations of the quality of heat treatment of steels from
magnetic parameters [21J, particularly the evaluation of hardness of various steels.
In most cases the magnetic properties were determined using a coercimeter and
correlations made between chemical composition, microstructure, heat treatment
and hardness and the principal magnetic parameter of interest, the coercivity.
Mikheev has written a review on the subject which may be used as an introduction
to his work [22].
372
Magnetic methods for materials evaluation
Kuznetosov and co-workers have also looked at the effects of heat treatments
such as quenching, hardening and tempering on the magnetic properties of steels
[23]. They have devised a method for determining the depth of case hardening.
Similar work on determination of quality of steel component heat treatment has
been carried out by Fridman et al., Konovalov et aI., Melgui et al., Katsevman and
Sandorskii and Gorkunov. Specific applications to sorting of components have
been described by Sandorskii and to determination of composition and
microstructure by Khavatov. Zatsepin et al. [24] investigated the variation of
coercivity with heat treatment and mechanical properties, while Rodigin and
Syrochkin were able to use the effect of stress on coercivity to check mechanical
hardness, thereby using hysteresis parameters as an accept -reject criterion for steel
components [25].
The effects of stress on the magnetic hysteresis properties of steels are of interest
because they have applications to nondestructive evaluation of structures such as
railroad lines and pipelines. Typically the hysteresis 'signature' is changed by stress
as shown in Fig. 16.6. Vekser et al. showed that it was possible to measure the stress
in rail steel from the permeability. Schcherbinin et al. were able to detect defects in
in-service railroad rails from inspection by fluxgate magnetometers [26]. Pyatunin
and Slavov investigated the effects of microstresses and texture in steel on the bulk
B
1.5T B
II
B
B
1.5T BII
1.25 T
BII
Fig. 16.6 Changes in magnetic hysteresis 'signatures' of mild steel with stress, after
Langman [27J
Methods for evaluation of intrinsic properties
373
magnetic properties and showed correlations between mechanical and magnetic
properties for specimens of similar texture.
The dependence of magnetic properties on static and dynamic stresses was the
subject of a study by Novikov and Fateev [28]. Similar work was performed by
Pravdin who drew distinctions between the effects of static and dynamic stresses.
The results revealed that dynamic loading changed the flux density B by different
amounts depending on the applied field H. This was very similar to the findings of
Jiles and Atherton described below.
The effects of elastic stress on hysteresis have been reported by a number of
investigators, including Jiles and Atherton [29], Burkhardt and Kwun [30] and
Polanschutz. The interpretation of results is difficult without a working
mathematical model of hysteresis, since none of the direct hysteresis parameters
such as coercivity, remanence, initial permeability or hysteresis loss is uniquely
related to a single physical property. This was realized by Davis who had NDE
applications of hysteresis for evaluation of steel quality in mind. Subsequent
attempts were made to utilize Davis's harmonic model by Willcock and Tanner
[31].
It is clear that any model which has a hope of being used for interpretation of
NDE results such as effects of stress, elastic and plastic strain, creep and fatigue
should have as few parameters as is reasonably possible. The stress or strain
dependence of these parameters can then be determined. Such a criterion
immediately excludes the Preisach model. Some success has been achieved recently
with the model of Jiles and Atherton [32]. Changes in the magnetic parameters
with stress have been determined empirically by Szpunar and Szpunar [33] and
from first principles by Sablik et al. and these have been used to model the
magnetic hysteresis properties under stress.
Hysteresis measurements have also been used by Abuku for evaluation of
residual strain in steel rods. Changes in magnetic properties with stress cycling
have been used to predict fatigue life of specimens by Sanford-Francis, and by
Shah and Bose [34]. It was found that the hardness of the specimens, which can be
inferred from coercivity measurements, began to change long before any crack
failure appeared. Coercivity measurements have been used by Jakel for evaluation
of quality control of steel components.
A brief review of several NDE techniques including hysteresis measurements for
on-line measurement of microstructure and mechanical properties of steel has
recently been given by Bussiere [35].
16.1.4 Residual field and remanent magnetization
This technique is closely related to the flux leakage method (see section 16.2.3).
However, whereas the flux leakage method is used to detect flaws from anomalies
in magnetic flux, the residual field method is usually aimed at detecting changes in
intrinsic properties such as strain, microstructure or heat treatment from
variations in the magnetic field close to the surface of a ferromagnetic structure or
374
Magnetic methods for materials evaluation
FLUX LINES ENTER AIR
NEAR REGION OF LOWER
PERMEABILITY CAUSING
FLUX ANOMALY
LOWER PERMEABILITY
MATERIAL
HIGH PERMEABILITY
MATERIAL
Fig. 16.7 Flux leakage into the air caused by a region oflower permeability within a piece of
steel.
component (Fig. 16.7). The magnetometers used with the technique are often
fluxgates (also known as ferroprobes), but Hall probes and induction coils are also
used and in cases where the coercivity needs to be measured, a coercimeter is used.
The measurement of field intensity was used by Lees et ai. for detecting the
accumulation of oxides within steel boiler tubing. Atherton and co-workers used
the same technique for detecting stresses in pipelines due to large-scale bending,
while Konovalov et al. determined the mechanical properties of steel pipes from a
combination of coercivity and remanence obtained from measurements of field
intensity close to the surface of pipes [36].
Suzuki et ai. [37] used measurements of remanent magnetization for detection
of stresses in pressure vessels, Langman detected and measured stress levels in steel
plates by a novel method of rotating the magnetic field H and noting any
differences caused by stress-induced anisotropy [38]. A detailed discussion of the
variation of magnetic field and flux density with stress has been given by Langman
in a subsequent paper [27]. A number of techniques for sorting steel components
on the basis of magnetic field measurements, usually from the determination of
coercivity, have been reported by Mikheev et al. [39] and by Tabachnik et ai.
No extensive reviews of the method have appeared in the literature. In general it
is very similar in nature to the magnetic flux leakage method. The only differences
lie in the interpretation of results, since the effect of stress on magnetic properties is
complicated. Another additional feature is the use of the coercimeter in a number
of instances. This device magnetizes the specimens to saturation in one direction,
using an electromagnet, and then determines He by reversing the field until the flux
density in the specimen is reduced to zero.
16.1.5
Magnetoelastic methods (magnetically induced velocity changes)
This method relies on the measurement of acoustic velocity in the presence of a
magnetic field to determine stress. Both magnetic field and applied stress change
6V(H)
~
_ _ cy=O
__-cy>O
cy<O
H
Fig. 16.8 Schematic diagram showing the change in ultrasonic velocity
magnetic field H under various stress levels.
~
V in steels with
10
5
10
'0
~
0
~
......
>
<1
040 Ksi
10Ksi
o OKsi
.-10Ksi
·-20Ksi
.-30Ksi
t.
-5
-10
L...-..-_---JL--_ _ _L--_ _---J
o
100
200
300
H (Oe)
FIg. 16.9 Variation of acoustic velocity with magnetic field strength H for various stress
levels applied parallel to the field, after K wun [43].
376
Magnetic methods for materials evaluation
the velocity of sound in ferromagnetic materials such as steel (Fig. 16.8). However
until quite recently few investigations of the effect of stress on the rate of change of
acoustic velocity with field (d VjdH), had been reported.
The first application of this method to ND E of stress in steels was by K wun and
Teller [40J who investigated the stress dependence of the velocity of ultrasonic
shear waves. The method has also been used by Namkung, Utrata and co-workers
[41,42]. One of the great advantages of this technique for NDE of stress is that it is
possible to detect residual uniaxial stress without reference to calibration data. The
results of Namkung and Utrata, who measured the slightly different parameter
(d V jdB), have shown that this is negative under coaxial compression but positive
under coaxial tension. The work has recently been extended to an investigation of
grain size and heat treatment of 4140 steel.
The most recent review of this technique has been given by Kwun [43]. Typical
variations of the acoustic velocity in steel with stress and magnetic field parallel are
shown in Fig. 16.9 and with stress and magnetic field perpendicular are shown in
Fig. 16.10.
10
5
10
'0
....
0
~
"
>
<]
610Ksi
-5
o OKsi
• -10Ksi
• -20 Ksi
• -30 Ksi
-10 L--_ _ _L--_ _ _L--_ _- - l
o
100
200
300
H(Oel
Fig. 16.10 Variation of acoustic velocity with magnetic field strength H for various stress
levels applied perpendicular to the field, after K wun [43].
Methods for detection of flaws and other inhomogeneities
16.2
377
METHODS FOR DETECTION OF FLAWS AND OTHER
INHOMOGENEITIES
This section reviews nondestructive evaluation of defects or flaws in ferromagnetic
steels. As in the first part of the chapter the objective is to provide a broad survey
of the existing techniques giving a comprehensive summary of earlier work but
without going into fine details, which have been covered in most cases by more
specialized reviews of the particular techniques.
The subject of flaw detection in materials using magnetic methods has a long
history, going back as far as the work of Saxby (1868). Systematic development of
testing techniques based on perturbations of the magnetic flux in iron and steel due
to the presence of defects did not begin however until after the chance discovery by
Hoke, that iron filings accumulated close to defects in hard steels while in the
process of being ground. The technique of magnetic particle inspection, which was
based on this discovery, was then developed by DeForest and Doane.
Later as the subject of flaw detection became more quantitative additional
methods were developed in which the leakage field in the vicinity of the flaw was
measured with a magnetometer. Once the field strengths of the leakage fields were
being measured on a routine basis it became desirable to relate these to flaw size
and shape, and therefore there arose the need for modelling the leakage fields from
different crack geometries.
16.2.1
Magnetic particle inspection
The technique of magnetic particle inspection (MPI) was the first magnetic NDE
method in widespread use. It was discovered accidentally by Hoke in 1918, but it
was left to DeForest to develop the method further for practical use. DeForest's
work involved devising methods of generating a magnetic field of sufficient
strength in any direction in a specimen. This was solved by using electrical contact
prods with heavy duty cables being used to pass large currents through test
specimens in desired directions. Furthermore it became apparent that better results
were obtained by using magnetic powders with uniform properties such as particle
shape, size and saturation magnetization in order to obtain more reliable results.
Particle sizes with diameters ranging from 0.3 micrometres up to 300 micrometres
are now used. DeForest and Doane formed the Magnaflux Corporation to exploit
the MPI method in 1934. This company remains one of the principal suppliers of
equipment for MPI in North America. In Europe the German company Tiede
Gmbh holds a comparable position.
The MPI method is very simple in principle. It depends on the leakage of
magnetic flux at the surface of a ferromagnetic material in the vicinity of surfacebreaking, or near-surface flaws, Fig. 16.11. There are now five main methods for
generating a magnetic field in a material. Three of these depend on the generation
of a magnetic field in the test specimen without necessarily inducing a current, and
378
Magnetic methods for materials evaluation
FLUX LINES ARE
PERTURBED ____
NEAR FLAW
---...
~=-
MAGNETIC PARTICLE
ACCUMULATION
MAGNETIC FWX
Fig. 16.11 Schematic diagram of leakage flux in the vicinity of a surface-breaking flaw
with accumulation of magnetic particles. In the presence of narrow cracks the particles
can form a 'bridge' across the crack. In the presence of wide cracks the particles accumulate
on both sides of the crack.
these are the yoke method, the encircling coil method and the internal conductor
method (also known as the 'threading bar' method). Two other methods depend
exclusively on the generation of high current densities in the test specimen, and
these are the 'prod' method, and the current induction method. These five methods
are depicted in Figs 16.12 and 16.13. In each case the best indication is given when
the magnetic field is perpendicular to the largest dimension of the flaw or crack.
Magnetic particle inspection is a very reliable method, when used correctly, for
finding surface flaws and gives a direct indication of the location and length of the
flaw. There is little or no limitation on the size or shape of component being tested,
although more care is needed in the application of the method to complex
geometries. Nevertheless the method does have some distinct limitations. It can of
course only be used on ferromagnetic materials, and in addition the magnetic field
must lie at a large angle to the direction of the flaw to give the best indication. An
angle of 90° gives optimum performance, but good indications can be obtained
with angles as low as 30°. Flaws can be overlooked if the angle is smaller. Finally
although the length of the flaw is easily found the depth is difficult, if not
impossible, to ascertain.
Various enhancements have been made to the original 'dry powder method'.
These include the used of water-borne and oil-borne suspensions or magnetic inks,
known as the 'wet method', and especially fluorescent powders or suspensions
which often give a clearer indication of small flaws when viewed under ultra-violet
light. Among these the water based suspensions of fluorescent particles are used
predominantly.
Another related method that has found use in detection of flaws in structural
components is the 'replica' or magnetographic method in which a magnetic tape is
placed over the area to be inspected. The tape is magnetized by the strong surface
field, and is then removed and inspected for magnetic anomalies. The magnetic
tape, which records an imprint of the flux leakage from the surfaces of components
can be inspected afterwards using magnetometers such as Hall probes or fluxgates.
The advantage of this method over the magnetic particle inspection technique is
Electromagnet
·C" core
Magnetic field lines
Test specimen
Magnetic field lines
Crack
Test specimen
---- ~'\
----
-----
-~agnetizo
coil
Flux coil leads
Crack
Test specimen
Magnetic field lines
Current flow
Fig. 16.12 Magnetization methods used in magnetic particle inspection based on
generation of fields in the specimen with (e.g. a.c. fields) or without (e.g. d.c. fields) generation
of associated currents in the specimen. (a) The magnetic yoke method; (b) the coil
magnetization method; (c) the internal conductor method.
380 Magnetic methods for materials evaluation
High current cable leads to prods
·Prod"
Magnetic field lines
Crack
Test specimen
Flux coil
Crack
Circulating induced
current
"
/
Flux coil leads
\
Laminated magnetic
core
Fig. 16.13 Magnetization methods used in magnetic particle inspection based on generation of fields in the specimen via generation of associated currents in the specimen. Top:
The current flow or 'prod' method. Below: The current induction method.
that the magnetograph (usually a magnetic tape) can be read using a magnetometer
to obtain a quantitative measure of flux leakage from regions where it would be
difficult to use a magnetometer, for example inside pipelines, or for underwater
applications.
16.2.2 Applications of the magnetic particle method
A particular application has been found for magnetic particle inspection in the
automotive industry where large numbers of parts (typically billions per year) have
to be inspected on a routine basis. In these cases multicircuit magnetization using
several of the principal techniques simultaneously has been found to be a reliable
method of defect detection for any orientation of flaw on parts of complex shape,
such as steering knuckles for cars.
Methodsfor detection offlaws and other inhomogeneities
381
Although it is possible to apply the MPI technique successfully to magnetically
harder materials using solely the remanent magnetization of a specimen and its
accompanying field, it is no longer possible to control the relative orientation of
field and flaw. Therefore it is preferable to apply a controlled magnetic field to the
specimen, using one of the five methods discussed above.
Research efforts in MPI have been directed towards establishing standard
procedures and conditions for applying the method. Gregory [44] showed that for
complex geometries the magnetic particle method is more difficult to apply and has
sometimes failed to reveal structural faults in aerospace components because of
low magnetization in some regions of those components. More recent developments using multicircuit magnetization methods described above have now
overcome this problem.
Optimum conditions for the application of MPI to inspection of welds were
described by Massa [45]. Once again the critical factor was to determine adequate
levels of magnetization in order to reveal the presence of defects. No simple
solution was found in that case but tables of values of critical field strength H for
various geometries were given.
Recent research efforts in magnetic particle inspection have been directed towards
modelling of the magnetic field throughout the whole of an object including the
leakage fields in the vicinity of flaws. These calculations use Maxwell's equations
and finite element methods of computation [46,47,48] which will be discussed in
detail in section 16.2.5. The computer aided design of magnetic particle inspection
systems, especially multi circuit units for inspection of automotive parts is now very
much state of the art.
Some standard procedures have been recommended in the UK where a field
strength of at least 2400 A/m (30 Oe) (British Standard 6072) was suggested for
using MPI on steels. However, opinions seem to differ over the necessity of the
recommendation. Work on MPI applications to pressure vessels and pipelines by
Raine, Robinson and Nolan has shown that the recommendations of BS 6072 do
not seem to be generally applicable. Their work indicated that lower field strengths
than 2400 A/m were quite adequate for satisfactory MPI indications. Edwards and
Palmer [49] have investigated procedures for applying the method to tubular
specimens threaded on a current carrying conductor and for a cylindrical bar using
prod magnetization. It was shown that care is needed to generate sufficient field
strength for adequate magnetization of the specimen in these cases.
It is sometimes assumed that the optimum magnetizing condition corresponds
to the maximum permeability, however this was not found to be true in the work of
Oehl and Swartzendruber [50]. They found that for cylindrical defects with square
cross sections the ratio of leakage field to applied field reached a maximum at an
applied field of between 800 and 2400 A/m in steels, depending on the lift off.
This was well removed from the maximum permeability of the material which
occurred at H = 120 A/m.
Recent enhancements of the MPI method include fully automated scanning of
steel components for crack indications using optoelectronic devices, followed by
382 Magnetic methods for materials evaluation
computer operated digital image processing techniques to enhance the results [51,
52]. The automation of the inspection method has the advantage of eliminating
sUbjective evaluation of results and is therefore desirable.
The 'wet method' in which a magnetic colloid (e.g. ferrofluid), or a suspension of
larger magnetic particles in a carrier fluid, is used has many similarities with the
Bitter pattern technique for magnetic domain observations. Because of the finer
particles used in the wet method it has some advantages in spatial resolution over
the dry powder method, and therefore it can be successfully used for detection of
smaller flaws. However in the case oflarger castings with relatively wide cracks the
coarse grained dry powder method remains the appropriate technique.
Present developments in the MPI technique are concentrated on improving
measurement techniques and control procedures for current, field strength and
light intensity in order to enhance the capabilities of fully automated inspection
systems. The next stage of the evolution of MPI should be the further development
of expert systems to improve further automated measurements and to develop
evaluation algorithms.
In conclusion therefore MPI is a well established technique and most of the
problems associated with its use appear to have been solved. The remaining
research effort is directed towards optimizing conditions for its use and increasing
automation. The standard reference work on MPI is Principles of Magnetic
Particle Testing by Betz [53], which contains much information on the subject,
including the history of its development, the underlying principles, methods of
generating fields in the materials and descriptions of the dry powder methods, the
wet, or magnetic ink, method and the fluorescent powder method. A more recent
reference work on MPI is to be found in volume six of the NDT Handbook [54].
16.2.3 Magnetic flux leakage
The magnetic flux leakage method is derived from the magnetic particle inspection
method. Both depend on the perturbation of magnetic flux in a ferromagnetic
material caused by surface or near-surface flaws. Whereas in the MPI method
detection relies on the accumulation of magnetic powder, or sometimes the use of a
magnetic recording tape, to indicate the presence of a defect, the flux leakage
technique utilizes a magnetometer. The use of a magnetometer for detecting
leakage fields was first suggested by Zuschlag. The magnetometer allows a
quantitative measurement of the leakage field in the vicinity of a flaw to be
obtained.
The field components in three directions, perpendicular and parallel to the flaw
and normal to the surface can be measured although in practice only components
parallel to the surface are usually measured. However the method only began to
gain wide acceptance after the design of a practical flux leakage measuring system
by Hastings [55]. This was capable of detecting surface and subsurface flaws on he
inner diameter of steel tubes, a location that was quite unsuitable for the MPI
Methods for detection of flaws and other inhomogeneities
383
method. The flux leakage technique has now been further developed so that it can
be used not only for flaw detection but also for their characterization. The
magnetometer probe is most often a Hall probe or induction coil. It is scanned
across the surface of the component looking for anomalies in the flux density which
indicate the location of a flaw, as in Fig. 16.14. The leakage flux as a function of
distance across a crack is shown in Fig. 16.15, together with a typical search coil
output as it is moved across the crack.
A drawback of this compared with the particle method is that while large areas
of material can be tested quickly using MPI, the scanning of a magnetometer over
the surface can be time consuming. Therefore the flux leakage method has
advantages in situations where the location offlaws is known or where the location
can be predicted with a reasonable chance of success. Under those circumstances a
careful magnetic inspection can be conducted over a confined area.
Another situation when flux leakage magnetometry has advantages over the
particle method is where the part to be tested is not easily accessible for a visual
inspection. An example is the inside surfaces of long tubes and pipelines. Once
MAGNETOMETER
SCANNING
SURFACE OF
COMPONENT
FLUX LINES ARE
PERTURBED NEAR
FLAW
..
..
..
FLUX LINES ENTER AIR
NEAR REGION OF LOWER
PERMEABILITY CAUSING
FWX ANOMALY
LOWER PERMEABILITY
MATERIAL
Fig. 16.14 Scanning the surface of a specimen with a magnetometer (a) to detect flaws and
(b) to detect regions of low permeability by the magnetic flux leakage method.
384 Magnetic methods for materials evaluation
§ 3000 APPARENT
::;
30
WIDTH
~
~
20 >
2000
E
...J
IL.
...J
o
lJJ
~~
10 u
1000
J:
~
IE<t
...J
o0
lJJ
2
IJ')
4
:::E
POSITION OF
COIL (mm)
SEARCH COIL
OUTPUT ________
~
-10 ~
!l.
I::::l
-20
o
-30
Fig. 16.15 Variation of flux leakage with distance across a crack.
again under these circumstances magnetic flux leakage magnetometry has been an
important and highly successful technique, both for detection of flaws and for
stresses. Subsequent work on detection offlux anomalies in pipelines indicated that
these could be related to stresses within the pipe wall. The method has been
particularly useful for the inspection of defects in underground pipelines as in the
work of Atherton [56] and Khalileev, Fridman and Grigorev [57].
As with the MPI method the flux leakage signal depends on the level of
magnetization within the material. Pashagin et al. have also shown that it depends
on the method magnetization, such as magnetizing by current flow or via an
electromagnet.
Introductory reviews of the flux leakage method and also a more advanced
review have been given by Forster [58]. These discuss both the experimental and
theoretical foundations of the method. The demagnetizing fields Hd of defects were
calculated for ellipsoidal flaws inside the material, and consequently the true
internal field in the defect Hi was calculated from the measured applied field Ha
using the equation
Hi =Ha-Hd·
Once again this procedure is equivalent to assuming that the defect behaves like
a simple magnetic dipole. Such an assumption is only valid as a first
approximation. If the permeability of the material is f.1iron and the permeability of
the flaw is f.1flaw and the demagnetizing factor which depends on the shape of the
Methods for detection of flaws and other inhomogeneities
385
3000
Gi
9
:I:
2000
.s:
go
~
u;
~
1000
4:
o
2
3
Height h (mm)
I--Inside Crcck--,+-,--Outside Crack-
I
120
~
100
:I:
.s:
0.
c:
80
2! 60
u;
"0
Qj
u::
40
20
0
o
2.5 2
2
3
Height h (mm)
/-Inside Crock
--t--- Outside Crack-
Fig. 16.16 Variation of leakage flux with position within and above a crack for various
crack widths, after Forster (1985). Upper diagram is for a carbon steel. Lower diagram is for
a chromium steel.
flaw is N d' then the internal field is
Hi = Ha{
/liron
/liron
+ N O(/liron + /lflaw)
}.
One of the difficulties here is that of course in general the magnetic induction in a
ferromagnet is not known as a function offield H, and hence /liron is not known as a
function of H. These reviews were completed with experimental results on the
variation ofleakage flux with lift-off position above the crack and also as a function
of depth within the crack as shown in Fig. 16.16. These showed that as the lift-off
distance increased the leakage field became insensitive to crack depths at lift-off
386 Magnetic methods for materials evaluation
distances above 1 mm for crack depths of 2.5 mm with varying widths of 0.2 to
2mm.
16.2.4
Applications of the flux leakage method
Stumm [59] has described several devices based on the flux leakage technique
which can be used for testing of ferromagnetic tubes for flaws. One of the problems
identified with these instruments was the difference in defect signal between an
external (near side) flaw and an internal (far side flaw). The signal from an internal
flaw was much smaller than the signal from a comparable external flaw, as
expected. The difficulty arises in deciding whether a given signal is due to a small
external flaw or a much larger internal flaw.
One device called the 'rotomat' enabled the tube to be passed through a rotating
magnetizing yoke which generated a circumferential magnetic field for detecting
longitudinal defects in the tube wall. The leakage flux was measured using a array
of Hall probes. Tube diameters of 20-450 mm could be inspected.
A second instrument, the 'tubomat' rotated the tube while maintaining the
detection system stationary. The detection and field generation systems were
identical to those used in the rotomat system. The method of magnetization was
however dependent upon the size ofthe pipe. For large pipes the pipe was threaded
with a central conductor which carried the high current required to obtain the
optimum field strength in the pipe wall. For smaller pipes the magnetic yoke
method was employed.
A third and somewhat different system described was the 'discomat' which was
used primarily for weld inspection. The detection system containing five Hall
probes was rotated on a disk at typically 50 rev /s and the tube was magnetized
transversely to the direction of the weld using a magnetizing yoke.
Owston [60] has reported on measurement of leakage flux from fatigue cracks
and artificial flaws such as saw slots in mild steel. These he attempted to interpret in
terms of a simple dipole model of the type described by Zatsepin and Schcherbinin.
The magnetic leakage field parallel to the surface and perpendicular to the cracks
was measured as a function of lift-off (i.e. distance from the field detector to the
surface of the test material). Results indicated that the leakage field increased
linearly with slot depths of up to 0.2/1, while the derivative d2 H/dz 2 , where H is the
magnetic field and z is the distance from the centre of the slot measured in the
surface of the specimen, was found to be proportional to 1/14 where I was the lift-off
distance.
Barton, Lankford and Hampton [61] investigated the flux leakage method for
testing of bearings, which revealed that magnetic signatures associated with pits,
voids and inclusions could be identified in the leakage fields of bearing races.
Forster has written a number of papers on the subject of flux leakage at a fairly
elementary level. The testing of round billets for cracks using leakage flux probes
has been reported using both the 'tubomat' and the 'rotomat' devices. The
locations of defects identified by these automated flux leakage detectors are
Methods for detection of flaws and other inhomogeneities
387
automatically marked with a paint spray for ease of identification. These systems
were capable of inspecting 12 metre long billets of diameter 90-230 mm in 16 s.
The most extensive review of the subject of magnetic flux leakage as an NDE
tool presently available is NDE Applications of Magnetic Leakage Field Methods
by Beissner, Matzkanin and Teller [62J and it must be considered to be the standard
reference on magnetic flux leakage at this time. It contains a history of the
development of the method, discusses the underlying theory, including the analysis
of leakage field results and defect characterization and finally a description of a
number of applications.
16.2.5 Leakage field calculations
Once quantitative measurements of leakage fields became a routine technique it
was natural to want to interpret the signals in terms of flaw size and shape.
Therefore leakage field calculations for specific flaw shapes began to be made, so
that the theoretical profiles could be compared with experimental observations
with the objective of characterizing flaws from their leakage fields. Two of the most
significant papers in the early development of this subject were by Schcherbinin and
Zatsepin [63,64], which were based on approximating surface defects by linear
magnetic dipoles and calculating the dipole magnetic field. In this way, expressions
were obtained for the normal and tangential components of the magnetic leakage
field due to a defect.
Zatsepin and Schcherbinin realized that the exact calculation of leakage fields
arising from real defects presented an extremely complicated problem, which was
intractable given the numerical methods and state of computer technology at that
time. They therefore looked for a relatively simple problem, the leakage field of
flaws which could be approximated as a point, line or strip dipole and calculated
the fields using analytical expressions. The results were then compared with
observations ofleakage fields due to defects as shown in Fig. 16.17. Notice that the
horizontal component of field across the flaw leads to a unipolar response, while
the normal component leads to a bipolar response. However these early papers did
not relate the leakage field to the internal field in the material, and therefore there
were limitations on its applicability to actual measured leakage fields.
Subsequent papers on leakage field calculations by Schcherbinin and Pashagin
were based on the same model. Experimental measurements were made on
specimens of carbon steel which had artificial flaws machined in the surfaces. These
defects were typically h = 0.2-3.0mm deep, 2b = 1.0mm wide and 21 = 1.030.0 mm long. The magnetic fields were generated using the magnetic yoke
method, with a maximum magnetic field of 12 kAjm. Measurements of the leakage
fields were made using a fluxgate (or 'ferroprobe') magnetometer. Field
components tangential to the surface of the specimen normal to the flaw, H x , and
normal to the surface of the specimen, H y , were measured.
It was found, for example, that the relationship between the field Hy and the
magnetizing field Ho remained almost linear, for any given length of flaw,
388
Magnetic methods for materials evaluation
Hx(Oe)
'E
CII
C
8;g
E.!!!
0UO)
_ 01
00
-.><
Co
00)
N_
.~
~o
50'
3D'
D·
.....
4
3
2
I
0
-I
Hy(Oe)
'E
2?::!2
OCII
a;<:
I
EO)
0
-0
-2
-3
-4
801
.2
8~ .....
..;:.
~
3
2
.... 0
-I
-10
+10
-10
+10
-10
tlO
Distance from crack (mm)
Fig. 16.17 Calculated leakage field due to a flat-bottomed flaw using the dipole model; (a),
(b) and (c) are horizontal field components, (d), (e) and (f) are vertical field components. The
spot is of length of length 4 mm, width 0.4 mm and depth 2 mm, after Zatsepin and
Shcherbinin (1966).
although the actual ratio Hy/Ho increased as the flaw length increased, and in fact
appeared to reach a saturation level as the length tended to infinity. The tangential
field Hx perpendicular to the flaw was also measured. For a given flaw it was found
to decay with displacement z from the centre of the flaw. Hx was also found to
increase with flaw length, although this variation was dependent upon the
magnetizing field Ho. The calculations of the leakage fields by Schcherbinin et al.
were analogous to the fringing field calculations for idealized magnetic recording
heads by Karlquist.
Significant progress in the calculation of leakage fields was made subsequently
by Hwang and Lord [65J using finite-element methods. This was the first attempt
to use numerical methods to find exact solutions for the field caused by defects and
represented a real breakthrough in the development of the subject because it
enabled the leakage fields to be calculated from the existing field and permeability
in the bulk of the material. This represents a landmark in the calculation ofleakage
fields. Some results of their calculations are shown in Fig. 16.18. The earlier work of
Zatsepin and Schcherbinin while commendable was not easily adaptable to the
range of shapes of defects encountered in practice. The leakage field profiles
obtained by Hwang and Lord for the case of a simple rectangular slot were in
excellent agreement with observation. This paper was therefore instrumental in
Methods for detection of flaws and other inhomogeneities
VU~
389
a b c
0.4
-0.4
0.4
-0.4
Fig. 16.18 Calculated leakage field due to a flaw using finite-element techniques, after
Hwang and Lord [66].
demonstrating how the finite-element technique could be used for modelling fields
of defects. It was clear from the work that the finite element method was sufficiently
flexible, so that its successful application in the case of a simple slot defect indicated
its likely successful application in the case of more complex defects, an extension to
which the Schcherbinin model was not capable.
This was followed by a series of papers of which the most significant were by
Lord and Hwang [66] and Lord et al. [67]. Lord and Hwang extended the
application of the finite-element method to a variety of more complicated flaw
shapes. It was deduced from this that the finite-element method provided the
possibility of defect characterization from the leakage field profile. It was found, for
example, that the peak-to-peak value of the leakage flux BN increased with
increasing flaw depth, while the separation between the peaks depended on flaw
width. These results were in agreement with experimental observation and
indicated the potential benefits of finite-element techniques for interpreting
leakage flux measurements with the objective of characterizing the defects.
Lord, Bridges, Yen and Palanisamy [67] have remarked that the cornerstone of
an approach to the application of magnetic flux leakage techniques to NDT is the
development of an adequate mathematical model for magnetic field-defect
interactions. It might also be added that the modelling of the B, H or hysteresis
characteristics in ferromagnetic materials adds another dimension to this problem,
390 Magnetic methods for materials evaluation
since even if the internal magnetic field is known there are a range of possible values
of the flux density inside the material. These different possible values of B will
certainly affect the leakage flux for a given type of defect with a given magnetizing
field Ho inside the material. Lord concluded that the complex defect geometries,
together with the nonlinear magnetization characteristics of ferromagnetic steels,
made closed form or analytical solutions of magnetic field defect interactions
virtually impossible. However, they were able to show a number of examples of
successful application of finite-element modelling and to conclude optimistically
about the potential for defect characterization for nondestructive testing.
Lord [68] has given a review of the application of the finite-element numerical
method for calculation of leakage fields arising from magnetic field-defect
interactions. In this he had indicated that much progress has been made in
theoretical modelling ofleakage field signatures using numerical methods, but that
there are many complex problems still to be solved. Perhaps the most significant of
these is taking into account the hysteresis in B, H characterisics of ferromagnetic
materials before applying the finite-element, or other numerical calculation of the
leakage field.
The work of Lord and co-workers has firmly established the use of numerical
methods such as finite-element calculations, as the technique with the most promise
for characterization of defects from leakage flux measurements. This has lead to
much interest in the area of numerical modelling of leakage fields, for example the
recent work of Atherton [69] and Brudar [70]. However it is realized that threedimensional finite-element calculations are ultimately desirable for more precise
characterization of real defects.
An analytical solution for the leakage field of surface breaking cracks has
recently been presented by Edwards and Palmer [71], in which the crack is
approximated as a semi-elliptic slot in the material. The advantages of an
analytical expression are that the fields due to defects can be rapidly calculated,
and in addition the equations can be differentiated to find the forces on magnetic
particles. From these results Edwards and Palmer calculated the magnetic field
strength H needed to detect 1 f.1,m and 10 f.1,m wide slots for a range of slot depths
and permeabilities. They concluded that fields in the range 10 2 -10 3 A/m are
required for crack detection, and these are in agreement with those used in
practice, for example BS 6072 which recommends a magnetic induction of 0.72 T,
corresponding to fields of 5700 A/m in a steel of relative permeability 100.
Reviews of the subject of magnetic leakage field calculations and the
interpretation of experimental measurements have been given by Holler and
Dobmann [72]. Dobmann [73] has dicussed the problems of both detection and
sizing of defects and has attempted to relate these to theoretical model predictions.
Theoretical work on flux leakage has lagged considerably behind experimental
development. Dobmann and Holler have reviewed the theoretical modelling of the
effects of various flaws on the leakage flux, and Forster has attempted to correlate
observed magnetic flux leakage measurements with expectations based on finite-
Methods for detection offlaws and other inhomogeneities
391
element model calculations. His results showed serious discrepancies. This
has made it difficult to quantitatively characterize flaws in some cases,
although qualitatively the explanations are relatively simple and have been
successful.
16.2.6 Eddy current inspection
Eddy current methods for nondestructive evaluation are not strictly magnetic
methods as such. That is they do not depend for their validity on any inherent
magnetic properties of the material under test. In fact they can be applied to any
conducting material. Furthermore the literature for eddy current techniques is
more extensive than that of all the other magnetic methods combined, and so it
would not be possible to give a comprehensive review of the subject in this paper.
Nevertheless in the interests of completeness eddy currents do deserve mention
since they are used on magnetic materials.
The eddy current inspection method depends on the change in impedance of a
search coil caused by the generation of electrical currents in the test specimen when
it is subjected to a time-varying magnetic field, Fig. 16.19. The response is usually
monitored in the form of a complex impedance plane map. Eddy currents can be
used for the detection of cracks and other defects, because the defects interrupt the
flow of the eddy currents generated in the material. This results in a different
complex impedance of the eddy current pick-up coil when it is positioned over the
flaw compared with when it is positioned over an undamaged region of the
material. Eddy currents can also be used for checking thickness of coatings,
determining permeability and conductivity, evaluating changes in heat treatment
and microstructure, estimating tensile strength, chemical composition and
ductility. However, whereas the interpretation of results in nonmagnetic matrials is
relatively straightforward, in ferromagnetic materials it is more difficult because
TIME VARYING
MAGNETIC FIELD H(t)
~-=
.-~t1
TIME DEPENDENT
CURRENT
I(t )
VI
EDDY CURRENTS
GENERATED IN
SOLID
Fig. 16.19 Generation of eddy currents in a conducting material by a timevarying field.
392 Magnetic methods for materials evaluation
the eddy current response depends on the permeability. In ferromagnetic materials
such as steels the permeability varies in a complicated way with the generating
field.
Much of the early work on development of eddy current techniques was due to
Forster [74]. A survey of the field until 1970 was given by Libby [75].
16.2.7 Applications of eddy currents inspection
Application of eddy current techniques to magnetic materials by Vroman [76] has
shown that the results are easier to interpret if the material is subjected to a
saturating dc field at the same time as the ac field which generates the eddy
currents. This insures that the prior state of magnetization of the material no
longer affects the observed results, while also preventing large changes in
permeability.
Cecco and Bax therefore saturated the magnetization within their steam
generator tubing while scanning for defects. In this way they reported that they
obtained improved sensitivity over earlier attempts to use eddy currents for
ferromagnetic tubes which had been hampered by inherent 'magnetic noise'. Deeds
and Dodd [77] have reported on detection of defects such as cracks, wall thinning
or holes in steam generator tubing using eddy currents. Dodd and Simpson
developed a low frequency eddy current system for accurate determination of small
changes in permeability of weakly magnetic materials, which was used to evaluate
the fabrication of austenitic stainless steels. This was able to determine the amount
of cold working in components and to determine the amount of delta ferrite in
welds, which reduces the incidence of hot cracking in the welds.
Detection of the depth of case hardening in steels by eddy current inspection has
been reported by Kuznetsov and Skripova. Clark and Junker [78] have
established the feasibility of an eddy current NDT approach for the detection of
temper embrittlement in low alloy steels, which was known to be difficult to detect
using other techniques. Eddy currents have also been used for mapping localized
changes in residual stress using automated scanning equipment by Clark and
Taszarek. However they concluded that the method was unable to discriminate
between tensile and compressive residual stresses, only the absolute magnitude of
stress could be ascertained. Lord and Palanisamy have also used eddy currents for
inspection of steam generator tubing and modelled the results using finite-element
techniques.
16.2.8 The remote field electromagnetic technique
The remote field electromagnetic technique is based on the diffusion of
electromagnetic energy through a pipe wall. This can be used for the detection of
corrosion or other forms of wall thinning in pipes and oil well castings. The
method was developed in the 1950s and early 1960s, but it is only in recent years
that it has suddenly received great attention as an NDE technique.
Methods for detection of flaws and other inhomogeneities 393
One of the earliest descriptions of the method was by Schmidt [79] who used it
for in situ detection of external casing corrosion in down hole inspection in oil
wells. This was an important breakthrough because there were no other techniques
available for detection of external corrosion. The advantages of the method over
conventional eddy current and ultrasonics inspection, which can also be used for
measuring wall thickness, include full 360 0 scanning of the pipe, rapid logging
speeds and its insensitivity to dirt or scale on the pipe.
Schmidt [80] has recently reviewed the remote field eddy current inspection
technique. In this he has shown that the development of improved spatial
resolution of these instruments has allowed isolated pits and cracks in the pipe wall
to be resolved, a result not possible with alternative techniques under the
conditions encountered in down hole inspection. The method has been used for
detection and classification of stress corrosion cracking.
The advantage of the remote field technique is that it detects only a flux which
has penetrated and been transmitted through the pipe wall. Even though both the
transmitter and receiver coils are located within the pipe there is very little direct
coupling between them. Therefore the detected signal contains information about
the pipe wall thickness. The variation of the detected signal with distance between
exciter and receiver coils is shown in Fig. 16.20.
The technique is equally sensitive to both interior and exterior metal loss, and
the observed phase lag was found to be linear with wall thickness, which is
a convenient property. Finally the problems that are normally encountered
10-1
360
(/)
~
10-2
t3
0
w
270 0::
>
w 10-3
(.!)
Cl
W
~
Cl
I-
:J 1(J"4
a.
«
0::
180
AMPLITUDE _
~
OUTERw.
:.tLL
10-5
~
90
~
I-
W
...J
10-6
(.!)
z
«
~
«J:
a.
w
Cl
10- 7
0
2
3
4
5
6
7
8
0
PIPE DIAMETERS FROM EXCITER COIL
Fig. 16.20 Variation of phase and amplitude of a detected signal as a function of distance
between exciter and receiver coils within the pipeline.
394 Magnetic methods for materials evaluation
ENERGY FLOW PATH
_1_+ ______
Fig. 16.21 Arrangement for the
remote-field electromagnetic inspection of pipes as described by
Atherton and Sullivan [81]
+_
ITTER COIL
WHEEL SUPPORT
with lift-off in eddy current inspection are less evident in the remote field
technique.
This method has recently been investigated by Atherton and Sullivan [81] for
inspecting zirconium-2.5% niobium nuclear reactor pressure tubes, which are
non-ferromagnetic, for the presence of defects. In this they operated the detection
coil at a distance of one pipe diameter from the transmitter coil as shown in
Fig. 16.21. They also have indicated that their results are consistent with Schmidt's
original suggestion that the signals are caused by a diffusion process through the
wall close to the transmitter coil, followed by propagation along the outside of the
tube, and then diffusion back through the pipe wall. Subsequently finite-element
calculations were used to model the propagation of the electromagnetic signals in
the remote field eddy current technique. The results ofthese calculations confirmed
the mode of propagation originally suggested.
16.3
CONCLUSIONS
A variety of magnetic methods for NDE are available and these have been critically
reviewed here. With the upsurge of interest in this field there must certainly be new
magnetic methods awaiting development in the near future. One of these may be
the use of magnetomechanical emissions of the type observed by Higgins and
Carpenter which would seem to be ideal for the detection of dynamic stresses in
steel, but which has not even begun to be exploited. Compared with ultrasound,
most of the magnetic methods are in their infancy and so offer wide scope for future
development and growth. But, also in comparison with ultrasound, the
instrumentation required for magnetic inspection is far simpler and therefore in
many cases easier to adapt to applications.
Three major classes of techniques for detection and characterization offlaws and
defects in ferromagnetic materials have been discussed. These are magnetic particle
inspection, magnetic flux leakage and eddy current inspection. The relative
advantages and disadvantages of each technique have been discussed, and a survey
of existing literature has been given. It is not possible to discuss each method in
detail here, nor it it even desirable. The references given however do provide access
to the specialized literature of each method and further details can be obtained
from them.
References
395
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NDT, London, September 1987, p. 2762 (eds J. M. Farley and R. W. Nichols), Pergamon Press, Oxford.
53. Betz, C. E. (1966) PrincipLes of Magnetic Particle Testing, Magnaflux Corporation.
54. Schmidt, J. T. and Skeie, K. (1989) NDT Handbook, Volume 6: Magnetic Particle
Testing (ed. P. McIntire) American Society for Nondestructive Testing.
55. Hastings, C. H. (1947) ASTM Proc., 47, 651.
56. Atherton, D. L. (1983) NDT International, 16, 145.
57. Grigorev, P. A., Fridman, L. A. and Khalileev, P. A. (1976) Sov. J. NDT, 12, 351.
58. Forster, F. (1982) Sov. J. NDT, 18, 841.
59. Stumm, W. (1974) Non-destructive testing, 7, 251.
60. Owston, C. N. (1974) Brit. J. NDT, 16, 162.
61. Barton, J. R., Lankford, J. and Hampton, P. L. (1972) Trans. Soc. Auto. Eng., 81, 681.
62. Beissner, R. E., Matzkanin, G. A. and Teller, C. M. (1980) SWRI Report Number
NTIAC-80-1.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
Zatsepin, N. N. and Schcherbinin, V. E. (1966) Sov. J. NDT, 2, 50.
Schcherbinin, V. E. and Zatsepin, N. N. (1966) Sov. J. N DT, 2, 59.
Hwang, J. H. and Lord, W. (1975) J. Test. and Eval., 3, 21.
Lord, W. and Hwang, 1. H. (1977) Brit. J. NDT, 19, 14.
Lord, W., Bridges, 1. M., Yen, W. and Palanisamy, S. (1978) Mater. Eval., 36, 47.
Lord, W. (1983) IEEE Trans. Mag., 19, 2437.
Atherton, D. L. (1988) Brit. J. NDT., 30, 159.
Brudar, B. (1985) NDT International, 18, 353.
Edwards, C. and Palmer, S. B. (1986) J. Phys. D. (Appl. Phys.), 19, 657.
Holler, P. and Dobmann, G. (1980) in Research Techniques in NDT, Vol.IV (ed. R. S.
Sharpe), Academic Press London.
Dobmann, G. (1985) in Electromagnetic Methods of NDT, (ed. W. Lord), Gordon and
Breach.
Forster, F. (1954) Z. Metallk., 45, 221.
Libby, H. L. (1971) Introduction to Electromagnetic Nondestructive Test Methods,
Wiley.
Vroman, 1. (1981) Brit. J. NDT, 23, 307.
Deeds, W. E. and Dodd, C. V. (1985)in Electromagnetic Methods ofNDT(ed. W. Lord),
Gordon and Breach.
Clark, W. G. and Junker, W. R. (1985) Mater. Eval., 43, 1546.
Schmidt, T. R. (1961) Corrosion, 17, 81.
Further reading
397
80. Schmidt, T. R. (1984) Mater Eval., 42, 225.
81. Atherton, D. L. and Sullivan, S. (1986) Mater. Eva/., 44, 1544.
FURTHER READING
Bezer, H. 1. (1964) Magnetic methods for NDT, Brit. J. NDT, 6, 85; and 6, 109.
Betz, C. E. (1966) Principles of Magnetic Particle Testing, Magnaflux Corporation.
Halmshaw, R. (1987) Non Destructive Testing, Edward Arnold Publishers, Ch. 5.
Jiles, D. C. (1988) Magnetic methods for non destructive evaluation, NDT International, 21,
311.
Jiles, D. C. (1990) Magnetic methods for non-destructive evaluation (Part 2), NDT
International, 23, 83.
Journals
Non-destructive Testing and Evaluation (formerly NDT Communications)
Journal of NDE
NDT International
Materials Evaluation (The official journal of the American Society of Nondestructive Testing)
British Journal of NDT (The official journal of the British Institute of NDT)
Solutions
Example 1.4 Magnetic field at the centre of a long solenoid. Using the result for
the field on the axis of a single turn 0 btained in exam pIe 1.1 consider the field due to
an elemental length of solenoid dx with n turns per unit length, each carrying a
current i
dB = ;~
sin 3 ~dx.
We now need to express either ~ in terms of x and integrate over the range x = to x = 00, or to express x in terms of ~ and integrate from ~ = - n to ~ = n.
tan~
00
a
=-
x
x=
acot~.
Therefore
dx = - acosec 2 ~d
- (
~)
sin 3 cosec 2 rJ.
= - (
~)
sin
dB =
Now integrating from
rJ.
=-
n to ~
~
d~
~ d~.
= +n
B = ni amps per metre.
Example 1.5 Force on a current-carrying conductor. (a) The magnetic field at a
distance a from a long conductor carrying a current i is
400 Solutions
The force per metre exerted on a current carrying conductor by a field His
F= J-loidl x H.
In this case Hand dl are perpendicular so that the force per unit length is
J-loi2
F= 2na'
When the conductors both carry 1 amp and are 1 metre apart
F = _(4_n_x_10_-_7 )_(1_)(_1)
2n
= 2 x 10 - 7 newtons per metre.
This is a surprisingly small force given that the currents seem so large.
(b) The force exerted, as given by the above equation, is
F=J-loidl x H
and in this case the current is perpendicular to the field, so
F = J-loilH
= (4n x 10- 7 )(5)(0.035)(160 x 103)
f = 0.0352 newton.
Example 1.6 Torque on a current-loop dipole.
J-lom x H
=mxB
't =
and if the m and B vectors are perpendicular and since m = ANi
t =
ANiB
= (4 x 10- 4 )(100)(1 x 10- 3 )(0.2)
=
8 x 10- 6 newton metre.
Example 2.3 Demagnetizing field calculation.
H in =
Happ - NdM
and for a sphere N d = 1/3. Since Ms = 1.69
X
106 amp/metre
Happ = H in + (1/3)(1.69 x 106 ) amp/metre
and since we are assuming N dM »Hin this gives
Happ = 5.62
X
10 5 amps/metre.
Example 2.4 Demagnetizing effects at different field strengths.
M=B/J-lo-H.
Solutions
401
At H=80kA/m
M = 7.16
10 5 - 0.8
X
X
10 5 Aim
10 5 Aim
= 6.36 x
The internal field is given by
= 8 X 104 - (0.02)(6.36) x
=
6.73 x 104 Aim.
=
8.75
10 5)
At 160kA/m
M
= 7.15
X
10 5 - 1.6
X
10 5 A/m
x 10 5 Aim
and
H in = 16.0
=
X
14.57
104 - (0.02)(7.15
X
X
105)
104 Aim.
Fractional errors obtained if fields are not corrected for demagnetizing effects are
at 80kA/m
at 160kA/m
Therefore demagnetizing effects become proportionally less of a problem at
higher field strengths.
Example 2.5 Flux density in an iron ring with and without an air gap. In the
continuous ring the flux density is given by
<I> = Ni/{l/I1A)
= 6.0
x 10- 4 weber.
When there is a saw cut in the ring the total reluctance is the sum of reluctance of
the iron and the air, so the relation becomes
where Ii = 0.4995 m, 1. = 0.0005 m,
f.1..= 12.57 x 1O- 7 H/m.
Ai = 2
X
10- 4 m 2,
l1i = 0.001885 Him,
402 Solutions
Under these conditions the flux will he,
<I> = (800)(2 x 10- 4 )
(397.77 + 264.98)
=2.4 x 1O- 4 weher.
The additional current required to restore the flux to its original value is
Nbi = <l>la
flaAa
bi = (6.0 x 1O- 4 )(0.000S)
(0.0002 x 411: x 10- 7 )
= 1.5 amps.
The total current needed to restore the flux density is therefore i + bi = 2.S amps.
Example 3.1
Torque magnetometer.
t=flom x H
= flomH sin e.
t
= (411: x 10- 7)(0.318)(14) newton metre
= S.6 x 10- 6 newton metre
t=(S.6 x 10- 6 ) sin 30 0
=
2.8 x 10 - 6 newton metre.
Example 3.2 Magnetic resonance. A paramagnet with S =! is a system with
one unpaired electron. Since this is a dilute paramagnet then we may consider the
electrons on neighbouring ionic sites as non-interacting. The magnetic moment of
a single electron is m = 9.27 x 10- 24 A m 2 , therefore since the gyro magnetic ratio
Y IS
F or an electron g = 2
'(9.27 x 10- 24)(2)(6.284)
y=
(6.63 x 10- 34)
= 1.76 x 1011 Hz/T.
Therefore the using the expression for the resonant frequency
vo=yB
=(1.76 x lOl1)(O.l)Hz
Vo
Solutions 403
1.76 X 10 10 Hz
= 17 600 MHz.
=
Example 3.3 Induction coil method. The applied field in amps per metre can be
found simply by multiplying the current in the windings by 400.
Reversing the current in the solenoid causes the flux density to reverse
completely, so ifthe prevailing value is Bo tesla the change in flux density caused by
,
reversing the field is 2Bo.
The flux density change is found by multiplying the deflection d (in mm)
0.17 x 10- 4 Wb/mm, and then dividing by the result by the product ofthe number
of turns, 40, and the cross-sectional area 0.196 x 10-4 m 2.
~B
H = 400iA/m
= 2B = 0.0217 d tesla
ir
(A)
Ha
(A/m)
d
(mm)
(A/m)
1.5
3.1
4.9
8.5
11.0
12.7
600
1240
1960
3400
4400
5080
24.0
49.2
77.6
103.7
107.5
109.1
0.26
0.53
0.84
1.12
1.17
1.18
B
From this table a plot of B against Ha gives the apparent magnetization
curve.
From charts of demagnetizing factors given in Chapter 2, we find that
N d = 0.0015 for a cylinder with length to diameter ratio 40: 1 and relative
permeability greater than 150.
The demagnetizing field Hd is given by
and hence the true internal field Hi is given by
404 Solutions
B
Ha
Hi
(T)
(A/m)
(A/m)
0.26
0.53
0.84
1.12
1.17
1.18
600
1240
1960
3400
4400
5080
291
609
960
2068
3009
3678
At B = 1 tesla, by linear interpolation, the permeability is 6.276 x 10 - 4 H m -1.
This corresponds to a relative permeability of 499.7.
Example 4.2 Use of initial magnetization curve to find flux in core. If the material
is in the form of a toroid with mean circumference 40 cm and cross-sectional area
4 cm 2 , with a coil of 400 turns, then the magnetic field generated by a current i in the
coils will be
H=Ni/1
= 400i/OA
=
1000iA/m.
This will correspond to a certain magnetic induction at each field strength. The
magnetic inductions B (in Wb/m 2 or equivalently tesla) can be read from the graph
of B against H for the calculated values of H
(A)
0.1
0.2
OJ
004
0.5
H
(A/m)
B
(T)
100
200
300
400
500
0.52
1.10
1.24
1.33
and since the flux <I> is given by
<I> = BA,
lAO
Solutions 405
where A = 4 x 10- 4 m 2 we arrive at the following values
(A)
B
(T)
(x 1O- 3 Wb)
0.1
0.2
0.3
0.4
0.5
0.52
1.10
1.24
1.33
1.40
0.208
0.440
0.496
0.532
0.560
<I>
Example 4.3 Calculation of atomic magnetic moment.
1 m 3 of iron at saturation will have a magnetic moment of 1.7 x 106 Am 2 • This
1 m 3 will have a mass of 7970 kg and therefore will contain N atoms
7970 x 6.025 x 1026
56
= 8.58 x 10 28 atoms.
N=-------
Therefore the magnetic moment per atom is m A given by
_ (1.7 x 106 ) Am 2
mA - (8.58 x 10 28 )
= 1.98 X 10- 23 Am 2
= 2.49 x 10- 29 J Aim
= 2.14 ttB.
Example 5.1 Determination of Rayleigh coefficients at low fields. Use the
Rayleigh equation to determine the coefficients from the first two data points
B=ttH+ vIP.
At H = 5 Aim, B = 0.0019 tesla
0.0019 = 5tt + 25v.
At H = 10 Aim, B = 0.0042 tesla
0.0042 = lOtt + 100v.
From these two equations we find tt = 3.4 x 10 - 4 H m - 1 and v = 8 x
10 - 6 H A - 1. Substituting these into the equation and using the data points
from the table it is apparent that the Rayleigh region does not extend to 40 Aim,
but that the Rayleigh approximation is valid at 20 Aim.
406 Solutions
The remanence BR in the Rayleigh region is given by
BR =!vHm 2
therefore at Hm = 10 Aim
BR =0.OOO4T
and at Hm = 20 Aim
BR = 0.0016 tesla.
The hysteresis loss in the Rayleigh region is given by
WH =1- vHm 2 •
So at Hm = 20 Aim
Example 5.2 M agnetostriction of terfenol without stress. Since the specimen of
terfenol does not have a preferred orientation and is not subject to stress we can use
the equation for the averaged magnetostriction of a randomly oriented polycrystal
As = (2/5)AIOO + (3/5)A 111 •
Consequently for the values given for this particular piece of terfenol
As = 36
X
10- 6 + 960
X
10- 6
= 996 X 10- 6 •
Now considering the observed magneto stricti on when the field is changed from
perpendicular to parallel to a given direction,
ASII - As-L = As + tAs = tAs
= 1494 x 10- 6 •
Example 5.3 Magnetostriction ofterfenol under compressive load. In this case the
uniaxial compressive stress causes all domain magnetic moments to align
perpendicular to the stress axis and hence to the field axis.
The magnetization will take place entirely by rotation and hence
M=MscosfJ
and therefore
A= tAs cos 2 fJ
A(M) = As (MI Ms)2
= (1494 X 10- 6)(MIMs)2.
t
Consequently, when M = Ms/2
A = 373.5
X
10- 6
Solutions 407
and when M
= Ms
A= 1494
X
10- 6 •
Example 6.3 Effects of anisotropy on rotation of magnetization. If the anisotropy
is such that the (100) directions are the magnetically easy axes, then the anisotropy
constant K will be positive. The anisotropy energy Ea can then be written
Ea =iK1 sin 22¢
where ¢ is the angle of the magnetization away from the nearest (l00) direction.
The field energy is EH
EH = - lloMoH cos e
= - lloMsH cos
e,
where () is the direction of the magnetization in any given domain away from the
(110) direction of the field.
Clearly as () becomes larger the anisotropy energy increases while the field
energy decreases.
After all domain-wall processes have been completed the magnetization in any
domain will lie along the (010) and (100) directions closest to the field direction
(110). Under these conditions we must have,
() = rc/4 - ¢
and
Etot =Ea +EH
=(:1
)sin22¢-lloMsHCOs(i-¢).
The magnetization will take up a direction such that dEtoJd¢ = 0
dEtot
.
2A,
A,
. (rc"4- ¢ )
d¢ = K1 sm 'l'cos2'1' - llo M sHsm
=0.
The component of magnetization in the field direction is of course
Consequently, we arrive at the following relation between magnetization M and
field H
H
= (~)
[sin 2¢ cos 2¢J
llo M s
.
sm
(rc A,)
--'I'
4
408 Solutions
=(~)Msin2</J[
/loMs
2cos2</J ]
. - - 'I'
sm
2
(n
Ms
2A.)
and since cos2</J = sin (n12 - 2</J)
H=
(~);
/loMs
2 sin 2</J
s
The field needed to saturate the magnetization under these conditions is
therefore
Example 6.4
Critical fields as determined by anisotropy.
The anisotropy energy in the case of uniaxial materials like cobalt is given by
while the field energy is
where </J is the direction of moments from the field direction. Given the situation we
have chosen f} + </J = nl2 and therefore
E tot
= Kul sin 2 f}
= Kul
sin 2
-
f} -
/loMsH cos </J
/loMsHsin f}.
The moments will reach equilibrium when
dE
.
tot
de
= 2Kul sm f}cos f} -
0= (2Kul sin
e
so either cos = 0, meaning
f}
dEtot/df}
°
so that
/loMsH cos f} = 0
e- /loMsH) cos e
= n12, or
e
2Kul sin = /loMsH
and hence
H
=
2Kul)'
=( - s m f}.
/loMs
Solutions 409
So when f} = n/2, meaning the magnetization is deflected into the base plane
H = 2Ku1 = 4.58
110Ms
and when f} =
X
10 5 A/m,
n/4, meaning the moments are at 45° to the unique axis, we have
H = KU1 j2 = 3.24 X 10 5 A/m.
110Ms
Example 6.5 Stress-induced anisotropy.
The stress anisotropy E" generated by a stress (1 will be
E,,= -iAs(1cOS2</>,
where </> is the angle between the magnetization vector and the stress direction.
The anisotropy energy E. is given by
E. = K 1 (cos 2 f}1 cos 2 f}2
+ cos 2 f}2 cos 2 f}3 + cos 2 f}3 cos 2 f}1),
where the f}i are the angles that the magnetization vector makes relative to the
crystal axes. We note that the anisotropy energies along certain axes are
t
E'(111) = K1
E'(110)=i K 1
E.(100) = 0,
and remember, K 1 is negative.
In order to rotate all the magnetization vectors into the plane perpendicular to
the stress axis, consider the case which requires the most energ.y. This will be when
the magnetization begins along an easy axis aligned coaxially with the stress.
To rotate this magnetization vector we must simply find the path of rotation,
between the easy axis along the stress direction and an easy axis perpendicular to
this axis, which has the smallest maximum anisotropy energy.
Clearly this energy maximum can not be any larger than Kd3 which is the
anisotropy energy along the hard axis. In fact if the rotation of magnetization is
restricted to one of the (100) planes the maximum anisotropy energy experienced
will be iKl
and we can write the anisotropy in this plane as a function of the angle f} from the
(100) axes or the angle </> from the stress axis which is coincident with the (111) axis
in this case
E.=iK1(sin 2 f}-1)
= iK1 [sin2(n/2 - </» -1]
=iK 1 (cos 2 </>-1),
410 Solutions
where the additional constant term K d4 has been included for convenience only to
ensure that the anisotropy along the (111) direction is zero. The moments must be
rotated to overcome the maximum anisotropy in this plane of rotation. Therefore
the stress anisotropy at the angle ¢ corresponding to Eamax must equal K d4
E(f(¢) =
Eamax
or
remembering that this is a compressive stress, and hence (J is negative.
Now substituting in the values of the saturation magneto stricti on and the
anisotropy constant
(J
= (3)(1067
x 10 6)
= - 6.25
106 Pa.
X
So (J must be compressive with magnitude greater than 6.25 MPa, which is
equivalent to 899 p.s.i.
Example 7.3 Critical dimensions for single-domain particles in nickel.
The magneto static energy per unit volume is EmslV
Emsl V =
f,uoHd dM
and for a sphere magnetized to saturation
Hd=tMs.
Therefore
EmsIV=~
M2
6
and V = 4nr 3 /3, therefore
The reduction of magnetostatic energy caused by dividing the particle into two
domains is
Solutions 411
A Bloch wall through the centre of the particle will have an area of nr2, so the
wall energy will be,
where y is the wall energy per unit area. The energy reduction caused by the
splitting of such a particle into two domains will be
AE = ynr 2 - nf1.o M s23
r
9
Ll
We therefore need to find the critical condition under which we just fail to save
energy by dividing into two domains. Let rc be the critical radius under this
condition
AE = yn
2 _
nf1.o
c
M2 3
s rc
9
=
0
9y
rC =--2
f1.o M s
=
1.92 x 10- 8 m.
Example 7.4 Calculation of wall energy and thickness from anisotropy energy,
saturation magnetization and exchange energy.
The material is barium ferrite. Following the equations given in the text we know
that
f1. fm 2 n2
° al
Y=
+ Kl
JS 2 n 2
=-al-+ Kl .
At equilibrium we must have dYldl = 0, and consequently,
_(J
S2n 2 )1/2
1- - Ka
=
86.5 x 1O- 1 °m.
The wall energy is
J S2n 2
y=-al-+ Kl
= 5.69 x 10- 3 J/m 2 •
412 Solutions
The critical radius for a spherical single-domain particle is then
9y
rc =
21lo M s
2
= 14.1 x 10- 8 m.
Example 7.5
Estimation of domain spacing in cobalt.
The magnetostatic energy per unit area of the surface is given as
Ems = 1.7
10- 7 M/d.
X
Suppose the other dimensions of the sample are x and y metres respectively, the
total magneto static energy will be
Ems = 1.7
X
10- 7 Ms2 xyd
The total number of walls N in a cuboid of sides x, y and d with domains
perpendicular to the xy plane and lying in the ly plane, is given by
Total wall energy is then
Ewall = Nlyy
lyxy
d'
The total energy is the sum of wall energy and magneto static energy
Etot = (1.7
X
10- 7 M/d
+ IYld)xy
dEtot
-7
2
2
(id= [1.7 x 10 Ms -lyld ] xy.
At equilibrium
dEtot=O
dd
and so
1. 7 x 10 - 7 Ms 2 - ly Id2 = 0
The equilibrium spacing of the domains therefore does not depend on x and y,
and is given by,
d=
[
~
1.7 x 10
= 1.49
X
7 M/
Jl~
1O- 4 (W/ 2 •
Solutions 413
And if 1= 1 cm, (0.01 m) then
d= 1.49 x 1O- 5 m.
Example 8.2 Magnetostatic energy associated with a void.
The magneto static energy per unit volume is given by
Ems = -110 fHdM.
In this case we have the demagnetizing field for a sphere, Hd = - (1/3) M
(92n)110Ms r .
2 3
Ems =
The volume of the sphere is (4/3)nr 3 , so the magneto static energy is
Ems =
(92n) 110Ms r .
2 3
We assume that the reduction in magneto static energy caused when a domain
wall intersects the void is ilE ms = (n/9)l1 oM/r 3 .
For iron with a spherical void of r = 5 x 10- 8 m
ilE ms =
(~
) 110Ms 2 r3
= 15.87
x 10- 17 joules.
The reduction in wall area is nr2
ilA=7.8 x 1O- 15 m2
and hence the reduction in wall energy is
ilEwall =(7.8 x 1O- 15 )y
where y is the wall energy per unit area
ilEwall = 1.57 x 10- 17 joules.
For a spherical void of radius 1O- 6 m
ilEms =
("9n) 110Ms r
2 3
= 1.27 X 10- 12 joules.
414 Solutions
The reduction in wall area is
AA = nr2
= 3.14 x 10- 12 m 2.
The reduction in wall energy is
AEwau=(2 x 10- 3)(3.12 x lO-12)m 2
= 6.28
x 10- 15 joules.
In both cases the reduction in magnetostatic energy is more significant.
Example 8.3 Reduction of domain wall energy by voids.
Let AEms be the reduction in magneto static energy when a domain wall
intersects a void of radius r.
AEms =
("9n) floMs
2 3
r .
If there are N such voids and the separation between the planar walls is d then each
domain wall will intersect nw voids
and since the wall separation is d, there will be lid walls per unit volume. Therefore
the total number of voids intersected by domain walls throughout the volume will
be n
N 2/3
n=-d-·
The total decrease in magneto static energy is therefore
AEms =
(~
) flo M/
r3 (
N~/3
).
The total energy associated with domain walls in this unit volume is
If the total reduction in magnetoelastic energy due to the voids is equal to the
increase in wall energy, we must have,
AEms = AEwall
Solutions 415
which gives
Example 8.4 Effect of stress on anhysteretic susceptibility.
If a 180 wall is moved through a distance dx the change in field energy will be
0
dE H = - 2JloMsH dx.
The change in magnetoelastic energy will be
dEme = ~Asd(J
=
3A saxdx.
At equilibrium we must have dETot/dx = 0
dEtot = dE H + dEme = 0
dx
dx
= -
dx
2JloMsH + 3Asax.
Consequently
The magnetization will be
M= 2AMsx
where A is the wall area
M = 4AJloMs2H
3A sa
and the initial susceptibility will therefore be
dM
dH
4AJloMs2
3As a
In determining the anhysteretic we will use the Frohlich-Kennelly equation
modified for the inclusion of the effects of stress. The original unstressed
anhysteretic magnetization can be modelled using the equation
IY.H
Man = 1 + /3H·
Since we know that Ms = 0.9 X 106 = IY.//3, and that the limit as H tends to zero of
dMan/dH = 1000 is equal to IY., we can determine the two coefficients IY., /3
IY.
= 1000
/3= 1.1 X 10- 3•
Now turning to the anhysteretic magnetization under stress we replace H by
416 Solutions
the effective field, giving
If we make the approximation A. = bM2 , then
dA.
dM=2bM
=
M
an
dMan
dB
a(B + 3baMan)
[1 + f3(B + 3baMan)]
a - f3Man
[1 + f3B - 3aba + 6f3baManJ'
so that in the limit as B approaches zero and Man approaches zero
dMan
dB
a
1- 3aba'
When a =0,
dMan = (0) = 1000.
dH
Xan
When a = - 20 MPa
dMan
dB
1000
1 +0.108
Xan(a) = 902.
Therefore the compressive stress reduces the susceptibility along the direction of
the stress of a ferromagnet with positive magnetostriction coefficient provided
dA./dM is positive.
Example 9.1
Paramagnetic susceptibility of oxygen.
Since the substance is a gas we can assume no interactions between the electrons
and neighbouring atoms, and hence no exchange interaction. According to the
classical Langevin theory of paramagnetism the magnetization M is given by
/lomB)
kBT
M=Nm [ coth ( - --] .
kBT
/lomB
Solutions 417
At 0 DC or 273 K we know that for the realistic values of H
J.lomH«kBT
and hence that
Consequently
NJ.lom2
X = 3kB T
(4n x 10- 7 )(2.69 x 10 25 )(2.78 x 10- 23 )2
X=
(3)(273)(1.38 x 10- 23 )
=
2.31 x 10 - 6 (dimensionless).
Example 9.2 Magnetic mean interaction field for iron.
In the Curie-Weiss law we have
M
X= H+aM
C
T- Tc
Tc=aC
and using the Langevin equation the Curie constant is given by
NJ.lom2
C=
3kB
T = aNJ.lom2
c
3kB '
So the value of a is given by
a
3kB Tc
NJ.lom2
a = (8.57 x
(3)(1.38 x 10- 23)(1043)
x 10- 7 )(2.2)2(9.27 x 10- 23 )2
10 28 )(4n
a=964
and since He = aM.
He = 1.64
X
109 A/m.
418
Solutions
Example 9.3 Critical behaviour of spontaneous magnetization.
The classical Weiss-type ferromagnet for a system with two possible microstates
leads to the following expression for magnetization.
M -- Mo tan h (JlOmrxMs) .
kBT
Close to the Curie point T;:::: Te , and so
JlomrxMs «.
1
kBT
The series expansion for tanh (x) is
x3
tanh (x) = x- 3
+ ....
Therefore using the first few terms of this series as the value of tanh (x) when x is
small
M = N [JlOmrxMs _ !(Jlo mrxMs)3 ]
s
m
3
kBT
kBT
.
For the tanh (x) expression the Curie temperature is given by
T =NJlom2rx
kB
e
Substituting these values into the above equation leads to,
Therefore
2
1 = (Te) _ (Te)M (Jlomrx
T
3T
s
2
kBT
T= T _ (Te)M (Jlomrx
e
3
s
kBT
)2
)2
Example 1004 Orbital and spin angular momentum ofan electron. (a) The principal
Solutions 419
quantum number n determines the shell of the electron, and hence its energy. Its
allowed values are n = 1,2,3, ....
The orbital angular momentum 1defines the orbital angular momentum Po of
the electron when multiplied by (h/2n). It can have values of 1= 0,1,2,3, ... , (n - 1),
where n is the principal quantum number.
The magnetic quantum number m1 gives the component of the orbital angular
momentum 1along the z-axis when a magnetic field is applied along that axis. Its
values are restricted to m1 = -l, ... , - 2, - 1,0,1,2, ... , + l.
The spin quantum number s defines the spin angular momentum of an electron.
The value of the spin quantum number is always 1/2 for an electron. The angular
momentum due to spin sh/2n is therefore always an integer mUltiple of h/4n. The
component or resultant of s in a magnetic field is represented by ms and is restricted
to ms = ± 1/2.
The various angular momenta of an electron can be calculated from the two
quantum numbers land s
Po = l(h/2n)
Ps = s(h/2n).
The total angular momentum j can then be calculated from the vector sum of 1
and s, remembering that for a single electron the magnitude of j must be half
integral.
Pj = j(h/2n).
The values of l, sand j differ from the classically expected values in practice and
this can be explained by quantum mechanics. The solution of the Schroedinger
equation in the simple case of a single electron orbiting a nucleus only permits
= y'[l(l + 1)]. A similar
solution (states) for which the angular momentum is
argument holds for s and hence j.
<1>
(b) The allowed solutions for the resultant orbital angular momentum L of the
vector sum of two orbital angular momentum vectors of length 2 and 3 are
L = 5,4, 3, 2, 1
The vector diagrams are shown in the accompanying figure.
L
5
4
3
2
(c) The Co2+ ion in a 3d 7 state has its electrons in the following state, 1= 2 for 3d
420 Solutions
electrons
occupancy
m1 2 1 0
ms 21 21 21
s i i i
-1
-2
2
1
1
1
1 0 1 2
1
2
2
-2
-2
i
i
!
!
So by summing the spins S = L ms = ~ and summing the components of
orbital angular momentum L = L ml = 3. The shell is more than half full so
J = IL + S I = 9/2.
Example 10.5 The Zeeman effect. The cadmium singlet observed at A= 643.8 nm
is a result of an electronic transition from 61D2 (L = 2, S = 0) to 51 P 1 (L = 1, S = 0).
On application of a magnetic field this will exhibit the normal Zeeman effect since
the net spin of the state is zero.
The shift in energy upon application of a magnetic field H is
!lE=(~)f.-OH
4nme
=f.-lBf.-lO H
= 5.8
x 10- 5 eV/T
=
0.9273 x 10- 23 l/T
=
1.165 x 1O- 29 1(A/m)-1
The equation for the energy shift with field is
flE = 1.165 x 10- 29 H;l
=
0.9273 x 10- 23 f.-loH 1.
=
0.464 x 10- 23 1.
When f.-loH = 0.5 T, this gives
flE
When f.-loH = 1.0 T
flE = 0.927 x 10- 23 1.
When f.-loH = 2.0 T
flE = 1.85 x 10- 23 1.
The shift in frequency can be calculated from the relation
flE = h(v - vo)
flv = flE
h
Solutions 421
=
.
23
(0.9273 x 10- ) S-1 T- 1
(6.626 x 10 34)
= 1.399 X 10 10s- 1 T- 1
and using the relation
AV =c,
where c is the speed of light
Llv
LlA
c
- ,12
,12
LlA = --Llv.
c
Substituting in the values for the shift in frequency at an induction of 1 T
LlA = -
(643.8 x 10- 9 )2(1.399 x 1010)
2.98 X 10 8
= - 1.946 X 10- 11 m.
At 1 T
LlA = - 0.01946 nm.
At 0.5T
LlA = - 0.009 730 nm.
At 2.0T
LlA = - 0.038 92 nm.
Example 10.6 Determination ofatomic angular momentum. The values of J can be
obtained by vector addition of Land S, remembering that if S is an integer then J is
an integer and if S is half integer then J is half integer.
(a) L = 2, S = 3. Possible values of J are 5,4,3,2,1.
(b) L = 3, S = 2. Possible values of J are 5, 4, 3, 2, 1.
(c) L = 3, S = 5/2.Possible values of J span the range from IL + SI to IL - SI, that is
11/2, 9/2, 7/2, 5/2, 3/2, 1/2.
(d) L = 2, S = 5/2. Possible values of J are 9/2, 7/2, 5/2, 3/2, 1/2.
The ground state of the carbon atom can be determined from Hund's rules.
Carbon has four electrons in the n = 2 shell. The s subshell is full with two electrons,
the p subshell, which has I = 1 has two electrons. The occupancy of available states
must be such as to maximize S by Hund's rules.
occupancy
ml 1 0
ms t t
s i i
-1
1
o
-1
422 Solutions
Therefore S = L ms = 1 and L = Lml = 1 and since the subshell is less than half
full J = IL- SI
J=O.
Example 11.1 The exchange interaction. The exchange interaction J arises from
the determination of the total energy of a two electron system in which the
electrons can never have the same set of quantum numbers by virtue of the Pauli
exclusion principle. (This means that the two electron wavefunction must be
anti symmetric).
Under these circumstances there arises an energy term which results from the
electrons 'exchanging' places. Classically this would not alter the energy of the
system (since the particles could occupy the same states in classical physics) but in
quantum mechanics the situation is different and the electrons therefore have an
additional energy which acts in many respects like a strong magnetic field.
Example 11.2 Magnetic moment of dysprosium ions. The dysprosium ion D y3 +
has 9 electrons in its 4f shell. For this shell n = 4 and 1=3 S0 the electron
distribution is
mel 3 2 1 0
s
ms
-2 -3
-1
2
3
i i i i
i
i
i
1
1
2
2
2
2
-2
-2
1
1
2
1
2
1
1
2
1
1
1
1 0
-1
-2 -3
1
Summing these leads to S = Lms = 5/2 and L= Lmi = 5. Since the shell is more
than halffull J = IL + S I = 15/2. To calculate the susceptibility of a salt containing
1 g mole of Dy 3 + at 4 K first calculate the magnetic moment per ion
m= -gP,BJ
= -
gP,BJ[J(J + 1)]
_ 1
g-
+
J(J + 1) + S(S + 1) - L(L + 1)
2J(J+l)
and in this case, when the values of L, Sand J are substituted in we obtain g = 4/3.
m= -1(9.27 x 1O- 24
= -
and
M
x=H
9.87
X
)J!¥
10- 23 Am 2
Solutions 423
(6.02 x 1023 )(4n x 10- 7 )(9.87 x 10- 23 )2
3(1.38 x 10 23)4
=4.45 x 10- 5 .
Example 11.3 Paramagnetism of S = 1 system. The magnetization can be expressed as a function of temperature and magnetic field using the Brillouin function
Blx) where the argument of the function x is JlomH/kBT
JlomH)
M=NmBJ ( kBT '
where N is the number of atoms per unit volume and m is the magnetic moment per
atom.
For a system with spin only we have g = 2 and J = S. In this case therefore J = 1.
Substituting these values into the Brillouin function
Blx) =
e
2J2;
= tcoth
1)Jcoth[X(2~+
1)J -
(2~)/
coth(~)
C;) - (I)
tcoth
and coth (x) = 1/x + x/3 for small x. If we make this approximation then, in this
case
So if JlomH/kBT« 1 we can make this approximation and consequently
BJ(JlomH) = 2JlomH
kBT
3kBT
and the magnetization M is given by
JlomH)
M=NmBJ ( kBT
Nm2JlomH
3kB T
2NJlom2H
3k B T .
Author Index
Abbundi, R., 56
Abrikosov, A. A., 350
Adey, R. A., 24
Allison, S. G., 376
Altpeter, I., 368
Ampere, A.M. (1785-1836), 4,108,109
Argyle, B., 116
Artman, 1. 0., 116
Asano, T., 359
Ashburn, 1. R., 348, 358
Ashtashenko, P. P., 372
Astie, B., 159, 161, 162
Atherton, D. 1., 158, 165, 169, 370, 373,
382, 388, 392
Back, E., 244
Bacon, D. 1., 149
Bacon, G. E., 197, 198,201,202
Bader, C. J., 58-59
Bardeen, J., 356
Barkhausen, H., 97, 113, 366
Barnier, Y., 134
Barton, 1. R., 385
Baruchel, J., 117, 137
Bashkirov, Y. P., 372
Bate, G., 77
Becker, 1. 1.,79,309,311,316-317
Becker, R., 94, 96, 138, 149
Bednorz, 1. G., 348, 358
Behrendt, D. R., 210
Beissner, R. E., 98, 369, 370, 385
Bernards,1. P. c., 78, 325
Bertram, H. N., 334, 342, 343
Bertraut, F., 289
Bethe, H., 253
Betz, C. E., 114,380
Bezer, H. 1., 380
Bida, G. v., 374
Biot, 1. B., 2
Bitter, F., 114
Bixio, A., 310
Blackie, G. N., 212
Bloch, F., 128, 199, 211
Bly, P. H., 207
Bobrov, E. S., 310
Bohr, N. (1881-1964),221
Boll, R., 270
Bonnebat, c., 329
Bose, M. S. c., 373
Bozorth, R. M., 96, 114, 212, 213
Bradbury, A., 344
Bradley, F. N., 309
Brebbia, C. A., 24, 25
Bridges, 1. M., 387, 388
Briggs, G. A. D., 98
Brillouin, 1., 256
Brown, G. V., 22
Brown, R. E., 48, 52
Brown, W. F., 31
Brudar, B., 388
Bull, S. A., 325
Burkhardt, G. 1., 171, 370, 373
Buschow, K. H. 1., 311
Bussiere, 1. F., 373
Buttle, D. 1., 98
Cable,1. W., 207, 208
Camras, M., 324, 327, 328
Carey, R., 115
Carlson, W. 1., 324
Carpenter, G. W., 324
Carpenter, S. H., 370
426
Author Index
Carrigan, R. A., 29
Casimir, H. B. G., 345
Celiotta, R. 1., 117
Chandrasekar, B. S., 134
Chang, T. T., 73,162
Chantrell, R. W., 344
Chari, M. V. K., 24
Charles, S. W., 344
Chen, C. W., 72, 130, 134,380
Chen, Y. F., 275
Chikazumi, S., 134, 140, 158, 309
Child, H. R., 207
Chin, G. Y., 75, 290
Chu, C. W., 348, 358
Clark, A. E., 56, 57
Clark, D. L., 324
Clark, W. G., 390
Clegg, A. G., 319
Cockcroft, 1. D., 21
Coleman, R. V., 119
Comstock, R. L., 338
Cooper, L. N., 356
Corliss, L. M., 207
Corner, W. D., 207
Craik, D. 1., 57
Crangle, 1., 31, 200, 342
Cribier, D., 203, 204
Croat, 1. 1., 79, 318
Cullity, B. D., 28, 56, 132, 134, 276, 300,
303
Curie, P. (1859-1906), 182, 185
Davis, M. E., 360
de Bie, R. W., 78, 325
De Mott, E. G., 48
Debye, P., 84
Decker, S. K., 114
Deeds, W. E., 390
DeForest, A. v., 114,377
Degauque, 1., 159, 161, 162
Degterev, A. P., 368
Del Mut, G., 310
Della Torre, E., 343
Demerdash, N. A., 25
Dietze, H. D., 158
Dijkstra, L. 1., 154
Dirac, P. A. M., 29, 251, 358
Dobmann, G., 389
Doclo, R., 208
Dodd, C. V., 390
Doring, W., 94, 96, 149
Dovnar, D. P., 372
Drillat, A., 137
Dunn, F. W., 380
Duplex, P., 162, 163, 164
Durst, K. D., 309
Edwards, c., 370, 379, 388
Elmen, G. W., 269
English, A. T., 171, 291
Enz, u., 74, 196
Erdelyi, E. A., 24
Ewing, 1. A. (1855-1935) 70, 108, 109
Fabry, M. P. A. (1867-1945), 21
Faraday, M., (1791-1867) 115
Fastnacht, R. A., 360
Fateev, I. G., 373
Ferrari, R. L., 24
Fert, c., 50-51
Flax, L., 22
Fleischer, R. L., 29
Foner, S., 52,208
Forrer, R., 191, 294
Forret, F., 289
Forster, F., 378, 382; 390
Fort, D., 137
Freeman, A. 1., 252
Freese, R. P., 331
Fridman, L. A., 382
Frohlich, 0., 94, 247
Fuchs, E. F., 24, 291, 292, 293
Fujimura, S., 318, 319
Fukutomi, M., 359
Fuller, H. W., 116
Fussel, R. L., 58-59
Galt,1. K., 130
Gao, L., 348, 358
Garcia, N., 117
Garikepati, P., 73
Gautier, P., 50-51
Gerlach, W., 236
Gerstein, B. c., 210
Giacolletto, L. 1., 31
Giauque, W. F., 84
Author Index
Gilbert, w. (1540-1603), 73
Ginzburg, W. L., 350, 355
Globus, A., 162, 163, 164
Golovko, A. S., 374
Gordon, D. I., 48, 52
Gorkov, L. P., 350
Gorter, C. 1., 345
Goudsmit, S., 234
Graham, C. D., Jr., 31, 132, 134, 207,
301
Green, H. S., 215
Green, R. W., 210
Greenough, R. D., 213
Gregory, C. A., 379
Griffel, M., 210
Grigorev, P. A., 382
Grossinger, R., 300
Guyot, M., 164
Haben, 1. F., 48
Hadjipanayis, G. c., 309
Hale, M. E., 116
Hall, R. c., 295
Halpern, 0., 200
Hampton, P. L., 385
Harker, 1. M., 325
Harle, M. E., 169
Hart, H. R., 29
Hart, P. 1.,22
Hartman, F., 60
Hartmann, u., 116
Hastings, C. H., 381
Hastings, 1. M., 207
Hayakawa, K., 117
Heck, c., 69, 81, 278
Hegland, D. E., 210
Heisenberg, W., 251, 267
Heisz, S., 300
Heitier, W., 248, 249
Hembree, G., 117
Henry, W. E., 258
Herbst, 1. F., 79, 318
Herman, A. M., 359
Herman, D., 116
Herring, c., 254
Hertzberg, G., 58-59, 232
Heyman, 1. S., 376
Higgins, F. P., 370
427
Hilscher, G., 300
Hilzinger, H. R., 73, 157, 158, 159
Hinton, E., 24
Hirosawa, S. 318
Hoffer, G., 111,316-317
Hofman, 1. A., 132, 134,210,252
Hoke, 114
Holler, P., 389
Holmes, L. c., 324
Holmes, V. L., 379
Honda, K., 204
Hoole, S. R. H., 25
Hopkins, N., 344
Hopkins, R. A., 31
Hor, P. H., 348, 358
Hoselitz, K., 140, 149
Hougen, D. R., 162
Huang, Z. 1., 348, 358
Hubert, A., 115
Hull, D., 149
Hummel, R., 300
Hund, F., 239
Hurst, C. A., 215
Hwang, c., 116
Hwang, 1. H., 24, 387
Ingersoll, L. P., 115
Irons, H. R., 58-59
Isaac, E. E., 115
Isci, c., 212, 213
Isin, A., 119
Ising, E., 110, 214
Islam, M. N., 212
Itozaki, H., 360
Iwasaki, S., 329
Jacobs, I. S., 29
Jacrot, B., 203, 204
Jakubovics, J. P., 98, 270
Jeanniot, D., 325
Jennings, L. D., 210
Jiles, D. c., 73, 158, 162, 165, 169, 170,
171, 212, 370, 373
Jin, S., 360
Johnson, M. H., 200
Jones, G. A., 115
Josephson, B. D., 361
Joule, J. P. (1818-89), 99
428
Author Index
Junker, W. R, 390
Kadar, G., 343
KamerIingh annes, H., 345
Kammlott, G. W., 360
Karjalainen, L. P., 368
KarIqvist
338
Kastin, V. N., 374
Kaverin, V. D., 372
Kaya, A., 204
Keith, H. D., 360
Kennelly, A. E., 94
Kerr, 1., 115
Kersten, M., 138, 139, 141, 149, 154, 158
Khalileev, P. A., 382
Kimura, H., 98, 370
King, 1. D., 370
Kittel, C, 101, 130, 131, 134, 138,242
Koehler, W. C, 197, 199, 207, 208
Koike, K., 117
Kojima, H., 316-317
Komura, I., 374
Kondorsky, E., 138, 149
Konovalov, a. S., 374
Kronmuller, H., 72, 157, 158, 159, 309
Kusanagi, H., 98, 370
Kuznetsov, I. A., 372
Kwun, H., 171, 373, 375, 376
a.,
Labusch, R, 158
Lamont, 96
Landau, L. D., 118, 350, 355
Landry, P. C, 207
Lang, A. R, 117
Langevin, P. (1872-1946),110,178,182,
183, 258
Langman, R., 372, 374
Lankford, 1., 385
Larmor, 1. (1857-1942), 223
Laughlin, D. E., 116
Layadi, A., 116
Lean, M.H., 25
Lee, E. W., 102, 103, 212
Lee, R W., 79, 318
Legvold, S., 210
Leslie, W. C, 171
Libby, H. L., 390
Lifschitz, E. M., 118
Lilley, B. A., 131, 134
Littmann, M. F., 274
Lodder, 1. C, 116
Lomaev, G. V., 368
London, F., 248, 249, 354
London, H., 354
Lord, A. E., 98, 370
Lord, W., 24, 387, 388
Lorentz, H. A. (1853-1928), 234
Lovesey, S. w., 197, 204
Luborsky, F. E., 76, 284, 311
Luitjens, S. B., 78, 325
Luthi, B., 212
McCaig, M., 79, 309, 311, 319
McClure, 1. C, 97, 369
McCurrie, R A., 312
McNiff, E. 1., 208
Maeda, M., 359
Maekawa, S., 212
Mallinson, 1. C., 324, 327
Malyshev, V. S., 368
Marabotto, R, 319
Marshall, w., 252, 253
Martin, D. H., 247
Martin, D. L., 311
Martin, Y, 117
Massa, G. M., 379
Matsuura, Y, 318, 319
Matsuyama, H., 117
Matzkanin, G. A., 98, 369, 370, 385
Maxwell, E., 356
Mayer, L., 117
Mayos, M., 368
Mee, CD., 94
Meissner, W., 349, 352
Meng, R L., 348, 358
Mikheev, M. N., 371, 374
Miller, R. E., 210
Mitchell, P. V., 116
Moilanen, M., 368
Molfino, P., 310
Montgomery, D. B., 20
Moran, T. 1., 212
Morozov, A. P., 371
Morozova, V. M., 371
Morrish, A. H., 242
Moskowitz, L. R, 79, 300
Author Index
Mountfield, K., 116
Muller, A., 348, 358
Namkung, M., 376
Neel, L., 136, 138, 141, 154, 194, 195
Nehl, T. W., 25
Neizvestnov, B. M., 371
Nesbitt, E. A., 300, 316-317
Nicklow, R. M., 196
Nicolas, J., 289
Nigh, H. E., 210
Nolan, P., 379
Novikov, V. F., 373
O'Grady, K., 344
Ochsenfeld, R., 349, 352
Oehl, C. L., 380
Oersted, H. c., (1777-1851) 1, 6
Olson, J., 79, 316-317
Ono, K., 98, 370
Orowan, E., 149
Osborne, J. A., 38
Ostertag, W., 79, 316-317
Ouchi, K., 329
Owen, D. R. J., 24
Owston, C. N., 384
Palanisamy, S., 387, 388
Palmer, S. B., 117, 137, 212, 213, 370,
379,388
Parette, G., 203, 204
Parker, R. J., 311, 313
Paschen, F., (1865-1947) 244
Paskin, A., 132, 210, 245, 252
Patterson, c., 137
Pauli, W., 182,251,262,263
Pauling, L., 265, 266
Pauthenet, R., 134
Pierce, D. T., 117
Pinkerton, F. E., 79, 309, 318
Pippard, A. B., 355
Poisson, S. D., (1781-1840), 107
Polanyi, M., 149
Poicarova, M., 117
Pollina, R. J., 212
Porteseil, J. L., 159, 161, 162
Potapova, N. A., 372
Poulsen, V., 324
Preis, K., 43
Preisach, F., 165, 343
Price, P. B., 29
Primdahl, H., 52
Puchalska, I., 115
Punchard, W. F. B., 310
Putignani, M., 368
Raine, G. A., 379
Ranjan, R., 162, 370
Rastogi, P. K., 370
Rautioaho, R., 368
Rave, W., 115
Rayleigh, L. (1842-1919), 95
Read, R. E., 196
Repetto, M., 310
Rhyne, J. J., 208
Richter, K. R., 43
Rieger, H., 157
Rimet, G., 134
Robinson, A. N., 379
Rodigin, N. M., 372
Roe, W. c., 207
Roehrs, R. J., 379
Roitman, V. I., 374
Rubinstein, H., 116
Russell, H. N., 240
Sablik, M. J., 171
Saenz, J. J., 117
Sagawa, M., 318, 319
Saito, y., 44
Sasaki, H., 98, 370
Sassik, H., 300
Sato, H., 134
Saunders, F. A., 240
Savage, H. T., 57
Savart, F., 3-4
Schafer, R., 115
Schcherbinin, V. E., 372, 385
Schlenker, M., 117
Schmidt, T. R., 391
Schneider, E., 368
Schrauwen, C. P. G., 78, 325
SchrieITer, J. R., 356
Schroeder, K., 97, 369
Schwall, R. E., 363
Schwarz, W. M., 29
429
430
Author Index
Schwee, H. R., 58-59
Scott, c., 119
Scruby, C. B., 98
Seeger, A., 157
Segalini, S., 368
Seiden, P. E., 58-59
Sells, R. L., 233
Shah, M. D., 373
Sheng, Z. Z., 359
Sherwin, C. W., 228
Sherwood, R. c., 360
Shibata, M., 98, 370
Shiozaki, M., 76
Shockley, W., 114
Shull, C. G., 199, 207
Silsbee, F. B., 352
Silvester, P. P., 24
Simkin, J., 25
Sixtus, K. J., 149
Skochdopole, R. E., 210
Slater, J. c., 253, 264, 265
Slick, P. I., 289
Smirnova, R. M., 372
Smith, D.O., 50-51
Snyder, J. E., 116
Sommerfeld, A., 222
Somova, V. M., 372
Spedding, F. H., 210
Squires, G. L., 203
Stablein, H., 316-317
Stanley, H. E., 215
Stanton, R. M., 210
Steinmetz, c., 271
Stephenson, E. T., 271
Stern, 0., 236
Stevens, D. W., 171
Stoffers, R. c., 291
Stogner, H., 43
Stoner, E. c., 165,264,305,344
Strandburg, D. L., 210
Strnat, K., 79, 316-317, 319
Stuart, R., 252, 253
Studders, R. J., 311, 313
Stumm, w., 384
Sullivan, S., 392
Sundstrom, 0., 367
Surin, G. V., 371
Suzuki, M., 374
Swartzendruber, L. J., 380
Swisher, J. H., 291, 292, 293
Syrochkin, V. P., 372
Szpunar, B., 373
Szpunar, J. A., 373
Tachiki, M., 212
Takahashi, H., 374
Tanaka, S., 360
Tanaka, Y., 359
Tanner, B. K., 207, 373
Tauer, K. J., 132, 210, 252
Taylor, G. I., 149
Taylor, K. N. R., 207
Tebble, R. S., 57, 367
Teller, C. M., 98, 393, 376, 385
Theiner, W. A., 368, 370
Thoelke, J. B., 170
Thomson, J. J., 158
Tiefel, T. H., 360
Tiitto, S., 367
Todokoro, H., 117
Togawa, N., 318
Tonks, L., 149
Torng, C. J., 348, 358
Torronen, K., 367
Trauble, H., 157
Trowbridge, C. W., 24
Trower, W. P., 29
Tsang, c., 114
Tsarkova, T. P., 374
Uhlenbeck, G. E., 234
Unguris, J., 117
Utrata, D., 376
Van Dover, R. B., 360
Van Vleck, J. H., 182
Van Wingerden, D. J., 309
Vergne, R., 159, 161, 162
Vicena, F., 157
Vigoureux, P., 31
Vlasov, V. V., 372
Vroman, J., 390
Wakabayaski, N., 196
Wakiyama, T., 213
Walker, S., 25
Author Index
Wang, Y. Q., 348, 358
Warburg, E., 70
Watson, R. E., 252
Webb, W., 64
Weber, W., (1804-1890) 107, 109
Weidner, R., 233
Weiss, P. (1865-1940), 94, 109, 185, 186,
191, 192, 193, 251, 252, 267, 294
Weiss, R. 1., 132, 197, 207, 210, 354
Wernick, 1. H., 75, 290, 300, 316-317
Wert, c., 154
Wexler, A., 25
White, R. M., 331
Wickramsinghe, H. K, 117
Wiesinger, G., 300
Wilkinson, M. K, 196, 207, 208
Willcock, S. N. M., 373
Willems, H. H., 370
431
Williams, H. 1., 114
Williams, M. L., 338
Winslow, A. M., 24
Wohlfarth, E. P., 69, 165,252,305, 344,
357
Wollan, E. 0., 199,207,208
Woods, R. T., 29
Wu, M. K, 348, 358
Wun Fogle, M., 57
Yamamoto, H., 318, 319
Yen, W., 387, 388
Zakharova, G. N., 371
Zatsepin, N. N., 372, 385
Zeeman, P. (1865-1943),58-59,232
Zieren, V., 78, 325
Zijlstra, H., 301
Subject Index
a.c. applications, materials for, 272
a.c. bias recording, 341
a.c. losses in transformers, 270
acoustic velocity, field dependence in
ferromagnets, 374
adiabatic demagnetization of
paramagnets, 84
alignment of magnetic moments, 113
alnico, 312
aluminum-iron alloys, 277
amorphous metals, 76, 284
Ampere's circuital law, 4
Ampere's hypothesis, 108
analytical balance, 62
angular momentum of electrons, 222,
223, 225
wave mechanical corrections to,
228
anhysteretic magnetization, 92
measurement of, 94
anisotropic materials, 102
anisotropy, 92, 122
constants, 124
cubic, 123
and domain rotation, 122, 305
energy of domain wall, 128
hexagonal, 122
antiferromagnetism, 136, 194
atomic force microscopy, 115
atomic magnetic moment, 70, 107,237
atomic orbital angular momentum, 238
atomic spin angular momentum, 239
atomic total angular momentum, 240
band theory of ferromagnetism, 263
band theory of magnetism, 261
band theory of paramagnetism, 262
Bardeen-Cooper-Schrieffer theory, 356
Barkhausen effect, 97, 113, 164, 366
for detection of stress, 366
for evaluation of microstructure, 366
BCS theory, 356
bending of domain walls, 141
Bethe-Slater curve, 253
Biot-Savart law, 2
bismuth-strontium-ca1cium-copper
oxide, 347
Bitter patterns, 114
Bloch walls, 128
boundary element technique, 23
Brillouin function, 256
bubble domain devices, 329
ceramic magnets, 81
chromium dioxide, 327
tapes, 77
circuits, 303
classical theory of diamagnetism, 179
classical theory of ferromagnetism, 189
classical theory of paramagnetism, 182
closure domains, 135
cobalt-chromium recording media, 325
cobalt-iron alloys, 296
cobalt-platinum, 415
cobalt-samarium, 318
cobalt surface-modified gamma iron
oxide, 327
coercimeter, 370, 373
coercivity, 72, 91, 269, 300
coherence length, 354
core losses, 76
Cotton-Mouton effect, 57
critical behaviour at the ordering
temperature, 209
434
Subject Index
critical current density, 352
critical field of superconductor, 348
critical magnetic field
under strong pinning, 154
under weak pinning, 155
critical temperature of superconductor,
346
cubic anisotropy, 123
Curie temperature, 73
and the mean field interaction, 193
Curie's law, 82, 182, 258
Curie-Weiss law, 185,259
d.c. applications of soft magnetic
materials, 290
demagnetization curve, of permanent
magnet, 302
demagnetizing factors, 38
demagnetizing fields, 36, 38
density of magnetic recording, 340
diamagnetism, theory of, 177
diamagnets, 33, 81, 85
susceptibility of, 33
Dipole, model for leakage field
calculation, 385
discontinuous magnetization, 97, 164
disks, magnetic recording, 325
domain
patterns and energy minimization,
118
processes, reversible and irreversible,
147
rotation, 120, 147
and anisotropy, 122
wall motion, 120, 137, 366
domain wall
bowing of, 141
defect interactions, 159
forces on, 138
motion
and the Barkhausen effect, 164, 366
and magnetoacoustic emission, 369
and magnetostriction, 165
pinning
by inclusions, 152
by strains, 149
planar displacement of, l39
strong pinning, 154
surface energy, l38
weak pinning, 155
domain walls, 127
180°, l34
anisotropy and exchange energies, l31
effects of stress on, l35
effects of weak fields on, 137
energy, 128
energy balance in, l37
Nee! walls, l36
non-180°, l34
thickness, 128, l30
domains, 107
and the magnetizing process, 120
nucleation, 119
observation of, 1l3, 115
single, 118
Weiss domain theory, 109
eddy current inspection methods
applications of, 390
for magnetic materials, 389
elastic constant anomalies at critical
temperatures, 212
electric motors, 272
electrical losses, 76
in transformers, 270
electromagnetic induction, 9
electromagnetic inspection, remote field
monitoring, 391
electromagnetic relays, 77
electromagnets, 74, 271
electron-electron interactions, 247
electron band theory of magnetism, 261
Electron microscopy
SEM,117
TEM,116
Electron spin
and exchange energy, 251
resonance, 79
electron states, occupancy according to
Hund's rules, 239
electronic energy levels, splitting by
magnetic field, 232
electronic magnetic moment, quantum
theory of, 221
electronic magnetic moments, 219
electronic orbital magnetic moment, 219
Subject Index
electronic spin magnetic moment, 220
electronic total magnetic moment, 221
energy gap in superconductors, 357
energy loss through wall pinning, 166
energy minimization and domain
structure, 118
energy product, 111, 301
typical values for permanent magnets,
78
energy states of magnetic moment
configurations, 112
exchange energy, 128
dependence on interatomic spacing,
253
and electron spin, 251
between electrons in filled shells, 252
values for various solids, 252
exchange integral, 249
exchange interaction, 249
excited states and spin waves, 208
Faraday effect, 57, 116
Faraday's law of induction, 9, 47
fatigue, effects of, on magnetic properties,
370
ferrimagnetism, 195
ferrites, 80, 81, 288, 315
hard, 81, 315
soft, 80, 288
ferrofluids, 114
ferromagnetism, 188
ferromagnets, 33, 69, 108
applications of, 74
ferroprobes (fluxgates), 374
field lines, 37
finite element techniques, 23
flaw detection using magnetic methods,
377
flux
coil
moving, 48
stationary, 48
leakage, 380
application of flux leakage method
to NDE, 380
lines, 36
pinning, in type II superconductor, 350
quantization, 64
435
quantum, 349
rate of change of, 47
trapping by toroid of superconductors,
353
fluxgate magnetometer, 52
fluxgates (ferro probes), 373
fluxmeter, 48
force on current-carrying conductor in a
field H, 8
free atoms, 237
Frohlich-Kennelly equation, 93
gamma iron oxide, 77, 324, 327
generators, 272
Ginzburg-London theory of
superconductivity, 355
Globus-Guyot model, 162
Hall coefficient, 53
Hall effect, 53
magnetometers, 53
hard magnetic materials, 74, 299
Heisenberg model of ferromagnetism, 251
Heitler-London approximation, 249
helimagnetism, 196
Helmholz coils, 16
hexagonal anisotropy, 122
hexagonal ferrites, 328
high-frequency applications, 288
Hund's rules, 239
hysteresis, 70, 89
coefficients and magnetic properties,
170
effects of microstructure and
deformation on, 171
loss, 91
macroscopic mean field theory of, 165
parameters, 90
stress dependence of, 370
hysteresis graphs, 48
inclusion theory of domain-wall pinning,
152
inductance cores, 80
induction coil methods, 47
initial permeability, 90
initial susceptibility
in the planar wall approximation, 141
436
Subject Index
in the wall bending approximation,
143
intensity of magnetization, 30
interatomic spacing, effect on exchange
energy, 253
iron-aluminum alloys, 277
iron-cobalt alloys, 296
iron and low-carbon steels, 290
iron-neodymium-boron, 319
iron-nickel alloys, 278, 292
iron oxide, 327
iron-silicon alloys, 272
irreversible magnetization changes, 166
Ising model, 161, 214
isotope effect, 356
itinerant electron model, 261
criticism of, 266
J-J coupling, 242
Josephson junction, 64
devices, 361
Kerr effect, 57, 115
Kundt's constant, 57
Langevin function, 84
Langevin theory
of diamagnetism, 178
of paramagnetism, 182
Langevin-Weiss theory, critique of, 187
laser magneto-optic microscope, 115
Law of approach to saturation, 96
Leakage fields, 377
calculation and modelling of, 385
finite element calculation of, 387
Lenz's law of induction, 9
localized atomic moments, 254
localized electron model, criticism of,
261
localized electron theory, 254
localized theory of electronic magnetic
moments, 254
London equations, 354
Lorentz microscopy, 116
magnetic bubble domain devices, 329
magnetic circuits, 36, 40, 303
magnetic dipole, 10
magnetic field
of circular coil, 13
definition, 1, 2
due to long conductor, 3
generation, 1
of long thin solenoid, 12
in magnetic materials, 43
numerical methods for calculation of, 23
patterns around conductor, 4
of a short thick solenoid, 19
of short thin solenoid, 19
of two coaxial coils, 16
magnetic flux leakage, 380
applications of, 384
instruments for automation of, 384
models for, 385
magnetic hysteresis, 370
applications in NDE, 371
magnetic induction, 6
definition of tesla, 7
lines of, 9
magnetic moment, 27, 107,219
of closed shell of electrons, 237
of electron, 219
due to orbital angular
momentum, 219
due to spin angular momentum, 220
total, 221
magnetic order, 108,204
magnetic properties
of free atoms, 237
microscopic, typical values of, 134
magnetic quantum numbers, 223
magnetic recording
history, 324
materials, 77
media, 323
materials for, 327
tapes, 324
magnetic particle inspection, 377
applications of, 379
'dry' method, 378
fluorescent particles, 379
optimum conditions for, 379
'wet' method, 379
magnetic resonance imaging, 363
magnetic structure, 197, 204
magnetic units, 11
Subject Index
magnetite (lodestone), 311
magnetization, 29
in materials with few defects, 162
microstructural effects, 157
relation to Band H, 30
relation to magnetic moment, 10
saturation, 31, 71
strain effects, 157
technical saturation, 101, 121
magneto-acoustic emission, 98, 369
stress dependence of, 369
magnetoelastic method for NDE, 374
magnetographic method of leakage flux
detection, 377
magnetometers, 47
magnetomotive force, 56
magneto-optic recording devices, 331
magnetoresistors, 56
magneto static energy, 40
magnetostriction, 98
at an angle 0 to the magnetic field, 101
field induced, 103
forced,101
polycrystalline, 103
saturation, 100
single-crystal magnetostriction
constant, cubic, 102
spontaneous, 99
in terms of single crystal
magneto stricti on constants, 102
transverse, 104
magnetostrictive devices, 56
maximum energy product, 78
mean field, 109, 190
Meissner effect, 352
metallic glasses, 284
metglas, 284
modelling of magnetization curves based
on pinning, 156
molecular field, 109
motors, 272
moving coil, 48
galvanometer, 48
MRI,363
mumetal,281
NDE, magnetic methods for, 365
nearest neighbour interactions, 192
437
Neel temperature, 195
Neel walls, 136
neodymium-iron-boron, 78, 319, 363
Neutron diffraction, 197
antiferromagnetic scattering, 201
Bragg diffraction peaks, 199
elastic scattering, 199
ferromagnetic scattering, 201
inelastic scattering, 203
magnetic diffraction peaks, 201
paramagnetic scattering, 201
topography, 115
nickel-iron alloys, 278, 292
ninety degree (90°) domain-wall motion,
369
niobium-tin, 359, 361
niobium-titanium, 361
niobium-zirconium, 361
non-integral atomic magnetic moments,
264
nuclear magnetic resonance (NMR) (see
magnetic resonance imaging), 60,
363
numerical methods for calculation of
magnetic fields, 123
orbital magnetic moment
of atom, 238
of electron, 219
orbital momentum quantum number, 222
orbital wave functions, two electron
system, Heitler-London model, 249
order in rare earth solids, 205
ordered magnetism, theories of, 188
ordering temperature, 73, 209
paramagnetism, 181
classical (Langevin) theory, 182
classical (Weiss) theory, 186
of 'free' electrons, Pauli theory, 262
theory of, 177, 182, 186
paramagnets, 33, 81
applications of, 84
field dependence of susceptibility, 83
properties of, 81
susceptibility of, 33
temperature dependence of
susceptibility, 82
438
Subject Index
Paschen-Back effect, 244
Pauli paramagnetism, 262
permalloy, 278, 292
permanent magnet steels, 311
permanent magnets, 79
applications, 309
materials, 311
properties, 299
stability, 310
permeability, 8, 32, 69, 108
differential, 32, 73
initial, 90
relative, 32
permeance coefficient, 303
permendur, 296
perpendicular recording media, 77, 329
pinning models, 156
pinning of domain walls
critical field, 154, 155
by inclusions, 152
by strains, 149
platinum-cobalt, 415
pole strength, 10
potential approximation for domain-wall
motion, 139
Preisach model, 343
use in magnetic recording industry, 343
principal quantum number, 221
proton precession magnetometers, 60
quenching of orbital angular momentum,
242
quantization
of angular momentum, 225
of electron spin, 232
of flux, 350
quantum mechanical exchange
interaction, 249
quantum number
1,222
mb 223
m., 225
samarium-cobalt, 78, 318
saturable coil magnetometers, 52
saturation, law of approach to, 96
saturation magnetization, 31, 71, 300
technical, 101, 121
search coil, 48
silicon-iron alloys, 272
Slater-Pauling curve, 265
soft ferrites, 288
soft iron, 290
soft magnetic materials, 74, 269
a.c. losses, 270
coercivity, 269
hysteresis loss, 270
n,221
s,223
quantum of flux, 349, 357
quantum theory of
electron-electron interactions, 247
electronic magnetic moments, 221
ferromagnetism, 259
paramagnetism, 255
Rayleigh's law, 95
recording density, 340
recording devices, 322
recording heads, 334
recording materials, 77
recording media, 77
recording process, 334
reading, 341
writing, 336, 338
relative permeability, 32
relays, 77, 271
reluctance, 40
remanence, 72, 90, 300, 373
remote field electromagnetic inspection
for NDE, 391
residual field, 373
resistance magnetometers, 56
resonance magnetometers, 60
retentivity, 70
reversible magnetization changes, 167
rigid band model of ferromagnetism
(Slater-Pauling), 265
rigid domain-wall motion, 139
initial susceptibility of, 141
rotating coil, 49
Russell-Saunders coupling, 241
permeability, 269
saturation magnetization, 270
solenoid
general formula for field of, 21
Subject Index
magnetic field
of a long thin, 12
of a short thick, 19
of a short thin, 19
optimization of geometry, 20
power considerations, 20
specific heat anomalies at critical
temperatures, 210
spin magnetic moment, 220
of atom, 239
spin quantum number, 223
spin waves, 208
spontaneous magnetization, 260
temperature dependence of, 260
SQUIDS, 64, 361
stability of permanent magnets, 310
steel, production, 365
Stern-Gerlach experiment, 236
Stoner-Slater (electron band) theory of
ferromagnetism, 263
Stoner-Wohlfarth model, 305, 344
strain theory of domain-wall pinning,
149
stress, effects of, on bulk magnetization,
171
strong magnetic fields, effect on electron
coupling, 244
superconducting
magnets, 360
solenoids, 360
superconductors, 85
applications, 358
conduction mechanism in, 359
development of improved materials,
358
basic properties, 345
supermalloy, 278
surface currents in superconductors, 354
susceptibility, 32, 108
anomalies at critical temperatures, 210
balance, 62
differential, 32
tapes, magnetic recording, 324
technical saturation magnetization, 31,
121
temperature dependent paramagnetic
439
susceptibility (Curie), 82,258
temperature independent paramagnetic
susceptibility (Pauli), 262
thalli um-barium -calcium -copper oxide,
360
theories of magnetic ordering, 188
thermal conductivity of superconductors,
357
thermal expansion anomalies at critical
temperature, 212
thin-film magnetometers, 57
three-d (3d) band electrons, magnetic
properties, 264
torque, 10
magnetometer, 60
torsion balance, 62
total atomic orbital angular momentum,
240
transformers, 76, 272
transverse magnetostriction, 104
type I superconductor, 349
type II superconductor, 349
ultrasonic velocity, field dependence in
steels, 374
units in magnetism, 11
vector model of the atom, 226
Verdet's constant, 57
vibrating coil, 49
magnetometer, 50
vibrating-sample magnetometer, 52
vortices in superconductor, 349
wall bowing approximation, 141
wave function of two-electron system,
247
including spin, 250
wave mechanical corrections to angular
momentum of electrons, 228
Weiss domain theory, 109
Weiss mean field theory, 109
consequences of, 187
critique of, 187
of ferromagnetism, 189
of paramagnetism, 186
Wiedemann-Franz ratio, 451
440
Subject Index
Winchester disk drive, 325
X-ray topography, 115
yttrium-barium-copper oxide, 358
critical current density, 358
critical fields, 358
critical temperatures, 358
Zeeman effect
anomalous, 234
normal,232
zero resistivity, 345
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